Some self-consistent two-state sliding filament models of muscle contraction.

The general formalism required to treat two-state sliding filament models of muscle contraction, including free energy considerations, is first reviewed and amplified. This formalism is then used to examine, and modify as needed, three models studied previously by Podolsky and Nolan, in which cross-bridge attachment-detachment and ATP turnover are not tightly coupled. No attempt is made here to establish an optimal, self-consistent model of this type because our interest is primarily in methadology rather than in fitting experimental results. But it appears from this preliminary study that such a model, with satisfactory mechanical and thermodynamic properties, could be found. An extremely simple but unrealistic two-state model is also studied which is of interest because it demonstrates the fact that it is possible, in principle at least, for sliding filament models to work with very high thermodynamic efficiencies (50-100 percent). An appendix is included that is concerned with the form of the dependence of certain first-order rate constants on the concentrations of ATP, ADP, and P.     

The theory of sliding filament models for muscle contraction. II. Biochemically-based models of the contraction cycle

A seven-state sliding filament model is proposed which differs from the model of Eisenberg & Greene. It is based on a simplified version of the in-vitro contraction cycle of Stein et al., and also has some desirable dynamical features of the empirical three-state model of Nishiyama & Murase. Appropriate x-dependences for all reaction rates are derived from the transition-state theory. The seventh-state is assumed to be a high-tension intermediate of A.M.ATP, from which direct but x-dependent dissociation can occur. If the final A.M.ATP state has a sufficiently lower tension than that of A.M.ADP.Pi, then the dominant escape path from the intermediate state is shown to be direct dissociation of the actin-myosin bond. This leads to an approximate five-state model for active and relaxed muscle in which A.M and the final A.M.ATP state are omitted.     

The theory of sliding filament models for muscle contraction. I. The two-state model

The mechanical and thermal properties of Huxley's 1957 sliding filament model for striated muscle contraction are reconsidered, with emphasis on general relationships between two-state models and muscle behaviour. New empirical forms for the rate functions f(x) and g(x) are proposed, which give an improved fit to experiments. A specific procedure, based on the shortening heat, for determining model parameters is described and applied to various choices of rate functions. The change in slope of the tension-velocity curve observed by Katz is shown to require at least one discontinuity in the binding rate f(x), and a general formula is given. The two-state model also gives a symmetric cusp in the ATP-rate vs velocity curve at v = 0. The theory is extended to time-varying situations and applied to transient responses following isometric activation by tetanus (including latency relaxation), and unloaded and after-loaded contractions. The early burst of heat production can also be fitted by assigning appropriate values to the heats of binding and dissociation.     

Step-size of net filament sliding of muscle contraction

The step-size defined as the movement between actin and myosin filaments per ATP split is an important issue in the research of muscle contraction. According to the conventional swinging-cross-bridge theory by Huxley, the step-size in this sense should be within 40 nm, a maximum possible stroke size of a myosin head, 20 nm, swing through an arc of 180 degrees . Since 1996, the estimated value over 40 nm has not been reported. Recently, however, there have been several discussions based on the one of the smallest values of step-size, 1.7 nm. We carefully re-evaluated the estimation process for the step-size, concluding that the values are still limited within 10-20 nm of conventional swinging-cross-bridge theory: 12.6 nm at no load for frog muscle.     

The sliding filament model of muscle contraction. I. Quantum mechanical formalism

The quantum mechanical analog of work is defined and discussed by using a simple hypothetical molecular machine, thus enabling the introduction of clearly defined ideas which are necessary for a molecular discussion of biological machines such as the contractile machinery in striated muscle. The problem of control of such quantum machines is discussed and shown to be possible using the concept of a stimulated transition. The problem of “reversibility” is also discussed and shown to have a satisfactory solution for the orders of magnitude of the forces and velocities involved in muscular contractile machinery.     

The sliding filament model of muscle contraction. III. Stability analysis and sinusoidal perturbations

A theoretical model of the energy transducing part of a single sarcomere, which has previously accounted for most of the steady state energetic and dynamical properties of striated muscle, is subjected to stability analysis. The steady states of isotonic and isometric contraction turn out to be stable and the magnitudes of the characteristic roots allow accurate reduction of the original seven state theory to a four state theory. Both versions possess a complex pair of roots giving a damped oscillatory character to the contraction velocity, the tension generated and the populations of the crossbridge states.     

