Strain-dependent cross-bridge cycle for muscle.

The cross-bridge cycle for actin, S1 myosin, and nucleotides in solution is applied to the sliding filament model for fully activated striated muscle. The cycle has attached and rotated isomers of each actomyosin state. It is assumed that these forms have different zero-strain conformations with respect to the filament and that strain-free rate constants are the nominal solution values. Only one S1 unit of heavy meromyosin is considered. Transition-state theory is used to predict the strain dependences of S1 binding to actin, the force-generating transition to rotated states, and the release/binding of nucleotide and phosphate. We propose that ADP release and ATP binding are blocked by positive strain and phosphate release by negative strain. At large strains, rapid dissociation of S1 nucleotide from actin is expected when the compliant element of the cross-bridge is strained in either direction beyond its elastic limits. The dynamical behavior of this model of muscle contraction is discussed in general terms. Its computed steady-state properties are presented in an accompanying paper.     

Kinetic and thermodynamic studies of the cross-bridge cycle in rabbit psoas muscle fibers.

The effect of temperature on elementary steps of the cross-bridge cycle was investigated with sinusoidal analysis technique in skinned rabbit psoas fibers. We studied the effect of MgATP on exponential process (C) to characterize the MgATP binding step and cross-bridge detachment step at six different temperatures in the range 5-30 degrees C. Similarly, we studied the effect of MgADP on exponential process (C) to characterize the MgADP binding step. We also studied the effect of phosphate (Pi) on exponential process (B) to characterize the force generation step and Pi-release step. From the results of these studies, we deduced the temperature dependence of the kinetic constants of the elementary steps and their thermodynamic properties. We found that the MgADP association constant (K0) and the MgATP association constant (K1) significantly decreased when the temperature was increased from 5 to 20 degrees C, implying that nucleotide binding became weaker at higher temperatures. K0 and K1 did not change much in the 20-30 degree C range. The association constant of Pi to cross-bridges (K5) did not change much with temperature. We found that Q10 for the cross-bridge detachment step (k2) was 2.6, and for its reversal step (k-2) was 3.0. We found that Q10 for the force generation step (Pi-isomerization step, k4) was 6.8, and its reversal step (k-4) was 1.6. The equilibrium constant of the detachment step (K2) was not affected much by temperature, whereas the equilibrium constant of the force generation step (K4) increased significantly with temperature increase. Thus, the force generation step consists of an endothermic reaction. The rate constant of the rate-limiting step (k6) did not change much with temperature, whereas the ATP hydrolysis rate increased significantly with temperature increase. We found that the force generation step accompanies a large entropy increase and a small free energy change; hence, this step is an entropy-driven reaction. These observations are consistent with the hypothesis that the hydrophobic interaction between residues of actin and myosin underlies the mechanism of force generation. We conclude that the force generation step is the most temperature-sensitive step among elementary steps of the cross-bridge cycle, which explains increased isometric tension at high temperatures in rabbit psoas fibers.     

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.     

Mathematical simulation of muscle cross-bridge cycle and force-velocity relationship

Muscle contraction underlies many essential functions such as breathing, heart beating, locomotion, regulation of blood pressure, and airway resistance. Active shortening of muscle is the result of cycling of myosin cross-bridges that leads to sliding of myosin filaments relative to actin filaments. In this study, we have developed a computer program that allows us to alter the rates of transitions between any cross-bridge-states in a stochastic cycle. The cross-bridge states within the cycle are divided into six attached (between myosin cross-bridges and actin filaments) states and one detached state. The population of cross-bridges in each of the states is determined by the transition rates throughout the cycle; differential equations describing the transitions are set up as a cyclic matrix. A method for rapidly obtaining steady-state exact solutions for the cyclic matrix has been developed to reduce computation time and avoid the divergence problem associated with numerical solutions. In the seven-state model, two power strokes are assumed for each cross-bridge cycle, one before the release of inorganic phosphate, and one after. The characteristic hyperbolic force-velocity relationship observed in muscle contraction can be reproduced by the model. Deviation from the single hyperbolic behavior at low velocities can be mimicked by allowing the rate of cross-bridge-attachment to vary with velocity. The effects of [ATP], [ADP], and [P(i)] are simulated by changing transition rates between specific states. The model has revealed new insights on how the force-velocity characteristics are related to the state transitions in the cross-bridge cycle.     

