MEDYAN: Mechanochemical Simulations of Contraction and Polarity Alignment in Actomyosin Networks

Exploring effects of platelet contractility on the kinetics, thermodynamics, and mechanisms of fibrin clot contraction

Mechanisms of blood clot contraction - platelet-driven fibrin network remodeling, are not fully understood. We developed a detailed computational ClotDynaMo model of fibrin network with activated platelets, whose clot contraction rate for normal 450,000/µl human platelets depends on serum viscosity η, platelet filopodia length l, and weakly depends on filopodia traction force f and filopodia extension-retraction speed v. Final clot volume is independent of η, but depends on v, f and l. Analysis of ClotDynaMo output revealed a 2.24 TJ/mol clot contraction free energy change, with ~67% entropy and ~33% internal energy changes. The results illuminate the "optimal contraction principle" that maximizes volume change while minimizing energy cost. An 8-chain continuum model of polymer elasticity containing platelet forces, captures clot contractility as a function of platelet count, η and l. The ClotDynaMo and continuum models can be extended to include red blood cells, variable platelet properties, and mechanics of fibrin network.

Polarity sorting of actin filaments by motor-driven cargo transport

During the active transport of cellular cargo, forces generated by cargo-associated molecular motors propel the cargo along cytoskeletal tracks. However, the forces impact not only the cargo, but also the underlying cytoskeletal filaments. To better understand the interplay between cargo transport and the organization of cytoskeletal filaments, we employ coarse-grained computer simulations to study actin filaments interacting with cargo-anchored myosin motors in a confined domain. We show that cargo transport can lead to the segregation of filaments into domains of preferred filament polarity separated by clusters of aggregated cargoes. The formation of polarity-sorted filament domains is enhanced by larger numbers of cargoes, more motors per cargo, and longer filaments. Analysis of individual trajectories reveals dynamic and heterogeneous behavior, including locally stable aggregates of cargoes that undergo rapid coalescence into larger clusters when sufficiently close. Our results provide insight into the impact of motor-driven organelle transport on actin filaments, which is relevant both in cells and in synthetic environments.

Morphological and Electrical Properties of Proteinoid-Actin Networks

Proteinoids, or thermal proteins, are produced by heating amino acids. Proteinoids form hollow microspheres in water. The microspheres produce oscillation of electrical potential. Actin is a filament-forming protein responsible for communication, information processing and decision making in eukaryotic cells. We synthesize randomly organized networks of proteinoid microspheres spanned by actin filaments and study their morphology and electrical potential oscillatory dynamics. We analyze proteinoid-actin networks' responses to electrical stimulation. The signals come from logistic maps, the Lorenz attractor, the Rossler oscillator, and the FitzHugh-Nagumo system. We show how the networks attenuated the signals produced by these models. We demonstrate that emergent logical patterns derived from oscillatory behavior of proteinoid-actin networks show characteristics of Boolean logic gates, providing evidence for the computational ability to combine different components through architectural changes in the dynamic interface. Our experimental laboratory study paves a base for generation of proto-neural networks and implementation of neuromorphic computation with them.

How to build the virtual cell with artificial intelligence: Priorities and opportunities

Cells are essential to understanding health and disease, yet traditional models fall short of modeling and simulating their function and behavior. Advances in AI and omics offer groundbreaking opportunities to create an AI virtual cell (AIVC), a multi-scale, multi-modal large-neural-network-based model that can represent and simulate the behavior of molecules, cells, and tissues across diverse states. This Perspective provides a vision on their design and how collaborative efforts to build AIVCs will transform biological research by allowing high-fidelity simulations, accelerating discoveries, and guiding experimental studies, offering new opportunities for understanding cellular functions and fostering interdisciplinary collaborations in open science.

Morphometric analysis of actin networks

The organization of cytoskeletal elements is pivotal for coordinating intracellular transport in eukaryotic cells. Several quantitative measures based on image analysis have been proposed to characterize morphometric features of fluorescently labeled actin networks. While helpful in detecting differences in actin organization between treatments or genotypes, the accuracy of these measures could not be rigorously assessed due to a lack of ground-truth data to which they could be compared. To overcome this limitation, we utilized coarse-grained computer simulations of actin filaments and cross-linkers to generate synthetic actin networks with varying levels of bundling. We converted the simulated networks into pseudofluorescence images similar to images obtained using confocal microscopy. Using both published and novel analysis procedures, we extracted a series of morphometric parameters and benchmarked them against analogous measures based on the ground-truth actin configurations. Our analysis revealed a set of parameters that reliably reports on actin network density, orientation, ordering, and bundling. Application of these morphometric parameters to root epidermal cells of Arabidopsis thaliana revealed subtle changes in network organization between wild-type and mutant cells. This work provides robust measures that can be used to quantify features of actin networks and characterize changes in actin organization for different experimental conditions.

How to Build the Virtual Cell with Artificial Intelligence: Priorities and Opportunities

The cell is arguably the most fundamental unit of life and is central to understanding biology. Accurate modeling of cells is important for this understanding as well as for determining the root causes of disease. Recent advances in artificial intelligence (AI), combined with the ability to generate large-scale experimental data, present novel opportunities to model cells. Here we propose a vision of leveraging advances in AI to construct virtual cells, high-fidelity simulations of cells and cellular systems under different conditions that are directly learned from biological data across measurements and scales. We discuss desired capabilities of such AI Virtual Cells, including generating universal representations of biological entities across scales, and facilitating interpretable in silico experiments to predict and understand their behavior using Virtual Instruments. We further address the challenges, opportunities and requirements to realize this vision including data needs, evaluation strategies, and community standards and engagement to ensure biological accuracy and broad utility. We envision a future where AI Virtual Cells help identify new drug targets, predict cellular responses to perturbations, as well as scale hypothesis exploration. With open science collaborations across the biomedical ecosystem that includes academia, philanthropy, and the biopharma and AI industries, a comprehensive predictive understanding of cell mechanisms and interactions has come into reach.

