The model shown in Figure 1 includes barbed end polymerization, pointed end depolymerization, capping, uncapping, branching near the leading edge, debranching, severing and annealing, without explicitly considering ATP hydrolysis or phosphate release (see Materials and methods and Table 1). We impose a constant network velocity vn⁢e⁢t with respect to the leading edge. We selected parameters corresponding to two frequently-studied cell systems, one for fast moving keratocyte cells and one for XTC or fibroblast cells. For keratocytes, vn⁢e⁢t corresponds to the rate of cell protrusion, since the actin network is almost stationary with respect to the substrate in experiments (Keren et al., 2008; Mueller et al., 2017; Yamashiro et al., 2018; Schaub et al., 2007). Lamellipodia of XTC cells are frequently studied in cells that do not crawl on the substrate, so vn⁢e⁢t provides the magnitude of the retrograde flow speed (Watanabe and Mitchison, 2002; Ryan et al., 2012). We considered uniform severing with rate constant ks⁢e⁢vu⁢n⁢i⁢f per filament length and enhanced end severing with a rate per filament ks⁢e⁢ve⁢n⁢d near the barbed end. If an oligomer is created from one of these severing events, we assume it can diffuse and anneal to a nearby filament end with rate constant ka⁢n⁢n⁢e⁢a⁢l.

Prior models of dendritic networks demonstrated how the ±35° orientation with respect to the direction of protrusion depends on the relationship between filament elongation velocity vp⁢o⁢l and relative extension rate vn⁢e⁢t. As shown in the results of the 2D model by Weichsel and Schwarz, 2010 in Figure 2A, for low vn⁢e⁢t/vp⁢o⁢l, the dominant orientation pattern has filaments branching at -70∘/0/70∘. Filaments oriented at angles larger than θc, for which cos⁡(θc)=vn⁢e⁢t/vp⁢o⁢l, lose contact with the membrane since they are not polymerizing quick enough to catch up. When this critical angle becomes smaller than 70°, the favored pattern is filaments with orientations centered at ±35°: the filament population around 35° can generate daughter branches at -35° and vice versa; thus the population sustains itself even as individual filaments get capped. The -70∘/0/70∘ was found to resume when vn⁢e⁢t/vp⁢o⁢l exceeds the critical angle for 35°; in this situation, only the 0° filaments polymerize quick enough to remain in the branching region. The above behavior can be quantified by the orientation order parameter, where a value 1 (or –1) indicates all filaments are in a –70°/0°/70° (or ±35°) pattern.

By contrast to the above 2D results, dendritic network models formulated in 3D have provided apparently contradictory results (Holz and Vavylonis, 2018). Atilgan et al., 2005 reported that obtaining the ±35° pattern requires restricting branching to occur primarily along the lamellipodium plane, which they attributed to structural constraints of the branching machinery at the leading edge. Schreiber et al., 2010 and Hu and Papoian, 2010 however did observe the ±35° pattern in 3D simulations, but the role of vn⁢e⁢t/vp⁢o⁢l in determining the pattern was not examined.

As we are interested in structural aspects of lamellipodia, we performed systematic simulations in both 2D and 3D to examine the filament orientation pattern as function of vn⁢e⁢t/vp⁢o⁢l. In the simulations of Figure 2 we consider the region close to the leading edge where severing, annealing and debranching does not influence the resulting structure, and varied the polymerization rate while keeping vn⁢e⁢t constant. We also kept the capping and branching rates kc⁢a⁢p and kb⁢r fixed as the orientation pattern is robust with respect to their values (Weichsel and Schwarz, 2010).

We reproduce the results of the Weichsel and Schwarz, 2010 simulations as a function of relative network growth speed, by imposing a tight planar branch restriction along the lamellipodial x⁢y plane (Figure 2A). Note that the transition among different orientation patterns is not abrupt at the critical angles, as a result of allowing fluctuations in branching angle and a finite size of the branching region. This is the reason for the largest difference occurring at high vn⁢e⁢t/vp⁢o⁢l: because only filaments at small angles can keep up with the leading edge under these conditions, our simulation evolves to a narrow comet-like branching structure where all branching is concentrated; this allows branching among ±35° and other orientations (e.g. -80∘/-10∘/60∘) to be maintained through double branching before filaments exit the branching region.

