Constrained actin dynamics emerges from variable compositions of actin regulatory protein complexes

Assemblies of actin and its regulators underlie the dynamic morphology of all eukaryotic cells. To begin to understand how diverse regulatory proteins work together to generate actin-rich structures we tracked the assembly of actin regulators and their relative proportions in a cell-free system that generates filopodia-like structures (FLS). We found that heterogeneous mixtures of regulators could give rise to morphologically similar structures and that the FLS actin bundles exhibited simple dynamic behaviour of growth and shrinkage. To explain these observations, we combined experiment with theory, and found that stochastic fluctuations between redundant actin regulatory subcomplexes can account for the actin dynamics. Comparing the localizations of a variety of endogenous actin regulators in Drosophila embryos and distributions of filopodia lengths yielded similar conclusions of heterogenous actin regulatory complexes and filopodia lengths governed by a stochastic growth process. Our results explain how weakly-associating assemblies of regulatory proteins can produce robust functional outcomes.


Introduction
The regulation of actin polymerization is crucial for numerous cell functions, including cell migration, adhesion and epithelial closure (1, 2) and is often disrupted in disease, such as cancer metastasis and intracellular infection by pathogens (3-6). Micron-scale structures of actin filaments and associated regulators provide the mechanical 20 infrastructure for the cell via transient membrane-bound complexes that somehow orchestrate large scale cytoskeletal remodelling (7,8). Filopodia, with their characteristic membrane-associated "tip complex" where new actin monomers are incorporated, are one such example (9, 10). Current models for filopodia formation posit key roles for either (1) formins in de novo nucleation processes (11-13), (2) a preexisting actin 25 network generated by the Arp2/3 complex and bundled by fascin (14)(15)(16) or (3) the clustering and outward projection of membrane-bound SH3 domain containing adaptor proteins that recruit Enabled (Ena)/ Vasodilator-stimulated phosphoprotein (VASP) proteins coupled to the wider actin machinery (17)(18)(19). One way to reconcile these models is to postulate the existence of subtypes of filopodia based on their mechanism 30 of formation (20-24). However, a unifying picture is still lacking, such that the nature, number and dynamics of putative filopodia subtypes remains in question.
Purified reconstituted systems using membranes, several signaling proteins and receptors have revealed that a semi-dynamic network of actin regulators can form from multivalent interactions, and that these can determine the locations of actin filament assembly (25-27). While we now appreciate how such condensates can be generated by 5 their underlying molecular characteristics, open questions remain. In particular, how the molecular nature and composition of the regulatory protein assemblies contribute to their functional output in producing large scale actin structures remains unknown (8, 28).
While some actin regulatory subcomplexes are known to have defined stoichiometry (for example the Wiskott-Aldrich syndrome protein family Verprolin-homologous protein 10 (WAVE) complex (29)), others such as neural Wiskott Aldrich syndrome protein (N-WASP) can participate at variable stoichiometries and be highly dynamic (25, 30). The complex of actin regulators such as those found at filopodia tips, presumably lies somewhere between a defined complex and a loose association of signaling and effector proteins. We do not know whether specific combinations and stoichiometries of actin 15 bundlers, nucleators, and elongators are needed to create an actin bundle and if there are, what these ratios would be.
We previously described a cell-free system comprised of Xenopus egg extracts and a phosphatidylinositol (4,5)-bisphosphate supported lipid bilayer that grows actin bundles 20 from a membrane-bound complex of actin regulators that is highly amenable to quantitative microscopy ( Fig. 1A) (19). Their compositional similarity to filopodia, and their ability to grow many microns long, led us to call them filopodia-like structures (FLS).
Although the complex of actin regulators is at the base of the FLS, in filopodia the complex of actin regulators is found at the tip; therefore, we refer to the complex of actin 25 regulators in FLS as the tip complex by analogy with filopodia. Both the tip complex and the actin bundle can be readily followed by timelapse fluorescence microscopy at the bilayer surface and up from it, in the z axis. FLS are not a strict filopodia mimic as membrane does not surround the shaft, and N-WASP is employed as an Arp2/3 complex activator rather than the closely related Scar/WAVE protein that may be fulfilling this role 30 in cells. Nonetheless, FLS contain bundled actin filaments and new actin monomers are incorporated at a membrane-localized complex that includes Ena, VASP and Diaphanous-related formin 3 (Diaph3). The actin bundling protein, fascin is also present throughout the shaft.
Here we tracked the assembly of the actin regulators to the FLS tip complex in time, as well as their intensities in relation to the actin bundle. We found that the tip complex 5 assembles cooperatively in an F-actin dependent manner, that the final complex is heterogeneous in composition, and that proteins can turn over within it. Despite their heterogeneity of accumulation, we find that some of the actin regulators comprising the tip complex form subcomplexes. However, the balance of assembly and disassembly of the actin gives rise to bundle morphologies that are independent of the precise

