  In this paper we propose a microscopic model to study the polymerization of
microtubules (MTs). Starting from fundamental reactions during MT's assembly
and disassembly processes, we systematically derive a nonlinear system of
equations that determines the dynamics of microtubules in 3D. %coexistence with
tubulin dimers in a solution. We found that the dynamics of a MT is
mathematically expressed via a cubic-quintic nonlinear Schrodinger (NLS)
equation. Interestingly, the generic 3D solution of the NLS equation exhibits
linear growing and shortening in time as well as temporal fluctuations about a
mean value which are qualitatively similar to the dynamic instability of MTs
observed experimentally. By solving equations numerically, we have found
spatio-temporal patterns consistent with experimental observations.
