Actin-Myosin Spatial Patterns from a Simplified Isotropic Viscoelastic Model

Abstract

F-actin networks are involved in cell mechanical processes ranging from motility to endocytosis. The mesoscale architecture of assemblies of individual F-actin polymers that gives rise to micrometer-scale rheological properties is poorly understood, despite numerous in vivo and vitro studies. In vitro networks have been shown to organize into spatial patterns when spatially confined, including dense spherical shells inside spherical emulsion droplets. Here we develop a simplified model of an isotropic, compressible, viscoelastic material continually assembling and disassembling. We demonstrate that spherical shells emerge naturally when the strain relaxation rate (corresponding to internal network reorganization) is slower than the disassembly rate (corresponding to F-actin depolymerization). These patterns are consistent with recent experiments, including a collapse of shells to a central high-density focus of F-actin when either assembly or disassembly is reduced with drugs. Our results demonstrate how complex spatio-temporal patterns can emerge without spatially distributed force generation, polar alignment of F-actin polymers, or spatially nonuniform regulation of F-actin by upstream biochemical networks.

Introduction

Actin forms polymers and dynamic networks in most eukaryotic cells (1). These networks are organized with the help of ∼130 accessory proteins (2) and have diverse architecture, each tuned for different cell functions including cell motility, division, mechanosensing, endocytosis, receptor-ligand binding, and maintaining cell coherence.

As a key participant in these cellular processes, and a primary determinant of overall cell mechanics (3), the mechanical properties of actin networks have been studied extensively. At the level of individual F-actin polymers, assembly is initiated by nucleators such as Arp2/3, which are activated by membrane-bound nucleation promoting factors such as WASp. Disassembly occurs both spontaneously and under regulation by depolymerization factors such as cofilin. Higher-order actin structures have been studied in vitro (4–7), where actin exhibits viscoelastic behavior. Unsurprisingly, given the divergent behaviors seen for in vivo actin networks, in vitro studies lead to estimates of parameters ranging over orders of magnitude, depending on polymerization setup, accessory molecules, and experimental timescale. For example, reported elastic moduli range in 0.8–30 Pa (4,6,8), small compared to reports of elastic moduli of live cells, which range in 20–40 Pa·s (3,9–12). The network’s internal viscosity also exhibits a large range, 10–100 Pa·s in vitro (13) to 100–700 Pa·s in live cells (3,9,11,12). In general, a viscoelastic material may not be easily characterized by a single well-defined elastic modulus and viscosity (7).

A major question is, therefore, what are the rheological properties of actin in vivo? In particular, what are the relative strengths of elastic stresses, effective viscosity, and myosin contractility? These properties are often necessary in quantitative models of cell processes (14–17), which would benefit from a mathematically simple model of actin-myosin rheology that nevertheless captures key phenomena.

It has been emphasized that acto-myosin constitutes an active, polar, viscoelastic gel (18,19). Although the bulk networks are energetically active and, microscopically, individual filaments are polar, it is unclear whether energetic activity and polarity are essential for explaining nontrivial spatial and temporal patterns observed at cellular scales.

Recently, Pinot et al. (20) reconstituted a dynamic, flowing F-actin network confined in emulsion droplets, shown in Fig. 1 A. The interior of the cell-sized droplets contains extract from meiotic Xenopus cells and thus ostensibly represents a full array of in vivo actin regulatory protein. In addition, the droplet boundary mimics the cell surface, e.g., recruiting actin nucleation promoting factors. Two curious observations arise from their study:

• 1.The highest density of actin is neither at the periphery nor the center, but at a concentric sphere, shown in Fig. 1 A, which the authors call “rings” and we refer to as “halos”; and
• 2.By using drugs that either reduce actin depolymerization (phalloidin) or reduce polymerization (cytochalasin), the halo collapses to a single maximum near the droplet center.

This biomimetic system provides a balance between live cell experiments (3,12) in which competing variables such as the cortex, the nucleus, and the larger, actively signaling organelles confound interpretation of experiments, and the in vitro reconstitution (4–7) that may lack physiological significance.

Figure 1.

