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

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.

Hierarchical modeling of force generation in cardiac muscle

Performing physiologically relevant simulations of the beating heart in clinical context requires to develop detailed models of the microscale force generation process. These models, however, may reveal difficult to implement in practice due to their high computational costs and complex calibration. We propose a hierarchy of three interconnected muscle contraction models-from the more refined to the more simplified-that are rigorously and systematically related to each other, offering a way to select, for a specific application, the model that yields a good trade-off between physiological fidelity, computational cost and calibration complexity. The three model families are compared to the same set of experimental data to systematically assess what physiological indicators can be reproduced or not and how these indicators constrain the model parameters. Finally, we discuss the applicability of these models for heart simulation.

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.

Acute Effects and Perceptions of Deep Oscillation Therapy for Improving Hamstring Flexibility

Context:Hamstring inflexibility is typically treated using therapeutic massage, stretching, and soft tissue mobilization. An alternative intervention is deep oscillation therapy (DOT). Currently, there is a lack of evidence to support DOT’s effectiveness to improve flexibility.Objective:To explore the effectiveness of DOT to improve hamstring flexibility.Design:Randomized single-cohort design.Setting:Research laboratory.Participants:Twenty-nine healthy, physically active individuals (self-reported activity of a minimum 200 min/wk).Interventions:All participants received a single session of DOT with randomization of the participant’s leg for the intervention. The DOT intervention parameters included a 1∶1 mode and 70% to 80% dosage at various frequencies for 28 minutes. Hamstring flexibility was assessed using passive straight leg raise for hip flexion using a digital inclinometer. Patient-reported outcomes were evaluated using the Copenhagen Hip and Groin Outcome Score and the Global Rating of Change (GRoC).Main Outcome Measure:The independent variable was time (pre and post). The dependent variables included passive straight leg raise, the GRoC, and the participant’s perceptions of the intervention. Statistical analyses included a dependentttest and a Pearson correlation.Results:Participants reported no issues with sport, activities of daily living, or quality of life prior to beginning the intervention study on the Copenhagen Hip and Groin Outcome Score. Passive straight leg raise significantly improved post-DOT (95% confidence interval, 4.48°–7.85°,P < .001) with a mean difference of 6.17 ± 4.42° (pre-DOT = 75.43 ± 21.82° and post-DOT = 81.60 ± 23.17°). A significant moderate positive correlation was identified (r = .439,P = .02) among all participants between the GRoC and the mean change score of hamstring flexibility. Participants believed that the intervention improved their hamstring flexibility (5.41 ± 1.02 points) and was relaxing (6.21 ± 0.86).Conclusions:DOT is an effective intervention to increase hamstring flexibility.

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.

Poorly understood aspects of striated muscle contraction

Muscle contraction results from cyclic interactions between the contractile proteins myosin and actin, driven by the turnover of adenosine triphosphate (ATP). Despite intense studies, several molecular events in the contraction process are poorly understood, including the relationship between force-generation and phosphate-release in the ATP-turnover. Different aspects of the force-generating transition are reflected in the changes in tension development by muscle cells, myofibrils and single molecules upon changes in temperature, altered phosphate concentration, or length perturbations. It has been notoriously difficult to explain all these events within a given theoretical framework and to unequivocally correlate observed events with the atomic structures of the myosin motor. Other incompletely understood issues include the role of the two heads of myosin II and structural changes in the actin filaments as well as the importance of the three-dimensional order. We here review these issues in relation to controversies regarding basic physiological properties of striated muscle. We also briefly consider actomyosin mutation effects in cardiac and skeletal muscle function and the possibility to treat these defects by drugs.

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.

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.

