Stochastic simulation of actin dynamics reveals the role of annealing and fragmentation

Emergence and maintenance of variable-length actin filaments in a limiting pool of building blocks

Actin is one of the key structural components of the eukaryotic cytoskeleton that regulates cellular architecture and mechanical properties. Dynamic regulation of actin filament length and organization is essential for the control of many physiological processes including cell adhesion, motility and division. While previous studies have mostly focused on the mechanisms controlling the mean length of individual actin filaments, it remains poorly understood how distinct actin filament populations in cells maintain different lengths using the same set of molecular building blocks. Here we develop a theoretical model for the length regulation of multiple actin filaments by nucleation and growth rate modulation by actin binding proteins in a limiting pool of monomers. We first show that spontaneous nucleation of actin filaments naturally leads to heterogeneities in filament length distribution. We then investigate the effects of filament growth inhibition by capping proteins and growth promotion by formin proteins on filament length distribution. We find that filament length heterogeneity can be increased by growth inhibition, whereas growth promoters do not significantly affect length heterogeneity. Interestingly, a competition between filament growth inhibitors and growth promoters can give rise to bimodal filament length distribution as well as a highly heterogeneous length distribution with large statistical dispersion. We quantitatively predict how heterogeneity in actin filament length can be modulated by tuning F-actin nucleation and growth rates in order to create distinct filament subpopulations with different lengths. SIGNIFICANCE: Actin filaments organize into different functional network architectures within eukaryotic cells. To maintain distinct actin network architectures, it is essential to regulate the lengths of actin filaments. While the mechanisms controlling the lengths of individual actin filaments have been extensively studied, the regulation of length heterogeneity in actin filament populations is not well understood. Here we show that the modulation of actin filament growth and nucleation rates by actin binding proteins can regulate actin length distribution and create distinct sub-populations with different lengths. In particular, by tuning concentrations of formin, profilin and capping proteins, various aspects of actin filament length distribution can be controlled. Insights gained from our results may have significant implications for the regulation of actin filament length heterogeneity and architecture within a cell.

Efficient Multiscale Models of Polymer Assembly

Protein polymerization and bundling play a central role in cell physiology. Predictive modeling of these processes remains an open challenge, especially when the proteins involved become large and their concentrations high. We present an effective kinetics model of filament formation, bundling, and depolymerization after GTP hydrolysis, which involves a relatively small number of species and reactions, and remains robust over a wide range of concentrations and timescales. We apply this general model to study assembly of FtsZ protein, a basic element in the division process of prokaryotic cells such as Escherichia coli, Bacillus subtilis, or Caulobacter crescentus. This analysis demonstrates that our model outperforms its counterparts in terms of both accuracy and computational efficiency. Because our model comprises only 17 ordinary differential equations, its computational cost is orders-of-magnitude smaller than the current alternatives consisting of up to 1000 ordinary differential equations. It also provides, to our knowledge, a new insight into the characteristics and functioning of FtsZ proteins at high concentrations. The simplicity and versatility of our model render it a powerful computational tool, which can be used either as a standalone descriptor of other biopolymers' assembly or as a component in more complete kinetic models.

Quantitative computational models of molecular self-assembly in systems biology

Molecular self-assembly is the dominant form of chemical reaction in living systems, yet efforts at systems biology modeling are only beginning to appreciate the need for and challenges to accurate quantitative modeling of self-assembly. Self-assembly reactions are essential to nearly every important process in cell and molecular biology and handling them is thus a necessary step in building comprehensive models of complex cellular systems. They present exceptional challenges, however, to standard methods for simulating complex systems. While the general systems biology world is just beginning to deal with these challenges, there is an extensive literature dealing with them for more specialized self-assembly modeling. This review will examine the challenges of self-assembly modeling, nascent efforts to deal with these challenges in the systems modeling community, and some of the solutions offered in prior work on self-assembly specifically. The review concludes with some consideration of the likely role of self-assembly in the future of complex biological system models more generally.

