Reconstitution of the equilibrium state of dynamic actin networks

Principles of regulation of actin network dimensions, fundamentally important for cell functions, remain unclear. We studied in vitro and in silico the effect of key parameters, actin density, ADF/Cofilin concentration and network width on the network length. In the presence of ADF/Cofilin, networks reached equilibrium and became globally treadmilling. At the trailing edge, the network disintegrated into large fragments. A mathematical model predicts the network length as a function of width, actin and ADF/Cofilin concentrations. Local depletion of ADF/Cofilin by binding to actin is significant, leading to wider networks growing longer. A single rate of breaking network nodes, proportional to ADF/Cofilin density and inversely proportional to the square of the actin density, can account for the disassembly dynamics. Selective disassembly of heterogeneous networks by ADF/Cofilin controls steering during motility. Our results establish general principles on how the dynamic equilibrium state of actin network emerges from biochemical and structural feedbacks.

Without ADF/Cofilin, the networks elongated steadily and did not disassemble -actin density 137 along the networks changed only slightly ( Figure 1A). Addition of ADF/Cofilin changed the networks' 138 dynamics: rather than growing steadily, the networks, after reaching a certain length, started to 139 disassemble at the trailing edge, so that a dynamic equilibrium is reached in which the network 140 length stayed roughly constant (Figure 2A). The   (actin density and architecture) but is unaffected by the network length. Thus, the equilibrium length 148 of the treadmilling network is determined by the length-dependent disassembly only: the longer 149 the network is, the faster is the disassembly at the trailing edge, and so the treadmilling length is 150 determined by the dynamic stable equilibrium, in which the trailing edge disassembly rate is equal 151 to the leading-edge growth rate. As the leading-edge growth rate is unaffected by ADF/Cofilin, our in 152 vitro assay allows to investigate the effect of the ADF/Cofilin-mediated disassembly on the network 153 length, without complications of feedbacks between disassembly and assembly.
154 155 Our data revealed that the equilibrium network length decreases with the ADF/Cofilin concentra- 156 tion and increases with the actin density (Figure 2A-B). Qualitatively, these results are very intuitive: 157 higher ADF/Cofilin concentration increases the disassembly rate, hence the equilibrium between 158 the leading edge growth and trailing edge disassembly is reached at shorter lengths. If the actin 159 network is denser at the leading edge, it takes a longer time to break such network down; during     If this rate stays constant, then the bound ADF/Cofilin density as a function of time and of distance from the network leading edge is the solution of the equation + = 0 , where is the rate of actin network growth at the leading edge. Since newly polymerized actin is free of ADF/Cofilin, we can assume ( = 0) = 0. In dynamic equilibrium, this equations yields the solution ( , ) = 0 , which can be easily understood: an actin spot takes time ∕ to drift a distance from the leading edge. As ADF/Cofilin binds with rate 0 , by that time the bound ADF/Cofilin density reaches the value of 0 . Assuming that the network falls apart when a critical amount of ADF/Cofilin per actin filament, ∕ = , is reached, this yields an equilibrium network length of * = 0 . This simple model predicts, that the equilibrium network length is proportional to the ADF/Cofilin 178 concentration in the solution, 0 , in qualitative agreement with the data (compare Figure 2A-B). In 179 Figure 2A-B, we also observe a clear correlation between the actin density and the network length. 180 Since denser networks also grow faster (Figure 1C), our estimate is again in qualitative agreement 181 with Figure 2A-B, however, it appears that the network growth rate increases only weakly with 182 the actin density, while the equilibrium network length increases dramatically, when the actin 183 filament density increases. Lastly, the simple model indicates that the equilibrium network length is 184 independent of the network width. 185 Equilibrium network length increases with the network width. 