Spreading out muscle mass within a Hill-type model: a computer simulation study

It is state of the art that muscle contraction dynamics is adequately described by a hyperbolic relation between muscle force and contraction velocity (Hill relation), thereby neglecting muscle internal mass inertia (first-order dynamics). Accordingly, the vast majority of modelling approaches also neglect muscle internal inertia. Assuming that such first-order contraction dynamics yet interacts with muscle internal mass distribution, this study investigates two questions: (i) what is the time scale on which the muscle responds to a force step? (ii) How does this response scale with muscle design parameters? Thereto, we simulated accelerated contractions of alternating sequences of Hill-type contractile elements and point masses. We found that in a typical small muscle the force levels off after about 0.2 ms, contraction velocity after about 0.5 ms. In an upscaled version representing bigger mammals' muscles, the force levels off after about 20 ms, and the theoretically expected maximum contraction velocity is not reached. We conclude (i) that it may be indispensable to introduce second-order contributions into muscle models to understand high-frequency muscle responses, particularly in bigger muscles. Additionally, (ii) constructing more elaborate measuring devices seems to be worthwhile to distinguish viscoelastic and inertia properties in rapid contractile responses of muscles.

STEP SIZE IN ACTIVATED RABBIT SARCOMERES IS INDEPENDENT OF FILAMENT OVERLAP

Investigations carried out on single cardiac and bumblebee myofibrils have shown stepwise sarcomere-length change of ~2.7 nm.1 We have carried out parallel measurements on single myofibrils from rabbit psoas muscle. Activated specimens were released or stretched using a motor-imposed ramp. With a high-resolution algorithm, we found that step sizes were always integer multiples of 2.7 nm, whether the length change was positive or negative, and independent of ramp velocity. Also, the influence of initial sarcomere length was studied, and found to be negligible. The value 2.7 nm, seen consistently, is equal to the linear repeat of actin monomers along the thin filament, a result that ties dynamical events to molecular structure, and places narrow constraints on any proposed molecular mechanism.

The extensor tibiae muscle of the stick insect: biomechanical properties of an insect walking leg muscle

We investigated the properties of the extensor tibiae muscle of the stick insect (Carausius morosus) middle leg. Muscle geometry of the middle leg was compared to that of the front and hind legs and to the flexor tibiae, respectively. The mean length of the extensor tibiae fibres is 1.41+/-0.23 mm and flexor fibres are 2.11+/-0.30 mm long. The change of fibre length with joint angle was measured and closely follows a cosine function. Its amplitude gives effective moment arm lengths of 0.28+/-0.02 mm for the extensor and 0.56+/-0.04 mm for the flexor. Resting extensor tibiae muscle passive tonic force increased from 2 to 5 mN in the maximum femur-tibia (FT)-joint working range when stretched by ramps. Active muscle properties were measured with simultaneous activation (up to 200 pulses s(-1)) of all three motoneurons innervating the extensor tibiae, because this reflects most closely physiological muscle activation during leg swing. The force-length relationship corresponds closely to the typical characteristic according to the sliding filament hypothesis: it has a plateau at medium fibre lengths, declines nearly linearly in force at both longer and shorter fibre lengths, and the muscle's working range lies in the short to medium fibre length range. Maximum contraction velocity showed a similar relationship. The force-velocity relationship was the traditional Hill curve hyperbola, but deviated from the hyperbolic shape in the region of maximum contraction force close to the isometric contraction. Step-like changes in muscle length induced by loaded release experiments characterised the non-linear series elasticity as a quadratic spring.

Determinants of force rise time during isometric contraction of frog muscle fibres

