Understanding the Topology of Microtubules

The phenomenon of dynamical instability in microtubules is of immense importance in understanding transport within the cell. In recent times, these polymeric proteins have been studied as mechanical lattices possessing a topological edge mode (Phys. Rev. Lett. 103, 248101 (2009)). We extend this idea by modelling the microtubule as a cylindrical lattice of dimers, with interactions between dimers modelled by a hopping Hamiltonian. We show the emergence of topological edge modes, and propose the modelling of dynamic instability as a phase transition. We explicitly determine the conditions for the existence of these edge modes by setting up appropriate difference equations. We speculate on the biological implications of these results, by mapping the dynamic instability of microtubules to the propagation of lattice defects, and also discuss possible methods of computational/experimental verification via microscopic network models of microtubules.