^{1}and Peter G. Wolynes

^{1,2}

### Abstract

Actomyosin networks are major structural components of the cell. They provide mechanical integrity and allow dynamic remodeling of eukaryotic cells, self-organizing into the diverse patterns essential for development. We provide a theoretical framework to investigate the intricate interplay between local force generation, network connectivity, and collective action of molecular motors. This framework is capable of accommodating both regular and heterogeneous pattern formation, arrested coarsening and macroscopic contraction in a unified manner. We model the actomyosin system as a motorized cat's cradle consisting of a crosslinked network of nonlinear elastic filaments subjected to spatially anti-correlated motor kicks acting on motorized (fibril) crosslinks. The phase diagram suggests there can be arrested phase separation which provides a natural explanation for the aggregation and coalescence of actomyosin condensates. Simulation studies confirm the theoretical picture that a nonequilibrium many-body system driven by correlated motor kicks can behave as if it were at an effective equilibrium, but with modified interactions that account for the correlation of the motor driven motions of the actively bonded nodes. Regular aster patterns are observed both in Brownian dynamics simulations at effective equilibrium and in the complete stochastic simulations. The results show that large-scale contraction requires correlated kicking.

The support of this work from the Center for Theoretical Biological Physics sponsored by the National Science Foundation (NSF) (Grant No. PHY-0822283) is gratefully acknowledged.

I. INTRODUCTION

II. THEORY

A. Quadratic expansion of the master equation: Effective equilibrium with modified potential

1. Fokker-Planck (FP)/Smoluchowski equation for Brownian particles

2. Master equation for motor-driven processes: Anti-correlated kicks

3. Generalized FP equation for motorized systems: Effective temperature and modified potential

B. Pair-level steady-state solution

C. Self-consistent phonon (SCP) calculation: Possibility of phase separation

III. SIMULATIONS

A. Simulation setup

B. Illustrations

1. Validity of an effective equilibrium and arrested phase separation

2. Effective attraction

3. Effective repulsion

4. Effect of motor activity and susceptibility on phase separation: Mean-field indications

IV. CONCLUSION AND DISCUSSION

### Key Topics

- Networks
- 34.0
- Phase separation
- 27.0
- Aggregation
- 24.0
- Fokker Planck equation
- 19.0
- Brownian dynamics
- 15.0

## Figures

Schematic illustration of the spatially anti-correlated kicks acting on motor-bonded node pairs. (Central image) A bipolar myosin minifilament pulls in slack locally, generating a pair of equal size (*l*) but oppositely directed displacements (red arrows) at the motor-bonded nodes (purple spheres) along their line of centers, where is a unit vector pointing from node *i* to node *j*. Upon zoom-out, this represents a typical functional unit (marked by a dashed circle in the top image) that generates incremental contractions within a crosslinked filamentous network. An enlarged view of the actin filament (bottom image) reveals its segmented structure. The size *l* of the subunits determines the magnitude of the relative node displacements due to contraction events of myosin sliding. *l* is thus taken to be the step size of anti-correlated kicks in our model.

Schematic illustration of the spatially anti-correlated kicks acting on motor-bonded node pairs. (Central image) A bipolar myosin minifilament pulls in slack locally, generating a pair of equal size (*l*) but oppositely directed displacements (red arrows) at the motor-bonded nodes (purple spheres) along their line of centers, where is a unit vector pointing from node *i* to node *j*. Upon zoom-out, this represents a typical functional unit (marked by a dashed circle in the top image) that generates incremental contractions within a crosslinked filamentous network. An enlarged view of the actin filament (bottom image) reveals its segmented structure. The size *l* of the subunits determines the magnitude of the relative node displacements due to contraction events of myosin sliding. *l* is thus taken to be the step size of anti-correlated kicks in our model.

Cartoon of the tensegrity structure composed of collapsed and stretched elements. In a crosslinked network of filaments, active filament sliding is stabilized by passive crosslinking. A tensegrity structure is formed once a global balance between local contraction and neighboring bond stretching is achieved. An initially homogeneous network then develops into dense floppy clumps (concentrated green wiggly lines) connected by highly stretched filaments (long red straight lines).

Cartoon of the tensegrity structure composed of collapsed and stretched elements. In a crosslinked network of filaments, active filament sliding is stabilized by passive crosslinking. A tensegrity structure is formed once a global balance between local contraction and neighboring bond stretching is achieved. An initially homogeneous network then develops into dense floppy clumps (concentrated green wiggly lines) connected by highly stretched filaments (long red straight lines).

