Rotating lamellipodium waves in polarizing cells

Abstract

Cellular protrusion-and lamellipodium waves are widespread for both non-motile and moving cells and observed for many cell types. They are involved in the cell's exploration of the substrate, its internal organization, as well as for the establishment of self-polarization prior to the onset of motion. Here we apply the recently developed phase field approach to model shape waves and their competition on the level of a whole cell, including all main physical effects (acto-myosin, cell membrane, adhesion formation and substrate deformation via traction) but ignoring specific biochemistry and regulation. We derive an analytic description of the emergence of a single wave deformation, which is of Burgers/Fisher-Kolmogorov type. Finally, we develop an amplitude equation approach to study multiple competing rotational waves and show how they allow the cell to transition from a non-moving state towards a polarized, steady moving state.

A simple model of Keratocyte membrane dynamics: The case of motionless living cell

“…4 confirm such approximation, since the numerical solutions of the both equations (25) and (30) almost coincide. Thus, the result of [14], obtained with diffusion terms neglected, is recovered from our more rigorous and precise equation (25). However for other values of parameters and when the initial curvature is large in some points the simulations of equations (25), (30) revel a significant deference see Fig Plots of the function f (R) in (17) for the values β = 1, 0.5 that manifest stable and unstable states, and for the value β = 0.3 that is nonphysical since it does not include any steady states.…”

“…In the present paper, we perform a more rigorous analysis of the problem and preserve the diffusion effects. We obtain a more general and exact description of the interface dynamics (25) that can be reduced to a Burgers-like equation in a certain limit, thus recovering the result of [14].…”

“…in agreement with [14]. Let us consider the fluctuation δR = R(θ, t) − R 0 1; then equation (29) takes the form,…”

“…Note that unlike [13] and [14], we do not assume that δ 1 = 0. Let us substitute (5)-(10) into (4a)-(4e), using the chain rule:…”

“…Recently, the same model has been used in [14] in order to describe lamellipodium waves dynamics in cells, where it was found that the cell interface dynamics could be described by the Burgers-like equation. It should be noted that in the course of the derivations, the diffusion effects, which constitute a very basic component of the model, were ignored.…”