Hugh E. Huxley: birth of the filament sliding model of muscle contraction

Hugh E. Huxley used a combination of X-ray diffraction and electron microscopy to define the structural basis for muscle contraction. He developed the entirely new concept of movement based on the relative motion of two elongated protein polymers powered by the action of independent molecular motors. The concept applies to a large number of cell motility mechanisms.     

The sliding filament model of muscle contraction. IV. The effect of variation of ATP concentration

A theory of the nature of the fundamental transducing unit in striated muscle is applied to some recent experimental results on the variation of isometric tension and stiffness resonance phenomena as functions of ATP concentration. Both are accounted for satisfactorily without the introduction of arbitrary curve fitting procedures.     

The sliding filament model of muscle contraction. II. The energetic and dynamical predictions of a quantum mechanical transducer model

A theoretical model of a molecular energy transducing unit designed for the production of mechanical work is constructed and its consequences examined and compared with the experimentally determined myothermal and dynamic properties of vertebrate striated muscle. The model rests on a number of independent assumptions which include: the almost instantaneous generation of mechanical force by the occurrence of a radiationless transition between vibronic states of the transducer (crossbridge) at a point of potential energy surface crossing; transmission of this force to the load via the active sites on the thin filament by means of non-bonding repulsive forces, no energy being required for detachment; “detachment” consists of a second radiationless transition at a lower energy point than the first force generating transition, the energy difference appearing largely as work. The method of force generation completely avoids problems such as the “force-rate dilemma” which occur repeatedly in any discussion where state populations are near-Boltzmann and also leads without further arbitrary assumptions to such concepts as “attached but non force-producing states” and strongly position dependent “attachment” and “detachment” rate constants since these can only be appreciable near potential energy surface crossings. The kinetics and energetics of a transducer of this type operating cyclically and converting ATP → ADP + P_i are considered and shown to lead to length-tension and energetic behaviour very similar to that exhibited by vertebrate striated muscle, both for contraction and stretching. The existence of a limiting tension for stretching is predicted by the model as is the decrease of the rate of enthalpy release rate below the isometric value. At the limiting tension the rate of enthalpy release by the transducers is virtually zero, as observed. However, the stretching only inhibits the ATP hydrolysis, the cyclic synthesis from ADP and work being impossible with this model. The response to rapid length step changes automatically contains the asymmetry observed experimentally (with respect to lengthening and shortening) and arbitrary assumptions over and above those giving adequate explanation of the steady-state properties are not required. The asymmetry arises mainly as a consequence of the non-bonded pushing action of the crossbridges. This same assumption predicts the occurrence of an asymmetric thermoelastic ratio for active muscle with respect to stretching and contraction. The quantitative aspects of the model are satisfactory as it simultaneously reconciles the numerical magnitudes of macroscopic quantities such as isometric tension, maximum contraction velocity, limiting tension sustainable on stretching, isometric heat rate and resting heat rate with molecular parameters such as the filament and crossbridge periodicities, molecular vibrational relaxation rates, recurrence times for the radiationless transitions occurring, etc. This is achieved without any parameter optimization and only a very much smaller number of unknown parameters than the number of observed results accounted for. Many of the entities occurring in the model cycle (vibronic states of crossbridges, ATP, etc.) appear to be in one-to-one correspondence with many of the kinetic entities postulated to account for the biochemical kinetic results obtained for the actomyosin ATPase system in vitro. Finally, the rigor state has to be viewed in a different way from the conventional one; on the basis that the present model states which are part of the contraction cycle but sparsely populated during the latter (and hence are of chemical kinetic but not dynamical importance) are heavily populated during the rigor state. The mechanical properties of the rigor state would then be determined by these molecular states which would be very short-lived during the contraction cycle. If this is correct the rigor state could yield much more information about inaccessible parts of the contraction cycle than is presently supposed. The model leads one to expect a rather different response to quick length step changes in the rigor state from that of the active state, in contrast to current interpretations in terms of a large number of attached crossbridges, unable to detach due to the absence of ATP.     

Fifty years of muscle and the sliding filament hypothesis.

This review describes the early beginnings of X-ray diffraction work on muscle structure and the contraction mechanism in the MRC Unit in the Cavendish Laboratory, Cambridge, and later work in the MRC Molecular Biology Laboratory in Hills Road, Cambridge, where the author worked for many years, and elsewhere. The work has depended heavily on instrumentation development, for which the MRC laboratory had made excellent provision. The search for ever higher X-ray intensity for time-resolved studies led to the development of synchrotron radiation as an exceptionally powerful X-ray source. This led to the first direct evidence for cross-bridge tilting during force generation in muscle. Further improvements in technology have made it possible to study the fine structure of some of the X-ray reflections from contracting muscle during mechanical transients, and these are currently providing remarkable insights into the detailed mechanism of force development by myosin cross-bridges.     

