Topological Phonon Modes and Their Role in Dynamic Instability of Microtubules

Microtubules (MTs) are self-assembled hollow protein tubes playing important functions in live cells. Their building block is a protein called tubulin, which self-assembles in a particulate 2 dimensional lattice. We study the vibrational modes of this lattice and find Dirac points in the phonon spectrum. We discuss a splitting of the Dirac points that leads to phonon bands with nonzero Chern numbers, signaling the existence of topological vibrational modes localized at MTs edges, which we indeed observe after explicit calculations. Since these modes are robust against the large changes occurring at the edges during the dynamic cycle of the MTs, we can build a simple mechanical model to illustrate how they would participate in this phenomenon.

Figure 1

The system can be switched between two equilibrium configurations by: (a) delivering a large amount of energy at once or (b) by attaching a $\mathrm{\text{spring}}+\mathrm{\text{weight}}$ and releasing the beads one by one. (c) The 2D lattice of dimers of primitive vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$. At equilibrium, the dimers are all parallel and orientated along an arbitrary direction. The $xy$ coordinate system indicates the plane of motion. The $y$ axis is along the dimer but the $x$ axis is arbitrary. (d) The interaction between the tubulin dimers is modeled by a network of springs. There are 7 distinct springs and corresponding $\widehat{\mathbf{e}}$ unit vectors. We also indicate second order neighbors whose harmonic interaction is also considered. (e) The degrees of freedom for the $xy$ planar motion. (f) Stretched configuration of a spring with the ends displaced by ${\mathbf{r}}_{1,2}$.

Figure 2

(a) The $\mathbf{k}$ values where $\Delta ϵ(\mathbf{k})=0$ (red [medium gray]) $w(\mathbf{k})=0$ (blue [dark gray]). (b) Plot of the phonon frequencies ${\omega }_{\pm }(\mathbf{k})$. ${\tilde{K}}_{1-7}$ were fixed at: 1.1, 0.7, 0.27, 0.75, 0.325, 0.27, 0.75, and $M=1$.

Figure 3

(a) The $\mathbf{k}$ values where $\Delta ϵ(\mathbf{k})=0$ (red [medium gray]), $\mathrm{Re}[w(\mathbf{k})]=0$ (blue [dark gray]) and $\mathrm{Im}[w(\mathbf{k})]=0$ (green [light gray]). The full/dashed lines are for $\gamma =0/0.5$. (b)–(d) The phonon spectrum for $\gamma =0$, 0.45 and 0.6. The last row shows the phonon spectrum of a ribbon as function of $k_{\parallel }$ for (e) periodic and (f) open boundary conditions.