Computational simulations reveal that Abl activity controls cohesiveness of actin networks in growth cones

INTRODUCTION

During embryonic development, neurons send out axonal projections to construct complex neural circuits (Araújo and Tear, 2003). The ability of axons to extend through the extracellular space in vivo is coordinated by specialized structures found at the tips of motile axons, called growth cones, that span tens of microns (Dent et al., 2011). Protrusive forces are generated in the growth cone through complex biochemical interactions between membrane-bound signaling molecules that intercept extracellular guidance cues (Netrins, Semaphorins, Slit, Ephrin, and others) (Huber et al., 2003; Hinck, 2004; Seiradake et al., 2016; Feinstein and Ramkhelawon, 2017) and the intracellular cascades that transmit these signals to force-producing cellular components, namely the cellular cytoskeleton. Thus, growth cones are made of actin and microtubule filaments along with a host of actin modifying proteins and signaling molecules such as Rho-family GTPases (Giniger, 2002; Huber et al., 2003; Gallo and Letourneau, 2004). Actin and microtubules play specific roles in the extension of axons, where actomyosin drives motility and guidance, while microtubules are thought to provide structural support and lock in the consequences of actin dynamics (Dent and Gertler, 2003; van Goor et al., 2012; Kahn and Baas, 2016; Turney et al., 2016). Many studies have tried to dissect the mechanism of axon growth by focusing on the morphological changes of growth cones or the cytoskeletal signaling processes that aid in growth (Witte and Bradke, 2008). Despite detailed understanding of various components involved in the process, however, we lack any clear picture of an emergent mechanism that can explain how the biochemically identified nanometer-level changes to signaling molecules and cytoskeletal binding proteins lead to micron-level changes to axonal architecture.

Studies focusing on growth cone morphology have investigated the relationship between structural dynamics of the cytoskeleton and the mechanism of axon extension. Fluorescent and electron microscopic imaging of neuronal cell cultures has revealed two major modes of axonal growth (Strasser et al., 2004; Sánchez-Soriano et al., 2010; Tamariz and Varela-Echavarría, 2015). Cultured neurons often display axons with flat, fan-shaped growth cones (Forscher and Smith, 1988; Schaefer et al., 2002; Gallo and Letourneau, 2004; Lowery and Vactor, 2009; Suter and Forscher, 2010). However, growing evidence from in vivo studies primarily reveals growth cones that are bulbous and only minimally adherent to the substratum, extending by the formation and selective stabilization of filopodial protrusions (O’Connor et al., 1990; Grabham et al., 2003; Omotade et al., 2017; Clarke et al., 2020a,b; Santos et al., 2020). The cytoskeletal organization and dynamics of this in vivo class of growth cones has received far less attention in the past, and it is the relationship between signaling and actin organization in growth cones such as these on which we focus in the current study.

The growth cone architecture during axonal protrusion is controlled by signal transduction molecules that dynamically alter kinetic fluxes of actin-binding proteins to modify cytoskeletal structure. One critical signaling module studied extensively in axons is the Abelson (Abl) nonreceptor tyrosine kinase. Abl is known to act downstream of many of the widespread, phylogenetically conserved axon guidance receptors (Jones et al., 2004). Genetic perturbations that alter Abl activity in vivo increase the probability of developmental errors due to axonal misrouting and stalling. Genetic and biochemical analyses have suggested that Abl alters growth cone organization by simultaneously stimulating activity of the branching actin nucleator, Arp2/3, while inhibiting the processive actin polymerase, Enabled (Ena) (Kannan and Giniger, 2017; Kannan et al., 2017). While the roles of these two critical downstream proteins on the actin cytoskeleton have been studied independently (Pollard, 2007; Winkelman et al., 2014; Harker et al., 2019; Ni and Papoian, 2019), the relationship and relative roles of the two are still poorly understood (Bear et al., 2002). Live in vivo imaging studies of an identified axon in the developing Drosophila wing, called TSM1, revealed that the baseline, wild-type (WT) level of Abl activity ensures spatial correlation and dynamic coherence of actin networks within the growth cone. Under both Abl overexpression and Abl knockdown conditions, growth cones are characterized by increased spatial fragmentation of actin and dynamic incoherence. Additionally, Abl regulates growth cone size, with Abl-overexpressing axons having longer growth cones, while axons from neurons with reduced Abl have shorter growth cones (Clarke et al., 2020a,b).

Despite knowledge of the biochemical targets of Abl and its consequences at the growth cone level, we do not understand how they are linked through the dynamics of actin. As it is not possible to resolve single actin filaments by in vivo imaging experiments in live tissue, here we use a sophisticated computational model called MEDYAN to simulate key cytoskeletal processes that result from Abl signaling and then compare those simulations to the effects of Abl that we have quantified previously in vivo (Clarke et al., 2020a,b). As biochemical and genetic experiments have identified Arp2/3 and Ena as crucial targets of Abl, we simulate networks with 20 µM actin and a wide range of Arp2/3 and Ena concentrations. Critically, we perform these simulations at spatial and temporal scales that are large enough to capture properties of the TSM1 growth cone. We find that Arp2/3 plays a crucial role in determining actin filament length distributions and actin packing in growth cones. We also see that the growth cone simulations capture essential features of actin distributions measured in live-imaging experiments and reveal evidence to explain the mechanism of actin fragmentation observed in Abl-overexpressing axons in vivo. Finally, we study the functional consequences of the actin architectures formed in response to various levels of Arp2/3 and Ena and show that Abl overexpression causes mechanochemical disconnection within the actin network. We discuss the potential consequences of fragmentation for the motility, growth, and guidance of axons such as TSM1.

