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On a Minimal Model for the Initiation of Cell Movement
Actin-driven motility of eucaryotic cells plays a crucial role in many biological processes and has therefore been under intense experimental and theoretical investigation throughout several decades. In [10], we introduced a minimal model for the preparation of movement in a symmetric resting cell on a flat substrate. This system consists of at least four hyperbolic conservation laws describing the evolution of densities of actin filament tips and at least one parabolic equation for the actin monomer concentration. For this coupled hyperbolic-parabolic system, we shall now formulate a free boundary problem to allow for actual motion of the cell. For this model with some specific boundary conditions, we will show short time well posedness and present several mechanisms by which the solutions might break down for large times. In particular, possible blow-up phenomena are described and investigated, both analytically and numerically. Moreover, we discuss how the cease of existence of solutions can be interpreted physically as the emergence of actin polymerization fronts. Finally, different possible boundary conditions are presented, and their biological meanings are explained. We furthermore reformulate the model under certain assumptions and derive a system of two parabolic equations describing the motion of two interacting species of filaments moving in opposite directions. This simplified model is investigated in part II where we ask for stability of particular steady states and construct traveling wave solutions. The existence of the latter can also be found in simulations, and we will discuss the type and velocity of the evolving wave profiles. Particular attention will be paid to the remarkable differences between different types of nonlinearities describing the mutual interaction. Of special interest are the deviations from the predictions about stability and the traveling wave solutions obtained from the linearization of the model around its equilibria. These predictions are met quite well by some versions of the nonlinear terms whereas for others they are missed significantly. We are thus dealing with a quite minimalistic system of reaction advection diffusion equations whose behavior cannot be predicted by linearization but strongly depends on the particular nonlinearity.
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510 Mathematik
510 Mathematics