Equilibrium muscle cross-bridge behavior. Theoretical considerations. II. Model describing the behavior of strongly-binding cross-bridges when both heads of myosin bind to the actin filament.

A model has been developed for characterizing the interaction between strongly-binding myosin cross-bridges and actin in muscle fibers under equilibrium conditions where both heads of the myosin cross-bridge bind to actin. The model, that of Anderson and Schoenberg (1987. Biophys. J. 52:1077-1082) is quite similar to that of Schoenberg (1985. Biophys. J. 48:467-475), except that explicit account is taken of the fact that each crossbridge has two heads which can bind to actin. The key assumption that allows this model to explain a large body of data unexplained by the Schoenberg (1985) model is that the two crossbridge heads are not totally independent of one another after attachment. After the first head attaches, the second head is then free to attach only to an actin site distal to the first head. This means that when the more distally attached head subsequently detaches and reattaches (as the heads continually do), it will not reattach in a position of lesser strain and reduce the force it supports, but instead will remain attached in its strained position until the proximally attached head also detaches. This model gives an explanation for two important and otherwise unexplained observations made previously: it explains why at ionic strengths in the range of 50-120 mM, (a) the rate constant of force decay after a small stretch is a sigmoidal function of nucleotide analogue concentration, and (b) why in the presence of analogues or in rigor the rate constant of force decay after a small stretch is significantly slower than the rate constant for myosin subfragment-1 detachment from actin in solution.     

Strain-dependent cross-bridge cycle for muscle. II. Steady-state behavior.

Quantitative predictions of steady-state muscle properties from the strain-dependent cross-bridge for muscle are presented. With a stiffness of 5.4 x 10(-4) N/m per head, a throw distance of 11 nm, and three allowed actin sites/head, isometric properties and their dependence on phosphate and nucleotide levels are well described if the tension-generating step occurs before phosphate release. At very low ATP levels, rigorlike states with negative strain are predicted. The rate-limiting step for cycling and ATP consumption is strain-blocked ADP release for isometric and slowly shortening muscle. Under rapid shortening, ATP hydrolysis on detached heads is the rate-limiting step, and the ratio of bound ATP to bound ADP.Pi increases by a factor of 7. At large positive strains, bound heads must be forcibly detached from actin to account for tension in rapid extension, but forced detachment in shortening has no effect without destroying isometric attached states. Strain-blocked phosphate release as proposed produces modest inhibition of the ATPase rate under rapid shortening, sufficient to give a maximum for one actin site per helix turn. Alternative cross-bridge models are discussed in the light of these predictions.     

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.     

Foraging behavior of ants: Theoretical considerations

This paper describes models of behavior for two ant species recruiting to very different sorts of resource. One species, Solenopsis geminata, recruits to sugar solution. The other, Pogonomyrmex occidentalis, recruits to patches of seeds. The models clarify the assumptions entering into the theoretical analysis of these behaviors and point to measurements that should be made for their experimental analysis.Both models assume that the colony gains energy when a forager gets a load of resource. Energy is lost during the trip to a patch of resource in a manner dependent upon the current physical conditions of the environment. The only factor counterbalancing an ever increasing net energy gain when the recruitment rates are increased is interference among workers at a patch. As the rate of recruitment is elevated this interference decreases the efficiency with which the resource can be used and, thus, sets an optimum at some intermediate rate of recruitment.The limitation of considerations in this paper to energetic efficiency is discussed and justified in terms of components of fitness and ant physiology. It is found that several qualitative predictions can be experimentally tested, but that quantitative predictions require fine measurement of metabolic costs to individual foragers and of complicated gains to the colony as a whole. Both models can be differentiated experimentally from corresponding time minimization models.     

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.     

Theoretical and experimental studies on cross-bridge migration during cell disaggregation.

A micromanipulation method is used to determine the adhesive energy density (gamma) between pairs of cytotoxic T cells (F1) and their target cells (JY: HLA-A2-B7-DR4,W6). gamma is defined as the energy per unit area that must be supplied to reduce the region of contact between a conjugated cell pair. Our analysis of the data indicates that the force applied by the micropipette on the cell is not uniformly distributed throughout the contact region as we had previously assumed (Sung, K. L. P., L. A. Sung, M. Crimmins, S. J. Burakoff, and S. Chien. 1986. Science (Wash. DC). 234: 1405-1408), but acts only at the edges of the contact region. We show that gamma is not constant during peeling but increases with decreasing contact area of the conjugated cell pairs F1-JY, F1-F1, and JY-JY in contrast to the constancy of gamma for typical engineering adhesives. This finding supports the notion that the cross-linking protein molecules slide towards the conjugated area across the leading edge of the separation while remaining attached to both cells. Our mathematical analysis shows that the elastic energy stored in the cross-links by the membrane tensions balances the diffusive forces that act against cross-bridge migration. The binding affinity between F1-JY is found to be approximately 15-20 times larger than the corresponding affinity for F1-F1. The number of binding sites of F1 for attachment to JY is approximately the same for binding F1 to another F1 and vary between 10(5) and 10(6).     

