Energetics and dynamics of constrained actin filament bundling

Liquid-like condensates mediate competition between actin branching and bundling

Cellular remodeling of actin networks underlies cell motility during key morphological events, from embryogenesis to metastasis. In these transformations, there is an inherent competition between actin branching and bundling, because steric clashes among branches create a mechanical barrier to bundling. Recently, liquid-like condensates consisting purely of proteins involved in either branching or bundling of the cytoskeleton have been found to catalyze their respective functions. Yet in the cell, proteins that drive branching and bundling are present simultaneously. In this complex environment, which factors determine whether a condensate drives filaments to branch or become bundled? To answer this question, we added the branched actin nucleator, Arp2/3, to condensates composed of VASP, an actin bundling protein. At low actin to VASP ratios, branching activity, mediated by Arp2/3, robustly inhibited VASP-mediated bundling of filaments, in agreement with agent-based simulations. In contrast, as the actin to VASP ratio increased, addition of Arp2/3 led to formation of aster-shaped structures, in which bundled filaments emerged from a branched actin core, analogous to filopodia emerging from a branched lamellipodial network. These results demonstrate that multi-component, liquid-like condensates can modulate the inherent competition between bundled and branched actin morphologies, leading to organized, higher-order structures, similar to those found in motile cells.

Liquid-like condensates mediate competition between actin branching and bundling

Cellular remodeling of actin networks underlies cell motility during key morphological events, from embryogenesis to metastasis. In these transformations there is an inherent competition between actin branching and bundling, because steric clashes among branches create a mechanical barrier to bundling. Recently, liquid-like condensates consisting purely of proteins involved in either branching or bundling of the cytoskeleton have been found to catalyze their respective functions. Yet in the cell, proteins that drive branching and bundling are present simultaneously. In this complex environment, which factors determine whether a condensate drives filaments to branch versus becoming bundled? To answer this question, we added the branched actin nucleator, Arp2/3, to condensates composed of VASP, an actin bundling protein. At low actin to VASP ratios, branching activity, mediated by Arp2/3, robustly inhibited VASP-mediated bundling of filaments, in agreement with agent-based simulations. In contrast, as the actin to VASP ratio increased, addition of Arp2/3 led to formation of aster-shaped structures, in which bundled filaments emerged from a branched actin core, analogous to filopodia emerging from a branched lamellipodial network. These results demonstrate that multi-component, liquid-like condensates can modulate the inherent competition between bundled and branched actin morphologies, leading to organized, higher-order structures, similar to those found in motile cells.

Interplay between Brownian motion and cross-linking kinetics controls bundling dynamics in actin networks

Morphology changes in cross-linked actin networks are important in cell motility, division, and cargo transport. Here, we study the transition from a weakly cross-linked network of actin filaments to a heavily cross-linked network of actin bundles through microscopic Brownian dynamics simulations. We show that this transition occurs in two stages: first, a composite bundle network of small and highly aligned bundles evolves from cross-linking of individual filaments and, second, small bundles coalesce into the clustered bundle state. We demonstrate that Brownian motion speeds up the first stage of this process at a faster rate than the second. We quantify the time to reach the composite bundle state and show that it strongly increases as the mesh size increases only when the concentration of cross-links is small and that it remains roughly constant if we decrease the relative ratio of cross-linkers as we increase the actin concentration. Finally, we examine the dependence of the bundling timescale on filament length, finding that shorter filaments bundle faster because they diffuse faster.

Toward the cellular-scale simulation of motor-driven cytoskeletal assemblies

The cytoskeleton - a collection of polymeric filaments, molecular motors, and crosslinkers - is a foundational example of active matter, and in the cell assembles into organelles that guide basic biological functions. Simulation of cytoskeletal assemblies is an important tool for modeling cellular processes and understanding their surprising material properties. Here, we present aLENS (a Living Ensemble Simulator), a novel computational framework designed to surmount the limits of conventional simulation methods. We model molecular motors with crosslinking kinetics that adhere to a thermodynamic energy landscape, and integrate the system dynamics while efficiently and stably enforcing hard-body repulsion between filaments. Molecular potentials are entirely avoided in imposing steric constraints. Utilizing parallel computing, we simulate tens to hundreds of thousands of cytoskeletal filaments and crosslinking motors, recapitulating emergent phenomena such as bundle formation and buckling. This simulation framework can help elucidate how motor type, thermal fluctuations, internal stresses, and confinement determine the evolution of cytoskeletal active matter.

