Cooperativity of myosin molecules through strain-dependent chemistry

Mechanisms of myosin II force generation: insights from novel experimental techniques and approaches

Myosin II is a molecular motor that converts chemical energy derived from ATP hydrolysis into mechanical work. Myosin II isoforms are responsible for muscle contraction and a range of cell functions relying on the development of force and motion. When the motor attaches to actin, ATP is hydrolyzed and inorganic phosphate (Pi) and ADP are released from its active site. These reactions are coordinated with changes in the structure of myosin, promoting the so-called "power stroke" that causes the sliding of actin filaments. The general features of the myosin-actin interactions are well accepted, but there are critical issues that remain poorly understood, mostly due to technological limitations. In recent years, there has been a significant advance in structural, biochemical, and mechanical methods that have advanced the field considerably. New modeling approaches have also allowed researchers to understand actomyosin interactions at different levels of analysis. This paper reviews recent studies looking into the interaction between myosin II and actin filaments, which leads to power stroke and force generation. It reviews studies conducted with single myosin molecules, myosins working in filaments, muscle sarcomeres, myofibrils, and fibers. It also reviews the mathematical models that have been used to understand the mechanics of myosin II in approaches focusing on single molecules to ensembles. Finally, it includes brief sections on translational aspects, how changes in the myosin motor by mutations and/or posttranslational modifications may cause detrimental effects in diseases and aging, among other conditions, and how myosin II has become an emerging drug target.

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.

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.

Coordinated force generation of skeletal myosins in myofilaments through motor coupling

In contrast to processive molecular motors, skeletal myosins form a large motor ensemble for contraction of muscles against high loads. Despite numerous information on the molecular properties of skeletal myosin, its ensemble effects on collective force generation have not been rigorously clarified. Here we show 4 nm stepwise actin displacements generated by synthetic myofilaments beyond a load of 30 pN, implying that steps cannot be driven exclusively by single myosins, but potentially by coordinated force generations among multiple myosins. The simulation model shows that stepwise actin displacements are primarily caused by coordinated force generation among myosin molecules. Moreover, the probability of coordinated force generation can be enhanced against high loads by utilizing three factors: strain-dependent kinetics between force-generating states; multiple power stroke steps; and high ATP concentrations. Compared with other molecular motors, our findings reveal how the properties of skeletal myosin are tuned to perform cooperative force generation for efficient muscle contraction.

Skeletal muscle myosin forms large ensembles to generate force against high loads. Using optical tweezers and simulation Kaya et al. provide experimental evidence for cooperative force generation, and describe how the molecular properties of skeletal myosins are tuned for coordinated power strokes.

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.

Velocity Fluctuations in Kinesin-1 Gliding Motility Assays Originate in Motor Attachment Geometry Variations

Motor proteins such as myosin and kinesin play a major role in cellular cargo transport, muscle contraction, cell division, and engineered nanodevices. Quantifying the collective behavior of coupled motors is critical to our understanding of these systems. An excellent model system is the gliding motility assay, where hundreds of surface-adhered motors propel one cytoskeletal filament such as an actin filament or a microtubule. The filament motion can be observed using fluorescence microscopy, revealing fluctuations in gliding velocity. These velocity fluctuations have been previously quantified by a motional diffusion coefficient, which Sekimoto and Tawada explained as arising from the addition and removal of motors from the linear array of motors propelling the filament as it advances, assuming that different motors are not equally efficient in their force generation. A computational model of kinesin head diffusion and binding to the microtubule allowed us to quantify the heterogeneity of motor efficiency arising from the combination of anharmonic tail stiffness and varying attachment geometries assuming random motor locations on the surface and an absence of coordination between motors. Knowledge of the heterogeneity allows the calculation of the proportionality constant between the motional diffusion coefficient and the motor density. The calculated value (0.3) is within a standard error of our measurements of the motional diffusion coefficient on surfaces with varying motor densities calibrated by landing rate experiments. This allowed us to quantify the loss in efficiency of coupled molecular motors arising from heterogeneity in the attachment geometry.

