Instabilities in the transient response of muscle

New paradigms in actomyosin energy transduction: Critical evaluation of non-traditional models for orthophosphate release

Release of the ATP hydrolysis product ortophosphate (Pi) from the active site of myosin is central in chemo-mechanical energy transduction and closely associated with the main force-generating structural change, the power-stroke. Despite intense investigations, the relative timing between Pi-release and the power-stroke remains poorly understood. This hampers in depth understanding of force production by myosin in health and disease and our understanding of myosin-active drugs. Since the 1990s and up to today, models that incorporate the Pi-release either distinctly before or after the power-stroke, in unbranched kinetic schemes, have dominated the literature. However, in recent years, alternative models have emerged to explain apparently contradictory findings. Here, we first compare and critically analyze three influential alternative models proposed previously. These are either characterized by a branched kinetic scheme or by partial uncoupling of Pi-release and the power-stroke. Finally, we suggest critical tests of the models aiming for a unified picture.

Theoretical efficiency limits and speed-efficiency trade-off in myosin motors

Muscle myosin is a non-processive molecular motor generates mechanical work when cooperating in large ensembles. During its cyle, each individual motor keeps attaching and detaching from the actin filament. The random nature of attachment and detachment inevitably leads to losses and imposes theoretical limits on the energetic efficiency. Here, we numerically determine the theoretical efficiency limit of a classical myosin model with a given number of mechano-chemical states. All parameters that are not bounded by physical limits (like rate limiting steps) are determined by numerical efficiency optimization. We show that the efficiency is limited by the number of states, the stiffness and the rate-limiting kinetic steps. There is a trade-off between speed and efficiency. Slow motors are optimal when most of the available free energy is allocated to the working stroke and the stiffness of their elastic element is high. Fast motors, on the other hand, work better with a lower and asymmetric stiffness and allocate a larger fraction of free energy to the release of ADP. Overall, many features found in myosins coincide with the findings from the model optimization: there are at least 3 bound states, the largest part of the working stroke takes place during the first transition, the ADP affinity is adapted differently in slow and fast myosins and there is an asymmetry in elastic elements.

Cooperativity of myosin II motors in the non-regulated and regulated thin filaments investigated with high-speed AFM

Skeletal myosins II are non-processive molecular motors that work in ensembles to produce muscle contraction while binding to the actin filament. Although the molecular properties of myosin II are well known, there is still debate about the collective work of the motors: is there cooperativity between myosin motors while binding to the actin filaments? In this study, we use high-speed AFM to evaluate this issue. We observed that the initial binding of small arrays of myosin heads to the non-regulated actin filaments did not affect the cooperative probability of subsequent bindings and did not lead to an increase in the fractional occupancy of the actin binding sites. These results suggest that myosin motors are independent force generators when connected in small arrays, and that the binding of one myosin does not alter the kinetics of other myosins. In contrast, the probability of binding of myosin heads to regulated thin filaments under activating conditions (at high Ca2+ concentration in the presence of 2 μM ATP) was increased with the initial binding of one myosin, leading to a larger occupancy of available binding sites at the next half-helical pitch of the filament. The result suggests that myosin cooperativity is observed over five pseudo-repeats and defined by the activation status of the thin filaments.

Simulated actin reorganization mediated by motor proteins

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

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

Distinct roles of nonmuscle myosin II isoforms for establishing tension and elasticity during cell morphodynamics

Nonmuscle myosin II (NM II) is an integral part of essential cellular processes, including adhesion and migration. Mammalian cells express up to three isoforms termed NM IIA, B, and C. We used U2OS cells to create CRISPR/Cas9-based knockouts of all three isoforms and analyzed the phenotypes on homogenously coated surfaces, in collagen gels, and on micropatterned substrates. In contrast to homogenously coated surfaces, a structured environment supports a cellular phenotype with invaginated actin arcs even in the absence of NM IIA-induced contractility. A quantitative shape analysis of cells on micropatterns combined with a scale-bridging mathematical model reveals that NM IIA is essential to build up cellular tension during initial stages of force generation, while NM IIB is necessary to elastically stabilize NM IIA-generated tension. A dynamic cell stretch/release experiment in a three-dimensional scaffold confirms these conclusions and in addition reveals a novel role for NM IIC, namely the ability to establish tensional homeostasis.