F-actin in biomimetic droplets exhibits maximum density at an internal shell. (A) Labeled F-actin in the cytoplasm extracted from meiotic Xenopus cells inside emulsion droplets, taken from Pinot et al. (20). (B) Line scan of F-actin intensity (red line in panel A). Background fluorescence was subtracted from the fluorescent intensity and normalized to the intensity at the droplet boundary. To see this figure in color, go online.

In this work, we propose a simple model and show that the halos and other experimentally observed behaviors can be understood as an interplay among four processes within the network: polymerization, compression, reorganization, and disassembly. Our simple model provides insights of relevance to acto-myosin dynamics in general: We find viscous, elastic, and active contractile stresses have similar magnitudes, although only viscous and elastic stresses are necessary for explaining these halos. Our model adds to a growing body of evidence (21,22) that nontrivial patterns can be generated in simple settings, in this case without invoking polarity or distributed myosin activity.

Model

We develop a continuum model that includes the processes depicted in Fig. 2 A: Assembly at the boundary, disassembly throughout the network, network reorganization, and compressional stress due to transient strain. We also include myosin contractility and distributed assembly (not depicted). In the Supporting Material, we provide further mathematical details.

Figure 2.

Compressible, viscoelastic model demonstrates F-actin interior shell pattern. (A) Schematic of model. F-actin filaments (green) assemble from G-actin monomers, primarily at the droplet edge. F-actin forms a compressible network. Compression leads to strain, which is eventually relaxed with timescale λ as crosslinkers (orange) are recycled. F-actin disassembly occurs throughout the network. (B) F-actin profile (red) and velocity profile (blue) predicted by the model with Γ = 10, V₀ = 1.5, α = 10⁻², M = 1, and B = 1. To see this figure in color, go online.

The density of F-actin, a, with units of μM, obeys the evolution equation

$$\frac{\partial a}{\partial t}+\nabla ·\left(\overrightarrow{v}a\right)=-\gamma a+\beta .$$ (1)

The left-hand side of the equation corresponds to flow of the F-actin structure due to internal stresses, considered below. The first term on the right-hand side corresponds to depolymerization. The final term represents polymerization in the bulk, which we assume is negligible (β = 0) for wild-type extract, but can be experimentally induced by the addition of soluble nucleation promoting factors (see Results).

Inertia is negligible and forces must balance, therefore ∇ · σ = 0, where σ is the total stress with units Pa, which is expressed as

$$\mathit{\sigma }=\left(\eta \left(\nabla \overrightarrow{v}+\nabla {\overrightarrow{v}}^T\right)+{\eta }^{\text{′}}\left(\nabla ·\overrightarrow{v}\mathbb{I}\right)\right)-G\left(e-1\right)\mathbb{I}+{\sigma }_M\mathbb{I}.$$ (2)

The three terms above correspond to the viscous, elastic, and active contractile stresses, respectively. The parameter η is the shear viscosity and η′ is the second coefficient of viscosity, both with units of Pa·s, although in spherically symmetric droplets only η impacts our results (see the Supporting Material). G is the bulk elastic modulus with units of Pa, e is the network volumetric strain, and σ_M is the strength of active stresses driven by myosin. Both the elastic and contractile stresses are assumed to be isotropic, and are therefore multiples of the identity tensor $\mathbb{I}$, an assumption we justify in the Supporting Material.

Because the dynamics of F-actin are on a timescale of minutes (R₀/v₀ ∼ 500 s), we assume that the forces due to the cytosolic fluid surrounding the F-actin network (which contains solutes including G-actin) are negligible. This allows us to avoid characterizing a full, two-phase model (23) in which the cytosol experiences the F-actin network as a porous medium with timescale ≈26 s (24). In the Supporting Material, we derive the equation for volumetric strain, e, as

$$\frac{\partial e}{\partial t}+\nabla ·\left(\overrightarrow{v}e\right)=-\lambda \left(e-1\right).$$ (3)

The left-hand side of this equation represents transport of local strain by the velocity of the actin network. The right-hand side represents strain relaxation. The underlying assumption is that the network reorganizes to an unstrained configuration (e = 1), and λ is the rate of the associated relaxation of strain within the network. Microscopically, we hypothesize that both strain relaxation (at rate λ) and viscosity (η) are due to the turnover of actin crosslinkers and untangling of filaments. However, even though they involve the same microscopic participants, they are distinct. In our macroscopic model, these processes are distinct because viscous stresses only arise when filaments are moved relative to each other, whereas strain relaxation will occur even if the material is stationary. We use the convention that e > 1 corresponds to compression. We assume the elastic modulus, G, scales linearly with actin density:

$$G\left(a\right)=g·a/a_0.$$ (4)

In Results and Fig. S1 in the Supporting Material, we explore more of the general dependencies between elastic modulus and local F-actin density, including superlinear scaling (4,6,25). We find that all our results are qualitatively the same for superlinear scaling.

