computational modeling of cell behavior in three-dimensional … · 2019-04-14 · don mohamed...

Computational Modeling of Cell Behaviorin Three-Dimensional Matrices

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of

Philosophy in Biomedical Engineering at the University of Zaragoza by

SEYED JAMALEDDIN MOUSAVI

Supervised by

Dr. MOHAMED HAMDY DOWEIDAR

Ph.D. Industrial Engineering

Presented at

Instituto de Investigación en Ingeniería de Aragón (i3A)

Universidad de Zaragoza

April 2015

Science is not only a disciple of reason but, also, one of romance and passion.

Science is not only a disciple of reason but, also, one of rororroror ∴ Stephen Hawking

Don Mohamed Hamdy Doweidar, Profesor Contratado Doctor del Departamento de Ingeniería Mecánica de la Escuela de Ingeniería y Arquitectura de la Universidad de Zaragoza,

CERTIFICA:

Que la memoria de Tesis Doctoral presentada por Don Seyed Jamaleddin Mousavicon el título “Computational Modeling of Cell Behavior in Three-Dimensional Matrices” ha sido realizada bajo su dirección en el Departamento de Ingeniería Mecánica de la Escuela de Ingeniería y Arquitectura de la Universidad de Zaragoza, y se corresponde con el Proyecto de Tesis aprobado por la Comisión de Doctorado en 2014, por lo que autoriza su presentación en la modalidad de compendio de publicaciones y con la Mención de “Doctor Internacional” cumpliendo por lo tanto las condiciones requeridas para que su autor pueda optar al grado de Doctora por la Universidad de Zaragoza.

Y para que así conste, firmado en Zaragoza a 10 de Febrero de 2015.

Fdo.: Mohamed Hamdy Doweidar

Acknowledgements

I would like to express my greatest gratitude to the persons who have helped and supported

me throughout this work.

Special thanks of mine goes to my supervisor, Dr. Mohamed Hamdy Doweidar, for his continu-

ous support during my study, from initial advice and contacts in the early stages of conceptual

inception and his encouragement to this day.

I am grateful to Professor Manule Doblaré who accepted me for PhD in GEMM group, without

his support this dissertation could not have been done.

I wish to appreciate Roxana, Iñaki, Raquel, Clara, for their kind support and help during my

study.

Also great thanks to my family, specially my mother, who devoted her life to her children and

tried her best to support me by giving a lot of encouragement during my schooling.

Words fail me to express my sincere appreciation to my wife Solmaz whose love, dedication and

persistent confidence in me, have taken the load off my shoulder. I owe for her great patience,

that made this dissertation possible.

v

The research developed in this Thesis has been supported by the Spanish Ministry of Economy

and Competitiveness and by the CIBER-BBN initiative to develop the Doctoral Thesis and to

obtain the title of Doctor, as well as by the institutions and projects detailed below:

• The Spanish Ministry of Economy and Competitiveness through project CYCIT DPI2010-

20399-C4-1.

• The Spanish Ministry of Economy and Competitiveness through project MINECOMAT2013-

46467-C4-3-R.

This thesis is presented as a compendium of articles published to obtain the

title of Doctor at the University of Zaragoza, following the agreement of the 20th

of December 2013 of the Governing Council of the University that approves the

Regulations on Doctoral Thesis.

The articles that are part of the thesis and that have been published in journals indexed in

the ISI are:

1. Mousavi SJ, Doweidar MH, Doblaré M. Computational modelling and analysis of

mechanical conditions on cell locomotion and cell-cell interaction. Comput Methods Biomech

Biomed Engin. 2014;17(6):678-93. (Journal Impact Factor: 1.79)

2. Mousavi SJ, Doweidar MH, Doblaré M. Cell migration and cell-cell interaction in the

presence of mechano-chemo-thermotaxis. Mol Cell Biomech. 2013;10(1):1-25.

3. Mousavi SJ, Doweidar MH, Doblaré M. 3D computational modelling of cell migra-

tion: A mechano-chemo-thermo-electrotaxis approach. J Theor Biol. 2013;329:64-73. (Journal

Impact Factor: 2.35)

4. Mousavi SJ, Doblaré M, Doweidar MH. Computational modelling of multi-cell mi-

gration in a multi-signalling substrate. Physical Biology. 2014;11(2):026002 (17pp). (Journal


5. Mousavi SJ, Doweidar MH. A Novel Mechanotactic 3D Modeling of Cell Morphology.

J Phys Biol. 2014;11(4):doi:10.1088/1478-3975/11/2/026002. (Journal Impact Factor: 3.14)

In addition to the articles previously listed and already published, two articles that are

under review are part of this thesis:

6. Mousavi SJ, Doweidar MH. Three-Dimensional Numerical Model of Cell Morphology

during Migration in Multi-Signaling Substrates. Plos One. 2015. accepted paper. (Journal


7. Mousavi SJ, Doweidar MH. Role of Mechanical Cues in Cell Differentiation and Pro-

liferation: A 3D Numerical Model. 2015. submitted. (Journal Impact Factor: 3.53)

viii

Esta tesis se presenta como un compendio de artículos publicados para optar

al título de Doctor en la Universidad de Zaragoza, siguiendo el acuerdo de 20 de

diciembre de 2013 del Consejo de Gobierno de la Universidad por el que se aprueba

el Reglamento sobre Tesis Doctorales.

Los artículos que forman parte de la tesis y que han sido publicados en revistas indexadas

en el ISI son:

1. Mousavi SJ, Doweidar MH, Doblaré M. Computational modelling and analysis of

mechanical conditions on cell locomotion and cell-cell interaction. Comput Methods Biomech

Biomed Engin. 2014;17(6):678-93. (Journal Impact Factor: 1.79)

2. Mousavi SJ, Doweidar MH, Doblaré M. Cell migration and cell-cell interaction in the

presence of mechano-chemo-thermotaxis. Mol Cell Biomech. 2013;10(1):1-25.

3. Mousavi SJ, Doweidar MH, Doblaré M. 3D computational modelling of cell migra-

tion: A mechano-chemo-thermo-electrotaxis approach. J Theor Biol. 2013;329:64-73. (Journal


4. Mousavi SJ, Doblaré M, Doweidar MH. Computational modelling of multi-cell mi-

gration in a multi-signalling substrate. Physical Biology. 2014;11(2):026002 (17pp). (Journal


5. Mousavi SJ, Doweidar MH. A Novel Mechanotactic 3D Modeling of Cell Morphology.

J Phys Biol. 2014;11(4):doi:10.1088/1478-3975/11/2/026002. (Journal Impact Factor: 3.14)

Junto a los artículos previamente enumerados y ya publicados, forman parte del desarrrollo

de esta tesis dos artículos que se encuentran en proceso de revisión:

6. Mousavi SJ, Doweidar MH. Three-Dimensional Numerical Model of Cell Morphology

during Migration in Multi-Signaling Substrates. Plos One. 2015. accepted paper. (Journal


7. Mousavi SJ, Doweidar MH. Role of Mechanical Cues in Cell Differentiation and Pro-

liferation: A 3D Numerical Model. 2015. submitted. (Journal Impact Factor: 3.53)

x

Abstract

Cell migration, differentiation, proliferation and morphology are essential processes in many

physiological and pathological processes ranging from wound healing to malignant diseases such

as cancer. Individual cell migration positions cells in tissues during morphogenesis and cancer,

or allows them to pass through the tissue, as seen by immune cells. On the other hand, col-

lective cell migration, such as neural crest, vasculature and many epithelial cells migration, is

another fundamental form of cell translocation which may relatively differ from individual cell

migration. During cell migration, amoeboid movement causes frequent changes in cell shape

due to the extension of protrusions in the cell front and retraction of cell rear. Regulation of

intracellular mechanics and cell’s physical interaction with its substrate rely on control of cell

shape during cell migration. Hence, it is of central importance to understand this process in

many biological processes ranging from morphogenesis to metastatic cancer cells.

Cell migration is governed by a complex network of signal transduction pathways. It has been

demonstrated that, in addition to mechanotaxis, cell migration can be also directed by chemical,

thermal and/or electrical stimuli. To achieve productive cell migration, each signaling passway

may be temporally and spatially effective in particular regions of the substrate in which the

cell migrates. Besides, experimental observations confirm that cells may undergo differenti-

ation and/or proliferation due to mechano-sensing process. For instance, mesenchymal stem

cells (MSCs) are susceptible to differentiate into different cell types such as neuroblast, chon-

drocyte, osteoblast and many others within substrates mimicking the stiffness of their native

Extracellular matrix (ECM).

In these concepts, numerical methods can effectively assist to better understand the physical

mechanism behind different aspects of cell behavior such as singular or collective cell migration,

cell morphological changes, cell differentiation and proliferation. However, they are also helpful

tool to design more efficient experimental setups. Therefore, through the present dissertation,

the main objective is to develop a numerical model that considers different features of single

and collective cell behavior in existence of different stimuli in their micro-environment. Guided

by experiential observations, the cell translocation is modeled by the equilibrium of effective

forces on cell body such as traction, protrusion, electrostatic and drag forces, assuming that the

cell traction forces are regulated by the cell internal deformations. To do so, the finite element

discrete methodology is employed in which the cell is represented by a group of finite elements

following two different strategies; a constant spherical cell shape and a free mood cell shape.

Our findings, which are qualitatively consistent with well-known related experimental obser-

vations, indicate that the cell migrates persistently towards stiffer or more fixed regions in its

neighbors. Any change in substrate stiffness and/or in the boundary conditions may affect the

path tracked by the cell and the cell final location. During cell migration within a substrate

with stiffness gradient, nodal traction forces increase while net traction force as well as cell

velocity decrease. In very soft or very stiff substrates, generated net traction forces may not

be strong enough to enable the cell anchors or penetrates further into the substrate. Besides,

during superficial cell migration on a substrate with restricted lower layer, the cell tendency is

to migrate towards lower depth locations of the substrate.

On the other hand, when another activation signal is added to the substrate, the overall cell

behavior changes. For instance, in the presence of chemotaxis and/or thermotaxis the cell final

location is quite sensitive to the imposed effective factors. Despite of the free boundary sur-

face, in the presence of thermotaxis and/or chemotaxis, the cell final location displaces towards

higher temperature and/or chemical concentration. Although the effect of these cues is negli-

gible on the cell local velocity, these signals remarkably affect the reorientation of the cell and

reduce its random motility. Moreover, it is well known that a typical cell may migrate towards

the cathode pole in the presence of exogenous electric field (EF). Amplification of the EF can

accelerate this phenomenon and causes the cell to move more directionally. Electrotaxis has a

dominant guidance role in directing cell migration compared to other stimuli.

During collective cell migration, the associated cell behavior is relatively different. For instance,

cells tend to create small slugs within the substrate and then these slugs aggregate in the middle

or end of the substrate depending on substrate characteristics (substrate stiffness and boundary

conditions). In addition, interaction between cells inside the substrate delays their movement

towards stiffer regions. This is because the signal coming from their inter-stretched region in-

duces the cells to move towards each other and maintain in contact. This interaction changes

the profile of average net traction force and average velocity of the cells. It is noteworthy to

mention that adding any new stimulus into the substrate displace cell aggregation centroid

towards this new cue. However, high electrical field strength can even cause the cell slug to be

flatten near the cathode pole.

xii

Similar to cell migration, cell shape can be regulated by cell internal deformation which is cou-

pled with the mechanical characteristics of the cell micro-environment. A resident cell within

an unconstrained soft (several kPa) or hard (greater than hundred kPa) substrate is unable to

adhere or penetrate into its substrate and keeps a spherical shape. When the substrate stiffness

is about tens of kPa (intermediate and rigid substrates), the cell can adequately adhere to the

substrate, increasing the traction forces, the cell elongation and Cellular Morphology Index

(CMI). Maximum cell elongation occurs in the middle of a constrained rigid substrate, which

decreases when the cell approaches a constrained surface. It can be concluded that the higher

the net traction force, the greater the cell elongation and the larger the cell membrane area

(CMI). Besides, the overall cell shape (cell elongation and CMI) may be changed by activation

of other stimuli. For instance, by adding chemotaxis, thermotaxis and/or electrotaxis to the cell

substrate, average cell elongation and CMI increases. However, the average cell elongation and

CMI are maximum in the presence of electrotaxis which confirm dominant role of electrotaxis.

In addition, using the present model, we can quantify cell differentiation and proliferation

due to mechanotaxis. MSC differentiates into neurogenic, chondrogenic or osteogenic lineage

specifications within soft (0.1-1 kPa), intermediate (20-25 kPa) or relatively hard (30-45 kPa)

substrates, respectively. When a MSC differentiate to a compatible phenotype, the average

net traction force depends on the substrate stiffness in such a way that it might increase in

intermediate and hard substrates but it would reduce in a soft matrix. However, the average

net traction force considerably increases at the instant of cell proliferation due to cell-cell in-

teraction. In addition, cell differentiation and proliferation can be accelerated by increasing

substrate the stiffness within the relative range.

xiii

Resumen

La migración, diferenciación, proliferación y morfología celular son aspectos esenciales en

numerosos procesos fisiológicos y patológicos, desde la cicatrización de heridas hasta enfer-

medades como el cáncer. La migración celular de una única célula es el proceso encargado de

posicionar a las células en el tejido durante la morfogénesis y el cáncer, que permite el paso

de las mismas a través del tejido, como se observa en las células inmunes. Por otro lado, la

migración celular colectiva es otra de las formas fundamentales de movimiento celular, como la

migración de las células mesenquimales estrechamente empaquetada y las láminas epiteliales.

Durante la migración celular, los movimientos ameboides causan frecuentes cambios en la forma

celular debido a la extensión de las protrusiones en la parte frontal y a la retracción de la parte

posterior de la célula. La regulación de la interacción célula-célula y la célula sustrato depende

de la forma celular durante la migración. Por lo tanto, es esencial entender este fenómeno y

su influencia en la mayoría de los procesos biológicos existentes desde la morfogénesis hasta la

metástasis en el cáncer. La migración celular está gobernada por una compleja red de señales

de transducción. Se ha demostrado que, además de la mecanotaxis, la migración celular puede

ser guiada por estímulos químicos, térmicos y/o eléctricos. Para lograr una migración celular

adecuada, cada señal debe ser espacial y temporalmente efectiva en regiones determinadas del

substrato en el que se encuentran embebidas. Adicionalmente, las observaciones experimentales

confirman que las células pueden diferenciarse o proliferar debido al proceso mecano-sensación.

Por ejemplo, las células madre mesenquimales (CMM) son susceptibles a diferenciar en difer-

entes tipos celulares tales como neuroblastos, condrocitos, osteoblastos y otros muchos tipos

celulares mediante el uso de sustratos que mimeticen la rigidez de la matriz extracelular (MEC)

nativa.

En estos aspectos, los métodos numéricos pueden ser herramientas útiles que ayuden al en-

tendimiento de los mecanismos físicos que hay detrás de los diferentes procesos que afectan

al comportamiento celular, como la migración individual o de forma colectiva, los cambios

morfológicos celulares, la diferenciación y proliferación celular. Así, en esta Tesis Doctoral,

el principal objetivo es desarrollar un modelo numérico que considere los diferentes aspectos

involucrados en la migración de una única célula y de una población celular cuando diferentes

estímulos son aplicados en el microambiente en el que se encuentran. Según numerosas obser-

vaciones experimentales, el movimiento celular se modela mediante el equilibrio de las fuerzas

efectivas en la célula como la tracción, protrusión, electrostática y fuerzas de arrastre, asum-

iendo que las fuerzas de tracción celular están reguladas por la deformación interna de la célula.

Para ello, un modelo discreto de elementos finitos ha sido desarrollado en el que la célula es

representada mediante un grupo de elementos finitos siguiendo dos estrategias diferentes; una

forma esférica constante y una forma libre que se modifica según las condiciones externas.

Los resultados conseguidos, consistentes cualitativamente con diversas observaciones experi-

mentales, indican que las células tienden a migrar hacia las zonas más rígidas. Los cambios

que tienen lugar en la rigidez de los substratos así como en las condiciones de contorno afectan

significativamente al camino migratorio de la célula y a su localización final. Durante el proceso

de migración celular a través de un substrato con gradiente de rigidez, las fuerzas de tracción

nodales se incrementan mientras que la fuerza de tracción neta y la velocidad de migración celu-

lar disminuyen. En substratos que presentan elevada rigidez o por el contrario, aquellos que son

muy blandos, se generan fuerzas de tracción netas que pueden no ser lo suficientemente fuertes

como para que el anclaje de la célula suceda o penetre en el substrato. Además, durante la

migración celular superficial sobre un sustrato cuya una superficie inferior restringida, la célula

muestra una clara tendencia a migrar hacia las zonas de menor profundidad de substrato.

Por otra parte, cuando se incorporan señales de activación adicionales en el substrato, el com-

portamiento celular se ve alterado. Por ejemplo, en presencia de agentes quimiotacticos y/o

termotacticos la localización final de la célula depende de forma directa de la intensidad de

dichos factores impuestos. A pesar de la existencia de una superficie libre, en presencia de

termotaxis y/o quimiotaxis, la célula se localiza finalmente en aquellas zonas de mayor temper-

atura y/o concentración química, respectivamente. Aunque el efecto de los agentes mencionados

es despreciable en la velocidad migratoria local de la célula, estas señales afectan en su reori-

entación y reducen su movilidad aleatoria. Además, la célula tiende a migrar hacia el polo

catódico en presencia de campo eléctrico exógeno. La amplificación de campo eléctrico puede

acelerar dicho fenómeno y causar que la célula manifieste un movimiento más direccionado.

La electrotaxis juega un papel dominante en el direccionamiento de la migración celular en

comparación con otros estímulos.

El comportamiento celular es diferente en el caso de la migración celular de un grupo de células.

Por ejemplo, las células tienden a crear pequeños agregaciones celulares en el substrato que se

xvi

sitúan en el medio o el final del mismo en función de las características de dicho substrato

(rigidez y condiciones de contorno del substrato). Además, las interacciones entre las células

retardan su movimiento hacia las zonas más rígidas porque la señal que proviene de la región

estirada provoca que las células se muevan hacia sus células vecinas para mantenerse en con-

tacto. La interacción celular cambia la fuerza de tracción neta y la velocidad de las células. Es

importante remarcar que la adición de nuevos estímulos en el substrato desplaza el centroide

de dicho agregado celular hacia el foco de la señal aplicada. Sin embargo, fuerzas eléctricas

elevadas creadas debido a los campos eléctricos conlleva los agregaciones celulares a adquirir

una forma aplanada en la zona cercana al polo catódico.

Al igual que en la migración celular, la forma celular puede ser regulada por la deformación

interna de la célula. Dicha deformación está directamente relacionada con las características

mecánicas del microambiente celular. Una célula residente en un substrato blando y sin re-

stricciones (varios kPa) o duro (más de cien kPa), es incapaz de adherirse o penetrar en dicho

substrato y se queda con una forma esférica. Cuando la rigidez del substrato es del orden de

diez kPa (substratos intermedios y rígidos), la célula se adhiere de forma adecuada, incremen-

tando para ello las fuerzas de tracción, la elongación celular y el índice Morfológico Celular

(IMC). La elongación celular máxima tiene lugar en el medio de substratos rígidos restringidos,

reduciendo su valor cuando la célula se aproxima a la superficie restringida. Se deduce que

conforme mayor sea la fuerza de tracción neta, mayor elongación y mayor área de membrana

celular tendrán lugar. Además, la forma celular (elongación celular y IMC) puede cambiar

mediante la activación de otros estímulos. Por ejemplo, añadiendo agentes quimiotácticos, ter-

motácticos y/o electrotácticos al substrato celular, la elongación celular media así como el CMI

se incrementan. Sin embargo, la elongación celular media y el IMC son máximos en presencia

de electrotaxis, indicando el papel dominante de dicho fenómeno. Además, usando el modelo

presentado, se puede cuantificar la diferenciación y proliferación celular debida a la mecan-

otaxis. La diferenciación de las CMM en linajes neurogénicos, condrogénicos u osteogénicos

tienen lugar en substratos blandos (0.1-1 kPa), intermedios (20-25 kPa) o relativamente rígidos

(30-45 kPa), respectivamente. Cuando CMM diferencian a un fenotipo compatible, la fuerza

celular media neta depende de la rigidez del substrato de forma que puede verse incrementada

en substratos rígidos e intermedios pero se reduce en matrices blandas. Sin embargo, en todos

los casos, la fuerza de tracción neta aumenta considerablemente cuando tiene lugar la prolif-

eración celular debido a la interacción celular. Adicionalmente, la diferenciación y proliferación

celular se aceleran con el incremento de la rigidez del substrato en un rango relativo.

xvii

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Migration process and cell behavior . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Cell effective stimuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Individual and collective cell migration . . . . . . . . . . . . . . . . . . . 8

1.2.3 Cell migration on 2D and within 3D matrices . . . . . . . . . . . . . . . 8

1.3 Numerical models of cell behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Methodology, Objective and Outline . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Publications 15

2.1 Computational modelling and analysis of mechanical conditions on cell locomo-

tion and cell-cell interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Cell migration and cell-cell interaction in the presence of mechano-chemo-thermotaxis.

2.3 3D computational modelling of cell migration: A mechano-chemo-thermo-electrotaxis

approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Computational modelling of multi-cell migration in a multi-signalling substrate.

2.5 A novel mechanotactic 3D modeling of cell morphology. . . . . . . . . . . . . . .

2.6 Three-dimensional numerical model of cell morphology during migration in multi-

signaling substrates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7 Role of mechanical cues in cell differentiation and proliferation: A 3D numerical

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Conclusions and Future Works

xix

CONTENTS

3.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xx

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

Introduction

Introduction

In this Chapter, a brief survey on experimental and numerical works of cell behavior is

presented. Specifically, the role of cell migration on a wide variety of critical processes, current

experimental knowledge on migration-related research and common approaches for mathemat-

ical and computational modeling of cell migration and morphology changes are addressed.

Afterward, the main objectives and a structure description of the employed methodologies are

presented.

1.1 Background

Cell migration is crucial for normal tissue development and morphogenesis of animal body

and organ systems. Therefore, over the last few decades, scientists have paid special attention

to investigate cell behavior in presence of different stimulating cues. It is well known that cell

migration plays a prominent role in numerous physiological and pathological processes, such as

morphogenesis [1–6], wound healing [7, 8], tumor metastasis [9, 10], tissue development [11, 12],

cell differentiation and proliferation [13–15].

During tissue or organ development, the structure of the generated tissue should be generated

with the appropriate cell type and in the correct 3D geometry. In most of cases, organs cell

type is specified during gastrulation, where they finally reside for its normal biological func-

tion. Abnormal cell migration during adult life causes pathological states such as invasion and

metastasis of cancer. Drosophila has employed this mode of cell migration during border cell

migration and tracheal development while in vertebrates, a key role for collective cell migration

has been noted in vascular sprout and pronephros development [16]. In such cases, groups of

cells migrate as tightly associated epithelial sheets or clusters (e.g., Drosophila border cells and

zebrafish lateral line primordium), or they possess a mesenchymal character as during gastru-

lation and neural crest migration [1].

Wound healing is an orchestrated multi-tissue response which is programmed through a defined

timetable in which each stage prepares the wound for subsequent stages that are required for

restoring the tissue. One of the most important stages in this timetable is cell migration by

which keratinocytes and fibroblasts migrate into the wound bed, after cutaneous injury. Ap-

proximately 7 days after injury, fibroblasts migrate towards the wounded site to improve matrix

structure and to modify wound contraction [17]. Subsequently, stimuli that regulate cell motil-

ity may change the rate of cell migration towards the wound bed and gradually improve the

rate of wound healing [18]. The other important stage in wound healing is cell morphology

changes. During this process, cells elongate during their migration towards wound locations [7]

3

1.1. Background

and change their shapes to cover the wound without leaving intercellular gaps [8]. In this step

the main cellular morphological alterations are observed around the wound edges.

Cancer cells migrate to invade the cancerous sites modifying their migration mechanisms in

response to different conditions. For instance, neoplastic cells follow this procedure to enter

lymphatic and blood vessels to disseminate into the circulation, causing metastatic growth in

distant organs [19]. Tumor cells migrate to spread throughout the tissues which is similar, if

not identical, to those that are observed in non-neoplastic cells (normal cells) during physiolog-

ical processes such as immune-cell motility, wound healing and embryonic morphogenesis [20].

Brain metastasis depends on multiple cell-cell and cell-matrix interactions, causing tumor cells

detachment from the primary tumor and transmigrating through the endothelial lining into the

parenchymal tissue. Therefore, obviously cancer cells follow a large spectrum of invasion and

migration mechanisms including both singular and collective cell migration strategies [20].

Stem cells have the potential of proliferation and are capable of self-maintenance and differen-

tiation towards different cell phenotypes. Both processes can be controlled by different signals

such as physico-chemical factors including specific mechanical mechanisms. Although the reg-

ulation of stem cell biology by mechanical cues is less understood, certain key themes have

been proven by experimental observations. Firstly, external mechanical forces exerted on stem

cells can drive and dictate stem cell differentiation into specific cell phenotype. Secondly, stem

cells are able to detect and respond to alterations in the stiffness of their surrounding micro-

environment through specific lineage specification. Finally, the differentiation process affects

the mechanical properties of the cells and its specific subcellular components. Combination

of these three intrinsic concepts allow to introduce a new theory for the maintenance of stem

cells and alternatively their differentiation. The stem cells retain their function as long as they

remain anchored to the supporting cells known as niche1. Their divisions occur in such a way

that one daughter cell keeps its contact with the supporting cell while the other one loses this

contact, migrates from the niche, and proceeds to generate terminally differentiated cells [22].

1An adult mammalian stem cell niche is defined as a micro-environment that facilitates the survival and

the self-renewing capacity of stem cells, as well as the production of actively dividing precursors leading to the

generation of a differentiated progeny [21].

4

Introduction

1.2 Migration process and cell behavior

To migrate, the cell maintain a defined direction and speed in response to environment

stimuli. Study of Pelham et al. [23] demonstrates that cells present different morphological

patterns and motility rates in presence of different substrate stiffnesses. Other findings indicate

that the cell determines the migration direction or destination by exerting contractile forces

and then interprets its internal deformation to sense its substrate mechanical properties [23–25].

These observations motivated the scientists to study cells capability of responding to substrate

stiffness through a true active tactile exploration process. Subsequently, for the first time, Lo

et al. [24] showed that cell migration can be guided by mechanical interactions at cell-substrate

interface. Their observations indicate that fibroblasts on collagen-coated polyacrylamide sub-

strates with a rigidity gradient can detect and respond to substrate stiffness by migration in

direction of increasing stiffness [24]. Similar results have been also verified through investiga-

tions of Ehrbar et al. [26] demonstrating that cell behavior strongly depends on its substrate

stiffness.

Cell migration is a highly integrated multi-step process during which amoeboid movement

causes frequent extension of protrusions in the cell front [27, 28], which is often termed pseu-

dopods or lamellipods, and cell rear retractions. It involves a number of coordinated and cyclic

processes, including the protrusion of pseudopodia, the formation of new adhesions, the move-

ment of cell body and the release of old adhesions (see Fig. 1.1) [24, 27–29]. Through this

process, the cell adheres to its substrate through the generated tension by the cytoskeleton

(CSK) which is created by different mechanisms including the concurrent acto-myosin forces

(traction forces) and the active polymerization of the actin network (random protrusion force)

[24, 27]. The effect of the cell shape on the balance of traction forces and its effect on the

cell behavior remains poorly understood. However experiments demonstrate that it depends

on the number and orientational distribution of stress-fibers formed within the cell. All these

parameters somehow depend on the magnitude and symmetry characteristics of the cell internal

deformation [30, 31] as well as the matrix stiffness [32].

A general pattern of tissue differentiation is also a challenging issue in tissue repair. However,

it is experimentally well known that cell differentiation and proliferation can be triggered by

mechano-sensing process and cell substrate interaction during cell migration [33–35]. To prove

this, the first attempt was made by Engler et al. [33] demonstrating that 2D matrix stiffness

(mechanotaxis) can guide the human mesenchymal cell (MSC) fate. In such a way when cells

are cultured on soft substrates mimicking the elasticity of brain tissue (a stiffness of 0.1-1 kPa)

5

1.2. Migration process and cell behavior

Figure 1.1: The cyclic four main steps of cell migration. a- Extension, b- Development of new

adhesion, c- translocation of cell body and d- de-adhesion.

they differentiate to neuronal precursors; on matrices with intermediate stiffness mimicking

muscle (a stiffness of 8-17 kPa) they induce myogenic commitment; while on relatively rigid

matrices like collagenous bone (a stiffness of 25-40 kPa) they differentiate to osteoblast. Sim-

ilar results were reported by Huebsch et al. [35] within a 3D hydrogel synthetic Extracellular

Matrix (ECM).

1.2.1 Cell effective stimuli

It is well known that besides mechanotaxis (durotaxis) [23, 24, 36], other stimuli such as

chemotaxis [37–41], thermotaxis [42, 43] and/or electrotaxis [44–46] can actively control cell

behavior. An accurate understanding of cell behavior requires a comprehensive understanding

of the role of aforementioned stimuli. Many experimental works [37, 39, 47, 48] address that

cells migrate directionally along even a shallow gradient of chemical substances such as growth

factors or attracting agents. Under a gradient of a chemoattractant source, cells are known to

polarise towards the positive or negative direction of the gradient depending on the chemotactic

signal type [37, 39, 40, 47]. Chemoattraction is thought to play a key role in guiding cells in

many immunobiological processes. For instance, leukocytes reach infection locations by means

of chemotaxis. However the mechanisms by which a cell transduces a chemotactic cue into a

certain movement still remain elusive [38].

In vivo, thermotaxis may be considered as a complementary cue to chemotaxis since each mech-

anism is active in a specific region where the other is ineffective [49]. For instance, trophoblasts

invade endometrium, the inner membrane of a uterus, by means of thermotaxis. These cells

6

Introduction

subjected to oxygen and thermal gradients do not migrate in response to oxygen gradient (a

chemotactic cue) but they migrate in response to thermal gradients of even less than 1 ◦C

towards the warmer locations [43]. Extravillous trophoblasts also migrate from the tips of the

anchoring villi that surround the developing blastocyst through the maternal deciduae to the

distal portions of the uterine spiral arteries.

Recent in vitro studies have demonstrated that the presence of endogenous or exogenous elec-

trotaxis is another guiding factor of cell migration and cell shape change [44, 50–54]. When

stationary cells are exposed to direct current EFs (dcEFs), they effectively migrate towards

cathode or anode poles depending on the cell type [51, 52]. For instance, Epithelia generate a

steady voltage across themselves, driving an electric current in the wounded sites [46, 51]. In

the rat cornea injury, an electric current about 10 μA/cm2 is measured and the skin of a finger

tip wound is able to create a lateral electric field (EF) in range of 40-200 mV/mm [46]. In

addition, a steady EF of 450-1600 mV/mm has been observed across the wall of the amphibian

neural tube during early neuronal development [55]. During wound healing, polarized epithelial

cells transport ions directionally and maintain Transepithelial Potentials (TEPs) [46]. When

an ulcer occurs it disrupts the epithelial barrier and consequently a short-circuit TEP is gener-

ated, called Endogenous Wound Electric Field (EWEF) [46]. The potential at the wound thus

drops, becoming negative compared to the potential underneath the unwounded epidermis that

is far from the wound site. This potential gradient drives the electrical current flow towards

the more negative site so that a laterally orientated wound EF is created (electrotaxis). At skin

and corneal wounds, electrical current flow is orientated towards the wound center from the

surrounding tissues and then out from the wound. Cells away from the wound keep transport-

ing of ions into wound to maintain the TEPs. Those cells keep driving the electrical currents

until the wound heals and the barrier is restored [46]. In the past few years, there has also been

a growing interest in the effects of an exogenous EF on cultured cells, postulating that calcium

ion, Ca2+, is involved in the electrical-field-induced cell response [53, 56–61]. A cell in natural

state have negative potential, exposing it to external dcEF causes extracellular Ca2+ influx

into intracellular through calcium gates on the cell membrane. Subsequently, in steady state

depending on Ca2+ content of intracellular, a typical cell may be charged negatively or posi-

tively [45]. Therefore, dcEF can be employed to stimulate cells and move directionally towards

the cathode or the anode [52, 62]. For instance, fish and human keratinocytes, human corneal

epithelials and dictyosteliums are attracted by the cathode [52, 62–65] while lens epithelial and

vascular endothelial cells are attracted by the anode [55, 63]. However there are some cells

which are not excitable by an exogenous electrotaxis, such as human dermal melanocytes [66].

7

1.2. Migration process and cell behavior

This may occur due to higher dcEF threshold of this cell type [60].

1.2.2 Individual and collective cell migration

Cells migrate either individually or as a cluster of cells. During the past few decades, single-

cell migration has been extensively studied in many experimental works [47, 67]. In contrast, it

has been recently appreciated that collective cell migration is highly relevant in many cell types,

especially those related to tumor cells [29], wound healing [68] and tissue remodeling [69]. It is

attributed to cell-cell attraction which has been observed in cell populations during collective

guidance. However, similar behavior of cell migration has been observed for fibroblasts which

are less cohesive cell types [52, 70].

Cell migration can be consider collective when two or more cells make contact and maintain

their cell-cell junctions during migration, at least sometimes [27]. In the presence of other

cells, cell-cell attraction may affect cell-substrate adhesion and may mediate cell-cell contacts.

Therefore, dynamics of collective cell migration results in complex changes in multicellular tis-

sue structures. Unlike single cell migration, collective cell migration serves to keep the tissue

intact during remodeling, allows mobile cells to carry other cells that are otherwise immobile,

and ensures appropriate distribution of cells within a tissue [71]. It has also been observed

that the overall migration pattern of such cohesive groups tends to be vastly different from

the migration characteristics of the individual cells constituting such groups, and it is rather

characterized by the type and strength of their reciprocal interaction [71].

Cell-cell interactions and cell motility coordination during multi-cell migration can be studied

from two viewpoints. First is how the cells affect one another, to what radius does a cell trans-

mit force and communicate for transmitting information? Second is, how cell-cell interactions

can affect their individual and collective behavior to answer the questions like; does collective

cell migration speed up or delay cell movement and how do cell slugs affect each other? Some

of these questions arisen may be answered via experimental works but, to profoundly answer

others, numerical investigation is inevitable.

1.2.3 Cell migration on 2D and within 3D matrices

Cell migration on 2D surfaces occurs during reepithelialization of wounds or the scanning

of leukocytes along the inner blood vessel wall or inner epithelial surfaces [27]. Although

8

Introduction

Figure 1.2: Cell elongation and morphology on 2D (a) and within 3D (b) substrate with identical

stiffness [3].

2D investigations have improved our insights in many contexts such as basic mechanisms by

which cells migrate, interact with the substrate, and change their speed or direction, they may

sometimes impose an artificial apical-based cell polarity that may not exist through 3D in vivo

processes. For example, studies of Hakkinen et al. [3] illustrated that cell morphology strongly

depends on substrate dimensionality since the cells tend to be less elongated and more spread

on 2D matrices than in 3D matrices (Fig. 1.2). It is attributed to the number of integrins and

receptors which are associated in cell-substrate interaction. Limited integrins and receptors of

the cell can participate in 2D cell-substrate adhesion. On the other hand, the ability of the

cells to move in a 3D substrate not only depends on the viscosity and stiffness of the substrate

but also the density of fibers. Therefore, to fully understand the underlying mechanisms by

which cells migrate in vivo, it is necessary to study the movement of cells in 3D environments

as well. This explains why experimental studies in 3D matrices have begun to grow gradually

in the past few years [26, 36].

1.3 Numerical models of cell behavior

Numerical modelling of cell motility has a relatively long history. For the first time, as early

as 1970, Keller et al. [72, 73] developed a model using partial differential equations to study

the biochemical regulation of bacterial movement. A decade later, in the 1980s, a distinguished

field was developed for research in which numerical models were proposed for the movement

of isolated individual cells. A key initial work in this direction is presented by Oster [74, 75].

9

1.3. Numerical models of cell behavior

Several works has been subsequently developed by different researchers [76–78]. Keller’s highly

effective equations became the basis of phenomenological models ranging from slime mould

slugs [79] to tumour angiogenesis [80] and wound healing [81].

Numerical models, which consider cell behavior, are recently increasing. For instance, several

numerical models have been recently developed to study behavior of cell populations [41, 82],

cell morphology [40, 41, 83–85] and cell fate [86, 87]. Each model has a sort of limitations and

there is no comprehensive model to simultaneously consider cell behavior in a multi-signaling

environment, collective cell migration and cell shape changes. Although computational models

such as [40, 88] simultaneously deal with mechanotactic and chemotaxtic cues they miss ther-

motactic and electrotactic effects. Some of numerical models only concentrate on migration

of specific cell type [89], consider 2D cell shape using a hybrid cellular Potts model [90] or

study 2D cell-cell interaction with defined cell configuration [89, 91, 92]. Each model is de-

veloped using a specific method. For example, there are energy based [85] and coarse-grained

[83, 84] models studying substrate rigidity influences on cell morphology and migration as well

as continuum mechanics models studying chemotactic effect on cell migration and cell shape

change [88]. In most of these models, actual forces acting on the cell body are not considered

[89, 91–93]. However the main shortcoming in many of them is that they have ignored the cell

mechano-sensing process which is an intrinsic feature in cell-substrate interaction [40, 89, 92]. In

addition, the previously mentioned 2D models, which simulate cell shape change, do not study

cell configuration in a free mode but they restrict the cell shape to a rigid ellipse by which

the cell shape change is represented by alteration of the aspect ratio of the ellipse (the ratio of

the major axis to the minor axis). A typical rigid mode of cell configuration is also assumed

in some 3D models [94] in which 3D epithelial cell interacts with a flat substrate. The main

limitation of this model is that the cell can have different rigid shapes such as hexagonal prism,

columnar, cuboidal and/or squamous. Vermolen et al. [41] presented a 3D phenomenological

numerical model to investigate the effect of chemotactic cues on cell morphology. Although in

their model the cell can take irregular shapes, their model is formulated in such a way that

the cell velocity intrinsically depends on the chemical gradient which is not sufficiently precise

according to many experimental works [24, 36, 95]. They also assumed that the cell volume

changes when the cell sends out pseudopods. This is unrealistic assumption since recent exper-

imental investigations [96–98] have demonstrated that overall cellular volume remains constant

as the cell shape changes. Han et al. [99] investigated the spatiotemporal dynamics of cell

migration using a bio-chemical-mechanical contractility model that incorporates the traction

forces developed by the cell with cell migration in 2D substrates. Unfortunately, their model

10

Introduction

does not include cell shape changes during cell migration. In their model, formation of a new

adhesion regulates a reactivation of stress fibre assembly within the cell and predicts the spatial

distribution of traction forces.

Besides of the issues discussed above, signaling mechanisms by which micro-environment stiff-

ness controls cell differentiation and proliferation are not computationally considered in cell

level. Many mechano-biological macro-level models have been developed to describe cell lin-

eage specification during fracture healing [86, 87, 100–105]. All of these numerical models are

able to predict the general patterns of tissue differentiation due to external mechanical stimuli

in macro-level. 2D Model presented by Stops et al. [87] considers cell differentiation and prolif-

eration in a collagen-glycosaminoglycan scaffold subjected to mechanical strain and perfusive

fluid flow. Their findings indicate that specific combinations of scaffold strains and inlet fluid

flows define the specific cell fate. Besides, the 2D model developed by Kang et al. [86] to

simulate bone fracture healing is formulated based on the density of each cell phenotype. It is

assumed that the cell differentiation and proliferation can be modulated according to the mag-

nitude and frequency of mechanical stimuli. According to their numerical results, bone healing

process can be improved when magnitude and frequency of mechanical stimuli are employed as

control factors of cell proliferation. To our knowledge, there is no numerical model to consider

cell differentiation and proliferation based on mechano-sensing process during cell migration.

1.4 Methodology, Objective and Outline

1.4.1 Methodology

Cell movement, similar to many other physical events, depends on the equilibrium of effective

forces acting on the cell body. Traction force which is generated due to cell internal deformation

is transmitted through the integrins at the focal adhesions to the ECM [8, 24]. This force is

the only directional force, in absence of other external cues, which guides the cell movement

accompanied with a random protrusion force. On the other hand, in presence of thermotactic

and chemotactic cues the direction of this force is modified according to the pointed direction

by those stimuli. In presence of electrotaxis the cell is exposed to an independent electrostatic

force. Therefore, in a multi-signaling environment all these effective forces on cell body should

be in equilibrium with opposing drag force. To simulate cell response to multiple signals

received from the cell environment we have developed a 3D computational model using the

finite element discrete methodology considering the cell as a group of finite elements. This

11

1.4. Methodology, Objective and Outline

allows us to predict the cell behavior and response when it is surrounded with different micro-

environmental characteristics.

1.4.2 Objective

In addition to the experimental investigations, theoretical studies and numerical models

also provide profound insights into cell behavior. Experimental investigations on cell behavior

are relatively time costly and expensive. Numerical models can save experimental resources

and time. These models not only associate the experimental results to the first principles, but

also describe the behavior and sensitivity of the systems as a function of each parameter. In

addition, mathematical models are helpful in identifying the key parameters that play the main

role in defining the overall behavior of the system, and thus lead to new and more effective in

vitro experiments.

So, the main objective of the present thesis is to cover different aspects of cell behavior such

as cell migration, differentiation, proliferation, shape changes and cell-cell interactions in a 3D

multi-signaling matrix. To achieve the aim, it is necessary to develop and extend an intensive

3D numerical model according to the different partial objectives. Hence, we first modeled the

mechano-sensing process of a single cell then it was extended to include individual cell migra-

tion, cell-cell interaction and collective cell migration due to pure mechanotaxis. Afterwards,

the effect of different signals is added to the model to consider the behavior of individual cell

and population of cells in a 3D multi-signaling matrix. To consider these processes, for the

sake of simplifications, initially a constant spherical cell shape is assumed. Then it is extended

to investigate the cell shape changes due to mechanotaxis, thermotaxis, chemotaxis and elec-

trotaxis. Finally, the basic model of cell migration was employed to include cell differentiation

and proliferation as a consequence of the mechano-sensing process.

1.4.3 Outline

This Thesis is presented as a compendium of publications. It complies with the requirements

established by the University of Zaragoza following the agreement of the 20th of December of

2013 of the Governing Council that approves the Regulations of Doctoral Theses. According

to this regulation, the present Thesis is composed of three chapters. The first one is a brief

introduction. While the second one, which represent the main core of the thesis, includes the

achieved publications. This chapter composed of five published papers and an accepted paper

in peer-reviewed journals (sections 1-6) and a paper submitted to peer-reviewed journals (sec-

12

Introduction

tion 7). The last one, chapter three, is the obtained conclusions. The flowing is a description

of every one of these chapters.

Chapter 1 (the present chapter) comprises an introduction that describes the prior investi-

gations and existent knowledge about cell behavior in addition to methodology, objective and

outline of the Thesis.

Chapter 2 is composed of seven sections which include

section 1 is the base of the present numerical model in which, considering constant cell shape,

the effects of substrate mechanical characteristics such as substrate stiffness and boundary con-

ditions on single and multi- cell migration are studied.

section 2 is to consider migration and interaction of two cells in the presence of mechanotaxis,

thermotaxis and chemotaxis.

section 3 presents a computational model of single cell migration with constant shape in a

multi-signaling micro-environment.

section 4 includes the simulation of multi-cell migration and cell-cell interaction in the presence

of multiple signals in the cell substrate.

section 5 is to study the effects of substrate mechanical characteristics (substrate stiffness and

boundary conditions) on the morphological changes of an individual cell.

section 6 is an extension of the model presented in section 4 to investigate influence of ther-

motactic, chemotactic and electrotactic stimuli on cell shape changes during individual cell

migration.

section 7 introduces an extension of the mechanotaxis model with constant cell shape to con-

sider the influence of substrate stiffness on cell differentiation, proliferation and apoptosis during

cell migration.

Chapter 3 concludes the main findings, conclusions and new insights achieved in the present

Thesis. Besides, several research lines for future work are proposed.

13

Chapter 2

Publications

Seyed Jamaleddin Mousavi, Mohamed Hamdy Doweidar and Manuel Doblaré

Computer methods in biomechanics and biomedical engineering

2014

Journal Impact Factor: 1.79

2.1 Computational modelling and analysis of mechanical

conditions on cell locomotion and cell-cell interaction.

Computational modelling and analysis of mechanical conditions on cell locomotion andcell–cell interaction

S.J. Mousavia, M.H. Doweidara,b,* and M. Doblarea,b

aGroup of Structural Mechanics and Materials Modelling (GEMM), Aragon Institute of Engineering Research (I3A),University of Zaragoza, Zaragoza, Spain; bBiomedical Research Networking center in Bioengineering, Biomaterials and Nanomedicine

(CIBER-BBN), Spain

(Received 8 May 2012; final version received 7 July 2012)

Between other parameters, cell migration is partially guided by the mechanical properties of its substrate. Although manyexperimental works have been developed to understand the effect of substrate mechanical properties on cell migration,accurate 3D cell locomotion models have not been presented yet. In this paper, we present a novel 3D model for cellsmigration. In the presented model, we assume that a cell follows two main processes: in the first process, it senses itsinterface with the substrate to determine the migration direction and in the second process, it exerts subsequent forces tomove. In the presented model, cell traction forces are considered to depend on cell internal deformation during the sensingstep. A random protrusion force is also considered which may change cell migration direction and/or speed. The presentedmodel was applied for many cases of migration of the cells. The obtained results show high agreement with the availableexperimental and numerical data.

Keywords: cell migration; 3D finite element simulation; cell mechanosensing; cell–substrate interaction; cell–cellinteraction

1. Introduction

Cell migration has recently received extensive attention due

to its important role in physiological, biological and

pathological processes such as tissue morphogenesis

(Juliano and Haskill 1993), cell differentiation (Pompe et al.

2009), cell proliferation (Behesti and Marino 2009), cancer

development (Suresh 2007; Ramis-Conde et al. 2008;

Levental et al. 2009; Ulrich et al. 2009), and wound healing

(Martin 1997), as well as in tissue engineering applications

(Ehrlich and Rajaratnam 1990).

The behaviour of cell during locomotion in a substrate is

not completely clear for scientists yet. However, it has been

conclusively demonstrated that biochemical, biophysical

and mechanical factors strongly affect cell migration

(Odell and Bonner 1986; Lo et al. 2000; Palsson 2001;

Penelope and Janmey 2005; Hadjipanayi et al. 2009;

Buxboim et al. 2010a, 2010b). In particular, mechanical

changes in cell substrate such as those on bound adhesive

ligands, topographical features and stiffness distribution

are thought to guide and control cell migration (Hadjipanayi

et al. 2009).

The physical process of cell migration involves a

number of coordinated events. There are three main

processes involved in cell migration (Shiu et al. 2004): the

first is sensing its environment, the second is the generation

of contractile forces by the actin–myosin apparatus in cell

cytoskeleton (CSK) driving forward the translocation of the

cell body and causing traction forces on its substrate, the last

one is the release of the adhesions at the rear (Lauffenburger

and Horwitz 1996). These steps involve the continuous

rearrangement of cytoskeletal elements and cell–extra-

cellular matrix (ECM) interactions. Briefly, these steps are

distinguished by the formation of new adhesions, the

development of traction, and the abandonment of old

adhesions (Lauffenburger and Horwitz 1996; Sheetz et al.

1997). Increasing intracellular forces or substrate stiffness

induces more stable cell–matrix adhesions, which explains

the cell tendency to move toward stiffer and/or more fixed

substrates (Chiquet et al. 2009).

The mechanisms behind network compaction are less

understood. Several works indicate that the traction

exerted during cell motility can continuously compact

the cell environment (Ehrlich and Rajaratnam 1990;

Andujar et al. 1992; Eastwood et al. 1996; Kanekar et al.

2000), others cite that matrix compaction is driven

primarily by cells exhibiting little locomotion (Roy et al.

1999). Studies of Shreiber et al. (2003) demonstrated

that depending on the compliance of the system, which

includes the mechanical properties of the gel as well as

mechanical restraints, both mechanisms of compaction

may be at play. They quantified cell migration by

modelling cell movement as a persistent random walk

q 2012 Taylor & Francis

*Corresponding author. Email: [email protected]

Computer Methods in Biomechanics and Biomedical Engineering, 2014

Vol. 17, No. 6, 678–693, http://dx.doi.org/10.1080/10255842.2012.710841

Publications

19

analysed with a generalised least squares regression

algorithm (Dickinson and Tranquillo 1993).

Experimental works by Lo et al. (2000) demonstrated

that a cell is able to determine substrate rigidity by

monitoring the magnitude of counter forces upon the

consumption of a given amount of energy, i.e. strong

mechanical feedback from the stiff substrate occurs after a

small reception of displacement. Since elastic energy is the

integration of forces along the distance, with the same

amount of energy consumption, soft substrates can

generate a weaker mechanical feedback but a longer

displacement (Lo et al. 2000). On the other hand, the

stronger mechanical feedback in stiffer substrates may

lead to the activation of stress-sensitive ion channels

(Lee et al. 1999). These responses may, in turn, regulate

the extent of protein tyrosine phosphorylation, the stability

of focal adhesions and the strength of contractile forces

(Pelham and Wang 1997).

Migration of high population of cells inside a substrate

is one of the main aspects of tissue formation (Dalton et al.

2001). There are several previous methods to model high

population of cells in a multicellular system, a good survey

can be found in Palsson (2001). One of the main problems

with some previous models is that they ignore to properly

balance the active movement forces generated by each

individual cell affecting cell–cell interaction (Savill and

Hogeweg 1997). Also some of them do not consider

effective forces such as traction, drag and protrusion forces

which act on a cell during migration (Nossal 1988).

Besides, most of them are 2D models (Graner and Glazier

1992; Brofland and Wiebe 2004; Brofland et al. 2007).

This study can be considered as an extension of the

previous model presented by Borau et al. (2010, 2011).

They developed a model to simulate single-cell migration

in 3D substrates but with several limitations which are

improved herein. It can be said that their model is

applicable only for a single cell. Besides, they considered

the cell as a hexahedral element, therefore, the cell has a

cubic form which is far from reality and may cause

inaccurate sensing and migration processes of the cell.

Moreover, reorientation of a cell is based on the projective

alignment with the resultant of the principal strain

directions. Reorientation according to minimum principal

strains or maximum principal stresses was not too accurate

in that paper, since the strains and stresses in the element

integration point were projected onto the nodes, being

therefore an average of the projected strains and stresses.

Here, for determination of the cell migration, we use a

criterion based on displacements of external nodes of the

cell with respect to its centroid (internal cell deformation).

In the presented model, not only is there no limitation for

the number of embedded cells and the number of elements

which represent the cell, but also a cell can have any shape

configuration. However, the presented simulations have

been carried out for a semi-sphere cell shape configuration

in case of surface locomotion while for a 3D migration a

spherical cell shape configuration is considered.

In this study, we investigate the effects of substrate

mechanical properties on cell migration as well as cell–

cell interactions. A 3D finite element model to simulate

cell migration within 3D substrates is developed and

extended to consider cell–cell interaction. Several

numerical experiments are presented to demonstrate the

predictive capability of the presented model for both cases

of single cell and of cell population. To verify our model,

the numerical experiments presented by Borau et al. (2010,

2011) were reproduced, the obtained results were totally

consistent with their results. Then, the presented model

was applied to a substrate with stiffness gradient and free

boundary surfaces (Hadjipanayi et al. 2009). The results

illustrate that the cell tendency is to migrate in the

direction of the stiffness gradient in a random path until

achieving an imaginary equilibrium plane (IEP) located

far from the free boundary surfaces. Once the cell arrives

to this IEP, it randomly moves around that plane.

Our model is also applicable for surface cell migration.

We used it to study the effect of substrate depth on cell

locomotion. The results demonstrate that the cell tends to

migrate towards minimum depth (Buxboim et al. 2010a,

2010b).

Since the phenomenon of cell–cell interaction is not

considered in previous numerical models, we extended our

model to represent the locomotion of cell population. In

case of only two cells, simulations show that there is a

tendency between them to migrate towards each other.

This phenomenon, in general, decreases their overall

migration velocity. Once they are in contact, they stay

together until the protrusion force changes their polaris-

ation direction and they may separate. This process is

frequently repeated while cells migrate towards stiffer

regions of the substrate. In the last experiment, 40 cells are

simultaneously embedded within a constant stiffness

substrate. As they start to move, their interaction causes

them to aggregate in slugs in the middle of the substrate.

This process can be considered as a step forward towards

tissue generation.

2. Model formulation

Cells have a special internal structure which is able to

sense the stiffness of the matrix in which they reside. For

instance, fibroblasts preferentially move towards stiffer

substrates (Ingber and Tensegrity 2003; Moreo et al.

2008). This phenomenon is known as mechanotaxis in

which a cell moves directionally following a mechan-

osensing process (Penelope and Janmey 2005). In the

mechanosensing step, the cell senses its substrate by

exerting a sensing force to diagnose its surroundings, and

consequently, gets some information about its substrate

rigidity. Once the cell has determined its surrounding

Computer Methods in Biomechanics and Biomedical Engineering 679

2.1. Computational modelling and analysis of mechanical conditions on cell locomotion and cell-cellinteraction.

20

mechanical conditions, it starts to pull itself towards the

stiffer and/or more fixed region.

The cellular elements with a relevant function in the

cell mechanosensing mechanics are the actin bundles, the

actomyosin contractile apparatus and the passive mech-

anical strength of the rest of the cell body, whose main

contribution is related to the action of the CSK

microtubules and the membrane (Figure 1) (Pelham and

Wang 1997; Palsson 2001; Penelope and Janmey 2005;

Moreo et al. 2008; Buxboim et al. 2010a). The cytoplasmic

CSK is linked with the external ECM through focal

adhesions and transmembrane integrins that are assumed

totally rigid for the present model. This scheme agrees

with the tensegrity hypothesis (Ingber and Tensegrity

2003), since deformation of external substrate is balanced

by tensile forces generated in the actin CSK. External

forces are also considered to be another possible cause of

the deformation of the substrate and cell.

The presented model is an extension of the one

presented by Borau et al. (2010, 2011). It can be used to

simulate adherent cells cultured on 2D substrates, cultured

in 3D hydrogels, on the surface of a scaffold or attached to

the ECM of a connective tissue, regardless of their real

environment. ECM and substrate will be used, therefore,

without distinction in this study.

In the same line of Borau et al. (2010, 2011), the 1D

model represented in Figure 2(a) can be particularised to

octahedral or hydrostatic stresses. Assuming that con-

tractile forces exerted by cells are isotropic (Moreo et al.

2008), the change in length of each element is then

interpreted as its corresponding volumetric strain

(Timoshenko and Goodier 1970). In such a case, the

characteristic spring constant can be identified with the

volumetric stiffness modulus of the representative

element. Then, the physical interpretation of the variables

in each branch of the model is as follows: Kpas, Kact and

Ksubs denote the stiffnesses of microtubules, myosin II and

substrate, respectively, e1 represents the minimum

volumetric strain, e2 the maximum volumetric strain of

the cell (Moreo et al. 2008), sact stands for the mean

contractile stress generated internally by the myosin II

machinery and transmitted through the actin bundles; spas

denotes the contractile stress supported by the passive

resistance of the cell, essentially corresponding to the CSK

microtubules and to the membrane; ss is the stress of the

ECM and f ext denotes the external forces. The effective

stress transmitted by the cell to the ECM, scell, is therefore

given by

scell ¼ sact þ spas: ð1ÞIt may be interpreted as this part of the active stress

that is not absorbed by the microtubules. Forces from both

microtubules, spas, and actin bundles, sact, are exerted to

the plaque proteins (Figure 1) (Moreo et al. 2008).

Thereby, scell can also be interpreted as the average cell

stress that bears the submembrane plaque in agreement

with the integrin-mediated mechanosensing hypothesis

(Bershadsky et al. 2003). In addition, e denotes local

volumetric strain. In the model herein, local strain is

computed from the deformation of the cell external nodes

along the direction of the traction force exerted at the

corresponding node. eact stands for the deformation of

the active contractile element. This deformation relates to

the fact that the real physical change in the overlap

between actin and myosin filaments occurs when active

forces are applied. Eventually, ea represents the

deformation of the actin bundles that promote the active

forces transmitted.

We approximated the cell unidimensional constitutive

behaviour by a simple linear elastic spring (Moreo et al.

2008), which is a reasonable simplification of cell–

substrate structure under moderate cell and substrate

Figure 1. Schematic diagram of the relevant mechanicalconstituents of a cell (Moreo et al. 2008).

Figure 2. Mechanosensing model of an adherent cell. (a)Mechanical model of the cell. Kact, Kpas and Ksubs denote thestiffness modulus of actin filaments, the passive components ofthe cell and the substrate, respectively. f ext stands for the externalforces applied to the cell or the substrate. (b) Dependence of thecontractile stress, sact, on the deformation of the contractileelement, eact. smax, stands for the maximum contractile stressexerted by the actin–myosin machinery, e1 and e2 are thecorresponding shortening and lengthening strains of thecontractile elements with respect to the unloaded length atwhich active stress becomes zero (Moreo et al. 2008).

S.J. Mousavi et al.680

Publications

21

strains. Besides, interaction between the actin and myosin

is considered as an active force arisen from a relative

sloping between actin and myosin filaments. It is

motivated by myosin cross-bridges on hydrolysis of

adenosine triphosphate (ATP). The reaction by which the

stored chemical energy transmits into high energy of

phosphoanhydride bonds in ATP (Spudich 2001) is

maximal for an optimal filament overlap and decreases

proportionally with a decrease in overlapping (Rassier

et al. 1999). Therefore, for calculation of the contractile

stress, sact, as a function of deformation of the contractile

elements, a simple piecewise linear constitutive model has

been used. As seen in Figure 2(b), if deformation of cell is

in the e1 2 e2 range, the active stress, sact, has effect on

scell, else it will be zero and scell will be equal to spas.

Therefore, the net stress, scell, transmitted to the ECM by a

single cell as a function of the ECM volumetric strain, e ,can be calculated as

scell ¼spas; e , e1 or e . e2;

spas þ sact;1; e1 # e # ~e ;

spas þ sact;2; ~e # e # e2;

8>><>>:

ð2Þ

where ~e ¼ smax=Kact and spas ¼ Kpase , while

sact;i ¼ Kactsmaxðe i 2 eÞKacte i 2 smax

; i ¼ 1; 2: ð3Þ

In the physical process of cell locomotion, the

contraction of the actin–myosin apparatus drives the

translocation of the cell body forward and causes traction

forces on the substrate (Lo et al. 2000; Shiu et al. 2004).

Actually, the movement of cells within a complex embryo

or organism is guided by a complex interplay between

chemical and physical signals like substrate stiffness (Lo

et al. 2000; Hadjipanayi et al. 2009), boundary conditions

and generated forces due to cell–cell and cell–substrate

interactions (Lo et al. 2000; Shiu et al. 2004). Anyway, the

simplified model described above is used here to predict

the cell migration as a function of cell internal

deformation.

The prominent aspect of the presented approach for

cell modelling is that the cell can have any shape and can

be represented by any number of finite elements. For this

purpose, an algorithm has been used to track the key

parameters required for migration at each time step

considering important processes for cell migration, such as

asymmetry of the cell, traction force and traction force

generation. Moreover, several aspects associated with the

substrate such as stiffness, boundary conditions and their

effects on the direction of cell locomotion have been taken

into account.

To calculate the velocity and the new position of the

cell within the substrate at each time step, the total net

force during cell movement is balanced. In this case, the

time step needed for model discretisation is equal to the

time of one cycle of cell migration, that is the time taken

by the cell to become sufficiently attached to its substrate

in its advancing front and simultaneously break the

adhesion with the substrate in the trailing back. It can

change regarding to the deformation of the cell, traction

and protrusion forces and the cell speed at every step. Time

of one complete cycle of cell migration for fibroblast,

epithelial or endothelial cells is around 600 s (Shiu et al.

2004).

The considered effective forces which act on cell and

substrate are traction forces, drag forces and protrusion

forces (Lo et al. 2000; Shiu et al. 2004; Penelope and

Janmey 2005). The traction forces are the result of traction at

the front and the rear of the cell and depend on the force per

ligand–receptor complex due to their different adhesive-

ness. These two forces at the back and front of the cell can be

mathematically represented by Zaman et al. (2005)

Ftracjf;b ¼ scellSzjf;b; ð4Þ

where scell, cell stress, can be calculated from Equation (2).

S stands for a proportionality model parameter with units of

area and z is the adhesiveness that takes into account

different numbers of receptors at the front and the rear of cell

and binding strength of these receptors to the ligand in the

ECM (Zaman et al. 2005) and is given by

z ¼ knc: ð5ÞThe value of z is different at the front and the rear of

the cell because there are different numbers of receptors

and the binding stress of each of those receptors may also

be different. k is the binding constant for the integrins at

the front and rear of the cell to the ligands in the ECM (in

mol21). For present simulations, we assume that the

binding constant is equal for both front and rear of the cell.

n is the total number of available receptors at the front or

back of the cell and it is assumed that nf . nb (in mol).

This means that when the cell polarizes the integrins will

be asymmetrically distributed on the cell surface (Zaman

et al. 2005). Finally, c represents the concentration of the

ligands at the leading edge of the cell in the ECM.

