Mechanochemical feedback between confinement and actin crosslinking drives the shape dynamics of liquid-like droplets

Domain: chemistry
Published: 2026-02-23
Editorial outcome: major_revision

Abstract
Several actin-binding proteins form phase-separated condensates that promote actin filament assembly and bundling. However, the mechanism by which crosslinker multivalency, actin growth, and condensate mechanics regulate actin organization and droplet shape is not well understood. Here, using a combination of agent-based simulations and experiments, we show that a dynamically deformable droplet interface enables the emergence of tightly-bundled actin rings and weakly-bundled actin discs. We find that crosslinked bundle thickness and droplet diameter follow a power law, consistent with measurements in condensates formed by vasodilator-stimulated phosphoprotein. In addition, the dynamics of droplet deformation exhibit a dynamic snapping behavior that depends on droplet surface tension and crosslinker binding kinetics. We assess the generalizability of these predictions in condensates formed by lamellipodin and RGG. Together, these results indicate that mechanochemical feedback between droplet interfacial mechanics and crosslinker multivalency tunes actin organization and controls the dynamics of droplet deformation driven by actin networks.

Introduction
Biomolecular condensates form membraneless organelles that function as dynamic compartments for organizing various cellular functions1–3. Recent discoveries have shown that several actin-binding proteins (ABPs) can form condensates, through liquid-liquid phase separation, both in vitro and in vivo3–12. Over 160 ABPs have been identified, and each of them plays a distinct role in transforming actin monomers and unorganized filaments into specialized structures that drive complex biological processes13,14. Despite their diversity, ABPs collectively regulate actin filaments through behaviors that can be broadly classified as tuning filament assembly and disassembly rates, branching, severing, annealing, and bundling, which lead to the formation of a variety of actin network architectures. These structures provide the mechanical framework for filopodia, lamellipodia, stress fibers, focal adhesions, contractile rings, and the actin cortex4,13,15,16. The reorganization of the actin cytoskeleton underlies a wide range of cellular processes that require cells to generate and transmit force, change shape, and respond to mechanical and chemical cues in their environment, including motility, signaling, and intracellular transport10,17–21. Dysregulation of these processes is implicated in many pathologies, notably in cancer and metastasis20. Importantly, ABP condensates have been observed in cells and are thought to provide a favorable environment for the recruitment of actin monomers, actin filaments, and other cytoskeletal proteins, and to promote nucleation of actin filaments12,16,22–27. However, a comprehensive understanding of the role of ABP condensates in actin network organization and remodeling is an active area of investigation.

Due to the high complexity of in vivo condensate assemblies, reconstituted systems have been employed to study interactions between actin and ABP condensates in a controlled environment to understand the underlying biophysical principles of actin organization and confinement28. Similar actin ring formation has been observed in experimental studies using giant unilamellar vesicles (GUVs)29,30 and vesicles31–33. Beyond the biochemical function of the constituent ABPs, the material and physical properties of condensates also regulate their function24,34,35. The same weak interactions that drive condensate formation also allow for the development of cohesive forces that endow the liquid droplet with viscoelastic properties such as surface tension and viscosity24,35–37. For actin filaments already within condensates, the surface tension drives confinement, which supports the accumulation of bending energy5,6,38. Condensate deformation occurs once the filament bending energy exceeds the surface energy6,39. Bending energy may accumulate from a combination of actin polymerization and increases in actin bundle rigidity mediated by ABP processes such as actin crosslinking. The resulting interplay between surface energy and bending energy plays a key role in regulating the resulting structures.

By combining in vitro and in silico approaches, we have been able to create a complementary framework where experiments provide physical constraints and simulations systematically explore the underlying mechanisms and generate new experimentally testable hypotheses. We have previously established how the spatial confinement of ABPs through phase separation affects actin filament assembly and bundling5,6,39,40. For example, we recently showed that vasodilator-stimulated phosphoprotein (VASP) forms phase-separated droplets wherein monomeric actin can assemble into filamentous actin networks6. Depending on the ratio of actin to VASP, the resulting actin networks can form shells, rings, or ellipsoidal discs6,39. Actin ring structures are tightly packed and aligned into an early 2D distribution, while actin disc structures are loosely aligned into a thick ellipsoidal structure. The formation of crosslinked actin networks (rings) was a precursor to droplet deformation, where spherical VASP droplets transitioned to ellipsoidal droplets and finally rods6,39. More recently, we also showed experimentally that actin bundling did not require native tetrameric strong bundling proteins such as VASP but was also possible due to proteins such as lamellipodin (Lpd), which bind actin but with a lower binding valency and without inherent polymerase activity16,41–43. We also showed that when Eps15, a condensate-forming protein without any known interaction with actin, is fused with Lifeact, a general F-actin binding motif, the resulting Eps15-Lifeact composite promotes actin filament assembly and bundling5. Using an agent-based modeling approach, we investigated how interactions between VASP and actin in rigid, non-deformable droplets can lead to the formation of rings versus shells39. We showed that rings of actin resulted from kinetic trapping, where, for a fixed filament growth rate, the binding and unbinding rate of VASP to actin allowed sufficient VASP residence time for binding and zippering additional filaments6. We also showed that systems with dynamic dimers and multimers, corresponding to Lpd and Eps15-Lifeact composites, respectively, formed rings versus shells resulting from kinetic trapping5. In the case of GUV-encapsulated actin, membrane-actin tethering is critical for ring formation31. These experimental observations suggest that general principles govern the assembly of actin shells, rings, and discs in protein droplets.

However, two questions remain: What is the role of mechanical feedback from deformable boundary confinement in tuning actin organization within the droplet? How do the biochemical properties of the crosslinker influence droplet deformation? To address these questions, we built a minimal model that captures the key elements of actin organization in phase-separated droplets using a combination of computational and experimental approaches. We modeled the ABP droplet using dynamically deformable ellipsoidal boundaries and represented both the ABP-ABP and ABP-actin kinetic interactions explicitly (Fig. 1). Our simulations revealed that the condensate environment tunes the nature of actin organization through kinetic trapping, thereby controlling droplet shape. We predicted that the droplet diameter scales with the actin bundle thickness required to deform a spherical droplet according to a power law. These predictions were validated by experiments. We then showed in simulations and experiments that capping protein tunes the filament length distribution and can alter the onset and extent of VASP droplet deformation. Finally, we characterized our observations throughout this study, which showed that the temporal behavior of droplet deformation has a dynamic snapping behavior depending on the droplet mechanical properties and the crosslinker kinetic properties. Thus, our study produces a comprehensive, general framework to understand the relationship between droplet deformation and the mechanochemical environment of the droplet.

Results
We constructed an agent-based model to investigate the interplay between the physical and chemical properties of the ABP droplet and the actin network. For these simulations, we chose to use the Cytosim framework, which simulates the chemical dynamics and mechanical properties of cytoskeletal filament networks44. This modeling framework allows us to explicitly simulate three distinct components: filaments, the droplet space, and crosslinkers.

Actin filaments are represented as inextensible fibers that elongate at a constant rate due to polymerization (Fig. 1A). We assume that the plus end elongates while the minus end is stable. These assumptions are consistent with the activity of VASP as a processive polymerase that mediates up to a three-fold increase in elongation rate at bound barbed ends compared to free barbed ends45, and based on previous models of actin filament elongation with a net elongation at the plus ends39. In this study, we introduce a method for implicitly modeling the effect of capping protein by stochastically stopping (capped state) and restarting (uncapped state) elongation via a capping reaction governed by the rate parameters k_cap and k_uncap (Fig. 1B). Finally, actin filaments are modeled as a series of linear segments of length 0.1 µm that are connected by hinge points to allow for bending (Fig. 1C). The bending stiffness of the fiber is captured by the flexural rigidity parameter k_bend46.

The ABPs are represented as spherical solids with a radius of 30 nm, with one of two different kinds of binding sites that either bind actin to promote bundling or bind to other ABP molecules to promote multimerization. Each binding site has a specified binding rate constant k_bind and binding distance [30 nm] that defines the radius within which targets are considered in range for binding to occur39. The unbinding rate, k_unbind, specifies the rate constant used, while the unbinding force F_0 represents the characteristic force used in the force-dependent kinetics (Bell’s law and Kramers’ theory) representation of slip bond unbinding kinetics47–52. We investigated the role of two distinct types of actin crosslinkers: tetrameric actin crosslinkers representing VASP39,40 and dynamically multimerizing actin crosslinkers representing the transient multivalent interactions between monomeric ABPs in the condensate environment5. VASP is modeled with four actin-binding sites that enable it to undergo a series of reversible reactions between five states that differ depending on the number of bound actin filaments (Fig. 1D)39. Additionally, we included modifications to the Cytosim code base to model dynamically multimerizing monomers based on a specified multimer formation rate k_form and splitting rate k_split, as introduced in Walker et al.5. These dynamic crosslinkers differ from static crosslinkers in that each crosslinker molecule is a monomer modeled with a single actin-binding site and two monomer-monomer binding sites to allow each monomer to bind with up to two other monomers to dynamically form multimers (such as dimers, trimers, tetramers, etc.). Due to the large number of possible states, we categorize each monomer unit by the number of other monomers it is bound to [0, 1, 2] and by whether or not it is bound to an actin filament [B] or free [F], thus resulting in six different states that each monomeric unit can cycle between through a series of actin or monomer binding and unbinding reactions (Fig. 1E).

The space representing the condensate modeled the 3D geometry confining where reactions, filament dynamics, and diffusion occur, as in previous work5,39,40. Whereas our previous models used a spherical geometry that was rigid or deformed in a quasi-static manner5,39,40, we now incorporate a deformable ellipsoid geometry that deforms dynamically in accordance with the forces acting on the boundary against the surface tension of the droplet (Fig. 1F, G). This framework in Cytosim was developed in Dmitrieff et al.53 to understand the role of cortical tension in microtubule assembly. By modeling the condensate geometry as a dynamic ellipsoid, we investigated the feedback between condensate properties and the actin network. The extent of deformation is calculated by the balance between the interfacial surface tension and the point forces exerted by the filaments on the boundary—subject to volume incompressibility constraints53. Point forces are calculated from the force exerted to keep actin filaments within the boundary, as modulated by the boundary repulsion stiffness (Supplementary Fig. 1). The speed at which deformation occurs is further modulated by an effective viscosity, µ_effective, which acts as a damping parameter to slow the deformation of the ellipse axes over time53.

Specific parameters used to set up the Cytosim model can be found in Supplementary Table 1, while the list of parameters varied in each set of simulations can be found in Supplementary Table 2. The Cytosim simulations were performed on the Triton Shared Computing Cluster (TSCC) at the San Diego Supercomputer Center (SDSC)54.

Our investigations are organized as follows: we first show that in droplets containing VASP, allowing for droplet deformation leads to the emergence of discs (weakly-bundled actin structures) that are oriented along the axis of deformation. We found that a power law relationship exists between the initial radius of the droplet and the number of filaments for a particular aspect ratio. Our simulations reveal that the kinetics of droplet deformation follow a dynamic snapping behavior, a snap-through of the shape of the droplet that occurs over time, depending on the organization of the actin network in the VASP droplet. We then predicted that actin filament length plays an important role in determining the onset and extent of droplet deformation; this prediction is verified by experimentally changing the capping protein concentration. We also show that all our observations are generalizable to ABPs that form dynamic multimers, establishing that actin bundling and droplet deformation do not require strong bundling proteins. Even in the absence of a specific ABP in the droplet, we show that actin filaments form discs that align along the axis of deformation, and therefore, concentrate actin in phase-separated containers is a sufficient condition for bundling and force generation. Finally, we show that the biochemical properties of actin crosslinkers drive actin structure formation by influencing the interplay between droplet shape and bundle alignment.

Deformable VASP droplets produce tightly-bundled actin rings and weakly-bundled discs
To consider the importance of VASP droplet deformation in actin assembly, we relaxed the assumption of a rigid droplet interface used in our previous work, where we simulated actin bundling in rigid VASP droplets5,39,40. Here, we model simple deformation of a droplet, consistent with experiments where we observed droplet deformation5,6,39,40, and ask whether the deformability of the droplet influences actin assembly. In our simulations, the droplet was allowed to deform with a given surface tension and a viscous dissipation parameter, along with a constant volume constraint53. To map the interaction between the droplet interface and the actin bundling, we simulated a range of binding and unbinding kinetics of VASP with actin filaments in deformable droplets (Fig. 2). In these deformable droplets, we observed the formation of weakly-bundled discs of actin filaments and rings; the weakly-bundled discs were seen in the regime of weak VASP binding kinetics (k_bind ≥ 0.1 and k_unbind < k_bind or k_bind < 0.1) (Fig. 2A).

This is different from the case of rigid droplets, where we observe shells and rings (Supplementary Fig. 2A and Chandrasekaran et al.)39. The disc-shaped bundles that emerged in deformable droplets appeared to align themselves along the axes of maximum droplet deformation to minimize the energetic cost of filament bending (Fig. 2A). We quantified the thickness of the discs by analyzing the surface area fraction of the droplet covered by actin39. The differences in surface area fraction for rings and discs are shown in Fig. 2B, where rings form with a low surface area fraction, while discs have a much larger surface area fraction similar to shells seen in rigid droplets (Fig. 2B, Supplementary Fig. 2B). Additionally, when comparing the fraction of VASP tetramers bound to actin filaments, almost no VASP tetramers were bound to any actin filaments in simulations with kinetic conditions that favored disc formation (Fig. 2C). This is in contrast to rings and shells (Supplementary Fig. 2C), where kinetic trapping by VASP tetramers was required to tightly bundle actin filaments into rings or tightly confine the actin filaments within the original spherical geometry of the droplet so that the filament network develops into shell-like shapes39.