RESULTS

To faithfully simulate axonal actin networks, we simulated contractile actin networks under cylindrical boundary conditions using Mechanochemical Dynamics of Active Matter (MEDYAN) (Popov et al., 2016), a powerful, active matter simulation software. In MEDYAN, individual actin filaments are represented as semiflexible polymers with experimentally observed persistence length (Popov et al., 2016). The actin-binding proteins (ABPs) such as nucleators, cross-linkers, motors, and capping agents are also represented explicitly when bound to the actin filaments. When actin monomers and ABPs are not associated with the filamentous actin phase, they are freely diffusing and are not spatially resolved. To ensure accurate computation of reaction propensities involving species of the filamentous and diffusing phase, MEDYAN divides the reaction volume into voxels of size 500 × 500 × 500 nm. Reactive species inside the voxel are assumed to be uniformly mixed and are used to calculate the propensity of relevant chemical reactions, including binding/unbinding, polymerization/depolymerization, and contractility. Diffusion is modeled as particle exchange between voxels. The spatially resolved diffusion and chemical reaction networks are evolved stochastically using the next reaction method (Gibson and Bruck, 2000), a variant of the Gillespie algorithm (Gillespie, 1977). The mechanical equilibration of the network is achieved using conjugate gradient minimization by taking advantage of the timescale separation between fast mechanical equilibration and slow chemical stress accumulation. In this study, we minimize the mechanical stress accumulated every 25 ms of chemical evolution. Finally, the mechanochemical rates (slip bond, catch bond, Brownian ratchet) are updated for the next chemical cycle using the mechanically equilibrated configurations. Please refer to the Supplemental Methods for a detailed discussion of the MEDYAN model and Supplemental Tables S1 and S2 for parameters relevant to this study.

As simulations over the entire length of a growing axon are computationally impractical, we restricted the bulk of our simulations to cylindrical reaction volumes of length 7.5 µm and radius 1 μm. This volume represents approximately 50% of the growth cone volume observed in vivo for TSM1, a growth cone with uniquely well-characterized actin distribution and dynamics that we take as our point of comparison throughout this analysis (Clarke et al., 2020a). A subset of conditions will also be investigated in a volume of the same diameter but 15 μm length to control for size-dependent features in the simulations. Axonal growth cone mimics were generated through stochastic evolution of actin networks initialized with 400 actin “seed” filaments, each 40 monomers long (∼108 nm), that were located and oriented randomly in the reaction volume, along with 20 μM actin. The seeds themselves provided ∼5.3% of the total initial actin in the volume, and >50% of the monomers in those seeds were exchanged by treadmilling over the first 300 s of the simulations (Supplemental Figure S26). Initialization with “seed” filaments allowed the simulations to bypass an otherwise lengthy lag phase for filament nucleation without biasing the further evolution or final conditions of the simulations. Additionally, this protocol allows us to generate actin networks with heterogeneous length distributions as found in cellular dendritic networks (Podolski and Steck, 1990; Lewis and Bridgman, 1992; Mongiu et al., 2007). Please refer to the Supplemental Methods for discussion on filament initialization. Actin filaments are restricted to the reaction volume through a reflective boundary that also slows the polymerization rates of filament ends that are close to the reaction volume boundary, in accordance with the Brownian ratchet model (Mogilner and Oster, 1996). The boundary cutoff distance was empirically arrived at by testing various distance cutoffs to ensure stable simulation of cytoskeletal networks that are enclosed within the reaction volume. Mole ratios of nonmuscle myosin minifilaments (NM-IIA) and α-actinin were chosen to be 0.1 and 0.01 relative to actin, respectively, to observe contractile behaviors similar to those in previous studies (Floyd et al., 2019; Li et al., 2020). Please refer to Methods for more information on MEDYAN and Supplemental Tables S1 and S2 for parameters used in the simulations employed for this study.

To understand the effect of Abl on actin architecture, we chose to model key downstream effectors of Abl, namely Arp2/3 and Ena. In MEDYAN, branched nucleation is modeled as a chemical event where freely diffusing Arp2/3 complex binds to a parent filament to produce an offspring filament at an angle of approximately 70° (Mullins et al., 1998; Pollard, 2007). In this study, we have also incorporated a finite lifetime and force-sensitive unbinding of Arp2/3 nucleated branches (Fujiwara et al., 2007; Pandit et al., 2020). Useful concentration limits for Arp2/3 and Ena were estimated using a minimalistic ordinary differential equation (ODE) model of actin network dynamics to identify the range over which network properties are sensitive to various concentrations (Supplemental Table S5 and Figure S25), and the utility of these ranges was then validated by trial simulations. Please refer to the Supplemental Methods for more details on the ODE model. To understand the effect of Abl on the actin cytoskeleton, we simulated networks at 1, 5, 10, 25, and 50 nM Arp2/3 and Ena each with six replicates of 2000-s-long trajectories and analyzed as shown below.

The rest of the Results is organized as follows. We start by analyzing the actin filament properties that are observed in our simulations when varying the two downstream Abl effectors, Arp2/3 and Ena. We then quantify micron-scale actin network organization in the simulations using metrics that correspond to properties measured in previous experimental analyses (actin spread and actin pair-correlation profile) or that illuminate the basis of those properties (modularity of actin networks). We then investigate the functional significance of network organization and axonal coherence with two perturbative experiments. Finally, we compare our computational results with experimental observations in vivo in TSM1 axons.

