EFFECT OF MICROTUBULE MOTORS ON MICROTUBULE MECHANICS

By

PARAG S. RANE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2017

© 2017 Parag S. Rane

I dedicate this to my family.

ACKNOWLEDGMENTS

I want to take this opportunity to thank all individuals who lead to the successful

completion of this work and made my life easier during this process. Firstly, I am indebted

to my advisors, Prof. Tony Ladd and Prof. Tanmay Lele for their limitless patience, invaluable

guidance, and unwavering support at all times with a much-needed room for growth. I would

like to thank Prof. Rich Dickinson for providing me his invaluable time particularly during

the summer of 2015 and for his lectures in biophysics which transformed my perspective

of this field. This work was enabled by the support of National Science Foundation (grant

no.1236616).

I would like to thank Dr. Ian Kent from Lele Lab who performed all the experiments

for the modeling project in chapter 2. My depth of knowledge in biology was improved

substantially because of the discussions with Samer Alam, Varun Agarwal and other members

in the Lele Lab group. I am thankful to Nandini Shekhar for answering the queries which

proved helpful in the completion of Chapter 4. I want to acknowledge my labmates, Dr. Virat

Upadhyay, Dr.Vitali Starchekno for making my workplace the place to be. I want to thank

Dr.Gaurav Mishra, who is a former student from Tony’s group for his helpful conversations

during my transition to the Ladd Lab.

It was a real treat and a valuable learning experience to be a teaching assistant for the

Chemical Reaction Engineering course taught by Prof. Jason Weaver and the Computer Model

Formulation course taught by Prof. Spyros Svoronos which has sparked my interest in teaching.

Shoutout to Sandaliya Sanbathula for the annual road trips where we explored this great and

diverse country. I will cherish my friendship with Mert Arca, Akshat Dimiri, Maxim Prokopenko

Rahul Rai, Varun Agarawal , Harsh Tanrang, and many others in times to come.

I would like to thank Shirley Kelly and other members of departmental staff for their kind

help. Thanks to Janis Mena for insightful discussions on nutrition. I am thankful to Janice

Ladd for her kindness and delicious thanksgiving dinners. I feel blessed to have my cousin

Sarang and his wife Yogini in my life, who made me feel at home during my visits to their

4

place. Lastly, I want to express my gratitude from the bottom my heart towards my parents,

Dr. Subhash Rane & Rekha Rane and my grandmother Kamlabai whose handvowen ”Godadi”

(blanket) has kept me warm during these years.

5

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION AND BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 Contributions of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Microtubule Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Force Acting on Microtubules . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 MODELING AND SIMULATION METHODS . . . . . . . . . . . . . . . . . . . . . 18

2.1 Model for Microtubule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.1 Discretization of Microtubule Shape . . . . . . . . . . . . . . . . . . . 182.1.2 Dynamic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.3 Internal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Model for Motor Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Continuous Motor Force Model for Dynein Forces . . . . . . . . . . . . 222.2.2 Discrete Motor Force Model for Dynein and Kinesin Forces . . . . . . . 24

2.2.2.1 Initialization and termination of motors . . . . . . . . . . . . 242.2.2.2 Linkage based force generation . . . . . . . . . . . . . . . . 26

2.2.3 Background Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.4 Pinning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.5 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 A MECHANISM FOR BENDING MICROTUBULES . . . . . . . . . . . . . . . . . 32

3.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Equation of Motion and and Selection of Model Parameters . . . . . . . . . . 343.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.1 Microtubules Bend against Transiently Immobilized Segments . . . . . 353.3.2 Dynein, but not Myosin is Involved in Formation of Local Bends under

Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.3 A Mechanism for Microtubule Bend Formation by Dynein-mediated

Transport towards a Pinning Point . . . . . . . . . . . . . . . . . . . . 373.3.4 Bending near the Microtubule Plus End Tip Leads to Tip Rotation . . . 38

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6

4 EXCESS LENGTH IN MICROTUBULES . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Objective and Background Information . . . . . . . . . . . . . . . . . . . . . 494.1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Data Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.1 Fourier Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.1 A Putative Role for Plus End Motors in Generating Excess Length . . . 52

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7

LIST OF TABLES

Table page

3-1 Model Parameters: Microtubule, Continous Motor Model . . . . . . . . . . . . . . 48

4-1 Model Parameters: Microtubule, Discrete Motor Model . . . . . . . . . . . . . . . 63

8

LIST OF FIGURES

Figure page

1-1 Microtubule Structure and Dynamic Instability. . . . . . . . . . . . . . . . . . . . . 16

1-2 Structure of Dynactin molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2-1 Model points representing microtubules. . . . . . . . . . . . . . . . . . . . . . . . 29

2-2 Stresses within the microtubule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2-3 Initial motor configuration leads to a shortlived compressive stress. . . . . . . . . . 31

3-1 Local bends develop by dynein-dependent translation of microtubule segments fromthe minus end. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3-2 Simulations of dynein mediated bend formation in microtubules. . . . . . . . . . . . 44

3-3 Rotation of the microtubule tip due to local bend formation. . . . . . . . . . . . . . 46

3-4 Effect of overhanging segment length on the rotation of the tip. . . . . . . . . . . . 47

4-1 Experimentally recorded tip trajectories of growing microtubules. . . . . . . . . . . 56

4-2 Microtubule straightening in absence of pinning under the action of dynein motor. . 56

4-3 Sine Waves corresponding to mode numbers in the Fourier mode spectrum. . . . . . 57

4-4 Simulations of growing microtubules subjected to motor forces. . . . . . . . . . . . 58

4-5 Motors binding near the tip change the orientation of the tip. . . . . . . . . . . . . 59

4-6 Comparison of Fourier modes of tip trajectories and simulated microtubules. . . . . 60

4-7 Average poly. speed estimated from tip trajectories of simulated microtubule. . . . . 61

4-8 Excess length in simulated microtubules. . . . . . . . . . . . . . . . . . . . . . . . 62

9

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

EFFECT OF MICROTUBULE MOTORS ON MICROTUBULE MECHANICS

By

Parag S. Rane

May 2017

Chair: Anthony J.C. LaddCochair: Tanmay LeleMajor: Chemical Engineering

Microtubules (MTs) and microtubule associated proteins (MAP’s), particularly motor

proteins are being extensively researched for two specific reasons, viz. anomalies in MAP’s is

the leading cause in many neurodegenerative diseases and for its potential application in cancer

treatment as MT and MAP’s form the machinery to separate chromosomes during cell division.

Microtubules have a persistence length of the order of millimeters in vitro, but inside cells, they

bend over length scales of microns which suggest non-thermal forces at work.

We modeled microtubules as elastic rods and associated motor proteins as elastic linkages

which attach and detach along its length to study the shape dynamics of microtubules in-vivo.

Our collaborative work with Lele lab shows that a majority of microtubule bending, in the cell

interior away from the cell boundary is primarily caused by dynein-mediated transport of MT

length towards its plus end in presence of cross-linkers. In absence of such linkers, we found

that MT was being straightened by dynein which led to the further question, how a nucleating

MT could retain its curvature required for MT bending in presence of dynein. We investigate

this question in chapter 3. Additionally, in some cases bend formation near the tip altered its

direction. We systematically investigated this mechanism in presence of cross-linkers and found

that this mechanism could influence microtubule tips growing at a slower rate.

We show that an ensemble of plus end-directed motors acting on a MT generates

observed quantified deflections in MT tips (variance) on dynein inhibition. Further, our

simulations show that the curvature along the MT can be introduced by the action of opposing

10

motor forces based on their distribution. This model allows a majority of MT to be under

tension with some of its sections under compression showing that centrosome centering

mechanism and microtubule bending away from periphery can co-exist together thereby tying

all the previous experimental observations in our group.

11

CHAPTER 1INTRODUCTION AND BACKGROUND

The eukaryotic cytoskeleton consists of three types of proteinaceous filaments- F-actin,

intermediate filaments and microtubules found in the cytoplasm. Polymerizing F-actin

filaments grow to push the cell membrane outward during cell motility and enable cell crawling.

Intermediate filaments are structurally stable networks, which provide shape stability to cells,

and also form the nuclear lamina which maintains a mechanically stable nucleus. Microtubules

are hollow biopolymers that participate in cell division, serve as traffic highways along which

motor proteins transport organelles, and generate mechanical forces to position the nucleus.

Given the mechanical functions of these three cytoskeletal structures in enabling cell motility,

intracellular transport, pulling chromatids apart during anaphase and maintaining cell shape in

homeostasis, there has been much interest in understanding the mechanical properties of these

cytoskeletal elements. This thesis focuses on the mechanical forces on interphase microtubules

generated by motor proteins in the cytoplasm.

When isolated in vitro, the microtubule has a persistence length of the order of millimeters

indicating that it is stiff to thermal fluctuations [1]. In comparison, F-actin has a persistence

of a few microns [1], while that of DNA is 30-50 nanometers [2]. Despite its stiffness, the

microtubule is bent with wavelengths of a few microns in cells [3–6]. Microtubules are also

dynamic in that they grow and shrink continuously in cells by cycles of polymerization and

depolymerization. This phenomenon is known as dynamic instability [7]. In cells, microtubules

are shown to deviate from their initial direction of growth [8]. Wavy shapes of microtubules

and dynamic polymerization and depolymerization are thought to allow sampling of cellular

space efficiently compared with a straight, static polymer. This has the benefit of efficient

sampling of cytoplasmic space [9], which is crucial for efficient transport of organelles.

What is the explanation for the bent shapes of microtubules? This question has been

of interest in the recent literature [3–5, 10]. The commonly accepted explanation is that

a growing microtubule pushing against an immobile barrier will bend by Euler buckling (or

12

modified Euler buckling if the microtubule is ’reinforced by connections with the cytoplasm

[4]). However, this explanation is challenged by experiments with laser ablation [11] in which

increased bending of severed microtubule fragments were observed instead of relaxation

of bending as would be predicted for a microtubule under compression. The laser ablation

experiments instead suggest that microtubules may be under tension along their lengths (this

does not argue against local compressive forces at a growing tip).