The strength of active stresses driven by myosin is

$${\sigma }_M=ma.$$ (5)

This form assumes myosin contractility is isotropic and proportional to F-actin density (26,27).

Polymerization is observed near the boundary. For simplicity, we assume F-actin is created at constant density a₀ in an unstrained state (see the Supporting Material), leading to boundary conditions

$$\overrightarrow{v}\left(r_0\right)=-v_0\hat{n},$$ (6)

$$a\left(r_0\right)=a_0,$$ (7)

$$e\left(r_0\right)=1,$$ (8)

where v₀ is the F-actin polymerization velocity and $\hat{n}$ is a unit vector normal to the boundary. To test sensitivity to the assumption of strain-free assembly, we varied strain near the boundary e(r₀) = 0.1 and e(r₀) = 10, and found negligible effect on flow velocities or the shape of F-actin density profile.

Equations 1–3 with the boundary condition equations, Eqs. 6–8, constitute a system of nonlinear, coupled partial differential equations. To facilitate their numerical solution and intuitive understanding, we perform nondimensionalization (rescaling) by choosing the droplet radius r₀, strain relaxation time 1/λ, and reference density a₀ as characteristic scales for space, time, and actin density, respectively. We report all findings in terms of A = a/a₀ and V = v/λr₀, the scaled actin density and velocity, respectively. There are five nondimensional parameters, listed in Table 1. We obtain nondimensional and radially symmetric versions of Eqs. 1–3. To numerically solve these, at every timestep we first use Eq. 2 to solve for the velocity field, and then update actin density and strain fields with Eqs. 1 and 3. Explicit equations and details of numerical methods are presented in the Supporting Material.

Table 1.

Nondimensional parameters

Symbol	Definition	Meaning
α	g/ηλ	Ratio of elastic to viscous stresses
Γ	γ/λ	Ratio of network decay rate to strain relaxation rate
V0	v0/λr0	Ratio of boundary polymerization rate to strain relaxation rate
M	ma0/ηλ	Ratio of active contractile stresses to viscous stresses
B	β/a0λ	Ratio of bulk polymerization rate to strain relaxation rate

Results

Isotropic model with stress reproduces halo pattern

We numerically solve the dynamic equations until they reach a steady-state pattern, shown in Fig. 2 B. Because of droplet-to-droplet variability observed by Pinot et al. (20), our goal is not precise quantitative agreement between our model and any single experimental profiles, but instead to elucidate the mechanical processes that underlie the observed behavior. We are able to reproduce approximate peak location and maximal density and central density relative to edge density, as well as the general character of the profile.

Halo can be understood as competition of F-actin decay and compression resistance

To gain a conceptual understanding of our model’s implications, we found numerical solutions over a range of three parameters, α, Γ, and V₀, corresponding to elastic resistance, disassembly, and assembly, respectively, with B = M = 0 for simplicity. The resulting patterns, shown in Fig. 3, can be understood as follows:

Figure 3.

(A–C) Location of maximal F-actin density depends on model parameters α, Γ, and V₀. For simplicity, we set B = M = 0. For α = 0 (A), F-actin profile either increases toward the middle if V₀ > 3Γ (i), or decays if V₀ < 3Γ (ii). For nonzero α (B), a nontrivial peak occurs when V₀ > 3Γ (iii). From this parameter set, if either Γ or V₀ are reduced ((iv) and (v), respectively), the peak moves toward the center. (vi–viii) Correspond to parameters shown in Fig. 4. To see this figure in color, go online.

