Emergent spatiotemporal dynamics of the actomyosin network in the presence of chemical gradients

Abstract

We used particle-based computer simulations to study the emergent properties of the actomyosin cytoskeleton. Our model accounted for biophysical interactions between filamentous actin and non-muscle myosin II and was motivated by recent experiments demonstrating that spatial regulation of myosin activity is required for fibroblasts responding to spatial gradients of platelet derived growth factor (PDGF) to undergo chemotaxis. Our simulations revealed the spontaneous formation of actin asters, consistent with the punctate actin structures observed in chemotacting fibroblasts. We performed a systematic analysis of model parameters to identify biochemical steps in myosin activity that significantly affect aster formation and performed simulations in which model parameter values vary spatially to investigate how the model responds to chemical gradients. Interestingly, spatial variations in motor stiffness generated time-dependent behavior of the actomyosin network, in which actin asters continued to spontaneously form and dissociate in different regions of the gradient. Our results should serve as a guide for future experimental investigations.

Insight.

We developed a computational platform for performing particle-based simulations of actomyosin networks. We used our platform to investigate how the biophysical and biochemical properties of the motor protein myosin contribute to the emergent behavior of actomyosin networks. We found that the ability of myosin to bind to actin filaments is responsible for the spontaneous emergence of actin asters, consistent with the punctate actin structures observed in various physiological contexts. Additionally, we investigated how spatial gradients of system parameters affect the behavior of the actomyosin network and discovered that in this situation spatiotemporal dynamics can occur, meaning actin asters can emerge and dissipate or asters emerge asymmetrically. Our approach allows for the development of new and testing of current hypotheses.

INTRODUCTION

The actomyosin portion of the cytoskeleton is comprised of two proteins, filamentous or F-actin and its associated motor protein, non-muscle myosin. F-actin exhibits many different organizational structures depending on the cell type and cellular function. For example, F-actin assembles into short, polymerizing branched filaments at the lamellipodial leading edge of protruding cells [1]. During cell migration, F-actin maintains its branched lamellipodial network at the leading edge and has aligned stress fibers in the rear of the cell [2]. Additionally, actin organizes into nodes or puncta in the body of migrating fibroblasts [2]. Similar concentrated actin structures are also seen during cytokinesis [3, 4], cell elongation in Xenopus laevis development [5], and apical constriction during Drosophila melanogaster development [6]. Therefore, understanding how these structures arise is an important problem in cell biology.

The motor protein non-muscle myosin II (referred to as myosin hereafter) binds to and reorganizes F-actin, by generating a contractile forces between adjacent actin filaments. It has been known for some time that myosin-based contractility is required for directed cell migration [1]. Surprisingly, however, it was recently shown that negative regulation of myosin activity by phosphorylation is required for migrating mesenchymal cells to orient their motion in the direction of chemical gradients [7]. This observation suggests that spatial regulation of myosin activity underlies chemotaxis in these cells.

The above observations led us to use computational modeling to investigate molecular interactions capable of generating emergent actomyosin structures and to identify possible mechanisms for spatially regulation. Our goal was to use computer simulations to understand how actomyosin structures initially emerge. We were particularly interested in emergent aster structures, because these structures are observed in the contractile puncta of F-actin in the cell body [5, 8]. Previous experimental and computational studies have shown that interactions between motors and polymer filaments are sufficient to generate asters and other network structures [9–15], while another study revealed that actin polymerization can also play a role in aster formation [16]. The computational model presented here represents an extension of our previously published model [17], and is used to determine which biophysical properties most strongly influence aster formation and understand how chemical gradients regulate the spatiotemporal dynamics of emergent actin structures. To answer these questions, we updated the previous model to include biophysical parameters representing the activation and bundling of myosin and used the new model to investigate the effects of spatial gradients on emergent actomyosin structures (see Table S1 for a detailed comparison between the previous model and the current model).

Our investigations are relevant to cells undergoing directed migration in response to chemotactic gradients. In particular, studies on fibroblasts migrating in response to a PDGF gradient found that regulation of myosin activity was critical for these mesenchymal cells to move in the direction of the gradient [7]. In particular, they found that noncanonical Ser1/2 phosphorylation of the regulatory light chain on myosin, which causes inactivation of myosin, was necessary for proper chemotaxis. This observation led the authors to hypothesize that PDGF signaling promotes localized phosphorylation of myosin regulatory light chain by PKCα and, therefore, cells in a PDGF gradient will experience more contractility of the actomyosin network on the side of the cell furthest away from the PDGF source.

While signal-dependent phosphorylation represents one mechanism of spatial regulation of myosin activity, other potential mechanisms also exist. For example, a recent study demonstrated that bipolar myosin filaments can contain multiple forms of myosin and the composition of the filament determines its biophysical properties [18]. This observation led the authors to suggest that regulating the makeup of myosin filaments represents a potential mechanism for controlling the biophysical properties of these filaments, their molecular interactions and where they are spatially located [18]. Although our computational models do not include fine molecular-level details, such as phosphorylation at a particular site or myosin filament composition, we are able to mimic these effects by altering the appropriate model parameters.

Our approach was to use particle-based models of actomyosin dynamics to investigate the minimal requirements for the emergence of actin asters and determine how their properties change when subjected to spatial regulation. Our models capture the basic biophysical properties of the F-actin and myosin mini-filament interactions as measured from in vitro and in vivo experimental work [11, 19–32]. To analyze our simulation results, we develop an automated method for identifying F-actin asters that can be adapted to analyze time-lapse fluorescent images from live cell experiments. By performing systematic simulation studies, we identified key parameters that control the emergence of F-actin asters. Finally, we investigated the response of our model to spatial gradients of model parameters to suggest potential spatial regulatory mechanisms. We found that spatial variations in myosin activation and inactivation rates significantly affected the emergent network structure. Interestingly, our simulations revealed actin puncta undergo dynamic spatiotemporal behavior when motor stiffness is spatially regulated. Our results suggest likely targets for spatial regulation of the actin network and should help to guide future experiments.

METHODS AND MATERIALS

Model development

We chose to model F-actin as cylindrical rods. Each actin filament has a fixed length, and a defined polarity: a plus- (barbed) and a minus- (pointed) end [17]. In the simulations, actin filaments move due to forces exerted on them by motile myosin mini-filaments and thermal diffusion (translational and rotational). Myosin mini-filaments also undergo translation Brownian motion. Myosin mini-filaments are restricted to bind to at most two actin filaments that are within a specified capture radius from the mini-filament. Once attached to two actin filaments, mini-filaments are modeled as Hookean springs that then exert the forces required to move the actin filaments.