Changes of thick filament structure during contraction of frog striated muscle.

The strongest myosin-related features in the low-angle axial x-ray diffraction pattern of resting frog sartorius muscle are the meridional reflections corresponding to axial spacings of 21.4 and 14.3 nm, and the first layer line, at a spacing 42.9 nm. During tetanus the intensities of the first layer line and the 21.4-nm meridional decrease by 62 and 80% respectively, but, when the muscle is fresh, the 14.3-nm meridional intensity rises by 13%, although it shows a decrease when the muscle is fatigued. The large change in the intensity of the 21.4-nm meridional reflection suggests that the projected myosin cross-bridge density onto the thick filament axis changes during contraction. The model proposed by Bennett (Ph.D. Thesis, University of London, 1977) in which successive cross-bridge levels are at 0,3/8, and 5/8 of the 42.9-nm axial repeat in the resting muscle, passing to 0, 1/3, and 2/3 in the contracting state, can explain why the 21.4-nm reflection decreases in intensity while the 14.3-nm increases when the muscle is activated. The model predicts a rather larger increase of the 14.3-nm reflection intensity during contraction than that observed, but the discrepancy may be removed if a small change of shape or tilt of the cross-bridges relative to the thick filament axis is introduced. The decrease of the intensity of the first layer line indicates that the cross-bridges become disordered in the plane perpendicular to the filament axis.     

Thermodynamic features of myosin filament suspensions: implications for the modeling of muscle contraction.

The analysis of myosin filament suspensions shows that these solutions are characterized by highly nonideal behavior. From these data a model is constructed that allows us to predict that 1) when subjected to an increasing protein osmotic pressure, myosin filaments experience an elastic deformation, which is not linearly related to the acting force; and 2) at constant protein osmotic pressure, when the cross-bridges of the myosin filaments are subjected to an external, nonosmotic force parallel to the filament axis, they are deformed and the water activity coefficient is altered. As a consequence, in muscle, passive and active shortening of the sarcomere is expected to promote the change of the water-water and of the water-protein interactions. We thus propose to depict muscle contraction as a chemo-osmoelastic transduction, where the analysis of the energy partition during the power stroke requires consideration of the osmotic factor in addition to the chemoelastic ones.     

Thin filament proteins and thin filament-linked regulation of vertebrate muscle contraction

Recent developments in the field of myofibrillar proteins will be reviewed. Consideration will be given to the proteins that participate in the contractile process itself as well as to those involved in Ca-dependent regulation of striated (skeletal and cardiac) and smooth muscle. The relation of protein structure to function will be emphasized and the relation of various physiologically and histochemically defined fiber types to the proteins found in them will be discussed.     

Analysis of models for the activation and contraction of muscle

Huxley (1957) proposed a sliding filament model of muscular contraction to which Julian (1969) added equations for the activation produced by cations. Each parameter in the combined Huxley-Julian model has been varied systematically to determine its effect on the predicted twitches. The slower rate constant for Ca activation has a predominant effect on the relaxation phase of the twitch. The series elasticity and the rate constants for the making and the breaking of cross-bridges all strongly affect the contraction phase of the twitch. Further experimental work is required to determine which factor is rate limiting under a given set of conditions.Taking the Fourier transform of the twitch gives a prediction for the frequency response of the model. The predicted frequency response curves are well-fitted by those of a simple, second order system, in agreement with recent experiments (Mannard &; Stein, 1973). The parameters of the best-fitting, second-order, frequency response curves vary experimentally with mean stimulus rate. This variation probably results from a saturation at higher stimulus rates of the pXocesses for reuptake of Ca into the sarcoplasmic reticulum. The saturation of Ca reuptake, together with the saturation of the myofilaments by Ca at higher stimulus rates, can account qualitatively for the sigmoid rate-tension curves found experimentally.     