Finally, we used principal component analysis (PCA) to classify the different actin network structures. PCA showed that for the data in Fig. 2B, C, which is governed by five independent parameters, the first three principal components (PCs) represent ~96% of the variance in the dataset (Supplementary Fig. 3A). To understand how each of the five parameters used to describe actin organization contributes to each PC, we then looked at the varimax loading parameter. We found that PC1 corresponded to information on the actin-covered surface area fraction and the fraction of VASP tetramers fully bound to four filaments, while PC2 captured information regarding the number of VASP tetramers bound to one and two filaments (Supplementary Fig. 3B). Then, using a K-means clustering algorithm on the first three PCs, we classified the resulting actin network shapes into four distinct clusters (Fig. 2D). We determined that four is the optimal number of clusters by choosing the largest silhouette coefficient among various cluster sizes (Supplementary Fig. 3C). The actin network shapes formed by each cluster categories corresponding to actin structures that are discs, rings, loose shells, or tight shells. The data for each snapshot is projected onto a scatter plot of the first two principal components (PC1 and PC2), and colored by clusters. Representative snapshots of each cluster are included above the plot. Please see Supplementary Table 3 for the corresponding cluster identities for each combination of kinetic parameters. E Ellipsoid span a, span b, and span c (a ≥ b ≥ c) are calculated from the eigenvalues of the gyration tensor and approximate the shape of the actin network as an ellipsoid. The distribution of each ellipsoid span is depicted as box plots and organized by cluster category. Each box plot displays the data points (gray), median (orange line), interquartile range (box), and 95% confidence interval (whiskers). Data was obtained from the last five snapshots of each replicate and sorted according to the cluster membership determined in (D): Discs (n = 590), Rings (n = 310), Loose Shells (n = 165), and Tight Shells (n = 185). For (B–E), 10 replicates are considered per condition. Source data are provided as a Source data file.

Actin bundle thickness and droplet diameter satisfy a power-law relationship
We next sought to systematically understand the dynamics of droplet deformation. We first analyzed how the aspect ratio of the droplet evolved over time under disc-forming conditions (weak VASP binding kinetics, Fig. 2, k_bind = 0.001 s−1) for a droplet of surface tension of σ_surface = 7 pN/µm. We observed that the aspect ratio evolved non-monotonically with a consistent time signature for all disc-forming conditions. Until the filament length reached the diameter of the droplet, the aspect ratio remained unchanged (L_fil = 2R µm, t ≈ 200 s). Once the filaments grew longer than the droplet diameter, the droplet aspect ratio increased until the filament length reached L_fil = πR µm (t ≈ 300 s). As the filaments bent around the periphery of the droplet, the aspect ratio decreased, indicating a partial recovery of the spherical shape after an ellipsoidal deformation. However, when the bundles of actin grew longer than 4πR/3 µm, the aspect ratio returned to growing monotonically (Fig. 3A, Supplementary Fig. 4).

From these dynamics, three interesting questions emerge: What is the relationship between droplet size, filament bending energy, and the extent of droplet deformation? What factors control the onset of droplet deformation? And what factors control the dynamics of droplet deformation? We systematically answer these questions below.

To understand the relationship between the droplet size and extent of droplet deformation, we recall that disc formation is governed by an interplay between droplet surface energy and filament bending energy6,29. Since the filament bending energy is proportional to the thickness of the filament bundle, varying the number of filaments within our simulations will modulate the accumulation of filament bending energy into the system. Varying the droplet size, on the other hand, will change the droplet surface energy, which is given by the product of the surface tension and the surface area of the droplet. A parameter sweep of the number of filaments versus the initial diameter of the droplet will thus allow us to study their effects on droplet deformation. To test this idea, we simulated actin bundling in deformable droplets of surface tension σ_surface = [2, 4, 8] pN/µm that ranged in size from an initial diameter of 0.5–4.0 µm and varied the number of filaments from 5 to 100 under fixed VASP tetramer concentration (~0.40 µM). The growth rate of each filament was adjusted so that the final filament length matched the circumference of the original droplet (2πR µm or πD µm) after 600 s for each droplet size. Our choice of surface tension values ensured that we were operating in the deformable regime established in our 30 filament, 2 µm diameter droplet simulations (Supplementary Fig. 4). We simulated 960 unique conditions (10 replicates each, parameters tested shown in Supplementary Table 2). For each case with a final aspect ratio between 1.5 and 2.0, we recorded the number of filaments as an indicator of the requisite number of filaments to produce a small droplet deformation. This range was held constant across all 960 conditions (surface tension values, droplet radii, and number of filaments) studied to provide a common threshold with a substantial sample size.

We first investigated the case of VASP crosslinkers with disc-forming kinetics (k_bind = 0.1 s−1, k_unbind = 1.0 s−1) and found that the droplet aspect ratio increased with the number of filaments. Furthermore, as the initial diameter of the droplet increased, more filaments were required to deform the droplet to a similar aspect ratio. We found that the number of filaments required to induce small deformations scaled with the initial droplet diameter according to a power law of the form y = ax^b, for all surveyed surface tensions for VASP crosslinkers with disc-forming kinetics (Fig. 3B). This power law relationship held even under ring-forming conditions (Fig. 3D).

This observation aligns with experimental findings that were shown in our previous work, wherein the thickness of the bundles that deformed actin rings to higher aspect ratios increased with droplet diameter6. VASP droplets that were deformed into higher aspect ratio structures tended to have thicker actin rings in comparison to droplets that remained spherical. In these experiments, a power law relationship was also seen between actin ring thickness and the effective diameter of the VASP droplets. We analyzed the experimental data from those prior experiments and compared them to the model predictions in this study (Fig. 3C, E). These experimental results corroborated the computational predictions, and together the model and experimental findings suggest that the thickness of the actin ring contributes to the deformation of the VASP droplets and follows a power law growth relationship with droplet diameter. We note that the existence of a power law relationship is also consistent with previous theoretical predictions29 and energetic arguments6. Differences between the power law exponents in the simulations and those predicted from theory arose from the aspect ratio range chosen for the simulations. The derivation of this power law relationship can be found in the Methods. Thus, using our agent-based simulations, we were able to confirm the existence of a power law relationship between the initial diameter of the droplet and the number of actin filaments for different droplet surface tensions.

Filament length determines the onset and extent of droplet deformation
We next investigated the role of filament length on droplet deformation. In our prior studies, we showed that the filament length controls both the onset of ring formation and the final aspect ratios as the rings deform the droplet6,39. However, the effect of the dynamics of non-deterministic actin elongation on ring formation remained unknown. Therefore, we investigated the relationship between filament length and droplet deformation under ring-forming conditions in the presence of actin capping protein, which arrests polymerization of filaments55. Specifically, we simulated the stochastic capping and uncapping of actin filaments (Supplementary Fig. 5) using an implicit capping and uncapping framework. In our simulations, plus ends were allowed to switch between capped (at rate k_cap) and uncapped (at rate k_uncap) states stochastically throughout the simulation. Once capped, the end could not grow until the end was uncapped. As a result, the filaments reached a distribution of lengths and were crosslinked by VASP tetramers to form rings (k_bind = 10.0 s−1, k_unbind = 1.0 s−1) within both rigid (Supplementary Fig. 5A–C) and deformable droplets (Fig. 4). In this implicit framework, while we specified the rate of filament (un)capping, we did not specify a total number of capping protein molecules.

We varied the capping rates in the range [k_cap = 1.0 s−1–11.314 s−1] for fixed uncapping rate k_uncap = 1.0 s−1 (Fig. 4A). These values were chosen because they produced the full range of VASP-crosslinked actin structures from linear bundles to rings (Supplementary Fig. 5A). We found that increasing capping rate (k_cap) resulted in shorter filaments and a narrower distribution of filament lengths (Fig. 4B). Additionally, decreasing capping rates in the range [k_cap = 0.088 s−1–1.0 s−1] also resulted in a narrower filament length distribution but centered around a larger filament length (Supplementary Fig. 5D). We then quantified the final aspect ratio of the droplet as a measure of the extent of the deformation. Increasing the capping kinetic rate resulted in lower aspect ratio droplets because shorter filaments cannot exert large deforming forces on the droplet surface (Fig. 4C). From these simulations, we predicted that when capping protein was added to VASP droplets with actin in them, the average droplet aspect ratio would decrease, and there would be fewer droplets with high aspect ratios.

To test this prediction in experiments, we incorporated capping protein, CapZ, into droplets composed of VASP. CapZ is an actin-binding protein that regulates filament dynamics by binding tightly to the barbed ends of actin filaments, thereby preventing further monomer addition and elongation and effectively controlling filament length56–58. Specifically, we formed droplets in a 10 µM solution of VASP in the presence of 3 µM actin with increasing concentrations of CapZ. Droplets were formed by mixing VASP and associated proteins with 3% (w/v) PEG 8000 as a crowding agent. PEG is often added in the study of LLPS to mimic the crowded environment in the cell cytoplasm59–61. As observed previously6,39,40, VASP droplets lacking CapZ deformed into rod-like shapes upon actin polymerization (Fig. 4D). As increasing concentrations of CapZ were added, the droplets became more spherical (Fig. 4D), and both the fraction of deformed condensates, defined as protein droplets with an aspect ratio greater than 1.2, and the average aspect ratio of droplets decreased. (Fig. 4E, F). Notably, phalloidin-stained actin did not arrange into rings at the inner surfaces of droplets in the presence of CapZ (Supplementary Fig. 5E), suggesting that CapZ addition reduced actin filament length such that there were no filaments long enough to exceed the condensate diameter.

The dynamics of droplet deformation exhibit a dynamic snapping behavior that depends on the droplet surface tension
We previously showed that the dynamics of droplet deformation for deformable VASP droplets can be non-monotonic in time (Fig. 3A). To investigate how these dynamics depend on VASP and droplet properties, we tracked the droplet deformation dynamics for different VASP binding/unbinding rates and for different values of droplet surface tension (σ_surface = 7 pN/µm (Fig. 5A–C) and σ_surface = 2 pN/µm (Fig. 5D–F)). Droplet deformation can be understood as the balance between the bending energy of the actin bundle and the surface energy of the droplet. The onset of droplet deformation occurs when the length of the bundle exceeds the diameter of the droplet because the linear actin bundles are able to overcome the surface energy of the droplet. However, the extent of droplet deformation depends on VASP-actin interaction kinetics. When k_bind > k_unbind, actin in ring-like states with strongly-bound VASP tetramers lead to minimal droplet deformation. These rings are kinetically trapped, which results in reduced deformation capacity of the actin networks (Fig. 5B, C, Supplementary Movie 1).

When k_bind ≤ k_unbind, we observed an increase in droplet deformation depending on the value of the surface tension. Under disc-forming conditions, where actin bundling is weak, the droplet begins to deform once the filaments reach L_fil > 2R (t ≈ 200 s). As the filaments grow longer, the filaments begin deforming the droplet linearly with time for low surface tension (σ_surface = 2 pN/µm) to form rod-like bundles of actin (Fig. 5D, E, Supplementary Movie 2). However, the variability increased with time in low surface tension and low binding and unbinding rates. In this case, we observed that out of ten replicates, six of them were rods while four of them snapped back into discs (Fig. 5D, left-side snapshots). On the other hand, when σ_surface = 7 pN/µm, the droplet aspect ratios evolve as a non-monotonic hump (t ≈ 200–400 s), after which they either experience continuous linear growth or remain stable. This nonlinear temporal behavior can be understood as follows: the aspect ratio of the droplet increases initially with filament growth as filaments are aligned along the ellipsoidal major axis (Supplementary Movie 3). This anisotropic filament distribution enables filament growth-driven droplet deformation. As the filament length reaches L_fil = πR µm at t ≈ 300 s, the filaments bend along the periphery of the droplet, resulting in spindle-shaped networks that trace the contour of the ellipsoidal droplet, forming a near-isotropic network (Supplementary Movie 3). In this state, actin does not drive droplet deformation as the filament segments are arranged isotropically along the droplet surface. The actin filaments align themselves and grow along the droplet surface; such tangential growth does not lead to substantial forces exerted by the actin bundle on the droplet interface. As a result, the droplet surface energy overcomes the bending energy, and the droplet tends towards a spherical state. Eventually (t > 400 s, L_fil = 4πR/3 µm), the filaments rearrange into discs that can overcome the surface energy, resulting in droplet deformation towards an ellipsoidal shape. This can be observed through the reduction in actin-covered surface area from ~300 to 400 s as shown in Fig. 2B. From ~200 to 400 s, the initial increase and decrease in aspect ratio forms a hump-like structure in the graph that can be understood in the context of a dynamic snapping behavior. Snap-through bifurcations are a common feature in mechanics when small changes to a control parameter can lead the system from one stable energy-minimizing state to another62–66. Albeit, in our case, we observe a dynamic snapping because the kinetic rearrangement of the actin filaments over time leads to a time-dependent response.

In the case of ring formation kinetics, the stronger actin binding-unbinding kinetics, specifically when k_bind ≈ k_unbind for k_bind ≥ 0.1 (Fig. 5B, C), resulted in actin bundles that have formed by the time that the filament length reaches the diameter of the droplet, L_fil = 2R µm (t ≈ 200 s). By forming a bundle early, nascent ring structures emerge before L_fil = πR µm (t ≈ 300 s) in the form of either a single main bundle or two primary bundles oriented 180° from each other. Here, the filaments are arranged into the bundle to minimize bending energy and, with the guidance of crosslinkers that favor bringing filaments together, are better able to concentrate force to deform the droplet.

Crucially, because the filaments have generally been confined into a highly-aligned, essentially 2D ring structure stabilized by crosslinkers, the bending energy of the ring is able to overcome the droplet surface energy to push the droplet toward higher aspect ratios without exhibiting an initial non-monotonic growth pattern as was the case for discs (compare rings in Fig. 5C and discs in Fig. 5A).

Deformable droplets made of ABPs that form dynamic multimers also exhibit actin bundling and dynamic snapping
We next investigated whether the relationship between droplet deformation and actin bundling was specific to VASP alone or generalizable to other ABPs. To do so, we investigated the role of monomeric ABPs that have homomeric interfaces. Through the formation of multimers, the actin-binding valency of the complex was increased, enabling crosslinking. We simulated ABPs as monomeric units capable of binding to a single actin filament and up to two other monomers, thus allowing for the formation of higher-order multimers (Fig. 1E)5. We prescribed ring-forming actin-binding kinetics, while modulating the kinetics of multimer formation and splitting, and found that actin rings and discs formed depending on the different kinetic parameters (Fig. 6A, Supplementary Fig. 6A). Our analysis of multimer lengths showed that rings only formed in kinetic conditions that favored multimerization of the crosslinkers, while discs formed under conditions in which the crosslinkers remained largely monomeric (Supplementary Fig. 6B, C).

