Defect Production in Compressed Filament Bundles

We discuss the response of biopolymer filament bundles bound by transient cross-linkers to compressive loading. These systems admit a mechanical instability at stresses typically below that of traditional Euler buckling. In this instability, there is thermally activated pair production of topological defects that generate localized regions of bending—kinks. These kinks shorten the bundle’s effective length, thereby reducing the elastic energy of the mechanically loaded structure. This effect is the thermal analog of the Schwinger effect, in which a sufficiently large electric field causes electron-positron pair production. We discuss this analogy and describe the implications of this analysis for the mechanics of biopolymer filament bundles of various types under compression.

Figure 1

Fluorescence microscopy images of a $\mathsf{z}$ bend (a) and a $\mathsf{u}$ bend (b) in a collagen bundle. (c) Two loops under compression form a $\mathsf{z}$ bend. (d) Two braids under compression form a $\mathsf{u}$ bend. (e) Angle $\varphi$ produced by a loop pair (blue, left axis) and energy difference between the looped and straight bundle as a function of dimensionless torque (red, right axis). (f) Angle $\varphi$ produced by a braid pair (blue, left axis) and the energy difference between braided and straight bundles as the function of dimensionless torque (red, right axis). (Images courtesy of E. Botvinick and Q. Hu.) (b) is reused from [6].

Figure 2

Dimensionless loop pair production rate with $\eta \approx 22.5$, $\tau =0.1$ [see Eq. (5)].

Figure 3

Numerical minimization of the energy (red dots) and analytical prediction [Eq. (8)] for a symmetric pseudobraid ${\kappa }_1={\kappa }_2$. The numerical solution of the more complex, three-filament braid with ${\kappa }_1=2{\kappa }_2$ (green circles) shows that the transition is shifted to higher compression.