Stabilization of microtubules due to microtubule-associated proteins: A simple model

A theoretical model of stabilization of a microtubule assembly due to microtubule-associated-proteins(MAP) is presented. MAPs are assumed to bind to the microtubule filaments, thus preventing their disintegration following hydrolysis and enhancing further polymerization. Using mean-field rate equations and explicit numerical simulations, we show that the density of MAP (number of MAP per tubulin in the microtubule) has to exceed a critical value $\rho_{c}$ to stabilize the structure against de-polymerization. At lower densities $\rho < \rho_{c}$, the microtubule population consists mostly of short polymers with exponentially decaying length distribution, whereas at $\rho > \rho_{c}$ the average length increases linearly with time and the microtubules ultimately extend to the cell boundary. Using experimentally measured values of various parameters, the critical ratio of MAP to tubulin required for unlimited growth is seen to be of the order of 1:100 or even smaller.