ddp stage-i report

8/2/2019 DDP Stage-I Report

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CHAPTER1: DEFINITION OF THE SUBJECT

There is a growing experimental evidence showing a profound influence of the stiffness of the

ECM on adhesion and contractility with increasing stiffness leading to stronger adhesion,

higher pre-stress, and a well spread cell[2,3].

Recently it has been demonstrated that trypsin-induced de-adhesion can be used as an assay for

probing cellular contractility [2]. Upon treatment with trypsin cellECM contacts rapidly

severed. As cellECM contacts are severed, tensile loads within the cytoskeleton suddenly

become unbalanced, causing the cell to rapidly contract to a rounded morphology. Kinetics of

this retraction was observed to obey sigmoidal kinetics with characteristic time constants, and

that the magnitude of these time constants closely track cellular elasticity as measured by AFM.

However ,while faster de-adhesion is observed with increased intra-cellular tension through

activation of cell contractility, faster de-adhesion is also observed upon treatment with higher

dose of trypsin leading to faster breakage of adhesion bonds. Thus de-adhesion is not only

influenced by cellular contractility but also by the rate of bond breakage making it difficult

distinguish between the contribution from these two sources.

To understand the relative importance of adhesion and contractility a formulation for computing

de-adhesion profiles 1-dimesional model of contractile cell of given mechanical properties

placed on an ECM of given mechanical properties and prescribed density of adhesion is

developed by Mandar Inamdar[1] . In this project I have extended the scope of the model by

formulating 2-dimesional model of contractile cell and also incorporating the effect of material

non-linearity and force dependence of bond breakage rate constant. The sensitivity of de-

adhesion profile on each parameter is assessed by varying each of these parameters while

keeping other parameters fixed and comparing de-adhesion profiles.


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CHAPTER2: INTRODUCTION

2.1 General aspects of cell adhesion

The human body consists of around 1013

cells, which can be classified into more than 200

different cell types. In order to function in the way we are used to, the human body has to fulfill

two seemingly contradicting principles. On the one hand, the cells in our body have to adhere to

each other, otherwise it would simply fall apart. On the other hand, they must be able to

reorganize quickly, for example when the body has to react to infection or injury. Nature has

evolved different strategies to cope with these conflicting requirements. On the molecular level,

biological adhesion is based on relatively weak (non-covalent) interactions with short lifetimes

of the order of seconds. In order to achieve long-lived assemblies, the cells in our body adhere

through clusters of adhesions bonds, which prolong lifetime both by large bond numbers and by

facilitating rebinding of single bonds. Because they are highly dynamic, biological adhesion

clusters can react quickly to new stimuli by association and dissociation. On the level of tissues,

cells build up an additional structure, the extracellular matrix (ECM), a network of protein

_laments (e.g. collagen in the connective tissue) which provides structural integrity to the tissue

as a whole. The ECM is secreted by cells during development or after injury and is continuously

remodeled by the cells. It provides structural coupling between the cells without preventing them

from dynamic rearrangements.

Cell adhesion to the extracellular matrix (ECM) has been of particular interest because it plays

important roles in many physiological processes wound healing and tissue regeneration, such as

the regulation of growth, differentiation, migration and survival of cells. The dynamic

equilibrium between a cell and its ECM is established through the balance between the

contractile forces exerted by the cell and the resistance to deformation offered by the ECM (i.e.,

ECM rigidity). This tensional homeostasis plays key regulatory roles in a wide variety of

cellular phenomena, including shape determination, migration, tissue assembly, and fate choice.

The mechanical properties of the substrate upon which cells are cultured have been shown to

influence a variety of cell properties including cell adhesion, spreading, protein expression and

differentiation. Moreover, there is growing evidence that changes in cellular mechanical

properties relevant to this force balance may serve as biomarkers for disease, including various

types of cancer. Many methods have been introduced to measure single-cell mechanical

properties, including atomic force microscopy, micropipette aspiration, traction force


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microscopy, optical tweezers, and intracellular particle tracking. While all of these methods

have proven extremely powerful in producing high-resolution measurements of cellular

viscoelastic properties, efforts to incorporate these methods into high throughput technologies

that might be used to diagnose disease or screen drug libraries based on changes in cell

mechanics have been limited in part by these methods inherently low throughput and low

proclivity toward automation. This in turn has spawned a new generation of high-throughput cell

mechanics technologies including optical stretchersand controlled detachment assays.