Growth-induced collective bending and kinetic trapping of cytoskeletal filaments

Growth and turnover of actin filaments play a crucial role in the construction and maintenance of actin networks within cells. Actin filament growth occurs within limited space and finite subunit resources in the actin cortex. To understand how filament growth shapes the emergent architecture of actin networks, we developed a minimal agent-based model coupling filament mechanics and growth in a limiting subunit pool. We find that rapid filament growth induces kinetic trapping of highly bent actin filaments. Such collective bending patterns are long-lived, organized around nematic defects, and arise from competition between filament polymerization and bending elasticity. The stability of nematic defects and the extent of kinetic trapping are amplified by an increase in the abundance of the actin pool and by crosslinking the network. These findings suggest that kinetic trapping is a robust consequence of growth in crowded environments, providing a route to program shape memory in actin networks.

Biophysical Modeling of Synaptic Plasticity

Dendritic spines are small, bulbous compartments that function as postsynaptic sites and undergo intense biochemical and biophysical activity. The role of the myriad signaling pathways that are implicated in synaptic plasticity is well studied. A recent abundance of quantitative experimental data has made the events associated with synaptic plasticity amenable to quantitative biophysical modeling. Spines are also fascinating biophysical computational units because spine geometry, signal transduction, and mechanics work in a complex feedback loop to tune synaptic plasticity. In this sense, ideas from modeling cell motility can inspire us to develop multiscale approaches for predictive modeling of synaptic plasticity. In this article, we review the key steps in postsynaptic plasticity with a specific focus on the impact of spine geometry on signaling, cytoskeleton rearrangement, and membrane mechanics. We summarize the main experimental observations and highlight how theory and computation can aid our understanding of these complex processes.

Discrete network models of endothelial cells and their interactions with the substrate

Endothelial cell monolayers line the inner surfaces of blood and lymphatic vessels. They are continuously exposed to different mechanical loads, which may trigger mechanobiological signals and hence play a role in both physiological and pathological processes. Computer-based mechanical models of cells contribute to a better understanding of the relation between cell-scale loads and cues and the mechanical state of the hosting tissue. However, the confluency of the endothelial monolayer complicates these approaches since the intercellular cross-talk needs to be accounted for in addition to the cytoskeletal mechanics of the individual cells themselves. As a consequence, the computational approach must be able to efficiently model a large number of cells and their interaction. Here, we simulate cytoskeletal mechanics by means of molecular dynamics software, generally suitable to deal with large, locally interacting systems. Methods were developed to generate models of single cells and large monolayers with hundreds of cells. The single-cell model was considered for a comparison with experimental data. To this end, we simulated cell interactions with a continuous, deformable substrate, and computationally replicated multistep traction force microscopy experiments on endothelial cells. The results indicate that cell discrete network models are able to capture relevant features of the mechanical behaviour and are thus well-suited to investigate the mechanics of the large cytoskeletal network of individual cells and cell monolayers.

Membrane curvature catalyzes actin nucleation through nano-scale condensation of N-WASP-FBP17

Actin remodeling is spatiotemporally regulated by surface topographical cues on the membrane for signaling across diverse biological processes. Yet, the mechanism dynamic membrane curvature prompts quick actin cytoskeletal changes in signaling remain elusive. Leveraging the precision of nanolithography to control membrane curvature, we reconstructed catalytic reactions from the detection of nano-scale curvature by sensing molecules to the initiation of actin polymerization, which is challenging to study quantitatively in living cells. We show that this process occurs via topographical signal-triggered condensation and activation of the actin nucleation-promoting factor (NPF), Neuronal Wiskott-Aldrich Syndrome protein (N-WASP), which is orchestrated by curvature-sensing BAR-domain protein FBP17. Such N-WASP activation is fine-tuned by optimizing FBP17 to N-WASP stoichiometry over different curvature radii, allowing a curvature-guided macromolecular assembly pattern for polymerizing actin network locally. Our findings shed light on the intricate relationship between changes in curvature and actin remodeling via spatiotemporal regulation of NPF/BAR complex condensation.

Adaptive nonequilibrium design of actin-based metamaterials: Fundamental and practical limits of control

The adaptive and surprising emergent properties of biological materials self-assembled in far-from-equilibrium environments serve as an inspiration for efforts to design nanomaterials. In particular, controlling the conditions of self-assembly can modulate material properties, but there is no systematic understanding of either how to parameterize external control or how controllable a given material can be. Here, we demonstrate that branched actin networks can be encoded with metamaterial properties by dynamically controlling the applied force under which they grow and that the protocols can be selected using multi-task reinforcement learning. These actin networks have tunable responses over a large dynamic range depending on the chosen external protocol, providing a pathway to encoding "memory" within these structures. Interestingly, we obtain a bound that relates the dissipation rate and the rate of "encoding" that gives insight into the constraints on control-both physical and information theoretical. Taken together, these results emphasize the utility and necessity of nonequilibrium control for designing self-assembled nanostructures.