1.1 Contributions of this Thesis

In this thesis, we performed computational simulations to explain the mechanical behavior

of static and growing microtubules in living cells. By comparing results of simulations and

experimental observations made in the Lele lab, we explain how bends develop in microtubules

in living cells in Chapter 2. In Chapter 3, we explain how microtubules bend during growth,

again through a comparison between simulation results and experimental observations.

1.2 Microtubule Structure

We now discuss microtubule structure, which is important to understand microtubule

mechanics. A single microtubule consists of 13 proto-filaments [12] that are parallel to one

another and arranged circumferentially, forming a hollow tube with an inner diameter of 12 nm

and outer diameter of 24 nm 1.3. Each protofilament is a polymer of a dimer of proteins, α

and β tubulin, each of which has a molecular mass of 50 kilo Dalton (kDa). The monomers

associate in a −(α − β)n− like configuration that induces polarity in the MT. The number

of protofilaments that make up the microtubule are determined by γ tubulin ring complex

(γ − TuRC). The ring complex acts as a scaffold for assembly for tubulin dimers and speeds

up the nucleation process. Most microtubules in interphase cells are anchored in this complex,

which forms the core functional unit of the structure known as centrosome. In a interphase,

most microtubules radiate out ward from centrosome. This network topology is peculiar to

microtubules.

The end of the microtubule that terminates in the γ − TuRC is called the minus

end while the other end which undergoes polymerization and depolymerization constantly

13

in cells is known as the plus end of the MT. The MT plus end cycles through phases of

polymerization and depolymerization and this phenomenon is termed as dynamic instability

[7]. In order to polymerize, tubulin needs to bind to GTP [13]. Upon binding in the cytoplasm,

the tubulin-GTP complex is stable [14, 15]. When GTP-tubulin subunits are incorporated

into an MT, the rate of hydrolysis of exposed GTP on β-tubulin increases significantly [16].

The GTP-tubulin hydrolyzes to GDP-tubulin and the resulting energy is stored within the

protofilament tube-like structure of the microtubule. This energy is available to do work during

depolymerization and is used to pull chromatids apart during anaphase [17]. Depolymerization

is triggered when there is a loss of GTP-tubulin at the plus end- GTP-tubulin on the plus end

is referred to as the GTP-cap. GTP-tubulin restricts the backward coiling of GDP-tubulin

protofilaments. GTP on the MT is continuously hydrolyzed but new GTP tubulin is added at

the tip maintaining the GTP cap. When the rate of GTP hydrolysis exceeds the recruitment of

GTP tubulin, depolymerization (also called catastrophe) takes place [18]. Additionally, evidence

has been found to support the presence of unhydrolyzed regions of GTP-tubulin away from the

plus end, which are likely to aid in rescue of polymerization following catastrophe [19–22].

1.3 Force Acting on Microtubules

When the plus end of a growing microtubule encounters a barrier, it may continue to

add monomer sub-units which allows the microtubule to push against the barrier [23, 24].

These polymerization forces are known to cause local buckling of the microtubule close to

the periphery [10, 25, 26]. Compressive forces generated by pushing against the barrier are

implicated in centering of the MTOC and spindle bodies [27–29] demonstrated in in-vitro

experiments [30, 31]. However, this mechanism seems viable in cells with small size as the

critical buckling force (F) for a microtubule of length (L) scales as F ∝ L−2 . Additionally, this

mechanism seem to function better in square shaped micro-fabricated chambers which is not a

typical geometry of a cell. Microtubule depolymerization releases energy stored in microtubule

lattice. This energy is stored in lattice due to hydrolysis of incorporated GTP tubulin units and

is used to generate forces on sister chromatids to pull them apart [32–37].

14

Of particular interest in this thesis the action of microtubule motors dynein and kinesin

family which processively walk along microtubules. Dynein walks toward the minus end of

microtubules while kinesin walks toward the plus end. These motors convert the energy of

ATP hydrolysis to motor along the microtubule, and are able to perform work by moving

organelles anchored to the other end of the motors in the crowded, viscous cytoplasm. Dynein

and kinesin may exert significant forces on the microtubule when they continue to motor

while their other end is bound to immobile objects [11]. For example, dynein can bind to

the nuclear surface [38–43]. The nucleus is a large object, and a single dynein motor that

is bound to the nucleus on one end and lengthens as it walks on the microtubule on the

other end will pull on the nucleus and on the microtubule. In the same way, dynein may walk

on microtubules in the cytoplasm, while being connected at the other end with dynactin.

Dynactin is a multisubunit protein complex found in eukarytoic cells that is required to

activate dynein[44, 45]. Dynactin increases the processivity of dynein, and allows a dynein

molecule to travel a larger distance on the microtubule contour [46–49]. The dynactin molecule

is 40 nm in length and 15 nm in height. Morphologically, dynactin can be divided into two

domains viz. Arp1 (Actin Related Protein 1) rod and the shoulder domain [50, 51]. Figure 1-2

shows a schematic representation of dynactin with various domains, including the p150Glued

which is located at the end of the shoulder domain. While dynein binds to the p150Glued

subunit of dynactin (corresponding to AA 217-548 of p150Glued), kinesin-2 binds to AA

600-811 of p150Glued. Schematic descriptions for dynein is included in Chapter 3.

15

Figure 1-1. Microtubule Structure and Dynamic Instability.(a) Microtubules are composed of stable / -tubulin heterodimers that are aligned ina polar head-to-tail fashion to form protofilaments. (b) The cylindrical and helicalmicrotubule wall typically comprises 13 parallel protofilaments in vivo. The 12-nmhelical pitch in combination with the 8-nm longitudinal repeat between / -tubulinsubunits along a protofilament generates the lattice seam (red dashed line). (c)Assemblypolymerization and disassemblydepolymerization of microtubules is drivenby the binding, hydrolysis and exchange of a guanine nucleotide on the -tubulinmonomer (GTP bound to tubulin is non-exchangeable and is never hydrolysed).GTP hydrolysis is not required for microtubule assembly per se but is necessary forswitching between catastrophe and rescue. (Adapted by permission from MacmillanPublishers Ltd: Nature Reviews Molecular Cell Biology, Anna Akhmanova andMichel O. Steinmetz. Tracking the ends: a dynamic protein network controls thefate of microtubule tips, 9(4):309322, 2008)[52]

16

Figure 1-2. Structure of Dynactin molecule.The p150glued arm has sites for attachment for microtubules as well as motors likedynein and heterotrimeric kinesin-2. The Arp1 rod provides binding for coupling ofthe complex with various cargoes,membranes, etc . We hypothesize that dynactinability to attach to immobile elements allows dynein to spend longer duration in itsstalled state. The Figure was taken from

17

CHAPTER 2MODELING AND SIMULATION METHODS

The objective of this work is to investigate the effect of motor forces on microtubule. In

Chapter 3, we show how microtubule bends illustrated in Figure 3-2 can result from motor

activity while in Chapter 4 we investigate how these forces can affect growing microtubules.

2.1 Model for Microtubule

A microtubule is modeled as infinitely thin linear object that is expected to behave like an

elastic, non-extensible rod. Typically a thin rod of finite thickness has 6 degrees of freedom.

Assuming the rod to be infinitely thin allows us to remove 2 degrees of freedom pertaining

to shear within the rod body and 1 degree of freedom pertaining to twisting motion. In most

experiments we observed the inplane motion of microtubule. As a result, we can neglect

out of plane bending of the microtubule. Thus, for our model we have 2 degrees of freedom

corresponding to the in-plane bending of the microtubule and motion of points along the

tangent direction due to compression/extensional forces, which is kept small by selecting a high

extensional modulus to mimic the non-extensible rod. The first step in the simulation is to

initialize the input shape which is described in the section below.

2.1.1 Discretization of Microtubule Shape

The initial microtubule shape is represented by N + 1 equidistant nodal points, mi where,

i = [0, 1, · · ·,N]. The distance between nodal points, referred to as segment length h and is

calculated as,

h =L

N + 1 + 0.5α(2–1)

where, the paramater α is the function of boundary conditions. The parameter α is given by,

α = O1 +O2 − 2 (2–2)

where, O1 and O2 determine the offsets that best account for boundary condition at the plus

and the minus end of the filament respectively. For instance, O1 is equal to 1, 0, -2 for free,

hinged and clamped boundary conditions respectively which is based on 2nd order convergence

18

of shapes and forces. The expected segment length h is determined during the initialization

step and we keep h constant for the rest of the entire simulation duration.

If the input shape is specified using a discrete set of co-ordinates (xinput , yinput) (not

necessarily equally spaced), we fix the origin at the minus end of the filament and orient it

along the x-axis. Cubic splines were used to store these co-ordinates as the function of s which

is the distance along the filament arc. The set of co-ordinates that defines the discretized input

shape are generated by reverse mapping at the points si along the microtubule contour.

si = (i + 0.5O1)h, i = [0, 1, 2, .., ·N]. (2–3)

Note that for a clamped boundary condition, s0 takes the value of -1 and the corresponding

co-ordinates are set to (−h, 0). This is shown in figure 2-1A This marks the completion of

the initialization step and we are ready to begin the simulation. The position vectors for mi

are denoted as ri . The bond vectors ri+1 − ri are denoted as ri ,i+1. Usually microtubule has N

segments including the ones that are part of the boundary condition. However when boundary

condition is free at both ends there are N + 1 segments.

2.1.2 Dynamic Instability

Microtubule length can change due to polymerization or depolymerization at the plus

end which differentiates one end from the other.The co-ordinate of the microtubule’s plus

end is mn. The instantaneous length of the microtubule Lt+δt is tracked independently and is

the updated at the end of each simulation step. For instance, for a microtubule undergoing

polymerization with velocity vpol ,

Lt+δt = Lt + vpolδt (2–4)

Additional nodal points are added or subtracted from plus end if,

|Lt+δt − Lt | >h

2. (2–5)

The co-ordinates for new nodal point/s are obtained by extrapolating past mn in

increments of h along the along rN−1,N which represents the direction of the tip. There is

19

a provision in the model for switching a polymerizing state to a depolymerizing one, with

probabilities derived form the rates of switching from polymerization to depolymerization

(catastrophe) and back again (recovery).