Flagellar Synchronization Is a Simple Alternative to Cell Cycle Synchronization for Ciliary and Flagellar Studies

Cilia and flagella are highly conserved antenna-like organelles that found in nearly all mammalian cell types. They perform sensory and motile functions contributing to numerous physiological and developmental processes. Defects in their assembly and function are implicated in a wide range of human diseases ranging from retinal degeneration to cancer. Chlamydomonas reinhardtii is an algal model system for studying mammalian cilium formation and function. Here, we report a simple synchronization method that allows detection of small changes in ciliary length by minimizing variability in the population. We find that this method alters the key relationship between cell size and the amount of protein accumulated for flagellar growth. This provides a rapid alternative to traditional methods of cell synchronization for uncovering novel regulators of cilia.

The unicellular green alga Chlamydomonas reinhardtii is an ideal model organism for studies of ciliary function and assembly. In assays for biological and biochemical effects of various factors on flagellar structure and function, synchronous culture is advantageous for minimizing variability. Here, we have characterized a method in which 100% synchronization is achieved with respect to flagellar length but not with respect to the cell cycle. The method requires inducing flagellar regeneration by amputation of the entire cell population and limiting regeneration time. This results in a maximally homogeneous distribution of flagellar lengths at 3 h postamputation. We found that time-limiting new protein synthesis during flagellar synchronization limits variability in the unassembled pool of limiting flagellar protein and variability in flagellar length without affecting the range of cell volumes. We also found that long- and short-flagella mutants that regenerate normally require longer and shorter synchronization times, respectively. By minimizing flagellar length variability using a simple method requiring only hours and no changes in media, flagellar synchronization facilitates the detection of small changes in flagellar length resulting from both chemical and genetic perturbations in Chlamydomonas. This method increases our ability to probe the basic biology of ciliary size regulation and related disease etiologies.

IMPORTANCE Cilia and flagella are highly conserved antenna-like organelles that found in nearly all mammalian cell types. They perform sensory and motile functions contributing to numerous physiological and developmental processes. Defects in their assembly and function are implicated in a wide range of human diseases ranging from retinal degeneration to cancer. Chlamydomonas reinhardtii is an algal model system for studying mammalian cilium formation and function. Here, we report a simple synchronization method that allows detection of small changes in ciliary length by minimizing variability in the population. We find that this method alters the key relationship between cell size and the amount of protein accumulated for flagellar growth. This provides a rapid alternative to traditional methods of cell synchronization for uncovering novel regulators of cilia.

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

Active matter systems, and in particular the cell cytoskeleton, exhibit complex mechanochemical dynamics that are still not well understood. While prior computational models of cytoskeletal dynamics have lead to many conceptual insights, an important niche still needs to be filled with a high-resolution structural modeling framework, which includes a minimally-complete set of cytoskeletal chemistries, stochastically treats reaction and diffusion processes in three spatial dimensions, accurately and efficiently describes mechanical deformations of the filamentous network under stresses generated by molecular motors, and deeply couples mechanics and chemistry at high spatial resolution. To address this need, we propose a novel reactive coarse-grained force field, as well as a publicly available software package, named the Mechanochemical Dynamics of Active Networks (MEDYAN), for simulating active network evolution and dynamics (available at www.medyan.org). This model can be used to study the non-linear, far from equilibrium processes in active matter systems, in particular, comprised of interacting semi-flexible polymers embedded in a solution with complex reaction-diffusion processes. In this work, we applied MEDYAN to investigate a contractile actomyosin network consisting of actin filaments, alpha-actinin cross-linking proteins, and non-muscle myosin IIA mini-filaments. We found that these systems undergo a switch-like transition in simulations from a random network to ordered, bundled structures when cross-linker concentration is increased above a threshold value, inducing contraction driven by myosin II mini-filaments. Our simulations also show how myosin II mini-filaments, in tandem with cross-linkers, can produce a range of actin filament polarity distributions and alignment, which is crucially dependent on the rate of actin filament turnover and the actin filament’s resulting super-diffusive behavior in the actomyosin-cross-linker system. We discuss the biological implications of these findings for the arc formation in lamellipodium-to-lamellum architectural remodeling. Lastly, our simulations produce force-dependent accumulation of myosin II, which is thought to be responsible for their mechanosensation ability, also spontaneously generating myosin II concentration gradients in the solution phase of the simulation volume.

Active matter systems have the distinct ability to convert energy from their surroundings into mechanical work, which gives rise to them having highly dynamic properties. Modeling active matter systems and capturing their complex behavior has been a great challenge in past years due to the many coupled interactions between their constituent parts, including not only distinct chemical and mechanical properties, but also feedback between them. One of the most intriguing biological active matter systems is the cell cytoskeleton, which can dynamically respond to chemical and mechanical cues to control cell structure and shape, playing a central role in many higher-order cellular processes. To model these systems and reproduce their behavior, we present a new modeling approach which combines the chemical, mechanical, and molecular transport aspects of active matter systems, all represented with equivalent complexity, while also allowing for various forms of mechanochemical feedback. This modeling approach, named MEDYAN, and software implementation is flexible so that a wide range of active matter systems can be simulated with a high level of detail, and ultimately can help to describe active matter phenomena, and in particular, the dynamics of the cell cytoskeleton. In this work, we have used MEDYAN to simulate a cytoskeletal network consisting of actin filaments, cross-linking proteins, and myosin II molecular motors. We found that these systems show rich dynamical behaviors, undergoing alignment and bundling transitions, with an emergent contractility, as the concentrations of myosin II and cross-linking proteins, as well as actin filament turnover rates, are varied.

How Cells Measure Length on Subcellular Scales

Cells are not just amorphous bags of enzymes, but precise and complex machines. With any machine, it is important that the parts be of the right size, yet our understanding of the mechanisms that control size of cellular structures remains at a rudimentary level in most cases. One problem with studying size control is that many cellular organelles have complex 3D structures that make their size hard to measure. Here we focus on linear structures within cells, for which the problem of size control reduces to the problem of length control. We compare and contrast potential mechanisms for length control to understand how cells solve simple geometry problems.

Combinative in vitro studies and computational model to predict 3D cell migration response to drug insult

The development of drugs to counter diseases related to cell migration has resulted in a multi-billion dollar endeavor. Unfortunately, few drugs have emerged from this effort highlighting the need for new methods to enhance assays to study, analyze and control cell migration. In response to this complex process, computational models have emerged as potent tools to describe migration providing a high throughput and low cost method. However, most models are unable to predict migration response to drug with direct application to in vitro experiments. In addition to this, no model to date has attempted to describe migration in response to drugs while incorporating simultaneously protein signaling, proteolytic activity, and 3D culture. In this paper, we describe an integrated computational approach, in conjunction with in vitro observations, to serve as a platform to accurately predict migration in 3D matrices incorporating the function of matrix metalloproteinases (MMPs) and their interaction with the Extracellular signal-related kinase (ERK) signaling pathway. Our results provide biological insight into how matrix density, MMP activity, integrin adhesions, and p-ERK expression all affect speed and persistence in 3D. Predictions from the model provide insight toward improving drug combinations to more effectively reduce both speed and persistence during migration and the role of integrin adhesions in motility. In this way our integrated platform provides future potential to streamline and improve throughput toward the testing and development of migration targeting drugs with tangible application to current in vitro assays.

Combined effects of temperature, pressure, and co-solvents on the polymerization kinetics of actin

In vivo studies have shown that the cytoskeleton of cells is very sensitive to changes in temperature and pressure. In particular, actin filaments get depolymerized when pressure is increased up to several hundred bars, conditions that are easily encountered in the deep sea. We quantitatively evaluate the effects of temperature, pressure, and osmolytes on the kinetics of the polymerization reaction of actin by high-pressure stopped-flow experiments in combination with fluorescence detection and an integrative stochastic simulation of the polymerization process. We show that the compatible osmolyte trimethylamine-N-oxide is not only able to compensate for the strongly retarding effect of chaotropic agents, such as urea, on actin polymerization, it is also able to largely offset the deteriorating effect of pressure on actin polymerization, thereby allowing biological cells to better cope with extreme environmental conditions.