Filament branching versus extension—Arp2/3-driven changes dominate filament length

MEDYAN simulations reveal that Arp2/3 and Enabled change the length distribution of actin filaments, but with Arp2/3 playing the predominant role. Representative final snapshots shown in Figure 1 show marked differences in actin network organization that result from changing Arp2/3 and Ena concentrations. Please refer to Supplemental Figures S1–S4 for enlarged images along each axis and Supplemental Movie M1 for visualization of representative trajectories for each condition studied. We quantified filament length distributions from the last 500 s of the trajectory to define the specific changes resulting from varying Arp2/3 and Ena (Figure 2A; corresponding probability density functions are shown in Supplemental Figure S5). As Arp2/3 concentration is increased, Arp2/3 kinetics causes increased branched filament nucleation (Figure 2B). As the total actin in the reaction volume is held constant, this results in an increased abundance of short filaments (<1 μm). We see two distinct trends that depend on [Arp2/3]. At the filament level, it is striking that at low levels of Arp2/3 (≤10 nM) we observe a bimodal distribution of long versus short actin filaments rather than a single, smooth distribution. At the actin level, we see that at low [Arp2/3], most of the actin is present in long filaments, and it shifts steadily toward short filaments at high [Arp2/3] (Supplemental Figure S6). We discuss this phenomenon in more detail in work presented elsewhere (Chandrasekaran et al., 2022). In brief, at low [Arp2/3], most of the actin is present in filaments extended from the original seeds (Supplemental Figure S6). As [Arp2/3] is increased, we see that the actin gets redistributed to newly nucleated filaments. Therefore, the resulting bimodal distribution consists of two distinct populations: submicron filaments constituted predominantly of Arp2/3 nucleated filaments and the longer filaments constituted predominantly of filaments extended from the seeds.

In contrast to the profound effects of varying Arp2/3, changing Ena concentration has a limited impact on filament length distribution. As Ena concentration increases, we see a steady increase in the fraction of filament plus ends stabilized by Ena (Figure 2C). At [Arp2/3] < 10 nM, addition of Ena caused Ena-driven filament extension, leading to an increased abundance of longer filaments (≥1 μm) and a decreased abundance of shorter filaments (<1 μm). At [Arp2/3] ≥10 nM, changing the Ena concentration has a limited impact on filament lengths (Figure 2A and Supplemental Figure S5).

To understand the role of reaction volume dimensions, we chose to simulate a subset of the Arp2/3 and Ena concentrations in a volume twice the length of that which we have investigated so far (15 µm) but the same radius (1 μm). As Abl signaling simultaneously promotes Arp2/3 activity and inhibits Ena activity (Kannan et al., 2017), we simulated actin networks at the conditions specified along the falling diagonal of Figure 1. We see consistent trends in filament length distribution from our simulations in 15-μm-long reaction volumes to that we observed in the shorter volume (Supplemental Figure S7; compare Figure 2A).

Arp2/3 and Ena alter network organization at different length scales

We next found that Arp2/3 also dominates the overall organization of the total actin network. To understand how differences in filament lengths alter network organization, we quantified actin concentration in 100-nm-thick disks along the cylindrical reaction volume axis (as shown in Figure 3A, schematic). Averaged one-dimensional (1D) profiles of actin distribution along the volume were obtained by using convolution to peak-align the actin profiles from the last 200 s of the trajectories (Figure 3B). Increasing Arp2/3 at any given Ena concentration leads to drastic changes in the final actin profiles, characterized by multiple peaks spread across an increasing fraction of the reaction volume. Looking at the time evolution of actin distribution, we initially observe uniform distribution of actin at all [Arp2/3] (Supplemental Figure S8). At t > 100 s, we observe loss of spatial homogeneity in the actin density distribution in [Arp2/3] ≤10 nM as myosin-driven contractility causes the actin to condense to the center of the volume. At higher [Arp2/3], however, we see that actin remains distributed rather equally across almost the entire reaction volume. Reduced effectiveness of contractility in this condition is also apparent as local actin concentration exceeds bulk only in small patches of the distribution. (The role of myosin in this evolution is studied in detail elsewhere (Chandrasekaran, et al., 2022.) In contrast to varying Arp2/3, increasing Ena at low Arp2/3 concentrations modestly broadens the width of a central peak in the actin distribution, while at higher Arp2/3, increasing Ena has little discernible effect.

To quantify the relative effects of Arp2/3 and Ena on the actin distribution profiles, we measured the spread of actin around the mean (Figure 4). Actin spread was defined as the sum of the square root of second moments on either side of the distribution mean (related to the SD of the distribution about its midpoint; Clarke et al., 2020a). We then quantified the significance of Arp2/3 and Ena on actin spread using the Wilcoxon rank-sum test. We find that in 94% (47/50) of pairwise comparisons, increasing Arp2/3 concentration at any particular Ena concentration increases the median spread of the actin profile at 0.05 significance. Similarly, at a given Arp2/3 concentration, increasing Ena increases actin spread in 86% (43/50) of comparisons. We note that at the two highest Arp2/3 concentrations, actin is already distributed roughly homogeneously across the entire volume, limiting the possibility of further spread. We also note, however, that increasing Arp2/3 increases actin spread over a multimicron-length scale at any given Ena concentration, whereas the range of actin spread from varying Ena is smaller than 1 µm in all cases. Thus, these results suggest that actin organization and distribution are dominated by Arp2/3, with modest effects due to changes in Ena.

Further analysis of the actin distribution was done in simulations at 15-μm-long reaction volume to understand the role of the reaction volume dimensions in determining the actin distributions. We see that under conditions that mimic elevated Abl expression, that is, under ([Arp2/3], [Ena]) pair values of (50,1) and (25,5), actin spreads homogeneously throughout both the short and long reaction volumes (Supplemental Figure S9), though in the short volume we do observe some pileup of actin at the ends of the interval. In contrast, in conditions that mimic low Abl activity, with Arp2/3 and Ena concentration pair values of (10,10), (5,25), and (1,50), we see that actin is highly condensed at positions in the middle part of the volume, evidenced by peaks in the axial actin distributions in both the short and the long volumes. Additionally, we also see that the widths of actin peaks are similar in actin distributions from both 7.5 and 15 µm reaction volumes. Finally, the overall distribution of actin within 15-µm-long reaction volumes under all conditions studied appears to approximate a concatenation of multiple copies of the actin profiles obtained at 7.5 µm length scales (albeit with some “filling-in” of the interval between actin peaks at the highest Ena concentration in the large volume). These similarities between simulations from the two length scales support the argument that the actin distributions obtained at 7.5 µm length scales largely display features that arise from the biophysical properties of the network, not from the spatial limits of the simulation volume, and that are also observed in volumes with length scales comparable to those of experimental growth cones.