The Synergic Role of Actomyosin Architecture and Biased Detachment in Muscle Energetics: Insights in Cross Bridge Mechanism Beyond the Lever-Arm Swing

Muscle energetics reflects the ability of myosin motors to convert chemical energy into mechanical energy. How this process takes place remains one of the most elusive questions in the field. Here, we combined experimental measurements of in vitro sliding velocity based on DNA-origami built filaments carrying myosins with different lever arm length and Monte Carlo simulations based on a model which accounts for three basic components: (i) the geometrical hindrance, (ii) the mechano-sensing mechanism, and (iii) the biased kinetics for stretched or compressed motors. The model simulations showed that the geometrical hindrance due to acto-myosin spatial mismatching and the preferential detachment of compressed motors are synergic in generating the rapid increase in the ATP-ase rate from isometric to moderate velocities of contraction, thus acting as an energy-conservation strategy in muscle contraction. The velocity measurements on a DNA-origami filament that preserves the motors’ distribution showed that geometrical hindrance and biased detachment generate a non-zero sliding velocity even without rotation of the myosin lever-arm, which is widely recognized as the basic event in muscle contraction. Because biased detachment is a mechanism for the rectification of thermal fluctuations, in the Brownian-ratchet framework, we predict that it requires a non-negligible amount of energy to preserve the second law of thermodynamics. Taken together, our theoretical and experimental results elucidate less considered components in the chemo-mechanical energy transduction in muscle.

Mechanosensitive self-assembly of myosin II minifilaments

Self-assembly and force generation are two central processes in biological systems that usually are considered in separation. However, the signals that activate nonmuscle myosin II molecular motors simultaneously lead to self-assembly into myosin II minifilaments as well as progression of the motor heads through the cross-bridge cycle. Here we investigate theoretically the possible effects of coupling these two processes. Our assembly model, which builds on a consensus architecture of the minifilament, predicts a critical aggregation concentration at which the assembly kinetics slows down dramatically. The combined model predicts that increasing actin filament concentration and force both lead to a decrease in the critical aggregation concentration. We suggest that due to these effects, myosin II minifilaments in a filamentous context might be in a critical state that reacts faster to varying conditions than in solution. We finally compare our model to experiments by simulating fluorescence recovery after photobleaching.

Tensile force-induced cytoskeletal remodeling: Mechanics before chemistry

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

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

The effects of inorganic phosphate on muscle force development and energetics: challenges in modelling related to experimental uncertainties

Muscle force and power are developed by myosin cross-bridges, which cyclically attach to actin, undergo a force-generating transition and detach under turnover of ATP. The force-generating transition is intimately associated with release of inorganic phosphate (Pi) but the exact sequence of events in relation to the actual Pi release step is controversial. Details of this process are reflected in the relationships between [Pi] and the developed force and shortening velocity. In order to account for these relationships, models have proposed branched kinetic pathways or loose coupling between biochemical and force-generating transitions. A key hypothesis underlying the present study is that such complexities are not required to explain changes in the force–velocity relationship and ATP turnover rate with altered [Pi]. We therefore set out to test if models without branched kinetic paths and Pi-release occurring before the main force-generating transition can account for effects of varied [Pi] (0.1–25 mM). The models tested, one assuming either linear or non-linear cross-bridge elasticity, account well for critical aspects of muscle contraction at 0.5 mM Pi but their capacity to account for the maximum power output vary. We find that the models, within experimental uncertainties, account for the relationship between [Pi] and isometric force as well as between [Pi] and the velocity of shortening at low loads. However, in apparent contradiction with available experimental findings, the tested models produce an anomalous force–velocity relationship at elevated [Pi] and high loads with more than one possible velocity for a given load. Nevertheless, considering experimental uncertainties and effects of sarcomere non-uniformities, these discrepancies are insufficient to refute the tested models in favour of more complex alternatives.

Spreading of perturbations in myosin group kinetics along actin filaments

Global changes in the state of spatially distributed systems can often be traced back to perturbations that arise locally. Whether such local perturbations grow into global changes depends on the system geometry and the spatial spreading of these perturbations. Here, we investigate how different spreading behaviors of local perturbations determine their global impact in 1-dimensional systems of different size. Specifically, we assessed sliding arrest events in in vitro motility assays where myosins propel actin, and simulated the underlying mechanochemistry of myosins that bind along the actin filament. We observed spontaneous sliding arrest events that occurred more frequently for shorter actin filaments. This observation could be explained by spontaneous local arrest of myosin kinetics that stabilizes once it spreads throughout an entire actin filament. When we introduced intermediate concentrations of the actin cross-linker filamin, longer actin was arrested more frequently. This observation was reproduced by simulations where filamin binding induces persistent local arrest of myosin kinetics, which subsequently spreads throughout the actin filament. A spin chain model with nearest-neighbor coupling reproduced key features of our experiments and simulations, thus extending to other linear systems with nearest-neighbor coupling the following conclusions: 1) perturbations that are persistent only once they spread throughout the system are more effective in smaller systems, and 2) perturbations that are persistent upon their establishment are more effective in larger systems. Beyond these general conclusions, our work also provides a theoretical model of collective myosin kinetics with a finite range of mechanical coupling along the actin filament.