Cosolvent and crowding effects on the polymerization kinetics of actin

Effects of cosolvents and macromolecular crowding agents on the G-to-F-transformation of actin are studied. Drastic and diverse changes in the lag phase and association rates of polymerizing actin are observed under different solvent conditions.

Quantitative analysis of approaches to measure cooperative phosphate release in polymerized actin

We use stochastic simulations that treat several experimental probes of actin dynamics to explore the extent to which phosphate dissociation in filamentous actin may be cooperative. Phosphate time-courses from polymerization and copolymerization experiments of ATP- and ADP-actin are studied, including the effects of variations in filament-number concentration as well as single-filament depolymerization time-courses. We find that highly cooperative models are consistent with the treated experimental data. We also find that some types of experiments that are believed to provide strong constraints on the cooperativity of actin hydrolysis models do not provide such constraints.

Polymerization of actin filaments coupled with adenosine triphosphate hydrolysis: Brownian dynamics and theoretical analysis

Polymerization dynamics of single actin filaments coupled with adenosine triphosphate (ATP) hydrolysis is investigated via both theoretical analysis and Brownian dynamics simulations. Brownian dynamics simulations have been applied recently to study the growth behaviors of long filaments as a function of the free actin monomer concentrations, C(T), which is found to be in agreement with the associated experiments. In the present study, both ATP cap length and length diffusivity are studied as a function of the free ATP-actin monomer concentrations, C(T). The exact analytical expressions are found to be in perfect consistency with Brownian dynamics simulations. Likewise, we find that the length diffusion coefficient is peaked near the critical concentration, C(T,cr). It is, therefore, expected that the dependence of length diffusivity on ATP-actin monomer concentrations is utilized to analyze the surprising experiments on the length fluctuations of individual actin filaments.

Fragmentation is crucial for the steady-state dynamics of actin filaments

Despite the recognition that actin filaments are important for numerous cellular processes, and decades of investigation, the dynamics of in vitro actin filaments are still not completely understood. Here, we follow the time evolution of the length distribution of labeled actin reporter filaments in an unlabeled F-actin solution via fluorescence microscopy. Whereas treadmilling and diffusive length fluctuations cannot account for the observed dynamics, our results suggest that at low salt conditions, spontaneous fragmentation is crucial.

Treadmilling of actin filaments via Brownian dynamics simulations

Actin polymerization is coupled to the hydrolysis of adenosine triphosphate (ATP) into adenosine diphosphate (ADP) and inorganic phosphate (P(i)). Therefore, each protomer within an actin filament can attain three different nucleotide states corresponding to bound ATP, ADP/P(i), and ADP. These protomer states form spatial patterns on the growing (or shrinking) filaments. Using Brownian dynamics simulations, the growth behavior of long filaments is studied, together with the associated protomer patterns, as a function of ATP-actin monomer concentration, C(T), within the surrounding solution. For concentrations close to the critical concentration C(T)=C(T,cr), the filaments undergo treadmilling, i.e., they grow at the barbed and shrink at the pointed end, which leads to directed translational motion of the whole filament. The corresponding nonequilibrium states are characterized by several global fluxes and by spatial density and flux profiles along the filaments. We focus on a certain set of transition rates as deduced from in vitro experiments and find that the associated treadmilling (or turnover) rate is about 0.08 monomers per second.

A theoretical analysis of filament length fluctuations in actin and other polymers

Control of the structure and dynamics of the actin cytoskeleton is essential for cell motility and for maintaining the structural integrity of cells. Central to understanding the control of these features is an understanding of the dynamics of actin filaments, first as isolated filaments, then as integrated networks, and finally as networks containing higher-order structures such as bundles, stress fibers and acto-myosin complexes. It is known experimentally that single filaments can exhibit large fluctuations, but a detailed understanding of the transient dynamics involved is still lacking. Here we first study stochastic models of a general system involving two-monomer types that can be analyzed completely, and then we report stochastic simulations on the complete actin model with three monomer types. We systematically examine the transient behavior of filament length dynamics so as to gain a better understanding of the time scales involved in reaching a steady state. We predict the lifetime of a cap of one monomer type and obtain the mean and variance of the survival time of a cap at the filament end, which together determine the filament length fluctuations.

