Cell crawling is a highly complex integrated process involving three distinct activities: protrusion adhesion and contraction, and also three players: the plasma membrane (car body), the actin network (engine) and the adhesion points (clutch). The actin network consists of actin polymers and many other types of molecules, e.g. molecular motors, which dynamically attach to and detach from the network, making it a biological gel. Furthermore, energy is consumed in the form of ATP due to both the activity of molecular motors and the polymerization at the filament tips; thus the system is far from thermodynamic equilibrium.

These characteristics make the above system unique and responsible for a wide range of phenomena (different force-velocity relationships) and behaviors (contraction, elongation, rotation, formation of dynamic structures). Like the story on the blind men and the elephant, previous works considered only parts of the complex process, neglecting other sub-processes, or using unrealistic assumptions. Our goal was to derive a mathematical model for the whole system that can predict the rich variety of behaviors. For this purpose we had to identify the major players and integrate previous works into a one coherent mathematical model with (almost) no arbitrary constraints, adding our own mathematical description where needed. The model we derived consists of several temporal and spatial scales, from molecular processes e.g. capping / branching to macro processes; furthermore, we used a hydrodynamic approach, hence could relate local dynamic events on the boundary to the bulk inside the domain.

We focused on the processes near the leading edge that drive the system, i.e. the complexity comes in the b.c., and termed this filaments-membrane dynamics “the polymerization machinery”.

In my talk I will describe the mathematical model we derived and its relation to previous works. I will also describe a proprietary numerical simulation we derived for a free-surface flow of complex fluid in arbitrary geometries. Finally I will discuss open questions and opportunities in this line of research**.**