The adhesion between cell and ECM is often localized to discrete contact regions called focal

adhesions (FAs), as illustrated in figure 1a. FAs usually evolve from small dot-like adhesions,

commonly referred to as focal complexes (FXs), which are continuously formed and turned over

under the protruding lamellipodia. Mature and stable FAs depend on the clustering of molecular

Figure 1: Schematic of focal contacts in cellECM adhesion based on specific binding between receptors and

complementary ligands. (a) CellECM adhesion localized to discrete focal contacts. (b) Actin bundles

anchored into an adhesion plaque that connects ECM through transmembrane molecular bonds. Focal

adhesions can be exposed to cytoskeletally generated contractile forces in actin bundles, as well as externally

applied loads outside of the cell[4].

bonds, creating an adhesion plaque of complex macromolecular assemblies in which many

cytoskeletal filaments are anchored (figure 1b). The recruitment of actin filaments and integrin

receptors to the contact regime is essential to FAs, and artificially mutated integrins that lack an

ability to connect with cytoskeletal filaments often fail to cluster and are unable to form stable

adhesions . Experimentally, the adhesion clusters between cell and matrix were found to have a

characteristic length scale in the order of a few micrometres.


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CHAPTER 3: MODEL FORMULATION

3.1 2-D viscoelastic model for Cell de-adhesion:

In this model cell is modeled as two dimensional (2-D) viscoelastic object adhered to viscoelastic

substrate by continuous distribution of bonds. In our model we assume that the traction force

applied to substrate is directly proportional to the displacement at that point with proportionality

constant being stiffness of the substrate. The basis for our assumption is that many times the

substrate is not modeled as continuous medium but consisting of a array pillars of viscoelastic

materials which can act like spring and damper system attached to a fixed base.

Figure 2:A schematic representation of 2D cell connected to substrate. The cell layer contracts and is resisted by a

distribution of linear spring-damper systems on the surface[5].

We measure displacements of any point with respect to resting position when contractile stressesin the cell are zero. Initially the cell is completely spread out and is in mechanical equilibriumwith substrate. In this configuration we assume displacement field uc0 (R) and 0(R) represent

bond density per unit area. At this time t=0, De-adhesion is initiated by time dependent

irreversible breakage of bonds at rate r(t) per unit time leading to cell retraction. We can useplane stress conditions with only radial displacements to write the equilibrium equation.

Uc0

U0

R
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Using Kelvin model of viscoelastic material[7], the stresses at any point R in the cell at any time

t can be written as

where Ec and Gc are elastic and viscous modulus of the cell, respectively.

As the stresses in the cell are in equilibrium with substrate force we can write radial equilibrium equation

for a small element as

Where Es and Gs are stiffness and viscosity, respectively, scaled with respect to thickness, and

is the density of adhesion along the radius of the cell. The substrate displacementUs(R,t)=U0 (R) - Uc(R,t).

To find the initial displacement uc0(r) we assume that the cell was stretched to initial

displacement u0(r) and then attached on substrate to attain an equilibrium condition with zero

radial stress on circumference as there is no external force. We solve following non-dimensionalized differential equation to get uc0(r)

Where r = R/R, u = U/ R0, r1 = Es/Es

Now we use uc0(r) thus found as the initial condition in finding time dependent displacement

uc(r,t). We obtain following simplified equation using 0=Gc/Ec and R0 as metrices of time andlength for nondimensionalization

where r2= r0, r3 = Gs/Gc.
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In the above equation we have assumed simple rate dependence of bond chopping given by

0(r, t) = 0(r)exp(-r2t).Time dependent displacement uc(r, t) can be obtained by solving the

above equation , subject to BCs uc(0,t) = 0, r(1,t) = 0 and initial condition uc(r,0) = uc0(r).

3.2 Simplified 1-D viscoelastic model for Cell de-adhesion:Displacements of any point are measured with respect to resting position when contractile

stresses in the cell are zero. Initially the cell is completely spread out and is in mechanical

equilibrium with substrate. In this configuration we assume displacement field uc0(R) and 0(R)

represent bond density per unit area. At this time t=0, De-adhesion is initiated by time dependent

irreversible breakage of bonds at rate r(t) per unit time leading to cell retraction.

Using Kelvin model of viscoelastic material, the stresses at any point R in the cell at any time t

can be written as

Equilibrium equation for the cell can be written as

We solve following non-dimensionalized differential equation to get uc0(r)

Where r = R/R, u = U/ R0, r1 = Es/Es

To obtain time dependent displacement uc(r,t) we obtain following simplified equation using

0=Gc/Ec and R0 as metrices of time and length for nondimensionalization

(

) where r2= r0, r3 = Gs/Gc.


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CHAPTER 4: RESULTS

To study the effect of different substrate properties on de-adhesion dynamics, we calculate area

A(t)as a function of time and as done in experiments[], we consider normalized area

A(t) = (A0-A(t))/ (A0- A).In our case

In the above model physical properties of the substrate are represented by two dimensionless parameters:

r1 representing the ratio of substrate stiffness to cell stiffness and r 3 representing ratio of substrate

viscosity to cell viscosity.