Growth-induced collective bending and kinetic trapping of cytoskeletal filaments

Growth and turnover of actin filaments play a crucial role in the construction and maintenance of actin networks within cells. Actin filament growth occurs within limited space and finite subunit resources in the actin cortex. To understand how filament growth shapes the emergent architecture of actin networks, we developed a minimal agent-based model coupling filament mechanics and growth in a limiting subunit pool. We find that rapid filament growth induces kinetic trapping of highly bent actin filaments. Such collective bending patterns are long-lived, organized around nematic defects, and arises from competition between filament polymerization and bending elasticity. The stability of nematic defects and the extent of kinetic trapping are amplified by an increase in the abundance of the actin pool and by crosslinking the network. These findings suggest that kinetic trapping is a robust consequence of growth in crowded environments, providing a route to program shape memory in actin networks.

SimService: a lightweight library for building simulation services in Python

SUMMARY

Integrative biological modeling requires software infrastructure to launch, interconnect, and execute simulation software components without loss of functionality. SimService is a software library that enables deploying simulations in integrated applications as memory-isolated services with interactive proxy objects in the Python programming language. SimService supports customizing the interface of proxies so that simulation developers and users alike can tailor generated simulation instances according to model, method, and integrated application.

AVAILABILITY AND IMPLEMENTATION

SimService is written in Python, is freely available on GitHub under the MIT license at https://github.com/tjsego/simservice, and is available for download via the Python Package Index (package name "simservice") and conda (package name "simservice" on the conda-forge channel).

Building the next generation of virtual cells to understand cellular biology

Cell science has made significant progress by focusing on understanding individual cellular processes through reductionist approaches. However, the sheer volume of knowledge collected presents challenges in integrating this information across different scales of space and time to comprehend cellular behaviors, as well as making the data and methods more accessible for the community to tackle complex biological questions. This perspective proposes the creation of next-generation virtual cells, which are dynamic 3D models that integrate information from diverse sources, including simulations, biophysical models, image-based models, and evidence-based knowledge graphs. These virtual cells would provide statistically accurate and holistic views of real cells, bridging the gap between theoretical concepts and experimental data, and facilitating productive new collaborations among researchers across related fields.

Dynamics of chromosome organization in a minimal bacterial cell

Computational models of cells cannot be considered complete unless they include the most fundamental process of life, the replication and inheritance of genetic material. By creating a computational framework to model systems of replicating bacterial chromosomes as polymers at 10 bp resolution with Brownian dynamics, we investigate changes in chromosome organization during replication and extend the applicability of an existing whole-cell model (WCM) for a genetically minimal bacterium, JCVI-syn3A, to the entire cell-cycle. To achieve cell-scale chromosome structures that are realistic, we model the chromosome as a self-avoiding homopolymer with bending and torsional stiffnesses that capture the essential mechanical properties of dsDNA in Syn3A. In addition, the conformations of the circular DNA must avoid overlapping with ribosomes identitied in cryo-electron tomograms. While Syn3A lacks the complex regulatory systems known to orchestrate chromosome segregation in other bacteria, its minimized genome retains essential loop-extruding structural maintenance of chromosomes (SMC) protein complexes (SMC-scpAB) and topoisomerases. Through implementing the effects of these proteins in our simulations of replicating chromosomes, we find that they alone are sufficient for simultaneous chromosome segregation across all generations within nested theta structures. This supports previous studies suggesting loop-extrusion serves as a near-universal mechanism for chromosome organization within bacterial and eukaryotic cells. Furthermore, we analyze ribosome diffusion under the influence of the chromosome and calculate in silico chromosome contact maps that capture inter-daughter interactions. Finally, we present a methodology to map the polymer model of the chromosome to a Martini coarse-grained representation to prepare molecular dynamics models of entire Syn3A cells, which serves as an ultimate means of validation for cell states predicted by the WCM.

Range and strength of mechanical interactions of force dipoles in elastic fiber networks

Mechanical forces generated by myosin II molecular motors drive diverse cellular processes, most notably shape change, division and locomotion. These forces may be transmitted over long range through the cytoskeletal medium - a disordered, viscoelastic network of biopolymers. The resulting cell size scale force chains can in principle mediate mechanical interactions between distant actomyosin units, leading to self-organized structural order in the cell cytoskeleton. Inspired by such force transmission through elastic structures in the cytoskeleton, we consider a percolated fiber lattice network, where fibers are represented as linear elastic elements that can both bend and stretch, and the contractile activity of myosin motors is represented by force dipoles. Then, by using a variety of metrics, we show how two such contractile force dipoles interact with each other through their mutual mechanical deformations of the elastic fiber network. As a prelude to two-dipole interactions, we quantify how forces propagate through the network from a single anisotropic force dipole by analyzing clusters of nodes connected by highly strained bonds, as well as through the decay rate of strain energy with distance from a force dipole. We show that predominant fiber bending screens out force propagation, resulting in reduced and strongly network configuration-dependent dipole interactions. On the other hand, stretching-dominated networks support longer-ranged inter-dipole interactions that recapitulate the predictions of linear elasticity theory. By characterizing the differences between tensile and compressive force propagation in the fiber network, we show how inter-dipole interaction depends on the dipoles' mutual separation and orientation. The resulting elastic interaction energy may mediate a force between multiple distant dipoles, leading to their self-organization into ordered configurations. This provides a potential pathway for active mechanical force-driven structural order in elastic biopolymer networks.