The bound ends of the myosin mini-filaments walk toward the plus-ends of the actin filaments. We assume that myosin processes along actin filaments at a constant velocity (v). In reality, myosin motors follow a cross-bridge cycle mechanism for binding to actin, cocking the head during the working stroke, unbinding from the actin, and recovering the head position so the whole sequence can repeat [21]. The speed of myosin ‘walking’ on F-actin can be estimated by the power stroke distance $\left(\delta \right)$ divided by the fraction of time the NM II head is attached to the F-actin filament $\left({\tau }_{on}\right)$. Myosin has a power stroke of 5 nm and an attached time of approximately 1 ms, which translates to a speed of $v=\frac{\delta }{{\tau }_{on}}=5\mathrm{\mu }\mathrm{m}/\mathrm{s}$[21]. Because it is unlikely that all myosin heads work cooperatively in the mini-filament, we assumed mini-filaments process along actin filaments with a speed of 3 μm/s in the standard model parameter set, and within a range of 0.75 μm/s to 12 μm/s for all instances where we vary the velocity. For reference, kinesin (a microtubule associated motor) obtains a top speed of 0.8 μm/s and in some cases myosin and dynein motors obtain speeds of 10 μm/s because of their low duty-ratio of 0.01 [21]. Furthermore, depending on the type of myosin, assays have found speeds between 0.04–3 μm/s [27, 28, 33]. Therefore, our simulations have captured a large range of physiologically relevant speeds for the processive movement of myosin.

Recent studies have demonstrated the importance of proper regulation of myosin activity in cellular process such as chemotaxis and morphogenesis [5, 7], in addition to being responsible for pulsatile actin contractions in mesenchymal, epithelial and sarcoma cells [8]. However, the mechanistic details for how this regulation occurs have not been elucidated. Therefore, we expanded the model to include rates for the activation/deactivation and bundling/unbundling of myosin mini-filaments to allow us to test potential regulatory mechanisms (Fig. 1). Individual myosin molecules can be in an inactive state, where they do not interact with each other or with actin filaments. The rate at which individual myosin molecules unfold and become active is given by the parameter activationRate. We chose to use a standard value of 10/s for the activation rate with a parameter range of 0.25/s to 400/s. We based these values on the large (16 orders of magnitude) variations in protein folding/unfolding rates [34]. After transitioning to an active state, two myosin molecules can bind to form a mini-filament whose heads can bind to actin filaments. Active and unfolded myosin motors bundle together based on the product of the bundling rate, bundleRate (0.2/s), and the number of single, active motors located within a search radius of length, motorBundleRadius (0.3 μm). Our rationale for using a 0.3 μm radius was that this length is the approximate length of an individual myosin molecule [28]. We have simplified mini-filament formation, by assuming only two individual myosin heads are required to bundle to form a mini-filament. However, within our modeling framework it is straightforward to include more detailed models of mini-filament formation in the future.

Figure 1.

Model kinetics diagram. Simulations begin with all myosin motors in an inactive state. The motors can then stochastically transition to an active state (activationRate). Once activated, a motor can bundle to a nearby active motor (bundleRate). An active myosin motor can also become inactive (inactivationRate). Bundled motors form mini-filaments that can bind to nearby actin filaments (p1) or become unbundled (unbundleRate). Once bound, mini-filaments walk towards the actin filament plus end (with speed, v), exerting a spring-like force on the pair of actin filaments, causing them to reorient. Actin-bound mini-filaments also can detach (p0).

Once two myosin motors are bundled to form an active mini-filament, they bind to F-actin at a rate of p1 (0.2/s) times the sum of actin filaments located within a radius of r (0.25 μm) of the mini-filament. If there are multiple mini-filaments or actin filaments that meet this criterion, then our algorithm randomly selects the ones that interact. We chose not to require that myosin and actin binding occurs at a preferential angle between the motor and filament. While it is reasonable to assume that multi-headed myosin (14–20 individual myosin’s bundled together [35]) would preferentially bind to actin when the myosin mini-filament and actin filament are nearly aligned, we do not believe this additional level of complexity would qualitatively change the behavior of the model.

We assumed the bundling rate and the rate of mini-filament attachment to actin take the same value (0.2/s). The closest biological analog for mini-filament attachment to actin would be an in vitro ATPase (s^-1) measurement from actomyosin motility assays under conditions with high ATP and filament concentrations. ATP binds to myosin, resulting in myosin detaching from actin, and the release of phosphate allows myosin to bind to actin. The ATPase measurements in vitro range from 0.8/s [36] to 20/s [21]. Since this rate is determined under in vitro conditions, we have assumed that for our model, where we are quantifying the rate of mini-filament attachment to actin, the rate would be lower for in vivo conditions with an ultimate range between 0.005/s and 8/s. We also assumed that the bundling rate of activated myosin is the same as the rate of myosin attachment to actin, based on the observation that myosin binding to actin is a non-covalent binding mechanism, and so is the binding of myosin tails to form the mini-filament [21].

Myosin mini-filaments are modeled as Hookean springs with stiffness k. If a mini-filament is stretched to a length greater than r, it will detach randomly from one of the two actin filaments. Our standard stiffness value for the mini-filament is 150 pN/μm and, because we also dictate that the maximum stretch of the mini-filament is 0.25 μm (r), the maximum force at any time a mini-filament can exert is $F_{\max }=\left(150pN/\mathrm{\mu }\mathrm{m}\right)\left(0.25\mathrm{\mu }\mathrm{m}\right)=37.5\mathrm{pN}$. Measured forces with F-actin and beads coated with NM II myosins are on the order of 1 pN to 10 pN, but this is a lower limit estimation of the actual forces between F-actin and NM II myosin because of the compliance, or flexibility, between the F-actin filaments and NM II myosin coated beads held in optical traps, so the max force in vivo should be greater than 10 pN [21]. Based on our range of motor stiffness (37.5 pN/μm to 600 pN/μm) and our range for maximum motor stretch (0.0625 μm to 1 μm), our simulations predict that the maximum force exerted by mini-filaments on F-actin filaments is between 9.375 pN and 150 pN. A range of stiffness values between 1.875 pN/μm and 1 250 pN/μm have been reported, depending on the type of myosin and species from which the myosin was derived [21–25]. A load dependence on the duty ratio in NM II has been experimentally measured [37]. For this iteration of the model, we chose not to include a load dependence on the mini-filament speed (v) or exerted mini-filament forces. However, including a load-dependent duty ratio is the next step in the model’s development and would allow for more potential regulatory mechanisms to be explored.

In the model, myosin mini-filaments detach from actin at the rate p0, mini-filaments dissociate into single, active motors at the rate unbundleRate, and individual motors become inactive at the rate inactivationRate. We chose the standard value for each of these rates to be 1/s and vary them in the range of 0.025/s to 40/s. The Sellers group measured the dissociation rate between actin and NM IIB sub-fragment 1 (S1) as $0.38\pm 0.09/\mathrm{s}$[38], which is lower than our standard value, but within our simulated range for this parameter. Furthermore, Sellers noted that NM IIB S1 has generally slower kinetics that other myosin II isoforms [38] and we are modeling a generic mini-filament which we would expect to have a larger range of possible kinetics.