There is little difference in the orientation order parameter between simulations that allow backward branching (angles greater than 80°, chosen to include angles that would be oriented away from the leading edge considering the variation of branching angle for both orientation patterns) and branching limited toward the leading edge (Figure 2A). Filaments that branch backward exit the branching region quicker than filaments that branched forward, decreasing the likelihood of branches off of backward filaments. Even though backward-facing filaments contribute to angles larger than 80°, they do not influence the order parameter that does not measure them. Since backward branching does not affect the filament orientation pattern, and since backward-facing filaments are not seen in electron microscopy images (Vinzenz et al., 2012; Mueller et al., 2017), backward branching was not allowed in the rest of our simulations.

Next, we allowed branching to occur uniformly in 3D (i.e. with equal probability along any azimuthal angle with respect to the axis of the parent filament) and studied the orientation order parameter as function of vn⁢e⁢t/vp⁢o⁢l (Figure 2B). The order parameter was measured using the angles of filaments projected along the lamellipodial x⁢y plane. Uniform branching led to low and weakly varying order parameter, unlike the sharp orientation pattern with distinct transitions observed in 2D. By comparison, a quasi-2D simulation with branching allowed to occur within 10° off the lamellipodial plane (close to the maximum off-plane angles observed in electron microscopy [Vinzenz et al., 2012] [A. Narita, personal communication, March 2018]) restored the behavior observed in 2D.

In the simulations described so far, filament elongation was assumed to stop when the polymerizing barbed end reached the top or bottom z plane. To investigate the influence of filament bending along the membrane, we also performed simulations with filament ‘kinking’, in which filaments were allowed to continue their elongation parallel to the plane representing the top or bottom cell membrane. Allowing kinking lead to a high concentration of filaments along the top and bottom plane of the lamellipodium but did not however significantly influence the orientation order parameter (Figure 2B).

To further visualize the network structure in simulations, Figure 2C and D (and Figure 2—figure supplement 1 for the case with kinking) show snapshots and filament orientation patterns at low and intermediate values of relative network growth speeds. The simulations with quasi-2D branching show clear –70°/0°/70° and ±35° respective orientation patterns. Interestingly, even though no prominent features are observed in simulations with uniform 3D branching at low relative network speeds (Figure 2C), intermediate relative network speeds do show features at ±35° (Figure 2D). The latter histogram is not very different from experimentally-measured distributions (Maly and Borisy, 2002; Vinzenz et al., 2012; Koseki et al., 2019).

We thus conclude that the ±35° pattern does occur within a broad range of relative network growth speeds with uniform 3D branching, however the peaks at ±35° are not very pronounced. Indeed, the parameters used by Atilgan et al., 2005 corresponded to vn⁢e⁢t=0.26-0.46, a parameter set that mostly lies outside the ±35° region; this is likely the reason why the ±35° was not observed in this study. Our results also suggest why Schreiber et al., 2010 who used vn⁢e⁢t≈0.37 and Hu and Papoian, 2010 who had vn⁢e⁢t=0.45-0.51, did observe a ±35° with uniform 3D branching.

Considering the simulation results as well as experimental evidence in electron tomograms for filaments oriented primarily along the lamellipodial plane (Vinzenz et al., 2012), for the rest of the simulations we proceed with the quasi-2D case where filament branching occurs within 10° of the lamellipodial plane and relative network growth speeds result in a ±35° orientation pattern.