Comparing models with one versus multiple myosin-binding sites per actin target zone: The power of simplicity

Mechanokinetic statistical models describe the mechanisms of muscle contraction on the basis of the average behavior of a large ensemble of actin-myosin motors. Such models often assume that myosin II motor domains bind to regularly spaced, discrete target zones along the actin-based thin filaments and develop force in a series of strain-dependent transitions under the turnover of ATP. The simplest models assume that there is just one myosin-binding site per target zone and a uniform spatial distribution of the myosin motor domains in relation to each site. However, most of the recently developed models assume three myosin-binding sites per target zone, and some models include a spatially explicit 3-D treatment of the myofilament lattice and thereby of the geometry of the actin-myosin contact points. Here, I show that the predictions for steady-state contractile behavior of muscle are very similar whether one or three myosin-binding sites per target zone is assumed, provided that the model responses are appropriately scaled to the number of sites. Comparison of the model predictions for isometrically contracting mammalian muscle cells suggests that each target zone contains three or more myosin-binding sites. Finally, I discuss the strengths and weaknesses of one-site spatially inexplicit models in relation to three-site models, including those that take into account the detailed 3-D geometry of the myofilament lattice. The results of this study suggest that single-site models, with reduced computational cost compared with multisite models, are useful for several purposes, e.g., facilitated molecular mechanistic insights.

Functionality of Disorder in Muscle Mechanics

A salient feature of skeletal muscles is their ability to take up an applied slack in a microsecond timescale. Behind this fast adaptation is a collective folding in a bundle of elastically interacting bistable elements. Since this interaction has a long-range character, the behavior of the system in force and length controlled ensembles is different; in particular, it can have two distinct order-disorder-type critical points. We show that the account of the disregistry between myosin and actin filaments places the elementary force-producing units of skeletal muscles close to both such critical points. The ensuing "double criticality" contributes to the system's ability to perform robustly and suggests that the disregistry is functional.

Reinterpretation of the Tension Response of Muscle to Stretches and Releases

We have reexamined the experimental time courses of tension in frog muscle after rapid length steps. The early tension recoveries are biexponential. After 3 nm/hs stretches and releases, the rates of the immediate rapid tension changes are similar but the subsequent tension fall after a stretch is much slower than the rise after a release. After 1.5 nm/hs length steps, the entire tension responses are more nearly mirror images. To identify the underlying processes, we used a model of the muscle cross-bridge cycle with two tension-generating (tensing) steps. Analysis of the time course of the tension, the rates of the steps in the cycle, and their contributions to tension provided insights into previously puzzling features of the experimental response. After a stretch, the initial rapid tension fall in the model is caused principally by the reversal of the first tensing step, but after a few milliseconds the tensing step resumes its forward direction. We conclude that the remaining response should not be included in phase 2, the period of early tension recovery. With this exclusion, T₂, the tension at the end of this period, rises with an increase of stretch. The rate of early tension recovery also increases with stretch size, showing that the reversal of the first tensing step is strain sensitive. After small length steps, the fast and slow components of the early tension recovery are both caused mainly by the first tensing step. The fast component is triggered by the initial sliding of the filaments, and the slow component is due to further sliding that occurs as the tension recovers. With small length steps (<0.5 nm/hs), the time course of the response to a stretch is the reverse of that to a release.

Chemomechanical regulation of myosin Ic cross-bridges: Deducing the elastic properties of an ensemble from single-molecule mechanisms

Myosin Ic is thought to be the principal constituent of the motor that adjusts mechanical responsiveness during adaptation to prolonged stimuli by hair cells, the sensory receptors of the inner ear. In this context myosin molecules operate neither as filaments, as occurs in muscles, nor as single or few molecules, as characterizes intracellular transport. Instead, myosin Ic molecules occur in a complex cluster in which they may exhibit cooperative properties. To better understand the motor’s remarkable function, we introduce a theoretical description of myosin Ic’s chemomechanical cycle based on experimental data from recent single-molecule studies. The cycle consists of distinct chemical states that the myosin molecule stochastically occupies. We explicitly calculate the probabilities of the occupancy of these states and show their dependence on the external force, the availability of actin, and the nucleotide concentrations as required by thermodynamic constraints. This analysis highlights that the strong binding of myosin Ic to actin is dominated by the ADP state for small external forces and by the ATP state for large forces. Our approach shows how specific parameter values of the chemomechanical cycle for myosin Ic result in behaviors distinct from those of other members of the myosin family. Integrating this single-molecule cycle into a simplified ensemble description, we predict that the average number of bound myosin heads is regulated by the external force and nucleotide concentrations. The elastic properties of such an ensemble are determined by the average number of myosin cross-bridges. Changing the binding probabilities and myosin’s stiffness under a constant force results in a mechanical relaxation which is large enough to account for fast adaptation in hair cells.

