Stochastic Ratcheting on a Funneled Energy Landscape Is Necessary for Highly Efficient Contractility of Actomyosin Force Dipoles

Current understanding of how contractility emerges in disordered actomyosin networks of nonmuscle cells is still largely based on the intuition derived from earlier works on muscle contractility. In addition, in disordered networks, passive cross-linkers have been hypothesized to percolate force chains in the network, hence, establishing large-scale connectivity between local contractile clusters. This view, however, largely overlooks the free energy of cross-linker binding at the microscale, which, even in the absence of active fluctuations, provides a thermodynamic drive towards highly overlapping filamentous states. In this work, we use stochastic simulations and mean-field theory to shed light on the dynamics of a single actomyosin force dipole—a pair of antiparallel actin filaments interacting with active myosin II motors and passive cross-linkers. We first show that while passive cross-linking without motor activity can produce significant contraction between a pair of actin filaments, driven by thermodynamic favorability of cross-linker binding, a sharp onset of kinetic arrest exists at large cross-link binding energies, greatly diminishing the effectiveness of this contractility mechanism. Then, when considering an active force dipole containing nonmuscle myosin II, we find that cross-linkers can also serve as a structural ratchet when the motor dissociates stochastically from the actin filaments, resulting in significant force amplification when both molecules are present. Our results provide predictions of how actomyosin force dipoles behave at the molecular level with respect to filament boundary conditions, passive cross-linking, and motor activity, which can explicitly be tested using an optical trapping experiment.

Popular Summary

Biological tissues have the amazing ability to generate significant force and respond to mechanical stimuli. In eukaryotic muscle cells, for example, neural activity can signal actomyosin fibers (long threads of proteins) to contract, generating tension from within the cell’s cytoskeleton that leads to muscle contraction. While nonmuscle cells can also produce significant contractile force with analogous sets of protein ingredients, this somehow takes place even in the absence of well-organized cytoskeletal structures that are thought to be key in generating contractility in muscle cells. This raises the question of whether the same chemical and physical principles are responsible for the contractile behavior and overall microscopic dynamics of nonmuscle cells. This question underlies understanding phenomena such as immune cell activation, wound healing, and cancer metastasis. Using theoretical analyses and computer simulations, we provide a novel explanation as to how contractile forces can be generated within general actomyosin networks with disordered architectures.

First, we show that fundamental contractile elements in actomyosin networks, called force dipoles, harness the thermodynamic free energy of an auxiliary cross-linking protein binding, independent of motor processes. Surprisingly, we discover that both energy- and nonenergy-consuming proteins in tandem can greatly amplify forces at a microscopic level because of a structural ratcheting of the motor by cross-linking proteins. Our work explains this synergistic phenomenon and provides scaling laws for how this force generation depends on the thermodynamic properties of the proteins, pointing to a nonequilibrium ratcheting mechanism as an important requirement for significant contractile force generation in nonmuscle actomyosin systems.

Having worked out the fundamental principles behind elementary actomyosin force dipoles, next we study how a large collection of such dipoles form mesoscopic-scale networks, which architecturally remodel themselves and show interesting, novel contractile behaviors at larger scales.

Figure 1

Actomyosin contractility mechanisms in a variety of cell types. (a) In a muscle sarcomere, filamentous actin (F-actin) are aligned in bands of opposite polarity such that a bipolar myosin II filament can walk towards the actin filament’s plus ends, generating maximal contractile force, shown as black arrows. The motor filaments also contain hundreds of heads, which are able to continuously generate force on the filaments and maintain attachment as they hydrolyze ATP to produce mechanical work. (b) In a nonmuscle actomyosin network, filaments are distributed in a random geometric fashion throughout the cytoskeleton. The myosin II filaments in a nonmuscle actomyosin system are also smaller (number of heads per side of bipolar filament $N_t\approx 30$ [8]) and highly transient (the duty ratio of bound to unbound states of the motor filament ${\rho }_m\ll 1$) compared to their muscle counterparts ($N_t\approx 500$, ${\rho }_m\approx 1$), but can form disordered arrangements of locally antiparallel actin filaments. Motor filaments of this nature are responsible for stress fiber formation via compression of a fragmented lamellipodial actin mesh.