4.1 Effect of Substrate Stiffness:

To find the effect of substrate stiffness on de-adhesion dynamics r 1 is varied over three orders ofmagnitude from 1 to 1000 keeping other variables constant (r2 = 1, r3 = 10,u0(r)= 0.5r, 0(r) = 6r

4).Under

these condition fastest de-adhesion was observed for r1 = 1 with half saturation time constant 1/2

of 2.3 units. 1/2 is time when A(1/2) =1/2.

Figure 3: Effect of substrate stiffness on de-adhesion. De-adhesion curves computed by varying stiffness ratio r1

over two orders of magnitude from 1 to 100 keeping other variables constant (r2 = 1, r3 = 10, u0(r) = 0.5r, 0(r)

= 6r4).

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

normalizedarea

A

t

r1


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These results illustrate that de-adhesion takes place faster on substrate with low stiffness than

substrate with relatively higher stiffness.

4.2 Effect of Substrate Viscosity:

Secondly, de-adhesion may also depend on the viscosity of the substrate. To study the effect ofsubstrate viscosity on de-adhesion profile r2 is varied over three orders of magnitude from 1 to

1000 keeping other variables constant (r1 = 10, r2 = 1, u0(r) = 0.5r, 0(r) = 6r4). As seen in the fig.4

fastest de-adhesion was observed for r3 = 1 with half saturation time constant 1/2 of 2.2 units.

Table 1:this is some table.

Figure 4: Effect of substrate viscosity on de-adhesion. De-adhesion curves computed by varying viscosity ratio

r3 over three orders of magnitude from 1 to 1000 keeping other variables constant (r1 = 10, r2 = 1, u0(r) = 0.5r,

0(r) = 6r4).

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

normalizedareaA

tr3


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4.3 Influence of distribution of adhesion:

We can expect that the strength of bonds formed between cell and substrate may also have an

influence on de-adhesion. To determine the influence of adhesion strength on de-adhesion we

computed the de-adhesion profile for different values of bond chopping rate r2 {0.5, 1, 5, 10,

50}.

Figure 5:Effect of bond breakage rate on de-adhesion. De-adhesion curves computed for bond breakage rate r 2

{0.5, 1, 5, 10, 50} keeping other variables constant (r1 = 10, r3 = 10, u0(r) = 0.5r, 0(r) = 5r4).

As seen in fig.5, r2 = 0.5 led to de-adhesion profile with half saturation period 1/2 = 3.5 units.A

tenfold incase in r2 led to significant faster de-adhesion response with half saturation period

1/2 = 1 units. Further increase in r2 led to faster de-adhesion with diminishing change in 1/2 for

r2>10.Moreover,the the shape of de-adhesion profile changed from sigmoidal curve to that of a

single exponential curve for high values of r2.

In above calculations we have kept the distribution of adhesions 0(r) the same. However

experimental observations indicate that cells form a discrete number of adhesions with their

substrates, with larger bond adhesions near peripheral region of the cell and smaller adhesions in

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

normalizedareaA

t

r2


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the central region. To study the effect of distribution of adhesion, we compare the de-adhesion

profiles arising from the different bond density distribution but having same total number of

bonds 0(r) {2, 4r2, 6r

4} while keeping all other variables constant (r1 = r3 =10, r2 = 1, u0(r) =

0.5r). In first case adhesion was uniform across entire area of the cell while other twodistributions were non-uniform but axially symmetric with strongest adhesions at the peripheral

region and no adhesion at the centre. As seen in fig.6 later two distributions yielded the same de-

adhesion profile suggesting that de-adhesion profile is independent of the adhesion distribution.

.

Figure 6:(b)Effect of distribution of adhesions on de-adhesion was assessed by comparing three different

adhesion distribution plotted in (a) while keeping all other variables unchanged.

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

5

6

r

or

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

normaliz

edareaA

t

(a) (b)


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4.4 Effect of non-linearity of material:

As experiments illustrate that bio-materials show linear characteristic for only small strains but

behave non-linearly for larger strains .To study the effect of different types of material on de-

adhesion profile we use simplified version of the above model, a1-D model. We calculate de-

adhesion profiles for different stress-strain characteristics linear, strain softening and strain

hardening. Further the stress-strain characteristics were chosen in such manner that stress-strain

curve for small stain is nearly the same .For linear characteristic we use , for strain hardeningwe use and for strain softening we use .The stress-stain characteristicsand de-adhesion profiles for them are plotted in fig.2.

Figure 7:Sketches of stress-strain characteristics of linear, strain-hardening and strain softening materials for(a) small strains upto 0.1(b) large strains upto 2 .De-adhesion profile with different materials for (c) initial

displacement u0=0.1r(d) initial displacement u0=2r.