For both cases of keratocytes and XTC cells, which correspond to different values for vn⁢e⁢t, we estimated the rates of polymerizing barbed end elongation vp⁢o⁢l, branching kb⁢r, and capping kc⁢a⁢p, that are needed for a dendritic network with the anticipated concentration, branch length, and ±35° filament orientation pattern (Table 1). We also assume that barbed ends can uncap with rates comparable to those in SiMS lifetime measurements of capping protein (Miyoshi et al., 2006). The availability of such uncapped barbed ends for annealing is an important assumption of this work. We further use debranching rates by considering the lifetime of Arp2/3 complex components in SiMS (Miyoshi et al., 2006) and measurements of Arp2/3 complex profiles in lamellipodia (Iwasa and Mullins, 2007; Lai et al., 2008; Ryan et al., 2012).

We also assumed that uncapped barbed ends away from the leading edge do not elongate or shrink and that free pointed ends depolymerize with a rate vd⁢e⁢p⁢o⁢l=5/s. The results we present below are robust with respect to small changes of these parameters, as long as the overall filament disassembly rate away from the leading edge is not reaching values comparable to vp⁢o⁢l. Maintaining a wide lamellipodium in the latter case would require a global treadmilling mechanism, which would contradict the evidence for distributed turnover. The assumption of slow barbed end dynamics away from the leading edge is consistent with the slow intensity increase in the back of lamellipodium after FRAP of actin (Smith et al., 2013) or after photoactivation of actin at the cell middle (Lai et al., 2008; Vitriol et al., 2015), as well as evidence that cofilin and twinfilin promote both barbed and pointed end depolymerization (Wioland et al., 2017; Johnston et al., 2015; Hakala et al., 2021; Shekhar et al., 2021).

A steady state with a finite lamellipodium depth is reached in the simulation whenever the net rate of depolymerization balances the net rate of polymerization at the leading edge. For example, in the case without enhanced end severing, annealing, or oligomer dissociation, the depth of the lamellipodium is determined by the rates of uniform severing, ku⁢n⁢i⁢fs⁢e⁢v, and vd⁢e⁢p⁢o⁢l (Figure 1—figure supplement 1): in this case, the fast growth of barbed ends at the leading edge is balanced by the slower depolymerization of a larger number of pointed ends created by severing.

Given the parameters in Table 1, this leaves three main unknown parameters related to oligomer dissociation: ku⁢n⁢i⁢fs⁢e⁢v, the rate of uniform severing along each filament; lm⁢a⁢xo⁢l⁢i⁢g, the longest length of a diffusing oligomer; ke⁢n⁢ds⁢e⁢v, the enhanced severing rate near the barbed end (within lm⁢a⁢xo⁢l⁢i⁢g of the end). We treated these three as fitting parameters and considered separately the cases in the presence or absence of annealing. In the absence of annealing, dissociating oligomers do not reincorporate into the network and are thus discarded from the simulation (corresponding to eventual disassembly into monomers, a process that we did not simulate).

We conducted a parameter search over the maximum oligomer size lm⁢a⁢xo⁢l⁢i⁢g as well as end and uniform severing rates ke⁢n⁢ds⁢e⁢v, ku⁢n⁢i⁢fs⁢e⁢v for keratocyte- and XTC-like parameters without annealing (Figure 3—figure supplement 1). We classified each parameter set in terms of how well it described the F-actin structure and concentration (including barbed end and branch concentration profiles), actin SiMS data (speckle lifetime distribution, appearance and disappearance profile), and if there was an increase in length between the filaments located in 0–1 µm and 3–4 µm region (see Materials and methods). Within a range of uniform and end severing rates, and short maximum oligomer lengths, the network structure and concentration of our simulation was close to that expected for keratocyte and XTC lamellipodium (Figure 3—figure supplement 1). However, as expected, we did not find any parameter sets that resulted in a length increase since a mechanism for an increase in length and remodeling is not included. We also did not find any parameter set with a sharp peak at short actin speckle lifetimes, though in some cases the actin speckle lifetime distribution has a peak at relatively short times (Figure 3—figure supplement 1).

