A mechanistic model of the actin cycle

Molecular mechanisms of inorganic-phosphate release from the core and barbed end of actin filaments

The release of inorganic phosphate (Pi) from actin filaments constitutes a key step in their regulated turnover, which is fundamental to many cellular functions. The mechanisms underlying Pi release from the core and barbed end of actin filaments remain unclear. Here, using human and bovine actin isoforms, we combine cryo-EM with molecular-dynamics simulations and in vitro reconstitution to demonstrate how actin releases Pi through a 'molecular backdoor'. While constantly open at the barbed end, the backdoor is predominantly closed in filament-core subunits and opens only transiently through concerted amino acid rearrangements. This explains why Pi escapes rapidly from the filament end but slowly from internal subunits. In a nemaline-myopathy-associated actin variant, the backdoor is predominantly open in filament-core subunits, resulting in accelerated Pi release and filaments with drastically shortened ADP-Pi caps. Our results provide the molecular basis for Pi release from actin and exemplify how a disease-linked mutation distorts the nucleotide-state distribution and atomic structure of the filament.

Actin polymerization and depolymerization in developing vertebrates

Development is a complex process that occurs throughout the life cycle. F-actin, a major component of the cytoskeleton, is essential for the morphogenesis of tissues and organs during development. F-actin is formed by the polymerization of G-actin, and the dynamic balance of polymerization and depolymerization ensures proper cellular function. Disruption of this balance results in various abnormalities and defects or even embryonic lethality. Here, we reviewed recent findings on the structure of G-actin and F-actin and the polymerization of G-actin to F-actin. We also focused on the functions of actin isoforms and the underlying mechanisms of actin polymerization/depolymerization in cellular and organic morphogenesis during development. This information will extend our understanding of the role of actin polymerization in the physiologic or pathologic processes during development and may open new avenues for developing therapeutics for embryonic developmental abnormalities or tissue regeneration.

Mechanics of cell-cell junctions

Tissue cells in epithelial or endothelial monolayers are connected through cell-cell junctions, which are stabilized by transmembrane E-cadherin bonds and intracellular actin filaments. These bonds and junctions play a crucial role in maintaining the barrier function of epithelia and endothelia and are believed to transmit forces between cells. Additionally, E-cadherin bonds can impact the shape of cell-cell junctions. In this study, we develop a continuum mechanical model of the cell-cell junction by explicitly incorporating the cell membrane, distributions of E-cadherin bonds, cytoplasmic fluid pressure, and F-actin dynamics. The static force-balanced version of the model is able to analyze the influences of cell cortical tension, actin dynamics, and cytoplasmic pressure on the junction shape and E-cadherin bonds. Furthermore, an extended model that incorporates fluid flow, across the cell boundary as well as around the cell, is also examined. This model can couple cell-shape changes with cell cortical tension and fluid flow, and predicts the additional effect of fluid motion on cell-cell junction mechanics. Taken together, our models serve as an intermediate link between molecular-scale models of cell-junction molecules and cell-scale models of tissue and epithelia.

Mechano-immunology in microgravity

Life on Earth has evolved to thrive in the Earth's natural gravitational field; however, as space technology advances, we must revisit and investigate the effects of unnatural conditions on human health, such as gravitational change. Studies have shown that microgravity has a negative impact on various systemic parts of humans, with the effects being more severe in the human immune system. Increasing costs, limited experimental time, and sample handling issues hampered our understanding of this field. To address the existing knowledge gap and provide confidence in modelling the phenomena, in this review, we highlight experimental works in mechano-immunology under microgravity and different computational modelling approaches that can be used to address the existing problems.

Compressive and Tensile Deformations Alter ATP Hydrolysis and Phosphate Release Rates in Actin Filaments

Actin filament networks in eukaryotic cells are constantly remodeled through nucleotide state controlled interactions with actin binding proteins, leading to macroscopic structures such as bundled filaments, branched filaments, and so on. The nucleotide (ATP) hydrolysis, phosphate release, and polymerization/depolymerization reactions that lead to the formation of these structures are correlated with the conformational fluctuations of the actin subunits at the molecular scale. The resulting structures generate and experience varying levels of force and impart cells with several functionalities such as their ability to move, divide, transport cargo, etc. Models that explicitly connect the structure to reactions are essential to elucidate a fundamental level of understanding of these processes. In this regard, a bottom-up Ultra-Coarse-Grained (UCG) model of actin filaments that can simulate ATP hydrolysis, inorganic phosphate release (Pi), and depolymerization reactions is presented in this work. In this model, actin subunits are represented using coarse-grained particles that evolve in time and undergo reactions depending on the conformations sampled. The reactions are represented through state transitions, with each state represented by a unique effective coarse-grained potential. Effects of compressive and tensile strains on the rates of reactions are then analyzed. Compressive strains tend to unflatten the actin subunits, reduce the rate of ATP hydrolysis, and increase the Pi release rate. On the other hand, tensile strain flattens subunits, increases the rate of ATP hydrolysis, and decrease the Pi release rate. Incorporating these predictions into a Markov State Model highlighted that strains alter the steady-state distribution of subunits with ADPPi and ADP nucleotide, thus identifying possible additional factors underlying the cooperative binding of regulatory proteins to actin filaments.

The role of mode switching in a population of actin polymers with constraints

In this paper, we introduce a stochastic model for the dynamics of actin polymers and their interactions with other proteins in the cellular envelop. Each polymer elongates and shortens, and can switch between several modes depending on whether it is bound to accessory proteins that modulate its behaviour as, for example, elongation-promoting factors. Our main aim is to understand the dynamics of a large population of polymers, assuming that the only limiting quantity is the total amount of monomers, set to be constant to some large N. We first focus on the evolution of a very long polymer, of size [Formula: see text], with a rapid switch between modes (compared to the timescale over which the macroscopic fluctuations in the polymer size appear). Letting N tend to infinity, we obtain a fluid limit in which the effect of the switching appears only through the fraction of time spent in each mode at equilibrium. We show in particular that, in our situation where the number of monomers is limiting, a rapid binding-unbinding dynamics may lead to an increased elongation rate compared to the case where the polymer is trapped in any of the modes. Next, we consider a large population of polymers and complexes, represented by a random measure on some appropriate type space. We show that as N tends to infinity, the stochastic system converges to a deterministic limit in which the switching appears as a flow between two categories of polymers. We exhibit some numerical examples in which the limiting behaviour of a single polymer differs from that of a population of competing (shorter) polymers for equivalent model parameters. Taken together, our results demonstrate that under conditions where the total number of monomers is limiting, the study of a single polymer is not sufficient to understand the behaviour of an ensemble of competing polymers.

Fully coupled numerical model of actin treadmilling in the lamellipodium of the cell

Cells rely on an interplay of subcellular elements for motility and migration. Certain regions of motile cells, such as the lamellipodium, are made of a complex mixture of actin monomers and filaments, which polymerize at the front of the cell, close to the cell membrane, and depolymerize at the rear. The dynamic actin turnover induces the so-called intracellular retrograde flow, and it is a fundamental process for cell motility. Apart from some comprehensive mathematical models, the computational modelling of actin treadmilling has been based on simpler biophysical models. Here, we adopt a highly detailed theoretical model of the actin treadmilling process and develop a coupled unsteady finite element formulation. We clearly describe the structure and implementation of the coupled problem within the finite element method. Our numerical results show an excellent correlation with experimental results from literature and with previous models. We include time dependent effects and convective transport terms, which expose puzzling dynamics in the retrograde flow. We propose several biological scenarios to analyze the behavior of the actin treadmilling along space and time. We observed response times of the main density variables in the order of seconds. Compared with previous analytical solutions, which make assumptions related to convective transport, transient dynamics, and actin fluxes, the generic solution can have significant influence on the retrograde flow. All together, our results unveil a promising applicability of classical finite element methods to derive an in silico testing platform for the actin treadmilling processes in motile cells, which could allow for an extension to other biophysical effects.

Cell mechanics: a dialogue

Under the microscope, eukaryotic animal cells can adopt a variety of different shapes and sizes. These cells also move and deform, and the physical mechanisms driving these movements and shape changes are important in fundamental cell biology, tissue mechanics, as well as disease biology. This article reviews some of the basic mechanical concepts in cells, emphasizing continuum mechanics description of cytoskeletal networks and hydrodynamic flows across the cell membrane. We discuss how cells can generate movement and shape changes by controlling mass fluxes at the cell boundary. These mass fluxes can come from polymerization/depolymerization of actin cytoskeleton, as well as osmotic and hydraulic pressure-driven flow of water across the cell membrane. By combining hydraulic pressure control with force balance conditions at the cell surface, we discuss a quantitative mechanism of cell shape and volume control. The broad consequences of this model on cell mechanosensation and tissue mechanics are outlined.