2.1.3 Internal Forces

The cellular environment is viscous and the inertial forces on the microtubule are

insignificant [53]. Internal forces arise the deviation of the microtubule shape from its stress

free state and the magnitude can be estimated from the resulting strain. The stresses in the

rod are assumed to be linear in the deviations in the strain fields. Note that the resulting stress

associated with extension Γ (s) and due to bending Ω(s) are independent, which is shown in

Figure 2-2. The symbol s is the distance along the filament arc. Thus the internal energy U

for the microtubule can be written as,

U = UΓ + UΩ. (2–6)

The extensional strain energy within the elastic filament of length L is given by line integral of

the strain within the filament.

UΓ =E

2

∫ L

0

Γ 2(s)ds, (2–7)

where, E is the extensional modulus of elasticity per micrometer. This equation can be written

in terms of position vectors for a discretized filament of segment length h,

UΓ =

N−1∑i=0

E

2

1

h

√(ri+1 − ri).(ri+1 − ri)− h)2. (2–8)

The corresponding force on each nodal point can be obtained by differentiating the above

equation,

F Γi = −∂UΓ

∂ ri. (2–9)

Note that every term except for the end points will have two terms, because they have they

have two neighbors.

20

The bending strain energy in a bent elastic filament of length L in absence of an external

couple can be written as a line integral of mean-square curvature [54].

UΩ =B

2

∫ L

0

C 2(s)ds (2–10)

where B is flexural rigidity per micrometer. The flexural rigidity B = EI , where I is the

moment of inertia along the filament axis. The equation can be discretized for a filament

composed of N equal segments of length h [55, 56].

UΩ =Bh

2

1

h2

N−1∑1

C 2

i . (2–11)

The curvature Ci is related to the angle between the adjacent segments and is given by

the formula,

C 2

i = 2(ri−1,i .ri−1,i − ri−1,i .ri ,i+1)/h4 (2–12)

The discrete approximation of UΩ can then be differentiated to find the elastic bending

force on each nodal point of the filament.

FΩi = −∂UΩ

∂ ri(2–13)

2.2 Model for Motor Forces

The motor body consists of 3 core structures, the cargo binding domain which connects

the motor to the cargo, the microtuble binding domain through which motor binds to a

microtubule segment and the linker domain which connects these two domains. A typical

microtubule-motor interaction can be broken down into 3 events. First, a motor binds to a

microtubule then it walks along the microtubule and at some point in time it unbinds. During

this interaction period, motor exerts force on the microtubule. For instance, consider the cargo

binding domain of a motor anchored to a relatively immobile surface like nucleus. As the motor

walks along the microtubule the force within the linkage increases and begins to exert tension.

This force is transferred to cargo end and the microtubule via linakge. In order to estimate the

motor force, we considered two different motor models in this work. In Chapter 3, we used the

21

model[11, 57] for dynein motors which computes the force exerted by a motor averaged over

all binding times. This is what we call as a continuous motor model. While in Chapter 4, we

simulated individual motors to determine to compute the motor force. This model is referred to

as a discrete motor model.

2.2.1 Continuous Motor Force Model for Dynein Forces

The speed of a dynein motor v−m attached to a microtubule under a linkage force f is [58],

v−m

v−max

= 1− t .f

f −max

(2–14)

where, v−max is the speed of the motor in a stress free state, f −max is the magnitude of the force

in the linkage along tangential direction when motor cannot walk any further which is known as

stall force and t is the unit tangent vector along the microtubule and directed towards the plus

end. Note that as t .f → f −max , vm → 0. The force on the linkage increases as dynein translates

with velocity vm inducing tension, where

vm = v−m (−t) (2–15)

We modeled the linkage as a spring with stiffness κ. If microtubule segment to which motor is

attached moves with a velocity v relative to the anchor point, the net rate of increase in the

linkage force is,

df

dt= −κ(v + vm), (2–16)

Substituting vm from Eq. 2–14 into Eq. 2–16,

df

dt= −κv + κv−

max t − κv−max(

t .f

f −max

)t (2–17)

Note that the equation needs to be integrated separately in the tangential and the normal

direction as the variable on the LHS is f while on the RHS is t .f . Assuming that linkage is in a

stress free state when motor binds to microtubule i.e. f (0) = 0, the time-dependent force f (t)

22

from a single motor obtained by integrating Eq. 2–17 is,

f (t) = f −max(1−v .t

v−max

)(1− e−t/τ )t − κv .(1− t t)t (2–18)

Linkages are assumed to dissociate with a first order rate constant ko (s−1)such that the

probability of a linkage existing at time t after its attachment,

Pb(t) = e−ko t . (2–19)

The mean force over the bond lifetime η is,

< f >=

∫ ∞

0

f (η)dPb(η) = f0(1−t.v

v−max

)t − κ

ko(1− t t)v (2–20)

where 1 denotes the unit tensor,

f0 = f −max/(f −maxko

κv−max

+ 1) (2–21)

is the average tangential force exerted by a dynein molecule on a stationary microtubule.

The expression for force per unit length exerted by dynein motors on the microtubule can be

obtained by multiplying eq. 2–20 with the dynein density per unit length, ρ.

K = ρf0(1−t.v

v−max

)t − ρκ

ko(1− t t)v (2–22)

Eq. 2–22 can be rewritten as,

K = ρf0t − ξ∥(v .t )t − ξ⊥(v .n)n (2–23)

where n is the unit normal vector. The coefficients ξ∥ and ξ⊥ represent the friction coefficients

in tangential and normal direction which resist the microtubule motion and are given by,

ξ∥ =ρf0vmax

(2–24)

ξ∥ =ρκ

ko(2–25)

23

The frictional nature arises from the transient nature of the linkages and is well known as

protein friction. The velocity of microtubule v can be obtained by solving the equation of

motion which is described in subsequent sections. We used this model to study small length

scale high amplitude transient bends observed in the area under the nucleus.

2.2.2 Discrete Motor Force Model for Dynein and Kinesin Forces

We model two families of microtubule motors that are likely to generate force on growing

microtubules. Motors that walk towards the minus end represent cytoplasmic dynein and the

ones that walk towards the plus end collectively represent members from kinesin family. Both

motor families operate similarly with individual set of specified parameters listed in Table XX.

Motor forces are included as linkages distributed along the length of the microtubule, as

shown in Figure 2-4. The linkages are modeled as springs with stiffness κ.

2.2.2.1 Initialization and termination of motors

We imagine that a particular segment of the microtubule is occasionally captured by a

motor attached to the background cytoskeleton and is bound for the times of the order of ko

where we take the motor off-rate to be of the order of 1 s−1 [57] for both motors. A motor

appears on a segment length h in a time step δt with a probability,

p = nmh ∗ (1− e−ko δt) (2–26)

where, nm is the combined linear density of both motor families. The number of motors Nb

that appear on the microtubule of length Lt at time t during the time step δt is determined by

drawing a sample from Possion distribution with mean µ = p ∗ Lt/h,

Nb = ⌊P(µ)⌋ (2–27)

Note that this also adds variability to number of motors that appear during every simulation

step. The initial assignment of parameters for each motor includes the position along the

microtubule contour Si(0), the microtubule-cytomatrix linkage bi(0) and the motor type

identifier Idxi . If Idxi = −1 then the motor then the motor is classified as dynein, if Idxi = +1

24

then the motor is classified as kinesin. The value is assigned using a random number X where,

X ∼ U[0, 1), (2–28)

via the formula,

Idxi = 2

⌊1− (nm − n+)

nm+ X

⌋− 1, (2–29)

where n+ is the density of the kinesin motors. Note that the resulting distribution is skewed

by the factor (nm − n+)/nm to obtain the desired motor densities. On average the binding

and unbinding rates will balance at the desired motor densities if each bound motor unbinds

with a probability (1 − e−ko δt). On an average it takes 1-2 seconds to saturate the long

initialized microtubule shape after the simulation begins. For instance, if relaxation of a

microtubule shape needs to be observed, the background friction is assigned a high value 100

Pa.S to keep the shape intact during this period saturation period. The position vector for a

motor xi(t = 0) is determined by interpolating the nodal vectors for the microtubule rj , rj+1

corresponding to Si(0). The initial distribution of directions of the linkage is isotropic. The

position vector for anchor point is initialized as, ai(0) = xi + bi(0). We model linkages as

springs with stiffness κ and assume that during the time of attachment the linkage is in a

stress-free state with rest linkage length lm.

|bi(0)| = lm (2–30)

κ and lm are kept identical for both motors for the sake of simplicity.

Motors are added and removed during the before executing the equation of motion during

each simulation step. Motors located on the microtubule in the previous step are assigned a

random number X which is then compared with unbinding probability 1 − e−ko δt . The motor

stays on the microtubule for the next simulation step if,

X > 1− e−ko δt (2–31)

25

Motors are also removed if they walk off the microtubule i.e,

Si(t) /∈ [0,Lt ] (2–32)

The motor model resembles that proposed by Nedelec and Foethke [59] The major

difference lies in our implementation where the motors only exist while they are bound to the

microtubule instead of being permanently bound to random positions in the cytoskeleton at all

times.

2.2.2.2 Linkage based force generation

A dynein motor walks towards the minus end with velocity vm,i 2–15 until the force within

the linkage reaches stall force or it unbinds. The equivalent expression for kinesin motors is,

vm,i = (v+max(1 +t.fif +max

))t (2–33)

where,f +max and v+max represent the stall force and stress free motor speed for kinesin motors.