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.

Implementing cell contractility in filament-based cytoskeletal models

Cells are known to respond over time to mechanical stimuli, even actively generating force at longer times. In this paper, a microstructural filament-based cytoskeletal network model is extended to incorporate this active response, and a computational study to assess the influence on relaxation behaviour was performed. The incorporation of an active response was achieved by including a strain energy function of contractile activity from the cross-linked actin filaments. A four-state chemical model and strain energy function was adopted, and generalisation to three dimensions and the macroscopic deformation field was performed by integration over the unit sphere. Computational results in MATLAB and ABAQUS/Explicit indicated an active cellular response over various time-scales, dependent on contractile parameters. Important features such as force generation and increasing cell stiffness due to prestress are qualitatively predicted. The work in this paper can easily be extended to encompass other filament-based cytoskeletal models as well.

The principle of conformational signaling

Specific conformations of signaling proteins can serve as “signals” in signal transduction by being recognized by receptors.

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.

Bidirectional bacterial gliding motility powered by the collective transport of cell surface proteins

The gliding motility of Flavobacterium johnsoniae is driven by moving surface adhesive proteins. Recently, these motility components were observed to travel along a closed loop on the cell surface. The mechanism by which such moving surface adhesins give rise to cell motion remains unknown. On the basis of the unique motility properties of F. johnsoniae, we present a generic model for bidirectional motion of rigidly coupled adhesins, which are propelled in opposite directions. Using analytical and numerical methods, we demonstrate that, for a sufficiently large adhesin speed, bidirectional motion arises from spontaneous symmetry breaking. The model also predicts that, close to the bifurcation point, a weak asymmetry in the binding dynamics is sufficient to facilitate directed motility, indicating that the direction of motion could be sensitively regulated internally in response to inhomogeneity of the environment.

Self-Regulation of Motor Force Through Chemomechanical Coupling in Skeletal Muscle Contraction

It is intriguing how the mechanics of molecular motors is regulated to perform the mechanical work in living systems. In sharp contrast to the conventional wisdom, recent experiments indicated that motor force maintains ∼6 pN upon a wide range of filament loads during skeletal muscle contraction at the steady state. Here we find that this rather precise regulation which takes place in an essentially chaotic system, can be due to that a “working” motor is arrested in a transitional state when the motor force is ∼6 pN. Our analysis suggests that the motor force can be self-regulated through chemomechanical coupling, and motor force homeostasis is a built-in feature at the level of a single motor, which provides insights to understanding the coordinated function of multiple molecular motors existing in various physiological processes. With a coupled stochastic-elastic numerical framework, the kinetic model for a Actin-myosin-ATP cycle constructed in this work might pave the way to decently investigate the transient behaviors of the skeletal muscle or other actomyosin complex structures.

A cross-bridge cycle with two tension-generating steps simulates skeletal muscle mechanics

We examined whether cross-bridge cycle models with one or two tension-generating steps can account for the force-velocity relation of and tension response to length steps of frog skeletal muscle. Transition-state theory defined the strain dependence of the rate constants. The filament stiffness was non-Hookean. Models were refined against experimental data by simulated annealing and downhill simplex runs. Models with one tension-generating step were rejected, as they had a low efficiency and fitted the experimental data relatively poorly. The best model with two tension-generating steps (stroke distances 5.6 and 4.6 nm) and a cross-bridge stiffness of 1.7 pN/nm gave a good account of the experimental data. The two tensing steps allow an efficiency of up to 38% during shortening. In an isometric contraction, 54.7% of the attached heads were in a pre-tension-generating state, 44.5% of the attached heads had undergone the first tension-generating step, and only 0.8% had undergone both tension-generating steps; they bore 34%, 64%, and 2%, respectively, of the isometric tension. During slow shortening, the second tensing step made a greater contribution. During lengthening, up to 93% of the attached heads were in a pre-tension-generating state yet bore elevated tension by being dragged to high strains before detaching.