Profile of the modified interaction given by the pair-level steady-state solution. We plot the effective interaction *U* _{eff} (Eq. (21)) scaled by effective temperature (Eq. (22)) for various motor activity (Δ) and susceptibility (*s*). At sufficiently high activity, load-resisting (*s* = −0.3) motors may yield a long-range effective repulsion, an energy barrier (indicated by a red arrow in panel d) thus appears at intermediate distances, suggesting the tendency for node separation and thus the bond stretching that occurs in aster formation. (a) and (c): *s* = 1, 0.5, 0, −0.2, and −0.3. (b) and (d): Δ = 0, 0.5, 1, 2, and 4. Common parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, and *P* _{ a } = 1.

Profile of the modified interaction given by the pair-level steady-state solution. We plot the effective interaction *U* _{eff} (Eq. (21)) scaled by effective temperature (Eq. (22)) for various motor activity (Δ) and susceptibility (*s*). At sufficiently high activity, load-resisting (*s* = −0.3) motors may yield a long-range effective repulsion, an energy barrier (indicated by a red arrow in panel d) thus appears at intermediate distances, suggesting the tendency for node separation and thus the bond stretching that occurs in aster formation. (a) and (c): *s* = 1, 0.5, 0, −0.2, and −0.3. (b) and (d): Δ = 0, 0.5, 1, 2, and 4. Common parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, and *P* _{ a } = 1.

Profile of the effective pair interaction (Eq. (26)) obtained by the self-consistent phonon calculation. (a) Susceptible motors (*s* = 1) with increasing activity Δ enhance the long-range attraction and strengthen the short-range effective attraction. (b) A zoom-in view of the small-*R* region close to the elasticity onset (dashed line) in panel (a) showing the absence of kink or inflection point in the potential profile. (c) Adamant motors (*s* = −0.3) weaken the long-range attraction. No stable α solution is found if the motor activity gets too high (Δ > 1). (d) Varying motor susceptibility does not affect the short-range effective attraction (inset), but increasingly susceptible motors (bottom to top) lead to a stronger long-range attraction. Common parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, *P* _{ a } = 1.

Profile of the effective pair interaction (Eq. (26)) obtained by the self-consistent phonon calculation. (a) Susceptible motors (*s* = 1) with increasing activity Δ enhance the long-range attraction and strengthen the short-range effective attraction. (b) A zoom-in view of the small-*R* region close to the elasticity onset (dashed line) in panel (a) showing the absence of kink or inflection point in the potential profile. (c) Adamant motors (*s* = −0.3) weaken the long-range attraction. No stable α solution is found if the motor activity gets too high (Δ > 1). (d) Varying motor susceptibility does not affect the short-range effective attraction (inset), but increasingly susceptible motors (bottom to top) lead to a stronger long-range attraction. Common parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, *P* _{ a } = 1.

Testing the validity of the effective equilibrium approximation: a comparison of three simulation schemes. Statistical characteristics and steady-state structures for a partially and randomly connected (*P* _{ c } = 0.4) network built on a simple cubic lattice driven by small-step (*l* = 0.03) susceptible (*s* = 1) motors are shown. (a) The potential energy; (b) the fraction of taut bonds; (c) the mean square node displacement; (d) main: the pair distribution function (PDF) averaged over a wide steady-state time window; inset: the aggregation strength, which is the height of the innermost peak of the PDF, versus simulation time; (e) initial (left) and steady-state (right) node configurations (upper row) and corresponding bond structures (lower row). The parameters chosen for illustration are *L* _{ e } = 1.2, βγ = 5, *P* _{ a } = 1, and κ = 1.

Testing the validity of the effective equilibrium approximation: a comparison of three simulation schemes. Statistical characteristics and steady-state structures for a partially and randomly connected (*P* _{ c } = 0.4) network built on a simple cubic lattice driven by small-step (*l* = 0.03) susceptible (*s* = 1) motors are shown. (a) The potential energy; (b) the fraction of taut bonds; (c) the mean square node displacement; (d) main: the pair distribution function (PDF) averaged over a wide steady-state time window; inset: the aggregation strength, which is the height of the innermost peak of the PDF, versus simulation time; (e) initial (left) and steady-state (right) node configurations (upper row) and corresponding bond structures (lower row). The parameters chosen for illustration are *L* _{ e } = 1.2, βγ = 5, *P* _{ a } = 1, and κ = 1.

The dependence of network tenseness and structure on motor concentration (*P* _{ a }) obtained by Monte Carlo simulations. The parameters were chosen such that the system is in the regime of arrested phase separation. (a) The fraction of taut bonds decreases as *P* _{ a } increases. A kink located around *P* _{ a } = 0.7 separates two descending branches: (I) *P* _{ a } = 0.5–0.7 and (II) *P* _{ a } = 0.8–1. (b) The aggregation strength exhibits a sharp peak at *P* _{ a } = 0.7. The error bars in (a) and (b) depict standard deviations from averages over a steady-state time window of 4 × 10^{6} Monte Carlo steps. (c) Bond structures and corresponding node configurations at various *P* _{ a } values are shown, from top to bottom *P* _{ a } = 0.2, 0.5, 0.6, 0.7, and 1. The arrow indicates the bond structure with the strongest aggregation. The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, *l* = 0.05, *s* = 1, and κ = 0.1.