186 We tested this last prediction experimentally for networks of widths 15, 30 and 90 , and the result 187 shows that this is not the case (Figure 2C-D). In  however, the rate of ADF/Cofilin binding decreased with starting time (Figure 3A-B), rather than 213 remaining constant. 214 ADF/Cofilin is locally depleted by binding to the growing actin network. 215 To confirm that the decrease of the ADF/Cofilin binding rate with time is due to the local ADF/Cofilin depletion, we analyzed the simplest model of the spatial-temporal ADF/Cofilin dynamics compatible with our observation. In the model, the density of free ADF/Cofilin molecules diffusing in the solute is denoted by ( , , ). Since the experimental chamber's depth in -direction is much smaller than all characteristic dimensions (including those of the actin network) in -and -directions, we use a 2D setting for modeling, with the space variables ( , ) ∈ ℝ 2 . In the simulations, an actin network of width and length ( ) = is positioned at  = [− ∕2, ∕2] × [0, ( )]. The model consists of the following equations: (2)    which implies Below, we show that the unbinding of ADF/Cofilin is very slow; besides, near the leading edge, ≈ 0, and so:̇ When the network grows, its length increases, and hence, as shown by these formulas, the local 231 concentration of free ADF/Cofilin near the actin network decreases with time, and so does the 232 rate of ADF/Cofilin binding at the leading edge, in agreement with the measurements (Figure 3B). 233 This provides a direct demonstration of the local depletion of ADF/Cofilin due to the diffusion and 234 binding to the network. We simulated the full 2D model (1)-(2) using parameters estimated from 235 our data and taken from the literature (details in the Appendix) and find a significant depletion ef-236 fect near the network where the free ADF/Cofilin concentration drops by as much as 50% ( Figure 3D). ) and modeled the network as a 2D ensemble of edges connected by nodes. We emphasize 254 that this representation is highly idealized, and the that the edges do not stand for individual 255 filaments, but rather represent actin filaments arrays; similarly, nodes are not individual physi-256 cal Arp2/3 complexes, but are abstracted crosslinking and/or branching points. We model the 257 disassembling effect of ADF/Cofilin by removing the nodes with certain rate, . Once a piece of 258 the network becomes disconnected from the main body of the network due to this edge removal, 259 we assume that this piece diffuses away and we delete it. Figure 4C illustrates how the model works. The key to the model behavior is setting rules that describe how the rate of breakage per node 262 varies spatially. It is natural to assume that this rate is a function of local densities of filamentous 263 actin and bound ADF/Cofilin. We also assume, for simplicity, that we can neglect a potentially 264 complex effect of sequential biochemical reactions preceding the breakage events. As we already   For comparison with data, it is useful to derive an analytical approximation of the discrete, stochastic model. In the Appendix, we introduce continuous deterministic densities of actin filament and of broken nodes in the network, derive differential equations for these densities and solve these equations. This continuous deterministic model allows deriving analytical expression for the equilibrium network length as the function of three parameters, average bound ADF/Cofilin density, , initial actin network density, 0 , and rate of the network growth at the leading edge, : (4) Figure 4D shows excellent agreement between the analytical approximation (4) and the correspond- to the ones predicted by formula (4) based on the measured values of parameters , 0 and . 293 We found that for any ∈ [1, 3] and ∈ [0.5, 1.2], we had 2 -values of over 0.7, and < 10 −7 . In the 294 following we use = 1, = 2 ( 2 = 0.72, < 10 −8 ). Figure 4F shows the quadratic dependence of the 295 equilibrium network length on the initial actin density. This fit suggests that rate of disassembly 296 of the effective network nodes is proportional to the bound ADF/Cofilin density and inversely 297 proportional to the square of the local actin density. We discuss implications of this finding below. 298 Balance between accumulation of ADF/Cofilin in longer networks and accumulation of 299 network-breaking events predicts equilibrium network length. 