Quantifying dissipation in actomyosin networks

Microscopic origin of tunable assembly forces in chiral active environments

Across a variety of spatial scales, from nanoscale biological systems to micron-scale colloidal systems, equilibrium self-assembly is entirely dictated by-and therefore limited by-the thermodynamic properties of the constituent materials. In contrast, nonequilibrium materials, such as self-propelled active matter, expand the possibilities for driving the assemblies that are inaccessible in equilibrium conditions. Recently, a number of works have suggested that active matter drives or accelerates self-organization, but the emergent interactions that arise between solutes immersed in actively driven environments are complex and poorly understood. Here, we analyze and resolve two crucial questions concerning actively driven self-assembly: (i) how, mechanistically, do active environments drive self-assembly of passive solutes? (ii) Under which conditions is this assembly robust? We employ the framework of odd hydrodynamics to theoretically explain numerical and experimental observations that chiral active matter, i.e., particles driven with a directional torque, produces robust and long-ranged assembly forces. Together, these developments constitute an important step towards a comprehensive theoretical framework for controlling self-assembly in nonequilibrium environments.

Computational simulations reveal that Abl activity controls cohesiveness of actin networks in growth cones

Extensive studies of growing axons have revealed many individual components and protein interactions that guide neuronal morphogenesis. Despite this, however, we lack any clear picture of the emergent mechanism by which this nanometer-scale biochemistry generates the multimicron-scale morphology and cell biology of axon growth and guidance in vivo. To address this, we studied the downstream effects of the Abl signaling pathway using a computer simulation software (MEDYAN) that accounts for mechanochemical dynamics of active polymers. Previous studies implicate two Abl effectors, Arp2/3 and Enabled, in Abl-dependent axon guidance decisions. We now find that Abl alters actin architecture primarily by activating Arp2/3, while Enabled plays a more limited role. Our simulations show that simulations mimicking modest levels of Abl activity bear striking similarity to actin profiles obtained experimentally from live imaging of actin in wild-type axons in vivo. Using a graph theoretical filament-filament contact analysis, moreover, we find that networks mimicking hyperactivity of Abl (enhanced Arp2/3) are fragmented into smaller domains of actin that interact weakly with each other, consistent with the pattern of actin fragmentation observed upon Abl overexpression in vivo. Two perturbative simulations further confirm that high-Arp2/3 actin networks are mechanically disconnected and fail to mount a cohesive response to perturbation. Taken together, these data provide a molecular-level picture of how the large-scale organization of the axonal cytoskeleton arises from the biophysics of actin networks.

Nucleation causes an actin network to fragment into multiple high-density domains

Actin networks rely on nucleation mechanisms to generate new filaments because spontaneous nucleation is kinetically disfavored. Branching nucleation of actin filaments by actin-related protein (Arp2/3), in particular, is critical for actin self-organization. In this study, we use the simulation platform for active matter MEDYAN to generate 2000 s long stochastic trajectories of actin networks, under varying Arp2/3 concentrations, in reaction volumes of biologically meaningful size (>20 μm3). We find that the dynamics of Arp2/3 increase the abundance of short filaments and increases network treadmilling rate. By analyzing the density fields of F-actin, we find that at low Arp2/3 concentrations, F-actin is organized into a single connected and contractile domain, while at elevated Arp2/3 levels (10 nM and above), such high-density actin domains fragment into smaller domains spanning a wide range of volumes. These fragmented domains are extremely dynamic, continuously merging and splitting, owing to the high treadmilling rate of the underlying actin network. Treating the domain dynamics as a drift-diffusion process, we find that the fragmented state is stochastically favored, and the network state slowly drifts toward the fragmented state with considerable diffusion (variability) in the number of domains. We suggest that tuning the Arp2/3 concentration enables cells to transition from a globally coherent cytoskeleton, whose response involves the entire cytoplasmic network, to a fragmented cytoskeleton, where domains can respond independently to locally varying signals.