In addition to the overall longitudinal profile of the actin distribution, it was essential to obtain a quantitative measure of the internal organization of the actin network. In our previous experimental analysis of TSM1 actin organization in vivo, we used wavelet analysis of the 1D actin profiles to determine the spatial distribution of actin within the mass of growth cone actin (Clarke et al., 2020b). We therefore performed a calculation that gives us an analogous relative measure of the internal distribution of actin density along the long axis of the simulation volume called, variously, the pair correlation profile or the radial distribution function of the actin profile. In brief, actin density is calculated in transverse slices at 50 nm intervals along the length of the volume. These allow us to calculate the pairwise correlation of density as a function of distance away from any reference point in the reaction volume, giving us a relative measure of the spatial distribution of actin density along the long axis of the simulation volume when compared with bulk linear density. (Please refer to Supplemental Figure S10 for a schematic explaining the calculation.) Thus, values above 1.0 indicate that actin abundance is above bulk density while values below 1.0 correspond to regions with actin below linear bulk density (moles of actin per nanometer). We performed this calculation for all the combinations of Arp2/3 and Ena concentrations studied (Figure 5).

We see that at Arp2/3 concentrations of 1 and 5 nM, the point-to-point correlation of F-actin density decreases monotonically with increasing distance. In other words, the slices of high actin density are clustered, with the probability of finding another high-density slice dropping monotonically with distance from the reference point along the longitudinal axis, consistent with the single peak profiles shown in Figure 3B. Additionally, as the Ena concentration is increased at these levels of Arp2/3, the pair correlation decays to 1.0 (where local density falls to bulk density values) at slightly larger length scales. This result is consistent with the finding that actin spread increases modestly with increasing Ena at low Arp, as discussed earlier (compare Figures 3B and 4A). At higher Arp2/3 concentrations, pair correlation shows reduced dependence on distance, consistent with the relatively flat axial actin distribution profiles across the reaction volume (Figure 5). Changing Ena has very small effects on the pair correlation function profiles in these conditions, again consistent with actin spread data.

In contrast, when the Arp2/3 concentration is increased at any given Ena concentration (Figure 5B), we see that linear actin densities are above bulk value (g(r) > 1) over longer spans, consistent with the increased actin spread observed earlier (Figure 4B). Specifically, the effect of Arp2/3 spans the entire reaction volume, consistent with the finding that Arp2/3 increases spread globally, across the entire volume, whereas Ena increases actin spread only locally, at short range. Finally, we see that at [Arp2/3] = 10 nM, the pair correlation function shows one auxiliary peak, suggesting multiple domains of actin organization. At higher Arp2/3 concentrations, actin profiles decay monotonically at a slower rate, with no significant features. This is consistent with the complex and highly variable collection of peaks observed in the 1D simulation profiles at high Arp2/3. In the next section, we establish that the patterns in order parameters measured at high Arp2/3 concentrations along the cylinder axis, such as actin spread and pair correlation function, are mechanistically a consequence of actin fragmentation (Chandrasekaran et al., 2022). Thus, changing Arp2/3 concentration has a profound effect on the internal organization of the actin network, with low Arp2/3 promoting consolidation of actin in a homogeneous mass and increasing Arp2/3 spreading actin into multiple masses. In the Discussion, we will analyze in detail the striking correspondence between these patterns of actin distribution in the simulations versus those observed experimentally by imaging of actin in TSM1 (Clarke et al., 2020b).

Networks corresponding to elevated Abl activity are composed of multiple, weakly interacting actin filament communities

We wished to understand why increasing the level of the branching actin nucleator, Arp2/3, expands the actin network. Quantitative investigation of network connectivity now revealed that increasing Arp2/3 causes the actin to self-assemble into multiple, internally connected domains that are only weakly associated with one another. We constructed graph representations of interfilament contacts by representing actin filaments as nodes and adding an edge whenever two filaments share a linker, motor, or brancher. We then used three methods to characterize the connectivity of these networks. We first computed the number of disconnected clusters in the network from the last 100 s of the trajectories. Figure 6A shows that the number of separate clusters in the network increases with [Arp2/3]. Second, to understand the differences in the interfilament contacts among the clusters, we performed graph burning studies (Bonato et al., 2016; Garcia-Diaz et al., 2022). The graph burning algorithm involves two steps. At each step, we choose a node at random and label it as “burned.” Additionally, we burn the neighbors of the nodes burned in the previous step. The number of iterations or the number of random node choices required to burn the entire graph informs us of the underlying connectivity of the graph and is called the burn sequence length. For example, a connected graph will have a lower burn sequence length compared with a graph made of multiple disconnected clusters. To understand how the graph architecture changes as a function of Arp2/3, we measured the burn sequence fraction of the filament–filament contact graph given by the ratio of burn sequence length to the total number of nodes in the graph (number of filaments). Figure 6B shows that the burn sequence fraction increases steadily with [Arp2/3] due to an increase in the number of clusters. Additionally, we see striking differences between [Arp2/3] = 1 and 5 nM where the number of clusters is fairly close but the burn fraction almost doubles, suggesting that the graph organization changes drastically due to Arp2/3.