Heterogeneity in actin regulators during FLS assembly
To track the assembly of the tip complex of the FLS, for example to determine whether there is a distinction between actin regulators that nucleate actin filaments, and those that bundle or extend actin filaments, we measured the intensites of actin regulatory 30 proteins with time as FLS form (Fig. 1A,B). To add fluorescently labelled proteins into the extracts, we expressed and purified Xenopus laevis or tropicalis Transducer of Cdc42 activation-1 (TOCA-1), the G-protein binding domain of N-WASP to monitor Cdc42•GTP (GBD), N-WASP, Ena, VASP, Diaph3 and fascin (domain structures are shown in Figure   1 supplement 1A, gels in B). TOCA-1, Ena and N-WASP were expressed as SNAPfusion proteins and chemically labelled with AlexaFluor (AF) 647 and AF 488. VASP was engineered to contain a cysteine close to the N-terminus for labelling with AF 568 and 5 AF 647 maleimide, as previously published (31). Fascin and Diaph3 were purified as Nterminal GFP-fusion proteins, and GBD as an N-terminal fusion protein with red fluorescent protein pmKate2. To assess the concentration of labelled protein added relative to the endogenous protein provided by the extracts, we measured the concentrations of TOCA-1, Ena, VASP, N-WASP and fascin in the high-speed 10 supernatant extracts using quantitative western blotting ( Fig. 1 supplement 1C-E). We obtained similar results to quantitative proteomics of Xenopus laevis egg cytoplasm (32).
We optimized our microscopy to add as little recombinant protein as possible while retaining sufficient signal-to-noise ratio in the images. The resulting reaction mix contained proteins at 1:1-1:2 labelled:unlabelled ratio. To monitor the simultaneous 15 accumulation of different actin regulators to the FLS tip complex together with elongation of the actin bundle, we used highly inclined and laminated optical sheet (HILO) illumination in up to 3 wavelengths and widefield illumination or spinning disc confocal optical sectioning in 3D. We employed rapid sequential timelapse imaging of actin and other proteins in different combinations to visualize bundle dynamics. We developed an 20 ImageJ plugin, "FLS Ace" that segments individual 3D actin structures from the confocal or widefield channel. Furthermore, it quantifies intensities from any additional HILO channels at the membrane underneath. This enabled us to extract protein intensities at the tip complex and along the shaft of actin bundles from the acquired sets of images.
Linear assignment between timepoints allows tracking of single FLS throughout their 25 lifetime, as well as the detection of actin regulators that accumulate at the membrane prior to actin polymerization ( Fig. 1 supplement 2).
These experiments revealed that the initiation of new FLSs occurs throughout the experiment with a peak at 3 min ( Fig. 1 supplement 3A). To be able to compare 30 individual FLSs irrespective of their time of initiation, we defined the timepoint at which a structure was first detected as t0 in "FLS-time" as compared to "experiment-time". While an FLS that appears later in the experiment can reach the same length as those initiated early, the size of their tip complex area is on average slightly smaller, probably due to limitations of either a protein component in the reaction mixture or physical space on the membrane ( Fig. 1 supplement 3B,C). Normalizing the data to FLS-time, we can see a cooperative assembly of the actin regulators to the tip complex, where the largest increase in fluorescence with time occurs at the same time as peak actin accumulation 5 just prior to t0, defined as the first detection of a 3D structure (Fig. 1B). The recruitment of the Cdc42•GTP probe, GBD, was retarded relative to the other proteins, which is likely due to competition with endogenous N-WASP (Fig. 1B). While there is very little variation in the timing of protein accumulation, considerable variability in the mean intensity at steady-state is evident in the data ( Fig. 1 supplement 3D). To test whether the assembly 10 of the complex is driven by F-actin (the central concept underlying the convergent elongation model of filopodia formation), we tested which of our proteins still assembled in the presence of actin monomer sequesting drug, Latrunculin B (LatB) to prevent the polymerization of actin into filaments. As observed previously, our experiments showed that TOCA-1 was still present, though is somewhat reduced, in the absence of F-actin 15 (19), with comparable reductions of Ena and N-WASP ( Fig. 1 supplement 4). The recruitment of fascin and VASP was prevented, similar to previous observations with Diaphanous 2 (19), and the presence of Cdc42•GTP at the membrane partially affected ( Fig. 1 supplement 4). This is in accordance with an initial slow rise in membrane/Cdc42•GTP interactors TOCA-1 and N-WASP laying a foundation for the 20 recruitment of actin and other regulatory proteins with cooperative assembly of the tip complex coinciding with the peak actin accumulation (Fig. 1B).
To assess the compositional heterogeneity of the final assembled FLS tip complex in more detail we took snapshot images 20 mins after starting the assay, including each 25 protein in combination with the others, in double and triple combinations. Otherwise indistinguishable FLS actin bundles emerge from tip complexes whose composition differs both qualitatively and quantitatively ( Fig. 1C shows example data from one combination). We looked at the correlation between each pair of actin regulators, and the correlation of each regulator with the actin intensity in the bundle (Fig. 1D). We found . The high correlation between Ena and VASP can be explained by their known ability to form heterotetramers (33) and VASP is also reported to cooperate with Cdc42 (17). Diaph3 is a SH3 domain binding partner of TOCA-1 previously observed by co-immunoprecipitations (34), and Cdc42•GTP is known to be a major input into N-WASP activation (35). Nearly all other pairs show weak positive correlations, with the exceptions of N-WASP/Diaph3 and Cdc42•GTP/Diaph3, which exhibit close to zero 5 correlation i.e. they distribute randomly relative to each other (Fig. 1D).
Low levels of fluorescence can result from both nonspecific protein-protein interactions and by specific interactions between proteins of low abundance. The difference between these two possibilities is obscured at low intensities, hence we verified our conclusions Thus we find a generally permissive but not totally promiscuous association between actin regulators. The lack of correlation of Diaph3 with Cdc42•GTP may be due to 30 competition with the GBD domain probe. However, in that case N-WASP should also be competitive with GBD; instead it shows a high correlation (Fig. 1D). The overall positive correlation between most proteins suggests a cooperativity with more proteins joining the assembly the more that are there, which agrees with the sigmoidal shape of protein assembly in the kinetic analysis (Fig. 1B). While some components are found to associate more frequently with each other, these data rule out the presence of discrete subtypes of FLS tip complex as in that scenario there would have been both strong positive and strong negative relationships. 5 We next examined the correlations between tip complex composition and FLS morphology (Fig. 1D). There was a low positive correlation between each of the regulators with the area of the complex, measured with fluorescent actin (Fig. 1D). FLS are somewhat wavy, though less so than the trajectories of Listeria comets in Xenopus 10 egg extracts (36), however there was little correlation between FLS straightness and protein intensity (Fig. 1D). Indeed, there was also little correlation of any single protein intensity parameter with FLS length (Fig. 1D). This is surprising as one would expect there to be a strong role for some regulators to determine bundle lengths. FLS simultaneously enriched with two or three of the observed regulators were slightly, but 15 significantly, longer than those with none or one (Fig. 1E, error bars show the 95% confidence interval). To determine whether we could discern any specifically short or long subtypes of FLS, which would be indicated by multimodal distributions, we plotted the length distributions. Instead, we found that the FLS lengths lay on a continuous exponential distribution ( Fig. 1F shows a quantile:quantile plot comparing the data to a 20 theoretical exponential distribution). Rather than a complex relationship between actin regulator compositions and lengths of FLS suggested by the diverse compositions of actin regulators, the exponential distribution points to a simple law governing FLS growth that is independent of the concentrations of actin regulators. Exponential distributions of lengths suggests a process dominated by stochastic incorporation of actin at functionally 25 equivalent yet molecularly diverse tip complexes.