Force-velocity (F-V) relationships were determined for single frog muscle fibres during the rise of tetanic contraction. F-V curves obtained using isotonic shortening early in a tetanic contraction were different from those obtained at equivalent times with isovelocity shortening, apparently because changing activation early in the contraction leads, in isovelocity experiments, to changing force and changing series elastic extension. F-V curves obtained with isotonic and with isovelocity shortening are similar if the shortening velocity in the isovelocity trials is corrected for series elastic extension. There is a progressive shift in the scaling of force-velocity curves along the force axis during the course of the tetanic rise, reflecting increasing fibre activation. The time taken for F-V curves to reach the steady-state position was quite variable, ranging from about 50 ms after the onset of contraction (1-3 degrees C) to well over 100 ms in different fibres. The muscle force at a fixed, moderately high shortening velocity relative to the force at this velocity during the tetanic plateau was taken as a measure of muscle activation. The reference velocity used was 60% of the maximum shortening velocity (V(max)) at the tetanic plateau. The estimated value of the fractional activation at 40 ms after the onset of contraction was used as a measure of the rate of activation. The rate of rise of isometric tension in different fibres was correlated with the rate of fibre activation and with V(max) during the plateau of the tetanus. Together differences in rate of activation and in V(max) accounted for 60-80% of the fibre-to-fibre variability in the rate of rise of isometric tension, depending on the measure of the force rise time used. There was not a significant correlation between the rate of fibre activation and V(max). The steady-state F-V characteristics and the rate at which these characteristics are achieved early in contraction are seemingly independent. A simulation study based on F-V properties and series compliance in frog muscle fibres indicates that if muscle activation were instantaneous, the time taken for force to rise to 50% of the plateau value would be about 60% shorter than that actually measured from living fibres. Thus about 60% of the force rise time is a consequence of the time course of activation processes and about 40% represents time taken to stretch series compliance by activated contractile material.

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.

Effect of temperature on the working stroke of muscle myosin

Muscle contraction is due to myosin motors that transiently attach with their globular head to an actin filament and generate force. After a sudden reduction of the load below the maximum isometric force (T0), the attached myosin heads execute an axial movement (the working stroke) that drives the sliding of the actin filament toward the center of the sarcomere by an amount that is larger at lower load and is 11 nm near zero load. Here, we show that an increase in temperature from 2 to 17 degrees C, which increases the average isometric force per attached myosin head by 60%, does not affect the amount of filament sliding promoted by a reduction in force from T0 to 0.7T0, whereas it reduces the sliding under low load by 2.5 nm. These results exclude the possibility that the myosin working stroke is due to the release of the mechanical energy stored in the initial endothermic force-generating process and show that, at higher temperatures, the working stroke energy is greater because of higher force, although the stroke length is smaller at low load. We conclude the following: (i) the working stroke is made by a series of state transitions in the attached myosin head; (ii) the temperature increases the probability for the first transition, competent for isometric force generation; and (iii) the temperature-dependent rise in work at high load can be accounted for by the larger free energy drop that explains the rise in isometric force.

Temporal Representation of the Delay of Iterated Rippled Noise in the Dorsal Cochlear Nucleus

It has been suggested that the dorsal cochlear nucleus (DCN) is involved in the temporal representation of both envelope periodicity and pitch. This hypothesis is tested using iterated rippled noise (IRN), which is generated by a cascade of delay and add [IRN(+)] or delay and subtract [IRN(−)] operations. The autocorrelation functions (ACFs) of the waveform and the envelope of IRN(+) have a first peak at the delay, which corresponds to the perceived pitch of the IRN. With the same delay, the pitch of IRN(−) is generally an octave lower than for IRN(+). This is reflected in a first peak at twice the delay in the ACF of the waveform for IRN(−). In contrast, for identical delays, the ACF of the envelope for both IRN(−) and IRN(+) is the same. Thus the use of IRN allows the distinction between envelope - or fine-structure sensitivity. Recordings were made from 135 single units (BFs <5 kHz) in the DCN of the anesthetized guinea pig using IRN with delays ranging from 1 to 32 ms. In our sample 42% were sensitive to the periodicity of IRN(+) and were tuned to a particular delay in their first-order interspike interval histograms (ISIHs). This tuning was highly correlated with their response to white noise. Most units with best frequencies (BFs) <500 Hz show a different all-order ISIH for IRN(+) and IRN(−), which corresponds to the perceived pitch difference, whereas units with higher BFs show a similar response to IRN(+) and IRN(−). The results indicate that low-frequency units (BF <500 Hz) in the DCN may be involved in the representation of the waveform fine structure, although units with BFs >500 Hz are able to encode only the envelope periodicity of broadband IRN in their temporal discharge characteristics.

