{"paper_id":"161e7b18-22f5-42cc-9f55-e6a51f065533","body_text":"1 \nFilament-resolved simulations reproduce  \nself-organization of lamellipodia and filopodia \n \nMasaya Fukui1, Yohei Kondo1, Nen Saito3,4,*, Honda Naoki1,2,3,5,* \n  \n1. Laboratory for Data-driven Biology, Nagoya University Graduate School of Medicine, Nagoya, \nJapan \n2. Laboratory for Data -driven Biology, Graduate School of Integrated Sciences for Life, \nHiroshima University, Higashihiroshima, Hiroshima, Japan \n3. Theoretical Biology Research Group, Exploratory Research Center on Life and Living Systems \n(ExCELLS), National Institutes of Natural Sciences, Okazaki, Aichi, Japan \n4. Life Science Center for Survival Dynamics, Tsukuba Advanced Research Alliance (TARA), \nUniversity of Tsukuba, Tsukuba, Japan \n5. Center for One Medicine Innovative Translational Research (COMIT), Nagoya University, \nNagoya, Aichi, Japan  \n \n* Corresponding author: Honda Naoki \nNagoya University Graduate School of Medicine 65, Tsurumai -cho, Showa-ku, Nagoya 466 -8550, \nJapan \nTel.: +81-52-744-1980 \nE-mail: honda.naoki.t1@f.mail.nagoya-u.ac.jp \n \n* Corresponding author: Nen Saito \nLife Science Center for Survival Dynamics, Tsukuba Advanced Research Alliance (TARA), \nUniversity of Tsukuba, Tsukuba, Japan \nE-mail: nensaito@tara.tsukuba.ac.jp \n \nAbstract \nThe dynamic assembly of actin filaments underlies diverse cellular morphologies such as lamellipodia, \nfilopodia, and reticulated networks. However, how filament -scale interactions among actin -binding \nproteins produce distinct actin architectures remains unclear. We developed a filament -resolved \ncomputational model of actin self-organization regulated by the Arp2/3 complex and fascin. Individual \nF-actin filaments are represented as elastic chains, and their stochastic polymerization, Arp2/3 -\nmediated branchin g, and fascin -mediated crosslinking and bundling are explicitly modeled. The \nsimulations reproduce three actin architectures observed in minimal reconstitution experiments, \nincluding lamellipodia -like branched networks, filopodia -like bundled protrusions, and reticulated \nmeshworks, as a function of Arp2/3 and fascin concentrations. We quantify these regimes using actin \ndensity, orientational order, and spikiness, which robustly separate the three morphologies across \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 2 \nconditions. To connect filament organization to shape change, we further couple the actin network to \nmembrane deformation using a phase -field formulation. This coupling shows how localized \nremodeling concentrates load to drive pseudopodial protrusions, whe reas highly branched networks \ndistribute stresses and stabilize rounded shapes. The model links molecular interactions to emergent \narchitecture and cell-scale morphodynamics. \n \n \nIntroduction \nCell morphology is largely determined by the cytoskeleton, especially the network of actin filaments \n(F-actins) beneath the plasma membrane  (Pollard & Borisy, 2003) . Through interactions with actin -\nbinding proteins (ABPs), other cytoskeletal components, and the membrane, actin filaments self -\norganize into higher -order architectures such as lamellipodia and filopodia (Blanchoin et al., 2014; \nSchaus et al., 2007; Svitkina et al., 2003; Vignjevic et al., 2006; Vinzenz et al., 2012; Welch et al., \n1997; Yang & Svitkina, 2011). Directly tracking individual filaments during morphogenesis remains \ntechnically challenging, which has limited mechanistic interpretation . Recent advances in live -cell \nimaging and super-resolution microscopy have begun to resolve actin assembly dynamics and the roles \nof key regulators including the Arp2/3 complex and formins (Ju et al., 2024; Mueller et al., 2017; \nVinzenz et al., 2012; Watanabe & Mitchison, 2002; Yamashiro et al., 2018) . Yet a central question \nremains: how do local filament -scale rules such as branching, crosslinking, and mechanical \ninteractions collectively produce distinct actin architectures at the cell scale, and what sets the \ntransition between lamellipodia-like and filopodia-like organizations (Carlier & Shekhar, 2017; Keren \net al., 2008; Tee et al., 2015).  \nTo isolate the essential molecular principles of actin -driven morphogenesis from the \ncomplexity of the intracellular environment , in vitro reconstitution systems using a minimal set of \ncytoskeletal components have been extensively employed. In particular, mixtures of actin monomers, \nthe Arp2/3 complex, fascin, and ATP generate a concentration -dependent phase diagram of F -actin \narchitectures (Haviv et al., 2006; Ideses et al., 2008) . High Arp2/3 complex levels yield branched \nnetworks reminiscent of lamellipodia , while high fascin favors  crosslinked, reticulated meshworks . \nNotably, at intermediate concentrations, spiny filopodia -like protrusions emerge, suggesting that \nfilopodia can arise from the interplay among F-actins, Arp2/3 complex, and fascin alone. These results \nimply that key aspects of cellular morphodynamics may be captured by a minimal set of ABPs. \nHowever, it remains unclear how filament -scale rules, e.g., branching, cross linking, and mechanics, \ncollectively generate these distinct regimes and set the boundaries between them. \nMany computational models have been developed to study actin -driven morphogenesis, yet a \nkey gap remains: no filament -resolved assembly model has reproduced the concentration -dependent \nF-actin repertoire observed in minimal reconstitution experiments —lamellipodia-like branched \nnetworks, filopodia-like bundled protrusions, and disordered reticulated networks —within a single \nframework (Haviv