FLS growth dynamics and theoretical framework
To draw a connection between the heterogeneity of actin regulators inside FLS tip 30 complexes and the observed exponential length distribution, we developed a theoretical framework to describe FLS length dynamics. We applied this to the lengths of large numbers (>100,000) of individual FLS measured with high time resolution ( Fig. 2A, B).
By analogy to a study of lamellipodia dynamics (37), after an initial rapid growth phase, the FLS length cycles between elongation and retraction phases (Video 1, Fig. 2A).
From these data, we calculated histograms of growth velocities, which had exponential tails both in positive (growth) and negative (shrinkage) directions and were largely symmetric. This bi-exponential shape is a Laplace distribution, which fits the data to a 5 remarkable degree (Fig. 2B), unlike others such as a normal or power-law distributions. This is especially striking because the Laplace distribution is controlled only by a single parameter (its variance).
We next sought to understand the simplicity of such a growth distribution apparently arising from the complex networks of molecular associations observed Our model for the growth velocity thus suggests that FLS transit between multiple redundant modes of growth and strictly interpreted suggests that two independent complexes consisting of two regulatory protein pairs (X 1 ,X 2 and X 3 ,X 4 ) can control the growth and shrinkage dynamics of actin structures. The data however indicates that there are at least three pairs of two regulators each, the strongest being N-5 WASP/Cdc42•GTP, TOCA-1/Diaph3 and Ena/VASP (Fig. 1D). We know biologically that there are other proteins such as the formin-like protein 3 (38) as well as the bundling protein fascin that are involved in filopodia assembly, plus the regulators of FLS disassembly that we have not measured. Reconciling with the complexity of the numbers of proteins and complexes that are present, we found that many combinations of