On Stretching, Bending, Shearing, and Twisting of Actin Filaments I: Variational Models

Mechanochemical simulations of actomyosin networks are traditionally based on one-dimensional models of actin filaments having zero width. Here, and in the follow up paper (arXiv, DOI 10.48550/arXiv.2203.01284), approaches are presented for more efficient modeling that incorporates stretching, shearing, and twisting of actin filaments. Our modeling of a semiflexible filament with a small but finite width is based on the Cosserat theory of elastic rods, which allows for six degrees of freedom at every point on the filament's backbone. In the variational models presented in this paper, a small and discrete set of parameters is used to describe a smooth filament shape having all degrees of freedom allowed in the Cosserat theory. Two main approaches are introduced: one where polynomial spline functions describe the filament's configuration, and one in which geodesic curves in the space of the configurational degrees of freedom are used. We find that in the latter representation the strain energy function can be calculated without resorting to a small-angle expansion, so it can describe arbitrarily large filament deformations without systematic error. These approaches are validated by a dynamical model of a Cosserat filament, which can be further extended by using multiresolution methods to allow more detailed monomer-based resolution in certain parts of the actin filament, as introduced in the follow up paper. The presented framework is illustrated by showing how torsional compliance in a finite-width filament can induce broken chiral symmetry in the structure of a cross-linked bundle.

A generalized Flory-Stockmayer kinetic theory of connectivity percolation and rigidity percolation of cytoskeletal networks

Actin networks are essential for living cells to move, reproduce, and sense their environments. The dynamic and rheological behavior of actin networks is modulated by actin-binding proteins such as α-actinin, Arp2/3, and myosin. There is experimental evidence that actin-binding proteins modulate the cooperation of myosin motors by connecting the actin network. In this work, we present an analytical mean field model, using the Flory-Stockmayer theory of gelation, to understand how different actin-binding proteins change the connectivity of the actin filaments as the networks are formed. We follow the kinetics of the networks and estimate the concentrations of actin-binding proteins that are needed to reach connectivity percolation as well as to reach rigidity percolation. We find that Arp2/3 increases the actomyosin connectivity in the network in a non-monotonic way. We also describe how changing the connectivity of actomyosin networks modulates the ability of motors to exert forces, leading to three possible phases of the networks with distinctive dynamical characteristics: a sol phase, a gel phase, and an active phase. Thus, changes in the concentration and activity of actin-binding proteins in cells lead to a phase transition of the actin network, allowing the cells to perform active contraction and change their rheological properties.

The actin cytoskeleton is a complex dynamic system, regulated by multiple proteins that bind to actin filaments. Some actin-binding proteins are crosslinkers, which can bind pairs of actin filaments, forming actin networks. Actin crosslinkers can be passive linkers, providing only structural integrity, or can be active linkers such as myosin motors, which exert forces on the network. Experiments have shown that crosslinked actin networks can behave viscously when the number of passive crosslinkers is low, but become elastic, when there are many crosslinkers. Motors can only lead to contraction of the network when there is an intermediate concentration of passive crosslinkers. The behavior of networks in the cell depends on the concentration and activity of several distinct crosslinkers, which have different binding sites, geometries, affinities, and concentrations. In this work we propose a simple analytical model based on chemical kinetics and the Flory-Stockmayer theory that gives us insight into how different crosslinkers interact with the actin filaments so as to give rise to the emergent mechanical behavior. This theory also allows us to compute analytically several crucial aspects of the development of the mechanical properties during network assembly.

Pervasive cytoquakes in the actomyosin cortex across cell types and substrate stiffness

The actomyosin cytoskeleton enables cells to resist deformation, crawl, change their shape and sense their surroundings. Despite decades of study, how its molecular constituents can assemble together to form a network with the observed mechanics of cells remains poorly understood. Recently, it has been shown that the actomyosin cortex of quiescent cells can undergo frequent, abrupt reconfigurations and displacements, called cytoquakes. Notably, such fluctuations are not predicted by current physical models of actomyosin networks, and their prevalence across cell types and mechanical environments has not previously been studied. Using micropost array detectors, we have performed high-resolution measurements of the dynamic mechanical fluctuations of cells' actomyosin cortex and stress fiber networks. This reveals cortical dynamics dominated by cytoquakes-intermittent events with a fat-tailed distribution of displacements, sometimes spanning microposts separated by 4 μm, in all cell types studied. These included 3T3 fibroblasts, where cytoquakes persisted over substrate stiffnesses spanning the tissue-relevant range of 4.3 kPa-17 kPa, and primary neonatal rat cardiac fibroblasts and myofibroblasts, human embryonic kidney cells and human bone osteosarcoma epithelial (U2OS) cells, where cytoquakes were observed on substrates in the same stiffness range. Overall, these findings suggest that the cortex self-organizes into a marginally stable mechanical state whose physics may contribute to cell mechanical properties, active behavior and mechanosensing.