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.     

Theoretical considerations on myofibril stiffness.

A discrete model of the interaction between individual myofilaments was developed to study the stiffness of a sarcomere for the case in which filament compliance is not negligible. Our model retains, in the limit, the characteristics of the previously published model by Ford et al. (Ford, L. E., A. F. Huxley, and R. M. Simmons. 1981. The relation between stiffness and filament overlap in stimulated frog muscle fibres. J. Physiol. 311:219-249). In addition, the model is able to model the interaction in cases in which few cross-bridges are attached, or when the distribution of attached cross-bridges is not uniform. Our results confirm previous indications that it might be impossible to calculate the number of attached cross-bridges by using only stiffness measurements in quick-stretch (or release) experiments.     

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.     

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.     

Resting myosin cross-bridge configuration in frog muscle thick filaments

Clear images of myosin filaments have been seen in shadowed freeze-fracture replicas of single fibers of relaxed frog semitendinosus muscles rapidly frozen using a dual propane jet freezing device. These images have been analyzed by optical diffraction and computer averaging and have been modelled to reveal details of the myosin head configuration on the right-handed, three-stranded helix of cross-bridges. Both the characteristic 430-A and 140-150-A repeats of the myosin cross-bridge array could be seen. The measured filament backbone diameter was 140-160 A, and the outer diameter of the cross-bridge array was 300 A. Evidence is presented that suggests that the observed images are consistent with a model in which both of the heads of one myosin molecule tilt in the same direction at an angle of approximately 50-70 degrees to the normal to the filament long axis and are slewed so that they lie alongside each other and their radially projected density lies along the three right-handed helical tracks. Any perturbation of the myosin heads away from their ideal lattice sites needed to account for x-ray reflections not predicted for a perfect helix must be essentially along the three helical tracks of cross-bridges. Little trace of the presence of non-myosin proteins could be seen.     

Myosin light chain 2 modulates calcium-sensitive cross-bridge transitions in vertebrate skeletal muscle.

We investigated the mechanism of the Ca2+ sensitivity of cross-bridge transitions that limit the rate of force development in vertebrate skeletal muscle. The rate of force development increases with Ca2+ concentration in the physiological range. We show here that at low concentrations of Ca2+ the rate of force development increases after partial extraction of the 20-kD light chain 2 subunit of myosin, whereas reconstitution with light chain 2 fully restores native sensitivity to Ca2+ in skinned single skeletal fibers. Furthermore, elevated free Mg2+ concentration reduces Ca2+ sensitivity, an effect that is reversed by extraction of the light chain but not by disruption of thin-filament activation by partial removal of troponin C, the Ca2+ binding protein of the thin filament. Our findings indicate that the Ca2+ sensitivity of the rate of force development in vertebrate skeletal muscle is mediated in part by the light chain 2 subunit of the myosin cross-bridge.     

Modeling rigor cross-bridge patterns in muscle I. Initial studies of the rigor lattice of insect flight muscle.

We have undertaken some computer modeling studies of the cross-bridge observed by Reedy in insect flight muscle so that we investigate the geometric parameters that influence the attachment patterns of cross-bridges to actin filaments. We find that the appearance of double chevrons along an actin filament indicates that the cross-bridges are able to reach 10--14 nm axially, and about 90 degrees around the actin filament. Between three and five actin monomers are therefore available along each turn of one strand of actin helix for labeling by cross-bridges from an adjacent myosin filament. Reedy's flared X of four bridges, which appears rotated 60 degrees at successive levels on the thick filament, depends on the orientation of the actin filaments in the whole lattice as well as on the range of movement in each cross-bridge. Fairly accurate chevrons and flared X groupings can be modeled with a six-stranded myosin surface lattice. The 116-nm long repeat appears in our models as "beating" of the 14.5-nm myosin repeat and the 38.5-nm actin period. Fourier transforms of the labeled actin filaments indicate that the cross-bridges attach to each actin filament on average of 14.5 nm apart. The transform is sensitive to changes in the ease with which the cross-bridge can be distorted in different directions.     