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A more specific prediction of our model of stochastic and redundant FLS heterogeneity is that knocking out one of the subcomplexes should lead to almost negligible effects on the shape of the length distribution since this would only reduce the number of sum terms by one (Fig. 2E). To test this prediction experimentally, we co-immunodepleted Ena and VASP from our extracts. We observed no effect on the number or length of 25 FLS, though adding additional Ena and VASP gave a very small yet significant increase in both length and number of FLS ( Fig. 4B-D). The key feature in terms of the model is that the exponential distribution of FLS lengths is maintained (Fig. 4E). These observations are consistent with the data fitted by the numerical description of a shift from M = 3 to M = 2 independent pathways and with a redundant mechanism underlying 30 FLS growth.

Heterogeneous tip complexes and exponentially distributed filopodia lengths in vivo in Drosophila
Since FLS are an in vitro model we next tested whether the molecular characteristics FLS were similar to filopodia within a native context. Overexpression of proteins is The redundancy in molecular composition described here may allow a variety of upstream and downstream components to intersect with control of actin, to bring the 15 power of actin remodelling to different biological needs, while conserving the overall properties of the cytoskeletal behaviour. A multi-component system could also ensure that signals regulating the cytoskeleton must be multiple and coincident, as only rarely will a single input be sufficient to cause an effect, and it takes an overexpression scenario to subvert the normal homeostatic mechanisms, such as fascin in cancer (4). 20 We show here that in spite of a dynamic and heterogeneous tip complex, it is possible for a constraint to emerge in the resulting activity such that actin cytoskeletal dynamics are similarly governed under a wide range of molecular states. These properties may be what allow the powerful actin machinery to be co-opted wherever necessary without altering its underlying properties.  In order to efficiently process our GFP-utrophin-CH-based high time resolution time 30 lapse Videos, we re-implemented the above algorithm in Python using algorithms from scikit-image.

