Network heterogeneity regulates steering in actin-based motility

The growth of branched actin networks powers cell-edge protrusions and motility. A heterogeneous density of actin, which yields to a tunable cellular response, characterizes these dynamic structures. We study how actin organization controls both the rate and the steering during lamellipodium growth. We use a high-resolution surface structuration assay combined with mathematical modeling to describe the growth of a reconstituted lamellipodium. We demonstrate that local monomer depletion at the site of assembly negatively impacts the network growth rate. At the same time, network architecture tunes the protrusion efficiency, and regulates the rate of growth. One consequence of this interdependence between monomer depletion and network architecture effects is the ability of heterogeneous network to impose steering during motility. Therefore, we have established that the general principle, by which the cell can modulate the rate and the direction of a protrusion, is by varying both density and architecture of its actin network.

C ell migration is an evolutionary conserved mechanism, essential for the proper development of living organisms 1 . A fundamental and still open question in biology is how cells direct their migration in response to external signals 2,3 . Much effort has been focused on understanding the mechanism of the first step in this process: membrane protrusion and its regulation 4 . Actin polymerization produces the intracellular force 5 that protrudes a thin and flat structure, called lamellipodium, which borders the leading edge of a motile cell over tens of micrometers [6][7][8] . The lamellipodial actin is a densely branched and dynamic meshwork [9][10][11] . Near the cell membrane, sustained Arp2/3-mediated dendritic nucleation 12  disassembly of the lamellipodial actin network are finely tuned in space and time through coordinated activities of regulatory factors 13,14 . Collectively, these processes generate cohesive branched actin networks 9,11,15 , along the leading edge that expand locally leading to directed motility in response to environmental cues 6,7,16,17 . Steering during motility is tightly linked to regulation of the Arp2/3-branching activity 18 . However, how actin-network organization and growth regulates steering is unclear 3 .
In vitro reconstituted propulsion of bacteria, viruses or small particles brought insights on how a minimal set of two molecular activities-Arp2/3 complex-driven nucleation and barbed-end capping by capping proteins-can result in the growth of protrusive actin organization [19][20][21][22][23][24][25] . The surface density of the nucleation-promoting factors (NPFs), the size and shape of the motile particles and the viscosity of the medium affect the velocity of propulsion 21,26,27 . In addition, a growing actin network is a mechanosensitive system that can respond and adapt to external forces 28 . However, we know little about how actin polymerization defines the rate of growth of a branched actin ultrastructure pushing against a load.
Here, we asked how the architecture of a branched actin network affects its growth and investigated the key parameters controlling speed and steering during motility. To achieve a high precision in controlling the organization of a growing branched actin network, we developed a methodology that combined contactless micropatterning of variable concentration of NPFs 29,30 , with an in vitro reconstituted actin-based motility assay 24 . Using this approach, we generated a diversity (in terms of size (geometry) and NPF concentration) of nucleation areas and studied their impact on the growth of branched actin networks. At the same time, we used quantitative fluorescence imaging to determine the density of the branched actin network and its relationship with network growth behavior. To explain the growth rate of the actin network, we developed a mathematical model relying on minimal assumptions. The model revealed that the local actinmonomer concentration at the site of active nucleation and the architecture of the branched network are the two fundamental parameters controlling motility in our experimental system. Our model was validated by a series of experiments where the growth behavior of the actin network was modulated by the geometry, density and composition of the nucleation area. In agreement with the model predictions, we reconstituted controlled steering of heterogeneous actin networks using NPFs patterned at a submicrometer scale. Therefore, the fine-tuning of only two parameters was sufficient to fully recapitulate the observed growing behavior of a branched actin network.

Results
Parameters controlling the actin network growth rate. To investigate how the organization of actin filaments modulates actin-based motility, we reconstituted in vitro branched actin networks with a diversity of nucleation geometries and characterized their growth dynamics. We assembled actin networks on functionalized micropatterned surfaces uniformly coated with NPFs, in the presence of a defined set of purified cellular factors ( Fig. 1a- 28,30,31 ). We imaged fluorescently labeled actin to follow actin-network assembly. This novel versatile method allowed actin assembly at a nucleation site and the growth of actin filaments at their barbed ends to be geometrically constrained (see two-color experiments Supplementary Fig. 1a), which in turn induced the growth of a cohesive actin network restricted in the extent of his growth by the presence of capping proteins ( Supplementary Figs 1b and 2a,  b). Hence the method was used to assemble thin and flat Arp2/3generated lamellipodium-like structures thereafter referred to as "LMs" (Supplementary Fig. 2a, b compared LMs with a classical bead comets Supplementary Fig. 2c, d).