To further demonstrate that the model without annealing cannot fit the experimental data, Figure 3 and Figure 3—figure supplement 2 contain examples of results for keratocyte and XTC parameters. In this and subsequent figures we color parameter sets black without end severing and red that include enhanced end severing; these curves correspond to the parameters of the scan that are highlighted with a thick frame of same color in Figure 3—figure supplement 1. For the case with moderate uniform severing and no end severing, both the XTC and keratocyte uniform severing cases have a concentration profile that is comparable to lamellipodium of their respective parameter set (Figure 3) as well as barbed end and branch distributions (Figure 3—figure supplement 2). However, the actin speckle lifetimes do not peak at short lifetimes. With the addition of enhanced end severing, as well as increase of uniform severing, a peak at short lifetimes is observed that is closer to the experimental SiMS curves; however, the lamellipodium becomes too narrow, there is a shortage of long speckle lifetimes compared to experiment, and the location of speckle appearances is restricted close the leading edge (Figure 3 and Figure 3—figure supplement 2 A,F). In conclusion, the model without annealing cannot reproduce the distributed turnover, structure of actin network and increase in filament length.

Next, we examined if inclusion of oligomer annealing might be able to provide an adequate fit to the structure, speckle, and length increase criteria. We performed another parameter search over the maximum oligomer length, end and uniform severing rates (lm⁢a⁢xo⁢l⁢i⁢g, ke⁢n⁢ds⁢e⁢v, ku⁢n⁢i⁢fs⁢e⁢v) for keratocyte and XTC parameter sets. For these simulations, we used an annealing rate constant measured in crowded surfaces in vitro (Popp et al., 2007). Similar to the parameter scan without annealing, a match to F-actin structure and concentration was obtained when uniform and end severing rates were within a certain range (Figure 4—figure supplement 1 and Figure 5—figure supplement 1). The addition of annealing improved the agreement between the speckle lifetimes, appearance and disappearance location profiles compared to SiMS data, resulting in parameter sets that agree with both structure and speckle dynamics (Figure 4—figure supplement 1 and Figure 5—figure supplement 1). We also see that with high enough severing and maximum oligomer lengths, the filament length increases in a region away from the leading edge (Figure 4—figure supplement 1 and Figure 5—figure supplement 1). Parameter sets where all three fitting criteria are satisfied exist for both keratocyte and XTC cases, and all these triple matches have a finite enhanced end severing rate.

Detailed results from our parameter scan for keratocytes are shown in Figure 4. This figure shows a case without enhanced end severing (black curves, double match in structure and speckles for keratocyte parameters) and with enhanced end severing (red, triple match in structure, speckles and filament length increase). Both parameter sets can reproduce the actin speckle dynamics as seen in the speckle lifetime, appearance and disappearance location distributions as well as the the structure as seen in the F-actin, barbed end, and branch concentration profiles (Figure 4A–F). However, only the case with enhanced end severing results in a simultaneous increase in length away from the leading edge (Figure 4G). The increase brings the filament length close to the estimated average of 800 nm in keratocytes (Schaub et al., 2007). The concentration profile of Arp2/3 complex branches (Figure 4F), snapshots (Figure 4H), and Video 1 of the optimized simulations with enhanced end severing clearly show more branches and shorter filaments in the region near the leading edge than away from the leading edge, similar to the electron micrographs of Svitkina et al., 1997. The ±35° filament orientation pattern is preserved throughout the lamellipodium in the presence of severing and annealing. Additional quantification in Figure 4—figure supplement 2 shows the steady state spatial distribution of capped and uncapped barbed ends, the approximate uniform distribution of oligomer sizes between 0 and lm⁢a⁢xo⁢l⁢i⁢g, and the profile of F-actin according the mechanism of assembly (polymerization as monomer at the leading edge versus annealing). Most of the F-actin at the back of the lamellipodium has undergone severing and annealing (Figure 4—figure supplement 2D.F), similar to an earlier particle model for distributed turnover (Smith et al., 2013).