Forecasting Avalanches in Branched Actomyosin Networks with Network Science and Machine Learning

We explored the dynamic and structural effects of actin-related proteins 2/3 (Arp2/3) on actomyosin networks using mechanochemical simulations of active matter networks. On the nanoscale, the Arp2/3 complex alters the topology of actomyosin by nucleating a daughter filament at an angle with respect to a mother filament. At a subcellular scale, they orchestrate the formation of a branched actomyosin network. Using a coarse-grained approach, we sought to understand how an actomyosin network temporally and spatially reorganizes itself by varying the concentration of the Arp2/3 complexes. Driven by motor dynamics, the network stalls at a high concentration of Arp2/3 and contracts at a low Arp2/3 concentration. At an intermediate Arp2/3 concentration, however, the actomyosin network is formed by loosely connected clusters that may collapse suddenly when driven by motors. This physical phenomenon is called an "avalanche" largely due to the marginal instability inherent to the morphology of a branched actomyosin network when the Arp2/3 complex is present. While embracing the data science approaches, we unveiled the higher-order patterns in the branched actomyosin networks and discovered a sudden change in the "social" network topology of actomyosin, which is a new type of avalanche in addition to the two types of avalanches associated with a sudden change in the size or shape of the whole actomyosin network, as shown in a previous investigation. Our new finding promotes the importance of using network theory and machine learning models to forecast avalanches in actomyosin networks. The mechanisms of the Arp2/3 complexes in shaping the architecture of branched actomyosin networks obtained in this paper will help us better understand the emergent reorganization of the topology in dense actomyosin networks that are difficult to detect in experiments.

Understanding cytoskeletal avalanches using mechanical stability analysis

Eukaryotic cells are mechanically supported by a polymer network called the cytoskeleton, which consumes chemical energy to dynamically remodel its structure. Recent experiments in vivo have revealed that this remodeling occasionally happens through anomalously large displacements, reminiscent of earthquakes or avalanches. These cytoskeletal avalanches might indicate that the cytoskeleton's structural response to a changing cellular environment is highly sensitive, and they are therefore of significant biological interest. However, the physics underlying "cytoquakes" is poorly understood. Here, we use agent-based simulations of cytoskeletal self-organization to study fluctuations in the network's mechanical energy. We robustly observe non-Gaussian statistics and asymmetrically large rates of energy release compared to accumulation in a minimal cytoskeletal model. The large events of energy release are found to correlate with large, collective displacements of the cytoskeletal filaments. We also find that the changes in the localization of tension and the projections of the network motion onto the vibrational normal modes are asymmetrically distributed for energy release and accumulation. These results imply an avalanche-like process of slow energy storage punctuated by fast, large events of energy release involving a collective network rearrangement. We further show that mechanical instability precedes cytoquake occurrence through a machine-learning model that dynamically forecasts cytoquakes using the vibrational spectrum as input. Our results provide a connection between the cytoquake phenomenon and the network's mechanical energy and can help guide future investigations of the cytoskeleton's structural susceptibility.

Detailed Balance Broken by Catch Bond Kinetics Enables Mechanical-Adaptation in Active Materials

Unlike nearly all engineered materials which contain bonds that weaken under load, biological materials contain "catch" bonds which are reinforced under load. Consequently, materials, such as the cell cytoskeleton, can adapt their mechanical properties in response to their state of internal, non-equilibrium (active) stress. However, how large-scale material properties vary with the distance from equilibrium is unknown, as are the relative roles of active stress and binding kinetics in establishing this distance. Through course-grained molecular dynamics simulations, the effect of breaking of detailed balance by catch bonds on the accumulation and dissipation of energy within a model of the actomyosin cytoskeleton is explored. It is found that the extent to which detailed balance is broken uniquely determines a large-scale fluid-solid transition with characteristic time-reversal symmetries. The transition depends critically on the strength of the catch bond, suggesting that active stress is necessary but insufficient to mount an adaptive mechanical response.

Mechanical Cues Affect Migration and Invasion of Cells From Three Different Directions