Association-dissociation process with aging subunits: Recursive solution

The coupling of stochastic growth and shrinkage of one-dimensional structures to random aging of the constituting subunits defines the simple association-dissociation-aging process which captures the essential features of the nonequilibrium assembly of cytoskeletal filaments. Because of correlations, previously employed mean-field methods fail to correctly describe filament growth. We study an alternative formulation of the full master equation of the stochastic process. An ansatz for the steady-state solution leads to a recursion relation which allows for the calculation of all emergent quantities with increasing accuracy and in excellent agreement with stochastic simulations.

Two-Phase Acto-Cytosolic Fluid Flow in a Moving Keratocyte: A 2D Continuum Model

The F-actin network and cytosol in the lamellipodia of crawling cells flow in a centripetal pattern and spout-like form, respectively. We have numerically studied this two-phase flow in the realistic geometry of a moving keratocyte. Cytosol has been treated as a low viscosity Newtonian fluid flowing through the high viscosity porous medium of F-actin network. Other involved phenomena including myosin activity, adhesion friction, and interphase interaction are also discussed to provide an overall view of this problem. Adopting a two-phase coupled model by myosin concentration, we have found new accurate perspectives of acto-cytosolic flow and pressure fields, myosin distribution, as well as the distribution of effective forces across the lamellipodia of a keratocyte with stationary shape. The order of magnitude method is also used to determine the contribution of forces in the internal dynamics of lamellipodia.

Actin age orchestrates myosin-5 and myosin-6 run lengths

Unlike a static and immobile skeleton, the actin cytoskeleton is a highly dynamic network of filamentous actin (F-actin) polymers that continuously turn over. In addition to generating mechanical forces and sensing mechanical deformation, dynamic F-actin networks serve as cellular tracks for myosin motor traffic. However, much of our mechanistic understanding of processive myosins comes from in vitro studies in which motility was studied on pre-assembled and artificially stabilized, static F-actin tracks. In this work, we examine the role of actin dynamics in single-molecule myosin motility using assembling F-actin and two highly processive motors, myosin-5 and myosin-6. These two myosins have distinct functions in the cell and travel in opposite directions along actin filaments [1-3]. Myosin-5 walks toward the barbed ends of F-actin, traveling to sites of actin polymerization at the cell periphery [4]. Myosin-6 walks toward the pointed end of F-actin [5], traveling toward the cell center along older segments of the actin filament. We find that myosin-5 takes 1.3- to 1.5-fold longer runs on ADP•Pi (young) F-actin, whereas myosin-6 takes 1.7- to 3.6-fold longer runs along ADP (old) F-actin. These results suggest that conformational differences between ADP•Pi and ADP F-actin tailor these myosins to walk farther toward their preferred actin filament end. Taken together, these experiments define a new mechanism by which myosin traffic may sort to different F-actin networks depending on filament age.

Two-tiered coupling between flowing actin and immobilized N-cadherin/catenin complexes in neuronal growth cones

Neuronal growth cones move forward by dynamically connecting actin-based motility to substrate adhesion, but the mechanisms at the individual molecular level remain unclear. We cultured primary neurons on N-cadherin-coated micropatterned substrates, and imaged adhesion and cytoskeletal proteins at the ventral surface of growth cones using single particle tracking combined to photoactivated localization microscopy (sptPALM). We demonstrate transient interactions in the second time scale between flowing actin filaments and immobilized N-cadherin/catenin complexes, translating into a local reduction of the actin retrograde flow. Normal actin flow on micropatterns was rescued by expression of a dominant negative N-cadherin construct competing for the coupling between actin and endogenous N-cadherin. Fluorescence recovery after photobleaching (FRAP) experiments confirmed the differential kinetics of actin and N-cadherin, and further revealed a 20% actin population confined at N-cadherin micropatterns, contributing to local actin accumulation. Computer simulations with relevant kinetic parameters modeled N-cadherin and actin turnover well, validating this mechanism. Such a combination of short- and long-lived interactions between the motile actin network and spatially restricted adhesive complexes represents a two-tiered clutch mechanism likely to sustain dynamic environment sensing and provide the force necessary for growth cone migration.