Post processing of snapshot data
We filter out segmented FLS with an effective diameter of less than 0.5 µm because base size information as well as fluorescent intensity measurements become unreliable at or below the resolution limit. When considering FLS length distributions, we truncated the distribution to lengths between 5 µm and 20 µm to exclude structures which have not  Live imaging and data analysis of Drosophila embryo filopodia Using ImageJ, a maximum projection of the respective stacks was applied for filopodia reconstruction. A combination of automated detection with some manual editing was used to track identified filopodia over time.          Output includes protein intensity information as well as shape parameters.         with the noise controlled by the parameter . We take the difference between concentration and baseline as ! = ! −˜!, hence the ! are normal distributed with zero mean in the long time limit.
One thus expects force to arise from a combination of protein concentration, as indicated 20 by the fact that we see weak, but non-zero correlations between FLS lengths and many proteins. The most minimalistic and generic version of such a model is to assume that each type of protein is independent of each other (uncorrelated random variables), but that they act on force in either an additive or a multiplicative way. For instance, if two proteins ! and ! act on the force via a common given complex, the output would be 25 expected to be multiplicative = ! ! , while if they interact in an independent pathway, the output would be expected to be additive = ! + ! . In general, we can thus see the output force as a sum of products of fluctuating protein concentrations: Strikingly an exact Laplacian force distribution can be achieved by the combination: This can be shown analytically. Indeed the product of two standard normal distributions = ! ! is distributed according to a modified Bessel function of the second kind: The sum of two products = ! + ! is then distributed according to a Laplace In order to simulate FLS growth, we generate = artificial protein concentration traces by integrating (1) using the Euler-Maruyama method with a time step of one second and combine them according to equation (2) to generate the force. We then arrive at FLS length trajectories ( ) by temporal integration of the equation Biochemical connection 15 In order to make the connection between fluctuating protein concentrations and instantaneous force / growth velocity more tangible, we present a chemical reaction network consisting of 2M species Ai, which interact in pairs to form M complexes Bj. The complexes are subject to decay. The reaction network is thus: The mass action rate equations for the single species are then given by with the (degenerate) steady state solutions ! = / . Therefore, we can write these concentrations in terms of the deviations (fluctuations) around the steady state: The rate equations for the complexes, expressed in terms of fluctuations are given by with the total velocity for an FLS reading: Thus, there are three contributions to the steady-state concentrations of the complex !

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(and conversely to FLS speed). The first contribution arises from the steady rate of production , which dictates the average complex concentration. The product of such terms gives rise to a constant non-zero velocity of FLS growth. However, as steady state, FLS net average velocity is close to zero, because of depolymerization processes we have not modelled so far. Monomer availability for instance is a straightforward way 15 to couple net polymerization and net depolymerization, ensuring that this first, non-zero average, contribution drops out. The second contribution !! !!!! has already been discussed in the manuscript, and involves the product of Gaussian fluctuations, yielding generically exponential tails for v (and in the limit of two sums, an exact Laplace distribution with each species !" strictly positive and obeying the modified stochastic dynamics mentioned above (as mentioned above, we assume that there are feedback mechanisms to ensure that the sum of all averages is zero, yielding steady-state 5 conditions for the force/velocity).
Finally, under the assumption that force / growth velocity generation is directly dependent on the concentrations of each of the regulatory complexes, we arrive at the sum of products rule mentioned in the main text: Video Legends Video 1. 15 Timelapse z-stack spinning disk confocal video of actin visualized with GFP-Utrophin CH domain probe at 20-30 min.
Representative video of 13 from 8 embryos.  Figure 1B and the temporal information in Figure 1 -supplement 3. Comma-separated table containing the data underlying the intensity and morphology correlation in Figure 1D, the relative FLS length data in Figure 1E and the FLS length distribution data in Figure 1F.

Data File -Figure 2 Panels A and B
Comma-separated table containing the time course information for the FLS segmentation of the high-timeresolution Utrophin-GFP timelapse movies. Used to generate Figure 2A and Figure 2B. shown in Figure 3B and Figure 3C.

Data File -Figure 4 Panel A
Comma-separated table containing the data used to generate the velocity-intensity 15 correlation graphs shown in Figure 4A.

Data File -Figure 4 Panels C, D and E
Comma-separated table containing the Ena/VASP depletion FLS quantification data used to generate Figure 4C, Figure 4D and Figure 4E.  Figure 5E and Figure 5F. shown in Figure 5G. and length data. The intensity data is shown in Figure 6B, and the intensity-length scatter plot is shown in figure 6D.
Data File - Figure 6 Panel C Comma-separated table containing FLS shaft fascin intensity to background ratios 5 shown in Figure 6C.

Data File -Figure 6 Panels E and F
Comma-separated table containing segmented Drosophila embryo dorsal closure leading edge cell filopodia lengths for wild-type control, GFP-fascin knock-in, mutant 10 fascin and fascin overexpression conditions, shown in Figure 6E and Figure 6F.