Quasi Steady-State Reduction of Non-linear PDE Models of Cellular Systems & Turing Analysis on Steady-State Pattern Formation

This summer I worked on several projects unrelated to each other. The first project was an extension of what I researched last summer with Dr. Keshet and Dr. Ward. Our group looked at several models of nonlinear motor dynamics in the cell, including kinesins along microtubules, as well as myosins along actin filaments. These models were essentially a collection of nonlinear coupled PDEs describing the advection, diffusion, and reaction components of the system states. The system itself represented the probability density of the motors as a vector with three states; right moving (or walking for myosins), left moving (or treadmilling for myosins), and freely diffusing. The boundary conditions of the system were no-flux, and there was ensured conservation in the system. The system of such equations for the kinesin model is shown below: