An open model of actin dendritic nucleation

Multi-condensate state as a functional strategy to optimize the cell signaling output

The existence of multiple biomolecular condensates inside living cells is a peculiar phenomenon not compatible with the predictions of equilibrium statistical mechanics. In this work, we address the problem of multiple condensates state (MCS) from a functional perspective. We combine Langevin dynamics, reaction-diffusion simulation, and dynamical systems theory to demonstrate that MCS can indeed be a function optimization strategy. Using Arp2/3 mediated actin nucleation pathway as an example, we show that actin polymerization is maximum at an optimal number of condensates. For a fixed amount of Arp2/3, MCS produces a greater response compared to its single condensate counterpart. Our analysis reveals the functional significance of the condensate size distribution which can be mapped to the recent experimental findings. Given the spatial heterogeneity within condensates and non-linear nature of intracellular networks, we envision MCS to be a generic functional solution, so that structures of network motifs may have evolved to accommodate such configurations.

Multi-condensate state as a functional strategy to optimize the cell signaling output

The existence of multiple biomolecular condensates inside living cells is a peculiar phenomenon not compatible with the predictions of equilibrium statistical mechanics. In this work, we address the problem of multiple condensates state (MCS) from a functional perspective. We combined Langevin dynamics, reaction-diffusion simulation, and dynamical systems theory to demonstrate that MCS can indeed be a function optimization strategy. Using Arp2/3 mediated actin nucleation pathway as an example, we show that actin polymerization is maximum at an optimal number of condensates. For a fixed amount of Arp2/3, MCS produces a greater response compared to its single condensate counterpart. Our analysis reveals the functional significance of the condensate size distribution which can be mapped to the recent experimental findings. Given the spatial heterogeneity within condensates and non-linear nature of intracellular networks, we envision MCS to be a generic functional solution, so that structures of network motifs may have evolved to accommodate such configurations.

A mechanism with severing near barbed ends andannealing explains structure and dynamics of dendriticactin networks

Single molecule imaging has shown that part of actin disassembles within a few seconds after incorporation into the dendritic filament network in lamellipodia, suggestive of frequent destabilization near barbed ends. To investigate the mechanisms behind network remodeling, we created a stochastic model with polymerization, depolymerization, branching, capping, uncapping, severing, oligomer diffusion, annealing, and debranching. We find that filament severing, enhanced near barbed ends, can explain the single molecule actin lifetime distribution, if oligomer fragments reanneal to free ends with rate constants comparable to in vitro measurements. The same mechanism leads to actin networks consistent with measured filament, end, and branch concentrations. These networks undergo structural remodeling, leading to longer filaments away from the leading edge, at the +/- 35𝑜 orientation pattern. Imaging of actin speckle lifetimes at sub-second resolution verifies frequent disassembly of newly-assembled actin. We thus propose a unified mechanism that fits a diverse set of basic lamellipodia phenomenology.

Modeling cell protrusion predicts how myosin II and actin turnover affect adhesion-based signaling

Orchestration of cell migration is essential for development, tissue regeneration, and the immune response. This dynamic process integrates adhesion, signaling, and cytoskeletal subprocesses across spatial and temporal scales. In mesenchymal cells, adhesion complexes bound to extracellular matrix mediate both biochemical signal transduction and physical interaction with the F-actin cytoskeleton. Here, we present a mathematical model that offers insight into both aspects, considering spatiotemporal dynamics of nascent adhesions, active signaling molecules, mechanical clutching, actin treadmilling, and nonmuscle myosin II contractility. At the core of the model is a positive feedback loop, whereby adhesion-based signaling promotes generation of barbed ends at, and protrusion of, the cell's leading edge, which in turn promotes formation and stabilization of nascent adhesions. The model predicts a switch-like transition and optimality of membrane protrusion, determined by the balance of actin polymerization and retrograde flow, with respect to extracellular matrix density. The model, together with new experimental measurements, explains how protrusion can be modulated by mechanical effects (nonmuscle myosin II contractility and adhesive bond stiffness) and F-actin turnover.

Mechanism of actin filament nucleation

We used computational methods to analyze the mechanism of actin filament nucleation. We assumed a pathway where monomers form dimers, trimers, and tetramers that then elongate to form filaments but also considered other pathways. We aimed to identify the rate constants for these reactions that best fit experimental measurements of polymerization time courses. The analysis showed that the formation of dimers and trimers is unfavorable because the association reactions are orders of magnitude slower than estimated in previous work rather than because of rapid dissociation of dimers and trimers. The 95% confidence intervals calculated for the four rate constants spanned no more than one order of magnitude. Slow nucleation reactions are consistent with published high-resolution structures of actin filaments and molecular dynamics simulations of filament ends. One explanation for slow dimer formation, which we support with computational analysis, is that actin monomers are in a conformational equilibrium with a dominant conformation that cannot participate in the nucleation steps.

Exploring the influence of cytosolic and membrane FAK activation on YAP/TAZ nuclear translocation

Membrane binding and unbinding dynamics play a crucial role in the biological activity of several nonintegral membrane proteins, which have to be recruited to the membrane to perform their functions. By localizing to the membrane, these proteins are able to induce downstream signal amplification in their respective signaling pathways. Here, we present a 3D computational approach using reaction-diffusion equations to investigate the relation between membrane localization of focal adhesion kinase (FAK), Ras homolog family member A (RhoA), and signal amplification of the YAP/TAZ signaling pathway. Our results show that the theoretical scenarios in which FAK is membrane bound yield robust and amplified YAP/TAZ nuclear translocation signals. Moreover, we predict that the amount of YAP/TAZ nuclear translocation increases with cell spreading, confirming the experimental findings in the literature. In summary, our in silico predictions show that when the cell membrane interaction area with the underlying substrate increases, for example, through cell spreading, this leads to more encounters between membrane-bound signaling partners and downstream signal amplification. Because membrane activation is a motif common to many signaling pathways, this study has important implications for understanding the design principles of signaling networks.

Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration.

Though the term noise usually carries negative connotations, it can also contribute positively to the characteristic dynamics of a system. In biological systems, where noise arises from the stochastic interactions between molecules, its study is usually confined to genetic regulatory systems in which copy numbers are small and fluctuations large. However, noise can have important roles when the number of signaling molecules is large. The extension of pseudopods and the subsequent motion of amoeboid cells arises from the noise-induced trigger of an excitable system. Chemoattractant signals bias this triggering thereby directing cell motion. To date, this paradigm has not been tested by mathematical models that account accurately for the noise that arises in the corresponding reactions. In this study, we employ a reaction-diffusion master equation approach to investigate the effects of noise. Using a modular approach and a three-dimensional cell model with specific subdomains attributed to the cell membrane and cortex, we explore the spatiotemporal dynamics of the system. Our simulations recreate many experimentally-observed cell behaviors thereby supporting the biased-excitable network hypothesis.

A spatial model of YAP/TAZ signaling reveals how stiffness, dimensionality, and shape contribute to emergent outcomes

YAP/TAZ is a master regulator of mechanotransduction whose functions rely on translocation from the cytoplasm to the nucleus in response to diverse physical cues. Substrate stiffness, substrate dimensionality, and cell shape are all input signals for YAP/TAZ, and through this pathway, regulate critical cellular functions and tissue homeostasis. Yet, the relative contributions of each biophysical signal and the mechanisms by which they synergistically regulate YAP/TAZ in realistic tissue microenvironments that provide multiplexed input signals remain unclear. For example, in simple two-dimensional culture, YAP/TAZ nuclear localization correlates strongly with substrate stiffness, while in three-dimensional (3D) environments, YAP/TAZ translocation can increase with stiffness, decrease with stiffness, or remain unchanged. Here, we develop a spatial model of YAP/TAZ translocation to enable quantitative analysis of the relationships between substrate stiffness, substrate dimensionality, and cell shape. Our model couples cytosolic stiffness to nuclear mechanics to replicate existing experimental trends, and extends beyond current data to predict that increasing substrate activation area through changes in culture dimensionality, while conserving cell volume, forces distinct shape changes that result in nonlinear effect on YAP/TAZ nuclear localization. Moreover, differences in substrate activation area versus total membrane area can account for counterintuitive trends in YAP/TAZ nuclear localization in 3D culture. Based on this multiscale investigation of the different system features of YAP/TAZ nuclear translocation, we predict that how a cell reads its environment is a complex information transfer function of multiple mechanical and biochemical factors. These predictions reveal a few design principles of cellular and tissue engineering for YAP/TAZ mechanotransduction.

Bridging from single to collective cell migration: A review of models and links to experiments

Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. We survey recent literature to summarize distinct computational methods (phase-field, polygonal, Cellular Potts, and spherical cells). We discuss models that bridge between levels of organization, and describe levels of detail, both biochemical and geometric, included in the models. We also highlight links between models and experiments. We find that models that span the 3 levels are still in the minority.

Thermodynamically consistent treatment of the growth of a biopolymer in the presence of a smooth obstacle interaction potential

We investigate the effect of filament-obstacle interactions on the force-velocity relation of growing biopolymers, via calculations explicitly treating obstacle diffusion and stochastic addition and subtraction of subunits. We first show that the instantaneous subunit on- and off-rates satisfy a rigorous thermodynamic relationship determined by the filament-obstacle interaction potential, which has been violated by several calculations in the literature. The instantaneous rates depend not only on the average force on the obstacle but also on the shape of the potential on the nanometer length scale. Basing obstacle-induced reduction of the on-rate entirely on the force, as previous work has often done, is thermodynamically inconsistent and can overestimate the stall force, sometimes by more than a factor of two. We perform simulations and analytic calculations of the force-velocity relation satisfying the thermodynamic relationship. The force-velocity relation can deviate strongly from the Brownian-Ratchet predictions. For shallow potential wells of depth ∼5k_{B}T, which might correspond to transient filament-membrane attachments, the velocity drops more rapidly than predicted by the Brownian-Ratchet model, in some cases by as much as a factor of 50 at an opposing force of only 1 pN. On the other hand, the zero-force velocity is much less affected than would be expected from naive use of the Boltzmann factor. Furthermore, the growth velocity has a surprisingly strong dependence on the obstacle diffusion coefficient even when the dimensionless diffusion coefficient is large. For deep potential wells, as might result from strong filament-membrane links, both the on- and off-rates are reduced significantly, slowing polymerization. Such potentials can sustain pulling forces while polymerizing but only if the attractive well is relatively flat over a region comparable to or greater than the monomer size. For double-well potentials, which have such a flat region, the slowing of polymerization by external pushing force is almost linear up to the stall force in some parameter ranges.

Building a dendritic actin filament network branch by branch: models of filament orientation pattern and force generation in lamellipodia

We review mathematical and computational models of the structure, dynamics, and force generation properties of dendritic actin networks. These models have been motivated by the dendritic nucleation model, which provided a mechanistic picture of how the actin cytoskeleton system powers cell motility. We describe how they aimed to explain the self-organization of the branched network into a bimodal distribution of filament orientations peaked at 35° and - 35° with respect to the direction of membrane protrusion, as well as other patterns. Concave and convex force-velocity relationships were derived, depending on network organization, filament, and membrane elasticity and accounting for actin polymerization at the barbed end as a Brownian ratchet. This review also describes models that considered the kinetics and transport of actin and diffuse regulators and mechanical coupling to a substrate, together with explicit modeling of dendritic networks.

A computational model of amoeboid cell motility in the presence of obstacles

Locomotion of amoeboid cells is mediated by finger-like protrusions of the cell body, known as pseudopods, which grow, bifurcate, and retract in a dynamic fashion. Pseudopods are the primary mode of locomotion for many cells within the human body, such as leukocytes, embryonic cells, and metastatic cancer cells. Amoeboid motility is a complex and multiscale process, which involves bio-molecular reactions, cell deformation, and cytoplasmic and extracellular fluid motion. Additionally, cells within the human body are subject to a confined 3D environment known as the extra-cellular matrix (ECM), which resembles a fluid-filled porous medium. In this article, we present a 3D, multiphysics computational approach coupling fluid mechanics, solid mechanics, and a pattern formation model to simulate locomotion of amoeboid cells through a porous matrix composed of a viscous fluid and an array of finite-sized spherical obstacles. The model combines reaction-diffusion of activator/inhibitors, extreme deformation of the cell, pseudopod dynamics, cytoplasmic and extracellular fluid motion, and fully resolved extracellular matrix. A surface finite-element method is used to obtain the cell deformation and activator/inhibitor concentrations, while the fluid motion is solved using a combined finite-volume and spectral method. The immersed-boundary methods are used to couple the cell deformation, obstacles, and fluid. The model is able to recreate squeezing and weaving motion of cells through the matrix. We study the influence of matrix porosity, obstacle size, and cell deformability on the motility behavior. It is found that below certain values of these parameters, cell motion is completely inhibited. Phase diagrams are presented depicting such motility limits. Interesting dynamics seen in the presence of obstacles but absent in unconfined medium, such as freezing or cell arrest, probing, doubling-back, and tug-of-war are predicted. Furthermore, persistent unidirectional motion of cells that is often observed in an unconfined medium is shown to be lost in presence of obstacles, and is attributed to an alteration of the pseudopod dynamics. The same mechanism, however, allows the cell to find a new direction to penetrate further into the matrix without being stuck in one place. The results and analysis presented here show a strong coupling between cell deformability and ECM properties, and provide new fluid mechanical insights on amoeboid motility in confined medium.

A master equation approach to actin polymerization applied to endocytosis in yeast

We present a Master Equation approach to calculating polymerization dynamics and force generation by branched actin networks at membranes. The method treats the time evolution of the F-actin distribution in three dimensions, with branching included as a directional spreading term. It is validated by comparison with stochastic simulations of force generation by actin polymerization at obstacles coated with actin “nucleation promoting factors” (NPFs). The method is then used to treat the dynamics of actin polymerization and force generation during endocytosis in yeast, using a model in which NPFs form a ring around the endocytic site, centered by a spot of molecules attaching the actin network strongly to the membrane. We find that a spontaneous actin filament nucleation mechanism is required for adequate forces to drive the process, that partial inhibition of branching and polymerization lead to different characteristic responses, and that a limited range of polymerization-rate values provide effective invagination and obtain correct predictions for the effects of mutations in the active regions of the NPFs.

Endocytosis is a dynamic process by which cells internalize substances from outside the cell. Especially in yeast, endocytosis is mechanically demanding due to the high pressure difference across the cell membrane, or turgor pressure. Polymerization of a branched actin network is the major process providing the mechanical force to overcome the turgor pressure. Understanding the kinetics of the actin network, and the mechanical interaction between the actin network and the cell membrane, is thus crucial for the study of endocytosis. We develop an efficient mathematical framework for actin dynamics that can realistically incorporate these two features, thus providing a practical method for quantitatively modeling actin dynamics during endocytosis. The resulting model mechanistically reveals that spontaneous nucleation at the center of the endocytic site is required for successful endocytosis, distinguishes the roles of branching and polymerization, and predicts several other experimentally testable outcomes. The accuracy and efficiency of the method, in describing both mechanics and chemistry, render it applicable to a broad field of membrane-bending processes.

Actin Turnover in Lamellipodial Fragments

Actin turnover is the central driving force underlying lamellipodial motility. The molecular components involved are largely known, and their properties have been studied extensively in vitro. However, a comprehensive picture of actin turnover in vivo is still missing. We focus on fragments from fish epithelial keratocytes, which are essentially stand-alone motile lamellipodia. The geometric simplicity of the fragments and the absence of additional actin structures allow us to characterize the spatiotemporal lamellipodial actin organization with unprecedented detail. We use fluorescence recovery after photobleaching, fluorescence correlation spectroscopy, and extraction experiments to show that about two-thirds of the lamellipodial actin diffuses in the cytoplasm with nearly uniform density, whereas the rest forms the treadmilling polymer network. Roughly a quarter of the diffusible actin pool is in filamentous form as diffusing oligomers, indicating that severing and debranching are important steps in the disassembly process generating oligomers as intermediates. The remaining diffusible actin concentration is orders of magnitude higher than the in vitro actin monomer concentration required to support the observed polymerization rates, implying that the majority of monomers are transiently kept in a non-polymerizable "reserve" pool. The actin network disassembles and reassembles throughout the lamellipodium within seconds, so the lamellipodial network turnover is local. The diffusible actin transport, on the other hand, is global: actin subunits typically diffuse across the entire lamellipodium before reassembling into the network. This combination of local network turnover and global transport of dissociated subunits through the cytoplasm makes actin transport robust yet rapidly adaptable and amenable to regulation.

