Cross-Link-Governed Dynamics of Biopolymer Networks

Recent experiments show that networks of stiff biopolymers cross-linked by transient linker proteins exhibit complex stress relaxation, enabling network flow at long times. We present a model for the dynamics controlled by cross-links in such networks. We show that a single microscopic time scale for cross-linker unbinding leads to a broad spectrum of macroscopic relaxation times and a shear modulus $G\sim {\omega }^{1/2}$ for low frequencies $\omega$. This model quantitatively describes the measured rheology of actin networks cross-linked with $\alpha$-actinin-4 over more than four decades in frequency.

Figure 1

A schematic of the frequency dependent shear modulus $G^{*}=G^{\prime }+ıG^{\prime \prime }$. Nonpermanent networks can exhibit a response ranging from a single time scale ($\tau$) Maxwell-like behavior (blue lines) to a power-law regime with an exponent $<1$ governed by a broad distribution of relaxation times ($>\tau$) (red lines). Upper inset: For times longer than the unbinding time ${\tau }_{\mathrm{off}}$, large scale conformational relaxation can occur via linker unbinding (open circle) and subsequent rebinding at a new location. Lower inset: For shorter times, only small-scale bend fluctuations between cross-links can relax, resulting in a plateau in $G^{\prime }$ for frequencies $>1/{\tau }_{\mathrm{off}}$.

Figure 2

(a) The simulated rheology for frequencies below ${\omega }_0$. The shear modulus is normalized by the elastic plateau value $G_0$. The inset shows the total power spectrum $C(\omega )$ (blue circles) of distance fluctuations, as well as the fraction $C_{\parallel }$ coming from effective stretch fluctuations originating in undulations on length scales shorter than the cross-linking distance. The distance fluctuations are determined over a length $16{\ell }_c$ of a polymer with a persistence length ${\ell }_p=32{\ell }_c$ and a total length $32{\ell }_c$. The solid black line represents our analytical mean-field CGD prediction. (b) Measured linear rheology of a $23.8\mu \mathrm{M}$ actin network cross-linked with various concentrations of $\alpha$-actinin-4. The low-frequency behavior is consistent with $G\sim (ı\omega {)}^{1/2}$. The solid and dashed lines are global fits utilizing our mean-field CGD model for the low-frequency regime together with the known high frequency response [3, 4].

Figure 3

The power spectrum $C_{\perp }(\omega )$ of longitudinal fluctuations originating from transverse undulations on length scales longer than ${\ell }_c$, multiplied with ${\omega }^{3/2}$ (a) and ${\omega }^{7/4}$ (b) for a range of polymer backbone compliances. (c) The simulated amplitude of the power spectrum $C(\omega )/\tau {\ell }_c^2$ plotted against the 2D mean-field prediction for a range of polymer lengths and bending rigidities.