Mathematical Biology

Cell-cell signaling between macrophages and mammary tumor cells

We continue the discussion from last time, and solve the polymer size distribution equations, which are linear in the case of constant monomer level.
In a distinct case, when monomer is depleted, we show that the size distribution evolves in two phases, where in the first, the entire distribution appears to satisfy a transport equation, and then, later on, once monomer is at its critical level, the process of length adjustment appears to be governed by
an effective diffusion (in size-class). Next, I introduce the problem of determining features of polymer assembly from experimental
polymerization versus time data. (Based on work by Flybjerg et al, this leads to an extended homework exercise carried out by the students.) Finally, I revisit microtubule growth and shrinking by discussing the Dogterom-Leibler equations and their steady state exponential solutions. I illustrate the use of XPP software to solve several problems in this lecture.

Size regulation in cell organelles

Reaction diffusion patterns

Positional information and gradients