Global treadmilling coordinates actin turnover and controls the size of actin networks

Various cellular processes (including cell motility) are driven by the regulated, polarized assembly of actin filaments into distinct force-producing arrays of defined size and architecture. Branched, linear, contractile and cytosolic arrays coexist in vivo, and cells intricately control the number, length and assembly rate of filaments in these arrays. Recent in vitro and in vivo studies have revealed novel molecular mechanisms that regulate the number of filament barbed and pointed ends and their respective assembly and disassembly rates, thus defining classes of dynamically different filaments, which coexist in the same cell. We propose that a global treadmilling process, in which a steady-state amount of polymerizable actin monomers is established by the dynamics of each network, is responsible for defining the size and turnover of coexisting actin networks. Furthermore, signal-induced changes in the partitioning of actin to distinct arrays (mediated by RHO GTPases) result in the establishment of various steady-state concentrations of polymerizable monomers, thereby globally influencing the growth rate of actin filaments.

Fragility of foot process morphology in kidney podocytes arises from chaotic spatial propagation of cytoskeletal instability

Kidney podocytes' function depends on fingerlike projections (foot processes) that interdigitate with those from neighboring cells to form the glomerular filtration barrier. The integrity of the barrier depends on spatial control of dynamics of actin cytoskeleton in the foot processes. We determined how imbalances in regulation of actin cytoskeletal dynamics could result in pathological morphology. We obtained 3-D electron microscopy images of podocytes and used quantitative features to build dynamical models to investigate how regulation of actin dynamics within foot processes controls local morphology. We find that imbalances in regulation of actin bundling lead to chaotic spatial patterns that could impair the foot process morphology. Simulation results are consistent with experimental observations for cytoskeletal reconfiguration through dysregulated RhoA or Rac1, and they predict compensatory mechanisms for biochemical stability. We conclude that podocyte morphology, optimized for filtration, is intrinsically fragile, whereby local transient biochemical imbalances may lead to permanent morphological changes associated with pathophysiology.

Model of turnover kinetics in the lamellipodium: implications of slow- and fast- diffusing capping protein and Arp2/3 complex

Cell protrusion through polymerization of actin filaments at the leading edge of motile cells may be influenced by spatial gradients of diffuse actin and regulators. Here we study the distribution of two of the most important regulators, capping protein and Arp2/3 complex, which regulate actin polymerization in the lamellipodium through capping and nucleation of free barbed ends. We modeled their kinetics using data from prior single molecule microscopy experiments on XTC cells. These experiments have provided evidence for a broad distribution of diffusion coefficients of both capping protein and Arp2/3 complex. The slowly diffusing proteins appear as extended 'clouds' while proteins bound to the actin filament network appear as speckles that undergo retrograde flow. Speckle appearance and disappearance events correspond to assembly and dissociation from the actin filament network and speckle lifetimes correspond to the dissociation rate. The slowly diffusing capping protein could represent severed capped actin filament fragments or membrane-bound capping protein. Prior evidence suggests that slowly diffusing Apr2/3 complex associates with the membrane. We use the measured rates and estimates of diffusion coefficients of capping protein and Arp2/3 complex in a Monte Carlo simulation that includes particles in association with a filament network and diffuse in the cytoplasm. We consider two separate pools of diffuse proteins, representing fast and slowly diffusing species. We find a steady state with concentration gradients involving a balance of diffusive flow of fast and slow species with retrograde flow. We show that simulations of FRAP are consistent with prior experiments performed on different cell types. We provide estimates for the ratio of bound to diffuse complexes and calculate conditions where Arp2/3 complex recycling by diffusion may become limiting. We discuss the implications of slowly diffusing populations and suggest experiments to distinguish among mechanisms that influence long range transport.

Simultaneous quantification of actin monomer and filament dynamics with modeling-assisted analysis of photoactivation

Photoactivation allows one to pulse-label molecules and obtain quantitative data about their behavior. We have devised a new modeling-based analysis for photoactivatable actin experiments that simultaneously measures properties of monomeric and filamentous actin in a three-dimensional cellular environment. We use this method to determine differences in the dynamic behavior of β- and γ-actin isoforms, showing that both inhabit filaments that depolymerize at equal rates but that β-actin exists in a higher monomer-to-filament ratio. We also demonstrate that cofilin (cofilin 1) equally accelerates depolymerization of filaments made from both isoforms, but is only required to maintain the β-actin monomer pool. Finally, we used modeling-based analysis to assess actin dynamics in axon-like projections of differentiating neuroblastoma cells, showing that the actin monomer concentration is significantly depleted as the axon develops. Importantly, these results would not have been obtained using traditional half-time analysis. Given that parameters of the publicly available modeling platform can be adjusted to suit the experimental system of the user, this method can easily be used to quantify actin dynamics in many different cell types and subcellular compartments.