Force can develop in the linkage due to change in position vector of the motor (point of motor

attachment to microtubule) and due to motion of anchor points. The position vector of motor

changes due its motion along the microtubule and the motion of microtubule itself.

xi(0) = xi(t) +

∫ t

0

(v+m,i + v)t (2–34)

The motor position Si(t) is updated in increments of vm,iδt at the end of every simulation

step corresponding to a small time δt and the position vectors are determined by interpolating

from the neighboring nodes. The velocity of the segment is found by interpolating the nodal

velocities which described in the next section. The position vector of the anchor points are

updated at the end of every simulation step,

ai(t + δt) = va,iδt + ai(t) (2–35)

where va,i is the velocity motor anchor points. This is added to model the motion background

cytoskelton which can result from actomyosin contractions within the cytoskeleton,retrograde

26

actin flow,etc. Note that va,i is significantly smaller than free motor velocity of myosin motors

because the cross linked network is expected to move slower than individual motors. In order to

compute the nodal velocities we need to compute the force exerted by the motor linkage/s on

the microtuble nodes. The force fi based on the extension in a linkage bi(t), where

bi(t) = ai(t)− xi(t) (2–36)

is

fi = max

(κ

(1− |bi0|

|bit|

), bi(t), 0

)(2–37)

This modification to the force eq. 2–37 is made to avoid motor exerting compressive elastic

force during initial stages of the walk due to its initial orientation see Figure 2-3. Such a state

is short lived and the motor will eventually walk to a position such that |bit| > |bi0|. This

is because one would expect the linear chains to buckle rather than compress under those

circumstances.

2.2.3 Background Friction

A substantial amount of background friction was observed by Wu et. al. [11] in cells

despite inhibiting dynein. This can possibly come from kinesins or other proteins that interact

with microtubules in cells. Wu et. al [11] observed straightening of microtubule near minus

end on laser ablation in dynein inhibited cells. However, this interaction is complicated by the

fact that tangential pulling from kinesin can straightens microtubules at the minus end despite

of lateral protein friction arising from linkages. As a result, we used the value of 10 Pa.S as

background friction which is lower than motor friction.

2.2.4 Pinning Points

Pinning points model the immobile sections observed in experiments listed in Chapter

3. The microtubule node closer the to the location of pinning point/s along the microtubule

contour is assigned a zero velocity in order to mimic the effect of pinning in simulations. The

list of nodes are stored in the list Pl .

27

2.2.5 Equation of Motion

The force balance on the microtubule in the overdamped limit where hamiltonian H ≈

U(internal energy)

− ∂H

∂ rn+ nhK + γbgvnh = 0; (vn = 0 for vn ∈ Pl , ) (2–38)

where K is the total motor force which takes nature of viscosity in continuous motor model

and γbg is the background friction. We do not require background friction in continuous motor

model because we get nodal velocity terms in the expression for average motor force. However,

we require in case of discrete motors. The difference in relaxation is insignificant because

typically motor friction is significantly higher in the magnitude.

28

The minus end has a clamped boundary condition i.e. the v0 and v1= 0. The plus end has

free boundary condition.

Figure 2-1. Stresses within the microtubule.A) A microtubule filament of length 0.15µm represented by 3 model points forvarious boundary conditions. The solid blue line represents the free filament lengthwhile the red dots represent the model points that model that free length by takingboundary condition into consideration. The blue line is not the part of freemicrotubule length. We do not use the model point that is not the part of freelength for estimating the properties of microtubule. B) The figure illustrates theprocess of adding nodal points for the polymerizing at 0.4µm.−1. Additional nodalpoints are added when the actual microtubule length and the representedmicrotubule length is more than half the segment length. The solid blue lineindicates the section of microtubule that constitutes the free length of themicrotubule.

29

Figure 2-2. Stresses within the microtubule.A) The figure shows the filament with no strains. The bond length for vectorsr0,1andr1,2 is equal to the segment length h. As a result, there is extenional strainwithin the microtubule. The bond vectors are parallel to one another as result thecurvature C = 0. Thus there is no bending strain this configuration. B) Thebond-vectors deviate from its rest length as a result there is extensional strainenergy associated with the configuration. Since, C = 0 there is no bending strainwithin the filament. C) The filament has no extensional strain but it has non-zerobending strain as the adjacent bond vectors are not parallel to one another.

30

Figure 2-3. Initial motor configuration leads to a shortlived compressive stress..The minus end is on the left and and + end is on the right which are marked intop most figure. A typical dynein linkage with a plausible initial configuration isshown here. Initially motors appears on the microtubule in a stress free state attime t=0 shown by the green color for the linkage. A plus end directed motorundergoes a shortlived state where the linkage is under compression due to itsinitial configuration. Motor does not produce any force on microtubule during thisstate because we assume that linkage chains would rather buckle and the resultingforce would be small. At time t=0.3 s motor linkage is under extension indicatedby red color. The motor is exerting some force on microtubule at this time.

31

CHAPTER 3A MECHANISM FOR BENDING MICROTUBULES

3.1 Background Information

Microtubules play critical roles in cell functions such as mitosis, intracellular transport, and

motility. They are the most rigid of the three cytoskeletal elements (microtubules, intermediate

filaments, and F-actin [53], and isolated microtubules are straight over length scales of

millimeters [1]. Microtubules can buckle by polymerizing against obstacles [4, 26, 31], and

their bent appearance in cells [6, 60, 61] is suggestive of a mechanical role in which they bear

compressive loads [62, 63]. In this way, microtubules may help stabilize and maintain cell shape

[64, 65].

However, microtubule bending does not only derive from polymerization against a barrier,

because local bends also develop when the tips are polymerizing freely. By freely polymerizing,

we mean that the microtubule tip moves to accommodate the additional length, which is

distinct from the situation when the polymerizing tip is immobilized by the cell periphery or

other obstacle. Moreover, polymerization forces cannot explain wavy microtubule growth from

the centrosome, which is observed in tip-tracking experiments [8, 66]. Bicek et al. [3] have

hypothesized that tangential forces generated by molecular motors can bend a microtubule by

transporting portions of it (referred to as segments from now on) toward an immobile point

on its contour. However, such pinning points have not been observed during the bending of

freely growing microtubules, nor have the motors that might push the segments towards these

pinning points been identified.

In vitro experiments with reconstituted microtubules have shown the feasibility of

microtubule bend formation due to the activity of myosin motors [5]; retrograde flow of

0 Reprinted under the Creative Commons Attribution (CC BY) license(http://creativecommons.org/licenses/by/4.0/) with changes to formatting from: KentIA, Rane PS, Dickinson RB, Ladd AJ, & Lele TP (2016) Transient Pinning and Pulling: AMechanism for Bending Microtubules. PLoS One 11(3):e0151322. Ian Kent and Tanmay Leleare credited with experiments.

32

actomyosin can also cause microtubule buckling [67, 68]. However, Bicek et al. [3] have

argued against a role for actomyosin contraction in driving anterograde flow of microtubule

bends, at least in LLC-PK1 epithelial cells [3]. Microtubule motors like kinesin [69] may

drive microtubule bending but others have suggested that dynein is the dominant motor

pulling along microtubule lengths [3, 11]. Thus the mechanisms for bend formation away

from the cell periphery remain unclear. Forces generated by dynein, kinesin, and actomyosin

suggest different outcomes for the direction of microtubule-segment transport during the

development of a bend. Dynein would be expected to translate segments from the minus end

to the plus end, while kinesin would translate segments from the plus end to the minus end

[3, 11]. Actomyosin contraction would bring in segments from both minus and plus ends. We

hypothesized that determining the predominant direction of microtubule motion during bend

formation, specifically the frequency of bends created by minus-end directed versus plus-end

directed transport, could help identify the motors and mechanisms responsible for the shapes of

microtubules in cells.

To detect the direction of transport of microtubule segments, photobleaching was used

to create fiduciary markers along the microtubule length (all experiments listed in this thesis

were performed by Dr. Ian Kent in the Lele laboratory), which could then be used to follow

individual segments as they reptate along the microtubule contour. We tracked the positions of

these markers in LLCPK-1 cells, porcine kidney epithelial cells stably expressing GFP--tubulin.

We observed that bends form primarily by plus-end directed transport of microtubule segments

toward stationary segments along the microtubule. These pinning points were found to be

transient, with a typical lifetime of less than 1 minute. When the pinning points released, the

stationary segments started to move and the bends relaxed. Dynein inhibition eliminated the

directional bias of microtubule transport and reduced the incidence of bend formation. We

propose a model based on these observations, which involves dynein motor forces pushing

segments towards a point. Simulations of the development of microtubule bends correlate well

33

with experimentally observed shapes and time scales. The model explains how bends forming

near a growing microtubule tip can cause changes in direction of the growing microtubule.

3.2 Equation of Motion and and Selection of Model Parameters

The detailed derivation for dynein mediated forces used in this section is included in

Chapter 2,2.2.1 [11, 57]. The force exerted by dynein per K is given by,

K = ρf0(1−t.v

v−max

)t − ρκ

ko(1− t t)v (3–1)

where v is the velocity of the segment the motor is bound to, t is the local tangent to the

microtubule contour, and 1 denotes the unit tensor. There are three parameters in this model

(Table 3-1): the maximum pulling (or stall) force of the motor f −max , the velocity of a force-free

dynein motor walking along the microtubule v−max , and the lateral friction from motor binding

and unbinding γ. Two of these parameters have established values in the literature: f −max = 8

pN and v−max = 0.8 µms−1 [58, 70]. The motor friction for nuclear-bound dynein motors, γ =

56 pNsµm−1 was determined by fitting to data on nuclear rotation rates [57]. Note that the

dynamics of bend relaxation in this model do not change significantly if we neglect background

friction as motor friction is substantially higher in the magnitude.

Microtubule dynamics was simulated by incorporating the model for dynein forces with

a standard model for the bending of an elastic filament [55, 56]. The details for estimating

internal forces can be found in in Chapter 2 2.1 In the over-damped limit, the model point

velocities vi are obtained from the balance between elastic and motor forces:

− ∂H

∂ ri+ nhKi = 0 (3–2)

where Ki is the motor force on model point mi , and n is the density of cytoskeletal-bound

dynein motors; as in previous work [11, 57], we take n = 2µm−1.

