A Model for the Transient and Steady-State Mechanical Behavior of Contracting Muscle

A model was developed which can simulate both the transient and steady-state mechanical behavior of contracting skeletal striated muscle. Thick filament cross-bridges undergo cycles of attachment to and detachment from thin filament sites. Cross-bridges can attach only while in the first of two stable states. Force is then generated by a transition to the second state after which detachment can occur. Cross-bridges are assumed to be connected to the thin filaments by an elastic element whose extension or compression influences the rate constants for attachment, detachment, and changes between states. The model was programmed for a digital computer and attempts made to match both the transient and the steady-state responses of the model to that of real muscle in two basic types of experiment: force response to sudden change in length and length response to sudden reduction of load from P_o. Values for rate constants and other parameters were chosen to try to match the model's output to results from real muscles, while at the same time trying to accommodate structural and biochemical information.     

A Phenomenological Theory of Muscular Contraction: I. Rate Equations at a Given Length Based on Irreversible Thermodynamics

A phenomenological theory for contracting muscle based on irreversible thermodynamics and the sliding filament theory is developed. The individual cross bridges, considered as subunits, are viewed as linear energy converters with constant transport coefficients. With this view of the subunits, phenomenological equations applicable to the whole muscle are obtained. The transport coefficients are shown to be a function of a single parameter which is the number of activated cross bridges at any instant. By requiring Hill's force-velocity relation (1) to be satisfied, the response of the muscle is related to the number of activated cross bridges. The resulting theory differs significantly from the theory developed by Caplan (2) and a comparison of the theories is presented. The theory is shown to correlate well with the heat data of Woledge (3) for a tortoise muscle and gives a value of Y (ratio of chemical affinity to enthalpy of reaction) equal to 0.945. The comparison of the theory with Hill's frog muscle data (1) and (4) is also encouraging. In part II of this series, length variations are considered and the resulting theoretical predictions are shown to be consistent with experimental data.     

Comparison of Caplan's Irreversible Thermodynamic Theory of Muscle Contraction with Chemical Data

Recently Caplan (1) applied the concepts of irreversible thermodynamics and cybernetics to contracting muscle and derived Hill's force-velocity relation. Wilkie and Woledge (2) then compared Caplan's theory to chemical rates inferred from heat data and concluded that the theory was not consistent with the data. Caplan defended his theory in later papers (3, 4) but without any direct experimental verifications. As Wilkie and Woledge (2) point out, the rate of phosphorylcreatine (PC) breakdown during steady states of shortening has not been observed because of technical difficulties. In this paper it is shown that the rate equations may be directly integrated with time to obtain relations among actual quantities instead of rates. The validity of this integration is based on experimental evidence which indicates that certain combinations of the transport coefficients are constant with muscle length. These equations are then directly compared to experimental data of Cain, Infante, and Davies (5) with the following conclusions: (a) The measured variations of ΔPC for isotonic contractions are almost exactly as predicted by Caplan's theory. (b) The value of the chemical rate ratio, ν_m/ν_o, obtained from these data was 3.53 which is close to the value of 3 suggested by Caplan (3). (c) The maximum value of the chemical affinity for PC splitting was found to be 10.6 k cal/mole which is as expected from in vitro measurements (2). Because of the excellent agreement between theory and experiment, we conclude that Caplan's theory definitely warrants further investigation.     

Optimal length of smooth muscle assessed by a microstructurally and statistically based constitutive model

Smooth muscle exhibits an optimal length at which it is able to generate a maximum amount of force. In this study, the optimal length is assessed by use of a microstructurally and statistically based constitutive model for smooth muscle. The model is based on the sliding filament theory, and a modified version of Hill's mechanical model was adopted. It was conjectured, that a variation in the overlap in the actomyosin contractile units together with a statistical dispersion in the size of the dense bodies are responsible for the optimal length characteristics. The influence of contractile unit length, dense body size and dense body compliance was investigated, and the model was fully able to predict experimental data. The results indicate that the compliance of the dense bodies does not contribute significantly to the total compliance of the contractile apparatus.     