For XTC cells, enhanced end severing is needed to match the actin speckle lifetime distribution, which further contributes to filament length increase. Results from our parameter scan for XTC cells in Figure 5 show a case without enhanced end severing (black curves, match in structure only) and a case with enhanced end severing (red, triple match in structure, speckles and filament length increase). The case with enhanced end severing provides a good overall fit to SiMS and structure (Figure 5A–G), demonstrating increase in length away from the leading edge (Figure 5G and H and Video 2). Figure 5B and F show that the model reproduces the narrower distribution of Arp2/3 complex as compared to F-actin in XTC cells (Ryan et al., 2012). Additional quantification in Figure 5—figure supplement 2 shows profiles of capped and uncapped end, oligomer size distribution, and origin of F-actin.

When performing the F-actin concentration and structure match for the XTC parameter sets, we used numbers in between those of electron tomograms in fibroblasts by Vinzenz et al., 2012 and the estimated F-actin concentration of 1000 µM for XTC cells (Watanabe, 2010). The branching rate in the simulations of Figure 5 corresponds to 0.05 µM s^-1, which is about half of the Arp2/3 complex nucleation rate of 0.11 µM s^-1 estimated by SiMS (Watanabe, 2010; Miyoshi et al., 2006). We checked that simulations with doubled the branching rate still provide a good fit to actin SiMS data as well as a length increase away from the leading edge, with F-actin concentration at the leading edge that was around 1,100 µM (Figure 5—figure supplement 3). We also tested that excluding end severing of polymerizing ends in Figures 4 and 5 did not modify our results for the optimized parameter sets (Figure 5—figure supplement 4).

The optimized parameter sets with speckle dynamics similar to SiMS experiments of Figures 4 and 5 also matched another observation from SiMS (Watanabe and Mitchison, 2002): the distribution of actin speckle lifetimes was weakly dependent on location of appearance with respect to the leading edge (Figure 4—figure supplement 3, Figure 5—figure supplement 5).

The simulations of Figures 4 and 5 implement a mechanism of local oligomer rebinding, which is needed to match actin FRAP and photoactivation data (Smith and Liu, 2013; Vitriol et al., 2015): as shown in Figure 4—figure supplement 3B and Figure 5—figure supplement 5B the distance travelled by oligomers before annealing is in the sub-µm range. As a further check of consistency of our simulations with SiMS results, simulated actin SiMS for parameters with enhanced end severing of Figure 4 (keratocytes) and Figure 5 (XTC) do resemble experimental images from Yamashiro et al., 2014 and Watanabe and Mitchison, 2002 (Videos 3 and 4). In Videos 3 and 4, when an appearing and disappearing speckle are near one another, this is typically a reannealing event occurring quickly over short distances. We confirmed that events that might be limited by spatial and temporal resolution or interpreted as blinking in SiMS experiments correspond to a very small fraction of speckle appearances in the simulation.

Finally, we note that even though the results of this section were obtained for a specific value of annealing rate constant and oligomer diffusion coefficient, they remain valid as long as the annealing of oligomers occurs over a sufficiently short distance. We find that this is the case even for annealing rate constants that are lower by nearly two orders of magnitude compared to the values of Table 1 (Figure 4—figure supplement 3C, Figure 5—figure supplement 5C).

An alternative mechanism to explain the short actin SiMS lifetimes (different to enhanced severing and annealing) is barbed end catastrophic disassembly by factors such as twinfillin (Wioland et al., 2017; Johnston et al., 2015; Hakala et al., 2021; Shekhar et al., 2021) or cooperative strand separation in the presence of cofilin, coronin and Aip1, depending on the cofactor concentrations (Kueh et al., 2008; Jansen et al., 2015; Tang et al., 2020).

To test such a mechanism, we implemented a model with stochastic transitions to rapid barbed end depolymerization (see Materials and methods, Video 5). To maintain the F-actin loss by such a disassembly process, a process of rapid regrowth must also be included. Using the insight gained from our parameter searches with severing and annealing, we can show that such a model can be tuned to come close to matching our three main experimental test criteria: agreement with SiMS data, an increase in filament length with distance from the leading edge, and broad F-actin concentration profile (Figure 4—figure supplement 4).