Consequently, Ftrac may be written as

Ftrac ¼ scellSðkncjf 2 kncjbÞ: ð6ÞThe second force which affects the cell is the resistive

force (drag force) which comes from the viscous resistance

to motility. In a Maxwell solid, the needed force for

deforming the ECM depends on the deformation rate and

accordingly the velocity. As the main objective here is to

imply a velocity-dependent opposing force associated

with the viscoelastic character of the cell surrounding

ECM, so, and for simplification, we assume the ECM as a

viscoelastic medium (Zaman et al. 2005). In this case, the



22

drag force can be defined as

Fdrag ¼ bmv; ð7Þwhere m is the effective viscosity of the viscoelastic

matrix, it is considered as a constant throughout the

substrate, and v denotes the velocity of cell relative to the

substrate. The constant b depends on the cell shape

(Zaman et al. 2005). In the ideal case of a spherical cell

moving through Newtonian, infinitely viscose medium, bcan be approximated as Zaman et al. (2005)

b ¼ 6pr; ð8Þwhere r is the radius of the cell.

It is necessary to note that if the cell migrates through a

purely elastic substrate, the force required for deforming

the matrix will not depend on the velocity. Therefore, a

more realistic representation of the opposing force would

be the summation of two contributions, one depending on

cell velocity and the other independent (James and Taylor

1969; Harris et al. 1980). This is why the protrusion force

has been introduced to be independent from cell velocity.

Therefore, an effective guidance system appears in which

cells send out local protrusions to probe the mechanical

properties of the environment (Lo et al. 2000). This force

is generated by actin polymerisation and cell or substrate

attachments at the new location of lamellipodia protrusion,

distinct from the cytoskeletal contractile force transmitted

to the ECM (Zaman et al. 2005). It is a random force

whose order of magnitude is the same to that of the traction

force and less than it at every time step (James and Taylor

1969; Ramtani 2004). Therefore, force equilibrium should

be satisfied during cell locomotion, hence

Fdrag ¼ Ftrac þ Fprot: ð9ÞThrough this presented work, neither degradation nor

remodelling of the ECM during cell locomotion is

considered.

2.1 Single-cell orientation

At every step, the cell exerts a sensing force to diagnose its

environment and hence it can determine the direction of

migration within the substrate. We suppose that this

sensing force is exerted at each external finite element

node of the elements that represent the cell towards the cell

centroid (Figure 3(b)). The deformed cell subjected to

those sensing forces is represented by dotted lines in

Figure 4. So, the internal strain of the cell for each external

node of the cell can be written as

ecell ¼ AB

OA: ð10Þ

Once the displacements of all nodes are calculated,

information needed to compute ecell is available. Using

Equation (2), scell at every external node of the cell can be

calculated. Therefore, at each external node of the cell,

Figure 3. (a) 3D sphere-shaped configuration of the cell. (b) Exerted sensing forces at every external node of the cell towards itscentroid.

Figure 4. Deformed cell (dashed line) subjected to sensingforces. epol stands for the unit vector of the polarisation directioncalculated via resultant traction force, Ftrac

R , and randomprotrusion force, Fprot. It is of interest to note that the effect ofthe resultant traction force on polarisation direction is moreimportant than the effect of the random protrusion force.


Publications

23

the traction force vector can be represented by

Ftraci ¼ Ftrac

i ei; ð11Þwhere Ftrac

i is the magnitude of the traction force

corresponding to the ith external node of the cell obtained

from Equation (6). ei is a unit vector standing along the

ith external node and the cell centroid which can be

obtained by

ei ¼ xo 2 xi

kxo 2 xik ; ð12Þ

where xo is the cell centroid position vector and xi is theposition vector of the ith external node of the cell.

Consequently, the resultant traction force, FtracR , can be

calculated as

FtracR ¼

Xni¼1

Ftraci ; ð13Þ

where n is the number of the external node of the cell.

To calculate the magnitude of the cell velocity, the

vector of the protrusion force should be estimated at every

step. It can be defined as

Fprot ¼ lerand ð14Þerand is a random unit vector. l is the magnitude of the

protrusion force which is estimated as

l ¼ kkFtracR k; ð15Þ

where k is a random number, 0 , k , 1. It is important to

note that the magnitude of the protrusion force, l, shouldbe less than the traction force (Lo et al. 2000; Zaman et al.

2005).

From Equations (7) and (9), the cell velocity can be

defined as

v ¼ kFtracR þ Fprotk

bm: ð16Þ

Thereby, at every time step (Figure 5), the distance

through which the cell migrates to locate in its new

position can be calculated. Hence, the displacement vector

of the cell can be defined as

d ¼ vtepol; ð17Þ

where t is the time step of simulation and epol is a unit

vector which represents the direction of cell migration. It is

important to note that, at every time step, the internal

deformation at every external node of the cell, caused by

the sensing force, is negative (cell exerts contraction

forces towards its centroid and always tries to compress

itself). Nodes with less internal deformation will have a

higher traction force (note that the cell internal

deformation has to be within the hatched area of

Figure 2(b)). As all the traction forces are acting towards

the cell centroid, the resultant of these traction forces will

have the direction of minimum internal deformation. So

that, the resultant of the traction force opposite direction

and random protrusion force presents the polarisation

direction of the cell. Therefore, the unit vector of cell

polarisation, epol, can be defined as

epol ¼ Fprot 2 FtracR

kFprot 2 FtracR k : ð18Þ

It is remarkable that according to the effect of the

random protrusion force, Fprot, the cell will move towards

stiffer and/or more fixed region of the substrate (minimum

deformation) in a random directed motion.

2.2 Cell–cell interaction

The same previous formulation is used to define traction

force, protrusion force, velocity and reorientation of each

individual cell. A model that defines the interaction of

cells will be presented along this section. Let us define rijas a vector passing through centroid of two cells i and j

(Figure 6(a)):

rij ¼ rj 2 ri: ð19ÞA useful simplification to avoid interference of two

cells is that the magnitude of rij should be greater than or

equal to the cell diameter. In reality, the cells inside a

multicellular system do not preserve a spherical shape but

deform to be tangent to each other and cover all parts of

Figure 5. Computational algorithm of the cell migration. First,the cell senses its environment. By information evaluated at thatsensing step, cell can polarise and migrate after calculatingtraction force and estimating protrusion force.



24

the matrix (Palsson 2001). In our case and for

discretisation, when two cells touch each other they can

have a maximum of four common nodes (Figure 6(b)).

In vivo, the cell drives out a pseudopod to sense its

environment better. Once the cell finds the stiffer region of

the substrate, it pulls up whole body in the direction of the

pseudopod (Taylor et al. 1982). Therefore, when two or

several cells touch each other, the common points of both

cells (for instance, nodes n1 : n4 in Figure 6(b)) are not

able to drive out the pseudopod to substrate (Taylor et al.

1982; Brofland and Wiebe 2004). Therefore, for two or

more cells, we assume that the cells do not exert any

sensing force at those nodes unless they get separated

again due to the protrusion force (see Figure 6(b)). It is

worth to note that in such a situation, these common nodes

do not have any role to sense their environment but traction

forces in those nodes are not zero.

3. Numerical experiments

During all the following experiments, a user subroutine

through the commercial software Simulia-ABAQUS FEA

6.10 was used (Hibbit et al.). The presented algorithm

(Figure 5) was run for about 100 steps with every

timescale approximately equal to 600 s, roughly the time

needed to complete one cell migration (Zaman et al. 2005).

It is important to note that the timescale is variable

depending on cell velocity. Of course, there exists the

possibility that during several steps, the velocity of the cell

is too slow so that the displacement vector will not be high

enough to move the cell to a new position, so the cell

remains in the same location. All the used parameters are

summarised in Tables 1 and 2. The considered substrate

dimensions are 400 £ 200 £ 200mm. A 3D regular mesh

of hexahedrons is used with 16,000 elements and 18,081

nodes. We assume a sphere-shaped configuration for the

cell (Figure 3) with a total number of external nodes of 24,

although it may change to fit any selected configuration of

the cell. The considered sensing force is about 1028 N,

which can create a measurable deformation in the

surrounding matrix in the range of 30–50mm radius,

depending on the substrate stiffness (Ramtani 2004). A

small sphere will visualise the cell centroid at every time

step to indicate the cell position in Figures 7, 8, 12, 14 and

Figure 6. (a) Distance between the centroids of two cells. (b) Interaction of two cells when they contact each other. The distancebetween their centroid is equal to the proposed cell diameter. Here, for assumed shape of the cell, two cells can have a maximum of fourcommon nodes in this situation.

Table 1. Cell parameters.

Description Symbol Value Refs

Stiffness of microtubules Kpas 2.8 kPa Schafer and Radmacher (2005)Stiffness of myosin II Kact 2 kPa Schafer and Radmacher (2005)Maximum strain of the cella e1 0.09 Ramtani (2004)Minimum strain of the cella e2 20.09 Ramtani (2004)Maximum contractile stress exerted byactin–myosin machinery

smax 0.1 kPa Oster et al. (1983),Penelope and Janmey (2005)

Surface area of the cell S 800mm2 Buxboim et al. (2010a)Binding constant at front and rear of the cell kf ¼ kb 108mol21 Zaman et al. (2005)Number of available receptors onthe front

nf 1.5 £ 105 Zaman et al. (2005)

Number of available receptors onthe rear

nb 105 Zaman et al. (2005)

Concentration of the ligands atrear and front of the cell

c 1025mol Zaman et al. (2005)

a These parameters depend on sensing force and substrate stiffness of the cell (Ramtani 2004).


Publications

25

16. In the case of high cell population, for better

visualisation, cells will visualise as a sphere with real

dimensions (Figure 18).

3.1 Two different stiffness substrates with differentboundary conditions

In this section, we reproduce the same experiments

proposed in Borau et al. (2010, 2011). The substrate is

divided into two parts with different stiffnesses. The

dimensions of both stiff and soft parts are the same

(200 £ 200 £ 200mm) but with three different boundary

conditions. Figure 7 shows the obtained results in full

agreement with their findings.

In Figure 7(a),(b), the surface perpendicular to the x-

axis in the stiffer side is fully constrained (zero

displacements) whereas other surfaces are free (zero

stress). Initially, the cell is located near to the free surfaces

in the soft part in which the cell senses maximum internal

deformation. When the cell starts to sense the substrate, it

recognises the stiffer part (left-hand side in Figure 7(a)). It

should be noted that the constrained surface increases the

apparent stiffness around it. Therefore, after the cell enters

into the stiffer region of the substrate, it moves towards the

constrained surface. Once the cell catches the constrained

surface, it keeps moving around it in random movement

and does not deviate far from this constrained surface.

In Figure 7(c),(d), perpendicular surfaces to the x-axis

in both soft and stiff regions were constrained. In such a

situation, an imaginary equilibrium plane (IEP) (the dotted

line in Figure 7(c),(d)) appears through where the cell

changes its migration behaviour. Around this plane, the

cell receives different signals from constrained surfaces or

from different stiffness regions. The location of this IEP

slightly changes between different simulations due to

random protrusion force. In this case, when the cell is

located close enough to the constrained surface in the

softer side (right-hand region of the IEP), the signal

coming from that constrained wall is higher, so it moves

towards this constrained boundary of the softer side

(Figure 7(c)). Alternatively, when the cell is placed on the

other side of the IEP, it firstly senses the stiffness of the

stiffer part of the substrate (left-hand region) so that it

migrates towards the stiffer part and then towards the

constrained surface in this side. In all the cases, when the

cell arrives to the constrained surface in the stiffer or softer

part, it keeps moving near to this surface randomly. If the

cell is initially positioned in the stiffer part, it never goes to

the softer part of the substrate. It is remarkable that if the

cell is placed close to the IEP, its behaviour depends on the

relative amplitude of the protrusion force in the initial

steps of its movement.

In Figure 7(e), when the cell is located close enough to

the free surfaces in the stiffer side of the substrate, it moves

away towards the interior of the substrate. Once the cell

finds the first IEP in the stiffer region, it keeps moving near

to it randomly. If the cell is initially positioned between

two IEPs (Figure 7(f)), no matter if in the stiffer or softer

parts of the substrate, it again moves towards the IEP in the

stiffer part. By contrast, if the cell is placed in the right-

hand side of the IEP, the migration of the cell changes and

the cell moves towards the constrained surface in the softer

side as seen in Figure 7(g).

3.2 Effect of stiffness gradient

To fully understand the effect of stiffness on cell

migration, we applied the proposed model to a substrate

with linear stiffness gradient which changes from 100 kPa

at x ¼ 0 to 200 kPa at x ¼ 400mm through which all the

substrate surfaces are considered free.

Hadjipanayi et al. (2009) analysed the effect of a 3D

substrate with linear stiffness gradient on cell migration.

They divided this substrate into soft, middle and stiff

regions. First, they inserted about the same number of cells

inside these zones. Since the number of cells was few, they

assumed that there was no interaction among cells. After

6 days, they observed a significant difference in cell

concentration in these three regions. They observed that

the higher cell concentration resulted in the stiffer part,

whereas the least concentration was identified in the softer

part.

In our simulation, and as expected when a cell is

placed in this substrate, the cell tendency is to migrate

towards the direction of the higher stiffness (Figure 8).

When all boundary surfaces are considered free, the cell

randomly moves around an IEP far enough from those free

surfaces located in the stiffer side. In this case, the results

depend on neither the initial location of the cell (Figure 8)

nor the stiffness gradient. Here, only the results of the

Table 2. Substrate properties.

Description Symbol Value Refs

Substrate elastic modulus (stiff part)a E1 200 kPa Zaman et al. (2005)Substrate elastic modulus (soft part) E2 100 kPa Zaman et al. (2005)Poisson ratio n 0.3 Akiyama and Yamada (1985), Ulrich et al. (2009)Viscosity m 1000 Pa s Akiyama and Yamada (1985), Zaman et al. (2005)

a This is an elastic modulus of the stiff part of the substrate when the substrate is composed of two stiff and soft parts.



26

simulation corresponding to two initial different locations

of the cell are presented (Figure 8). This simulation was

repeated for several values of the stiffness gradient, and

the initial position of the cell and all results were

consistent and in agreement with the experimental results

(Hadjipanayi et al. 2009).

While the cell migrates from a soft part to a stiffer part,

the nodal traction forces at external nodes increase. This

Figure 7. Cell migration in two different stiffness substrates. In (a) and (b), the surface x ¼ 0 is fully constrained. The cell starts to movefrom the corner with free surfaces and moves towards constrained plane at stiff part. For (c) and (d), both surfaces perpendicular to the x-axisare constrained. In this case, there exists an IEP through where the cell changes the direction of migration. In (e), (f) and (g), the surface in thesofter part has been restrained whereas the rest are free. Two IEPs appear, one in the stiffer part and the other in the softer part.


Publications

27

happens because in stiffer regions, internal strain of cell

decreases while cell stress increases (hatched area in

Figure 2(b)). In Figure 9, the nodal traction force

corresponding to one of the external nodes of the cell

has been plotted. On the contrary, the net traction force

affecting cell migration decreases (Figure 10). This

happens because while the nodal traction forces increase,

differences between them decrease and consequently the

net traction force decreases. This explains why the cell in

stiffer parts generally remains round and symmetric

whereas it exerts higher nodal traction forces. This result is

corroborated by the findings of Ehrbar et al. (2011). It is

worth to remark that curve fluctuation in Figures 9 and 10

is due to the effect of the cell protrusion force.

Figure 11 shows how the overall cell velocity

decreases during cell migration towards stiffer regions.

This means that the low net traction force in stiffer zones

causes low cell speed which can even stop cell migration

in very dense substrates (Zaman et al. 2005; Ehrbar et al.

2011). On the other hand, the oscillations of the cell

velocity in Figure 11 seem to have higher amplitude than

that of nodal traction force and net traction force in Figures

9 and 10. This is because the protrusion force affects not

only the direction of cell migration but also the magnitude

of the cell velocity. It is clear that when the cell approaches

the IEP, after about 80 time steps, the studded parameters

tend to be more stable since the local strains of the cell do

not change too much.

Figure 8. Cell migration in a substrate with a stiffness gradient and free boundary surfaces. In the first case (a) and (b), cell has beenembedded in one of the corners of the substrate (softer side), it migrates towards the IEP (dotted line) far enough from free surfaces andkeeps moving randomly around it. In the second case (c) and (d), the cell starts to migrate from the stiffer side towards the softer one untilit reaches the IEP and keeps moving around it.

Figure 9. The effect of substrate stiffness on the nodal tractionforce of the cell during migration.



28

3.3 Effect of substrate depth

The proposed model was applied for a substrate with a

slope as seen in Figure 12 to study the effect of substrate

depth on cell migration. Here, the sloped surface has been

constrained and the rest of surfaces are free. Elastic

modulus of the substrate is assumed to be 100 kPa. In this

experiment, the cell moves over the substrate surface.

Because the sloped surface is constrained, the cell will

sense less internal deformation for lower depth. The cell

started the migration process in a point on the substrate

surface that has maximum depth. When the cell exerts

sensing forces, it recognises the direction of the slope,

therefore, it migrates towards minimum depth in a random

path. Figure 13 demonstrates an intense difference of

sensing domain between the substrate surface and

substrate depth. Here, since the cell exerts sensing force

at the substrate surface, the z-component of the sensing

force is close to zero, so that the cell cannot sense too

deep. Consequently, due to this limitation, the speed of

propagation of the cell on the surface is less than that of the

cell inside 3D domains. This explains why the cell moves

more randomly in surface migration (Lo et al. 2000).

3.4 Interaction between two cells

As cited in previous sections, the developed model can

simulate cell migration with any number of cells in the

populations. Hence, to fully understand the interaction

between two cells, a substrate with the same dimensions of

the one with stiffness gradient is employed to simulate the

behaviour of two cells in the same substrate (Figure 14).

All boundary surfaces of the substrate are considered free.

The substrate elastic modulus is constant and equal to

100 kPa. Near to the free surfaces, one cell is located in the

top corner of the substrate and another in the bottom corner

of the substrate. It should be again mentioned that red and

yellow spheres represent the centroid of the each cell. As

we observe in Figure 14, both cells migrate in a random

path towards each other. Once they sense each other in the

middle of the substrate, they keep moving around each

other randomly. As mentioned earlier, the cell exerts

contraction sensing forces at its external nodes towards its

centroid to feel its environment. So, when cells are

sufficient near to each other, the region between them will

be under tension (Figure 15). Consequently, cells will feel

less internal deformation in this direction. They will detect

this zone as stiffer region of the substrate and migrate

towards each other. Once they contact each other as long

as their polarisation direction is either towards each other

or in the same direction they remain in contact. When their

polarisation directions become different, they separate

until they regain contact and so on.

To better understand the effect of this phenomenon of

cell migration, simulations were repeated for the previous

substrate with stiffness gradient. The results are in

agreement with migration of one cell inside the substrate

with constant stiffness as seen in Figure 16. First, the cells

try to reach each other and after crossing the first quarter of

the x-axis, they continue to move together towards the

stiffer region of their substrate. Again, and as in the case of

individual cell, there exists an IEP where cells maintain

Figure 11. The effect of substrate stiffness on the cell migrationvelocity.

Figure 10. The effect of substrate stiffness on the net tractionforce of the cell during migration. Figure 12. Migration of the cell due to the variance of substrate

depth. The sloped surface is constrained.


Publications

29

moving together randomly around it. Figure 17 shows the

curve of cell velocity for the migration of one cell and two

cells. To compare the cell velocities in both cases, we fitted

an average curve for each cell velocity to eliminate the

velocity fluctuation due to the protrusion force. The

simulations were repeated several times and the result was

similar. The figure shows that the overall cell velocity in

case of single-cell migration is less than two-cell

migration. This is because in the case of two cells, the

nodes near to the stretched area between the two cells

experience less deformation (Figure 15). On the contrary,

the nodes that are away from this area in the other side of

the cell will experience more deformation and less traction

force. The local cell velocity for two-cell migration is

higher than the local cell velocity for single-cell migration.

In this case, the number of steps needed by the cell to catch

the IEP was about 130 time steps which is higher than in

case of one cell.

3.5 Cell population

In this experiment, the proposedmodel is used to simulate a

cell population with 40 cells simultaneously embedded

inside a substrate with the same dimensions as that of the

previous experiment. All boundary surfaces of the substrate

are considered free. Elastic modulus is considered to be

constant throughout the substrate (100 kPa). At the

beginning, the cells are randomly distributed near

the boundary surfaces (Figure 18(a)). When cells exert

Figure 13. Stress and deformation in the x-direction during sensing process.

Figure 14. Interaction between two cells inside a substrate withconstant elastic modulus. All boundary surfaces of the substrateare considered free. Cells start to move from two corners of thesubstrate. They migrate towards the centre of the substrate andkeep moving around each other randomly.

Figure 15. von Mises stresses due to mechanosensing of twocells during migration. The stretched zone between two cells isclear because of their interaction.



30

the sensing force to check their environment, they recognise

the middle of the substrate as a more stable region. Figure

18 represents locomotion of the cells for different time

steps. Here, for better representation of cell –cell

interaction and their contacts, we represented the proposed

sphere-shaped configurationof the cells in each step.During

migration, there exist several stretched zones between cells

that affect cell locomotion. So that, in primary steps of

migration, they aggregate in small groups (Figure 18).

Joining these slugs to each other, several big slugs of the

cells are formed. Afterwards, all these slugs contact each

other in the middle of the substrate. Internal cells which are

enveloped by several cells stop moving, but boundary cells

may move around enveloped cells. In this simulation, the

cells respond similarly and consistently as previous

experimental (Odell and Bonner 1986) and numerical

works (Palsson 2001).

4. Conclusion

We extended a previous 2Dmodel of cell migration to take

into account features such as 3D movement, cell shape and

cell–cell interactions with the initial goal of analysing how

substrate stiffness guides cell migration and how cell–cell

interaction affects this process. The presented results are

qualitatively validated by experimental observations (Odell

and Bonner 1986; Lo et al. 2000; Penelope and Janmey

2005; Hadjipanayi et al. 2009; Buxboim et al. 2010b) and

with previous numerical models (Palsson 2001; Borau et al.

2010, 2011; Buxboim et al. 2010a).

We extended our simulations to understand the role of

the boundary conditions, stiffness gradient, substrate depth

and cell–cell interaction in cell migration. As observed in

the results shown above, any change in the boundary

conditions may change the final position of the cell and

locomotion path. When there are constrained surfaces, the

primary position of the cell becomes important. In general,

the cell migrates towards stiffer or more fixed parts in its

neighbours. But, as observed in the presented experiments,

in some cases, the cell cannot move towards the stiffer

region of the substrate. For example, when the initial

location of the cell is close to a constrained wall in the

softer part, the signal from this wall may be higher than

that coming from the stiffer region. In such situations and

depending on the boundary conditions, it may appear one

(Figure 7(c),(d)) or two (Figure 7(e),(g)) IEPs which

separate different parts in which the cell displays different

behaviours.

For a substrate with stiffness gradient, there exists an

IEP towards which the cell always tends to migrate. Once

the cell catches it, it keeps moving around it far from free

boundary surfaces. In this case, the primary position of the

cell is not important. Again and as exception, the cell tends

to migrate from stiffer to softer regions (Figure 8(c),(d)).

The obtained results demonstrate that during cell

migration from a softer towards a stiffer region, nodal

traction forces increase while net traction force and cell

velocity decrease. Reduction in net traction force causes

the cell to be more rounded and symmetric (Ehrbar et al.

2011). This finding describes why the cell stability

Figure 16. Interaction between two cells inside a substrate with stiffness gradient. All the boundary surfaces of the substrate areconsidered free. Two cells start to move from the two corners of the substrate.

Figure 17. Comparison of cell migration velocities for two cellsand single cell.


Publications

31

enhances in stiffer regions. In very stiff substrates,

generated net traction forces may not be enough to move

the cell to a new position.

Applying the proposed model for an individual cell

migrating over the matrix surface, the results demonstrate

that the cell tendency is to migrate towards minimum

depth (Figure 12). However, its sensing radius on surface

movement is higher in comparison with its depth feeling

(Figure 13), notably depending on several factors as cell

type, sensing forces, depth of substrate and matrix

stiffness. Our finding in this case is also qualitatively

consistent with previous experimental (Buxboim et al.

2010a) and numerical results (Buxboim et al. 2010b).

The obtained results demonstrate that interaction

between two cells inside a substrate causes decay in

their mean velocity towards stiffer regions due to the

tendency of cells to maintain contact. This phenomenon

occurs due to the presence of a stretched region between

Figure 18. Interaction between 40 cells inside a substrate with free boundary surfaces. During the locomotion, cells tend to contactforming small slugs and then they attach each other in centre the of the substrate to create an aggregation of cells.



32

the two cells created by contraction forces exerted by cells.

As expected, this process also happens for cell populations

with a higher number of cells. When there are several cells

inside a substrate, if the zone between one cell and the

other in its neighbourhood is stretched enough such that

the signal coming from this stretched region becomes

higher than that coming from any other mechanical

conditions, cells tend to be gathered forming small slugs.

In the case of free boundary substrate, these created slugs

will then attach to each other and aggregate in the middle

of substrate (Figure 18). This process can be seen in

sorting of different cells (Palsson 2001) or in modulation

of epithelial tissue (Dalton et al. 2001).

As the magnitude of the cell velocity, nodal traction

force and net traction force depend on the cell type and the

elastic modulus of matrix, the presented model can be a

helpful tool to predict all these parameters. Besides, it is

capable to predict the cell behaviour for any kind of cell

shape and substrate. So, we think that the proposed model

can be used to simulate in vivo or in vitro experiments to

anticipate the behaviour of single or high population of

cells. In fact, more sophisticated experiments not only can

verify or refute predictions of numerical models such as

the one described here, but also can determine other

effective factors that may act on cell migration in vivo.

However, additional experimental works are needed to

fully understand the exact role of the mechanical

conditions on cell behaviour and cell–cell interaction.

Acknowledgements

The authors gratefully acknowledge the financial support fromthe Spanish Ministry of Science and Technology (CYCITDPI2010-20399-C04-01) and CIBER-BBN initiative. CIBER-BBN is an initiative funded by the VI National R&D&I Plan2008–2011, Iniciativa Ingenio 2010, Consolider Program,CIBER Actions and financed by the Instituto de Salud CarlosIII with the assistance from the European Regional DevelopmentFund.

References

Akiyama SK, Yamada KM. 1985. The interaction of plasmafibronectin with fibroblastic cells in suspension. J Biol Chem.260:4492–4500.

Andujar MB, Melin M, Guerret S, Grimaud JA. 1992. Cellmigration influences collagen gel contraction. J SubmicroscCytol Pathol. 24:145–154.

Behesti H, Marino S. 2009. Cerebellar granule cells: insights intoproliferation, differentiation, and role in medulloblastomapathogenesis. J Appl Physiol. 41:435–445.

Bershadsky AD, Balaban NQ, Geiger B. 2003. Adhesion-dependent cell mechanosensitivity. Annu Rev Cell Dev Biol.19:677–695.

Borau C, Doweidar MH, Garcia-Aznar JM. 2010. Modeling ofcell migration and mechano-sensing in 3d. 17th Congress ofthe European Society of Biomechanics. Edinburgh, UK.

Borau C, Kamm RD, Garcıa-Aznar JM. 2011. Mechano-sensingand cell migration: a 3D model approach. J Phys Biol. 8:1078–1088.

Brofland GW, Wiebe CJ. 2004. Mechanical effects of cellanisotropy on epithelia. Comput Methods Biomech BiomedEng. 7:91–99.

Brofland GW, Viens D, Veldhuis JH. 2007. A new cell-basedFE model for the mechanics of embryonic epithelia.Comput Methods Biomech Biomed Eng. 10:121–128.

Buxboim A, Ivanovska IL, Discher DE. 2010a. Matrix elasticity,cytoskeletal forces and physics of the nucleus: how deeply docells ‘feel’ outside and in? J Cell Sci. 123:297–308.

Buxboim A, Rajagopal K, Brown AEX, Discher DE. 2010b. Howdeeply cells feel: methods for thin gells. J Phys CondensMatter. 22(19):194116(10pp).

Chiquet M, Gelman L, Lutz R, Maier S. 2009. Wound healing:aiming for perfect skin regeneration. Biochim Biophys ActaJ. 1973:911–920.

Dalton BA, Walboomers XF, Dziegielewski M, Evans MDM,Taylor S, Jansen JA, Steele JG. 2001. Modulation ofepithelial tissue and cell migration by microgrooves.Phil Trans Roy Soc London. 56:195–207.

Dickinson RB, Tranquillo RT. 1993. Optimal estimation of cellmovement indices from the statistical analysis of celltracking data. Am Inst Chem Eng J. 39:1995–2010.

Eastwood M, Porter R, Khan U, McGrouther G, Brown R. 1996.Quantitative analysis of collagen gel contractile forcesgenerated by dermal fibroblasts and the relationship to cellmorphology. J Cell Physiol. 166:33–42.

Ehrbar M, Sala A, Lienemann P, Ranga A, Mosiewicz K,Bittermann A, Rizzi SC, Weber FE. 2011. Elucidating therole of matrix stiffness in 3D cell migration and remodeling.Biophys J. 100:284–293.

Ehrlich HP, Rajaratnam JB. 1990. Cell locomotion forces versuscell contraction forces for collagen lattice contraction: anin vitro model of wound contraction. Tissue Cell. 22:407–417.

Graner F, Glazier J. 1992. Simulations of biological cell sortinusing a two-dimensional extended potts model. Phys RevLett. 69:2013–2016.

Hadjipanayi E, Mudera V, Brown RA. 2009. Guiding cellmigration in 3D: a collagen matrix with graded directionalstiffness. Cell Motil Cytoskeleton. 66:435–445.

Harris AK, Wild P, Stopak D. 1980. Silicone rubber substrata: anew wrinkle in the study of cell locomotion. Science.208:177–179.

Hibbit HD, Karlson BI, Sorensen P. 2006. ABAQUS TheoryManual, Version 6.10. Pawtucket, Rhode Island, USA.

Ingber DE, Tensegrity I. 2003. Cell structure and hierarchicalsystems biology. J Cell Sci. 116:1157–1173.

James DW, Taylor JF. 1969. The stress developed by sheets ofchick fibroblasts in vitro. Exp Cell Res. 54:107–110.

Juliano RL, Haskill S. 1993. Signal transduction from theextracellular matrix. J Cell Biol. 120:577–585.

Kanekar S, Borg TK, Terracio L, Carver W. 2000. Modulation ofheart fibroblast migration and collagen gel contraction byIGF-I. Cell Adhes Commun. 7:513–523.

Lauffenburger DA, Horwitz AF. 1996. Cell migration: aphysically integrated molecular process. Cell. 84:359–369.

Lee J, Ishihara A, Oxford G, Johnson B, Jacobson K. 1999.Regulation of cell movement is mediated by stretch-activatedcalcium channels. Nature. 400:382–386.

Lo CM, Wang HB, Dembo M, Wang YL. 2000. Cell movementis guided by the rigidity of the substrate. Biophys J. 79:144–152.


Publications

33

Levental KR, Yu H, Kass L, Lakins JN, Egeblad M, Erler JT,Fong SF, Csiszar K, Giaccia A, Weninger W, Yamauchi M,Gasser DL, Weaver VM. 2009. Matrix cross linking forcestumor progression by enhancing integrin signaling.Cell. 139:891–906.

Martin P. 1997. Wound healing: aiming for perfect skinregeneration. Science. 276:75–81.

Moreo P, Garcia-Aznar JM, Doblare M. 2008. Modelingmechanosensing and its effect on the migration andproliferation of adherent cells. Acta Biomater. 4:613–621.

Nossal R. 1988. On the elasticity of cytoskeletal networks.Biophys J. 53:349–359.

Odell GM, Bonner JT. 1986. How the dictyostelium discoideumgrex crawls. Phil Trans Roy Soc London. 312:487–525.

Oster GF, Murray JD, Harris AK. 1983. Mechanical aspects ofmesenchymal morphogenesis. J Embryol Exp Morph. 78:83–125.

Palsson E. 2001. A three-dimensional model of cell movement inmulticellular systems. Future Gen Comput Syst. 17:835–852.

Pelham RJ, Jr, Wang YL. 1997. Cell locomotion andfocal adhesions are regulated by substrate flexibility.PNAS. 94:13661–13665.

Penelope CG, Janmey PA. 2005. Cell type-specific responseto growth on soft materials. J Appl Physiol. 98:1547–1553.

Pompe T, Glorius S, Bischoff T, Uhlmann I, Kaufmann M,Brenner S, Werner C. 2009. Dissecting the impact of matrixanchorage and elasticity in cell adhesion. Biophys J. 97:2154–2163.

Ramis-Conde I, Drasdo D, Anderson ARA, Chaplain M. 2008.Modeling the influence of the e-cadherin-b-catenin pathwayin cancer cell invasion: a multiscale approach. Biophys J. 95:155–165.

Ramtani S. 2004. Mechanical modelling of cell/ECM andcell/cell interactions during the contraction of a fibroblast-populated collagen microsphere: theory and model simu-lation. J Biomech. 37:1709–1718.

Rassier DE, MacIntosh BR, HerzogW. 1999. Length dependenceof active force production in skeletal muscle. J Appl Physiol.86:1445–1457.

Roy P, Petroll WM, Cavanagh HD, Jester JV. 1999. Exertion oftractional force requires the coordinated up-regulation ofcell contractility and adhesion. Cell Motil Cytoskeleton. 43:23–34.

Savill N, Hogeweg P. 1997. Modeling morphogenesis: fromsingle cells to crawling slugs. J Theor Biol. 184:229–235.

Schafer A, Radmacher M. 2005. Influence of myosin ii activityon stiffness of fibroblast cells. Acta Biomater. 1:273–280.

Sheetz MP, Felsenfeld D, Galbraith CG, Choquet D. 1997.Cell migration as a five-step cycle. Biochem Soc Symp. 65:233–243.

Shiu YT, Li S, Marganski W, Uami S, Schwartz MA, Wang YL,Dembo M, Chien S. 2004. Rho mediates the shear-enhancement of endothelial cell migration and tractionforce generation. Biophys J. 86:2558–2565.

Shreiber DI, Barocas VH, Tranquillo RT. 2003. Temporalvariations in cell migration and traction during fibroblast-mediated gel compaction. Biophys J. 84:4102–4114.

Spudich JA. 2001. The myosin swinging cross-bridge model.Nat Rev Mol Cell Biol. 2:387–392.

Suresh S. 2007. Biomechanics and biophysics of cancer cells.Acta Biomater. 3:413–438.

Taylor DL, Heiple J, Wang YL, Luna EJ, Tanasugarn L, Brier J,Swanson J, Fechheimer M, Amato P, Rockwell M, Daley G.1982. Cellular and molecular aspects of amoeboid move-ment. CSH Symp Quant Biol. 46:101–111.

Timoshenko S, Goodier JN. 1970. Theory of elasticity.New York: McGraw-Hill Publishing Co.

Ulrich TA, Pardo EMDJ, Kumar S. 2009. The mechanicalrigidity of the extracellular matrix regulates the structure,motility, and proliferation of glioma cells. Cancer Res. 69:4167–4174.

Zaman MH, Kamm RD, Matsudaira P, Lauffenburgery DA.2005. Computational model for cell migration in three-dimensional matrices. Biophys J. 89:1389–1397.



34


Journal of Molecular & Cellular Biomechanics

2013

2.2 Cell migration and cell-cell interaction in the presence

of mechano-chemo-thermotaxis.

CCell Migration and Cell-Cell Interaction in the Presence ofMechano-Chemo-Thermotaxis

S.J. Mousav M.H. Doweida M. Doblar

Abstract Although there are several computational models to explain the trajec-tory that cells take during migration, little attention has been paid to consider thecell migration in a multi-signalling system. A generalized model of cell migrationand cell-cell interaction based on multi-signal analysis is presented. Besides, weinvestigate the spatiotemporal cell-cell interaction by means of mechano-chemo-thermo tactic cues. It is assumed that formation of a new adhesion generatestraction forces which are proportional to the transmitted stresses by the cell tothe extracellular matrix. Thus, the cell velocity and polarization direction are cal-culated based on equilibrium of effective forces through cell motility. It is assumedthat existence of chemotactic and thermotactic cues beside of mechanotaxis cuecontrol the direction of resultant traction force without any change in its magni-tude. The model enable us to predict the trajectory of migrating cells, spatial andtemporal distribution of net traction force and cell velocity. Results indicate thatthe tendency of the cells is firstly to reach each other and then migrate towards anImaginary Equilibrium Plane located near the source of the signals. The positionof the this plane is sensitive to the gradient slope and the corresponding efficientfactors. The cells come into contact and separate several times during migrationbut adding another effective cue to the substrate (such as chemotaxis and/or ther-motaxis) delays their primary contact. Moreover, in all states, the average localvelocity and the net traction force of the cells decrease while the cells approachthe cues source. Our findings are qualitatively consistent with experimental obser-vations reported in the related literatures.

Keywords cell migration cell-cell interaction mechanotaxis chemotaxisthermotaxis numerical simulation finite element method.

Group of Structural Mechanics and Materials Modelling (GEMM), Aragón Institute ofEngi-neering Research (I3A), University of Zaragoza, Spain. · Centro de InvestigaciónBiomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Spain.∗E-mail: [email protected]

MCB, vol.10, no.l, pp.l-25, 2013

Publications

37

2 Copyright © 2013 Tech Science Press MCB, vol.10, no.l, pp.l-25, 2013

wound healing,immune response as well as cancer metastasis [1, 2, 3, 4, 5]. A living cell can bestimulated by many types of cues such as mechanotaxis (duro-taxis) [6, 7, 8],chemotaxis [9, 10, 11] and thermotaxis [12, 13]. Each of these cues can activate thecell pseudopodia and lamellipodia (pseudopods) to steer the cell towards moreeffective signal [9, 14].Experiments demonstrate that the cell can recognize stiffer region of the Extra-cellular Matrix (ECM) in which it resides [15, 16, 17]. For instance, fibroblast cellspreferentially move towards stiffer substrates [18, 19, 20]. This phenomenon isknown as mechanotaxis which refers to the directed movement of cell motility dueto mechanical cues. During this process, the cell senses the stiffness of its substrateby exerting a sensing force which is in general smaller than the traction force(mechano-sensing process). The generated traction forces by the cell as a resultantof the combined effort of the actin bundles, the actomyosin contractile machineryand the passive mechanical apparatus change the cell shape and propel it towardsstiffer region [16, 17, 20, 21].Chemotaxis is another key signal that can regulate the direction of cell migration.Experiments [9, 10, 22, 23] address that a cell reorientates its migration direc-tionwhen it is subjected to a chemoattractant cue. For instance, exposing a cell to cyclicAndenosine Monophosphate (cAMP) gradient activates cAMP receptors and theirassociated G-proteins which cause the cell reorientation [24]. Howevercomprehensive reason of this phenomenon is unknown, some authors attribute it toa "compass" that the cell may has [22, 23]. The exact meaning of the compass variesbut it is implied that there is a simple "compass needle" inside the cell, which is alocalized signal in the direction of the chemotaxis. In contrast, some researchersreject this theory and address that this internal compass does not exist but the cellorients itself simply by its pseudopods [11]. Till now, there are few computationalmodels that consider the cell motility associated to chemotaxis. For instance,Neilson et al. [25] modeled cell movement in absence and in presence of achemoattractant cue by means of the Level Set Method. The weakest point of theirmodel is that they ignored the effect of the mechanical conditions of the substrateon cell motility. Moreover, in the absence of chemotaxis, they considered a totallyrandom cell movement which is not accurate enough.Cell thermotaxis has been known for many years in trophoblast cells [13]. In vivo,thermotaxis is a complementary to chemotaxis. For example, in oviduct (a longreagin of female genital) the cells are guided by thermotaxis while in fertiliza-tionsites (a short area) they are steered by means of chemotaxis [26]. Moreover,trophoblast cells move towards the inner membrane of a uterus (warmer region) dueto thermotaxis [26]. Extravillous trophoblasts also migrate from the tips of theanchoring villi that surround the developing blastocyst through the mater-naldeciduae to the distal portions of the uterine spiral arteries [27]. In addition,thermotaxis is described by individual amoebae of Dictyostelium discoideum on athermal gradient. These amoebae show positive thermotaxis at temperaturesbetween 14 ◦C and 28 ◦C shortly (3 hr) after food depletion [28]. Experiments byHigazi et al. [13] demonstrated that cultured human trophoblasts are influenced bythermal gradient. They reported that human trophoblastic cells sense differences ofless than 1 ◦C above or below physiological temperature.The present work is an extension of previous mechanotactic model [20]. The pro-posed new model is capable of predicting the cell behavior through a 3D substrate inthe existence of all these effective signals simultaneously. In this model it is as-

1 Introduction

2.2. Cell migration and cell-cell interaction in the presence of mechano-chemo-thermotaxis.

38

Cell Migration and Cell-Cell Interaction 3

sumed that boundaries of the cells (cells membrane) are connected to cell centroidvia elastic-linear springs. This provides a straightforward formulation which en-ables us to develop a 3D finite element model based on equilibrium of traction andassociated effective forces acting on cell motility. Several numerical experimentsare presented to demonstrate the predictive capability of the presented model fortwo interacting cells in a substrate with gradient stiffness. The obtained resultsare verified by the available experimental results [15, 9, 13]. Some of our findingsmatch with results of experimental studies and others provide new insights formore experimental considerations.

2 Constitutive model

To devise a finite element model, the biological processes occurring in the cellmust link with the formation and dissociation of the stress fibers, as well as theassociated generation of tension and contractility. The relevant role of each cellcomponent is discussed and the effective forces acting on the cell body are calcu-lated. Therefore, velocity and polarization direction of the cell are computed basedon equilibrium of calculated effective forces.

2.1 Net cell stress transmitted by a single cell to the ECM

The cellular elements with a relevant function on the cell motility mechanism canbe divided to two main parts, active and passive mechanical elements. The formeris due to change of the overlap between myosin and actin elements while the latteris related to the action of the cytoskeleton (CSK) microtubules and the membrane.In a very simplified way, the constitutive behavior of each element of the cell can beapproximated by a linear-elastic spring. Therefore, the effective stress transmittedby the cell to ECM can be calculated by [19, 20]

σcell =

⎧⎪⎨⎪⎩

Kpasεcell εcell < εmin or εcell > εmaxKactσmax(εmin−εcell)

Kactεmin−σmaxεmin ≤ εcell ≤ ε

Kactσmax(εmax−ε)Kactεmax−σmax

ε ≤ εcell ≤ εmax

(1)

where Kpas and Kact stand for the stiffness of passive and active cellular elementsrespectively, and εcell and σmax denote the internal strain of the cell and the max-imum contractile stress exerted by the actin-myosin machinery respectively, whileε = σmax/Kact.

2.2 Effective forces on translocation of the cell

We suppose that the cell exerts sensing forces at each finite element node of thecell membrane towards its centroid. The cell subjected to these sensing forces getsstrained so that effective cell stress, σcell, corresponding to each finite elementnode can be obtained. Traction forces are generated due to the contraction of theactin-myosin apparatus which are proportional to the effective stress transmitted

Publications

39


by the cell to ECM and the cell area, S. Therefore, the nodal traction force actingat ith finite element node of the cell membrane towards the cell centroid can beobtained [20]

Ftraci = σcellSknrψei (2)

where k, is the binding constant of the integrins, nr, the total number of availablereceptors, and ψ the concentration of the ligands at the leading edge of the cell.ei is a unit vector passing from ith node towards the cell centroid. So, the nettraction force acting on whole cell body, F trac

net , is calculated by

Ftracnet =

n∑i=1

Ftraci (3)

where n is the number of the cell nodes associated on the cell membrane.On the contrary, the drag force comes from the viscous resistance of the ECM tocell motility. We assume a linear viscoelastic ECM and the cell as a sphere movingthrough Newtonian infinitely viscose medium [20, 29]. So, the drag force can beexpressed as

Fdrag = 6πrηv (4)

where r is the cell radius, η stands for the effective viscosity of the matrix, and vdenotes the cell velocity.Moreover, the cell spreads and retracts a kind of force that is generated by localprotrusions to probe the substrate cues. It is generated by actin polymerizationwhich is different from the cytoskeletal contractile force transmitted to the ECM[29]. This force is here implemented by the protrusion force which is a random inboth direction and magnitude. It is of the same order of the traction force butnormally lower [30, 20, 31]. Therefore it is randomly estimated in function of thenet traction force as

Fprot = κF tracnet erand (5)

where erand is a random unit vector and κ is a random number, 0 ≤ κ < 1 [20].It is reported that the presence of chemotaxis and thermotaxis independentlyregulated the cell migration [32, 5]. As the cell exerts traction force to move itsbody through the substrate, the mechanotactic cues always exist in presence orabsence of other cues. In contrast, Chemotaxis and thermotaxis cues only affect thecell polarization direction by modulating the direction of pseudopods. Therefore,regulation of the cell polarization direction by these cues, on average, causes thecell to reorient towards these signals.Consequently, the force equilibrium on cell locomotion system yields

Fdrag + Feff + Fprot = 0 (6)

where Feff is the effective force due to mechanotaxis, chemotaxis and thermotaxiswhose magnitude and direction depend on the net traction force and the directionof each cue respectively. It will be derived in detail in the next section.


40


2.3 Migration direction and velocity

In the sensing process the cell is subjected to sensing forces that are acting ateach finite element node of the cell membrane towards its centroid (Fig. 1-b). Thedeformed cell subjected to those sensing forces is represented by dashed lines inFig. 1-c. Therefore, the cell internal strain can be written as

εcell =AB

OA(7)

Once the internal strain of the cell is calculated, using Eq. 1 and Eq. 3, σcell andthe net traction force can be obtained respectively.It is important to note that the internal deformation created by the sensing forcesat each node of the cell membrane is negative (cell exerts contraction forces towardits centroid and always compresses itself). Therefore, nodes with lower internaldeformation will have a higher traction force [20]. As all the traction forces areacting towards the cell centroid, the resultant of these traction forces will have thedirection of minimum internal deformation (Eq. 3). Consequently, the oppositedirection of the net traction force presents the mechanotaxis reorientation of thecell [20]. Therefore, the unit vector of the mechanotaxis that reorients the cell,emech, can be defined as

emech = − Ftracnet

‖ Ftracnet ‖

(8)

In the presence of other cues such as chemotaxis and thermotaxis, the reorientationof the cell depends not only on the cell mechano-sensing but also on these cues.Let us assume ech and eth represent unit vectors associated to cell reorientationdue to chemical and thermal gradients respectively. Therefore

ech =∇C

‖ ∇C ‖ (9)

eth =∇T‖ ∇T ‖ (10)

where ∇ is the gradient operator, while C and T are the chemoattractant concen-tration and the temperature respectively.An activation signal in the presence of chemotaxis and/or thermotaxis triggersactin polymerization and myosin phosphorylation. Whereas the properties of atypical cell as well as the parameters of its ECM will not change in the presenceof these cues, it is assumed that the magnitude of the net traction force exertedby the cell is independent of these two cues, while it is a function of cell andsubstrate characteristics. Therefore, the presence of these signals only changes theeffective direction of the previously calculated net traction force. Besides, we con-sider that the realignment of this force in presence of these cues is proportional tothe chemotactic and thermotactic gradients and their associated effective factors.Consequently, from Eq. 6 and according to Fig. 1-c, the effective force, Feff , inthe presence of the chemotaxis and thermotaxis can be defined as

Feff = F tracnet (μmechemech + μchech + μtheth) (11)

Publications

41


(a)

(b) (c)

Fig. 1 Calculation of the internal deformation and the reorientation of the cell. a- Sphericalcell shape with finite element nodes is proposed. b- Exerted sensing forces at each node ofthe cell membrane. c- Deformed cell (dashed line) due to mechano-sensing in the presenceof mechanotaxis, chemotaxis and thermotaxis where emech, ech and eth indicate the unitvectors in the direction of each cue respectively and μmech, μch and μth their associatedefficient factors. F trac

net is the magnitude of the net traction force while Fprot is the randomprotrusion force.


42


where μmech, μch, and μth are the efficient factors associated to mechanotaxis,chemotaxis, and thermotaxis cues respectively, such that μmech + μch + μth = 1.Moreover, the local velocity, v, and net polarization direction, epol, of the cell canbe calculated by

v =‖ Fdrag ‖6πrη

(12)

epol = − Fdrag

‖ Fdrag ‖ (13)

Using Eq. 12 and 13, the displacement vector of an individual cell over the timeincrement, τ , is obtained by

d = vτepol (14)

2.4 Cell-cell interaction

During interaction of two cells, the same previous formulas are valid to calculatethe acting forces on the cell and define reorientation of each individual cell. Fordiscretization purposes and to avoid interference of two cells we assume

‖ rj − ri ‖≥ 2r (15)

where ri and rj are position vector of each cell centroid (Fig. 2-a). In reality thecells inside a multicellular system do not maintain a spherical shape but deform tobe tangent to each other and occupy the entire matrix [20, 33]. Therefore, we haveassumed that when two or several cells touch each other (see Fig. 2-b), the commonpoints of cells are not able to send out any pseudopod to substrate (such as fourcommon nodes n1 : n4 in the assumed cell configuration) [20, 34, 35]. Therefore,for two interacting cells, we assume that the cells do not exert any sensing force atthose nodes unless they again get separated. Note that however there is no sensingforce in these nodes, their corresponding nodal traction force are not zero.

3 Numerical experiments and results

The model presented above has been implemented into the commercial FE soft-ware ABAQUS [36] using a defined user subroutine. Fig. 3 presents the algorithmof that routine. Throughout the following simulations, we applied the model to a400×200×200 μm substrate with linear stiffness gradient which changes from 80kPa at x = 0 to 100 kPa at x = 400 μm. All the substrate surfaces are consid-ered to be free. The substrates are meshed by 16000 regular hexahedral elementswith 18081 nodes, while there is no external force acting on the substrate. Theuser subroutine is run for about 200 steps with time step approximately equal to10 minutes which is relatively enough to complete one cell migration [29]. It isof interest to mention that depending on the cell type, mechanical and boundaryconditions of the substrate and existent cues, the cell can complete the migrationprocess sooner than 200 steps. This can be referred to the existence of a strongsignal towards a cue that reduces the cell random movement.Even any cell shape can be considered, we assume a spherical shape with 24 finite

Publications

43


(a) (b)

Fig. 2 a- Calculation of position vector of each cell and distance between the centroid of twocells. The distance between their centroids is equal to the proposed cell diameter. b- Interactionof two cells when they contact each other. Here, for the assumed cell shape, two cells can havemaximum four common nodes when they are in contact.

Table 1 Substrate and cell properties.

Description Symbol Value Ref.Poisson ratio ν 0.3 [37, 17]Viscosity μ 1000 Pa.s [37, 29]Cell radius r 20 μm [38]Stiffness of microtubules Kpas 2.8 kPa [39]Stiffness of myosin II Kact 2 kPa [39]Maximum strain of the cell εmax 0.09 [20, 31]Minimum strain of the cell εmin -0.09 [20, 31]Maximum contractile stress ex-erted by actin-myosin machin-ery

σmax 0.1 kPa [40, 21]

Binding constant at the rearand front of the cell

kf = kb 108 mol−1 [29]

Number of available receptorsat the rear and front of the cell

nf = nb 105 [29]

Concentration of the ligands atrear and front of the cell

ψ 10−5 mol [29]

element nodes on its membrane (Fig. 1-a). The properties of the cell and the sub-strate are listed in table 1. During migration, the centroid of each cell is visualizedby a small sphere at each time step to indicate the cell position.

3.1 Cell migration and cell-cell interaction in existence of pure mechanotaxis

Fig. 4 presents migration trajectory of two cells due to pure mechanotaxis inthe above mentioned substrate. The cells is firstly located in different corners ofthe substrate where the substrate stiffness is minimum (80 kPa). As expected,


44


Fig. 3 Computational algorithm of cell migration and cell-cell interaction considering mechan-otaxis, chemotaxis, and/or thermotaxis.

the cells tendency is to migrate in the direction of the stiffness gradient towardshigher stiffness. Because of the extended region between the two cells, they feelless internal deformation in this zone. Therefore, they firstly migrate towards eachother to be in contact at x = 85 μm. Once their polarization direction changesdue to protrusion force they get separated and migrate in the direction of thestiffness gradient. This phenomenon (contact and separation of the cells) can beinfinitely repeated during their migration. However maximum stiffness gradientof the substrate occurs at x = 400 μm, the cells do not migrate towards endof the substrate but they move around an Imaginary Equilibrium Plane (IEP)located at x = 325 μm. Because the cells recognize the free surface of the substrateas a soft and unstable region despite of maximum stiffness that it has [15, 16].It is remarkable that any increase in the stiffness gradient slope can move theIEP towards the end of the substrate. In Figs. 5 and 6 the average net tractionforce and velocity of the cells are plotted versus average translocation of the cellsrespectively. The results demonstrate that during cells migration towards morestable and stiffer region, the average net traction force and velocity of the cellsreduce. It means that the cell tends to adhere to stiffer substrates and stay almost

Publications

45


(a) (b)

Fig. 4 Cell migration and interaction due to pure mechanotaxis. It is considered that thereis a linear stiffness gradient which changes from 80 kPa at x = 0 to 100 kPa at x = 400 μm.Firstly, the cells are located in different corners of the substrate near to x=0 where substratestiffness is minimum. The cells tendency is to contact each other at x = 80 μm then theymigrate in the direction of the stiffness gradient towards higher stiffness. Finally, the cellsmove around an IEP located far from free surfaces at x = 325 μm.

Fig. 5 Average magnitude of the net traction force versus average translocation of the cellsdue to pure mechanotaxis.

without any locomotion there [15, 20]. Reduction of net traction force causes thecell to be more round and symmetric [41]. These results demonstrate why thecell stability rises in stiffer regions. It is remarkable that in very stiff substrates,generated net traction forces may not be enough to move the cells to a new position.Besides, comparison of the appearance of the both curves with those of a singlecell migration [20] demonstrates that presence of two cells simultaneously in asubstrate with stiffness gradient increases fluctuation of the curves. Because thecontact and separation of the cells suddenly changes the average traction force andvelocity of them in a point.


46


Fig. 6 Average local velocity versus average translocation of the cells due to pure mechano-taxis.

3.2 Cell migration and cell-cell interaction in existence of thermotaxis

Higazi et al. [13] observed that trophoblast cells migrate towards warmer sites bymeans of thermotaxis in the absence of chemotaxis. Generally, in vivo, thermotaxisis a complementary cue to chemotaxis [26]. Therefore this experiment is designed toinvestigate thermotactic effect on cell migration and cell-cell interaction (Fig. 7). Itis assumed that there is a thermal gradient throughout the substrate in x directionwhich uniformly increases from 35 ◦C at x=0, to 38 ◦C at x=400 μm [13]. Wehave assumed μth=0.2. The results show that the existence of thermotaxis delayscontact of the cells until x=115 μm (7-a). Because beside of the force pointing thecells towards each other (mechanotactic force) a part of net traction force directedby thermotactic gradient persistently guides the cells towards the warmer siteswhich inhibits the cells to come in contact. In this case, as mechanotaxis, contactand separation of the cells can be infinitely repeated until they achieve warmerplaces. The migration of the cells towards warmer sites is consistent with findingsof Higazi et al. [13] who reported that trophoblastic cells migrate towards warmerzones in the presence of thermal gradient. It should be noted that due to theexistence of free boundary surface in the end of the substrate, the mechanotacticsignal received by the cells dissuades them to move towards the warmest region ofthe substrate at x=400 μm. Consequently, thermotaxis causes the IEP to slightlymove towards warmer site of the substrate and to locate at x=335 μm. Oncethe cells achieve this IEP they get contact and separate near the IEP. Moreover,comparison of Figs. 7 with 4 indicates that the random movement of the cellsthrough the substrate in existence of thermotaxis gradient is relatively less thanthat of pure mechanotactic one.Figs. 8 and 9 display the average magnitude of the net traction force and the localvelocity of the cells versus average translocation of the cells respectively. As seen,during the cell migration towards stiffer and warmer regions of the substrate boththe magnitude of the net traction force and the cell local velocity decrease, havingthe same trend as cell migration due to pure mechanotaxis.

Publications

47


(a) (b)

Fig. 7 Cell migration and interaction in presence of thermotaxis. It is considered that there isa linear thermal gradient along the x direction where x=400 μm has the maximum temperature(38 ◦C). The thermal efficient factor is assumed to be 0.2. The cells are firstly placed in oppositecorners of the substrate near x=0 with minimum temperature (35 ◦C). They migrate towardswarmer sites in the thermal gradient direction. Finally the cell keeps moving around an IEPlocated at x=335 μm.

Fig. 8 Average magnitude of the net traction force versus average translocation of the cellsin presence of thermotaxis.

3.3 Cell migration and cell-cell interaction in existence of chemotaxis

In this experiment, a chemotactic cue is added to the same substrate with stiffnessgradient to assay chemoattractant effect on cell migration and cell-cell interaction.It is assumed that there is a uniform chemical gradient along the x-axis increasingfrom zero at x =0 to 10−4 M at x = 400 μm. As before, the cells are located at dif-ferent corners of the substrate near x =0 where the chemoattractant concentrationis null. We assume that chemotactic efficient factor is higher than thermotacticone due to its higher gradient slope [9]. Fig. 10 present the trajectory of the cellsfor μch = 0.3 and μch = 0.4. In this experiment the results indicate that existenceof chemoattractant cue in the substrate with gradient stiffness delays the cellscontact even more than thermotaxis due to higher efficient factor. In this case,there exists an IEP as well that the cells finally move towards. The location of this


48


Fig. 9 Average local velocity versus average translocation of the cells in presence of thermo-taxis.

IEP also is sensitive to the chemotactic efficient factor. As seen from Fig. 10-a, forμch=0.3, the IEP displaces towards the chemoattractant source (x = 355 μm) andachieves more the end of the substrate (x = 375 μm) by increasing the chemo-tactic efficient factor to 0.4 (Fig. 10-c). It is notable that the more IEP reachesto the end of the substrate (higher efficient factor), the more contact of the cellsdelays. The existence of chemotaxis beside of mechanotaxis simultaneously in asubstrate decreases the random movement of the cells throughout the substraterather than that of pure mechanotaxis or mechano-thermotaxis (compare Figs. 4and 7 with 10). This occurs, because the received signals by pseudopods of thecells strongly steer the cells in the chemotactic direction [22]. Consequently, aspreviously explained, the cell polarization direction which is persistently towardsthe chemoattractant source sums to the mechanotaxis effects and hence increasesthe cell global velocity. These results are in agreement with the experimental in-vestigations of Andrew and Haaster [22, 42] who observed that in the presence ofchemoattractant, depending on the signal strength of the chemotaxis, the cell canactively move towards chemoattractant sources. Moreover, similar cell behaviorwas reported by Bosgraaf et al. [9]. In addition, our findings are consistent withnumerical results of Neilson et al. [25] as well.Figs. 11 and 12 present the average net traction force and local velocity of thecells versus average translocation of the cells during chemotactic process. In bothcases of μch=0.3 and μch=0.4, similar to pure mechanotaxis migration, the cell nettraction force and the cell local velocity decrease. This occurs because the magni-tude of the net traction force is relatively independent of chemotactic cue but it isfunction of the mechanical properties and boundary conditions of the substrate.On the other hand, the presence of the chemotaxis (as thermotaxis) affects thedrag force which in turn can slightly reduce the cell velocity, although this reduc-tion is negligible. Therefore during cell migration towards stiffer region and highchemoattractant concentration behavior of the cells is similar to pure mechano-taxis process in terms of average local velocity and magnitude of net traction force.

Publications

49


(a) (b)

(c) (d)

Fig. 10 Migration and interaction of the cells in presence of chemotaxis. It is assumed thata chemoattractant source at x=400 μm creates a linear chemoattractant gradient through thesubstrate in x direction. Firstly, the cells are located in different corners of the substrate nearx=0 where chemoattractant concentration is null. Cells migration is presented for two differentchemoattractant efficient factors. In the both cases the cells migrate in the direction of thechemoattractant gradient towards an IEP. When μch=0.3 the IEP is located at x =355 μm (aand b) while for μch=0.4 the IEP displaces to x =375 μm (c and d).

Fig. 11 Average magnitude of the net traction force versus average translocation of the cellsin presence of thermotaxis for μch=0.3 and μch=0.4.


50


Fig. 12 Average local velocity versus average translocation of the cells in presence of chemo-taxis for μch=0.3 and μch=0.4.

3.4 Cell migration and cell-cell interaction in existence of chemotaxis andthermotaxis

The last experiment is designed to consider the effect of all stimuli together oncell-cell interaction. It is assumed that there exist maximum chemoattractant con-centration and temperature at x=400 μm which yield to a linear chemotactic andthermotactic gradient through the x axis. The cells are placed in opposite cornersof the substrate near x=0. We assumed μch=0.3 and μth=0.2. The results demon-strate that the existence of chemotaxis and thermotaxis together in the substratebeside to gradient stiffness delays the cells contact even more than previous exper-iments (the cells first contact at x=265 μm as seen at 13-a). Because the existenceof chemotactic and thermotactic gradient at the same direction amplifies the sig-nals received by the cells so that it encourages the cells to migrate towards thesecues instead of each other. Therefore, cell migration towards chemotactic and ther-motactic stimuli dominate cell-cell contact. As seen in Fig. 13, the IEP disappearsand the cells migrate towards the end of the substrate despite of the free boundarysurface. The presence of all stimuli together in the substrate causes that overallrandom movement of the cells through the substrate decreases more than that ofprevious experiments (compare Figs. 13 with 4, 7, 10). This occurs, because thereceived signals by cell pseudopods strongly steer the cells in the direction of thegradients [22]. These results are in agreement with the chemotactic [42, 22] andthermotactic [26, 13] experimental investigations.Figs. 14 and 15 present the average net traction force and local velocity of thecells versus average translocation of the cells during mechano-chemo-thermotacticprocess. Similar to previous experiments, in the first interval the average net trac-tion force and local velocity of the cells decrease. Because, as said, the magnitudeof the net traction force is relatively independent of chemotactic and thermotacticsignals. In the contrary, drag force which is affected by mechanotactic, chemotac-tic and thermotactic efficient force, Feff , cause to slightly reduce the cell velocity,however it is negligible. In the last interval of these curves the average net tractionforce and local velocity of the cells increases due to the cells migration towardsfree boundary surface (soft region).