300 We can now combine the results from two models -for ADF/Cofilin binding and for network disassembly -to understand how the ADF/Cofilin dynamics and network fragmentation determine the equilibrium network length. In light of the relation all that remains is to use the model from the previous section to estimate the average density of bound ADF/Cofilin and substitute the value into Eq. (5). In the Appendix, we derived the following analytical estimate, based on the analysis of Eqs. (3) and (1):   ADF/Cofilin density as the increasing function of the network length ( Figure 5A). Together, these 321 two equations constitute an algebraic system of equations for two variables -and -that has a 322 unique solution for each value of four parameters 0 , 0 , , given graphically by the intersection 323 of two curves for the relations ( ) given by Eq. (6) and Eq. (5) as shown in Figure 5A. In particular, 324 since these two curves will always intersect, the network will reach some equilibrium length for 325 any parameter combination. The effect of varying individual factors can now easily be understood 326 ( Figure 5B) and allows to elucidate the experimental observations from Figure 2: Increasing the 327 ADF/Cofilin concentration leads to more bound ADF/Cofilin and thereby shorter networks ( Figure 5B, 328 top). Interestingly increasing network density leads to less disassembly on the one hand, but also to 329 more depletion. However, overall, the increased stability dominates over the depletion effect, and 330 denser networks grow longer (Figure 5B, bottom). Figure 2B shows very good agreement between 331 the model and the measurements. In the second experiment in Figure 2C To simulate the steering heterogeneous network, we modeled the two networks as two elastic beams growing side-by-side. The networks had different densities and different growth speeds; we took the values of those from the data ( Figure 6D). We used the result of Gardel et al. (2004) that the actin network elasticity scales with actin density as ∝ 2.5 . We therefore modeled the networks as two attached beams of width , growing at speeds 1 and 2 , with elastic moduli 1 and 2 . In the absence of ADF/Cofilin we can assume that the densities and hence elasticities stay constant along the network. In the Appendix we demonstrated, that in mechanical equilibrium, the heterogeneous network forms a bent shape with constant curvature (Figure 6A-C): growing with speed: Here = 2 ∕ 1 is the ratio of the elastic moduli. The dependencies of the curvature and combined 348 network length on parameter are depicted in Figure 6B  To asses the effect of ADF/Cofilin on heterogeneous networks, we used the model from the 360 previous sections to calculate the equilibrium lengths of the two sub-networks and simulate the 361 heterogeneous networks. Since the two sub-networks compete for the same pool of ADF/Cofilin, 362 we need to adjust the depletion factor in Eq. (6). As described in the Appendix we can determine 363 two equilibrium lengths 1 and 2 of the sub-networks. Effectively both networks will reach longer 364 lengths together than in isolation, since there is more local depletion of ADF/Cofilin in the combined 365 network. In addition, the sparser network is affected more by the depletion, as the denser networks 366 'uses up' disproportionately more ADF/Cofilin. Also, the network densities are not constant along 367 the sub-networks, thereby leading to varying elasticities along the network. In terms of the model, 368 this means that the parameter becomes a function of the distance from the leading edge. Finally, 369 the sparser sub-network has a trailing edge much closer to the leading edge than the dense one.  sub-network is the only one remaining, and it continues to grow straight. Figure 6E shows We imaged the curving heterogeneous networks ( Figure 6D) and found that indeed increased 378 ADF/Cofilin concentration straightens the combined network ( Figure 6F) due to selective disassem-379 bly of the sparser sub-network and relieving the elastic constrain on the denser sub-network. The 380 data not appear qualitatively like the predicted shapes, the measurements of the average curvatures 381 give the same values as those predicted by the model (Figure 6F). Note, that the curvature changes 382 very little on average when ADF/Cofilin concentration is increased from 250 to 500 because in 383 both cases the sparser sub-network is almost completely disassembled.  