Mathematical Biology Seminar

Jun Allard
Department of Mathematics, University of California, Irvine

A non-local Maxwell condition for mechano-chemical traveling waves in cells

Wednesday, September 14, 2016, at 3:05pm
LCB 225

It is increasingly understood that cells of many different types exhibit traveling waves. These waves can be pulses of biochemical factors (diffusing proteins or metabolites) and also mechanical factors (such as the cell cortex). One example of mechanical traveling wave is offered by cellular blebs, pressure-driven “bubbles” on the cell surface implicated in cell division, apoptosis and cell motility. Blebs exhibit a range of behaviors including contracting in place, travel around the cell’s periphery, or repeated blebbing, making them biophysically interesting. Mechanical traveling waves are naturally modeled using non-local integro-PDEs, which lack the theoretical tools available for reaction-diffusion waves. This lack obfuscates simple questions such as what determines if a bleb will travel or not, and, if it travels, what determines its velocity? These questions are readily answered for reaction-diffusion waves. We present results in two parts: First, we develop a simple model of the cell surface describing the membrane, cortex, and adhesions, including the slow timescale cortical healing (treating implicitly the fast timescale of fluid motion). We find traveling and stationary blebs, which we characterize through numerical simulation. In the second part, we review the classical Maxwell condition for reaction-diffusion systems that determines whether an excitation will travel or recover in place. We present our progress in deriving an analogue of the Maxwell condition for non-local integro-PDEs suitable for our cell surface model. This condition allows the theoretical (simulation-free) elucidation of blebbing including bleb travel.