We also examined the relationship between droplet size and the number of filaments. We chose the multimerization condition that maximizes multimer size (Supplementary Fig. 6C, k_form = 10.0 s−1, k_split = 0.01 s−1). We found that the initial radius of the droplet and the number of filaments required to deform it also followed a power law relationship (Fig. 6B) similar to stable tetramers studied earlier (Fig. 3B, D). To test our model prediction that dynamic multimers of ABPs were sufficient to bundle actin filaments and sustain droplet deformation within protein condensates, experiments were performed using condensates made of an EGFP-tagged C-terminal IDR region of the Lpd protein. This protein, henceforth called monomer mini-Lpd as it is a native monomer, was shown to phase separate and facilitate actin filament assembly5. To test if there was a power law relationship between actin ring thickness and condensate diameter, we performed similar experiments to those done previously with VASP (Fig. 3C, E)6 with condensates composed of monomer mini-Lpd. Specifically, condensates were formed from a 5 µM solution of monomer mini-Lpd and 3 w/v% PEG 8000. Monomeric actin (G-actin) was added to the condensates at a concentration of 3 µM and allowed to assemble prior to filamentous actin staining with phalloidin-594 (Fig. 6C). Actin rings appeared at the droplet periphery. The thickness of the actin ring and the effective diameter of the encapsulating mini-Lpd condensate were quantified as described in the Methods. We then plotted ring thickness as a function of effective condensate diameter and extracted a power law scaling relationship (Fig. 6D). Ring thicknesses somewhat below the optical diffraction limit were estimated by measuring the Gaussian intensity profile for actin at the droplet edge and subtracting the microscope point spread function. In line with model predictions, condensates formed from monomer mini-Lpd also displayed a power law relationship between condensate diameter and actin ring thickness, displaying similar values in the power law fit to those displayed in VASP condensates. Together, the experimental results and modeling predictions suggest that this power law relationship is generalizable to any condensate-forming protein that facilitates actin filament assembly.

In simulations, analysis of the deformation kinetics of these droplets showed that dynamic multimer crosslinkers promoted a higher final aspect ratio than the analogous condition for stable tetramers (VASP) (compare Fig. 6E and Fig. 5C, k_unbind = 1.0 s−1). Furthermore, while all conditions experienced droplet deformation, the conditions that formed rings had larger final aspect ratios than the conditions that formed discs (Fig. 6E, Supplementary Fig. 6D). Thus, dynamic multimers form robust rings with additional modes of reorganization, which facilitates the bending energy dissipation resulting in higher aspect ratios. This also alters the dynamics of droplet deformation, as reflected by the lack of a pronounced hump in the aspect ratio time series (Fig. 6E, Supplementary Fig. 6D).

Confinement of growing actin filaments in liquid droplets is sufficient for filament bundling and droplet deformation in the absence of crosslinkers
Our analysis of VASP and Lpd droplets and their interactions with actin shows that weakly-bundled actin discs emerge when the droplet boundary is allowed to deform, even when crosslinker interactions with actin filaments are minimal. The lack of appreciable crosslinker activity suggests that disc formation might be primarily driven by the material properties of the droplet itself, rather than the crosslinkers. Therefore, we next investigated how droplet properties impacted actin bundling. Specifically, we varied σ_surface and µ_effective in a deformable droplet in the absence of any crosslinking interactions. We found that actin filaments formed loose discs when the droplet surface tension was sufficiently low (Fig. 7A). The fraction of the droplet surface covered by actin was larger for shells than for discs, and the transition from discs to shells was clear as surface tension increased (Supplementary Fig. 7A, B). We also observed that shell structures were recovered as surface tension increased, indicating that droplet deformation could drive assembly of weakly-bundled actin discs. As expected, changes in the effective viscosity had a very muted effect on the final aspect ratio, as this parameter alters only the time necessary to deform the droplet and not the final state of deformation within the deformable ellipsoid framework53. Thus, we focused on the σ_surface for the next steps.

We next investigated whether droplet deformation in the absence of crosslinkers also exhibits a power law relationship between the number of actin filaments and droplet diameter. As before, we varied the number of actin filaments and the initial diameter of the droplet and extracted a power law relationship for nonspecific crosslinkers (Fig. 7B). Again, we observed that actin elongation alone was sufficient to deform the droplet. This result led us to predict that an arbitrary condensate-forming protein with a weak affinity for actin filaments would be sufficient to form these discs. To experimentally test this prediction, we formed condensates from the RGG domain of LAF-1, a well-characterized phase-separating protein that lacks actin-binding domains34, which has a high concentration of positively charged arginine residues and is therefore likely to interact non-specifically with negatively charged actin. We examined how the addition of actin impacted RGG condensate shape and filament organization. RGG droplets were formed from a 20 µM solution of the protein and 3 w/v% PEG 8000. Increasing concentrations of actin were added to the droplets after they formed, as described above. As predicted by the simulations, RGG condensates were progressively deformed into rod-like morphologies as the concentration of actin increased (Fig. 7C). Additionally, both the fraction of deformed condensates, defined as protein droplets with an aspect ratio greater than 1.2 (Fig. 7D), and the average aspect ratio of droplets increased with increasing actin concentration (Supplementary Fig. 7C). Notably, phalloidin staining revealed that actin filaments were arranged into rods and peripheral rings at the inner surfaces of the droplets (Supplementary Fig. 7D). These experimental observations suggested that actin filament assembly within condensates was alone sufficient to promote the bundling of actin filaments into discs, and that asymmetries in the condensate boundary could alone facilitate bundling of actin filaments into discs. We note that the RGG droplets were diffuse at larger condensate diameters. As ring thicknesses across a range of droplet diameters are necessary to derive a power law trend, we were unable to quantify the relationship between droplet diameter and ring thickness in RGG droplets. Finally, when we increased the surface tension in the simulations, we observed that the droplets deformed to lower aspect ratios. The dynamics of droplet deformation transitioned from dynamic snapping to a slow monotonic increase over time (Fig. 7E, Supplementary Fig. 7B).

Feedback between droplet shape and bundle alignment is tuned by crosslinker properties
Throughout our simulations, we found different dynamics for droplet deformation depending on the crosslinker kinetics and multivalency and droplet surface tension (Figs. 3A, 4C, 5, 6E, 7E, Supplementary Figs. 4C, 6D, 7B). Based on these observations, we hypothesized that there exists a feedback between the droplet shape and filament alignment. To test this hypothesis, we first simulated actin filament elongation and bundling within rigid ellipsoidal droplets without any crosslinkers (Fig. 8A). We defined an alignment angle of the actin bundle as the angle formed between ellipsoid span a and span c, obtained from the eigenvalues of the gyration tensor as in Fig. 2E and described in the Methods. A filament alignment angle approaching 0° describes a flat distribution of actin, as expected for a ring, while an alignment angle approaching 45° describes a perfectly spherical distribution of actin, as expected for a spherical shell. We found a strong relationship between the droplet aspect ratio and filament angle (Fig. 8). To understand the role of boundary shape and actin organization, we simulated actin networks within rigid ellipsoidal droplets fixed at various aspect ratios (Fig. 8A). We found that the actin filaments aligned with the major axis of the ellipsoid at all aspect ratios studied (Fig. 8A). When we repeated these calculations with deformable droplets without crosslinkers, we found that decreasing surface tension resulted in both higher aspect ratio droplets and lower filament alignment values (Fig. 8B). Further, we observed that small changes to alignment angle resulted in large changes to aspect ratios (Fig. 8B, inset). This observation further validates our hypothesis that the filament alignment angle controls the extent of droplet deformation. When we analyzed the relationship between filament alignment angle and aspect ratio in the presence of dynamic multimers under ring-forming conditions, we saw a shift towards more aligned filaments and higher aspect ratios as multimer formation became more favorable (Fig. 8C, D). When we used stable tetrameric VASP as a crosslinker, we saw that, depending on the binding kinetics, we could achieve the same alignment angle for a wider range of aspect ratios (Fig. 8E, F). Comparing time profiles of aspect ratio and alignment angle evolution between ring-forming dynamic multimeric (Fig. 8D) and stable, tetrameric (Fig. 8F) crosslinkers, we observed that the stability of a multimeric unit also tuned the maximum aspect ratio achieved during deformation. Thus, we concluded that, while the material properties of the droplet determine the extent of filament alignment and droplet deformation, the biochemical properties of ABPs controlled the dynamic emergence of different actin networks.

Methods
Model development
Chemical and mechanical framework employed in Cytosim. We constructed a minimal computational model to explore the emergent properties of actin networks in liquid droplets. Our simulations were performed in Cytosim (https://gitlab.com/f-nedelec/cytosim), an agent-based modeling framework that simulates the chemical dynamics and mechanical properties of cytoskeletal filament networks. Filament dynamics and diffusing species are modeled by numerically solving a constrained Langevin framework in a viscous medium at short time intervals. Additionally, actin filaments are composed of a series of inextensible rigid linear segments of length 0.1 µm that are connected by hinge points to allow for bending. Cytosim computes the bending energy of the fiber using the specified flexural rigidity k_bend in the input parameters. Tetrameric VASP crosslinkers are modeled as a single spherical solid of radius 30 nm with four actin-binding sites. Mini-Lpd monomers are modeled as spherical solids of radius 30 nm and function according to the dynamic multimerization model with a single actin-binding site and two monomer-monomer binding sites. Actin binding is governed by rate parameters k_bind and k_unbind. Each binding site has a specified binding distance that specifies the radius of the spherical binding volume within which binding partners are considered as part of the binding reaction. Unbinding reactions are governed by Bell’s law and Kramers’ theory model representation of slip bond unbinding kinetics, where the rate constant is given by the unbinding rate and the characteristic force is given by the unbinding force. Steric repulsion potentials between diffusing crosslinkers and filaments are employed to avoid spatial overlap of species.

Representation of actin elongation dynamics
Within the existing actin elongation framework, actin filaments elongated deterministically at a constant rate k_grow. Each filament is represented by a series of segments, each with a maximum length L_seg = 0.1 µm. Filament elongation rates are scaled to the size of the droplet space such that the filaments grow, if unperturbed, to the length of the circumference of the original droplet, L_fil = 2πR µm, by the end of the simulation at 600 s. As such, for a droplet with a radius R = 1 µm where growth is deterministic, each filament will grow at a rate k_grow = 0.0103 µm/s to reach L_fil = 2π µm in length.

Dynamics of capping protein
On top of the existing framework for filament elongation dynamics, the effect of capping protein is modeled implicitly by functionalizing the growing end of filaments with the ability to stochastically stop or restart elongation using a similar underlying framework as binding reactions in Cytosim. The capping reaction is governed by the rate parameters k_cap and k_uncap. The implementation of our implicit capping protein model required edits to the Cytosim source code to simulate filament capping and uncapping, and has been made available on the GitHub repository alongside this publication.

Underlying physics of dynamic ellipsoidal deformation model
The dynamically deformable ellipsoid framework for Cytosim was developed in Dmitrieff et al.53 as a computationally expedient method for investigating the role of cortical tension in microtubule assembly within red blood cells. Whereas our previous models used a spherical geometry with a rigid boundary, we now incorporate a deformable ellipsoid geometry that dynamically adjusts the space and axes with each time step to model the condensate surface as a simple, continuously deformable surface. Deformation forces are calculated by the force balance between pseudoforces associated with the interfacial surface tension, σ_surface, the point forces exerted by the filaments on the boundary, and the pseudoforces associated with the pressure53. Pressure is represented as a Lagrange multiplier to the constant volume constraint used in this model. Time evolution casts the force balance along the three axes of the ellipsoid and calculates the speed at which deformation occurs as modulated by the effective viscosity, µ_effective, which acts as a damping parameter to slow the deformation of the ellipse axes over time without affecting the final droplet shape53. Note that µ_effective (units pNs/µm) sets the timescale of droplet deformation and is a distinct parameter from the droplet viscosity (units pNs/µm2), which we hold constant at 0.5 pNs/µm2 throughout all simulations. A full formulation of this model can be found in Dmitrieff et al.53.

Position evolution
Using the Langevin equation, a stochastic differential equation for describing Brownian motion, Cytosim is able to calculate the time evolution of 3D discretized points for the position of actin filaments and crosslinkers throughout the entire simulation space and duration of the simulation. For a system of N particles, there are 3N coordinates associated with the position of each particle i in 3D space as given by x_i = (x_i1, x_i2, x_i3). Each position x_i is then evolved along each dimension j as governed by the following representation of the Langevin equation:

dx_ij(t) = µf_tot_ij(t)dt + dB_j(t)

Here, µ is the viscosity of the solvent (not to be confused with µ_effective). The first term on the right-hand side, f_tot_ij(t), is the total force acting on each particle as a function of time. The second term on the right-hand side, B_j(t), is the noise term for diffusion, which is a randomly sampled variable drawn from a normal distribution centered around a mean of 0 and with a standard deviation of sqrt(2D_i dt). The diffusion constant D is given by the Einstein relation D = µk_B T, where µ is the solvent viscosity, k_B is the Boltzmann constant, and T is the temperature.

Dynamic multimerization model
We explore the role of multivalent interactions using the dynamic multimerization model, wherein independently diffusing monomers are simulated as solids that each have a single actin-binding site and two multimerization sites. These two multimerization sites allow each monomer to bind with up to two other monomers to dynamically form multimers such as dimers, trimers, tetramers, etc. This model introduces additional input parameters that describe the kinetics of multimer formation, k_form, and multimer splitting, k_split, and differs from the previous simulations, where static crosslinkers are prescribed as a single solid with multiple actin-binding sites. The monomer-monomer binding and unbinding reactions are modeled similarly to actin-crosslinker binding interactions already described. The implementation of our dynamic multimerization model required edits to the Cytosim source code to simulate multimerization reactions, as detailed in a previous publication5.