For networks that are completely inelastic (α = 0; Fig. 3 A), the formation of actin halos is not possible. The F-actin profile is determined entirely by a competition between assembly and disassembly. If V₀ > 3Γ (Fig. 3 (i)), polymerization dominates, the network is driven inward, and actin concentration increases toward the middle. If V₀ < 3Γ (Fig. 3 (ii)), network depolymerization dominates and the actin profile decays toward the center. The factor of 3 is a geometric constant that arises in spherical coordinates.

For elastic networks (α ≠ 0; Fig. 3, B and C), a nontrivial peak generally occurs when V₀ > 3Γ and Γ > 1 (Fig. 3 (iii)). To elucidate the nature of this peak, we explore the stress profile for three parameter sets with only Γ varying (profiles in Fig. 4 correspond to parameters (vi–viii) in Fig. 3).

Figure 4.

(A) Elastic resistance leads to domain separation of primarily viscous stress (orange) near the edge and primarily elastic stress near the center (green), computed using the first and second terms in Eq. 2. Total stress (black) is constant throughout the material, as required by force-balance. (B) The switch from viscous to elastic stress corresponds to the peak in F-actin (vi). This peak results from a competition between strain decay (relaxation) and material decay (disassembly): if Γ > 1 (λ < γ) as in (vi), F-actin decays toward the center, resulting in an F-actin density peak. If Γ < 1 (λ > γ) as in (viii), strain decays toward the center, which is compensated by increasing F-actin density. To see this figure in color, go online.

We find the material separates into two domains: in the outer domain, viscous stresses dominate, whereas in the central domain, elastic stresses dominate. In this central domain, because viscous stresses are small, the elastic stresses must balance either by having constant material (Fig. 4 (vii)), higher density but lower strain (Fig. 4 (viii)), or higher strain but lower density (Fig. 4 (vi)). Which case prevails is decided by a competition between strain decay (relaxation) and material decay (depolymerization), and is determined solely by Γ = γ/λ, the ratio of two decay rates. The latter case, with faster material decay, leads to the peak reported experimentally.

Four experimental perturbations validate model

A counterintuitive result reported by Pinot et al. (20) is that drugs that either reduce actin polymerization (cytochalasin) or reduce depolymerization (phalloidin) result in shell collapse to a single maximum near the droplet center. Our model reproduces this behavior: From the parameter set shown in Fig. 3 (iii), if either Γ or V₀ are reduced (Fig. 3, (iv) and (v), respectively), the peak moves toward the center. Qualitatively, we can understand this behavior as two separate effects. Velocity reduction delays the actin buildup, allowing it to move closer to the center before elastic stresses accumulate. When the depolymerization rate is reduced sufficiently below the strain decay rate, the network relieves elastic stress by reorganization rather than depolymerization, leading to a higher concentration of actin in the center.

Moreover, our model reproduces two other experimental perturbations. When myosin is inhibited (using blebbistatin), the flow velocity is reduced by ∼30%. We varied the myosin contractility parameter M to find what value perturbs the velocity profile by this amount, shown in Fig. 5 A. We find that from M = 1 to M = 0, the velocity profile changes by approximately one-third, giving a rough order-of-magnitude estimate that M ∼ 1. If M > 10, the flow velocity reverses sign inside the shell. This reversal is driven by a positive feedback loop built into our model: Contractility tends to concentrate actin, and higher concentrations of actin lead to higher concentrations of myosin, furthering contraction. Our model thus predicts that large enough myosin contractility will reverse the flow inside the ring, directing it outward.

Figure 5.

(A) Myosin contractility increases flow velocity, until myosin is strong enough to reverse the flow. Numerical solutions computed for M = 0, 1, 2, 5 (blue solid lines) and M = 10 (dashed line). Note in this plot, V > 0 corresponds to inward flow, and V < 0 is outward flow. The resulting F-actin density profiles (not shown) become increasingly more sharply peaked. (B) Distributed F-actin assembly leads to an approximately uniform density profile. To see this figure in color, go online.

Pinot et al. (20) engineered a diffusible actin nucleation promoting factor called Scar-WA. The addition of such distributed F-actin assembly led to a uniform density of F-actin, with apparent lower density near the edges (although this may be due to microscopy effects). Our model with distributed assembly β > 0 → B > 0 recapitulates this result, shown in Fig. 5 B.