Kinetic model of a single muscle contraction

A model of single muscle contraction is proposed. The kinetics of two reactions have been used: interaction between the excitation mediator and the muscle membrane receptor, and the reaction of enzymatic mediator splitting. The muscle contraction parameters have been correlated with the concentration of the mediator-receptor complex.     

The myosin cross-bridge cycle and its control by twitchin phosphorylation in catch muscle

The anterior byssus retractor muscle of Mytilus edulis was used to characterize the myosin cross-bridge during catch, a state of tonic force maintenance with a very low rate of energy utilization. Addition of MgATP to permeabilized muscles in high force rigor at pCa > 8 results in a rapid loss of some force followed by a very slow rate of relaxation that is characteristic of catch. The fast component is slowed 3-4-fold in the presence of 1 mM MgADP, but the distribution between the fast and slow (catch) components is not dependent on [MgADP]. Phosphorylation of twitchin results in loss of the catch component. Fewer than 4% of the myosin heads have ADP bound in rigor, and the time course (0.2-10 s) of ADP formation following release of ATP from caged ATP is similar whether or not twitchin is phosphorylated. This suggests that MgATP binding to the cross-bridge and subsequent splitting are independent of twitchin phosphorylation, but detachment occurs only if twitchin is phosphorylated. A similar dependence of detachment on twitchin phosphorylation is seen with AMP-PNP and ATPgammaS. Single turnover experiments on bound ADP suggest an increase in the rate of release of ADP from the cross-bridge when catch is released by phosphorylation of twitchin. Low [Ca(2+)] and unphosphorylated twitchin appear to cause catch by 1) markedly slowing ADP release from attached cross-bridges and 2) preventing detachment following ATP binding to the rigor cross-bridge.     

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.     

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.     

Kinetic analysis of a molecular model of the budding yeast cell cycle

The molecular machinery of cell cycle control is known in more detail for budding yeast, Saccharomyces cerevisiae, than for any other eukaryotic organism. In recent years, many elegant experiments on budding yeast have dissected the roles of cyclin molecules (Cln1-3 and Clb1-6) in coordinating the events of DNA synthesis, bud emergence, spindle formation, nuclear division, and cell separation. These experimental clues suggest a mechanism for the principal molecular interactions controlling cyclin synthesis and degradation. Using standard techniques of biochemical kinetics, we convert the mechanism into a set of differential equations, which describe the time courses of three major classes of cyclin-dependent kinase activities. Model in hand, we examine the molecular events controlling "Start" (the commitment step to a new round of chromosome replication, bud formation, and mitosis) and "Finish" (the transition from metaphase to anaphase, when sister chromatids are pulled apart and the bud separates from the mother cell) in wild-type cells and 50 mutants. The model accounts for many details of the physiology, biochemistry, and genetics of cell cycle control in budding yeast.     

An enhanced model of cross-bridge operation with internal elasticity

A recent study has shown (as reported by Rosenfeld, Eur Biophys J 41:733–753, 2012) that an apparatus consisting of a cycling pump, a lever, and charged beads is able to generate force in accordance with Hill’s force–velocity relation. Here, we show that a spring integrated into this microscopic model of a myosin motor allows reproducing, in general terms, the muscle fiber responses to sudden changes in fiber length. The time course of relaxation is governed by the same hindering force that determines the maximal value of muscle contraction velocity. Any single one of the exceptionally simple parts of the proposed model device corresponds to some element of the real myosin head and interacts with any other part in accordance with the laws of Newton, Coulomb, and Hooke. In essence, the model demonstrates that Coulomb repulsion should be understood as the physical source of muscle force. Accordingly, some fictitious master equation with ad hoc postulated rate constants is not needed to explain the essential mechanical characteristics of a muscle. The current model still contains no mechanism that could account for superfast relaxations within periods of about 0.1 ms.     