Computational model of integrin adhesion elongation under an actin fiber

Cells create physical connections with the extracellular environment through adhesions. Nascent adhesions form at the leading edge of migrating cells and either undergo cycles of disassembly and reassembly, or elongate and stabilize at the end of actin fibers. How adhesions assemble has been addressed in several studies, but the exact role of actin fibers in the elongation and stabilization of nascent adhesions remains largely elusive. To address this question, here we extended our computational model of adhesion assembly by incorporating an actin fiber that locally promotes integrin activation. The model revealed that an actin fiber promotes adhesion stabilization and elongation. Actomyosin contractility from the fiber also promotes adhesion stabilization and elongation, by strengthening integrin-ligand interactions, but only up to a force threshold. Above this force threshold, most integrin-ligand bonds fail, and the adhesion disassembles. In the absence of contraction, actin fibers still support adhesions stabilization. Collectively, our results provide a picture in which myosin activity is dispensable for adhesion stabilization and elongation under an actin fiber, offering a framework for interpreting several previous experimental observations.

Myosin and [Formula: see text]-actinin regulation of stress fiber contractility under tensile stress

Stress fibers are actomyosin bundles that regulate cellular mechanosensation and force transduction. Interacting with the extracellular matrix through focal adhesion complexes, stress fibers are highly dynamic structures regulated by myosin motors and crosslinking proteins. Under external mechanical stimuli such as tensile forces, the stress fiber remodels its architecture to adapt to external cues, displaying properties of viscoelastic materials. How the structural remodeling of stress fibers is related to the generation of contractile force is not well understood. In this work, we simulate mechanochemical dynamics and force generation of stress fibers using the molecular simulation platform MEDYAN. We model stress fiber as two connecting bipolar bundles attached at the ends to focal adhesion complexes. The simulated stress fibers generate contractile force that is regulated by myosin motors and [Formula: see text]-actinin crosslinkers. We find that stress fibers enhance contractility by reducing the distance between actin filaments to increase crosslinker binding, and this structural remodeling ability depends on the crosslinker turnover rate. Under tensile pulling force, the stress fiber shows an instantaneous increase of the contractile forces followed by a slow relaxation into a new steady state. While the new steady state contractility after pulling depends only on the overlap between actin bundles, the short-term contractility enhancement is sensitive to the tensile pulling distance. We further show that this mechanical response is also sensitive to the crosslinker turnover rate. Our results provide new insights into the stress fiber mechanics that have significant implications for understanding cellular adaptation to mechanical signaling.

Organization and dynamics of cross-linked actin filaments in confined environments

The organization of the actin cytoskeleton is impacted by the interplay between physical confinement, features of cross-linking proteins, and deformations of semiflexible actin filaments. Some cross-linking proteins preferentially bind filaments in parallel, although others bind more indiscriminately. However, a quantitative understanding of how the mode of binding influences the assembly of actin networks in confined environments is lacking. Here we employ coarse-grained computer simulations to study the dynamics and organization of semiflexible actin filaments in confined regions upon the addition of cross-linkers. We characterize how the emergent behavior is influenced by the system shape, the number and type of cross-linking proteins, and the length of filaments. Structures include isolated clusters of filaments, highly connected filament bundles, and networks of interconnected bundles and loops. Elongation of one dimension of the system promotes the formation of long bundles that align with the elongated axis. Dynamics are governed by rapid cross-linking into aggregates, followed by a slower change in their shape and connectivity. Cross-linking decreases the average bending energy of short or sparsely connected filaments by suppressing shape fluctuations. However, it increases the average bending energy in highly connected networks because filament bundles become deformed, and small numbers of filaments exhibit long-lived, highly unfavorable configurations. Indiscriminate cross-linking promotes the formation of high-energy configurations due to the increased likelihood of unfavorable, difficult-to-relax configurations at early times. Taken together, this work demonstrates physical mechanisms by which cross-linker binding and physical confinement impact the emergent behavior of actin networks, which is relevant both in cells and in synthetic environments.

Characterizing emerging features in cell dynamics using topological data analysis methods

Filament-motor interactions inside cells play essential roles in many developmental as well as other biological processes. For instance, actin-myosin interactions drive the emergence or closure of ring channel structures during wound healing or dorsal closure. These dynamic protein interactions and the resulting protein organization lead to rich time-series data generated by using fluorescence imaging experiments or by simulating realistic stochastic models. We propose methods based on topological data analysis to track topological features through time in cell biology data consisting of point clouds or binary images. The framework proposed here is based on computing the persistent homology of the data at each time point and on connecting topological features through time using established distance metrics between topological summaries. The methods retain aspects of monomer identity when analyzing significant features in filamentous structure data, and capture the overall closure dynamics when assessing the organization of multiple ring structures through time. Using applications of these techniques to experimental data, we show that the proposed methods can describe features of the emergent dynamics and quantitatively distinguish between control and perturbation experiments.

The dawn of interoperating spatial models in cell biology

Spatial simulations are becoming an increasingly ubiquitous component in the cycle of discovery, experimentation, and communication across the sciences. In cell biology, many researchers share a vision of developing multiscale models that recapitulate observable behaviors spanning from atoms to cells to tissues. For this dream to become a reality, however, simulation technologies must provide a means for integration and interoperability as they advance. Already, the field has developed numerous methods that span scales of length, time, and complexity to create an extensive body of effective simulation approaches, and although these approaches rarely interoperate, they collectively cover a large spectrum of knowledge that future models may handle in a more unified manner. Here, we discuss the importance of making the data, workflows, and outputs of spatial simulations shareable and interoperable; and how democratization could encourage diverse biologists to participate more easily in developing models to advance our understanding of biological systems.

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.

Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics

Many emergent, non-fundamental models of complex systems can be described naturally by the temporal evolution of spatial structures with some nontrivial discretized topology, such as a graph with suitable parameter vectors labeling its vertices. For example, the cytoskeleton of a single cell, such as the cortical microtubule network in a plant cell or the actin filaments in a synapse, comprises many interconnected polymers whose topology is naturally graph-like and dynamic. The same can be said for cells connected dynamically in a developing tissue. There is a mathematical framework suitable for expressing such emergent dynamics, "stochastic parameterized graph grammars," composed of a collection of the graph- and parameter-altering rules, each of which has a time-evolution operator that suitably moves probability. These rule-level operators form an operator algebra, much like particle creation/annihilation operators or Lie group generators. Here, we present an explicit and constructive calculation, in terms of elementary basis operators and standard component notation, of what turns out to be a general combinatorial expression for the operator algebra that reduces products and, therefore, commutators of graph grammar rule operators to equivalent integer-weighted sums of such operators. We show how these results extend to "dynamical graph grammars," which include rules that bear local differential equation dynamics for some continuous-valued parameters. Commutators of such time-evolution operators have analytic uses, including deriving efficient simulation algorithms and approximations and estimating their errors. The resulting formalism is complementary to spatial models in the form of partial differential equations or stochastic reaction-diffusion processes. We discuss the potential application of this framework to the remodeling dynamics of the microtubule cytoskeleton in cortical microtubule networks relevant to plant development and of the actin cytoskeleton in, for example, a growing or shrinking synaptic spine head. Both cytoskeletal systems underlie biological morphodynamics.

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.

F-actin bending facilitates net actomyosin contraction By inhibiting expansion with plus-end-located myosin motors

Contraction of actomyosin networks underpins important cellular processes including motility and division. The mechanical origin of actomyosin contraction is not fully-understood. We investigate whether contraction arises on the scale of individual filaments, without needing to invoke network-scale interactions. We derive discrete force-balance and continuum partial differential equations for two symmetric, semi-flexible actin filaments with an attached myosin motor. Assuming the system exists within a homogeneous background material, our method enables computation of the stress tensor, providing a measure of contractility. After deriving the model, we use a combination of asymptotic analysis and numerical solutions to show how F-actin bending facilitates contraction on the scale of two filaments. Rigid filaments exhibit polarity-reversal symmetry as the motor travels from the minus to plus-ends, such that contractile and expansive components cancel. Filament bending induces a geometric asymmetry that brings the filaments closer to parallel as a myosin motor approaches their plus-ends, decreasing the effective spring force opposing motor motion. The reduced spring force enables the motor to move faster close to filament plus-ends, which reduces expansive stress and gives rise to net contraction. Bending-induced geometric asymmetry provides both new understanding of actomyosin contraction mechanics, and a hypothesis that can be tested in experiments.

CellDynaMo–stochastic reaction-diffusion-dynamics model: Application to search-and-capture process of mitotic spindle assembly

We introduce a Stochastic Reaction-Diffusion-Dynamics Model (SRDDM) for simulations of cellular mechanochemical processes with high spatial and temporal resolution. The SRDDM is mapped into the CellDynaMo package, which couples the spatially inhomogeneous reaction-diffusion master equation to account for biochemical reactions and molecular transport within the Langevin Dynamics (LD) framework to describe dynamic mechanical processes. This computational infrastructure allows the simulation of hours of molecular machine dynamics in reasonable wall-clock time. We apply SRDDM to test performance of the Search-and-Capture of mitotic spindle assembly by simulating, in three spatial dimensions, dynamic instability of elastic microtubules anchored in two centrosomes, movement and deformations of geometrically realistic centromeres with flexible kinetochores and chromosome arms. Furthermore, the SRDDM describes the mechanics and kinetics of Ndc80 linkers mediating transient attachments of microtubules to the chromosomal kinetochores. The rates of these attachments and detachments depend upon phosphorylation states of the Ndc80 linkers, which are regulated in the model by explicitly accounting for the reactions of Aurora A and B kinase enzymes undergoing restricted diffusion. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of the chromosome connections, that adding chromosome arms to kinetochores improve the accuracy by slowing down chromosome movements, that Aurora A and kinetochore deformations have a small positive effect on the attachment accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy of spindle assembly.

The CellDynaMo package models, in 3D, any cellular subsystem where sufficient detail of the macromolecular players and the kinetics of relevant reactions are available. The package is based on the Stochastic Reaction-Diffusion-Dynamics model that combines the stochastic description of chemical kinetics, Brownian diffusion-based description of molecular transport, and Langevin dynamics-based representation of mechanical processes most pertinent to the system. We apply the model to test the Search-and-Capture mechanism of mitotic spindle assembly. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of chromosome connections, that chromosome arms improve the attachment accuracy by slowing down chromosome movements, that Aurora A kinase and kinetochore deformations have small positive effects on the accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy.

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.

Simulated actin reorganization mediated by motor proteins

Cortical actin networks are highly dynamic and play critical roles in shaping the mechanical properties of cells. The actin cytoskeleton undergoes significant reorganization in many different contexts, including during directed cell migration and over the course of the cell cycle, when cortical actin can transition between different configurations such as open patched meshworks, homogeneous distributions, and aligned bundles. Several types of myosin motor proteins, characterized by different kinetic parameters, have been involved in this reorganization of actin filaments. Given the limitations in studying the interactions of actin with myosin in vivo, we propose stochastic agent-based models and develop a set of data analysis measures to assess how myosin motor proteins mediate various actin organizations. In particular, we identify individual motor parameters, such as motor binding rate and step size, that generate actin networks with different levels of contractility and different patterns of myosin motor localization, which have previously been observed experimentally. In simulations where two motor populations with distinct kinetic parameters interact with the same actin network, we find that motors may act in a complementary way, by tuning the actin network organization, or in an antagonistic way, where one motor emerges as dominant. This modeling and data analysis framework also uncovers parameter regimes where spatial segregation between motor populations is achieved. By allowing for changes in kinetic rates during the actin-myosin dynamic simulations, our work suggests that certain actin-myosin organizations may require additional regulation beyond mediation by motor proteins in order to reconfigure the cytoskeleton network on experimentally-observed timescales.