Myosin mini-filaments can dissociate from actin filaments when they reach the plus-end or if the actin filament turns over. For simplicity, we have chosen not to explicitly simulate polymerization or depolymerization dynamics of actin filaments. However, we do include a parameter, p2, which represents the rate at which actin filaments turn over. The standard rate for this parameter is 0.7/s. When an actin filament turns over in the simulation, it is immediately removed. We assume that the actin filament concentration remains constant, so that when a filament is removed a new actin filament appears at a random location within the simulation domain. The range we consider for p2 is 0/s to 2.8/s.

Myosin motors that are inactive, active or bundled and not attached to actin filaments undergo thermal diffusion with a diffusion coefficient given by the Einstein–Stokes relation for spherical particles assuming a radius of 0.15 μm:

$$D=\frac{k_{B}T}{6\pi {\eta }_d\left(0.15\mathrm{\mu }\mathrm{m}\right)}=0.0029{\mathrm{\mu }\mathrm{m}}^2/\mathrm{s}$$ (1)

where $k_{B}T=0.005{\mathrm{\mu }\mathrm{m}}^2\mathrm{m}\mathrm{g}/{\mathrm{s}}^2$ and ${\eta }_d=0.6\mathrm{Pa}\cdot \mathrm{s}$ is the dynamic viscosity. In each time step of the simulation, the position of the jth myosin molecule (x_j, y_j) is updated according to the equations:

$$\begin{array}{cc}{x}_j\left(t+dt\right)\hfill & =x_j\left(t\right)+\sqrt{2\times dt\times D}\times Z_{xj}\hfill \\ y_j(t+dt)\hfill & =y_j(t)+\sqrt{2\times dt\times D}\times Z_{yj}\hfill \end{array}$$ (2)

where Z_xi and Z_yi are independent standard normal random variables.

Actin filaments move due to thermal diffusion and forces exerted by myosin mini-filaments. To compute diffusion coefficients for the actin filaments, we assume these filaments have a cylindrical shape. There are three different diffusion coefficients corresponding to perpendicular (D_perp), parallel (D_par), and rotational (D_rot) movements determined from the ratio of $k_{B}T$ to translational drag coefficients for a cylinder:

$$\begin{array}{cc}{D}_{\mathrm{perp}}\hfill & =\frac{k_{B}T}{{\mathrm{\Gamma }}_{\mathrm{perp}}}\hfill \\ D_{\mathrm{par}}\hfill & =\frac{k_{B}T}{{\mathrm{\Gamma }}_{\mathrm{par}}}\hfill \\ D_{\mathrm{rot}}\hfill & =\frac{k_{B}T}{{\mathrm{\Gamma }}_{\mathrm{rot}}}\hfill \end{array}$$ (3)

We calculated the total translational drag coefficients for a cylinder in an unbounded fluid (Γ_perp, Γ_par, and Γ_rot) with approximations of the end-correction terms $\left({\gamma }_{\mathrm{perp}}=0.84,{\gamma }_{\mathrm{par}}=-0.2,\mathrm{and}{\gamma }_{\mathrm{rot}}=-0.662\right)$ given by Tirado and de la Torre [39].

$$\begin{array}{cc}{\Gamma }_{\mathrm{perp}}\hfill & =\frac{4\pi \eta L}{\log p+{\gamma }_{\mathrm{perp}}}\hfill \\ {\Gamma }_{\mathrm{par}}\hfill & =\frac{2\pi \eta L}{\log p+{\gamma }_{\mathrm{par}}}\hfill \\ {\Gamma }_{\mathrm{rot}}\hfill & =\frac{\frac{1}{3}\pi \eta L^3}{\log p+{\gamma }_{\mathrm{rot}}}\hfill \end{array}$$ (4)

We made a distinction between dynamic viscosity, ${\eta }_d$, used to calculate thermal diffusion of myosin molecules and the effective viscosity, $\eta$, used to calculate diffusion of the actin filaments. The dynamic viscosity is an order of magnitude smaller than the effective viscosity $\left({\eta }_d=0.6Pa\cdot s\mathrm{versus}\eta =5Pa\cdot s\right)$. The difference stems from the additional drag actin filaments are likely to feel in crowded, in vivo conditions. Our value of $5Pa\cdot s$ is 5000 times the viscosity of water. While this value may appear high, Valberg and Feldman reported intracellular viscosities on the order of a million times that of water [40]. In our simulations, $\eta$ ranged from $1.75Pa\cdot s$ to $20Pa\cdot s$, which is within the range of reported values. The parameter p is the ratio of filament/cylinder length (L = 1 μm) to diameter (di = 0.008 μm; p = L/di).

We simulated an actin filament length of 1 μm in the standard parameter set simulations. During our parametric analysis, we varied filament length from 0.25 μm to 2 μm. There is a wide variation in reported F-actin lengths coming from different experimental set ups (in vivo versus in vitro) and cell types encompassing a range from 6 nm to 10 μm [19]. Since actin filament polymerization is critical to many different actomyosin morphologies [41–43], we likewise expect that varying actin filament lengths in our simulation will yield different emergent morphologies.

To compute the movement of actin filaments, we determine the force applied by the myosin mini-filaments, and then decompose the forces on each actin filament into components that are parallel and perpendicular to the actin filament length (Fig. 2). Details for how we perform this transformation are given in our previous manuscript [17]. For example, suppose a mini-filament indexed by j, is attached to two actin filaments indexed by i and h. Let the center of mass of actin filament i be given by ($x_i,y_i$) and the center of mass of actin filament h be given by ($x_h,y_h$). We denote the orientation of actin filament i as ${\theta }_i$, and the orientation of actin filament h as ${\theta }_h$. Additionally, we denote the distance between the center of mass of actin filaments i and h and the attachment of the mini-filament j as $le{n}_{ji}$ and $le{n}_{jh}$, respectively. For the example illustrated (Fig. 2), $le{n}_{jh}$ will be negative, because the attachment of mini-filament j on actin filament h is between the actin filament’s minus end and center of mass, whereas $le{n}_{ji}$ is positive because the attachment of mini-filament j on actin filament i is between the actin filament’s center of mass and the plus end. Knowing these quantities allows us to determine the position ($a_{ji},b_{ji}$) of the jth mini-filament end attached to actin filament i. The position ($a_{jh},b_{jh}$) of the jth mini-filament end attached to actin filament h is also calculated using.

Figure 2.

Diagram of forces exerted on two actin filaments by a single myosin mini-filament. Actin filaments (red) are modeled as cylindrical rods with a polarity (plus-end shown by arrow), and angular orientation (θ). Myosin mini-filaments (green) are modeled as springs that are able to bind to a pair of actin filaments at a location ($a_{ji},b_{ji}$) and (a_jh, b_jh), where the center of masses of the actin filaments are denoted by ($x_i,y_i$) and (x_h, y_h), and the length between the center of mass and the mini-filament attachment point is given by ($le{n}_{ji}$) and (len_jh). The force vector created by the spring’s force exerted on the actin filament (dark green) is transformed to a new coordinate system and decomposed into respective x’ and y’ components during the calculations.

