Topological phonon modes in filamentary structures

This work describes a class of topological phonon modes, that is, mechanical vibrations localized at the edges of special structures that are robust against the deformations of the structures. A class of topological phonons was recently found in two-dimensional structures similar to that of microtubules. The present work introduces another class of topological phonons, this time occurring in quasi-one-dimensional filamentary structures with inversion symmetry. The phenomenon is exemplified using a structure inspired from that of actin microfilaments, present in most live cells. The system discussed here is probably the simplest structure that supports topological phonon modes, a fact that allows detailed analysis in both time and frequency domains. We advance the hypothesis that the topological phonon modes are ubiquitous in the biological world and that living organisms make use of them during various processes.

Figure 1

The ends of the microfilaments oscillate while pushing against the membrane, exposing their ends to the pool of available ATP-actin. The ATP-actin is then able to attach to the exposed ends and continue the filament’s polymerization.

Figure 2

The adiabatic transport is carried over the paths $\gamma$ and ${\gamma }^{\prime }$.

Figure 3

The structure consists of a periodic array of dimers, connected by four different springs. The diagram depicts the system at equilibrium.

Figure 4

If $K_{++}=K_{--}$, the structure is mapped into itself by the symmetry operations shown in this diagram.

Figure 5

The diagram introduces the notations used in the text. The array of dimers is shown at the equilibrium (faded) and at some arbitrary snapshot in time. $x_n$ represents the position of the center of mass of the $n$th dimer and ${\varphi }_n$ represents the angle of rotation of the $n$th dimer relative to the vertical axis. In the equilibrium configuration, all ${\varphi }_n$ are equal to ${\varphi }_0$.

Figure 6

The phonon bands for (a) $K_{--}=K_{++}=0.5$ and $K_{-+}=K_{+-}=2$ and (b) $K_{--}=K_{++}=0.5$ and $K_{-+}=K_{+-}=4$.

Figure 7

The spectral gap opening for (a) $K_{--}=K_{++}=0.5$, $K_{-+}=3$, and $K_{+-}=5$ and (b) $K_{--}=0.3$, $K_{++}=0.7$, and $K_{-+}=K_{+-}=4$.

Figure 8

Each vertical sequence of dots represent the frequencies of the normal modes of a chain made of 100 dimers arranged in a circle. The spring constants $K$ of the springs connecting two dimers were gradually weakened by making the substitution $K\to \mathit{wK}$ and letting $w$ vary from 1 to 0. The frequencies were recomputed for many $w$ values. Panel (a) refers to the topological case, and panel (b) to the nontopological case.

Figure 9

(Top diagram) Representation of the dimer chain and the forced motion of the rightmost dimer. (Main panel) The trajectories covered by the dimers after $t=500$, 1000, and 2000. Each subpanel contains three plots, representing, from the top to bottom, the actual trajectories of the dimers, the trajectories of the horizontal displacements $x_n(t)$, and of the angular displacements ${\varphi }_n(t)$. Plots (a), (b), and (c) represent the behavior of the dimers in the topological case where $K_{--}=K++=0.5$, $K_{+-}=3$, and $K_{-+}=5$. Plots (a${}^{\prime }$), (b${}^{\prime }$), and (c${}^{\prime }$) represent the behavior of the dimers in the nontopological case where $K_{--}=0.7$, $K_{++}=0.3$, and $K_{-+}=K_{+-}=4$.