How mathematics help us understand cell motion

The motion of animal cells is a complex and important process that affects growth, development, wound healing, as well as disease processes (such as cancer). Cell motility is known to depend on protrusion, adhesion, and retraction of parts of the cell, which, in turn, stem from both chemical and mechanical changes in a structure called the cytoskeleton. Polymerization of components of this structure (usually actin) is one of the important processes underlying motility. Generation of force by "molecular motors" such as myosin is also important. Mathematical models (ordinary and partial differential equations) and simulations (in 2D and 3D) can help to understand the roles of the components, their interactions, and how they are controlled to produce motion in the cell.

I will present a mechanochemical analysis of a crawling cell and describe a finite element model wherein (a) localized protein polymerization and bundling generate the force for extension, and (b) energy stored in the gel formed from the polymers at the leading edge is subsequently used to produce the contraction that pulls the rear of the cell forward. While this model has features of general interest, I apply it to a specific example, the crawling of the nematode sperm cell. These cells crawl using a specialized "major sperm protein", rather than actin, in their cytoskeleton. Their simplicity provides a 'stripped down' version of a crawling cell in which to examine the basic mechanism of cell locomotion, independent of other cellular functions. I show how results of the models and simulations, based on realistic values of known biological parameters agree with the experimental observations.