Role of ATP-hydrolysis in the dynamics of a single actin filament

We study the stochastic dynamics of growth and shrinkage of single actin filaments taking into account insertion, removal, and ATP hydrolysis of subunits either according to the vectorial mechanism or to the random mechanism. In a previous work, we developed a model for a single actin or microtubule filament where hydrolysis occurred according to the vectorial mechanism: the filament could grow only from one end, and was in contact with a reservoir of monomers. Here we extend this approach in two ways--by including the dynamics of both ends and by comparing two possible mechanisms of ATP hydrolysis. Our emphasis is mainly on two possible limiting models for the mechanism of hydrolysis within a single filament, namely the vectorial or the random model. We propose a set of experiments to test the nature of the precise mechanism of hydrolysis within actin filaments.

Mechanisms controlling cell size and shape during isotropic cell spreading

Cell motility is important for many developmental and physiological processes. Motility arises from interactions between physical forces at the cell surface membrane and the biochemical reactions that control the actin cytoskeleton. To computationally analyze how these factors interact, we built a three-dimensional stochastic model of the experimentally observed isotropic spreading phase of mammalian fibroblasts. The multiscale model is composed at the microscopic levels of three actin filament remodeling reactions that occur stochastically in space and time, and these reactions are regulated by the membrane forces due to membrane surface resistance (load) and bending energy. The macroscopic output of the model (isotropic spreading of the whole cell) occurs due to the movement of the leading edge, resulting solely from membrane force-constrained biochemical reactions. Numerical simulations indicate that our model qualitatively captures the experimentally observed isotropic cell-spreading behavior. The model predicts that increasing the capping protein concentration will lead to a proportional decrease in the spread radius of the cell. This prediction was experimentally confirmed with the use of Cytochalasin D, which caps growing actin filaments. Similarly, the predicted effect of actin monomer concentration was experimentally verified by using Latrunculin A. Parameter variation analyses indicate that membrane physical forces control cell shape during spreading, whereas the biochemical reactions underlying actin cytoskeleton dynamics control cell size (i.e., the rate of spreading). Thus, during cell spreading, a balance between the biochemical and biophysical properties determines the cell size and shape. These mechanistic insights can provide a format for understanding how force and chemical signals together modulate cellular regulatory networks to control cell motility.

A mathematical model of actin filament turnover for fitting FRAP data

A novel mathematical model of the actin dynamics in living cells under steady-state conditions has been developed for fluorescence recovery after photobleaching (FRAP) experiments. As opposed to other FRAP fitting models, which use the average lifetime of actins in filaments and the actin turnover rate as fitting parameters, our model operates with unbiased actin association/dissociation rate constants and accounts for the filament length. The mathematical formalism is based on a system of stochastic differential equations. The derived equations were validated on synthetic theoretical data generated by a stochastic simulation algorithm adapted for the simulation of FRAP experiments. Consistent with experimental findings, the results of this work showed that (1) fluorescence recovery is a function of the average filament length, (2) the F-actin turnover and the FRAP are accelerated in the presence of actin nucleating proteins, (3) the FRAP curves may exhibit both a linear and non-linear behaviour depending on the parameters of actin polymerisation, and (4) our model resulted in more accurate parameter estimations of actin dynamics as compared with other FRAP fitting models. Additionally, we provide a computational tool that integrates the model and that can be used for interpretation of FRAP data on actin cytoskeleton.

An integrative simulation model linking major biochemical reactions of actin-polymerization to structural properties of actin filaments

We report on an advanced universal Monte Carlo simulation model of actin polymerization processes offering a broad application panel. The model integrates major actin-related reactions, such as assembly of actin nuclei, association/dissociation of monomers to filament ends, ATP-hydrolysis via ADP-Pi formation and ADP-ATP exchange, filament branching, fragmentation and annealing or the effects of regulatory proteins. Importantly, these reactions are linked to information on the nucleotide state of actin subunits in filaments (ATP hydrolysis) and the distribution of actin filament lengths. The developed stochastic simulation modelling schemes were validated on: i) synthetic theoretical data generated by a deterministic model and ii) sets of our and published experimental data obtained from fluorescence pyrene-actin experiments. Build on an open-architecture principle, the designed model can be extended for predictive evaluation of the activities of other actin-interacting proteins and can be applied for the analysis of experimental pyrene actin-based or fluorescence microscopy data. We provide a user-friendly, free software package ActinSimChem that integrates the implemented simulation algorithms and that is made available to the scientific community for modelling in silico any specific actin-polymerization system.