34

3.3 Result

3.3.1 Microtubules Bend against Transiently Immobilized Segments

Microtubules commonly form local bends, even when the microtubule plus end is freely

polymerizing (Figure 3-1A). Thus, polymerization against an obstruction such as the cell

membrane is not necessary for formation of local bends. The bends have a characteristic oxbow

shape in which the curvature changes sign twice. We observed the direction of tangential

motion of microtubules during bend formation by photo-bleaching fiduciary markers onto

microtubules under the nucleus. Before photobleaching fiduciary markers, cells were imaged to

select growing microtubules. This procedure allowed for an unambiguous identification of the

growing plus end, because free minus ends never polymerize in cells [61, 71–75]. The nuclear

region was selected because it offered a dark background against which the microtubules were

clearly visible and was far from the cell periphery (Materials and Methods and Figure 3-1B).

During bend formation, the bleached microtubule segments were consistently observed to

transport towards the bend from only one side (kymograph in Figure 3-1C), while remaining

stationary on the other side. Hence, translation of the microtubule towards a stationary region,

or pinning point, appeared to be a consistent characteristic of local bend formation. After some

time the bend relaxed, either through motion of the pinning point (Figure 3-1C) or motion of

the bend while the pinning point itself remained stationary (Figure 3-1D).

Having established that tangential motion toward stationary segments produces bends

in microtubules, we counted the frequency of bends formed by translation toward the plus

end and by translation toward the minus end. The majority of bends under the nucleus

(87.54.8%SE) were formed by translation of microtubule length from the minus end rather

than from the plus end (Figure 3-1E). This argues against kinesin and myosin as dominant

drivers of local bend formation in these cells and supports the hypothesis that dynein is

involved in bend formation. It also implies that the additional microtubule length required to

form bends does not come from plus-end polymerization.

35

3.3.2 Dynein, but not Myosin is Involved in Formation of Local Bends underNucleus

To determine whether dynein is involved in local bend formation under the nucleus,

we inhibited dynein activity by over-expression of the fluorescently tagged CC1 domain of

p150 (Glued), which is the dynein-binding domain of the dynactin complex [11, 47, 76–

79]. Inhibition of dynein activity was validated by dispersion of the Golgi apparatus (S1

Fig). In cells expressing DsRed-CC1, the percentage of bends caused by translation of the

microtubule from the minus end was reduced significantly, to 55.2 9.2% (2 = 10.16, p =

0.001; Figure 3-1E). Therefore, the bias in the direction of microtubule translation is due to

dynein force generation. Inhibiting myosin activity by treating cells with Y-27632, a Rho kinase

inhibitor [80], had no effect on the directional bias of microtubule translation (Figure 3-1E),

in agreement with previously reported results in this cell type that argue against a role for

actomyosin contraction in bend formation [3].

To determine the effect of dynein inhibition on the probability of bend formation, the area

under the nucleus was bleached, and microtubules growing back into the area were observed

over 3 minutes. New segments of microtubule that polymerized to a length of at least 5 m

under the nucleus over the observation period exhibited a reduced frequency of bend formation

in dynein-inhibited cells compared to control cells, from 36.5 3.7% in control to 17.9 4.7%

in CC1-expressing cells (2 = 7.72; p = 0.005; Figure 3-1F). This result adds support to the

idea that dynein is acting to form bends from translation of microtubule segments from the

minus end rather than to inhibit bend formation from translation of segments from the plus

end. Inhibiting dynein not only equalized the probability of bending in the anterograde and

retrograde directions, but also decreased the overall probability that a microtubule will bend.

On the other hand, treating cells with Y-27 to disrupt myosin contractility did not decrease the

frequency of bend formation, consistent with myosin activity not being a major cause of local

bending (Figure 3-1F).

36

3.3.3 A Mechanism for Microtubule Bend Formation by Dynein-mediated Transporttowards a Pinning Point

We applied a mathematical model [11, 57] for dynein force generation on microtubules

(Figure 3-2A) to explain bend formation near a pinning point. In this model, an ensemble of

dynein motors linking the microtubule to the surrounding cytoplasmic structures [11] or to the

nucleus [57] exert a tangential force directed towards the plus end of the microtubule. The

microtubule is modeled as an elastic filament subjected to tangential pulling forces from the

dynein motors, and a lateral viscous force due to protein friction arising from transient dynein

linkages [11].Figure 3-2B shows results of a simulation in which the microtubule is pinned

at the minus end (green circle) and also pinned (red circle) at a significant distance (8µm)

from the free plus end. This configuration represents a centrosome-bound microtubule with

a (transient) pinning point at a distance from the growing tip. The microtubule to the right

of the purple line represents the region lying within the field of view of the microscope. We

assume there must be sufficient curvature to provide the necessary excess length (the difference

between the contour length of the microtubule and the linear distance between the endpoints)

to form a bend, since the minus end is anchored at the centrosome and microtubules are

essentially inextensible.

Initially (t = 0), the microtubule in view appears straight, but as tangential dynein forces

drive the excess length into the region of view, a bend begins to form near the pinning point

(Figure 3-2B ). Using the estimated motor friction for nuclear-bound dynein [57] simulations

predict that bends are formed in about 5-10 s, which is consistent with the timescales of the

experimental observations.

Simulations support the conclusion that bends are formed when the transport of excess

length by dynein from the minus end direction is halted by stationary (pinned) segments. The

resulting bend shapes depend on the amount of excess length, but are insensitive to the shape

of the microtubule segments acting as the source of excess length (i.e. the region to the left of

37

the field of view in Figure 3-2B ). Typical microtubule bends arising from various excess lengths

are shown in Figure 3-2C; these are similar to shapes observed experimentally (Figure 3-2E).

Microtubule bends formed in the simulations are not static but undergo limit cycles

consisting of slow oscillations between different bend orientations (Figure 3-2D). The cycle

time varies inversely with excess length; from tens of seconds for large excess lengths (> 3µm)

to several minutes for small excess lengths (< 1µm). Although bends in experiments were

sometimes observed to change orientation, we did not observe limit cycles with multiple periods

because pinning points were too short-lived.

3.3.4 Bending near the Microtubule Plus End Tip Leads to Tip Rotation

When local bends form far from the microtubule tip, the shape changes did not propagate

very far from the bend. For example, Figure 3-3A shows a typical experiment when a bend

formed far from the tip. The bend formed and relaxed over a period of about 25 s, but the

microtubule shape and position on either side of the bend remained nearly the same, and the

initial shape was restored after the bend relaxed. However, when a bend formed near the tip

of a growing microtubule (Figure 3-3B), the direction of polymerization changed and the new

direction persisted even after the bend relaxed.

Bend formation generates a bending moment at the pinning point, as illustrated in Figure

3-4A. When the segment on the plus-end side of the pinning point is sufficiently short, the

resulting moment causes the tip to rotate to a new direction for subsequent growth. However,

when the segment is long, the large lateral friction prevents rotation and the initial orientation

is preserved. These qualitative insights are confirmed by numerical simulations, illustrated in

Figure 3-3C and Figure 3-3D, where the motion of growing microtubules with either a long or a

short segment past the pinning point were tracked through a cycle of pinning and release. The

time evolution of the microtubule shape and tip orientation closely resembles the corresponding

dynamics observed experimentally. The microtubule pinned far from the tip (Figure 3-3C)

continues growing in the same direction after release, whereas the one pinned near the tip

changes its direction of growth. Thus, our model for dynein force generation accounts for the

38

observed correlation between the formation of bends and a sharp change in the direction of a

growing microtubule tip; furthermore it explains why the bend must be close to the growing tip

in order to cause a change of direction.

We repeated the simulations shown in Figure 3-3C and Figure 3-3D with different

segment lengths between the pinning point and the tip, which we refer to as the overhang.

The resulting tip orientations are plotted versus time in Figure 3-4B. When the distance

between the pinning point and the tip is small (blue line) the tip rotates rapidly as the bend

forms. However, after pinning, the overhang acquires additional frictional resistance because of

polymerization, which opposes the relaxation in orientation after the pinning point is released.

This leads to a substantial rotation of the tip, even 25 s or more after the pinning point is

released, but when the pinning is more than about 1 µm from the microtubule tip (green and

red lines), there is no significant change of direction.

When the overhang is small (< 1µm), microtubule polymerization contributes significantly

to the net rotation of the tip. In Figure 3-4C the rotation of the tip of non-polymerizing

microtubules is shown. Here the short (0.5µm) overhang rotates rapidly after pinning, but

when the bend relaxes it tends to return to its initial direction because the friction remains

small. The longer (1.5µm) overhang now rotates significantly (because its friction remains

constant during bend formation) but it also relaxes once the pinning point it released, although

more slowly than the 0.5µm overhang. The longest overhang has sufficient friction that it

scarcely rotates, even when the tip is not growing.

3.4 Discussion

Polymerization forces from a growing microtubule can cause compressive buckling

if the tip is pinned at the cell periphery, but it is difficult to see how this mechanism can

generate local bends far from the tip or how they can propagate back from the periphery. In

addition, such a model cannot explain how local bends form when the microtubule tips are

free. However, tangential force generation by molecular motors can transport curved segments

towards any stationary point (including the cell periphery). In this paper we focused on bends

39

that develop under the nucleus, where individual microtubules could be easily seen and where

they were far from the cell periphery. Tracking of fiduciary markers along the microtubule

length showed that transport of length during the formation of a bend occurs primarily from

the minus-end direction; the bend develops because excess length is transported toward pinned

microtubule segments; and dynein activity increases the frequency of bend formation and helps

drive excess length preferentially from the minus end. Simulations that account for dynein

activity and microtubule mechanics correctly predict the observed shapes and time scales of

bend formation, supporting our physical explanation for how dynein generates local bends in

microtubules. Dynein activity also explains previously reported transport of microtubule bends

toward the cell periphery [3].

While bends occur primarily by translation of microtubule segments from the minus

end direction, in a small but significant fraction of the observed bend formations, the bends

developed by transport from the plus end direction. In dynein-inhibited cells, the overall

frequency of bending decreased and the probability of bending in either direction was equalized.

This suggests that other mechanisms exist by which microtubule bends can form, potentially

including kinesin motoring [69] and actomyosin contraction [4, 5].While we do not rule out

these other mechanisms contributing to the bending, the primary mechanism in LLC-PK1 cells

appears to be dynein mediated forces.

