Nonequilibrium Scaling Behavior in Driven Soft Biological Assemblies

Measuring and quantifying nonequilibrium dynamics in active biological systems is a major challenge because of their intrinsic stochastic nature and the limited number of variables accessible in any real experiment. We investigate what nonequilibrium information can be extracted from noninvasive measurements using a stochastic model of soft elastic networks with a heterogeneous distribution of activities, representing enzymatic force generation. In particular, we use this model to study how the nonequilibrium activity, detected by tracking two probes in the network, scales as a function of the distance between the probes. We quantify the nonequilibrium dynamics through the cycling frequencies, a simple measure of circulating currents in the phase space of the probes. We find that these cycling frequencies exhibit power-law scaling behavior with the distance between probes. In addition, we show that this scaling behavior governs the entropy production rate that can be recovered from the two traced probes. Our results provide insight into how internal enzymatic driving generates nonequilibrium dynamics on different scales in soft biological assemblies.

Figure 1

Schematic illustrating soft viscoelastic networks with heterogeneous driving for various types of cellular systems. (a) Chromosome, (b) red blood cell membrane, (c) cytoskeletal network with (d)–(f) associated bead-spring models with heterogeneous active driving. The color of the bead indicates the intensity of activity, representing the variance (increasing from blue to red) of the associated active noise process.

Figure 2

Reduced system of tracked probed. (a) Schematic of two fluorescently labeled probe beads in a larger system. (b) Elastic force acting on bead $j$ obtained at different time steps of a simulation of the Langevin dynamics of the full system (blue points) and the effective linear force ${\mathbf{A}}_{\text{eff}}{\mathbf{x}}_{\mathrm{r}}$ from analytical calculations (light blue plane). (c) Probability density (color map) and probability current (white arrows) calculated analytically from the effective 2D system, together with results from simulating the full system in the inset. (d) The nonconservative part of the effective force field, $[({\mathbf{A}}_{\text{eff}}-{\mathbf{A}}_{\text{eff}}^T)/2]{\mathbf{x}}_{\mathrm{r}}$ (black arrows), can contribute to the rotation in phase space in nonequilibrium systems. Note, for ${\alpha }_i=\alpha \forall i$ (effective equilibrium scenario), ${\mathbf{A}}_{\text{eff}}$ becomes symmetric.

Figure 3

Spatial scaling behavior of cycling frequencies. (a) Steady-state current cycles in phase space of the displacements (along the lattice direction) of two tracer beads for a nearby pair of probes (left) and distant pair of probes (right). (b) Scaling behavior of the cycling frequencies $\sqrt{⟨{\omega }^2(r)⟩}$ of a pair of probes beads as a function of their spatial distances, obtained for a 1D chain and different activity distributions, as indicated in the legend. Black dashed line is obtained from Eq. (10). (c) Scaling behavior of the cycling frequencies $\sqrt{⟨{\omega }^2(r)⟩/⟨{\omega }^2(1)⟩}$, obtained for different lattices and a folded Gaussian activity distribution. Triangular and square markers represent triangular and square or cubic lattices, respectively. Light (dark) blue triangles represent triangular networks with zero (finite) rest length springs. In both (b) and (c), we used $\overline{\alpha }/T=0.15$.

Figure 4

Spatial scaling behavior of the average entropy production rate $⟨{\mathrm{\Pi }}_{\mathrm{r}}^{(2)}⟩$ of a pair of probe beads as a function of their spatial distance $r$, obtained for different lattices, and a folded Gaussian activity distribution with $\overline{\alpha }/T=0.15$. Note the entropy production rate of the reduced system is scaled by the total entropy production rate of the whole network ${\mathrm{\Pi }}_{\text{tot}}$. Triangular and square markers represent triangular and square or cubic lattices, respectively. Light (dark) blue triangles represent triangular networks with zero (finite) rest length springs.