Rheology of Semiflexible Bundle Networks with Transient Linkers

We present a theoretical and computational analysis of the rheology of networks made up of bundles of semiflexible filaments bound by transient cross-linkers. Such systems are ubiquitous in the cytoskeleton and can be formed in vitro using filamentous actin and various cross-linkers. We find that their high-frequency rheology is characterized by a scaling behavior that is quite distinct from that of networks of the well-studied single semiflexible filaments. This regime can be understood theoretically in terms of a length-scale-dependent bending modulus for bundles. Next, we observe new dissipative dynamics associated with the shear-induced disruption of the network at intermediate frequencies. Finally, at low frequencies, we encounter a region of non-Newtonian rheology characterized by power-law scaling. This regime is dominated by bundle dissolution and large-scale rearrangements of the network driven by equilibrium thermal fluctuations.

Figure 1

Example of a bundle network as prepared for rheological measurements. The simulation cube ($V=216\mu {\mathrm{m}}^3$) has periodic boundary conditions and 360 filaments ($c_f=4\mu \mathrm{M}$) that are in chemical equilibrium with a solution of linkers at a concentration $c_l\approx 0.07\mu \mathrm{M}$. Singly bound linkers (blue) and doubly bound linkers (red) are shown.

Figure 2

Comparison of simulated rheological spectra to experiments. Simulation parameters are $c_f=4\mu \mathrm{M}$, $c_l/c_f=0.017$, $k_{\text{on}}=90{\mathrm{s}}^{-1}$, and $k_{\text{off}}=\{2,0.07\}{\mathrm{s}}^{-1}$. We encounter three distinct regimes. At high frequencies ($\omega >{\omega }_1$), we see scaling behaviors $G^{\prime }\sim {\omega }^{0.93}$ and $G^{″}\sim {\omega }^{0.55}$. The inset shows $G^{\prime }$ and $G^{″}$ of the same system without linkers. The expected ${\omega }^{3/4}$ scaling relation is recovered. At intermediate frequencies (${\omega }_2<\omega <{\omega }_1$), there is a local maximum in $G^{″}$ and large sample-to-sample variations. At low frequencies, there is again a power-law regime.

Figure 3

The distribution of distances between pairs of initially cross-linked binding sites during oscillatory shear at frequencies of $\omega \in [20000\pi ;0.02\pi ]\mathrm{rad}/\mathrm{s}$ showing that the network becomes more sparsely cross-linked at low frequencies. Inset: Binding site pairs (red and blue) with highlighted distances (green) between ruptured cross-links at sites of bundle-bundle detachment for the $\omega =0.02\pi \mathrm{rad}/\mathrm{s}$ case.

Figure 4

(a) Shear stress relaxation and fluctuations after a step strain at $t=0\mathrm{s}$. The $G^{\prime }\sim G^{″}\sim {\omega }^{1/2}$ scaling relation predicts a $1/t^{1/2}$ stress decay. For $t>200\mathrm{s}$, spontaneous large-amplitude fluctuations overwhelm the mean stress and are accompanied by subdiffusion of the principle axes of the moment of inertia tensor (inset). (b) Frequency-dependent mean square amplitude of the stress fluctuations (red squares) in the low-frequency regime at $k_{\text{off}}=0.01{\mathrm{s}}^{-1}$ compared to the prediction of equilibrium fluctuation from the fluctuation-dissipation theorem [Eq. (4)] (dashed blue line). (c) The nonequilibrium fluctuation enhancement disappears at an increased $k_{\text{off}}=0.1{\mathrm{s}}^{-1}$.