Publications

51


(a) (b)

Fig. 13 Migration and interaction of the cells in presence of chemotaxis and thermotaxis.It is assumed that chemoattractant concentration and temperature at x=400 μm are maxi-mum. It simultaneously creates a linear chemotactic and thermotactic gradient through thesubstrate in the x direction. Firstly, the cells are located in the different corners of the sub-strate near x=0 where chemoattractant concentration is null and temperature is minimum.μch=0.3 and μth=0.2. In this case the IEP disappears and the cells migrate towards maximumchemoattractant concentration and temperature.

Fig. 14 Average magnitude of the net traction force versus average translocation of the cellsin presence of chemotaxis and thermotaxis. μch=0.3 and μth=0.2.

4 Conclusions

Here, previous computational model [20] is extended to investigate interaction oftwo cells migrating across a 3D substrate in the presence of mechanotactic, chemo-tactic and thermotactic cues. The model is based on equilibrium of the effectiveforces on cell motility. The generated traction force during a cell migration isincorporated by activated signals such as chemotaxis and thermotaxis beside ofmechanotaxis. We focused on the interaction of two cells during their migrationthrough the substrate with linear stiffness gradient.During pure mechanotactic process, the cells tendency is firstly to reach eachother and then migrate in the direction of stiffness gradient towards an IEP[15, 16, 20, 43]. The IEP is located far from free boundary surfaces, because thecells feel more stable and adhere better to substrate in this region [20]. The cells


52


Fig. 15 Average local velocity versus average translocation of the cells in presence of chemo-taxis and thermotaxis. μch=0.3 and μth=0.2.

contact and separate several times during migration in the direction of stiffnessgradient, which can continue infinitely after achievement to the IEP.Study of chemotactic and/or thermotactic effects on cell migration demonstratethat the IEP location is sensitive to corresponding efficient factors. In existence ofthermotaxis, the IEP slightly displaces towards warmer sites but adding chemo-taxis can farther displace the IEP to the end of the substrate. Besides, IEP caneven vanish when there exist chemotactic and thermotactic cues together in thesubstrate with stiffness gradient. In other words, presence of chemotaxis and ther-motaxis simultaneously in the substrate can dominate mechanotactic signals andpropel the cell towards the chemoattractant source and maximum temperature.It is remarkable that the effect of these cues is negligible on the cell local velocitybecause the magnitude of the net traction force is function of the surrounding me-chanical conditions and substrate properties. But their effect on the reorientationof the cells is not connivance. In addition, presence of chemotactic and/or ther-motactic signals delays contact of the cells because these cues persistently steerthe cells towards high chemoattractant concentration and/or warmer sites respec-tively. Therefore, the higher efficient factors of cues, the more delays contact of twocells. These findings are in agreement and consistent with chemotaxis [9, 22, 42]and thermotaxis [13, 26] experimental works.Submitted results by the model are qualitatively consistent with experimentalworks [9, 13, 15, 16]. The results demonstrate that cells behavior is relativelychanged in the presence of thermotaxis and/or chemotaxis.Consequently, the presented model can answer this important question: how isthe overall cell behavior when there exist simultaneously two or several signalsin the surrounding substrate? These supplementary features enable the model topredict a wide range of experimental observations. The model is highly flexible insight of cell shape, number of cells and multi-signal analysis of cell migration. Theobtained results depend on the choice of the parameters such as chemotactic andthermotactic efficient factors which can be calibrated with further experimentalworks. Moreover, the present mechano-chemo-thermotactic model can be used toexamine a wider range of efficient signals affecting cell migration such as durotaxis,haptotaxis and topotaxis. It is remarkable that a note of caution is necessary due

Publications

53


to inaccessibility to the some experimental data to quantitatively validate or refusesome of obtained results via the model.

Acknowledgements

The authors gratefully acknowledge the financial support from the Spanish Min-istry of Science and Technology (CYCIT DPI2010-20399-C04-01) and CIBER-BBN initiative. CIBER-BBN is an initiative funded by the VI National R&D&IPlan 2008-2011, Iniciativa Ingenio 2010, Consolider Program, CIBER Actions andfinanced by the Instituto de Salud Carlos III with assistance from the EuropeanRegional Development Fund.

References

1. H. Behesti and S. Marino. Cerebellar granule cells: Insights into proliferation,differentiation, and role in medulloblastoma pathogenesis. Journal AppliedPhysiology, 41:435–445, 2009.

2. B.A. Dalton, X.F. Walboomers, M. Dziegielewski, M.D.M. Evans, S. Taylor,J.A. Jansen, and J.G. Steele. Modulation of epithelial tissue and cell migrationby microgrooves. Phil. Trans. Roy. Soc. London, 56:195–207, 2001.

3. T. Pompe, S. Glorius, T. Bischoff, I. Uhlmann, M. Kaufmann, S. Brenner, andC. Werner. Dissecting the impact of matrix anchorage and elasticity in celladhesion. Biophysical Journal, 97:2154–2163, 2009.

4. S. Suresh. Biomechanics and biophysics of cancer cells. Acta Biomaterialia,3:413–438, 2007.

5. M. Zhao. Electrical fields in wound healing-an overriding signal that directscell migration. Seminars in Cell and Developmental Biology, 20:674–82, 2009.

6. R. Allena and D. Aubry. run-and-tumble or look-and-run ? a mechani-cal model to explore the behavior of a migrating amoeboid cell. Journal ofTheoretical Biology, 306:15–31, 2012.

7. B.K. Chirag, R.P. Shelly, and J.P. Andrew. Intrinsic mechanical properties ofthe extracellular matrix affect the behavior of pre-osteoblastic mc3t3-e1 cells.Am J Physiol Cell Physiol, 290:C1640–C1650, 2006.

8. R.J. Pelham, Jr., and Y.L. Wang. Cell locomotion and focal adhesions areregulated by substrate flexibility. PNAS, 94:13661–13665, 1997.

9. L. Bosgraaf and P.J.M. Van Haastert. Navigation of chemotactic cells by par-allel signaling to pseudopod persistence and orientation. PLoS ONE, 4:e6842,2009.

10. O. Hoeller and R.R. Kay. Chemotaxis in the absence of pip3 gradients. CUR-RENT BIOLOGY, 17:813–817, 2007.

11. M.P. Neilson, D.M. Veltman1, P.J.M. van Haastert, S.D. Webb, J.A. Macken-zie, and R.H. Insall. Chemotaxis: A feedback-based computational model ro-bustly predicts multiple aspects of real cell behaviour. PLoS Biol, 9:e1000618,2011.

12. B.H. Choo, R.F. Donna, , and L.P. Kenneth. Thermotaxis of dictyosteliumdiscoideum amoebae and its possible role in pseudoplasmodial thermotaxis.Proc. Natl. Acad. Sci. USA, 80:5646–5649, 1983.


54


13. A.A. Higazi, D. Kniss, J. Manuppelloand E.S. Barnathan, and D.B.Cines.Thermotaxis of human trophoblastic cells. Placenta, 17:683–7, 1996.

14. Y.T. Maeda, J. Inose, M.Y. Matsuo, S. Iwaya, and M. Sano. Ordered patternsof cell shape and orientational correlation during spontaneous cell migration.PLoS ONE, 3:e3734, 2008.

15. E. Hadjipanayi, V. Mudera, and R.A. Brown. Guiding cell migration in 3d:A collagen matrix with graded directional stiffness. Cell Motility and theCytoskeleton, 66:435–445, 2009.

16. C.M. Lo, H.B. Wang, M. Dembo, and Y.L. Wang. Cell movement is guidedby the rigidity of the substrate. Biophysical Journal, 79:144–152, 2000.

17. T.A. Ulrich, E.M. De Juan Pardo, and S. Kumar. The mechanical rigidity ofthe extracellular matrix regulates the structure, motility, and proliferation ofglioma cells. Cancer Research, 69:4167–4174, 2009.

18. D.E. Ingber and I. Tensegrity. Cell structure and hierarchical systems biology.Journal Cell Science, 116:1157–1173, 2003.

19. P. Moreo, J.M. Garcia-Aznar, and M. Doblaré. Modeling mechanosensing andits effect on the migration and proliferation of adherent cells. Acta Biomate-rialia, 4:613–621, 2008.

20. S.J. Mousavi, M.H. Doweidar, and M. Doblaré. Computational mod-elling and analysis of mechanical conditions on cell locomotion and cell-cell interaction. Comput Methods Biomech Biomed Engin., page DOI:10.1080/10255842.2012.710841, 2012.

21. C.G. Penelope and P.A. Janmey. Cell type-specific response to growth on softmaterials. Journal Applied Physiology, 98:1547–1553, 2005.

22. P.J.M. Van Haastert. Chemotaxis: insights from the extending pseudopod. JCell Sci., 123:3031–37, 2010.

23. K.F. Swaney, C.H. Huang, and P.N. Devreotes. Eukaryotic chemotaxis: a net-work of signaling pathways controls motility, directional sensing, and polarity.Annu Rev Biophys, 39:265–289, 2010.

24. S. Zhang, P.G. Charest, and R.A. Firtel. Spatiotemporal regulation of rasactivity provides directional sensing. Curr. Biol, 18:1587–93, 2008.

25. M.P. Neilson, J.A. Mackenzie, S.D. Webb, and R.H. Insall. Modeling cellmovement and chemotaxis using pseudopod-based feedback. SIAM Journalon Scientific Computing, 33:1035–1057, 2011.

26. M. Eisenbach A. Bahat. Sperm thermotaxis. Molecular and Cellular En-docrinology, 252:115–119, 2006.

27. G.B. Young. Thermography. CRC Critical Reviews of Radiological Science,1:593–638, 1970.

28. C.B. Hong, D.R. Fontanat, and K.L. Pofft. Thermotaxis of dictyosteliumdiscoideum amoebae and its possible role in pseudoplasmodial thermotaxis.Proc. Natl. Acad. Sci, 80:5646–49, 1983.

29. M.H. Zaman, R.D. Kamm, P. Matsudaira, and D.A. Lauffenburgery. Compu-tational model for cell migration in three-dimensional matrices. BiophysicalJournal, 89:1389–1397, 2005.

30. D.W. James and J.F. Taylor. The stress developed by sheets of chick fibrob-lasts in vitro. Experimental Cell Research, 54:107–110, 1969.

31. S. Ramtani. Mechanical modelling of cell/ecm and cell/cell interactions dur-ing the contraction of a fibroblast-populated collagen microsphere:theory andmodel simulation. Journal of Biomechanics, 37:1709–1718, 2004.

Publications

55


32. R.C. Gao, X.D. Zhang, Y.H. Sun, Y. Kamimura, A. Mogilner, P.N. Devreotes,and M. Zhao. Different roles of membrane potentials in electrotaxis and chemo-taxis of dictyostelium cells. EUKARYOTIC CELL, 10:doi: 10.1128/EC.05066–11, 2011.

33. E. Palsson. A three-dimensional model of cell movement in multicellular sys-tems. Future Generation Computer System, 17:835–852, 2001.

34. G.W. Brofland and C.J. Wiebe. Mechanical effects of cell anisotropy on ep-ithelia. Computer Methods Biomech. Biomed. Engin., 7:91–99, 2004.

35. D.L. Taylor, J. Heiple, Y.L. Wang, E.J. Luna, L. Tanasugarn, J. Brier, J. Swan-son, M. Fechheimer, P. Amato, M. Rockwel, and G. Daley. Cellular and molec-ular aspects of amoeboid movement. CSH Symp. Quant. Biol., 46:101–111,1982.

36. Hibbit, Karlson, and Sorensen. Abaqus-Theory manual, 6.3 edition edition.37. S.K. Akiyama and K.M. Yamada. The interaction of plasma fibronectin with

fibroblastic cells in suspension. Journal of Biological Chemistry, 260:4492–4500, 1985.

38. A. Buxboim, I.L. Ivanovska, and D.E. Discher. Matrix elasticity, cytoskeletalforces and physics of the nucleus: how deeply do cells ’feel’ outside and in?Journal of cell science, 123:297–308, 2010.

39. A. Schäfer and M. Radmacher. Influence of myosin ii activity on stiffness offibroblast cells. Acta Biomaterialia, 1:273–280, 2005.

40. G.F. Oster, J.D. Murray, and A.K. Harris. Mechanical aspects of mesenchymalmorphogenesis. journal Embryol Exp Morph, 78:83–125, 1983.

41. M. Ehrbar, A. sala, P. Lienemann, A. Ranga, K. Mosiewicz, A. Bittermann,S. C. Rizzi, and F. E. Weber. Elucidating the role of matrix stiffness in 3dceell migration and remodeling. Biophysical Journal, 100:284–293, 2011.

42. N. Andrew and R.H. Insall. Chemotaxis in shallow gradients is mediated in-dependently of ptdlns 3-kinase by biased choices between random protrusions.Nature Cell Biology, 9:193–200, 2007.

43. D.K. Schlüter, I.R.C., and M.A.J. Chaplain. Computational modeling ofsingle-cell migration: The leading role of extracellular matrix fibers. Biophys-ical Journal, 103:1141–51, 2012.


56


Journal of Theoretical Biology

2013


2.3 3D computational modelling of cell migration: A mechano-

chemo-thermo-electrotaxis approach.

3D computational modelling of cell migration:A mechano-chemo-thermo-electrotaxis approach

Seyed Jamaleddin Mousavi a,b,c, Mohamed Hamdy Doweidar a,b,c,n, Manuel Doblaré a,b,c

a Group of Structural Mechanics and Materials Modelling (GEMM), Aragón Institute of Engineering Research (I3A), University of Zaragoza, Spainb Mechanical Engineering Department, School of Engineering and Architecture (EINA), University of Zaragoza, Spainc Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Spain

H I G H L I G H T S

� 3D Computational Modelling of single Cell Migration.� A Mechano-Chemo-Thermo-Electrotaxis Approach.� Predicting cell migration behavior under different conditions and substrate properties.� The presence of the chemotaxis, thermotaxis and/or electrotaxis reduce the random component of cell movement.� The results are qualitatively in agreement with well-known experimental ones.

a r t i c l e i n f o

Article history:Received 30 November 2012Received in revised form14 February 2013Accepted 22 March 2013Available online 6 April 2013

Keywords:Single cell migrationMechanotaxisChemotaxisThermotaxisElectrotaxis

a b s t r a c t

Single cell migration constitutes a fundamental phenomenon involved in many biological events such aswound healing, cancer development and tissue regeneration. Several experiments have demonstratedthat, besides the mechanical driving force (mechanotaxis), cell migration may be also influenced bychemical, thermal and/or electrical cues. In this paper, we present an extension of a previous model ofthe same authors adding the effects of chemotaxis, thermotaxis and electrotaxis to the initialmechanotaxis model of cell migration, allowing us to predict cell migration behaviour under differentconditions and substrate properties. The present model is based on the balance of effective forces duringcell motility in the presence of the several guiding cues. This model has been applied to severalnumerical experiments to demonstrate the effect of the different drivers onto the cell path and finallocation within a certain three-dimensional substrate with heterogeneous properties. Our findingsindicate that the presence of the chemotaxis, thermotaxis and/or electrotaxis reduce, in general, therandom component of cell movement, being this reduction more important in the case of electrotaxisthat can be considered a dominating signal during cell migration (besides the underlying mechanicaleffects). These results are qualitatively in agreement with well-known experimental ones.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Cell migration is an essential phenomenon in a vast variety ofbiological processes ranging from tissue development and woundhealing to malignant diseases such as cancer (Behesti and Marino,2009; Dalton et al., 2001; Pompe et al., 2009; Suresh, 2007; Zhao,2009). To migrate, a cell needs to polarise and decide a migrationdirection which are complex biological processes (Etienne-

Manneville, 2008; Tanos and Rodriguez-Boulan, 2008). Cell migra-tion can be regulated by mechanical properties of its substrate(mechanotaxis or durotaxis) (Allena and Aubry, 2012; Chirag et al.,2006; Pelham and Wang, 1997), chemical gradient (chemotaxis)(Bosgraaf and VanHaastert, 2009; Hoeller and Kay, 2007; Neilsonet al., 2011), thermal gradient (thermotaxis) (Choo et al., 1983;Higazi et al., 1996) and/or electric field (electrotaxis or galvano-taxis) (Maria and Djamgoz, 2004; Zhao, 2009). In such cases, thecell senses its surroundings and identifies its polarisation directionby sending out pseudopodia and lamellipodia (pseudopods)(Bosgraaf and VanHaastert, 2009; Maeda et al., 2008).

The cell has a special internal structure which enables it tosense the stiffness of the Extracellular Matrix (ECM) in which itresides. For instance, fibroblast cells preferentially move towards

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/yjtbi

Journal of Theoretical Biology

0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jtbi.2013.03.021

n Corresponding author at: Mechanical Engineering Department, School ofEngineering and Architecture (EINA), University of Zaragoza María de Luna s/n,Edificio Betancourt, 50018 Zaragoza, SPAIN. Tel.: +34 876555210.

E-mail addresses: [email protected],[email protected] (M.H. Doweidar).

Journal of Theoretical Biology 329 (2013) 64–73

Publications

59

stiffer substrates (Ingber and Tensegrity, 2003; Moreo et al., 2008;Mousavi et al., 2012). This phenomenon is known as mechanotaxiswhich refers to the directed movement of cell motility viamechanical cues. In this mechano-sensing process, the cell sensesits substrate by exerting a sensing force, developed by thecombined effort of the actin bundles, the actomyosin contractilemachinery and the passive mechanical apparatus. The result ofthis action unchains a cascade of biochemical reaction and thereshaping of the cell to pull itself towards stiffer regions (Mousaviet al., 2012; Penelope and Janmey, 2005).

Besides, cell migration due to chemotaxis is also an importantbiological process as well. Many experimental works may be foundin the literature with regards to this phenomenon (Bosgraaf andVanHaastert, 2009; Van Haastert, 2010; Hoeller and Kay, 2007;Swaney et al., 2010) while only a few theoretical works have beenpresented (Elliott et al., 2012; Neilson et al., 2011). Most of theseexperiments demonstrate that a cell reorientates when subjectedto a chemoattractant gradient. Recently, many authors haveaddressed that chemotaxis is driven by a “compass” produced byprocessing the extracellular signal into an intracellular mechanismthat points the cell in the direction of the gradient (Van Haastert,2010; Swaney et al., 2010). Till now, there are few computationalmodels that consider the cell motility associated to chemotaxis.For instance, Neilson et al. (2011) modelled cell movement inabsence and in presence of a chemoattractant cue by means of theLevel Set Method. The effect of the mechanical conditions of thesubstrate on cell motility is not presented in their model. More-over, in the absence of chemotaxis, they considered a totallyrandom cell movement which is not accurate enough.

In vivo, thermotaxis may be complementary to chemotaxis,with each mechanism works in regions where the other isineffective (Eisenbach and Bahat, 2006). For instance, after culture,trophoblasts invade endometrium, the inner membrane of auterus. Extravillous trophoblasts also migrate from the tips ofthe anchoring villi that surround the developing blastocystthrough the maternal deciduae to the distal portions of the uterinespiral arteries. It is remarkable that in all the experimentallythermotactic observations of the cell motility, the cell migratestowards warmer sites (Eisenbach and Bahat, 2006; Higazi et al.,1996; Young, 1970).

Through wound healing process, cell migration is an essentialprocess. During this process, cells must recognise when and inwhich direction they have to migrate (Zhao, 2009). Polarisedepithelial cells transport ions directionally and maintain Transe-pithelial Potentials (TEPs) (Zhao, 2009). When a ulcer occurs itdisrupts the epithelial barrier and consequently a sort-circuit TEPis generated, called Endogenous Wound Electric Field (EWEF)(Zhao, 2009). The potential at the wound thus drops, becomingnegative in relation to the potential underneath the unwoundedepidermis that is away from the wound. This potential gradientdrives the electrical current flow towards the more negative site sothat a laterally orientated wound electric fields (EFs) is created(electrotaxis). At skin and corneal wounds, electrical current flowis orientated towards the wound center from the surroundingtissues and then out from the wound. Cells away from the woundkeep transporting of ions into wound to maintain the TEPs. Thosecells keep driving the electrical currents until the wound heals andthe barrier gets restored (Zhao, 2009). Besides, the majority of cellsare sensitive to exogenous EFs. Direct-current EFs (dcEFs) are usedto stimulate cells and move them directionally towards thecathode or the anode in wound healing and its associatedprocesses (Cooper and Shliwa, 1986; Nishimura et al., 1996).

In our previous work (Mousavi et al., 2012), a computationalmechanosensing model describing the motility of a single cell andof cell population was developed. Using a 3D finite element model,cell migration and cell–cell interaction in 3D substrates were

considered. For both cases, several experiments were simulatedto demonstrate the predictive capability of the model. It wasdemonstrated how a cell senses mechanical conditions of itssubstrate, gets strained and consequently is polarised towardsstiffer and/or fixed regions. This model is based on pure mechan-ical signals (mechanotaxis effects) acquired by a migrating cellfrom its surrounding substrate.

In the present paper, we extend the previous model to take intoaccount other effective cues such as chemotaxis, electrotaxis andthermotaxis. A 3D finite element model to simulate cell migrationwithin 3D substrates that consider the previously mentioned cuesis developed. Several numerical experiments are presented todemonstrate the predictive capability of the presented model forsingle cell. The obtained results are verified by the availableexperimental results (Bosgraaf and VanHaastert, 2009; Higaziet al., 1996; Nishimura et al., 1996). The presented model has aflexible potential to simultaneously consider the effects of one orseveral cues on the cell migration. Moreover, it can provide aqualitatively useful information to anticipate the cell migrationin vivo or in vitro experiments.

2. Model description

2.1. Effective stress transmitted by the cell to the ECM

The cellular elements with a relevant function on the cellmotility mechanism can be divided into two main parts, activecontractile and passive mechanical strength elements. The formeris generated by the actin filaments and the myosin II (machineryAM) while the latter is associated to the rest of the cell body,essentially microtubules and the cell membrane. Therefore theeffective stress transmitted by the cell to the ECM, scell, may beexpressed as (Moreo et al., 2008; Mousavi et al., 2012)

scell ¼ sact þ spas ð1Þwhere sact stands for the mean active contractile stress and spas

denotes the passive resistance of the rest of the cell body. In a verysimplified way, the constitutive behaviour of each element of thecell can be approximated by a simple linear-elastic spring. Thebehaviour of the active elements mainly depends on the cellminimum, ϵmin, and maximum internal strain, ϵmax. When celldeformation is out of the range ϵmax−ϵmin, the active stresstransmitted to the cell is equal to zero. Therefore, the mean activecontractile stress can be calculated by (Moreo et al., 2008; Mousaviet al., 2012)

sact ¼

0 ϵoϵmin or ϵ4ϵmax

Kactsmaxðϵmin−ϵÞKactϵmin−smax

ϵmin ≤ϵ≤ ~ϵ

Kactsmaxðϵmax−ϵÞKactϵmax−smax

~ϵ ≤ϵ≤ϵmax

8>>>>><>>>>>:

ð2Þ

where Kact, ϵ and smax denote the stiffness of active elements, theinternal strain of the cell and the maximum contractile stressexerted by the actin-myosin machinery respectively, while~ϵ ¼ smax=Kact .

On the other hand, the passive stress transmitted by the cell tothe ECM can be obtained by

spas ¼ Kpasϵ ð3Þwhere Kpas stands for the stiffness of passive cellular elements.

2.2. Effective forces on propelling the cell body

The cell migration herein developed model is based on equili-brium of acting forces on the cell body which are traction,

S.J. Mousavi et al. / Journal of Theoretical Biology 329 (2013) 64–73 65

2.3. 3D computational modelling of cell migration: A mechano-chemo-thermo-electrotaxis approach.

60

protrusion and drag forces. The contraction of the actin-myosinapparatus drives forward the translocation of the cell body andgenerates traction forces on the substrate. It is proportional to theeffective stress transmitted by the cell to ECM (Mousavi et al.,2012). So, it can be obtained as

Ftrac ¼ scellSζ ð4Þwhere S denotes the cell area and ζ is a parameter that depends onthe cell type. It is proportional to the binding constant of theintegrins, k, the total number of available receptors, nr, and theconcentration of the ligands at the leading edge of the cell, ψ .Therefore, it can be defined as

ζ¼ knrψ ð5ÞIt should be noted that the value of ζ for a asymmetric cell isdifferent in the front and rear. This traction force corresponds tothe nodal traction force that acts at each external finite elementnode of the cell towards the cell centroid. So, the net traction forceacting on whole cell body, Fnettrac, is calculated by (Mousavi et al.,2012)

Ftracnet ¼ ∑

n

i ¼ 1Ftraci ei ð6Þ

where n is the number of the external nodes of the cell and ei isthe unit vector from each external node towards the cell centroid.

On the contrary, the drag force comes from the viscousresistance to cell motility. In a Maxwell solid, the required forcefor deforming the ECM depends on the deformation rate as well asthe relative velocity. Whereas, the main objective here is toconsider a velocity dependent but opposing force associated tothe viscoelastic character of the cell ECM, for simplification, weassume the ECM as a linear viscoelastic medium and the cell as asphere moving through Newtonian infinitely viscose medium(Mousavi et al., 2012; Zaman et al., 2005). So, the drag force canbe expressed as

Fdrag ¼ 6πrηv ð7Þ

where r is the cell radius, η stands for the effective viscosity of thematrix, and v denotes the cell velocity.

In an effective guidance system, a cell sends out local protru-sions to probe the substrate cues which is here implemented bythe protrusion force which is a random in both direction andmagnitude. It is generated by actin polymerisation which isdifferent from the cytoskeletal contractile force transmitted tothe ECM (Zaman et al., 2005). It is of the same order of the tractionforce but normally lower (James and Taylor, 1969; Mousavi et al.,2012; Ramtani, 2004). Therefore it should randomly be estimatedin function of the net traction force as

Fprot ¼ κFtracnet erand ð8Þ

where erand is a random unit vector and κ is a random number,0≤κo1 (Mousavi et al., 2012).

In the presence of other cues such as electrotaxis, chemotaxisand thermotaxis it is worth to distinguish the role of each one oncell motility. Gao et al. (2011) and Zhao (2009) reported that eachof these signals indeed regulated independently the cell motility.Chemotaxis and thermotaxis cues only affect the cell polarisationdirection by modulating the direction of pseudopods. Therefore,correction of the cell polarisation direction by these cues, onaverage, causes the cell to reorient towards these signals.

On the other hand, electrotaxis affects cell direction and cellvelocity due to electrostatic forces. Upon exposing a cell to an EF,the cell membrane potentials change. In an EF, the plasmamembrane facing the cathode depolarises while the membranefacing the anode hyperpolarises (Gao et al., 2011; Mycielska andDjamgoz, 2004; Nuccitelli, 2003). In most studied cells, this is

thought to depend on changes in Ca2+ (Mycielska and Djamgoz, 2004).It has been proposed that the changes in the membrane potentialsmay imply electrotaxis. A simple cell in the resting state has anegative membrane potential (Mycielska and Djamgoz, 2004;Nuccitelli, 2003). In case of a cell with negligible voltage-gatedconductance, the hyperpolarised membrane facing the anodeattracts Ca2+ by passive electrochemical diffusion. This side ofthe cell then contracts, thereby propelling the cell toward thecathode. This process continues until the voltage-gated Ca2+

channels (VGCCs) near the cathodal side open (depolarised),thereby allowing Ca2+ influx (Fig. 1). Intracellular Ca2+ level willrise on both anodal and cathodal sides of the cell (Gao et al., 2011;Mycielska and Djamgoz, 2004). The cell net movement, if any, inthis situation would depend on the balance between the opposingmagnetic contractile forces produced by the EF (Mycielska andDjamgoz, 2004). This causes some cell types to reorient towardsthe cathode, such as embryo fibroblasts (Onuma and Hui, 1985),human keratinocytes (Sheridan et al., 1996), fish epidermal cells(Cooper and Shliwa, 1986), human retinal pigment epithelial cells(Sulik et al., 1992), while other types towards the anode, likehuman granulocytes (Rapp et al., 1988), metastatic human breastcancer cells (Fraser et al., 2002).

Once the cell is positioned in an EF, there exists a significantpassive influx of Ca2+ ions which causes the cell to be charged. It isknown in electrostatics that the experienced force in an EF by anindividual charged cell can be obtained by

FEF ¼ EΩSeEF ð9Þ

where E stands for uniform EF, Ω denotes surface charge density ofa body exposed to uniform EF and eEF is the unit vector of thedirection of the EF force, depending on the cell type it can betowards cathode or anode.

Consequently, the force equilibrium on cell locomotion systemyields

Fdrag ¼−ðFeff þ Fprot þ FEF Þ ð10Þ

where Feff is the effective force due to mechanotaxis, chemotaxisand thermotaxis whose magnitude and direction depend on thenet traction force and the direction of each cue. It will be derivedin detail in the next section.

2.3. Cell deformation and reorientation

Fig. 2 shows how the internal deformation and the reorienta-tion of a cell is calculated. We suppose that the sensing forces areexerted at each external finite element node of the cell elementstowards its centroid (Fig. 2a). The deformed cell subjected to thosesensing forces is represented by dotted lines in Fig. 2b. Therefore,the cell internal strain in each external node of the cell can be

Fig. 1. Response of a cell to dcEFs. A simple cell in a resting state has a negativemembrane potential (Mycielska and Djamgoz, 2004). When a cell with a negligiblevoltage-gated conductance is exposed to a dcEF, its membrane near the anode willbe hyperpolarised so that the cell attracts Ca2+ due to passive electrochemicaldiffusion. Consequently, this side of the cell contracts, thereby propelling the celltowards the cathode. Therefore, VGCCs near to cathode are depolarised allowingCa2+ influx. In such a case, intracellular Ca2+ level will rise in both sides. Therefore,the net direction of cell movement depends on the balance between the opposingmagnetic contractile forces by both poles (Mycielska and Djamgoz, 2004).

S.J. Mousavi et al. / Journal of Theoretical Biology 329 (2013) 64–7366

Publications

61

written as

ϵcell ¼ABOA

ð11Þ

Once the internal strain of the cell is calculated, using Eqs. (1) and(6), scell and the net traction force can be obtained respectively.

It is important to note that the internal deformation created bythe sensing forces at every external node of a cell is negative (cellexerts contraction forces toward its centroid and always com-presses itself). Therefore, nodes with lower internal deformationwill have a higher traction force (Mousavi et al., 2012). As all thetraction forces are acting towards the cell centroid , the resultantof these traction forces will have the direction of minimuminternal deformation towards the cell centroid (Eq. (6)). So that,the opposite direction of the net traction force presents themechanotaxis reorientation of the cell (Mousavi et al., 2012).Therefore, the unit vector of the mechanotaxis reorientation ofthe cell, emech, can be defined as

emech ¼−Ftracnet

∥Ftracnet ∥

ð12Þ

In the presence of other cues such as chemotaxis and thermo-taxis, the reorientation of the cell depends not only on the cellmechano-sensing but also on the gradient of these cues. Let usassume ech and eth represent unit vectors associated to cellreorientation due to chemical and thermal gradients respectively.Therefore

ech ¼∇C∥∇C∥

ð13Þ

eth ¼∇T∥∇T∥

ð14Þ

where ∇ is the gradient operator, while C and T are the chemoat-tractant concentration and the temperature respectively.

We assume that the properties of a typical cell as well as theparameters of its ECM will not change in the presence of thesecues. Therefore, the net traction force exerted by the cell isindependent of these two cues, while it is a function of cell andsubstrate characteristics. Therefore, the presence of these signalsonly changes the effective direction of the previously calculatednet traction force. Besides, we consider that the realignment ofthis force in presence of these cues is proportional to thechemotaxis and thermotaxis gradients. Therefore, a part of thenet traction force is guided by mechanotaxis cue while the rest are

guided by chemotaxis and thermotaxis cues (Fig. 2b). It is ofinterest to remark that in the absence of these cues the nettraction force will act along the mechanotaxis direction. Conse-quently, according to Fig. 2b, the effective force, Feff , in thepresence of the chemotaxis and thermotaxis in Eq. (10) can bedefined as

Feff ¼ Ftracnet ðμmechemech þ μchech þ μthethÞ ð15Þ

where μmech, μch, and μth are the efficient factors associated tomechanotaxis, chemotaxis, and thermotaxis cues respectively,such that μmech þ μch þ μth ¼ 1. Likewise, the cell local velocitycan be calculated as

v¼ ∥Fdrag∥6πrη

ð16Þ

The cell local velocity is the cell velocity at each time step whilethe cell global velocity is the cell velocity in direction of the cuegradient. The later strongly depends on the cell random move-ment. In this case, the net polarisation direction of the cell can beobtained by

epol ¼−Fdrag

∥Fdrag∥ð17Þ

Using Eq. (16) and (17), the displacement vector of an individualcell over the time increment, τ, is given by

d¼ vτepol ð18Þ

3. Numerical experiments and discussion

The model presented above has been implemented into thecommercial FE software ABAQUS (Hibbit et al.,) using a userdefined subroutine. First, the cell nodes are defined and then thesensing forces are applied at each node towards cell centroid.Using the FE approach, the cell internal strain can be calculatedwhich can be used to get the cell traction force. If there is anyother stimuli in the substrate, the corresponding effective force or/and electrical force should be calculated. Once the previous forcesare calculated, the drag force and hence the cell polarisationdirection and velocity can be calculated. After defining the cellnew position, the process can be again resumed till the cellachieves its destination.

Throughout the following simulations, we applied the model toa 400� 200� 200 μm substrate with linear stiffness gradient

Fig. 2. Calculation of the internal deformation and the reorientation of the cell. (a) Sensing forces at each external node and (b) Deformed cell (dashed line) due to mechano-sensing in the presence of mechanotaxis, chemotaxis, thermotaxis and electrotaxis where emech , ech , eth , eEF indicate the unit vectors in direction of each cue respectively.Fprot is the random protrusion force. The coefficients μmech , μch , and μth are respective the efficient factors of mechanotaxis, chemotaxis, and thermotaxis cues respectivelywhile Ftrac

net is the magnitude of the net traction force.



62

which changes from 100 kPa at x¼0 to 200 kPa at x¼ 400 μm. Allthe substrate surfaces are considered to be free. The substrates aremeshed by 16,000 regular hexahedral elements with 18,081nodes, while there is no external force acting on the substrate.The user subroutine is run for about 100 steps with every timestep approximately equal to 600 s which is relatively enough tocomplete one cell migration (Zaman et al., 2005). It is of interest tomention that depending on the cell type, mechanical and bound-ary conditions of the substrate and existent cues, the cell cancomplete the migration process sooner than 100 steps. This can bereferred to the existence of a strong signal towards a cue thatreduces the cell random movement. To evaluate the randomnessof the average cellular translocation in different regimes, the anglebetween net polarisation direction and the imposed gradient orelectrical filed direction, θ, is calculated at each time step. There-fore the Random Index (RI) can be calculated as

RI¼ ∑Ni ¼ 1cos θi

Nð19Þ

where N is the number of required time steps during which thecell needs to reach the destination. RI is +1 if the cell migratesdirectly towards the stimuli (maximum stiffness, chemical con-centration, temperature or/and negative pole), and −1 if the cellmigrates directly in the opposite direction. Consequently thecloser RI to +1, the lower the cell random motility.

Even any cell shape can be considered, we assume a sphericalshape for the cell with 24 external finite element nodes. Table 1shows the properties of the cell and the substrate. During migra-tion, the cell centroid is visualised by a small sphere at each timestep to indicate the cell position.

Each simulation has been repeated 20 times to evaluateconsistency of the results. The obtained results for all the experi-ments are compared with the previously presented work in whichthe cell migration due to mechanotactic cues was studied(Mousavi et al., 2012). In that work, it was demonstrated that inthe substrate with stiffness gradient while all boundary surfacesare considered free, the cell randomly moves towards an imagin-ary equilibrium plane (IEP) located at stiff part (x¼ 355 μm) andkeeps moving around it.

3.1. Cell migration in the presence of chemotaxis

Zhang et al. (2008) applied cyclic Andenosine Monophosphate(cAMP) to create a shallow chemoattractant gradient in thesurrounding substrate. This activates the cells cAMP receptorsand their associated G-proteins to reorient the cell, although thefull understanding of this mechanism still remains elusive. More-over, Hoeller and Kay (2007) observed that chemotactic neutro-phils and dictyostelium amoebae produce signalling lipidgradients of Phosphatidylinositol (3,4,5)-Triphosphate (Ptdlns(3,4,5)P3) in their plasma membranes, which are orientated withthe external chemotactic gradient. These signals have been pro-posed to act as a cell internal compass that is guiding its move-ment (Hoeller and Kay, 2007).

Here, we investigate the cell migration in the previouslymentioned substrate in the presence of a linear chemoattractantgradient along the x-axis. It is assumed that there is a chemoat-tractant source (5�10−5 M) at x¼ 400 μm which creates a uni-form chemical gradient over the x-axis. The cell is firstly located ata corner of the substrate where the chemoattractant concentrationis null. In this case there exists also an IEP that the cell randomlymoves towards and then keeps moving around it. In the presenceof chemotaxis the location of the IEP also depends on thechemotaxis efficient factor. Fig. 3 shows the cell migration in thepresence of chemotaxis for two different chemotaxis effectivefactors. As seen in Fig. 3a and b, for μch ¼ 0:35, the IEP slightlydisplaces towards the chemoattractant source (x¼ 375 μm). Onthe contrary, the IEP disappears for μch ¼ 0:4 and the cell migratestowards the maximum chemoattractant concentration at the endof the substrate. The corresponding RI of each chemotacticefficient factor is presented in Table 2. The results demonstratethat the cell random movement through the substrate, is relativelylower in the presence of chemoattractant source than that of puremechanotaxis. This occurs due to the received signals by cellpseudopods are in the direction of the chemotaxis (Van Haastert,2010). Therefore, the cell polarisation direction is persistentlytowards the chemoattractant source which sums to the mechan-otaxis effects and hence increases the cell global velocity. Theseresults are consistent with findings of Neilson et al. (2011) as well

Table 1Substrate and cell properties.

Symbol Description Value Ref.

ν Poisson ratio 0.3 (Akiyama and Yamada, 1985; Ulrich et al., 2009)μ Viscosity 1000 Pa s (Akiyama and Yamada, 1985; Zaman et al., 2005)r Cell radius 20 μm (Buxboim et al., 2010)Kpas Stiffness of microtubules 2.8 kPa (Schäfer and Radmacher, 2005)Kact Stiffness of myosin II 2 kPa (Schäfer and Radmacher, 2005)ϵmax Maximum strain of the cell 0.09 (Mousavi et al., 2012; Ramtani, 2004)ϵmin Minimum strain of the cell −0.09 (Mousavi et al., 2012; Ramtani, 2004)smax Maximum contractile stress exerted by actin-myosin machinery 0.1 kPa (Oster et al., 1983; Penelope and Janmey, 2005)kf ¼ kb Binding constant at the rear and front of the cell 108 mol−1 (Zaman et al., 2005)nf ¼ nb Number of available receptors at the rear and front of the cell 105 (Zaman et al., 2005)ψ Concentration of the ligands at rear and front of the cell 10−5 mol (Zaman et al., 2005)Ω Order of surface charge density of a cell 10−4 C/m2 (Huang et al., 2011)E The range of applied electric field 0–100 mV/mm (Mycielska and Djamgoz, 2004; Nuccitelli, 2003)

Table 2Average RI of single cell migration trough a multisignalling substrate. The deviations present the maximum difference for 20 computational runs.

Mechanotaxis Thermotaxis μth ¼ 0:2 Chemotaxis Electrotaxis

μch ¼ 0:35 μch ¼ 0:4 E¼10 mV/mm E¼100 mV/mm

RI 0.5170.04 0.5670.06 0.5970.05 0.670.06 0.6670.04 0.9170.02


Publications

63

as experimental investigations of Andrew and Insall (2007) andVan Haastert (2010) who observed that in the presence ofchemoattractant, depending on the signal strength of the che-moattractant source, the cell can actively move towards chemoat-tractant sources. Moreover, similar cell behaviour was reported byBosgraaf and VanHaastert (2009).

Figs. 4 and 5 present the cell local velocity and net tractionforce versus the cell displacement in chemotaxis processes respec-tively, to demonstrate the trend of each one in the presence of newstimuli. In both cases, similar to pure mechanotaxis migration(Mousavi et al., 2012), the cell net traction force and the cell localvelocity decrease. This occurs because the magnitude of the nettraction force is independent of chemotactic cue but it is functionof the mechanical conditions of the substrate. On the other hand,the presence of the chemotaxis affects the drag force which in turncan slightly affect the cell velocity although this effect on the cellvelocity can be neglected compared to the mechanotaxis effect.Therefore during cell migration towards stiffer region and highchemoattractant concentration the cell behaviour is similar topure mechanotaxis process in terms of local velocity and nettraction force.

3.2. Cell migration in presence of thermotaxis

Eisenbach and Bahat (2006) reported that, in vivo, thermotaxiswhich affects at long distances is complementary to chemotaxiswhich acts in short ones. Experiments by Higazi et al. (1996)demonstrated that cultured human trophoblasts are influenced bythermal gradient. They reported that human trophoblastic cellssense differences of less than 1 1C above or below physiologicaltemperature.

Fig. 6 shows the designed experiment to elucidate the effect ofthe thermotactic cue on cell migration. We suppose a thermal

Fig. 3. Cell migration in presence of chemotaxis. It is considered that there is a chemoattractant source at x¼ 400 μm which creates a chemotaxis gradient through thesubstrate in the x direction. First, the cell is located in one of the corners of the substrate near to x¼0 where chemoattractant concentration is minimum. The results of twodifferent chemoattractant efficient factors have been presented. For both cases the cell migrates in the direction of the chemoattractant gradient. When μch ¼ 0:35 the cellmoves around an IEP located at x¼ 375 μm (a and b). Increasing the chemoattractant efficient factor to μch ¼ 0:4 cause the IEP to disappear so that the cell migrates towardsthe end of the substrate with high chemoattractant concentration (c and d).

Fig. 4. Magnitude of the net traction force in the presence of the chemotaxis versuscell displacement.

Fig. 5. Cell local velocity in the presence of the chemotaxis versus celldisplacement.



64

gradient throughout the substrate in x direction. At x¼0, thetemperature is equal to 36 1C and is linearly increasing to achieve39 1C at x¼ 400 μm (Higazi et al., 1996). Because the thermotaxisgradient has relatively small slope, we have assumed μth ¼ 0:2which is less than the chemotaxis efficient factor. In this case theIEP is located at x¼ 365 μm which has been slightly movedtowards the end of the substrate rather than pure mechanotaxis(Mousavi et al., 2012). Once the cell achieves this IEP it randomlymoves around it. This result is consistent with findings of Higaziet al. (1996) who reported that trophoblastic cells migrate towardswarmer sites in the presence of thermal gradient.

Comparing the obtained RI in existence of thermotactic andchemotactic cues, (Table 2) demonstrates that the random move-ment of the cell through the substrate with thermotaxis gradient isrelatively higher than that of chemotaxis for the proposed efficientfactors. It is remarkable that in both cases the cell random

movement is lower than that of pure mechanotaxis. Figs. 7 and8 show the net traction force and the cell local velocity versus thecell displacement respectively. As it can be shown, during the cellmigration towards stiffer and warmer regions of the substrateboth the net traction force and the cell local velocity decrease,having the same trend as the cell migration due to pure mechan-otaxis and mechano-chemo taxis.

3.3. Cell migration in the presence of electrotaxis

In cornea wound, EFs of 42 mV/mm are measured (Chianget al., 1992; Zhao, 2009). In addition, lateral potential gradient alsodrops from 140 mV/mm at the wound edge to 0 mV/mm just3 mm away from ulcer in a Guinea pig skin (Barker et al., 1982;Illingworth and Barker, 1980; Jaffe and Vanable, 1984; Zhao, 2009).Experiments show that exposing the cell to an exogenous dcEf inthe most cases causes that the cell moves towards the cathode;such as metastatic rat prostate cancer cells (Djamgoz et al., 2001),embryo fibroblasts (Onuma and Hui, 1985), human keratinocytes(Sheridan et al., 1996), fish epidermal cells (Cooper and Shliwa,1986), human retinal pigment epithelial cells (Sulik et al., 1992),epidermal and human skin cells (Nuccitelli, 2003). However, thereare some cell types that move towards the anode; for example,human granulocytes (Rapp et al., 1988), rabbit corneal endothelialcells (Chang et al., 1996), metastatic human breast cancer cells(Fraser et al., 2002). Although Grahn et al. (2003) reported thathuman dermal melanocytes are not excitable to an exogenousdcEF of 100 mV/mm which can be due to their higher dcEFthreshold (Onuma and Hui, 1988). These observations altogetherdemonstrate that different cell types may show different electro-taxis behaviour.

To study the influence of the electrotaxis on cell migration inabsence of chemotactic and thermotactic cues, we assumed thatthe cell is exposed to a dcEF inside the substrate. In this example,the cathode is located at x¼ 400 μm and the anode at x¼ 0 μm.Furthermore, we consider that the cell type is such that to beattracted by the cathode (such as embryo fibroblasts Onuma andHui, 1985). The cell is firstly placed in one of the corners of thesubstrate as well. The simulation is repeated for E¼10 and100 mV/mm. The results for both weak and strong EF are pre-sented in Fig. 9. It is obvious that for both cases the cellpersistently migrates towards the cathode pole and for both weakand strong EF there is no IEP. Besides, when the cell arrives the endof the substrate and keeps very near the cathode, it does notrandomly move as the previous cases. Because the generatedelectrical force dominates the other forces, it points the celltowards the cathode and the cell keeps its position near the freesurface. It is worth to mention that the magnitude of Feff do notchanges with the changing of μmech, μch nor μth. Changes of μmech,μch and μth can only change the direction of Feff (see Eq. (15)).

Fig. 6. Cell migration in presence of thermotaxis. It is considered that there is a linear thermal gradient along the x direction where x¼ 400 μm has the maximumtemperature (39 1C). The thermal efficient factor is assumed to be 0.2. When the cell is firstly placed in one of the corners of the substrate with minimum temperature(36 1C) it migrates towards high temperature in the thermal gradient direction. Finally the cell keeps moving around an IEP located at x¼ 365 μm.

Fig. 7. Magnitude of the net traction force in the presence of thermotaxis versuscell displacement.

Fig. 8. Cell local velocity in presence of the thermotaxis versus cell displacement.


Publications

65

Consequently, changes of these cues do not affect the magnitudeof Fdrag. This is not the case when the EF changes, variation of theEF can affect the magnitude and direction of Fdrag (see Eqs. (9) and(10)). This can justify how EF can dominate other effective cuesduring cell motility as reported by Illingworth and Barker (1980),Maria and Djamgoz (2004) and Zhao (2009).

Moreover, cell random motility decreases in existence ofelectrotaxis (Table 2) even more than other cues. Besides, increas-ing the EF from 10 to 100 mV/mm remarkably increase the cell RI.This is because any amplification of the EF increases the magni-tude of the drag force proportionally and affects the cell reorienta-tion (see Eq. (17)). Consequently, the effect of the randomprotrusion force on cell reorientation reduces. This result isconsistent with findings of Nishimura et al. (1996) who reportedthat for 10 mV/mm of the exogenous EF, 10% of the keratinocytesmigrate towards cathode while this amount increases up to 98%when the EF is increased to 100 mV/mm. Moreover, experimentalresults presented by McCaig et al. (2002) demonstrated thatneuron growth is strongly aligned by EF and exhibits randomorientation when the EF reduces.

According to Fig. 10, during cell motility, the net traction forcereduces since it is function of surrounding mechanical conditionsand independent of the EF. On the contrary, at the end of the cellmigration it increases due to cell migration toward the cathodepole which is located at the free surface of the substrate. The celllocal velocity is plotted versus the cell displacement in Fig. 11.From it, we note that firstly the cell local velocity reduces. Becausein this region the exerted force by EF to the cell is not enough tocompensate the reduction of the net traction force due to cellmigration towards stiffer region. In contrast, in the intermediateinterval, the cell local velocity does not significantly change. In thisinterval, the exerted force by EF can compensate the net tractionforce reduction. In the last interval of Fig. 11, the local velocityagain increases due to higher net traction force (Fig. 10) due to theexerted force by the EF. This means that the net traction force andEF force collaborate to increase the cell local velocity, so that thecell migrates towards the cathode even in existence of the freesurface. These findings are consistent with the results of Zhao(2009) who reported that exposing a cell to an EF can overrideother cues.

Fig. 9. Cell migration in presence of the electrotaxis. It is considered that there is a dcEF in the x direction. The anode is located at x¼0 and the cathode at x¼400 mm. It issupposed that the cell type is embryo fibroblast which is directed to the cathode. The cell is firstly placed in one of the corners of the substrate, far from the cathode pole. Thecell exhibits a very low randommovement towards the negative pole much lower than in the previous experiments. This randommovement decreases when the EF strengthincreases from 10 mV/mm (a and b) to 100 mV/mm (c and d).

Fig. 10. Magnitude of the net traction force in presence of the EF versus celldisplacement.

Fig. 11. Cell local velocity in presence of the EF versus cell displacement.



66

Fig. 12 shows the relation between all the considered cues andthe RI. The results show that the increase of chemotactic orthermotactic efficient factors as well as the cell charge density ordcEF decrease the cell random movement (increase the RI). It isclear that effect of electrotaxis on reduction of the cell randommotility is higher than that of chemotaxis or thermotaxis.

4. Conclusions

We have extended the 3D mechanotaxis model of cell migra-tion based on force equilibrium (Mousavi et al., 2012) to investi-gate the influence of chemotaxis, thermotaxis and electrotaxiscues on cell motility. The proposed model allows predicting thecell behaviour in different environmental conditions. The modelcan deliver results that are in a reasonable qualitative agreementwith experimental works (Bosgraaf and VanHaastert, 2009; Higaziet al., 1996; Nishimura et al., 1996).

The cell behaviour in a substrate with stiffness gradient duringpure mechanotaxis had been presented (Mousavi et al., 2012). Inthe absence of other signals, we observed that the cell tendency isto migrate in the direction of the stiffness gradient towards an IEPlocated far from free surfaces at stiffer region of the substrate(Mousavi et al., 2012; Schlüter et al., 2012). The extended modelwas applied for the same substrate to study the effect of other cueson cell migration. When another activation signal was added tothe substrate, the overall cell behaviour changes. For instance, inthe presence of chemotaxis and/or thermotaxis the location of theIEP is sensitive to their efficient factors. In existence of tempera-ture gradient (thermotaxis), the IEP displaces towards highertemperature. In addition, when there is a chemoattractant source(chemotaxis) the IEP approaches the end of the substrate and evendisappears for high efficient factors, i.e. chemoattractant efficientfactor can dominate mechanotaxis signals and propel the celltowards the chemoattractant source. It is interesting to note thatthe effect of these cues is negligible on the cell local velocity sincethe magnitude of the net traction force depends on the surround-ing mechanical conditions (Figs. 5 and 8). In contrast, these signalsremarkably affect the reorientation of the cell and reduce the cellrandom motility (see Table 2). Since chemotaxis or thermotaxissignals persistently point the cell towards high chemoattractantconcentration or warmer sites respectively, the cell global velocityincreases. These findings are in agreement and consistent withchemotaxis (Andrew and Insall, 2007; Bosgraaf and VanHaastert,2009; Van Haastert, 2010) and thermotaxis (Eisenbach and Bahat,2006; Higazi et al., 1996) experimental works.

Moreover, the results demonstrate that a typical cell migratestowards the cathode pole in the presence of exogenous dsEF.We observed that amplification of the EF can accelerate this

phenomenon that is consistent with findings of Nishimura et al.(1996). The existence of the electrostatic force also drives the cellto track more straightly towards the electrotactic cue (Fig. 9). Inthis case, the RI increases (Table 2) which means reduction on thecell randommovement and consequently increasing the cell globalvelocity. Since the cell reorientation points strongly towards thecathode pole, due to the domination of the generated force by theEF over the other forces, the cell remains near the end of thesubstrate without almost any movement. This was observed byMcCaig et al. (2002) for the alignment of neuron outgrowths in thepresence of an EF. Furthermore, despite the free surface at thecathode site, the cell migrates towards the end of the substrate.This indicates that electrotaxis may dominate other signals (heremechanotaxis). This explains why EFs play a dominant guidancerole in directing cell migration in epithelial wound healing (Zhao,2009). Besides, as seen from Fig. 12, combining different signals tothe cell substrate reduces its random movement (increasing thecell RI). This reduction is significant with electrotactic stimulus. Aspreviously mentioned, this occurs because the drag force notablyincreases due to electrical charge attraction.

The proposed model, beside providing a better understandingof cell migration, has a potential to simultaneously consider theeffects of several cues on cell motility in different biologicalprocesses. The obtained results through the simulations are inagreement with experimental observations. However moresophisticated experiments are a key to illustrate the preciseresponse of the cell for different environmental signals. So, thepresented model can provide useful insights for the design ofthese experiments. Moreover, the obtained results can provide aqualitatively useful information to predict cell migration bothin vivo and in vitro experiments.

Acknowledgements

The authors gratefully acknowledge the financial support fromthe Spanish Ministry of Science and Technology (CYCIT DPI2010-20399-C04-01) and CIBER-BBN initiative. CIBER-BBN is an initia-tive funded by the VI National R&D&I Plan 2008-2011, IniciativaIngenio 2010, Consolider Program, CIBER Actions and financed bythe Instituto de Salud Carlos III with assistance from the EuropeanRegional Development Fund.

Appendix A. Supplementary material

Supplementary data associated with this article can be found inthe online version of http://dx.doi.org/10.1016/j.jtbi.2013.03.021.

References

Akiyama, S.K., Yamada, K.M., 1985. The interaction of plasma fibronectin withfibroblastic cells in suspension. J. Biol. Chem. 260, 4492–4500.

Allena, R., Aubry, D., 2012. Run-and-tumble or look-and-run? A mechanical modelto explore the behavior of a migrating amoeboid cell. J. Theor. Biol. 306, 15–31.

Andrew, N., Insall, R.H., 2007. Chemotaxis in shallow gradients is mediatedindependently of ptdlns 3-kinase by biased choices between random protru-sions. Nat. Cell Biol. 9, 193–200.

Barker, A.T., Jaffe, L.F., Vanable Jr., J.W., 1982. The glabrous epidermis of caviescontains a powerful battery. Am. J. Physiol. 242, R358–R366.

Behesti, H., Marino, S., 2009. Cerebellar granule cells: insights into proliferation,differentiation, and role in medulloblastoma pathogenesis. J. Appl. Physiol. 41,435–445.

Bosgraaf, L., Van Haastert, P.J.M., 2009. Navigation of chemotactic cells by parallelsignaling to pseudopod persistence and orientation. PLoS ONE 4, e6842.

Buxboim, A., Ivanovska, I.L., Discher, D.E., 2010. Matrix elasticity, cytoskeletal forcesand physics of the nucleus: how deeply do cells ‘feel’ outside and in?. J. Cell Sci.123, 297–308.

Fig. 12. Variation of the RI with chemotaxis, thermotaxis and electrotaxis cues.


Publications

67

Chang, P.C.T., Soong, H.G., Sulik, G.L., Parkinson, W.C., 1996. Galvanotropic andgalvanotactic responses of corneal endothelial cells. Formos. Med. Assoc. J. 95,623–627.

Chiang, M., Robinson, K.R., Vanable, J.W., 1992. Electrical fields in the vicinity ofepithelial wounds in the isolated bovine eye. Exp. Eye Res. 54, 999–1003.

Chirag, B.K., Shelly, R.P., Andrew, J.P., 2006. Intrinsic mechanical properties of theextracellular matrix affect the behavior of pre-osteoblastic mc3t3-e1 cells. Am.J. Physiol. Cell Physiol. 290, C1640–C1650.

Choo, B.H., Donna, R.F., Kenneth, L.P., 1983. Thermotaxis of dictyostelium discoi-deum amoebae and its possible role in pseudoplasmodial thermotaxis. Proc.Natl. Acad. Sci. USA 80, 5646–5649.

Cooper, M.S., Shliwa, M., 1986. Motility of cultured fish epidermal cells in thepresence and absence of direct current electric fields. J. Cell Biol. 102,1384–1399.

Dalton, B.A., Walboomers, X.F., Dziegielewski, M., Evans, M.D.M., Taylor, S., Jansen, J.A.,Steele, J.G., 2001. Modulation of epithelial tissue and cell migration by micro-grooves. Phil. Trans. R. Soc. London 56, 195–207.

Djamgoz, M.B.A., Mycielska, M., Madeja, Z., Fraser, S.P., Korohoda, W., 2001.Directional movement of rat prostate cancer cells in directcurrent electricfield: involvement of voltage-gated Na+ channel activity. J. Cell Sci. 114,2697–2705.

Eisenbach, M., Bahat, A., 2006. Sperm thermotaxis. Mol. Cell. Endocrinol. 252,115–119.

Elliott, Ch.M., Stinner, B., Venkataraman, Ch., 2012. Modelling cell motility andchemotaxis with evolving surface finite elements. J. R. Soc. Interface 9 (76),3027–3044.

Etienne-Manneville, S., 2008. Polarity proteins in migration and invasion. Oncogene27, 6970–6980.

Fraser, S.P., Diss, J.K.J., Mycielska, M.E., Coombes, R.C., Djamgoz, M.B.A., 2002.Voltage-gated sodium channel expression in human breast cancer: possiblefunctional role in metastasis. Breast Cancer Res. Treat. 76, S142.

Gao, R.C., Zhang, X.D., Sun, Y.H., Kamimura, Y., Mogilner, A., Devreotes, P.N., Zhao, M.,2011. Different roles of membrane potentials in electrotaxis and chemotaxis ofdictyostelium cells. Eukaryotic Cell 10, http://dx.doi.org/10.1128/EC.05066-11.

Grahn, J.C., Reilly, D.A., Nuccitelli, R.L., Isseroff, R.R., 2003. Melanocytes do notmigrate directionally in physiological dc electric fields. Wound Repair Regen.11, 64–70.

Van Haastert, P.J.M., 2010. Chemotaxis: insights from the extending pseudopod. J.Cell Sci. 123, 3031–3037.

Hibbit, Karlson, Sorensen, 2011. Abaqus-Theory Manual, 6.11-3 edition.Higazi, A.A., Kniss, D., Manuppelloand, J., Barnathan, E.S., Cines, D.B., 1996.

Thermotaxis of human trophoblastic cells. Placenta 17, 683–687.Hoeller, O., Kay, R.R., 2007. Chemotaxis in the absence of pip3 gradients. Curr. Biol.

17, 813–817.Huang, Yao-X., Zheng, X.J., Kang, L.L., Chen, X.Y., Liu, W.J., Wu, Zh.-J., Huang, B.T.,

2011. Quantum dots as a sensor for quantitative visualization of surface chargeson single living cells with nano-scale resolution. Biosensors Bioelectronics 26,2114–2118.

Illingworth, C.M., Barker, A.T., 1980. Measurement of electrical currents emergingduring the regeneration of amputated finger tips in children. Clin. Phys. Physiol.Meas. 1, 87.

Ingber, D.E., Tensegrity, I., 2003. Cell structure and hierarchical systems biology. J.Cell Sci. 116, 1157–1173.

Jaffe, L.F., Vanable, J.W., 1984. Electric fields and wound healing. Clin. Dermatol. 2,34–44.

James, D.W., Taylor, J.F., 1969. The stress developed by sheets of chick fibroblastsin vitro. Exp. Cell Res. 54, 107–110.

Maeda, Y.T., Inose, J., Matsuo, M.Y., Iwaya, S., Sano, M., 2008. Ordered patterns of cellshape and orientational correlation during spontaneous cell migration. PLoSONE 3, e3734.

Maria, E.M., Djamgoz, M.B.A., 2004. Cellular mechanisms of direct-current electricfield effects: galvanotaxis and metastatic disease. J. Cell Sci. 117, 1631–1639.

McCaig, C.D., Rajnicek, A.M., Song, B., Zhao, M., 2002. Has electrical growth coneguidance found its potential? Trends Neurosci. 25, 354–359.

Moreo, P., Garcia-Aznar, J.M., Doblaré, M., 2008. Modeling mechanosensing and itseffect on the migration and proliferation of adherent cells. Acta Biomaterialia 4,613–621.

Mousavi, S.J., Doweidar, M.H., Doblaré, M., 2012. Computational modelling andanalysis of mechanical conditions on cell locomotion and cell–cell interaction.Comput. Methods Biomech. Biomed. Eng. page http://dx.doi.org/10.1080/10255842.2012.710841.

Mycielska, M.E., Djamgoz, M.B., 2004. Cellular mechanisms of directcurrent electricfield effects: galvanotaxis and metastatic disease. J. Cell Sci. 117, 1631–1639.

