Analytical solution for steady-state populations in the self-assembly of microtubules from nucleating sites

In addition to the biological importance of microtubules, which form a portion of the cellular cytoskeleton and a network for intracellular transport, the kinetics of microtubule self-assembly have generated great interest because individual microtubules exist in growing and decaying phases, with randomly occurring interconversions between them. Although a great deal is known concerning the microscopic details of these growth and decay processes, no description is available for the steady-state microtubule concentrations that are observed experimentally when microtubules are grown from nucleating sites. We generalize Hill's two-state model to include the dependence of the rates on tubulin concentration for systems where microtubules are grown from nucleating sites. An analytic solution is provided here to the resulting nonlinear, doubly infinite set of kinetic equations for the steady-state concentrations of both the growing and decaying phase microtubules as a function of the degree n of self-assembly and of the tubulin concentration. We also discuss the conditions for the stability of the steady state.