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.

A mathematical model for eph/ephrin-directed segregation of intermingled cells

Eph receptors, the largest family of receptor tyrosine kinases, control cell-cell adhesion/de-adhesion, cell morphology and cell positioning through interaction with cell surface ephrin ligands. Bi-directional signalling from the Eph and ephrin complexes on interacting cells have a significant role in controlling normal tissue development and oncogenic tissue patterning. Eph-mediated tissue patterning is based on the fine-tuned balance of adhesion and de-adhesion reactions between distinct Eph- and ephrin-expressing cell populations, and adhesion within like populations (expressing either Eph or ephrin). Here we develop a stochastic, Lagrangian model that is based on Eph/ephrin biology: incorporating independent Brownian motion to describe cell movement and a deterministic term (the drift term) to represent repulsive and adhesive interactions between neighbouring cells. Comparison between the experimental and computer simulated Eph/ephrin cell patterning events shows that the model recapitulates the dynamics of cell-cell segregation and cell cluster formation. Moreover, by modulating the term for Eph/ephrin-mediated repulsion, the model can be tuned to match the actual behaviour of cells with different levels of Eph expression or activity. Together the results of our experiments and modelling suggest that the complexity of Eph/ephrin signalling mechanisms that control cell-cell interactions can be described well by a mathematical model with a single term balancing adhesion and de-adhesion between interacting cells. This model allows reliable prediction of Eph/ephrin-dependent control of cell patterning behaviour.

Allosteric regulation of DNA cleavage and sequence-specificity through run-on oligomerization

SgrAI is a sequence specific DNA endonuclease that functions through an unusual enzymatic mechanism that is allosterically activated 200- to 500-fold by effector DNA, with a concomitant expansion of its DNA sequence specificity. Using single-particle transmission electron microscopy to reconstruct distinct populations of SgrAI oligomers, we show that in the presence of allosteric, activating DNA, the enzyme forms regular, repeating helical structures characterized by the addition of DNA-binding dimeric SgrAI subunits in a run-on manner. We also present the structure of oligomeric SgrAI at 8.6 Å resolution, demonstrating the conformational state of SgrAI in its activated form. Activated and oligomeric SgrAI displays key protein-protein interactions near the helix axis between its N termini, as well as allosteric protein-DNA interactions that are required for enzymatic activation. The hybrid approach reveals an unusual mechanism of enzyme activation that explains SgrAI's oligomerization and allosteric behavior.

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.

Theoretical analysis of microtubules dynamics using a physical-chemical description of hydrolysis

Microtubules are cytoskeleton multifilament proteins that support many fundamental biological processes such as cell division, cellular transport, and motility. They can be viewed as dynamic polymers that function in nonequilibrium conditions stimulated by hydrolysis of GTP (guanosine triphosphate) molecules bound to their monomers. We present a theoretical description of microtubule dynamics based on discrete-state stochastic models that explicitly takes into account all relevant biochemical transitions. In contrast to previous theoretical analysis, a more realistic physical-chemical description of GTP hydrolysis is presented, in which the hydrolysis rate at a given monomer depends on the chemical composition of the neighboring monomers. This dependence naturally leads to a cooperativity in the hydrolysis. It is found that this cooperativity significantly influences all dynamic properties of microtubules. It is suggested that the dynamic instability in cytoskeleton proteins might be observed only for weak cooperativity, while the strong cooperativity in hydrolysis suppresses the dynamic instability. The presented microscopic analysis is compared with existing phenomenological descriptions of hydrolysis processes. Our analytical calculations, supported by computer Monte Carlo simulations, are also compared with available experimental observations.

Mathematical modeling of eukaryotic cell migration: insights beyond experiments

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

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.

Deconstructing signal transduction pathways that regulate the actin cytoskeleton in dendritic spines

Dendritic spines are the sites of most excitatory synapses in the central nervous system. Recent studies have shown that spines function independently of each other, and they are currently the smallest known processing units in the brain. Spines exist in an array of morphologies, and spine structure helps dictate synaptic function. Dendritic spines are rich in actin, and actin rearrangements are critical regulators of spine morphology and density. In this review, we discuss the importance of actin in regulating dendritic spine morphogenesis, and discuss the upstream signal transduction pathways that either foster or inhibit actin polymerization. The understanding of actin regulatory pathways is best conceptualized as a hierarchical network in which molecules function in discrete levels defined by their molecular distance to actin. To this end, we focus on several classes of molecules, including guanine nucleotide exchange factors, small GTPases, small GTPase effectors, and actin binding proteins. We discuss how individual proteins in these molecular classes impact spine morphogenesis, and reveal the biochemical interactions in these networks that are responsible for shaping actin polymerization. Finally, we discuss the importance of these actin regulatory pathways in neuropsychiatric disorders.