Cell shape is dictated by a scaffolding network called the cytoskeleton. Actin filaments, a main component of the cytoskeleton, are found predominantly at the periphery of the cell, where they organize into different patterns in response to various stimuli, such as progression through the cell cycle. The actin filament reorganizations are mediated by motor proteins from the myosin superfamily. Using a realistic stochastic model that simulates actin filament and motor protein dynamics and interactions, we systematically vary motor protein kinetics and investigate their effect on actin filament organization. Using novel measures of spatial organization, we quantify conditions under which motor proteins, either alone or in combination, can produce the different actin filament organizations observed in vitro and in vivo. These results yield new insights into the role of motor proteins, as well as into how multiple types of motors can work collectively to produce specific actomyosin network patterns.

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.

A computational modeling of invadopodia protrusion into an extracellular matrix fiber network

Invadopodia are dynamic actin-rich membrane protrusions that have been implicated in cancer cell invasion and metastasis. In addition, invasiveness of cancer cells is strongly correlated with invadopodia formation, which are observed during extravasation and colonization of metastatic cancer cells at secondary sites. However, quantitative understanding of the interaction of invadopodia with extracellular matrix (ECM) is lacking, and how invadopodia protrusion speed is associated with the frequency of protrusion-retraction cycles remains unknown. Here, we present a computational framework for the characterization of invadopodia protrusions which allows two way interactions between intracellular branched actin network and ECM fibers network. We have applied this approach to predicting the invasiveness of cancer cells by computationally knocking out actin-crosslinking molecules, such as α-actinin, filamin and fascin. The resulting simulations reveal distinct invadopodia dynamics with cycles of protrusion and retraction. Specifically, we found that (1) increasing accumulation of MT1-MMP at tips of invadopodia as the duration of protrusive phase is increased, and (2) the movement of nucleus toward the leading edge of the cell becomes unstable as duration of the retractile phase (or myosin turnover time) is longer than 1 min.

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.

Simulations of dynamically cross-linked actin networks: Morphology, rheology, and hydrodynamic interactions

Cross-linked actin networks are the primary component of the cell cytoskeleton and have been the subject of numerous experimental and modeling studies. While these studies have demonstrated that the networks are viscoelastic materials, evolving from elastic solids on short timescales to viscous fluids on long ones, questions remain about the duration of each asymptotic regime, the role of the surrounding fluid, and the behavior of the networks on intermediate timescales. Here we perform detailed simulations of passively cross-linked non-Brownian actin networks to quantify the principal timescales involved in the elastoviscous behavior, study the role of nonlocal hydrodynamic interactions, and parameterize continuum models from discrete stochastic simulations. To do this, we extend our recent computational framework for semiflexible filament suspensions, which is based on nonlocal slender body theory, to actin networks with dynamic cross linkers and finite filament lifetime. We introduce a model where the cross linkers are elastic springs with sticky ends stochastically binding to and unbinding from the elastic filaments, which randomly turn over at a characteristic rate. We show that, depending on the parameters, the network evolves to a steady state morphology that is either an isotropic actin mesh or a mesh with embedded actin bundles. For different degrees of bundling, we numerically apply small-amplitude oscillatory shear deformation to extract three timescales from networks of hundreds of filaments and cross linkers. We analyze the dependence of these timescales, which range from the order of hundredths of a second to the actin turnover time of several seconds, on the dynamic nature of the links, solvent viscosity, and filament bending stiffness. We show that the network is mostly elastic on the short time scale, with the elasticity coming mainly from the cross links, and viscous on the long time scale, with the effective viscosity originating primarily from stretching and breaking of the cross links. We show that the influence of nonlocal hydrodynamic interactions depends on the network morphology: for homogeneous meshworks, nonlocal hydrodynamics gives only a small correction to the viscous behavior, but for bundled networks it both hinders the formation of bundles and significantly lowers the resistance to shear once bundles are formed. We use our results to construct three-timescale generalized Maxwell models of the networks.

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.

Membrane-MEDYAN: Simulating Deformable Vesicles Containing Complex Cytoskeletal Networks

The plasma membrane defines the shape of the cell and plays an indispensable role in bridging intra- and extracellular environments. Mechanochemical interactions between plasma membrane and cytoskeleton are vital for cell biomechanics and mechanosensing. A computational model that comprehensively captures the complex, cell-scale cytoskeleton-membrane dynamics is still lacking. In this work, we introduce a triangulated membrane model that accounts for the membrane's elastic properties, as well as for membrane-filament steric interactions. The corresponding force-field was incorporated into the active biological matter simulation platform, MEDYAN ("mechanochemical dynamics of active networks"). Simulations using the new model shed light on how actin filament bundling affects generation of tubular membrane protrusions. In particular, we used membrane-MEDYAN simulations to investigate protrusion initiation and dynamics while varying geometries of filament bundles, membrane rigidities and local G-Actin concentrations. We found that the bundles' protrusion propensities sensitively depend on the synergy between bundle thickness and inclination angle at which the bundle approaches the membrane. The new model paves the way for simulations of biological systems involving intricate membrane-cytoskeleton interactions, such as those occurring at the leading edge and the cortex, eventually helping to uncover the fundamental principles underlying the active matter organization in the vicinity of the membrane.