Because the geometry of the nucleation sites could be altered with the patterning method, we compared the configuration of LMs from a functionalized NPF-bar-shaped pattern of 3 × 15 µm 2 (Fig. 1a, c and Supplementary Movie 1) and thin-tail branched networks from NPF-spots of <1 µm 2 (Fig. 1b, c and Supplementary Movie 1). Interestingly, the growth rate of the "restricted" networks (i.e. a restriction imposed by the presence of capping proteins) varied with the geometry of the nucleation area. LMs from 15-µm bars grew significantly slower than those from small spots (Fig. 1c). This difference was not due to a dependency of actin assembly on the geometry of the nucleation area because, regardless of the nucleation area and its geometry (bar vs. dot, Fig. 1d, e and Supplementary Movie 2), with "unrestricted" actin networks (i.e. in the absence of capping protein), the network growth rate was not statistically different from the rate measured for individual (free) actin filaments (Fig. 1f, g).
The rate of free actin filament elongation was expressed by the canonical equation for actin filament elongation, whereby, V 0 = k on × δ × C, where k on ≈ 10 µM −1 s −1 is the polymerization rate constant, δ≈ 0.003 µm is the half-size of actin monomer, and C is the local actin monomer concentration (Supplementary Methods 32 ). This equation predicted a free polymerization rate ≈ 0.03 µm µM −1 s −1 , which was in agreement with the measurements for single filaments and unrestricted networks. In comparison to single filament assembly, quantitative analysis of the restricted network revealed a 6-and 1.6-fold decrease in Fig. 1 Nucleation geometry controls the actin network growth rate. a, b The growth of different restricted actin organizations. a Actin network emerging from 3 × 15 µm 2 GST-pWA-coated bar. Conditions: 6 µM G-actin Alexa-568 labeled, 18 µM Profilin, 120 nM Arp2/3 complex, 25 nM CP. b Same as a but for GST-pWA-coated sub-micron dot. c The growth rate of actin networks was calculated in µm s −1 µM −1 of available [G-actin]. Red bars represent mean speed values. Error bars show mean s.d. for n = 20 LMs from four experiments a and n = 13 LMs from five experiments b. ****p < 0.0001, multiple comparison Šídák method. d, e The growth of different unrestricted actin networks. d Actin assembly from 3 × 15 µm 2 GST-pWA-coated bar. Conditions: 1 µM G-actin Alexa-568 labeled, 3 µM Profilin, 80 nM Arp2/3 complex e same as d but for 1 × 1 µm 2 GST-pWA-coated dot. f The assembly of single actin filaments. Conditions: 1 µM G-actin Alexa-568 labeled, 3 µM Profilin. In f each filament is identified by a star. In a, b, c and d: purple shows the nucleation sites; in b, speckles in the comet tail indicated by colored stars were used for speed measurement. g For homogeneity and for each condition, the growth rate of actin filaments or actin organizations was calculated in µm s −1 µM −1 of available G-actin. Red bars represent mean speed values provided above. Error bars show mean s.d. for n patterns per condition, n = 26 from four experiments d, n = 11 from three experiments e and n = 26 from four experiments f. Adjusted p values were computed according to Šídák method to assess statistically significant difference in mean growth rates measured for each condition. h Cartoon for the reconstituted flat LMs: LMs grow from NPF-coated patterns (light red rectangle) printed on Silane-PEG-coated slide (darkened glass). The height between glass and coverslip was controlled by two pieces of calibrated tape (yellow walls). Schematic of actin assembly and the parameters used in the model equation: Network growth rate "V" depends on (i) the barbed end rate of assembly (k on ), (ii) the monomer size (δ), (iii) the local monomer concentration (C, factor 1) at the nucleation site, and (iv) a geometry/mechanical factor Φ that integrates the pulling force (black arrows, factor 2) and filament orientation with respect to the load that modulates the pushing force (yellow arrows, factor 3). Grey dots: actin monomers and subunits. Grey gradient: gradient of actin monomers (darker color for higher (C)). Red dots: transient actin attachment to the load through the NPFs-Arp2/ 3 complex-actin filament ternary complex (see also related Supplementary Figs 1 and 2). Scale bars in a, d are 15 µm. Scale bars in b, e, f are 10 µm the growth rate of LMs and the thin actin tails, respectively. Based on these results and previous experimental and theoretical work 5, 22-24 , we formulated a minimal mathematical equation (the actin network growth-rate model) that best described the growth rate of LMs (Fig. 1h). In this equation, the network growth rate is a function of: (i) the barbed end rate of assembly (k on ); (ii) the monomer half-size (δ); (iii) the local monomer concentration at the nucleation site, which was calculated by solving a set of differential equations for monomer diffusion and assembly using an experimentally determined diffusion coefficient (Supplementary Fig. 3 and Supplementary Methods); and (iv) a geometry/mechanical factor that resists against network growth. This latter geometry/mechanical factor Φ integrates the impacts of (i) the angle of actin filaments impinging the nucleation site (geometry/architecture factor), and (ii) the transient tethering 22 of de novo nucleated filaments (mechanical factor) ( Fig. 1h and Supplementary Methods). Mathematical estimates of network-growth rates using this equation showed unambiguously that the effect of local monomer depletion was negligible for thin tails formed on an NPF-spot ( Fig. 1b) compared those formed on broad LMs (Fig. 1a). This allowed for a direct experimental measurement of the geometry/mechanical factor that resists against network growth: Φ = V thin tail /V free polymerization = 0.7 (Supplementary Methods). Given our expectation that the network architectures and dynamics of actin tethering were the same in both thin tails and broad LMs, the value of Φ was kept constant. Thus, according to the actin network growth-rate model, the significant slowdown of the growth rate measured for broad LMs compared with the thin actin tails was due to either a local decrease in monomer concentration because of monomer consumption at the site of active assembly, or due to mechanical friction in the actin network, or due to both.