We can then calculate the resulting force for each myosin mini-filament j on the actin filament. In some situations, more than one mini-filament is attached to filament h or i.

We update the positions of the actin filaments’ center of masses based on parallel, perpendicular, and rotational components of the applied forces and thermal diffusion.

$$\begin{array}{cc}X{R}_{n_i}\left(1\right)\hfill & =X{R}_i\left(1\right)+dt\times \frac{1}{{\mathrm{\Gamma }}_{\mathrm{par}}}\sum _{j}F{R}_j\left(1\right)+\sqrt{2\times dt\times D_{\mathrm{par}}}\times \mathrm{randn}\hfill \\ X{R}_{n_i}\left(2\right)\hfill & =X{R}_i\left(2\right)+dt\times \frac{1}{{\mathrm{\Gamma }}_{\mathrm{perp}}}\sum _{j}F{R}_j\left(2\right)+\sqrt{2\times dt\times D_{\mathrm{perp}}}\times \mathrm{randn}\hfill \\ {\theta }_{n_i}\hfill & ={\theta }_i+dt\times \frac{1}{{\mathrm{\Gamma }}_{\mathrm{rot}}}\sum _{j}le{n}_j\times F{R}_j\left(2\right)+\sqrt{2\times dt\times D_{\mathrm{rot}}}\times \mathrm{randn}\hfill \end{array}$$ (5)

Finally, we transform the updated positions of the actin filaments to the original coordinate system using the inverse matrix.

It is difficult to estimate the relative concentrations of myosin and actin. In our simulations, we include 5000 individual myosin molecules and 1000 actin filaments, each 1 μm long. There are approximately 26 G-actin in a 72 nm actin filament [21], so our simulation has approximately 300 000 G-actin monomers, even though we do not explicitly model each monomer. This produces a ratio of myosin to actin of 1:60. The actual value of this ratio could vary greatly in different regions of the cell and physiological contexts. For example, in the vicinity of the leading edge, F-actin is more abundant by about 1–2 orders of magnitude as represented in our simulation [44]. We also note that using the standard parameter values with 1000 actin filaments and 5000 myosin molecules, at steady state there are on average 243 free active myosin monomers and 45 inactive myosin monomers. The other myosin molecules exist as dimers in free mini-filaments (653) and mini-filaments interacting with actin cables (1 703).

Our computational actomyosin network was previously validated in a ‘by-eye’ comparison to sparse or dense actomyosin morphologies from experimental studies (see Fig. 1 in [17]). The approach we took in that study of varying parameters and observing final actin network morphologies is similar to the approach we employ here; however, we have greatly expanded our quantification of actomyosin morphologies and investigate spatial regulatory control and dynamic properties of the network.

Automated aster detection

Because aster formation is stochastic process, it is important to run multiple simulations to generate statistically significant results. To efficiently perform averages over multiple runs, we developed a novel algorithm to detect and count asters. Our automated method for aster detection makes use of the divergence of the filament density. Details for numerically computing the divergence can be found in our previous publication [17]. We divide the computational domain into squares of length L/8, where L is the length of the filaments (1 μm for standard parmeter sets). For actin filament plus-ends located within a L/8 x L/8 square, we sum the orientation vectors (direction from actin filament minus- to plus-end) to determine a summed orientation vector per square. Then, we determine the divergence of each square using a second-order finite difference method (for details see [17]). The results are plotted with a pixelated, ‘hot’ colormap (Fig. 3A) where negative or low divergence is black and positive or high divergence is white.

Figure 3.

Method for aster identification. (A) The two-dimensional divergence of actin filaments represented by a hot color map within boxes of size L/8 x L/8 for the same square simulation in Fig. 4A. (B) We used the signal-to-noise ratio as a threshold criterion and applied a two-dimensional Gaussian smoothing function to the data to enhance the areas of sharpest contrast between high and low divergence, i.e. locations of actin filament asters. (C) We tiled a small square ROI over the entire image to determine local maxima (upper 25% threshold of the z-score standardized image after Gaussian smoothing; magenta ‘+’), and local minima (lower 25% threshold of the z-score standardized image after Gaussian smoothing; white ‘+’). The local maxima and minima are filtered for candidates that are at least 2.5 standard deviations away from the mean divergence. We then calculated the shortest distance between white and magenta ‘+’s (teal lines). (D) If the distance is less than an actin filament length, we colored the local maxima and minima with filled circles. Our algorithm counts the number of pairs to determine how many asters were detected in the morphology.

Once we obtained a pixelated divergence map, we calculated the signal to noise ratio of the divergence map ($SNR={\mathrm{\mu }}_{\mathrm{Divergence}}/{\sigma }_{\mathrm{Divergence}}$; ${\mathrm{\mu }}_{\mathrm{Divergence}}=\frac{1}{A}\sum _i\sum _{j}DI{V}_{i,j}$, ${\sigma }_{\mathrm{Divergence}}=\sqrt{\frac{1}{A-1}\sum _i\sum _j{\left|{\mathrm{DIV}}_{i,j}-{\mathrm{\mu }}_{\mathrm{Divergence}}\right|}^2}$, where A is the number of L/8 squares required to cover the entire simulation domain). We used the SNR value as a threshold for post-processing the spatial divergence map. If the absolute value of the divergence within a square was greater than the SNR (meaning the signal was greater than noise or background), then we left its value, otherwise we set it to zero. Because numerical calculations of filament divergence can fluctuate significantly from simulation to simulation, we applied a 2D Gaussian smoothing filter to the results to aid in the identification of asters for various simulations. The goal was to use the smoothing filter to amplify the extreme low and high divergences, because actin filament asters were located where there was a switch from low to high divergence in a short distance. The 2D Gaussian filter took into account values over a 10×10 grid using a standard deviation of $\sigma =2$. Elements of the filter were calculated as follows:

$$\begin{array}{cc}{f}_{i,j}\hfill & =e^{\frac{-x_{i,j}^2}{2{\sigma }^2}-\frac{-y_{i,j}^2}{2{\sigma }^2}}\hfill \\ F_{i,j}\hfill & =\frac{f_{i,j}}{{\sum }_i{\sum }_j{f}_{i,j}}\hfill \end{array}$$ (6)

We then performed a two-dimensional convolution of the divergence matrix with the 2D Gaussian smoothing filter and plotted the color heat map (Fig. 3B).

