# The universal dynamics of cell spreading

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 $R ~ t^{1/2}$ and (2) a sub-diffusive region with $R ~ t^{1/4}$.

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 $R_c$. The change in contact area is related to the rate of change in the radius by $\frac{dA}{dt} = R \frac{dR}{dt}$. 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: $J R \frac{dR}{dt}$. Assuming that the cell cortex has a width w everywhere, the strain is of the order $\frac{dR/dt}{w}$. This strain acts over a volume approximately equal to $R^2 w$. The product of the strain, volume, and viscosity leads to an energy dissipation of $\eta (\frac{dR}{dt} \frac{1}{w})$. By setting this equal to the change in the adhesion energy, we can get the following power law: $R = C \big(\frac{J w}{\eta}\big)^{1/2} t^{1/2}$.

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 $J R \frac{dR}{dt} = \eta_c \big(\frac{dR}{dt} \frac{1}{w}\big)^2 R^2 w_c$. This can be re-arranged to the scaling law: $R ~ \big(\frac{J R_c^3}{\eta_c}\big)^{1/4} t^{1/4}$.

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.

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.