Muscle as a metamaterial operating near a critical point

The passive mechanical response of skeletal muscles at fast time scales is dominated by long range interactions inducing cooperative behavior without breaking the detailed balance. This leads to such unusual "material properties" as negative equilibrium stiffness and different behavior in force and displacement controlled loading conditions. Our fitting of experimental data suggests that "muscle material" is finely tuned to perform close to a critical point which explains large fluctuations observed in muscles close to the stall force.

The interrelation between mechanical characteristics of contracting muscle, cross-bridge internal structure, and the mechanism of chemomechanical energy transduction

The cross-bridge working stroke is regarded as a continuous (without jumps) change of myosin head internal state under the action of a force exerted within the nucleotide-binding site. Involvement of a concept of continuous cross-bridge conformation enables discussion of the nature of the force propelling muscle, and the Coulomb repulsion of like-charged adenosine triphosphate (ATP) fragments ADP(2-) and P (i) (2-) can quite naturally be considered as the source of this force. Two entirely different types of working stroke termination are considered. Along with the fluctuation mechanism, which controls the working stroke duration t (w) at isometric contraction, another interrupt mechanism is initially taken into account. It is triggered when the lever arm shift amounts to the maximal value S ≈ 11 nm, the back door opens, and P(i) crashes out. As a result, t (w) becomes inversely proportional to the velocity v of sliding filaments t (w) ≈ S/v for a wide range of values of v. Principal features of the experimentally observed dependences of force, efficiency, and rate of heat production on velocity and ATP concentration can then be reproduced by fitting a single parameter: the velocity-independent time span t (r) between the termination of the last and beginning of the next working stroke. v becomes the principal variable of the model, and the muscle force changes under external load are determined by variations in v rather than in the tension of filaments. The Boltzmann equation for an ensemble of cross-bridges is obtained, and some collective effects are discussed.

Crossbridge and filament compliance in muscle: implications for tension generation and lever arm swing

The stiffness of myosin heads attached to actin is a crucial parameter in determining the kinetics and mechanics of the crossbridge cycle. It has been claimed that the stiffness of myosin heads in the anterior tibialis muscle of the common frog (Rana temporaria) is as high as 3.3 pN/nm, substantially higher than its value in rabbit muscle (~1.7 pN/nm). However, the crossbridge stiffness measurement has a large error since the contribution of crossbridges to half-sarcomere compliance is obtained by subtracting from the half-sarcomere compliance the contributions of the thick and thin filaments, each with a substantial error. Calculation of its value for isometric contraction also depends on the fraction of heads that are attached, for which there is no consensus. Surprisingly, the stiffness of the myosin head from the edible frog, Rana esculenta, determined in the same manner, is only 60% of that in Rana temporaria. In our view it is unlikely that the value of such a crucial parameter could differ so substantially between two frog species. Since the means of the myosin head stiffness in these two species are not significantly different, we suggest that the best estimate of the stiffness of the myosin heads for frog muscle is the average of these data, a value similar to that for rabbit muscle. This would allow both frog and rabbit muscles to operate the same low-cooperativity mechanism for the crossbridge cycle with only one or two tension-generating steps. We review evidence that much of the compliance of the myosin head is located in the pliant region where the lever arm emerges from the converter and propose that tension generation ("tensing") caused by the rotation and movement of the converter is a separate event from the passive swinging of the lever arm in its working stroke in which the strain energy stored in the pliant region is used to do work.