We implemented our custom algorithm to identify actin filament asters based on the L/8 squares from the divergence calculations. An actin filament aster is characterized by a sharp change in low to high divergence, or visually, a black region next to a white region in a small area. We tiled the entire domain in regions L/2xL/2 (or four by four squares of the divergence regions of L/8). In order to identify local maxima and minima within these tiled regions, we applied a z-score standardization to the values of divergence, after the Gaussian smoothing filter was applied, to extrapolate out extrema within the set of divergence values. The z-score values were calculated by subtracting the mean of the population of divergence values from each individual divergence value, then dividing by the standard deviation of the divergence value population $\left(z=\frac{x-\mathrm{\mu }}{\sigma }\right)$. To establish a criterion for identifying asters, we first took the range between the highest and lowest divergence values after z-score standardization. Within the tiled regions, local maxima were observed if the z-score was within the upper 25% threshold of the range of values; for local minima, the value had to fall within the lower 25% threshold of the range (Fig. 3C). The local z-score extrema observed were only counted as potential locations for asters if the z-score value was above 2.5 for maxima and below −2.5 for minima – that is, if the divergence value of a local extrema was at least 2.5 standard deviations away from the mean divergence across the entire domain. Once all candidate minima and maxima had been identified, the distance between each pair of maxima-minima was calculated. If a maximum was within a distance less than that of the length of a filament from a respective minimum, the point was included in the count, otherwise, the point was excluded from the count (Fig. 3D). The number of asters in the simulation is thus determined by the smallest value of either the number of included maxima or the number of included minima.

RESULTS AND DISCUSSION

Computational simulation shows emergent F-actin asters

To initialize the simulations, we randomly placed 1000 actin filaments and 5000 inactive myosin motors in a square (4 μm × 4 μm) computational domain. We assumed that the boundaries of the computational domain were reflective for myosin motors and mini-filaments, meaning that molecules encountering the boundary are reflected back into the computational domain. Enforcing a reflective boundary for actin filaments is more of a challenge, so instead we imposed boundary conditions such that if an entire actin filament moved out of the computational domain, a new actin filament would appear in a random location inside the computational domain. In this way, a constant concentration of actin filaments was maintained for the duration of the simulation. Table 1 provides a summary of the standard parameter values used in the simulations. A complete discussion of the rationale for these values is given in the Materials and Methods section. Our initial simulations revealed the formation of four aster structures located symmetrically within the computational domain (Fig. 4A). To investigate if the aster formation was an artifact of the computational domain size or boundary conditions, we increased the size of the computational domain, and increased the abundance of actin filaments and myosin motors to maintain the same concentration. Simulations in which the area of the computational domain was doubled also displayed aster formation (Fig. 4B). The symmetry of the asters in the larger domain was less apparent, because the influence of the boundaries is diminished. We also ran simulations in which we quadrupled the system size (Fig. 4C). Again, aster formation was observed, but now the positions of the asters appeared significantly more random. These results demonstrate that aster formation is an emergent property of the system and not dictated by the size of the computational domain or boundary effects (additional evidence see Fig. S1), however the symmetry of aster location seen in the smallest domain size is an artifact of the boundary.

Table 1.

Parameters, values and description for the model.

Parameter description	Symbol	Value
Simulation parameters
Number of steps in time to run the simulation	time	2000
Time step size	dt	0.01 s
Number of actin filaments	N	1000
Number of myosin motors	M	5000
Computational domain parameters
Domain size	–	4 μm × 4 μm
Effective viscosity of media in the domain	η	5 Pa s
Actin filament parameters
Length of actin filaments	L	1 μm
Diameter of actin filaments	di	0.008 μm
Rate at which myosin mini-filaments detach from actin filaments	p0	1 s−1
Rate at which myosin mini-filaments to actin filaments	p1	0.2 s−1
Actin filament turn over rate	p2	0.7 s−1
Myosin motor and mini-filament parameters
Maximum allowed stretch of myosin mini-filaments when attached to a pair of actin filaments	r	0.25 μm
Speed at which myosin mini-filaments move along actin filaments	v	3 μm/s
Stiffness of myosin mini-filaments	k	150 pN/μm
Rate at which active, bundled myosin mini-filaments unbundle into active single active motors	unbundleRate	1 s−1
Rate at which two active myosin motors bundle together to form a mini-filament	bundleRate	0.2 s−1
Rate at which single myosin motors activate	activationRate	10 s−1
Rate at which active myosin motors become inactive	inactivationRate	1 s−1
Search radius for individual active motors	motorBundleRadius	0.3 μm
Myosin motor diffusion coefficient	D	0.0029 μm2/s
Dynamic viscosity for diffusing myosin motors	${\eta }_d$	0.6 Pa s

Figure 4.

Simulation results for domains of increasing size. (A) Actin filaments (red), mini-filaments (green), and a merged image are shown for a representative simulation using the standard parameters. The emergence of the symmetric four-aster morphology is shown over the first 3.25 seconds of the simulation, but all simulations were run to a minimum of 20 seconds. (B) Final morphology for actin filaments (red) and mini-filaments (green) when the domain size and molecular abundances are doubled. (C) Representative final morphology for actin filaments (red) and mini-filaments (green) when the domain size and molecular abundances are quadrupled.

Our model consists of 13 parameters that define myosin and actin filament properties (Table 1). To investigate how the behavior of the system depends on the choice of parameter values, we systematically increased (2× and 4×) and decreased (1/4× and 1/2×) the value of each parameter away from its standard value. Our measure for how a parameter affected the behavior of the model was based on the structure of the actin filaments at T = 20 seconds. It is possible that in some cases, the system did not reach ‘steady state’ by this time. However, we do not believe this choice of final time point significantly affects our results. Our simulations generated network structures that ranged from no asters to more than four asters (Fig. S2).

The identification of asters ‘by eye’ is time consuming and subject to human bias. Therefore, we developed an automated computational method for quantifying the number of actin asters. Our method is based on numerically computing the divergence of the vector field defined by the density and orientation of the actin filaments (Fig. 3; see Methods and Materials for details). To account for variability in simulation results arising from the stochastic nature of our model, we ran 100 simulations for each parameter set and calculated the fraction of simulations that generated a given number of asters. We used these fractions to compute a mean aster number for each parameter set (Table S2). Using this method of analysis, we were able to identify parameters that had a significant effect on the actin network structure.

Actin filament length, myosin mini-filament properties and effective network viscosity affect aster formation

Our analysis identified five parameters that had a large impact on the number of asters formed by the system (Fig. 5). We defined ‘large impact’ as a change in the mean number of asters of greater than two as the parameter is varied. The parameters that generated such changes were actin filament length (L), maximum mini-filament stretch (r), myosin speed (v), mini-filament stiffness (k) and effective viscosity (η). Results for other parameters that did not significantly affect the actin filament morphology are included in the supplement (motor bundle radius and filament turnover Fig. S3; number of filaments and number of motors, i.e. concentration Fig. S4). In Fig. 5, the mean number of resulting asters is shown with a bar for each parameter value (1/4×, 1/2×, 1×, 2× and 4×, see Table S3 for specific parameter values). For each parameter variation, we ran 100 simulations and report the number of simulations that ended with 0, 1, 2, 3, 4 or more than 4 asters inside the corresponding box in the plots.

Figure 5.