Conformational changes of a Swi2/Snf2 ATPase during its mechano-chemical cycle

Remodelling protein nucleic acid interfaces is an important biological task, which is often carried out by nucleic acid stimulated ATPases of the Swi2/Snf2 superfamily. Here we study the mechano-chemical cycle of such an ATPase, namely the catalytic domain of the Sulfolobus solfataricus Rad54 homologue (SsoRad54cd), by means of fluorescence resonance energy transfer (FRET). The results of the FRET studies show that the enzyme can be found in (at least) two different possible conformations in solution. An open conformation, consistent with a recently reported crystal structure, is converted into a closed conformation after DNA binding. Upon subsequent binding of ATP no further change in conformation can be detected by the FRET measurements. Instead, a FRET detectable conformational change occurs after ATP hydrolysis and prior to ADP release, suggesting a powerstroke that is linked to phosphate release. Based on these data we will present a new model for the mechano-chemical cycle of this enzyme. This scheme in turn provides a working model for understanding the function of other members of the Swi2/Snf2 family.     

Kinetic modeling of [(18)F]FDG in skeletal muscle by PET: a four-compartment five-rate-constant model

Various modeling strategies have been developed to convert regional [(18)F]fluorodeoxyglucose ([(18)F]FDG) concentration measured by positron emission tomography (PET) to a measurement of physiological parameters. However, all the proposed models have been developed and tested mostly for brain studies. The purpose of the present study is to select the most accurate model for describing [(18)F]FDG kinetics in human skeletal muscle. The database consists of basal and hyperinsulinemic-euglycemic studies performed in normal subjects. PET data were first analyzed by an input-output modeling technique (often called spectral analysis). These results provided guidelines for developing a compartmental model. A new model with four compartments and five rate constants (5K model) emerged as the best. By accounting for plasma and extracellular and intracellular kinetics, this model allows, for the first time, PET assessment of the individual steps of [(18)F]FDG kinetics in human skeletal muscle, from plasma to extracellular space to transmembrane transport into the cell to intracellular phosphorylation. Insulin is shown to affect transport and phosphorylation but not extracellular kinetics, with the transport step becoming the main site of control. The 5K model also allows definition of the domain of validity of the classic three-compartment three- or four-rate-constant models. These models are candidates for an investigative tool to quantitatively assess insulin control on individual metabolic steps in human muscle in normal and physiopathological states.     

Kinetic model of regulation of muscle protein activity.

The widely accepted steric model of calcium regulation of actin-myosin interactions in vertebrate muscles has to be completed to fit the kinetic data. It should be supposed that: (1) the thin filaments consist of functionally independent units, containing seven actin sites regulated by one troponin-tropomyosin complex; (2) actin sites become available for myosin heads only due to fluctuations of tropomyosin position; (3) binding of calcium to troponin results either in the shift of the tropomyosin equilibrium position or in the weakening of its interactions with actin strand so that the probability of effective fluctuations increases; (4) link formation between myosin head and some of the available actin site fixates the tropomyosin in such a position that the other six actin sites of the same functional unit become available for myosin too.The model gives linear kinetic scheme for the transitions of a functional unit between nine states (a “turned off” state, and eight “turned on” ones with different occupancy by myosin heads). The dependences of the apparent rate constants of actomyosin formation and dissociation upon the myosin head and substrate concentrations are obtained from the Lymn-Taylor scheme. The frequency of the actomyosin complexes dissociation is assumed to give the ATPase rate.The model fits the kinetic data on the ATP hydrolysis by myosin subfragment-1 with regulated or unregulated actin as a cofactor under various conditions. It shows a sharp dependence of activation upon the apparent affinity of the actin and myosin sites. Therefore, the model appears to be applicable to myosin controlled systems.     