Modeling spatiotemporally varying protein-protein interactions in CyLaKS, the Cytoskeleton Lattice-based Kinetic Simulator

Interaction of cytoskeletal filaments, motor proteins, and crosslinking proteins drives important cellular processes such as cell division and cell movement. Cytoskeletal networks also exhibit nonequilibrium self-assembly in reconstituted systems. An emerging problem in cytoskeletal modeling and simulation is spatiotemporal alteration of the dynamics of filaments, motors, and associated proteins. This can occur due to motor crowding, obstacles along the filament, motor interactions and direction switching, and changes, defects, or heterogeneity in the filament binding lattice. How such spatiotemporally varying cytoskeletal filaments and motor interactions affect their collective properties is not fully understood. We developed the Cytoskeleton Lattice-based Kinetic Simulator (CyLaKS) to investigate such problems. The simulation model builds on previous work by incorporating motor mechanochemistry into a simulation with many interacting motors and/or associated proteins on a discretized lattice. CyLaKS also includes detailed balance in binding kinetics, movement, and lattice heterogeneity. The simulation framework is flexible and extensible for future modeling work and is available on GitHub for others to freely use or build upon. Here we illustrate the use of CyLaKS to study long-range motor interactions, microtubule lattice heterogeneity, motion of a heterodimeric motor, and how changing crosslinker number affects filament separation.

Protein friction and filament bending facilitate contraction of disordered actomyosin networks

We use mathematical modeling and computation to investigate how protein friction facilitates contraction of disordered actomyosin networks. We simulate two-dimensional networks using an agent-based model, consisting of a system of force-balance equations for myosin motor proteins and semiflexible actin filaments. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use repeated simulations of disordered networks to confirm that both protein friction and actin filament bending are required for contraction. We then use simulations of elementary two-filament systems to show that filament bending flexibility can facilitate contraction on the microscopic scale. Finally, we show that actin filament turnover is necessary to sustain contraction and prevent filament aggregation. Simulations with and without turnover also exhibit contractile pulses. However, these pulses are aperiodic, suggesting that periodic pulsation can only arise because of additional regulatory mechanisms or more complex mechanical behavior.

Different translation dynamics of β- and γ-actin regulates cell migration

β- and γ-cytoplasmic actins are ubiquitously expressed in every cell type and are nearly identical at the amino acid level but play vastly different roles in vivo. Their essential roles in embryogenesis and mesenchymal cell migration critically depend on the nucleotide sequences of their genes, rather than their amino acid sequences; however, it is unclear which gene elements underlie this effect. Here we address the specific role of the coding sequence in β- and γ-cytoplasmic actins’ intracellular functions, using stable polyclonal populations of immortalized mouse embryonic fibroblasts with exogenously expressed actin isoforms and their ‘codon-switched’ variants. When targeted to the cell periphery using β-actin 3′UTR; β-actin and γ-actin have differential effects on cell migration. These effects directly depend on the coding sequence. Single-molecule measurements of actin isoform translation, combined with fluorescence recovery after photobleaching, demonstrate a pronounced difference in β- and γ-actins’ translation elongation rates in cells, leading to changes in their dynamics at focal adhesions, impairments in actin bundle formation, and reduced cell anchoring to the substrate during migration. Our results demonstrate that coding sequence-mediated differences in actin translation play a key role in cell migration.

Most mammalian cells make both β- and γ-actin, two proteins which shape the cell’s internal skeleton and its ability to migrate. The molecules share over 99% of their sequence, yet they play distinct roles. In fact, deleting the β-actin gene in mice causes death in the womb, while the animals can survive with comparatively milder issues without their γ-actin gene. How two similar proteins can have such different biological roles is a long-standing mystery.

A closer look could hold some clues: β- and γ-actin may contain the same blocks (or amino acids), but the genetic sequences that encode these proteins differ by about 13%. This is because different units of genetic information – known as synonymous codons – can encode the same amino acid. These ‘silent substitutions’ have no effect on the sequence of the proteins, yet a cell reads synonymous codons (and therefore produces proteins) at different speeds.

To find out the impact of silent substitutions, Vedula et al. swapped the codons for the two proteins, forcing mouse cells to produce β-actin using γ-actin codons, and vice versa. Cells with non-manipulated γ-actin and those with β-actin made using γ-actin codons could move much faster than cells with β-actin. This suggested that silent substitutions were indeed affecting the role of the protein.

Vedula et al. found that cells read γ-codons – and therefore made γ-actin – much more slowly than β-codons: this also affected how quickly the protein could be dispatched where it was needed in the cell. Slower production meant that bundles of γ-actin were shorter, which allowed cells to move faster by providing a weaker anchoring system. Overall, this work provides new links between silent substitutions and protein behavior, a relatively new research area which is likely to shed light on other protein families.

Comparison of explicit and mean-field models of cytoskeletal filaments with crosslinking motors

In cells, cytoskeletal filament networks are responsible for cell movement, growth, and division. Filaments in the cytoskeleton are driven and organized by crosslinking molecular motors. In reconstituted cytoskeletal systems, motor activity is responsible for far-from-equilibrium phenomena such as active stress, self-organized flow, and spontaneous nematic defect generation. How microscopic interactions between motors and filaments lead to larger-scale dynamics remains incompletely understood. To build from motor-filament interactions to predict bulk behavior of cytoskeletal systems, more computationally efficient techniques for modeling motor-filament interactions are needed. Here, we derive a coarse-graining hierarchy of explicit and continuum models for crosslinking motors that bind to and walk on filament pairs. We compare the steady-state motor distribution and motor-induced filament motion for the different models and analyze their computational cost. All three models agree well in the limit of fast motor binding kinetics. Evolving a truncated moment expansion of motor density speeds the computation by [Formula: see text]-[Formula: see text] compared to the explicit or continuous-density simulations, suggesting an approach for more efficient simulation of large networks. These tools facilitate further study of motor-filament networks on micrometer to millimeter length scales.