Parameters affecting aster formation. Results are based on 100 simulations for each parameter value (1/4×, 1/2×, 1× or standard parameter set, 2×, and 4×; specific values listed in Table S3), and the number of asters computed at T = 20 seconds. Shown are the mean aster number (colored bars) and frequency of each aster number for each parameter value. Results are shown for (A) filament length, (B) maximum mini-filament stretch, (C) myosin speed, (D) mini-filament stiffness, and (E) effective viscosity. We use a consistent color scheme throughout the paper and supplement: filament length (L)—orange, maximum NM II motor stretch (r)—sky blue, NM II mini-filament velocity (v)—reddish purple, NM II mini-filament stiffness (k)—dark blue, and viscosity (η)—red.

We first investigated the effect of varying the actin filament length, L. The ability of F-actin to regulate its length is extremely important for the mechanics and formation of lamellipodial protrusions [1, 45–48], and various pharmacological perturbations that disrupt actin polymerization dynamics are known to result in alterations to the F-actin network morphology [5]. Consistent with these observations that changes in actin filament length influence the network structure, we found that when actin filaments are short in comparison to the domain size, the system generates more than four asters (1/4×, 1/2×; Fig. 5A). At the standard parameter value (L = 1 μm; 1×), four asters typically emerge, but occasionally fewer than four asters are seen, and increasing the actin filament length leads to the formation of a single aster. Our results demonstrate that short filaments promote the formation of multiple asters, and there is a rapid transition to a single aster morphology as filament length is increased. Based on this result, we hypothesize that the size and number of punctuated actin asters within cells are correlated to individual F-actin filament lengths or polymerization activity.

The function of myosin mini-filaments is to bind actin filaments and exert forces to move the actin filaments. For our model, the parameters responsible for altering the mini-filaments ability to exert force are its stiffness (k) and maximum stretch (r). The parameter r is the maximum length of the mini-filament allowed before myosin heads detach from the actin filament. This parameter also defines the interaction radius for unbound mini-filaments to bind nearby actin filaments. The maximum force a mini-filament can exert on actin filaments is $F=kr$. Therefore, a consequence of increasing r is that bound mini-filaments are able to exert larger forces on the actin filaments. Varying k and r produced qualitatively similar results (Fig. 5B and D). For small k and r values, a one- or two-aster morphology was dominant. However, as these parameters were increased, the system underwent transitions where either three- or four-aster morphologies were favored. At high values, aster formation became unlikely. We did notice, however, that large values of r do not produce any asters, while large values for the motor stiffness, k, produce asters just over 10% of the time. We believe this difference is because the parameter r also defines the search radius for actin binding events while there are no additional roles for k.

We found a similar behavior in aster formation for variations in the effective viscosity felt by the actin filaments as with variation in mini-filament stiffness (Fig. 5E). Based on the model equations of motion, we find a ratio relationship between mini-filament stiffness and effective viscosity ($\frac{k}{\eta }$; see Methods and Materials) where increasing mini-filament stiffness is similar to the effect of decreasing effective viscosity. It is possible that variations in protein density in different regions of the cell could lead to a change in the effective viscosity. For example, in dense regions of the cell, the effective viscosity felt by actin filaments will be larger, because molecular crowding effects inhibit the motion of the filaments.

Although the speed at which a mini-filament walks on an actin filament is not a variable in the force equation, it is related to the change in work performed by the system $\left(\frac{dW}{dt}=k\frac{dr}{dt}v\right)$, thus affecting the amount of work done by mini-filaments to move actin filaments instead of directly affecting the ability of the mini-filament to exert force on actin filaments. We observed that the speed at which myosin mini-filaments move along actin filaments sharply impacts aster morphology. Slow moving mini-filaments resulted in single asters, whereas faster moving mini-filaments resulted in four-asters (Fig. 5C). The shape of this transition appears to be sigmoidal (see Fig. S5). One consequence of changing myosin speed is that it affects the time mini-filaments are attached to pairs of actin filament. Slower speeds increase this interaction time resulting in the formation of a larger single aster, or a highly connected cluster of actin filaments with no well-defined aster morphology. Our results are consistent with measurements made using in vitro actomyosin systems [19] and with our previous simulation results that were based on an older version of the model and did not include averaging over 100 realizations of the system [17].

Inhibiting myosin mini-filament-actin filament binding disrupts network morphology

Recently, we found that negative regulation of myosin activity through phosphorylation of the Serine 1 and 2 sites on the regulatory light chain was required for gradient sensing by fibroblasts responding to PDGF [7]. While the mechanistic details of how this negative regulation operates are unknown, expression of a dominant mutant where the phosphorylation sites were changed to non-phosphorylatable alanine (S1AS2A) rendered cells incapable of following chemical gradients. However, cells expressing this construct were still able to migrate. Our functional data supports a model in which inactivation of myosin II on the up-gradient side of the cell via the PLC-PKC-RLC pathway is required for chemotaxis. Consistent with this model, we observed that the myosin organization in chemotacting cells follows a distinct pattern (see Movie S1). A pool of myosin II accumulates into puncta at the leading edge while the majority of the myosin II was localized into stress fibers at the trailing end [49]. In S1AS2A-expressing cells, stress fibers and puncta were randomly localized with no correlation to gradient direction.

These observations lead us to perform computer simulations to investigate which biochemical step in myosin activation or mini-filament formation (Fig. 1) is likely affected by phosphorylation of Serine 1 and 2. Our rationale was that if varying the value of a kinetic parameter involved in myosin interactions (activationRate, inactivationRate, bundleRate, unbundleRate, p1, and p0) affected aster formation, then that parameter would be a good candidate for regulatory control. When we varied the values of these parameters, we found that the rates at which mini-filaments bind/unbind actin (p0, p1) showed slight variation in the number of asters formed at high values (4×) and low values (1/4×), respectively, while varying the other parameters did not produce significant changes in aster numbers (Fig. 6). To determine if larger changes in these parameters would affect the average aster number, we performed simulations with ten times smaller and larger than the previous smallest and largest parameter values (1/40×, 40×). For these simulations, again, the only parameters that produced a significant change in the mean aster number were the mini-filament unbinding rate (p0) and binding rate (p1), in which case the mean aster number dropped to less than one as p0 increased or as p1 decreased (Fig. 6). Based on these results, we conclude these rates are good candidates for negative regulation through phosphorylation. Previous studies have also found that myosin binding rates have to be high enough or unbinding rates low enough for different myosin isoforms’ functions to bind to, and to move on actin filaments in in vitro motility assays as fast as wild types [50, 51].

Figure 6.

Effect of myosin activation and mini-filament formation on emergent morphology. Results for varying the rates associated with myosin activation, mini-filament formation and interactions with actin filaments. We ran 100 simulations for each parameter at each value (1/40×, 1/4×, 1/2×, 1×, 2×, 4×, 40× of standard value). Plots show the mean number of asters (colored bars) and the frequency of each ending morphology.