Myosin molecules are biological nanomachines that transduce chemical energy into mechanical work and thus produce directed motion in living cells. These molecules proceed through cyclic reactions in which they change their conformational states upon the binding and release of nucleotides while attaching to and detaching from filaments. The myosin family consists of many distinct members with diverse functions such as muscle contraction, cargo transport, cell migration, and sensory adaptation. How these functions emerge from the biophysical properties of the individual molecules is an open question. We present an approach that integrates recent findings from single-molecule experiments into a thermodynamically consistent description of myosin Ic and demonstrate how the specific parameter values of the cycle result in a distinct function. The free variables of our description are the chemical input and external force, both of which are experimentally accessible and define the cellular environment in which these proteins function. We use this description to predict the elastic properties of an ensemble of molecules and discuss the implications for myosin Ic’s function in the inner ear as a tension regulator mediating adaptation, a hallmark of biological sensory systems. In this situation myosin molecules cooperate in an intermediate regime, neither as a large ensemble as in muscle nor as a single or a few molecules as in intracellular transport.

Three-dimensional stochastic model of actin-myosin binding in the sarcomere lattice

The effect of molecule tethering in three-dimensional (3-D) space on bimolecular binding kinetics is rarely addressed and only occasionally incorporated into models of cell motility. The simplest system that can quantitatively determine this effect is the 3-D sarcomere lattice of the striated muscle, where tethered myosin in thick filaments can only bind to a relatively small number of available sites on the actin filament, positioned within a limited range of thermal movement of the myosin head. Here we implement spatially explicit actomyosin interactions into the multiscale Monte Carlo platform MUSICO, specifically defining how geometrical constraints on tethered myosins can modulate state transition rates in the actomyosin cycle. The simulations provide the distribution of myosin bound to sites on actin, ensure conservation of the number of interacting myosins and actin monomers, and most importantly, the departure in behavior of tethered myosin molecules from unconstrained myosin interactions with actin. In addition, MUSICO determines the number of cross-bridges in each actomyosin cycle state, the force and number of attached cross-bridges per myosin filament, the range of cross-bridge forces and accounts for energy consumption. At the macroscopic scale, MUSICO simulations show large differences in predicted force-velocity curves and in the response during early force recovery phase after a step change in length comparing to the two simplest mass action kinetic models. The origin of these differences is rooted in the different fluxes of myosin binding and corresponding instantaneous cross-bridge distributions and quantitatively reflects a major flaw of the mathematical description in all mass action kinetic models. Consequently, this new approach shows that accurate recapitulation of experimental data requires significantly different binding rates, number of actomyosin states, and cross-bridge elasticity than typically used in mass action kinetic models to correctly describe the biochemical reactions of tethered molecules and their interaction energetics.

Actomyosin based contraction: one mechanokinetic model from single molecules to muscle?

Bridging the gaps between experimental systems on different hierarchical scales is needed to overcome remaining challenges in the understanding of muscle contraction. Here, a mathematical model with well-characterized structural and biochemical actomyosin states is developed to that end. We hypothesize that this model accounts for generation of force and motion from single motor molecules to the large ensembles of muscle. In partial support of this idea, a wide range of contractile phenomena are reproduced without the need to invoke cooperative interactions or ad hoc states/transitions. However, remaining limitations exist, associated with ambiguities in available data for model definition e.g.: (1) the affinity of weakly bound cross-bridges, (2) the characteristics of the cross-bridge elasticity and (3) the exact mechanistic relationship between the force-generating transition and phosphate release in the actomyosin ATPase. Further, the simulated number of attached myosin heads in the in vitro motility assay differs several-fold from duty ratios, (fraction of strongly attached ATPase cycle times) derived in standard analysis. After addressing the mentioned issues the model should be useful in fundamental studies, for engineering of myosin motors as well as for studies of muscle disease and drug development.

Sensitivity of small myosin II ensembles from different isoforms to mechanical load and ATP concentration

Based on a detailed crossbridge model for individual myosin II motors, we systematically study the influence of mechanical load and adenosine triphosphate (ATP) concentration on small myosin II ensembles made from different isoforms. For skeletal and smooth muscle myosin II, which are often used in actomyosin gels that reconstitute cell contractility, fast forward movement is restricted to a small region of phase space with low mechanical load and high ATP concentration, which is also characterized by frequent ensemble detachment. At high load, these ensembles are stalled or move backwards, but forward motion can be restored by decreasing ATP concentration. In contrast, small ensembles of nonmuscle myosin II isoforms, which are found in the cytoskeleton of nonmuscle cells, are hardly affected by ATP concentration due to the slow kinetics of the bound states. For all isoforms, the thermodynamic efficiency of ensemble movement increases with decreasing ATP concentration, but this effect is weaker for the nonmuscle myosin II isoforms.