As the above metrics did not consider the degree of coupling (number of contacts between any two filaments), we employed the modularity optimization technique to uncover the underlying communities. Modularity is a graph theoretical measure of networks to quantify whether and how network components are divided into modules, and such graph theoretical metrics have proven helpful for defining network architecture (Floyd et al., 2019; Eliaz et al., 2020). Modules or communities identified by optimizing the Louvain-Modularity metric (Blondel et al., 2008) are sets of actin filaments that interact more closely with other filaments within their own set than with the rest of the filaments in the network. We begin by constructing a weighted graph where the edge weight corresponds to the number of cross-linkers, motors, and branchers between any two filaments in the network. We identify communities as connected units made of multiple actin filaments (≥5) and containing total monomers that can span more than ∼2 μm (i.e., that contain at least ∼2000 monomers, 0.25% of the total actin in the simulation). More details about the modularity metric can be found in the Supplemental Methods. We find that actin is organized into distinguishable communities that are only weakly connected and that increasing the Arp2/3 concentration increases the number of communities and decreases community size (Figure 6, C and D). This altered organization of actin filaments also reduces the amount of actin in each cluster as Arp2/3 increases, as shown in Figure 6D (and visualized by the color coding of the communities in Supplemental Figures S11–S15). These results are also observed in simulations at 15 μm length scales (Supplemental Figure S16). Thus, we believe that the findings from our study reflect the inherent connectivity of the networks formed under these conditions and are not due to confinement resulting from reaction volume choices. Time evolution of the number of communities shows us that the networks initially consist of multiple, disconnected clusters in all conditions (Supplemental Figure S17). At [Arp2/3] ≤ 10 nM, the number of clusters drops quickly over time, converging on a more densely connected steady state value by ∼500 s, while at higher [Arp2/3], networks persist in the fragmented state.

Actin communities in elevated Arp2/3 networks are mechanochemically disconnected

Functional analysis of mechanical connectivity and mechanochemical response to perturbation reveal mechanical disconnection of actin networks as the Arp2/3 level is increased. First, we asked whether the weakly coupled communities that we find in high-Arp2/3 networks are mechanically disconnected. We employed the final configurations of the 2000-s-long simulations presented above as the initial conditions for this perturbative study. To examine just the mechanical response of the system, we prevented chemical dynamics in filaments (polymerization, depolymerization, branching, and unbranching), thereby freezing any myosin and cross-linker molecules in the bound state. Localized myosin and cross-linker activity were then allowed only in a single 500 nm “active zone” that was chosen to lie at the region of highest actin density. This localized activity was allowed to proceed for 500 s (full parameters of the simulation can be found in Supplemental Table S2). This simulation was inspired by previous experimental studies investigating spatial correlation in linear actin networks (Linsmeier et al., 2016). Looking at the velocity of filaments from the last 100 s of trajectories, we see that filaments move within the active zone and in a region adjacent to it (Figure 7). Specifically, in low-Arp2/3 networks, actin filament movement extended in regions flanking the active zone over longer distances than filaments in high-Arp2/3 networks. The difference in velocity decay rates between low- and high-Arp2/3 networks demonstrates that the increasing number and decreasing size of filament–filament contact-based communities are associated with a decreased range of mechanical connection of a given zone of the actin network to the remainder of the network.

We next complemented this analysis of the structural connectivity of a static network by investigating the mechanochemical response of an active actin network to mechanical perturbation. Again using the final configuration of filaments, linkers, motors, and branchers from our earlier simulations as the initial conditions, we exert an external force on the actin network using a mimic of a functionalized “Atomic Force Microscope (AFM) tip.” The AFM tip mimic is functionalized with 50 stabilized actin filaments of 540 nm length (200 monomers). It is introduced at one end of the simulated growth cone network, close to a region of high actin density, and allowed to interact and become incorporated into that network through cross-linker and myosin driven mechanochemical couplings. With its associated stabilized actin filaments, the AFM tip mimic was then moved by a distance of 1 µm over a 5.0 s timescale. As a result, tensile forces are transmitted across the actin network through contacts between stabilized filaments in the probe and the actin filaments in the network. Chemical interactions such as filament treadmilling and interfilament chemical interactions from branching, myosin, and cross-linker kinetics remained active throughout the reaction volume. We then visualize the response of the network to the force transmitted through the motions of the AFM probe. Networks are subject to three pull events, and Figure 8 shows snapshots before and after each displacement of the AFM probe. In addition, we also show the “after” snapshot with each filament colored by the net displacement that it undergoes during the pull event. We observe stark differences in how actin networks at various Arp2/3 concentrations respond to this perturbation. We see that networks with [Arp2/3] = 1 and 5 nM move coherently as a single connected unit in response to the external force. We also see that the AFM tip couples effectively with the rest of the network after the first pull event leading to significant filament displacements in the subsequent two pull events. On the other hand, in networks generated with [Arp2/3] = 10 and 25 nM, AFM pulling reveals a functional disconnect between actin domains, preventing effective transmission of tension across the entire network. The displacement visualization also suggests that very few filaments responded to the local probe displacement at these higher Arp2/3 concentrations. To control for the effects of local actin concentration, we repeated these AFM pull experiments by placing the probe mimic in regions selected to have the same starting F-actin concentration (30 μM) and found consistent results (Supplemental Figures S19–S21). Taken together, the results from the perturbative simulations show that elevated Arp2/3 levels result in a network architecture fragmented into domains that are mechanochemically disconnected.