Feedback mechanisms in a mechanical model of cell polarization

Directed cell migration requires a spatially polarized distribution of polymerized actin. We develop and treat a mechanical model of cell polarization based on polymerization and depolymerization of actin filaments at the two ends of a cell, modulated by forces at either end that are coupled by the cell membrane. We solve this model using both a simulation approach that treats filament nucleation, polymerization, and depolymerization stochastically, and a rate-equation approach based on key properties such as the number of filaments N and the number of polymerized subunits F at either end of the cell. The rate-equation approach agrees closely with the stochastic approach at steady state and, when appropriately generalized, also predicts the dynamic behavior accurately. The calculated transitions from symmetric to polarized states show that polarization is enhanced by a high free-actin concentration, a large pointed-end off-rate, a small barbed-end off-rate, and a small spontaneous nucleation rate. The rate-equation approach allows us to perform a linear-stability analysis to pin down the key interactions that drive the polarization. The polarization is driven by a positive-feedback loop having two interactions. First, an increase in F at one side of the cell lengthens the filaments and thus reduces the decay rate of N (increasing N); second, increasing N enhances F because the force per growing filament tip is reduced. We find that the transitions induced by changing system properties result from supercritical pitchfork bifurcations. The filament lifetime depends strongly on the average filament length, and this effect is crucial for obtaining polarization correctly.

Membrane related dynamics and the formation of actin in cells growing on micro-topographies: a spatial computational model

Background

Intra-cellular processes of cells at the interface to an implant surface are influenced significantly by their extra-cellular surrounding. Specifically, when growing osteoblasts on titanium surfaces with regular micro-ranged geometry, filaments are shorter, less aligned and they concentrate at the top of the geometric structures. Changes to the cytoskeleton network, i. e., its localization, alignment, orientation, and lengths of the filaments, as well as the overall concentration and distribution of key-actors are induced. For example, integrin is distributed homogeneously, whereas integrin in activated state and vinculin, both components of focal adhesions, have been found clustered on the micro-ranged geometries. Also, the concentration of Rho, an intracellular signaling protein related to focal adhesion regulation, was significantly lower.

Results

To explore whether regulations associated with the focal adhesion complex can be responsible for the changed actin filament patterns, a spatial computational model has been developed using ML-Space, a rule-based model description language, and its associated Brownian-motion-based simulator. The focus has been on the deactivation of cofilin in the vicinity of the focal adhesion complex. The results underline the importance of sensing mechanisms to support a clustering of actin filament nucleations on the micro-ranged geometries, and of intracellular diffusion processes, which lead to spatially heterogeneous distributions of active (dephosphorylated) cofilin, which in turn influences the organization of the actin network. We find, for example, that the spatial heterogeneity of key molecular actors can explain the difference in filament lengths in cells on different micro-geometries partly, but to explain the full extent, further model assumptions need to be added and experimentally validated. In particular, our findings and hypothesis referring to the role, distribution, and amount of active cofilin have still to be verified in wet-lab experiments.

Conclusion

Letting cells grow on surface structures is a possibility to shed new light on the intricate mechanisms that relate membrane and actin related dynamics in the cell. Our results demonstrate the need for declarative expressive spatial modeling approaches that allow probing different hypotheses, and the central role of the focal adhesion complex not only for nucleating actin filaments, but also for regulating possible severing agents locally.

Modeling the synergy of cofilin and Arp2/3 in lamellipodial protrusive activity

Rapid polymerization of actin filament barbed ends generates protrusive forces at the cell edge, leading to cell migration. Two important regulators of free barbed ends, cofilin and Arp2/3, have been shown to work in synergy (net effect greater than additive). To explore this synergy, we model the dynamics of F-actin at the leading edge, motivated by data from EGF-stimulated mammary carcinoma cells. We study how synergy depends on the localized rates and relative timing of cofilin and Arp2/3 activation at the cell edge. The model incorporates diffusion of cofilin, membrane protrusion, F-actin capping, aging, and severing by cofilin and branch nucleation by Arp2/3 (but not G-actin recycling). In a well-mixed system, cofilin and Arp2/3 can each generate a large pulse of barbed ends on their own, but have little synergy; high synergy occurs only at low activation rates, when few barbed ends are produced. In the full spatially distributed model, both synergy and barbed-end production are significant over a range of activation rates. Furthermore, barbed-end production is greatest when Arp2/3 activation is delayed relative to cofilin. Our model supports a direct role for cofilin-mediated actin polymerization in stimulated cell migration, including chemotaxis and cancer invasion.

Bidirectional coupling between integrin-mediated signaling and actomyosin mechanics explains matrix-dependent intermittency of leading-edge motility

A physicochemical model is used to describe the coupling of adhesion, cytoskeletal, and signaling dynamics during cell migration. Analysis of stochastic simulations predicts relationships between measurable quantities that reflect partitioning of stress between F-actin–bound adhesions, which act as a molecular clutch, and retrograde F-actin flow.