Statistical mechanics of the Huxley-Simmons model

The chemomechanical model of Huxley and Simmons (HS) [A. F. Huxley and R. M. Simmons, Nature 233, 533 (1971)NATUAS0028-083610.1038/233533a0] provides a paradigmatic description of mechanically induced collective conformational changes relevant in a variety of biological contexts, from muscles power stroke and hair cell gating to integrin binding and hairpin unzipping. We develop a statistical mechanical perspective on the HS model by exploiting a formal analogy with a paramagnetic Ising model. We first study the equilibrium HS model with a finite number of elements and compute explicitly its mechanical and thermal properties. To model kinetics, we derive a master equation and solve it for several loading protocols. The developed formalism is applicable to a broad range of allosteric systems with mean-field interactions.

Rigidity generation by nonthermal fluctuations

Active stabilization in systems with zero or negative stiffness is an essential element of a wide variety of biological processes. We study a prototypical example of this phenomenon and show how active rigidity, interpreted as a formation of a pseudowell in the effective energy landscape, can be generated in an overdamped stochastic system. We link the transition from negative to positive rigidity with time correlations in the additive noise, and we show that subtle differences in the out-of-equilibrium driving may compromise the emergence of a pseudowell.

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.

Ensemble velocity of non-processive molecular motors with multiple chemical states

We study the ensemble velocity of non-processive motor proteins, described with multiple chemical states. In particular, we discuss the velocity as a function of ATP concentration. Even a simple model which neglects the strain dependence of transition rates, reverse transition rates and nonlinearities in the elasticity can show interesting functional dependencies, which deviate significantly from the frequently assumed Michaelis-Menten form. We discuss how the order of events in the duty cycle can be inferred from the measured dependence. The model also predicts the possibility of velocity reversal at a certain ATP concentration if the duty cycle contains several conformational changes of opposite directionalities.

The kinetics of mechanically coupled myosins exhibit group size-dependent regimes

Naturally occurring groups of muscle myosin behave differently from individual myosins or small groups commonly assayed in vitro. Here, we investigate the emergence of myosin group behavior with increasing myosin group size. Assuming the number of myosin binding sites (N) is proportional to actin length (L) (N = L/35.5 nm), we resolve in vitro motility of actin propelled by skeletal muscle myosin for L = 0.2-3 μm. Three distinct regimes were found: L < 0.3 μm, sliding arrest; 0.3 μm ≤ L ≤ 1 μm, alternation between arrest and continuous sliding; L > 1 μm, continuous sliding. We theoretically investigated the myosin group kinetics with mechanical coupling via actin. We find rapid actin sliding steps driven by power-stroke cascades supported by postpower-stroke myosins, and phases without actin sliding caused by prepower-stroke myosin buildup. The three regimes are explained: N = 8, rare cascades; N = 15, cascade bursts; N = 35, continuous cascading. Two saddle-node bifurcations occur for increasing N (mono → bi → mono-stability), with steady states corresponding to arrest and continuous cascading. The experimentally measured dependence of actin sliding statistics on L and myosin concentration is correctly predicted.

Stochastic force generation by small ensembles of myosin II motors

Forces in the actin cytoskeleton are generated by small groups of nonprocessive myosin II motors for which stochastic effects are highly relevant. Using a cross-bridge model with the assumptions of fast power-stroke kinetics and equal load sharing between equivalent states, we derive a one-step master equation for the activity of a finite-sized ensemble of mechanically coupled myosin II motors. For constant external load, this approach yields analytical results for duty ratio and force-velocity relation as a function of ensemble size. We find that stochastic effects cannot be neglected for ensemble sizes below 15. The one-step master equation can be used also for efficient computer simulations with linear elastic external load and reveals the sequence of buildup of force and ensemble rupture that is characteristic for reconstituted actomyosin contractility.

