Elementary mechano-chemical models for cell motility and adhesion dynamics

Understanding the processes of Life requires all tools and toys available for us as natural scientists -- and mathematicians. As such we have learned to play with elementary ingredients and their suitable combinations, in order to invent dynamical systems and make them work and function in a proper and useful way. What else can we see in Nature, where on the evolutionary time scale functionable and useful dynamical systems have been invented and properly developed during millions of cell generations.

Here we consider, as prototypical example, the ubiquitous phenomenon of crawling cell motility, with the reactive and contractile cytoplasm system as elementary ingredient in combination with associated crosslinker and plasma membrane proteins. Their mechano-chemical properties and mechanisms span a large range of spatio-temporal scales, from molecular events as polymerization or adhesion receptor binding (nm; sec) over cytoplasmic flow and cell deformation (µm; min) up to effective cell translocation and path generation (mm; h). Since an important task of modelling is to reconstruct, simulate and understand the essential mechanisms behind the observed modes of Life, the invented mathematical models have to be treatable and accessible for further analysis.

Therefore, in this talk we present simplified 1-D approximations of a general 2-D hyperbolic-elliptic-parabolic stochastic PDE system with free boundary conditions describing the dynamics of single cell adhesion and motility. The basic problem is to reproduce the observed transitions between a non-polarized cell with on-spot motility and a polarized cell with directionally persistent cell locomotion.