Our simulations show that tangential forces pushing mobile microtubule segments against

stationary pinning points are mechanically sufficient to give rise to the bends observed in

experiments. The time scale of bend formation depends on the lateral friction from transient

motor linkages. The fact that the timescales of bend formation can be predicted with the same

motor parameters that were estimated from a separate study on nuclear rotation [57] increases

our confidence in the proposed model.

Microtubules in cells do not grow in straight lines [66]. In previous experiments, we have

shown that the waviness in microtubule growth is primarily due to dynein activity [8]. Tip

rotation due to bend formation may be a means whereby a growing microtubule amplifies

40

existing excess length through a succession of pinning and unpinning events. A pinning event

near the tip, together with bend formation, causes a change in the direction of growth and

therefore generates additional excess length because the microtubule is following a wavier path.

This would appear as random fluctuations in a tip-tracking experiment, because the developing

bend behind the tip remains unobserved. We note that this mechanism requires preexisting

excess length, but its origin remains unclear because microtubules can grow along wavy paths

even when dynein, myosin, or kinesin are inhibited [8].

In summary, the experimental results and simulations presented here support the existence

of two related mechanisms of microtubule bending by dynein: bend formation near pinning

points and resulting reorientation of the direction of polymerization. These findings continue to

point to the critical role of motor forces and protein friction in governing microtubule bending

dynamics in vivo.

41

Figure 3-1. Local bends develop by dynein-dependent translation of microtubule segments fromthe minus end. Figure description is on the next page.

42

(A) Image sequence shows bending of a freely growing microtubule, demonstrating thatpinning of the plus end is not required for bends to develop. (B) Experimental Technique.Fluorescent microtubules under the cell nucleus are photobleached to allow for easier analysis ofindividual microtubules. Only newly polymerized segments of microtubules are fluorescent afterbleaching. Microtubule plus ends are identified by their polymerization, since minus ends donot polymerize in cells. Once new microtubules grow to a sufficient length, fiduciary markersare bleached onto them so we may observe the lengthwise translation of different regions alongtheir length. (C) The top panel shows a microtubule that polymerizes to the right over thecourse of 31 seconds and is bleached with a pattern of dashes at 66 seconds. A local bendthen forms (white frames in bottom panel). During development of the bend, the minus end(left) side of the microtubule moves towards the plus end, and buckles against a stationarymicrotubule region, indicating pinning. The local bend then maintains its shape for some time(yellow frame) before relaxing. Bend relaxation (cyan frames) occurs by movement of thepreviously stationary right side toward the plus end, indicating unpinning of the microtubule,while the left side is stationary. Vertical white lines are provided to help visualize the movementof fiduciary markers. A cartoon trace of the microtubules shape evolution is provided below,in which the pinning point is marked with an x and translation of different regions of themicrotubule during development and relaxation of the local bend is marked with red andcyan arrows, respectively. (D) The top panel shows that the microtubule polymerizes to theright over the course of 28 seconds and is then pattern bleached. A local bend develops bytranslation of the plus end (right) side of the microtubule toward the minus end, the oppositeof what occurs in C. Also different from the microtubule in C, the bend relaxes away fromthe pinning point (in this case back toward the plus end), with the pinning point stayingintact throughout. All colors and symbols are the same as in part C. (E) Plot showing thepercentage of bends that formed by microtubule translation towards the plus and minus ends incontrol (n = 48 bends from 21 cells), dynein inhibited (CC1; n = 29 bends from 16 cells), andmyosin-inhibited (Y-27; n = 45 bends from 18 cells) cells. Error bars indicate standard error.(F) Proportion of visible microtubules that bent in three minutes following photobleaching ofthe area under the nucleus in control (n = 167 microtubules from 9 cells), dynein inhibited(CC1; n = 67 microtubules from 6 cells), and myosin-inhibited (Y-27; n = 83 microtubulesfrom 7 cells) cells. Error bars indicate standard error.

43

Figure 3-2. Simulations of dynein mediated bend formation in microtubules. Proceedto next page for figure description

44

A) The cartoon illustrates a model for the dynein-generated force on a microtubule. Anet tangential force is experienced by the microtubule due to collective motor activity. B)Simulated bend development by tangential forces using the model represented in A. Thered dot is the location of the pinning site, the green dot indicates the minus end of themicrotubule, and the blue dot is a point that marks the beginning of the microtubule segmentthat would be observed in a typical experiment. The magenta colored line marks initial positionof the blue dot. The figure shows microtubule translation due to cytoskeletal dynein motors.The excess length behind the blue marker translates into the viewing window by motor activity,leading to a pronounced bend on the minus side of the pinning point. The excess length isequal to the maximum displacement of the blue marker from its original position. C) Simulatedshapes from varying excess lengths, generated after 10 seconds. D) Snapshots of a segment ofa microtubule at different times, showing how different shapes can develop if the microtubuleis pinned for a long time (excess length of 2.5 µm). E) Microtubule shapes observed in livingcells.

45

Figure 3-3. Rotation of the microtubule tip due to local bend formation.A) Images show that the microtubule shape on either side of the bend remainedalmost the same, and the initial shape was ultimately restored after the bendrelaxed (initial and final shapes marked in red). B) An example of a microtubulethat changes direction of polymerization due to a local bend near the tip. Initiallythe filament is aligned at an angle of roughly 30 degrees to the horizontal (markedby red dash), but as soon as the bend starts to form, the tip starts to rotatetoward the horizontal direction. The plus end continues to grow as the benddevelops over the course of about three seconds. The bend relaxes after about 10seconds, but the tip keeps growing in the newly acquired direction (marked inyellow). C) Simulation of local bend formation when the pinning point is far fromthe tip (as in the experiment shown in panel A). The red dot indicates the pinningsite which is initially at a distance of 8 µm from the tip. A bend forms within thefirst 5 seconds, after which the pinning point is released. The segment on the plusside of the pinning point does not rotate because of the large lateral friction alongits length. As a result, the direction of microtubule growth remains unchanged. D)Local bend formation when the pinning point is close (0.5 µm) to the tip (as in theexperiment shown in panel B). The short microtubule segment to the plus side ofthe pinning point changes direction during bend formation, but microtubule growthduring the pinning event increases the lateral friction and prevents the tip relaxingback to its original direction after the pinning point is released. There is a netchange in the tip direction even after 30 seconds.

46

Figure 3-4. Effect of overhanging segment length on the rotation of the tip.A) The sketch shows how a local increase in curvature during bend formationdrives tip rotation. A local bend (a) generates a bending moment across thepinning point (b). Motion of the overhanging segment is opposed by friction, butgiven enough time, the tip rotates to straighten the segment (c). B) Orientation ofa growing microtubule tip as a function of time. The microtubule is initially pinnedat different distances from the tip: 0.5 µm (blue), 1.5 µm (green) and 2.5 µm(red) and unpinned after 15 seconds (black dot). C) Orientation of anon-polymerizing microtubule tip as a function of time. The microtubule is initiallypinned at different distances from the tip: 0.5 µm (blue), 1.5 µm (green) and 2.5µm (red) and unpinned after 15 seconds (black dot).

47

Table 3-1. Model Parameters: Microtubule and Dynein Motors

Parameter Symbol ValueModulus of Elasticity E 1.75 ∗ 104 pN.µm−2

Flexural Rigidity B 25 pN.µm2

Polymerization Velocity vpol 0.1µ.s−1

Number of motors ρ 2µ−1

Dynein Stall force f −max 8 pNDynein stress free speed vmax

− 8µm.s−1

Lateral friction coefficient γ 56 pN.s.µm−1

48

CHAPTER 4EXCESS LENGTH IN MICROTUBULES

4.1 Objective and Background Information

4.1.1 Objective

While Chapter 3 showed that microtubule bends develop by transport of contour length

of initially curved microtubules, it did not address the origin of this initial curved shape. Here

we explore how this contour length- the length which is ’excess’ over the shortest line segment

joining two end points on the microtubule- could arise. Quantifying this length in microtubules

from experimental images is a daunting task because microtubules frequently overlap with each

other and other structures, which makes it difficult to trace a complete microtubule in the cell.

Since, this question is not easily addressed experimentally, we chose to explore this question

with computations.

4.1.2 Background Information

As shown in the Chapter 3, microtubule segments are transported toward the plus end

of microtubules, which give rise to local, dynamic bends in the microtubule. Naturally, this

transport cannot occur if the microtubule is straight to begin with as microtubule behaves

like a non-extensible rod. We term the length along the curved microtubule contour which

is available for transport and results in local bends, ’excess’ length. Shekhar et al. [8] and

Brangwynne et al. [66] have previously shown that microtubules do not grow along straight

paths. Rather, as the microtubule tip moves outward from the centrosome due to microtubule

polymerization, it deviates from its initial trajectory. Shekhar et al. [8] quantified this deviation

by measuring the motion of the microtubule tips labeled with GFP-EB1 (green fluorescent

protein that labels End Binding protein-1). They plotted the tip trajectories such that

the initial orientation of all tip trajectories was along the positive x-axis. As expected, the

trajectories deviated from the x-axis, resulting in a plot like Figure 4-1.

As seen in Figure 4-1B, Shekhar et al. [8] further showed that inhibiting dynein reduced

the spread of the trajectories. They also reported that inhibiting myosin had only a modest

49

effect, while kinesin-1 inhibition did not affect the spread. These results implicated dynein as

the major motor protein involved in deviating microtubule tips from their original directions.

Yet, simulations of growing microtubules under dynein-forces tangential to the contour using a

model similar to the one in Chapter 3 do not predict such a spread. Rather dynein forces are

predicted to straighten out any curvature in the growing microtubules, because dynein pulls

toward the plus end which generates tension in the centrosome bound microtubules. This can

be seen in Figure 4-2.

To explain how dynein could potentially deviate tips from their straight path, Shekhar

et al. [8] explored the possibility that single dynein motors binding at random angles to the

microtubule contour could exert lateral (normal) force on it. In addition, the anchor point

to which the dynein is bound could move in a random direction due to local actomyosin

contraction in the cytoplasm [81], exerting further lateral forces on the microtubule.