Actin organization within in silico networks resembles that of WT Drosophila axons

The simulations above were designed to model the properties of actomyosin networks. A key question, however, is how similar these are to actin distributions observed experimentally. We have previously used live fluorescent imaging to quantify the actin distribution in the growth cone of a growing axon in vivo in the developing Drosophila wing (TSM1 axon). To obtain representations of those experimental actin distributions comparable with the simulation results shown in Figure 3, we normalized the intensity profiles obtained from fluorescent imaging of actin in single WT axons over 30 min (sampling frequency = 3 min), aligned the profiles by the position of maximum actin intensity in each time point, and averaged the intensity for a 7.5 µm span around that peak position. This procedure was repeated for 14 WT axons (presented as a gallery in Figure 9, showing the mean and SD for each axon; Supplemental Figure S22 shows actin profiles from MEDYAN simulations at all conditions studied alongside the experimental WT profiles). Comparing the experimentally observed actin to the simulated profiles presented in Figure 3, we see that the experimental profiles show a distribution similar to that observed in the simulations at [Arp2/3] ≤ 10 nM, with a dominant central peak surrounded by a steady decrease in actin intensity to either side. Moreover, expanding the analysis of the experimental data to a total length of 15 μm reveals the appearance of subsidiary peaks spaced a few microns from the central peak in many trajectories and bears a marked resemblance to the additional peaks observed in simulated actin profiles in calculations performed in 15-μm-long cylinders with [Arp2/3] ≤ 10 nM (Supplemental Figure S23). Thus, even though our simulations approximate Abl signaling by just the downstream effector concentrations and do not include any neuronal polarization mechanisms, the actin profiles obtained from our simulations at [Arp] ≤ 10 nM bear a striking similarity to the experimentally observed features of actin organization discussed above. This similarity is also evident in overlays of the individual profiles from the constituent experimental and simulation profiles (Supplemental Figure S24). In contrast, it is clear from inspection of Figure 3 that simulations with higher levels of Arp2/3 do not bear any substantial resemblance to the experimentally observed axial actin distributions. Beyond the similarity in the overall profile of the experimental and simulated actin profiles, quantitative analysis reveals strikingly similar features in the internal organization of the actin density and in the response of both profile and organization to perturbations of Abl-Arp2/3 signaling. These are summarized in Supplemental Table S3 and will be considered in detail in the Discussion.

DISCUSSION

Axon growth and guidance are mediated by cytoskeletal dynamics that result from signaling cascades. In Drosophila, as in vertebrates, Abl is a crucial signaling molecule that controls axon growth by activating the actin nucleator, Arp2/3, and inhibiting the actin polymerase, Ena. Current methods, however, do not allow us to directly image the dynamics of individual actin filaments in growth cones that are embedded in living tissue. Therefore, to understand actin organization within an Abl-guided axon, we employed simulations that approximated the growth cone as a cylinder and varied Arp2/3 and Ena levels independently and calculated their predicted effects on actin architecture. These showed that Arp2/3 and Ena affect actin organization at different length scales and magnitudes. Specifically, we found that Arp2/3 dynamics determine overall actin distribution, as it is a much more potent modulator of filament length and network organization than is Ena. While both Arp2/3 and Ena increase the spread of the actin network within the reaction volume, Ena locally broadens features of the actin distribution by tenths of a micron by extending individual actin filaments, whereas Arp2/3 globally spreads the actin distribution across multiple microns by fragmenting the actin into separate, weakly coupled domains. The functional consequences of this fragmentation are revealed by two perturbative simulations showing that actin networks become mechanochemically decoupled under elevated Abl-Arp2/3 activity into separate domains of actin that fail to act cohesively. Finally, we show that our computational networks have a marked resemblance to the actin distributions observed in our previous quantitative imaging of the Drosophila TSM1 growth cone in vivo. Below, we present quantitative comparisons that further support the concordance between these simulations and previous live-imaging data. We also discuss how the simulation results can account mechanistically for the actin distributions observed experimentally in both WT axons and axons with altered Abl activity and why the potential for variable fragmentation and condensation of actin that we demonstrate here may be essential to the normal dynamics of axonal actin in the WT context and how they may contribute to mutant phenotypes such as axon misrouting in the case of Abl-overexpressing axons.

As Abl steers the growth cone through changes to both Arp2/3 (promotion) and Ena (inhibition), how do we understand the relative role of each of the two downstream effector molecules on F-actin? The pronounced sensitivity of actin networks to changes in Arp2/3 over that of Ena is evident from our analysis of the spread and internal organization of the actin distributions. We define the spread of the distribution as the SD of actin density about the midpoint of the volume. While increasing Ena at any given Arp2/3 concentration spreads the actin by no more than 0.1–0.3 µm, changing Arp2/3 at any Ena spreads actin distribution on the order of several microns. Consistent results are obtained with profiles from 7.5- and 15-micron-long reaction volumes, suggesting that these observations are not due to actin-boundary interactions. The internal organization of the actin, in contrast, is a measure of how homogeneously actin is packed in the volume. By calculating the pairwise correlation of actin density as a function of distance along the long axis of the cylindrical volume, we can assess whether all of the actin is concentrated in a single, continuous domain or is split into local concentrations of actin separated by regions of reduced concentration. Pair correlation of actin along the cylindrical axis shows that Ena-driven spread has limited impact on domain organization, while Arp2/3-driven spread could arise from the separation of the underlying actin into discrete domains. An independent graph theoretical modularity analysis confirmed these results, suggesting fragmentation of actin due to Arp2/3 activity. Thus, we see that the network architecture reflects the presence of a variable number of domains, dependent on Arp2/3 activity and characterized by a combination of domain size and domain–domain spacing. We were initially surprised to find that Ena, the most thoroughly characterized Abl effector, had such a modest effect on actin organization in our simulations. However, an independent experimental study, carried out in parallel, led to the same conclusion based on in vivo imaging of actin in the Drosophila TSM1 growth cone (Fang et al., 2022).