Simulating the formation of keratin filament networks by a piecewise-deterministic Markov process

Keratin intermediate filament networks are part of the cytoskeleton in epithelial cells. They were found to regulate viscoelastic properties and motility of cancer cells. Due to unique biochemical properties of keratin polymers, the knowledge of the mechanisms controlling keratin network formation is incomplete. A combination of deterministic and stochastic modeling techniques can be a valuable source of information since they can describe known mechanisms of network evolution while reflecting the uncertainty with respect to a variety of molecular events. We applied the concept of piecewise-deterministic Markov processes to the modeling of keratin network formation with high spatiotemporal resolution. The deterministic component describes the diffusion-driven evolution of a pool of soluble keratin filament precursors fueling various network formation processes. Instants of network formation events are determined by a stochastic point process on the time axis. A probability distribution controlled by model parameters exercises control over the frequency of different mechanisms of network formation to be triggered. Locations of the network formation events are assigned dependent on the spatial distribution of the soluble pool of filament precursors. Based on this modeling approach, simulation studies revealed that the architecture of keratin networks mostly depends on the balance between filament elongation and branching processes. The spatial distribution of network mesh size, which strongly influences the mechanical characteristics of filament networks, is modulated by lateral annealing processes. This mechanism which is a specific feature of intermediate filament networks appears to be a major and fast regulator of cell mechanics.

Growing actin networks form lamellipodium and lamellum by self-assembly

Many different cell types are able to migrate by formation of a thin actin-based cytoskeletal extension. Recently, it became evident that this extension consists of two distinct substructures, designated lamellipodium and lamellum, which differ significantly in their kinetic and kinematic properties as well as their biochemical composition. We developed a stochastic two-dimensional computer simulation that includes chemical reaction kinetics, G-actin diffusion, and filament transport to investigate the formation of growing actin networks in migrating cells. Model parameters were chosen based on experimental data or theoretical considerations. In this work, we demonstrate the system's ability to form two distinct networks by self-organization. We found a characteristic transition in mean filament length as well as a distinct maximum in depolymerization flux, both within the first 1-2 microm. The separation into two distinct substructures was found to be extremely robust with respect to initial conditions and variation of model parameters. We quantitatively investigated the complex interplay between ADF/cofilin and tropomyosin and propose a plausible mechanism that leads to spatial separation of, respectively, ADF/cofilin- or tropomyosin-dominated compartments. Tropomyosin was found to play an important role in stabilizing the lamellar actin network. Furthermore, the influence of filament severing and annealing on the network properties is explored, and simulation data are compared to existing experimental data.

A stochastic analysis of actin polymerization in the presence of twinfilin and gelsolin

We develop an efficient stochastic simulation algorithm for analyzing actin filament growth and decay in the presence of various actin-binding proteins. The evolution of nucleotide profiles of filaments can be tracked and the resulting feedback to actin-binding proteins is incorporated. The computational efficiency of the new method enables us to focus on experimentally realistic problems, and as one example we use it to analyze the experimental data of Helfer et al. [(2006). Mammalian twinfilin sequesters ADP-G-actin and caps filament barbed ends: implications in motility. EMBO J. 25, 1184-1195] on the capping and G-actin sequestering activity of twinfilin. We show that the binding specificity of twinfilin for ADP-G-actin is crucial for the observed biphasic evolution of the filament length distribution in the presence of twinfilin, and we demonstrate that twinfilin can be an essential part of the molecular machinery for regulating filament lengths after a short burst of polymerization. Significantly, our simulations indicate that the pyrenyl-actin fluorescence experiments would fail to report the emergence of large filaments under certain experimental conditions.