Crawling cells, including the white blood cells that patrol your body in search of infections, display several distinct dynamical patterns, driven by both biochemistry (diffusion and reactions between chemical species) and mechanics (physical forces between the components inside cells). Our understanding of these spatiotemporal patterns has been aided by mathematical modeling using techniques including partial differential equations (PDEs). Recently, traveling waves have been observed in the protein actin, which powers certain cells’ ability to crawl. Following experimental observation of one type of crawling cell, we hypothesized that traveling waves are excitable waves arising from interactions of three components and developed a mathematical model formulated as a system of PDEs with a nonlocal integral term. Numerical solutions lead to a number of predictions, confirmed in further experiments. Our model also reveals a role for tension in the membrane that surrounds the cell, which would otherwise be difficult to observe directly by experiment.
In this talk I will use a classic moving boundary problem of fluid dynamics to offer some insight into the kind of questions and results researchers in nonlinear partial differential equations are interested in.
I will discuss some situations when uncertainty in model parameters motivates modeling in terms of random functions. Moreover, some about what is involved in the analysis of such problems. In the first example I consider a problem in mathematical finance, while in the second I consider a problem regarding waves propagating through very complex media.