Cardiac myosin binding protein C and its phosphorylation regulate multiple steps in the cross-bridge cycle of muscle contraction

Cardiac myosin binding protein C (c-MyBPC) is a thick filament protein that is expressed in cardiac sarcomeres and is known to interact with myosin and actin. While both structural and regulatory roles have been proposed for c-MyBPC, its true function is unclear; however, phosphorylation has been shown to be important. In this study, we investigate the effect of c-MyBPC and its phosphorylation on two key steps of the cross-bridge cycle using fast reaction kinetics. We show that unphosphorylated c-MyBPC complexed with myosin in 1:1 and 3:1 myosin:c-MyBPC stoichiometries regulates the binding of myosin to actin (K(D)) cooperatively (Hill coefficient, h) (K(D) = 16.44 ± 0.33 μM, and h = 9.24 ± 1.34; K(D) = 11.48 ± 0.75 μM, and h = 3.54 ± 0.67) and significantly decelerates the ATP-induced dissociation of myosin from actin (K(1)k(+2) values of 0.12 ± 0.01 and 0.22 ± 0.01 M(-1) s(-1), respectively, compared with a value of 0.42 ± 0.01 M(-1) s(-1) for myosin alone). Phosphorylation of c-MyBPC abolished the regulation of the association phase (K(1)k(+2) values of 0.32 ± 0.02 and 0.33 ± 0.01 M(-1) s(-1) at 1:1 and 3:1 myosin:c-MyBPC ratios, respectively) and also accelerated the dissociation of myosin from actin (K(1)k(+2) values of 0.23 ± 0.01 and 0.29 ± 0.01 M(-1) s(-1) at a 1:1 and 3:1 myosin:c-MyBPC ratios, respectively) relative to the dissociation of myosin from actin in the presence of unphosphorylated c-MyBPC. These results indicate a direct effect of c-MyBPC on cross-bridge kinetics that is independent of the thin filament that together with its phosphorylation provides a mechanism for fine-tuning cross-bridge behavior to match the contractile requirements of the heart.     

Kinetic model describing the action of tryptophanyl-tRNA-synthetase]

Reaction rates for ATP-PPi isotope exchange (vex) and tryptophanyl-tRNA formation (vaa) catalysed concomitantly in one incubation mixture by beef pancreas tryptophanyl-tRNA synthetase (trsase) have been examined as a function of substrate concentrations. Comparison of the vex/vaa ratio found experimentally with the ratio predicted theoretically conforms the mechanism suggested earlier and permits to describe it in more detail. I. At least two reaction routes exist in which an ATP-PP: exchange is allowed. These routes are interconnected with each other via the stage at which tRNA binds to the enzyme. 2. In both these routes the low molecular weight substrates bind with enzyme in the order ATP first, tryptophan second. 3. Enzyme-aminoacyladenylate complex is an intermediate in the reaction of aminoacyl-tRNA formation. Pyrophosphate is detached from the enzyme prior to tRNA. 4. The enzyme releases AMP and tryptophanyl-tRNA in a random fashion. All the aformentioned properties are common both for trigger mechanism and Yarus-Berg mechanism which up to now were considered in literature independently.     

Dynamics of actomyosin interactions in relation to the cross-bridge cycle

Transient kinetic measurements of the actomyosin ATPase provided the basis of the Lymn-Taylor model for the cross-bridge cycle, which underpins current models of contraction. Following the determination of the structure of the myosin motor domain, it has been possible to introduce probes at defined sites and resolve the steps in more detail. Probes have been introduced in the Dicytostelium myosin II motor domain via three routes: (i) single tryptophan residues at strategic locations throughout the motor domain; (ii) green fluorescent protein fusions at the N and C termini; and (iii) labelled cysteine residues engineered across the actin-binding cleft. These studies are interpreted with reference to motor domain crystal structures and suggest that the tryptophan (W501) in the relay loop senses the lever arm position, which is controlled by the switch 2 open-to-closed transition at the active site. Actin has little effect on this process per se. A mechanism of product release is proposed in which actin has an indirect effect on the switch 2 and lever arm position to achieve mechanochemical coupling. Switch 1 closing appears to be a key step in the nucleotide-induced actin dissociation, while its opening is required for the subsequent activation of product release. This process has been probed with F239W and F242W substitutions in the switch 1 loop. The E706K mutation in skeletal myosin IIa is associated with a human myopathy. To simulate this disease we investigated the homologous mutation, E683K, in the Dictyostelium myosin motor domain.