Model is robust to elastic modulus scaling with F-actin density

Measurements of in vitro F-actin networks have reported the scaling between elastic modulus and F-actin density as G ∼ a^7/5 (4) and G ∼ a^11/5 (25). To test whether our results are robust to the scaling law, we explore dependencies that are more general between elastic modulus and local F-actin density, including superlinear scaling G ∼ a^7/5. We find that all our results are qualitatively the same for superlinear scaling. To further explore the importance of the density dependence, we simulated constant G (independent of actin density a) and were unable to reproduce assembly/disassembly perturbation experiments.

Dynamics without spherical symmetry and merging droplets

The viscoelastic model we propose can be generalized to higher dimensions without imposing spherical symmetry. To explore the behavior without imposed symmetry, we solve the model equations in a two-dimensional square with F-actin assembly on the four straight edges. We find a large regime of parameters result in steady states with halo structures. An example steady state for α = 0.1, Γ = 2.5, V₀ = 1, and M = B = 0 is shown in Fig. 6 and Movie S1 in the Supporting Material. Fourfold symmetry in the halo intensity is an artifact of our square boundary.

Figure 6.

F-actin density from a two-dimensional simulation of actin confined inside a square with assembly on the outer edges. Time sequence of F-actin density profiles as two steady-state halos are placed in a larger square domain. The final frame shows steady state. Time is in scaled units of the strain relaxation time λ⁻¹. Fourfold symmetry is an artifact of the square boundary. To see this figure in color, go online.

When two droplets, each containing an F-actin halo, contact each other, surface tension induces a rapid merger into a single oil droplet. After the droplet merging, the F-actin halos merge into a single halo with a merging timescale of minutes. We explore the dynamics predicted by our viscoelastic model during network merging. To separate the physics of the droplet merger, which is beyond the scope of this work, with the physics of F-actin network merger, we simulate in rectangular geometry. We first find the steady state of an individual network confined to a square with edge length r₀/2 at parameters that produce halo patterns. We then simulate a larger square with edge length r₀ and use as initial condition two identical copies of the smaller halo, show in Fig. 6, t = 0. The resulting merger is shown in Movie S2.

These simulations demonstrate the following:

• 1.Our results qualitatively carry over to different dimensions and symmetry assumptions; e.g., we did not detect instabilities in circumferential directions, and
• 2.The halos are not metastable, and merge into a single halo on a timescale comparable to the timescale of halo merger reported experimentally.

Discussion

F-actin networks are complex materials, exhibiting phenomena depending on at least five properties:

• 1.They are viscoelastic (10);
• 2.They continually assemble and disassemble;
• 3.Both polymerization and depolymerization generate force throughout the network (28);
• 4.In general, the polarity of individual polymers can lead to anisotropic network behavior; and
• 5.The network is under spatially nonuniform regulation by an upstream biochemical networks (29).

The primary result of this work is that nontrivial spatial patterns can arise only from properties 1 and 2. These critical features are consistent with a large class of constitutive laws, and are not limited to our specific functional form. For example, polarity (18,27) and more-complex viscous and elastic constitutive laws may be included; but with our four basic ingredients, we expect halo formation.

We observe the emergence of halo patterns, in which the maximal F-actin density is neither at the boundary nor in a single focus, but rather in a spherical shell. The phase diagrams in Fig. 3 lead to a conceptual description of halo formation under two conditions:

• 1.If assembly dominates over disassembly, the network must become more dense as it flows inward from the boundary, thus the maximal density will not be at the boundary; and
• 2.A competition arises between the timescale of material strain decay (the viscoelastic timescale) and material decay (F-actin depolymerization).

If material decay is faster than material strain decay, the network depolymerizes at the center but strain increases. In Fig. S1, we show that the dimensionality and precise geometry are unimportant by solving the equations in a one-dimensional strip, where we find qualitatively similar behavior. The phase diagrams also recapitulate the counterintuitive observation that the halo pattern collapses to a central focus under either cytochalasin, which decreases F-actin assembly, or phalloidin, which decreases disassembly. Conceptually, the cytochalasin collapse results from a competition between assembly and disassembly, whereas the phalloidin collapse results from competition between disassembly and strain decay.