with the small ADF/Cofilin-free gap near that edge (Reymann et al., 2011). Just like our model,  , 2011), and so this is unrelated to the effect that 475 we report in this study. 476 Novelty of our findings and relevance to in vivo networks 477 We established a simple formula that allows estimating the network length, , as a function of a wide range of geometric and biochemical parameters: Actin filament density at the leading edge, We found that diffusion of ADF/Cofilin in the solution and binding to the growing actin network 498 can locally deplete the cytoplasmic ADF/Cofilin, which makes wider and denser actin networks 499 grow longer ( Figure 5D). Quantitatively, whether the depletion is significant or not, is determined by 500 the magnitude of the non-dimensional quantity : if this factor is smaller than 1 (e.g. when 501 the network width and length are small enough), there is no significant depletion; otherwise, there is. 502 503 Using ≈ 0.01∕( ) (Tania et al., 2013), we estimated that ≈ 1∕ (F-actin density is ≈ 100 for observed branched networks, and so we use this parameter for all estimates), and ≈ 10 2 ∕ (Tania et al., 2013). Thus, for actin tails propelling intracellular pathogens and organelles, for which ≈ 1 and ≈ 3 , we have: ≈ 0.3, and the depletion of ADF/Cofilin is present but moderate. In this case, the length of the tails can be estimated by the simple formula: as observed. We used ≈ 0.1 ∕ (Theriot et al., 1992) and assumed that the ADF/Cofilin concen-504 tration in most of animal cells is on the order of 20 (Pollard et al., 2000). 505 506 The diffusion-limited depletion of ADF/Cofilin is also relevant for very large cells, i.e. oocytes, muscle cells, nerve cells, megakaryosytes, with size on the order of hundreds of microns. At such size scale, the ADF/Cofilin diffusion time is on the order of minutes to many minutes, comparable to the characteristic time of the network treadmill. In these cases, assuming the characteristic dimensions of the flat Arp2/3-controlled networks, ≈ 10 and on the order of 10 , we have ∼ 10, and there is sizable depletion effect for ADF/Cofilin. In this case, we predict that ≈ 2 and so we predict that: We can also use our formulas to estimate the lamellipodial length in usual animal motile cells of intermediate size, like keratocytes and fibroblasts. In those cases, as the characteristic cell size is on the order of tens of microns, the characteristic diffusion time for ADF/Cofilin molecules is tens of seconds, less than or equal to the characteristic time of the network treadmill, so the local ADF/Cofilin depletion has a small effect. However, the lamellipodial actin represents a significant fraction of the total cell actin (Barnhart et al., 2011; Ofer et al., 2011), and so a significant fraction of all ADF/Cofilin is bound to the lamellipodial actin filaments. Thus, the global ADF/Cofilin depletion is of a major importance: our estimates of the binding and unbinding rates suggest that majority of ADF/Cofilin molecules would be bound to the lamellipodial actin. Then, assuming that the ADF/Cofilin concentration is on the order of 20 (Pollard et al., 2000) and taking into account characteristic volumes of the whole cell and of its lamellipodial part, we estimate that the bound ADF/Cofilin in the lamellipodial network has concentration on the order of ≈ 100 . Then, the lamellipodial length which is the order of magnitude found in experiments. Our study leads to the important general conclusion that the cell is able to control the dynamic actin 572 network length by adjusting either geometric, structural, or biochemical parameters, as needed. 573 For example, if the network's width is dictated by the environment around the cell, then network's 574 length can be regulated by tuning ADF/Cofilin concentration (Figure 5C-F). On the other hand, if 575 the ADF/Cofilin concentration has to be tuned for timely disassembly of other actin structures, 576 then the branched network's density or width can be changed in order to achieve necessary length 577 (Figure 5C-F). In other words, there are multiple ways to set the dynamic balance of the biochemical 578 and transport pathways regulating the global actin treadmill. This gives the cell a sufficient flexibility 579 in the control of the cytoskeletal geometry, without compromising requirements for mechanical 580 and biochemical parameters to control multiple cytoskeletal functions.