Pre-power-stroke cross-bridges contribute to force transients during imposed shortening in isolated muscle fibers

When skeletal muscles are activated and mechanically shortened, the force that is produced by the muscle fibers decreases in two phases, marked by two changes in slope (P₁ and P₂) that happen at specific lengths (L₁ and L₂). We tested the hypothesis that these force transients are determined by the amount of myosin cross-bridges attached to actin and by changes in cross-bridge strain due to a changing fraction of cross-bridges in the pre-power-stroke state. Three separate experiments were performed, using skinned muscle fibers that were isolated and subsequently (i) activated at different Ca²⁺ concentrations (pCa²⁺ 4.5, 5.0, 5.5, 6.0) (n = 13), (ii) activated in the presence of blebbistatin (n = 16), and (iii) activated in the presence of blebbistatin at varying velocities (n = 5). In all experiments, a ramp shortening was imposed (amplitude 10%L_o, velocity 1 L_o•sarcomere length (SL)•s⁻¹), from an initial SL of 2.5 µm (except by the third group, in which velocities ranged from 0.125 to 2.0 L_o•s⁻¹). The values of P₁, P₂, L₁, and L₂ did not change with Ca²⁺ concentrations. Blebbistatin decreased P₁, and it did not alter P₂, L₁, and L₂. We developed a mathematical cross-bridge model comprising a load-dependent power-stroke transition and a pre-power-stroke cross-bridge state. The P₁ and P₂ critical points as well as the critical lengths L₁ and L₂ were explained qualitatively by the model, and the effects of blebbistatin inhibition on P₁ were also predicted. Furthermore, the results of the model suggest that the mechanism by which blebbistatin inhibits force is by interfering with the closing of the myosin upper binding cleft, biasing cross-bridges into a pre-power-stroke state.

Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models

We present a detailed analysis of the dynamical instabilities appearing in two kinetic theories for the collective behavior of molecular motors: the rigid two-state model and the two-state crossbridge (or power-stroke) model with continuous binding sites. We calculate force-velocity relations, discuss their stability, plot a diagram that summarizes the oscillation regimes, identify the location of the Hopf bifurcation with a memory effect, discuss the oscillation frequency and make a link with single-molecule experiments. We show that the instabilities present in these models naturally translate into non-linearities in force-displacement relations, and at linear order give forces that are similar to the delayed stretch activation observed in oscillating muscles. We also find that instabilities can appear for both apparent load-decelerated and load-accelerated detachment rates in a 3-state crossbridge model.

Muscle contraction: A mechanical perspective

In this paper we present a purely mechanical analog of the conventional chemo-mechanical modeling of muscle contraction. We abandon the description of kinetics of the power stroke in terms of jump processes and instead resolve the continuous stochastic evolution on an appropriate energy landscape. In general physical terms, we replace hard spin chemical variables by soft spin variables representing mechanical snap-springs. This allows us to treat the case of small and even disappearing barriers and, more importantly, to incorporate the mechanical representation of the power stroke into the theory of Brownian ratchets. The model provides the simplest non-chemical description for the main stages of the biochemical Lymn-Taylor cycle and may be used as a basis for the artificial micro-mechanical reproduction of the muscle contraction mechanism.

Mechanics of the power stroke in myosin II

Power stroke in skeletal muscles is a result of a conformational change in the globular portion of the molecular motor myosin II. In this paper we show that the fast tension recovery data reflecting the inner working of the power stroke mechanism can be quantitatively reproduced by a Langevin dynamics of a simple mechanical system with only two structural states. The proposed model is a generalization of the two state model of Huxley and Simmons. The main idea is to replace the rigid bistable device of Huxley and Simmons with an elastic bistable snap spring. In this setting the attached configuration of a cross bridge is represented not only by the discrete energy minima but also by a continuum of intermediate states where the fluctuation induced dynamics of the system takes place. We show that such soft-spin approach explains the load dependence of the power stroke amplitude and removes the well-known contradiction inside the conventional two state model regarding the time scale of the power stroke.

Coordination and collective properties of molecular motors: theory

Many cellular processes require molecular motors to produce motion and forces. Single molecule experiments have led to a precise description of how a motor works. Under most physiological conditions, however, molecular motors operate in groups. Interactions between motors yield collective behaviors that cannot be explained only from single molecule properties. The aim of this paper is to review the various theoretical descriptions that explain the emergence of collective effects in molecular motor assemblies. These include bidirectional motion, hysteretic behavior, spontaneous oscillations, and self-organization into dynamical structures. We discuss motors acting on the cytoskeleton both in a prescribed geometry such as in muscles or flagella and in the cytoplasm.

Twirling motion of actin filaments in gliding assays with nonprocessive Myosin motors

We present a model study of gliding assays in which actin filaments are moved by nonprocessive myosin motors. We show that even if the power stroke of the motor protein has no lateral component, the filaments will rotate around their axis while moving over the surface. Notably, the handedness of this twirling motion is opposite from that of the actin filament structure. It stems from the fact that the gliding actin filament has target zones, where its subunits point toward the surface and are therefore more accessible for myosin heads. Each myosin head has a higher binding probability before it reaches the center of the target zone than afterwards, which results in a left-handed twirling. We present a stochastic simulation and an approximative analytical solution. The calculated pitch of the twirling motion depends on the filament velocity (ATP concentration). It reaches approximately 400 nm for low speeds and increases with higher speeds.