Animal cell migration is a complex process characterized by the coupling of adhesion, cytoskeletal, and signaling dynamics. Here we model local protrusion of the cell edge as a function of the load-bearing properties of integrin-based adhesions, actin polymerization fostered by adhesion-mediated signaling, and mechanosensitive activation of RhoA that promotes myosin II–generated stress on the lamellipodial F-actin network. Analysis of stochastic model simulations illustrates how these pleiotropic functions of nascent adhesions may be integrated to govern temporal persistence and frequency of protrusions. The simulations give mechanistic insight into the documented effects of extracellular matrix density and myosin abundance, and they show characteristic, nonnormal distributions of protrusion duration times that are similar to those extracted from live-cell imaging experiments. Analysis of the model further predicts relationships between measurable quantities that reflect the partitioning of stress between tension on F-actin–bound adhesions, which act as a molecular clutch, and dissipation by retrograde F-actin flow.

Spatiotemporal dynamics of actomyosin networks

Rhodamine-phalloidin-labeled actin filaments were visualized gliding over a skeletal heavy meromyosin (HMM)-coated surface. Experiments at low filament densities showed that when two filaments collided, their paths were affected in a manner that depended on collision angle. Some collisions resulted in complete alignment of the filament paths; in others, the filaments crossed over one another. Filament crossover or alignment was equally probable at ∼40° contact angle. Filaments often underwent significant bending during collision and analysis of filament shape indicated an energy requirement of ∼13 kBT. Experiments were performed over a wide range of HMM surface density and actin filament bulk concentration. Actin filament gliding speed and path persistence plateaued above a critical HMM surface density, and at high (micromolar) actin filament concentrations, filament motion became dramatically aligned in a common direction. Spatiotemporal features of alignment behavior were determined by correlation analysis, supported by simulations. The thermal drift of individual filament tracks was suppressed as the population became more oriented. Spatial correlation analysis revealed that long-range alignment was due to incremental recruitment rather than fusion of locally ordered seed domains. The global alignment of filament movement, described by an "order parameter," peaked at optimal actin concentrations and myosin surface densities, in contrast to previous predictions of a critical phase transition. Either hydrodynamic coupling or exchange of filaments between the surface bound and adjacent bulk phase layers might degrade order at high actin filament concentration, and high HMM surface densities might decrease alignment probability during collisions. Our results are compatible with generation of long-range order from mechanical interaction between individual actin filaments. Furthermore, we show that randomly oriented myosin motors align relatively short, submicrometer actin filaments into motile surface domains that extend over many tens of micrometers and these patterns persist for several minutes.

There is more than one way to model an elephant. Experiment-driven modeling of the actin cytoskeleton

Mathematical modeling has established its value for investigating the interplay of biochemical and mechanical mechanisms underlying actin-based motility. Because of the complex nature of actin dynamics and its regulation, many of these models are phenomenological or conceptual, providing a general understanding of the physics at play. But the wealth of carefully measured kinetic data on the interactions of many of the players in actin biochemistry cries out for the creation of more detailed and accurate models that could permit investigators to dissect interdependent roles of individual molecular components. Moreover, no human mind can assimilate all of the mechanisms underlying complex protein networks; so an additional benefit of a detailed kinetic model is that the numerous binding proteins, signaling mechanisms, and biochemical reactions can be computationally organized in a fully explicit, accessible, visualizable, and reusable structure. In this review, we will focus on how comprehensive and adaptable modeling allows investigators to explain experimental observations and develop testable hypotheses on the intracellular dynamics of the actin cytoskeleton.

Distributed actin turnover in the lamellipodium and FRAP kinetics

Studies of actin dynamics at the leading edge of motile cells with single-molecule speckle (SiMS) microscopy have shown a broad distribution of EGFP-actin speckle lifetimes and indicated actin polymerization and depolymerization over an extended region. Other experiments using FRAP with the same EGFP-actin as a probe have suggested, by contrast, that polymerization occurs exclusively at the leading edge. We performed FRAP experiments on XTC cells to compare SiMS to FRAP on the same cell type. We used speckle statistics obtained by SiMS to model the steady-state distribution and kinetics of actin in the lamellipodium. We demonstrate that a model with a single diffuse actin species is in good agreement with FRAP experiments. A model including two species of diffuse actin provides an even better agreement. The second species consists of slowly diffusing oligomers that associate to the F-actin network throughout the lamellipodium or break up into monomers after a characteristic time. Our work motivates studies to test the presence and composition of slowly diffusing actin species that may contribute to local remodeling of the actin network and increase the amount of soluble actin.

A comparison of computational models for eukaryotic cell shape and motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.

Stoichiometry of Nck-dependent actin polymerization in living cells

Regulation of actin dynamics through the Nck/N-WASp (neural Wiskott-Aldrich syndrome protein)/Arp2/3 pathway is essential for organogenesis, cell invasiveness, and pathogen infection. Although many of the proteins involved in this pathway are known, the detailed mechanism by which it functions remains undetermined. To examine the signaling mechanism, we used a two-pronged strategy involving computational modeling and quantitative experimentation. We developed predictions for Nck-dependent actin polymerization using the Virtual Cell software system. In addition, we used antibody-induced aggregation of membrane-targeted Nck SH3 domains to test these predictions and to determine how the number of molecules in Nck aggregates and the density of aggregates affected localized actin polymerization in living cells. Our results indicate that the density of Nck molecules in aggregates is a critical determinant of actin polymerization. Furthermore, results from both computational simulations and experimentation support a model in which the Nck/N-WASp/Arp2/3 stoichiometry is 4:2:1. These results provide new insight into activities involving localized actin polymerization, including tumor cell invasion, microbial pathogenesis, and T cell activation.