582
Protein production and labeling 583 Actin was purified from rabbit skeletal-muscle acetone powder (Spudich and Watt, 1971). Actin was 584 labeled on lysines with Alexa-568 (Isambert et al., 1995). Labeling was done on lysines by incubating 585 actin filaments with Alexa-568 succimidyl ester (Molecular Probes). All experiments were carried 586 out with 5% labeled actin. The Arp2/3 complex was purified from bovine thymus (Egile et al., 1999). 587 GST-pWA is expressed in Rosettas 2 (DE3) pLysS and purified according to Boujemaa-Paterski et al. 588 (2017). Human profilin is expressed in BL21 DE3 pLys S Echerichia coli cells and purified according 589 to Almo et al. (1994). Mouse capping protein is purified according to Finally, parameter = 0.942 characterizes the critically dense network, which completely obstructs the diffusion. A conservative estimate can be made by assuming a dense actin network with = 300∕ 2 . This gives estimates of = 0.042 and eff = 0.97 . Thus, the effect of even a dense actin network on the ADF/Cofilin diffusion coefficient is but a few per cent and can be neglected. Determining initial ADF/Cofilin binding rate 816 To determine the ADF/Cofilin binding rate (see Sec. Spatio-temporal ADF/Cofilin dynamics and its local depletion in the main text), we used the experimentally measured concentrations of ADF/Cofilin and actin. We focused on the changes of bound ADF/Cofilin concentration at the beginning the network growth, since in this early stage we can neglect both ADF/Cofilin unbinding and depletion of free ADF/Cofilin. This means that we can assume that: where ( , 0 ) is the binding rate we would like to determine. Since we know the network growth speed, we can measure the increase of bound ADF/Cofiliṅ in moving patches of actin, i.e. we can directly measure ( , 0 ). First, we examined networks with similar actin densities , and found a strong correlation ( = 0.69, < 10 −3 ) between the binding ratė and the concentration of the ADF/Cofilin in the solute 0 (App. Fig. 1A). Next, we examined networks in the experiments with similar solute ADF/Cofilin concentrations 0 , but with varying actin densities , and found a strong correlation betweeṅ and the actin density ( = 0.69, < 10 −5 ) (App. Fig. 1B). Finally, we examined networks with varying values of parameters 0 and , and found that indeed the binding ratė ∝ 0 ( = 0.51, < 10 −5 , App. Fig. 1C), justifying the use of the proposed mathematical form for the binding rate ( , 0 ) = 0 at the beginning of the network growth. In the main text, we show, by comparison with the data, that in fact the form ( , 0 ) = , i.e. the rate of binding being limited by the local, not initial, concentration of ADF/Cofilin in the solute, , leads to the model predictions that fit the data very well.
In the experiment, networks grow in a large, several square millimeter sized chambers, so that the total amount of ADF/Cofilin is not limiting, and moreover, over the time of the experiment, about 60 minutes, diffusion is not fast enough to diminish the ADF/Cofilin concentration in the solute farther than a few hundred microns from the growing network (for relevant estimates, see Boujemaa-Paterski et al. (2017).) For this reason, we performed the simulations in the area 1 millimeter in size, smaller that the size of the whole experimental chamber, but large enough so that the concentration of free ADF/Cofilin at its boundary is almost identical to the initial ADF/Cofilin solute concentration. This justifies using Dirichlet boundary conditions also for , rather than no flux boundary conditions. For the actin density, we assume ( , , ) ≡ 0 whenever ( , ) ∉  . Within the network, we use two scenarios: The actin density is constant, or the actin density is a function of only, i.e.