Modeling morphodynamic phenotypes and dynamic regimes of cell motion

Many cellular processes and signaling pathways converge onto cell morphology and cell motion, which share important components. The mechanisms used for propulsion could also be responsible for shape changes, if they are capable of generating the rich observed variety of dynamic regimes. Additionally, the analysis of cell shape changes in space and time promises insight into the state of the cytoskeleton and signaling pathways controlling it. While this has been obvious for some time by now, little effort has been made to systematically and quantitatively explore this source of information. First pioneering experimental work revealed morphodynamic phenotypes which can be associated with dynamic regimes like oscillations and excitability. Here, we review the current state of modeling of morphodynamic phenotypes, the experimental results and discuss the ideas on the mechanisms driving shape changes which are suggested by modeling.

Individual actin filaments in a microfluidic flow reveal the mechanism of ATP hydrolysis and give insight into the properties of profilin

A novel microfluidic approach allows the analysis of the dynamics of individual actin filaments, revealing both their local ADP/ADP-Pi-actin composition and that Pi release is a random mechanism.

The hydrolysis of ATP associated with actin and profilin-actin polymerization is pivotal in cell motility. It is at the origin of treadmilling of actin filaments and controls their dynamics and mechanical properties, as well as their interactions with regulatory proteins. The slow release of inorganic phosphate (Pi) that follows rapid cleavage of ATP gamma phosphate is linked to an increase in the rate of filament disassembly. The mechanism of Pi release in actin filaments has remained elusive for over 20 years. Here, we developed a microfluidic setup to accurately monitor the depolymerization of individual filaments and determine their local ADP-Pi content. We demonstrate that Pi release in the filament is not a vectorial but a random process with a half-time of 102 seconds, irrespective of whether the filament is assembled from actin or profilin-actin. Pi release from the depolymerizing barbed end is faster (half-time of 0.39 seconds) and further accelerated by profilin. Profilin accelerates the depolymerization of both ADP- and ADP-Pi-F-actin. Altogether, our data show that during elongation from profilin-actin, the dissociation of profilin from the growing barbed end is not coupled to Pi release or to ATP cleavage on the terminal subunit. These results emphasize the potential of microfluidics in elucidating actin regulation at the scale of individual filaments.

Actin proteins assemble into microfilaments that control cell shape and movement by polymerizing or depolymerizing. These actin monomers can bind ATP or ADP molecules. The incorporation of an ATP-actin monomer into a growing filament results in rapid cleavage of ATP into ADP and inorganic phosphate (Pi), followed by a slower release of Pi. As a consequence, actin filaments are composed mainly of ADP- and ADP-Pi-actin subunits, which have different depolymerization kinetics and mechanical properties, and can be targeted specifically by regulatory proteins that affect filament function. Hence, the understanding of many cellular processes requires a knowledge of the ADP/ADP-Pi composition of actin filaments at a molecular scale. This has so far remained elusive because traditional studies rely on measuring an average over many filaments in solution. To address this issue, we developed a microfluidics setup to monitor individual filaments with light microscopy while rapidly changing their chemical environment. We find that depolymerization accelerates progressively and corresponds to an exponential ADP-Pi-actin profile in the filament, meaning that each subunit releases its Pi with the same rate. Our method also provides novel insight into the function of profilin, a protein important for regulation of actin dynamics in cells, thus demonstrating the method's potential in the functional analysis of actin regulators.

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.