Although flow velocities and network densities can be observed directly in experiment, internal stress and strain remains difficult to measure in bulk content of cells and in vitro networks. Thus, one benefit of mathematical models is their ability to relate observable network properties to these unobservable properties. Here, the major conclusion from our model is that the internal stress in the F-actin network in Pinot et al. (20) is approximately evenly shared between elastic stress, viscous stress, and myosin contractile stress. These stresses are, however, not uniformly shared in space. Instead, we predict the stress is predominantly viscous in an outer region (outside the halo) and predominantly elastic in an inner region (within the halo), as shown in Fig. 4.

The simplicity of our nondimensional model equations allows description of spatial patterns and the effect of drug perturbations without specifying precise values of rheological parameters, for which quantitative values remain elusive (see Introduction). However, connecting qualitative features of halos with our model allows for inference of parameters. The strain relaxation rate λ is related to the reorganization timescale of the F-actin network, itself related to the kinetics of F-actin cross-linking proteins such as filamin and fascin (6–8). This timescale also determines the characteristic scale above which the network may be assumed to be approximately fluidlike, and below which the network is primarily elastic, and has been estimated to be 0.1–10 s in cells and 10–100 s in vitro when crosslinked by ADP-myosin (13). Our model suggests that the network in Pinot et al. (20) has a strain relaxation time slightly below the disassembly time γ, which is straightforward to observe experimentally, for example with fluorescence recovery after photobleaching (30).

Nonetheless, we expect our nondimensional parameters to be consistent with the range of reported (physical) parameters. The behavior we report occurs over a range of nondimensional parameters, including α ∼ 10⁻¹, Γ ∼ 10, and V₀ ∼ 10⁻¹. Among many possibilities, these parameters are consistent with λ ∼ 0.1 s⁻¹, γ ∼ 1 s⁻¹ (see previous paragraph), η ∼ 10³ Pa·s (3,9,11,12), and G ranging from 10 Pa at the boundary to 100 Pa at maximal density (3,9–12), assuming R₀ = 25 μm and v₀ = 250 nm/s.

We have developed a simple continuum model of a compressible viscoelastic material with assembly/disassembly. The material we describe is similar to Oldroyd-B and upper-convected Maxwell materials because it has fluid behavior at long timescales, and an effective memory at intermediate timescales. These models have been extended to allow for compressibility (31–33). Our model equations are simple enough to allow for efficient numerical solution over a large parameter regime: in the case of no myosin or distributed assembly, the model has three nondimensional parameters that we characterize completely (Fig. 3). In addition, it is straightforward to add more complex assembly and disassembly kinetics by modifying Eq. 1.

Relatively recent experiments (34,35) and theories (28,36) have demonstrated that disassembly can generate stress, including at the rear of motile cells. This could be included in our modeling framework as a term in the stress equation, e.g., σ = +E(a – a₁), a > a₁, where a₁ is the preferred density and the positive sign indicates a contractile stress. Full exploration of disassembly stress is beyond the scope of this work. However, we speculate it would not be sufficient to generate halos because the stress would decay at the F-actin disassembly rate; we find that halos emerge only when the material disassembly rate and stress decay rate are different.

Previous experiments (4,37,38) demonstrate that stabilized (depolymerization inhibited) F-actin confined in droplets or vesicles form rings when the diameter of the ring was below the persistence length of the filaments (∼15 μm). We hypothesize that these patterns emerge from distinct mechanisms from the halos reported by Pinot et al. (20), because of the following:

• 1.The oil droplets are much larger, with diameters ∼100 μm, sizes at which rings were not observed in previous experiments (4,37,38);
• 2.The halos are consistently not at the confinement boundary in Pinot et al. (20), whereas most rings occurred at the confinement boundary (with exceptions (38)); and
• 3.The phalloidin and cytochalasin-D experiments in Pinot et al. (20) demonstrate these halos depend on continual, unperturbed assembly and disassembly.

Our model makes several testable predictions, including the flow profiles shown Fig. 2 (blue dashed curve) and how they vary under experimental perturbations. By increasing myosin contractility strength, we predict that above a threshold strength, the F-actin peak density (i.e., the halo’s location in the droplet) will move out, and at this myosin strength, the flow inside the halo will reverse direction. This prediction could be tested with drugs that increase myosin contractility, e.g., calyculin A.

Supporting Citations

References (39,40) appear in the Supporting Material.