et al., 2006; Ideses et al., 2008). Existing approaches broadly fall into two classes. \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 3 \nOne class captures cellular morphological dynamics by coupling membrane deformation to \nintracellular reaction–diffusion systems (Camley et al., 2017; Cao et al., 2019; Marth & Voigt, 2014; \nNajem & Grant, 2013; Nonaka et al., 2011; Shao et al., 2012), including our previous work reproducing \ndiverse migration modes  (Imoto et al., 2021) . However, such continuum models typically do not \nexplicitly represent force generation and mechanical interactions at the level of individual filaments. \nA second class incorporates filament elasticity and network mechanics (Kim et al., 2009; Ma & Berro, \n2018; Nedelec & Foethke, 2007; Popov et al., 2016; Weichsel & Schwarz, 2010) , but these studies \noften emphasize local assembly mechanics or specific structures rather than the emergence of cell -\nscale morphology (Chandrasekaran et al., 2024; X. Chen et al., 2020) . Developing a model that \ncaptures the emergence of these diverse structures from filament-level interactions would offer crucial \ninsights into the physical principles underlying cellular morphogenesis. \nHere we present a filament -resolved computational model in which branching by the Arp2/3 \ncomplex and bundling by fascin are sufficient to generate the concentration-dependent phase behavior \nobserved in minimal reconstitution experiments. The simulations re produce three distinct actin \narchitectures, such as lamellipodia -like branched networks, filopodia -like bundled protrusions, and \nreticulated meshworks, and quantify them using actin density and filament orientation. Extending the \nsame framework with a phas e-field membrane captures reciprocal coupling between cytoskeletal \nremodeling and membrane deformation, linking filament-scale rules to cell-scale morphodynamics. \n \nResults \nFilament-resolved mechanochemical model of actin assembly \nWe developed a mathematical model of self -organization of F -actin assembly to examine how the \nthree different structures, i.e., Network structure, Filopodia -like structure, and Lamellipodia -like \nstructure, emerge. In this model, a single F -actin was addres sed as an elastic filament, which was \ncoarse-grained by a one-dimensional connected node. In addition, F -actins were branched by Arp2/3 \ncomplex, and cross-linked by fascin (Fig. 1). The dynamics of the F-actin assembly can be described \nas  \n𝑑𝑟𝑖\n𝑑𝑡 = − 𝜆 𝑑𝑈\n𝑑𝑟𝑖\n \n𝑈 = 𝑈tension({𝑟𝑖}) + 𝑈bending({𝑟𝑖}) + 𝑈branch({𝑟𝑖}) + 𝑈fascin({𝑟𝑖}) + 𝑈bandle({𝑟𝑖}) \n \nwhere 𝑟𝑖 indicates position of  𝑖 -th node; 𝑈 ({𝑟𝑖}) indicates potential energy including tension \nenergy, bending energy, crosslink energy, and branching energy 𝑈∗∗∗({𝑟𝑖}) . 𝑈tension({𝑟𝑖}) \nrepresents the generation of tensile force between adjacent nodes, which contributes to maintaining \nthe length of F -actin, whereas 𝑈bending({𝑟𝑖}) represents the generation of bending force, which \ncontributes to maintaining the straightness of F-actin (see Methods in detail).  \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 4 \nWe performed simulations in 2D to observe the process of actin self-organisation. This means F-actins \ncan cross each other in our simulations.  \n In the model, F -actins involves several reactions: polymerization/depolymerization, and \nbinding/unbinding with arp2/3 complex and fascin (see Methods in details ). Polymerization and \ndepolymerization occur at both ends (Kuhn & Pollard, 2005). The net elongation rate of F-actin at each \nbarbed and pointed end is given by 𝑘𝑝\n+[G − actin] − 𝑘𝑑\n+ and 𝑘𝑝\n− [G − actin] − 𝑘𝑑\n− , respectively. Note \nthat we considered a situation that the net elongation rates were positive with assumption that G-actins \nexist abundant in the bulk. Arp2/3 complex associates on F -actin at the rate of  𝑘𝑓Arp [Arp2/3] \nwhereas associated Arp2/3 complex cannot dissociate in the model  (Rutkowski & Vavylonis, 2021). \nFascin associates to and dissociates from F-actin at the rates of 𝑘𝑓Fas[Fascin] and 𝑘𝑏Fas, respectively. \nIn the model, we assumed that bindings of Arp2/3 complex and fascin were mutually exclusive, namely \neach actin node can bind either Arp2/3 complex and fascin. Nodes bound by fascin can be connected \nif they are within the reaction distance.  \nWe assumed that the concentrations are well-mixed. That is, these chemical reaction events can occur \nwith equal probability in all regions in our simulations. We also consider nucleation of F-actin in our \nsimulations. The rate of nucleation is formulated in the same way as for the polymerization of actin. \n(see Methods in detail) \n \n \nFigure 1: Mathematical model for self-organization of F-actins \n(a) Our model incorporates key processes governing F -actin dynamics, including actin polymerization and \ndepolymerization, nucleation, branching mediated by the Arp2/3 complex, and bundling facilitated by Fascin. (b) The \nmodel accounts for the various mechanical properties of the actin filament system, specifically filament tension, bending, \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 5 \nbundling, elasticity at branch points maintaining specific angles between two F-actins. (c) Chemical reactions underlying \nF-actin self -organization. The model includes polymerization and depolymerization of actin monomers, binding and \nbranching induced by the Arp2/3 complex, and cross-linking by fascin. It is assumed that the binding of the Arp2/3 complex \nand fascin to the same filament segment is mutually exclusive.  \n \nSelf-organization dynamics of actin networks in filament-resolved simulations \nHere, we performed the simulation of the self -organizing process of F -actin networks ( Fig. 2 a). \nInitially, several F-actin segments were located for seeds of growing F-actin network (Fig. 2a-1). After \na short interval, the F -actins elongate by polymerization, generate the branched F -actins by Arp2/3 \ncomplex and increase the number of total F -actins ( Fig. 2 a-2). When the F -actins elongate and \napproach each other, the F-actins can be cross-linked by Fascin (Fig. 2a-3). During the growth process \nof the F-actin network, the number of F-acins exponentially increased (Fig. 2b) and concentrations of \nG-actin, Arp2/3 complex, and Fascin monotonically decreased ( Fig. 2c). To validate this simulation, \nwe confirmed that the potential energy decreases during mechanical relaxation of F -actins, Arp2/3 \ncomplex and Fascin ( Fig. 2d). In this simulation, we visualized spatial locations of Arp2/3 complex \nand Fascin ( Fig. 2e), respectively, and spatial distribution of tension  and bending forces ( Fig. 2f), \nrespectively. Thus, our model recapitulated the basic growing dynamics of the self -organized F-actin \nnetworks.  \n \n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 6 \nFigure 2: Simulations of self-organizing F-actin networks \n(a) Growing F-actin network in time. In this simulation, the total number of actin particles is set to 2000, with 800 Arp2/3 \ncomplexes and 4000 fascin molecules available as resources. (b) Time series of the number of F -actin constituting the \nstructure. (c) Time-series of the resources of unbound actin monomer, Arp2/3 complex, and Fascin. (d) Time-series of the \ntotal potential energy. (e) Distribution of Arp2/3 complex and Fascin in filopodia-like structure. (f) Visualization of tensile \nand bending forces on the F-actin network. All images were generated by introducing three new particles between adjacent \nF-actin particles, rasterizing the F-actin particles to the pixelated image, and then applying a Gaussian filter for a realistic \nvisualization comparable to experimental imaging. All scale bars indicate 20 μm. \n \nSimulations reproduce three F-actin structures across Arp2/3–fascin conditions  \nUsing the model, we performed simulations, varying concentrations of Arp2/3 complex and Fascin. \nThen, the model successfully reproduced three distinct types of F -actin structures. With high \nconcentration of Arp2/3 complex and low concentration of Fascin, t he F-actin assembly grew into a \nlocalized round shape, whose structure corresponds to lamellipodia -like branched networks (Fig. 3a-\n1). This is due to the rapid polymerization from the Arp2/3 -induced branching of initially existing F-\nactins and uniform grow th of actin assembly to the surrounding area. On the other hand, with low \nconcentration of Arp2/3 complex and high concentration of Fascin, network structures are generated \n(Fig. 3a -3). In this condition, the localized round shaped -structure did not emerge due to the low \nconcentration of Arp2/3 complex, and then the nucleation of F-actin happened uniformly in space. As \na result, F -actins assembled like a network by cross -linking of fas cin. In the intermediate condition \nbetween two conditions above (i.e., in termediate concentrations of Arp2/3 complex and Fascin), F -\nactin assembly grew into a localized star shape, which corresponds to a filopodia -like structure (Fig. \n3a-2). This structure is generated through a process that Arp2/3 complex initially generated the \nbranched-F-actin network, which are then cross-linked by fascin to form bundles. These three emerged \nstructures were consistent with previous in vitro experiments (Haviv et al., 2006; Ideses et al., 2008).  \n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 7 \n \nFigure 3: Three types of F-actin structures in simulation \n(a) Representative simulated image of filopodia -like, lamellipodia-like, and Network structure. The number of actin \nparticles is 2000 in all simulations. Arp2/3 resources are 80 for filopodia -like, 240 for lamellipodia -like, and 640 for \nnetwork structures, re spectively. Fascin resources are 300000, 1000, and 400 for the same structures, respectively. (b) \nGrowth process of three types of F-actin structures. All scale bars indicate 20 μm. \n \nQuantitative evaluations of three types of F-actin structures  \nWe summarized F-actin structures in a phase diagram of Arp2/3 and Fascin concentrations ( Fig. 4a). \nTo quantitatively evaluate F-actin structures, we characterized them by three different features: F-actin \ndensity, orientation order parameter, and degree of spikiness.   \n F-actin density was calculated by the number of actin nodes within an arbitrary circle, where \nits radius was selected to ensure that its radius was fixed across all conditions to enable consistent \ncomparison between different structures and concentrations (see Methods). We computed the heatmap \nof F-actin density in space of the concentrations of Arp2/3 and Fascin (Fig. 4b). We found that F-actin \ndensity was dominantly determined by the concentration of Arp2/3. Under the conditions of \nlamellipodia-like structures (i.e., high Arp2/3 and low Fascin) and filopodia -like structures (medium \nArp2/3 and high Fascin), the F-actin density tends to be high. On the other hand, under the condition \nof network structure (i.e., low Arp2/3 and high Fascin), the density became low. Thus, the density \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 8 \nfeature can separate the lamellipodia/filopodia -like and the network structures, and its boundary was \nconsistent with previous studies.  \n The orientation order parameter was calculated by the local filament angle (see Methods) (Fig. \n4c). This order parameter varies depending on the concentration of Fascin and increases continuously \nin the order of lamellipodia -like, filopodia-like, and network structures. Thus, the orientation order \nparameter can characterize the three types of structures.   \nThe degree of spikiness is evaluated by the variance of angle of actin nodes, where the positions \nof actin nodes were represented in the polar coordinate system from the center of F-actin structures. In \nsituations of lamellipodia-like structures (i.e. high Arp2/3, low Fascin) and network structures (i.e. low \nArp2/3, low Fascin), the spikiness tends to smaller angular dispersion. Conversely, conditions similar \nto filopodia-like structures (i.e. medium Arp2/3, low Fascin) correlate with an increased degree of  \nspikiness. Thus, the degree of spikiness distinctly characterizes filopodia -like structures, which is \nconsistent with the phase diagram of F -actin structures observed in previous in vitro study  (Fig. 4d). \nTaken together, three types of F -actin structures can be quantitatively separated by three features: F -\nactin density, orientation order parameter, and the degree of spikiness.  \n  \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 9 \n \n \nFigure 4: Quantitative evaluations of three types of F-actin structures  \n(a) Phase diagram for the F-actin structures varying concentrations of Fascin and Arp2/3 complex. (b,c) Heat maps for the \ndensity (b) the orientation order parameter (c) density of F-actin structures. (d) Phase diagram of F-actin structures formed \nwith 7 µM G-actin at varying concentrations of fascin and Arp2/3. Structural types are indicated as follows: flamellipodia-\nlike structure (circles), filopodia -like structure (triangles), and network structure (squ ares). Blue and red lines represent \nboundaries three types. Adapted from Ideses Y et al., PLOS ONE, 2008. \n \nMembrane coupling links actin architectures to protrusive morphodynamics \nSo far, we have modeled the mechanism of self -organization of F-actin structures in vitro. However, \nthis model cannot discuss cell morphogenesis because the membrane is not present in this model. \nTherefore, it is necessary to construct a framework to simul ate the dynamics of F -actins surrounded \nby the cell membrane, rather than simulating only F -actins. Here, we propose a model consisting of \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 10 \ntwo processes: (1) F-actin reorganization within the cell membrane and (2) cell membrane deformation \n(Fig 3-a). As a method to describe the cell membrane, we applied the phase-field method proposed in \nprevious studies (Camley et al., 2017; Cao et al., 2019; Imoto et al., 2021; Marth & Voigt, 2014; Najem \n& Grant, 2013; Shao et al., 2012; Taniguchi et al., 2013).  \nWe consider the interaction between the cell membrane and intracellular F -actins. We \nrepresent the cell membrane using a continuous phase -field field 𝜙 and the F -actin as a chain of \ndiscrete sets of points r described earlier. The purpose of this section is to obtain the time evolution \nequation for some 𝑖 -th vertex 𝑟𝑖 and cell 𝜙. Let 𝑟𝑖 be the position of the particle of interest that \nconstitutes the filament. In this study, we represent the filament as a continuous field by considering \nthat this point has a width and can be viewed as a distribution such that 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟) in space. \n𝜏 𝜕𝜙\n𝜕𝑥 = − 𝜕ℋ\n𝜕𝜙 = 𝛾 ( ∇ 2𝜙 − 𝐺 ′\n𝜖2 ) − 2𝜇 ( ∫ ℎ(𝜙)𝑑𝑟− 𝑉𝑜) ℎ′(𝜙) + 𝑔𝑜ℎ′(𝜙) ∑ 𝒩(𝑟|𝑟𝑖, 𝜎𝑟)\n𝑖\n \nwhere 𝐺 (𝜙) = 18𝜙2(1 − 𝜙)2 and ℎ(𝜙) = 𝜙2(3 − 2𝜙). The equation of motion of the particle center \n𝑟𝑖 due to the force exerted by the field 𝜙(𝑟) on the particle center 𝑟𝑖 at position 𝑟 is given by \n𝜏𝑟\n𝑑\n𝑑𝑡𝑟𝑖 = ∫ 𝑔𝑜(1 − ℎ(𝜙))∇ 𝑔𝑖𝑑𝑟 \nBy solving numerically for these time evolutions, the mechanical interactions between the membrane \nand F-actins can be explicitly treated. \nIn our simulations, a distinct membrane deformation was identified by varying the Arp2/3 and \nFascin concentrations and by adjusting the membrane tension parameters . When Arp2/3 \nconcentration was high and Fascin  concentration was low, local aggregation of F -actin resulted in a \nrounded morphology, forming an approximately circular membrane structure ( Fig. 5a-1). This is due \nto the branching of F-actin mediated by Arp2/3, which reduces the individual force exerted by each F-\nactin chain on the membrane (Fig. 5b).  \nIn contrast, under conditions of moderate concentrations of Arp2/3 and Fascin, pseudopodia \nwere extended and the membrane was dynamically deformed into complex forms  (Fig. 5a-2). The \nbundled and rigid F -actin exerts a strong force  to push the membrane forward  (Fig. 5 c). This \nconfiguration led to the emergence of complex structures, including filopodia that demonstrated \nexpansion and contraction. Therefore, our mathematical model of cytoskeletal organization within the \ndynamic membrane is capable of generating both lamellipodia and filopodia formation in a \ncomprehensive manner.  \n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 11 \n \n \nFigure 5: Two types of cell morphology in simulation coupled with membrane dynamics  \n(a) Simulated cell morphology under conditions favoring lamellipodia-like and filopodia-like structures. The heatmap \nrepresents the load exerted by F-actin on the cell membrane. Actin is shown in red, Fascin in green, and the Arp2/3 complex \nin pink. The number of actin particles is 1000 in all simulations. Arp2/3 resources are 80 for filopodia-like and 640 for \nlamellipodia-like structures, respectively. Fascin resources are 2000 and 640 for the same structures, respectively. (b) \nSnapshots of morphological changes in a cell featuring only lamellipodia. As F -actin presses against the membrane, the \nload it exerts on the cell membrane gradually reduces. All scale bars represent 20 μm.  (c) Snapshots of morphological \nchanges in a cell exhibiting pseudopodia. Over time, these pseudopodia are observed to fuse. \n \nDiscussion \nIn this study, we developed a computational model that captures the self -organization of F -actins \nregulated by Arp2/3 and fascin, and demonstrated its ability to reproduce three distinct cytoskeletal \nstructures—lamellipodia-like, filopodia-like, and mesh-like networks—previously observed in vitro. \nBy incorporating filament elasticity, stochastic binding dynamics of regulatory proteins, and \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 12 \nmembrane deformation using a phase-field approach, our model offers a unified framework to explore \nhow local filament-level interactions give rise to diverse cellular morphologies. \nA central contribution of this study is the construction of a physically grounded model that \nlinks molecular-scale interactions to the emergence of large -scale F-actin structures. Specifically, we \nshowed that varying the concentrations of Arp2/3 and fascin leads to the formation of different F-actin \narchitectures, consistent with experimental observations. This result highlights the critical role of these \ntwo actin-binding proteins in regulating cytoskeletal morphogenesis and demonstrates that complex \nstructural transitions can arise from a minimal set of molecular components. Furthermore, the model \nintegrates filament elasticity and dynamic assembly mechanisms, which are often simplified or omitted \nin previous models. The ability to simulate lamellipodia -like, filopodia -like, and network -like \nconfigurations within a single framework marks a significant step forward in computational modeling \nof the actin cytoskeleton. \nBy incorporating membrane dynamics through a phase -field formulation, the model also \ncaptures the reciprocal interactions between cytoskeletal assembly and cell shape. Unlike traditional \nmesh-based methods that impose geometric constraints, the phase -field approach enables seamless \ncoupling between internal filament dynamics and membrane deformation. This methodological choice \nprovides a flexible and extensible platform for simulating cell morphogenesis, with potential for future \nintegration of biochemical gradients, reaction-diffusion systems, or mechanical signaling pathways. \nNotably, the model remains relatively simple in terms of parameterization, enhancing its \ninterpretability and facilitating further biological applications. \nThe model is based on biologically plausible assumptions. First, we assume a quasi -two-\ndimensional system where overlapping filaments can pass through one another. This assumption is \nsupported by previous studies that compared two-dimensional and thin three-dimensional simulations, \nconcluding that two -dimensional modeling sufficiently captures essential dynamics. Second, we \nassume that each site on a filament can be bound by only one actin -binding protein at a time. This is \nconsistent with biochemical evidence showing mutually exclusive binding among certain regulators, \nsuch as between Arp2/3 and capping proteins. Finally, we propose a novel assumption that fascin -\nmediated bundling occurs preferentially on filaments with limited branching density. While this  has \nyet to be experimentally verified, our simulations suggest that such a mechanism could account for the \ndistinct spatial separation of branched and bundled domains, warranting further experimental \ninvestigation.  \nA number of computational models have been proposed to investigate F-actin dynamics. Early \nmodels often neglected filament elasticity  (Nonaka et al., 2011) , while more recent efforts, such as \nMEDYAN (Ni & Papoian, 2021; Popov et al., 2016), incorporated mechanical properties like bending \nand tension, focusing on phenomena such as actin comet tails and contractile rings. However, few \nmodels have succeeded in recapitulating the in vitro self-organization of actin structures under minimal \nconditions. While some models included Arp2/3 -mediated branching and fascin -mediated bundling, \nthey did not explore the concentration -dependent transitions among lamellipodia-like, filopodia-like, \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 13 \nand network morphologies. On the other hand, phase -field models have been used to simulate cell \nmorphogenesis and membrane deformation (Camley et al., 2017; Cao et al., 2019; Imoto et al., 2021; \nMarth & Voigt, 2014; Najem & Grant, 2013; Shao et al., 2012) . These models typically focused on \nmacroscopic shape dynamics and did not explicitly incorporate filament -level binding rules or \ncytoskeletal mechanical feedback. Our approach bridges these gaps by integrating microscopic \nfilament dynamics with mesoscopic  membrane behavior, thus providing a multiscale modeling \nframework not previously achieved in this domain. \nDespite its strengths, our model has several limitations. For example, the phase-field approach \nused here does not currently incorporate focal adhesion and actomyosin, which are crucial for linking \nintracellular actin structures to the extracellular matrix (ECM) and for generating traction forces during \nmigration (Even-Ram & Yamada, 2005; Kuo, 2013) . Incorporating such mechanisms would require \nadditional modeling frameworks or hybrid approaches. Additionally, while our current model operates \neffectively in two dimensions, extension to three-dimensional