Simulations without specific crosslinkers
We explore the role of condensate deformation in directing filament organization in the absence of specific crosslinkers by constructing a model of actin polymerization within a liquid condensate where crosslinkers are omitted and only actin filaments are present within the droplet. This model serves as an analog to droplets composed of RGG, a protein that has only non-specific interactions with actin filaments.

Theoretical scaling considerations for the droplet diameter and filament thickness
Here, we present the salient arguments laid out in Limozin et al.29 and rederived in our earlier study6. Consider a liquid droplet of radius R with an actin bundle of length L. The actin bundle is present below the interface between the droplet and the surrounding medium. We assume that the thermal fluctuations of the bundle dominate the droplet shape. The coordinate of the bundle centerline subject to fluctuations can be written as,

r(s) = R + u(s)

u(s) = R u(θ), s = Rθ, u(θ) = Σ_m u_m cos(mθ)

Here, s represents arclength and r(s) represents the radial coordinate of the bundle centerline as a function of arclength. The radial coordinate is represented as fluctuations u(s) around the droplet radius.

The bending energy of the bundle with flexural rigidity k_b and local curvature κ(s) is given by,

E_bend = 1/2 k_b L ∫_0^L κ(s)^2 ds

The flexural rigidity is related to the persistence length L_p of the bundle by k_b = L_p k_B T, where T is the temperature and k_B is the Boltzmann constant. The local curvature can be written in polar coordinates as,

κ(s) = sqrt((x''(s))^2 + (y''(s))^2)

where

s = Rθ, x = r cos(θ), y = r sin(θ)

Substituting, we get,

κ(s)^2 = (r'' + r/R^2)^2 + (2r'^2/R^2)

where, r'(s) = Σ_m u_m sin(mθ/R), r''(s) = Σ_m m^2 u_m cos(mθ/R).

Now, we assume that the different modes (m with amplitude u_m) are uncorrelated, and ignore cross-correlations to write <κ(s)^2>,

<κ(s)^2> = 1/R^2 + 2/R^3 Σ_m <u_m cos(mθ/R)> (1+m^2) + 1/R^4 Σ_m (1+m^4) <u_m^2 cos^2(mθ/R)> + 2 <u_m^2 sin^2(mθ/R)>

Please note that the primary contribution to <κ(s)^2> comes from the global curvature (droplet curvature) 1/R while the fluctuations contribute to higher order terms.

Substituting Eq. (8) in (4), we get,

E_bend = 1/2 k_b L [1/R^2 + Σ_m (2+m^4) <u_m^2>/R^4]

The primary contribution to the bending energy comes from the droplet radius. The fluctuation above that baseline energy ΔE_bend is given by,

ΔE_bend = Σ_m 1/2 k_b L <u_m^2> (2+m^4)/R^4

Applying the equipartition theorem, each mode contributes k_B T/2 energy, where k_B is the Boltzmann constant and temperature T as given by,

k_B T/2 = 1/2 k_b f(m) L <u_m^2>/R^4

Here, we represent the mode-dependent terms by the function f(m). Assuming that the ring thickness is proportional to the Fourier amplitude u_m, for a bundle of length L = 2πR, we get,

u_m ∝ R^1.5 / sqrt(L_p)

Through this, we show that the bundle thickness has a power law relationship with droplet radius.

Experimental methods
Reagents. Tris base, NaCl, Hellmanex III, Tris(2-carboxyethyl)phosphine (TCEP), poly-L-lysine, Atto 594 maleimide, and Atto 488 maleimide were purchased from Sigma-Aldrich. mPEG SVA was purchased from Laysan Bio. Phalloidin-iFluor594 was purchased from Abcam. Rabbit muscle actin was purchased from Cytoskeleton. Capping Protein (CapZ) was purchased from Hypermol.

Plasmids. A pET vector encoding the “cysteine light” variant of human VASP (pET-6xHis-TEV-KCK-VASP(CCC-SSA)) and monomer mini-Lpd (his-EGFP-Lpd(aa850-1250)) were gifts from Scott Hansen.

Protein purification. The pET-His-KCK-VASP(CCC-SSA) plasmid was transformed into Escherichia coli BL21(DE3) competent cells (NEB, Cat. no. C2527). Cells were grown at 30°C to an optical density (OD) of 0.8. Protein expression was performed using the following steps, which were adapted from a method previously described but with minor alterations5,6,41. Expression of VASP was induced with 0.5 mM isopropyl thiogalactoside (IPTG), and cells were shaken at 200 rpm at 12°C for 24 h. The rest of the protocol was carried out at 4°C. Cells were pelleted from 2 L cultures by centrifugation at 4785g (5000 rpm in Beckman JLA-8.100) for 20 min. Cells were resuspended in 100 mL lysis buffer (50 mM sodium phosphate pH 8.0, 300 mM NaCl, 5% glycerol, 0.5 mM TCEP, 10 mM imidazole, 1 mM phenylmethyl sulfonyl fluoride (PMSF)) plus EDTA-free protease inhibitor tablets (1 tablet per 50 mL, Roche, Cat. no. 05056489001), 0.5% Triton X-100, followed by homogenization with a dounce homogenizer and sonication (4 × 2000 J). The lysate was clarified by ultracentrifugation at 125,171g (40,000 rpm in Beckman Ti45) for 30 min. The clarified lysate was then applied to a 10 mL bed volume Nickel nitrilotriacetic acid (Ni-NTA) agarose (Qiagen, Cat. no. 30230) column, washed with 10 column volumes of lysis buffer plus EDTA-free protease inhibitor tablets (1 tablet per 50 mL), 20 mM imidazole, 0.2% Triton X-100, followed by washing with 5 × CV of lysis buffer plus 20 mM imidazole. The protein was eluted with elution buffer (50 mM Tris, pH 8.0, 300 mM NaCl, 5% glycerol, 250 mM imidazole, 0.5 mM TECP, EDTA-free protease inhibitor tablets (1 tablet per 50 mL)). The protein was further purified by size exclusion chromatography with Superose 6 resin. The resulting purified KCK-VASP was eluted in storage buffer (25 mM HEPES pH 7.5, 200 mM NaCl, 5% glycerol, 1 mM EDTA, 5 mM DTT). Single-use aliquots were flash-frozen using liquid nitrogen and stored at −80°C until the day of an experiment.

The mini-Lpd (his-Z-EGFP-LZ-Lpd(aa850-1250)) was transformed into Escherichia coli BL21 (NEB, Cat. no. C2527H) and grown at 30°C to an OD of 0.8. Protein expression was performed using the following steps, which were adapted from a method previously described but with minor alterations5. The bacteria were then cooled to 12°C and induced for 24 h with 1 mM IPTG. The rest of the protocol was performed at 4°C. Cells were pelleted from a 2 L culture by centrifugation at 4785g (5000 rpm in Beckman JLA-8.100) for 20 min. Pellets were resuspended in 100 mL of lysis buffer (50 mM sodium phosphate pH 8.0, 300 mM NaCl, 10 mM imidazole, 0.5 mM TCEP, 0.2% Triton X-100, 10% glycerol, 1 mM PMSF, and EDTA free protease inhibitor tablets (1 tablet per 50 mL) (Roche, Cat. no. 05056489001)) followed by sonication on ice for 4 × 2000 J with amplitude at 10 (Sonicator Qsonica LLC, Q700). The lysate was clarified by centrifugation at 48,384g (20,000 rpm in Beckman JA25.50) for 30 min at 4°C before being applied to a 10 mL bed volume Nickel nitrilotriacetic acid (Ni-NTA) agarose (Qiagen, Cat. no. 30230) column, and washed with 10 column volumes (CVs) of lysis buffer to which imidazole had been added to a final concentration of 20 mM. The column was then washed with 5 column volumes of lysis buffer containing 20 mM Imidazole but lacking Triton X-100 and protease inhibitor tablets. The protein was eluted with elution buffer (50 mM Tris-HCl, pH 7.5, 300 mM NaCl, 10% glycerol, 400 mM imidazole, 1 mM TECP, 1 mM PMSF, and EDTA-free protease inhibitor tablets (1 tablet per 50 mL). The protein was concentrated using Amicon Ultra-15, 30K MWCO (Millipore, Cat. no. UFC903024) to 5 mL, and clarified by ultracentrifugation for 5 min at 68,000g (40,000 rpm with Beckman optimal MAX-E Ultracentrifuge and TLA 100.3 rotor). The protein was further purified by size exclusion chromatography with Superose 6, and ion exchange chromatography with SP Sepharose Fast Flow (GE Healthcare, Cat. no. 17-0729-01), and stored as liquid nitrogen pellets at −80°C.

Expression and purification of his-LAF-1 RGG, containing amino acids 1–168 of LAF-1 and a C-terminal 6xHis tag, was performed using the following steps, which were adapted from a method previously described, including starting plasmids97. In brief, RGG was overexpressed in E. coli BL21 (DE3) cells. One-liter cultures were induced with 0.5 mM IPTG overnight at 18°C and 220 rpm, and pellets of cells expressing his-LAF-1 RGG were harvested from the cultures via centrifugation when the OD600 reached 0.8. The pellets were then lysed in a buffer containing 20 mM Tris, 500 mM NaCl, 20 mM imidazole, and one EDTA-free protease inhibitor tablet (Sigma-Aldrich) for 5 min on ice and then sonicated. The cell lysates were centrifuged at 134,000g for 40 min, after which his-LAF-1 RGG resided in the supernatant. The supernatant was then mixed with Ni-NTA resin (G-Biosciences) for 1 h, settled in a glass column, and washed with a buffer containing 20 mM Tris, 500 mM NaCl, 20 mM imidazole (pH 7.5). The bound proteins were eluted from the Ni-NTA resin with a buffer containing 20 mM Tris, 500 mM NaCl, 500 mM imidazole (pH 7.5). The purified proteins were then buffer exchanged into 20 mM Tris, 500 mM NaCl (pH 7.5) using Amicon Ultra centrifugal filters. Small aliquots of the protein were frozen in liquid nitrogen at a protein concentration of approximately 120 mM and stored at −80°C. To promote solubility of the LAF-1 RGG protein, the entire purification process was performed at room temperature, except for the cell lysis process, which was done on ice.

Protein labeling. The VASP used in this study is a previously published “cysteine light” mutant that replaced the three endogenous cysteines with two serines and an alanine. A single cysteine was then introduced at the N-terminus of the protein to allow selective labeling with maleimide dyes. This mutant was found to function in an indistinguishable manner from the wild-type proteins98. Thus, VASP was labeled at the N-terminal cysteine using maleimide-conjugated dyes. VASP was buffer exchanged into 20 mM Tris (pH 7.4) 150 mM NaCl buffer to remove DTT from the storage buffer and then incubated with dye for 2 h at room temperature. Free dye was then removed by applying the labeling reaction to a spin column packed with Sephadex G-50 Fine DNA Grade (GE Healthcare GE17-0573-01) hydrated with TNT buffer (20 mM Tris pH 7.4, 150 mM NaCl, and 5 mM TCEP). The labeled protein was then centrifuged at 100,000g for 10 min at 4°C to remove aggregates before being flash-frozen in single-use aliquots.

Monomeric actin was labeled using maleimide-conjugated dyes. Dyes were incubated with G-actin for 2 h at room temperature before being separated from the labeled protein by applying the labeling reaction to a spin column packed with Sephadex G-50 Fine DNA Grade (GE Healthcare GE17-0573-01) hydrated with a buffer (5 mM Tris-HCL (pH 8), 0.2 mM ATP, and 0.5 mM DTT, pH 8). The labeled protein was then centrifuged at 100,000g for 10 min at 4°C to remove aggregates before being flash-frozen in single-use aliquots.

RGG was labeled using amine-reactive NHS Ester dyes. RGG was buffer exchanged into 25 mM HEPES (pH 7.4), 500 mM NaCl prior to labeling with dyes. After labeling, RGG was separated from unconjugated dye and buffer exchanged back to 20 mM Tris (pH 7.4) 500 mM NaCl buffer using 3K Amicon columns (MilliporeSigma, Burlington, MA). Small aliquots of the protein were frozen in liquid nitrogen and stored at −80°C.

Protein droplet formation and actin polymerization. Condensates composed of VASP, monomer mini-Lpd, or RGG were formed using the following steps, which were adapted from a method previously described with minor alterations5,6. The given concentrations of the respective proteins were mixed with 3% (w/v) PEG-8000 in 20 mM Tris, pH 7.4, 5 mM TCEP, and either 150 mM NaCl (for VASP condensates) or 50 mM NaCl (for monomer mini-Lpd and RGG condensates). PEG was added last to induce droplet formation after the protein was well-mixed into the solution. All protein concentrations refer to the monomeric concentrations. For actin polymerization assays within protein droplets, condensates were allowed to form for 10 min following PEG addition, after which Atto-594-labeled G-actin was mixed into the droplet solution. Following actin addition, the sample was incubated at room temperature for 15 min prior to imaging to allow for actin polymerization and assembly. For phalloidin-actin assays, unlabeled actin monomers were added to protein droplets and allowed to polymerize for 15 min. Phalloidin-iFluor594 (Abcam) was then added and allowed to stain filamentous actin for 15 min prior to imaging.

Microscopy. Samples were prepared for microscopy in 3.5 mm diameter wells formed using biopsy punches to create holes in 1.6 mm thick silicone gaskets (Grace Biolabs) on Hellmanex III cleaned, no. 1.5 glass coverslips (VWR). Coverslips were passivated using poly-L-lysine conjugated PEG chains (PLL-PEG). After the addition of samples to these wells, an additional small coverslip was placed on top of the gasket to seal the imaging chamber and prevent evaporation during imaging. Fluorescence microscopy was done using an Olympus SpinSR10-Yokogawa SoRa spinning disk confocal microscope with a Hamamatsu Orca Flash 4.0 V3 Scientific CMOS camera. FRAP was done using the Olympus FRAP unit 405 nm laser.

PLL-PEG was prepared using the following steps, which were adapted from a method previously described with minor alterations5,99. Amine-reactive mPEG succinimidyl valerate was conjugated to poly-L-lysine at a molar ratio of 1:5 PEG to PLL. The conjugation reaction takes place in a 50 mM sodium tetraborate solution, pH 8.5, and is allowed to react overnight at room temperature while continuously stirring. The final product is then buffer exchanged to PBS pH 7.4 using 7000 MWCO Zeba spin desalting columns (Thermo Fisher) and stored at 4°C.