A review of models of fluctuating protrusion and retraction patterns at the leading edge of motile cells

A characteristic feature of motile cells as they undergo a change in motile behavior is the development of fluctuating exploratory motions of the leading edge, driven by actin polymerization. We review quantitative models of these protrusion and retraction phenomena. Theoretical studies have been motivated by advances in experimental and computational methods that allow controlled perturbations, single molecule imaging, and analysis of spatiotemporal correlations in microscopic images. To explain oscillations and waves of the leading edge, most theoretical models propose nonlinear interactions and feedback mechanisms among different components of the actin cytoskeleton system. These mechanisms include curvature-sensing membrane proteins, myosin contraction, and autocatalytic biochemical reaction kinetics. We discuss how the combination of experimental studies with modeling promises to quantify the relative importance of these biochemical and biophysical processes at the leading edge and to evaluate their generality across cell types and extracellular environments.

Cease-fire at the leading edge: new perspectives on actin filament branching, debranching, and cross-linking

Membrane protrusion at the leading edge of migrating cells is driven by the polymerization of actin. Recent studies using advanced imaging techniques raised a lively controversy about the morphology of these filaments; however, common ground between the two sides now appears to have been found. Here we discuss how the controversy has led to a deeper consideration of the architecture of actin networks underlying cell migration, and has helped define new challenges that lie ahead.

How does the antagonism between capping and anti-capping proteins affect actin network dynamics?

Actin-based cell motility is essential to many biological processes. We built a simplified, three-dimensional computational model and subsequently performed stochastic simulations to study the growth dynamics of lamellipodia-like branched networks. In this work, we shed light on the antagonism between capping and anti-capping proteins in regulating actin dynamics in the filamentous network. We discuss detailed mechanisms by which capping and anti-capping proteins affect the protrusion speed of the actin network and the rate of nucleation of filaments. We computed a phase diagram showing the regimes of motility enhancement and inhibition by these proteins. Our work shows that the effects of capping and anti-capping proteins are mainly transmitted by modulation of the filamentous network density and local availability of monomeric actin. We discovered that the combination of the capping/anti-capping regulatory network with nucleation-promoting proteins introduces robustness and redundancy in cell motility machinery, allowing the cell to easily achieve maximal protrusion speeds under a broader set of conditions. Finally, we discuss distributions of filament lengths under various conditions and speculate on their potential implication for the emergence of filopodia from the lamellipodial network.

Signaling pathways that control cell migration: models and analysis

Dissecting the intracellular signaling mechanisms that govern the movement of eukaryotic cells presents a major challenge, not only because of the large number of molecular players involved, but even more so because of the dynamic nature of their regulation by both biochemical and mechanical interactions. Computational modeling and analysis have emerged as useful tools for understanding how the physical properties of cells and their microenvironment are coupled with certain biochemical pathways to actuate and control cell motility. In this focused review, we highlight some of the more recent applications of quantitative modeling and analysis in the field of cell migration. Both in modeling and experiment, it has been prudent to follow a reductionist approach in order to characterize what are arguably the principal modules: spatial polarization of signaling pathways, regulation of the actin cytoskeleton, and dynamics of focal adhesions. While it is important that we 'cut our teeth' on these subsystems, focusing on the details of certain aspects while ignoring or coarse-graining others, it is clear that the challenge ahead will be to characterize the couplings between them in an integrated framework.

Quantifying a pathway: kinetic analysis of actin dendritic nucleation

Progress in uncovering the reaction networks that underlie important cell functions is laying the groundwork for quantitative identification of protein-interaction pathways. Since direct measurement of rate constants is not always feasible, the parameters are often inferred from multiple pieces of data using kinetic analyses based on appropriate mathematical models. The success of this approach relies on the sufficiency of available experimental data for a unique parameterization of the network. The concept of a rate-limiting step is applied to the analysis of experimental data that are usually used to quantify a pathway of actin dendritic nucleation, the Arp2/3-mediated mechanism that enables rapid changes of cell shape in response to external cues. The method yields analytical descriptions of the dynamics of polymerized actin and provides insights into how the experimental curves should be analyzed. It is shown that dynamics measured by pyrene-labeled actin assays with varying Arp2/3 concentrations are equally well described by two different rate-limiting steps: 1), binding of a nucleating complex to the side of a preexisting filament; or 2), its subsequent activation. To distinguish between the alternatives, we propose experiments with varying concentrations of actin monomers, taking advantage of the fact that the number of branches in the two cases depends differently on the initial monomer concentration. The idea is tested by simulating the proposed experiments with the use of spatial stochastic modeling.

Modeling capping protein FRAP and CALI experiments reveals in vivo regulation of actin dynamics

To gain insights on cellular mechanisms regulating actin polymerization, we used the Virtual Cell to model fluorescence recovery after photobleaching (FRAP) and chromophore-assisted laser inactivation (CALI) experiments on EGFP-capping protein (EGFP-CP). Modeling the FRAP kinetics demonstrated that the in vivo rate for the dissociation of CP from actin filaments is much faster (approximately 0.1 s(-1)) than that measured in vitro (0.01-0.0004 s(-1)). The CALI simulation revealed that in order to induce sustainable changes in cell morphology after CP inactivation, the cells should exhibit anticapping ability. We included the VASP protein as the anticapping agent in the modeling scheme. The model predicts that VASP affinity for barbed ends has a cooperative dependence on the concentration of VASP-barbed end complexes. This dependence produces a positive feedback that stabilizes the complexes and allows sustained growth at clustered filament tips. We analyzed the range of laser intensities that are sufficient to induce changes in cell morphology. This analysis demonstrates that FRAP experiments with EGFP-CP can be performed safely without changes in cell morphology, because, the intensity of the photobleaching beam is not high enough to produce the critical concentration of free barbed ends that will induce filament growth before diffusional replacement of EGFP-CP occurs.