Figure 2

One-dimensional “force dipole” actomyosin model schematic. Two filaments with plus ends facing outward and midpoints $x_r$ and $x_l$ are both connected to springs with stiffness $K_t$, initially with an equilibrium filament position of $x_r^0$ and $x_l^0$. When no cross-linkers are bound between the two filaments, they separately undergo tethered Langevin motion. While the filament overlap $l_o$ is large enough, cross-linkers can transiently bind and unbind according to their kinetic rates as in Eq. (2), which arrests the filaments if the number of cross-linkers bound is nonzero.The number of available binding sites $n_p$ varies with actin filament overlap. Motor filaments can (un)bind and walk stochastically on the pair of actin filaments, generating force via a time-varying filament overlap potential.

Figure 3

Cross-linker driven contraction of the passive force dipole in one dimension. (a) Average force $⟨F_D⟩$ at ${\tau }_{\mathrm{lab}}=200\mathrm{s}$ for $K_t=0.01\mathrm{pN}/\mathrm{nm}$, as a function of cross-linker binding energy $\epsilon$. The analytically predicted $F_D^{\mathrm{ss}}$ according to the thermodynamic limit in Eq. (7) is shown as the black dotted curve, with the numerical solution to Eq. (6) as solid black; these solutions diverge at ${\epsilon }_{\mathrm{lab}}\approx 6k_{b}T$. This corresponds to the transition ${\tau }_{\mathrm{lab}}\ll {\tau }_{\mathrm{ss}}$ caused by increasing binding strength of cross-links. Increasing speed of unbinding shifts curves to the predicted thermodynamic limit. The inset displays stochastic trajectories for the same cross-linker binding energies. (b) Log-log plot of the average filament overlap $⟨l_o⟩$ vs time for $\epsilon$ values around ${\epsilon }_{\mathrm{lab}}$ shows a linear force-generating regime $l_o\propto \epsilon t/(1+v{e}^{-\epsilon /k_{b}T})$ followed by an exponential relaxation to steady state: $l_o^{\mathrm{ss}}-l_o\propto e^{-t/{\tau }_{\mathrm{ss}}}$, with ${\tau }_{\mathrm{ss}}\propto (1+v^{-1}{e}^{\epsilon /k_{b}T}{)}^{l_o/\mathrm{\Delta }}$. Trajectories above ${\epsilon }_{\mathrm{lab}}$ are far from equilibrium at ${\tau }_{\mathrm{lab}}=200\mathrm{s}$, as shown by a significant flattening of the $l_o$ vs $t$ curve at long times for increasing binding energy.

Figure 4

The free-energy landscape that results from cross-linker binding in a force dipole. The total free energy $G_D$ is a summation of cross-link ($G_{\mathrm{cl}}$) and tether ($U_t$) contributions. $G_{\mathrm{cl}}$ is a linear function of overlap, and the slope of this line can be thought of as the average free-energy difference per addition of a single binding site in the dipole (since the number of binding sites $n_p\propto l_o/\mathrm{\Delta }$). The dipole’s time scale of approach to steady-state overlap (dipole overlap shown as red dots in $G-l_o/\mathrm{\Delta }$ diagram) is in general an exponential function of cross-link binding affinity such that a dipole can become kinetically arrested on the laboratory time scale if $\epsilon >{\epsilon }_{\mathrm{lab}}$, and thus ${\tau }_{\mathrm{lab}}\ll {\tau }_{\mathrm{ss}}$. Otherwise, the dipole can reach the steady-state free-energy minimum overlap $l_o^{\mathrm{ss}}$.

