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.     

Computer simulation of flagellar movement. IV. Properties of an oscillatory two-state cross-bridge model.

A stochastic computational method is used to examine the properties of a simple two-state cross-bridge model, of a type which has been shown previously to self-oscillate without requiring any feedback control of the active process. The force transients obtained with this model show the major features observed with oscillatory insect fibrillar flight muscle. The effects of viscosity and cross-bridge detachment rate on the frequency of oscillation of the model resemble the effects of viscosity and ATP concentration on flagellar oscillation, and the relationship between turnover rate and frequency of oscillation is also consistent with observations on flagella. However, the amplitude of oscillation of the model does not show the degree of frequency-independence which is typical of flagella.     

Modelling concentric contraction of muscle using an improved cross-bridge model.

Despite its overwhelming acceptance in muscle research, the cross-bridge theory does not account for all phenomena observed during muscular contractions. A phenomenon which has received much attention in the biomechanics literature, but has evaded convincing explanation and is not accounted for in the formulation of the classic cross-bridge theory, is the persistent aftereffects of muscular length changes on force production. For example, following muscle shortening, the isometric force of a muscle is depressed for a long time period ( > 5 s) compared to the corresponding isometric force following no length change. In the present study, the classic cross-bridge model was modified in two ways in an attempt to account for the force depressions following muscle shortening. First, the steady-state force depressions following shortening were described by a single scalar variable: the work performed by the muscle during shortening; and second, the dynamic, history-dependent cross-bridge properties were described using a fading memory function. The proposed model was developed and tested for shortening of the cat soleus at constant speeds ranging from 4 to 32 mm/s, for shortening at changing speeds, and for shortening of different magnitudes ranging from 2 to 10 mm. The history-dependent forces during shortening and the steady-state force depressions following shortening were well captured with the modified cross-bridge model. The present model contains two mathematically simple adaptations to the classic cross-bridge model, and is the first such model to account for the long-lasting force depressions following muscle shortening using a single scalar variable.     

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.     

Skeletal muscle myosin cross-bridge cycling is necessary for myofibrillogenesis

A major stimulus affecting myofibrillogenesis in both embryonic and mature striated muscle is contractile activity. There are two major signals associated with contractile activity: a physiological signal, the transient increase in intracellular calcium, and a physical signal, the transient increase in tension production. However, dissociating these two signals to examine their relative contributions to myofibrillogenesis has proven difficult. In this study, we have used two different myosin inhibitors to determine the importance of myosin cross-bridge cycling in sarcomere assembly. We find that the small-molecule inhibitor 2,3-butanedione monoxime (BDM), which inhibits myosin ATPase, disrupts myofibrillogenesis in amphibian myocytes, consistent with results from avian studies. However, BDM is a weak myosin inhibitor and it is non-specific; concentrations that inhibit contraction and disrupt myofibrillogenesis also disrupt calcium signaling. Therefore, we also used the recently identified skeletal muscle myosin II inhibitor, N-benzyl-p-toluenesulphonamide (BTS), which has high affinity and specificity for skeletal muscle fast myosin. BTS inhibits contraction and results in myofibrillar disruption that phenocopies our results with BDM. However, BTS does not affect either spontaneous or induced calcium transients. Furthermore, BTS is reversible and does not significantly affect the expression levels of myosin or actin. Thus, our convergent results with BDM and BTS suggest that sarcomere assembly depends on active regulation of tension in the forming myofibril.     

The myosin swinging cross-bridge model

No biological system has been studied by more diverse approaches than the actin-based molecular motor myosin. Biophysics, biochemistry, physiology, classical genetics and molecular genetics have all made their contributions, and myosin is now becoming one of the best-understood enzymes in biology.     

A cross-bridge model that is able to explain mechanical and energetic properties of shortening muscle.

The responses of muscle to steady and stepwise shortening are simulated with a model in which actin-myosin cross-bridges cycle through two pathways distinct for the attachment-detachment kinetics and for the proportion of energy converted into work. Small step releases and steady shortening at low velocity (high load) favor the cycle implying approximately 5 nm sliding per cross-bridge interaction and approximately 100/s detachment-reattachment process; large step releases and steady shortening at high velocity (low load) favor the cycle implying approximately 10 nm sliding per cross-bridge interaction and approximately 20/s detachment-reattachment process. The model satisfactorily predicts specific mechanical properties of frog skeletal muscle, such as the rate of regeneration of the working stroke as measured by double-step release experiments and the transition to steady state during multiple step releases (staircase shortening). The rate of energy liberation under different mechanical conditions is correctly reproduced by the model. During steady shortening, the relation of energy liberation rate versus shortening speed attains a maximum (approximately 6 times the isometric rate) for shortening velocities lower than half the maximum velocity of shortening and declines for higher velocities. In addition, the model provides a clue for explaining how, in different muscle types, the higher the isometric maintenance heat, the higher the power output during steady shortening.     

The cross-bridge cycle and skeletal muscle fatigue.