An exact analysis of the sliding motion in contractile system: A simplified model of muscle system

A system involving two kinds of sliding filaments is analysed with special attention to the actomyosin system. Rigorous results are obtained about the statistical effect originating from many active sites distributed on both filaments. It is necessary for the occurrence of smooth motion in sliding filament that the spatial periods of active sites on both filaments are relatively incommensurable, and that the number of active sites on each filament is large enough. Sufficient conditions for smooth contraction are derived under the assumption that both filaments are rigid; this is called rigid rod approximation in the present paper. The elastic mode of the filaments, during the sliding process, is analysed by perturbation theory based on the rigid rod approximation. A stochastic theory is briefly discussed in reference to the cooperative generation of contractile force, which is concerned in Hill's relation of muscle contraction.     

The Effect of Myofilament Compliance on Kinetics of Force Generation by Myosin Motors in Muscle

We use the inhibitor of isometric force of skeletal muscle N-benzyl-p-toluene sulfonamide (BTS) to decrease, in a dose dependent way, the number of myosin motors attached to actin during the steady isometric contraction of single fibers from frog skeletal muscle (4°C, 2.1 μm sarcomere length). In this way we can reduce the strain in the myofilament compliance during the isometric tetanus (T₀) from 3.54 nm in the control solution (T_0,NR) to ∼0.5 nm in 1 μM BTS, where T₀ is reduced to ∼0.15 T_0,NR. The quick force recovery after a step release (1-3 nm per half-sarcomere) becomes faster with the increase of BTS concentration and the decrease of T₀. The simulation of quick force recovery with a multistate model of force generation, that adapts Huxley and Simmons model to account for both the high stiffness of the myosin motor (∼3 pN/nm) and the myofilament compliance, shows that the increase in the rate of quick force recovery by BTS is explained by the reduced strain in the myofilaments, consequent to the decrease in half-sarcomere force. The model estimates that i), for the same half-sarcomere release the state transition kinetics in the myosin motor are five times faster in the absence of filament compliance than in the control; and ii), the rate of force recovery from zero to T₀ is ∼6000/s in the absence of filament compliance.     

On the molecular dynamics of the chemo-mechanical conversion in muscle contraction

The molecular dynamics of energy conversion by the actomyosin system in muscle contraction is studied by comparing two different types of model on the motion of crossbridge on thin filament. The motion is associated with a transition between two stable states in Huxley and Simmons' model while in Shimizu et al.'s model with a transition from an unstable to a stable state. The rate of the transition, which is proportional to the velocity of shortening of muscle in steady state, is calculated by representing the motion of crossbridge by that of a Brownian particle moving on a one-dimensional linear potential. In the case of the Huxley-Simmons model the energy conversion process is essentially a thermal one and the velocity of shortening depends sharply on the number of crossbridges on muscular filament, which is proportional to the overlapping length between thin and thick filaments. On the other hand, in the case of the Shimizu model the energy conversion process is a deterministic one which means that muscle is able to shorten smoothly and that the velocity of shortening is almost independent of the overlapping length. Experimental observations by Gordon et al. are consistent with the latter model.     

A simplified sliding-filament muscle model for simulation purposes

A muscle model based on the sliding filament concept is put forward. The model is simplified in such a way that it may easily be simulated on a computer (analog or digital). The model shows quite a few muscle properties rather well e.g. Hill's relation, quick-stretch and quick-release series elasticity and some dynamic transfer properties. It is found that, using an antagonistic muscle model pair and an appropriate load, realistic arm movements can be simulated.     

An X-Ray Diffraction Study of Contracting Molluscan Smooth Muscle

The living anterior byssus retractor muscle of Mytilus (ABRM), a smooth, “catch” muscle, has been studied by X-ray diffraction while relaxed and while tonically contracted. X-ray reflections were observed from the actin and paramyosin filaments and from the α-helical substructure of the paramyosin filaments. No differences in spacings or relative intensities were observed when the relaxed and contracting muscle patterns were compared. This result is consistent with a sliding filament mechanism involving an interaction between actin and paramyosin filaments.     

Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function

Striated muscle contraction requires intricate interactions of microstructures. The classic textbook assumption that myosin filaments are compressed at the meshed Z-disc during striated muscle fibre contraction conflicts with experimental evidence. For example, myosin filaments are too stiff to be compressed sufficiently by the muscular force, and, unlike compressed springs, the muscle fibres do not restore their resting length after contractions to short lengths. Further, the dependence of a fibre''s maximum contraction velocity on sarcomere length is unexplained to date. In this paper, we present a structurally consistent model of sarcomere contraction that reconciles these findings with the well-accepted sliding filament and crossbridge theories. The few required model parameters are taken from the literature or obtained from reasoning based on structural arguments. In our model, the transition from hexagonal to tetragonal actin filament arrangement near the Z-disc together with a thoughtful titin arrangement enables myosin filament sliding through the Z-disc. This sliding leads to swivelled crossbridges in the adjacent half-sarcomere that dampen contraction. With no fitting of parameters required, the model predicts straightforwardly the fibre''s entire force–length behaviour and the dependence of the maximum contraction velocity on sarcomere length. Our model enables a structurally and functionally consistent view of the contractile machinery of the striated fibre with possible implications for muscle diseases and evolution.     

A continuous-binding cross-linker model for passive airway smooth muscle

Although the active properties of airway smooth muscle (ASM) have garnered much modeling attention, the passive mechanical properties are not as well studied. In particular, there are important dynamic effects observed in passive ASM, particularly strain-induced fluidization, which have been observed both experimentally and in models; however, to date these models have left an incomplete picture of the biophysical, mechanistic basis for these behaviors. The well-known Huxley cross-bridge model has for many years successfully described many of the active behaviors of smooth muscle using sliding filament theory; here, we propose to extend this theory to passive biological soft tissue, particularly ASM, using as a basis the attachment and detachment of cross-linker proteins at a continuum of cross-linker binding sites. The resulting mathematical model exhibits strain-induced fluidization, as well as several types of force recovery, at the same time suggesting a new mechanistic basis for the behavior. The model is validated by comparison to new data from experimental preparations of rat tracheal airway smooth muscle. Furthermore, experiments in noncontractile tissue show qualitatively similar behavior, suggesting support for the protein-filament theory as a biomechanical basis for the behavior.     

Hugh E. Huxley: The Compleat Biophysicist

The sliding filament model of muscle contraction, put forward by Hugh Huxley and Jean Hanson in 1954, is 60 years old in 2014. Formulation of the model and subsequent proof was driven by the pioneering work of Hugh Huxley (1924–2013). We celebrate Huxley’s integrative approach to the study of muscle contraction; how he persevered throughout his career, to the end of his life at 89 years, to understand at the molecular level how muscle contracts and develops force. Here we show how his life and work, with its focus on a single scientific problem, had impact far beyond the field of muscle contraction to the benefit of multiple fields of cellular and structural biology. Huxley introduced the use of x-ray diffraction to study the contraction in living striated muscle, taking advantage of the paracrystalline lattice that would ultimately allow understanding contraction in terms of single molecules. Progress required design of instrumentation with ever-increasing spatial and temporal resolution, providing the impetus for the development of synchrotron facilities used for most protein crystallography and muscle studies today. From the time of his early work, Huxley combined electron microscopy and biochemistry to understand and interpret the changes in x-ray patterns. He developed improved electron-microscopy techniques, thin sections and negative staining, that enabled answering major questions relating to the structure and organization of thick and thin filaments in muscle and the interaction of myosin with actin and its regulation. Huxley established that the ATPase domain of myosin forms the crossbridges of thick filaments that bind actin, and introduced the idea that myosin makes discrete steps on actin. These concepts form the underpinning of cellular motility, in particular the study of how myosin, kinesin, and dynein motors move on their actin and tubulin tracks, making Huxley a founder of the field of cellular motility.     