The effects of spatial regulation on network morphologies

Our ultimate goal is to understand the role of spatial regulation of the actomyosin cytoskeleton, especially in the context of cellular gradient sensing. Therefore, we investigated how spatial variations in parameter values affect the emergent actin network morphology. Guided by the results of our investigations using spatially homogenous parameters, we focused on the rate at which mini-filaments bind actin, p1, actin filament length, L, the maximum mini-filament stretch, r, myosin speed, v, and mini-filament stiffness, k.

To investigate the effect of spatial variations in these parameter values, we imposed a linear gradient in the y-direction with the minimum value of the parameter at the top of the computational domain. Gradient values for various parameters were chosen by selecting over the range of values where we observed a significant change in the mean number of asters. For the case of the mini-filament binding rate p1, the gradient ranged between values of 0.01 to 0.05 s⁻¹ (Fig. 7A). In this case, asters emerged at the bottom of the computational domain, but did not form at the top. These simulations demonstrate that network structure can be spatially controlled by regulating the mini-filament binding rate even when changes in the rate is relatively modest. To further quantify the results, we ran 100 simulations of the spatial gradient and determined the mean number of asters and plotted the average aster location for each time point (see Table S4 and Fig. S6). In this case, knowing the mean number of asters present was less useful as a metric than seeing where asters formed. We observed that asters uniformly appear at the bottom of the computational domain but then move into a more central location over time (Fig. S6A) and that typically two asters are present at any given time. We also imposed a gradient in the rate at which myosin unbinds to actin, but we did not observe significant changes in aster morphology (Fig. S7). These results suggest that the actomyosin binding interactions act like a switch with only small spatial gradients needed to affect large changes in actin structure. Additionally, our results revealed that spatial regulation of the rate at which mini-filaments bind to F-actin generates network morphologies consistent with previous experimental observations. For migrating mesenchymal cells subjected to a gradient of PDGF, myosin is inactive at the leading edge (i.e. top of the simulation domain), and motor activity increases at the rear of the cell (see Movie S2) [7].

Figure 7.

Emergent actin morphologies in spatial gradients. All parameters vary linearly in space with the indicated minimum and maximum values on the left side by the arrow to show the direction of increasing gradient. Actin filaments are shown in red and mini-filaments in green. (A) Results for variations in the mini-filament binding rate (p1). In this simulation, asters tend to form at the bottom of the domain where p1 is greatest. (B) Two examples of emergent morphologies when the actin filament length (L) is varied. The observed morphologies depend on the steepness of the gradient. (C) An example of an emergent morphology for maximum mini-filament stretch (r). (D) Time courses for actin filaments and myosin mini-filaments when mini-filament stretch (r) is spatially varied. These results reveal dynamic behavior in which asters form at the bottom of the domain and move upward. (E) Time courses for actin filaments and mini-filaments in a gradient of motor speed (v). In this case, four symmetric asters form initially, but eventually resolve into two asters located in the region of slow motor speed.

Next, we simulated a gradient of actin filament length. In these simulations, the length was varied based on the location of the plus-end of the actin filament. We assumed actin filaments are more likely to be shorter at the top of the domain and longer at the bottom. The length of the filament is determined by the location of its plus end in the simulation domain. Plus ends located at the top of the domain are short, and as the short filament moves due to diffusion or myosin generated forces, the filament will increase in length if the plus end moves down, or decrease in length if the plus end moves up. Our simulations revealed that depending on the mean length of the actin filaments and steepness of the gradient different emergent behaviors were possible (Fig. 7B). The range of actin filament lengths we investigated ranged from 0.1 μm to 2 μm, which are biologically realistic values. It is reasonable to assume that actin filament length varies inside a cell when the different roles of F-actin from the lamellipodial leading edge to stress fiber formation behind the lamella are considered. The location of asters can be tuned by altering the steepness of the gradient with more asters emerging where actin filaments are shorter in shallow gradients (Fig. S6B). The left panel of Fig. 7B shows the scenario in which asters are located in the middle of the domain where actin filaments are longer, but there are also smaller asters in the top of the domain composed of shorter filaments. Alternately, in the right panel, there are two short actin filament asters at the top of the domain and clusters of longer actin filaments, but not quantifiable asters, in the middle of the domain (Fig. 7B; Fig. S6B). It was surprising to not observe any asters when filaments are longer, based on our results with spatial homogeneous longer asters, but it appears that the longer actin filaments serve as ‘tracks’ for mini-filaments to follow and travel towards the shorter actin filaments where they are able to do the work of clustering actin filaments and generating asters.

As previously discussed, the ability of cells to regulate actin filament length is important for different cortical actin morphologies [5] and lamellipodial mechanics and protrusions [1]. Our simulation results for varying actin filament length are consistent with these experimental results [52, 53], and also predict that to observe morphological changes in the actomyosin network a relatively sharp gradient in actin filament length is required. This suggests that experiments investigating the dose-dependent effects of drugs that affect actin (de)polymerization rates might yield important insights into cortical actin dynamics. We also hypothesize that if F-actin length were quantified within a cell, the shortest F-actin filaments would be at the puncta in the cell body, and the longest F-actin filaments would be at lamellipodial protrusions.

Our simulations with spatially homogenous parameters revealed that the maximum mini-filament stretch, r, significantly affected the structure of the actin network (Fig 5B). There was an optimal stretch value needed to achieve a four-aster morphology and varying r higher or lower resulted in no asters formed. When r varies in space with a sufficiently shallow gradient, our simulations generated an emergent gradient morphology that consisted of two static asters forming in regions of large r (Fig. 7C; Fig. S6C). This aster morphology arises when mini-filaments can stretch farther to search for F-actin and exert a higher maximum force. The ability of the mini-filaments to exert force results in the remodeling of the actin filament network which then results in directed ‘tracks’ for bound motors to walk along [17]. For a steeper gradient, our simulations produced significant variability in the transient behavior of the system, where asters formed in regions of large r, the same result as previously (Fig. 7C), but were more dynamic in their spatial movement through the domain (Fig. 7D; Fig. S6D).

For the case of myosin motor speed v, we simulated a gradient ranging between 0.25 and 6 μm/s (Fig. 7E). In this case, multiple asters emerged throughout the computational domain; however, asters at the top of the domain, where mini-filaments walked slower, were larger and tended to migrate toward the center of the domain. The smaller asters that initially formed at the bottom of the domain eventually dissipated. Because the larger asters grew in size during the simulation, we believe the larger asters attracted mini-filaments and actin filaments from the smaller asters. One advantage we find with slower motor speeds is that it allows sufficient time for the mini-filaments to move actin filaments and generate well-defined aster structures. When we considered the average aster location for 100 simulations (Fig. S6E), we were unable to discern a definitive pattern in aster location, indicating that gradients of motor speed generate highly dynamic behavior. These simulations demonstrate that network structure can be spatially controlled by regulating motor speed. The formation of mini-filaments containing myosin isoforms with different biophysical properties could be a mechanism of spatially regulating mini-filament properties in cells [18, 54].