Neilson, M.P., Veltman1, D.M., van Haastert, P.J.M., Webb, S.D., Mackenzie, J.A.,Insall, R.H., 2011. Chemotaxis: a feedback-based computational model robustlypredicts multiple aspects of real cell behaviour. PLoS Biol. 9, e1000618.

Nishimura, K.Y., Isseroff, R.R., Nuccitelli, R., 1996. Human keratinocytes migrate tothe negative pole in direct current electric fields comparable to those measuredin mammalian wounds. J. Cell Sci. 109, 199–207.

Nuccitelli, R., 2003. A role for endogenous electric fields in wound healing. Curr. TopDev. Biol. 58, 1–26.

Onuma, E.K., Hui, S.W., 1985. A calcium requirement for electric field-induced cellshape changes and preferential orientation. Cell Calcium 6, 281–292.

Onuma, E.K., Hui, S.W., 1988. Electric field-directed cell shape changes, displace-ment, and cytoskeletal reorganization are calcium dependent. J. Cell Biol. 106,2067–2075.

Oster, G.F., Murray, J.D., Harris, A.K., 1983. Mechanical aspects of mesenchymalmorphogenesis. J. Embryol. Exp. Morph. 78, 83–125.

Pelham, R.J., Wang, Y.L., 1997. Cell locomotion and focal adhesions are regulated bysubstrate flexibility. Proc. Natl. Acad. Sci. 94, 13661–13665.

Penelope, C.G., Janmey, P.A., 2005. Cell type-specific response to growth on softmaterials. J. Appl. Physiol. 98, 1547–1553.

Pompe, T., Glorius, S., Bischoff, T., Uhlmann, I., Kaufmann, M., Brenner, S., Werner,C., 2009. Dissecting the impact of matrix anchorage and elasticity in celladhesion. Biophys. J. 97, 2154–2163.

Ramtani, S., 2004. Mechanical modelling of cell/ECM and cell/cell interactionsduring the contraction of a fibroblast-populated collagen microsphere: theoryand model simulation. J. Biomech. 37, 1709–1718.

Rapp, B., Boisfleury-Chevance, A., Gruler, H., 1988. Galvanotaxis of human granu-locytes dose–response curve. Eur. Biophys. J. 16, 313–319.

Schäfer, A., Radmacher, M., 2005. Influence of myosin II activity on stiffness offibroblast cells. Acta Biomaterialia 1, 273–280.

Schlüter, D.K.I.R.C., Chaplain, M.A.J., 2012. Computational modeling of single-cellmigration: the leading role of extracellular matrix fibers. Biophys. J. 103, 1141–1151.

Sheridan, D.M., Isseroff, R.R., Nuccitelli, R., 1996. Imposition of a physiologic dcelectric field alters the migratory response of human keratinocytes on extra-cellular matrix molecules. J. Invest. Dermatol. 106, 642–646.

Sulik, G.L., Soong, H.K., Chang, P.C., Parkinson, W.C., Elner, S.G., Elner, V.M., 1992.Effects of steady electric fields on human retinal pigment epithelial cellorientation and migration in culture. Acta Ophthalmol. (Copenh.) 70, 115–122.

Suresh, S., 2007. Biomechanics and biophysics of cancer cells. Acta Biomaterialia 3,413–438.

Swaney, K.F., Huang, C.H., Devreotes, P.N., 2010. Eukaryotic chemotaxis: a networkof signaling pathways controls motility, directional sensing, and polarity. Annu.Rev. Biophys. 39, 265–289.

Tanos, B., Rodriguez-Boulan, E., 2008. The epithelial polarity program: machineriesinvolved and their hijacking by cancer. Oncogene 27, 6939–6957.

Ulrich, T.A., De Juan Pardo, E.M., Kumar, S., 2009. The mechanical rigidity of theextracellular matrix regulates the structure, motility, and proliferation ofglioma cells. Cancer Res. 69, 4167–4174.

Young, G.B., 1970. Thermography. CRC Crit. Rev. Radiol. Sci. 1, 593–638.Zaman, M.H., Kamm, R.D., Matsudaira, P., Lauffenburgery, D.A., 2005. Computa-

tional model for cell migration in three-dimensional matrices. Biophys. J. 89,1389–1397.

Zhang, S., Charest, P.G., Firtel, R.A., 2008. Spatiotemporal regulation of Ras activityprovides directional sensing. Curr. Biol 18, 1587–1593.

Zhao, M., 2009. Electrical fields in wound healing—an overriding signal that directscell migration. Semin. Cell Dev. Biol. 20, 674–682.



68

Seyed Jamaleddin Mousavi, Manuel Doblaré and Mohamed Hamdy Doweidar

Physical Biology

2014


2.4 Computational modelling of multi-cell migration in a

multi-signalling substrate.

Physical Biology

Phys. Biol. 11 (2014) 026002 (17pp) doi:10.1088/1478-3975/11/2/026002

Computational modelling of multi-cellmigration in a multi-signalling substrate

Seyed Jamaleddin Mousavi1,2,3, Manuel Doblare1,2,3

and Mohamed Hamdy Doweidar1,2,3

1 Group of Structural Mechanics and Materials Modelling (GEMM), Aragon Institute of EngineeringResearch (I3A), University of Zaragoza, Spain2 Mechanical Engineering Department, School of Engineering and Architecture (EINA), University ofZaragoza, Spain3 Centro de Investigacion Biomedica en Red en Bioingenierıa, Biomateriales y Nanomedicina(CIBER-BBN), Spain

E-mail: [email protected]

Received 8 January 2014, revised 6 February 2014Accepted for publication 17 February 2014Published 17 March 2014

AbstractCell migration is a vital process in many biological phenomena ranging from wound healing totissue regeneration. Over the past few years, it has been proven that in addition to cell–cell andcell-substrate mechanical interactions (mechanotaxis), cells can be driven by thermal,chemical and/or electrical stimuli. A numerical model was recently presented by the authors toanalyse single cell migration in a multi-signalling substrate. That work is here extended toinclude multi-cell migration due to cell–cell interaction in a multi-signalling substrate underdifferent conditions. This model is based on balancing the forces that act on the cell populationin the presence of different guiding cues. Several numerical experiments are presented toillustrate the effect of different stimuli on the trajectory and final location of the cell populationwithin a 3D heterogeneous multi-signalling substrate. Our findings indicate that althoughmulti-cell migration is relatively similar to single cell migration in some aspects, theassociated behaviour is very different. For instance, cell–cell interaction may delay single cellmigration towards effective cues while increasing the magnitude of the average net celltraction force as well as the local velocity. Besides, the random movement of a cell within acell population is slightly greater than that of single cell migration. Moreover, higher electricalfield strength causes the cell slug to flatten near the cathode. On the other hand, as with singlecell migration, the existence of electrotaxis dominates mechanotaxis, moving the cells to thecathode or anode pole located at the free surface. The numerical results here obtained arequalitatively consistent with related experimental works.

Keywords: cell–cell interaction, multi-cell migration, mechano–chemo–thermo-electrotaxis,numerical simulation, finite element method

S Online supplementary data available from stacks.iop.org/PhysBio/11/026002/mmedia

1. Introduction

Cell migration is receiving increasing attention not only in thefield of mechanobiology but also in other fields of biomedicalengineering. Several research works have demonstrated thatcell migration is a key factor in several cellular processessuch as proliferation [8, 78], differentiation [8, 29], woundhealing [67, 85], inflammatory diseases and tumour evolution[62, 72, 75]. Several signals received by the cell direct its

leading edge protrusions and orchestrate its motility. Thisis a complex process which involves the formation of newprotrusions (pseudopods), releasing old adhesions at theleading edges and the formation of new adhesions at the front[3]. This coordinated process occurs in a wide range of celltypes such as epithelial and mesenchymal cells. It mediates celltranslocation, membrane adhesion to the migratory substrateand the coordinated dynamics of the cytoskeleton (CSK).Although the underlying mechanotransducing mechanisms

1478-3975/14/026002+17$33.00 1 © 2014 IOP Publishing Ltd Printed in the UK

Publications

71

Phys. Biol. 11 (2014) 026002 S J Mousavi et al

behind this process are less well understood, it is well knownthat, among other cues, a cell can be guided by mechanotaxis[4, 15, 28, 41, 59, 70], chemotaxis [10, 17, 34, 52, 79],thermotaxis [35, 32] and/or electrotaxis [24, 43, 85].

Cell motility, like many other mechanical events, relieson the equilibrium of effective forces acting on the celland is due to generated internal stresses within the actinCSK, which are transmitted via the integrins at the focaladhesions to the extracellular matrix (ECM) [41, 72].Cells ‘choose’ the migration direction by active mechano–chemo–thermo-electrosensing of their micro-environment.Recent investigations have demonstrated that cells migratepreferentially towards stiffer regions [15, 41], where they needless energy to move.

On the other hand, cells migrate directionally alonggradients of chemical factors such as growth factors orattracting agents. Under a gradient of a chemoattractant source,cells are known to polarise towards the positive or negativedirection of the gradient depending on the chemotactic signaltype [10, 27, 34, 52]. Chemoattraction is thought to playa key role in guiding cells to the main action site inmany immunobiological processes. For instance, it enablesleukocytes to reach infection locations, allows wounds to healand forms embryonic patterns [39]. Furthermore, artificialchemotaxis via drugs may help to alleviate diseases, includingcancer metastasis [39]. However the mechanisms by which acell transduces a chemotactic cue into a certain movement stillremain elusive [17]. Recently, several authors have proposedthat chemotaxis is driven by a ‘compass’ [27, 73]. Thisis produced by processing the extracellular signal into anintracellular mechanism which can steer the cell in the gradientdirection towards a chemoattractant cue. However, some otherauthors reject this theory suggesting that this internal compassdoes not exist but that the cell orients itself simply byits pseudopods [52]. Several numerical models have beenproposed to analyze chemotactic effects on single- [17, 19, 51]and multi-cell [58, 79] migration. In these works, not only hasthe multi-signalling stimulus of the cell not been consideredbut also in most cases the crucial effect of mechanotaxis oncell motility has been ignored [51, 58].

On the other hand, there is evidence that some cellsmigrate towards warmer locations by means of thermotaxis[1, 32]. Thermotaxis may be considered as a complementarycue to chemotaxis. It is active over long distances comparedwith chemotaxis [83]. Higazi et al [32] reported thattrophoblast cells migrate towards maternal uterine arteries(warmer sites) via thermotaxis, where there is no chemotacticsignal.

Moreover, endogenous and exogenous electric fields(EFs) play a significant role in major biological processessuch as embryogenesis, wound healing and tissue regeneration[50, 53, 68, 85]. For instance, in most injuries, epitheliallayers of the skin or surrounding organs disrupt. Consequently,transporting ions of epithelia become directionally polarizedto maintain transepithelial potentials (TEPs). This generates ashort-circuit TEP known as the endogenous wound electricfield. The potential of the EF at the wound thus drops,becoming negative in comparison with the potential of

unwounded locations. This potential gradient drives theelectric current flow towards the more negative position sothat a laterally orientated wound EF forms the so calledelectrotaxis. It is reported that epithelia generate a steadyvoltage across themselves, driving an electric current in thewounded sites [50, 85]. For instance, in the rat cornea andthe skin of a finger tip wound, the generated electric currentis about 10 μA/cm2 which can create a lateral EF in therange of 40–200 mV mm−1 [85]. Besides, a steady EF of450–1600 mV mm−1 has been measured across the wall ofthe amphibian neural tube during early neuronal development[7]. Furthermore, when stationary cells are exposed to directcurrent EFs (dcEFs), they effectively migrate towards cathodeor anode poles depending on the cell type [50, 53]. Nishimuraet al [53] reported that human keratinocytes follow thedirection of the convex edge, while elongated cells displaythree general types of movement depending on the strengthof the imposed EF: (1) under low dcEF strengths (<100 mVmm−1), the cells orient parallel to the direction of the current,extend themselves towards the cathode, then retract the trailingend; (2) some elongated cells will pivot in place due toactive lamellipodia at each end moving in opposite directions;(3) in EF strengths above 100 mV mm−1, elongated cells tendto align with their long axis perpendicular to the field and movetowards the cathode at a steep angle of 45–90◦ with respect totheir long axis.

Many cell types, especially those related to tumourcells, wound healing and tissue regeneration, migrate as cellpopulations through the ECM, unlike discrete cells. Migrationof cell populations, in turn, arises from collective guidancecues of cell neighbours and a dynamic interaction betweenindividually migratory cell responses to surrounding cues [45].Cell–cell adhesiveness has been observed for cell populationsduring collective guidance so that some cell populations tendto migrate as a group. This has been reported for epithelial[23], dictyostelium discoideum [58] and metastatic [30] cells.Besides, similar behaviour of cell migration has been observedfor fibroblasts which are less cohesive cell types [9, 53, 54, 65].

Although, several existing models can be applied tostructured cell populations [20, 26, 33, 79, 80, 82], thereis a lack of comprehensive multi-signalling models dealingwith the multicellular migration process. In the previous work[48], we developed a numerical model to simulate single cellmigration in multi-signalling substrates. The main purpose ofthe present work is to extend the previous model to include andto study the influence of cell–cell interaction in the presenceof chemotaxis, thermotaxis and electrotaxis on the migrationof cell populations.

2. Model formulation

To study the cell response to electrotaxis, chemotaxis andthermotaxis as well as mechanotaxis, it is important todistinguish the effect of each one on cell motility. Durotaxisonly depends on the mechanical properties of the substrate[41, 70], such as stiffness, that intrinsically exist during celland substrate interaction. Therefore, mechanotactic cell forceswhich are exerted during cell motility in the presence or

2

2.4. Computational modelling of multi-cell migration in a multi-signalling substrate.

72


absence of other stimuli depend on the cell type and thephysical properties of the migratory substrate. Chemotaxisand thermotaxis affect only the cell polarization directionby modulating the direction of pseudopods, in such a waythat the cell reorients itself towards the positive gradientof these signals. In contrast, in the presence of electrotaxisthe electrophoresis force affects both the velocity and themigration direction of the cell.

The model is developed with the notion that each stimulusdirects the cells independently and does not remove or changethe effect of other stimulus/stimuli. This argument in conceptof chemo-electrotaxis was reported by Gao et al [24]. Even anycell shape can be considered, for the sake of simplicity, it isassumed that cells have spherical shape. During cell migration,the actin CSK provides the driving force at the cell front,while the microtubule network assumes a regulatory functionin coordinating rear retraction [81]. The changes of actin ormicrotubule network are not considered in this work.

2.1. Stress transmitted by each individual cell to the ECM

There are two main cellular elements which affect cell motility.The first, is formed by actin filaments and myosin II (AMmachinery) which generate the active contractile stress. Thebehaviour of cell active elements mainly depends on theminimum, εmin, and maximum, εmax, internal strains [47, 49].When cell deformation is out of the εmin–εmax range, theactive stress transmitted to the cell is equal to zero. The meanactive contractile stress that is transmitted from the cell to thesubstrate can be calculated as [46–49]

σact =

⎧⎪⎨⎪⎩

0 εcell < εmin or εcell > εmax

Kactσmax(εmin−εcell)

Kactεmin−σmaxεmin � εcell � ε

Kactσmax(εmax−εcell)

Kactεmax−σmaxε � εcell � εmax

(1)

where Kact, σmax and εcell represent the stiffness of activecellular elements, the maximum contractile stress exerted bythe actin–myosin machinery and the internal strain of the cell,respectively, while ε = σmax/Kact.

The second is basically formed by the microtubulesand the cell membrane which generally define the passivemechanical strength which can be calculated by [46–49]

σpas = Kpasεcell (2)

where Kpas is the stiffness of the passive cellular elements.Consequently, the net cell stress transmitted to the ECM, σcell,is the sum of the passive and active stresses.

σcell = σact + σpas. (3)

2.2. Effective mechanical force

There are two main mechanical forces during cell motility: thetraction force and the drag force. The former is a consequenceof the contraction of the actin–myosin apparatus which drivesforward the cell body and is transmitted to the substrate. Thisforce is proportional to the stress transmitted by the cell tothe ECM. Assuming the cell is represented by a connectedgroup of finite elements, the nodal traction force that acts at

each finite element node of the cell membrane towards the cellcentroid can be expressed as [46, 47]

Ftraci = σcellSζei (4)

where ei is a unit vector from the ith node of the cell membranetowards the cell centroid. S denotes the cell membrane areaand ζ is ‘adhesivity’, which is a dimensionless parametermeasuring the binding strength of the receptors to the ligandsin the ECM [47, 84]. This depends on the cell type and isproportional to the binding constant of the cell integrins, k, thetotal number of available receptors, nr, and the concentrationof the ligands at the leading edge of the cell, ψ . Therefore, itcan be defined as [47, 84]

ζ = knrψ. (5)

Consequently, the net traction force acting on the whole cellbody, Ftrac

net , can be calculated as [47]

Ftracnet =

n∑i=1

Ftraci (6)

where n is the number of cell membrane nodes.By contrast, the drag force resists the cell motility. The

main objective here is to define a velocity-dependent opposingforce associated with the linear viscoelastic character of thecell-surrounding ECM, so, for simplification, we have assumedthe ECM to be a viscoelastic medium [84]. At the microscale,the inertial resistance of a viscose fluid is small enough tobe negligible, while the viscous resistance dominates. Sincea rigorous evaluation of the viscous resistance around anarbitrary shape is quite complicated, we have assumed aspherical cell shape. Hence, the drag force on a sphere withradius r, moving with velocity v in a viscous fluid can becalculated by [47, 84]

Fdrag = 6πrηv. (7)

2.3. Protrusion force

During cell migration, cells send out local protrusions to probetheir environment by exerting a protrusion force. This force isgenerated by actin polymerization and must be distinguishedfrom the cytoskeletal contractile force transmitted to the ECM[4, 84]. This causes cells to move along a directed random pathfollowing the effective signal. The direction and magnitude ofthe protrusion force is chosen randomly at each time step. It isworth noting that the order of the protrusion force magnitude isthe same as that of the traction force but with a lower amplitude[38, 47, 63, 84]. Therefore, we describe it by


where erand represents a random unit vector and F tracnet is the

magnitude of the net traction force while κ is a random number,0 � κ < 1 [47, 48].

2.4. Effective electrical force in the presence of electrotacticcues

It has been suggested that cellular Ca2+ influx plays a key rolein the electrotactic response of the cell [50, 53]. However,

3

Publications

73


Figure 1. Response of a cell to a dcEFs. A simple cell in the restingstate has a negative membrane potential [50]. When a cell with anegligible voltage-gated conductance is exposed to a dcEF, itshyperpolarized membrane near the anode attracts Ca2+ due topassive electrochemical diffusion. Consequently, this side of the cellcontracts, propelling the cell towards the cathode. Therefore,voltage-gated Ca2+ channels (VGCCs) near the cathode(depolarized side) open and a Ca2+ influx occurs. In such a cell, theintracellular Ca2+ level rises in both sides. The direction of cellmovement, then, depends on the difference in the opposing magneticcontractile forces which are exerted by cathode and anode [50].

this is still a controversial question. For instance, Ca2+

dependence of electrotaxis has been observed in neural crestcells, embryo mouse fibroblasts, fish and human keratocytes[35, 24, 50, 54, 55]. By contrast, Ca2+ independent electrotaxishas been reported in mouse fibroblasts [12]. The exactmechanism behind the Ca2+ role in electrotaxis is largelyunknown. A simple cell at rest has a negative membranepotential [50]. If this cell is exposed to a dcEF, the sideof the plasma membrane that faces the cathode depolarizeswhile the other side that faces the anode hyperpolarizes[24, 50, 54]. In the case of a cell with negligible voltage-gated conductance, the hyperpolarized membrane side attractsCa2+ by passive electrochemical diffusion. This side of thecell then contracts and propels the cell towards the cathode.This process continues until the voltage-gated Ca2+ channels(VGCCs) near the cathodal side open (depolarized) to allowCa2+ influx (figure 1). Consequently, the intracellular Ca2+

level rises on both anodal and cathodal sides of the cell. Theresultant of the force generated in this case would depend onthe balance between the opposing magnetic contractile forces[50]. That is why some cells reorient towards the cathode, suchas embryo fibroblasts [55], human keratinocytes [53, 69], fishepidermal cells [35] and human retinal pigment epithelial cells[71], while others reorient towards the anode, such as humangranulocytes [64] and metastatic human breast cancer cells[22].

Let us assume that a cell is located in a uniform dcEF.As a consequence, it is ionized and acquires an electrostaticcharge. The force experienced by this individual charged cellcan be obtained by

FEF = ESeEF (9)

where E represents uniform dcEF strength and denotes thesurface charge density of the cell. eEF is a unit vector in the

direction of the dcEF towards the cathode or anode, dependingon the cell type. The time course of the translocation responseduring exposing a cell to a dcEF demonstrates that the cellvelocity versus translocation varies depending on the dcEFstrength. Experiments conducted by Nishimura et al [53] onhuman keratinocytes indicate that the net migration velocity ismaximal when the dcEF strength is about 100 mV mm−1 whileit decreases as the dcEF strength is reduced. They reportedthat increasing the dcEF strength to 400 mV mm−1 does nothave a marked effect on the net migration velocity of thecell. As previously stated, it is thought that the Ca2+ influxmay play a role in this process [6, 35, 42, 50, 53, 68]. Inother words, increasing the concentration of intracellular Ca2+

correlates with the magnitude of the imposed dcEF. Therefore,the cell surface charge is directly proportional to the imposeddcEF strength [50, 53]. Consequently, we assume a linearrelationship between the cell surface charge and the applieddcEF strength as

=⎧⎨⎩

satur

EsaturE E � Esatur

satur E > Esatur

(10)

where satur is the saturation value of the surface charge andEsatur is the maximum dcEF strength that causes Ca2+ influx.

2.5. Deformation and reorientation of the cell

Although the present model is applicable for any cellconfiguration, we assume a spherical configuration for the sakeof simplicity. It is considered that a cell firstly exerts sensingforces to diagnose its surrounding substrate. These forces actat each finite element node of the membrane towards the cellcentroid. The deformed cell resulting from those sensing forcesis represented by dotted lines in figure 2. Therefore, the internalstrain at each finite element node of the cell membrane can bewritten as

εcell = AB

OA. (11)

A cell exerts contraction forces towards its centroidcompressing itself so that the internal deformation created bythese forces on each finite element node of the cell membraneis negative. Hence, nodes with less internal deformation willhave higher traction (see [47] for more details). The resultantof these contraction forces, the traction force, will take thedirection of the minimum internal deformation of the cell(equation (6)). Therefore, the opposite direction of the nettraction force presents the mechanotaxis reorientation of thecell [47]. The unit vector of the mechanotaxis reorientation ofthe cell, emech, is defined as

emech = − Ftracnet

‖Ftracnet ‖

. (12)

In the presence of chemotaxis or thermotaxis, the cellpolarization direction depends not only on the mechanotacticsignal but also on these additional stimuli. We assume that thepresence of these cues does not change either the physical orthe mechanical properties of a typical cell, nor its surroundingsubstrate. Since the traction forces exerted by the cell dependon the cell type and the mechanical properties of the substrate,

4


74


Figure 2. Calculation of the internal deformation and thereorientation of a cell. Dashed line presents a deformed cell due tomechano-sensing in the presence of mechanotaxis, chemotaxis,thermotaxis and electrotaxis. emech, ech, eth, eEF indicate the unitvectors in the direction of each cue, respectively. The coefficientsμmech, μch, and μth are effective factors of mechanotaxis,chemotaxis and thermotaxis cues, respectively. Ftrac

net is themagnitude of the net traction force, Fprot is the random protrusionforce, FEF

tot represents total electrical force which is exerted by dcEFand neighbouring cells on a cell, and Fdrag represents drag force.

the presence of these cues does not affect the magnitudeof the net traction force but changes its direction. Underthermal and/or chemical gradients, the unit vectors associatedto the thermotactic and chemotactic stimuli can be writtenrespectively as [48]

eth = ∇T

‖∇T‖ (13)

ech = ∇C

‖∇C‖ (14)

where ∇ is the gradient operator and T and C arethe temperature and the chemoattractant concentration,respectively. We also assume that the realignment of the nettraction force in the presence of these cues depends on thedirection of chemotaxis and thermotaxis gradients. Thus, asshown in figure 2, a part of the net traction force is guided bymechanotaxis while the rest is guided by chemotaxis and/orthermotaxis cues. Consequently, the effective force due tomechanotaxis, chemotaxis and thermotaxis, Feff, is calculatedas


where μmech, μch and μth are the effective factors formechanotaxis, chemotaxis and thermotaxis cues, respectively,μmech +μch +μth = 1. Since the drag force always acts againstthe cell motion, the force balance yields the definite equationof the cell motion as

Fdrag = −(Feff + Fprot + FEF

tot

)(16)

where FEFtot is the net electrostatic force partly due to the effect

of the dcEF on each individual cell, FEF, and partly because of

(b)(a)

Figure 3. (a) Two cells in contact. They do not exert sensing forcesin common nodes. (b) Position vector and distance between thecentroids of two cells.

cell–cell electrostatic interaction which will be derived in thenext section.

The instantaneous velocity of the cell may therefore bewritten as

v = ‖Fdrag‖6πrη

(17)

with the net polarization direction

epol = − Fdrag

‖Fdrag‖ . (18)

Finally, the incremental translocation vector of an individualcell over a certain small time increment τ is calculated as

d = vτepol. (19)

2.6. Cell–cell interaction

Cells in a multicellular system deform as they become tangentto each other when trying to occupy all the matrix [47, 58]. Asa consequence, in this work we assume that when two or morecells touch each other (see figure 3(a)), the sides (nodal point)in common are not able to send out any pseudopods to sensethe substrate (see the four common nodes n1 : n4 in the cellconfiguration in figure 3(a)) [11, 47, 77]. Therefore we assumethat those cells do not exert any sensing force at those nodesuntil they are separated again. Note that even though there is nosensing force in these nodes, the corresponding nodal tractionforces are not zero. Besides, to avoid interference of two cellswe assume

‖r j − ri‖ � 2r (20)

where ri and r j are position vectors of each cell centroid(figure 3(b)).

On the other hand, in the presence of dcEF, cells facea cell–cell electrostatic force due to the cell charge which isdifferent from the dcEF force. Therefore, the generated forcebetween jth and ith cells, FEF

i j , can be expressed as

FEFi j = ke

εr

(S

‖ri j‖)2

ei j (21)

where ke is the coulomb constant in vacuum and ri j is thevector joining the centroids of these two cells (figure 3(b)). εr

is the relative permittivity (dielectric constant) of the medium

5

Publications

75


Table 1. Substrate and cell properties.


ν Substrate Poisson ratio 0.3 [2, 78]η Substrate viscosity 1000 Pa s [2, 84]r Cell radius 20 μm [4, 13]Kpas Stiffness of microtubules 2.8 kPa [66]Kact Stiffness of myosin II 2 kPa [66]εmax Maximum strain of the cell 0.09 [47, 63]εmin Minimum strain of the cell −0.09 [47, 63]σmax Maximum contractile stress exerted by actin–myosin machinery 0.1 kPa [57, 60]k f = kb Binding constant at back and front of the cell 108 mol−1 [84]nrb = nr f Number of available receptors at the back and front of the cell 105 [84]ψ Concentration of the ligands at back and front of the cell 10−5 mol [84]satur Saturation value of surface charge density 10−4 C m−2 [36]Esatur Maximum dcEF causing Ca2+ influx 100 mV mm−1 [50, 53, 54]ke Coulomb’s constant 9 × 109 m2C−2 –εr Dielectric constant 107 [21]

and, finally, ei j is the direction of the force generated betweentwo cells. It is known that due to the electrostatic charge andfor one cell type in the same substrate, cells tend to repel eachother. Thus

ei j = − ri j

‖ri j‖ . (22)

Therefore, the resulting electrostatic force exerted by a groupof cells on the ith cell can be calculated as

FEFip =

nc−1∑j=1

FEFi j (23)

where nc is the number of cells. Consequently, the effectiveelectrostatic force on a cell in the presence of a dcEF and othercells is obtained as

FEFtot = FEF + FEF

ip . (24)

3. Application to multi-cell migration

We have implemented the present model in the commercial FEsoftware ABAQUS [31] through a coupled user subroutine.The corresponding algorithm used to implement multi-cell migration within a multi-signalling ECM is presentedin figure 4. In the following numerical experiments wedemonstrate cell population behaviour by applying the modelto 50 resident cells within a 800×400×400 μm3 substrate witha linear stiffness gradient increasing from 10 kPa at x = 0 to100 kPa at x = 800 μm. There is no external force acting on thesubstrates while all the substrate surfaces are considered to befree. The substrate is meshed by 128 000 regular hexahedralelements and 136 161 nodes. In experiments with stiffnessgradient, 50 cells are uniformly distributed in the first quarterof the substrate while in case of constant stiffness cells areuniformly distributed in the middle of the substrate. Thecalculation time is about 120 h for 500 time steps, each stepcorresponding to approximately 10 min of real cell migration[47, 48, 84]. Note that depending on cell interaction, existentstimuli and cell random movement, the cells may reach theirdestination sooner or later.

The cells are assumed to be of spherical shape with aradius of 20 μm, although any configuration can be considered.

Figure 4. Computational algorithm of multi-cell migrationconsidering mechanotaxis, chemotaxis, thermotaxis and/orelectrotaxis.

Every cell is represented by 24 finite element nodes located onits membrane. The properties of the cell and the substrate areenumerated in table 1.

6


76


(a)

(b)

Figure 5. Multi-cell migration due to pure mechanotaxis in asubstrate with stiffness gradient that linearly changes along thex-direction from 10 kPa at x = 0 to 100 kPa at x = 800 μm[4, 41, 84]. Initially, cells are distributed in the first quarter ofthe substrate. They first migrate towards each other to createsmall slugs and then move in the direction of higher stiffness toform a cell aggregation around xCSC = 598 ± 10 μm. (a)Intermediate step. (b) Final step. See movie 1 (available atstacks.iop.org/PhysBio/11/026002/mmedia).

To measure the randomness of the average cellulartranslocation in different regimes, the angle between the netpolarization direction and the imposed gradient or electricalfield direction, θ , is calculated at each time step. Therefore therandom index (RI) can be calculated as

RI =∑N

i=1 cos θi

N(25)

where N is the number of time steps during which the cellreaches its destination. If the RI is equal to +1, the cell migratesdirectly towards the stimuli, while a RI equal to −1 means thatthe cell migrates in the opposite direction. Consequently, thecloser RI to +1, the lower the cell random motility.

One of our aims is to probe and investigate the centroidof cell slug/aggregation at each time step during multi-cellmigration. Therefore, the position vector of the cell slugcentroid (CSC) is simply obtained by

XCSC =∑nc

i=1 Xi

nc(26)

where Xi presents the position vector of each cell centroid.

3.1. Multi-cell migration in the presence of mechanotacticcues

Single cell mechanotaxis was first observed by Lo et al [41] for2D cell migration with an elastic boundary between juxtaposedstiff and soft substrates. They observed that the cell moves inthe direction of the stiffer region [41]. Furthermore, recentexperiments by Hadjipanayi et al [28] demonstrated that,in the presence of a stiffness gradient, the cell populationmigrates towards stiffer regions. Figure 5 (movie 1 availableat stacks.iop.org/PhysBio/11/026002/mmedia) shows an

Figure 6. CSC trajectory in the substrate with stiffness gradient foreach experiment. Although experiments are run ten times in order tocheck the consistency of the results, the CSC trajectory for only oneof the experiments is shown here.

experiment designed to elucidate the cell response in thepresence of mechanotactic cues. As expected, in the presentsimulation the cells initially distributed in the first quarter ofthe substrate, preferentially migrate towards each other, eventhose located in the stiffer regions. The first location of thecell interaction mostly depends on the initial distribution ofthe cells. This is due to the stretch exerted by cells whichfirst dominate multi-cell migration. In contrast, as seen infigure 5(b), the final migration direction of the cells is alongthe stiffness gradient towards the stiffer region regardless ofthe primary distribution of the cells. Despite the maximumstiffness at the end of the substrate, the cells do not reach itdue to the existence of a free boundary surface which causesthe cells to ‘feel’ a less compliant region. In all the presentexperiments the imposed gradients are in the x direction, so,we consider the x-component of the CSC (see figure 6 for theCSC trajectory). In this case, xCSC = 598 ± 10 μm which canslightly vary from one run to another. Here, all the experimentshave been run ten times in order to check that the results areconsistent. It is worth noting that cell aggregation in the middleof a substrate with uniform stiffness had been observed in theprevious work [47]. The cell aggregation in the stiff regionis similar to the single cell migration [47, 48]. Aggregationof the cell population in the stiff zone is consistent with theexperimental observations of Hadjipanayi et al [28]. Moreover,the simulation demonstrates that multi-cell interaction maydelay cell migration such that it is completed in 500 time steps.For single cell migration, the cell achieves its destination inabout 150 time steps in the same substrate [48]. Figures 7 and 8show the average cell velocity and net traction force at everytime step versus average cell translocation, respectively. As canbe seen, in the first interval both average cell velocity and thenet traction force decrease. When cells accumulate in a slug,only those cells located on the outer surface can move whilethe cells enveloped inside the slug cannot send out protrusionsand remain immobile. Consequently, high cell density in thesubstrate decreases the overall cell velocity during multi-cellmigration.

Once cells come into contact with each other, they cannotexert more sensing forces in the common nodes (as explainedin section 2.6) so that their net traction force decreases.Therefore the average cell net traction force as well as thevelocity diminish until the cells move in the direction of thegradient stiffness. Each slug behaves like a single cell. Theformed slugs cannot sense each other because of the large

7

Publications

77


Figure 7. Average cell net traction force at every time step versusaverage cell translocation during multi-cell migration through asubstrate with stiffness gradient due to pure mechanotaxis.

Figure 8. Average cell velocity at every time step versus average celltranslocation during multi-cell migration through a substrate withstiffness gradient due to pure mechanotaxis.

interspace between them. Small slugs start to migrate alongthe gradient direction which increases traction forces. Since allslugs migrate towards stiffer regions, they approach each otherand the interspaces reduce. This enables the slugs to sense eachother and migrate towards each other to form an aggregationof cells around xCSC = 598 ± 10 (figures 5(b) and 6).Consequently, in the last interval of multi-cell migration, theaverage cell net traction force and velocity decrease again(figures 7 and 8).

A comparison of these curves with those of individualcell migration reveals a notable difference. As explained in[47, 48], the cell net traction force and the velocity of single cellmigration do not have peaks but uniformly decrease with somefluctuations during cell migration towards the stiffer regions.It is noteworthy that, for the same substrate, the magnitudeof the average net traction force and the local velocity in thecase of multi-cell migration are higher than those of single cellmigration due to cell–cell interaction.

In this case, the RI is computed as the average of tenruns. The mean standard deviation of these runs is presentedin figure 9. Comparing the RI of multi-cell migration with thatof individual cell migration [48], a reduction can be seen in the

Figure 9. Average RIs for different experiments with error barscorresponding to the mean standard deviation.

case of multi-cell migration. It is notable that this reductiondoes not come from the greater randomness of multi-cellmigration but is a consequence of cells migrating towards eachother in the first steps instead of migrating along the stiffnessgradient. This significantly reduces cell RI in these steps sothat this reduction does not refer to the protrusion force but totraction forces.

3.2. Multi-cell migration in the presence of thermotactic cues

The migration of eukaryotic organisms towards warmer sites,thermotaxis, has been reported by Poff et al and Tawadaet al [61, 76], while mammalian cells have received littleattention [40]. Higazi et al [32] demonstrated that trophoblastsrespond to the thermal gradients even less than 1 ◦C above orbelow the physiological temperature. Their findings indicatethat the response begins within minutes and is reversed by achange in the direction of the thermal gradient. Moreover,their data suggest that cell–cell interaction may delay orinhibit cell migration towards higher temperatures. Hong et al[35] observed that the individual amoeba of dictyosteliumdiscoideum migrates towards higher temperatures. Theseamoebae show positive thermotaxis at temperatures between14 and 28 ◦C shortly (3 h) after food depletion. In some othercases; cell may migrate towards the lower temperature awayfrom heated regions; such as burn traumas, influenza or somewild cell types [74].

Here, a thermotactic cue is added to the substrate witha stiffness gradient to check the effect of a thermotacticstimulus on multi-cell migration. It is assumed that besidesthe stiffness gradient there is a linear thermal gradient in thex-direction throughout the substrate. The temperatureincreases from 35 ◦C at x = 0, to 39 ◦C atx = 800 μm [32]. Figure 10 (movie 2 availableat stacks.iop.org/PhysBio/11/026002/mmedia) presents theresults of multi-cell migration in the presence of thermotaxisconsidering μth = 0.2. Again, the cells first tend to create smallslugs in the substrate. As before, the primary position of cell

8


78


(a)

(b)

Figure 10. Multi-cell migration in the presence of thermotaxis in asubstrate with gradient stiffness. It is assumed that there is a linearthermal gradient along the x-axis. The thermotactic efficiency factoris assumed to be equal to 0.2. Maximum temperature (39 ◦C) isassigned to the end of the substrate (x = 800 μm) and minimumtemperature at x = 0 is equal to 35 ◦C. Initially, cells are distributedin the first quarter of the substrate. First, they tend to comeinto contact and afterwards migrate towards warmer sites toaggregate at around xCSC = 641 ± 10 μm. (a) Intermediatestep. (b) Final step. See movie 2 (available atstacks.iop.org/PhysBio/11/026002/mmedia).

contact depends on the primary distribution of the cells overthe substrate as well as the effect of the protrusion force oneach cell. Multi-cell migration towards warmer sites occursafter cell contact. It is of interest to mention that the existenceof free boundary surface at the end of the substrate impedes thecell migration towards the end of the substrate despite havingmaximum stiffness and temperature (figure 10(b)). Comparedwith the previous experiment, the CSC moves slightly closertowards the end of the substrate due to the thermotactic effectas can be seen in figure 6 (xCSC = 641 ± 10 μm).

Comparing individual [48] and multi-cell migration froma temporal point of view demonstrates that the required numberof time steps for multi-cell migration is higher than that forindividual cell migration. This indicates that cell populationdelays cell migration, which is consistent with the findings ofHigazi et al [32].

The average net traction force and local velocity of cellsat every time step are plotted versus average cell translocationin figures 11 and 12, respectively. It can be seen that addingthermotactic stimulus to the substrate with a stiffness gradientdoes not significantly change either the magnitude of theaverage cell velocity or the net traction force. This is becausetheir magnitudes are independent of the thermal gradient. Incontrast, the value of the average cell translocation (maximumof the curve) shifts to the right. This is because, despite theextended zone among the cells which steers them towardseach other (mechanotactic force), a part of the individualcell net traction force directed by the thermotactic stimulus(thermal gradient) persistently guides the cells towards the

Figure 11. Average cell net traction force at every time step versusaverage cell translocation during multi-cell migration through asubstrate with stiffness gradient in the presence of thermotaxis.

Figure 12. Average cell velocity versus average cell translocationduring multi-cell migration through a substrate with stiffnessgradient in the presence of thermotaxis.

warmer sites. This event inhibits cell contact temporally andspatially, shifting the peak of the curves to the right. On theother hand, the signals received by the cells due to thermotaxiscauses that the cell RI to increase slightly as can be seen infigure 9. However, it is still less than that of individual cellmigration under thermotaxis [48].

3.3. Multi-cell migration in the presence of chemotactic cues

Chemotaxis allows neutrophils or dictyostelium amoebae tofind their bacterial prey and also allows amoebae, whenstarved, to aggregate into multicellular masses using cyclicAMP signals. Chemotaxis is a key mechanism by whichtissues and organs become organized during development.Chemotactic ability can lie dormant in fibroblasts, but isevoked when a wound needs to be healed. In metastasis ofcancer cells, it would be desirable to suppress chemotaxiscompletely since it promotes the escape and spread of cellsfrom the primary tumour that makes cancer so intractable [39].

This experiment is planned to clarify the role ofchemotaxis in multi-cell migration as well as in cell–cellinteraction concurrent with mechanotaxis. Although it isassumed that the cells which are in contact cannot sense the

9

Publications

79


(a)

(b)

Figure 13. Multi-cell migration due to chemotaxis with μch = 0.3 ina substrate with stiffness gradient. It is assumed that a chemotacticsource is located at x = 800 μm (10−4 M), which creates a linearchemical gradient along the x-direction. Initially, cells aredistributed in the first quarter of the substrate. First they tend tocome into contact and then migrate towards the chemotactic sourceand the stiffer region. In this case the CSC is displaced towardsthe end of the substrate and the cells accumulate around xCSC =688 ± 5 μm. (a) Intermediate step. (b) Final step. See movie 3(available at stacks.iop.org/PhysBio/11/026002/mmedia).

substrate stiffness in common nodes, in these simulationswe do not consider depletion of chemoattractant agent orshielding of a cell by its neighbours. In other words, a cellthat is behind other cells still feels a chemotactic signal.It is assumed that a chemoattractant source is located at x= 800 μm creating a linear chemical gradient along thex-axis of a substrate with a stiffness gradient (increasingfrom zero at x = 0 to 10−4 M at x = 800 μm). As inthe previous experiments, the cells are uniformly distributedin the first quarter of the substrate. To elucidate the effectof the chemotaxis efficiency factor on cell–cell interaction,we repeated this experiment for μch = 0.3 (figure 13) andμch = 0.4 (figure 14). See movie 3 and 4, respectively(available at stacks.iop.org/PhysBio/11/026002/mmedia). Theoverall cell behaviour is similar to that of the previousexperiments. However, the primary position of the cellcontact is shifted towards the end of the substrate evenmore than that in the case of thermotaxis because of thehigher efficiency factor of chemotaxis. The initial positionof cell interaction not only depends on the first distributionof the cells and the exerted random protrusion force ofeach cell but also on the chemotaxis efficiency factor.Consequently, the higher the efficiency factor the slower theaggregation process. Once the cells come into contact inthe substrate, they move towards the chemotactic source.Since the stiffness and chemotactic gradients are in the samedirection, each amplifies the effect of the other. Consequently,for μch = 0.3 and μch = 0.4 the CSCs are located atxCSC = 688 ± 5 μm and xCSC = 715 ± 8 μm, respectively (seefigures 6, 13(b) and 14(b)) . Once cells reach these positionsthey form a big slug. As in the previous experiments, the cells

(a)

(b)

Figure 14. Multi-cell migration due to chemotaxis with μch = 0.4 ina substrate with stiffness gradient. It is assumed that a chemotacticsource is located at x = 800 μm (10−4 M), which creates a linearchemical gradient through the x-direction. Initially, cells aredistributed in the first quarter of the substrate. First, they tend tocome into contact and then migrate towards the chemotactic sourceand the stiffer region. Finally, the cells aggregate at xCSC = 715 ±8 μm. In this case the CSC is displaced further towards the end ofthe substrate than in the cases of mechanotaxis and thermotaxis.(a) Intermediate step. (b) Final step. See movie 4 (available atstacks.iop.org/PhysBio/11/026002/mmedia).

enveloped inside the slug are frozen, i.e. keeping immobile anddo not exert any more sensing forces, while the cells located onthe surface of the cell aggregation move around the slug due tothe random protrusion force as well as the higher chemotacticsignal coming from the end of the substrate. The aggregation ofcells around the chemotactic source was observed by Andrewand Haaster [5, 27]. Besides, Bosgraaf et al [10] reported thatin the presence of chemoattractant, the cells actively movetowards the chemoattractant sources, depending on the signalstrength. This is also consistent with the numerical results ofNeilson et al [51]. Migration of individual cell is again fasterthan that of the cell population in this experiment [48].

Figures 15 and 16 present the average net traction forceand local velocity of the cells at every time step versus theaverage cell translocation for both chemical efficiency factors(μch = 0.3 and μch = 0.4). As can be seen, although themagnitudes of the average cell velocity and the net tractionforce follow the same trend as in the previous experiments, thepeak of the curves shifts to the right in comparison with thethermotaxis experiment due to the higher chemical efficiencyfactor. Here, the cell RI slightly increases depending on thechemical efficiency factor (see figure 9). This means thatin the presence of chemotaxis, depending on the chemicalefficiency factor, the random movement of the cells decreases,in agreement with the findings of Haastert [27]. Moreover, itis worth noting that the average chemotaxis RIs for multi-cellmovement are lower than those of the corresponding individualcell migration [48].

10


80


Figure 15. Average cell net traction force at every time step versusaverage cell translocation during multi-cell migration through asubstrate with stiffness gradient in the presence of chemotaxis.

Figure 16. Average cell velocity at every time step versus averagecell translocation during multi-cell migration through a substratewith stiffness gradient in the presence of chemotaxis.

3.4. Multi-cell migration in the presence of electrotactic cues

In the presence of electrotactic cues, cells move witha directional preference towards the cathode or anode[6, 35, 42, 50, 53, 68]. The preferential direction of migrationvaries between cell types. For instance, corneal rat epithelialcells, human keratinocytes, osteoblasts, rat prostate cancercells, lymphocyte and xenopus neurons migrate towards thecathode [35, 18, 54, 55, 69, 71], whereas corneal stromalfibroblasts, osteoclasts, human granulocyte and macrophagemigrate towards the anode [14, 22]. In addition, Grahnet al [25] reported that human dermal melanocytes are notstimulated by an exogenous dcEF of 100 mV mm−1. Onumaet al [56] argue that this kind of cell may have a differentdcEF threshold. Even for the same cell type, cells derivedfrom different species migrate in opposite directions in thepresence of dcEFs. Bovine vascular endothelial cells migratetowards the cathode, whereas human vascular endothelial cellsmigrate towards the anode. Furthermore, lens epithelial cellschange their migration direction depending on the applied EFstrength [85].

The aim of this experiment is to study the influence ofthe electrotaxis on multi-cell migration as well as cell–cell

(a)

(b)

Figure 17. Multi-cell migration due to electrotaxis in a substratewith stiffness gradient. In this case, it is assumed thatE = 10 mV mm−1. Cells are attracted by the cathode located atx = 800 μm. Initially, cells are distributed in the first quarter of thesubstrate. They then migrate towards the cathode pole and arefinally located at around xCSC = 736 ± 6 μm. (a) Intermediatestep. (b) Final step. See movie 5 (available atstacks.iop.org/PhysBio/11/026002/mmedia).

interaction. Therefore, we consider similar cell types (suchas embryo fibroblasts [55]) which are attracted by the cathodethat is located at x = 800 μm. The initial cell distribution in thesubstrate is similar to those of the previous experiments. Thesimulation is repeated for 10 and 100 mV mm−1 EF strength.

A comparison of cell motility in the presence of dcEFwith that of pure mechanotaxis illustrates that the randommotion of the cells noticeably diminishes (figure 9). This is inagreement with the experimental results of McCaig et al [44]and Nishimura et al [53]. It is also interesting to mention thatthe RIs of multi-cell migration for both low and high dcEFsare lower than those of individual cell migration [48] due tothe cell–cell interaction, which is consistent with the findingsof Nishimura et al [53].

The multi-cell migration patterns for both low and highdcEFs are presented in figures 17 and 18, respectively. Asdiscussed in section (2.6), in the presence of dcEF, multi-cell migration is relatively different from that of individualcell migration in terms of the total electrical force acting oneach cell. Due to cell–cell electrostatic forces (FEF

ip ), cellsrepel each other during multi-cell migration in the presenceof dcEF. This is not the case for individual cell migration[48]. As in the previous experiments, the cells firstly migratetowards each other. When the cells come into contact, theymay separate not only because of the protrusion force butalso because of repelling electrostatic forces which are at amaximum when the cells come into contact. Besides, sincethe cell charge is linearly proportional to the dcEF strength,increasing the dcEF strength progressively increases repellingelectrostatic forces. Consequently, although in both cases cellscreate a slug, for E = 10 mV mm−1 (figure 17(b)) the slugis round and compacted while for E = 100 mV mm−1 it is

11

Publications

81


(a)

(b)

Figure 18. Multi-cell migration due to electrotaxis in a substratewith stiffness gradient. In this case, it is assumed that E =100 mV mm−1. Cells are attracted by the cathode located atx = 800 μm. Initially, cells are distributed in the first quarterof the substrate. They then migrate towards the cathode poleand are finally located around xCSC = 763 ± 4 μm.(a) Intermediate step. (b) Final step. See movie 6 (available atstacks.iop.org/PhysBio/11/026002/mmedia).

flattened (figure 18(b)) (movie 5 and 6, respectively (availableat stacks.iop.org/PhysBio/11/026002/mmedia)). This meansthat the asymmetric distribution of cells is increased byincreasing the dcEF strength, as seen in the results ofNishimura et al [53]. Moreover, as can be seen infigure 6, for a dcEF strength of 10 mV mm−1 xCSC =736 ± 6 μm while for a dcEF strength of 100 mV mm−1

xCSC = 763 ± 4 μm. As can be seen, increasing thedcEF strength to the saturation value (Esatur) causes thetotal electrotactic force to persistently pull the cells towardsthe cathode and to reduce the mechanotactic effect in theintermediate zone (figure 18(b)). Consequently, the dcEFplays an overriding guiding role in directing cell migration insubstrates with a stiffness gradient. This is consistent with thefindings of Nishimura et al [53] who reported that for a dcEFstrength of 10 mV mm−1, 10% of the keratinocytes migratetowards the cathode while this amount increases to 98% whenthe dcEF rises to 100 mV mm−1, despite the free boundaries.Furthermore, several experimental works have reported thedominant effect of dcEF on multi- and individual cell migration[37, 42, 43, 50, 85]. It should be noted that in the case of lowdcEFs, the cells located around the surface of the slug may havesome motility when the cells reach their destination. However,increasing the EF strength significantly decreases this motility.Therefore, in the case of E = Esatur the cells do not move afterarriving at the cathode. This is also observed in the case ofsingle cell migration [48].

In this experiment, the behaviour of the net tractionforce curve is similar to that of pure mechanotaxis since thetraction force is proportional to the mechanical conditionsand independent from the dcEF, according to figure 19. Itis worth mentioning that, in contrast with pure mechanotaxis,the average net traction force increases in the last interval since

Figure 19. Average cell net traction force at every time step versusaverage cell translocation during multi-cell migration through asubstrate with stiffness gradient in the presence of dcEF.

Figure 20. Average cell velocity at every time step versus averagecell translocation during multi-cell migration through a substratewith stiffness gradient in the presence of dcEF.

cells migrate towards the free boundary surface (specificallyin the case of E = 100 mV mm−1). For E = 10 mV mm−1

and E = 100 mV mm−1, the average cell local velocity ateach time step is plotted versus the average cell translocationin figure 20. From this figure, we note that the local velocityrange is higher than that of the previous experiments. As aresult of the presence of electrotaxis, an independent force(electrical force) is added to the drag force (equation (16))which collaborates with the net traction force to amplify themagnitude of the local cell velocity. This increase in the cellvelocity during multi-cell migration towards the cathode hasalso been observed in experimental works by Nishimura et al[53]. These explanations can further justify the overridingeffect of dcEF reported by Zhao [85] and some others[37, 42, 43, 50]. Adding electrotaxis to the substrate with astiffness gradient shifts the peak of the curves to the right(figures 19 and 20).

Besides the cell type that is considered in the previousexperiments, another cell type that is attracted towards anodeis considered in the same substrate. It is assumed that theanode is located at x = 0 and the cathode at x = 800 μm.Initially, the cells are distributed in the middle of the substratewith a uniform stiffness (100 kPa), far from anode and cathode

12


82


Figure 21. Migration of two cell types due to electrotaxis in asubstrate with uniform stiffness (100 kPa). It is assumed thatE = 10 mV mm−1 where the anode is located at x = 0 and thecathode at x = 800 μm. Blue cells are typical cells attracted byanode and red cells by cathode. Initially, the cells are distributed inthe middle of the substrate far from anode and cathode poles.Because the EF strength is not sufficient to separate the two celltypes, they accumulate in the middle of the substrate. See movie 7(available at stacks.iop.org/PhysBio/11/026002/mmedia).

Figure 22. Average cell net traction force at every time step versusaverage cell translocation. Two cell types, one type attracted towardsthe anode and the other towards the cathode, are initially distributedin the middle of the substrate with uniform stiffness(100 kPa) in the presence of dcEF, E = 10 mV mm−1.

poles. In figures 21 and 24, blue cells are typical cells that areattracted by anode while red cells are attracted by cathode.For an EF strength less than 160 mV mm−1, the blue and redcells are barely separated from each other. This is because,in this condition, the intensity of the EF is not enough tocompensate mechanotactic interaction and electrical attractionbetween blue and red cells. The aggregation of blue and redcells can be seen in figure 21 for E = 10 mV mm−1 (movie 7available at stacks.iop.org/PhysBio/11/026002/mmedia). Theaverage cell net traction force and cell velocity versusaverage cell translocation are separately computed for eachgroup of the blue and red cells (figures 22 and 23).Because the cells have stuck in each other and arealmost immobile in middle of the substrate there are notfluctuation in the computed values as much as previousexperiments. Increasing poles EF strength to 160 mV mm−1

increases the electrical force over cells to dominate cell–cell mechanical and electrical attraction, separating blue andred cells from each other (figure 24, also movie 8 availableat stacks.iop.org/PhysBio/11/026002/mmedia). Average cell

Figure 23. Average cell velocity at every time step versus averagecell translocation. Two cell types, one type attracted towards theanode and the other towards the cathode, are initially distributed inthe middle of the substrate with uniform stiffness (100 kPa) in thepresence of dcEF, E = 10 mV mm−1.

Figure 24. Migration of two cell types due to electrotaxis in asubstrate with uniform stiffness (100 kPa). It is assumed that E =160 mV mm−1 where the anode is located at x = 0 and the cathodeat x = 800 μm. Blue and red cells are typical cells attracted byanode and cathode, respectively. Initially, the cells are distributed inthe middle of the substrate far from anode and cathode poles. Seemovie 8 (available at stacks.iop.org/PhysBio/11/026002/mmedia).

Figure 25. Average cell net traction force at every time step versusaverage cell translocation. Two cell types, one type attracted towardsthe anode and the other towards the cathode, are initially distributedin the middle of the substrate with uniform stiffness(100 kPa) in the presence of dcEF, E = 160 mV mm−1.

net traction force and cell velocity versus average celltranslocation for each group of the blue and red cells are shownin figures 25 and 26, respectively. For each group, there is apeak near the anode (blue cells) or cathode (red cells).

13

Publications

83


Figure 26. Average cell velocity at every time step versus averagecell translocation. Two cell types, one type attracted towards theanode and the other towards the cathode, are initially distributed inthe middle of the substrate with uniform stiffness (100 kPa) in thepresence of dcEF, E = 160 mV mm−1.

Figure 27. Variation of the cell average RI versus chemotaxis,thermotaxis and electrotaxis cues.

3.5. Multi-cell migration in multi-signalling substrate

Finally, to simultaneously elucidate the effect of all the cueson multi-cell migration in a substrate with a stiffness gradient,we have designed 49 different experiments with differentthermotactic and chemotactic efficiency factors as well asdifferent EF strengths. The change in the average RI of the cellpopulations versus the combination of cues is summarized infigure 27. The results demonstrate that increasing the value ofeach stimulus increases the cell RI, i.e. it reduces the randommigration of the cells. It can be seen in this figure that the slopeof the curve is higher in the direction of the electrotactic axis(E.), which indicates its dominant role. Moreover, increasingthe EF strength over the saturation value does not significantlychange the average cell RI. A comparison of this figure with thecorresponding curve of individual cell migration [48] confirmsthat cell–cell interaction slightly reduces the cell RI.

4. Conclusions

Most cells respond in vitro to mechanotaxis [41], thermotaxis[35], chemotaxis [73] and electrotaxis [85] by directionalmigration. To predict the behaviour of an individual cell inthe presence of a multi-signalling environment, we previouslypresented a computational model [48] which is here extendedto include cell–cell interaction and multi-cell migration. It canbe seen that in all the experiments, cells tend to come intocontact before migrating towards the most effective stimulus.By adding a new cue to the substrate with a stiffness gradient,the overall behaviour of the cell population changes. Forinstance, adding thermotaxis, chemotaxis and/or electrotaxiscues to the substrate with a stiffness gradient pulls the cellslug centroid (CSC) towards the end of the substrate due tocollaboration of these cues with the stiffness gradient. Thisis consistent with the experimental works of several authors[27, 28, 35, 53]. In all the cases analysed here, the peaks ofthe average net traction force and local velocity curves shift tothe right. In addition, thermotaxis and chemotaxis noticeablychange the reorientation of the cells, delay cell contact andincrease the average random index (RI) of the cells (figure 9).However, their effect on the magnitude of the average nettraction force and local velocity of the cells is negligiblesince the magnitude of the traction force is considered tobe independent of these cues, being dependent only on themechanical properties of the substrate. Besides, the localvelocity is directly proportional to the drag force which, inturn, is a function of the traction force.

The results demonstrate that the cells aggregate near thecathode in the presence of exogenous dsEF. Amplification ofthe direct current EF (dcEF) linearly increases the cell chargedue to Ca2+ influx enhancement. This, depending on the dcEFstrength, increases the electrical force generated on the cellpopulation and displaces the CSC towards the cathode. Theseresults are consistent with the findings of Nishimura et al[53]. On the other hand, for high dsEF, cells are flattenednear the cathode (figure 18). It is noticeable that for low dcEF(10 mV mm−1) the cells located on the external surface of theslug may have some little motility but when the dcEF increasesto 100 mV mm−1 the outer cells near the cathode are almostcompletely immobilized. Furthermore, for 100 mV mm−1

dcEF the cells migrate to the end of the substrate, despitethe fact that the cathode is located at the free surface. Thisindicates that electrotaxis dominates other signals (in thiscase mechanotaxis) and explains why dcEF plays a dominantguiding role in directing cell migration in epithelial woundhealing, as reported in Zhao et al [85]. figure 27 shows RI inthe presence of different stimuli. It becomes clear that addingany new cue to the substrate increases the cell RI, reducingrandom migration of the cells. This is especially significantunder an electrotactic stimulus. As previously mentioned, thisoccurs because the dcEF directly affects the drag force andnotably reorientates the cells towards the cathode.

A comparison of multi- and single cell migration [48]illustrates that, from some viewpoints, migration of a cellpopulation is different from single cell migration. For instance,in multi-cell migration cells do not migrate immediately

14


84


towards the final destination. They first tend to move towardseach other due to the stretched region generated by the forcesexerted by the cells (mechanotactic effect). Some while afterthey come into contact, they migrate towards the effectivecues. On the other hand, cell–cell interaction not only maydelay multi-cell migration but it also slightly decreases theaverage RI of the cells. Besides, cell aggregation causes theaverage net traction force and local velocity of the cells toincrease and decrease instantaneously. This was not observedin the case of single cell migration [48].

The model presented is a robust tool for simulating multi-cell migration in a multi-signalling substrate and is sufficientlyflexible to consider any cell shape. In addition, consideringdifferent cell properties in the present model enables usto investigate the behaviour of s large range of cell types.The results obtained here demonstrate that alterations of thecell environment stimuli can modify the spontaneous cellmigration behaviour, thus providing an insight into how thecell would potentially react to an external cue. It enablesus to simultaneously consider cell–cell interaction in themulti-signalling substrates associated with different complexbiological processes. Therefore, this model could be veryeffective in helping to increase cell migration in such casesas wound healing or to decrease it in other cases suchas cancer growth. In addition, using this formalism andemploying appropriate stimulus/stimuli within the substratein the particular location/locations, one can think of trying todisperse certain types of cells, for instance tumour cells, or letthese cells converge such that a physician is able to removethe tumour and such that the tumour might not seed out, orto prevent endothelial cells (as blood capillaries) to migratetowards the tumour by chemo/mechanotaxis and/or changingthe electrical environment locally. The numerically obtainedresults are qualitatively consistent with previous experimentalobservations. However, more sophisticated experiments arenecessary to illustrate the accuracy of the results obtainedand to calibrate the model. We believe that the computationalmodel described here can be employed to design more efficientexperiments for the prediction of the various aspects of multi-cell migration within a multi-signalling extracellular matrix.

Acknowledgments

The authors gratefully acknowledge financial support fromthe Spanish Ministry of Science and Technology (CYCITDPI2010-20399-C04-01) and the CIBER-BBN initiative.CIBER-BBN is funded by the VI National R&D&I Plan 2008-2011, Iniciativa Ingenio 2010, Consolider Program, CIBERActions and financed by the Instituto de Salud Carlos III withassistance from the European Regional Development Fund.