Monomer depletion occurs by assembly at the nucleation site.
To distinguish between the effects of actin monomer depletion or mechanical friction on network growth rates, we compared the growth rates of two similar but physically independent networks when they grew distant (25 µm apart) or proximal (6 µm apart) from each other ( Fig. 2a and Supplementary Movie 3). We observed a significant drop in the growth rate of the two proximal networks compared with the two distal networks (Fig. 2b). One explanation for this drop is that the two distal networks (Fig. 2a, two bars separated by 25 µm) use monomers from two separate areas around them, whereas the two proximal networks use monomers from areas that overlap (Fig. 2a, two bars separated by 6 µm) and therefore the overlapping areas lead to a higher depletion in the local monomer concentration at the site of active nucleation.
To further explore the relationship between the size of the nucleation area and the extent of local monomer depletion, we analyzed the growth rates of LM of different widths generated on bar-shaped patterns of increasing size (15, 30 and 90 µm) in "2D" (Fig. 2c, d and Supplementary Movie 4). The growth rate of the LMs decreased as the size of the nucleation area increased (Fig. 2d, e, black symbols). The processing of the data using the actin network growth-rate model (Fig. 2e, red symbols) and keeping the geometry/mechanical Φ factor constant for these LMs (Supplementary Methods) revealed that reduced growth rate was due to a greater depletion of monomers at the site of active nucleation of wider LMs.
To confirm the above relationship, we calculated the local concentration of actin monomers at the nucleation sites when the LMs were assembled in a 2D configuration. We solved equations for monomer diffusion and actin assembly based on the controlled parameters of our reconstituted systems (Supplemen- tary   . c-f Actin assembly was followed for LMs growing from NPF-coated nucleation bars of 3 µm width and increasing length, as mentioned. The growth was followed for 2D-like configuration in small volume with 4.5 µm space between glass and coverslip c, d, e, or for 3D-like configuration in a larger reconstituted volume with 70 µm space between glass and coverslip f, g, h.  Fig. 4a, b). Moreover, we found that this depletion effect depended on the size of the nucleation site ( Supplementary Fig. 5a, b), and on the distance between nucleation sites ( Supplementary Fig. 6). After 20 min of actin assembly, monomer concentration drastically dropped to 32, 22, 12% of the initial concentration for the 15, 30 and 90 µm patterns, respectively, in a 5 µm-wide border around the nucleation site ( Supplementary Fig. 5a, b). By processing these data using the actin network growth-rate model and using the parameters described in Fig   Indeed, given the steady-state cumulative length of LMs, our assays contained a total amount of actin monomers approximately 13 orders of magnitude larger than the number of actin subunits assembled in the F-actin networks. Similarly, quantitative estimates show that the consumption of Arp2/3 complex or capping protein by the networks were not significant enough to deplete the local concentrations of these molecules (Supplementary Methods). Thus, our results demonstrated that the sustained assembly at the nucleation site established diffusive gradients that led to local monomer depletion.
