Nonequilibrium dynamics of probe filaments in actin-myosin networks

Active dynamic processes of cells are largely driven by the cytoskeleton, a complex and adaptable semiflexible polymer network, motorized by mechanoenzymes. Small dimensions, confined geometries, and hierarchical structures make it challenging to probe dynamics and mechanical response of such networks. Embedded semiflexible probe polymers can serve as nonperturbing multiscale probes to detect force distributions in active polymer networks. We show here that motor-induced forces transmitted to the probe polymers are reflected in nonequilibrium bending dynamics, which we analyze in terms of spatial eigenmodes of an elastic beam under steady-state conditions. We demonstrate how these active forces induce correlations among the mode amplitudes, which furthermore break time-reversal symmetry. This leads to a breaking of detailed balance in this mode space. We derive analytical predictions for the magnitude of resulting probability currents in mode space in the white-noise limit of motor activity. We relate the structure of these currents to the spatial profile of motor-induced forces along the probe polymers and provide a general relation for observable currents on two-dimensional hyperplanes.

Figure 1

Sketch of a probe filament in a motor activated network. A probe filament, e.g., a carbon nanotube (white-blue, foreground), is introduced into a crosslinked network of actin filaments (gray, background), with myosin motors (red barbell-shapes). The probe is assumed to be fixed to the network. Motor action (red arrows) results in external forces $f_{\text{M}}$ impinging on the probe along its contour $s$. Both myosin and actin are drawn opaque as they would not be visible in fluorescence microscopy experiments.

Figure 2

Parametrization of the probe filament shape.

Figure 3

The breaking of Onsager's time-reversal symmetry results in different nonzero normalized cross-correlations of the mode pair $(j,k)=(2,4)$ (red) and $(j,k)=(4,2)$ (blue). Continuous lines represent correlations inferred from Brownian dynamics simulations, while analytical predictions from Eq. (16) are shown as dashed lines. In all panels, modes with relaxational time scales ${\tau }_2\approx 0.58\mathrm{s}$ and ${\tau }_4\approx 0.12\mathrm{s}$ were considered. The time scale of the motor noise $f_{\text{M}}(s,t)$ was set to (a) ${\tau }_{\text{M}}=0.375$ s, (b) ${\tau }_{\text{M}}=0.0375$ s, and (c) ${\tau }_{\text{M}}=0.000375$ s. At long lag times, the error increases due to limited sampling statistics, which causes noise in the correlation function.

Figure 4

(a) Temporal average of upward (blue) and downward (red) motor-induced force $\langle f_{\text{M}}(s,t)\rangle$. The peaks are shown with a finite width for illustration purposes. In the two top rows, 12 interactions points $s_n$ were chosen from an equal distribution with density $\rho \left(s_n\right)=1/L$, and half of them were assigned negative force coefficients $f_{-}=-f_{+}$. The magnitude of the force was fixed at $f_{+}=\left|f_{-}\right|=10$ pN, $L$ was set to $10\mu \text{m}$, and ${\tau }_{\text{off}}/{\tau }_{\text{on}}=1/3$. In the two bottom rows, the interaction points were distributed periodically with varying density, such that the average total force and torque were zero. (b) The corresponding coupling matrices ${\mathbf{F}}_{q,w}$ for the first 11 modes.

Figure 5

Correlation matrices calculated from Eq. (24) for the same sets of random interaction points $\left\{s_n\right\}$ as in Fig. 4.

Figure 6

Analytical predictions of the currents for (a) mode pair $a_2$ and $a_3$ and (b) mode pair $a_2$ and $a_6$. All parameters are chosen to match the scenario presented in Fig. 7.

Figure 7

Probability fluxes in normal mode phase space inferred from simulations. In both cases the arrow scale is $0.2{\text{s}}^{-1}$. (a) Phase space of mode pair $a_2$ and $a_3$ with no significant probability currents. (b) Phase space of mode pair $a_2$ and $a_6$ with significant currents in accordance with our theory. The indices 2, 3, and 6 describe the free-end mode numbers and correspond to wave vectors of $q\left(2\right)\approx 0.73\mu {\text{m}}^{-1}, q\left(3\right)\approx 1.1\mu {\text{m}}^{-1}$, and $q\left(6\right)\approx 2.04\mu {\text{m}}^{-1}$, respectively. Interaction points $s_n$ were chosen in a periodic pattern as in the third row from the top in Fig. 4. Motor parameters are chosen to be the same as in Fig. 4, with the exception of ${\tau }_{\text{off}}=0.0005\mathrm{s}, {\tau }_{\text{on}}=0.0015\mathrm{s}$. Filament parameters were set to model microtubules in a typical cytoskeletal actin gel: $\kappa =24\times {10}^{-24}{\mathrm{Nm}}^2, L=10\mu \mathrm{m}, G_0=10$ Pa, and $\eta =1$ Pa s.

Figure 8

The apparent cycling frequency depends on the choice of the observed plane. An observer of the plane ${\tilde{a}}_q\times {\tilde{a}}_v$ will measure the frequency ${\tilde{\omega }}_{q,v}={\tilde{\mathrm{\Omega }}}_{q,v}$, while ${\tilde{\omega }}_{q,w}={\tilde{\mathrm{\Omega }}}_{q,w}$ would be measured in the plane ${\tilde{a}}_q\times {\tilde{a}}_w$.

Figure 9

(a) The first 11 mode amplitudes $a_q\left(t\right)$ inferred from the first 3000 steps of the simulation. (b) Comparison of net cycling frequencies calculated analytically using Eq. (28) (black $x$) for isolated pairs of modes of the same parity and numerically in a 15-dimensional mode space from the matrix elements ${\tilde{\mathrm{\Omega }}}_{j,k}$ (red squares) in correlation-identity coordinates as described in Eq. (27). The blue circles are frequencies inferred from the same Brownian simulation data as in Fig. 7 using Eq. (29). We excluded all data points with a radius $\sqrt{{\tilde{a}}_q^2+{\tilde{a}}_w^2}<0.1$, where small steps across the origin would lead to large angular changes. Error bars indicate the standard error of the mean. (c) Distributions of angular displacements on a log scale for a few mode pairs.