geometries will be necessary to capture \nmore complex cellular behaviors and morphologies. Finally, the reaction rules and kinetic parameters \nare based on idealized conditions, and refinements will be necessary to fully align the model with \nexperimental systems. In the future, this modeling framework could be extended to include additional \nactin regulators (e.g., cofilin, Ena/VASP)  (X. J. Chen et al., 2014) , signaling pathways, or \nmechanochemical feedback loops  (Hannezo & Heisenberg, 2019) . Moreover, the integration of this \nmodel with experimental studies may provide a powerful platform for data -driven discovery of \ncytoskeletal organization principles in cell morphogenesis. \n \nMethods \nMechanical model  \nWe developed a mechanical model of the F-actin network represented as a chain of particles connected \nby springs (Baschnagel et al., 2016) (Fig. 1). The positions of the 𝑖 -th particle, 𝑟𝑖 = (𝑥𝑖, 𝑦𝑖)⊤ , evolve \nfollowing dynamics: \n𝑑𝑟𝑖\n𝑑𝑡 = − 𝜆 𝑑𝑈\n𝑑𝑟𝑖\n \n𝑈 = 𝑈tension({𝑟𝑖}) + 𝑈bending({𝑟𝑖}) + 𝑈branch({𝑟𝑖}) + 𝑈fascin({𝑟𝑖}) + 𝑈bandle({𝑟𝑖}) \nwhere {𝑟} represents the set of  𝑟𝑖 (𝑖= 1, 2, … ) ; 𝜆 is a positive constant;  𝑈 ({𝑟}) is an energy \npotential consisting of contributions from tension, bending, branching, and bundling. Each energy term \nwas modelled below: \n \n𝑈tension({𝑟}) = ∑ 𝑘𝑡𝑒(∥𝑟𝑖 − 𝑟𝑗∥− 𝑟𝑜)2\n{𝑖,𝑗}∈𝐶\n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 14 \nwhere 𝐶  is the set of connected particle pairs,  𝑘𝑡𝑒 is the tension stiffness, and 𝑟0  is natural length \nof the spring  (Gittes et al., 1993; Wisanpitayakorn et al., 2022) . The bending energy penalizes \ndeviations from a straight configuration: \n \n𝑈bending({𝑟}) = ∑ 𝑘𝑏𝑒cos (∠ (𝑒𝑖,𝑗, 𝑒𝑗,𝑘))\n{𝑖,𝑗},{𝑗,𝑘}∈𝐶\n \nwhere 𝑘𝑏𝑒 is the bending stiffness, ∠ (𝑥, 𝑦) represent the angle between unit vectors  𝑥 and 𝑦, and  \n𝑒𝑖,𝑗 = (𝑟𝑗 − 𝑟𝑖)/|𝑟𝑗 − 𝑟𝑖|. \n \n𝑈branch({𝑟}) = ∑ 𝑘𝑏𝑒cos (∠ (𝑒𝑖,𝑗, 𝑒𝑗,𝑘 ) − 70𝜋/180)\n𝑖∈𝐵\n \nwhere 𝐵  is the set of particles at the branching point; the 𝑖 -th particle is the branching point \nconnected to the  𝑗 -th and k th participles toward the barbed end direction; 𝑘𝑏𝑟 is the bending \nstiffness at branching points; 70π/180 represents natural angle of 70 degrees which is based on well-\nestablished experimental evidence that the Arp2/3 complex nucleates new actin branches at an angle \nof approximately 70 degrees relative to the mother filament (Amann & Pollard, 2001; Dyche Mullins \net al., 1998). \n \n𝑈fascin({𝑟}) = ∑ 𝑘𝑓 (‖𝑟𝑖 − 𝑟𝑘 ‖− 𝑟𝑓 )2\n{𝑖,𝑘}∈𝐹\n \nwhere 𝐹  is the set of particle pairs connected by fascin across different filaments; 𝑘𝑓 is the tension \nstiffness of fascin; 𝑟𝑓 is the natural length of the fascin (Gong et al., 2025; Jansen et al., 2011). \n \n𝑈bundle({𝑟}) = ∑ 𝑘𝑏𝑢min [cos(∠ (𝑒𝑖,𝑗, 𝑒𝑘,𝑙)) , cos(∠ (𝑒𝑖,𝑗, 𝑒𝑘,𝑙) − 𝜋)]\n{𝑖,𝑘}∈𝐹\n \nwhere 𝑘𝑏𝑢 is modulus of the fascin-induced force promoting parallel alignment of F -actins.; the 𝑖 -\nth and 𝑘 -th particles connected to the  𝑗 -th and 𝑙 -th participles within the same filament toward \nthe barbed end direction.   Note that min operator represents that if the + ends of the two filaments \norient the same direction, the angle formed by the two filaments relaxes to 0 degrees, and if the + ends \nof the two filaments orient different directions, the angle formed by the two filaments relaxes to 180 \ndegrees. Mechanical model parameters are summarized in Table 1.  \n \nReaction model  \nF-actin undergo stochastic processes such as polymerization, depolymerization, nucleation, and \nassociation/dissociation with regulatory proteins including Arp2/3 and Fascin. These dynamics are \nimplemented using probabilistic rules based on the respective reaction rates below.  \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 15 \nPolymerization and depolymerization are assumed to occur at both the barbed and pointed ends of \nfilaments. The probabilities of polymerization (  𝑃𝑏\n+  and 𝑃𝑝\n+  ) and depolymerization (  𝑃𝑏\n−  and \n𝑃𝑝\n−  ) at each end are given by: \n \n𝑃𝑏\n+ = 𝑘𝑏\n+[G − actin] ⋅Δ 𝑡, \n𝑃𝑝\n+ = 𝑘𝑝\n+[G − actin] ⋅Δ 𝑡, \n𝑃𝑏\n− = 𝑘𝑏\n− ⋅Δ 𝑡, \n𝑃𝑝\n− = 𝑘𝑝\n− ⋅Δ 𝑡. \n \nwhere the superscripts “ b” and “p” denote the plus and minus ends of the filament, respectively;  𝑘𝑏\n± \nand 𝑘𝑝\n± represent the polymerization and depolymerization rate constants;  [ ⋅ ] denotes molecular \nconcentration; and Δ 𝑡 is the simulation time step. Each polymerization event consumes one actin \nmonomer, while each depolymerization event releases one monomer. \nActin nucleation is modeled as a stochastic process with the following probability: \n \n𝑝nu = 𝑘𝑛𝑢 [G − actin] ⋅Δ 𝑡, \n \nwhere 𝑘𝑛𝑢  is the nucleation rate constant. Each nucleation event consumes two actin monomers and \ninitiates a new filament consisting of two connected particles. \nThe probability of Arp2/3 -mediated branch formation is modeled by Hill -type saturation \nkinetics: \n𝑝Arp\n+ = 𝑉max ,𝑎[Arp2/3]𝑛\n𝐾 𝑚 ,𝑎 + [Arp2/3]𝑛 ⋅Δ 𝑡 \n \nUpon binding, the Arp2/3 complex remains stably associated with the mother filament and nucleates \na daughter filament at an angle of 70° relative to the axis of the mother filament. Branching is restricted \nto a two -dimensional plane, with the daughter fila ment extending at either +70° or −70° with equal \nprobability. The occurrence of a branching event is determined stochastically according to the \nprobability 𝑃𝑏\n+ and 𝑃𝑝\n+. \nThe probabilities of Fascin binding and unbinding is also modeled using Hill kinetics: \n \n𝑝Fas\n+ = 𝑉max ,𝑓[Fascin]𝑛\n𝐾 𝑚 ,𝑓 + [Fascin]𝑛 ⋅Δ 𝑡, \n \n𝑝Fas\n− = 𝑘𝑓\n− ⋅Δ 𝑡 \n \nFascin and Arp2/3 compete for the same binding site on each actin particle, allowing only one of them \nto bind at a given time ( Fig. 2c). Upon binding, fascin