The functional correlates of fatigue observed in both animals and humans during exercise include a decline in peak force (P0), maximal velocity, and peak power. Establishing the extent to which these deleterious functional changes result from direct effects on the myofilaments is facilitated through understanding the molecular mechanisms of the cross-bridge cycle. With actin-myosin binding, the cross-bridge transitions from a weakly bound low-force state to a strongly bound high-force state. Low pH reduces the number of high-force cross bridges in fast fibers, and the force per cross bridge in both fast and slow fibers. The former is thought to involve a direct inhibition of the forward rate constant for transition to the strong cross-bridge state. In contrast, inorganic phosphate (Pi) is thought to reduce P0 by accelerating the reversal of this step. Both H+ and Pi decrease myofibrillar Ca2+ sensitivity. This effect is particularly important as the amplitude of the Ca2+ transient falls with fatigue. The inhibitory effects of low pH and high Pi on P0 are reduced as temperature increases from 10 to 30 degrees C. However, the H+-induced depression of peak power in the slow fiber type, and Pi inhibition of myofibrillar Ca2+ sensitivity in slow and fast fibers, are greater at high compared with low temperature. Thus the depressive effects of H+ and Pi at in vivo temperatures cannot easily be predicted from data collected below 25 degrees C. In vitro, reactive oxygen species reduce myofibrillar Ca2+ sensitivity; however, the importance of this mechanism during in vivo exercise is unknown.     

Conversion of two-state to multi-state folding kinetics on fusion of two protein foldons

Chymotrypsin inhibitor 2 (CI2) is the archetypal single-foldon protein that folds in simple two-state kinetics without the accumulation of a folding intermediate. To model the effects of fusion of single foldons to give a multi-foldon protein, we engineered a "double-CI2" protein, in which another CI2 polypeptide was inserted into the loop region of the parent CI2. CD and HSQC spectra demonstrated that while the double-CI2 protein adopted two kinds of native conformations, CI2-like structure was almost preserved in both the domains of double-CI2. In the folding kinetic studies, double-CI2 exhibited a remarkable rollover of the observed folding rates at low denaturant concentrations, indicating that double-CI2 accumulated a kinetic folding intermediate. The different folding mechanisms between WT-CI2 and double-CI2 support the present view that protein size or number of domains is an important determinant for formation of folding intermediates.     

Kinetic mechano-chemical model of a muscle with a six-stage cross-bridge cycle]

A kinetic scheme of the mechano-chemical cycle of the cross-bridges and a mathematical model based on this scheme are proposed. The main assumptions accepted in the scheme are: the step of the inorganic phosphate release precedes the force-generating step of a cross-bridge; the rate-limiting step of the ATP hydrolysis is isomerization of the actomyosin-ADP complex. It is shown that the model well describes the mechanical and biochemical transients initiated by the temperature jump and flash photolysis of the caged compounds in skinned muscle fibres.     

Direct tests of muscle cross-bridge theories: predictions of a Brownian dumbbell model for position-dependent cross-bridge lifetimes and step sizes with an optically trapped actin filament.

Force and displacement events from a single myosin molecule interacting with an actin filament suspended between optically trapped beads (Finer, J. T., R. M. Simmons, and J. A. Spudich. 1994. Nature. 368:113-119) can be interpreted in terms of a generalized cross-bridge model that includes the effects of Brownian forces on the beads. Steady-state distributions of force and displacement can be obtained directly from a generalized Smoluchowski equation for Brownian motion of the actin-bead "dumbbell," and time series from Monte Carlo simulations of the corresponding Langevin equation. When the frequency spectrum of Brownian motion extends beyond cross-bridge transition rates, the inverse mean lifetimes of force/displacement pulses are given by cross-bridge rate constants averaged over a Boltzmann distribution of Brownian noise. These averaged rate constants reflect the strain-dependence of the rate constants for the stationary filament, most faithfully at high trap stiffness. Hence, measurements of the lifetimes and displacements of single events as a function of the resting position of the dumbbell can provide a direct test of different cross-bridge theories of muscle contraction. Quantitative demonstrations are given for Huxley models with 1) faster binding or 2) slower dissociation at positive cross-bridge strain. Predictions for other models can be inferred from the averaging procedure.     