While we cannot fully exclude such a dynamic-instability-like mechanism, we note that: (1) it would require additional controls or homeostatic mechanisms to balance disassembly and reassembly away from the leading edge (while maintenance of F-actin mass is ensured by a severing and annealing mechanism), (2) ATP hydrolysis associated with ATP-actin monomer polymerization would be energetically more costly, and (3) reassembly of rapidly diffusing monomers away from the leading edge to recover bursting would also be less consistent with FRAP or photoactivation experiments (Smith et al., 2013; Vitriol et al., 2015).

The actin speckle lifetime distribution of the models with severing and annealing of Figures 4–5 characteristically peaks at t=0. This is a general feature of a barbed-end disassembly mechanism where newly polymerized monomers are the ones that disassemble at higher rates, being closer to the barbed end. The experimental SiMS lifetimes in Figures 3—5 used a temporal resolution of 1 s (Watanabe and Mitchison, 2002; Yamashiro et al., 2018). To probe the kinetics at even shorter lifetimes, SiMS of Dylight actin was repeated on XTC cells using the methods of Yamashiro et al., 2014, with a temporal resolution of 0.1 s. We measured actin speckles appearing near the leading edge, which should represent polymerization of actin monomers rather than oligomer annealing. Approximately 16% and 30% of speckles disappeared within within 0.5 and 1 s, respectively (Figure 6A). This large amount of short speckle lifetimes strongly supports frequent disassembly of newly polymerized F-actin near the barbed end.

To compare to simulations, Figure 6B contains speckle lifetimes within 1 µm of the leading edge for the XTC parameter set with enhanced end severing and annealing of Figure 5 (the uniform severing case of Figure 5 did not provide as good a match to SiMS measurements and is not shown). The speckle distribution extended to short lifetimes, similar to the experiments of Figure 6A, with the percentage of lifetimes within 0.5 s and 1 s being 3% and 7 %, respectively. Doubling the enhanced end severing rate while also choosing parameters that reproduced speckle dynamics, lamellipodial structure and an overall increase in length (kunifsev=0 /sub/s; lmaxolig=150 sub, XTC parameters, Table 1), doubled the percentage of speckle short lifetimes within 0.5 s and 1 s to 7% and 14%, respectively, closer to the experimental percentages of Figure 6A. We conclude the short lifetimes of Figure 6A are within the range of what is expected by an enhanced end severing mechanism, which may even occur at a rate 2–4 times faster than the estimate of Figure 5.

Kueh et al., 2008 observed that actin disassembly in lamellipodia was inhibited in the presence of cytochalasin D (CD), a barbed end capper that also inhibits binding of cofilin to F-actin at high concentrations (Shoji et al., 2012). We tested the effect of CD treatment on XTC cells using SiMS (Figure 6C). We confirmed CD’s inhibitory effect on filament disassembly by observing a larger fraction of actin speckles that survive after CD treatment. These results provide further support for rapid disassembly of newly polymerized actin at barbed ends (either through ‘bursting’ as proposed in Kueh et al., 2008 or enhanced severing).

To check if tropomyosin (TPM) might be involved in the reorganization of lamellipodial networks by debranching and stabilizing progressively the longest filaments, we observed SiMS of TPM-EGFP, as described previously in Higashida et al., 2008, in lamellipodia of XTC cells ~30 min after cell spreading. In this early phase, the density of TPM SiMS was substantially lower in lamellipodia than in lamella (Figure 6—figure supplement 1A, Video 6). Within lamellipodia, the density of TPM SiMS becomes gradually higher toward the base of lamellipodia. TPM SiMS are scarce near the leading edge, suggesting that TPM may not have a significant effect on actin turnover near the leading edge.