Regulation of contraction and thick filament assembly-disassembly in glycerinated vertebrate smooth muscle cells

Isolated smooth muscle cells and cell fragments prepared by glycerination and subsequent homogenization will contract to one-third their normal length, provided Ca++ and ATP are present. Ca++- independent contraction was obtained by preincubation in Ca++ and ATP gamma S, or by addition of trypsin-treated myosin light chain kinase (MLCK) that no longer requires Ca++ for activation. In the absence of Ca++, myosin was rapidly lost from the cells upon addition of ATP. Glycerol-urea-PAGE gels showed that none of this myosin is phosphorylated. The extent of myosin loss was ATP- and pH-dependent and occurred under conditions similar to those previously reported for the in vitro disassembly of gizzard myosin filaments. Ca++-dependent contraction was restored to extracted cells by addition of gizzard myosin under rigor conditions (i.e., no ATP), followed by addition of MLCK, calmodulin, Ca++, and ATP. Function could also be restored by adding all these proteins in relaxing conditions (i.e., in EGTA and ATP) and then initiating contraction by Ca++ addition. Incubation with skeletal myosin will restore contraction, but this was not Ca++- dependent unless the cells were first incubated in troponin and tropomyosin. These results strengthen the idea that contraction in glycerinated cells and presumably also in intact cells is primarily thick filament regulated via MLCK, that the myosin filaments are unstable in relaxing conditions, and that the spatial information required for cell length change is present in the thin filament- intermediate filament organization.     

Invited Review: pathophysiology of cardiac muscle contraction and relaxation as a result of alterations in thin filament regulation.

Cardiac muscle contraction depends on the tightly regulated interactions of thin and thick filament proteins of the contractile apparatus. Mutations of thin filament proteins (actin, tropomyosin, and troponin), causing familial hypertrophic cardiomyopathy (FHC), occur predominantly in evolutionarily conserved regions and induce various functional defects that impair the normal contractile mechanism. Dysfunctional properties observed with the FHC mutants include altered Ca(2+) sensitivity, changes in ATPase activity, changes in the force and velocity of contraction, and destabilization of the contractile complex. One apparent tendency observed in these thin filament mutations is an increase in the Ca(2+) sensitivity of force development. This trend in Ca(2+) sensitivity is probably induced by altering the cross-bridge kinetics and the Ca(2+) affinity of troponin C. These in vitro defects lead to a wide variety of in vivo cardiac abnormalities and phenotypes, some more severe than others and some resulting in sudden cardiac death.     

X-ray evidence for radial cross-bridge movement and for the sliding filament model in actively contracting skeletal muscle

Equatorial X-ray diffraction patterns have been studied from muscles at rest, during contraction and in rigor. It is confirmed that the relative intensity ($\text{I}\text{1,0}\text{I}\text{1,1}$) of the two main equatorial reflections depends both on the sarcomere length and on the state of the muscle; in any one state the ratio $\text{I}\text{1,0}\text{I}\text{1,1}$ increases as the sarcomere length of the muscle increases, while at any fixed sarcomere length the ratio is smaller for contracting muscle than for resting muscle and smaller still for rigor muscles. The change of $\text{I}\text{1,0}\text{I}\text{1,1}$ with change of state at constant sarcomere length is interpreted as being due to radial movement of cross-bridges: the average movement during contraction being about 40% of that in rigor.Over the whole range of sarcomere length studied (between 1.8 and 2.7 μm) there was no evidence for any change in lattice spacing when a muscle contracts isometrically.Muscles were studied generating tension after they had shortened actively against a load. The lattice spacings and intensity ratio $\text{I}\text{1,0}\text{I}\text{1,1}$ both changed during active shortening in a way entirely consistent with the sliding filament theory of contraction.     

On the theory of muscle contraction: filament extensibility and the development of isometric force and stiffness.

The newly discovered extensibility of actin and myosin filaments challenges the foundation of the theory of muscle mechanics. We have reformulated A. F. Huxley's sliding filament theory to explicitly take into account filament extensibility. During isometric force development, growing cross-bridge tractions transfer loads locally between filaments, causing them to extend and, therefore, to slide locally relative to one another. Even slight filament extensibility implies that 1) relative displacement between the two must be nonuniform along the region of filament overlap, 2) cross-bridge strain must vary systematically along the overlap region, and importantly, 3) the local shortening velocities, even at constant overall sarcomere length, reduce force below the level that would have developed if the filaments had been inextensible. The analysis shows that an extensible filament system with only two states (attached and detached) displays three important characteristics: 1) muscle stiffness leads force during force development; 2) cross-bridge stiffness is significantly higher than previously assessed by inextensible filament models; and 3) stiffness is prominently dissociated from the number of attached cross-bridges during force development. The analysis also implies that the local behavior of one myosin head must depend on the state of neighboring attachment sites. This coupling occurs exclusively through local sliding velocities, which can be significant, even during isometric force development. The resulting mechanical cooperativity is grounded in fiber mechanics and follows inevitably from filament extensibility.