Quantitative analysis of aspartate receptor signaling complex reveals that the homogeneous two-state model is inadequate: development of a heterogeneous two-state model

The two-state model of receptor activation, in which a receptor population exists in equilibrium between a single on-state and a single off-state, has long been considered a viable model for the signaling behavior of bacterial chemoreceptors. Here, we show that this simple, homogeneous two-state model is adequate for a pure receptor population with just one adaptation state, but fails to account quantitatively for the observed linear relationship between the apparent attractant affinity (K(1/2)) and kinase activity (V(obs)(apo)) as the adaptation state is varied. Further analysis reveals that the available data are instead consistent with a heterogeneous two-state model in which covalent modification of receptor adaptation sites changes the microscopic properties of the on-state or off-state. In such a system, each receptor molecule retains a single on-state and off-state, but covalent adaptation generates a heterogeneous population of receptors exhibiting a range of different on-states or off-states with different microscopic parameters and conformations. It follows that covalent adaptation transforms the receptor from a simple, two-state toggle switch into a variable switch. In order to identify the microscopic parameters most sensitive to covalent adaptation, six modified, two-state models were examined in which covalent adaptation alters a different microscopic parameter. The analysis suggests that covalent adaptation primarily alters the ligand-binding affinity of the receptor off-state (K(D1)). By contrast, models in which covalent adaptation alters the ligand-binding affinity of the receptor on-state, the maximal kinase stimulation of the on-state or off-state, cooperative interactions between receptors, or the assembly of the receptor-kinase signaling complex are inconsistent with the available evidence. Overall, the findings support a heterogeneous two-state model in which modification of the receptor adaptation sites generates a population of receptors with heterogeneous off-states differing in their attractant affinities.In the process of testing the effects of covalent adaptation on the assembly of the receptor-kinase signaling complex, a new method for estimating the stoichiometric ratio of receptor and CheA in the ternary signaling complex was devised. This method suggests that the ratio of receptor dimers to CheA dimers in the assembled complex is 6:1 or less.     

The smooth muscle cross-bridge cycle studied using sinusoidal length perturbations

The mechanical characteristics of smooth muscle can be broadly defined as either phasic, or fast contracting, and tonic, or slow contracting (, Pharmacol. Rev. 20:197-272). To determine if differences in the cross-bridge cycle and/or distribution of the cross-bridge states could contribute to differences in the mechanical properties of smooth muscle, we determined force and stiffness as a function of frequency in Triton-permeabilized strips of rabbit portal vein (phasic) and aorta (tonic). Permeabilized muscle strips were mounted between a piezoelectric length driver and a piezoresistive force transducer. Muscle length was oscillated from 1 to 100 Hz, and the stiffness was determined as a function of frequency from the resulting force response. During calcium activation (pCa 4, 5 mM MgATP), force and stiffness increased to steady-state levels consistent with the attachment of actively cycling cross-bridges. In smooth muscle, because the cross-bridge states involved in force production have yet to be elucidated, the effects of elevation of inorganic phosphate (P(i)) and MgADP on steady-state force and stiffness were examined. When portal vein strips were transferred from activating solution (pCa 4, 5 mM MgATP) to activating solution with 12 mM P(i), force and stiffness decreased proportionally, suggesting that cross-bridge attachment is associated with P(i) release. For the aorta, elevating P(i) decreased force more than stiffness, suggesting the existence of an attached, low-force actin-myosin-ADP- P(i) state. When portal vein strips were transferred from activating solution (pCa 4, 5 mM MgATP) to activating solution with 5 mM MgADP, force remained relatively constant, while stiffness decreased approximately 50%. For the aorta, elevating MgADP decreased force and stiffness proportionally, suggesting for tonic smooth muscle that a significant portion of force production is associated with ADP release. These data suggest that in the portal vein, force is produced either concurrently with or after P(i) release but before MgADP release, whereas in aorta, MgADP release is associated with a portion of the cross-bridge powerstroke. These differences in cross-bridge properties could contribute to the mechanical differences in properties of phasic and tonic smooth muscle.     

The ADP release step of the smooth muscle cross-bridge cycle is not directly associated with force generation.

When smooth muscle myosin subfragment 1 (S1) is bound to actin filaments in vitro, the light chain domain tilts upon release of MgADP, producing a approximately 3.5-nm axial motion of the head-rod junction (Whittaker et al., 1995. Nature. 378:748-751). If this motion contributes significantly to the power stroke, rigor tension of smooth muscle should decrease substantially in response to cross-bridge binding of MgADP. To test this prediction, we monitored mechanical properties of permeabilized strips of chicken gizzard muscle in rigor and in the presence of MgADP. For comparison, we also tested psoas and soleus muscle fibers. Any residual bound ADP was minimized by incubation in Mg2+-free rigor solution containing 15 mM EDTA. The addition of 2 mM MgADP, while keeping ionic strength and free Mg2+ concentration constant, resulted in a slight increase in rigor tension in both gizzard and soleus muscles, but a decrease in psoas muscle. In-phase stiffness monitored during small (<0.1%) 500-Hz sinusoidal length oscillations decreased in all three muscle types when MgADP was added. The changes in force and stiffness with the addition of MgADP were similar at ionic strengths from 50 to 200 mM and were reversible. The results with gizzard muscle were similar after thiophosphorylation of the regulatory light chain of myosin. These results suggest that the axial motion of smooth muscle S1 bound to actin, upon dissociation of MgADP, is not associated with force generation. The difference between the present mechanical data and previous structural studies of smooth S1 may be explained if geometrical constraints of the intact contractile filament array alter the motions of the myosin heads.