Equilibrium muscle crossbridge behavior: The interaction of myosin crossbridges with actin

The interaction of myosin crossbridges with actin under equilibrium conditions is reviewed. Similarities and differences between the weakly- and strongly-binding interactions of myosin crossbridges with actin filaments are discussed. A precise, narrow definition of weakly- binding crossbridges is given. It is postulated that the fundamental interaction of crossbridges with actin is that the crossbridge heads are mobile after attachment in the first case but not in the second. It is argued that because the weakly-binding crossbridge heads are mobile after attachment, the heads appear to function independently of each other. The lack of head mobility in attached strongly-binding crossbridges makes the strongly-binding crossbridge heads appear to act cooperatively. This model of the strongly-binding crossbridge gives an explanation for two important and otherwise unexplained observations. It explains why the rate constant of force decay after a small stretch is a sigmoidal function of nucleotide analogue concentration, and why, in the presence of analogues or in rigor, the rate constant of force decay after a small stretch is often significantly slower than the rate constant for myosin subfragment-1 detachment from actin in solution. The model of the weakly-binding crossbridge accurately describes the behavior of the myosin·ATP crossbridge.     

Optical depolarization changes in single, skinned muscle fibers. Evidence for cross-bridge involvement.

Optical ellipsometry studies of single, skinned muscle fibers conducted on the diffraction orders have yielded spectra that are sensitive to the state of the fiber. The linearly polarized light field vector becomes elliptically polarized as it passes through the fiber and may be collected at the diffraction orders. Fibers that have been subjected to extraction of myosin (0.6 M KCl) retain a weak diffraction pattern and exhibit a substantially decreased depolarization of incident linearly polarized light. A significant decrease in polarization is seen in skinned fibers that are subject to an increase in pH from 7.0 to 8.0. This increase in pH results in a decrease of approximately 30% in the depolarization angle of single fibers. The major decrease in depolarization angle that we observe at pH 8.0 is consistent with the notion that as cross-bridges move out from the shaft of the thick filament, their ability to cause depolarization of the incident linearly polarized light decreases. This interpretation is also consistent with the work of Ueno and Harrington where the decrease in the ability to cross-link S-1 and S-2 to the thick filament at pH 8.2 suggests cross-bridge movement away from the thick filament. A large decrease in birefringence, seen after treatment of skinned fibers with alpha-chymotrypsin, appears to be related to the breakdown of myosin into rod, S-1, heavy meromyosin, and light meromyosin.     

Characterization of cross-bridge elasticity and kinetics of cross-bridge cycling during force development in single smooth muscle cells

Force development in smooth muscle, as in skeletal muscle, is believed to reflect recruitment of force-generating myosin cross-bridges. However, little is known about the events underlying cross-bridge recruitment as the muscle cell approaches peak isometric force and then enters a period of tension maintenance. In the present studies on single smooth muscle cells isolated from the toad (Bufo marinus) stomach muscularis, active muscle stiffness, calculated from the force response to small sinusoidal length changes (0.5% cell length, 250 Hz), was utilized to estimate the relative number of attached cross-bridges. By comparing stiffness during initial force development to stiffness during force redevelopment immediately after a quick release imposed at peak force, we propose that the instantaneous active stiffness of the cell reflects both a linearly elastic cross-bridge element having 1.5 times the compliance of the cross-bridge in frog skeletal muscle and a series elastic component having an exponential length-force relationship. At the onset of force development, the ratio of stiffness to force was 2.5 times greater than at peak isometric force. These data suggest that, upon activation, cross-bridges attach in at least two states (i.e., low-force-producing and high-force-producing) and redistribute to a steady state distribution at peak isometric force. The possibility that the cross-bridge cycling rate was modulated with time was also investigated by analyzing the time course of tension recovery to small, rapid step length changes (0.5% cell length in 2.5 ms) imposed during initial force development, at peak force, and after 15 s of tension maintenance. The rate of tension recovery slowed continuously throughout force development following activation and slowed further as force was maintained. Our results suggest that the kinetics of force production in smooth muscle may involve a redistribution of cross-bridge populations between two attached states and that the average cycling rate of these cross-bridges becomes slower with time during contraction.