Image analysis. ImageJ was used to quantify the distribution of condensate characteristics. Specifically, condensates were selected using thresholding in the brightest channel and shape descriptors (i.e., diameter, aspect ratio, etc.), and protein fluorescent intensities were measured using the built-in analyze particles function. For aspect ratio analysis, condensates that had come into contact with other condensates were removed from the analysis to avoid any skewing of data from misrepresentation of single condensate deformation.

For actin ring thickness and power law fitting analysis, quantification of the ring thickness within the protein condensates was performed by measuring the phalloidin-actin intensity across a radial line bisecting the ring, taking the full-width at half-maximum intensity of the actin ring using a Gaussian fit of the intensity profile across one side of the actin ring, and then subtracting the diffraction limit to get an approximate ring thickness. To compare droplet diameters among aspherical droplets with AR > 1, we calculated an effective spherical diameter for the droplets based on their major and minor axis lengths: D = 3 sqrt(a b^2), where a refers to the major axis, and b refers to the minor axis lengths.

Data analysis
Filament length probability density function calculation. The lengths of each filament are calculated by adding up the length of each segment at each timestep. A filament length distribution is then created by sorting the lengths into 130 bins of equal width 0.05 µm. Since the filament length distribution changes with each timestep, we then smooth the distribution over time by calculating a moving average of filament lengths over a period of ±5 timesteps (±5 s) at each timestep. Then we feed this into a kernel density estimation to form a Gaussian approximation of the probability density function of filament lengths at each timestep.

Actin-covered surface fraction calculation. As the actin filaments grow to form shells, discs, and rings, they will move to occupy space near the surface of the droplet. Since the droplet boundary is modeled as a dynamically deformable ellipsoid, the droplet shape may be different at each time point. As such, at each time point, we discretize the surface of the droplet to an icosphere that is then deformed according to the current state of the ellipsoid. The actin-covered surface fraction calculation then proceeds as in our previous study39. Actin filaments are discretized to the monomer level and, using a threshold distance of 0.1 µm from the surface of the deformed icosphere, each monomer is assigned to the nearest surface triangle. These assignments are then used to calculate the surface density via the fraction of occupied triangles at each time point.

Filament alignment angle calculation. Throughout the duration of our simulations, actin elongation within the confinement of a droplet drives the rearrangement of individual filaments, which leads to the emergence of actin network structures such as shells, discs, and rings. These structures differ in how filaments are organized with respect to one another. As such, we define an alignment angle as a metric that describes the extent to which the actin network structure has collapsed from a 3D spherical shell to a tight 2D ring, inclusive of intermediary shapes. At each time point, we first discretize the actin filaments to the monomer level and use the resulting filament positions to construct the gyration tensor for the filament network. We next extract the eigenvalues of the gyration tensor to compute the ellipsoidal spans, thus providing us with the axis lengths for an ellipsoidal shape approximation that best explains the distribution of filaments in the network. The alignment angle is thus defined as the angle between the major axis (span a) and minor axis (span c) of the ellipsoidal approximation. As such, the filament alignment angle varies from 0° (perfectly flat) to 45° (perfectly spherical).