crosslinks the nearest actin particle from a \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 16 \nneighboring filament located within a predefined interaction distance (0.1 μm(100nm)). Fascin \ndissociation is modeled as a stochastic event that occurs with a constant probability 𝑝Fas\n−  per time \nstep. Reaction model parameters are summarized in Table 1. \n \nParameter Description Value (in vitro) Value (in vivo) \nActin filaments    \n The total number of actin particles 10000 2000 \n𝑟𝑜 Natural length of the spring [𝜇m] 1.2 1.2 \n𝑘𝑡𝑒 Tension stiffness [pN/𝜇m] 100.0 100.0 \n𝑘𝑏𝑒 Bending stiffness 10.0 10.0 \n𝑘𝑏\n+ Barbed end polymerization rate 2.0×10\n-6\n 2.0×10\n-3\n \n𝑘𝑝\n+ Pointed end polymerization rate 0.0 0.0 \n𝑘𝑏\n−  Barbed end depolymerization rate 0.0 0.0 \n𝑘𝑝\n−  Pointed end depolymerization rate 0.0 0.0 \n𝑘𝑛𝑢  Nucleation rate 1.0×10\n-5\n 1.0×10\n-3\n \nArp2/3 complex    \n The total number of Arp2/3 complex Varied Varied \n𝑘𝑏𝑟 Bending stiffness at branching points 10.0 10.0 \n𝑉max ,𝑎 Maximum Response 5.0×10\n-3\n 1.0 \n𝐾 m ,a Hill Constant 160 200 \n𝑛 Hill coefficient 2 2 \nFascin    \n The total number of Fascin Varied Varied \n𝑙0 Crosslinker spring eq. dist. [𝜇m] 0.1 0.1 \n𝑘𝑓 Crosslinker spring constant [pN/𝜇m] 50.0 50.0 \n𝑘𝑏𝑢 Crosslinker bending modulus [pN/𝜇m] 10.0 10.0 \n𝑉max ,𝑓 Maximum Response 1.0×10\n-2\n 1.0 \n𝐾 𝑚 ,𝑓 Hill Constant 480 700 \n𝑛 Hill coefficient 1 1 \n𝑘𝑓\n−  Dissociation rate of Fascin [1/s] 4.0×10\n-4\n 4.0×10\n-4\n \nTable 1: Filament-resolved computational model parameters \n \nSimulation protocol \nTo achieve high computational efficiency, all simulations were implemented in the C programming \nlanguage. At each time step, key physical quantities —including particle positions, binding states, \nfilament connectivity, and membrane morphology —were recorded.  Post-processing and data \nvisualization were conducted using custom Python scripts, allowing flexible analysis of filament \ndynamics and structural evolution. \nSimulations were performed in a two -dimensional square domain with periodic boundary \nconditions. The system was initialized with 40 short F -actin fibers, each consisting of two connected \nparticles generated via nucleation. These filaments were randomly pla ced within a circular region \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 17 \ncentered in the domain, with a radius of 2μm. The concentrations of free G-actin, Arp2/3 complex, and \nFascin were assumed to be spatially uniform throughout the simulation, under the assumption that their \ndiffusion is sufficiently fast relative to the timescale of filament dynamics. \nEach actin particle was represented as a C -language structure containing the following \nattributes: a particle index, spatial coordinates, binding states with regulatory proteins (e.g., Arp2/3, \nFascin), indices of neighboring connected particles, and crosslinking information mediated by Fascin. \nIn addition, each particle could form connections at up to three distinct sites: (i) to a neighboring \nparticle in the barbed-end (+) direction, (ii) to a neighboring particle in the pointed -end (−) direction, \nand (iii) to an additional neighboring particle in the barbed-end (+) direction in the case of branching. \nThese connection states were stored as a three-element binary vector (0 or 1), indicating the presence \nor absence of a connection at each corresponding site.  \nBiochemical events such as nucleation, polymerization, depolymerization, and \nbinding/dissociation of Arp2/3 and Fascin were modeled as stochastic processes based on reaction \nprobabilities. Upon nucleation, two actin monomers were consumed; polymerization a nd \ndepolymerization altered the amount of free G -actin accordingly. Similarly, when Arp2/3 or Fascin \nbound to or dissociated from filaments, the corresponding molecule counts were updated to reflect \nthese changes. In the simulation, concentrations of all d iffusive components were internally \nrepresented as discrete molecular copy numbers. When multiple events —such as polymerization, \nArp2/3 binding, and Fascin binding—could potentially occur at the same actin particle during a given \ntime step, the actual outcome was sampled based on the relative probabilities of all possible events. \n \nFeature Quantification for Structural Classification  \nThe simulated F-actin structures were quantitatively evaluated by two feature quantities: density and \norientation order parameter. These metrics were used to construct a phase diagram (Fig. 5).  \nDensity is defined as the local concentration of F-actins around the center. It is defined as: \n \ndensity =\n∑ 𝐼(|𝑟𝑖| < 𝑅 𝑜)𝑖\n𝜋𝑅 𝑜\n2  \nwhere 𝑟𝑖 denotes the position vector of 𝑖 -th actin particle relative to the center of the domain, and  \n𝐼(|𝑟𝑖| < 𝑅 𝑜) is an indicator function that takes the value 1 if  |𝑟𝑖| < 𝑅 𝑜, and 0 otherwise. Here, 𝑅 𝑜 \ndefines the radius of the region of interest. \nThe orientation order parameter (Mottram & Newton, 2014; Steinhardt et al., 1983)  was \ncalculated to quantify the angular correlation among neighboring F -actin. This parameter reflects the \ndegree of filament alignment, with higher values indicating more ordered, bundled structures such as \nfilopodia. To compute this value, the region of i nterest was divided into a fine lattice grid. For each \nlattice site, we calculated the angular correlation among F -actins present within that site, and then \naveraged the values across all lattice sites to obtain the global order parameter 𝑆 as \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 18 \n𝑆 = ∑ 𝑆𝑖𝑗\n𝑛𝑘>2\n \n \nwhere 𝑛𝑘>2 indicates the number of lattice sites containing two or more filaments. 