( , , ) = ( ), where we use the measured actin density along the network, averaged over its width and fitted using a smoothing spline. The smoothing avoids potential numerical problems when solving partial differential equations due to the roughness of the measured data. Since large density gradients can be expected only near the network, we used a triangular FEM-mesh that is much finer on and near the network than far away from it. Overall, the number of mesh triangles was between 2000 and 5000 for each simulation. The mesh itself was time-independent, avoiding having to re-mesh at each time step. Next, we employed a forward-Euler finite-difference scheme for the transport term + , with shifting the calculated values of and subsequent interpolation onto the elements of the mesh. In our discrete network model, we describe the network as a collection of nodes and edges in 2D. Each edge represents an array of actin filaments, each note represents cross-linking or branching points. Our discrete network model follows ideas presented in Carlsson (2007); Michalski and Carlsson (2010Carlsson ( , 2011, however we allow our breakage rate to depend on local actin density and present a new analytical approximation (see Sec. Derivation of analytical model of network fragmentation). At its fully connected state each inner node is connected to four edges. We represent the whole network as a graph, i.e. for each node, we track the nodes to which this given node is connected to. During each time step, the discrete network model is updated in four steps: 1. Remove individual nodes. Given an actin density for a given node (step 4 below), we determine a breakage rate per node and time = , where parameter is a constant. The node breakage follows a Poisson process with rate , and we determine the probability of breakage at each time step as 1 − − Δ . 2. Remove edges and network pieces. There are two ways edges can be removed: Individually -this happens if both nodes an edge is connected to are removed. On the other hand, a larger network piece could become disconnected as a consequence of step 1. We considered network segments to be disconnected if they have no connection to the leading edge (i.e. there is no path of edges connecting the given piece to the leading edge) and assume disconnected network pieces diffuse away quickly. 3. Grow network. This step simply adds rows of nodes and edges at the leading edge proportional to the network growth speed . 4. Calculate local actin densities. In our model, the local actin density depends on the number of edges present, not the number of nodes, i.e. if a node is removed within an otherwise fully connected patch, the actin density would not be affected. To calculate the local actin density at a node, we count the number of missing edges within a square patch around the node with a of nod = 2 (in units of the edge length), i.e. we are considering the 24 nodes or 40 edges around the given node. Finally, the determined fraction of the unbroken edges was multiplied by the model parameter 0 , the initial actin density. Implementation and Parameters. 920 We performed numerical tests and found that as long as the node number along the leading edge, , is larger than ≈ 20, the equilibrium length is barely affected by the choice of (as noted also in Carlsson (2007)). We therefore decided to use = 30, 60 and 180 for 15 , 30 and 90 networks respectively. We used the time step of Δ = 1min. For Figure 4D we varied the initial actin density 0 and the exponent . For illustration purposes, we also changed the parameter and used = 0.25, 2.5, 25 and 250 for = 0.5, 1, 1.5 and 2 respectively (otherwise the obtained network lengths would differ by orders of magnitudes, making the visualization less clear). For the simulation in main Figure 4B we used width= 90 , = 2, 0 = 100 , = 194 2 ∕min, = 0.8 /min, for the comparison shown for the network in Figure 4E we used width= 15 , = 2, 0 = 400 , = 250 2 ∕min, = 1.5 /min. We represented the network as an undirected graph using Matlab routines, allowing to quickly determine connected components, which can be a time-consuming step. Edges that are connected to only one node can be represented as a loop, i.e. we formally connect both edge ends to the same node. Finally, if a node has two or three edges that are connected to only this node, this can be accounted for by assigning a weight of two or three to that edge. This is necessary to keep edges unique in the graph-based description. For example, a weight of two means that this edge counts twice when determining actin densities.