References

[1] Eisenbach M and Bahat A 2006 Sperm thermotaxis Mol. Cell.Endocrinol. 252 115–9

[2] Akiyama S K and Yamada K M 1985 The interaction ofplasma fibronectin with fibroblastic cells in suspensionJ. Biol. Chem. 260 4492–500 (PMID: 3920218)

[3] Allena R 2013 Cell migration with multiple pseudopodia:temporal and spatial sensing models Bull. Math. Biol.75 288–316

[4] Allena R and Aubry D 2012 Run-and-tumble or look-and-run?a mechanical model to explore the behavior of a migratingamoeboid cell J. Theor. Biol. 306 15–31

[5] Andrew N and Insall R H 2007 Chemotaxis in shallowgradients is mediated independently of ptdlns 3-kinase bybiased choices between random protrusions Nature CellBiol. 9 193–200

[6] Armon L and Eisenbach M 2011 Behavioral mechanismduring human sperm chemotaxis: involvement ofhyperactivation PLoS ONE 6 e28359

[7] Bai H, McCaig C D, Forrester J V and Zhao M 2004 Dcelectric fields induce distinct preangiogenic responses inmicrovascular and macrovascular cells Arterioscler.Thromb. Vasc. Biol. 24 1234–9

[8] Behesti H and Marino S 2009 Cerebellar granule cells:insights into proliferation, differentiation, and role inmedulloblastoma pathogenesis J. Appl. Physiol. 41 435–45(PMID: 18755286)

[9] Bindschadler M and McGrath J L 2007 Sheet migration bywounded monolayers as an emergent property of single-celldynamics J. Cell Sci. 120 876–84

[10] Bosgraaf L and Van Haastert P J M 2009 Navigation ofchemotactic cells by parallel signaling to pseudopodpersistence and orientation PLoS ONE 4 e6842

[11] Brofland G W and Wiebe C J 2004 Mechanical effects of cellanisotropy on epithelia Comput. Methods Biomech. Biomed.Eng. 7 91–9 (PMID: 15203957)

[12] Brown M J and Loew L M 1994 Electric field-directedfibroblast locomotion involves cell surface molecularreorganization and is calcium independent J. Cell Biol.127 117–28

[13] Buxboim A, Ivanovska I L and Discher D E 2010 Matrixelasticity, cytoskeletal forces and physics of the nucleus:how deeply do cells ‘feel’ outside and in? J. Cell Sci.123 297–308

[14] Chang P C, Sulik G L, Soong H K and Parkinson W C 1996Galvanotropic and galvanotaxic responses of cornealendothelial cells J. Formos. Med. Assoc. 95 623–7(PMID: 8870433)

[15] Chirag B K, Shelly R P and Andrew J P 2006 Intrinsicmechanical properties of the extracellular matrix affect thebehavior of pre-osteoblastic mc3t3-e1 cells Am. J. Physiol.Cell Physiol. 290 C1640–50

[16] Cooper M S and Shliwa M 1986 Motility of cultured fishepidermal cells in the presence and absence of direct currentelectric fields J. Cell Biol. 102 1384–99

[17] Deshpande V S, McMeeking R M and Evans A G 2006 Abio-chemo-mechanical model for cell contractility Proc.Natl Acad. Sci. USA 103 14015–20

[18] Djamgoz M B A, Mycielska M, Madeja Z, Fraser S Pand Korohoda W 2001 Directional movement of rat prostatecancer cells in direct current electric field: involvement ofvoltage-gated Na+ channel activity J. Cell Sci.114 2697–705 (PMID: 11683396)

[19] Drasdo D and Hohme S 2005 A single-cell-based model oftumor growth in vitro: monolayers and spheroids Phys.Biol. 2 133–47

[20] Ducrot A, Foll F L, Magal H M, Pasquier P J and Webb G F2011 An in vitro cell population dynamics modelincorporating cell size, quiescence, and contact inhibitionMath. Models Methods Appl. Sci. 21 871–92

[21] Foster K R and Schwan H P 1989 Dielectric properties oftissues and biological materials: a critical review Crit. Rev.Biomed. Eng. 17 25–104 (PMID: 2651001)

15

Publications

85


[22] Fraser S P et al 2005 Voltage-gated sodium channel expressionand potentiation of human breast cancer metastasis ClinCancer Res. 11 5381–9 (PMID: 16061851)

[23] Friedl P, Hegerfeldt Y and Tusch M 2004 Collective cellmigration in morphogenesis and cancer Int. J. Dev. Biol.48 441–9

[24] Gao R C, Zhang X D, Sun Y H, Kamimura Y, Mogilner A,Devreotes P N and Zhao M 2011 Different roles ofmembrane potentials in electrotaxis and chemotaxis ofdictyostelium cells Eukaryot. Cell 10 1251–6

[25] Grahn J C, Reilly D A, Nuccitelli R L and Isseroff R R 2003Melanocytes do not migrate directionally in physiologicaldc electric fields Wound Repair Regen. 11 64–70

[26] Gyllenberg G and Heijmans H 1987 An abstractdelay-differential equation modelling size dependent cellgrowth and division SIAM J. Math. Anal. 18 74–88

[27] Van Haastert P J M 2010 Chemotaxis: insights from theextending pseudopod J. Cell Sci. 123 3031–37

[28] Hadjipanayi E, Mudera V and Brown R A 2009 Guiding cellmigration in 3D: a collagen matrix with graded directionalstiffness Cell Motil. Cytoskeleton 66 435–45

[29] Harrison N C, del Corral R D and Vasiev B 2011 Coordinationof cell differentiation and migration in mathematical modelsof caudal embryonic axis extension PLoS ONE 6 e22700

[30] Hegerfeldt Y, Tusch M, Brocker E B and Friedl P 2002Collective cell movement in primary melanoma explants:plasticity of cell–cell interaction, beta1-integrin function,and migration strategies Cancer Res. 62 2125–30(PMID: 11929834)

[31] Hibbit, Karlson and Sorensen 2011 Abaqus-Theory Manual6.11 3rd edn (Providence, RI: Hibbit, Karlson andSorensen)

[32] Higazi A A, Kniss D, Manuppelloand J, Barnathan E Sand Cines D B 1996 Thermotaxis of human trophoblasticcells Placenta 17 683–7

[33] Hoehme S and Drasdo D 2010 A cell-based simulationsoftware for multi-cellular systems Bioinformatics26 2641–2

[34] Hoeller O and Kay R R 2007 Chemotaxis in the absence ofpip3 gradients Curr. Biol. 17 813–7

[35] Hong C B, Fontana D R and Poff K L 1983 Thermotaxis ofdictyostelium discoideum amoebae and its possible role inpseudoplasmodial thermotaxis Proc. Natl Acad. Sci. USA80 5646–49

[36] Huang Y-X, Zheng X J, Kang L L, Chen X Y, Liu W J,Wu Z-J and Huang B T 2011 Quantum dots as a sensor forquantitative visualization of surface charges on single livingcells with nano-scale resolution Biosens. Bioelectron.26 2114–8

[37] Illingworth C M and Barker A T 1980 Measurement ofelectrical currents emerging during the regeneration ofamputated finger tips in children Clin. Phys. Physiol. Meas.1 87

[38] James D W and Taylor J F 1969 The stress developed bysheets of chick fibroblasts in vitro Exp. Cell Res. 54 107–10

[39] Kay R R, Langridge P, Traynor D and Hoeller O 2008Changing directions in the study of chemotaxis Nature Rev.Mol. Cell Biol. 9 455–63

[40] Kesslet J O, Jarvik L F, Fu T K and Matsuyama S S 1979Thermotaxis, chemotaxis and age Age 2 5–11

[41] Lo C M, Wang H B, Dembo M and Wang Y L 2000 Cellmovement is guided by the rigidity of the substrate Biophys.J. 79 144–52

[42] Long H, Yang G and Wang Z 2011 Galvanotactic migration ofea.hy926 endothelial cells in a novel designed electric fieldbioreactor Cell Biochem. Biophys. 61 481–91

[43] Maria E M and Djamgoz M B A 2004 Cellular mechanisms ofdirect-current electric field effects: galvanotaxis andmetastatic disease J. Cell Sci. 117 1631–39

[44] McCaig C D, Rajnicek A M, Song B and Zhao M 2002 Haselectrical growth cone guidance found its potential? TrendsNeurosci. 25 354–9

[45] Miller E D, Li K, Kanade T, Weiss L E, Walker L Mand Campbell P G 2011 Spatially directed guidance of stemcell population migration by immobilized patterns ofgrowth factors Biomaterials 32 2775–85

[46] Moreo P, Garcia-Aznar J M and Doblare M 2008 Modelingmechanosensing and its effect on the migration andproliferation of adherent cells Acta Biomaterialia 4 613–21

[47] Mousavi S J, Doweidar M H and Doblare M 2014Computational modelling and analysis of mechanicalconditions on cell locomotion and cell–cell interactionComput. Methods Biomech. Biomed. Eng. at press

[48] Mousavi S J, Doweidar M H and Doblare M 2013 3Dcomputational modelling of cell migration: amechano–chemo–thermo-electrotaxis approach J. Theor.Biol. 329 64–73

[49] Mousavi S J, Doweidar M H and Doblare M 2013 Cellmigration and cell–cell interaction in the presence ofmechano–chemo-thermotaxis Mol. Cell. Biomech. 10 1–25(PMID: 24010243)

[50] Mycielska M E and Djamgoz M B 2004 Cellular mechanismsof direct current electric field effects: galvanotaxis andmetastatic disease J. Cell Sci. 117 1631–9

[51] Neilson M P, Mackenzie J A, Webb S D and Insall R H 2011Modeling cell movement and chemotaxis usingpseudopod-based feedback SIAM J. Sci. Comput.33 1035–57

[52] Neilson M P, Veltman D M, van Haastert P J M, Webb S D,Mackenzie J A and Insall R H 2011 Chemotaxis: afeedback-based computational model robustly predictsmultiple aspects of real cell behaviour PLoS Biol.9 e1000618

[53] Nishimura K Y, Isseroff R R and Nuccitelli R 1996 Humankeratinocytes migrate to the negative pole in direct currentelectric fields comparable to those measured in mammalianwounds J. Cell Sci. 109 199–207 (PMID: 8834804)

[54] Nuccitelli R 2003 A role for endogenous electric fields inwound healing Curr. Top. Dev. Biol. 58 1–26

[55] Onuma E K and Hui S W 1985 A calcium requirement forelectric field-induced cell shape changes and preferentialorientation Cell Calcium 6 281–92

[56] Onuma E K and Hui S W 1988 Electric field-directed cellshape changes, displacement, and cytoskeletalreorganization are calcium dependent J. Cell Biol.106 2067–75

[57] Oster G F, Murray J D and Harris A K 1983 Mechanicalaspects of mesenchymal morphogenesis J. Embryol Exp.Morph 78 83–125 (PMID: 6663234)

[58] Palsson E 2001 A three-dimensional model of cell movementin multicellular systems Future Gener. Comput. Syst.17 835–52

[59] Pelham R J Jr and Wang Y L 1997 Cell locomotion and focaladhesions are regulated by substrate flexibility Proc. NatlAcad. Sci. 94 13661–5

[60] Penelope C G and Janmey P A 2005 Cell type-specificresponse to growth on soft materials J. Appl. Physiol.98 1547–53

[61] Poff K L and Skokut M 1977 Thermotaxis bypseudoplasmodia of dictyostelium discoideum Proc. NatlAcad. Sci. USA 74 2007–10

[62] Ramis-Conde I, Drasdo D, Anderson A R Aand Chaplain M A J 2008 Modeling the influence of thee-cadherin-β-catenin pathway in cancer cell invasion: amultiscale approach Biophys. J. 95 155–65

[63] Ramtani S 2004 Mechanical modelling of cell/ECM andcell/cell interactions during the contraction of a

16


86


fibroblast-populated collagen microsphere:theory andmodel simulation J. Biomech. 37 1709–18

[64] Rapp B, Boisfleury-Chevance A and Gruler H 1988Galvanotaxis of human granulocytes. dose–response curveEur. Biophys. J. 16 313–9

[65] Rorth P 2007 Collective guidance of collective cell migrationTrends Cell Biol. 17 575–9

[66] Schafer A and Radmacher M 2005 Influence of myosin IIactivity on stiffness of fibroblast cells Acta Biomaterialia1 273–80

[67] Schneider L et al 2010 Directional cell migration andchemotaxis in wound healing response to PDGF-AA arecoordinated by the primary cilium in fibroblasts Cell Phys.Biochem. 25 279–92

[68] Shanley L J, Walczysko P, Bain M, MacEwan D J and Zhao M2006 Influx of extracellular Ca2+ is necessary forelectrotaxis in dictyostelium J. Cell Sci. 4741–48

[69] Sheridan D M, Isseroff R R and Nuccitelli R 1996 Impositionof a physiologic dc electric field alters the migratoryresponse of human keratinocytes on extracellular matrixmolecules J. Invest. Dermatol. 106 642–6

[70] Stefanoni F, Ventre M, Mollica F and Netti P A 2011 Anumerical model for durotaxis J. Theor. Biol. 280 150–8

[71] Sulik G L, Soong H K, Chang P C, Parkinson W C, Elner S Gand Elner V M 1992 Effects of steady electric fields onhuman retinal pigment epithelial cell orientation andmigration in culture Acta Ophthalmol. (Copenh.)70 115–22

[72] Suresh S 2007 Biomechanics and biophysics of cancer cellsActa Biomaterialia 3 413–38

[73] Swaney K F, Huang C H and Devreotes P N 2010 Eukaryoticchemotaxis: a network of signaling pathways controlsmotility, directional sensing, and polarity Annu. Rev.Biophys. 39 265–89

[74] Tanizawa Y, Kuhara A, Inada H, Kodama E, Mizuno Tand Mori I 2006 Inositol monophosphatase regulateslocalization of synaptic components and behavior in the

mature nervous system of c. elegans Genes Dev.20 3296–310

[75] Tanos B and Rodriguez-Boulan E 2008 The epithelial polarityprogram: machineries involved and their hijacking bycancer Oncogene 27 6939–57

[76] Tawada K and Miyamoto H 1973 Sensitivity of parameciumthermotaxis to temperature change J. Protozool 20 289–92

[77] Taylor D L et al 1982 Cellular and molecular aspects ofamoeboid movement Cold Spring Harb. Symp. Quant. Biol.46 101–11

[78] Ulrich T A, De Juan Pardo E M and Kumar S 2009 Themechanical rigidity of the extracellular matrix regulates thestructure, motility, and proliferation of glioma cells CancerRes. 69 4167–74

[79] Vermolen F J and Gefen A 2013 A phenomenological modelfor chemico-mechanically induced cell shape changesduring migration and cell–cell contacts Biomech. ModelMechanobiol. 12 301–23

[80] Webb G 2008 Population models structured by age, size andspatial position Structured Population Models in Biologyand Epidemiology (Lecture Notes in Mathematics vol 1936)(Berlin: Springer) pp 1–49

[81] Wehrle-Haller B and Imhof B A 2003 Actin, microtubules andfocal adhesion dynamics during cell migration Int. J.Biochem. Cell Biol. 35 39–50

[82] Yamao M, Naoki H and Ishii S 2011 Multi-cellularlogistics of collective cell migration PLoS ONE6 e27950

[83] Young G B 1970 Thermography CRC Crit. Rev. Solid StateMater. Sci. 1 593–638

[84] Zaman M H, Kamm R D, Matsudaira P andLauffenburgery D A 2005 Computational model for cellmigration in three-dimensional matrices Biophys. J.89 1389–97

[85] Zhao M 2009 Electrical fields in wound healing—anoverriding signal that directs cell migration Semin. CellDev. Biol. 20 674–82

17

Publications

87

Seyed Jamaleddin Mousavi and Mohamed Hamdy Doweidar

Physical Biology

2014


2.5 A novel mechanotactic 3D modeling of cell morphol-

ogy.

A novel mechanotactic 3D modeling of cellmorphology

Seyed Jamaleddin Mousavi1,2,3 and Mohamed Hamdy Doweidar1,2,3,4

1Group of Structural Mechanics and Materials Modelling (GEMM), Aragón Institute of EngineeringResearch (I3A), University of Zaragoza, Spain.2Mechanical Engineering Department, School of Engineering and Architecture (EINA), University ofZaragoza, Spain.3 Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina(CIBER-BBN), Spain.

E-mail: [email protected]

Received 15 May 2014, revised 6 June 2014Accepted for publication 10 June 2014Published 22 July 2014

AbstractCell morphology plays a critical role in many biological processes, such as cell migration, tissuedevelopment, wound healing and tumor growth. Recent investigations demonstrate that, amongother stimuli, cells adapt their shapes according to their substrate stiffness. Until now, thedevelopment of this process has not been clear. Therefore, in this work, a new three-dimensional(3D) computational model for cell morphology has been developed. This model is based on aprevious cell migration model presented by the same authors. The new model considers thatduring cell–substrate interaction, cell shape is governed by internal cell deformation, which leadsto an accurate prediction of the cell shape according to the mechanical characteristic of itssurrounding micro-environment. To study this phenomenon, the model has been applied todifferent numerical cases. The obtained results, which are qualitatively consistent with well-known related experimental works, indicate that cell morphology not only depends on substratestiffness but also on the substrate boundary conditions. A cell located within an unconstrainedsoft substrate (several kPa) with uniform stiffness is unable to adhere to its substrate or to sendout pseudopodia. When the substrate stiffness increases to tens of kPa (intermediate and rigidsubstrates), the cell can adequately adhere to its substrate. Subsequently, as the traction forcesexerted by the cell increase, the cell elongates and its shape changes. Within very stiff (hard)substrates, the cell cannot penetrate into its substrate or send out pseudopodia. On the other hand,a cell is found to be more elongated within substrates with a constrained surface. However, thiselongation decreases when the cell approaches it. It can be concluded that the higher the nettraction force, the greater the cell elongation, the larger the cell membrane area, and the lessrandom the cell alignment.

S Online supplementary data available from stacks.iop.org/pb/11/046005/mmedia

Keywords: cell morphology, cell migration, mechanotaxis, cell-substrate interaction, 3D finiteelement model

(Some figures may appear in colour only in the online journal)

1. Introduction

Adherent cells can sense the mechanical properties of theirsubstrates through the traction forces transferred to the

extracellular matrix (ECM) through its focal adhesions andintegrins. Traction forces exerted by the cell influence the cellcytoskeleton (CSK) and consequently change the cell shape.These changes are important in many biological processessuch as morphogenesis [5, 54], wound healing [19, 30],tumour growth [24, 62], tissue development [26, 46], cell

Physical Biology

Phys. Biol. 11 (2014) 046005 (13pp) doi:10.1088/1478-3975/11/4/046005

4 Author to whom any correspondence should be addressed.

1478-3975/14/046005+13$33.00 © 2014 IOP Publishing Ltd Printed in the UK1

Publications

91

differentiation and proliferation [13, 31, 51]. For instance,during morphogenesis, cells participate effectively in tissueremodelling and organism shaping by adapting themselves tothe mechanical characteristics of their surrounding micro-environment [26]. Matsuda et al [30] observed that in thewound healing process, after injury, cells change their shapesto cover the wound without leaving intercellular gaps. Duringthis process, cellular morphologic alterations are mainlyobserved near the wound edges. Cell elongation has beenobserved during this process when the cells migrate towardswound locations [19, 30]. A study by McBeath et al [31]demonstrated that cell morphology affects human mesench-ymal stem cell (hMSC) differentiation by modulating endo-genous RhoA activity. Moreover, changes of cell shapecontrol the geometry of subsequent cell division, which iscritical for embryo development and tissue integrity [13, 52].Cell morphology is derived through a cyclic coordinatedmulti-step process that occurring in a wide range of cell typessuch as epithelial and mesenchymal cells [7, 40]. It takesplace as a consequence of several main events during cellmigration, such as leading-edge protrusion, formation of newadhesions near the front, contraction, releasing old adhesionsand rear retraction [3, 32, 49]. Many scientists have studiedexperimentally the parameters that influence the process ofcell shape alteration [23, 27, 43, 55]. They found that, besidesother cues, the potential player in this process is themechanical cue. For example, a typical cell in stiff substratesis less rounded and more extended than the same cell in softsubstrates [23, 27]. Findings of Peyton et al [43] show that thecells are rounded on both two-dimensional (2D) and three-dimensional (3D) soft (448-5804 Pa) substrates. Moreover,the cells become stretched by increasing substrate stiffness. Inthis concept, investigations of cell behaviour on 2D and in 3Dsubstrates have shown that the mechanical properties of theECM influence cell migration, spreading and morphology.However, cellular mechano-sensing and behaviour in 3Dmatrices may differ from 2D substrates. For example, studiesof Hakkinen et al [16] illustrated that the cell morphologystrongly depends on substrate dimensionality since the cellstend to be less elongated and more spread on 2D matricesthan in 3D matrices. Dikovsky et al [9] used a 3D hydrogelscaffold to study optimal combinations of synthetic poly-ethylene glycol fibrinogen and endogenous fibrinogen on cellbehavior in smooth muscle cell culture. Their experimentalobservations demonstrate a qualitative relationship betweenthe molecular architecture of the matrix and the cellularmorphology. Unfortunately, precise mechanisms by whichthe cycle of cell motility is related to specific biological sig-nalling are not completely clear yet [25]. In this context,mathematical models are useful tools for providing someexplanation of such process. Several computational workshave studied cell morphology in 2D substrates [8, 22, 38, 39].Ni et al [39] developed a 2D energy-based mathematicalmodel to study how substrate rigidity influences cell mor-phology and migration. They assumed that the morphology ofan adherent cell is characterized by the competition betweencell, substrate and cell periphery energies. Therefore, the finalcell configuration is determined by minimization of the total

free energy of the cell–substrate system. Besides, Kabaso et al[22] provided a 2D model to explore the dynamics of cellularshapes driven by coupling the cytoskeletal forces with themembrane. The key point of their model is the role of curvedmembrane proteins that recruit the cytoskeleton forces to themembrane unlike the model presented by Satulovsky et al[47] in which the actual forces acting on the cell body are notexplicitly calculated. The model presented by Kabaso is acoarse-grained model similar to the other theoretical modelproposed by De et al [8]. Using a simple theoretical model,they predicted the dynamics and orientation of cells in boththe absence and presence of applied stresses. This modelincluded the forces generated due to cells mechano-sensingand the elasticity of the matrix. Their model focuses onneedle-like cells with bipolar morphologies, so that in the caseof cells with more isotropic morphologies, the model shouldbe extended to consider all of the dipolar components. Neil-son et al [38] have presented a 2D numerical model to studycell shape changes in the presence of chemotaxis. A majorshortcoming of their model is that they have ignored the cellmechano-sensing process, which is an intrinsic aspect ofcell–substrate interaction. Although 2D models provide somenotions on cell motility and shape, according to the compre-hensive experimental investigations of Hakkinen et al [16], inmany concepts cell behaviour in 2D substrates is distinct from3D cell–substrate interaction, particularly for the study of cellshape changes. In addition, the previously mentioned 2Dmodels do not study cell configuration in a free mode, butrestrict the cell shape to a rigid ellipse by which the cell shapechange is represented by alteration of the aspect ratio of theellipse (the ratio of the major axis to the minor axis). A typicalrigid mode of cell configuration is also assumed in some 3Dmodels. For instance, Hannezoa et al [17] proposed a theo-retical description of 3D epithelial cells that interact with flatsubstrates. Similar to Ni et al [39], their energy-based modelconsiders the following parameters: the contribution ofcell–substrate energy, which is proportional to the basal area;cell–cell lateral energy, which is proportional to the lateralarea; and energy associated with the tension of the apicalactomyosin belt, which is proportional to the apical perimeter.Thus the cell base length and height is obtained by mini-mizing the energy function. The main limitation of theirmodel is that the cell can have different rigid shapes (hex-agonal prism, columnar, cuboidal and/or squamous). Ver-molen et al [56] presented a 3D phenomenological numericalmodel to investigate the effect of chemotactic cues on cellmorphology. Although in their model the cell can take irre-gular shapes, their model is formulated in such a way that thecell velocity intrinsically depends on a chemical gradient,which according to many experimental works is not suffi-ciently precise [15, 28, 55]. They also assumed that the cellvolume changes when the cell sends out pseudopods. This isunrealistic since experimental investigations [2, 4, 6] havedemonstrated that overall cellular volume remains constant asthe cell shape changes. Therefore, here, the main objective isnot only to study the cell morphology in 3D substrates, but isalso to obviate the existing difficulties and weaknesses of theprevious 3D models. To simulate cell migration within a 3D

2

Phys. Biol. 11 (2014) 046005 S J Mousavi and M H Doweidar

2.5. A novel mechanotactic 3D modeling of cell morphology.

92

substrate, a numerical model was previously presented by thepresent authors considering a constant spherical cell config-uration during cell migration [34–37]. Guided by the above-mentioned experimental observations [23, 27, 55] and in lightof the authors previous model, a novel 3D computationalmodel is here developed to accurately describe the evolutionof cell morphology associated with mechanotactic cell–matrixinteraction, using discrete methodology considering the cellas a group of finite elements. This group of elements can bereordered in a free mode to represent the cell efficiently. In thepresent model, it is assumed that cell shape is governed byinternal deformation, which leads to an accurate prediction ofthe cell shape according to the mechanical characteristic of itssurrounding micro-environment. In the subsequent sections,the fundamental equations of the model are derived, and itsefficiency is tested by its application in several numericalcases.

2. Model description

2.1. Transmission of cell internal stress to the substrate

It is well known that active (actin filaments and AMmachinery) and passive (microtubules and cell membrane)cellular elements play the principal role in generating cellstress, which is transmitted by integrins to the substrate. Theactive cellular elements mainly regulate cell stress accordingto minimum, ϵmin, and maximum, ϵmax, internal strains. Out-side the ϵmin–ϵmax range, the active stress transmitted to thesubstrate is zero. Therefore, the mean contractile stress raisedby the active cellular element of the cell can be obtained by[34–36]

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

σ

ϵ ϵ ϵ ϵσ ϵ ϵ

ϵ σϵ ϵ ϵ

σ ϵ ϵϵ σ

ϵ ϵ ϵ

=

< >−

−⩽ ⩽ ˜

−− ˜ ⩽ ⩽

K

K

K

K

0 or( )

( )(1)act

cell min cell max

act max min cell

act min maxmin cell

act max max cell

act max maxcell max

where Kact, σmax and ϵcell represent the stiffness of the activecellular elements, the maximum contractile stress exerted bythe actin–myosin machinery and the internal strain of the cell,respectively, while ϵ σ˜ = K/max act. On the other hand, stressgenerated by passive cellular elements is basically propor-tional to their stiffness, Kpas, as [35, 36]

σ ϵ= K (2)pas pas cell

Consequently, the net cell stress transmitted to the ECM canbe defined as

σ σ σ= + (3)cell act pas

2.2. Effective mechanical forces

To migrate, a cell needs to extend protrusions in the directionof migration, adhere to its substrate and pull itself forward.Adhesion is thought to implement this process by providing

sufficient traction required for efficient translocation [41].Cell shape is directly related to the equilibrium of forcesexerted on the cell body [26, 33]. Therefore, to reveal themechanisms of shape formation we need to understand howthe cell regulates its body according to exerted forces. Thereare two main mechanical forces acting on a cell: traction forceand drag force. Traction force is a consequence of the con-traction of the actin–myosin apparatus, which drives forwardthe cell body and is transmitted to the substrate. This force isproportional to the stress transmitted by the cell to the ECM.Assuming the cell is represented by a connected group offinite elements, the nodal traction force exerted by the cell toits substrate at each finite element node of the cell membranepassing by the cell centroid can be expressed as [35]

σ ζ= S tF e( ) (4)itrac

icell

where ei is a unit vector from the ith node of the cell mem-brane towards the cell centroid. S(t) is the cell membrane area,which varies with time. Although the cell shape and cellmembrane area change, it is assumed that the cell volume isconstant during cell motility [2, 4, 6]. ζ is the adhesivity,which is a dimensionless parameter proportional to thebinding strength of the receptors to the ligands in the ECM[35, 61]. It depends on the binding constant of the cellintegrins, k, the total number of available receptors, nr, andthe concentration of the ligands at the leading edge of the cell,ψ. Therefore, it can be defined as [35–37]

ζ ψ= kn (5)r

ζ depends on the cell type and has different values in the frontand rear of the cell. Consequently, to correctly include it, weneed to distinguish between the anterior and posterior parts ofthe cell. The definition is given in the following sections.During migration, a cell exerts contraction forces (nodaltraction forces) towards its centroid, compressing itself. As aresult, each finite element node belonging to the cell mem-brane that has less internal deformation will have a highertraction force [35]. Consequently, the net traction force actingon the whole cell body due to cell–substrate interaction can becalculated by [35]

∑= −=

F F (6)inettrac trac

i

n

1

where n is the number of the cell membrane nodes. Bycontrast, the drag force resists the cell motility. To define avelocity-dependent opposing force associated with the linearviscoelastic character of the ECM surrounding the cell, wehave assumed for the sake of simplification that the ECM is aviscoelastic medium [61]. At a micro-scale, the viscousresistance dominates, while the inertial resistance of a viscosefluid is small enough to be ignored. Stokes [50] described thedrag force of a sphere at the limit of negligible convection as

β=F v (7)Ds

where v is the relative velocity and β is often referred to as theStokes’ drag regime for a small spherical object moving

3


Publications

93

slowly through a viscous fluid expressed as

β π η= ( )r E6 (8)sub

where r is the object radius. η E( )sub is the effective mediumviscosity. In a substrate with a stiffness gradient we assumethat it is linearly proportional to the substrate stiffness, Esub, ateach point. Therefore it can be calculated as

η η λ= +( )E E (9)sub min sub

where ηmin is the minimum viscosity of the substrate corre-sponding to minimum stiffness and λ is the proportionalitycoefficient. The viscosity coefficient may eventually besaturated with substrate rigidity; however, this saturationoccurs outside the substrate rigidity range suitable for fibro-blasts [10]. The drag of non-spherical solid particles willdepend on the degree of non-sphericity as well as theirorientation to the flow (since the drag will generally be ani-sotropic with respect to direction). As cell morphologychanges during migration, equation (7) will not deliver anaccurate representation of the drag force. Its definition in thecase of irregular particles is further complicated by the ran-domness of the shapes and dynamics. However, a review ofexperimental studies of the mean drag of irregularly-shapedparticles suggests that it is reasonable and appropriate to use ashape factor, fshape, to moderate the Stokes expression as[29, 53]

=F f F (10)Ds

drag shape

Nevertheless, it is expected that only approximate and prob-abilistic predictions are possible for highly irregular particles.A wide variety of shape-characterizing parameters has beensuggested for irregular particles, the most common and mostsuccessful being the Corey Shape Factor (CSF). This shapefactor employs three lengths of a particle in mutually per-pendicular directions, being representative of cell surface areachanges [29]; the cellʼs longest dimension, lmax, the shortestdimension, lmin, and the intermediate or medium dimension,lmed. Herein, we take advantage of a convenient version ofCSF used by Loth [29] to estimate the shape factor as

⎛⎝⎜

⎞⎠⎟=f

l l

l(11)shape

max med

min2

0.09

This is nearly identical to the expression proposed by Dressel[11] except that he used a 0.1 exponent. Other shape factorshave also been suggested to characterize the shape irregu-larity, but the impact of the max-med-min area factor tends tobe the strongest [21, 29]. It is notable that this shape factor inthe case of a spherical cell shape delivers a value of 1.

2.3. Protrusion force

During cell migration, cells send out local protrusions toprobe their environment by exerting a protrusion force. Thisforce is generated by actin polymerization and must be dis-tinguished from the cytoskeletal contractile force [3, 61]. Itarises from cell–matrix attachments at the new sites oflamellipodia and filopodia development, which have a

stochastic nature during cell migration [44]. This causes cellsto move along a directed random path towards the effectivesignals. The direction and magnitude of the protrusion force ischosen randomly at each time step. It is remarkable that theorder of the protrusion force magnitude is the same as that ofthe traction force but with lower amplitude[20, 35, 37, 45, 61]. Therefore, we randomly estimate it as

κ= FF e (12)nettrac

prot rand

where erand represents a random unit vector and Fnettrac is the net

traction force while κ is a random number, such thatκ⩽ <0 1, [35, 36]. It is assumed that there is neither

degradation nor remodelling of the ECM during cell motility.As previously mentioned, the inertial force is negligible sothat the force balance yields the actual equation of cell motionas

+ + =F F F 0 (13)nettrac

prot drag

2.4. Deformation and reorientation of the cell

We initially assume a spherical cell configuration (solid linein figure 1). It is considered that a cell first exerts sensingforces on the membrane to diagnose its surrounding substrate(mechano-sensing). These forces act at each finite elementnode of the membrane towards the cell centroid. The celldeformation resulting from these sensing forces is representedby dashed lines in figure 1. Therefore, the cell internal strainat each finite element node of the cell membrane can berepresented by

ϵ = MN

OM(14)cell

According to equation (13) the net cell polarisation direction

Figure 1. (a) Calculation of the cell internal deformation due to cellmechano-sensing. An initially spherical cell (solid line) is deformed(dashed line) due to cell mechano-sensing. (b) Calculation of cellpolarisation direction, epol, for the reoriented cell. Fnet

trac, Fprot and Fdrag

represent the net traction force, protrusion force and drag force,respectively.

4



94

can be calculated by (figure 1)

= −∥ ∥

eF

F(15)pol

drag

drag

Likewise, substituting equation (10) in equation (13) andrearranging the elements, the cell velocity can be defined as

π η=

∥ ∥( )

vr E

F

6(16)

drag

sub

2.5. Cell morphology during cell migration

Cell migration within a 3D substrate is a coordinationbetween several cyclic cellular processes. At the level of thelight microscope, some authors address that this cycle can bedivided into five steps: (1) extension of leading edges; (2)adhesion to matrix contacts; (3) contraction of the cytoplasm;(4) release from contact sites; and (5) recycling of membranereceptors from the rear to the front of the cell [32, 49].However, some authors summarize these steps in four [49] oreven three [3] steps. At the leading edge, the tension exertedby protrusions adhering to a substrate pulls the cells forward,while at the trailing end the cortical tension pushes orsqueezes the cytoplasm in the direction of migration [41, 57].In line with the aforementioned experimental observations,we have developed a numerical model of cell–substrateinteraction using the notion of mechanotaxis to investigate thecell morphology during migration. The regulatory processbehind the cell shape changes described above is simplifiedhere as the cyclic repetition of the cell membrane extension atthe anterior and its retraction at the posterior. Consider thatthe working domain is represented by Λ⊂R3 (figure 2). Let usdenote the global coordinates by X and the local cell

coordinates by ′X located in the cell centroid (figure 1). Theinitial cell domain can be defined as

Ω Λ′ = ′ ′ ′ ′ ∈ ∀ ∥ ′ ∥ ⩽{ }( ) ( ) rx X x X x: (17)

while ′Ω∂ represents the cell membrane where cell–substrateinteraction occurs. Therefore, the substrate domain can bedetermined by

|Ω Λ Ω= ∈ ∉ ′x X x X x X{ ( ) ( ) , ( ) } (18)

It is worth mentioning that while the cell migrates, both of thedomains ′Ω and Ω vary such that ∪′Ω Ω Λ= and∩′Ω Ω = ∅. Since the cell volume is considered constant,

the number of cell finite elements during changes in the cellshape is constant. We analysed the dominant modes of cellshape changes and their association to traction forces, con-sidering retraction of the cell body at the rear and its extensionat the front. To achieve this, it is essential to define theanterior and posterior parts of the cell at each time step. Let usassume χ is a plane passing by the cell centroid, O, with unitnormal vector n that is parallel to epol, and ′s X( ) that is aposition vector of the cell membrane nodes located on ′Ω∂(figure 2), projection of s on n submits δ as

δ = ·n s (19)

So, nodes with positive δ are located at the cell front on Ω∂ ′+

and nodes with negative δ are located at the cell rear on Ω∂ ′−,where

∪Ω Ω Ω∂ ′ = ∂ ∂′+ ′− (20)

We assume that the cell is retracted from the weakest point onΩ∂ ′−, where the cell experiences the highest deformationduring the mechano-sensing process. To determine the cellextension points, let us denote an arbitrary finite element nodelocated on Ω∂ ′+ by P (figure 2) and has a position vectorrepresented by ′v X( ). The angle between the cell polarisationdirection, epol, and v, can be calculated as

α =·

∥ ∥−cose v

v(21)1 pol

Accordingly, the minimum value of α delivers the membranenode from which the cell must be extended. We assume

′Ω∈ere is the finite element that the cell should retract andΩ∈eex is the finite element that the cell should extend. To

consider cell shape changes and cell migration more simply,ere is eliminated from the ′Ω domain and is added to the Ωdomain. In contrast, eex is removed from the Ω domain and isadded to the ′Ω domain.

Wessels et al [58, 59] show that a cell can extend about10 % of its total volume as pseudopodia, depending on thecell type and the characteristics of the substrate in which itresides. In accordance with these observations, we assumethat the cell can extend a maximum of 10 % of its wholevolume as pseudopodia. Consequently, in the present modelthe cell is not allowed to become infinitely thin.

Figure 2. Definition of extension and retraction points, anterior andposterior parts of the cell at each time step. Cell domain, Ω′, andsubstrate domain, Ω, in the working space, Λ⊂R3. X represents theglobal coordinates and ′X represents the local cell coordinateslocated in the cell centroid, O. χ is a plane passing by the cellcentroid with unit normal vector n parallel to the cell polarisationdirection, epol. P denotes a finite element node located on the front ofthe cell membrane, Ω∂ ′+.

5


Publications

95

3. Finite element implementation

The present model is implemented with the commercial FEsoftware ABAQUS [18] through a coupled user subroutine.The corresponding algorithm is presented in figure 3. Wehave applied the model in several numerical examples wherethe cell is embedded within a μ× ×400 200 200 m substrate.The results are compared with those of correspondingexperimental works reported in the literature [12, 23, 27, 55].It is assumed that there is no external force acting on thesubstrate. The substrate is meshed by 16 000 regular hex-ahedral elements and 18 081 nodes, while the cell is repre-sented by 536 elements. The calculation time is about 1 min.for each time step in which each step corresponds toapproximately 10 min of real cell–substrate interaction[35, 36, 61]. Initially, the cell is assumed to have a sphericalshape. The properties of the cell and the substrate areenumerated in table 1. To quantify the cell shape, cell

elongation, ϵelong, and cell morphological index (CMI) arerespectively calculated as

ϵ = −l l

l1 (22)elong

min med

max

=tS t

SCMI( )

( )(23)

in

where Sin is the cell membrane area in the initial step. Inequation (22) the second term describes the ratio of thegeometric mean over cell length. It is clear that for a sphericalcell shape, ϵelong is zero, while for a bipolar spindle and per-fectly elongated cell shape, ϵelong will be close to 1 [27]. Toanalyse the cellular random alignment in substrates with astiffness gradient or substrates with constrained boundarysurfaces, the angle between the net polarisation direction ofthe cell and the stiffness gradient direction, θ, is defined ateach time step. Therefore, the Random Index (RI) can becalculated by

θ=∑= cos

NRI (24)i

Ni1

where N is the number of time steps during which the cellreaches steady state (state in which the cell elongation nolonger changes considerably). RI=-1 indicates totally randomalignment of the cell while RI=+1 represents perfect align-ment of the cell in the stiffness gradient direction. Therefore,in the presence of a stiffness gradient, the closer RI is to +1,the lower the cellʼs random orientation.

4. Numerical examples

Although the exact manner in which the substrate stiffnessinfluences cell morphology remains elusive, there is 2D [60]

Figure 3. Computational algorithm of cell morphology changes dueto mechano-taxis.

Table 1. General parameters used in the model except where othervalues are specified.


ν Substrate Poisson ratio 0.3 [1, 55]ηmin Minimum substrate viscosity 1000 Pa.s [1, 61]

Kpas Stiffness of microtubules 2.8 kPa [48]

Kact Stiffness of myosin II 2 kPa [48]ϵmax Maximum strain of the cell 0.09 [35, 45]ϵmin Minimum strain of the cell -0.09 [35, 45]σmax Maximum contractile stress

exerted by actin–myosinmachinery

0.1 kPa [40, 42]

kf = kb Binding constant at rear andfront of the cell

108

mol−1[61]

nr f Number of available receptors atthe front of the cell

×1.5 105 [61]

nrb Number of available receptors atthe back of the cell

105 [61]

ψ Concentration of the ligands atrear and front of the cell

10−5 mol [61]

6



96

and 3D [23, 27] experimental evidence indicating that, amongother factors [14, 16], the cell morphology can be regulatedby substrate stiffness. Our objective here is to study thedependency of cell morphology on substrate stiffness andboundary conditions. Therefore, the main questions that wewill answer here are the following. Does a cell sense andrespond to any change in the substrate stiffness? What is theminimum and maximum substrate stiffness that a cell canadhere to and penetrate into? How does the cell shape changewithin 3D substrate in different conditions? To obtain reliableand consistent answers to these questions, several numericalcases have been studied, all of them were repeated at least 20times.

4.1. Cell morphology in unconstrained substrates with differentuniform stiffnesses

Experimental works by Lee et al [27] demonstrated thatfibroblasts exhibit different morphological trends in 3D sub-strates with different stiffnesses. They reported that fibroblastsembedded within soft substrates (several kPa), similar to thestiffness of adipose tissue, do not spread and remain rounded,while resident fibroblasts in intermediate and rigid substrates(tens of kPa) can spread in and adhere to the matrix. How-ever, the overall elongation of cells within rigid substrates is

substantially higher than that within intermediate matrices. Inaddition, Ehrbar et al [12] claim that in hard (very stiff) gels,cells generally remain round with faded filopodia, inhibitingthe cells from penetrating and adhering to hard substrates.However, it is evident that a matrix may be considered hardfor one type of cell while it is rigid, intermediate or even softfor another cell type.

Since our aim is to provide a phenomenologicaldescription of a single cell within 3D substrates, we havedesigned four different examples to investigate the cell mor-phology in an unconstrained substrate with different uniformstiffnesses. At the beginning, as mentioned above, the cell hasa spherical shape. Figure 4 shows the results obtained of thecell morphology in substrates with different stiffnesses (soft,intermediate, rigid and hard). The cell located in the softsubstrate is not able to adhere to the substrate since theaverage net traction force generated by the cell is trivial(figure 5), so it remains almost rounded (figure 4(a)) (movie 1(stacks.iop.org/pb/11/046005/mmedia)). In this case, as seenin figure 6, the mean cell elongation is ∼0.07 and the cellmembrane area does not noticeably change, remaining almostconstant (CMI∼1), as can be seen in figure 7. Since thesubstrate is unconstrained, the cell should be reorientedtowards the substrate centre (positive RI) [35]. For this case,

Figure 4. Cell morphology in an unconstrained substrate with stiffness of a- 1 kPa (soft), the cell is not able to adhere to the substratemaintaining its initial spherical shape (movie 1, supplementary data); b- 10 kPa (intermediate), the cell has capability to migrate and getselongated (movie 2, supplementary data); c- 100 kPa (rigid), the cell has more capability to migrate and gets more elongated (movie 3,supplementary data); d- 200 kPa (hard), the cell is not able to penetrate into the substrate maintaining its initial spherical shape (movie 4,supplementary data).

7


Publications

97

as seen in figure 8, the mean RI is ∼ − 0.37, which meanstotal random alignment of the cell. This is because the celltraction forces are weaker than that needed to align the celltowards stiffer regions (substrate centre), highlighting theprotrusion force participation on the cell orientation. Theresults do not depend on the initial location of the cell whilethe substrate stiffness is uniform. On the other hand,figure 4(b) (movie 2 (stacks.iop.org/pb/11/046005/mmedia))shows that the cell is able to elongate in a substrate withintermediate stiffness (∼10 kPa). This is because increasingthe substrate stiffness enables the cell to generate strongertraction forces (see figure 5). The mean cell elongation in this

case is ∼0.31, which can slightly vary for different runs,presented as mean standard deviation in figure 6. For a cellresident in a substrate with intermediate stiffness, it can beseen in figure 7 that the average CMI is ∼1.23 in a steadystate, which means the increase in the substrate stiffnessincreases the cell membrane area. It is of interest to mentionthat an increase in the substrate stiffness causes the cell toconsiderably align towards the substrate centre, enhancingthe mean RI to ∼0.3 as seen in figure 8. Increasing the sub-strate stiffness to ∼100 kPa causes further cell shape defor-mation (figure 4(c)) (movie 3 (stacks.iop.org/pb/11/046005/mmedia)). Accordingly, the cell displays a more elongatedmorphology (ϵelong =0.42) within a rigid substrate than in asubstrate with intermediate stiffness (figure 6). In this case,the mean CMI is equal to 1.34, even higher than that within asubstrate with intermediate stiffness (see figure 7). As seen infigure 5, in a rigid substrate the average net traction force ofthe cell is maximum while the cell morphology changes. Themean RI for the rigid substrate is equal to ∼0.5, which is morethan that of the intermediate stiffness substrate (figure 8).Increasing the substrate stiffness more than ∼100 kPadecreases the cell elongation, CMI and average net traction

Figure 5. Average net traction force, Fnettrac, generated by the cell in

unconstrained (UN) and constrained (CO) substrates with differentstiffnesses. The error bars represent the mean standard deviation ofdifferent runs. The cell traction forces exerted in soft and hardsubstrates are minimum while it is maximum in constrained substratewith 100 kPa stiffness.

Figure 6. Cell elongation, ϵelong, versus the cell longest dimension,lmax, for a cell resident within unconstrained (UN) and constrained(CO) substrates with uniform stiffness. The error bars represent themean standard deviation of different runs. The cell is unable toelongate in soft and hard substrates while its elongation is maximumin the middle of the constrained substrate.

Figure 7. CMI with its mean standard deviation for a cell residentwithin constrained (CO) and unconstrained (UN) substrates withdifferent stiffnesses.

Figure 8.Mean RI of the cell in unconstrained (UN) and constrained(CO) substrates with different stiffnesses. The error bars representthe mean standard deviation of different runs.

8



98

force. Consequently, in the substrates stiffer than ∼200 kPa(hard substrates) the cell is unable to send out pseudopodia,acquiring its initial shape as seen in figure 4(d) (movie 4(stacks.iop.org/pb/11/046005/mmedia)). Therefore, the cellcannot penetrate into the substrate where ϵ ∼elong 0 (figure 6)and CMI∼1 (figure 7). In this case, the average net tractionforces generated by the cell are negligible (figure 5). Althoughthe cell cannot penetrate into hard substrates, according tofigure 8 its polarisation direction persistently points the celltowards stiffer regions, showing a different trend in the termsof RI in comparison with soft substrates. This demonstratesthat although the cell is not able to generate sufficiently strongtraction force to penetrate into the hard substrate, the cellorientation is towards the substrate centre (stiff region) and itsalignment is less random than that in soft substrates. In thecase of a cell resident within soft, intermediate and rigidsubstrates, our findings are qualitatively consistent with theobservations of Lee et al [27]. In the case of hard substrates,the results obtained are qualitatively consistent with theexperimental report of Ehrbar et al [12].

4.2. Cell morphology in a constrained substrate with uniformstiffness

Experimental studies by Karamichos et al [23] illustrate that,in 3D substrates, the substrate boundary conditions affect cellmorphology. Their observations indicate that corneal fibro-blasts in unconstrained matrices were less deformed and lesselongated than those in constrained matrices. Moreover, cellalignment in unconstrained substrates was more random thanthat within constrained matrices. In order to consider theeffect of boundary conditions on cell morphology in a sub-strate with a uniform stiffness, the surface x = 400 μm wasfully constrained (figure 9). Initially, a spherical-shaped cell islocated far from the constrained surface. A low elastic mod-ulus (∼1 kPa) is assigned to the substrate. Similar to theprevious example (figure 4(a)), the cell is unable to adhere tothe substrate and cannot send out pseudopodia due to weaktraction forces. Although the substrate surface is constrained

and the cell should feel more rigidity to generate strongertraction forces in comparison with the unconstrained sub-strate, the cell is unable to sense this because of the longdistance from the constrained surface. Increasing the substratestiffness to tens of kPa causes the cell to be deformed andelongated. Figure 9 shows two states of cell morphologies inthe middle of a substrate with 100 kPa stiffness and near theconstrained surface (movie 5 (stacks.iop.org/pb/11/046005/mmedia)). As the cell approaches the middle of the substrate,the cell elongation increases along the x-axis (substrate longaxis). The cell elongation is maximum in the middle of thesubstrate and decreases as the cell approaches the constrainedsurface (figure 6). When the cell reaches the constrainedsurface, it tends to adhere to that surface since it ”feels” moresubstrate stability there. Accordingly, the cell is spread overthe constrained surface and never separates from it. The trendof the cell membrane area is the same as the cell elongation atthe coincident positions, whereas the CMI is maximum in themiddle of the substrate and decreases as the cell comes closeto the constrained surface (figure 7). It should be noted that, inthis case, neither the mean cell elongation nor the CMIbecomes less than those within the unconstrained substratewith similar substrate stiffness (see figures 6 and 7). Inaddition, it is interesting that the cell RI in a constrainedsubstrate is notably higher than that in an unconstrainedsubstrate, which means less random motility of the cell (seefigure 8). These findings are in agreement with the experi-mental observations of Karamichos et al [23].

4.3. Cell morphology in an unconstrained substrate withstiffness gradient

To fully demonstrate the effect of matrix stiffness on cellmorphology, we applied the present model to an unconstrainedsubstrate with a linear stiffness gradient along the long axis ofthe substrate (x-axis), which changes from 1 kPa at =x 0 to300 kPa at =x 400 μm. This range covers the stiffness of thesoft–hard substrates in the examples described above. The cellis initially located in the soft region of the substrate. Although

Figure 9. Cell morphology in a constrained substrate with uniform stiffness of 100 kPa. The boundary surface at x = 400 μm is fullyconstrained. (a) cell morphology in middle; (b) cell morphology near the constrained surface. The cell elongation is maximum in the middleof the substrate and decreases as the cell approaches the constrained surface. When the cell reaches the constrained surface, it tends to adhereto and spread over that surface (movie 5, supplementary data).

9


Publications

99

in the unconstrained substrate with uniform stiffness the cellwas unable to adhere to the soft substrate (see figure 4(a), inthis case it is able to adhere to the substrate and to be deformeddue to the presence of the stiffness gradient between the frontand the rear of the cell. This is due to the fact that even in thesoft regions, the presence of the stiffness gradient causes aconsiderable difference between the generated traction forcesin the front and rear of the cell membrane, resulting in a greaternet traction force than that generated within matrices withuniform stiffness. The cell morphology is shown in figure 10 insequential time steps (movie 6 (stacks.iop.org/pb/11/046005/mmedia)). While the cell approaches the middle of the sub-strate (rigid regions), the cell elongation increases in the stiff-ness gradient direction (x-direction). In contrast, as the cellcomes close to the hard region of the substrate (end of sub-strate), the elongation decreases and the cell becomes morerounded, consistent with the claim of Ehrbar et al [12], (seefigures 10 and 11). This is because the cell is unable topenetrate further into the hard substrate, as seen in the previousexample (figure 4(d)). It is notable that in this case the cellnever penetrates into regions stiffer than ∼200 kPa. In addition,according to figure 11, the CMI and ϵelong curves follow thesame trends, being maximum in the middle of the substratewhere the substrate stiffness is ∼100 kPa. Consequently, as thecell becomes elongated or rounded, the CMI increases ordecreases, respectively. As seen in figure 8, there is no

considerable difference between the mean RI in this case andthe other case of the constrained substrate, whereas in bothcases the cell is aligned along the x-axis towards the substrateend where the cell ”feels” more rigidity.

Figure 10. Cell morphology in an unconstrained substrate with stiffness gradient changing from 1 kPa at =x 0 to 300 kPa at =x 400 μm.The presence of the stiffness gradient causes a considerable difference between the generated traction forces in the front and rear of the cellmembrane, which increases the net traction force. As the cell approaches the middle of the substrate (rigid regions), the cell elongationincreases. In contrast, when the cell becomes close to the hard region of the substrate, the elongation decreases and the cell becomes morerounded (movie 6, supplementary data).

Figure 11. Cell elongation, ϵelong (left axis), and CMI (right axis)versus the substrate local stiffness for a cell resident within anunconstrained substrate with stiffness gradient. The cell is able toadhere to the substrate and to be deformed. While the cellapproaches the middle of the substrate (rigid regions) ϵelong and CMIincrease. In contrast, as the cell comes close to the hard region of thesubstrate ϵelong and CMI decreases, the cell becomes more rounded.

10



100

5. Conclusions

Previous experimental studies [12, 23, 27, 55] have shownthat the stiffness of the substrates can lead to considerablechanges in cell morphology. This effect varies for differentcell types and depends on the nature of the adhesionreceptors by which the cells bind to their substrate. In thisstudy, we have extended the previous work presented by thesame authors [34–37] to develop a numerical model todescribe how the cell morphology is linked to the substratestiffness as well as cell migration. Consistent with experi-mental observations reported in the related literature[23, 27], our model demonstrates that the morphology of anadherent cell is governed by substrate stiffness and othercues [14, 16]. Our findings indicate that within soft and hardsubstrates, the cell morphological changes are trivial and thecell remains mainly rounded with negligible elongation.Likewise, in these cases the cell membrane area is constant,such that CMI∼1. We figured out that increasing the ECMstiffness in unconstrained substrates decreases the cell ran-dom motility. Interestingly, in the presence of a stiffnessgradient, the cell random motility decreases. However, thismotility is minimal for constrained substrates. It is worthnoticing that in the case of constrained substrates the RIvalues near the constrained surface were similar to those inthe middle of the substrate. Accordingly, the cell undergoesconsiderable elongation within intermediate and rigid sub-strates (tens of kPa) where the cell membrane area followsthe same trend, proportional to cell elongation. Theseparameters are enhanced when a stiffness gradient isemployed in cell substrates and are maximized for con-strained substrates. It should be noted that in the constrainedsubstrates, the cell elongation decreases near the constrainedsurface (not less than that within unconstrained substrates)and is perpendicular to the long axis of the substrate(figures 6 and 9(b)). This occurs because the cell persistentlypoints towards the constrained surface. Since the cell cannotmove beyond the constrained surface, it spreads on thesurface and becomes elongated in the direction of the y- or/and z-axis. Consequently, it seems that the lower stiffness inthe unconstrained ECM results in a smaller cell elongationand unchanged membrane area, weaker average net tractionforces and a more random pattern of cell spreading. We havepresented a 3D numerical model to investigate cell mor-phology during cell–matrix interaction. The model includesa 3D viscoelastic system, stress–strain behaviour and thecontractile apparatus of the cell. The main factors thatinfluence cell behaviour are taken into account, including (i)the active signals triggered from actin polymerization andmyosin phosphorylation, (ii) the tension-dependent assem-bly of actin and myosin into stress fibers, and (iii) the bal-ance of cellular traction, protrusion and drag forces. Takentogether, these factors enable us to model realistic cellbehaviour, leading to a deeper understanding of cellresponses to signals received from the substrate.

Acknowledgements

The authors gratefully acknowledge financial support fromthe Spanish Ministry of Economy and Competitiveness(MINECO MAT2013-46467-C4-3-R) and the CIBER-BBNinitiative. CIBER-BBN is financed by the Instituto de SaludCarlos III with assistance from the European RegionalDevelopment Fund.

References

[1] Akiyama S K and Yamada K M 1985 The interaction ofplasma fibronectin with fibroblastic cells in suspensionJournal of Biological Chemistry 260 4492–500 PMID:39.20218

[2] Albrecht-Buehler G 1982 Does blebbing reveal the convulsiveflow of liquid and solutes through the cytoplasmicmeshwork? Cold Spring Harbor Symposia on QuantitativeBiology 46 45–49

[3] Allena R and Aubry D 2012 Run-and-tumble or look-and-run? a mechanical model to explore the behavior of amigrating amoeboid cell Journal of Theoretical Biology306 15–31

[4] Charras G T, Yarrow J C, Horton M A, Mahadevan L andMitchison T J 2005 Non-equilibration of hydrostaticpressure in blebbing cells Nature 435 365–69

[5] Cintron C, Covington H and Kublin C L 1983 Morphogenesisof rabbit corneal stroma Invest Ophthalmol Vis Sci. 24543–56 PMID: 6841000

[6] Casey Cunningham C 1995 Actin polymerization andintracellular solvent flow in cell surface blebbing J Cell Biol.129 1589–99

[7] Dalton B A, Walboomers X F, Dziegielewski M,Evans M D M, Taylor S, Jansen J A and Steele J G 2001Modulation of epithelial tissue and cell migration bymicrogrooves Phil. Trans. Roy. Soc. London 56 195–207

[8] De R, Zemel A and Safran S A 2007 Dynamics of cellorientation Nat. Phys. 3 655–9

[9] Dikovsky D, Bianco-Peled H and Seliktar D 2006 The effect ofstructural alterations of PEG-fibrinogen hydrogel scaffoldson 3-d cellular morphology and cellular migrationBiomaterials 27 1496–506

[10] Dokukina I V and Gracheva M E 2010 A model of fibroblastmotility on substrates with different rigidities Biophys. J. 982794–803

[11] Dressel M 1985 Dynamic shape factors for particle shapecharacterization Part. Charact. 2 62–6

[12] Ehrbar M, Sala A, Lienemann P, Ranga A, Mosiewicz K,Bittermann A, Rizzi S C and Weber F E 2011 Elucidatingthe role of matrix stiffness in 3D cell migration andremodeling Biophys. J. 100 284–93

[13] Gong Y, Mo C and Fraser S E 2004 Planar cell polaritysignalling controls cell division orientation during zebrafishgastrulation Nature 430 689–93

[14] van Haastert P J M 2010 Chemotaxis: insights from theextending pseudopod J Cell Sci. 123 3031–37

[15] Hadjipanayi E, Mudera V and Brown R A 2009 Guiding cellmigration in 3d: A collagen matrix with graded directionalstiffness Cell Motility and the Cytoskeleton 66 435–45

[16] Hakkinen K M, Harunaga J S, Doyle A D and Yamada K M2011 Direct comparisons of the morphology, migration, celladhesions, and actin cytoskeleton of fibroblasts in fourdifferent three-dimensional extracellular matrices Tissue EngPart A 17 713–24

11


Publications

101

[17] Hannezo E, Prost J and Joanny J F 2014 Theory of epithelialsheet morphology in three dimensions Proc. of the NationalAcademy of Sciences current issue 111

[18] Hibbit, Karlson, and Sorensen 2011 Abaqus-Theory manual6.11-3 edition

[19] Hoppenreijs V P, Pels E, Vrensen G F, Oosting J andTreffers W F 1992 Effects of human epidermal growthfactor on endothelial wound healing of human corneas InvestOphthalmol Vis Sci. 33 1946–57 PMID: 1582800

[20] James D W and Taylor J F 1969 The stress developed by sheetsof chick fibroblasts in vitro Experimental Cell Research 54107–10

[21] Jimenez J A and Madsen O S 2003 A simple formula toestimate settling velocity of natural sediments J. Waterw.Port Coast Ocean Eng. 129 70–8

[22] Kabaso D, Shlomovitz R, Schloen K, Stradal T and Gov N S2011 Theoretical model for cellular shapes driven byprotrusive and adhesive forces PLoS Comput Biol. 71001127

[23] Karamichos D, Lakshman N and Petroll W M 2007 Regulationof corneal fibroblast morphology and collagenreorganization by extracellular matrix mechanical propertiesInvest Ophthalmol Vis Sci 48 5030–37

[24] Kirkeeide E K, Pryme I F and Vedeler A 1992 Morphologicalchanges in Krebs II ascites tumour cells induced by insulinare associated with differences in protein composition andaltered amounts of free, cytoskeletal-bound and membrane-bound polysomes Molecular and Cellular Biochemistry 116131–40

[25] Lasheras J C, Alonso-Latorre B, Bastounis E, Meili R,del Alamo J C and Firtel R A 2011 Distribution of tractionforces and intracellular markers associated with shapechanges during amoeboid cell migration I.J. Trans.Phenomena 12 3–12 (arXiv:1309.2686)

[26] Lecuit T and Lenne P F 2007 Cell surface mechanics and thecontrol of cell shape tissue patterns and morphogenesis Nat.Rev. Mol. Cell. Biol. 8 633–44

[27] Lee Y, Huang J, Wangc Y and Lin K 2013 Three-dimensionalfibroblast morphology on compliant substrates of controllednegative curvature Integr. Biol. 5 1447–55

[28] Lo C M, Wang H B, Dembo M and Wang Y L 2000 Cellmovement is guided by the rigidity of the substrate Biophys.J. 79 144–52

[29] Loth E 2008 Drag of non-spherical solid particles of regularand irregular shape Powder Technology 182 342–53

[30] Matsuda M, Sawa M, Edelhauser H F, Bartels S P,Neufeld A H and Kenyon K R 1985 Cellular migration andmorphology in corneal endothelial wound repair InvestOphthalmol Vis. Sci. 26 443–9 (PMID: 3980166)

[31] McBeath R, Pirone D M, Nelson C M, Bhadriraju K andChen C S 2004 Cell shape, cytoskeletal tension, and rhoaregulate stem cell lineage commitment Developmental Cell 6483–95

[32] Meili R, Alonso-Latorre B, del Alamo J C, Firtel R A andLasheras J C 2010 Myosin II is essential for thespatiotemporal organization of traction forces during cellmotility Mol. Biol. Cell 21 405–17

[33] Montell D J 2008 Morphogenetic cell movements: diversityfrom modular mechanical properties Science 322 1502–05

[34] Mousavi S J, Doblaré M and Doweidar M H 2014Computational modelling of multi-cell migration in a multi-signalling substrate J. Phys. Biol. 11 026002

[35] Mousavi S J, Doweidar M H and Doblaré M 2012Computational modelling and analysis of mechanicalconditions on cell locomotion and cell-cell interactionComput Methods Biomech Biomed Engin 17 678–93(PMID: 24010243)

[36] Mousavi S J, Doweidar M H and Doblaré M 2013 3Dcomputational modelling of cell migration: a mechano-

chemo-thermo-electrotaxis approach J. Theor. Biol. 32964–73

[37] Mousavi S J, Doweidar M H and Doblaré M 2013 Cellmigration and cell-cell interaction in the presence ofmechano-chemo-thermotaxis Mol Cell Biomech 10 1–25

[38] Neilson M P, Veltman1 D M, van Haastert P J M, Webb S D,Mackenzie J A and Insall R H 2011 Chemotaxis: afeedback-based computational model robustly predictsmultiple aspects of real cell behaviour PLoS Biol. 9e1000618

[39] Ni Y and Chiang M Y M 2007 Cell morphology and migrationlinked to substrate rigidity Soft Matter 3 1285–92

[40] Oster G F, Murray J D and Harris A K 1983 Mechanicalaspects of mesenchymal morphogenesis J. Embryol Exp.Morph 78 83–125 (PMID: 6663234)

[41] Paluch E and Heisenberg C P 2009 Biology and physics of cellshape changes in development Current Biology 19 R790–9

[42] Penelope C G and Janmey P A 2005 Cell type-specificresponse to growth on soft materials Journal AppliedPhysiology 98 1547–53

[43] Peyton S R, Kim P D, Ghajar C M, Seliktar D and Putnam A J2008 The effects of matrix stiffness and RhoA on thephenotypic plasticity of smooth muscle cells in a 3-Dbiosynthetic hydrogel system Biomaterials 29 2597–607

[44] Ponti A, Machacek M, Gupton S L, Waterman-Storer C M andDanuser G 2004 Two distinct actin networks drive theprotrusion of migrating cells Science 305 1782–6

[45] Ramtani S 2004 Mechanical modelling of cell/ECM and cell/cell interactions during the contraction of a fibroblast-populated collagen microsphere: theory and modelsimulation Journal of Biomechanics 37 1709–18

[46] Rauzi M and Lenne P F 2011 Cortical forces in cell shapechanges and tissue morphogenesis Curr Top Dev Biol 9593–144

[47] Satulovsky J, Lui R and Wang Y L 2008 Exploring the controlcircuit of cell migration by mathematical modeling BiophysJ 94 36713683

[48] Schäfer A and Radmacher M 2005 Influence of myosin IIactivity on stiffness of fibroblast cells Acta Biomaterialia 1273–80

[49] Sheetz M P, Felsenfeld D, Galbraith C G and Choquet D 1999Cell migration as a five-step cycle Biochem Soc Symp. 65233–43 (PMID: 10320942)

[50] Stokes G G 1951 On the effect of the inertial friction of fluidson the motion of pendulums Trans. Camb. Phil. Soc. 9 8–21

[51] Théry M and Bornens M 2006 Cell shape and cell divisionCurrent Opinion in Cell Biology 18 648–57

[52] Thery M, Racine V, Pepin A, Piel M, Chen Y, Sibarita J B andBornens M 2005 The extracellular matrix guides theorientation of the cell division axis Nat Cell Biol 7 947–53

[53] Tran-Cong S, Gay M and Michaelides E E 2004 Dragcoefficients of irregularly shaped particles PowderTechnology 139 21–32

[54] Trelstad R L and Coulombre A J 1971 Morphogenesis of thecollagenous stroma in the chick cornea J Cell Biol. 50840–58

[55] Ulrich T A, de Juan Pardo E M and Kumar S 2009 Themechanical rigidity of the extracellular matrix regulates thestructure, motility, and proliferation of glioma cells CancerResearch 69 4167–74

[56] Vermolen F J and Gefen A 2013 A phenomenological modelfor chemico-mechanically induced cell shape changes duringmigration and cell-cell contacts Biomech ModelMechanobiol. 12 301–23

[57] Wehrle-Haller B and Imhof B A 2003 Actin microtubules andfocal adhesion dynamics during cell migration TheInternational Journal of Biochemistry and Cell Biology 3539–50

12



102

[58] Wessels D, Titus M and Soll D R 1996 A dictyosteliummyosin I plays a crucial role in regulating the frequency ofpseudopods formed on the substratum Cell MotilCytoskeleton 33 64–79

[59] Wessels D, Voss E, von Bergen N, Burns R, Stites J andSoll D R 1998 A computer-assisted system forreconstructing and interpreting the dynamic three-dimensional relationships of the outer surface nucleus andpseudopods of crawling cells Cell Motil. Cytoskeleton 41225–46

[60] Yeung T, Georges P C, Flanagan L A, Marg B, Ortiz M,Funaki M, Zahir N, Ming W, Weaver V and Janme P A2005 Effects of substrate stiffness on cell morphologycytoskeletal structure, and adhesion Cell Motility and theCytoskeleton 60 24–34

[61] Zaman M H, Kamm R D, Matsudaira P and Lauffenburgery D A2005 Computational model for cell migration in three-dimensional matrices Biophys. J. 89 1389–97

[62] Zink D, Fischer A H and Nickerson J A 2004 Nuclear structurein cancer cells CANCER 4 677–87 PMID: 15343274

13


Publications

103


Plos One

2015

Accepted paper


2.6 Three-dimensional numerical model of cell morphology

during migration in multi-signaling substrates.