Force propagation and force generation in cells

Determining how forces are produced by and propagated through the cytoskeleton (CSK) of the cell is of great interest as dynamic processes of the CSK are intimately correlated with many molecular signaling pathways. We are presenting a novel approach for integrating measurements on cell elasticity, transcellular force propagation, and cellular force generation to obtain a comprehensive description of dynamic and mechanical properties of the CSK under force loading. This approach uses a combination of scanning force microscopy (SFM) and Total Internal Reflection Fluorescence (TIRF) microscopy. We apply well-defined loading schemes onto the apical cell membrane of fibroblasts using the SFM and simultaneously use TIRF microscopy to image the topography of the basal cell membrane. The locally distinct changes of shape and depth of the cytoskeletal imprints onto the basal membrane are interpreted as results of force propagation through the cytoplasm. This observation provides evidence for the tensegrity model and demonstrates the usefulness of our approach that does not depend on potentially disturbing marker compounds. We confirm that the actin network greatly determines cell stiffness and represents the substrate that mediates force transduction through the cytoplasm of the cell. The latter is an essential feature of tensegrity. Most importantly, our new finding that, both intact actin and microtubule networks are required for enabling the cell to produce work, can only be understood within the framework of the tensegrity model. We also provide, for the first time, a direct measurement of the cell's mechanical power output under compression at two femtowatts.

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.

How depolymerization can promote polymerization: the case of actin and profilin

Rapid polymerization and depolymerization of actin filaments in response to extracellular stimuli is required for normal cell motility and development. Profilin is one of the most important actin-binding proteins; it regulates actin polymerization and interacts with many cytoskeletal proteins that link actin to extracellular membrane. The molecular mechanism of profilin has been extensively considered and debated in the literature for over two decades. Here we discuss several accepted hypotheses regarding the mechanism of profilin function as well as new recently emerged possibilities. Thermal noise is routine in molecular world and unsurprisingly, nature has found a way to utilize it. An increasing amount of theoretical and experimental research suggests that fluctuation-based processes play important roles in many cell events. Here we show how a fluctuation-based process of exchange diffusion is involved in the regulation of actin polymerization.

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.

Accelerators, Brakes, and Gears of Actin Dynamics in Dendritic Spines

Dendritic spines are actin-rich structures that accommodate the postsynaptic sites of most excitatory synapses in the brain. Although dendritic spines form and mature as synaptic connections develop, they remain plastic even in the adult brain, where they can rapidly grow, change, or collapse in response to normal physiological changes in synaptic activity that underlie learning and memory. Pathological stimuli can adversely affect dendritic spine shape and number, and this is seen in neurodegenerative disorders and some forms of mental retardation and autism as well. Many of the molecular signals that control these changes in dendritic spines act through the regulation of filamentous actin (F-actin), some through direct interaction with actin, and others via downstream effectors. For example, cortactin, cofilin, and gelsolin are actin-binding proteins that directly regulate actin dynamics in dendritic spines. Activities of these proteins are precisely regulated by intracellular signaling events that control their phosphorylation state and localization. In this review, we discuss how actin-regulating proteins maintain the balance between F-actin assembly and disassembly that is needed to stabilize mature dendritic spines, and how changes in their activities may lead to rapid remodeling of dendritic spines.

Structural plasticity in actin and tubulin polymer dynamics

Actin filaments and microtubules polymerize and depolymerize by adding and removing subunits at polymer ends, and these dynamics drive cytoplasmic organization, cell division, and cell motility. Since Wegner proposed the treadmilling theory for actin in 1976, it has largely been assumed that the chemical state of the bound nucleotide determines the rates of subunit addition and removal. This chemical kinetics view is difficult to reconcile with observations revealing multiple structural states of the polymer that influence polymerization dynamics but that are not strictly coupled to the bound nucleotide state. We refer to these phenomena as "structural plasticity" and discuss emerging evidence that they play a central role in polymer dynamics and function.

Actin polymerization and depolymerization coupled to cooperative hydrolysis

The hydrolysis of adenosine triphosphate (ATP) during actin (de)polymerization consists of two subprocesses, ATP cleavage and phosphate (P_{i}) release, which involve three nucleotide states of each actin protomer. A new theoretical model that explicitly incorporates these different subprocesses and states is introduced and compared with recent experimental data for actin depolymerization. These data can be explained by strongly cooperative ATP cleavage followed by strongly cooperative P_{i} release but are incompatible with random and/or vectorial subprocesses as proposed previously.