Spontaneous oscillations of a minimal actomyosin system under elastic loading

Spontaneous mechanical oscillations occur in various types of biological systems where groups of motor molecules are elastically coupled to their environment. By using an optical trap to oppose the gliding motion of a single bead-tailed actin filament over a substrate densely coated with myosin motors, we mimicked this condition in vitro. We show that this minimal actomyosin system can oscillate spontaneously. Our finding accords quantitatively with a general theoretical framework where oscillatory instabilities emerge generically from the collective dynamics of molecular motors under load.

Hysteresis in cross-bridge models of muscle

A dynamical system is said to exhibit hysteresis if its current state depends on its history. Muscle shows hysteretic properties at constant length, such as residual force enhancement after stretch. There is no generally accepted explanation for residual force enhancement. Here we examine a very simple kinetic model for the interaction between actin and myosin, the two main proteins involved in muscle contraction. We demonstrate that this model shows hysteresis at constant force. Since muscle is not a continuum but rather a group of repeating elements, called sarcomeres, arranged in series, we perform simulations of three sarcomeres. These simulations show hysteresis at constant length. This result is the first time that residual force enhancement has been demonstrated using an experimentally motivated kinetic model and multi-sarcomere simulations without passive elastic elements, damping and/or force-length relationships. We conclude by suggesting some experiments to test the model's predictions. If these experiments support the model, it becomes important to understand multiple sarcomere systems, since their behavior may be very different from most current simulations that neglect the coupling between sarcomeres.

Mechanistic role of movement and strain sensitivity in muscle contraction

Tension generation can be studied by applying step perturbations to contracting muscle fibers and subdividing the mechanical response into exponential phases. The de novo tension-generating isomerization is associated with one of these phases. Earlier work has shown that a temperature jump perturbs the equilibrium constant directly to increase tension. Here, we show that a length jump functions quite differently. A step release (relative movement of thick and thin filaments) appears to release a steric constraint on an ensemble of noncompetent postphosphate release actomyosin cross-bridges, enabling them to generate tension, a concentration jump in effect. Structural studies [Taylor KA, et al. (1999) Tomographic 3D reconstruction of quick-frozen, Ca(2+)-activated contracting insect flight muscle. Cell 99:421-431] that map to these kinetics indicate that both catalytic and lever arm domains of noncompetent myosin heads change angle on actin, whereas lever arm movement alone mediates the power stroke. Together, these kinetic and structural observations show a 13-nm overall interaction distance of myosin with actin, including a final 4- to 6-nm power stroke when the catalytic domain is fixed on actin. Raising fiber temperature with both perturbation techniques accelerates the forward, but slows the reverse rate constant of tension generation, kinetics akin to the unfolding/folding of small proteins. Decreasing strain, however, causes both forward and reverse rate constants to increase. Despite these changes in rate, the equilibrium constant is strain-insensitive. Activation enthalpy and entropy data show this invariance to be the result of enthalpy-entropy compensation. Reaction amplitudes confirm a strain-invariant equilibrium constant and thus a strain-insensitive ratio of pretension- to tension-generating states as work is done.

Myosin V passing over Arp2/3 junctions: branching ratio calculated from the elastic lever arm model

Myosin V is a two-headed processive motor protein that walks in a hand-over-hand fashion along actin filaments. When it encounters a filament branch, formed by the Arp2/3 complex, it can either stay on the straight mother filament, or switch to the daughter filament. We study both probabilities using the elastic lever arm model for myosin V. We calculate the shapes and bending energies of all relevant configurations in which the trail head is bound to the actin filament before Arp2/3 and the lead head is bound either to the mother or to the daughter filament. Based on the assumption that the probability for a head to bind to a certain actin subunit is proportional to the Boltzmann factor obtained from the elastic energy, we calculate the mother/daughter filament branching ratio. Our model predicts a value of 27% for the daughter and 73% for the mother filament. This result is in good agreement with recent experimental data.