To further validate the monomer-depletion hypothesis, we extended our model to consider diffusive monomer gradients in a 3D configuration ( Supplementary Fig. 4c, d). The model predicts that the monomer flow towards the nucleation site should be higher in 3D than in 2D (12-fold in the case of Fig. 2f  monomer depletion and nucleation area held in the 3D configuration (Fig. 2h). Importantly, the results obtained from the comparison between the 2D and 3D configurations argue against a strong effect during LM growth of the friction of the filaments against the wall of the experimental chamber. On the contrary, the difference in growth rates between 2D and 3D can be fully accounted for by the difference in the local monomerdepletion effect, without changing the geometry/mechanical factor in the equation for the growth rate. Hence, we reasoned that increasing viscosity by adding methylcellulose would reduce monomer diffusion but should have a minimal effect on the mechanical friction (Supplementary Methods). In this regime and as expected, LM growth rate was slightly lower with higher viscosity (Supplementary Fig. 5f).
A new experimental setup to control growing actin network.
To uncouple the contribution on actin growth of the orientation of filaments within the network from the positions at which they are tethered within the nucleation site, we developed a novel and versatile experimental method that allows the precise control of the spacing between the nucleation spots (i.e. spot densities) in the nucleation area (Fig. 3). To this end, we used a pulsed UV laser to print nucleation patterns that consist of arrays of nucleation spots of a predefined density ( Fig. 3a and Supplementary Movie 5). We only used spot densities that led to the reconstitution of continuous LMs on the patterns. As the branching reaction is confined to the surface of the nucleation spot (300 nm in diameter) and the actin filaments extend outside the spot (reflecting the continuous aspect of the LMs), we hypothesized that the distance between spots controls both the orientation of actin filament within the network and the density of filaments tethering in the nucleation area. Hence, by varying the density of the nucleation spots and/or the amount of NPFs grafted to these spots, we were able to modulate the geometrical organization and density of an actin network (Fig. 3b). For every spot density evaluated, the amount of NPFs grafted per nucleation spot remained constant for each given NPF concentration ( Fig. 3c and Supplementary Fig. 7). Accordingly, the density of the spots correlated well with the concentration of NPFs (Supplementary Fig. 7). Moreover, the density of spots correlated well with the fluorescence intensity of the LMs (Fig. 3c). Therefore, this method appeared suitable to fine-tune the overall filament organization and density of the LMs.
Actin filament density modulates LM growth rate. Using the method described above, we explored how actin-filament density controlled the growth rate of the branched networks ( Fig. 4a and Supplementary Movie 6). Using patterns with high-spot densities and various concentrations of NPFs, LMs were generated with different filament densities, which were quantified by assessing their fluorescence intensity as function of the NPF concentration (Fig. 4b). We analyzed the LM growth rate as a function of LM fluorescence, which in turn is dependent on filament density (Fig. 4c). Our results showed that LMs generated with low-NPF concentrations contained lower densities of filaments and had higher growth rates than the LMs generated by higher NPF  (Fig. 4c, black dots). This result was consistent with the local monomer depletion hypothesis, in that the LMs with higher filament densities will consume more actin monomers than those with lower filament densities. Processing the data through the actin network growth-rate model using the parameters derived from the preceding experiments, the decrease in the LM growth rate was satisfactorily simulated as a function of the increased density of the network (Fig. 4c, red line). We therefore concluded that for nucleation areas, which have the same spot densities, the growth of higher filament density networks leads to higher local monomer depletion, which in turn slows down the network-growth rate.
Actin network architecture dependence of LM growth rate. We then hypothesized that actin filament arrays tethered by nucleation spots were more effective at developing pushing forces from elongation than non-tethered actin arrays (see cartoons with schematic network growth in Fig. 5a). To address this hypothesis, we compared the growth rate of LMs generated by two different (low and high) densities of spots in the same-sized nucleation area and with the same NPF concentration per spot ( Fig. 5a and Supplementary Movie 7). We confirmed that the distance between the nucleation spots controlled the density of LMs (quantified by LM fluorescence ; Fig. 5b). Unexpectedly and seemingly in contradiction with the above results (Fig. 4), LMs  Fig. 6 Density and geometrical organization interplay to control the steering of heterogeneous networks. a Cartoon predicting heterogeneous actinnetwork-growth behavior. If the two densities are generated by the same pattern family (i.e. two positions along the same curve; Case A), the steering is predicted to be always towards the denser network; if the two densities are generated by two different pattern families (two positions from two different curves Case B), the steering can be towards the sparser network. Solid curve refers to networks generated by high-spot-density patterns; dotted curve refers to networks generated by low-spot-density patterns. Double-density branched actin networks were polymerized on heterogeneous patterns. Patterns consisted either in an array of spots of the same density, with one half coated with high and the other half with low amounts of NPFs per spot b, or in an array of spots of two distinct densities both coated with the same amount of NPFs per spot c. Network curvatures, represented by the