An open model of actin dendritic nucleation

The availability of quantitative experimental data on the kinetics of actin assembly has enabled the construction of many mathematical models focused on explaining specific behaviors of this complex system. However these ad hoc models are generally not reusable or accessible by the large community of actin biologists. In this work, we present a comprehensive model that integrates and unifies much of the in vitro data on the components of the dendritic nucleation mechanism for actin dynamics. More than 300 simulations have been run based on compartmental and three-dimensional spatial versions of this model. Several key findings are highlighted, including an explanation for the sharp boundary between actin assembly and disassembly in the lamellipodia of migrating cells. Because this model, with the simulation results, is "open source", in the sense that it is publicly available and editable through the Virtual Cell database (http://vcell.org), it can be accessed, analyzed, modified, and extended.

Nonequilibrium actin polymerization treated by a truncated rate-equation method

Actin polymerization time courses can exhibit rich nonequilibrium dynamics that have not yet been accurately described by simplified rate equations. Sophisticated stochastic simulations and elaborate recursion schemes have been used to model the nonequilibrium dynamics resulting from the hydrolysis and subsequent exchange of the nucleotide bound within the actin molecules. In this work, we use a truncation approach to derive a set of readily accessible deterministic rate equations which are significantly simpler than previous attempts at such modeling. These equations may be incorporated into whole-cell motility models which otherwise quickly become computationally inaccessible if polymerization of individual actin filaments is stochastically simulated within a virtual cell. Our equations accurately predict the relative concentrations of both monomeric and polymerized actin in differing nucleotide hydrolysis states throughout entire polymerization time courses nucleated via seed filaments. We extend our model to include the effects of capping protein. We also detail how our rate-equation method may be used to extract key parameters from experimental data.

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.

Mathematical models and simulations of cellular processes based on actin filaments

Actin filaments help to maintain the physical integrity of cells and participate in many processes that produce cellular movements. Studies of the processes that depend on actin filaments have progressed to the point where mathematical models and computer simulations are an essential part of the experimental toolkit. These quantitative models integrate knowledge about the structures of the key proteins and the rate and equilibrium constants for the reactions for comparison with a growing body of quantitative measurements of dynamic processes in live cells. Models and simulations are essential because it is impossible to appreciate by intuition alone the properties that emerge from a network of coupled reactions, particularly when the system contains many components, and force is one of the parameters.

Effect of profilin on actin critical concentration: a theoretical analysis

To explain the effect of profilin on actin critical concentration in a manner consistent with thermodynamic constraints and available experimental data, we built a thermodynamically rigorous model of actin steady-state dynamics in the presence of profilin. We analyzed previously published mechanisms theoretically and experimentally and, based on our analysis, suggest a new explanation for the effect of profilin. It is based on a general principle of indirect energy coupling. The fluctuation-based process of exchange diffusion indirectly couples the energy of ATP hydrolysis to actin polymerization. Profilin modulates this coupling, producing two basic effects. The first is based on the acceleration of exchange diffusion by profilin, which indicates, paradoxically, that a faster rate of actin depolymerization promotes net polymerization. The second is an affinity-based mechanism similar to the one suggested in 1993 by Pantaloni and Carlier although based on indirect rather than direct energy coupling. In the model by Pantaloni and Carlier, transformation of chemical energy of ATP hydrolysis into polymerization energy is regulated by direct association of each step in the hydrolysis reaction with a corresponding step in polymerization. Thus, hydrolysis becomes a time-limiting step in actin polymerization. In contrast, indirect coupling allows ATP hydrolysis to lag behind actin polymerization, consistent with experimental results.

Actin-binding proteins: how to reveal the conformational changes

Actin is the most abundant protein in eukaryotes. Under physiological conditions, it can polymerize into polarized filaments. At the heart of these processes are actin-binding proteins that stimulate actin assembly. Most of them are composed of multiple domains that perform both regulatory and signaling functions. Many actin-binding proteins, including WASP and formin family proteins, are auto-inhibited through intramolecular interactions that mask the actin-regulating sites of these proteins. The large flexible molecules of formins have so far eluded crystallization, and have been crystallized only partially. The information from the available crystal structures is valuable, but somewhat difficult to interpret without a larger framework on which to pose the actin-binding mechanism. Single-particle electron microscopy and electron tomography could provide such a large framework with the full-length structures of protein complexes. The recent advances in determining the molecular interactions in protein complexes predict that the molecular modeling and molecular dynamics methods could be employed to study conformational changes in molecules.

Mathematics of cell motility: have we got its number?

Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biological and mathematical mentalities continue to hinder the scientific progress. Here we focus on one area-cell motility-characterized by an unusually high modeling activity, largely due to a vast amount of quantitative, biophysical data, 'modular' character of motility, and pioneering vision of the area's experimental leaders. In this review, after brief introduction to biology of cell movements, we discuss quantitative models of actin dynamics, protrusion, adhesion, contraction, and cell shape and movement that made an impact on the process of biological discovery. We also comment on modeling approaches and open questions.

Actin polymerization overshoots and ATP hydrolysis as assayed by pyrene fluorescence

We investigate via stochastic simulation the overshoots observed in the fluorescence intensity of pyrene-labeled actin during rapid polymerization. We show that previous assumptions about pyrene intensity that ignore the intensity differences between subunits in different ATP hydrolysis states are not consistent with experimental data. This strong sensitivity of intensity to hydrolysis state implies that a measured pyrene intensity curve does not immediately reveal the true polymerization kinetics. We show that there is an optimal range of hydrolysis and phosphate release rate combinations simultaneously consistent with measured polymerization data from previously published severing and Arp2/3 complex-induced branching experiments. Within this range, we find that the pyrene intensity curves are described very accurately by the following average relative intensity coefficients: 0.37 for F-ATP actin; 0.55 for F-ADP + P(i) actin; and 0.75 for F-ADP actin. Finally, we present an analytic formula, which properly accounts for the sensitivity of the pyrene assay to hydrolysis state, for estimation of the concentration of free barbed ends from pyrene intensity curves.

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

Recent observations of F-actin dynamics call for theoretical models to interpret and understand the quantitative data. A number of existing models rely on simplifications and do not take into account F-actin fragmentation and annealing. We use Gillespie's algorithm for stochastic simulations of the F-actin dynamics including fragmentation and annealing. The simulations vividly illustrate that fragmentation and annealing have little influence on the shape of the polymerization curve and on nucleotide profiles within filaments but drastically affect the F-actin length distribution, making it exponential. We find that recent surprising measurements of high length diffusivity at the critical concentration cannot be explained by fragmentation and annealing events unless both fragmentation rates and frequency of undetected fragmentation and annealing events are greater than previously thought. The simulations compare well with experimentally measured actin polymerization data and lend additional support to a number of existing theoretical models.

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.

Regulation of actin filament assembly by Arp2/3 complex and formins

This review summarizes what is known about the biochemical and biophysical mechanisms that initiate the assembly of actin filaments in cells. Assembly and disassembly of these filaments contribute to many types of cellular movements. Numerous proteins regulate actin assembly, but Arp2/3 complex and formins are the focus of this review because more is known about them than other proteins that stimulate the formation of new filaments. Arp2/3 complex is active at the leading edge of motile cells, where it produces branches on the sides of existing filaments. Growth of these filaments produces force to protrude the membrane. Crystal structures, reconstructions from electron micrographs, and biophysical experiments have started to map out the steps through which proteins called nucleation-promoting factors stimulate the formation of branches. Formins nucleate and support the elongation of unbranched actin filaments for cytokinesis and various types of actin filament bundles. Formins associate processively with the fast-growing ends of filaments and protect them from capping.

Polymerization kinetics of ADP- and ADP-Pi-actin determined by fluorescence microscopy

We used fluorescence microscopy to determine how polymerization of Mg-ADP-actin depends on the concentration of phosphate. From the dependence of the elongation rate on the actin concentration and direct observations of depolymerizing filaments, we measured the polymerization rate constants of ADP-actin and ADP-P(i)-actin. Saturating phosphate reduces the critical concentration for polymerization of Mg-ADP-actin from 1.8 to 0.06 microM almost entirely by reducing the dissociation rate constants at both ends. Saturating phosphate increases the barbed end association rate constant of Mg-ADP-actin 15%, but this value is still threefold less than that of ATP-actin. Thus, ATP hydrolysis without phosphate dissociation must change the conformation of polymerized actin. Analysis of depolymerization experiments in the presence of phosphate suggests that phosphate dissociation near the terminal subunits is much faster than in the interior. Remarkably, 10 times more phosphate is required to slow the depolymerization of the pointed end than the barbed end, suggesting a weak affinity of phosphate near the pointed end. Our observations of single actin filaments provide clues about the origins of the difference in the critical concentration at the two ends of actin filaments in the presence of ATP.