September 22, 2017, Friday

# Cooperativity Between Cell Contractility And Adhesion

## Context/biological phenomenon:

Cells can sense and adapt to their external environment, change shape and move through the Extra Cellular Matrix (ECM). Contractile cells like fibroblasts or cardiomyocytes can change their internal stiffness and contractibility in response to external mechanical stimuli. Lymphocytes crawl on the ECM by throwing and anchoring membrane extensions. Their sensors are the focal adhesions, a group of molecules assembled in a structure that connect them to the ECM. From these focal adhesions and through a polymerization process arise fiber like structures: actin stress fibers for fibroblasts or myofibrils for cardiomyocytes, with different contractibility that will help the cell move or contract. The polymerization processes of the different monomers (the building blocks that are integrin for focal adhesions and actin for stress fibers) into focal adhesions or stress fibers are known to be promoted by the level of stress on the membrane or inside the cell. It has also been observed that focal adhesions are usually concentrated on high curvature regions of the cell’s membrane. In the figure below from The simulation of stress fibre and focal adhesion development in cells on patterned substrates by Amit Pathak, Vikram S Deshpande, Robert M McMeeking and Anthony G Evans, fibroblasts are attached to ligands forming letters, and the actin and integrin of stress fibers (on the left hand side of the picture) and focal adhesions (on the right hand side) are stained:

In Cooperativity between Cell contractility and Adhesion, Igor L. Novak, Boris M. Slepchenko, Alex Mogilner, and Leslie M. Loew develop a mathematical model that explains the migration of focal adhesions toward the periphery of the cell in high curvature region, driven by the diffusion and the forces generated by the stress fibers.

## Mathematical model

In this model, three quantities are described inside a cell of geometry $\Omega$:

-the local density of integrins bounded to the ECM (adhesions) $\rho ({\mathbf {X}},t)$:

-the local density of unbounded integrins $\rho ^{{*}}({\mathbf {X}},t)$

-the force per unit area generated by stress fibers ${\mathbf {F}}({\mathbf {X}},t)$.

The rate of formation of adhesions depends on the concentration of unbounded integrins and on the force generated by stress fibers. It is a reversible reaction, and they disassemble with a constant rate $k_{{-1}}$. Unbounded integrins diffuse inside the cell with a diffusion coefficient $D$. The rate of generation of force form the stress fibers is the rate of formation of stress fibers $k_{2}$(which is proportional to the adhesion size) times the force generated by one stress fiber $f_{0}$. Stress fibers also disassemble with a constant rate $k_{{-2}}$. This is summarized in the following model:

${\frac {\partial \rho }{\partial t}}=\left[k_{0}+k_{1}\left|{\mathbf {F}}({\mathbf {X}},t)\right|\right]\rho ^{{*}}-k_{{-1}}\rho$(1)

${\frac {\partial \rho ^{{*}}}{\partial t}}=-\left[k_{0}+k_{1}\left|{\mathbf {F}}({\mathbf {X}},t)\right|\right]\rho ^{{*}}+k_{{-1}}\rho +D\bigtriangledown ^{{2}}\rho ^{{*}}$(2)

${\frac {\partial {\mathbf {F}}}{\partial t}}=k_{{2}}f_{{0}}\underbrace {\rho }_{{\rho {\mbox{at point X}}}}\underbrace {\int _{{\Omega }}\rho ({\mathbf {X'}},t){\frac {{\mathbf {X'}}-{\mathbf {X}}}{\left|{\mathbf {X'}}-{\mathbf {X}}\right|}}d^{2}X}_{{{\mbox{total density of integrins}}}}-k_{{-2}}{\mathbf {F}}$(3)

$k_{0}$ is the rate of assembly of integrins independent of the forces generated by stress fibers. Equations (1) and (2) are the mass conservation of bounded and free integrins, equation (3) is the kinetic law for the forces generated by the stress fibers. The model is then nondimensionalized as follow:

-${\bar {\rho }}$ is the average density of free and bounded integrins

-$\left|\Omega \right|$ is the surface area

-$c=\rho /{\bar {\rho }}$

-$c^{{*}}=\rho ^{{*}}/{\bar {\rho }}$

-${\mathbf {f}}={\mathbf {F}}k_{{-2}}/\left|\Omega \right|$

-${\mathbf {x}}={\mathbf {X}}/{\sqrt {\left|\Omega \right|}}$

-$\tau =tk_{{-2}}$

-$\beta =k_{{-1}}k_{{-2}}/k_{{1}}k_{{2}}f_{{0}}{\bar {\rho }}^{2}\left|\Omega \right|$

-$\alpha =k_{{-1}}/\beta k_{{-2}}$

-$\alpha _{{0}}=\beta k_{{0}}/k_{{-1}}$

-${\tilde {D}}=D/\left|\Omega \right|k_{{-2}}$

equations (1),(2) and (3) become:

${\frac {1}{\alpha }}{\frac {\partial c}{\partial \tau }}=(\alpha _{{0}}+\left|{\mathbf {f}}\right|)c^{{*}}-\beta c$(4)

${\frac {1}{\alpha }}{\frac {\partial c^{{*}}}{\partial \tau }}=-(\alpha _{{0}}+\left|{\mathbf {f}}\right|)c^{{*}}+\beta c+{\frac {{\tilde {D}}}{\alpha }}\bigtriangledown ^{{2}}c^{{*}}$ (5)

${\frac {\partial {\mathbf {f}}}{\partial \tau }}=c\int _{{\Omega }}c({\mathbf {x'}},t){\frac {{\mathbf {x'}}-{\mathbf {x}}}{\left|{\mathbf {x'}}-{\mathbf {x}}\right|}}d^{2}x-{\mathbf {f}}$(6)

## Analysis of the steady state in the simplified case $D=\infty$ and large $k_{{2}}$ and $k_{{-2}}$

With $D=\infty$ and large $k_{{2}}$ and $k_{{-2}}$, the diffusion of free integrins is instantaneous as well as the equilibrium between stress fibers and adhesions (bounded integrins). The steady state gives us:

$c={\frac {1}{\beta }}(\alpha _{{0}}+\left|{\mathbf {f}}\right|)c^{{*}}$(7)

${\frac {{\tilde {D}}}{\alpha }}\bigtriangledown ^{{2}}c^{{*}}=0$(8)

${\mathbf {f}}=c\int _{{\Omega }}c({\mathbf {x'}},t){\frac {{\mathbf {x'}}-{\mathbf {x}}}{\left|{\mathbf {x'}}-{\mathbf {x}}\right|}}d^{2}x$(9)

equation (8) gives us a uniform $c^{{*}}$ (of the form $A{\mathbf {x}}+B$). Let us define the unit force per integrin ${\mathbf {E}}=({\mathbf {f}}/c)$:

${\mathbf {E}}=\int _{{\Omega }}c({\mathbf {x'}},t){\frac {{\mathbf {x'}}-{\mathbf {x}}}{\left|{\mathbf {x'}}-{\mathbf {x}}\right|}}d^{2}x$

The term ${\frac {{\mathbf {x'}}-{\mathbf {x}}}{\left|{\mathbf {x'}}-{\mathbf {x}}\right|}}$ (= sgn(x'-x) in 1D) makes ${\mathbf {E}}$ monotonically increasing from a point inside the cell (${\mathbf {x}}=0$) to the membrane (${\mathbf {x}}\in \partial \Omega$) for any positive distribution $c({\mathbf {x}})$. In this simplified case, the concentration of bounded integrins now only depend on the force generated by the stress fibers. Since we showed that $c^{{*}}$ is uniform, and that the force per integrin is maximum on the membrane of the cell, it follows from (7) that the density of bounded integrins $c$ is maximum on the membrane of the cell, confirming experimental results.

## Full analysis of the steady state in the 1D case

In the full analysis case, the density of adhesion can rise to infinity, so instead of following the density, the mass variable $M$ is introduced since it is bounded with $c=\partial M/\partial x$ and $M(x)={\frac {1}{2}}\int _{{0}}^{{1}}c(x')sgn(x-x')dx'$ which is a monotonic variable. From equation (8) we still have a uniform distribution of $c^{{*}}$, and with the introduction of the variable $M$, equations (7) and (9) can be rewritten and combined as follows:

$(x-{\frac {1}{2}})\alpha _{{0}}+M\left|M\right|={\frac {\beta M}{1+2M(0)}}$(10)

We now solve equation (10) for our new variable M, considering the case $\alpha _{{0}}=0$, which means that there are no spontaneous generation of bounded integrins without a force from the stress fibers. In that case, in order for $M$ to satisfy equation (10), it must satisfy:

$M(x)=0$(11)

or

$\left|M(x)\right|={\frac {\beta }{1+2M(0)}}$(12)

So if $M\neq 0$, it is fully determined by $M$ at the boundary i.e. $M(0)$. Let us find $M(0)$:

from equation (10), and with $\alpha _{{0}}=0$ we have:

$-M(0)\left|M(0)\right|-{\frac {1}{2}}\left|M(0)\right|+{\frac {\beta }{2}}M(0)=0$(14)

and $M(0)=-{\frac {1}{2}}\int _{{0}}^{{1}}cdx<0$

so (14) becomes $M(0)^{2}+{\frac {1}{2}}M(0)+{\frac {\beta }{2}}M(0)=0$(15)


which yields

$M(0)=0$(16)

or

$M(0)={\frac {1}{4}}(-1\pm {\sqrt {1-8\beta }})$(17)

### case $\beta <1/8$: bistable region

If $M(x)$ satisfies (12) and $M(0)$ satifies (17), or $M(x)$ satisfies (11) and $M(0)$ satifies (16) the solution is a continuously derivable function, the adhesions are almost evenly distributed throughout the cell and the cell is considered as being in a inactive state.

If $M(x)$ satisfies (11) and $M(0)$ satifies (17), then $\lim _{{x\rightarrow 0}}M(x)\neq M(0)$ so $\lim _{{x\rightarrow 0}}{\frac {M(x)-M(0)}{x}}={\frac {\partial M(0)}{\partial x}}=c(0)=\infty$ and the density of adhesion is discontinuous on the the boundaries $x=0$ or $x=1$ (by symmetry). This is the active state, where all the adhesions are localized on the cell's membrane.

### case $\beta >1/8$: inactive region

In this case, since the solutions of (15) are complex, M(x) and M(0) can only satisfy equations (11) and (16), i.e. $M(x)=M(0)=0$. The solution is continuous, so only the inactive state exists in the case $\beta >1/8$.

This is summarized in fig2:

Figure 2a shows the bistable and the inactive regions, and fig2b shows the distribution of focal adhesions in the active and inactive states. In the bistable region, the discrimination between active or inactive state comes form the initial amount of bounded integrins: the cell will be active is the initial amount of adhesions is greater than a critical amount. This critical amounts drops as $\beta$ becomes smaller, i.e. as the rate of binding of the integrins becomes higher.

## Numerical results and analysis for the dynamics of the 1D system

Fig3 illustrates the dynamics of the system in 1D: at t=0, an high initial density of adhesions is set up at the center. This peak splits in two since the net force is null at the center. Then at the rear of the peak, adhesions disassemble into free intergrins, which has for effect a rise in the density of free integrins between the two peaks (red line in fig3.a, .b and .c). There is a steep gradient in the density of free integrins between the rear and the front of the peaks since the rates of binding at rear and the front are different (because the stress fiber forces increase from the center to the boundaries), which drives the free integrins to the front of the peak where they are immediately binded into adhesions. In this treadmill manner, the peaks move toward the boundaries and reaches steady state. Fig3:

## Estimation of the adhesion migration speed

An estimate of the speed of the adhesions moving to the cell periphery can be obtained by substituting a traveling wave solution (for example of the form $c=c(x+vt)$ for $c$) into equations (4), (5) and (6). In their paper, the authors find the propagation velocity of the wave is determined by the diffusion time of the integrins $v\propto {\tilde {D}}c_{{int}}^{{*2}}/\beta$ with $c_{{int}}^{{*}}$ being the concentration of free integrins between the two peaks (the higher it is, the greater the gradient between the rear and the front of the peaks is and the faster the adhesions migrate). The speed of propagation also increases as $\beta$ decreases, i.e. as the binding rate of integrins increases.

## Numerical results and analysis for the 2D case

The same observations can be made for the 2D case as for the 1D case: a high concentration of integrins is initially deposited at the center of the cell. Since all the stress fibers cross each other at the center, the resulting force is null whereas at the periphery stress fibers are all oriented toward the center so the resulting force is much higher. This drives the adhesions toward the periphery of the cell with the same mechanism as explained for the 1D case. The picture below shows how the numerical results compare and matches experimental results for the density of adhesions:

From this image can also be explained why high curvature regions have a higher adhesion density: at the corners of the cell, the orientation disparity of the stress fibers is much smaller then on the sides of the cell, resulting in a higher net force and therefore a higher adhesion concentration. It is also important to notice the feedback mechanism between ahesions and stress fibers: adhesion move toward the stress fiber high net force regions, and the stress fibers grow on adhesions: these two phenomenons fuel each other in a feedback mechanism.

## Conclusion

In this paper the authors show that the high concentration of adhesions in the high curvature regions of the cell is the result of a feedback mechanism between stress fibers and adhesions. They also showed that the time necessary for adhesions to migrate toward the cell's periphery is mostly determined by the diffusion constant of integrins.