Head Rotation or Dissociation?: A Study of Exponential Rate Processes in Chemically Skinned Rabbit Muscle Fibers When MgATP Concentration Is Changed

The mechanical response of fully activated muscle bundles (one to five fibers) to sinusoidal length perturbation (~0.4% L₀) was studied as a function of MgATP concentration. The frequency response (0.25-167 Hz; corresponding to 1 ms time resolution) of chemically skinned rabbit muscle fibers was resolved into three exponential rate processes, (A), (B), and (C). At 20°C, the apparent rate constants associated with the fast exponential lead (2πc = 388-588 s^-1) and the oscillatory work (2πb = 59-116 s^-1) both increase with increment of the MgATP concentration from 1 to 5 mM, and they both saturate for further increase. Over the whole range of MgATP concentrations the slow exponential lead (2πa = 9-7 s^-1) remains constant. The effect of MgATP on processes (B) and (C) can be interpreted in the context of the biochemical evidence, in which MgATP enters the cross-bridge cycle after the desorption of the product, and the binding of MgATP to rigorlike cross-bridges promotes a rapid dissociation of actomyosin (Lymn and Taylor, 1971. Biochemistry. 10:4617-4624.). The effect is not predicted by a model for force generation in which head rotation dominates the fast component (“stage 2” of Huxley and Simmons, 1971. Nature (Lond.). 233:533-538. and 1973. Cold Spring Harbor Symp. Quant. Biol. 37:669-680.), and head dissociation dominates the slow component (“phase 4” of Huxley, 1974. J. Physiol. (Lond.). 243:1-43; Julian et al., 1974. Biophys. J. 14: 546-562.).     

Hugh E. Huxley: The Compleat Biophysicist

The sliding filament model of muscle contraction, put forward by Hugh Huxley and Jean Hanson in 1954, is 60 years old in 2014. Formulation of the model and subsequent proof was driven by the pioneering work of Hugh Huxley (1924–2013). We celebrate Huxley’s integrative approach to the study of muscle contraction; how he persevered throughout his career, to the end of his life at 89 years, to understand at the molecular level how muscle contracts and develops force. Here we show how his life and work, with its focus on a single scientific problem, had impact far beyond the field of muscle contraction to the benefit of multiple fields of cellular and structural biology. Huxley introduced the use of x-ray diffraction to study the contraction in living striated muscle, taking advantage of the paracrystalline lattice that would ultimately allow understanding contraction in terms of single molecules. Progress required design of instrumentation with ever-increasing spatial and temporal resolution, providing the impetus for the development of synchrotron facilities used for most protein crystallography and muscle studies today. From the time of his early work, Huxley combined electron microscopy and biochemistry to understand and interpret the changes in x-ray patterns. He developed improved electron-microscopy techniques, thin sections and negative staining, that enabled answering major questions relating to the structure and organization of thick and thin filaments in muscle and the interaction of myosin with actin and its regulation. Huxley established that the ATPase domain of myosin forms the crossbridges of thick filaments that bind actin, and introduced the idea that myosin makes discrete steps on actin. These concepts form the underpinning of cellular motility, in particular the study of how myosin, kinesin, and dynein motors move on their actin and tubulin tracks, making Huxley a founder of the field of cellular motility.     

Phosphate and ADP Differently Inhibit Coordinated Smooth Muscle Myosin Groups

Actin filaments propelled in vitro by groups of skeletal muscle myosin motors exhibit distinct phases of active sliding or arrest, whose occurrence depends on actin length (L) within a range of up to 1.0 μm. Smooth muscle myosin filaments are exponentially distributed with ≈150 nm average length in vivo—suggesting relevance of the L-dependence of myosin group kinetics. Here, we found L-dependent actin arrest and sliding in in vitro motility assays of smooth muscle myosin. We perturbed individual myosin kinetics with varying, physiological concentrations of phosphate (P_i, release associated with main power stroke) and adenosine diphosphate (ADP, release associated with minor mechanical step). Adenosine triphosphate was kept constant at physiological concentration. Increasing [P_i] lowered the fraction of time for which actin was actively sliding, reflected in reduced average sliding velocity (ν) and motile fraction (f_mot, fraction of time that filaments are moving); increasing [ADP] increased the fraction of time actively sliding and reduced the velocity while sliding, reflected in reduced ν and increased f_mot. We introduced specific P_i and ADP effects on individual myosin kinetics into our recently developed mathematical model of actin propulsion by myosin groups. Simulations matched our experimental observations and described the inhibition of myosin group kinetics. At low [P_i] and [ADP], actin arrest and sliding were reflected by two distinct chemical states of the myosin group. Upon [P_i] increase, the probability of the active state decreased; upon [ADP] increase, the probability of the active state increased, but the active state became increasingly similar to the arrested state.     