The analyses discussed above provide rich and quantitative metrics of computational actin distributions at length scales similar to experimental actin intensity measurements that we have made previously in the TSM1 neuron in the developing Drosophila wing. These metrics allow us to compare values and trends derived for corresponding properties in the computational and experimental distributions and provide strong evidence that the computational model captures fundamental properties of the actin network that exists in a growth cone in vivo. Experimental evidence showed that the growth cone actin network expands progressively over a range of several microns as Abl activity increases, from the Abl-knockdown condition, to WT, to Abl overexpression (Figure 5D in Clarke et al., 2020b), just as the simulations presented here show the multimicron spread of the actin network with increasing activity of the dominant Abl effector, Arp2/3 (Figure 4). Similarly, quantifying the internal organization of the actin network reveals parallel trends from experiment versus simulation. The pair-correlation analysis of actin organization described above queries the internal distribution of actin density in the simulated actin networks in a manner that mirrors the wavelet decomposition of experimental TSM1 growth cone actin distributions that we used previously to characterize the internal organization of experimental actin distribution in TSM1. For the in vivo data, the wavelet analysis revealed that the Abl loss-of-function condition induced an enhanced contribution of short-length-scale actin spacings to the actin distribution when compared with WT (higher-order wavelets; 0.5–4 μm spacing: Figure 5 in Clarke et al., 2020b), consistent with the substantial enhancement of short-range (<2 μm) pairwise actin correlations observed computationally at the lowest Arp2/3 levels (Figure 5A, [Arp2/3] < 5 nM). Conversely, under Abl gain-of-function conditions, wavelet analysis reveals enhanced contribution from long length scales (lower-order wavelets: 4–32 μm spacing), consistent with the enhancement of computational pairwise actin correlations >4 μm as simulated Arp2/3 levels were increased in the simulations (please refer to Supplemental Table S3 for more details). Having said this, it is important to note that it would be an overstatement to call any particular condition from our concentration matrix a unique “model” for the WT condition of TSM1. Rather, both the simulations here and the quantitative analyses in our previous work (Clarke et al., 2020a,b; Chandrasekaran et al., 2022) suggest that the WT condition encompasses substantial stochastic fluctuations that populate a range of actomyosin network architectures. Nonetheless, the quantitative analyses of actin spread and actin internal organization strongly support the conclusion that the trends observed in the computational simulations presented here capture essential properties of the space of actin distributions populated by TSM1 in vivo both in WT and in experimentally perturbed conditions. It is also important to note that the concentration bounds of Arp2/3 and Ena used in this work were chosen based on a deterministic model built with published kinetic parameters and were selected before comparing the simulation results with our previous measurements of growth cone actin distributions. Therefore, our observations about the network-level properties of actin reflect inherent emergent properties of the network under standard experimental conditions and were not tuned to match our experimental observations.

The data presented here show that activation of Arp2/3 (and, to a lesser degree, suppression of Ena activity) shortens the lengths of actin filaments and that this filament shortening is associated with fragmentation of actin network connectivity and broadening of the overall volume spanned by an actin network. But how are we to understand in detail the links between nanometer-scale biochemistry, filament-level network structure, and multimicron-scale cell biological consequences? The binding of Arp2/3 results in local nucleation around the parent filament while unbinding creates a filament that can (de)polymerize at both ends, both of which lead to a larger number of shorter filaments. On the other hand, Ena binding to plus ends enhances extension kinetics, resulting in longer filaments. The filament length distributions presented in Figure 2 stem from the additive effects of these individual events. Thus, at low Arp2/3 concentrations, Ena-driven polymerase activity alters the filament length distributions slightly toward longer filaments. However, as the Arp2/3 concentration increases, nucleation activity dominates, boosting the abundance of short filaments and reducing both average filament length and the diversity in lengths. As a result, Ena-driven filament length changes saturate in an Arp2/3-dependent manner. The mechanism explaining how the shift in filament distribution to shorter lengths leads to fragmentation of the actin network is presented in detail elsewhere (Chandrasekaran et al., 2022). In brief, however, the network organization of such filaments into distinct clusters is due to stochastic binding, unbinding, and walking of contractile elements. As in any network, nodes and edges self-assemble stochastically into communities that appear as domains with enhanced local actin concentration. If we now consider the effect of filament length distribution, long filaments can bridge between separate domains, allowing them to coalesce under the action of myosin contractility. When short filaments dominate the distribution, the lack of such linking elements causes the network to break down into small, local domains, each of which is internally contractile, but that are only weakly coupled to one another, as seen here. Over time, these spread out randomly through the available volume by a combination of treadmilling and myosin-driven forces. The loss of long-range connectivity also limits the range over which the network can transmit force, that is, the network becomes mechanochemically disconnected.

While our computational analysis reproduces many properties of Abl function, particularly under WT and overexpressed conditions, the correlation with Abl loss of function is somewhat less complete. At the lowest levels of Arp2/3, simulated actin networks showed condensation of actin, consistent with the actin distribution in Abl loss-of-function conditions in vivo. However, in the experimental study, we also observe enhanced actin disorder in the most highly condensed actin networks (assayed as Fisher information content). This feature was not reproduced in the simulations. The reason for this is not apparent. It may be that a more detailed model of Enabled will illuminate this question. For example, the actin-bundling activity of Ena tetramers has been found to be crucial in filopodia (Winkelman et al., 2014; Harker et al., 2019) and also genetically in vivo (Gates et al., 2009), so incorporating this activity of Ena in our model may improve the correlation with experimental observations. Additionally, in this study, we investigate the organization of nonpolarized actin networks. Polarization aids in directional force generation and may also contribute to the organization of the actin network. Finally, in this study, we have explored the effect of Abl signaling under constant actin conditions, but axon growth is also driven by the dynamic production of actin and other cytoskeletal proteins within the growth cone. Given the evidence that Abl can regulate processes linked to translation (Kannan et al., 2014; Kannan and Giniger, 2017), it may be that such effects also modulate Abl-dependent actin organization. Nonetheless, it is remarkable that the highly simplified model presented here, incorporating only a handful of actin regulatory proteins and a minimal set of biophysical processes, is sufficient to recreate such a rich variety of the actin properties and dynamics observed experimentally in living axons and thereby not only make strong predictions for how other signaling mechanisms may modulate actomyosin organization but also identify unexplained phenomena for further investigation.