Three-Dimensional Numerical Model of Cell Morphologyduring Migration in Multi-Signaling Substrates

Seyed Jamaleddin Mousavi1,2,3, Mohamed Hamdy Doweidar1,2,3,*

1 Group of Structural Mechanics and Materials Modeling (GEMM), Aragon Instituteof Engineering Research (I3A), University of Zaragoza, Zaragoza, Spain.

2 Mechanical Engineering Department, School of Engineering and Architecture(EINA), University of Zaragoza, Zaragoza, Spain.

3 Centro de Investigacion Biomedica en Red en Bioingenierıa, Biomateriales yNanomedicina (CIBER-BBN), Zaragoza, Spain.

*Corresponding authorEmail: [email protected] (MHD)

Abstract

Cell Migration associated with cell shape changes are of central importance inmany biological processes ranging from morphogenesis to metastatic cancer cells. Cellmovement is a result of cyclic changes of cell morphology due to effective forces on cellbody, leading to periodic fluctuations of the cell length and cell membrane area. It iswell-known that the cell can be guided by different effective stimuli such asmechanotaxis, thermotaxis, chemotaxis and/or electrotaxis. Regulation of intracellularmechanics and cell’s physical interaction with its substrate rely on control of cell shapeduring cell migration. In this notion, it is essential to understand how each natural orexternal stimulus may affect the cell behavior. Therefore, a three-dimensional (3D)computational model is here developed to analyze a free mode of cell shape changesduring migration in a multi-signaling micro-environment. This model is based onprevious models that are presented by the same authors to study cell migration with aconstant spherical cell shape in a multi-signaling substrates and mechanotaxis effecton cell morphology. Using the finite element discrete methodology, the cell isrepresented by a group of finite elements. The cell motion is modeled by equilibriumof effective forces on cell body such as traction, protrusion, electrostatic and dragforces, where the cell traction force is a function of the cell internal deformations. Tostudy cell behavior in the presence of different stimuli, the model has been employedin different numerical cases. Our findings, which are qualitatively consistent withwell-known related experimental observations, indicate that adding a new stimulus tothe cell substrate pushes the cell to migrate more directionally in more elongated formtowards the more effective stimuli. For instance, the presence of thermotaxis,chemotaxis and electrotaxis can further move the cell centroid towards thecorresponding stimulus, respectively, diminishing the mechanotaxis effect. Besides, thestronger stimulus imposes a greater cell elongation and more cell membrane area. Thepresent model not only provides new insights into cell morphology in a multi-signaling

PLOS ONE 1/32

Publications

107

micro-environment but also enables us to investigate in more precise way the cellmigration in the presence of different stimuli.

Introduction

Cell shape change during cell migration is a key factor in many biological processessuch as embryonic development [1–3], wound healing [4–6] and cancer spread [7–9].For instance, during embryogenesis the head-to-tail body axis of vertebrates elongatesby convergent extension of tissues in which cells intercalate transversely between eachother to form narrower and long body [1]. Besides, after an injury in the cornea, thehealing process is followed by epithelial shape changes during cell migration.Epithelials near the wound bed change their shape to cover the defect without leavingintercellular gaps. The greatest cellular morphological alterations are observed aroundthe wound edges. Remote cells from wounded regions migrate towards the woundcenter and are elongated during migration in the migration direction, increasing theirmembrane area. As the healing proceeds, the cell original pattern is changed which isrecovered after wound healing [4]. Invasion of cancerous cells into surrounding tissueneeds their migration which is guided by protrusive activity of the cell membrane,their attachment to the extracellular matrix and alteration of their micro-environmentarchitecture [9]. Many attempts have been made to explain cell shape changesassociated with directed cell migration, but the mechanism behind it is still not wellunderstood. However, it is well-known that cell migration is fulfilled via successivechanges of the cell shape. It is incorporated by a cyclic progress during which a cellextends its leading edge, forms new adhesions at the front, contracts its cytoskeleton(CSK) and releases old adhesions at the rear [10,11]. A key factor of thedevelopmental cell morphology is the ability of a cell to respond to directional stimulidriving the cell body. Several factors are believed to control cell shape changes andcell migration including intrinsic cue such as mechanotaxis or extrinsic stimuli such aschemotaxis, thermotaxis and electrotaxis.

For the first time Lo et al. [12] demonstrated that cell movement can be guided bypurely physical interactions at the cell-substrate interface. After, investigations ofEhrbar et al. [13] illustrated that cell behavior strongly depends on its substratestiffness. During cell migration in consequence of mechanotaxis, amoeboid movementcauses frequent changes in cell shape due to the extension of protrusions in the cellfront [14,15], which is often termed pseudopods or lamellipods, and retraction of cellrear. Therefore, during this process, protrusions develop different cell shapes that arecrucial for determination of the polarization direction, trajectory, traction forces andcell speed.

In addition to mechanotaxis, gradient of chemical substance or temperature in thesubstrate gives rise to chemotactic [16, 17] or thermotactic [18, 19] cell shape changesduring migration, respectively. Existent chemical and thermal gradients in thesubstrate regulate the direction of pseudopods in such a way that the cell migrates inthe direction of the most effective cues [19, 20]. However, it is actually myosin-basedtraction force (a mechanotactic tool) that provides the force driving the cell bodyforward [12,21]. Recently, a majority of authors have experimentally considered cellmovement in the presence of chemotactic cue [17, 20] demonstrating that a shallowchemoattractant gradient guides the cell in the direction of imposed chemical gradientsuch that the extended pseudopods and cell elongation are turned in the direction ofthe gradient [20]. In contrast, some cells such as human trophoblasts subjected tooxygen and thermal gradients do not migrate in response to oxygen gradient (a

PLOS ONE 2/32

2.6. Three-dimensional numerical model of cell morphology during migration in multi-signaling substrates.

108

chemotactic cue) but they elongate and migrate in response to thermal gradients ofeven less than 1 ◦C towards the warmer locations [19]. However, there are some othercases such as burn traumas, influenza or some wild cell types that cell may migratetowards the lower temperature, away from warm regions [22].

Recent in vitro studies have demonstrated that the presence of endogenous orexogenous electrotaxis is another factor for controlling cell morphology and guidingcell migration [23–28]. Influence of endogenous Electric Fields (EFs) on cell responsewas first studied by Verworn [29]. Experimental evidences reveal important role ofendogenous electrotaxis in directing cell migration during wound healing processduring which the cell undergoes crucial shape changes [30,31]. In the past few years,there has also been a growing interest in the effects of an exogenous EF on cells inculture, postulating that calcium ion, Ca2+, is involved in electrotactic cellresponse [27,32–37]. A cell in natural state have negative potential that exposing it toan exogenous direct current EF (dcEF) causes extracellular Ca2+ influx intointracellular through calcium gates on the cell membrane. Subsequently, in steadystate, depending on intracellular content of Ca2+, a typical cell may be chargednegatively or positively [38]. This is the reason that many cells such as fish andhuman keratinocytes, human corneal epithelials and dictyostelium are attracted by thecathode [26,39–42] while some others migrate towards the anode, e.g. lens epithelialand vascular endothelial cells [39, 43]. Although, experiments of Grahn et al. [44]demonstrate that human dermal melanocyte is unexcitable by dcEFs, it may occurdue to its higher EF threshold [36].

To better understand how each natural biological cue or external stimulusinfluences the cell behavior, several kinds of mathematical and computational modelshave been developed [17,45–54]. Some of these models commonly simulate the effectof only one effective cue on cell migration [50,52,55] while some others at most dealwith mechanotactic and chemotactic cues, simultaneously [17,51]. There are severalenergy based mathematical models considering the effect of substrate rigidity on cellshape changes [52,56]. They assumed that the cell morphology is changed by theenergy stored in cell-substrate system, thus, minimization of the total free energy ofthe system defines the final cell configuration [52]. 2D model presented by Neilson etal. [51] simulates eukaryotic cell morphology during cell migration in presence ofchemotaxis by employing a system of non-linear reaction-diffusion equations. The cellboundary is characterized using an arbitrary Lagrangian-Eulerian surface finiteelement method. The main advantage of their model is prediction of the cell behaviorwith and without chemotactic effect although it has two key objections: (i) the cellmovement is totally random in absence of chemotactic stimulus, missingmechano-sensing process; (ii) the study of the cell configurations is limited to ellipticalmodes. In addition, numerical model presented by Han et al. [49] predicts thespatiotemporal dynamics of cell behavior in presence of mechanical and chemical cueson 2D substrates. Considering constant cell shape, they assume that the formation ofa new adhesion regulates the reactivation of the assembly of fiber stress within a celland defines the spatial distribution of traction forces. Their findings indicates that thestrain energy is produced by the traction forces which arise due to a cyclic relationshipbetween the formation of a new adhesion in the front and the release of old adhesionat the rear.

Altogether, although, available models provide significant insights about cellbehavior, they include several main drawbacks: (i) most of the present modelsincorporate signals received by the cell with mechanics of actin polymerization, myosincontraction and adhesion dynamics but do not deal with the traction forces exerted bythe cell during cell movement [57–60]; (ii) some of available models simply simulatecell migration with constant cell configuration [57,61]; (iii) models considering cell

PLOS ONE 3/32

Publications

109

morphology only concentrate on the dynamics of cellular shapes which are not easilyapplicable for temporal and spatial investigation of cell shape changes coupled withcell movement [52,62–65]; (iv) models predicting cell morphology are restricted to afew rigid cellular configurations [52, 62]; (v) some of existent models overlookmechanotactic process of cell migration [17,50, 51] which is inseparable fromcell-matrix interaction [12]. Apart from this shortages, most of the models dealingwith cell migration and cell shape changes are developed in 2D [17,52,55, 57–60] thataccording to the comprehensive experimental investigations of Hakkinen et al. [63], inmany concepts cell behavior, particularly as for cell morphology, on 2D substratesstrongly differs from that within 3D substrates. However in many viewpoints, 2Dmodels improve our notions on cell motility and cellular configuration. Above allshortcomings mentioned before, to our knowledge, there is no comprehensive model toinvestigate cell shapes changes during cell-matrix interactions within multi-signalingenvironments (mechano-chemo-thermo-electrotaxis).

We have previously developed a 3D numerical model of cell migration within a 3Dmulti-signaling matrix with constant cell configuration [66, 67]. In addition, a novelmechanotactic 3D model of cell morphology is recently presented by the sameauthors [68]. The objective of the present work is to extend previously presentedmodels [66–68] to investigate cell shape changes during cell migration in a 3Dmulti-signaling micro-environment. The model takes into account the fundamentalfeature of cell shape changes associated in cell migration in consequence of cell-matrixinteraction. It relies on equilibrium of forces acting on cell body which is able topredict key spatial and temporal features of cell such as cell shape changesaccompanied with migration, traction force exerted by the cell and cell velocity in thepresence of multiple stimuli. Some of the results match with findings of experimentalstudies while some others provide new insights for performing more efficientexperimental investigations.

Model description

Transmission of cell internal stresses to thesubstrate

Recent investigations have demonstrated that active (actin filaments and AMmachinery) and passive (microtubules and cell membrane) cellular elements play a keyrole in generating the cell contractile stress which is transmitted to the substratethrough integrins. The former, which generates active cell stress, basically depends onthe minimum, εmin, and maximum, εmax, internal strains, which is zero outside ofεmax-εmin range, while the latter, which generates passive cell stress, is directlyproportional to stiffness of passive cellular elements and internal strains. Therefore,the mean contractile stress arisen due to incorporation of the active and passivecellular elements can be presented by [66–69]

σ =

⎧⎪⎨⎪⎩

Kpasεcell εcell < εmin or εcell > εmaxKactσmax(εmin−εcell)Kactεmin−σmax

+Kpasεcell εmin ≤ εcell ≤ εKactσmax(εmax−εcell)Kactεmax−σmax

+Kpasεcell ε ≤ εcell ≤ εmax

(1)

where Kpas, Kact, εcell and σmax represent the stiffness of the passive and activecellular elements, the internal strain of the cell and the maximum contractile stress

PLOS ONE 4/32


110

exerted by the actin-myosin machinery, respectively, while ε = σmax/Kact.

Effective mechanical forces

A cell extends protrusions in leading edges in the direction of migration andadheres to its substrate pulling itself forward in direction of the most effective signal.The cell membrane area is as tiny as to produce strong traction force due to cellinternal stress, consequently, adhesion is thought to compensate this shortage byproviding the sufficient traction required for efficient cell translocation [3]. Theequilibrium of forces exerted on the cell body should be satisfied by cell migration andcell shape changes [70,71]. In the meantime, two main mechanical forces act on a cellbody: traction force and drag force. The former is exerted due to the contraction ofthe actin-myosin apparatus which is proportional to the stress transmitted by the cellto the ECM by means of integrins and adhesion. Representing the cell by a connectedgroup of finite elements, the nodal traction force exerted by the cell to the surroundingsubstrate at each finite element node of the cell membrane can be expressed as [69]

Ftraci = σiS(t)ζei (2)

where σi is the cell internal stress in ith node of the cell membrane and ei represents aunit vector passing from the ith node of the cell membrane towards the cell centroid.S(t) is the cell membrane area which varies with time. During cell migration, it isassumed that the cell volume is constant [72–74], however the cell shape and cellmembrane area change. ζ is the adhesivity which is a dimensionless parameterproportional to the binding constant of the cell integrins, k, the total number ofavailable receptors, nr, and the concentration of the ligands at the leading edge of thecell, ψ. Therefore, it can be defined as [66–68]

ζ = knrψ (3)

ζ depends on the cell type and can be different in the anterior and posterior parts ofthe cell. Its definition is given in the following sections. Thereby, the net traction forceaffecting on the whole cell because of cell-substrate interaction can be calculatedby [69]

Ftracnet = −

n∑i=1

Ftraci (4)

where n is the number of the cell membrane nodes. During migration, nodal tractionforces (contraction forces) exerted on cell membrane towards its centroid compressingthe cell. Consequently, each finite element node on the cell membrane, which has lessinternal deformation, will have a higher traction force [69]. On the contrary, the dragforce opposes the cell motion through the substrate that depends on the relativevelocity and the linear viscoelastic character of the cell substrate. At micro-scale theviscous resistance dominates the inertial resistance of a viscose fluid [75]. AssumingECM as a viscoelastic medium and considering negligible convection, Stokes’ dragforce around a sphere can be described as [76]

F sD = 6πrη(Esub)v (5)

where v is the relative velocity and r is the spherical object radius. η(Esub) is theeffective medium viscosity. Within a substrate with a linear stiffness gradient, we

PLOS ONE 5/32

Publications

111

assume that effective viscosity is linearly proportional to the medium stiffness, Esub,at each point. Therefore it can be calculated as

η(Esub) = ηmin + λEsub (6)

where λ is the proportionality coefficient and ηmin is the viscosity of the mediumcorresponding to minimum stiffness. Although, the viscosity coefficient may be finallysaturated with higher substrate stiffness, this saturation occurs outside the substratestiffness range that is proper for some cells [58].

Eq. 5 was developed by Stoke to calculate the drag force around a spherical shapeobject with radius r. This typical equation was employed in our previous works forcell migration with constant spherical shape [66,69]. In the present work, according toEqs. 17-19, an inaccurate calculation of the drag force may affect considerably thecalculation accuracy of the cell velocity and polarization direction. So that, accordingto [77,78], a shape factor is appreciated to moderate the Stokes’ drag expression to besuitable for irregular cell shape. The drag of irregular solid objects depends on thedegree of non-sphericity and their relative orientation to the flow. Therefore for anirregular object shape the drag is basically anisotropic compared to movementdirection. Since here the objective is to investigate cell migration while cell morphologychanges, calculation of the drag force using Eq. 5 will not be precise enough. Due tothe randomness of the cell shapes and dynamics, description of drag force for objectswith irregular shape is extremely complicated. It is thought that only probabilisticand approximate predictions can be reasonable and useful to describe drag force forhighly irregular particles [77, 78]. Therefore, referring to experimental observations, anappropriate shape factor, fshape, is appreciated to moderate the Stokes’ dragexpression for highly irregularly-shaped objects which is accurate enough [68,77,78]

Fdrag = fshapeFsD (7)

A wide variety of shape-characterizing parameters has been suggested for irregularparticles. Here we have employed Corey Shape Factor (CSF) which is the mostcommon and accurate shape factor. It appreciates three main lengths of an objectthat are mutually perpendicularly to each other as

fshape = (lmaxlmed

l2min

)0.09 (8)

where lmax, lmed and lmin are the cell’s longest, intermediate and the shortestdimensions, respectively, which are representative of cell surface area changes [77]. Inthe case of a spherical cell shape, this shape factor delivers 1. Although other shapefactors have been proposed to characterize the shape irregularity, using themax-med-min length factor leads to reliable results [77, 79].

Protrusion force

To migrate, cells extend local protrusions to probe their environment. This is theduty of protrusion force generated by actin polymerization which has a stochasticnature during cell migration [80]. It should be distinguished from the cytoskeletalcontractile force [68,75]. The order of the protrusion force magnitude is the same asthat of the traction force but with lower amplitude [69, 75,81–83]. Therefore, werandomly estimate it as


where erand is a random unit vector and F tracnet is the magnitude of the net traction

force while κ is a random number, such that 0 ≤ κ < 1, [66, 68].

PLOS ONE 6/32


112

Electrical force in presence of electrotacticcue

Exogenous EFs imposed to a cell have been proposed as a directional cue thatdirects the cells to migrate in cell therapy. Besides, studies in the last decade haveprovided convincing evidence that there is a role for EFs in wound healing [6].Significantly, this role is highlighted more than expected due to overriding other cuesin guiding cell migration during wound healing [6, 31]. Experimental worksdemonstrate that Ca2+ influx into cell plays a significant role in the electrotactic cellresponse [25,26,28]. Although this is still a controversial open question, Ca2+

dependence of electrotaxis has been observed in many cells such as neural crest cells,embryo mouse fibroblasts, fish and human keratocytes [23, 25,27,30,40]. On the otherhand, Ca2+ independent electrotaxis has been observed in mouse fibroblasts [32]. Theprecise mechanism behind intracellular Ca2+ influx during electrotaxis is notwell-known. A simple cell at resting state maintain a negative membrane potential [25]so that exposing it to a dcEF causes that the side of the plasma membrane near thecathode depolarizes while the the other side hyperpolarizes [23, 25,30]. For a cell withtrivial voltage-gated conductance, the membrane side which is hyperpolarized attractsCa2+ due to passive electrochemical diffusion. Therefore, this side of the cell contractsand propels the cell towards the cathode which causes to open the voltage-gated Ca2+

channels (VGCCs) near the cathode (depolarised) and allows intracellular Ca2+ influx(Fig. 1). So, on both anodal and cathodal sides of the cell, intracellular Ca2+ levelenhances. Balance between the opposing magnetic forces defines the resultantelectrical force affecting the cell body [25]. That is the reason that some cells tend toreorient towards the anode, like metastatic human breast cancer cells [84], humangranulocytes [85], while some others do towards the cathode, such as humankeratinocytes [26, 86], embryo fibroblasts [27], human retinal pigment epithelialcells [87] and fish epidermal cells [40].

A single cell embedded within a uniform EF will be ionized and charged. Thereforethe electrical force experienced by this individual cell can be obtained by

FEF = E Ω(E)S(t)eEF (10)

where E is uniform dcEF strength and Ω(E) stands for the surface charge density ofthe cell. eEF is a unit vector in the direction of the dcEF toward the cathode or anode,depending on the cell type. The time course of the translocation response duringexposing a cell to a dcEF demonstrates that the cell velocity versus translocationvaries depending on the dcEF strength. Experiments of Nishimura et al. [26] onhuman keratinocytes indicate that the net migration velocity raises by increase thedcEF strength to about 100 mV/mm while further increase the dcEF strength doesnot affect the cell net migration velocity. Since the Ca2+ influx into intracellular mayplay a role in this process [25, 26, 28,88–90], it is thought that the imposed dcEFregulates the concentration of intracellular Ca2+. Therefore, it can be deduced thatthe cell surface charge is directly proportional to the imposed dcEF strength [25, 26].Consequently, we assume a linear relationship between the cell surface charge and theapplied dcEF strength as

Ω(E) =

{Ωsatur

EsaturE E ≤ Esatur

Ωsatur E > Esatur(11)

where Ωsatur is the saturation value of the surface charge and Esatur is the maximumdcEF strength that causes Ca2+ influx into intracellular.

PLOS ONE 7/32

Publications

113

Fig 1. Response of a cell to a dcEFs. A simple cell in the resting state has anegative membrane potential [25]. When a cell with a negligible voltage-gatedconductance is exposed to a dcEF, it is hyperpolarised membrane near the anodeattracts Ca2+ due to passive electrochemical diffusion. Consequently, this side of thecell contracts, propelling the cell towards the cathode. Therefore, voltage-gated Ca2+

channels (VGCCs) near cathode (depolarised side) open and a Ca2+ influx occurs. Insuch a cell, intracellular Ca2+ level rises in both sides. The direction of cell movement,then, depends on the difference of the opposing magnetic contractile forces, which areexerted by cathode and anode [25].

Deformation and reorientation of the cell

Solid line in Fig. 2 shows a spherical cell configuration which is initially considered.It is assumed that the cell first exerts mechano-sensing forces on the membrane toprobe its surrounding micro-environment which is named mechano-sensing process.Thus, the cell internal strain at each finite element node of the cell membrane along eican be calculated by

εcell = ei : εi : eiT (12)

where εi is the strain tensor of ith node located on cell membrane due tomechano-sensing process.

A cell exerts contraction forces towards its centroid compressing itself so that thecell internal deformation, εcell, created by these forces on each finite element node ofthe cell membrane is negative. Hence, according to Eqs. 1 and 2 nodes with a lessinternal deformation experience a higher internal stress and traction force. Therefore,the net traction forces, Ftrac

net , points towards the direction of minimum cell internaldeformation (Eq. 4), presenting the mechanotaxis reorientation of the cell [69].Consequently, the unit vector of the mechanotactic reorientation of the cell, emech,reads

emech =Ftracnet

‖ Ftracnet ‖ (13)

In presence of thermotaxis or chemotaxis, the cell polarisation direction will becontrolled by all the existent stimuli. It is assumed that the presence of bothadditional cues does not affect either the physical or the mechanical properties of atypical cell, nor its surrounding ECM. Traction forces exerted by a typical cell dependon the mechanical apparatus of the cell and the mechanical properties of thesubstrate [21]. Therefore, the mechanotactic tool practically drives the cell bodyforward while the presence of chemotaxis and/or thermotaxis cues only changes thecell polarisation direction such that a part of the net traction force is guided bymechanotaxis and the rest is guided by these stimuli (Fig. 2). Consequently, underchemical and/or thermal gradients, the unit vectors associated to the chemotactic andthermotactic stimuli can be represented, respectively, as [66, 67]

ech =∇C

‖ ∇C ‖ (14)

eth =∇T

‖ ∇T ‖ (15)

where ∇ denotes the gradient operator while C and T represent the chemoattractantconcentration and the temperature, respectively. As mentioned above, the realignmentof the net traction force under these cues is affected by the direction of chemical and

PLOS ONE 8/32


114

Fig 2. Calculation of the cell reorientation. a- A initially spherical cell (solidline) is deformed (dashed line) during mechano-sensing process. emech is mechanotaxisreorientation of the cell. b- A cell is reoriented due to exposing to chemotaxis,thermotaxis and electrotaxis where ech, eth and eEF denote the unit vector in thedirection of each cue, respectively. The coefficients μmech, μch, and μth are effectivefactors of mechanotactic, chemotactic and thermotactic cues, respectively. F trac

net is themagnitude of the net traction force, Fprot is the random protrusion force, FEF

represents the electrical force that is exerted by dcEF and Fdrag stands for drag force.epol represents the net polarisation direction of a cell in a multi-signaling environment.

thermal gradients, so that the effective force, Feff, which incorporates mechanotactic,chemotactic and thermotactic effects can be defined as


where μmech, μch and μth are the effective factors of mechanotaxis, chemotaxis, andthermotaxis cues respectively, μmech + μch + μth = 1. It is assumed that there isneither degradation nor remodeling of the ECM during cell motility. Having in accountthat the inertial force is negligible, the cell motion equation delivers drag force as

Fdrag + Feff + Fprot + FEF = 0 (17)

Thereby, using Eq. 7, the instantaneous velocity of the cell is defined as

v =‖ Fdrag ‖

fshape 6πrη(Esub)(18)

with the net polarisation direction

epol = − Fdrag

‖ Fdrag ‖ (19)

Cell morphology and cell remodeling duringcell migration

Cell migration composed of several coordinated cyclic cellular processes. At thelight microscope level, many authors summarize this process into several steps such asleading-edge protrusion, formation of new adhesions near the front, contraction,releasing old adhesions and rear retraction [11,91]. At the trailing end the cortical

PLOS ONE 9/32

Publications

115

tension squeezes or presses the cytoplasm in the direction of migration while at theleading edge, the tension generated due to protrusions drives the cells forward [3, 92].

Guided by the aforementioned experimental observations, the regulatory processbehind the cell shape during cell migration is here simplified to analyze cell shapechanges coupled with the cell traction forces. Therefore, we model the dominantmodes of cell morphological changes considering the cell body retraction at the rearand extension at the front. Referring to Fig. 3, the initial domain of the cell, which islocated within the working space of Λ ⊂ R3 with the global coordinates of X, may bedescribed as

Ω′ = {x′(X′)|x′(X′) ∈ Λ : ∀‖x′‖ � r} (20)

where X′ denotes the local cell coordinates located in the cell centroid. Accordinglythe cell membrane can be represented by ∂Ω′. Thereby, the substrate domain can bedefined as

Ω = {x(X)|x(X) ∈ Λ, x(X) /∈ Ω′} (21)

During cell migration, both domains Ω′ and Ω vary such that Ω′ ∪ Ω = Λ andΩ′ ∩ Ω = ∅.

To correctly incorporate adhesivity, ζ, of cell in the cell front and rear, it isessential to define the cell anterior and posterior during cell motility. Assuming χ is aplane passing by the cell centroid, O, with unit normal vector n, parallel to epol, ands(X′) is a position vector of an arbitrary node located on ∂Ω′ (Fig. 3), projection of son n can be defined as

δ = n · s (22)

Consequently, nodes with positive δ are located on the cell membrane at the front,∂Ω′+, while nodes with negative δ belong to the cell membrane at the cell rear, ∂Ω′−,where ∂Ω′ = ∂Ω′+ ∪ ∂Ω′− should be satisfied.

We assume that the cell extends the protrusion from the membrane vertex whoseposition vector is approximately in the direction of cell polarisation, on the contrary, itretracts the trailing end from the membrane vertex whose position vector is totally inthe opposite direction of cell polarisation. Thus, the maximum value of δ delivers themembrane node located on ∂Ω′+ from which the cell must be extended while theminimum value of δ represents the membrane node located on ∂Ω′− from which thecell must be retracted. Assume eex ∈ Ω is the finite element that the membrane nodewith the maximum value of δ belongs to its space and ere ∈ Ω′ is the finite elementthat the membrane node with the minimum value of δ belongs to its space. Tointegrate cell shape changes and cell migration, simply, ere is moved from the Ω′

domain to the Ω domain, in contrast, eex is eliminated from the Ω domain and isincluded in the Ω′ domain [68].

In the present model the cell is not allowed to obtain infinitely thin shape duringmigration. Therefore, consistent with the experimental observation of Wessels etal. [93, 94], it is considered that the cell can extend approximately 10% of its wholevolume as pseudopodia.

Finite element implementation

The present model is implemented through the commercial finite element (FE)software ABAQUS [95] using a coupled user element subroutine. The correspondingalgorithm is presented in Fig. 4.

The model is applied in several numerical examples to investigate cell behavior inthe presence of different stimuli. It is assumed that the cell is located within a

PLOS ONE 10/32


116

Fig 3. Definition of extension and retraction points as well as anterior andposterior parts of the cell at each time step. Λ ⊂ R3, Ω and Ω′ represent the 3Dworking space, matrix and cell domains, respectively. X stands for the globalcoordinates and X′ represents the local cell coordinates located in the cell centroid, O.χ is a plane passing by the cell centroid with unit normal vector n parallel to the cellpolarisation direction, epol. P denotes a finite element node located on the cellmembrane, ∂Ω. ∂Ω′+ and ∂Ω′− are the finite element nodes located on the front andrear of the cell membrane, respectively.

Fig 4. Computational algorithm of migration and cell morphology changesin a multi-signaling environment.

400×200×200 μm matrix without any external forces. The matrix is meshed by128,000 regular hexahedral elements and 136,161 nodes while the cell is represented by643 elements. The calculation time is about one minute for each time step in whicheach step corresponds to approximately 10 minutes of real cell-matrix interaction [68].

PLOS ONE 11/32

Publications

117

Initially the cell is assumed to have a spherical shape as shown in Fig. 2-a. In table 1,the properties of the matrix and the cell are enumerated.

For each simulation it is of interest to quantify the cell shape. Therefore, twoparameters are calculated to quantify the cell shape changes during cell migration in3D multi-signaling matrix: Cell Morphological Index (CMI)

CMI(t) =S(t)

Sin(23)

where Sin denotes the initial area of cell membrane (spherical cell shape); and the cellelongation

εelong = 1−√

lminlmed

lmax(24)

Here the second term of the equation represents the ratio of the geometric meanover the cell length. εelong is a representative value of cell elongation. It is calculatedto evaluate a spherical cell shape versus an elongated cell configuration according tothe experimental work of Lee et al [96]. According to Eq. 24, εelong=0 for a sphericalcell configuration, in contrast for a highly elongated cell, εelong �1. This means thatthe cell length in one direction is much higher than that of other two mutualperpendicular directions. On the other hand, CMI is another parameter to show howthe cell surface area changes during cell migration. In our cases study, we assume thatthe cell initially has a spherical shape (CMI=1). This value goes to increase while cellmigrates. Therefore, although there is no direct relation between εelong and CMI, theymay follow the same trend during cell migration. So, both parameters are minimumfor a spherical cell shape and maximum for an elongated cell shape. These variablesare probed versus cell position (the cell centroid translocation) in each step to see howthe cell elongation and surface area change during cell migration in presence ofdifferent stimuli.

In addition, the cellular random alignment in a 3D matrix with a cue gradient(stiffness, thermal and/or chemical gradients) or dcEF can be assessed by the anglebetween the net polarisation direction of the cell and the imposed gradient direction orEF direction, θ. Therefore, the Random Index (RI) can be described by

RI =

N∑i=1

cosθi

N(25)

where N represents the number of time steps during which the cell elongation does notchange considerably (the cell reaches steady state). RI=-1 indicates totally randomalignment of the cell while RI=+1 represents perfect alignment of the cell in directionof the cue gradient or EF direction. Consequently, in the presence of a cue gradient ordcEF, the closer RI to +1, the lower the cell random orientation.

Numerical examples and results

During cell migration, amoeboid mode of cells causes frequent changes in cell shapeas a result of the extension and retraction of protrusions [20]. To consider this, fourdifferent categories of numerical examples have been represented to consider cellbehavior in presence of different stimuli. All the stimuli such as thermotaxis,chemotaxis and electrotaxis are considered within the matrix with a linear stiffnessgradient and free boundary surfaces. It is assumed that, initially, the cell has aspherical configuration. Each simulation has been repeated at least 10 times toevaluate the results consistency.

PLOS ONE 12/32


118

Table 1. 3D matrix and cell properties.

Symbol Description Value Ref.ν Poisson ratio 0.3 [97,98]μ Viscosity 1000 Pa·s [75, 97]r Cell radius 20 μm [99]Kpas Stiffness of microtubules 2.8 kPa [100]Kact Stiffness of myosin II 2 kPa [100]εmax Maximum strain of the cell 0.09 [69,83]εmin Minimum strain of the cell -0.09 [69,83]σmax Maximum contractile stress exerted by

actin-myosin machinery0.1 kPa [101,102]

kf = kb Binding constant at the rear and at thefront of the cell

108 mol−1 [75]

nf = nb Number of available receptors at therear and at the front of the cell

105 [75]

ψ Concentration of the ligands at the rearand at the front of the cell

10−5 mol [75]

Ω Order of surface charge density of thecell

10−4 C/m2 [24]

E Range of applied electric field 0-100 mV/mm [25,30]

Cell behavior in a 3D matrix with a puremechanotaxis

Experimental investigations demonstrate that cells located within 3D matrixactively migrate in direction of stiffness gradient towards stiffer regions [103]. Inaddition, it has been observed that during cell migration towards stiffer regions, thecell elongates and subsequently the cell membrane area increases [13, 96].

To consider the effect of mechanotaxis on cell behavior, it is assumed that there isa linear stiffness gradient in x direction which changes from 1 kPa at x=0 to 100 kPaat x = 400 μm. The cell is initially located at a corner of the matrix near theboundary surface with lowest stiffness. Fig. 5 and Fig. 6 show the cell configurationand the trajectory tracked by the cell centroid within a matrix with stiffness gradient,respectively. As expected, independent from the initial position of the cell, when thecell is placed within a substrate with pure stiffness gradient it tends to migrate indirection of the stiffness gradient towards the stiffer region and it becomes graduallyelongated. The cell experiences a maximum elongation in the intermediate region ofthe substrate since it is far from unconstrained boundary surface which is discussed inthe previously presented work [66]. As the cell approaches the end of the substrate thecell elongation and CMI decrease (see Fig. 7). Despite the boundary surface atx = 400 μm has maximum elastic modulus, due to unconstrained boundary, the celldoes not tend to move towards it and maintains at a certain distance from it. The cellmay extend random protrusions to the end of the substrate but it retracts again andmaintains its centroid around an imaginary equilibrium plane (IEP) located far fromthe end of the substrate at x = 351± 5μm (see Fig. 8) [69]. Therefore, the cell neverspread on the surface with the maximum stiffness. It is worth noting that thedeviation of the obtained IEP coordinates is due to the stochastic nature of cellmigration (random protrusion force). Fig. 8 represents cell RI for the imposedstiffness gradient slope. The simulation was repeated for several initial positions of the

PLOS ONE 13/32

Publications

119

cell and several values of the gradient slope, all the obtained results were consistent.However, change in the gradient slope can change the cell random movement andslightly displace the IEP position (results of different gradient slopes are not shownhere). Cell behavior within the substrate with stiffness gradient is in agreement withexperimental observations [13,96, 103] and the results of the previous works presentedby the same authors in which a constant spherical configuration has been consideredfor the cell [67, 69]. It is worth mentioning that the net cell traction force and velocitycurves are not presented here since they roughly follow the same trend as the previouswork [67].

Fig 5. Shape changes during cell migration within a substrate with a linearstiffness gradient. The substrate stiffness changes linearly in x direction from 1 kPaat x=0 to 100 kPa at x=400 μm. At the beginning the cell is located at the corner ofthe substrate near the soft region. The results demonstrate that the cell migrates inthe direction of stiffness gradient and the cell centroid finally moves around an IEPlocated at x = 351± 5μm. a- The cell at the middle of the substrate, b- the cell finalposition (see also S1 Video).

Fig 6. Trajectory of the cell centroid within a substrate with stiffnessgradient in presence of different stimuli. Examples are run 10 times in order tocheck consistency of the results. The slop of the cell centroid trajectory reflects theattractivity of every cue to the cell.

PLOS ONE 14/32


120

Fig 7. Cell elongation, εelong (left axis), and CMI (right axis) versus thecell centroid translocation within a substrate with a pure stiffness gradient.As the cell approaches the intermediate regions of the substrate (rigid regions) boththe εelong and CMI increase. On the contrary, they decrease near the surface withmaximum stiffness because the cell retracts protrusions due to unconstrainedboundary surface.

Fig 8. Mean RI (left axis) and IEP position (right axis) of cell in thepresence of different cues. The error bars represent mean standard deviationamong different runs. Adding a new stimulus to the substrate with stiffness gradientdecreases the cell random alignment (increases mean RI) and moves the cell towardsthe end of the substrate.

Cell behavior in presence of thermotaxis

Several experimental studies [18, 19] have demonstrated that, in vivo, different celltypes are affected by thermal gradient. Here, employing the present model, weinvestigate that how the cell can sense and respond to the presence of thermal gradientin its substrate. To do so, a thermal gradient is added to the aforementioned substratewith stiffness gradient. It is assumed that the temperature at x=0 is equal to 36 ◦Cand at x=400 μm is 39 ◦C [19], while μth=0.2. This creates a linear thermal gradientthroughout the substrate along x axis. At the beginning, the cell is located at one ofthe corners of the substrate near the boundary surface with minimum temperature.

PLOS ONE 15/32

Publications

121

The results indicate that the cell gradually elongates and migrates towards warmerzone in direction of the thermal gradient by means of thermotaxis (Fig. 9). Fig. 6demonstrates the trajectory that is tracked by the cell centroid. In this case also thereis an IEP located at x = 359± 3μm (Fig. 8) that the cell centroid finally move aroundit. Comparing the trajectory of the cell centroid in the presence of thermotaxis withthat of pure mechanotaxis indicates that the cell centroid slightly moves towards theend of the substrate with greater temperature. Once the cell achieves IEP, it extendsprotrusions randomly in different directions maintaining the position of the cellcentroid near the IEP. These findings are independent from the initial cell positionand are consistent with experimental findings of Higazi et al. [19] who demonstratedthat trophoblasts migrate towards warmer locations due to thermal gradient.Comparing RIs of mechanotaxis and thermotaxis cases in Fig. 8 illustrates thatadding thermotaxis cue to the substrate with stiffness gradient causes decrease in cellrandom motility (increase in RI). Because mechanical and thermal gradients, whichare in the same directions, contribute with each other to more directionally guide thecell. Both the cell elongation and the CMI follow the same trend as mechanotaxisexample but in average there is an increase in their amount, which means thecontribution of mechanotaxis and thermotaxis increases the cell elongation and theCMI (Fig. 10). The thermal gradient imposed here may be considered as themaximum biological gradient, which is applicable in cell environment. We haverepeated the simulation for mild thermal gradients but there is no considerabledeviation in results (results not shown). Therefore, it can be deduced that thevariation of gradient slope in thermotaxis do not dramatically affect the final resultsbecause in biological ranges, sharp thermal gradients are not applicable. However, it isnoticeable that the cell does not exhibit significant thermotactic response to a verymild thermal gradients (when difference between maximum and minimumtemperatures is less than 0.2 ◦C in the substrate).

Cell behavior in the presence of chemotaxis

Many experimental investigations have demonstrated that the cell has a directionalmigratory capability in presence of a shallow chemoattractant gradient within 3Dsurrounding substrates [16, 104]. In vitro, observations indicate that cells include astrong basal pseudopod cycle by which pseudopod extension occurs along chemicalgradient at the close side of the cell to the higher chemical concentration [20]. Thismeans that the cell elongates its body in direction of chemical gradient towards thehigher concentration of chemoattractant substance.

Here, to consider effect of chemotaxis on cell behavior, a chemical gradient is addedinto the same substrate with stiffness gradient. It is assumed that a chemoattractantsubstance with concentration of 5×10−5 M exists at x =400 μm while chemoattractantconcentration at x =0 μm is null. This creates a linear chemical gradient along the xaxis. The evolution of shape changes during cell migration in the presence ofchemotaxis is presented in Fig. 11 for two different chemotaxis effective factors,μch=0.35 and μch=0.4. In Fig. 6, the trajectory, which is tracked by the cell centroid,is compared with that of the previous experiments. It implies that the cell centroidultimately moves around an IEP located at x = 368± 3μm and x = 374± 4μm forμch=0.35 and μch=0.4, respectively, (Fig. 8). Therefore, it can be deduced thatadding a chemotactic stimulus to the substrate moves the final position of the cellcentroid towards the chemoattractant source, of course depending on the employedchemotactic effective factor. Similar behavior of cell motility has been observed in thepreviously presented work by the same authors in which the cell has been represented

PLOS ONE 16/32


122

Fig 9. Shape changes during cell migration within a substrate withconjugate linear stiffness and thermal gradients. It is assumed that there is alinear thermal gradient in x direction (as stiffness gradient) which changes from 36 ◦Cat x=0 to 39 ◦C at x=400 μm. At the beginning the cell is located at a corner of thesubstrate near the surface with lower temperature. The results demonstrate that thecell migrates along the thermal gradient towards warmer region. Finally, the cellcentroid moves around an IEP located at x = 359± 3μm. When the cell centroid isnear the IEP the cell may send out and retract protrusions but it maintains theposition around IEP. a- The cell at the middle of the substrate, b- the cell finalposition (see also S2 Video).

Fig 10. Cell elongation, εelong (left axis), and CMI (right axis) versus thecell centroid translocation in the presence of thermotaxis. The cell elongationand CMI are maximum in the intermediate regions of the substrate and decreases asthe cell approaches the unconstrained surface with higher temperature.

by a constant spherical shape [67]. In both cases, when the cell is near to thechemoattractant source, it may extend or retract protrusions in random directions, nocell tendency to leave the IEP. It is clear from Fig. 12 that for both cases the cellfollows the same trend as that of the previous examples in terms of the cell elongationand CMI. However, here, the peak of the cell elongation and CMI slightly increases incomparison with mechanotaxis and/or thermotaxis. In the presence of chemotaxis, thecell tends to spread on the surface on which chemoattractant source is located. It

PLOS ONE 17/32

Publications

123

causes cell elongation and CMI increase in perpendicular direction to the imposedchemical gradient, which is considerable in case of greater chemotaxis effective factor(see Figs. 11-d and 12-a). Because of the higher chemotaxis effective factor, the cellreceives stronger chemotactic signal to spread more on the surface withchemoattractant source. Besides, the cell random movement relatively decreases forboth cases in comparison with either mechanotaxis or thermotaxis example (Fig. 8).

Cell migration towards chemoattractant source is qualitatively consistent withmany experimental [20, 105,106] and numerical [17, 51,107] studies. Besides, cellelongation and shape change during migration is consistent with finding of Maeda etal. [108] implying that gradient sensing and polarization direction of the cell are linkedto the cell shape changes and accompanied with motility length of pseudopods.

Cell behavior in presence of electrotaxis

As mentioned above, endogenous EF is developed around wounds during tissuesinjury, causing cell migration towards wound cites. Experiments show that in a Guineapig skin injury just 3 mm away from wound, lateral potential drops to 0 from 140mV/mm at the wound edge [6, 109–111]. Besides, in cornea ulcer, an EF equal to 42mV/mm is measured [6, 112]. The cell movement can be also directed and acceleratedvia exposing it to an exogenous dcEF depending on cell phenotype. In this process,both calcium ion release from and influx into intracellular are generally associatedwith cell polarisation direction. For instance, human granulocytes [85], rabbit cornealendothelial cells [113], metastatic human breast cancer cells [84] are attracted byanode. Unlike metastatic rat prostate cancer cells [114], embryo fibroblasts [27],human keratinocytes [86], fish epidermal cells [40], human retinal pigment epithelialcells [87], epidermal and human skin cells [30] that move towards cathode. Therefore,altogether, different cell phenotypes may present different electrotactic behavior.

To consider the influence of the electrotaxis on cell behavior, it is considered thatthe cell is exposed to a dcEF through which the anode is located at x=0 μm and thecathode at x=400 μm. It is assumed that the cell phenotype is such that to beattracted by the cathode, such as human keratinocytes [86] or embryo fibroblasts [27].First, the cell is located near the anode at x=0. To demonstrate effect of dcEFstrength on cell behavior the simulation is repeated for two different dcEF strength,E=10 mV/mm and E=10 100 mV/mm. Cell migration and shape change in thepresence of both weak and strong EF are presented in Fig. 13. In response of an EF,the cell re-organizes its side that is facing the cathode, and migrates directionallytowards the cathode. The presence of the EF can dominate mechanotaxis effect andmove the cell to the end of the substrate even more than previous cases where the cellcentroid locates around IEP at x = 379± 3μm and x = 383± 2μm for the weak andstrong EF strengths, respectively, (Fig. 6 and Fig. 8). Besides, the presence of the EFdecreases considerably the random movement of the cell (see Fig. 8). Near thecathode pole in the presence of weak EF the cell may extend many protrusions indifferent directions but the change of the cell centroid position is trivial. This is notthe case in presence of strong EF, the position of the cell centroid remains constantdue to the domination EF role. The cell is even unable to send out any protrusion.This takes place because the strong EF provides a dominant directional signal to guidethe migrating cell towards the cathode, dominating the effect of other forces. This isconsistent with previous work presented by the same authors assuming constantspherical cell shape [67] where the cell became immobile when it reaches the cathodein the presence of stronger EF strength. EF induces morphological change in themigrating cell where for both cases the average cell elongation and CMI are higher

PLOS ONE 18/32


124

Fig 11. Shape changes during cell migration in presence of chemotaxiswithin a substrate with stiffness gradient. It is assumed that there is achemoattractant substance with concentration of 5×10−5 M at x =400 μm, whichcreates a linear chemical gradient across x direction. At the beginning the cell islocated at one of the corners of the substrate near the surface of null chemoattractantsubstance. Two chemotaxis effective factors are considered; μch=0.35 (a and b) andμch=0.4 (c and d). The results demonstrate that, for both cases, the cell migratesalong the chemical gradient towards the higher chemoattractant concentration.Depending on chemical effective factor, the ultimate position of the cell centroid willbe different, for μch=0.35 the cell centroid keeps moving around an IEP located atx = 368± 3μm (b) while for higher chemical effective factor, μch=0.4, the position ofthe IEP moves towards chemoattractant source to locate at x = 374± 4μm (d). It isremarkable that in both cases the IEP displaces further towards the end of substratein comparison with thermotaxis case (see also S3 and S4 Videos for low and highchemical effective factors, respectively).

than those of all the previous cases (Fig. 14). In presence of electrotaxis the cellachieves the maximum elongation sooner than the other cases and it maintains the

PLOS ONE 19/32

Publications

125

Fig 12. Cell elongation, εelong (left axis), and CMI (right axis) versus thecell centroid translocation in the presence of chemotaxis as well asmechanotaxis. a- μch=0.35 and b- μch=0.40. For both cases, the cell elongation andCMI are maximum in the intermediate regions of the substrate and decreases as thecell approaches the unconstrained surface with chemoattractant source. Because, whenthe cell reaches the surface with maximum chemoattractant concentration, it tends toadhere to and spread over that surface. However, in the case of chemotaxis cue withhigher effective factor the cell again elongates in perpendicular direction to theimposed chemical gradient.

maximum amounts until it reaches the end of substrate. Therefore, a flat region canbe seen in the fitted elongation and CMI curves (Fig. 14). For both cases, near thecathode, the cell elongation and CMI decrease, because in the presence of dcEF thecell tends to spread on the surface where the cathodal pole is located. However, incase of strong EF the cell elongation and CMI again increases because the electricalforce acting on the cell body is strong enough to cause the cell elongationperpendicularly to dcEF direction, leading increase in the cell elongation and CMI. Itis noteworthy mentioning that for both cases the ultimate cell elongation and CMI aregreater that all previous studied cases.

PLOS ONE 20/32


126

Fig 13. Shape changes during cell migration in presence of electrotaxiswithin a substrate with stiffness gradient. A cell is exposed to a dcEF where theanode is located at x=0 and the cathode at x=400 μm. It is supposed that the cell isattracted to the cathode pole. At the beginning, the cell is placed in one of the cornersof the substrate near the anode and far from the cathode pole. Two EF strength areconsidered; E=10 mV/mm (a and b) and E=100 mV/mm (c and d). For both cases,the cell migrates along the dcEF towards the surface in which the cathode pole islocated. Depending on EF strength, the ultimate location of the cell centroid will bedifferent so that for E=10 mV/mm the cell centroid keeps moving around an IEPlocated at x = 379± 3μm (b) while for saturation EF strength, E=100 mV/mm, theposition of the IEP moves further to the cathode pole to locate at x = 383± 2μm (d).In the case of saturation EF strength (E=100 mV/mm) the cell perfectly elongates onthe surface of cathode pole without extending any protrusion (see also S5 and S6Videos for low and high EF strengths, respectively).

Cell shape change in Multi-signalling substrate

Finally, to simultaneously evaluate the effect of different stimuli on cell shapechange during cell migration, we have designed 30 different cases through which

PLOS ONE 21/32

Publications

127

Fig 14. Cell elongation, εelong (left axis), and CMI (right axis) versus thecell centroid translocation in the presence of electotaxis as well asmechanotaxis. a- E=10 mV/mm and b- E=100 mV/mm. The cell elongation andCMI reaches a maximum amount sooner than previous cases and are aproximatelyconstant until the cell reaches the cathode pole. The cell elongation and CMI decreasewhen the cell reaches the surface on which the cathode pole is located but they neverdiminish less than those of other stimuli. However, in the case of higher EF strengththe cell elongation and CMI again increase. The cell elongation and CMI aremaximum in this case compared to the other previous cases.

different thermotaxis and chemotaxis effective factors as well as different EF strengthsare applied. The maximum cell elongation, εelong, and CMI versus the combination ofstimuli, which occur in the intermediate area of the substrate, are summarized in Figs.15 and 16, respectively. Our findings indicate that the increase of each stimulus effectincreases both the cell elongation and CMI. Obviously, Figs. 15 and 16 illustrate thatthe rate of changes in the cell elongation and CMI is greater in the direction of theelectrotactic axis (E.Ω) than that of other cues (μch+μthμmech

), indicating dominant role ofelectrotaxis. Moreover, increasing the EF strength more than the saturation valuedoes not remarkably affect the cell elongation and CMI. It should be mentioned that,generally, the greater the cell elongation and CMI the less cell random movement. Thedominant role of the electrotaxis on cell directional movement is already discussed inthe previous work in which a constant spherical cell shape was considered [67].

PLOS ONE 22/32


128

Fig 15. Variation of the maximum cell elongation, εelong, versusthermotaxis, chemotaxis and electrotaxis stimuli.

Fig 16. Variation of the maximum CMI versus thermotaxis, chemotaxisand electrotaxis stimuli.

Conclusions

In this study, our objective is to qualitatively characterize cell shape changescorrelated with cell migration in the presence of multiple signals. Therefore, previouslydeveloped models of cell migration with constant spherical cell shape [67, 69] andmechanotactic effect on cell morphology [68] are here extended. The present 3D modelis developed base on force equilibrium on cell body using finite element discretemethodology. This model allows predicting the cell behavior when it is surrounded bydifferent micro-environmental cues. The results obtained here are qualitativelyconsistent with those of corresponding experimental works reported in theliterature [13,19,20,26,96,106].

In absence of external stimuli, the cell elongates along the stiffness gradient andmigrates towards the surface of maximum stiffness. Although the cell may randomlyextend different pseudopods, it retracts those pseudopods in subsequent steps andmaintains its body in determinated distance from the surface of maximum elasticmodulus, due to its unconstrained state. This is observed in the previous works of cellmigration with a constant spherical shape as well [67,69]. This causes a decrease in thecell elongation and CMI once the cell centroid is around IEP. The overall cell behaviorand cell shape may be changed by activation of other signals in the cell environment.For instance, by adding chemotaxis and/or thermotaxis to the micro-environment, themaximum cell elongation and CMI increase and the location of the cell centroid movestowards the end of the substrate despite of the unconstrained boundary surface. Asthe cell migrate along chemical gradient, the cell elongates in gradient direction but

PLOS ONE 23/32

Publications

129

when it is near the end of the substrate, the cell elongation and CMI decrease. Oncethe cell reaches the surface of maximum chemoattractant concentration, it extendspseudopods in the vertical direction of chemical gradient. Afterward, because the cellextends pseudopods in the vertical direction of chemical gradient, the cell elongationand CMI slightly increases, which is more obvious for greater chemical effective factor(Fig. 12). The ultimate location of the cell centroid is sensitive to the chemotacticeffective factors whereas employing of a higher chemoattractant effective factor causesthat the cell centroid moves further to the end of the substrate. In other words, agreater chemoattractant effective factor dominates mechanotaxis signal and drives thecell towards the chemoattractant source. The cell movement to the end of thesubstrate is more critical in presence of electrotaxis. Since our study focuses on atypical cell migrating towards the cathode, EF significantly reorientates the celltowards the cathodal pole. This reorientation can be even considerably affected byincrease of EF strength, in agreement with experimental observations [26].

So, generally, the stronger signal imposes a greater cell elongation and CMI that isbecause of directional cell polarisation towards the more effective stimulus. Becauseadding any new stimulus to the cell substrate will affect the cell polarization directionby increase of directional motility of the cell so that all signals directionally guide thecell towards the source of stimuli (warmer position, chemoattractant source, cathodalpole), diminishing the cell random movement (see Fig. 8). In particular, in presence ofthe saturated EF there is a considerable increase in cell elongation and CMI due toexposing the cell to a greater electrostatic force. As a general remark, consistent withexperimental observations, our findings indicate that electrotaxis effect is a dominantcue (see Figs. 15 and 16). Because, for both the thermotactic and chemotactic signals,the variation of μth and μch parameters has trivial effect on the magnitude of effectiveforce (Eq. 16), however it may considerably change the cell polarisation direction [67].Therefore, changes of thermotaxis and chemotaxis slightly affect the magnitude ofdrag force in contrast to electrotaxis, which is an independent force from others, itsmagnitude can be directly controlled by the EF strength. Consequently, according toEq. 17 electrotaxis can affect both magnitude and direction of drag force. Takingtogether, this can clearly justify how electrotaxis is the most effective guidingmechanism of the cell elongation, CMI and the cell RI, which dominates othereffective cues during cell motility, reported in many experimental works [6, 38, 110].

In summary, this study characterizes, for the first time, cell shape changeaccompanied with the cell migration change within 3D multi-signaling environments.We believe that it provides one step forward in computational methodology tosimultaneously consider different features of cell behavior which are a concern invarious biological processes. Although more sophisticated experimental works arerequired to calibrate quantitatively the present model, general aspects of the resultsdiscussed here are qualitatively consistent with documented experimental findings.

Acknowledgments

The authors gratefully acknowledge the support from the Spanish Ministry ofEconomy and Competitiveness and the CIBER-BBN initiative.

PLOS ONE 24/32


130

References

1. Keller R. Shaping the vertebrate body plan by polarized embryonic cellmovements. Science. 2002;298(5600):1950–4.

2. Kurosaka S, Kashina A. Cell Biology of Embryonic Migration. Birth DefectsRes C Embryo Today. 2008;84(2):102–22.

3. Paluch E, Heisenberg CP. Biology and Physics of Cell Shape Changes inDevelopment. Current Biology. 2009;19:R790–9.

4. Matsuda M, Sawa M, Edelhauser HF, Bartels SP, Neufeld AH, Kenyon KR.Cellular migration and morphology in corneal endothelial wound repair. InvestOphthalmol Vis Sci. 1985;26(4):443–9.

5. Schneider L, Cammer M, Lehman J, Nielsen SK, Guerra CF, Veland IR, et al.Directional Cell Migration and Chemotaxis in Wound Healing Response toPDGF-AA are Coordinated by the Primary Cilium in Fibroblasts. Cell PhysiolBiochem. 2010;25:279–292.

6. Zhao M. Electrical fields in wound healing-An overriding signal that directscell migration. Seminars in Cell and Developmental Biology. 2009;20:674–82.

7. Ramis-Conde I, Drasdo D, Anderson ARA, Chaplain MAJ. Modelingtheinfluence of the E-cadherin–b-catenin pathway in cancer cell invasion: amultiscale approach. Biophysical Journal. 2008;95:155–165.

8. Suresh S. Biomechanics and biophysics of cancer cells. Acta Biomaterialia.2007;3:413–438.

9. Tanos B, Rodriguez-Boulan E. The epithelial polarity program: Machineriesinvolved and their hijacking by cancer. Oncogene. 2008;27:6939–6957.

10. Lauffenburger DA, Horwitz AF. Cell migration: a physically integratedmolecular process. Cell. 1996;84:359–369.

11. Sheetz MP, Felsenfeld D, Galbraith CG, Choquet D. Cell migration as afive-step cycle. Biochem Soc Symp. 1999;65:233–43.

12. Lo CM, Wang HB, Dembo M, Wang YL. Cell Movement Is Guided by theRigidity of the Substrate. Biophysical Journal. 2000;79:144–152.

13. Ehrbar M, sala A, Lienemann P, Ranga A, Mosiewicz K, Bittermann A, et al.Elucidating the role of matrix stiffness in 3D cell migration and remodeling.Biophysical Journal. 2011;100:284–293.

14. Friedl P, Wolf K. Plasticity of cell migration: a multiscale tuning model. J CellBiol. 2009;188:11–9.

15. Lammermann T, Sixt M. Mechanical modes of amoeboid cell migration. CurrOpin Cell Biol. 2009;21:636–44.

16. Hoeller O, Kay RR. Chemotaxis in the absence of PIP3 gradients. CURRENTBIOLOGY. 2007;17:813–817.

17. Neilson MP, Veltman1 DM, van Haastert PJM, Webb SD, Mackenzie JA,Insall RH. Chemotaxis: A Feedback-Based Computational Model RobustlyPredicts Multiple Aspects of Real Cell Behaviour. PLoS Biol. 2011;9:e1000618.