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.

Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks

In developmental biology as well as in other biological systems, emerging structure and organization can be captured using time-series data of protein locations. In analyzing this time-dependent data, it is a common challenge not only to determine whether topological features emerge, but also to identify the timing of their formation. For instance, in most cells, actin filaments interact with myosin motor proteins and organize into polymer networks and higher-order structures. Ring channels are examples of such structures that maintain constant diameters over time and play key roles in processes such as cell division, development, and wound healing. Given the limitations in studying interactions of actin with myosin in vivo, we generate time-series data of protein polymer interactions in cells using complex agent-based models. Since the data has a filamentous structure, we propose sampling along the actin filaments and analyzing the topological structure of the resulting point cloud at each time. Building on existing tools from persistent homology, we develop a topological data analysis (TDA) method that assesses effective ring generation in this dynamic data. This method connects topological features through time in a path that corresponds to emergence of organization in the data. In this work, we also propose methods for assessing whether the topological features of interest are significant and thus whether they contribute to the formation of an emerging hole (ring channel) in the simulated protein interactions. In particular, we use the MEDYAN simulation platform to show that this technique can distinguish between the actin cytoskeleton organization resulting from distinct motor protein binding parameters.

Towards understanding the messengers of extracellular space: Computational models of outside-in integrin reaction networks

The interactions between cells and their extracellular matrix (ECM) are critically important for homeostatic control of cell growth, proliferation, differentiation and apoptosis. Transmembrane integrin molecules facilitate the communication between ECM and the cell. Since the characterization of integrins in the late 1980s, there has been great advancement in understanding the function of integrins at different subcellular levels. However, the versatility in molecular pathways integrins are involved in, the high diversity in their interaction partners both outside and inside the cell as well as on the cell membrane and the short lifetime of events happening at the cell-ECM interface make it difficult to elucidate all the details regarding integrin function experimentally. To overcome the experimental challenges and advance the understanding of integrin biology, computational modeling tools have been used extensively. In this review, we summarize the computational models of integrin signaling while we explain the function of integrins at three main subcellular levels (outside the cell, cell membrane, cytosol). We also discuss how these computational modeling efforts can be helpful in other disciplines such as biomaterial design. As such, this review is a didactic modeling summary for biomaterial researchers interested in complementing their experimental work with computational tools or for seasoned computational scientists that would like to advance current in silico integrin models.

Insights from graph theory on the morphologies of actomyosin networks with multilinkers

Quantifying the influence of microscopic details on the dynamics of development of the overall structure of a filamentous network is important in a number of biologically relevant contexts, but it is not obvious what order parameters can be used to adequately describe this complex process. In this paper we investigated the role of multivalent actin-binding proteins (ABPs) in reorganizing actin filaments into higher-order complex networks via a computer model of semiflexible filaments. We characterize the importance of local connectivity among actin filaments, as well as the global features of actomyosin networks. We first map the networks into local graph representations and then, using principles from network-theory order parameters, combine properties from these representations to gain insight into the heterogeneous morphologies of actomyosin networks at a global level. We find that ABPs with a valency greater than 2 promote filament bundles and large filament clusters to a much greater extent than bivalent multilinkers. We also show that active myosinlike motor proteins promote the formation of dendritic branches from a stalk of actin bundles. Our work motivates future studies to embrace network theory as a tool to characterize complex morphologies of actomyosin detected by experiments, leading to a quantitative understanding of the role of ABPs in manipulating the self-assembly of actin filaments into unique architectures that underlie the structural scaffold of a cell relating to its mobility and shape.

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.

The Actin Cytoskeleton as an Active Adaptive Material

Actin is the main protein used by biological cells to adapt their structure and mechanics to their needs. Cellular adaptation is made possible by molecular processes that strongly depend on mechanics. The actin cytoskeleton is also an active material that continuously consumes energy. This allows for dynamical processes that are possible only out of equilibrium and opens up the possibility for multiple layers of control that have evolved around this single protein.Here we discuss the actin cytoskeleton from the viewpoint of physics as an active adaptive material that can build structures superior to man-made soft matter systems. Not only can actin be used to build different network architectures on demand and in an adaptive manner, but it also exhibits the dynamical properties of feedback systems, like excitability, bistability, or oscillations. Therefore, it is a prime example of how biology couples physical structure and information flow and a role model for biology-inspired metamaterials.

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.

Dynamic motions of molecular motors in the actin cytoskeleton

During intracellular transport, cellular cargos, such as organelles, vesicles, and proteins, are transported within cells. Intracellular transport plays an important role in diverse cellular functions. Molecular motors walking on the cytoskeleton facilitate active intracellular transport, which is more efficient than diffusion-based passive transport. Active transport driven by kinesin and dynein walking on microtubules has been studied well during recent decades. However, mechanisms of active transport occurring in disorganized actin networks via myosin motors remain elusive. To provide physiologically relevant insights, we probed motions of myosin motors in actin networks under various conditions using our well-established computational model that rigorously accounts for the mechanical and dynamical behaviors of the actin cytoskeleton. We demonstrated that myosin motions can be confined due to three different reasons in the absence of F-actin turnover. We verified mechanisms of motor stalling using in vitro reconstituted actomyosin networks. We also found that with F-actin turnover, motors consistently move for a long time without significant confinement. Our study sheds light on the importance of F-actin turnover for effective active transport in the actin cytoskeleton.