Dynamic aster behavior results from gradients of mini-filament stiffness

Our most interesting results were achieved when a spatial gradient of myosin mini-filament stiffness, k, was imposed. This gradient caused asters to move from regions of low k to the center and bottom of the domain where k was the highest (Fig. 8, Movie S3, and Fig. S8). Interestingly, the system appeared to show the emergence of a new aster around T = 20 seconds, so we extended the simulation to T = 100 seconds. The extended simulation revealed the formation of two asters from the breaking apart of an existing aster (Fig. 8, shown with an asterisk at T = 34.42 s; Fig. S8 panel T = 100 seconds shows asters more likely where k is high). We consistently see aster movement up the gradient to where motors are the stiffest (k is high, bottom of the spatial domain; shown in Fig 8 by bold white arrow pointing down. See also Movie S3 and Fig. S8). We also observed multiple different actomyosin morphologies throughout the time course of this simulation (Movie S3). We saw changes from four, three, two, and one aster morphologies, in addition to a linear clustering of motors at the bottom of the domain when an aster has dissipated (see T = 46.58, T = 50.07 and T = 76.23 seconds). Previous simulations showed the emergence of filament asters, but not the dissipation and emergence of new asters.

Figure 8.

Dynamic emergent behavior in a gradient of myosin mini-filament stiffness. Actin filaments (red) and myosin mini-filaments (green) are shown over time for a gradient of mini-filament stiffness (k). Less stiff mini-filaments are at the top of the domain while stiff mini-filaments are at the bottom of the domain. New asters are marked with an asterisk, asters that are splitting are shown with a double-headed arrow, and asters that move up the gradient to regions of stiff mini-filaments are shown with a larger, single-headed arrows. Disassembly of an aster is not marked by a symbol, however, when these events occur the mini-filaments form a ‘bar’ structure instead puncta (see T = 46.58, T = 50.07 and T = 76.23).

Previously, we discussed potential mechanisms for regulating aster formation and disassembly through switching parameter values at set time points [17]. In vivo the actomyosin cytoskeleton dynamically contracts and relaxes over periods of 90 seconds, but our previous model was only capable of generating static asters [17]. Here, we demonstrate that gradients in myosin mini-filament stiffness generate continuous aster formation and deformation without the need for additional regulation (Fig. 8). It is worth pointing out that the spatial gradient values for k to achieve this phenomenon varied by over an order of magnitude. The standard value used in our simulations for this parameter is k = 150 pN/μm. We assume this is the effective stiffness from approximately 15 bundled motors (each motor has two heads capable of binding to F-actin and each head has a stiffness of around 5 pN/μm [21–25]). For our simulated spatial gradient of myosin mini-filament stiffness, the minimum stiffness value of 20 pN/μm is analogous to two motors bundled together to form the mini-filament and the maximum stiffness of 270 pN/μm is analogous to 27 bundled motors. Typical mini-filaments are bundled with 14–20 motors [35], but it is not unreasonable to assume that any individual motor can be regulated so our simulated gradient captures a range of approximately 2–27 myosin motors. Furthermore, recent work from the Sellers lab has identified a Myosin 18 isoform which decreases the processivity of NM II through the creation of hybrid myosin mini-filaments and they hypothesize that hybrid myosins could have variability in stiffness [18].

Although our initial question about a spatial gradient stemmed from the ability of the PLC-PKC-RLC pathway to spatially inactivate myosin, spatial regulation of mini-filament stiffness could arise from a spatial signaling pathway, or spatially located myosin isoforms capable of creating hybrid myosin mini-filaments. We propose that the next step to validate our findings would be to link data of spatial localization of myosin isoforms (for example, Myosin 18 Aα is frequently found under the nucleus and in the lamella [18]) to biochemical and biophysical studies of the properties of these isoforms. Furthermore, our computer simulation results point to the required stiffness of hybrid myosin filaments needed to generate dynamic actomyosin morphologies.

CONCLUSIONS

We used computational simulations to investigate the requirements for organized structures to emerge in the actomyosin cytoskeleton and how these structures can be spatially regulated. Our initial simulations revealed that these additions to the model were sufficient to spontaneously induce the formation of aster structures with no additional regulation required. Our simulations also revealed that aster formation was independent of the computational domain size, demonstrating that aster formation was an intrinsic property of the system and not driven by boundary conditions.

Next, we performed systematic parameter studies to determine which parameters controlled aster formation. Our parametric analysis demonstrated that aster formation was sensitive to actin filament length and parameters related to the myosin mini-filament’s ability to exert force. These results are consistent with published experimental results where Cytochalasin D, Latrunculin B, or Latrunculin A cause decreased F-actin filament lengths and show cortical actin morphologies with small asters composed of short filaments [5, 55–57]. Jasplakinolide stabilizes F-actin and yields large stable arrays of cortical F-actin [5, 58], analogous to our simulation in which longer filaments yielded larger asters. The small molecule ROCK inhibitor, Y-27 632, reduces myosin contraction generating cortical actin networks that look more sparse [5, 59], while treatment with Calyculin A causes myosin motors to contract more resulting in denser cortical actin networks [5]. Additionally, we identified the rate at which myosin mini-filaments bind to actin as a candidate target for negative regulation of myosin function. Lastly, we examined the behavior of our model in a spatial gradient and observed actomyosin structures emerging throughout the domain. In particular, when we imposed a spatial gradient of mini-filament stiffness, the model generated time-dependent behavior in which asters continuously form and disassemble.

Some of the parameter gradient results could have been anticipated from the results of the simulations using constant parameter values (see Table S5), however, the gradient simulations also revealed emergent behavior that could not have been predicted solely from the constant parameter simulations. Specifically, the ability of motors to bind to actin filaments in a spatially graded fashion can generate asymmetric aster geometries. Simulations using gradients of filament length of different steepness show large variability in the final structure of the actomyosin network suggesting cells might regulate filament length to control the structure of the actomyosin network during cell migration. Finally, spatial regulation of myosin stiffness can generate time-dependent behavior of the actomyosin network.

While relatively simple, our model generated results consistent with several experimental observations on actomyosin dynamics during cell migration. Because we kept the number of model parameters to a minimum, we were able to perform a systematic parameter study. Our analysis identified likely regulatory targets and should serve as a guide for future experimental investigations into the role of actomyosin dynamics in gradient sensing. To investigate the necessary requirements for the formation of actin structures other than asters, we plan to expand the model to include the effects of actin cross linkers and other actin-associated proteins. Our results also suggest that including more biological details in our treatment of myosin and the clustering of motors into mini-filaments could create dynamic spatial morphologies without the need for spatial regulation. Finally, we note that our current simulation platform provides a framework for connecting models for actomyosin dynamics with spatiotemporal regulation by signaling pathways.

Funding

This work was supported by National Institutes of Health grant U01-EB018816, and a grant through the Army Research Office (Proposal Number: 67117-MA).