traces of the boundary between high-and low-density sides, were reported at 40 min of assembly in d, e, referring to b, c respectively. The red dashed line in d, e represents the predicted curvature according to the model. f Complex heterogeneous patterns were generated. A gradient of high-to low-spot density (left panel) or a symmetric and alternate low-high-low spot density (right panel). g LMs were polymerized on nucleation zones as indicated, and their curvature measured. High and low densities were 6.6 and 2.5 spots µm −2 , respectively. Gradually sparse patterns were a series of 3 × 5 µm 2 rectangles of decreasing spot density (from 8.3 to 2 spots µm −2 ). The "low-high-low" patterns were a series of 3 × 10, 3 × 6, 3 × 10 µm 2 rectangles of 2.5, 6.6, 2.5 spots µm −2 , respectively. ***p < 0.001, multiple comparison Šídák method. Scale bars are 15 µm (see also Supplementary Fig. 9) generated by nucleation areas with a high spot density had an overall growth rate 1.3-fold greater (statistically significant) than LMs generated by nucleation areas with a low spot density. Interestingly, when these growth rates were plotted as a function of LM filament density (i.e. actin fluorescence intensity; Fig. 5c), the growth rates for LMs with identical filament densities were greater for LMs generated by nucleation areas with a high-spot density than for those generated by nucleation areas with a lowspot density (Fig. 5c, black dots above open dots). We attributed the lower growth rate with nucleation areas of low spot density (Fig. 5c) to the contribution of the geometrical organization (architecture) of actin filaments (see cartoons in Fig. 5a). We hypothesized that LMs comprised two sub-populations of actin filaments: a population that is more effective for protrusion (i.e.   Mechanochemical model for growing branched actin network. a Network growth rate (green box) can be summarized by three terms; (i, white box) V 0 , free polymerization rate of barbed ends; (ii, yellow boxes) the monomer depletion factor, which is inversely proportional to (χ × η × Φ), where χ represents monomer consumption due to polymerization corrected by diffusion, η is the network density, and Φ is the network geometry factor of 0.7 calculated in Fig. 1; and, (iii, blue boxes). As depicted in yellow boxes, because soluble proteins were maintained constant throughout the experiments, higher nucleation density (pink to red), translates into higher filament density and yields higher local monomer depletion, thus lower growth rate. Blue boxes summarize the influence of the geometry factor according to which filaments push more efficiently against the load when they are in contact with the NPF area. Accordingly, actin filaments not in contact with the nucleation area do not produce as much force and are more prone to bending and/or capping. b Rationale for the control of speed and steering during actin network growth according to two scenarios based on actin filaments density and network architecture. The first is, from (1) to (2), where actin networks evolved by a geometrical reorganization of the network at a constant network density, or from (1) to (3) where actin networks evolved by maintaining the geometry of the network constant but increasing its density. As depicted in cartoons, for the compacting networks-the (1) to (2) case-as the network changes from low (1) to high (2) organization, the density is constant but the efficiency of the network pushing against the load is higher. Therefore, the actin network in (2) is growing faster than the network in (1). For concentrating networks-the (1) to (3) case-individual networks generated by nucleation surface unit (pink to red dots) become denser, without changing their organization. Therefore, the actin network in (3) is growing slower than the network in (1). According to these simple rules, heterogeneous networks will steer (or turn) towards the less growth-efficient actin network overall developing pushing forces) because the filaments are tethered at NPFs spots; and a population that is less effective for protrusion because the filaments are situated between NPFs spots. Based on this hypothesis, we assumed a 2-fold decrease in the geometry/ mechanical factor Φ between LMs generated by low vs. high spot density nucleation areas. Using this assumption in the actin network growth-rate model, the LM growth rates were satisfactorily simulated (Fig. 5c, red dotted and solid lines; Supplementary  Fig. 8). Accordingly, we concluded that the LM-growth rate is dependent on filament density-via the extent of local monomer depletion-and on filament orientation and tethering that controls the efficiency by which filaments develop pushing forces and hence protrusions.
Density and architecture control steering of actin network. To investigate how the local regulation of LM growth rates could impact the steering of protrusions, we generated heterogeneous LMs made of different nucleation-spot densities. We tested two different conditions to manipulate the heterogeneity of the growing actin network. First, the two halves of the nucleation area differed in the NPF concentrations per nucleation spot but had the same nucleation-spot density (Fig. 6a, Case A and Supplementary Movie 8 left); and second, the two halves of the nucleation area differed in nucleation-spot densities but had the same NPF concentration per spot (Fig. 6a, Case B and Supplementary Movie 8 right). In Case A (Fig. 6a), we predicted that when the nucleation architecture is constant (Φ is constant), the filament density within the network controls the growth rate as a function of the local monomer concentration. Therefore, the denser side of the network depletes more monomers than the sparser side ( Supplementary Fig. 9a-d). In accordance with the prediction, the growth rate at the side of the nucleation area with the denser network of actin filaments was lower than at the side with a sparser network of actin filaments and, the overall direction of network growth was deflected (i.e. steered) towards the denser side (Fig. 6b, d).