Figure 5

Actomyosin ratcheting due to transient cross-linking in the one-dimensional active force dipole. (a) Variation of average dipole force $⟨F_D⟩$ produced on ${\tau }_{\mathrm{lab}}=200\mathrm{s}$ shows similar biphasic behavior compared to a PFD, but is amplified greatly due to the presence of the stochastic motor against a $0.01\mathrm{pN}/\mathrm{nm}$ tether. Amplifications from the transient motor force are shown ($\approx 0.3\mathrm{pN}$). Inset shows the effect of varying external stiffness of the diplole $K_t$ for a fixed $k_u^{\mathrm{cl}}=1/\mathrm{s}$. The catch-bond nature of myosin II is apparent in the overall increase in generated force at high stiffness. (b) Stochastic force production of the dipole at $\epsilon =5k_{b}T$ shows a steplike approach to steady state. Increasing kinetic rates maximize overlap generated on ${\tau }_{\mathrm{lab}}$. (c) The corresponding state trajectory of the same force dipole with $k_u^{\mathrm{cl}}=0.1/\mathrm{s}$, which correspond to contraction and extension events in (b). $m=0$, 1 represents the unbound and bound states of the motor, respectively. $n=0$ and $n\ne 0$ represent the cross-linker state. (d) The proposed ratcheting process in an active force dipole. (1) The process starts at a stable configuration $n\ne 0$. The motor can be either bound or unbound ($m=0$ or $+1$). (2) Cross-linkers then unbind from the dipole ($n=0$). If $m=+1$, a contraction event occurs, contracting the filaments to a new overlap $l_o^f>l_o^i$. (3) If $m=0$, an extension event occurs in which the filaments lose overlap such that $l_o^f<l_o^i$. (4) Cross-linkers rebind ($n\ne 0$), stabilizing the $l_o^f$ achieved in (1)–(3) in a contracted or extended configuration. This process repeats as pairwise overlap is created between the antiparallel pair, generating contractile force.

Figure 6

Effective interfilament velocity in the one-dimensional active force dipole as a function of pairwise overlap, with $k_u^{\mathrm{cl}}=10/\mathrm{s}$ against $K_t=0.01\mathrm{pN}/\mathrm{nm}$ tethers. This plot can be interpreted as the contractile force-velocity relation of the structural element, since pairwise filament overlap $l_o$ is directly proportional to the restoring force on filaments due to external tethers. The effective interfilament velocity of the dipole (i.e., the rate at which filament overlap increases) is calculated by combining the change in overlap and frequency of contraction and extension events: $V_{\mathrm{eff}}(l_o)=\chi (l_o){\omega }_{\chi }(l_o)-\xi (l_o){\omega }_{\xi }(l_o)$. Cross-linking in an optimal range amplifies forces produced by the dipole and is apparent in the effective contractile velocity of the filament pair at large overlaps. $\chi (l_o)$ and $\xi (l_o)$ are also shown in the inset. Analytic approximations for the expected value of these events are showed as filled lines.

Figure 7

A three-dimensional force dipole simulated using medyan, an active matter reaction-diffusion model. (a) A schematic of the simulation setup, very similar to the one-dimensional model shown in Fig 2. Two filaments (red), now containing three-dimensional bending and stretching modes, are oriented so that their plus ends [${\mathbf{x}}_l^p=(0,0,0)\mu \mathrm{m}$ and ${\mathbf{x}}_r^p=(4,0,0)\mu \mathrm{m}$, respectively, at $t=0$] are connected to elastic, stationary tethers at ${\mathbf{x}}_l^t$ and ${\mathbf{x}}_r^t$ (initially ${\mathbf{x}}_{l,r}^p={\mathbf{x}}_{l,r}^t$). As contraction is produced by myosin II filaments (blue) walking towards the plus ends of each actin filament, cross-linkers (green) and actin filaments can bend and stretch according to polymer potentials outlined in Appendix pp4. (b) The averaged dipole force after 200 s, $⟨F_D⟩$, is shown as a function of $\epsilon$. The steady-state data follow the power laws $F_D^{\mathrm{ss}}\propto (\epsilon P_o^{\mathrm{cl}}{)}^{\alpha }$, where $\alpha =0.4$, 0.24, and 0.05, respectively, for increasing $K_t$. The probability of a single-site occupancy $P_o^{\mathrm{cl}}$ is given by $(1+v{e}^{-\epsilon /k_{b}T}{)}^{-1}$. A diminishing effect of cross-linker ratcheting is explained by myosin II’s ability to increase its binding lifetime under stress, allowing for more stable walking and force generation at $K_t=1\mathrm{pN}/\mathrm{nm}$. (c) Fluctuations of bound cross-linkers (defined as ${\sigma }_n/⟨n⟩=⟨n-⟨n⟩⟩/⟨n⟩$) show a sharp onset of kinetic arrest at $\epsilon ={\epsilon }_{\mathrm{lab}}$, similar to the one-dimensional case. (d) Average dipole force produced for varying initial overlap $l_o^i=1000$, 500, 200, 0 nm, with $l_o^i=0$ as the top curve. Increasing initial overlap $l_o^i$ will decrease contractile force produced in simulation (${\tau }_{\mathrm{lab}}=200\mathrm{s}$) due to kinetic friction.