Bidirectional transport by molecular motors: enhanced processivity and response to external forces

Intracellular transport along cytoskeletal filaments is often mediated by two teams of molecular motors that pull on the same cargo and move in opposite directions along the filaments. We have recently shown theoretically that this bidirectional transport can be understood as a stochastic tug-of-war between the two motor teams. Here, we further develop our theory to investigate the experimentally accessible dynamic behavior of cargos transported by strong motors such as kinesin-1 or cytoplasmic dynein. By studying the run and binding times of such a cargo, we show that the properties of biological motors, such as the large ratio of stall/detachment force and the small ratio of superstall backward/forward velocity, are favorable for bidirectional cargo transport, leading to fast motion and enhanced diffusion. In addition, cargo processivity is shown to be strongly enhanced by transport via several molecular motors even if these motors are engaged in a tug-of-war. Finally, we study the motility of a bidirectional cargo under force. Frictional forces arising, e.g., from the viscous cytoplasm, lead to peaks in the velocity distribution, while external forces as exerted, e.g., by an optical trap, lead to hysteresis effects. Our results, in particular our explicit expressions for the cargo binding time and the distance of the peaks in the velocity relation under friction, are directly accessible to in vitro as well as in vivo experiments.

Robust transport by multiple motors with nonlinear force-velocity relations and stochastic load sharing

Transport by processive molecular motors plays an important role in many cell biological phenomena. In many cases, motors work together to transport cargos in the cell, so it is important to understand the mechanics of the multiple motors. Based on earlier modeling efforts, here we study effects of nonlinear force-velocity relations and stochastic load sharing on multiple motor transport. We find that when two or three motors transport the cargo, then the nonlinear and stochastic effects compensate so that the mechanical properties of the transport are robust. Similarly, the transport is insensitive to compliance of the cargo-motor links. Furthermore, the rate of movement against moderate loads is not improved by increasing the small number of motors. When the motor number is greater than 4, correlations between the motors become negligible, and the earlier analytical mean-field theory of the multiple motor transport holds. We predict that the effective diffusion of the cargo driven by the multiple motors under load increases by an order of magnitude compared to that for the single motor. Finally, our simulations predict that the stochastic effects are responsible for a significant dispersion of velocities generated by the 'tug-of-war' of the multiple opposing motors.

Drug effect unveils inter-head cooperativity and strain-dependent ADP release in fast skeletal actomyosin

Amrinone is a bipyridine compound with characteristic effects on the force-velocity relationship of fast skeletal muscle, including a reduction in the maximum shortening velocity and increased maximum isometric force. Here we performed experiments to elucidate the molecular mechanisms for these effects, with the additional aim to gain insight into the molecular mechanisms underlying the force-velocity relationship. In vitro motility assays established that amrinone reduces the sliding velocity of heavy meromyosin-propelled actin filaments by 30% at different ionic strengths of the assay solution. Stopped-flow studies of myofibrils, heavy meromyosin and myosin subfragment 1, showed that the effects on sliding speed were not because of a reduced rate of ATP-induced actomyosin dissociation because the rate of this process was increased by amrinone. Moreover, optical tweezers studies could not detect any amrinone-induced changes in the working stroke length. In contrast, the ADP affinity of acto-heavy meromyosin was increased about 2-fold by 1 mm amrinone. Similar effects were not observed for acto-subfragment 1. Together with the other findings, this suggests that the amrinone-induced reduction in sliding velocity is attributed to inhibition of a strain-dependent ADP release step. Modeling results show that such an effect may account for the amrinone-induced changes of the force-velocity relationship. The data emphasize the importance of the rate of a strain-dependent ADP release step in influencing the maximum sliding velocity in fast skeletal muscle. The data also lead us to discuss the possible importance of cooperative interactions between the two myosin heads in muscle contraction.