939
In this section we describe an analytical approximation of the network length and actin density along the network of the discrete network model of Sec. Stochastic fragmentation model: details and simulation. In the discrete network model, we describe the whole network as a collection of nodes and edges, representing branches and connecting actin filaments respectively. At its fully connected state each (inner) node is connected to four edges. Each nodes is being broken with a probability per node and time, that will depend on local properties of the network. An edge (i.e. actin filament segment) is removed only if either both of the nodes it is connected to are broken, or, if it is removed as part of a larger patch that is being disconnected. The analytical results can be summarized as follows: If the breakage rate per node is = , where is the (constant) concentration of bound ADF/Cofilin in the network, ( ) is the actin density, is the network growth speed and is a dimensional constant proportionality coefficient, then the equilibrium network length is given by: Note that this integral is finite for any choice of . If 0 is the initial actin density, then the actin density along the network in equilibrium is given by where ( ), the fraction of broken nodes along the network, is the solution of the ordinary differential equation: A: In the absence of removal of larger pieces of the network, the deletion of a node will only affect edges that are connected to this very node, and only if those edges are unconnected at the other ends. The expected number of such edges for an unbroken node is: no. of edges per node × prob. that the node at the other end is broken = 4 0 .
We model continuous densities and using the following equations: The second equation is simply stating that the rate of edge removal is equal to the rate of the node removal times the expected number of edges connected to the node being removed. In the first equation, expression ( 0 − ) accounts for the node breakage with rate . Factor 1 + ∕ 0 (1− ∕ 0 ) 2 in this equation accounts for the factor B: if the network connectedness is low, then per each removed node, more nodes could be removed. This factor is equal to 1 for very low density of broken nodes, and has to be an increasing function of the variable ∕ 0 . Rather than using theoretical arguments to try to find this function, we simply used a few tens of simulations of the discrete stochastic model to estimate numerically the average number of the nodes removed for each randomly removed node at any given density of the broken nodes. The function 1 + ∕ 0 (1− ∕ 0 ) 2 approximated the numerical data well for 0 < ∕ 0 < 0.7. For larger ∕ 0 the network is already largely falling apart. We found that using more complicated functions to approximate the behavior hardly affects the predictions of network length and density. Adding transport effects 996 We introduce the space and time dependent fraction of broken nodes = ∕ 0 and the rescaled actin density = ∕ 0 , and note that values of and are connected by the relation = ∕ √ 0 , where 1∕ √ 0 is the approximate edge length in 2D. Since both edges and nodes are being transported within the network at speed , we can replace Eqs. (12) by the following PDE system for densities ( , ) and ( , ), where is the distance along the network: Explicitly calculating the network equilibrium length. 1007 In equilibrium, system (13) takes the form: Using the separations of variables, we can rewrite Eq. (14) and find the following equation for the equilibrium length : The final result depends on the choice of how the breakage rate depends on the actin density and the average amount of bound ADF/Cofilin . We assume to be constant and use the breakage rate in the form: where is the proportionality constant. This implies that the equilibrium length is given by: The integral can be evaluated exactly: This is the formula used to compare the simulated equilibrium length to the calculated one in the main Figure 4D. Note that, given , there are no free parameters, i.e. the lengths are determined exactly. As described in the main text, we used the measured equilibrium lengths, actin densities and amounts of bound ADF/Cofilin to determine exponents and , and found good agreement for = 2, = 1. This lead to main Eq. (5) and is one of the ingredients used to calculate the equilibrium length below in Sec. Equilibrium lengths of homogeneous and heterogeneous networks.
First, we estimate the average amount of bound ADF/Cofilin in a network of a given length . We use the estimate for the density of ADF/Cofilin in the solute in the vicinity of the network, derived in the main text (Eq. (3)), Substitution of this expression into the equation for bound ADF/Cofilin (Eq. (10)) yields: where we define the depletion factor ( ) as: In our case, ≈ 0.3∕ min, ≈ 1.5 ∕ , ≈ 30 , ≈ 600 2 ∕ , ≈ 100 , ≈ 0.5∕ ∕ , and so ≈ 10 −2 . Hence we approximate the amount of bound ADF/Cofilin by the limit → 0, yielding: The average amount of bound ADF/Cofilin, calculated as 1 ∫ 0 ( , , ) d , is therefore given in Eq. (6): From the network fragmentation model (see Sec. Derivation of analytical model of network fragmentation) and the data, we found that for a given amount of bound ADF/Cofilin , the network length is given by: where = 0.0669 3 × is the proportionality constant found using fitting to the data. Solving Eqs. (18)-(19) for and gives the equilibrium length * : This is the formula used for all equilibrium length predictions for Figure 2. Note that since the expression under the square root is always positive, the model predicts that the networks always reach an equilibrium length. Before the equilibrium length is reached, the length is simply given by = , which explains the plateaus in Figure 2B and Figure 2D: According to the model, the networks had not yet reached equilibrium length at = 20 min and = 36 min.
