  The microtubule is a cylindrical biological polymer that plays key roles in
cellular structure, transport, and signalling. In this work, based on studies
of electronic properties of polyacetelene and mechanical properties of
microtubules themselves (see Phys. Rev. Lett. 103, 248101), we explore the
possibility that microtubules could act as topological insulators that are
gapped to electronic excitations in the bulk but possess robust electronic
bounds states at the tube ends. Through analyses of structural and electronic
properties, we model the microtubule as a cylindrical stack of
Su-Schrieffer-Heeger chains (originally proposed in the context of
polyacetylene) describing electron hopping between the underlying dimerized
tubulin lattice sites. We postulate that the microtubule is mostly uniform,
dominated purely by GDP-bound dimers, and is capped by a disordered regime due
to the presence of GTP-bound dimers as well. In the uniform region, we identify
the electron hopping parameter regime in which the microtubule is a topological
insulator. We then show the manner in which these topological features remain
robust when the hopping parameters are disordered. We briefly mention possible
biological implications for these microtubules to possess topologically robust
electronic bound states.