In Case B and according to our prediction, the geometry/ mechanical factor Φ would control the growth rate as a consequence of the different nucleation spot densities (Fig. 6a,  c). Indeed, the network growth rate at the side with the lower spot density was lower than that at the side with the higher spot density, and hence the growth of the heterogeneous network steered towards the side with a lower spot density (Fig. 6c, e and Supplementary Fig. 9e-h). Moreover, the side with the lower spot density also generated a filament density that was lower than at the side with the higher spot density. Therefore, these results show that the direction of network growth can be modulated by the architecture of the branched network, in addition to and concomitant with the density of the actin filaments. To further examine this general rule about steering control during LM growth, we evaluated more complex patterns of nucleation areas consisting of a graded density of nucleation spots (Fig. 6f, left panel and Supplementary Movie 9) or of a central area of high-spot density surrounded by two areas of low-spot density (Fig. 6f, right panel). Interestingly, the actin growth-rate model could account quantitatively for the steering of this complex actin network (Fig. 6g, red symbols and Supplementary Fig. 9d, h). Therefore, we conclude that the density of the nucleation spots and the resulting architecture of the branched actin network determine the growing properties of LMs and emerge as critical factors in controlling the steering of LM growth.

Discussion
This study has established how the heterogeneity in a branched actin network can control its growth rate and the orientation of this growth (i.e. steering). Specifically, by combining experimental observation and theoretical modeling, we have demonstrated that actin-monomer depletion and the architecture of the actin filaments at the site of assembly are critical to this control during LM growth. Therefore, we propose that the fine-tuning of these two parameters within the cell enable a diversity of branched actin network growth behaviors that are fundamental to controlling cell motility and its steering.
The most dramatic effect on the rate of growth of the experimental actin networks was obtained when the size and/or the NPF density of the nucleation area were increased (Figs 1a, b, 2d,  4a, 5a). How can these two related variables affect the rate of growth? At the point of contact with the patterned NPFs, actinfilament nucleation and elongation consume rapidly the available local pool of actin monomers. This generates a local depletion of available monomers slowing down filament elongation and therefore the growth rate of the LM. A variation in the density of filaments in contact with the nucleation area will have therefore a direct effect on the LM growth rate via monomer depletion (Fig. 7a, "Filament density"). Indeed, a local increase of NPF concentration tends to generate a greater local depletion of monomers and thus to locally slow down filament elongation forcing the direction of network growth overall to turn towards such regions where the filament density is high (Fig. 7b, "concentrating" scenario). According to this description of actin-based motility, the dynamic localization of actin monomers will provide a potential spatiotemporal mechanism to regulate the protrusion efficiency during cell locomotion. This view is supported by early theoretical work 33 and a recent in vivo study on neuronal motility 34 . In this latter study, the modulation of the expression of thymosin β4, a monomer sequestering protein, regulates the local pool of actin monomers at the leading edge of the cell and the underlying LM protrusion and growth cone motility 34 . To maximize the protrusion, cells may locally adopt a denser and more homogeneous distribution of the nucleation promoting complexes (Fig. 7, solid curve in the plot of V vs. η), ensuring thus an optimal filament density, leading to optimal network stiffness in order to resist the membrane tension and induce protrusion, but with a limited effect on the local monomer depletion.
Our results demonstrate that the NPF distribution at the site of nucleation directly impacts LM growth rate (Fig. 5). We propose that two populations of actin filaments are present in contact with the site of nucleation (Fig. 7a "Geometrical organization"). One population that is effective at force production during LM growth because it contains actin filaments transiently tethered with NPF spots 22 ; and a second population that is not effective at force production because it contains actin filaments present between NPFs spots and not directly tethered to them 22 . Indeed, a local increase of NPF packing will generate a denser network with a higher pushing efficiency, forcing the network to turn away from the region where the filaments are tethered and potentially at high density (Fig. 7b, "compacting" scenario). The dependence of the rate of growth with the architecture of actin branched network is consistent with the relationship between LM architecture and protrusion behavior 3,35 . However, in the cellular context the contribution of actin filaments within the LM generated by additional factors including formins or ENA/VASP, introduces another level of complexity in the regulation of protrusion speed and force generation 36,37 .