Statistics and reproducibility. Due to the stochastic nature of the interactions in our simulations, 10 replicate simulations were performed per condition to yield a sufficiently large sample size for statistical analyses of the results. All Cytosim trajectories were initialized with a random distribution of filaments and crosslinker species. For power law fitting of simulation data, initial diameter values with fewer than 10 samples were excluded to limit the influence of sparsely sampled regions of the parameter space. This criterion was applied uniformly and affected 2 of 12 fits, from which only 2–3 data points per fit were excluded. All other fitting and analyses were performed using the full datasets. Experiments with condensates composed of VASP, monomer mini-Lpd, and RGG were repeated independently three times per condition. All data analysis for simulations was performed on Python 3.11.5.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work by Mansour et al., combines biochemistry, microscopy, and agent-based modeling to answer some interesting biophysical questions related to the organization of actin structures in the presence of actin-binding protein condensates. This work is incremental, building upon previous studies and observations made by the same group – as such, it builds on an established literature with significant implications for increasing our understanding of the general principles underlying cytoskeletal organization and the generation of complex actomyosin networks in biological and synthetic systems. While the paper is incremental, building on observations with rigid geometries, the introduction of deformable droplets in silico, along with the biochemical assays and imaging performed in this work do provide some significant advancement to our understanding of the biophysical mechanisms by which actin structures can assemble at different scales within cells. Furthermore, the authors mention modifications to the code of a commonly-used agent-based simulation engine, Cytosim, which is employed by many researchers working on cytoskeletal and cell biology questions. These modifications, which introduce dynamically multimerizing monomers based on a specified multimer formation and splitting rates, could be of benefit to the larger community of researchers working with Cytosim.\n\nIt is my recommendation that this manuscript be considered for publication, but only after some textual concerns are addressed. The only significant weakness I noticed with this work is that not enough detail is provided in the introduction to establish a biological context to the significance of the observations being made. The authors introduce that similar observations to theirs have been made for actin organization in GUVs (but the authors do not define for their reader what a GUV is) and fail to provide a context for actin-binding proteins forming biological condensates within cells until the end of the discussion… there, the authors reference work demonstrating that focal adhesion proteins can and do form condensates on membranes, which provides a clear context in which the organization of actin within droplets bears biological significance. Until this point, my reading of the paper was hindered as the interesting biophysical observations being made did not contain a clear tie-in to any real biology. I therefore strongly recommend that the initial reference to focal adhesion condensates be moved to the introduction.\n\nMinor things to address:\n1. “GUVs” are not defined in the introduction.\n2. In the methods the authors mention their Cytosim code modifications and Python code will be made available upon publication. The code and scripts should be available already as reviewers may want or need to inspect it for validity. Further, this seems to contradict a statement from the supplemental methods which states: “The implementation of our dynamic multimerization model required edits to the Cytosim source code to simulate multimerization reactions and have previously been made available on a GitHub repository alongside a previous publication.”\n3. Italics needed for ’in vivo’ and ‘in vitro’ in several places.\n4. Confinement of actin within the condensate is an important component of these models and observations. In the in vitro condensate system actin is clearly confined. Is this true of any biological condensate? Also, according to the parameter table, actin confinement within the deformable space was 200 pN/um, is this based on some biological observation, or was there a parameter sweep performed to select this number? What would happen if this number was increased?\n5. Figure 2D - A visual example of each PCM paradigm would be useful to understand what the morphological difference is between the four outcomes. Minimally, their occurrence in A could be highlighted by bordering the simulations with the same color scheme.\n6. Besides focal adhesion proteins, do other actin-binders form condensates with a known purpose within cells?\n7. “Actin rings appeared at the droplet periphery. Actin ring thickness within mini-Lpd condensates was quantified by Gaussian fitting of the fluorescence intensity profile across one side of each ring to measure the full-width at half-maximum, subtracting the diffraction limit to approximate the true thickness of the actin rings. The effective spherical condensate diameter for asymmetrical condensates was calculated by plotting ring thickness as a function of condensate diameter (Fig. 6D).” This is a method explanation in the results. Move to the methods section.\n8. In Figure 6, why are there rings of mini-Lpd when neither VASP nor RGG also formed rings? This should be explained.\n9. Figure 8C and 8D are too busy. These figures lack clarity and should probably be split into simpler panels.", "reviewer_2": "Mansour et al. follow up on their prior work that showed that actin binding proteins (like VASP, Lamellipodin and even synthetic chimeras) can form phase separated condensates that recruit G-actin and promote its assembly into F-actin, with or without intrinsic polymerization activity. Interestingly, depending on the experimental conditions and to some extent stochastically, the outcome of these experiments are very variable when filaments are allowed to grow beyond the droplet radius: initially spherical droplets may remain spherical with a shell of F-actin at the surface, deform into an oblate spheroid with a compact ring or somewhat less compact disk that aligns with its major axes or extend all the way into rod shape. Their prior work using simulations and in vitro experiments has already established that the length of actin filaments as well as the multivalency of the crosslinker VASP plays a role in determining the outcome via kinetic trapping. Here they add a layer of sophistication to their simulations by making droplets deformable with varying material properties. For the sphere to spheroid transition, this allows them to ask how droplet deformation affects actin networks within and inversely how actin networks with different properties affect droplet deformation kinetics. They find that the relationship between droplet size and actin bundle thickness generally follows a power law in vitro as well as in their simulations. At least in simulations, the droplet deformation follows an unexpected non-monotonous path in certain conditions during actin filament elongation. For as much as I can assess as a molecular biologist without much insight on simulations, the work seems methodologically correct. I however find that the interpretation and comparison of the theoretically derived, simulated and experimentally determined power laws is a bit superficial and thus adds little to what I might intuitively have expected. While the quality of the simulations seems impressive to me and the mechanics of this system are fascinating, the relevance for biology in general remains a bit vague.\n\nMajor Concerns:\n1. One of the main claims in this work is that elongating actin filaments can cause non-monotonous deformation behavior while filaments grow. There seems to be no validation of that prediction, though. I assume that the precise actin filament length is difficult to assess in vitro, but if the time scale of filament growth on the order of minutes in the simulations is reproducible in vitro, an early aspect ratio “hump” with a magnitude of around 1.3 should be observable in time course experiments if it exists. In case it is not observed, is it because it is impossible to create condensates/crosslinkers with the predicted required properties or would it suggest that there is an unaccounted mechanism that allows bypassing the hump?\n2. The other main claim of the manuscript is that filament number or bundle diameter required to deform a droplet as a function of droplet radius universally follows a power law. This is shown for simulations with rigid, dynamic and unspecific bundling as well as in vitro for the first two cases. The exponents match particularly well for the VASP/rigid crosslinking (considering a factor 2 between number and diameter), but much less so for the multivalent Lpd example. For the unspecific case, no in vitro data is shown. A theoretical derivation in the supplement yields a power law as well. It is, however, not entirely clear to me what the significance of power laws with rather variable exponents would be. I think this should be discussed in more detail. Is there anything we could learn from the coefficients or exponents or that it generally follows power laws? Could the conditions of the simulation be changed to match the power laws that were experimentally determined (e.g. in case of LPD, maybe for RGG or just variations of in vitro conditions) and would that allow us to learn something about the biochemistry of the condensates and crosslinkers? It feels like there might be some missed opportunities given the sophisticated simulation framework that the authors have built.\n3. The biological relevance of this work is almost not at all covered in the text. The authors mention focal adhesions as an example where feedback between actin network and condensate deformation might play a role but remain extremely vague. It would help to elaborate more on why such feedback might matter in that case or other potential cases. Also, it might be good to mention in the introduction that condensates of VASP and other ABPs have been observed to recruit F-actin in cells (e.g. https://doi.org/10.1038/s41586-022-05084-3 or https://doi.org/10.1016/j.ejcb.2009.02.185 might be useful citations. I am not aware of in vivo examples of filaments bending along the condensate surface, but if the authors know any, that will help too.\n\nMinor points:\n1. In the legends of figures 2, 4 and 6 as well as table S2, the units of viscosity are given in pN s/μm. Should it be pN s/μm2 as in table S1 or am I confused about the meaning of effective viscosity?\n2. In the first paragraph on page 11: “However, when the bundles of actin grew longer than πR μm, the aspect ratio returned to growing monotonically” should it say 4R or 4 πR/3 instead?\n3. In figure 3 (or maybe a supplementary figure), would it be possible to convert the number of filaments to thickness in μm, such that the simulation data would be more comparable to the in vitro data?\n4. In figure 6D, the power law fit does not look particularly convincing. It seems to me that it fits small droplets well, but since there are not so many that are bigger than 3 μm, that area seems to not have contributed much to determining the fitted curve. Would there be a way of fitting the data that is not affected by the scarcity of the data points in a particular region of the plot? I think a better fit might give a power law with an exponent that is more comparable to the ones in the simulations.\n5. In the same figure, are the actin densities (as measured by fluorescence intensity) in the thicker rings comparable to thinner rings in smaller droplets? If it is variable, might a integration of the fluorescent signal give a better approximation of filament number than thickness?\n6. Fig. 7. Could similar power laws be determined for the in vitro data? It looks like there are less rings here. Do the droplets extend into rods without going through ring shaped intermediates? Is anything like that observed in the simulations with particular parameters and would that allow determining material properties of the condensates?"}, "author_response": "We thank the reviewer for their positive evaluation, which acknowledges that our work provides a significant advancement in our understanding of the biophysical mechanisms underlying actin structure assembly. We understand that the reviewer might perceive this work as possibly incremental because of our group’s contribution to the field. However, this work is holistic because, through a series of systematically implemented computational models, we seek to identify shared principles of actin network organization in phase-separated droplets of actin-binding proteins. Indeed, we were quite pleased to note that the reviewer highlights the computational advances of the work, specifically the introduction of deformable droplets and the resulting significant advances that emerge from combining modeling and experiments. We have elaborated on the novelty of the work in the revision in addition to addressing all the reviewer’s remarks.\n\nWe thank the reviewer for noting these significant technical improvements to Cytosim. We have made the code publicly available on GitHub (https://github.com/RangamaniLabUCSD/abp-deformable-droplet-Mansour-2025.git ) so that this codebase is of use to the larger community.\n\nWe have extensively revised the introduction to provide a biological context for our model system and findings. The changes to the text are as follows:\nBiomolecular condensates form membraneless organelles that function as dynamic compartments for organizing various cellular functions1–3. Recent discoveries have shown that several actin-binding proteins (ABPs) can form condensates, through liquid-liquid phase separation, both in vitro and in vivo3–12. Over 160 ABPs have been identified, and each of them plays a distinct role in transforming actin monomers and unorganized filaments into specialized structures that drive complex biological processes13,14. Despite their diversity, ABPs collectively regulate actin filaments through behaviors that can be broadly classified as tuning filament assembly and disassembly rates, branching, severing, annealing, and bundling, which leads to the formation of a variety of actin network architectures. These structures provide the mechanical framework for filopodia, lamellipodia, stress fibers, focal adhesions, contractile rings, and the actin cortex4,13,15,16. The reorganization of the actin cytoskeleton underlies a wide range of cellular processes that require cells to generate and transmit force, change shape, and respond to mechanical and chemical cues in their environment, including motility, signaling, and intracellular transport10,17–21. Dysregulation of these processes is implicated in many pathologies, notably in cancer and metastasis20. Importantly, ABP condensates have been observed in cells and are thought to provide a favorable environment for the recruitment of actin monomers, actin filaments, and other cytoskeletal proteins, and to promote nucleation of actin filaments12,16,22–27. However, a comprehensive understanding of the role of actin condensates in actin network organization and remodeling is an active area of investigation.\nReconstituted systems have been employed to study interactions between actin and ABP condensates in a controlled environment to understand principles of actin organization and confinement28. Similar actin ring formation has been observed in experimental studies using giant unilamellar vesicles (GUVs)29,30 and vesicles31–33. Beyond the biochemical function of the constituent ABPs, the material and physical properties of condensates also regulate their function24,34,35. The same weak interactions that drive condensate formation also allow for the development of the cohesive forces, which endow the liquid droplet with viscoelastic properties such as surface tension and viscosity24,35,36. For actin filaments already within condensates, the surface tension drives confinement, which supports the accumulation of bending energy5,6,37. Condensate deformation occurs once the filament bending energy exceeds the surface energy6,38. Bending energy may accumulate from a combination of actin polymerization and increases in actin bundle rigidity mediated by ABP processes such as actin crosslinking. The resulting interplay between surface energy and bending energy plays a key role in regulating the resulting structures.\n\nWe thank the reviewer for bringing this oversight to our attention. We have now expanded the definition of GUV in the text. The changes to the text are as follows:\nSimilar actin ring formation has been observed in experimental studies using giant unilamellar vesicles (GUVs)29,30 and vesicles31–33.\n\nWe have made the Cytosim code modifications and Python analysis scripts publicly available on a dedicated GitHub repository for this publication. We note that we had developed the dynamic multimerization model for a previous publication1. In this work, we integrated the dynamic multimerization model1 with the actin network model2 and introduced capping kinetics. Finally, we adopted the deformable ellipsoidal boundary formulation from Dmitrieff et al. 20173 to create a computational workflow that can capture all the necessary elements of the different cases considered here.\n\nAction Taken:\nWe have added the Data Availability and Code Availability statements to the manuscript, which includes a link to the GitHub repository containing the Cytosim source code with the additional integrated functionality for this publication. The changes to the text are as follows:\nData Availability\nThe Cytosim input files and source data for experiments and simulations are available at https://github.com/RangamaniLabUCSD/abp-deformable-droplet-Mansour-2025.git.\nCode Availability\nThe customized Cytosim source code used to generate trajectories and the Python scripts used to analyze the Cytosim trajectories are available on GitHub at https://github.com/RangamaniLabUCSD/abp-deformable-droplet-Mansour-2025.git.\nThe implementation of our dynamic multimerization model required edits to the Cytosim source code to simulate multimerization reactions, as detailed in a previous publication5.\n\nWe have now italicized each instance of in vivo and in vitro in the text. The changes to the text are as follows:\nRecent discoveries have shown that several actin-binding proteins (ABPs) can form condensates, through liquid-liquid phase separation, both in vitro and in vivo3–12.\nIn this work, we use a combination of agent-based modeling and in vitro biophysical experiments to systematically dissect how the biochemical and physical properties of liquid-like droplets of ABPs modulate actin filament organization and droplet shape.\n\nThe reviewer is right in that for the systems focused on in our study, actin is confined to the droplet. To our knowledge, while actin can be confined in many different biological condensates, we cannot say with certainty that any biological condensate will confine actin. As an example, recent studies of synapsin-actin interactions show that actin can both be confined within the droplet and also form networks that connect multiple droplets4. One advantage of the systems we are studying is that we can focus on actin networks that emerge due to confinement in droplets, allowing us to integrate modeling and experiments to draw insights. Ongoing work in our group is focused on understanding the conditions under which actin filaments may “escape” from a droplet made of actin-binding proteins.\n\nAction Taken:\nWe have discussed other configurations of actin-droplet interactions in the discussion section of the revised manuscript. The changes to the text are as follows:\nBiomolecular condensates are emerging as a key regulatory mechanism of cellular phenomena. The idea that containment of actin in a droplet can lead to feedback between bundling and condensate deformation can have implications in the organization of actin across a wide variety of biomolecular condensates. While we have shown such feedback in VASP condensates, these principles may be generalizable and apply to other condensates containing actin. For example, synapsin, a protein responsible for regulating synaptic vesicle clustering in the presynapse of neurons, has been shown to form condensates that recruit and organize actin12,76. Additionally, it has been recently shown in live cells that α-synuclein can regulate the material properties of synapsin droplets and alter their ability to cluster synaptic vesicles77. While many condensates form spherical three-dimensional droplets, recent studies have shown that two-dimensional condensates can form on the surface of the plasma membrane where transmembrane functions necessitate coupling between the condensate and the membrane. For condensates that assemble on surfaces, such as in focal adhesions and tight junctions, wetting and prewetting interactions with the membrane emerge from interfacial energy minimization, which depends on the material properties and characteristics of the condensate and surface4,26,27,74,78–81. These properties determine how the condensate spreads over the membrane, the droplet size, and the spatial patterning on the membrane75,82,83. This feedback between bundling and condensate deformation shapes how actin networks assemble and remodel around focal adhesions in cells as well. Focal adhesion proteins such as talin and vinculin have been shown to form 2D condensate structures on the cellular membrane7,8,80. The phase separation of focal adhesion and focal adhesion adapter proteins forms a dynamic and flexible assembly that clusters integrins and couples them to actin filaments, thus mediating the mechanical coupling between the actin cytoskeleton and the extracellular environment7,80. Beyond confinement, in instances where the surface energy of the droplet is much smaller than the bending energy of the actin filaments, condensates can mediate actin crosslinking driven primarily by wetting filaments via capillary bridge interactions24,73,84,85. Additionally, N-WASP forms condensates on the plasma membrane that function as focus points for Arp2/3 activation. N-WASP condensates form as a result of a prewetting transition whereby condensation occurs once N-WASP reaches a critical surface concentration79. Thus, force generation at the membrane may result from feedback between droplet properties, actin bundling, and membrane properties86–88.