The dependence of network tenseness and structure on motor concentration (*P* _{ a }) obtained by Monte Carlo simulations. The parameters were chosen such that the system is in the regime of arrested phase separation. (a) The fraction of taut bonds decreases as *P* _{ a } increases. A kink located around *P* _{ a } = 0.7 separates two descending branches: (I) *P* _{ a } = 0.5–0.7 and (II) *P* _{ a } = 0.8–1. (b) The aggregation strength exhibits a sharp peak at *P* _{ a } = 0.7. The error bars in (a) and (b) depict standard deviations from averages over a steady-state time window of 4 × 10^{6} Monte Carlo steps. (c) Bond structures and corresponding node configurations at various *P* _{ a } values are shown, from top to bottom *P* _{ a } = 0.2, 0.5, 0.6, 0.7, and 1. The arrow indicates the bond structure with the strongest aggregation. The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, *l* = 0.05, *s* = 1, and κ = 0.1.

An illustration of the motor-induced effective attraction for a non-percolating network (*P* _{ c } = 0.2) at various motor susceptibilities. (a) The potential energy; (b) the aggregation strength; (c) initial (upper left) and later node configurations and bond structures for a control run with pure thermal motion (upper right), and for motorized systems with *s* = 1 (lower left), *s* = 0 (lower middle), and *s* = −0.5 (lower right). Despite having different dynamics, similar steady-state structures with isolated floppy clumps are reached in each case, regardless of the motor susceptibility. The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ a } = 1, *l* = 0.03, and κ = 0.1.

An illustration of the motor-induced effective attraction for a non-percolating network (*P* _{ c } = 0.2) at various motor susceptibilities. (a) The potential energy; (b) the aggregation strength; (c) initial (upper left) and later node configurations and bond structures for a control run with pure thermal motion (upper right), and for motorized systems with *s* = 1 (lower left), *s* = 0 (lower middle), and *s* = −0.5 (lower right). Despite having different dynamics, similar steady-state structures with isolated floppy clumps are reached in each case, regardless of the motor susceptibility. The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ a } = 1, *l* = 0.03, and κ = 0.1.

An illustration of the motor-induced effective repulsion caused by load-resisting (*s* = −0.5) motors for various network connectivities and motor kicking rates. (a) Initial (upper) and steady-state (lower) bond structures are shown at low (left: *P* _{ c } = 0.2) and at high (right: *P* _{ c } = 0.6) connectivity with κ = 1. (b) Node configurations (upper) and corresponding bond structures (lower) at various motor kicking rates (left to right: κ = 0.1, 0.5, and 1) with *P* _{ c } = 0.4. The common set of simulation parameters are given by *L* _{ e } = 1.2, βγ = 5, *P* _{ a } = 1, and *l* = 0.05.

An illustration of the motor-induced effective repulsion caused by load-resisting (*s* = −0.5) motors for various network connectivities and motor kicking rates. (a) Initial (upper) and steady-state (lower) bond structures are shown at low (left: *P* _{ c } = 0.2) and at high (right: *P* _{ c } = 0.6) connectivity with κ = 1. (b) Node configurations (upper) and corresponding bond structures (lower) at various motor kicking rates (left to right: κ = 0.1, 0.5, and 1) with *P* _{ c } = 0.4. The common set of simulation parameters are given by *L* _{ e } = 1.2, βγ = 5, *P* _{ a } = 1, and *l* = 0.05.

Mean-field predictions of the effect of the concentration (*P* _{ a }) of load-resisting (*s* = −0.5) motors on long-range interactions. (a) The localization strength α of individual nodes. The localization at large separation *R* is considerably suppressed as *P* _{ a } increases. (b) The effective potential. Increasing *P* _{ a } weakens the long-range attraction; the potential profile actually flattens out (red arrow) at *P* _{ a } = 0.8 indicating a vanishing restoring force. (c) The overall tension (−*p*) vanishes at large *R* (red arrow) for high *P* _{ a }. This suggests the tendency for contraction is counterbalanced by a motor-induced long-range repulsion. The simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, and Δ = 1.