PLOS ONE 25/32

Publications

131

18. Bahat A, Eisenbach M. Sperm thermotaxis. Molecular and CellularEndocrinology. 2006;252:115–119.

19. Higazi AA, Kniss D, Barnathan JMES, Cines DB. Thermotaxis of humantrophoblastic cells. Placenta. 1996;17:683–7.

20. Haastert PJMV. Chemotaxis: insights from the extending pseudopod. J CellSci. 2010;123:3031–37.

21. Giannone G, Sheetz MP. Substrate rigidity and force define form throughtyrosine phosphatase and kinase pathways. Trends Cell Biol.2006;16(4):213–23.

22. Tanizawa Y, Kuhara A, Inada H, Kodama E, Mizuno T, Mori I. Inositolmonophosphatase regulates localization of synaptic components and behaviorin the mature nervous system of C. elegans. Genes Dev. 2006;20(23):3296–310.

23. Gao RC, Zhang XD, Sun YH, Kamimura Y, Mogilner A, Devreotes PN, et al.Different Roles of Membrane Potentials in Electrotaxis and Chemotaxis ofDictyostelium Cells. EUKARYOTIC CELL. 2011;10:doi:10.1128/EC.05066–11.

24. Huang YX, Zheng XJ, Kang LL, Chen XY, Liu WJ, B T Huang ZJW.Quantum dots as a sensor for quantitative visualization of surface charges onsingle living cells with nano-scale resolution. Biosensors and Bioelectronics.2011;26:2114–2118.

25. Mycielska ME, Djamgoz MB. Cellular mechanisms of directcurrent electricfield effects: galvanotaxis and metastatic disease. J Cell Sci.2004;117:1631–1639.

26. Nishimura KY, Isseroff RR, Nuccitelli R. Human keratinocytes migrate to thenegative pole in direct current electric fields comparable to those measured inmammalian wounds. JCell Sci. 1996;109:199–207.

27. Onuma EK, Hui SW. A calcium requirement for electric field-induced cellshape changes and preferential orientation. Cell Calcium. 1985;6:281–292.

28. Shanley LJ, Walczysko P, Bain M, MacEwan DJ, Zhao M. Influx ofextracellular Ca2+ is necessary for electrotaxis in Dictyostelium. J Cell Sci.2006;p. 4741–48.

29. Verworn M. Die polare Erregung der Protisten durch den galvanischen Strom.Pfluegers Arch Gesante Physiol. 1889;45:1–36.

30. Nuccitelli R. A role for endogenous electric fields in wound healing. Curr TopDev Biol. 2003;58:1–26.

31. Zhao M, Song B, Pu J, et al TW. Electrical signals control wound healingthrough phosphatidylinositol-3-OH kinase-g and PTEN. Nature.2006;442(7101):457–60.

32. Brown MJ, Loew LM. Electric field-directed fibroblast locomotion involves cellsurface molecular reorganization and is calcium independent. J Cell Biol.1994;127:117–128.

33. Cooper MS, Keller RE. Perpendicular orientation and directional migration ofamphibian neural crest cells in dc electrical fields. Proceedings of the NationalAcademy of Sciences USA. 1984;81:160–64.

PLOS ONE 26/32


132

34. Erickson CA, Nuccitelli R. Embryonic fibroblast motility and orientation canbe influenced by physiological electric fields. Journal of Cell Biology.1984;98:296–307.

35. Lee J, Ishihara A, Oxford G, Johnson B, Jacobson K. Regulation of cellmovement is mediated by stretch-activated calcium channels. Nature.1999;400:382–386.

36. Onuma EK, Hui SW. Electric field-directed cell shape changes, displacement,and cytoskeletal reorganization are calcium dependent. J Cell Biol.1988;106:2067–2075.

37. Orida N, Feldman JD. Directional protrusive pseudopodial activity andmotility in macrophages induced by extracellular electric fields. Cell Motility.1982;2:243–55.

38. Maria EM, Djamgoz MBA. Cellular mechanisms of direct-current electric fieldeffects: galvanotaxis and metastatic disease. Journal of Cell Science.2004;117:1631–39.

39. cc M, Pu J, Forrester JV, McCaig CD. Membrane lipids, EGF receptors, andintracellular signals colocalize and are polarized in epithelial cells movingdirectionally in a physiological electric field. FASEB J. 2002;16(8):857–9.

40. Cooper MS, Shliwa M. Motility of cultured fish epidermal cells in the presenceand absence of direct current electric fields. J Cell Biol. 1986;102:1384–99.

41. Zhao M, Agius-Fernandez A, Forrester JV, McCaig CD. Directed migration ofcorneal epithelial sheets in physiological electric fields. Invest Ophthalmol VisSci. 1996;37(13):2548–58.

42. Zhao M, McCaig CD, Agius-Fernandez A, Forrester JV, Araki-Sasaki K.Human corneal epithelial cells reorient and migrate cathodally in a smallapplied electric field. Curr Eye Res. 1997;16(10):973–84.

43. Bai H, McCaig CD, Forrester JV, Zhao M. DC electric fields induce distinctpreangiogenic responses in microvascular and macrovascular cells.Arteriosclerosis, Thrombosis, and Vascular Biology. 2004;24:1234–39.

44. Grahn JC, Reilly DA, Nuccitelli RL, Isseroff RR. Melanocytes do not migratedirectionally in physiological DC electric fields. Wound Repair Regen.2003;11:64–70.

45. Allena R, Aubry D. Run-and-tumble or look-and-run? A mechanical model toexplore the behavior of a migrating amoeboid cell. Journal of TheoreticalBiology. 2012;306:15–31.

46. Allena R, Aubry D. A purely mechanical model to explore amoeboid cellmigration. Comput Methods Biomech Biomed Engin. 2012;15:14–6.

47. Allena R. Cell migration with multiple pseudopodia: temporal and spatialsensing models. Bull Math Biol. 2013;75(2):288–316.

48. Elliott CM, Stinner B, Venkataraman C. Modelling cell motility andchemotaxis with evolving surface finite elements. Journal of the Royal SocietyInterface. 2012;p. DOI: 10.1098/rsif.2012.0276.

PLOS ONE 27/32

Publications

133

49. Han SJ, Sniadecki NJ. Simulations of the contractile cycle in cell migrationusing a bio-chemical-mechanical model. Computer Methods in Biomechanicsand Biomedical Engineering. 2011;14(5):459–68.

50. Kim MC, Neal DM, Kamm RD, Asada HH. Dynamic modeling of cellmigration and spreading behaviors on fibronectin coated planar substrates andmicropatterned geometries. PLoS Comput Biol. 2013;9(2):e1002926.

51. Neilson MP, Mackenzie JA, Webb SD, Insall RH. Modeling Cell Movementand Chemotaxis Using Pseudopod-Based Feedback. SIAM Journal onScientific Computing. 2011;33:1035–1057.

52. Ni Y, Chiang MYM. Cell morphology and migration linked to substraterigidity. Soft Matter. 2007;3:1285–1292.

53. Savill N, Hogeweg P. Modeling morphogenesis: from single cells to crawlingslugs. Journal of Theoret Biol. 1997;184:229–235.

54. Vroomans RM, Maree AF, de Boer RJ, Beltman JB. Chemotactic migration ofT cells towards dendritic cells promotes the detection of rare antigens. PLoSComput Biol. 2012;8(11):e1002763.

55. van Oers RF, Rens EG, LaValley DJ, Reinhart-King CA, Merks RM.Mechanical cell-matrix feedback explains pairwise and collective endothelialcell behavior in vitro. PLoS Comput Biol. 2014;10(8):e1003774.

56. Hannezo E, Prost J, Joanny JF. Theory of epithelial sheet morphology inthree dimensions. Proceedings of the National Academy of Sciences currentissue. 2014;111(1):10.1073/pnas.1312076111.

57. DiMilla PA, Barbee K, Lauffenburger DA. Mathematical model for the effectsof adhesion and mechanics on cell migration speed. Biophys J.1991;60(4):15–37.

58. Dokukina IV, Gracheva ME. A Model of Fibroblast Motility on Substrateswith Different Rigidities. Biophysical Journa. 2010;98:2794–803.

59. Gracheva ME, Othmer HG. A continuum model of motility in ameboid cells.Bull Math Biol. 2004;66(1):167–93.

60. Maree AF, Jilkine A, Dawes A, Grieneisen VA, Edelstein-Keshet L.Polarization and movement of keratocytes: a multiscale modelling approach.Bull Math Biol. 2006;68:1169–211.

61. Palsson E. A three-dimensional model of cell movement in multicellularsystems. Future Generation Computer System. 2001;17:835–852.

62. De R, Zemel A, Safran SA. Dynamics of cell orientation. Nature Physics.2007;3:655–659.

63. Hakkinen KM, Harunaga JS, Doyle AD, Yamada KM. Direct comparisons ofthe morphology, migration, cell adhesions, and actin cytoskeleton of fibroblastsin four different three-dimensional extracellular matrices. Tissue Eng Part A.2011;17(5-6):713–24.

64. Kabaso D, Shlomovitz R, Schloen K, Stradal T, Gov NS. Theoretical Modelfor Cellular Shapes Driven by Protrusive and Adhesive Forces. PLoS ComputBiol. 2011;7(5):e1001127.

PLOS ONE 28/32


134

65. Satulovsky J, Lui R, Wang YL. Exploring the control circuit of cell migrationby mathematical modeling. Biophys J. 2008;94:36713683.

66. Mousavi SJ, Doblare M, Doweidar MH. Computational modelling of multi-cellmigration in a multi-signalling substrate. Physical Biology. 2014;11(2):026002(17pp).

67. Mousavi SJ, Doweidar MH, Doblare M. 3D computational modelling of cellmigration: A mechano-chemo-thermo-electrotaxis approach. J Theor Biol.2013;329:64–73.

68. Mousavi SJ, Doweidar MH. A Novel Mechanotactic 3D Modeling of CellMorphology. J Phys Biol. 2014;11(4):doi:10.1088/1478–3975/11/2/026002.

69. Mousavi SJ, Doweidar MH, Doblare M. Computational modelling and analysisof mechanical conditions on cell locomotion and cell-cell interaction. ComputMethods Biomech Biomed Engin. 2014;17(6):678–93.

70. Lecuit T, , Lenne PF. Cell surface mechanics and the control of cell shape,tissue patterns and morphogenesis. Nat Rev Mol Cell Biol. 2007;8:633–44.

71. Montell DJ. Morphogenetic cell movements: diversity from modularmechanical properties. Science. 2008;322:1502–05.

72. Albrecht-Buehler G. Does blebbing reveal the convulsive flow of liquid andsolutes through the cytoplasmic meshwork? Cold Spring Harbor Symposia onQuantitative Biology. 46(1);1982:45–49.

73. Charras GT, Yarrow JC, Horton MA, Mahadevan L, Mitchison TJ.Non-equilibration of hydrostatic pressure in blebbing cells. Nature.2005;435:365–69.

74. Cunningham CC. Actin polymerization and intracellular solvent flow in cellsurface blebbing. J Cell Biol. 1995;129(6):1589–99.

75. Zaman MH, Kamm RD, Matsudaira P, Lauffenburgery DA. ComputationalModel for Cell Migration in Three-Dimensional Matrices. Biophysical Journal.2005;89:1389–1397.

76. Stokes GG. On the effect of the inertial friction of fluids on the motion ofpendulums. Trans. Camb. Phil. Soc. 9; 1951.

77. Loth E. Drag of non-spherical solid particles of regular and irregular shape.Powder Technology. 2008;182:342–53.

78. Tran-Cong S, Gay M, Michaelides EE. Drag coefficients of irregularly shapedparticles. Powder Technology. 2004;139:21–32.

79. Jimenez JA, Madsen OS. A simple formula to estimate settling velocity ofnatural sediments. J Waterw Port Coast Ocean Eng. 2003;p. 70–8.

80. Ponti A, Machacek M, Gupton SL, Waterman-Storer CM, Danuser G. Twodistinct actin networks drive the protrusion of migrating cells. Science.2004;305(5691):1782–6.

81. James DW, Taylor JF. The stress developed by sheets of chick fibroblasts invitro. Experimental Cell Research. 1969;54:107–110.

PLOS ONE 29/32

Publications

135

82. Mousavi SJ, Doweidar MH, Doblare M. Cell migration and cell-cell interactionin the presence of mechano-chemo-thermotaxis. Mol Cell Biomech.2013;10(1):1–25.

83. Ramtani S. Mechanical modelling of cell/ECM and cell/cell interactions duringthe contraction of a fibroblast-populated collagen microsphere:theory andmodel simulation. Journal of Biomechanics. 2004;37:1709–1718.

84. Fraser SP, Diss JKJ, Mycielska ME, Coombes RC, Djamgoz MBA.Voltage-gated sodium channel expression in human breast cancer: possiblefunctional role in metastasis. Breast Cancer Res Treat. 2002;76:S142.

85. Rapp B, Boisfleury-Chevance A, Gruler H. Galvanotaxis of humangranulocytes. Dose-response curve. Eur Biophys J. 1988;16:313–19.

86. Sheridan DM, Isseroff RR, Nuccitelli R. Imposition of a physiologic DCelectric field alters the migratory response of human keratinocytes onextracellular matrix molecules. J Invest Dermatol. 1996;106:642–646.

87. Sulik GL, Soong HK, Chang PC, Parkinson WC, Elner SG, Elner VM. Effectsof steady electric fields on human retinal pigment epithelial cell orientation andmigration in culture. Acta Ophthalmol (Copenh). 1992;70:115–22.

88. Armon L, Eisenbach M. Behavioral Mechanism during Human SpermChemotaxis: Involvement of Hyperactivation. PLoS ONE. 2011;6:e28359.

89. Hong CB, Fontana DR, Poff KL. Thermotaxis of Dictyostelium discoideumamoebae and its possible role in pseudoplasmodial thermotaxis. Proc NatlAcad Sci USA. 1983;80:5646–49.

90. Long H, Yang G, Wang Z. Galvanotactic migration of EA.Hy926 endothelialcells in a novel designed electric field bioreactor. Cell Biochem Biophys.2011;61:481–91.

91. Meili R, Alonso-Latorre B, del Alamo JC, Firtel RA, Lasheras JC. Myosin IIis essential for the spatiotemporal organization of traction forces during cellmotility. Mol Biol Cell. 2010;21:405–417.

92. Wehrle-Haller B, Imhof BA. Actin, microtubules and focal adhesion dynamicsduring cell migration. The International Journal of Biochemistry & CellBiology. 2003;35(1):39–50.

93. Wessels D, Titus M, Soll DR. A Dictyostelium myosin I plays a crucial role inregulating the frequency of pseudopods formed on the substratum. Cell MotilCytoskeleton. 1996;33(1):64–79.

94. Wessels D, Voss E, Bergen NV, Burns R, Stites J, Soll DR. Acomputer-assisted system for reconstructing and interpreting the dynamicthree-dimensional relationships of the outer surface, nucleus, and pseudopodsof crawling cells. Cell Motil Cytoskeleton. 1998;41:225–46.

95. Hibbit, Karlson, Sorensen. Abaqus-Theory manual;.

96. Lee Y, Huang J, Wangc Y, Lin K. Three-dimensional fibroblast morphology oncompliant substrates of controlled negative curvature. Integr Biol. 2013;p.10.1039/C3IB40161H.

PLOS ONE 30/32


136

97. Akiyama SK, Yamada KM. The interaction of plasma fibronectin withfibroblastic cells in suspension. Journal of Biological Chemistry.1985;260:4492–4500.

98. Ulrich TA, Pardo EMDJ, Kumar S. The mechanical rigidity of theextracellular matrix regulates the structure, motility, and proliferation ofglioma cells. Cancer Research. 2009;69:4167–4174.

99. Buxboim A, Ivanovska IL, Discher DE. Matrix elasticity, cytoskeletal forcesand physics of the nucleus: how deeply do cells ’feel’ outside and in? Journalof cell science. 2010;123:297–308.

100. Schafer A, Radmacher M. Influence of myosin II activity on stiffness offibroblast cells. Acta Biomaterialia. 2005;1:273–280.

101. Oster GF, Murray JD, Harris AK. Mechanical aspects of mesenchymalmorphogenesis. journal Embryol Exp Morph. 1983;78:83–125.

102. Penelope CG, Janmey PA. Cell type-specific response to growth on softmaterials. Journal Applied Physiology. 2005;98:1547–1553.

103. Hadjipanayi E, Mudera V, Brown RA. Guiding cell migration in 3D: Acollagen matrix with graded directional stiffness. Cell Motility and theCytoskeleton. 2009;66:435–445.

104. Zhang S, Charest PG, Firtel RA. Spatiotemporal regulation of Ras activityprovides directional sensing. Curr Biol. 2008;18:1587–93.

105. Andrew N, Insall RH. Chemotaxis in shallow gradients is mediatedindependently of Ptdlns 3-kinase by biased choices between randomprotrusions. Nature Cell Biology. 2007;9:193–200.

106. Bosgraaf L, Haastert PJMV. Navigation of chemotactic cells by parallelsignaling to pseudopod persistence and orientation. PLoS ONE. 2009;4:e6842.

107. Vermolen FJ, Gefen A. A phenomenological model for chemico-mechanicallyinduced cell shape changes during migration and cell-cell contacts. BiomechModel Mechanobiol. 2013;12(2):301–23.

108. Maeda YT, Inose J, Matsuo MY, Iwaya S, Sano M. Ordered patterns of cellshape and orientational correlation during spontaneous cell migration. PLoSONE. 2008;3:e3734.

109. Barker AT, Jaffe LF, Vanable JJW. The glabrous epidermis of cavies containsa powerful battery. Am J Physiol. 1982;242:R358–366.

110. Illingworth CM, Barker AT. Measurement of electrical currents emergingduring the regeneration of amputated finger tips in children. Clin Phys PhysiolMeasure. 1980;1:87.

111. Jaffe LF, Vanable JJW. Electric fields and wound healing. Clin Dermatol.1984;2:34–44.

112. Chiang M, Robinson KR, Vanable JJW. Electrical fields in the vicinity ofepithelial wounds in the isolated bovine eye. Exp Eye Res. 1992;54:999–1003.

113. Chang PCT, Sulik SHG G L, Parkinson WC. Galvanotropic and galvanotacticresponses of corneal endothelial cells. Formos Med Assoc J. 1996;95:623–27.

PLOS ONE 31/32

Publications

137

114. Djamgoz MBA, Mycielska M, Madeja Z, Fraser SP, Korohoda W. Directionalmovement of rat prostate cancer cells in directcurrent electric field:involvement of voltage-gated Na+ channel activity. J Cell Sci.2001;114:2697–2705.

PLOS ONE 32/32


138


Plos One

Submitted paper


2.7 Role of mechanical cues in cell differentiation and pro-

liferation: A 3D numerical model.

Role of Mechanical Cues in Cell Differentiation andProliferation: A 3D Numerical Model

Seyed Jamaleddin Mousavi1,2,3, Mohamed Hamdy Doweidar1,2,3,*

1 Group of Structural Mechanics and Materials Modeling (GEMM), Aragon Instituteof Engineering Research (I3A), University of Zaragoza, Spain.

2 Mechanical Engineering Department, School of Engineering and Architecture(EINA), University of Zaragoza, Spain.

3 Centro de Investigacion Biomedica en Red en Bioingenierıa, Biomateriales yNanomedicina (CIBER-BBN), Spain.

*Corresponding authorEmail: [email protected] (MHD)

Abstract

Cell differentiation, proliferation and migration are essential processes in tissueregeneration. Experimental evidence confirms that cell differentiation or proliferationcan be regulated according to the extracellular matrix stiffness. For instance,mesenchymal stem cells (MSCs) can differentiate to neuroblast, chondrocyte orosteoblast within matrices mimicking the stiffness of their native substrate. However,the precise mechanisms by which the substrate stiffness governs cell differentiation orproliferation are not well known. Therefore, a mechano-sensing computational modelis here developed to elucidate how substrate stiffness regulates cell differentiationand/or proliferation during cell migration. In agreement with experimentalobservations, it is assumed that internal deformation of the cell (a mechanical signal)together with the cell maturation state directly coordinates cell differentiation and/orproliferation. Our findings indicate that MSC differentiation to neurogenic,chondrogenic or osteogenic lineage specifications occurs within soft (0.1-1 kPa),intermediate (20-25 kPa) or hard (30-45 kPa) substrates, respectively. These resultsare consistent with well-known experimental observations. Remarkably, when a MSCdifferentiate to a compatible phenotype, the average net traction force depends on thesubstrate stiffness in such a way that it might increase in intermediate and hardsubstrates but it would reduce in a soft matrix. However, in all cases the average nettraction force considerably increases at the instant of cell proliferation because ofcell-cell interaction. Moreover cell differentiation and proliferation accelerate withincreasing substrate stiffness due to the decrease in the cell maturation time. Thus,the model provides insights to explain the hypothesis that substrate stiffness plays akey role in regulating cell fate during mechanotaxis.

PLOS ONE 1/23

2.7. Role of mechanical cues in cell differentiation and proliferation: A 3D numerical model.

142

Introduction

Cell differentiation, proliferation, apoptosis and migration play an important rolein the early stages of the tissue regeneration process. The ability of a stem cell todifferentiate into different cell types allows it to generate different tissues. Forinstance, mesenchymal stem cells (MSCs) have the ability to differentiate intofibroblasts, chondrocytes, osteoblasts, neuronal precursors, adipocytes and manyothers [1–4]. Although, on the one hand, the multi-lineage differentiation potential ofstem cells is an advantage, on the other hand, it can be a disaster if they differentiateat the wrong time, at an undesirable place or to an inappropriate cell type. This maylead to a pathophysiologic state or non-functional tissue construction. To overcomesuch abnormalities, stem cells have been particularized in such a way as todifferentiate in response only to appropriate biological cues.

Cell differentiation and proliferation are governed by a combination of chemical [5]and mechanical [6,7] cues, although biologists have frequently reported that other cuessuch as growth factors and cytokines may be involved in the regulation of stem celldifferentiation [5, 8]. Recent observations have demonstrated that cell differentiationand proliferation can be significantly influenced by mechanical cues [6, 9].Experimental studies have shown that mechanical factors, including substrate stiffness,nanotopography of the adhesion surface, mechanical forces, fluid flow and cell colonysizes can direct stem cell fate even in the absence of biochemical factors [3, 4, 7]. Manyexperimental studies [1,2,4,6,7,9–11] have been dedicated to investigating the effect ofmechanical cues on cell differentiation and proliferation in tissue regeneration. Forinstance, Pauwels [11] mentioned that distortional shear stress is a specific stimulus forMSCs to differentiate into fibroblasts for fibrous tissue generation. Hydrostaticcompression is a specific stimulus for MSCs to differentiate into chondrocytes incartilage formation while MSCs differentiate into the osteogenic pathway (ossification)only when the strain felt by the cell is below a defined threshold.

Cells actively sense and react to their micro-environment mechanical conditions(mechano-sensing) through their focal adhesions [4, 6, 7, 9, 12, 13]. For instance, it hasbeen observed that the variation of matrix stiffness from soft to relatively rigid candirect MSC fate [1, 2, 10]. Engler et al. [1] investigated, for the first time, the key roleof matrix stiffness on the fate of human MSCs (hMSCs). To study the influence ofvarious matrix stiffnesses on hMSCs, they built artificial matrices ranging from soft torigid for surface cell attachment. They inferred that matrix stiffness dictates hMSCcommitment: cells cultured on soft substrates comparable with brain tissue (a stiffnessof 0.1-1 kPa) generated neuronal precursors; matrices with intermediate stiffnessresembling the elasticity of muscle tissue (a stiffness of 8-17 kPa) induced myogeniccommitment while relatively hard matrices mimicking collagenous bone (a stiffness of25-40 kPa) committed to an osteogenic lineage specification. The effect of substratestiffness on mouse MSC lineage specification has also been studied by Huebsch etal. [2] within 3D substrates which are physiologically more a relevant environment ascell substrate. They showed that matrix stiffness plays a significant role in MSClineage specification where adipogenic commitment was seen in relatively softermicro-environments (a stiffness of 2.5-5.0 kPa) while osteogenic specificationpredominated in substrates with intermediate elasticity (a stiffness of 11-30 kPa).Their findings indicate that the effect of matrix stiffness on cell phenotype in 3Dmatrices is generally consistent with 2D experimental observations [1, 10]. Besides,matrix stiffness controls the proliferation of the self-renewal of adult stem cells. Forinstance, muscle stem cells cultured on intermediate substrates resembling theelasticity of muscle tissue have proliferation potential while they are unable to

PLOS ONE 2/23

Publications

143

proliferate on rigid substrates [10].The signaling mechanisms by which micro-environment stiffness controls cell

lineage is still an open question. Many mechano-biological models have been developedto describe cell differentiation during fracture healing [14–21]. For example, Stops etal. [21] simulated cell differentiation and proliferation in a collagen-glycosaminoglycanscaffold subjected to mechanical strain and perfusive fluid flow. They assumed thatthe responses of the representative cells depend on the level of scaffold strain and theinlet fluid velocity. They demonstrated that according to MSC differentiation patterns,specific combinations of scaffold strains and inlet fluid flows cause phenotypeassemblies dominated by single cell types. Besides, Kang et al. [20] developed a modelto simulate bone fracture healing corresponding to cell differentiation, proliferationand apoptosis. Their model is formulated based on the cell density of each phenotypeassuming that cell differentiation and proliferation can be modulated according to themagnitude and frequency of the mechanical stimuli. According to their numericalresults, the bone healing process can be improved when the magnitude and frequencyof the mechanical stimuli are employed as controlling factors of cell proliferation.

All of the numerical models mentioned above are able to predict the generalpatterns of tissue differentiation due to external mechanical stimuli, confirmed byexperimental observations of mechano-regulated tissue differentiation. Althoughdefining a general patterns of cell differentiation for the tissue repairing is achallenging issue, as previously discussed, cell differentiation and proliferation can betriggered by mechano-sensing process and cell substrate interaction during cellmigration [1, 2, 10]. To the best of our knowledge, there is no numerical model thatconsiders cell differentiation and proliferation based on mechano-sensing processduring cell migration. In previous works presented by the same authors [22–24], anumerical model was developed to simulate cell migration in substrates with differenteffective cues. The main purpose of the present work is to extend the previous modelto study the influence of substrate mechanical conditions on cell differentiation,proliferation and apoptosis during migration.

Model formulation

A discrete finite element approach has been chosen to formulate cell migration,differentiation, proliferation and apoptosis in defined substrates. This approachprovides flexibility in the definition of migration direction without the need to remeshthe substrate and allows both deterministic and stochastic modelling of cellbehaviour [24, 25].

Cell migration

Stress transmitted by each individual cell to the ECM

During cell translocation, the actin cytoskeleton (CSK) controls the driving forcesat the cell front while the microtubule network regulates the rear retraction of thecell [26–28]. Active stress generated by actin filaments and myosin II, active cellularelements, basically depends on the maximum, εmax, and the minimum, εmin, internalcell strains. Besides, passive stress is related to the microtubules and the cell

PLOS ONE 3/23


144

membrane, passive cellular elements [24,25]. The cell stress which is transmitted tothe ECM can be defined as the sum of the passive and active stresses by [24,29]

σcell =

⎧⎪⎨⎪⎩

Kpasεcell εcell < εmin or εcell > εmaxKactσmax(εmin−εcell)Kactεmin−σmax

+Kpasεcell εmin ≤ εcell ≤ εKactσmax(εmax−εcell)Kactεmax−σmax

+Kpasεcell ε ≤ εcell ≤ εmax

(1)

where Kpas, Kact, εcell and σmax denote the stiffness of passive and active cellularelements, the internal strain of the cell and the maximum contractile stress exerted bythe actin-myosin machinery, respectively, while ε = σmax/Kact.

Effective mechanical force

During cell migration two main mechanical forces affect the cell body, the tractionforce and the drag force. The former, which is transmitted to the substrate throughintegrins, is generated due to the contraction of the actin-myosin apparatus. Thisforce drives the cell body forward and is directly proportional to the cell stress, σcell.Representing the cell by a connected group of finite elements, the nodal traction forceof the cell can be represented as [22,24]

Ftraci = σcellSζei (2)

where ei denotes a unit vector passing through the ith node of the cell membranetowards the cell centroid (Fig. 1-a). S represents the cell membrane area and ζ is adimensionless parameter named ”adhesivity” which is directly proportional to theconcentration of the ligands at the leading edge of the cell, ψ, the total number ofavailable receptors, nr, and the binding constant of the cell integrins, k. Therefore, itcan be defined as [24,30]

ζ = knrψ (3)

Consequently, the net effective traction force on the cell body is calculated as [24]

Ftracnet =

n∑i=1

Ftraci (4)

where n is the number of cell membrane nodes.In contrast, the drag force refers to the force which acts in the opposite direction of

the motion of the cell. Our aim here is to specify a velocity-dependent resisting forceproportional to the linear viscoelastic character of the substrate. Thus it is assumedthat the ECM is a viscoelastic medium [30]. Note that, at microscale, the inertialresistance of the medium can be neglected because it is sufficiently small incomparison with the viscous resistance. Therefore, referring to Stokes’ drag regime,the drag force on a small sphere with radius r, moving with velocity v within amedium with viscosity η can be defined by [24,30]

Fdrag = 6πrηv (5)

Protrusion force

To migrate, cells extend local protrusions by exerting a protrusion force to evaluatetheir surrounding substrate. This refers to the actin polymerization and differs from

PLOS ONE 4/23

Publications

145

the cytoskeletal contractile force transmitted to the substrate [22,29,30]. It is arandom force that causes cells to move along a directed random path towards theeffective cue. It is remarkable that the order of the protrusion force magnitude is thesame as that of the traction force but with lower amplitude [24,30–32]. Therefore, itcan be described at each time step as


where κ is a random number, 0 ≤ κ < 1, and F tracnet denotes the magnitude of the net

traction force while erand represents a random unit vector [23,24]. It is assumed thatneither degradation nor remodelling of the ECM occurs during cell migration. Asmentioned above, the inertial force is negligible so that the force balance reads

Ftracnet + Fprot + Fdrag = 0 (7)

Cell deformation and reorientation

For the sake of simplicity a spherical cell shape is considered here (solid line in Fig.1-b). However, any cell shape can be considered using the present model [29]. In themechano-sensing stage, a cell firstly exerts sensing forces at each finite element nodelocated on the cell membrane towards the cell centroid to probe its surroundingenvironment. The cell deformation resulting from the mechano-sensing step is shownby dashed lines in Fig. 1-b. Therefore, the internal deformation of the cell at eachfinite element node of the cell membrane can be defined as

εcell =MN

OM(8)

Subsequently, referring to Eq. 7 the net cell polarisation direction can be calculated by(Fig. 1-b)

epol = − Fdrag

‖ Fdrag ‖ (9)

From Eq. 5 and Eq. 7 the cell velocity can be defined as

v =‖ Fdrag ‖6πrη

(10)

Using Eq. 10, during time step, τ , the translocation vector of the cell through whichthe cell migrates to locate in a new position can be defined as

d = vτepol (11)

Cell-cell interaction

In the presence of two or more cells in a substrate, the traction force, protrusionforce, velocity and reorientation of each individual cell can be calculated using theprevious formulation. According to Fig. 2, a vector passing through the centroid oftwo cells i and j can be obtained by

xij = xj − xi (12)

PLOS ONE 5/23


146

Figure 1. Cell mechanosensing. a- Spherical configuration of the cell in whichsensing forces are exerted at each membrane node towards the cell centroid(mechano-sensing process). b- Calculation of the cell internal deformation due to cellmechano-sensing. Deformed cell due to mechano-sensing. epol stands for polarisationdirection of the reoriented cell while Ftrac

net , Fprot and Fdrag represent the net tractionforce, protrusion force and drag force, respectively.

where xi and xj are position vectors of ith and jth cells. In reality cells inside amulticellular system do not preserve a spherical shape but deform to be tangent toeach other [33]. Here, a useful simplification to avoid interference of two cells is‖xij‖ ≤ 2r.

For the assumed cell configuration, when two cells come into contact they have amaximum of four common nodes (n1:n4 in Fig. 2). In vivo, the cell pushes out apseudopod to get a better sense its environment. Once the cell locates the desiredregion of the substrate, it pulls itself in the direction of the pseudopod [34]. Therefore,when two or more cells come into contact with each other, the common points of bothcells (for instance nodes n1:n4 in Fig. 2) are not able to send out the pseudopod to thesubstrate [34,35]. Therefore, for two or more cells, we assume that cells do not exertany sensing force at their common nodes unless they become separated again due tothe random protrusion force. It worth noting that although in such a situation thenodes in contact do not have any role in the mechano-sensing process, the tractionforces are not zero in those nodes [22,24,25].

Cell differentiation, proliferation and apopto-sis

The importance of sensing the mechanical properties of the ECM has been reportedin many experimental studies for different cell types [3, 36,37]. Cells may respond tothe mechanical signals received from their micro-environment by differentiation orapoptosis [38–41]. A specific deformation range experienced by a cell is shown to leadto a specific differentiation [15,42,43]. This diversity may arise from differences intheir tissue origin, in the magnitude and duration of the mechanical signal sensed bythe cell and in the degree of preconditioning. Although the precise effect of mechanicalcues on cell apoptosis is still poorly understood, there are experimental worksreporting that cell death may occur due to the deformation which a typical cell canbear [38, 40]. Experimental observations of Kearney et al. [40] indicate that tensilestrain induced on MSCs mediates cell apoptosis. Cell death depends on the internal

PLOS ONE 6/23

Publications

147

Figure 2. Interaction of two cells in contact. For the assumed cell configuration,two cells can have four common nodes (n1:n4). xi and xj are position vectors of theith and the jth cells, respectively, while xij is a vector passing by the centroids of theith and jth cells. The distance between their centroids (Oi and Oj) is equal to orgreater than the proposed cell diameter, ‖xij‖ ≤ 2r.

stain felt by a cell whereby specific strain ranges in 2D MSC cultures exhibitsignificant apoptosis while maximal apoptosis occurs in response to 10% tensile cyclicstrain when applied continuously. Gladman et al. [38] used an in vitro approach toexamine the apoptosis of adult dorsal root ganglion cells. Their finding illustrated thatmechanical injury beyond 20% of tensile cyclic strain led to significant neuronal celldeath which was also proportional to the duration of the imposed strain.

Let us assume that MSCs are able to differentiate into a certain cell type i, wherei ∈ {s, c, l} represents lineage specifications such as osteoblasts, s, chondrocytes, c, andneuroblasts, l. Mechano-regulation of differentiation is introduced in terms of themechanical signal (cell internal deformation) in the cell polarisation direction.Deformation of each node located on the cell membrane in the cell polarisationdirection can be calculated as

γi = epol : εi : eTpol (13)

where εi is the strain tensor of ith node located on the cell membrane due to themechano-sensing process. Therefore, cell internal deformation in the cell polarisationdirection can be obtained by

γ(x, t) =n∑i=1

γi (14)

where n is the number of the cell membrane nodes and x is the position vector of cellcentroid at the time t. γl ≤ γ(x, t) ≤ γu varies spatially and temporally during cellmigration [15, 20,44] where γl and γu are lower and upper bounds of cell internaldeformations leading to cell differentiation, respectively.

Based on experimental observations [12,45, 46], not only cell differentiation ismechano-biologically dependent, but it is also time-dependent. For instance, thisargument indicates that MSCs [12] and chondrocyte [46] need to be at a certain levelof maturity before they undergo differentiation or proliferation. In this context, the

PLOS ONE 7/23


148

cell differentiation depends on a maturation time which is the time that the cell needsto be active in the differentiation stage [20, 45]. The mechanical signals received by acell can regulate the cell maturation period, being different for each cell type.Although stronger mechanical signals (less internal deformation) can decrease the cellmaturation time, after cell culture even within substrates producing the strongestmechanical signals, cells need a minimum time period to start differentiation orproliferation [45]. Therefore, we assume that a strong mechanical signal will increasethe differentiation rate while it decreases the cell maturation time linearly as

tmat(γ, t) = tmin + tpγ(x, t) (15)

where tmin is the minimum time needed by the cell for differentiation and tp is a timeproportionality. Therefore, beside lineage specifications i ∈ {s, c, l}, each cell is alsorepresented via a maturation index (MI) which can be described as

MI =

{ttmat

t ≤ tmat

1 t > tmat(16)

MI=1 indicates that a cell (MSCs, osteoblasts, chondrocytes and neuroblasts) iscompletely mature and is prone for differentiation or proliferation in presence ofproper mechanical signal. MI=0 corresponds to young cell, which means that the cellis not yet able to start the differentiation or proliferation process, even if mechanicalstimulus is appropriate. We assume that the evolution of cell MI is an irreversibleprocess. This means that once a specific cell phenotype adheres to the substrateduring cell migration, the cell MI cannot be reduced except when the cell phenotypechanges due to differentiation. It is of interest to mention that although the cell MI isan irreversible parameter during cell migration, the cell maturation time, tmat, candecrease or increase depending on the mechanical signal strength received by the cell.Considering these previously mentioned conditions, the process of MSC differentiationand apoptosis related to mechanical signals and maturation can be represented by [20]

Cell phenotype =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

s γl < γ ≤ γs & MI = 1c γs < γ ≤ γc & MI = 1l γc < γ ≤ γu & MI = 1

apoptosis γapop < γno diffrentiation otherwise

(17)

It should be noted that small strains exerted cyclically on a typical cell may causefatigue apoptosis of the cell [40] that we have not considered here.

Cell proliferation is the process of producing two daughter cells from a mother. Innormal tissues, this generally refers to cells that replenish the tissue by cell growthfollowed by cell division. Cell proliferation occurs in defined steps including the firstgrowth phase, the synthesis phase, the second growth phase and the mitosis phase,respectively [47, 48]. During the first growth phase, known as the G1 phase, the cellsynthesizes a huge content of biological material. As soon as the G1 phase iscompleted the cell enters the synthesis phase, the S phase, to replicate its DNA. Atthe end of the S phase it starts the second growth phase, G2, that finally leads to themitosis phase, the M phase. Subsequently, reorganization of the cell chromosomes isfollowed by the cell division so that a mother cell is divided into two daughter cells.This is a critical instant because some cells temporarily stop proliferation by enteringinto the quiescence state which is called the G0 phase [47,48].

Our objective here is to model the proliferation process using a biologicallyappropriate method. It is hypothesized that there are no concerns about shortage ofoxygen or nutrients for the cells in culture. Therefore, we intend to model the

PLOS ONE 8/23

Publications

149

dominant cell division cycle through two main steps. It is assumed that during the G1,S and G2 phases the cell grows and matures such that when the cell maturation isachieved, depending on the strength of the mechanical signal received by the cell, onemature mother cell may enter into the mitosis phase and divide into two non-maturedaughters. Thus, in the present model, the cell is either under maturation or in theproliferation phase. In other words, each cell is in the quiescence phase unless itdelivers two daughter cells. Accordingly, cell growth can be considered as

Cell growth =

⎧⎨⎩

1 mother cell → 2 daughter cells γ ≤ γprofi

& MI = 1no cell division otherwise

(18)

where i ∈ {m, s, c, l} and γprofi < γu is the mechanical signal that defines theproliferation limit of the ith cell [20]. Here, m represents the MSC phenotype. When amother cell is divided into two daughter cells, it is assumed that one of the daughter

cells is located in the same position as the mother cell, x(1)daut = xmoth, while the other

is located in the vicinity of the mother cell as

x(2)daut = xmoth + 2rerand (19)

where ”moth” and ”daut” subscripts denote mother and daughter cells, respectivelywhile erand represents a random unit vector.

Finite element implementation

The present model is implemented within the commercial FE softwareABAQUS [49] through a coupled user subroutine. The corresponding algorithm ispresented in Fig. 3.

We have applied the model for several numerical cases where the cell is embeddedwithin a 400×200×200 μm substrate with different ranges of stiffness. It is assumedthat there is no external force acting on the substrate and all of the boundary surfacesare considered free. The substrate is meshed by 16,000 regular hexahedral elementsand 18,081 nodes while the cell has a constant spherical shape with 24 nodes on itsmembrane. The calculation time is about one minute for each time step in which eachstep corresponds to approximately 6 hr of real cell-substrate interaction. Theproperties of the cells and the substrate are enumerated in table 1.

Numerical examples

As discussed before, MSCs can be differentiated into neurogenic, chondrogenic orosteogenic cell types by varying the magnitude of the matrix stiffness to mimic that ofthe native tissue [1, 2, 10]. However, not all cell types are sensitive to substratestiffness or have a similar mechano-sensitive response to changes in substratestiffness [55]. Our objective here is to study the dependency of cell fate on substratestiffness according to available experimental observations. The model presented herewill be used to predict cell fate within soft (0.1-1 kPa), intermediate (20-25 kPa) andhard (30-45 kPa) substrates that are comparable with neurogenic, chondrogenic andosteogenic tissue, respectively. This section provides an insight into the role ofsubstrate stiffness in MSC proliferation in order to examine how cells are able to

PLOS ONE 9/23


150

Table 1. General parameters employed in the model. General parametersemployed in the model except where other values are specified.

Symbol Description Value Ref.ν Substrate Poisson ratio 0.3 [41,50]η Minimum substrate viscosity 1000 Pa.s [30, 50]Kpas Stiffness of microtubules 2.8 kPa [51]Kact Stiffness of myosin II 2 kPa [51]εmax Maximum strain of the cell 0.9 [24, 32]εmin Minimum strain of the cell -0.9 [24, 32]σmax Maximum contractile stress exerted by

actin-myosin machinery0.1 kPa [52,53]

kf = kb Binding constant at rear and front ofthe cell

108 mol−1 [30]

nrf Number of available receptors at thefront of the cell

1.5× 105 [30]

nrb Number of available receptors at theback of the cell

105 [30]

ψ Concentration of the ligands at rear andfront of the cell

10−5 mol [30]

tmin Minimum time needed for cell prolifer-ation

4 days [20,45]

tp Time proportionality 200 days [20,45]γl Lower bound of cell internal deforma-

tion leading to osteoblast differentia-tion

0.005 [20,54]

γs Upper bound of cell internal deforma-tion leading to osteoblast differentia-tion

0.04 [20,54]

γc Upper bound of cell internal deforma-tion leading to chondrocyte differentia-tion

0.1 [20]

γu Upper bound of cell internal deforma-tion leading to neuroblast differentia-tion

0.5

γapop Cell internal deformation leading to cellapoptosis

1 [20]

γprofm Limit of MSC proliferation 0.2 [20]

PLOS ONE 10/23

Publications

151

Figure 3. Computational algorithm of cell mechano-sensing and consequentcell fate due to mechanotaxis.

detect and respond to alterations in the stiffness of their surroundingmicro-environment via induction of lineage-specific differentiation. To obtain reliableand consistent results all the numerical cases have been repeated at least 20 times.

MSC proliferation and differentiation

The fate decision of MSCs can be influenced by the microenvironment in whichthey reside. Their coordinated interactions with the ECM and neighbour cells providebiomechanical signals that direct them to survive, migrate, proliferate ordifferentiate [3]. Here, using the presented model, we investigate MSC proliferationand differentiation in substrates with different uniform stiffnesses. MSC behaviorwithin a hard substrate of 45 kPa stiffness is represented in Fig. 4 (results for soft andintermediate substrates are not shown here). Initially, it is assumed that a MSC islocated in one of the corners of a hard substrate. Regardless the substrate stiffness,the initial cell tendency is to migrate towards the middle of the substrate where itfeels less internal deformation and more stability (discussed previously in [24]). Duringcell migration over time, the cell becomes mature, however the cell maturation ratedepends on the substrate stiffness. After cell maturation (Fig. 4-b), depending on themechanical signal received by MSC, γ, one mature mother MSC may proliferate anddeliver two daughter cells in a non-mature state (Fig. 4-c). It is worth noting that anincrease in the substrate stiffness decreases the time needed for cell maturation (seeFig. 6). This occurs because the increase in the substrate stiffness decreases theinternal deformation of the cell which in turn decreases the MSC maturation time (seeEq. 15). Accordingly, the shorter maturation times of MSCs, the higher theirproliferation rate. This has been suggested by the experimental findings of Evans etal. [56] which revealed that cell growth is increased as a function of substrate stiffness.Besides, during cell migration towards the middle of the substrate, the net tractionforce generated by the cell decreases which means that the cell can be adhered to thesubstrate consuming less energy (discussed previously in [23–25]). Fig. 5 demonstratesthe average cell traction force versus time within a hard substrate of 45 kPa stiffness

PLOS ONE 11/23


152

during MSC proliferation and differentiation. Point A represents the instant of MSCproliferation (coincident with Fig. 4-a) which causes a considerable jump in theaverage net traction force. This occurs because upon MSC proliferation, cell-cellinteraction causes an asymmetric distribution of the internal cell deformation andsubsequently the nodal traction force in the membrane nodes (see [24,25] for moredetails about the effect of cell-cell interaction on average traction force). Point B isthe instant of MSC differentiation to osteoblast (coincident with Fig. 4-b) leading toan enhancement of the average net traction force which is consistent with theobservations of Fu et al. [57].

Fate decision of MSCs in hard substrates (os-teogenic tissue)

This numerical example is designed to study the lineage specification of MSCs inhard substrates. Here, to avoid repetition of the proliferation process of MSCsdiscussed above, we will present the results starting from the instant of celldifferentiation. It is assumed that the stiffness of osteogenic tissue is in the range of30-45 kPa [58]. In order to investigate the influence of matrix stiffness on osteogeniclineage specification, simulations were repeated for lower (30 kPa) and upper (45 kPa)bounds of the hard substrates. The commitment of MSCs to osteogenic lineagespecification in substrates of 30 kPa and 45 kPa stiffnesses is presented in Figs. 7-aand 7-b, respectively. During migration, the cell gradually matures and once the MSCis completely mature (MI=1) it differentiates into osteoblast within both hardsubstrates. Osteoblast lineage specification of MSCs within a substrate with a stiffnessequivalent to that of osteogenic tissue is supported by the experimental observations ofEngler [1] and Huebsch [2].The MSC is mature and differentiates into osteoblastwithin a substrate of 30 kPa stiffness after ∼7 days while it becomes mature anddifferentiates within a substrate of 45 kPa stiffness after ∼5.5 days. Therefore,consistent with the findings of Fu et al. [57], an increase in the substrate stiffnessexpedites MSC differentiation to osteoblast. In addition, this is in agreement with theobservations of Evans et al. [56] who show that terminal osteogenic differentiation ofEmbryonic SCs (ESCs) is enhanced on stiff substrates compared with soft substrates.Although cell migration towards the middle of the substrate causes the net tractionforce to decrease [24,25], MSC differentiation to osteoblast instantly leads to a greatermagnitude of the net traction force (points A in Fig. 8). A strong correlation betweenthe traction force and the ultimate lineage specification of MSCs has been observed byFu et al. [57], while their observations indicate that the osteogenic lineage specificationof MSCs demonstrates higher traction force than that of the progenitor MSCs. Thiscan be attributed to mechanical coupling between the extracellular matrix (ECM) andinternal CSK organization, according to the suggestion of Zemel et al. [59] whichindicates that a perfect alignment of stress fibers in the direction of the cellpolarisation occurs when the cell and matrix stiffness are similar due to thedifferentiation of MSCs into osteoblasts. After MSC differentiation to osteoblastwithin hard substrates, new cell phenotypes can be proliferated depending on thestrength of the mechanical signal received by the cell and its maturation state. AfterMSC differentiation, osteoblast becomes fully mature and starts to proliferate withinsubstrates of 30 kPa and 45 kPa stiffnesses after ∼ 14.5 and ∼ 12 days, respectively.So each new mature osteoblast within substrates of 30 kPa and 45 kPa stiffnesses canbe proliferated to many osteoblasts, as shown in Figs. 7-a and 7-b, respectively. Thenormalized density of each cell phenotype versus substrate stiffness is shown in Fig. 9

PLOS ONE 12/23

Publications

153

Figure 4. MSC proliferation and differentiation within a substrate of 45kPa stiffness. a- beginning of MSC proliferation after almost 5.5 days, b- the firstcommitment of a mature mother MSC to osteogenic lineage specification afterapproximately 32 days, c- continuing differentiation and proliferation of MSCs andosteoblasts.

for identical times. Comparing the corresponding Figs. 7-a and 7-b and taking intoaccount the results in Fig. 9, it can be concluded that, similar to MSC differentiation

PLOS ONE 13/23


154

Figure 5. Average cell traction force within a hard substrate of 45 kPastiffness. Average cell traction force, F trac

net , versus time within a hard substrate of 45kPa stiffness during MSC proliferation and differentiation. Point A represents theinstant of MSC proliferation which causes a considerable jump in the average nettraction force while point B is the initial instant of MSC differentiation to osteoblastleading to an enhancement of the average net traction force.

Figure 6. MI of MSCs within substrates of different uniform stiffnesses. Erepresents substrate elasticity modulus.

to osteoblast, the proliferation of osteoblasts is accelerated by an increase in matrixstiffness, due to the decrease in the maturation time. This is in agreement with thefindings of Fu et al. [57]. Moreover, during osteoblast proliferation the averagemagnitude of the net traction force considerably increases (points B in Fig. 8), due tocell-cell interaction which causes an asymmetric nodal traction forcedistribution [24, 25].

PLOS ONE 14/23

Publications

155

Figure 7. Osteoblast proliferation in hard substrates. a- 30 kPa and b- 45 kPastiffness.

Figure 8. Average cell traction force within a hard substrate of 30 kPa and45 kPa stiffness. Average cell traction force, F trac

net , versus time within hardsubstrates during MSC differentiation and osteoblast proliferation. Points A representthe instant of MSC differentiation to osteoblast which instantly causes a traction forceincrease while points B are the initial instant of osteoblast proliferation causing ajump in the average net traction force.

Fate decision of MSCs in intermediate sub-strates (chondrogenic tissue)

To study the lineage specification of MSCs in substrates with intermediate stiffness,simulations are repeated for substrates mimicking the elasticity of chondrogenic tissue.Again, to avoid repetition of the proliferation process of MSCs, the results will berepresented starting from the instant of cell differentiation. The stiffness ofchondrogenic tissue is assumed to be in the range of 20-25 kPa [58]. MSCs locatedwithin substrates of 20 kPa and 25 kPa stiffnesses are completely mature and start todifferentiate into chondrocyte after ∼ 10.5 and ∼ 8 days, respectively. So, similar tothe previous numerical case, an increase in substrate stiffness in the range of

PLOS ONE 15/23


156

Figure 9. Normalized density of each cell phenotype. Normalized density ofeach cell phenotype versus substrate stiffness during identical times as a consequenceof MSC differentiation and proliferation of each cell phenotype. The error barsrepresent mean standard deviation of different runs.

intermediate tissue assists MSC differentiation to chondrocyte. MSC differentiation tochondrocyte in substrates of 20 kPa and 25 kPa stiffnesses is presented in Figs. 10-aand 10-b, respectively. This is consistent with the findings of Burke and Kelly [60]indicating that MSC differentiation along either a chondrogenic, osteogenic oradipogenic lineage specification is regulated by the stiffness of the local substrate andthe local oxygen tension. According to their results, chondrogenesis of MSCs occurswithin matrices that mimic the micro-environmental elasticity of chondrogenic tissue.Furthermore, MSC differentiation to chondrocyte causes the net traction force exertedby the chondrocyte to increase (points A in Fig. 11). However its amplification is lessthan that of the osteoblast traction force within hard substrates. Depending on thematuration state and mechanical signal received by the new cell phenotype, it can beproliferated to many similar cell types. After ∼ 21 and ∼ 18 days, chondrocytesdifferentiated from MSCs located within substrates of 20 kPa and 25 kPa stiffnesses,respectively, are sufficiently mature to proliferate. As seen in Figs. 10-a and 10-b,maturated chondrocytes can be proliferated to many chondrocytes within substrates of20 kPa and 25 kPa stiffnesses, respectively. Our findings indicate that like thedifferentiation process, the proliferation of chondrocytes is advanced by an increase inthe matrix stiffness in the range of chondrogenic tissue. The chondrocyte density overan identical time period is higher for a substrate with greater stiffness according to theresults shown in Figs. 9, 10-a and 10-b. In addition, chondrocyte proliferation causesthe average magnitude of the net traction force to increase as a consequence of cell-cellinteraction (points B in Fig. 11).

Fate decision of MSCs in soft substrates (neu-rogenic tissue)

Experimental observations confirm that neural precursor cells can be obtained fromMSCs by culturing them in substrates having a very low stiffness (0.1-1 kPa),comparable to that of the neurogenic tissue [1, 58,61]. As with the earlier examples,

PLOS ONE 16/23

Publications

157

Figure 10. Chondrocyte proliferation in intermediate substrates. a- 20 kPaand b- 25 kPa stiffness.

Figure 11. Average cell traction force within intermediate substrates of 20kPa and 25 kPa stiffness. Average cell traction force, F trac

net , versus time withinintermediate substrates during MSC differentiation and chondrocyte proliferation.Point A represents the moment of MSC differentiation to chondrocyte which instantlycauses the traction force to increase while point B is the initial moment of chondrocyteproliferation causing a jump in the average net traction force.

we will represent the results starting from the instant of cell differentiation to avoidrepetition of the MSC proliferation process. MSCs located within substrates of 0.1kPa and 1 kPa stiffnesses are mature and initiate the differentiation process after ∼28.5 and ∼ 20.5 days, respectively. Figs. 12-a and 12-b represent the cell response tomatrix stiffness in substrates of 0.1 kPa and 1 kPa stiffnesses, respectively. Theacceleration of MSC differentiation by an increase in the substrate stiffness withinrelatively soft substrates is supported by the experimental observations of Fu etal. [57] for adipoblasts, differentiating in substrates of 2.5-5 kPa stiffness. Within aneurogenic medium, MSC is more contractile than neuroblast which causes a suddendecrease in the traction force at the instant of MSC differentiation to neuroblast(points A in Fig. 13). This is consistent with the findings of Fu et al. [57] for MSCdifferentiation in soft substrates. After MSC differentiation to neuroblast, a new cell

PLOS ONE 17/23


158

phenotype proliferates depending on its maturation state and the mechanical signalreceived by the neuroblast. Therefore, MSC differentiation is followed by fullmaturation of neuroblasts within substrates of 0.1 kPa and 1 kPa stiffnesses after ∼ 57and ∼ 41 days, respectively. Subsequently, the neuroblast may proliferate to severalcells as seen in Figs. 12-a and 12-b which indicates that neuroblast proliferationwithin a substrate of 1 kPa stiffness is quicker than that within a substrate of 0.1 kPastiffness. The same conclusion can be drawn from the normalized cell density over anidentical time period, as shown in Fig. 9. This occurs because a higher substratestiffness advances the instant of cell maturation. In this case, as in the previous onesthe cell-cell interaction increases the average magnitude of the net traction force dueto neuroblast proliferation (points B in Fig. 13).

Figure 12. Neuroblast proliferation in soft substrates a- 0.1 kPa and b- 1 kPastiffness.

Conclusions

The comprehensive signaling mechanisms by which micro-environmental stiffnesscontrols the lineage specification of MSCs is still unknown [3]. Recently, a number ofdifferent hypotheses have been proposed about how mechanical signals govern cellfate [3, 15]. To acquire accurate control over cell differentiation and proliferation, it isessential to elucidate and quantify the contribution of mechanical factors to cellresponse. In this paper, we study the influence of substrate stiffness on cell fatethrough a new 3D numerical model. This model can be considered as a first steptowards the interpretation of existing knowledge about the effect of cellularmicro-environment on cell fate. However, we believe that further research onmechanical and physical factor, such as cell shape, topographic changes, externalmechanical forces and colony size, can provide a broader understanding of thedetermination of the cell fate. In this study, we have investigated how a 3D substratewith soft, intermediate and hard stiffnesses influences the differentiation of MSCstowards neurogenic, chondrogenic or osteogenic lineage, respectively. Our findingscorrelated with experimental observations [1, 2, 56, 57,59] indicate that matrix stiffnesscan govern the lineage type of MSCs. The traction force generated by a specific cellphenotype can increase (osteoblasts and chondrocytes) or decrease (neuroblast) during

PLOS ONE 18/23

Publications

159

Figure 13. Average cell traction force within soft substrates of 0.1 kPa and1 kPa stiffness. Average cell traction force, F trac

net , versus time within soft substratesduring MSC differentiation and neuroblast proliferation. Points A represent theinstant of MSC differentiation to neuroblast which instantly causes the traction forceto decrease while points B are the initial instant of neuroblast proliferation causing ajump in the average net traction force.

differentiation [57]. In contrast, in all cases the proliferation of a typical cellconsiderably increases the average cell traction force due to the cell-cell interactionthat causes an asymmetric distribution of the nodal traction force [22]. In addition, anincrease in the matrix stiffness accelerates cell proliferation and differentiation [56]. Fora typical cell, internal deformation and focal adhesions developed through integrins arekey molecular mechanisms in the mechano-sensing process. For instance, Friedland etal. [62] reported that α5β1-integrin could switch between relaxed and tensioned statesin response to traction forces generated by a cell. Therefore, any changes in the matrixstiffness causes alteration in the cell internal deformation which accordingly regulatesthe cell differentiation or proliferation. On the other hand, the cell behavior in term ofcell differentiation or proliferation depends on the cell maturation. Interestingly,according to Fig. 6, cells might require a longer time to become fully mature withinsoft substrates compared to hard and intermediate substrates. This is consistent withthe observations of Hera et al. [61]. Taken together, the results of the model presentedhere and the earlier experimental observations [1, 2, 56, 57,59] show that matrixstiffness plays a significant role in controlling the fate decision of MSCs. Accordingly,the present 3D numerical model can successfully predict essential aspects of cellmaturation, differentiation, proliferation and apoptosis during regenerative events.

Acknowledgments

The authors gratefully acknowledge the support from the Spanish Ministry ofEconomy and Competitiveness and the CIBER-BBN initiative.

PLOS ONE 19/23


160

References

1. Engler AJ, Sen S, Sweeney HL, Discher DE. Matrix elasticity directs stem celllineage specification. Cell. 2006;126:677–89.

2. Huebsch N, Arany PR, Mao AS, Shvartsman D, Ali OA, Bencherif SA, et al.Harnessing traction-mediated manipulation of the cell/matrix interface tocontrol stem-cell fate. Nat Mater. 2010;9:518–26.

3. Li D, Zhou J, Chowdhury F, Cheng J, Wang N, Wang F. Role of mechanicalfactors in fate decisions of stem cells. Regen Med. 2011;6(2):229–40.

4. Palomares KT, Gleason RE, Mason ZD, Cullinane DM, Einhorn TA,Gerstenfeld LC, et al. Mechanical stimulation alters tissue differentiation andmolecular expression during bone healing. J Orthop Res. 2009;27(9):1123–32.

5. Pittenger MF, Mackay AM, Beck SC, Jaiswal RK, Douglas R, Mosca JD, et al.Multilineage potential of adult human mesenchymal stem cells. Science.1999;284(5411):143–7.

6. Maul TM, Chew DW, Nieponice A, Vorp DA. Mechanical stimuli differentiallycontrol stem cell behavior: morphology, proliferation, and differentiation.Biomech Model Mechanobiol. 2011;p. 939–53.

7. Subramony SD, Dargis BR, Castillo M, Azeloglu EU, Tracey MS, Su A, et al.The guidance of stem cell differentiation by substrate alignment and mechanicalstimulation. Biomaterials. 2013;34(8):1942–53.

8. Pierce JH, Marco ED, Cox GW, Lombardi D, Ruggiero M, Varesio L, et al.Macrophage-colony-stimulating factor (CSF-1) induces proliferation,chemotaxis, and reversible monocytic differentiation in myeloid progenitor cellstransfected with the human c-fms/CSF-1 receptor cDNA. Proc Natl Acad Sci.1990;87(15):5613–7.

9. Lee DA, Knight MM, Campbell JJ, Bader DL. Stem cell mechanobiology. JCell Biochem. 2011;112(1):1–9.

10. Gilbert PM, Havenstrite KL, Magnusson KE, Sacco A, Leonardi NA, Kraft P,et al. Substrate elasticity regulates skeletal muscle stem cell self-renewal inculture. Science. 2010;329:1078–81.

11. Pauwels F. A new theory concerning the influence of mechanical stimuli on thedifferentiation of the supporting tissues. Biomechanics of the LocomotorApparatus. 1980;p. 375–407.

12. Delaine-Smith RM, Gwendolen CR. Mesenchymal stem cell responses tomechanical stimuli. Muscles Ligaments Tendons J. 2012;2(3):169–80.

13. Lee Y, Huang J, Wangc Y, Lin K. Three-dimensional fibroblast morphology oncompliant substrates of controlled negative curvature. Integr Biol. 2013;p.10.1039/C3IB40161H.

14. Carter DR, Blenman PR, Beaupre GS. Correlations between mechanical stresshistory and tissue differentiation in initial fracture healing. Journal ofOrthopaedic Research. 1988;7:398–407.

PLOS ONE 20/23

Publications

161

15. Claes LE, Heigele CA. Magnitudes of local stress and strain along bony surfacespredict the coarse and type of fracture healing. Journal of Biomechanics.1999;32:255–66.

16. GERIS L, OOSTERWYCK HV, SLOTEN JV, DUYCK J, NAERT I.Assessment of Mechanobiological Models for the Numerical Simulation of TissueDifferentiation around Immediately Loaded Implants. Comput MethodsBiomech Biomed Engin. 2003;6:277–88.

17. Kelly DJ, Prendergast PJ. Mechano-regulation of stem cell differentiation andtissue regeneration in osteochondral defects. Journal of Biomechanics.2005;38(7):1413–22.