We also compared the dissociation rate of tropomyosin between lamellipodia and lamella. In lamellipodia of XTC cells 30 min after cell spreading, TPM SiMS dissociated much faster in the lamellipodia than in the lamella (Figure 6—figure supplement 1B). TPM-EGFP was localized to actin stress fibers after cells were grown for 24 hr in the presence of serum (Figure 6—figure supplement 1C). The localization and kinetics of TPM SiMS are in agreement with slow assembly kinetics of TPM along the actin filament in vitro (Schmidt et al., 2015). We thus conclude that at least under our observation conditions, it is unlikely that TPM stabilizes actin filaments near the tip of XTC lamellipodia.

We developed a three-dimensional stochastic simulation of the actin network within the lamellipodium (Figure 1). Actin filaments are represented as straight lines without excluded volume, within the simulation box. We work in the coordinate system where the leading edge is at rest. For the purposes of this work, we do not consider explicitly the effects of ATP hydrolysis, diffusion of the actin monomer pool, or excluded volume interactions among filaments. The leading edge at y = 0, as well as the lamellipodium top and bottom at z=0 and z=0.2μm are hard boundaries. There is no boundary at y→-∞, allowing the mechanisms and rate constants of the system to determine the length of the lamellipodium. Periodic boundary conditions with length at least 1 µm are applied in the x-direction. A constant relative velocity, vn⁢e⁢t, is imposed between the network and the leading edge. The network is initialized by filament seeds of 5 subunits in length at random orientations near the leading edge. The probability of a reaction event (polymerization, depolymerization, capping, uncapping, branching, severing, and annealing) is calculated using the corresponding rates or rate constants as described below and in Table 1. The time step was d⁢t=0.002 s.

To compare our simulations to the actin speckle appearance and disappearance in SiMS, we tagged and tracked 1% of actin monomers that add to the network through polymerization at the leading edge. Each such polymerization corresponds to a speckle appearance. Disappearance events occur when the tagged monomers depolymerize off a pointed end or when they become part of an oligomer. Annealing of an oligomer carrying a tagged monomer is an appearance. The speckle lifetime is the time between appearance and disappearance events.

We counted the number of filaments having a segment within the first micrometer of the leading edge and defined the order parameter similar to Weichsel and Schwarz, 2010 and Mueller et al., 2017: O=(N-70∘/0∘/70∘-2⁢N±35∘)/(N-70∘/0∘/70∘+2⁢N±35∘), where N-70∘/0∘/70∘ and N±35∘ are the number of filaments oriented between -20° to 20° and 60° to 80° in either direction, or between 25° - 45° in either direction, respectively. The angle is measured with respect to the axis of network growth. For 3D simulations, we used the angles of the filaments projected along the x⁢y lamellipodial plane.

SiMS experiments using Dylight 550 labeled actin introduced to XTC cells by electroporation was performed as in Yamashiro et al., 2014. Cells adhered on a poly-lysine-coated glass coverslip were observed by epi-fluorescence microscopy. For measurements of speckle lifetime distribution, the exposure time was 0.1 s/frame. The lifetime data were normalized for photobleaching as in Watanabe and Mitchison, 2002. In Figure 6A experiments, the leading edge was manually marked by a line which connects the centers of furthermost speckles in several consecutive images. Only actin SiMS that appeared within ~300 nm from the line were analyzed for lifetimes. Lifetimes of 0.1 s were omitted as being beyond the temporal resolution limit. Disassembly of actin speckles in lamellipodia of XTC cells were also observed after treatment with 5 µM cytochalasin D, added after several seconds from the start of observation, at 2 s intervals.

To estimate parameters for uniform and enhanced end severing rates (ku⁢n⁢i⁢fs⁢e⁢v, ke⁢n⁢ds⁢e⁢v), we performed a parameter search over these values as well as the maximum oligomer length lm⁢a⁢xo⁢l⁢i⁢g. To summarize the results, we classified each set of these parameters in terms of their ability to match the actin network structure and concentration, actin speckle dynamics, and if there was an increase in length between the front and back of the lamellipodium (Figure 3—figure supplement 1, Figure 4—figure supplement 1, and Figure 5—figure supplement 1):

Capping and branching rates were determined by comparing to the the barbed end, branch and filament number of electron micographs of fibroblast cells in Vinzenz et al., 2012 (used for the XTC parameter set) and barbed end, pointed end and filament number from Mueller et al., 2017 for the keratocyte parameter set. In all instances of data comparison for the barbed end and pointed end number of Mueller et al., 2017, the average value of two 0.106 µm bins was compared to simulation results of 0.212 µm bins.

