# The universal dynamics of cell spreading

Original entry: Naveen Sinha, APPHY 226, Spring 2009

In numerous biological experiments, cells are plated onto an artificial adhesive surface to study them under a microscope. After being deposited on the surface, they flatten and spread outwards along the surface. This process incorporates many biological reactions, including the diffusion of adhesion receptors and the polymerization of the scaffolding-like actin molecules. However, Prof. Mahadevan and his collaborators show how a variety of cells and surface exhibit a power-law behavior in the contact radius of the colony of cells as a function of time. This can be understood when the cells are modeled as a viscous shell of liquid with a much less viscous interior.

The basic biological system is shown below. Cells are placed on a surface and Reflectance Contrast Imaging Microscopy is used to measure the contact area as a function of time.

The length measure used to characterize the system is the radius of a circle with the same area. On a log-log plot, the power law behavior becomes apparent.

There appear to be two different regimes: (1) an initial diffusive regime with <math>R ~ t^{1/2}</math> and (2) a sub-diffusive region with <math>R ~ t^{1/4}</math>.

Despite all the biochemical complexities of the spreading process, a simple model that treated the cells as a shell of viscous liquid is sufficient to describe most of the behavior.

Initially, the power-law can be derived by setting the chance in adhesion energy equal to the viscous energy dissipation. This approximation is valid for when the radius of the contact angle is less than te initial cell radius <math>R_c</math>. The change in contact area is related to the rate of change in the radius by <math>\frac{dA}{dt} = R \frac{dR}{dt}</math>. This can related to the rate of energy gain by multiplying by the adhesion energy per unit area, J, which is the product of the area of each bond and the energy per bond: <math>J R \frac{dR}{dt}</math>. Assuming that the cell cortex has a width w everywhere, the strain is of the order <math>\frac{dR/dt}{w}</math>. This strain acts over a volume approximately equal to <math>R^2 w</math>. The product of the strain, volume, and viscosity leads to an energy dissipation of <math>\eta (\frac{dR}{dt} \frac{1}{w})</math>. By setting this equal to the change in the adhesion energy, we can get the following power law: <math>R = C \big(\frac{J w}{\eta}\big)^{1/2} t^{1/2}</math>.

At longer times, when cell has flattened out (i.e. the radius of contact is larger than the initial radius), the viscous energy dissipation occurs throughout the whole volume of the cell. The power law is modified to <math>J R \frac{dR}{dt} = \eta_c \big(\frac{dR}{dt} \frac{1}{w}\big)^2 R^2 w_c</math>. This can be re-arranged to the scaling law: <math>R ~ \big(\frac{J R_c^3}{\eta_c}\big)^{1/4} t^{1/4}</math>.

One prediction of this model is that the properties of the cellular membrane are far more important than the interior of the cell in the cell spreading. One way to test this is by creating two mutants: one in which the microtubules molecules within the cell are unable to polymerize and another in which the actin shell in the cortex is altered.

Sung Hoon's comment: In the Fig. 4 (a), the Cytochalasin D-treated cell showed linear behavior. What does the treatment do so that the treated cells spread more quickly?

Has anyone actually created these mutant cells to test this theory? Is that what you're showing below? Maybe it's better to write your own captions that get right to your point? --Lidiya 03:06, 18 February 2009 (UTC)

This work is important by providing unity to the plethora of information about the molecular mechanics for cell motility and adhesion. It allows future researchers to focus on the aspects of cell structure that are most relevant for thee processes and make refinements to this basic model.

Second Entry: Nick Chisholm, AP 225, Fall 2009

## Contents

## General Information

**Authors**: D. Cuvelier, M. Thery, Y-S. Chu, S. Dufour, J-P. Thiery, M. Bornens, P. Nassoy, and L. Mahadevan

**Publication**: Current Biology **17** 694 (2007)

## Soft Matter Keywords

Adhesion, Cell, Polymerization, Viscous

## Summary

The authors explore the early stages of cell spreading. In particular, they look at the situation where cells are plated onto artificial adhesive surfaces; they first flatten and deform extensively as they spread. This article presents experimental probing of the dynamics of this spreading process using quantitative visualization and biochemical manipulation with a variety of cell types, adhesive surfaces, adhesion receptors, and cytoskeleton-altering drugs. Surprisingly, the authors find that the adhesion dynamics of cells follow a universal power-law behavior (i.e. the contact radius is proportional to elapsed time to a certain power), regardless of cell type or adhesion surface. The authors then provide a theoretical model that justifies their experimental results.