For heterogeneous networks we need to determine both equilibrium lengths 1 , 2 for two sub-networks, i.e. we need to formulate Eqs. (18) and (19) separately for each sub-network. We denote by 0,1 and 0,2 the initial actin densities at the leading edge and by 1 and 2 the sub-network growth speeds. Since the networks are competing for the same pool of diffusing ADF/Cofilin, we assume that there is a common depletion factor, which we denote by ℎ ( 1 , 2 ) and model by the expression: ℎ ( 1 , 2 ) = + ( 0,1 1 + 0,1 2 ) , which takes into account the different densities and network lengths. Now we replace Eq. (18) by: All that remains is to solve the four equations (20)-(21) for 1 , 2 , ,1 and ,2 . We find the relation The expressions for the equilibrium lengths are very similar to the homogeneous network case -in fact if 1 = 2 and 0,1 = 0,2 , they simplify to the case of one single network with width 2 .
In this section we model the shape of a network consisting of two sub-networks having different actin densities and/or growth speeds. We start by assuming that their material properties are constant along the network. We denote by 1 > 2 the growth speeds of subnetworks 1 and 2 respectively. As the network assembles, two sub-networks are effectively 'glued' together. In our simple model, we assume the sub-networks to be elastic. We model each sub-network segment by two springs, placed at a distance (the network width) from each other (App. Fig. 2). The springs at the interface Γ between the sub-networks are connected and forced to have the same length (representing the 'glued' together condition). The differences in sub-network growth speeds lead to different resting lengths 1 and 2 , proportional to the respective speeds. To account for elastic effects, we assume that the differences in density lead to different elastic moduli 1 and 2 , and hence to different spring constants 1 ∝ 1 ∕ 1 and 2 ∝ 2 ∕ 2 . We call the length of the outermost and innermost spring 1 and 2 respectively; the length of the two springs at the interface therefore has to be ( 1 + 2 )∕2. Results. 1146 To obtain the equilibrium lengths of the sub-networks, we minimize the elastic energy of each segment, which is given by adding the potential energies for each of the four springs: pot = 1 2 ( 1 − 1 ) 2 + 1 + 2 2 − 1 2 + 2 2 ( 2 − 2 ) 2 + 1 + 2 2 − 2 2 .
These are the formulas used to calculate the curves shown in Figure 6B,C in the main text. Note that: In other words, if one of the sub-networks is much stiffer that the other one, the heterogeneous network will become straight and grow with the speed of the stiffer sub-network.
Since our discussion so far concerned local properties of the sub-networks, we can account for changes in density along the combined network simply by making the elasticity ratio , and hence the curvature , a function of arc length along the network . Then, curve Γ( ) at the interface of the two sub-networks can be parametrized as ↦ Γ( ) = (Γ 1 ( ), Γ 2 ( )) (App. Fig. 2), where: In case of constant material properties along each sub-network (as shown in Figure 6A in the main text), these expressions simplify to: Γ 1 ( ) = 1 − cos( ) , Γ 2 ( ) = sin( ) .
Since * 1 > * 2 , we can approximate the elasticity ratio of the two sub-networks as: These are the formulas used to calculate the network shapes shown in Figure 6E in the main text. Note that in the figure, networks have been rotated to match the experimental set-up.