Cell migration and invasion is a key driving factor for providing essential cellular functions under physiological conditions or the malignant progression of tumors following downward the metastatic cascade. Although there has been plentiful of molecules identified to support the migration and invasion of cells, the mechanical aspects have not yet been explored in a combined and systematic manner. In addition, the cellular environment has been classically and frequently assumed to be homogeneous for reasons of simplicity. However, motility assays have led to various models for migration covering only some aspects and supporting factors that in some cases also include mechanical factors. Instead of specific models, in this review, a more or less holistic model for cell motility in 3D is envisioned covering all these different aspects with a special emphasis on the mechanical cues from a biophysical perspective. After introducing the mechanical aspects of cell migration and invasion and presenting the heterogeneity of extracellular matrices, the three distinct directions of cell motility focusing on the mechanical aspects are presented. These three different directions are as follows: firstly, the commonly used invasion tests using structural and structure-based mechanical environmental signals; secondly, the mechano-invasion assay, in which cells are studied by mechanical forces to migrate and invade; and thirdly, cell mechanics, including cytoskeletal and nuclear mechanics, to influence cell migration and invasion. Since the interaction between the cell and the microenvironment is bi-directional in these assays, these should be accounted in migration and invasion approaches focusing on the mechanical aspects. Beyond this, there is also the interaction between the cytoskeleton of the cell and its other compartments, such as the cell nucleus. In specific, a three-element approach is presented for addressing the effect of mechanics on cell migration and invasion by including the effect of the mechano-phenotype of the cytoskeleton, nucleus and the cell's microenvironment into the analysis. In precise terms, the combination of these three research approaches including experimental techniques seems to be promising for revealing bi-directional impacts of mechanical alterations of the cellular microenvironment on cells and internal mechanical fluctuations or changes of cells on the surroundings. Finally, different approaches are discussed and thereby a model for the broad impact of mechanics on cell migration and invasion is evolved.

Tensile force-induced cytoskeletal remodeling: Mechanics before chemistry

Understanding cellular remodeling in response to mechanical stimuli is a critical step in elucidating mechanical activation of biochemical signaling pathways. Experimental evidence indicates that external stress-induced subcellular adaptation is accomplished through dynamic cytoskeletal reorganization. To study the interactions between subcellular structures involved in transducing mechanical signals, we combined experimental data and computational simulations to evaluate real-time mechanical adaptation of the actin cytoskeletal network. Actin cytoskeleton was imaged at the same time as an external tensile force was applied to live vascular smooth muscle cells using a fibronectin-functionalized atomic force microscope probe. Moreover, we performed computational simulations of active cytoskeletal networks under an external tensile force. The experimental data and simulation results suggest that mechanical structural adaptation occurs before chemical adaptation during filament bundle formation: actin filaments first align in the direction of the external force by initializing anisotropic filament orientations, then the chemical evolution of the network follows the anisotropic structures to further develop the bundle-like geometry. Our findings present an alternative two-step explanation for the formation of actin bundles due to mechanical stimulation and provide new insights into the mechanism of mechanotransduction.

Remodeling the cytoskeletal network in response to external force is key to cellular mechanotransduction. Despite much focus on cytoskeletal remodeling in recent years, a comprehensive understanding of actin remodeling in real-time in cells under mechanical stimuli is still lacking. We integrated tensile stress-induced 3D actin remodeling and 3D computational simulations of actin cytoskeleton to study how the actin cytoskeleton form bundles and how these bundles evolve over time upon external tensile stress. We found that actin network remodels through a two-step process in which rapid alignment of actin filaments is followed by slower actin bundling. Based on these results, we propose a “mechanics before chemistry” model of actin cytoskeleton remodeling under external tensile force.

The role of the Arp2/3 complex in shaping the dynamics and structures of branched actomyosin networks

Actomyosin networks give cells the ability to move and divide. These networks contract and expand while being driven by active energy-consuming processes such as motor protein walking and actin polymerization. Actin dynamics is also regulated by actin-binding proteins, such as the actin-related protein 2/3 (Arp2/3) complex. This complex generates branched filaments, thereby changing the overall organization of the network. In this work, the spatiotemporal patterns of dynamical actin assembly accompanying the branching-induced reorganization caused by Arp2/3 were studied using a computational model (mechanochemical dynamics of active networks [MEDYAN]); this model simulates actomyosin network dynamics as a result of chemical reactions whose rates are modulated by rapid mechanical equilibration. We show that branched actomyosin networks relax significantly more slowly than do unbranched networks. Also, branched networks undergo rare convulsive movements, "avalanches," that release strain in the network. These avalanches are associated with the more heterogeneous distribution of mechanically linked filaments displayed by branched networks. These far-from-equilibrium events arising from the marginal stability of growing actomyosin networks provide a possible mechanism of the "cytoquakes" recently seen in experiments.