\n\nWe originally chose 200 pN/μm, as was done in our previous work1,2,5 and in Dmitrieff et al. 20173. In response to this question, we conducted additional simulations for two different values of boundary repulsion stiffness (100 pN/μm and 400 pN/μm). Overall, for this range of values, the qualitative behavior of our observations did not change, but there were some quantitative differences. When we increased the actin confinement (400 pN/μm), we observed greater droplet deformation in our simulations. We note that the role of the confinement term and the surface tension in Cytosim is different. The boundary repulsion stiffness is the parameter in the simulation that determines the strength of the inward-facing repulsive force applied to the actin filament to enforce the droplet boundary. An equal and opposite outward-facing force is then incorporated as a point force acting on the boundary. The point forces are balanced by the forces associated with surface tension, and subjected to volume incompressibility, in order to determine ellipsoid deformation. As such, for a fixed surface tension, a larger boundary repulsion stiffness increases the magnitude of the point forces, and thus deformation. For this reason, we keep the boundary repulsion stiffness constant throughout all of our simulations, as we are interested in the material properties of the condensate.\n\nAction Taken:\nWe have included a supplementary figure that compares what happens when we increase or decrease the actin boundary repulsion stiffness in deformable VASP condensates under conditions that favor the formation of A) Discs and B) Rings. The changes to the text are as follows:\nPoint forces are calculated from the force exerted to keep actin filaments within the boundary, as modulated by the boundary repulsion stiffness (Supplementary Fig. 1). This specifies the spring stiffness that acts on the discretized points of each actin filament and crosslinking molecule if the point lies outside the specified boundary. The force on each point is dependent on the distance that it lies beyond the confines of the boundary. Additionally, for actin filaments, these forces are used to calculate the point forces acting on the boundary that drive droplet deformation51.\n\nSupplementary Figure 1: Magnitude of condensate aspect ratio varies with filament boundary repulsion stiffness. A) Time series showing the mean (solid line) and standard deviation (shaded area) of droplet aspect ratio for each actin boundary repulsion condition for simulations of VASP that have disc-forming kinetics (k = 0.1 s-1, k = 1.0 s-1). B) Time bind unbind series showing the mean (solid line) and standard deviation (shaded area) of droplet aspect ratio for each actin boundary repulsion condition for simulations of VASP that have ring-forming kinetics (k = 10.0 s-1, k = 1.0 s-1). Representative final snapshots of each condition are bind unbind included above each plot.\n\nWe thank the reviewer for their suggestions on how to improve Figure 2D. We have made modifications to include representative snapshots for each of the four clusters we identified from our principal component analysis. Additionally, we have included a new Supplementary Table 3, which details the clusters that correspond to each VASP kinetic condition. The changes to the text are as follows:\nD) K-means clustering was performed on the last five snapshots from each replicate (data shown in B and C), which revealed four cluster categories corresponding to actin structures that are discs, rings, loose shells, or tight shells. The data for each snapshot is projected onto a scatterplot of the first two principal components (PC1 and PC2), and colored by clusters. Representative snapshots of each cluster are included above the plot. Please see Supplementary Table 3 for the corresponding cluster identities for each combination of kinetic parameters.\n\nThe actin network shapes formed by each combination of kinetic parameters are given in Supplementary Table 3.\nSupplementary Table 3: Table of actin network shapes for each (k , k ) condition bind unbind\n\nYes, many actin-binding proteins form condensates that serve a known function within cells. For example, as mentioned in the response to this reviewer’s comment 4a, synapsin forms condensates that help regulate the synaptic vesicle cluster in the presynapse of neurons. Additionally, N-WASP forms condensates on the plasma membrane that function as focus points for Arp2/3 activation.\n\nAction Taken:\nWe have expanded the introduction and discussion to provide greater biological context and add additional references to other condensates that bind actin.\n\nWe have removed this passage from the results section and ensured its placement in the image analysis section of the supplementary methods. The changes to the text are as follows:\nActin rings appeared at the droplet periphery. The thickness of the actin ring and the effective diameter of the encapsulating mini-Lpd condensate were quantified as described in the Supplementary Methods. We then plotted ring thickness as a function of effective condensate diameter and extracted a power law scaling relationship (Fig. 6D).\n\nWe believe that the reviewer is referring to Figure 6C. We would like to thank the reviewer for pointing out this lack of clarification. The “ring” like shape that is seen in the mini-Lpd condition, we define as a toroidal morphology due to the evacuation of the condensate phase from the center of the actin ring. We have a recent publication that explains that these toroidal shapes likely occur due to the balance between nucleation and elongation of actin filaments6. Toroidal morphologies appear to arise when there is less nucleation of new filaments, such that the actin filaments are trapped in a ring state, and the condensate deforms to accommodate the thickening actin ring. Due to actin being in a ring state, the filaments are unable to exert asymmetric pressure on the condensate boundary. As such, the sphere collapses into a disc and then a toroid rather than undergoing deformation into a rod-like morphology. This toroidal morphology is very infrequent in the conditions shown (<1% of condensates), and toroidal structures were not included in the ring thickness analysis.\n\nAction Taken:\nTo avoid confusion for the readers, we have updated the images in Figure 6C with ones that are more representative of our analysis and lack a toroidal shape, as shown below:\n\nWe thank the reviewer for making this suggestion and have made changes to simplify the panels and improve clarity.\n\nAction Taken:\nWe have removed extraneous elements, improved the clarity of the plotted points, and split the panels such that the scatter plots and line plots have their own separate panels. The changes to the text are as follows:\nFigure 8: Feedback between droplet deformation and actin filament alignment. A) Alignment angle for rigid ellipsoidal droplets with fixed but varied aspect ratios. The error bars represent the standard deviation. Inset images are final representative snapshots of each condition. B) Phase diagram plotting the final alignment angle and final aspect ratio across all replicates for simulations without crosslinkers. Each σ condition is indicated by color. The surface inset graph shows the time series profile showing the mean (solid line) and standard deviation (shaded area) of the filament alignment angle (blue) and the droplet aspect ratio (red) for the condition indicated by a red circle (σ = 4 pN/µm). C) Phase diagram plotting the final surface alignment angle and final aspect ratio across all replicates for simulations with dynamically multimerizing crosslinkers. Each k condition is indicated by color, and each k condition is form split indicated by marker shape. D) Time series profile showing the mean (solid line) and standard deviation (shaded area) of the filament alignment angle (blue) and the droplet aspect ratio (red) for the condition indicated by the red circle in panel C (k = 10.0 s-1, k = 10.0 s-1). E) Phase form split diagram plotting the final alignment angle and final aspect ratio across all replicates for simulations with VASP crosslinkers. Each k condition is indicated by color, and each k bind unbind condition is indicated by marker shape. F) Time series profile showing the mean (solid line) and standard deviation (shaded area) of the filament alignment angle (blue) and the droplet aspect ratio (red) for the condition indicated by the red circle in panel E (k = 10.0 s-1, k = 10.0 bind unbind s-1).\n\nWe thank the reviewer for their positive evaluation of our work and for acknowledging the sophistication of our simulations. In the revised version, and as noted in the response to Reviewer 1, we have extensively edited the introduction and the discussion to emphasize the biological significance of our work.\n\nUnfortunately, due to technical limitations associated with the image acquisitions in the experiments, we have not yet been able to visualize droplet deformation dynamics that are non-monotonic. To ensure that the light exposure does not cause the condensates to form gels and completely alter their material properties, we restrict our imaging to intervals of 10-15 s. Since extended exposure causes the droplets to become stiffer, our attempts to acquire a time-lapse of droplet deformation have only worked for small droplets, where the non-monotonicity of deformation would be difficult to resolve. However, in the simulations, we do not have such constraints. An additional element at play here is that we cannot measure or control for the number of actin filaments internalized by the droplets in experiments, but there are likely hundreds of filaments within each droplet. In contrast, in the simulations, we do not simulate hundreds of filaments to ensure that the system does not blow up out of combinatorial complexity. While it is possible that the hump in the data is a computational artifact, similar dynamic snapping behaviors are well known to occur in viscoelastic systems undergoing mechanical deformations7–11.\n\nAction Taken:\nWe have noted these points as a limitation of our work in the discussion. The changes to the text are as follows:\nWhile current imaging modalities were not able to capture the snapping behavior in experiments, the common principle driving droplet deformation dynamics across all cases we studied is that the droplet deforms when the energy exerted by the filament bundle overcomes the interfacial energy of the droplet.\n\nWe agree with the reviewer that there was a larger discrepancy for the power law fits with the mini-Lpd conditions. Therefore, we conducted additional experiments to obtain more data for larger mini-Lpd droplets. We then refit the data to the equation y = axb and obtained a better fit (R2 = 0.8251) and an exponent (b = 1.9910) that is more comparable to the simulation results as well as the results from previous experiments. In response to this reviewer's comment, we also attempted to quantify ring thicknesses for the nonspecific case. However, a combination of the RGG condensate properties and passive physical filament bundling resulted in actin rings that were too diffuse to provide highly quantitative data across a broad range of droplet diameters, especially for larger condensate diameters, as would be required to map the power law dependency for this case.\n\nAction Taken:\nWe updated Figure 6D with the additional data and new power law fit. We have noted our inability to resolve a clear trend in RGG condensates as a minor limitation of our study. The changes to the text are as follows:\nD) Power law scaling relation between the measured actin ring thickness and the diameter of monomer mini-Lpd condensates (R2 = 0.8251, n = 228). The inset images correspond to the denoted points on the graph and depict examples of the difference in measured actin ring thickness. Scale bars, 1 µm.\nWe note that the RGG droplets were diffuse at larger condensate diameters. As ring thicknesses across a range of droplet diameters are necessary to derive a power law trend, we were unable to quantify the relationship between droplet diameter and ring thickness in RGG droplets.\n\nThe power laws throughout the manuscript show that as the droplet size increases, the thickness of the actin ring required to deform the droplet increases nonlinearly according to a power law y = axb, where a is the prefactor and b is the exponent. This general principle holds true across all cases tested in experiments and simulations, and is also consistent with experiments in other systems12. From the simulations, we note that increasing the surface tension of the droplet increases the prefactor (Fig. 3B, Fig. 3D, Fig. 6B, and Fig. 7B), which represents the number of filaments required to deform a droplet with an initial diameter of 1.0 µm, according to the mathematical fit of the simulation data. As such, across all of our simulations, we observe that higher surface tension droplets require a greater number of filaments, and therefore greater force, to deform to the same extent as lower surface tension droplets. The nonlinear relation between the number of filaments and the initial diameter of the droplets is described by the exponent and affected by the properties of the crosslinkers. However, interpreting trends in the values of the exponent alone is trickier. If we were to fix the value of the prefactor at 1.0 (a = 1.0), then we would expect to see the exponent increase as we increase the surface tension. However, fixing the prefactor is an unrealistic scenario because more filaments are required to deform stiffer droplets. As such, it is a combination of the prefactor and the exponent that describes the data. Beyond these observations, we do not have the experimental or computational resolution to draw insights into the specific values of the exponents at this time.\n\nAction Taken:\nWe have included a more detailed discussion of the significance of the power laws obtained from simulations in the discussion section of the revised manuscript. The changes to the text are as follows:\nIn both cases, the number of filaments needed to deform droplets scales as a power law with respect to droplet radius, indicating that larger droplets need more forces from actin filaments to deform. This general principle is true across all the cases tested in experiments and in simulations, and is also consistent with experiments in other systems29. The prefactor of the power law depends on the interfacial properties of the droplet. Increasing the surface tension increases the prefactor, indicating that, for droplets with high surface tension, a higher number of filaments, and therefore a greater force, is needed to deform a droplet of the same diameter.\n\nAs noted in response to comment 2b, the entire power law formulation indicates a nonlinear scaling between the thickness of the actin ring and droplet diameter. The prefactor represents the number of filaments required to deform a droplet with an initial diameter of 1.0 µm, while the exponents describe the nonlinear relation and are affected by the properties of the crosslinkers. As such, the terms of the power law are representative of the material and biochemical properties of the droplet and constituent crosslinkers. Since force generation in our model depends on the filaments, we can, in principle, generate predictions that link surface tension in simulations to experiments. However, we must acknowledge that the complexity of the physical reality of the condensate environment limits our ability to generate precise values. Our simulations are based upon a minimal physics model of actin crosslinking and organization in deformable condensates, where deformation is implemented with a simple boundary deformation model restricted to ellipsoidal geometries. Additionally, our model does not explicitly represent molecular crowding or include hydrodynamic interactions. In a real biomolecular condensate, variations in local interfacial conditions may result in varied surface tension along the interface and a variety of more complex geometries. While we appreciate the idea of matching the exponents between the simulations and experiments, we note that this is a rather challenging exercise in parameter variation due to its combinatorial complexity. We agree that there is an opportunity to explore this further in the future.\n\nAction Taken:\nWe have noted these points as a limitation of our work in the discussion. The changes to the text are as follows:\nWhile our findings have broad implications for the interactions between the cytoskeleton and protein droplets, we acknowledge that we have constructed a simple model that reproduces the minimal physics of actin organization within condensates. To improve upon our current simulations, future models require the incorporation of hydrodynamic interactions between the actin filaments and the fluid droplet89–92.\n\nAs noted in our response to Reviewer 1, we have extensively revised the introduction and discussion of our manuscript to address the biological relevance of our work.\n\nThe changes to the introduction are as follows:\nBiomolecular condensates form membraneless organelles that function as dynamic compartments for organizing various cellular functions1–3. Recent discoveries have shown that several actin-binding proteins (ABPs) can form condensates, through liquid-liquid phase separation, both in vitro and in vivo3–12. Over 160 ABPs have been identified, and each of them plays a distinct role in transforming actin monomers and unorganized filaments into specialized structures that drive complex biological processes13,14. Despite their diversity, ABPs collectively regulate actin filaments through behaviors that can be broadly classified as tuning filament assembly and disassembly rates, branching, severing, annealing, and bundling, which leads to the formation of a variety of actin network architectures. These structures provide the mechanical framework for filopodia, lamellipodia, stress fibers, focal adhesions, contractile rings, and the actin cortex4,13,15,16. The reorganization of the actin cytoskeleton underlies a wide range of cellular processes that require cells to generate and transmit force, change shape, and respond to mechanical and chemical cues in their environment, including motility, signaling, and intracellular transport10,17–21. Dysregulation of these processes is implicated in many pathologies, notably in cancer and metastasis20. Importantly, ABP condensates have been observed in cells and are thought to provide a favorable environment for the recruitment of actin monomers, actin filaments, and other cytoskeletal proteins, and to promote nucleation of actin filaments12,16,22–27. However, a comprehensive understanding of the role of actin condensates in actin network organization and remodeling is an active area of investigation.\nReconstituted systems have been employed to study interactions between actin and ABP condensates in a controlled environment to understand principles of actin organization and confinement28. Similar actin ring formation has been observed in experimental studies using giant unilamellar vesicles (GUVs)29,30 and vesicles31–33. Beyond the biochemical function of the constituent ABPs, the material and physical properties of condensates also regulate their function24,34,35. The same weak interactions that drive condensate formation also allow for the development of the cohesive forces, which endow the liquid droplet with viscoelastic properties such as surface tension and viscosity24,35,36. For actin filaments already within condensates, the surface tension drives confinement, which supports the accumulation of bending energy5,6,37. Condensate deformation occurs once the filament bending energy exceeds the surface energy6,38. Bending energy may accumulate from a combination of actin polymerization and increases in actin bundle rigidity mediated by ABP processes such as actin crosslinking. The resulting interplay between surface energy and bending energy plays a key role in regulating the resulting structures.\n\nThe changes to the discussion are as follows:\nBiomolecular condensates are emerging as a key regulatory mechanism of cellular phenomena. The idea that containment of actin in a droplet can lead to feedback between bundling and condensate deformation can have implications in the organization of actin across a wide variety of biomolecular condensates. While we have shown such feedback in VASP condensates, these principles may be generalizable and apply to other condensates containing actin. For example, synapsin, a protein responsible for regulating synaptic vesicle clustering in the presynapse of neurons, has been shown to form condensates that recruit and organize actin12,76. Additionally, it has been recently shown in live cells that α-synuclein can regulate the material properties of synapsin droplets and alter their ability to cluster synaptic vesicles77. While many condensates form spherical three-dimensional droplets, recent studies have shown that two-dimensional condensates can form on the surface of the plasma membrane where transmembrane functions necessitate coupling between the condensate and the membrane. For condensates that assemble on surfaces, such as in focal adhesions and tight junctions, wetting and prewetting interactions with the membrane emerge from interfacial energy minimization, which depends on the material properties and characteristics of the condensate and surface4,26,27,74,78–81. These properties determine how the condensate spreads over the membrane, the droplet size, and the spatial patterning on the membrane75,82,83. This feedback between bundling and condensate deformation shapes how actin networks assemble and remodel around focal adhesions in cells as well. Focal adhesion proteins such as talin and vinculin have been shown to form 2D condensate structures on the cellular membrane7,8,80. The phase separation of focal adhesion and focal adhesion adapter proteins forms a dynamic and flexible assembly that clusters integrins and couples them to actin filaments, thus mediating the mechanical coupling between the actin cytoskeleton and the extracellular environment7,80. Beyond confinement, in instances where the surface energy of the droplet is much smaller than the bending energy of the actin filaments, condensates can mediate actin crosslinking driven primarily by wetting filaments via capillary bridge interactions24,73,84,85. Additionally, N-WASP forms condensates on the plasma membrane that function as focus points for Arp2/3 activation. N-WASP condensates form as a result of a prewetting transition whereby condensation occurs once N-WASP reaches a critical surface concentration79. Thus, force generation at the membrane may result from feedback between droplet properties, actin bundling, and membrane properties86–88.