𝑆𝑙 represents the \nangular correlation among neighboring F-actins within the 𝑙 -th lattice as \n𝑆𝑙 = 1\n𝐾 𝑙\n∑ cos{2(𝜃𝑙,𝑘 −  〈𝜃𝑙〉)}\n𝐾 𝑙\n𝑘=1\n \nwhere 𝑘 is the index of the F -actin, 𝐾 𝑙 is the total number of filaments within the  𝑙 -th lattice \nsite, 𝜃𝑙,𝑘  is the orientation angle of the  𝑘 -th filament, and  < 𝜃𝑙 > is the average angle of all \nfilaments within that lattice site (i.e., 〈𝜃𝑙〉= (1/2) ⋅atan(∑ sin 2𝜃𝑙,𝑘𝑘 , ∑ cos 2𝜃𝑙,𝑘𝑘 ) ). Note that the \norientation angle 𝜃 represents the direction of F -actin regardless of polarity, i.e., filaments that are \nparallel but point in opposite directions are regarded as having the same angle. \n \nPhase-field models \nThe total energy of the system is given by the following Hamiltonian: \nℋ = 𝛾 ∫ ( 𝜖\n2 |∇ 𝜙|2 + 𝐺\n𝜖) 𝑑𝑟+ 𝜇 ( ∫ ℎ(𝜙)𝑑𝑟− 𝑉𝑜)\n2\n+ 𝑔𝑜(1 − ℎ(𝜙)) ∑ 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟)\n𝑖\n \nIn this formulation, the first term represents the membrane surface tension, where  𝛾 is the surface \ntension coefficient and ϵ determines the characteristic thickness of the interface. The potential   \n𝐺 (𝜙) = 18𝜙2(1 − 𝜙)2 is a Landau-type double-well potential that stabilizes the interior and exterior \nregions of the cell. The second term imposes an area constraint using a smooth indicator function   \nℎ(𝜙) = 𝜙2(3 − 2𝜙), with 𝜇 as the strength of the constraint and 𝑉𝑜 as the target area. The third term \nrepresents a repulsive interaction between the membrane and F -actins, where 𝑖 -th particle of each \nfilament at position 𝑟𝑖 is modeled by a Gaussian  kernel 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟) with mean of 𝑟𝑖 and variance \nof 𝜎𝑟 . The coefficient 𝑔𝑜 determines the strength of this interaction.  \nThe membrane field 𝜙 evolves according to a reaction–diffusion-type equation obtained from \nthe variational derivative of the total energy functional ℋ  with respect to 𝜙: \n𝜏 𝜕𝜙\n𝜕𝑡= − 𝛿ℋ\n𝛿𝜙, \n𝛿∫ ( 𝜖\n2|∇ 𝜙|2 + 𝐺\n𝜖) 𝑑𝑟= ∫ ( − 𝜖∇ 2𝜙 + 𝐺 ′\n𝜖 ) 𝛿𝜙𝑑𝑟= − 𝜖∇ 2𝜙 + 𝐺 ′\n𝜖 , \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 19 \n𝛿( ∫ ℎ(𝜙)𝑑𝑟− 𝑉𝑜)\n2\n= 2 ( ∫ ℎ(𝜙)𝑑𝑟− 𝑉𝑜) ⋅∫ ℎ′(𝜙)𝛿𝜙𝑑𝑟= 2 ( ∫ ℎ(𝜙)𝑑𝑟− 𝑉𝑜) ℎ′(𝜙), \n𝛿{(1 − ℎ(𝜙)) ∑ 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟)\n𝑖\n} = − ℎ′(𝜙) ∑ 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟)\n𝑖\n= − 𝑔𝑜ℎ′(𝜙) ∑ 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟),\n𝑖\n \n𝜏 𝜕𝜙\n𝜕𝑡= − 𝜕ℋ\n𝜕𝜙 = 𝛾 ( ∇ 2𝜙 − 𝐺 ′\n𝜖2 ) − 2𝜇 ( ∫ ℎ(𝜙)𝑑𝑟− 𝑉𝑜) ℎ′(𝜙) + 𝑔𝑜ℎ′(𝜙) ∑ 𝒩(𝑟|𝑟𝑖, 𝜎𝑟)\n𝑖\n. \nThis equation incorporates three physical effects. The first term corresponds to the \nminimization of membrane curvature via surface tension. The second term restores deviations from \nthe target area, maintaining approximate volume conservation. The third te rm introduces mechanical \nfeedback from the cytoskeleton by representing how the presence of filament particles locally deforms \nthe membrane. Taken together, these terms define a mechanochemical membrane model that \ndynamically adapts to intracellular cytoskeletal activity.  \n \nInteraction between F-actin and membrane  \nThe influence of membrane deformation on actin dynamics was incorporated by introducing feedback \nfrom the membrane field 𝜙(𝑟, 𝑡) to individual F-actins. Each F-actin is modeled as a chain of discrete \nparticles 𝑟𝑖, which interact with the membrane through the same Gaussian kernel 𝑔(𝑟− 𝑟𝑖) as \ndefined in the energy functional. \nThe force exerted by the membrane on each actin particle is derived by taking the functional \nderivative of the total energy  ℋ  with respect to the particle position. This results in the following \nequation of motion for particle 𝑟𝑖: \n𝑓𝑖(𝑟) = − 𝛿\n𝛿𝑟𝑖\nℋ (𝒩 (𝑟|𝑟𝑖, 𝜎𝑟)) = 𝛿ℋ\n𝛿𝒩 (𝑟|𝑟𝑖, 𝜎𝑟) ∇ 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟) = 𝑔𝑜(1 − ℎ(𝜙))∇ 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟), \n𝜏𝑟\n𝑑\n𝑑𝑡𝑟𝑖 = ∫ 𝑔𝑜(1 − ℎ(𝜙))∇ 𝒩 (𝑟|𝑟𝑖, 𝜎𝑟)𝑑𝑟. \nThe repulsive interaction is restricted to regions in close proximity to the membrane interface. The \nterm (1 − ℎ(𝜙)) ensures that the force vanishes in the cell interior and increases as the particle \napproaches the membrane. Parameters are summarized in Table 2. \n \nParameter Description Value \nMembrane   \n𝜏  Viscous friction coefficient [pN/𝜇𝑚!] 1.0 \n𝜎 Spatial scale of F-actin [𝜇m] 0.6 \n𝑉0 Cell area [𝜇𝑚!] 600.00 \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted March 18, 2026. ; https://doi.org/10.64898/2026.03.15.711798doi: bioRxiv preprint \n\n 20 \n𝑀 𝑉  Area conservation constraint [pN/𝜇𝑚\"] 0.1 \n𝑔0 Protrusion force [pN/μm (conc.)-1]  2.0 \n𝛾 Surface tension [pN] 2.0 \n𝜖 Spatial scale of the phase boundary [𝜇m] 2.0 \n𝜏𝑟 Viscous friction coefficient [pN/𝜇𝑚!] 10.0 \nTable 2: Phase-field model parameters \n \nAcknowledgements \nThis study was supported in part by the Moonshot R&D –MILLENNIA Program [grant number \nJPMJMS2024-9 to H.N.] by Japan Science and Technology Agency (JST), Grant -in-Aid for \nTransformative Research Areas (B) [grant number 21H05170 to H.N.], Grant -in-Aid for Sc ientific \nResearch (B) [grant number 21H03541 to H.N.] and JSPS KAKENHI [25H01364 and 25K07242 to \nN.S.] from the Japan Society for the Promotion of Science (JSPS), Cooperative Study Program of \nExploratory Research Center on Life and Living Systems (ExCELLS)  [program number 19 -102 to \nH.N.], and Joint Research of the Exploratory Research Center on Life and Living Systems (ExCELLS) \n[ExCELLS program No. 25EX603]. \n \nCode Availability \nSimulations are mainly implemented in C and performed under Python 3.1 1.0, and the code will be \ndistributed through GitHub after publication.is distributed through GitHub.  \n  \nAuthor Contributions \nH.N. conceived the project. 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