We use a simplified 2D dendritic nucleation model, without severing and annealing, to estimate the capping and branching rates for our simulations of keratocyte and XTC cases. We assume linear actin filaments, all oriented ±35° relative to the leading edge with the -y direction toward the center of the cell. These filaments polymerize, branch, cap and move with respect to the position of the leading edge due to retrograde flow or cell protrusion. Filaments with free barbed ends (uncapped) remain at the leading edge and continue to polymerize. As they cap, they stop polymerization and move toward the center of the cell with retrograde flow. For the purposes of this section, we do not consider uncapping of already capped filaments. Branches form at the leading edge and also move away with retrograde flow. Denoted by l the length of a filament with barbed end located at y, the system of equations for the number of uncapped nu⁢(l,t), capped nc⁢(l,y,t) barbed ends and branches nb⁢r⁢(y,t) are:(1)∂nu(l,t)∂t=kbrδ(l)−kcapnu(l,t)−vpol,35∂nu(l,t)∂l,(2)∂nc(l,y,t)∂t=kcapnu(l,t)δ(y)−vnet∂nc(l,y,t)∂y,(3)∂nbr(y,t)∂t=kbrδ(y)−vnet∂nbr(y,t)∂y,

The first term on the right hand side of Equation 1 represents the branching source term which creates a new filament at 0 length. The last term of Equation 1 is polymerization of the filaments with vn⁢e⁢t=vp⁢o⁢l⁢cos⁡(35∘). The second term of Equation 1 and first term of Equation 2 represents the loss of uncapped filaments and addition of capped filaments. The second term of Equation 2 represents retrograde flow of capped filaments. Equation 3 accounts for the generation of branches and their motion with retrograde flow toward the center of the cell.

In steady-state, the solutions of the uncapped and capped ends to be exponential and the number of branches is:(4)nu(l)=kbrvpole−l/l¯(5)nc(l,y)=kcapkbrvnetvpol,35e−l/l¯(6)nbr(y)=kbrvnet

where l¯=vp⁢o⁢l,35/kc⁢a⁢p. To compare to electron tomogram quantifications we calculate the number of branches, barbed ends and the number of filaments in bins of increasing distance to the leading edge.

Integrating Equation 6 over y from the minimum to maximum of the bin we find the number of branches within the ith bin, Nb⁢r,i.(7)Nbr,i=kbrΔyvnet,   i=1,2,3,...

where Δ⁢y is the bin width centered at y=(1/2-i)⁢Δ⁢y such that i=1 is the bin that includes the leading edge.

The number of barbed ends within a bin centered at y depends on the total number of capped and uncapped filaments. We integrate over all filament lengths l and the distance from the leading edge y:(8)NBE(y)=∫0∞nu(l)dl∫y−Δy/2y+Δy/2δ(y)dy+∫0∞∫y−Δy/2y+Δy/2nc(l,y)dydl.

We find the solution for the number of barbed ends in the the ith bin to be:(9)NBE,i=kbrkcapδi,1+kbrvnetΔy,   i=1,2,3,...

In the electron tomograms, the filament number was measured by counting the number of filaments crossing the middle plane of the measured bins. We calculate the equivalent filament number by integrating the sum of the capped and uncapped filaments that are long enough to cross the distance in y at the center of such bins:(10)Nfil(y)=∫−y/cos⁡(35)∞nu(l)dl+∫0−yds∫s/cos⁡(35)∞nc(l,y−s)dl.

Integrating, we find,(11)Nfil(y)=kbrkcap(2eykcapvnet−1)

which can be evaluated at bin i using y=(1/2−i)Δy.