This model successfully predicted the experimentally observed spread in tip trajectories

(more specifically the amplification of the spread by dynein) for anchor-point speeds that were

closer to lower end for myosin walking speeds as crosslinked cytoskelton is hypothesized to

move slower that individual motors pulling on it. While the previously proposed explanation [8]

is plausible, it suffers from the obvious weakness that it cannot explain the experimentally

observed variance of tip trajectories in dynein inhibited cells [8]. In these experiments,

dynein was inhibited by over expression of the CC1-domain (coiled coil 1) of dynactin, which

competitively binds to dynein and disengages it from the dynactin complex. Therefore, no

lateral force is expected due to dynein, yet there is a significant spread observed in microtubule

trajectories, far in excess of that expected by thermal forces. It would appear that a mechanism

separate from dynein operates in cells to impose deviations in tip growth. Another weakness

of the model by Shekhar et al. [8] is that there is no direct evidence that dynein anchor points

move and thereby exert lateral forces on microtubules.

50

4.2 Data Analysis Methods

4.2.1 Fourier Mode Analysis

Fourier mode analysis was performed to quantitatively investigate the extent of bending

in the simulated microtubules. Figure 4-1 shows that the ensemble of microtubules span into

a divergent V shape with the tip of V locate at origin. Keeping this geometry in mind we

characterized the shape of the microtubule as sum of sine waves of imaginary frequency. We

computed frequencies corresponding to quarter waves and the half waves.

θ(s) =

√2

L

∞∑n=1

ansin

(2πns

4L

)(4–1)

The shapes for sin waves are for n=1,2..7, are show shown in Figure 4-3 with a fix

amplitude A=0.25 units for demonstrative purpose. The amplitude for actual modes goes down

as n increases. From a set of N coordinates (xk , yk) of a trajectory, the length ∆sk and the

angles k for (N − 1) segments that connect these co-ordinates was calculated.

∆sk = [(xk+1 − xk)2 + (yk+1 − yk)

2]1

2 (4–2)

θk = tan−1[yk+1 − yk

xk+1 − xk

](4–3)

The amplitudes computed by taking the Fourier inverse of eq. 4–1,

an =

√2

L

N−1∑k=1

θk∆sksin(2πnsmid

k

L) (4–4)

where,

L =

N−1∑k=1

∆sk (4–5)

and

smidk = ∆s1 +∆s2 + ....∆sk−1 +

1

2∆sk (4–6)

4.2.2 Variance

Simulated trajectories are oriented along the x-axis to begin with. We recorded the

co-ordinates for the last node that represents the microtubule contour as the position of the

51

tip. Length dependent variance was estimated for the simulated tip trajectories at points along

the trajectory contour (s) corresponding to the ones reported experimentally. The variance is

computed as y(s)2.

4.3 Results

4.3.1 A Putative Role for Plus End Motors in Generating Excess Length

Recent literature suggests that kinesin can generate opposing forces to dynein, resulting in

a ’tug-of-war’ between the two motors [82–87]. The ’tug-of-war’ is commonly associated with

organelle transport. Here we expand this idea from the perspective of the force balance on the

microtubule and its consequence on deviation of microtubule tips and excess length within the

microtubule.

While Shekhar et al. [8] ruled out kinesin-1 as being involved in deviating tips, there are

other kinesins (such as kinesin-2) that they did not explore experimentally. We reasoned that

plus-end directed motors like those belonging to the kinesin family could potentially generate

tangential forces on interphase microtubules directed toward the minus end. It is intuitively

obvious that such forces could bend an initially straight microtubule anchored at its minus

end, and thereby introduce a curvature in the microtubule. A strong case can be made for

heterotrimeric kinesin-2 given its ability to bind to dynactin and its expression in the NIH3T3

cells used in experiments conducted by Shekhar et al. [8]. Dynactin is shown to be necessary

for dynein function.

We therefore performed computations of growing microtubules under the action of

plus-end directed motors. Microtubules were nucleated with a polymerization velocity of 0.4

µm.s−1 [8]. Motors were allowed to bind to the growing microtubule at random locations

and orientations consistent with a specified motor density (e.g. 1.5 µm−1), and allowed to

motor toward the plus end. Motor force was chosen to be 5 pN (similar to force generated by

heterotrimeric kinesin-2) [88]. Details for the model are listed in section 2.2.2 of Chapter 2.

Shown in Figure 4-4 are results of simulations first with + end motors alone (Figure

4-4B) and then with both + and - end motors (Figure 4-4C). In Figure 4-4B, we computed

52

variances in an ensemble of growing microtubules at different densities. As seen in Figure

4-4B, the computed variances compare reasonably well with experimental measurements of

variances of tip trajectories in dynein-inhibited cells (blue squares in Figure 4-4B). As + end

motors were able to produce the variance in dynein inhibited cells in the above calculations, we

next examined if the action of minus-ended motors could amplify this ’background’ variance

produced by plus-end motors. To do this, we performed simulations that incorporated both

plus-end and minus-end directed motors, and calculated the spread in tip trajectories. We

chose 1.5 µm−1 as the density for + end motors because it appeared to closely match the length-

depedent variance in dynein inhibited cells. As seen in Figure 4-4C, including both - end and + end

motors in the simulation adequately predicts variance in tip trajectories for a density of 3 µm−1 and

1.5 µm−1 respectively.

The deflection of the tip is influenced by motors binding close to the tip. This can be seen in

Figure 4-5 where individual motor bound near the tip is able to change the orientation of the tip.

We see that both plus and minus end motor are capable of deflecting tips.The length of

microtubule is kept short because another linkage would appear considering the combined linear

density on the actual microtubule that would offer resistance to motion. We can expect a slight

higher deflection as the resisting motor may act more like a hinge than a clamped boundary

condition for the microtubule here. We computed Fourier modes for an ensemble of 8 µm long

simulated microtubules (+ end motor density 1.5 µm−1) and compated it with the tip trajectories

in dynein inhibited cells. The fourier modes agree remarkably for this case which can be seen in

Figure 4-6A. Later we compared the Fourier modes for control cells with microtubules subjected to

+ and - end motors with density 1.5 and 3 µm−1 respectively. We did get find that 1st 3 modes

were in close agreement with experiment (see Figure 4-6B).

At this length microtubule shapes are fairly similar to their tip trajectories. Fourier modes

for simulated trajectories with + end motor density of 1.5 µm−1 agree remarkably with the

experiments. The case corresponding to control cells i.e with plus and minus end directed motors

were in agrement with the experiment. It is only in the case of longer microtubules

53

in kinesin only case (dynein inhibited case) they differ significantly because motor force from

kinesin alone tends to buckle the microtubules.

Additionally, we found that the mean polymerization speed estimated from tip trajectories

was lower in case where microtubules were sujected to plus end motors only in comparison

microtubules subjected to plus and minus end motors despite of the fact that the polymerization

velocities are identical in simulations seen in Figure 4-7. This is also reported in Shekhar et al.

This apparent lowering is observed because as microtubules become longer they tend to buckle

when only kinesin. As a result tip appears to move slower. It is a minor observation that this

model is able to explain based on the microtubule force balance.

4.4 Discussion

Here we presented a simulation study of a growing microtubule under forces generated

by stochastically binding plus and minus-ended motors binding along the length of the

microtubule. We showed that plus ended motors deflect the microtubule tip as it grows.

We tuned the parameters such that the variance of the growing tip is similar to experimental

measurements in dynein-inhibited cells. Addition of minus ended motors to the growing

microtubule amplified the variance, similar to experimental observations. By qualitative

comparison with experimental measurements of variance, we estimated the density of plus

ended motors at 1.5 µm−1m, and the density of minus-ended motors at 3 µm−1. The average

net force of 17.5 pN.µ−1

n+f+

max − n−f−max = 17.5pN.µm−1 (4–7)

toward the plus end. This force is consistent with the net force per micron of the microtubule

estimated in Wu et al.[11]. There, a force of 16 pNµm−1 was shown to predict the increased

bending of minus ended microtubule fragments that were created by laser severing. A force

of 16 pNµm−1 was also found necessary to predict the timescales of centrosome centering

[11]. This consistency in parameter estimates which were deduced from disparate experimental

observations- variance of growing tips, time scales of microtubule bending in laser severing

experiments, and centrosome centering time scales- increase our confidence in the model.

54

In the model, because motors bind stochastically, it can happen that parts of the

microtubule are under tension and parts of the microtubule are under compression as motors

work against each other. At the same time, we predict the average force on the microtubule

to be tensile with a magnitude of roughly 17.5 pNµm−1. Experiments that inhibit kinesin-2 or

other kinesin motors, and quantify the spread in microtubule trajectories will be necessary to

test our model. The list of all parameters can be found in Table 4-1.

The mechanism by which dynein amplifies the spread in microtubule tip trajectories

created by plus end motors is similar to that proposed by Sekhar et al. [8]. That is, the binding

of dynein at random angles and extensions and subsequent walking and finally stalling, exerts a

lateral force on the microtubule. This force ensures a spread despite the tendency for tangential

dynein forces to straighten out bends. Such a mechanism is able to explain the accumulation of

contour length in the microtubule as shown in Figure 4-8. The curved contour of microtubules

created by this mechanism results in occasional transport of curved lengths next to pinned

regions, which gives rise to dynamic bends in the microtubule as shown in Chapter 2. As

argued in Chapter 2, it is also possible that pinning by itself could give rise to a spread in

the microtubules, but we found that speed of microtubule growth and our preliminary studies

suggest that pinning under the action of dynein motors can amplify the excess length but fails

to nucleate and sustain the excess length within the microtubule.

55

Figure 4-1. Experimentally recorded tip trajectories of growing microtubules.An ensemble of microtubule tip trajectories recorded in control cells. A) and indynein inhibited cells. B) Tip trajectories were oriented to have the same initialdirection.

Figure 4-2. Microtubule straightening in absence of pinning under the action of dynein motor.The black dot represents the centrosome. The microtuble has an excess length of 1µm at 0 sec and it is subjected to dynein forces using the continuous motor modelfrom Chapter 2. Tangential dynein forces transport microtubule length lengthtowards plus end as seen from microtubule shape at 3 sec and and 6 sec. Thisresults into a straight microtubule in absence of pinning.