Properties of the kinesin-3 NcKin3 motor domain and implications for neck function

Members of the Kinesin-3 family are microtubule motors involved in the transport of membranous cargo. NcKin3 from the fungus Neurospora crassa is dimeric but inactivates one of its motor heads to generate nonprocessive motility. To determine how one of the heads is inactivated, we investigated truncated monomeric constructs. None of the constructs generated processive single-molecule motility, and multimotor velocities depended linearly on the number of residues remaining in the neck. The kinetic analysis suggests futile ATP hydrolysis cycles, because a representative monomer showed a faster ATP turnover than the dimer while supporting slower motility. The K(0.5,MT) was 70-fold lower, the microtubule-bound portion of the kinetic cycle eight-fold longer and the microtubule detachment rate almost 15-fold slower than that of the dimer. Moreover, the monomer's microtubule-dependent ADP release occurred three-fold to four-fold faster than k(cat) (125 versus 34 s(-1)), whereas phosphate release was approximately equally fast (29 s(-1)). A dimeric construct containing a structure-breaking insert between motor head and neck showed a similar behaviour. These data suggest that the heads of the wild-type NcKin3 motor are strictly coupled via the neck domain, and that the dimeric structure is required for proper detachment after one ATPase cycle. This is the first direct comparison of a monomeric Kinesin-3 with its dimeric full-length counterpart, and the kinetic changes observed here may also apply to other Kinesin-3 motors.

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.

Tug-of-war as a cooperative mechanism for bidirectional cargo transport by molecular motors

Intracellular transport is based on molecular motors that pull cargos along cytoskeletal filaments. One motor species always moves in one direction, e.g., conventional kinesin moves to the microtubule plus end, whereas cytoplasmic dynein moves to the microtubule minus end. However, many cellular cargoes are observed to move bidirectionally, involving both plus- and minus-end-directed motors. The presumably simplest mechanism for such bidirectional transport is provided by a tug-of-war between the two motor species. This mechanism is studied theoretically using the load-dependent transport properties of individual motors as measured in single-molecule experiments. In contrast to previous expectations, such a tug-of-war is found to be highly cooperative and to exhibit seven different motility regimes depending on the precise values of the single motor parameters. The sensitivity of the transport process to small parameter changes can be used by the cell to regulate its cargo traffic.

A homotetrameric kinesin-5, KLP61F, bundles microtubules and antagonizes Ncd in motility assays

BACKGROUND

Mitosis depends upon the cooperative action of multiple microtubule (MT)-based motors. Among these, a kinesin-5, KLP61F, and the kinesin-14, Ncd, are proposed to generate antagonistic-sliding forces that control the spacing of the spindle poles. We tested whether purified KLP61F homotetramers and Ncd homodimers can generate a force balance capable of maintaining a constant spindle length in Drosophila embryos.

RESULTS

Using fluorescence microscopy and cryo-EM, we observed that purified full-length, motorless, and tailless KLP61F tetramers (containing a tetramerization domain) and Ncd dimers can all cross-link MTs into bundles in MgATP. In multiple-motor motility assays, KLP61F and Ncd drive plus-end and minus-end MT sliding at 0.04 and 0.1 microm/s, respectively, but the motility of either motor is decreased by increasing the mole fraction of the other. At the "balance point," the mean velocity was zero and MTs paused briefly and then oscillated, taking approximately 0.3 microm excursions at approximately 0.02 microm/s toward the MT plus end and then the minus end.

CONCLUSIONS

The results, combined with quantitative analysis, suggest that these motors could act as mutual brakes to modulate the rate of pole-pole separation and could maintain a prometaphase spindle displaying small fluctuations in its steady-state length.

Engineering Cooperativity in Biomotor-Protein Assemblies

A biosynthetic approach was developed to control and probe cooperativity in multiunit biomotor assemblies by linking molecular motors to artificial protein scaffolds. This approach provides precise control over spatial and elastic coupling between motors. Cooperative interactions between monomeric kinesin-1 motors attached to protein scaffolds enhance hydrolysis activity and microtubule gliding velocity. However, these interactions are not influenced by changes in the elastic properties of the scaffold, distinguishing multimotor transport from that powered by unorganized monomeric motors. These results highlight the role of supramolecular architecture in determining mechanisms of collective transport.