Chemotaxis and haptotaxis cues as well as signaling feedback loops are known to either promote or silence Arp2/3 complexmediated branching during the steering of cell motility 3,6,7 . In the case of haptotaxis, cells can sense differences in extracellular matrix (ECM) composition and modulate their Arp2/3 complexdependent nucleation to adapt and migrate up the ECM gradient 7, 38 . An explanation on how these signals may act on the LM organization to control steering comes from the fact that these inputs can modulate the amount of NPFs as well as their distribution along the membrane, leading to networks that are more or less efficient at protruding. Accordingly, the heterogeneity of the actin network can control steering during cell motility depending on the filament densities within the network and the degree of membrane tethering, and is sufficiently responsive to enable the cell to adjust its motility in a changing environment. Therefore, our actin growth-rate model provides a general framework to describe how the steering is controlled during cell locomotion and how this is an emergent property of the heterogeneity of actin networks in the LM.

Methods
Protein production and labeling. Actin was purified from rabbit skeletal-muscle acetone powder 39 . Monomeric Ca-ATP-actin was purified by gel-filtration chromatography on Sephacryl S-300 40  GST-WA, GST-pWA 43 are expressed in Rosettas 2 (DE3) pLysS. Fusion protein was purified by glutathione-Sepharose affinity chromatography (Amersham) and stored in Buffer PWA (20 mM Tris pH 8, 150 mM NaCl, 1 mM DTT, 0.5 mM ethylenediaminetetraacetic acid (EDTA)). Human profilin 44 is expressed in BL21 DE3 pLys S Escherichia coli cells. Culture is grown in LB medium + 100 µg ml −1 carbenicilin to OD of 0.6 at 600 nm, then 0.5 mM isopropyl β-D-1thiogalactopyranoside (IPTG) is added and cultures are grown for four more hours at 37°C. Pelleted cells are resuspended in Buffer P (20 mM Tris pH 8.0, 150 mM KCl, 0,2 mM DTT, 1 mM EDTA) + 2 M Urea. Following sonication and centrifugation the clarified extract is loaded on a polyproline sepharose column equilibrated in buffer 1 + 2 M Urea. Resin is washed with four volumes of buffer P + 3 M Urea. Profilin is eluted with Buffer P + 8 M Urea. Pooled fractions are dialyzed extensively to remove urea in storage buffer (20 mM Tris pH 8.0, 1 mM EDTA, 1 mM DTT). Protein is centrifuged at 150,000×g for 30 min to remove precipitate. Protein aliquots are stored at 4°C for 6 weeks, or flash frozen in liquid nitrogen and stored at −80°C, and mouse CP (α/β) 45 is cloned in a pRFSDuet-1 plasmid (Novagen) containing two cloning sites. The full length CP is a 6× His tagged at the N-terminus of the α subunit. CP is expressed in Rosetta2 DE3 pLys S in LB carbenicillin (100 µg ml −1 ). Culture is grown until OD is 0.6 at 600 nm. Induction is achieved by addition of 0.5 mM IPTG at 26°C overnight. Cells are pelleted and suspended in Buffer CP (20 mM Tris pH 8.0, 250 mM NaCl, 10 mM Imidazole, 5% Glycerol, 1 mM DTT, 1 mM EDTA) + protease inhibitors cocktail tablet. Cells are then sonicated and centrifuged at 39,000×g. Supernatant is applied to 1 ml of Ni sepharose fast flow resin (GE Healthcare). After 1 h at 4°C under gentle rotation, resin is washed with 20 volumes of Buffer CP containing 20 mM Imidazole. Elution is performed with buffer CP + 300 mM Imidazole. Purified protein is dialyzed overnight against a storage buffer (20 mM Tris pH 8.0, 1 mM DTT, 1 mM EDTA, 0.2 mM CaCl 2 ), flash frozen in liquid nitrogen and stored at −80°C. GST-pWA constructs attached to glutathione beads were labeled by incubating 1 ml of a 50% resin suspension overnight at 4°C with 7 excess molar ratio of Alexa-488 (Molecular Probes) in TBSE (10 mM Tris-HCl pH 8.0, 100 mM NaCl and 1 mM EDTA) 46 .