Efficient simulation of thermally fluctuating biopolymers immersed in fluids on 1-micron, 1-second scales

The combination of fluid-structure interactions with stochasticity, due to thermal fluctuations, remains a challenging problem in computational fluid dynamics. We develop an efficient scheme based on the stochastic immersed boundary method, Stokeslets, and multiple timestepping. We test our method for spherical particles and filaments under purely thermal and deterministic forces and find good agreement with theoretical predictions for Brownian Motion of a particle and equilibrium thermal undulations of a semi-flexible filament. As an initial application, we simulate bio-filaments with the properties of F-actin. We specifically study the average time for two nearby parallel filaments to bundle together. Interestingly, we find a two-fold acceleration in this time between simulations that account for long-range hydrodynamics compared to those that do not, suggesting that our method will reveal significant hydrodynamic effects in biological phenomena.

Assembly kinetics determine the architecture of α-actinin crosslinked F-actin networks

The actin cytoskeleton is organized into diverse meshworks and bundles that support many aspects of cell physiology. Understanding the self-assembly of these actin-based structures is essential for developing predictive models of cytoskeletal organization. Here we show that the competing kinetics of bundle formation with the onset of dynamic arrest arising from filament entanglements and cross-linking determine the architecture of reconstituted actin networks formed with α-actinin cross-links. Cross-link mediated bundle formation only occurs in dilute solutions of highly mobile actin filaments. As actin polymerization proceeds, filament mobility and bundle formation are arrested concomitantly. By controlling the onset of dynamic arrest, perturbations to actin assembly kinetics dramatically alter the architecture of biochemically identical samples. Thus, the morphology of reconstituted F-actin networks is a kinetically determined structure similar to those formed by physical gels and glasses. These results establish mechanisms controlling the structure and mechanics in diverse semi-flexible biopolymer networks.

A mechanochemical model explains interactions between cortical microtubules in plants

Microtubules anchored to the two-dimensional cortex of plant cells collide through plus-end polymerization. Collisions can result in rapid depolymerization, directional plus-end entrainment, or crossover. These interactions are believed to give rise to cellwide self-organization of plant cortical microtubules arrays, which is required for proper cell wall growth. Although the cell-wide self-organization has been well studied, less emphasis has been placed on explaining the interactions mechanistically from the molecular scale. Here we present a model for microtubule-cortex anchoring and collision-based interactions between microtubules, based on a competition between cross-linker bonding, microtubule bending, and microtubule polymerization. Our model predicts a higher probability of entrainment at smaller collision angles and at longer unanchored lengths of plus-ends. This model addresses observed differences between collision resolutions in various cell types, including Arabidopsis cells and Tobacco cells.

Modeling the formation of in vitro filopodia

Filopodia are bundles of actin filaments that extend out ahead of the leading edge of a crawling cell to probe its upcoming environment. In vitro experiments (Vignjevic et al. in J Cell Biol 160:951-962, 2003) have determined the minimal ingredients required for the formation of filopodia from the dendritic-like morphology of the leading edge. We model these experiments using kinetic aggregation equations for the density of growing bundle tips. In mean field, we determine the bundle size distribution to be broad for bundle sizes smaller than a characteristic bundle size above which the distribution decays exponentially. Two-dimensional simulations incorporating both bundling and cross-linking measure a bundle size distribution that agrees qualitatively with mean field. The simulations also demonstrate a nonmonotonicity in the radial extent of the dendritic region as a function of capping protein concentration, as was observed in experiments, due to the interplay between percolation and the ratcheting of growing filaments off a spherical obstacle.

From branched networks of actin filaments to bundles

Cross-linking proteins can mediate the emergence of rigid bundles from a dense branched network of actin filaments. To enable their binding, the filaments must first bend towards each other. We derive an explicit criterion for the onset of bundling, in terms of the initial length of filaments L, their spacing b, and cross-linker concentration f, reflecting the balance between bending and binding energies. Our model system contains actin, the branching complex Arp2/3 and the bundling protein fascin. In the first distinct stage, during which only actin and Arp2/3 are active, an entangled aster-like mesh of actin filaments is formed. Tens of seconds later, when filaments at the aster periphery are long and barely branched, a sharp transition takes place into a star-like structure, marking the onset of bundling. Now fascin and actin govern bundle growth; Arp2/3 plays no role. Using kinetic Monte Carlo simulations we calculate the temporal evolution of b and L, and predict the onset of bundling as a function of f. Our predictions are in good qualitative agreement with several new experiments that are reported herein and demonstrate how f controls the aster-star transition and bundle length. We also present two models for aster growth corresponding to different experimental realizations. The first treats filament and bundle association as an irreversible sequence of elongation-association steps. The second, applicable for low f, treats bundling as a reversible self-assembly process, where the optimal bundle size is dictated by the balance between surface and bending energies. Finally, we discuss the relevance of our conclusions for the lamellipodium to filopodia transition in living cells, noting that bundles are more likely nucleated by "tip complex" cross-linkers (e.g. mDia2 or Ena/VASP), whereas fascin is mainly involved in bundle maintenance.