Mean-field predictions of the effect of the concentration (*P* _{ a }) of load-resisting (*s* = −0.5) motors on long-range interactions. (a) The localization strength α of individual nodes. The localization at large separation *R* is considerably suppressed as *P* _{ a } increases. (b) The effective potential. Increasing *P* _{ a } weakens the long-range attraction; the potential profile actually flattens out (red arrow) at *P* _{ a } = 0.8 indicating a vanishing restoring force. (c) The overall tension (−*p*) vanishes at large *R* (red arrow) for high *P* _{ a }. This suggests the tendency for contraction is counterbalanced by a motor-induced long-range repulsion. The simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, and Δ = 1.

The effect of motor activity (Δ) and susceptibility (*s*) on phase separation. Shown are the calculated localization strength α (upper row) and the tension (−*p*) (bottom row) as a function of the mean separation *R* for (a) *s* = 1 with various motor activities, Δ = 0, 0.5, 1, 2, and 4 (bottom to top), and (b) Δ = 1 with various motor susceptibilities, *s* = −0.3, −0.2, 0, 0.5, and 1 (bottom to top). The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, and *P* _{ a } = 1.

The effect of motor activity (Δ) and susceptibility (*s*) on phase separation. Shown are the calculated localization strength α (upper row) and the tension (−*p*) (bottom row) as a function of the mean separation *R* for (a) *s* = 1 with various motor activities, Δ = 0, 0.5, 1, 2, and 4 (bottom to top), and (b) Δ = 1 with various motor susceptibilities, *s* = −0.3, −0.2, 0, 0.5, and 1 (bottom to top). The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, and *P* _{ a } = 1.

The role of motor kicking rate and effective attraction in aggregation. (a)–(c) Statistical measures for the dynamic and structural development at various motor kicking rates. Steady-state bond structures (upper) and node configurations (lower) are shown at increasing motor kicking rates (d): from left to right κ = 0.1, 0.2, 0.5, and 1 (converted into *T* _{eff}/*T*), and for corresponding models without motor-induced short-range attraction at κ = 1 (e). The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, *P* _{ a } = 1, *s* = 1, and *l* = 0.03.

The role of motor kicking rate and effective attraction in aggregation. (a)–(c) Statistical measures for the dynamic and structural development at various motor kicking rates. Steady-state bond structures (upper) and node configurations (lower) are shown at increasing motor kicking rates (d): from left to right κ = 0.1, 0.2, 0.5, and 1 (converted into *T* _{eff}/*T*), and for corresponding models without motor-induced short-range attraction at κ = 1 (e). The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, *P* _{ c } = 0.4, *P* _{ a } = 1, *s* = 1, and *l* = 0.03.

The stability diagram at various motor susceptibilities. The colored lines represent the stability boundaries, solid red for *s* = 1, dashed grey for *s* = 0, and dotted blue for *s* = −0.5. Below the stability boundaries the pressure exhibits a non-monotonic dependence on particle separation indicating the tendency toward phase separation. The instability region (shaded area) extends to lower *P* _{ c } and higher *P* _{ a } as *s* increases, suggesting that susceptible motors promote phase separation. The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, and Δ = 1.

The stability diagram at various motor susceptibilities. The colored lines represent the stability boundaries, solid red for *s* = 1, dashed grey for *s* = 0, and dotted blue for *s* = −0.5. Below the stability boundaries the pressure exhibits a non-monotonic dependence on particle separation indicating the tendency toward phase separation. The instability region (shaded area) extends to lower *P* _{ c } and higher *P* _{ a } as *s* increases, suggesting that susceptible motors promote phase separation. The remaining simulation parameters are *L* _{ e } = 1.2, βγ = 5, and Δ = 1.

Patterns of behavior for susceptible (*s* > 0) and load-resisting (*s* < 0) motors at a high motor concentration. Typical structures generated by simulations are shown for each situation. The horizontal axis indicates increasing network connectivity from left to right. The vertical line locates the percolation threshold. (a) For susceptible motors, effective Brownian dynamics simulations and Monte Carlo simulations give similar results. At intermediate connectivity above the percolation threshold, arrested phase separation occurs. (b) For load-resisting motors, (anti-) correlation plays a key role in the active patterning. Macroscopic contraction occurs only in the presence of anti-correlation in motion, otherwise connected asters form that cannot collapse.

Patterns of behavior for susceptible (*s* > 0) and load-resisting (*s* < 0) motors at a high motor concentration. Typical structures generated by simulations are shown for each situation. The horizontal axis indicates increasing network connectivity from left to right. The vertical line locates the percolation threshold. (a) For susceptible motors, effective Brownian dynamics simulations and Monte Carlo simulations give similar results. At intermediate connectivity above the percolation threshold, arrested phase separation occurs. (b) For load-resisting motors, (anti-) correlation plays a key role in the active patterning. Macroscopic contraction occurs only in the presence of anti-correlation in motion, otherwise connected asters form that cannot collapse.

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