18. Lacroix D, Prendergast PJ. A mechano-regulation model for tissuedifferentiation during fracture healing: analysis of gap size and loading. Journalof Biomechanics. 2002;35:1163–71.

19. Lacroix D, Prendergast PJ, Li G, Marsh D. Biomechanical model of simulatetissue differentiation and bone regeneration: application to fracture healing.Medical and Biological Engineering and Computing. 2002;40:14–21.

20. Kang KT, Park JH, Kim HJ, Lee HM, Lee KI, Jung HH, et al. Study onDifferentiation of Mesenchymal Stem Cells by Mechanical Stimuli and anAlgorithm for Bone Fracture Healing. Tissue Engineering and RegenerativeMedicine. 2011;8(4):359–70.

21. Stops AJ, Heraty KB, Browne M, O’Brien FJ, McHugh PE. A prediction of celldifferentiation and proliferation within a collagen-glycosaminoglycan scaffoldsubjected to mechanical strain and perfusive fluid flow. J Biomech.2010;43(4):618–26.

22. Mousavi SJ, Doblare M, Doweidar MH. Computational modelling of multi-cellmigration in a multi-signalling substrate. Physical Biology. 2014;11(2):026002(17pp).

23. Mousavi SJ, Doweidar MH, Doblare M. 3D computational modelling of cellmigration: A mechano-chemo-thermo-electrotaxis approach. J Theor Biol.2013;329:64–73.

24. Mousavi SJ, Doweidar MH, Doblare M. Computational modelling and analysisof mechanical conditions on cell locomotion and cell-cell interaction. ComputMethods Biomech Biomed Engin. 2014;17(6):678–93.

25. Mousavi SJ, Doweidar MH, Doblare M. Cell migration and cell-cell interactionin the presence of mechano-chemo-thermotaxis. Mol Cell Biomech.2013;10(1):1–25.

26. Allena R, Aubry D. Run-and-tumble or look-and-run? A mechanical model toexplore the behavior of a migrating amoeboid cell. Journal of TheoreticalBiology. 2012;306:15–31.

27. Allena R. Cell migration with multiple pseudopodia: temporal and spatialsensing models. Bull Math Biol. 2013;75(2):288–316.

28. Wehrle-Haller B, Imhof BA. Actin, microtubules and focal adhesion dynamicsduring cell migration. The International Journal of Biochemistry & Cell Biology.2003;35(1):39–50.

PLOS ONE 21/23


162

29. Mousavi SJ, Doweidar MH. A Novel Mechanotactic 3D Modeling of CellMorphology. J Phys Biol. 2014;11(4):doi:10.1088/1478–3975/11/2/026002.

30. Zaman MH, Kamm RD, Matsudaira P, Lauffenburgery DA. ComputationalModel for Cell Migration in Three-Dimensional Matrices. Biophysical Journal.2005;89:1389–1397.

31. James DW, Taylor JF. The stress developed by sheets of chick fibroblasts invitro. Experimental Cell Research. 1969;54:107–110.

32. Ramtani S. Mechanical modelling of cell/ECM and cell/cell interactions duringthe contraction of a fibroblast-populated collagen microsphere:theory and modelsimulation. Journal of Biomechanics. 2004;37:1709–1718.

33. Palsson E. A three-dimensional model of cell movement in multicellular systems.Future Generation Computer System. 2001;17:835–852.

34. Taylor DL, Heiple J, Wang YL, Luna EJ, Tanasugarn L, Brier J, et al. Cellularand molecular aspects of amoeboid movement. CSH Symp Quant Biol.1982;46:101–111.

35. Brofland GW, Wiebe CJ. Mechanical effects of cell anisotropy on epithelia.Computer Methods Biomech Biomed Engin. 2004;7(2):91–99.

36. Discher DE, Janmey P, Wang YL. Tissue cells feel and respond to the stiffnessof their substrate. Science. 2005;310:1139–43.

37. Paszek MJ, Zahir N, Johnson KR, Lakins JN, Rozenberg GI, Gefen A, et al.Tensional homeostasis and the malignant phenotype. Cancer Cell.2005;8:241–54.

38. Gladman SJ, Ward RE, Michael-Titus AT, Knight MM, Priestley JV. Theeffect of mechanical strain or hypoxia on cell death in subpopulations of ratdorsal root ganglion neurons in vitro. Neuroscience. 2010;171(2):577–87.

39. Harrison NC, del Corral RD, Vasiev B. Coordination of Cell Differentiation andMigration in Mathematical Models of Caudal Embryonic Axis Extension. PLoSONE. 2011;6:e22700.

40. Kearney EM, Prendergast PJ, Campbell VA. Mechanisms of strain-mediatedmesenchymal stem cell apoptosis. J Biomech Eng. 2008;130(6):061004.

41. Ulrich TA, Pardo EMDJ, Kumar S. The mechanical rigidity of the extracellularmatrix regulates the structure, motility, and proliferation of glioma cells. CancerResearch. 2009;69:4167–4174.

42. Kurpinski K, Chu J, Hashi C, Li S. Anisotropic mechanosensing bymesenchymal stem cells. Proc Natl Acad Sci USA. 2006;103:16095–100.

43. Wozniak MA, Chen CS. Mechanotransduction in development: a growing rolefor contractility. Nat Rev Mol Cell Biol. 2009;10:34–43.

44. Prokharaua PA, Vermolena FJ, Garcia-Aznar JM. A mathematical model forcell differentiation, as an evolutionary and regulated process. Comput MethodsBiomech Biomed Engin. 2014;17(10):1051–70.

45. Cullinane DM, Salisbury KT, Alkhiary Y, Eisenberg S. Effects of the localmechanical environment on vertebrate tissue differentiation during repair: doesrepair recapitulate development. J Exp Biol. 2003;206:2459–71.

PLOS ONE 22/23

Publications

163

46. Wu QQ, Chen Q. Mechanoregulation of Chondrocyte Proliferation, Maturation,and Hypertrophy: Ion-Channel Dependent Transduction of Matrix DeformationSignals. Exp Cell Res. 2000;256(2):383–91.

47. Fouliarda S, Benhamidaa S, Lenuzzab N, Xaviera F. Modeling and simulationof cell populations interaction. Mathematical and Computer Modelling.2009;49(11):2104–8.

48. Thery M, Bornens M. Cell shape and cell division. Current Opinion in CellBiology. 2006;18(6):648–57.

49. Hibbit, Karlson, Sorensen. Abaqus-Theory manual; 2011.

50. Akiyama SK, Yamada KM. The interaction of plasma fibronectin withfibroblastic cells in suspension. Journal of Biological Chemistry.1985;260(7):4492–4500.

51. Schafer A, Radmacher M. Influence of myosin II activity on stiffness offibroblast cells. Acta Biomaterialia. 2005;1:273–280.

52. Oster GF, Murray JD, Harris AK. Mechanical aspects of mesenchymalmorphogenesis. J Embryol Exp Morphol. 1983;78:83–125.

53. Penelope CG, Janmey PA. Cell type-specific response to growth on softmaterials. Journal Applied Physiology. 2005;98:1547–1553.

54. Isaksson H, Wilson W, van Donkelaar CC, Huiskes R, Ito K. Comparison ofbiophysical stimuli for mechano-regulation of tissue differentiation duringfracture healing. J Biomech. 2006;39(8):1507–16.

55. Solon J, Leventa I, Sengupta K, Georges PC, Janmey PA. FibroblastAdaptation and Stiffness Matching to Soft Elastic Substrates. Biophys J.2007;93(12):4453–61.

56. Evans ND, Minelli C, Gentleman E, LaPointe V, Patankar SN, Kallivretaki M,et al. Substrate stiffness affects early differentiation events in embryonic stemcells. Eur Cell Mater. 2009;18:1–14.

57. Fu J, Wang YK, Yang MT, Desai RA, Yu X, Liu Z, et al. Mechanical regulationof cell function with geometrically modulated elastomeric substrates. NatureMethods. 2010;7:733–736.

58. Buxboim A, Ivanovska IL, Discher DE. Matrix elasticity, cytoskeletal forces andphysics of the nucleus: how deeply do cells ’feel’ outside and in? Journal of cellscience. 2010;123:297–308.

59. Zemel A, Rehfeldt F, Brown AE, Discher DE, Safran SA. Optimal matrixrigidity for stress fiber polarization in stem cells. Nat Phys. 2010;6:468–473.

60. Burke DP, Kelly DJ. Substrate Stiffness and Oxygen as Regulators of Stem CellDifferentiation during Skeletal Tissue Regeneration: A MechanobiologicalModel. PLoS ONE. 2012;7(7):e40737.

61. Hera GJ, Wub HC, Chenc MH, Chene MY, Change SC, Wanga TW. Control ofthree-dimensional substrate stiffness to manipulate mesenchymal stem cell fatetoward neuronal or glial lineages. Acta Biomaterialia. 2013;9(2):5170–80.

62. Friedland JC, Lee MH, Boettiger D. Mechanically activated integrin switchcontrols α5β1 function. Science. 2009;323:642–44.

PLOS ONE 23/23


164


166

Chapter 3

Conclusions and Future Works


3.1 Conclusions

Cells respond in vitro as well as in vivo to mechanotactic [24], thermotactic [42], chemotac-

tic [48] and/or electrotactic [46] signals through different types of reactions such as migration,

differentiation and proliferation. To predict the behavior of an individual cell and/or cell pop-

ulation initially a computational model with constant cell shape was designed. Afterwards, it

was extended to consider cell shape changes as a reaction of the previously mentioned signals.

This model has been developed based on force equilibrium on cell body using the Finite Ele-

ment (FE) discrete methodology and considering the cell as a group of finite elements.

The obtained results show the important role of the boundary conditions, stiffness gradient,

substrate depth and cell-cell interaction in cell migration. Any change in the boundary con-

ditions may change the cell migration path and final position. For a substrate composed of

two parts, soft and stiff, in the existence of constrained surfaces, the call initial location play

an important role on cell behavior. Generally, the cell migrates towards stiffer or more fixed

region in its neighborhoods. For instance, when the cell initial location is near the constrained

boundary surface in the soft part of substrate, the cell do not migrate towards the stiffer region

of the substrate, because the signal received from the constrained wall is stronger than that of

coming from the stiffer region [106]. Additionally, depending on the boundary conditions, one

or two Imaginary Equilibrium Planes (IEPs) around which the cell displays different behaviors

may appear. In the case of a substrate with a stiffness gradient, the cell tends to migrate

towards an IEP, located far from free boundary surfaces, regardless its initial position. The ob-

tained results demonstrate that during cell migration towards the stiffer region, nodal traction

force increases while net traction force as well as cell velocity decrease. However, in very stiff

substrates, generated net traction forces may not be enough to drive the cell to a new position.

The studied numerical cases for superficial single cell migration on a substrate with constrained

bottom surface demonstrates that the cell tends to migrate towards the minimum depth [106].

In this case, depending on cell type, depth of substrate and matrix stiffness, the cell superficial

sensing radius is greater than its depth feeling, consistent with previous experimental [107] and

numerical results [108].

When other activation signals are added to the substrate, the overall cell behavior changes.

For instance, in a substrate with stiffness gradient, in presence of chemotaxis and/or ther-

motaxis the location of the IEP is sensitive to their effective factors. In both cases, the IEP

displaces towards higher temperature and/or greater chemoattractant concentration. Even,

high chemotactic effective factor may dominate mechanotaxis cues and propels the cell towards

169

3.1. Conclusions

the chemoattractant source. However, the effect of these signals is negligible on the cell trac-

tion force and local velocity. This is because the magnitude of the net traction force depends

on the surrounding mechanical conditions rather these cues. On the contrary, these signals

remarkably increase the cell Random Index (RI), reducing the cell random motility. Because

they persistently direct the cell towards high chemoattractant concentration and/or warmer

location [109]. These findings are in agreement with chemotaxis [37, 47, 110] and thermotaxis

[43, 49] experimental works. In the presence of exogenous direct current electric field (dcEF),

a typical cell migrates towards the cathode pole. Consistent with findings of Nishimura et al.

[52], increasing the EF strength may significantly accelerate the cell migration. Besides, the

presence of electrotaxis reduces the cell random movement even more than that of thermotaxis

and chemotaxis. Since the cell reorientation is strongly towards the cathode pole, located on

unconstrained substrate wall, the cell remains near cathodal pole without almost any movement

[109]. This is due to the domination of the generated force by the EF over the other forces

[111]. This also explains why EFs play a dominant guidance role in directional migration of

epithelial cells during wound healing [46].

Interaction between two cells within a substrate with stiffness gradient decreases their global

speed towards stiffer regions because they tend to interact each other and to maintain their

contact for a long time. This is due to the stretched region between two cells that is generated

in consequence of exerted contractile forces [106]. This phenomenon is also noted for collec-

tive cell migration which may be found in sorting of different cells [82] or in modulation of

epithelial tissue [112]. In these cases, cells tend to come into contact before migrating towards

the stiffest or most fixed regions and then aggregate near stiffest region [113]. Probing the cell

slug centroid (CSC) is one of the key parameters to evaluate the behavior of cell population

during collective cell migration. Adding thermotaxis, chemotaxis and/or electrotaxis cues to

the substrate with a stiffness gradient displaces the CSC towards the corresponding signal, even

if it is located on the unconstrained wall of the substrate. It occurs due to the collaboration

of these cues with the stiffness gradient which is consistent with the earlier experimental ob-

servations [36, 42, 47, 52]. Contrary to single cell migration, during collective cell migration a

peak appears in the average net traction force and local velocity curves in the cell accumulation

instance [113]. Activation of another signal in the substrate, of course depending on strength

of the signal, shifts the peaks of these curves in direction of the imposed gradient, delaying the

first contact instance of the cells. In the presence of exogenous dcEF, the cells aggregate near

the cathode. Amplification of the dcEF linearly increases the cell charge due to Ca2+ influx

enhancement [113]. This, depending on the the dcEF strength, increases the electrical force

170


exerted on the cell population and displaces the CSC towards the cathode, consistent with the

findings of Nishimura et al. [52]. Besides, for low dcEF (10 mV/mm) the external cells of the

slug may have some motility but when the dcEF strength increases to the saturation value (100

mV/mm) they are almost immobilized while cell aggregation are flattened near the cathode

[113]. This again recalls the electrotaxis dominant role over other signals [46]. Cells RI in a

multi-signaling micro-environment (combination of cell RI) demonstrates that collective cell

migration is significantly directional under the electrotactic stimulus. A comparison of collec-

tive and single cell migration illustrates that, from some viewpoints, collective cell migration

differs from single cell migration. For instance, in collective cell migration, cells do not migrate

immediately towards the final destination but they first tend to make contact (mechanotactic

effect) and then aggregate near the most effective cue/s. Moreover, cell-cell interaction not only

may delay collective cell migration but it also slightly decreases the average cell RI. Besides,

cell aggregation causes the average net traction force and local velocity of the cells to increase

and decrease instantaneously while this is not the case during single cell migration [109, 113].

The presented 3D model also enables us to investigate cell morphology changes due to mechan-

otaxis, thermotaxis, chemotaxis and/or electrotaxis [114, 115]. Depending on the nature of

the adhesion receptors through which the cells bind to their substrate, the substrate stiffness

may lead to considerable changes in cell morphology [26, 95, 116, 117]. The presented model

demonstrates that the morphology of an adherent cell is, between others, regulated by substrate

stiffness [3, 47]. Specifically, within soft (several kPa) and hard (200 kPa) substrates, the cell

shape change is trivial and it remains mainly round. The incapability of the cell to penetrate

prevents the cell elongation and causes the cell membrane area to remain constant, where the

Cell Membrane Index (CMI) keeps equal to 1. On the contrary, the cell undergoes considerable

elongation within intermediate and rigid substrates (tens of kPa) where the cell membrane area

follows the same trend as the cell elongation. These parameters increase when a stiffness gradi-

ent is employed in cell substrates and are maximum for substrates with a constrained surface.

However, within constrained substrates the cell elongation decreases near the constrained sur-

face. This is because the cell polarization direction persistently points towards the constrained

surface while the cell cannot move beyond the constrained surface. In this sense, it spreads on

the constrained surface and becomes elongated in the vertical direction to the gradient direction

[114]. The overall cell shape may be changed by adding another stimulus in the cell substrate.

For instance, by activation of thermotaxis and/or chemotaxis in the substrate, the maximum

cell elongation and CMI increase and the location of the cell centroid displaces towards the

warmer region and/or chemical source. In the presence of chemotaxis, the cell elongates in

171

3.1. Conclusions

chemical gradient direction but near the chemoattractant source the cell elongation decreases

and cell pseudopods are extended perpendicular to chemical gradient. The cell elongation and

CMI slightly increase, in the case of greater chemical effective factor. Even so, they are maxi-

mum in presence of electrotaxis [115]. This is due to the addition of electrostatic force to the

mechanotactic force, in agreement with experimental observations [52]. So, it can be deduced

that the stronger the signaling the greater the cell elongation and CMI.

Recently, different hypotheses which propose that the mechanical signals may regulate cell fate

have been presented [101, 118]. Consistent with experimental observations [32, 33, 35, 119, 120],

the findings obtained here indicate that Mesenchymal Stem Cells (MSC) differentiation to neu-

rogenic, chondrogenic or osteogenic lineage specifications occurs within soft (0.1-1 kPa), inter-

mediate (20-25 kPa) or hard (30-45 kPa) substrates, respectively. The traction forces generated

by a specific cell phenotype may increase (osteoblasts and chondrocytes) or decrease (neurob-

last) when cell undergoes differentiation [120]. On the contrary, cell proliferation of a typical

cell suddenly increases the average net traction force due to an asymmetric distribution of the

nodal traction forces in consequence of cell-cell interaction. Moreover, increasing the matrix

stiffness, in consistent range, can accelerate cell proliferation and differentiation [119]. Because

any changes in the matrix stiffness causes alteration in the cell internal deformation which

accordingly regulates the cell differentiation and/or proliferation [121]. On the other hand, the

cell differentiation or proliferation depends on the cell maturation so, cells probably need a

longer time to undergo full maturation within soft substrates in comparison with intermediate

and hard substrates, [122]. The results of the model presented here and the earlier experimental

observations [32, 33, 35, 119, 120] show that matrix stiffness plays a significant role to control

the fate decision of pluripotent MSCs.

Taken all together, the results obtained here are qualitatively consistent with those of cor-

responding experimental works reported in the literature [26, 32, 33, 35, 37, 43, 47, 52, 117,

119, 120]. However, more sophisticated experiments are a key tool to demonstrate the precise

response of the cell for different environmental signaling pathways, to illustrate the accuracy of

the obtained results and for the calibration of the presented model. Therefore, the presented

model exposes useful insights to predict the cell behavior for any cell shape and substrate as

well as to help in the design of laboratory experiments for single cell or population of cells.

In addition, as the magnitude of the cell velocity, nodal traction forces and net traction force

depend on the cell type and the matrix stiffness, the presented model is a helpful alternative

tool to estimate all these parameters. In summary, the proposed model provides a better un-

derstanding of the mechanism behind single and collective cell migration in presence of different

172


stimuli. It has the potential to simultaneously consider the cell shape changes in multi-signaling

substrates and accordingly enable us to successfully predict essential aspects of cell matura-

tion, proliferation, differentiation and apoptosis during regenerative processes. It provides one

step forward in computational methodology to simultaneously consider different features of cell

behavior which are a concern in various biological processes.

3.2 Future works

� Time cost remains a challenging problem in finite element discrete models. Because some-

times to acquire a precise results, it is essential to increase the number of domain elements.

Unfortunately, increase of the element number progressively increases simulation time cost.

Model order reduction (MOR) is one of the best alternatives to reduce time cost of numeri-

cal models. In MOR, the high-dimension original model is replaced by a system of smaller

dimension so that the time cost of simulation remarkably reduces. Therefore, using MOR, it

is possible to acceptably approximate the time-costing of the original model with an enough

accuracy. In the presented model, to increase the number of cell elements, it is necessary to

increase the element number of whole domain that results increase of time involving. There-

fore, by coupling of the presented model with a suitable MOR methodology we may reduce

simulation time when it is needed.

� Through the numerical model presented here, for cell differentiation and proliferation, the

effect of external forces acting on cell-substrate system doesn’t studied. Experimental works

demonstrate that one of the important alternatives which can activate signaling pathways of

differentiation and proliferation of smooth muscle cells (SMCs) is cyclic stretch of the substrate

[118]. Therefore, it would be interesting to extend the presented model of cell differentiation

to predict the effect of cyclic external forces on SMC and hMSC growth.

� Tissue construction in vitro remains conventional in 3D scaffolds. Recent advances in 3D

cell culture open new perspectives in the context of in in vitro reconstruction at the micro- and

macro-scales. Multiple parallel multi-scale tissue engineering studies are ongoing; however, no

comprehensive tissue engineer for most of organs that is appropriate for clinical use has yet

been realized. The integration of multi-scale tissue engineering studies seems to be essential

for further understanding of tissue reconstruction strategies. Therefore, it would be interesting

to link the present micro-scale model to an appropriate macro-scale model to predict tissue

generation in macro-level.

� Experimental observations demonstrate that when cell migrates through physically confined

173

3.2. Future works

spaces, water permeation is a major mechanism of cell migration in such confined microenvi-

ronments. For instance, tumor cells which are confined in a micro channels establish ions and

water pump through the cell membrane. This phenomenon leads to cell translocation due to

an inflow of ions and water at the cell leading edges as well as an outflow of ions and water at

the trailing edges. Therefore, it would be interesting to numerically study this mechanism of

cell migration which is regulated by water permeation through cell membrane.

� Tumors may be characterized by their high interstitial fluid pressure. There are specific

rates of tumor-associated interstitial fluid flow which lead to cell migration in the upstream

directions. Despite numerous studies in this field, due to the complexity of this phenomenon,

the biophysical mechanism behind it remains remarkably incomplete. However, experimental

investigations propose that stresses imposed by interstitial fluid flow may affect matrix adhe-

sions due to generated drag force. Regulation of interstitial fluid pressure in such a situations is

a fundamental key factor to control cell migration within porous extracellular matrices. There-

fore, numerical simulations may help to study the effect of shear stresses on cell migration in

such conditions.

� Cells contain several mechano-sensing components that transduce mechanical signals into

biochemical cascades. During cell-matrix adhesion, a complex network of molecules mechani-

cally couples the extracellular matrix (ECM), cytoskeleton (CSK), nucleus and nucleoskeleton.

Loss of cytoskeletal tension during cell migration and/or transmission of unexpected force at

adhesions via this components may result in cell nuclear deformation which can ultimately lead

to an alteration of gene expression and other cellular changes. Therefore, it would be inter-

esting to consider cytoskeletal stress distribution to evaluate nuclear deformation due to force

transmission through CSK under different cell micro-environmental characteristics.

174

Conclusiones

Las células responden, tanto in vitro como in vivo, a diferentes señales mecanotácticas [24],

termotácticas [42], quimiotácticas [48] y/o electrotácticas [46] mediante diferentes tipos de reac-

ciones tales como la migración, diferenciación y proliferación. Para predecir el comportamiento

de una célula individual, así como de una población celular, inicialmente se diseñó un mod-

elo computacional de células con forma fija. Posteriormente, dicho modelo fue extendido para

poder considerar los cambios que tienen lugar en la morfología celular como respuesta a las di-

versas señales externas, anteriormente mencionadas. Finalmente, el modelo fue extendido para

investigar la diferenciación y la proliferación celular como consecuencia de diferentes procesos

mecano-sensores. Dichos modelos han sido desarrollados basándonos en el equilibrio de fuerzas

que tiene lugar en el cuerpo de una célula, usando para ello el método de los Elementos Finitos

(EF) y considerando la célula como un grupo de elementos finitos. Los resultados obtenidos

muestran la gran influencia de las condiciones de contorno, gradientes de rigidez, profundidad

del substrato e interacción célula-célula en la migración celular. Cualquier cambio que tenga

lugar en las condiciones de contorno puede desencadenar cambios en la migración celular y,

consecuentemente, en la posición final de la célula. Para un substrato compuesto de dos partes,

una rígida y otra más blanda, la posición inicial de la célula juega un papel determinante en su

comportamiento. Por lo general, las células migran hacia zonas de mayor rigidez. Por ejemplo,

cuando la localización inicial de la célula es cercana a la zona que limita la parte blanda del

substrato, esta no migra hacia la región rígida, ya que la señal recibida de la pared limítrofe es

más fuerte que aquella que proviene de la parte rígida [106]. Además, en substratos compuestos

por dos regiones, una rígida y otra blanda, dependerá de las condiciones de contorno, aparezcan

uno o dos Planos Imaginarios de Equilibrio (PIE) alrededor de los cuales las células muestran

diferentes comportamientos celulares.

175

En el caso de un substrato con un gradiente de rigidez, las células tienden a migrar hacia

aquellas zonas con un PIE, localizado lejos de las superficies de contorno libres, sin importar

la posición inicial de dichas células. Los resultados obtenidos demuestran que durante la mi-

gración celular hacia aquellas zonas de mayor rigidez, las fuerzas de tracción nodales aumentan

mientras la fuerza de tracción neta y la velocidad celular se reducen. Sin embargo, en substratos

muy rígidos, las fuerzas de tracción netas generadas pueden no ser suficientes para dirigir a la

célula a una nueva posición. Los casos estudiados numéricamente de la migración superficial

de células individuales en un sustrato con superficie inferior restringido demuestra que la célula

tiende a migrar hacia la mínima profundidad [108]. En este caso, dependerá del tipo celular,

la profundidad del substrato y la rigidez de la matriz, el radio de sensibilidad superficial de la

célula es mayor que su sensibilidad en profundidad, en consistencia con experimentos previos

[107] y resultados numéricos [108].

Cuando otra señal de activación se añade al substrato, el comportamiento general de la célula

cambia. Por ejemplo, en un substrato con un gradiente de rigidez, en presencia de quimio-

taxis y/o termotaxis, la localización del PIE es sensible a sus factores de eficacias. En ambos

casos, el PIE se desplaza hacia zonas con temperatura más elevada y/o mayor concentración

quimiotáctica. Incluso, factores quimiotácticos elevados pueden dominar cuestiones mecanotác-

ticas y llevar a la célula hacia fuentes quimiotácticas. Sin embargo, el efecto de estas señales

no es considerable en la fuerza de tracción celular y en la velocidad local, ya que la magnitud

de la fuerza de tracción celular neta depende del ambiente mecánico en mayor medida que de

otras cuestiones. Por el contrario, estas señales aumentan considerablemente el índice Aleatorio

Celular (IAC), reduciendo la movilidad celular aleatoria, ya que estas dirigen la célula hacia

concentraciones quimiotácticas elevadas y/o localizaciones cuya temperatura es mayor [109].

Estos descubrimientos están en concordancia con experimentos quimiotácticos [37, 47, 110] y

termotácticos [43, 49]. En presencia directa de Campos Eléctricos Exógenos (CEE), las célu-

las suelen migrar hacia los polos catódicos. Según investigaciones realizadas por Nishimura et

al. [52], un incremento de la fuerza de los CEE puede acelerar significativamente la migración

celular. Además, la presencia de electrotaxis reduce el movimiento celular aleatorio incluso

más que la termotaxis y la quimiotaxis. Esto es debido a que la reorientación celular es mayor

hacia el polo catódico, localizado en paredes no restringidos del substrato, la celula permanece

cerca del polo catódico prácticamente sin realizar ningún movimiento [109]. Esto es debido a

que la fuerza generada por el CEE tiene una mayor influencia en la célula que otro tipo de

fuerzas [111]. Esto también explica por qué el CEE juega un papel dominante a la hora de

direccionar la migración de las células epiteliales en la cicatrización de heridas [46]. Además,

176

Conclusiones

la combinación de las diferentes señales que guían a la célula a través del substrato ilustra el

gran papel que tiene la electrotaxis en la reducción del movimiento aleatorio de la célula en

comparación con otros estímulos.

La interacción entre dos células localizadas en un substrato con gradiente de rigidez, reduzca la

velocidad de migración de las mismas hacia la zona de mayor rigidez, debido a que las células

tienden a migrar hacia donde se encuentran otras células y mantener el contacto por periodos

largos de tiempo. Esto se debe a la región alargada que se genera entre dos células como con-

secuencia de las fuerzas contráctiles que estas ejercen [106]. Este fenómeno se observa también

en la migración de una población celular de diferentes tipos celulares [82] así como en la modu-

lación del tejido epitelial [112]. Además, en el caso de la migración de una población celular, las

células tienden a entrar en contacto entre las células vecinas antes de migrar hacia zonas más

rígidas del substrato [113]. Es bien sabido que la localización del Centroide de dichos Agrega-

dos Celulares (CAC) es uno de los parámetros clave a la hora de evaluar el comportamiento

de la población celular durante la migración colectiva. Añadir termotaxis, quimiotaxis y/o

electrotaxis al substrato con un gradiente de rigidez supone el desplazamiento del CAC hacia

la señal correspondiente, incluso si está localizada en la pared libre del substrato. Esto se debe

al efecto conjunto de dichas señales con el gradiente de rigidez del substrato, esto es consistente

con diversas observaciones experimentales [36, 42, 47, 52]. Al contrario a lo que ocurre en la

migración celular de una única célula, durante la migración celular colectiva aparece un pico

en la fuerza de tracción neta y en las curvas de velocidad local en el momento en el que se pro-

duce la acumulación de diferentes células [113]. La activación de otras señales en el substrato,

dependiendo de su intensidad, controla dichos picos guiándolos en la dirección del gradiente

impuesto, retrasando así el primer contacto entre células. En presencia de un campo exógeno,

se producen los agregados celulares en las zonas cercanas al cátodo. La ampliación de dicho

campo produce un incremento lineal de los cambios celulares debido al influjo de Ca2+ [113].

Esto produce un incremento de la fuerza eléctrica ejercida en la población celular y desplaza el

CAC hacia el cátodo, consistentemente con las investigaciones realizadas por Nishimura et al.

[52]. Además, para CEE pequeños (10 mV/mm) las células externas al agregado deben tener

alguna movilidad, pero cuando la fuerza del CEE aumenta al valor de saturación (100 mV/mm),

estas están prácticamente inmovilizadas mientras que los agregados celulares se mueven cerca

del cátodo [113]. Esto pone nuevamente en evidencia el papel predominante de la electrotaxis

frente a otro tipo de señales [46]. EL IAC en un microambiente multi-señal (combinación de

IACs) demuestra que la migración de una población celular es significativamente direccional

bajo estímulos electrotácticos. La comparación entre la migración de una población celular y

177

la migración de una única célula ilustra que, desde algunos puntos de vista, la migración de

la población celular difiere de la segunda. Por ejemplo, en la migración colectiva, las células

no tienden a migrar inmediatamente hacia la posición final, sino que en primer lugar tratan

de agregarse unas con otras (efecto mecanotáctico) y entonces unirse cerca de las señales más

efectivas. Además, la interacción célula-célula no sólo retrasa la migración celular colectiva

sino que también reduce el IAC medio. Además, la agregación celular produce el incremento y

reducción instantáneo de la fuerza de tracción neta media y de la velocidad local de las células,

mientras que esto no ocurre en la migración de una célula individual [109, 113].

El modelo presente también nos permite investigar los cambios en la morfología celular causa-

dos por la mecanotaxis, termotaxis, quimiotaxis y/o electrotaxis [114, 115]. Dependiendo de

la naturaleza de los receptores de adhesión a través de los cuales las células se unen al sub-

strato, la rigidez del substrato puede influir en los cambios que tienen lugar en la morfología

celular [26, 95, 116, 117]. El modelo demuestra que la morfología de las células adherentes

está, entre otros, regulada por la rigidez del substrato [3, 47]. Específicamente, en substratos

blandos (varios kPa) y duros (200 kPa), el cambio en la forma celular es trivial y se mantiene

principalmente redonda. La incapacidad de las células de penetrar evita la elongación celular y

causa que el área de la membrana celular se mantenga constante, donde el índice de Membrana

Celular (IMC) se mantiene igual a 1. Por el contrario, la célula sufre elongaciones considerables

en substratos intermedios y rígidos (decenas de kPa) donde el área de membrana celular sigue

la misma tendencia que la de la elongación celular. Estos parámetros se incrementan cuando

existen gradientes de rigidez en el substrato y son máximos para aquellos substratos con una

superficie restringida. Sin embargo, en substratos con superficies restringida la elongación celu-

lar se reduce cerca de estas superficies. Esto se debe a que la dirección de polarización celular se

produce hacia superficies restringidas mientras que las células no pueden moverse hacia dichas

superficies. En este sentido, se extienden hacia la superficie restringida y se alargan en dirección

vertical respecto a la dirección del gradiente [114]. La forma celular puede cambiar al añadir

otro tipo de estímulos in el substrato. Por ejemplo, mediante la activación de termotaxis y/o

quimiotaxis en el substrato, la máxima elongación celular y el IMC aumenta y la localización

del centroide celular se desplaza hacia las regiones más cálida y/o hacia la fuente química. En

presencia de quimiotaxis, la célula se alarga en la dirección del gradiente químico pero cerca de

la fuente quimiotáctica, la elongación celular se reduce y los seudópodos celulares se extienden

perpendicularmente al gradiente químico. La elongación celular y el IMC se reducen ligera-

mente, en el caso de factores efectivos químicos mayores. Incluso son máximos en presencia de

electrotaxis [115]. Esto se debe a la adición de fuerzas electrostáticas a aquellas mecanotácticas,

178

Conclusiones

en concordancia con observaciones experimentales [52]. Así pues, se deduce que cuanto más

intensa sea la señal, mayor será la elongación celular y el IMC.

Recientemente, diferentes hipótesis que proponen que las señales mecánicas pueden regular el

destino celular ha sido presentadas [101, 118]. En consistencia con observaciones experimen-

tales [32, 33, 35, 119, 120], estas indican que la diferenciación de Células Madre Mesenquimales

(CMM) a linajes neurogénicos, condrogénicos u osteogénicos tienen lugar en substratos blandos

(0.1-1 kPa), intermedios (20-25 kPa) o duros (30-45 kPa), respectivamente. Las fuerzas de trac-

ción generadas por un fenotipo celular específico pueden aumentar (osteoblastos y condrocitos)

o reducirse (neuroblastos) cuando se produce la diferenciación celular [120]. Por el contrario,

la proliferación celular de una célula típica aumenta la fuerza de tracción neta debido a la

distribución asimétrica de las fuerzas de tracción nodales como consecuencia de la interacción

célula-célula. Además, el incremento de la rigidez de la matriz, en un rango consistente, puede

acelerar la proliferación y diferenciación celular [119]. Esto se debe a que cualquier modificación

en la rigidez del substrato produce alteración en la deformación celular interna, la cual regula la

diferenciación y/o la proliferación [121]. Por otro lado, ambos procesos celulares dependen de la

maduración celular, las células probablemente necesitan más tiempo para madurar por completo

en substratos blandos en comparación con substratos intermedios o duros, [122]. Los resultados

del modelo aquí presentado y las incipientes observaciones experimentales [32, 33, 35, 119, 120]

muestran que la rigidez de la matriz juega un papel determinante a la hora de controlar el

destino de las CMM pluripotenciales.

Teniendo en cuenta todo ello, los resultados obtenidos están en consistencia con aquellos recogi-

dos de la literatura en lo referente a trabajos experimentales similares [26, 32, 33, 35, 37, 43,

47, 52, 117, 119, 120]. Sin embargo, se requieren experimentos más sofisticados para cuan-

tificar la respuesta celular con mayor precisión en presencia de diferentes señales, para poder

así demostrar la precisión de los resultados obtenidos con las simulaciones y para determinar

la precisión del modelo aquí presentado. Además, el modelo presentado ofrece grandes avances

a la hora de predecir el comportamiento celular de cualquier forma celular y en cualquier sub-

strato. Asimismo es de gran utilidad para el diseño de experimentos de laboratorio para una

única célula o población celular. Adicionalmente, como la magnitud de la velocidad celular,

fuerzas de tracción nodales y fuerzas de tracción netas dependen del tipo celular y de la rigidez

de la matriz, el modelo presentado, es una herramienta alternativa, útil a la hora de estimar to-

dos estos parámetros. En resumen, el modelo propuesto proporciona un mayor entendimiento

de los mecanismos que se esconden detrás de la migración de una única célula así como de

una población celular en presencia de diferentes estímulos. Considera de forma simultánea los

179

cambios de forma celulares en substratos bajo multi-señales y nos permite predecir aspectos

esenciales de la maduración celular, proliferación, diferenciación y apoptosis en procesos regen-

erativos. Además, va un paso más allá en cuanto a metodología computacional para considerar

de forma simultanea diferentes aspectos del comportamiento celular involucrados en numerosos

procesos biológicos.

180

Bibliography

[1] A. Aman and T. Piotrowski. Cell migration during morphogenesis. Dev Biol, 341(1):20–

33, 2010.

[2] C. Cintron, H. Covington, and C.L. Kublin. Morphogenesis of rabbit corneal stroma.

Invest Ophthalmol Vis Sci, 24(5):543–56, 1983.

[3] K.M. Hakkinen, J.S. Harunaga, A.D. Doyle, and K.M. Yamada. Direct comparisons of

the morphology, migration, cell adhesions, and actin cytoskeleton of fibroblasts in four

different three-dimensional extracellular matrices. Tissue Eng Part A, 17(5-6):713–24,

2011.

[4] G.F. Oster, J.D. Murray, and A.K. Harris. Mechanical aspects of mesenchymal morpho-

genesis. J Embryol Exp Morphol, 78:83–125, 1983.

[5] R.L. Trelstad and A.J. Coulombre. Morphogenesis of the collagenous stroma in the chick

cornea. J Cell Biol, 50(3):840–58, 1971.

[6] N. Savill and P. Hogeweg. Modeling morphogenesis: from single cells to crawling slugs.

J Theor Biol, 184:229–35, 1997.

[7] V.P. Hoppenreijs, E. Pels, G.F. Vrensen, J. Oosting, and W.F. Treffers. Effects of hu-

man epidermal growth factor on endothelial wound healing of human corneas. Invest

Ophthalmol Vis Sci, 33(6):1946–57, 1992.

[8] M. Matsuda, M. Sawa, H.F. Edelhauser, S.P. Bartels, A.H. Neufeld, and K.R. Kenyon.

Cellular migration and morphology in corneal endothelial wound repair. Invest Ophthal-

mol Vis Sci, 26(4):443–9, 1985.

[9] E.K. Kirkeeide, I.F. Pryme, and A. Vedeler. Morphological changes in krebs II ascites

tumour cells induced by insulin are associated with differences in protein composition

181

BIBLIOGRAPHY

and altered amounts of free, cytoskeletal-bound and membrane-bound polysomes. Mol

Cell Biochem, 118(2):131–40, 1992.

[10] D. Zink, A.H. Fischer, and J.A. Nickerson. Nuclear structure in cancer cells. Nat Rev

Cancer, 4(9):677–87, 2004.

[11] T. Lecuit, , and P.F. Lenne. Cell surface mechanics and the control of cell shape, tissue

patterns and morphogenesis. Nat Rev Mol Cell Biol, 8:633–44, 2007.

[12] M. Rauzi and P.F. Lenne. Cortical forces in cell shape changes and tissue morphogenesis.

Curr Top Dev Biol, 95:93–144, 2011.

[13] Y. Gong, C. Mo, and S.E. Fraser. Planar cell polarity signalling controls cell division

orientation during zebrafish gastrulation. Nature, 430(7000):689–93, 2004.

[14] R. McBeath, D.M. Pirone, C.M. Nelson, K. Bhadriraju, and C.S. Chen. Cell shape,

cytoskeletal tension, and rhoa regulate stem cell lineage commitment. Dev Cell, 6(4):483–

95, 2004.

[15] M. Théry and M. Bornens. Cell shape and cell division. Curr Opin Cell Biol, 18(6):648–

57, 2006.

[16] C. Ribeiro, V. Petit, and M.Affolter. Signaling systems, guided cell migration, and organo-

genesis: insights from genetic studies in drosophila. Dev Biol, 260(1):1–8, 2003.

[17] J.J. Tomasek, G. Gabbiani, and B. Hinz. Myofibroblasts and mechano-regulation of

connective tissue remodelling. Nat Rev Mol Cell Biol, 3(5):349–63, 2002.

[18] R.M. Castilho, C.H. Squarize, and K. Leelahavanichkul. Rac1 is required for epithelial

stem cell function during dermal and oral mucosal wound healing but not for tissue

homeostasis in mice. PLoS One, 5(1):e10503, 2010.

[19] A.F. Chambers, A.C. Groom, and I.C. MacDonald. Dissemination and growth of cancer

cells in metastatic sites. Nat Rev Cancer, 2(8):563–72, 2002.

[20] P. Friedl and E.B. Brocker. The biology of cell locomotion within three-dimensional

extracellular matrix. Cell Mol Life Sci, 57(1):41–64, 2000.

[21] E. Fuchs, T. Tumbar, and G. Guasch. Socializing with the neighbors: stem cells and

their niche. Cell, 116(6):769–78, 2004.

182

BIBLIOGRAPHY

[22] M.M. Nava, M.T. Raimondi, and R. Pietrabissa. Controlling self-renewal

and differentiation of stem cells via mechanical cues. J Biomed Biotechnol,

2012:doi:10.1155/2012/797410, 2012.

[23] R.J. Pelham, Jr., and Y.L. Wang. Cell locomotion and focal adhesions are regulated by

substrate flexibility. Proc Natl Acad Sci USA, 94(24):13661–5, 1997.

[24] C.M. Lo, H.B. Wang, M. Dembo, and Y.L. Wang. Cell movement is guided by the rigidity

of the substrate. Biophys J, 79(1):144–52, 2000.

[25] M.P. Sheetz, D. Felsenfeld, C.G. Galbraith, and D. Choquet. Cell migration as a five-step

cycle. Biochem Soc Symp, 65:233–43, 1997.

[26] M. Ehrbar, A. sala, P. Lienemann, A. Ranga, K. Mosiewicz, A. Bittermann, S.C. Rizzi,

and F.E. Weber. Elucidating the role of matrix stiffness in 3D cell migration and remod-

eling. Biophys J, 100(2):284–93, 2011.

[27] P. Friedl and K. Wolf. Plasticity of cell migration: a multiscale tuning model. J Cell

Biol, 188(1):11–9, 2009.

[28] T. Lammermann and M. Sixt. Mechanical modes of amoeboid cell migration. Curr Opin

Cell Biol, 21(5):636–44, 2009.

[29] D.A. Lauffenburger and A.F. Horwitz. Cell migration: a physically integrated molecular

process. Cell, 84(3):359–69, 1996.

[30] P. Moreo, J.M. Garcia-Aznar, and M. Doblaré. Modeling mechanosensing and its effect

on the migration and proliferation of adherent cells. Acta Biomater, 4(3):613–21, 2008.

[31] M.H. Zaman, R.D. Kamm, P. Matsudaira, and D.A. Lauffenburgery. Computational

model for cell migration in three-dimensional matrices. Biophys J, 89:1389–1397, 2005.

[32] A. Zemel, F. Rehfeldt, A.E.X Brown D.E. Discher, and S.A. Safran. Optimal matrix

rigidity for stress fiber polarization in stem cells. Nat Phys, 6:468–73, 2010.

[33] A.J. Engler, S. Sen, H.L. Sweeney, and D.E. Discher. Matrix elasticity directs stem cell

lineage specification. Cell, 126(4):677–89, 2006.

[34] P.M. Gilbert, K.L. Havenstrite, and et al. Substrate elasticity regulates skeletal muscle

stem cell self-renewal in culture. Science, 329(5995):1078–81, 2010.

183

BIBLIOGRAPHY

[35] N. Huebsch, P.R. Arany, and et al. Harnessing traction-mediated manipulation of the

cell/matrix interface to control stem-cell fate. Nat Mater, 9:518–26, 2010.

[36] E. Hadjipanayi, V. Mudera, and R.A. Brown. Guiding cell migration in 3D: A collagen

matrix with graded directional stiffness. Cell Motil. Cytoskeleton, 66(3):435–45, 2009.

[37] L. Bosgraaf and P.J.M. Van Haastert. Navigation of chemotactic cells by parallel signaling

to pseudopod persistence and orientation. PLoS ONE, 4(8):e6842, 2009.

[38] V.S. Deshpande, R.M. McMeeking, and A.G. Evans. A bio-chemo-mechanical model for

cell contractility. PNAS, 103(38):14015–20, 2006.

[39] O. Hoeller and R.R. Kay. Chemotaxis in the absence of PIP3 gradients. Curr Bio,

17(9):813–7, 2007.

[40] M.P. Neilson, D.M. Veltman1, P.J.M. van Haastert, S.D. Webb, J.A. Mackenzie, and R.H.

Insall. Chemotaxis: A feedback-based computational model robustly predicts multiple

aspects of real cell behaviour. PLoS Biol, 9(5):e1000618, 2011.

[41] F.J. Vermolen and A. Gefen. A phenomenological model for chemico-mechanically

induced cell shape changes during migration and cell-cell contacts. Biomech Model

Mechanobiol, 12(2):301–23, 2013.

[42] B.H. Choo, R.F. Donna, , and L.P. Kenneth. Thermotaxis of dictyostelium discoideum

amoebae and its possible role in pseudoplasmodial thermotaxis. Proc. Natl. Acad. Sci.

USA, 80(18):5646–9, 1983.

[43] A.A. Higazi, D. Kniss, J. Manuppelloand E.S. Barnathan, and D.B.Cines. Thermotaxis

of human trophoblastic cells. Placenta, 17(8):683–7, 1996.

[44] R.C. Gao, X.D. Zhang, and et al. Different roles of membrane potentials in electrotaxis

and chemotaxis of dictyostelium cells. Eukaryot Cell, 10(9):1251–6, 2011.

[45] E.M. Maria and M.B.A. Djamgoz. Cellular mechanisms of direct-current electric field

effects: galvanotaxis and metastatic disease. J Cell Sci, 117(9):1631–9, 2004.

[46] M. Zhao. Electrical fields in wound healing-An overriding signal that directs cell migra-

tion. Semin Cell Dev Biol, 20(6):674–82, 2009.

[47] P.J.M. Van Haastert. Chemotaxis: insights from the extending pseudopod. J Cell Sci,

123(18):3031–37, 2010.

184

BIBLIOGRAPHY

[48] K.F. Swaney, C.H. Huang, and P.N. Devreotes. Eukaryotic chemotaxis: a network of

signaling pathways controls motility, directional sensing, and polarity. Annu Rev Biophys,

39:265–89, 2010.

[49] M. Eisenbach A. Bahat. Sperm thermotaxis. Mol Cell Endocrino, 252(1-2):115–9, 2006.

[50] Y.X. Huang, X.J. Zheng, and et al. Quantum dots as a sensor for quantitative visu-

alization of surface charges on single living cells with nano-scale resolution. Biosens

Bioelectron, 26(8):2114–8, 2011.

[51] M.E. Mycielska and M.B. Djamgoz. Cellular mechanisms of directcurrent electric field

effects: galvanotaxis and metastatic disease. J Cell Sci, 117(9):1631–9, 2004.

[52] K.Y. Nishimura, R.R. Isseroff, and R. Nuccitelli. Human keratinocytes migrate to the

negative pole in direct current electric fields comparable to those measured in mammalian

wounds. J Cell Sci, 109(1):199–207, 1996.

[53] E.K. Onuma and S.W. Hui. A calcium requirement for electric field-induced cell shape

changes and preferential orientation. Cell Calcium, 6(3):281–92, 1985.

[54] L.J. Shanley, P. Walczysko, M. Bain, D.J. MacEwan, and M. Zhao. Influx of extracellular

Ca2+ is necessary for electrotaxis in dictyostelium. J Cell Sci, 119(22):4741–8, 2006.

[55] H. Bai, C.D. McCaig, J.V. Forrester, and M. Zhao. Dc electric fields induce distinct

preangiogenic responses in microvascular and macrovascular cells. Arterioscler Thromb

Vasc Biol, 24(7):1234–9, 2004.

[56] M.J. Brown and L.M. Loew. Electric field-directed fibroblast locomotion involves cell

surface molecular reorganization and is calcium independent. J Cell Biol, 127(1):117–28,

1994.

[57] M.S. Cooper and R.E. Keller. Perpendicular orientation and directional migration of

amphibian neural crest cells in dc electrical fields. Proc Natl Acad Sci USA, 81(1):160–4,

1984.

[58] C.A. Erickson and R. Nuccitelli. Embryonic fibroblast motility and orientation can be

influenced by physiological electric fields. J Cell Biol, 98(1):296–307, 1984.

[59] J. Lee, A. Ishihara, G. Oxford, B. Johnson, and K. Jacobson. Regulation of cell movement

is mediated by stretch-activated calcium channels. Nature, 400(6742):382–6, 1999.

185

BIBLIOGRAPHY

[60] E.K. Onuma and S.W. Hui. Electric field-directed cell shape changes, displacement, and

cytoskeletal reorganization are calcium dependent. J Cell Biol, 106(1):2067–75, 1988.

[61] N. Orida and J.D. Feldman. Directional protrusive pseudopodial activity and motility in

macrophages induced by extracellular electric fields. Cell Motil, 2(3):243–55, 1982.

[62] M.S. Cooper and M. Shliwa. Motility of cultured fish epidermal cells in the presence and

absence of direct current electric fields. J Cell Biol, 102(4):1384–99, 1986.

[63] M. cc, J. Pu, J.V. Forrester, and C.D. McCaig. Membrane lipids, egf receptors, and

intracellular signals colocalize and are polarized in epithelial cells moving directionally in

a physiological electric field. FASEB J, 16(8):857–9, 2002.

[64] M. Zhao, A. Agius-Fernandez, J.V. Forrester, and C.D. McCaig. Directed migration

of corneal epithelial sheets in physiological electric fields. Invest Ophthalmol Vis Sci,

37(13):2548–58, 1996.

[65] M. Zhao, C.D. McCaig, A. Agius-Fernandez, J.V. Forrester, and K. Araki-Sasaki. Human

corneal epithelial cells reorient and migrate cathodally in a small applied electric field.

Curr Eye Res, 16(10):973–84, 1997.

[66] J.C. Grahn, D.A. Reilly, R.L. Nuccitelli, and R.R. Isseroff. Melanocytes do not migrate

directionally in physiological dc electric fields. Wound Repair Regen, 11(1):64–70, 2003.

[67] P.J. Van Haastert and P.N. Devreotes. Chemotaxis: signalling the way forward. Nat Rev

Mol Cell Biol, 5(8):626–34, 2004.

[68] R. Riahi, Y. Yang, D.D. Zhang, and P.K. Wong. Advances in wound-healing assays for

probing collective cell migration. J Lab Autom, 17(1):59–65, 2012.

[69] O. Ilina and P. Friedl. Mechanisms of collective cell migration at a glance. J Cell Sci,

122(18):3203–8, 2009.

[70] R. Nuccitelli. A role for endogenous electric fields in wound healing. Curr Top Dev Biol,

58:1–26, 2003.

[71] S.R. Vedula, A. Ravasio, C.T. Lim, and B. Ladoux. Collective cell migration: a mecha-

nistic perspective. Physiology (Bethesda), 28(6):370–9, 2013.

[72] E.F. Keller and L.A. Segel. Initiation of slide mold aggregation viewed as an instability.

J Theor Biol, 1970:399–415, 26(3).

186

BIBLIOGRAPHY

[73] E.F. Keller and L.A Segel. Model for chemotaxis. J Theor Biol, 30(2):225–34, 1971.

[74] G. Oster. On the crawling of cells. J Embryol Exp Morphol, 83:329–64, 1984.

[75] G. Oster and A. Perelson. Cell spreading and motility: a model lamellipod. J Math Biol,

21:383–8, 1985.

[76] D.C. Bottino and L.J. Fauci. A computational model of ameboid deformation and loco-

motion. Eur Biophys J, 27(5):532–9, 1998.

[77] D. Bottino, A. Mogilner, T. Roberts, M. Stewart, and G. Oster. How nematode sperm

crawl. J Cell Sci, 115(2):367–84, 2002.

[78] H.G. Othmer and A. Stevens. Aggregation, blowup and collapse: the ABC’s of generalized

taxis. SIAM J Appl Math, 57(4):1044–81, 1997.

[79] T. Hofer, J.A. Sherratt, and P.K. Maini. Dictyostelium discoideum: cellular self-

organisation in an excitable biological medium. Proc Biol Sci, 259(1356):249–57, 1995.

[80] M.A.J. Chaplain and A.M. Stuart. A model mechanism for the chemotactic response of

endothelial cells to tumor angiogenesis factor. IMA J Math Appl Med Biol, 10(3):149–68,

1993.

[81] G.J. Pettet, H.M. Byrne, D.L.S. Mcelwain, and J. Norbury. A model of wound-healing

angiogenesis in soft tissue. Math Biosci, 136(1):35–63, 1996.

[82] E. Palsson. A three-dimensional model of cell movement in multicellular systems. Future

Generation Computer System, 17:835–52, 2001.

[83] R. De, A. Zemel, and S.A. Safran. Dynamics of cell orientation. Nature Physics, 3:655–9,

2007.

[84] D. Kabaso, R. Shlomovitz, K. Schloen, T. Stradal, and N.S. Gov. Theoretical model

for cellular shapes driven by protrusive and adhesive forces. PLoS Comput Biol,

7(5):e1001127, 2011.

[85] Y. Ni and M.Y.M. Chiang. Cell morphology and migration linked to substrate rigidity.

Soft Matter, 3:1285–92, 2007.

187

BIBLIOGRAPHY

[86] K.T. Kang, J.H. Park, and et al. Study on differentiation of mesenchymal stem cells

by mechanical stimuli and an algorithm for bone fracture healing. TERM, 8(4):359–70,

2011.

[87] A.J. Stops, K.B. Heraty, M. Browne, F.J. O’Brien, and P.E. McHugh. A prediction of cell

differentiation and proliferation within a collagen-glycosaminoglycan scaffold subjected

to mechanical strain and perfusive fluid flow. J Biomech, 43(4):618–26, 2010.

[88] M.P. Neilson, J.A. Mackenzie, S.D. Webb, and R.H. Insall. Modeling cell movement and

chemotaxis using pseudopod-based feedback. SIAM Journal on Scientific Computing,

33(3):1035–57, 2011.

[89] M. Yamao, H.N. mail, and S. Ishii. Multi-cellular logistics of collective cell migration.

PLoS ONE, 6(12):e27950, 2011.

[90] R.F. van Oers, E.G. Rens, D.J. LaValley, C.A. Reinhart-King, and R.M. Merks. Me-

chanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in

vitro. PLoS Comput Biol, 10(8):e1003774, 2014.

[91] A. Ducrot, F.L. Foll, H.M. Magal, P.J. Pasquier, and G.F. Webb. An in vitro cell

population dynamics model incorporating cell size, quiescence, and contact inhibition.

Math Models Methods Appl Sci, 21(1):871–892, 2011.

[92] G. Gyllenberg and H. Heijmans. An abstract delay-differential equation modelling size

dependent cell growth and division. SIAM J Math Anal, 18(1):74–88, 1987.

[93] J. Satulovsky, R. Lui, and Y.L. Wang. Exploring the control circuit of cell migration by

mathematical modeling. Biophys J, 94(9):3671–83, 2008.

[94] E. Hannezo, J. Prost, and J.F. Joanny. Theory of epithelial sheet morphology in three

dimensions. Proceedings of the National Academy of Sciences current issue, 111(1):27–32,

2014.

[95] T.A. Ulrich, E.M. De Juan Pardo, and S. Kumar. The mechanical rigidity of the extra-

cellular matrix regulates the structure, motility, and proliferation of glioma cells. Cancer

Res, 69(10):4167–74, 2009.

[96] G. Albrecht-Buehler. Does blebbing reveal the convulsive flow of liquid and solutes

through the cytoplasmic meshwork? Cold Spring Harb Symp Quant Biol, 1982:45–9,

46(1).

188

BIBLIOGRAPHY

[97] C. Casey Cunningham. Actin polymerization and intracellular solvent flow in cell surface

blebbing. J Cell Biol, 129(6):1589–99, 1995.

[98] G.T. Charras, J.C. Yarrow, M.A. Horton, L. Mahadevan, and T.J. Mitchison. Non-

equilibration of hydrostatic pressure in blebbing cells. Nature, 435:365–69, 2005.

[99] S.J. Han and N.J. Sniadecki. Simulations of the contractile cycle in cell migration using a

bio-chemical-mechanical model. Comput Methods Biomech Biomed Engin, 14(5):459–68,

2011.

[100] D.R. Carter, P.R. Blenman, and G.S. Beaupré. Correlations between mechanical stress

history and tissue differentiation in initial fracture healing. J Orthop Res, 6(5):736–48,

1988.

[101] L.E. Claes and C.A. Heigele. Magnitudes of local stress and strain along bony surfaces

predict the coarse and type of fracture healing. J Biomech, 32(3):255–66, 1999.

[102] L. Geris, H. Van Oosterwyck, J. Vander Sloten, J. Duyck, and I. Naert. Assessment of

mechanobiological models for the numerical simulation of tissue differentiation around

immediately loaded implants. Comput Methods Biomech Biomed Engin, 6(5-6):277–88,

2003.

[103] D.J. Kelly and P.J. Prendergast. Mechano-regulation of stem cell differentiation and

tissue regeneration in osteochondral defects. J Biomech, 38(7):1413–22, 2004.

[104] D. Lacroix and P.J. Prendergast. A mechano-regulation model for tissue differentiation

during fracture healing: analysis of gap size and loading. J Biomech, 35(9):1163–71, 2002.

[105] D. Lacroix, P.J. Prendergast, G. Li, and D. Marsh. Biomechanical model of simulate

tissue differentiation and bone regeneration: application to fracture healing. Med Biol

Eng Comput, 40(1):14–21, 2002.

[106] S.J. Mousavi, M.H. Doweidar, and M. Doblaré. Computational modelling and analysis

of mechanical conditions on cell locomotion and cell-cell interaction. Comput Methods

Biomech Biomed Engin, 17(6):678–93, 2014.

[107] A. Buxboim, I.L. Ivanovska, and D.E. Discher. Matrix elasticity, cytoskeletal forces and

physics of the nucleus: how deeply do cells ’feel’ outside and in? J Cell Sci, 123(3):297–

308, 2010.

189

BIBLIOGRAPHY

[108] A. Buxboim, K. Rajagopal, A.E.X. Brown, and D.E. Discher. How deeply cells feel:

methods for thin gells. J Phys Condens Matter, 22(19):194116, 2010.

[109] S.J. Mousavi, M.H. Doweidar, and M. Doblaré. 3D computational modelling of cell

migration: A mechano-chemo-thermo-electrotaxis approach. J Theor Biol, 329:64–73,

2013.

[110] N. Andrew and R.H. Insall. Chemotaxis in shallow gradients is mediated independently

of ptdlns 3-kinase by biased choices between random protrusions. Nature Cell Biology,

9:193–200, 2007.

[111] C.D. McCaig, A.M. Rajnicek, B.Song, and M.Zhao. Has electrical growth cone guidance

found its potential? Trends Neurosci, 25(7):354–9, 2002.

[112] B.A. Dalton, X.F. Walboomers, and et al. Modulation of epithelial tissue and cell migra-

tion by microgrooves. J Biomed Mater Res, 56(2):195–207, 2001.

[113] S.J. Mousavi, M. Doblaré, and M.H. Doweidar. Computational modelling of multi-cell

migration in a multi-signalling substrate. Phys Biol, 11(2):026002, 2014.

[114] S.J. Mousavi and M.H. Doweidar. A novel mechanotactic 3D modeling of cell morphology.

J Phys Biol, 11(4):046005, 2014.

[115] S.J. Mousavi and M.H. Doweidar. Three-dimensional numerical model of cell morphology

during migration in multi-signaling substrates. Plos One, accepted, 2015.

[116] D. Karamichos, N. Lakshman, and W.M. Petroll. Regulation of corneal fibroblast mor-

phology and collagen reorganization by extracellular matrix mechanical properties. Invest

Ophthalmol Vis Sci, 48(11):5030–7, 2007.

[117] Y. Lee, J. Huang, Y. Wang, and K. Lin. Three-dimensional fibroblast morphology on

compliant substrates of controlled negative curvature. Integr Biol (Camb), 5(12):1447–55,

2013.

[118] D. Li, J. Zhou, F. Chowdhury, J. Cheng, N. Wang, and F. Wang. Role of mechanical

factors in fate decisions of stem cells. Regen Med, 6(2):229–40, 2011.

[119] N.D. Evans, C. Minelli, and et al. Substrate stiffness affects early differentiation events

in embryonic stem cells. Eur Cell Mater, 18:1–13, 2009.

190

BIBLIOGRAPHY

[120] J. Fu, Y.K. Wang, M.T. Yang, R.A. Desai, X. Yu, Z. Liu, and C.S. Chen. Mechanical

regulation of cell function with geometrically modulated elastomeric substrates. Nat

Methods, 7(9):733–6, 2010.

[121] J.C. Friedland, M.H. Lee, and D. Boettiger. Mechanically activated integrin switch con-

trols α5β1 function. Science, 323:642–44, 2009.

[122] G.J. Hera, H.Ch. Wub, M.H. Chenc, M.Y. Chene, Sh.Ch. Change, and T.W. Wanga.

Control of three-dimensional substrate stiffness to manipulate mesenchymal stem cell

fate toward neuronal or glial lineages. Acta Biomater, 9(2):5170–80, 2013.

191

computational modeling of cell behavior in three-dimensional … · 2019-04-14 · don mohamed...

Documents