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.

Elastically coupled two-dimensional Brownian motors

Many workers have theoretically and numerically studied Brownian motors (ratchet models) as models for molecular motors for many years. Their studies have been mainly concerned about one-dimensional motion of a particle or coupled particles experiencing spatially asymmetric interaction force. In this paper, we introduce a coupled flashing ratchet model, that is, elastically coupled Brownian particles with their own easily advancing directions under the on-and-off (or occasional) influence of two-dimensional interaction force asymmetric in the easily advancing direction of each particle. The dynamics of the model is investigated by computer simulation and characteristic motion of the particles in the two-dimensional space is observed. Our model is also studied as that for molecular motors in muscle contraction and the simulation results are compared with those of in vitro biological experiments in motility assays of the molecular motors. We succeed in reproducing the experimental results qualitatively with our model.

Multi-scale modelling and the IUPS physiome project

We review the development of models of cellular and tissue function and in particular address issues of multi-scale modelling, including the transition from stochastic models to continuum models and the incorporation of cell and tissue structure. The heart is used as an example of linking models at the molecular level to cell, tissue and organ level function.

Conformational spread: the propagation of allosteric states in large multiprotein complexes

The phenomenon of allostery is conventionally described for small symmetrical oligomeric proteins such as hemoglobin. Here we review experimental evidence from a variety of systems-including bacterial chemotaxis receptors, muscle ryanodine receptors, and actin filaments-showing that conformational changes can also propagate through extended lattices of protein molecules. We explore the statistical mechanics of idealized linear and two-dimensional arrays of allosteric proteins and show that, as in the analogous Ising models, arrays of closely packed units can show large-scale integrated behavior. We also discuss proteins that undergo conformational changes driven by the hydrolysis of ATP and give examples in which these changes propagate through linear chains of molecules. We suggest that conformational spread could provide the basis of a solid-state "circuitry" in a living cell, able to integrate biochemical and biophysical events over hundreds of protein molecules.

Stepwise sliding of single actin and Myosin filaments

Dynamics of sliding were explored in isolated actin and myosin filaments. Sliding occurs in steps. The steps are integer multiples of 2.7 nm, which is equal to the monomeric repeat along the actin filament. When filaments were forced to slide in the reverse direction, the size paradigm was the same. This size paradigm is parallel to that seen in the kinesin-microtubule system, where step size is an integer multiple of the tubulin repeat along the microtubule.

Instabilities in the transient response of muscle

We investigate the isometric transient response of muscle using a quantitative stochastic model of the actomyosin cycle based on the swinging lever-arm hypothesis. We first consider a single pair of filaments, and show that when values of parameters such as the lever-arm displacement and the cross-bridge elasticity are chosen to provide effective energy transduction, the T(2) curve (the tension recovered immediately after a step displacement) displays a region of negative slope. If filament compliance and the discrete nature of the binding sites are taken into account, the negative slope is diminished, but not eliminated. This implies that there is an instability in the dynamics of individual half sarcomeres. However, when the symmetric nature of whole sarcomeres is taken into account, filament rearrangement becomes important during the transient: as tension is recovered, some half sarcomeres lengthen whereas others shorten. This leads to a flat T(2) curve, as observed experimentally. In addition, we investigate the isotonic transient response and show that for a range of parameter values the model displays damped oscillations, as recently observed in experiments on single muscle fibers. We conclude that it is essential to consider the collective dynamics of many sarcomeres, rather than the dynamics of a single pair of filaments, when interpreting the transient response of muscle.