56

Figure 4-3. Sine Waves corresponding to mode numbers in the Fourier mode spectrum.Sine waves were used for calculating the amplitudes of fourier modes based on theboundary conditions. The tip trajectories spread around its initial direction in a ’V’shape with tip at origin. Wave shapes corresponding to the modes are plotted witha constant amplitude of 0.25 units for demonstrative purpose as the spectrum iscombination of quarter waves ad half waves.

57

Figure 4-4. Simulations of growing microtubules subjected to motor forces.A) Simulated microtubules that grew to about 8 µm length is shown here. Duringthe entire duration of growth it was subjected to randomly binding + end motors. The density of motors along its length is 1.5 µm−1 on average. As seen, the microtubule bends significantly during growth. B) Variance calculated from tip trajectories of an ensemble of growing microtubules a different motor densities. 0.5 to 2 µm−1 densities are reasonable in predicting the variance in dynein inhibited cells (blue squares, experimental data). We chose a density of 1.5 µm−1 as an optimal density. C) Using 1.5 µm−1 as the density for +end motors, we simulated growth, but now under different densities of - end motors. As seen, + and - end motors collectively predict the variance appropriately. Simulated trajectories with minus end directed motor density of 3 µm−1 in addition to plus end directed motors was found to be in close agreement with experimentally measured values as seen in Figure 4-4C.

58

Figure 4-5. Motors binding near the tip change the orientation of the tip.A) A typical kinesin motor appears on the MT at 0.9 L and starts walking towardsthe plus end thereby changing the orientation of the tip. B) Direction of tip altered by - end directed motor dynein. The extent of change in tip orientation will be higher than what is observed her because of the small lenght of the microtubule with clamped boundary condition. A small length of microtubule is selcted to account for motors that would actually be present on a typical microtubule. However, those motors are expected to impose a hinge like boundary condition than the clamped one.

59

Figure 4-6. Comparison of Fourier modes of tip trajectories and simulated microtubules.Fourier modes for 8 µm long simulated microtubules (circle) and 8 µm long tip trajectories (squares) were calculated using the method described in the earlier section on page 57. A) Fourier modes for simulated microtubules grown with stochastically binding + end directed motors with linear density of 1.5 µm−1 are compared with tip trajectories in dynein inhibited cells. Apart from the 2nd Fourier mode corresponding to half sine wavelength they match remarkably with one another. B) Simulated microtubules experienced + and - end directed motors with linear density of 3 µm−1 and 1.5 µm−1 respectively. First 2 Fourier modes show reasonable agreement with the experiments, however the 4th Fourier mode for simulated trajectories corresponding to a complete sine wave is significantly lower ( 0.03 µm) and shows poor agreement with the experiment.

60

Figure 4-7. Average poly. speed estimated from tip trajectories of simulated microtubule.The actual polymerization speed for both cases is 0.4µm.s−1. Microtubules buckle under the action of plus end motors alone. This leads to lower measured speed. Sections of microtubules buckle in when + and - end motors but buckling is transient and not persistent due to the dynein dominated force balance. Hence the value for estimated polymerization speed when + and - end motors are active is close to actual polymerization velocity.

61

Figure 4-8. Excess length in simulated microtubules.An ensemble of 100 microtubules was simulated for calculation of excess lengthdistribution in the histogram above. The histogram data is collected by selection ofthe maximum excess length for a given microtubules during the entire simulation.About 25% of them have excess length more than 0.5 µm. Figure 3-2C in Chapter3 has an example of microtubule with 0.5 µm that generates a small length highcurvature bend.

62

Table 4-1. Model Parameters: Microtubule, + and - End Directed Motors

Parameter Symbol ValueModulus of Elasticity E 1.75 ∗ 104 pN.µm−2

Flexural Rigidity B 25 pN.µm2

Polymerization Velocity vpol 0.4µ.s−1

- end motor stall force f −max 8 pN+ end motor stall force f +max 5 pN

- end motor stress free speed vmax− 8µm.s−1

+ end motor stress free speed vmax+ 0.5µm.s−1

- end motor density n− 3µ−1

+ end motor density n+ 1.5µ−1

motor off-rate ko 1 s−1

motor spring constant κ 1000 pN.µm−1

Background Friction γbg 10Pa.s

63

CHAPTER 5OUTLOOK

In this thesis, we explained how microtubules undergo local bending in the cytoplasm. In

Chapter 2, we showed that microtubule bending occurs by the plus-end-directed transport of

microtubule segments toward stationary microtubule segments termed pinning points. This

transport causes the microtubule to display local oxbow type bent shapes. Such shapes have

been posited to be caused by lateral forces due to myosin activity. But such forces cannot

explain the directional bias in bending. We solved a computational model that incorporates

tangential forces generated by dynein, frictional forces due to dynein dissociation, and frictional

forces due to cytoplasmic background. We showed how local bends develop in microtubules on

time scales similar to those observed in experiments.

A corollary of Chapter 2 was to explain how dynein forces can cause lateral deviation of

growing microtubule tips. Dynein inhibition has been experimentally shown in the Lele lab to

decrease the spread of growing microtubules. However, as we showed in Chapter 3, dynein

activity is expected to straighten out bends in a short growing microtubule, not increase them.

Moreover, dynein inhibition in experiments does not eliminate variance in tip trajectories. Using

a stochastic model of motor binding and dissociation, we computationally showed in Chapter

3 that the action of plus-ended directed motors can produce bends in a growing microtubule.

Adding dynein motors to this model amplifies variance. Our computational model explains how

dynein amplifies variance, and how excess length is generated by plus-end directed motors.

Collectively, this thesis has brought clarity to the mechanisms by which microtubules bend

along their lengths, and the mechanism by which the microtubule deviates from originally

straight trajectories.

Our thesis is significant because there is little appreciation in the literature of the fact that

tensile forces can be generated by motor forces along the lengths of microtubules in interphase

cells. Most researchers assume that microtubules exist under simple compression due to

immobile but growing tips. Tensile forces are thought to exist on microtubules only in mitosis

64

when chromosomes separate due to microtubules that shorten at their kinetochore-bound

ends. That dynein can generate tension on microtubules has largely escaped attention in the

literature. Yet, such tensile forces are primarily responsible for centering the centrosomal array

of microtubules, and therefore crucial in the establishment of cell polarity, a function critical for

processes like wound healing and development.

Looking to the future, many questions remain which will need to be answered by a similar

cross-talk between experimental and computational efforts. First, what is the mechanism by

which the microtubule gets pinned at pinning points? We hypothesize that molecular motors

like dynein or kinesin may pin the microtubule if they lose their ability to process (i.e. dead

motors). Other possibilities include protein complexes at the growing tip which might connect

microtubules to dynactin or other complexes.

Second, what is the functional significance of microtubule bending? The dynamic bending

of microtubules, and its growth in wavy paths in cells is a fascinating feature of microtubules,

yet this phenomenon largely lacks a (known) function. Does random microtubule bending

allow efficient sampling of the neighborhood? Is this perhaps for transport of cargo to a larger

space than possible by a straight microtubule? These questions could potentially be explored

computationally, but perhaps also experimentally. Alternatively, microtubule bending and

bending during growth might just be an intrinsic property of thermally driven motor driven

bending of randomly growing, stiff microtubules.

Our computational model suggests that plus-ended motors might be important in

the mechanical behavior of growing microtubules. This prediction awaits experimental

confirmation. If it is found that kinesin plays no role in bending microtubules, then the

question of how the spread in the tips of growing microtubules develops will need to be

tackled. As we have mentioned in Chapter 3, an alternative hypothesis is that anchor points of

dynein motors move due to random dipoles in the cytoplasm generated by actomyosin forces.

Experimentally testing this hypothesis requires the imaging of the anchor points of individual

dynein proteins. While this may be experimentally challenging, it also represents an exciting

65

opportunity to understand how dynein motors function in the complex environment of the cell

cytoplasm, and impact microtubule behavior.

Also important in testing the predictions of our model will be to experimentally measure

the density of dynein and kinesin molecules as they walk along microtubules. In particular, our

model requires estimates of these proteins while they are bound to relatively immobile objects

(i.e. not that fraction of motors that transport organelles). It is when the anchor points are

bound to immobile objects that the motors are predicted to generate measureable forces on the

microtubule. Admittedly, such an experiment will be greatly challenging. In particular, if force

sensors are developed that can measure the force on a microtubule as the dynein motor walks,

that would be the ultimate test of the mechanical models in this thesis.

66

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BIOGRAPHICAL SKETCH

Born and raised in the town of Jalgaon, in India Parag Rane is the son of Subhash Rane

and Rekha Rane. From his childhood, his parents nurtured his curiosity by providing him with

ample books and visits to various places like Nehru planetarium at Mumbai, botanical gardens

at Bangalore and Singadh fort in Pune to name a few. This played a vital part is his strong

inclination towards STEM fields.

For his undergraduate education, Parag landed at Institute of Chemical Technology (ICT),

Mumbai in fall of 2007 where he earned his Bachelor of Chemical Engineering (B.Chem Engg.)

degree in spring of 2011. During this time he was thankful for having the opportunity to lead

the undergraduate technical festival at the institute. He interned at Vega Chemicals. Ltd.

during summer of 2010. He is grateful for Prof.V.G. Gaiker, Prof.S.D. Samant and his parents

who played a vital role in his decision to persue higher education. He joined University of

Florida in fall of 2011 for doctorate studies and was fortunate enough to find a chance to work

with Prof. Tony Ladd, Prof. Rich Dickinson and Prof. Tanmay Lele. They refined Parag’s

sense of science, how it is done at the highest level and how it must be communicated in its

simplest form. He feels lucky to have a very strong PhD committee with whom he was able to

interact with quite frequently.

In his leisure time, Parag likes to sing, run and play computer games on his system which

he assembled during Christmas in 2012. He has keen interest in machine learning, physics,

and American politics. He is looking forward to make his transition to the field of data science

which he not only finds fascinating but it also compliments his skill set that he has developed

during the doctoral program.

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