Thermodynamic Control of Activity Patterns in Cytoskeletal Networks

Biological materials, such as the actin cytoskeleton, exhibit remarkable structural adaptability to various external stimuli by consuming different amounts of energy. In this Letter, we use methods from large deviation theory to identify a thermodynamic control principle for structural transitions in a model cytoskeletal network. Specifically, we demonstrate that biasing the dynamics with respect to the work done by nonequilibrium components effectively renormalizes the interaction strength between such components, which can eventually result in a morphological transition. Our work demonstrates how a thermodynamic quantity can be used to renormalize effective interactions, which in turn can tune structure in a predictable manner, suggesting a thermodynamic principle for the control of cytoskeletal structure and dynamics.

Figure 1

Structural transition between asters and bundles. (a) Schematic of two filaments (blue) connected by a motor (orange). The motor is modeled as a Hookean spring with rigidity $k$. The motor force $|{\overrightarrow{f}}_m|$ is proportional to motor rigidity $k$. Each motor head (dark orange) binds to one filament and moves toward the barbed ($+$) end with velocity ${\overrightarrow{v}}_m$ [Eq. (1)]. When the motor is bound to two filaments, the rate of work done by the motor spring ($\dot{w}$) is computed as the sum of ${\overrightarrow{f}}_m·{\overrightarrow{v}}_m$ over the two motor heads. (b) Tuning $k$ induces the structural transition between asters and bundles. (c) Color map of $⟨\dot{w}⟩$. Asters and bundles are respectively associated with positive and negative values of $⟨\dot{w}⟩$. (d) Color map of the largest cluster size. A system with no cluster larger than 40 is considered to be isotropic. (e) Color map of radius of gyration $R_g$, of the largest cluster. The boundary between the bundle and aster regions $R_g=0.125$. The boundary of the isotropic region is the same as in (d).

Figure 2

Dynamical bias effectively renormalizes the motor rigidity $k$. (a) Probability distribution obtained by biasing the dynamics of a given order parameter $\psi$ (with free energy $\mathcal{F}(\psi )=-a\psi +b{\psi }^2-c{\psi }^3+d{\psi }^4$) with respect to $\psi$. Parameters: $a=4d=-b=-30c=1$. Biasing parameter: $\alpha =0$ (unbiased dynamics, blue), 0.005 (green), 0.1 (orange), and 0.3 (purple). (b) Snapshots of unbiased simulations with changing motor rigidity $k$. (c) Snapshots of biased simulations with fixed motor rigidity $k=3$. (d) The statistics of structures from biased dynamics match that from an unbiased simulation at a different $k$. The order parameter $\sin \theta$ is calculated from angles between neighboring filaments and averaged across nearest neighbors. Matching the distribution of $\sin \theta$ from biased (black lines with the error bar shown with gray area) and unbiased simulations (red and blue) results in defining an effective rigidity $k_{\mathrm{eff}}$. (e) Effective rigidity $k_{\mathrm{eff}}$ as a function of bias parameter $\alpha$ at $v_0=0.8$ (black), 1.0 (magenta), and 1.2 (green). The two green curves correspond to $k=3$ and 2.8 analyzed for $v_0=1.2$. Filled points are obtained by matching structures from biased and unbiased simulations. Hollow points are analytical predictions derived from the two-state model [Eq. (7)].