Modeling polymerization of microtubules: A semi-classical nonlinear field theory approach

In this paper, for the first time, a three-dimensional treatment of microtubules’ polymerization is presented. Starting from fundamental biochemical reactions during microtubule’s assembly and disassembly processes, we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in three dimensions. We found that the dynamics of a microtubule is mathematically expressed via a cubic-quintic nonlinear Schrödinger (NLS) equation. We show that in 3D a vortex filament, a generic solution of the NLS equation, exhibits linear growth/shrinkage in time as well as temporal fluctuations about some mean value which is qualitatively similar to the dynamic instability of microtubules. By solving equations numerically, we have found spatio-temporal patterns consistent with experimental observations.