In Figure 1, one can see an image of a cell spreading on a substrate. The image is taken using reflection interference contrast microscopy (RICM).

## Soft Matter Discussion

As is clear by Figure 1, the spreading is isotropic, the authors can define the spreading radius as: <math>R = \sqrt{\frac{A}{\pi}}</math>.

The contact radius was found as a function of time for various cells and adhesive surfaces; the results are plotted in Figure 2 (careful: it's a logarithmic plot!). From this plot, the authors note two regimes: an initial diffusive regime summarized by the scaling law <math>R \propto t^{1/2}</math>, and later a subdiffusive regime summarized by the scaling law <math>R \propto t^{1/4}</math>. To be completely honest, I'm not quite sure how accurate these scaling laws are, if only derived from the plot in Figure 2. However, I would assume (or, at least, hope) that these plots were also examined without a logarithmic scale, and then fitted using these power-law forms.

In any case, the authors have also developed a theory which would predict these particular scaling laws. I will briefly outline the physical motivation behind this theory, and then derive the scaling laws. Physically, the authors assume that for short times (i.e. times where the radius of the adhesive patch is smaller than the size of the cell), the cell encompasses a viscous shell enclosing a liquid. For long times, where the radius of the adhesive patch is comparable to the size of the cell, the cell is a homogeneous viscous drop. See Figure 3 for details. Note that this model makes sense: for short times, only part of the cell will be touching the surface, whereas for long times most of the cell has already spread, and thus is substantially flattened.

For short times, the contact zone will be disk-like, and the rate of change of the contact area will be: <math>\frac{dA}{dt} \approx R \frac{dR}{dt}</math>. The rate of energy gain is then <math>JR \frac{dR}{dt}</math>, where <math>J</math> is the adhesion energy per unit area. Using Figure 3, one then realizes that there is a characteristic shear strain of order <math>\frac{dR/dt}{w}</math>. The energy dissipation rate due to the viscous flow in the shell is proportional to the product of the shell viscosity (<math>\eta</math>), the square of the shear strain, and the volume over which dissipation occurs (see Figure 3; <math>V = R^{2}w</math>. Thus, it is proportional to: <math>\eta (\frac{dR}{dt} \frac{1}{w})^{2}R^{2}w</math>. By balancing this with the adhesive power, one find the scaling law for the contact radius for short times: <math>R = C(\frac{Jw}{\eta})^{\frac{1}{2}}t^{\frac{1}{2}}, R \leq R_{C}</math>, where <math>C</math> is a dimensionless constant and <math>R_{C}</math> is the initial radius of the cell. This agrees with the experimental results.

For long times, viscous dissipation occurs in the whole cell (since the contact radius is comparable to the cell size). Now, for a cell of initial height <math>w_{C}</math> and radius <math>R_{C}</math>, we notice that <math>R_{C}^{3} \propto R^{2}w_{C}</math> by volume conservation. The balance of adhesive and viscous power becomes: <math>JR \frac{dR}{dt} \cong \eta_{C} (\frac{dR}{dt} \frac{1}{w_{C}})^{2}R^{2}w_{C}</math>, where <math>\eta_{C}</math> is the effective cell viscosity. The scaling law is thus: <math>R \propto (\frac{JR_{C}^{3}}{\eta_{C}})^{\frac{1}{4}}t^{\frac{1}{4}}, R > R_{C}</math>. Again, this agrees with the experimental results.

In addition, in order to test the theory, the authors performed another experiment. For this experiment, they altered the geometry and mechanical structure of the shell of the cell, which lead to a drastic change in the scaling law obtained above. I think this experiment is much more complicated than the value of its explanation is worth, but the results are shown in Figure 4. Note that the black line is for a cell with a normal shell, and the red line is for a cell with a "patchy" shell (meaning parts of it missing). They conclude that their model is justified.

I think the next step would be to test the altering of the geometric and mechanical structure of the shell of the cell in more than just one (as shown in Figure 4), in order to more accurately conclude that the authors' theory is justified. I'm not entirely sure how one can do this, but certainly they could have done further experiments on different types of cells to ensure that this behavior was universal.

Although this paper was quite interesting, I have a hard time trying to think of any relevant application.

## Reference

[1] D. Cuvelier, M. Thery, Y-S. Chu, S. Dufour, J-P. Thiery, M. Bornens, P. Nassoy, and L. Mahadevan, "The universal dynamics of cell spreading," Current Biology **17** 694 (2007).