Nonequilibrium noise emerging from broken detailed balance in active gels

TL;DR

The paper develops a minimal mesoscopic model of an active gel formed by elastic elements connected with transient crosslinkers whose binding/unbinding breaks detailed balance. Through coarse-graining, it derives fluctuating hydrodynamics with an explicit active noise term, linking molecular activity to mesoscopic fluctuations and predicting a fluctuation-dissipation-activity relation. It provides concrete predictions for stress fluctuations and tracer dynamics in microrheology, including an anisotropic noise component arising from nematic order. The framework offers a path to extend the fluctuation-dissipation theorem to active systems and informs experimental tests in reconstituted networks and living cells.

Abstract

In thermodynamic equilibrium, the fluctuation-dissipation theorem links thermal fluctuations and dissipation. Biological systems, however, are driven out of equilibrium by internal processes that produce additional, active fluctuations. Despite being relevant for biological functions such as intracellular transport, predicting the statistical properties of active fluctuations remains challenging. Here, we address this challenge in a minimal model of an active gel as a network of elastic elements connected by transient crosslinks. The crosslinkers' binding and unbinding rates break detailed balance, which drives the system out of equilibrium. Through coarse-graining, we derive fluctuating hydrodynamic equations including an active noise term, which emerges explicitly from the breaking of detailed balance. Finally, we provide predictions for the stochastic motion of a tracer particle embedded in the active gel, which enables comparisons with microrheology experiments both in synthetic active gels and in cells. Overall, our work provides an explicit link between the statistical properties of active fluctuations and the molecular breaking of detailed balance. Thus, it paves the way toward complementing the fluctuation-dissipation theorem with a fluctuation-activity relation in active systems.

Figures (6)

• Figure 1: (a) Cell cortex and mitotic spindle, the relevant cellular structures where our model applies. Filaments are interconnected by active molecular motors/crosslinks. (b) Coarse-graining volume $V$ containing the active gel model. The background filaments establish the nematic order ${\mathbf Q}$, while an external shear rate ${\mathbf v}$ is applied. Each bound molecular motor is characterized by a mesoscopic strain ${\mathbf u}$ and an orientation $\mathbf{q}$, independent of other motors. (c) Energy landscape of the bound state. We illustrate the contributions to the detailed balance relation \ref{['kakd']}, setting $\Omega=0$ for simplicity. In our model, the activation energy for unbinding is constant, making the unbinding rate ${k}_\text{u}$ independent of the bound state.
• Figure 2: Activity and external drivings shift the stress probability distribution. Steady-state distribution of the $\sigma_{11}$ component of the stress tensor in equilibrium (a), in the presence of activity (b), and of activity and external driving (c). The stress distribution results from the distribution of bound linkers given in \ref{['ss-10']} (see text). Each bound linker contributes a stress ${\boldsymbol \sigma}^{\text{linker}}({\mathbf u},\mathbf{q})$ as defined in \ref{['sigma-1-def']}. (a) Isotropic stress distribution in thermal equilibrium ($\Omega=0$ in \ref{['kakd-new']}). The distribution is symmetric around $\sigma_{11}=0$ with vanishing mean stress. (b) Effect of activity on the stress distribution. The distribution has contributions from thermal and active fluctuations; the latter arise from the breaking of detailed balance in \ref{['kakd']}. For the active part, we consider the molecular activity $\Omega({\mathbf u},\mathbf{q})$ presented in \ref{['sec-specific']}, which promotes binding at certain orientation $\mathbf{q}={\mathbf Q}$. We take $Q_{11}=1$. Because we also choose a positive elastonematic coupling $D>0$ (which couples ${\mathbf u}$ and $\mathbf{q}$, see free energy \ref{['free-energy']}), the linker is biased to binding at states with positive $u_{11}$. Thus, the stress $\sigma_{11}\sim \mu u_{11}+Dq_{11}$ is also biased towards positive values. Overall, the mean stress becomes positive because of the symmetry-breaking molecular activity. (c) Shearing the system at a rate $v_{11}>0$ additionally shifts the distribution in the direction of $v_{11}$, as encoded in \ref{['ss-10']}. Parameter values, chosen for illustration purposes, are given in \ref{['params-fig1d']}. See \ref{['app-plots']} for more details, and the distribution function's visualisation in two dimensions.
• Figure 3: Visualisation of the response function $\chi(\omega)$ from \ref{['chi-final']} and the fluctuation spectrum ${\rm S}^R(\omega)$ from \ref{['SR-final']}. The latter is scaled using the FDT \ref{['FDT']} to highlight the departure from equilibrium. The parameters are ${k}_\text{u}={\rm k}_{\rm B}T=\mu=\chi=\Omega_0=a=1$, $D=0.8$, $\rho=A=0.15$ and $|{\mathbf Q}|=1$ with $z$-axis of nematic order. The dashed line indicates the equilibrium case $\Omega=0$, when all three curves coincide. Note that the dashed line is above the response function (blue solid line), because the viscosity $\eta$ also depends on activity.
• Figure 4: This figure depicts the Ornstein-Uhlenbeck colored noise characterized by $\mathopen{}\mathclose{\left\langle \hat{\eta}(t) \hat{\eta}(t')\right\rangle = \frac{k}}{2} \mathrm{e}^{-k |t-t'|}$. The left panel shows steady-state trajectories of $Y(t)$, while the right panel presents trajectories conditioned on $Y_0=1$. The red solid line in the right panel represents the conditional average of these trajectories at later times, and the black dashed line indicates the trajectory of the average: $Y_t = \mathrm{e}^{-t}$.
• Figure 5: At steady state, the states of the bound linkers are distributed according to $n_{\rm ss}({\mathbf u}, \mathbf{q})$ defined in \ref{['ss-10']}. Each bound linker contributes to the stress tensor $\sigma({\mathbf u},\mathbf{q})$ as defined in \ref{['sigma-1-def']}. Here, we visualize the distribution of the stress tensor in $d=2$ dimensions. The stress tensor has two independent components, $\sigma_{11}$ and $\sigma_{12}$, which are taken as axes. The tangent of the black streamlines represents the principal axis of the corresponding stress tensor(the directions where there is no shear stress). For example, the principal axis of the stress tensor in the right diagram is at 45 degrees, the corresponding point on the left plots is $(\sigma_{11},\sigma_{12}) = (0,1)$, where the tangent of the streamline is also at 45 degrees. Bright spots on the plots indicate regions of high probability density. Similar to \ref{['fig-sigmadist-1d']}, we set the activity $\Omega$ to promote binding to states with orientation $\mathbf{q} \approx {\mathbf Q} = 100-1$. For the left plot, we set vanishing external drives, ${\mathbf v}=\dot{\mathbf Q}=0$, and we assume that the activity is much stronger than the thermal contribution in \ref{['kakd']}; latter is responsible for the less bright spot at the center. As the elastonematic coupling $D=0.8 \ne 0$, there is a high probability density at $\sigma \propto {\mathbf Q}$. The black dot marks the average of the distribution, and the red arrow indicates the probability shift due to activity. In the right plot, we introduce shear with rate ${\mathbf v}= 0110$ in the orthogonal direction in contrast to the \ref{['fig-sigmadist-1d']}. This distorts the stress distribution. The blue arrow indicates the shift in the average of the distribution induced by the external shear rate ${\mathbf v}$. Parameter values, chosen for illustration purposes, are given in \ref{['params-fig2d']}.