Mechanism of tension generation in muscle: an analysis of the forward and reverse rate constants

Tension generation in muscle occurs during the attached phase of the ATP-powered cyclic interaction of myosin heads with thin filaments. The transient nature of tension-generating intermediates and the complexity of the mechanochemical cross-bridge cycle have impeded a quantitative description of tension generation. Recent experiments performed under special conditions yielded a sigmoidal dependence of fiber tension on temperature--a unique case that simplifies the system to a two-state transition. We have applied this two-state analysis to kinetic data obtained from biexponential laser temperature-jump tension transients. Here we present the forward and reverse rate constants for de novo tension generation derived from analysis of the kinetics of the fast laser temperature-jump phase tau(2) (equivalent of the length-jump phase 2(slow)). The slow phase tau(3) is temperature-independent indicating coupling to rather than a direct role in, de novo tension generation. Increasing temperature accelerates the forward, and slows the reverse, rate constant for the creation of the tension-generating state. Arrhenius behavior of the forward and anti-Arrhenius behavior of the reverse rate constant is a kinetic signature of multistate multipathway protein-folding reactions. We conclude that locally unfolded tertiary and/or secondary structure of the actomyosin cross-bridge mediates the power stroke.

Oscillations in molecular motor assemblies

Autonomous oscillations in biological systems may have a biochemical origin or result from an interplay between force-generating and visco-elastic elements. In molecular motor assemblies the force-generating elements are molecular engines and the visco-elastic elements are stiff cytoskeletal polymers. The physical mechanism leading to oscillations depends on the particular architecture of the assembly. Existing models can be grouped into two distinct categories: systems with a delayed force activation and anomalous force-velocity relations. We discuss these systems within phase plane analysis known from the theory of dynamic systems and by adopting methods from control theory, the Nyquist stability criterion.

Elastic lever-arm model for myosin V

We present a mechanochemical model for myosin V, a two-headed processive motor protein. We derive the properties of a dimer from those of an individual head, which we model both with a four-state cycle (detached; attached with ADP.Pi; attached with ADP; and attached without nucleotide) and alternatively with a five-state cycle (where the powerstroke is not tightly coupled to the phosphate release). In each state the lever arm leaves the head at a different, but fixed, angle. The lever arm itself is described as an elastic rod. The chemical cycles of both heads are coordinated exclusively by the mechanical connection between the two lever arms. The model explains head coordination by showing that the lead head only binds to actin after the powerstroke in the trail head and that it only undergoes its powerstroke after the trail head unbinds from actin. Both models (four- and five-state) reproduce the observed hand-over-hand motion and fit the measured force-velocity relations. The main difference between the two models concerns the load dependence of the run length, which is much weaker in the five-state model. We show how systematic processivity measurement under varying conditions could be used to distinguish between both models and to determine the kinetic parameters.

Dynamics of myosin-driven skeletal muscle contraction: I. Steady-state force generation

Skeletal muscle contraction is a canonical example of motor-driven force generation. Despite the long history of research in this topic, a mechanistic explanation of the collective myosin force generation is lacking. We present a theoretical model of muscle contraction based on the conformational movements of individual myosins and experimentally measured chemical rate constants. Detailed mechanics of the myosin motor and the geometry of the sarcomere are taken into account. Two possible scenarios of force generation are examined. We find only one of the scenarios can give rise to a plausible contraction mechanism. We propose that the synchrony in muscle contraction is due to a force-dependent ADP release step. Computational results of a half sarcomere with 150 myosin heads can explain the experimentally measured force-velocity relationship and efficiency data. We predict that the number of working myosin motors increases as the load force is increased, thus showing synchrony among myosin motors during muscle contraction. We also find that titin molecules anchoring the thick filament are passive force generators in assisting muscle contraction.

Three-dimensional bead position histograms reveal single-molecule nanomechanics

We describe a method to investigate the structure and elasticity of macromolecules by a combination of single molecule experiments and kinematic modeling. With a photonic force microscope, we recorded spatial position histograms of a fluctuating microsphere tethered to full-length myosin-II. Assuming only that the molecule consists of concatenated rigid segments, a model derived from robot kinematics allows us to relate these histograms to the molecule's segment lengths and bending stiffnesses. Both our calculated position distributions and the experimental data show an asymmetry characteristic of a mixed entropic-enthalpic spring. Our model that fits best to experimental line profiles has two intramolecular hinges, one at the bound head domain, and another about 50 nm down the myosin tail, with a summed bending stiffness of about 3 k(B)T/rad.

Exact solution of a linear molecular motor model driven by two-step fluctuations and subject to protein friction

We investigate by analytical means the stochastic equations of motion of a linear molecular motor model based on the concept of protein friction. Solving the coupled Langevin equations originally proposed by Mogilner et al. [Phys. Lett. A 237, 297 (1998)], and averaging over both the two-step internal conformational fluctuations and the thermal noise, we present explicit, analytical expressions for the average motion and the velocity-force relationship. Our results allow for a direct interpretation of details of this motor model which are not readily accessible from numerical solutions. In particular, we find that the model is able to predict physiologically reasonable values for the load-free motor velocity and the motor mobility.