Mathematical Biology

First, the universal features of polarizing cells are listed, and details of the Mori-Jilkine wave-pinning model are assembled and discussed biologically and mathematically. A short review of thelocal pulse analysis is provided to indicate the usefulness of this method of analysis. Then, I discuss the survey of polarizing cells from a paper by Jilkine and LEK (2011) that appeared in PLoS Comput Biol 7(4): e1001121. Here, common and distinct attributes of different cell types and of several models for cell polarization are compared. The responses of models to a set of computational perturbations mimicking stimuli protocols are described.

This lecture introduces the topic of 2D cell motility simulations, but focuses on one specific method, the CPM (as implemented by Maree et al in Bull Math Biol (2006), 68(5):1169-1211 and PLoS Comput Biol (2012) 8(3): e1002402). I explain the details of the method, the biological facts that were included (signaling from GTPases and phosphoinositides to actin assembly and myosin contraction). I illustrate typical results, and then discuss some of the technical aspects of the method, emphasizing its link to the (previously discussed) Metropolis-Hastings algorithm. I also show how Stan Maree was able to chose CPM parameters to phenomenologically mimic the known relationship between actin filament ends and cell protrusion speed.

In this lecture I describe a model by Grimm et al (2003) Eur Biophys J 32: 563-577. The authors ask what processes might account for a parabolic density profile of actin seen across the front edge of a keratocyte. In an elegant coupled PDE model, they show that right and left growing actin filaments, competing for the actin branching complex Arp2/3 have solutions with the appropriate profile. I here
consider one of the cases, that of local competition and slow capping of filaments, where the equations are fully analytically solvable in closed form.

Data Analysis Methods