Arp2/3 branched actin network mediates filopodia-like bundles formation in vitro

During cellular migration, regulated actin assembly takes place at the cell leading edge, with continuous disassembly deeper in the cell interior. Actin polymerization at the plasma membrane results in the extension of cellular protrusions in the form of lamellipodia and filopodia. To understand how cells regulate the transformation of lamellipodia into filopodia, and to determine the major factors that control their transition, we studied actin self-assembly in the presence of Arp2/3 complex, WASp-VCA and fascin, the major proteins participating in the assembly of lamellipodia and filopodia. We show that in the early stages of actin polymerization fascin is passive while Arp2/3 mediates the formation of dense and highly branched aster-like networks of actin. Once filaments in the periphery of an aster get long enough, fascin becomes active, linking the filaments into bundles which emanate radially from the aster's surface, resulting in the formation of star-like structures. We show that the number of bundles nucleated per star, as well as their thickness and length, is controlled by the initial concentration of Arp2/3 complex ([Arp2/3]). Specifically, we tested several values of [Arp2/3] and found that for given initial concentrations of actin and fascin, the number of bundles per star, as well as their length and thickness are larger when [Arp2/3] is lower. Our experimental findings can be interpreted and explained using a theoretical scheme which combines Kinetic Monte Carlo simulations for aster growth, with a simple mechanistic model for bundles' formation and growth. According to this model, bundles emerge from the aster's (sparsely branched) surface layer. Bundles begin to form when the bending energy associated with bringing two filaments into contact is compensated by the energetic gain resulting from their fascin linking energy. As time evolves the initially thin and short bundles elongate, thus reducing their bending energy and allowing them to further associate and create thicker bundles, until all actin monomers are consumed. This process is essentially irreversible on the time scale of actin polymerization. Two structural parameters, L, which is proportional to the length of filament tips at the aster periphery and b, the spacing between their origins, dictate the onset of bundling; both depending on [Arp2/3]. Cells may use a similar mechanism to regulate filopodia formation along the cell leading edge. Such a mechanism may allow cells to have control over the localization of filopodia by recruiting specific proteins that regulate filaments length (e.g., Dia2) to specific sites along lamellipodia.

Actin bundling: initiation mechanisms and kinetics

Bundling of rapidly polymerizing actin filaments underlies the dynamics of filopodial protrusions that play an important role in cell migration and cell-cell interaction. Recently, the formation of actin bundles has been reconstituted in vitro, and two scenarios of bundle initiation, involving binding of two filament tips and, alternatively, linking of the tip of one filament to the side of the other, have been discussed. A first theoretical analysis is presented indicating that the two mechanisms can be distinguished experimentally. While both of them result counterintuitively in comparable numbers of bundles, these numbers scale differently with the average bundle length. We propose an experiment for determining which of the two mechanisms is involved in the in vitro bundle formation.

First-contact time to a patch in a multidimensional potential well

The escape of a diffusing particle from a potential well is an important aspect of many dynamic processes in chemistry, physics, and biology, and such an escape process often involves finding a restricted region or patch in a multidimensional potential well. We study an idealized model of this process via simulation and analytic theory. By combining results from special cases having either high symmetry or zero potential, we obtain a simple formula for the first-contact time for a particle moving to a boundary patch in an arbitrary number of dimensions. We apply this formula in two, three, and six dimensions. The predicted dependences of the first-contact time on the well depth and patch size are compared to results from simulations, and close agreement is found. We extend the theory to calculate the first-contact time between two particles in separate harmonic potential wells. As an application of this extended theory, we calculate the first-contact time for two parallel semiflexible biopolymer filaments and compare these results to previous simulations.

Lattice model for self-assembly with application to the formation of cytoskeletal-like structures

We introduce a stochastic approach for self-assembly in systems far from equilibrium. The building blocks are represented by a lattice of discrete variables (Potts-like spins), and physically meaningful mechanisms are obtained by restricting transitions through spatially local rules based on experimental data. We use the method to study nucleation of filopodia-like bundles in a system consisting of purified actin, fascin, actin-related protein 2/3 , and beads coated with Wiskott-Aldrich syndrome protein. Consistent with previous speculation based on static experimental images, we find that bundles derive from Lambda-precursor-like patterns of spins on the lattice. The ratcheting of the actin network relative to the surface that represents beads plays an important role in determining the number and orientation of bundles due to the fact that branching is the primary means for generating barbed ends pointed in directions that allow rapid filament growth. By enabling the de novo formation of coexisting morphologies without the computational cost of explicit representation of proteins, the approach introduced complements earlier models of cytoskeletal behavior in vitro and in vivo.