Development of an apparatus to control load by electromagnet for a motility system in vitro

We developed an electromagnet to perform quick changes in load in the motility system consisting of myosin molecules attached to a magnetizable bead and actin filaments The electromagnet was combined with an inverted microscope and load could be quickly changed under optical observation. The magnetic field was generated by high electric current (6V, 0-125A) and the maximum field was 8,000 Oe. The maximum force exerted on a bead was 80pN at 2.5mm distance from a magnet. The change in force was 0.48% at the distance of 5.0mm from the magnet when a bead moved longitudinally for 30microm. The time to change load was about 20ms. The movements of a bead in water were recorded by video when step changes in magnetic field were applied and it was shown that a bead exactly followed the change in force. This apparatus is very much useful to analyze the transient changes in the movement of a bead, if the movement is relatively slow as in the interaction between actin and myosin from molluscan smooth muscle.

Synchronization of active mechanical oscillators by an inertial load

Motivated by the operation of myogenic (self-oscillatory) insect flight muscle, we study a model consisting of a large number of identical oscillatory contractile elements joined in a chain, whose end is attached to a damped mass-spring oscillator. When the inertial load is small, the serial coupling favors an antisynchronous state in which the extension of one oscillator is compensated by the contraction of another, in order to preserve the total length. However, a sufficiently massive load can synchronize the oscillators and can even induce oscillation in situations where isolated elements would be stable. The system has a complex phase diagram displaying quiescent, synchronous and antisynchronous phases, as well as an unusual asynchronous phase in which the total length of the chain oscillates at a different frequency from the individual active elements.

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.

The size and the speed of the working stroke of muscle myosin and its dependence on the force

Myosin II is the motor protein that produces force and shortening in muscle by ATP-driven cyclic interactions of its globular portion, the head, with the actin filament. During each interaction the myosin head undergoes a conformational change, the working stroke, which, depending on the mechanical conditions, can generate a force of several piconewtons or an axial displacement of the actin filament toward the centre of the sarcomere of several nanometres. However, the sizes of the elementary force and length steps and their dependence on the mechanical conditions are still under question. Due to the small fraction of the ATPase cycle time myosin II spends attached to actin, single molecule mechanics failed to produce definitive measurements of the individual events. In intact frog muscle fibres, however, myosin II's working stroke can be synchronised in the few milliseconds following a step reduction in either force or length superimposed on the isometric contraction. Here we show that with 150 micros force steps it is possible to separate the elastic response from the subsequent early rapid component of filament sliding due to the working stroke in the attached myosin heads. In this way we determine how the size and the speed of the working stroke depend on the clamped force. The relation between mechanical energy and force provides a molecular basis for muscle efficiency and an estimate of the isometric force exerted by a myosin head.

Fundamental step size in single cardiac and skeletal sarcomeres

In attempting to deduce the size of the elementary molecular translation step, recent experiments using single myosin molecules translating over actin filaments have shown a consistent step size of 5.4 nm (10, 21). We have carried out parallel measurements on single myofibrils from rabbit cardiac muscle and bumblebee flight muscle. Activated specimens were released or stretched with a motor-imposed ramp, and the time course of length of individual sarcomeres was measured by projecting the image of the striations onto a linear photodiode array and tracking the spacing between A-band centroids. We confirmed the 5.4-nm step. With subnanometer precision, however, we find that this value is two times that of a more fundamental step size of 2.7 nm. Step sizes were always integer multiples of 2.7 nm, whether the length change was positive or negative. This value is equal to the linear repeat of actin monomers along the thin filament, a result that ties dynamic events to molecular structure and places narrow constraints on any proposed molecular mechanism.