Nanoablation station. Inverted microscope (TE2000-E, Nikon) equipped with a CFI S-Fluor oil objective (×100, NA 1.3, Nikon), a perfect focus system (Nikon), motorized stage (Marzhauser), and a dual-axis galvanometer that focalizes the laser beam on the sample on the field of the camera, including a telescope that adjust the laser focalization with the image focalization, and polarizer to control the laser power (iLasPulse device, Roper Scientific). The microscope uses a pulsed laser passively Q-switched laser (STV-E, TeamPhotonics) that delivers 300 ps pulses at 355 nm (energy per pulse 1.2 µJ, peak power 4 kW, variable repetition rate 0.01-2 kHz, average power ≈ 100 mW). The laser was scanned throughout the region of interest, ROI, with a power set to 300 nJ. The ROIs (or patterns) used in this study were rectangles of usually 3 µm width and 15 µm long or as indicated. The laser displacement that defines the laser spot density, the distance between the patterns, and the number of repetition of patterned rectangles, as well as the laser exposure time were controlled using Metamorph software (Universal Imaging Corporation).
The microscope is moreover equipped with a fluorescence illumination system X-Cite 120PC Q (Lumen Dynamics) and QuantEM:512SC camera (Photometrics) to monitor the laser printing procedure.
Functionalization of laser-patterned surfaces. 20 × 20 mm 2 coverslips and cover glasses (Agar Scientific) were extensively cleaned, oxidized with oxygen plasma (3 mn, 30 W, Harrick Plasma, Ithaca, NY, USA) and incubated with 1 mg ml −1 of Silane-PEG overnight. Patterns of the desired area were printed on Silane-PEGcoated surfaces using the nanoablation station.
For patterns homogeneously coated with the same concentration of NPFs, immediately after laser-patterning patterned coverslips were coated with a solution of the nucleation promoting factor GST-pWA at the appropriate concentration (typically between 100 and 1000 nM) for 15 min 47 . When needed, the fluorescence density of the NPFs density was quantified before the assembly of actin on patterns.
For patterns of the same spot density but with two concentrations of NPFs, half of the patterns were printed with a 6.6 spot −1 µm −2 , coated with 300 nM GST-pWA, the excess of GST-pWA was wash out, and the surface was dried. The same procedure was then repeated to print the second pattern halves on the coverslips with pre-coated halves, the second round of coating was performed with 300 nM GST-pWA, the excess of GST-pWA was washed out and the surface was dried, ready to assess actin assembly 47 .
In order to control the reconstitution chamber height, we used BSA-coated 4.5 µm carboxylate polystyrene microspheres (4.5 µm diameter, 2.6% solids-latex suspension, Polysciences, Inc) as pillars. Briefly, beads were incubated for 15 min at 20°C on thermoshaker in X buffer solution containing 1% BSA, then pelleted and stored in ice for 48 h in X buffer-0.1% BSA.
To normalize actin network fluorescence between assays we used in the polymerization medium 0.2 µm fluorescent beads (Molecular Probes), at a dilution allowing for the presence of around 10 tiny beads per observation field. The network fluorescence at a given time of assembly was the average fluorescence measured in a 5 × 5 µm 2 ROI in the LMs at 10 µm from the nucleation pattern edge. For each polymerization assay, the maximum fluorescence of beads was then taken as a reference to normalize network fluorescence.
Growth rate were calculated using ImageJ software. The 2D-growth rate at a given time was calculated according to the network elongation during the last 4 min. When the LMs were elongated in a large reconstitution volume and grew in the Z-direction, we used the Simple Neurite Tracer plugins of ImageJ that allows for the visualization of the image stack through the XZ, ZY and XY planes. Points taken along the LM trace in the Z-stack at the proximal and the distal LM edges permit the calculation of the LM length. Thus, the 3D-growth rate at a given time t was then calculated according the elongation of LMs (total length t minus total length at t−2 min) during the last 2 min. We use the view through the XY plane to calculate network fluorescence as described above.
Image acquisition. For the 2D growth of reconstituted LMs, image acquisition was performed using an upright Axioimager M2 Zeiss microscope equipped with an EC Plan-Neofluar dry objective (×20, NA 0.75), a computer controlled fluorescence microscope light source X-Cite 120PC Q (Lumen Dynamics), a motorized XY stage (Marzhauser) and an ORCA-ER camera (Hamamatsu). For the 3D growth of reconstituted LMs, image acquisition was performed using an Eclipse TI-E Nikon inverted microscope equipped with a CSUX1-A1 Yokogawa confocal head, an Evolve EMCCD camera (Roper Scientific), a CFI Plan APO VC oil objective (×60/ NA 1.4; Nikon), a CFI Plan Fluor oil objective (×40/NA 1.3 and ×100/1.45; Nikon), and a motorized stage MS 2000 (ASI imaging). Both stations were driven by MetaMorph software (Universal Imaging Corporation). The use of the motorized stage allowed acquiring actin dynamics of several networks assembled either on beads or on micropatterns under exactly the same biochemical conditions.