Gibbs free energy change of a discrete chemical reaction event

In modeling the interior of cells by simulating a reaction-diffusion master equation over a grid of compartments, one employs the assumption that the copy numbers of various chemical species are small, discrete quantities. We show that, in this case, textbook expressions for the change in Gibbs free energy accompanying a chemical reaction or diffusion between adjacent compartments are inaccurate. We derive exact expressions for these free energy changes for the case of discrete copy numbers and show how these expressions reduce to traditional expressions under a series of successive approximations leveraging the relative sizes of the stoichiometric coefficients and the copy numbers of the solutes and solvent. Numerical results are presented to corroborate the claim that if the copy numbers are treated as discrete quantities, then only these more accurate expressions lead to correct behavior. Thus, the newly derived expressions are critical for correctly computing entropy production in mesoscopic simulations based on the reaction-diffusion master equation formalism.

Turnover versus treadmilling in actin network assembly and remodeling

Actin networks are highly dynamic cytoskeletal structures that continuously undergo structural remodeling. One prominent way to probe these processes is via Fluorescence Recovery After Photobleaching (FRAP), which can be used to estimate the rate of turnover for filamentous actin monomers. It is thought that head-to-tail treadmilling and de novo filament nucleation constitute two primary mechanisms underlying turnover kinetics. More generally, these self-assembly activities are responsible for many important cellular functions such as force generation, cellular shape dynamics, and cellular motility. In what relative proportions filament treadmilling and de novo filament nucleation contribute to actin network turnover is still not fully understood. We used an advanced stochastic reaction-diffusion model in three dimensions, MEDYAN, to study turnover dynamics of actin networks containing Arp2/3, formin and capping protein at experimentally meaningful length- and time-scales. Our results reveal that, most commonly, treadmilling of older filaments is the main contributor to actin network turnover. On the other hand, although turnover and treadmilling are often used interchangeably, we show clear instances where this assumption would not be justified, for example, finding that rapid turnover is accompanied by slow treadmilling in highly dendritic Arp2/3 networks.

Mechanical and kinetic factors drive sorting of F-actin cross-linkers on bundles

In cells, actin-binding proteins (ABPs) sort to different regions to establish F-actin networks with diverse functions, including filopodia used for cell migration and contractile rings required for cell division. Recent experimental work uncovered a competition-based mechanism that may facilitate spatial localization of ABPs: binding of a short cross-linker protein to 2 actin filaments promotes the binding of other short cross-linkers and inhibits the binding of longer cross-linkers (and vice versa). We hypothesize this sorting arises because F-actin is semiflexible and cannot bend over short distances. We develop a mathematical theory and lattice models encompassing the most important physical parameters for this process and use coarse-grained simulations with explicit cross-linkers to characterize and test our predictions. Our theory and data predict an explicit dependence of cross-linker separation on bundle polymerization rate. We perform experiments that confirm this dependence, but with an unexpected cross-over in dominance of one cross-linker at high growth rates to the other at slow growth rates, and we investigate the origin of this cross-over with further simulations. The nonequilibrium mechanism that we describe can allow cells to organize molecular material to drive biological processes, and our results can guide the choice and design of cross-linkers for engineered protein-based materials.

Remarkable structural transformations of actin bundles are driven by their initial polarity, motor activity, crosslinking, and filament treadmilling

Bundled actin structures play a key role in maintaining cellular shape, in aiding force transmission to and from extracellular substrates, and in affecting cellular motility. Recent studies have also brought to light new details on stress generation, force transmission and contractility of actin bundles. In this work, we are primarily interested in the question of what determines the stability of actin bundles and what network geometries do unstable bundles eventually transition to. To address this problem, we used the MEDYAN mechano-chemical force field, modeling several micron-long actin bundles in 3D, while accounting for a comprehensive set of chemical, mechanical and transport processes. We developed a hierarchical clustering algorithm for classification of the different long time scale morphologies in our study. Our main finding is that initially unipolar bundles are significantly more stable compared with an apolar initial configuration. Filaments within the latter bundles slide easily with respect to each other due to myosin activity, producing a loose network that can be subsequently severely distorted. At high myosin concentrations, a morphological transition to aster-like geometries was observed. We also investigated how actin treadmilling rates influence bundle dynamics, and found that enhanced treadmilling leads to network fragmentation and disintegration, while this process is opposed by myosin and crosslinking activities. Interestingly, treadmilling bundles with an initial apolar geometry eventually evolve to a whole gamut of network morphologies based on relative positions of filament ends, such as sarcomere-like organization. We found that apolar bundles show a remarkable sensitivity to environmental conditions, which may be important in enabling rapid cytoskeletal structural reorganization and adaptation in response to intracellular and extracellular cues.

Multi-resolution simulations of intracellular processes

This is an introduction to the Thematic Issue of Interface Focus containing papers by speakers of the Theo Murphy International Scientific Meeting on ‘Multi-resolution simulations of intracellular processes’