Control of actin filament treadmilling in cell motility

Recent advances in structural, biochemical, biophysical, and live cell imaging approaches have furthered our understanding of the molecular mechanisms by which regulated assembly dynamics of actin filaments drive motile processes. Attention is focused on lamellipodium protrusion, powered by the turnover of a branched filament array. ATP hydrolysis on actin is the key reaction that allows filament treadmilling. It regulates barbed-end dynamics and length fluctuations at steady state and specifies the functional interaction of actin with essential regulatory proteins such as profilin and ADF/cofilin. ATP hydrolysis on actin and Arp2/3 acts as a timer, regulating the assembly and disassembly of the branched array to generate tropomyosin-mediated heterogeneity in the structure and dynamics of the lamellipodial network. The detailed molecular mechanisms of ATP hydrolysis/Pi release on F-actin remain elusive, as well as the mechanism of filament branching with Arp2/3 complex or that of the formin-driven processive actin assembly. Novel biophysical methods involving single-molecule measurements should foster progress in these crucial issues.

Mechano-chemical feedbacks regulate actin mesh growth in lamellipodial protrusions

During cell motion on a substratum, eukaryotic cells project sheetlike lamellipodia which contain a dynamically remodeling three-dimensional actin mesh. A number of regulatory proteins and subtle mechano-chemical couplings determine the lamellipodial protrusion dynamics. To study these processes, we constructed a microscopic physico-chemical computational model, which incorporates a number of fundamental reaction and diffusion processes, treated in a fully stochastic manner. Our work sheds light on the way lamellipodial protrusion dynamics is affected by the concentrations of actin and actin-binding proteins. In particular, we found that protrusion speed saturates at very high actin concentrations, where filament nucleation does not keep up with protrusion. This results in sparse filamentous networks, and, consequently, high resistance forces on individual filaments. We also observed maxima in lamellipodial growth rates as a function of Arp2/3, a nucleating protein, and capping proteins. We provide detailed physical explanations behind these effects. In particular, our work supports the actin-funneling-hypothesis explanation of protrusion speed enhancement at low capping protein concentrations. Our computational results are in agreement with a number of related experiments. Overall, our work emphasizes that elongation and nucleation processes work highly cooperatively in determining the optimal protrusion speed for the actin mesh in lamellipodia.

The effects of filament aging and annealing on a model lamellipodium undergoing disassembly by severing

We use numerical simulations to study the properties of a model lamellipodium as it disassembles by filament severing. The growing lamellipodium is modeled as a 2D or 3D periodic lattice of crosslinked actin filaments. At each time step a new layer of actin filaments is added at the membrane, existing filaments are severed stochastically and disconnected sections of the network are removed. Filament aging is modeled by including several different filament chemical states. Filament annealing is included by allowing existing filaments to grow new filaments. The properties of the model are studied as functions of the number of states and the severing and annealing rates. The network width is proportional to the sum of the average lifetimes of the states, and is well modeled by a simple kinetic theory. The length of the network scales linearly with actin concentration and has a finite width even at high severing protein concentrations. The edge of the growing network becomes sharper as either the number of states or the dimensionality is increased. Annealing increases the average length of the network, and the network length diverges at a critical annealing rate.

Inside view of cell locomotion through single-molecule: fast F-/G-actin cycle and G-actin regulation of polymer restoration

The actin cytoskeleton drives cell locomotion and tissue remodeling. The invention of live-cell fluorescence single-molecule imaging opened a window for direct viewing of the actin remodeling processes in the cell. Since then, a number of unanticipated molecular functions have been revealed. One is the mechanism of F-actin network breakdown. In lamellipodia, one third of newly polymerized F-actin disassembles within 10 seconds. This fast F-actin turnover is facilitated by the filament severing/disrupting activity involving cofilin and AIP1. Astoundingly fast dissociation kinetics of the barbed end interactors including capping protein suggests that F-actin turnover might proceed through repetitive disruption/reassembly of the filament near the barbed end. The picture of actin polymerization is also being revealed. At the leading edge of the cell, Arp2/3 complex is highly activated in a narrow edge region. In contrast, mDia1 and its related Formin homology proteins display a long-distance directional molecular movement using their processive actin capping ability. Recently, these two independently-developed projects converged into a discovery of the spatiotemporal coupling between mDia1-mediated filament nucleation and actin disassembly. Presumably, the local concentration fluctuation of G-actin regulates the actin nucleation efficiency of specific actin nucleators including mDia1. Pharmacological perturbation and quantitative molecular behavior analysis synergize to reveal hidden molecular linkages in the actin turnover cycle and cell signaling.

Use of virtual cell in studies of cellular dynamics

The Virtual Cell (VCell) is a unique computational environment for modeling and simulation of cell biology. It has been specifically designed to be a tool for a wide range of scientists, from experimental cell biologists to theoretical biophysicists. The models created with VCell can range from the simple, to evaluate hypotheses or to interpret experimental data, to complex multilayered models used to probe the predicted behavior of spatially resolved, highly nonlinear systems. In this chapter, we discuss modeling capabilities of VCell and demonstrate representative examples of the models published by the VCell users.