Experimental determination of sarcomere force–length relationship in type-I human skeletal muscle fibers

The objectives of this study were to measure the active and passive force–length (F–L) relationships in type-I human single muscle fibers and to compare the results to predictions from the sliding filament model (the “standard model”). We measured isometric forces in chemically skinned fibers at different sarcomere lengths (SLs) in separate maximal activations. The experimental tolerance interval for optimal SL was calculated to be (2.37, 2.95 μm), which included the prediction by the standard model (2.64, 2.81 μm). Average passive slack length was 2.22±0.08 μm, and the passive F–L relationship was well described by an exponential function. Best fit lines were used to estimate the ascending and descending limbs from the active F–L data using the average SL obtained from a digital image of the fiber. The experimental descending limb was also estimated using the shortest SL to address the possible effects of sarcomere inhomogeneity (SI). The experimental slopes of the ascending and descending limbs, 0.42 F_o/μm and ?0.52 F_o/μm (vs. ?0.55 F_o/μm with the shortest SL) respectively, F_o being the maximal isometric force, were significantly less in magnitude than those from the standard model. These results suggested that the difference between experimental and standard models was not fully explained by SI and other factors could be important. The broader experimental F–L curve compared to the standard model implies that human muscle has functionally a wider operating length range where its force-generating capacity is not compromised.     

Myosin S2 Origins Track Evolution of Strong Binding on Actin by Azimuthal Rolling of Motor Domain

Myosin crystal structures have given rise to the swinging lever arm hypothesis, which predicts a large axial tilt of the lever arm domain during the actin-attached working stroke. Previous work imaging the working stroke in actively contracting, fast-frozen Lethocerus muscle confirmed the axial tilt; but strongly bound myosin heads also showed an unexpected azimuthal slew of the lever arm around the thin filament axis, which was not predicted from known crystal structures. We hypothesized that an azimuthal reorientation of the myosin motor domain on actin during the weak-binding to strong-binding transition could explain the lever arm slew provided that myosin’s α-helical coiled-coil subfragment 2 (S2) domain emerged from the thick filament backbone at a particular location. However, previous studies did not adequately resolve the S2 domain. Here we used electron tomography of rigor muscle swollen by low ionic strength to pull S2 clear of the thick filament backbone, thereby revealing the azimuth of its point of origin. The results show that the azimuth of S2 origins of those rigor myosin heads, bound to the actin target zone of actively contracting muscle, originate from a restricted region of the thick filament. This requires an azimuthal reorientation of the motor domain on actin during the weak to strong transition.     

A Finite Element Approach for Skeletal Muscle using a Distributed Moment Model of Contraction

Abstract

The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.     

A MELTING POINT FOR THE BIREFRINGENT COMPONENT OF MUSCLE

The A filament of the striated muscle sarcomere is an ordered aggregate of one or a few species of proteins. Ordering of these filaments into a parallel array is the basis of birefringence in the A region, and loss of birefringence is therefore a measure of decreased order. Heating caused a large decrease in the birefringence of glycerinated rabbit psoas muscle fibers over a narrow temperature range (~3°C) and a large decrease in both the birefringence and optical density of the A region of Drosophila melanogaster fibrils. These changes were interpreted as a loss of A filament structure and were used to define a transition temperature (T_tr) as a measure of the stability of the A region. Since the transition temperature was sensitive to pH, ionic strength, and urea, solvent conditions which often affect protein structure, it is an experimentally useful indicator for factors affecting the structure of the A filament. Fibers from glycerinated frog muscle were less stable over a wide pH range than fibers from glycerinated rabbit muscle, a fact which demonstrates a species difference in structure. Glycerinated rabbit fibrils heated to 70°C shortened to about 40% of their initial length. The extent of shortening was not correlated with the loss of birefringence, and phase-contrast microscopy showed that this shortening occurred in the I region as well as in the A region. This response may be useful for studying the I filament and actin in much the same way that the decrease in birefringence was used for studying the A filament and myosin. The observations presented show that some properties of muscle proteins can be studied essentially in situ without the necessity of first dispersing the structure in solutions of high or low ionic strength.