Finally, the critical question motivating this work is how changes in architecture due to Abl signaling may alter functional aspects of the growth cone. While other studies have also employed graph theoretical approaches to characterizing actin organization (Eliaz et al., 2020; Floyd et al., 2021), the physical and potential biological consequences of such findings have sometimes been elusive. In this study, we designed two perturbative simulations to investigate the effects of Abl-driven changes of actin dynamics in a growth cone–like geometry. The first investigates the transmission of force within static networks built under low- versus high–Abl-Arp2/3 conditions, while the second investigates the spatial reorganization of fully dynamic low- and high–Abl-Arp2/3 networks in response to perturbation. For the first study, we considered the final axonal architectures from the simulations discussed so far and studied their response to introducing myosin-driven forces exclusively in the densest part of the network. This protocol is akin to published experimental and computational studies that locally activate myosin 2 in an otherwise frozen actomyosin network (Linsmeier et al., 2016). We see that effective transmission of forces across the growth cone occurs under low-Arp2/3 concentrations, while at high-Arp2/3 concentrations, force transmission is effective only over short (submicron) distances. This is consistent with the weak structural coupling we observed above between actin domains in high-Arp2/3 networks. For the second study, we restored the complete, force-sensitive mechanochemical activity of the network to investigate how it responds to short timescale tensile forces. We embedded mimics of AFM tips functionalized with stable actin filaments in our actin networks and then moved them to exert forces on the network. Here again, we found that at low Arp2/3 levels (equivalent to low Abl), the actin network was displaced as a single unit, while under elevated levels of Arp2/3 (equivalent to enhanced Abl), the weak coupling between actin domains hampers effective force transmission across the network. As a result, only fragments of the overall network move in response to a local pulling force, and the growth cone actin network splits apart. Together, these computational experiments illuminate how the actin cytoskeleton in the growth cone organizes to move as a single entity under WT conditions and perhaps how loss of information- and force-coupling upon Abl overexpression may disrupt growth cone integrity to impair effective guidance.

Our study reveals that Abl signaling can modulate actin organization between connected and fragmented states. We suggest that switching between these states might lend critical mechanistic opportunities to the growth cone as it advances through the developing tissue. The connected state helps axons generate and transmit protrusive forces throughout the network, which helps advance the growth cone and keeps it acting as a single unit. On the other hand, one can suppose that growth cone turning requires sampling of information in multiple directions followed by coalescence of structure and force in a single direction. Therefore, we hypothesize that spatial gradients of Abl may create dynamic networks in the growth cone with a range of spatial architectures that switch between globally connected and locally fragmented states to respond directionally to localized extracellular signals.

Axon guidance arises from the action of dozens of neuronal signaling networks, consisting of hundreds or thousands of proteins linked in a wide variety of ways. However, to produce growth cone motility, those signals converge on just a handful of elementary transformations of the actomyosin cytoskeleton. In this work, we have used only the Abl network to manipulate these basic biophysical processes and only one neuron, TSM1, as our biological point of comparison, but the analysis we describe does not depend on those specific choices. Instead, as many cytoskeletal signaling cascades of growth and protrusion terminate on actin polymerization and nucleation as their outputs, we suggest that the mechanistic insights obtained here, especially the fundamental organizational modes, are likely to be broadly relevant to other signaling pathways that regulate growth cone architecture and dynamics. We suggest, therefore, that the perspective we describe here should apply to a wide variety of signaling systems in axon growth and guidance across animal phylogeny.

METHODS

Simulation methods: MEDYAN

To computationally simulate actin networks under various levels of Abl, we used MEDYAN (Popov et al., 2016), a powerful, freely downloadable (www.medyan.org) active matter simulation software. MEDYAN simulates chemically active reaction networks and enables force-sensitive, stochastic simulations of active filamentous networks, such as actin and microtubules. In MEDYAN, actin filaments are explicitly represented with monomer resolution as a series of nondeformable cylinders coupled at hinge points. Bending deformations around hinge points are allowed, corresponding to a persistence length of 16.7 µm (Ott et al., 1993). Myosin and α-actinin molecules are represented as Hookean springs with equilibrium lengths based on experimental measurements (Redowicz et al., 1999; Ferrer et al., 2008). As a result, chemical reactions such as cross-linker (un)binding, motor (un)binding, and motor walking events are defined between filaments that satisfy the appropriate distance thresholds. Branched nucleation is represented as a che­mical event leading to the formation of an offspring filament at an angle of approximately 70° with respect to the parent filament. Ena tetramers are modeled as single molecules that bind to the tip of a free F-actin barbed end, thereby enhancing the polymerase activity.

The simulation space is composed of reaction-diffusion compartments overlaid with the filamentous actin phase. Diffusing reactive molecules (G-actin, unbound cross-linkers, myosins, Arp2/3, and Ena) are considered to be homogeneously distributed throughout the reaction-diffusion compartments. Thus, a diffusion reaction is modeled as a hopping reaction between two neighboring compartments. Reaction propensities are defined over the compartment phase, and stochastic trajectories are generated by evolving the chemical reaction network using the next reaction method (Gibson and Bruck, 2000). Chemical events in MEDYAN can lead to mechanical stress accumulation. MEDYAN alternates between short bursts of chemical evolution (25 ms) and conjugate gradient mechanical equilibration to generate physically realistic trajectories. Additionally, mechanochemical sensitivity of α-actinin unbinding, myosin walking, and unbinding are incorporated in the model. In this study, we restrict our discussions to actin networks in the growth cone. Please refer to Supplemental Table S1 and S4 for further details, including all parameters used in this study.

Data and software availability

MEDYAN software and documentation are freely downloadable at www.medyan.org. Results of all simulations and analyses and MATLAB code used for all calculations presented here are available on Github: https://github.com/achansek/MEDYANArp23_2021.

FOOTNOTES

ACKNOWLEDGMENTS

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