As seen from the fig.4 for large strains de-adhesion takes place faster in strain hardening material

and slower in strain softening materials than linear material. However near saturation i.e. r =0.99

0.00 0.02 0.04 0.06 0.08 0.10

0.00

0.02

0.04

0.06

0.08

0.10

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

r

0.0 0.5 1.0 1.5 2.0

0

1

2

3

4

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

r

(a) (c)

(b)(d)


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de-adhesion takes place faster for strain softening material and slower for strain hardening

material than for linear material. From the above study we can say that material non-linearity has

very little significant effect on de-adhesion profile and linear material description of the cell is

sufficient to capture most of the experimentally observed phenomena.


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4.5 Effect Force dependent bond breakage rate:

In all the above analysis we had assumed that the rate constant of bond breakage r2 remains the

same throughout the process of de-ahesion. However the specific receptorligand bonds are

often considered a lock-and-key mechanism, which can transit stochastically between a closed

(binding) state and an open (broken) state. A single closed receptorligand bond in cell adhesion

has a binding energy of 1025 kBT (kBT: the product of Boltzmann constant and absolute

temperature) and can undergo a transition from the original closed state to an open state owing to

thermally activated bond dissociation even in the absence of an external force[4]. The process

of bond dissociation is often regarded as thermally assisted escape over a potential energy

barrier. Application of an external force changes the energy landscape and therefore influences

the rupture process. For time-independent loading, both theories and experiments have indicated

that the dissociation rate koff of a closed bond increases exponentially with a force F acting on

the bond as [6]: koff= k0exp (FxB/kbT)where k0 is the spontaneous dissociation rate in theabsence of the force, xb is the distance between the minimum of the binding potential and the

transition state barrier and kBT is the unit of thermal energy.

To consider the force dependence of the bond breakage we use bond breaking rate given by

Where f(t) is the force in a bond. We use 1-D contracting cell model with adhesion bonds concentrated

only on the extreme edges. By solving coupled differential equations we get de-adhesion profiles as

shown in Fig.8.

Figure 8: Effect Force dependent bond breakage rate blue line shows de-adhesion profile for constant rate

constant during the de-adhesion while red line shows de-adhesion profile for rate constant varying

exponential withforce in a bond during the de-adhesion

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

r


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CHAPTER 5: CONCLUSION

In the above study we have addressed the influence of substrate properties and cell substrate

adhesions in regulating the de-adhesion dynamics. We have shown that stiffness ratio, viscosity

ratio and bond-breakage rate all play important role in setting de-adhesion time scales. We also

shown that linear material description of the cell is sufficient to capture most of the

experimentally observed phenomena and non-linearity of the material has very little significant

effect on de-adhesion profiles.


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Bibliography

[1] M.Inamdar, S.Sen, Interplay of substrate properties and cell-substrate adhesion in regulation

of de-adhesion dynamics of adherent cells.

[2] S.Sen, W.P.Ng, S.Kumar, Contractility Dominates Adhesive Ligand Density in RegulatingCellular De-adhesion and Retraction Kinetics, Annals of Biomedical Engineering,(2011).

[3]S.Sen and S, Kumar, Cell. Mol. Bioeng. (2009)

[4] Huajian Gao, Jin Qian and Bin Chen, Probing mechanical principles of focal contacts in

cell-matrix adhesion with a coupled stochastic -elastic modeling framework, J. R. Soc. Interface

(2011)

[5] Carina Edwards and Ulrich S. Schwarz, Force Localization in Contracting Cell Layers,

PRL 107, 128101 (2011)

[6]Bell, G. I. 1978 Models for specific adhesion of cells to cells Science 200, 618627.[7]A.D. Mesquita, H.B. Coda, A simple Kelvin and Boltzmann viscoelastic analysis of three-

dimensional solids by the boundary element method,Engineering Analysis with Boundary

Elements 27 (2003)


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Appendix

Mathematica code for 2-D contractile cell model:

v=0.5;uo[r_]:=0.5 r;

o[r_]:=6r^4;r1=10;

Tmax=15.;

{r2,r3}={1,10};


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0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.05

0.100.15

0.20

0.25

0.30

0.35

r

ucor

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

r

usor

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Time t 0

normalizedareaA

t


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Mathematica code for 1-D contractile cell model

r1=10;uo[r_]:=.1r;

o[r_]:=6r4;

Tmax=15.;{r2,r3}={1,1};


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Mathematica code for 1-D cell with different material characteristics:

r1=1;uo[r_]:=2 r;

o[r_]:=5r4;

Tmax=15.;{r2,r3}={1,1};

ddp stage-i report

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