\n\nWe agree with the reviewer that it is important to mention F-actin recruitment into condensates of ABPs in cells. We thank the reviewer for recommending additional important references to include in our manuscript. We have now included these references as reference 22 and reference 23.\n\nAction Taken:\nAs noted in our response to this reviewer’s major concern 3a, we have made extensive edits to the introduction and discussion. Additionally, we have included an explicit mention in the introduction that ABP condensates recruit actin filaments in cells. The changes to the text are as follows:\nImportantly, ABP condensates have been observed in cells and are thought to provide a favorable environment for the recruitment of actin monomers, actin filaments, and other cytoskeletal proteins, and to promote nucleation of actin filaments12,16,22–27.\n\nEssentially, there are two different viscosity parameters in our simulations. The viscosity referenced in Supplementary Table 1, with units pN s/μm2, describes the resistance to displacement felt by diffusing species within the droplet medium and sets the timescale of diffusion within our simulations. The effective viscosity (µ ) that is referenced in effective Figure 2, Figure 4, Figure 6, and Supplementary Table 2 is a distinct parameter that describes the resistance to changes in the displacement of the three axes of the deformable elipsoid and sets the timescale of droplet surface deformation. The units for µ are worked out from the formula for the time evolution of deformation, described effective in Dmitrieff et al. 20173 as,\n1 ( 𝑘 𝑘 𝑘)\n𝑎̇ = ϕ + ϕ + Σϕ (1)\n𝑘 µ 𝑃 σ 𝑓\n𝑘\nwhere, for each direction 𝑘, 𝑎̇ is the rate of deformation (units µm/s) and ϕ are the 𝑘 effective forces (units pN). This relation yields the units of pN s/µm for µ .\neffective\n\nAction Taken:\nWe have added text to clarify the distinction between µ and the droplet viscosity. effective The changes to the text are as follows:\nTime evolution casts the force balance along the three axes of the ellipsoid and calculates the speed at which deformation occurs as modulated by the effective viscosity, µ , which acts as a damping parameter to slow the deformation of the effective ellipse axes over time without affecting the final droplet shape51. Note that µ (units effective pN s/µm) sets the timescale of droplet deformation and is a distinct parameter from the droplet viscosity (units pN s/µm2), which we hold constant at 0.5 pN s/µm2 throughout all simulations. A full formulation of this model can be found in Dmitrieff et al. 201751.\n\nWe thank the reviewer for pointing out this error. The monotonic increase in aspect ratio resumed at around 400 s, which corresponds with an actin filament length of about 4πR/3 µm. The text has been updated to reflect the correct length of actin filaments.\nThe changes to the text are as follows:\nHowever, when the bundles of actin grew longer than 4πR/3 µm, the aspect ratio returned to growing monotonically (Fig. 3A, Supplementary Fig. 4).\n\nUnfortunately, we were unable to convert the number of filaments to a thickness in the simulations. Unlike previous 2D simulations13, increasing the number of filaments does not necessarily scale the thickness of the rings. In 3D, filaments can be arranged and packed in numerous ways to form a variety of actin structures that are not necessarily uniform in thickness or even ring-like in shape. As such, determinations of ring thickness in our simulations would not give a good quantitative match with experiments. However, common to both of these simulations is that the forces generated by actin scale with the number of filaments. For these reasons, we chose the number of actin filaments in simulations as the most comparable to the ring thicknesses measured in our experiments.\n\nIn response to this question, we conducted additional experiments and refit the data to obtain a better power law fit that is more comparable to results from simulations and previous experiments. Additional details are included in the response to this reviewer’s major concern 2a. The revised figure is included below:\nD) Power law scaling relation between the measured actin ring thickness and the diameter of monomer mini-Lpd condensates (R2 = 0.8251, n = 228). The inset images correspond to the denoted points on the graph and depict examples of the difference in measured actin ring thickness. Scale bars, 1 µm.\n\nThe reviewer is correct that an integration under the Gaussian fit would likely give a better approximation of the actin density within rings. However, in our experiments with phalloidin-stained actin filaments, phalloidin is added to the top of the imaging well after the condensate-actin mix is added in order to allow for actin assembly and settling to the coverslip. This leads to variations in the intensity of the phalloidin channel across the well as the phalloidin mixes and diffuses through the sample. Due to such variation, we are not confident that the integration of the Gaussian intensity profile would be completely accurate as a comparison tool; therefore, we have used the thickness of the actin rings instead.\n\nIn response to this reviewer's comment, we attempted to quantify ring thicknesses for the nonspecific case shown in Figure 7. However, the passive physical filament bundling that occurred using the nonspecific crosslinkers resulted in actin rings that were too diffuse at larger condensate diameters to provide highly quantitative data across a broad range of droplet diameters. As such, we were unable to obtain the data needed to identify a power law trend from RGG condensates. Accordingly, we have added a qualification to the text, which is cited below for ease of reference.\nWe note that the RGG droplets were diffuse at larger condensate diameters. As ring thicknesses across a range of droplet diameters are necessary to derive a power law trend, we were unable to quantify the relationship between droplet diameter and ring thickness in RGG droplets."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors undertook a significant rewrite of the introduction and discussion sections of the manuscript. More biological context was added to provide significance and improve interest from a wider audience. Overall, I believe the authors addressed all of my major concerns with the manuscript and took several suggestions on improving figure clarity and legibility. The resultant manuscript feels more polished and ready for acceptance.\n\n(Remarks on code availability)\nAll the necessary code to run their modifications is there and appears to compile fine. Unfortunately, the way the authors branched from the main Cytosim branch, it is unclear where the major code modifications are. It might be useful for their commit comments to point to the specific files where modifications were made over \"vanilla\" Cytosim.", "reviewer_2": "I appreciate the authors' efforts to address all my previous comments and write a comprehensive response letter listing them in detail. While some of my major concerns were only partially fixed, the authors justified clearly why parts of my suggestions were hard to implement due to experimental limitations and adjusted the text accordingly. The added general biological context to the introduction and discussion is a great improvement in my opinion. I still sense a bit of a conceptual jump from the new part of the introduction to the motivation for the specific experiments performed in this manuscript and similarly I am missing a cohesive argument about how the results are relevant to the in vivo condensate examples that are now provided in the discussion. I think the manuscript can be published with minor edits, but the relevance to a biological audience could still be further improved by polishing those parts. Here are some specific things I noticed during rereading:\n1. Lines 785-6: \"Previous studies have shown that actin confinement within rigid spheres38,39\nand inter-filament attraction are necessary conditions for bundling28,31\" To the best of my knowledge, actin filaments can be bundled without confinement. Did you mean to say \"sufficient\" or is this under specific circumstances?\n2. Lines 823-4: I would mention more explicitly that one of the cited manuscripts observes actin organized along the surface of condensates, reminiscent of the simulations in this work.\n3. Lines 826-49: I agree that FAs, their maturation from nascent dynamic into mature more layered structures and particularly how forces produced by the actin cytoskeleton contribute to that process are a potentially exciting case of mechanical functions of condensates. But how is this related to the work presented in this manuscript? I do not understand the sentence on lines 835-6 that suggests a connection."}, "author_response": "Response to Reviews for the submission NCOMMS-25-54600-T\nMechanochemical feedback between confinement and actin crosslinking drives\nthe shape dynamics of liquid-like droplets\nAuthors: Daniel Mansour, Dominique Jordan, Caleb Walker, Aravind Chandrasekaran,\nChristopher T. Lee, Kristin Graham, Jeanne Stachowiak, Padmini Rangamani\nReference numbers within direct text from the revised manuscript correspond to the\nreference list in the manuscript document.\nReviewer #1:\n1.1 The authors undertook a significant rewrite of the introduction and discussion\nsections of the manuscript. More biological context was added to provide significance\nand improve interest from a wider audience. Overall, I believe the authors addressed all\nof my major concerns with the manuscript and took several suggestions on improving\nfigure clarity and legibility. The resultant manuscript feels more polished and ready for\nacceptance.\n\nAuthor Response:\nWe thank the reviewer for their positive evaluation and for their acknowledgement of our\nextensive edits. We agree that the quality of the manuscript has been greatly improved\nby providing greater biological context and improving figure clarity.\n\n1.2 All the necessary code to run their modifications is there and appears to compile\nfine. Unfortunately, the way the authors branched from the main Cytosim branch, it is\nunclear where the major code modifications are. It might be useful for their commit\ncomments to point to the specific files where modifications were made over \"vanilla\"\nCytosim.\n\nAuthor Response:\nWe have imported the commit history to the GitHub repository, which includes the\nmodifications made by the authors as well as the preexisting development that\nproduced the underlying Cytosim version. In addition, we have included a README.md\nfile in the Source_Code directory that points to the specific files where the major\nmodifications build over base Cytosim. The contents of this file are provided below:\nOrigin\nThe base Cytosim version can be found at https://gitlab.com/f-nedelec/cytosim. The\nmodifications introduced by the authors to the dynamic_ellipse fork of Cytosim are\ndocumented in the commit history from 2024 onward and build upon the extensive\npreexisting development that produced the underlying Cytosim version.\n\nCapping Protein\nCapping protein was implemented implicitly as a special type of growing fiber introduced\nin this work. The following files are the major source code modifications:\n● Cytosim/src/sim/fibers/capping_fiber.cc\n● Cytosim/src/sim/fibers/capping_fiber.h\n● Cytosim/src/sim/fibers/capping_fiber_prop.cc\n● Cytosim/src/sim/fibers/capping_fiber_prop.h\n\nDynamic Ellipsoidal Deformation\nThe implementation of dynamic ellipsoidal deformation was built as part of the\ndynamic_ellipse fork of the base Cytosim version and presented in Dmitrieff et al. 2017.\nThe following files are the major modifications that introduced the simple deformable\nboundary to Cytosim:\n● Cytosim/src/sim/spaces/space_dynamic_ellipse.cc\n● Cytosim/src/sim/spaces/space_dynamic_ellipse.h\n● Cytosim/src/sim/spaces/space_dynamic_prop.cc\n● Cytosim/src/sim/spaces/space_dynamic_prop.h\n\nDynamic Multimerization\nDynamically multimerizing crosslinkers was implemented as an extension of the\ndynamic dimers introduced in Walker et al. 2025. The following files are the major\nsource code modifications:\n● Cytosim/src/sim/hands/dimerizer.cc\n● Cytosim/src/sim/hands/dimerizer.h\n● Cytosim/src/sim/hands/dimerizer_prop.cc\n● Cytosim/src/sim/hands/dimerizer_prop.h\n2.1 I appreciate the authors' efforts to address all my previous comments and write a\ncomprehensive response letter listing them in detail. While some of my major concerns\nwere only partially fixed, the authors justified clearly why parts of my suggestions were\nhard to implement due to experimental limitations and adjusted the text accordingly. The\nadded general biological context to the introduction and discussion is a great\nimprovement in my opinion.\n\nAuthor Response:\nWe thank the reviewer for their positive evaluation of the extensive modifications made\nto the text. We agree that the quality of the manuscript has been greatly improved by\naddressing the previous remarks. We appreciate the reviewer for understanding our\nexperimental limitations.\n\n2.2 I still sense a bit of a conceptual jump from the new part of the introduction to the\nmotivation for the specific experiments performed in this manuscript and similarly I am\nmissing a cohesive argument about how the results are relevant to the in vivo\ncondensate examples that are now provided in the discussion. I think the manuscript\ncan be published with minor edits, but the relevance to a biological audience could still\nbe further improved by polishing those parts. Here are some specific things I noticed\nduring rereading:\n\nAuthor Response:\nWe thank the reviewer for their feedback and have made edits to the introduction and\ndiscussion in response. In essence, we wanted to communicate that in vivo condensate\nsystems are very complex, and our approach has been to build a minimal model that\ncaptures the key elements of actin organization within phase-separated droplets using a\ncombination of in vitro and in silico systems.\n\nThe changes to the text of the introduction are as follows:\nDue to the high complexity of in vivo condensate assemblies, reconstituted systems\nhave been employed to study interactions between actin and ABP condensates in a\ncontrolled environment to understand the underlying biophysical principles of actin\norganization and confinement28.\n\nBy combining in vitro and in silico approaches, we have been able to create a\ncomplementary framework where experiments provide physical constraints and\nsimulations systematically explore the underlying mechanisms and generate new\nexperimentally testable hypotheses. We have previously established how the spatial\n3\nconfinement of ABPs through phase separation affects actin filament assembly and\nbundling5,6,39,40.\n\nHowever, two questions remain: What is the role of mechanical feedback from\ndeformable boundary confinement in tuning actin organization within the droplet? How\ndo the biochemical properties of the crosslinker influence droplet deformation? To\naddress these questions, we built a minimal model that captures the key elements of\nactin organization in phase-separated droplets using a combination of computational\nand experimental approaches.\n\nThe changes to the text of the discussion are as follows:\nThe feedback between bundling and condensate deformation presented in this work\nmay have implications for how surface-associated condensates respond to mechanical\nload, where shape changes and interfacial geometry are important for force\ntransmission. In focal adhesions, proteins such as talin and vinculin form dynamic and\nflexible two-dimensional condensate assemblies on the cellular membrane that cluster\nintegrins and couple them to actin filaments, thus mediating the mechanical coupling\nbetween the actin cytoskeleton and the extracellular environment7,8,82. These\nassemblies undergo maturation under tension that is driven by actin filaments, which is\naccompanied by the reorganization of the actin architecture and changes in condensate\nshape, suggesting a mechanically analogous coupling between actin-generated forces\nand condensate deformation. Beyond confinement, in instances where the surface\nenergy of the droplet is much smaller than the bending energy of the actin filaments,\ncondensates can mediate actin crosslinking driven primarily by wetting filaments via\ncapillary bridge interactions24,75,86,87. Additionally, N-WASP forms condensates on the\nplasma membrane that function as focus points for Arp2/3 activation. N-WASP\ncondensates form as a result of a prewetting transition whereby condensation occurs\nonce N-WASP reaches a critical surface concentration81. Together, these examples\nillustrate how feedback between condensate material properties, actin organization, and\nmembrane properties may contribute to force generation and cytoskeletal remodeling at\nthe membrane88–90.\n1. Lines 785-6: \"Previous studies have shown that actin confinement within rigid\nspheres38,39 and inter-filament attraction are necessary conditions for\nbundling28,31\" To the best of my knowledge, actin filaments can be bundled\nwithout confinement. Did you mean to say \"sufficient\" or is this under specific\ncircumstances?\n\nAuthor Response:\nWe thank the reviewer for this suggestion. We agree that “sufficient” is a better choice of\nwords in the context of the surrounding discussion. The changes to the text are as\nfollows:\nPrevious studies have shown that actin confinement within rigid spheres39,40 and\ninter-filament attraction are sufficient conditions for bundling28,31.\n\nReviewer Comment:\n2. Lines 823-4: I would mention more explicitly that one of the cited manuscripts\nobserves actin organized along the surface of condensates, reminiscent of the\nsimulations in this work.\n\nAuthor Response:\nWe thank the reviewer for this suggestion. We agree that more explicitly highlighting the\nsimilarities between these works strengthens the applicability of our findings to other\ncondensate systems. The changes to the text are as follows:\nFor example, synapsin, a presynaptic protein that regulates synaptic vesicle clustering,\nhas been shown to form condensates that recruit and organize actin along the\ncondensate surface in a manner similar to the condensates described in this work12,78.\n\nReviewer Comment:\n3. Lines 826-49: I agree that FAs, their maturation from nascent dynamic into\nmature more layered structures and particularly how forces produced by the actin\ncytoskeleton contribute to that process are a potentially exciting case of\nmechanical functions of condensates. But how is this related to the work\npresented in this manuscript? I do not understand the sentence on lines 835-6\nthat suggests a connection.\nWe thank the reviewer for their feedback. We have revised the text to clarify the\nconceptual connection between focal adhesions and the feedback between actin\nbundling and condensate deformation identified here, specifically that focal adhesions\nrespond to mechanical load. This comparison is intended to highlight focal adhesion\nmaturation under tension and accompanying actin reorganization as mechanically\nanalogous to the coupling between actin-generated forces, cytoskeletal remodeling, and\ncondensate deformation. The changes to the text are as follows:\nWhile many condensates form spherical three-dimensional droplets, recent studies\nhave shown that two-dimensional condensates can form on the surface of the plasma\nmembrane where transmembrane functions necessitate coupling between the\ncondensate and the membrane. For condensates that assemble on surfaces, such as in\nfocal adhesions and tight junctions, wetting and prewetting interactions with the\nmembrane emerge from interfacial energy minimization, which depends on the material\nproperties and characteristics of the condensate and surface4,26,27,76,80–83. These\nproperties determine how the condensate spreads over the membrane, the droplet size,\nand the spatial patterning on the membrane77,84,85. The feedback between bundling and\ncondensate deformation presented in this work may have implications for how\nsurface-associated condensates respond to mechanical load, where shape changes\nand interfacial geometry are important for force transmission. In focal adhesions,\nproteins such as talin and vinculin form dynamic and flexible two-dimensional\ncondensate assemblies on the cellular membrane that cluster integrins and couple them\nto actin filaments, thus mediating the mechanical coupling between the actin\ncytoskeleton and the extracellular environment7,8,82. These assemblies undergo\nmaturation under tension that is driven by actin filaments, which is accompanied by the\nreorganization of the actin architecture and changes in condensate shape, suggesting a\nmechanically analogous coupling between actin-generated forces and condensate\ndeformation. Beyond confinement, in instances where the surface energy of the droplet\nis much smaller than the bending energy of the actin filaments, condensates can\nmediate actin crosslinking driven primarily by wetting filaments via capillary bridge\ninteractions24,75,86,87. Additionally, N-WASP forms condensates on the plasma membrane\nthat function as focus points for Arp2/3 activation. N-WASP condensates form as a\nresult of a prewetting transition whereby condensation occurs once N-WASP reaches a\ncritical surface concentration81. Together, these examples illustrate how feedback\nbetween condensate material properties, actin organization, and membrane properties\nmay contribute to force generation and cytoskeletal remodeling at the membrane88–90.\n6"}]

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