Cooperative behavior of molecular motors

Both experimental evidence and theoretical models for collective effects in the working mechanism of molecular motors are reviewed at three different levels, namely: (i) interaction between the two heads of double-headed motors, particularly in processive motors like kinesin, myosin V and myosin VI, (ii) cooperative regulation of muscle thin filaments by accessory proteins and the Ca2+ level, and (iii) collective dynamic effects stemming from the mechanical coupling of molecular motors within macroscopic structures such as muscle thick filaments or axonemes. We aim to bridge the gap between structural information at the molecular level and physiological data with accompanying specific models on the one hand, and general stochastic physical models for the action of molecular motors on the other hand. An underlying assumption is that while, ultimately, the function of molecular motors will be explainable by a quantitative description of specific intramolecular dynamics and intermolecular interactions, for some coarse grained larger scale dynamic features it will be sufficient and illuminating to construct physical models that are simplified to the bare essentials.

Atomic force microscopy of height fluctuations of fibroblast cells

We investigated the nanometer scale height fluctuations of 3T3 fibroblast cells with the atomic force microscope under physiological conditions. A correlation between these fluctuations and lateral cellular motility can be observed. Fluctuations measured on leading edges appear to be predominantly related to actin polymerization-depolymerization processes. We found fast (5 Hz) pulsatory behavior with 1-2 nm amplitude on a cell with low motility showing emphasized structure of stress fibers. Myosin driven contractions of stress fibers are thought to induce this pulsation.

Synchronous oscillations of length and stiffness during loaded shortening of frog muscle fibres

1. A study was made of the damped oscillations in fibre length that are observed when isolated muscle fibres from the frog are released during the plateau of an isometric tetanus to shorten against a constant load (force clamp recording) near the isometric level (temperature, 1.0-11.0 degrees C; initial sarcomere length, 2.25 microm). 2. The oscillatory length changes of the whole fibre were associated with similar length changes of marked consecutive segments along the fibre. The segmental length changes were initially in synchrony with the whole-fibre movements but became gradually more disordered. At the same time the length oscillation of the whole fibre was progressively damped. 3. The fast length step that normally occurs at the outset of the load-clamp manoeuvre was essential for initiating the oscillatory behaviour. Accordingly, no length oscillation occurred when the load clamp was arranged to start as soon as the selected tension level was reached during the rising phase of the tetanus. 4. The instantaneous stiffness was measured as the change in force that occurred in response to a high-frequency (2-4 kHz) length oscillation of the fibre. During the load-clamp manoeuvre, when the tension was kept constant, the stiffness underwent periodic changes that correlated well in time with the damped oscillatory changes in fibre length. However, there was a phase shift between the stiffness oscillation and the oscillation of shortening velocity, the latter being in the lead of the stiffness response by 21.4 +/- 0.8 ms (n = 19) at 1.8 +/- 0.1 degrees C. 5. A mechanism is proposed to explain the oscillatory behaviour of the muscle fibre based on the idea that the quick length step at the outset of the load clamp leads to synchronous activity of the myosin cross-bridges along the length of the fibre.

Visualizing myosin's power stroke in muscle contraction

The long-standing swinging crossbridge or lever arm hypothesis for the motor action of myosin heads finds support in recent results from 3-D tomograms of insect flight muscle (IFM) fast frozen during active contraction and from both fluorescence polarization and X-ray diffraction during rapid stretches or releases of isometrically contracting fibers. The latter provide direct evidence for lever arm movements synchronous with force changes. Rebuilding the atomic model of nucleotide-free subfragment 1 (S1) to fit fast-frozen, active IFM crossbridges suggests a two-stage power stroke in which the catalytic domain rolls on actin from weak to strong binding; this is followed by a 5-nm lever arm swing of the light chain domain, which gives a total interaction distance of approx. 12 nm. Comparison of S1 crystal structures with in situ myosin heads suggests that actin binding may be necessary in order to view the full repertoire of myosin motor action. The differing positions of the catalytic domains of actin-attached myosin heads in contracting IFM suggest that both the actin-myosin binding energy and the hydrolysis of ATP may be used to cock the crossbridge and drive the power stroke.