Cross-bridge model of muscle contraction. Quantitative analysis

Two elementary models for the regulation of skeletal muscle contraction by calcium

It is shown by use of an extremely simple explicit two-state model that two basic ideas may be sufficient to understand at least qualitatively the sensitive activation of isometric muscle contraction by Ca2+. (a) Ca2+ binds much more strongly on troponin if myosin is already attached to actin. The steady state analogue of this is that the single rate constant (in the two-state model) for myosin attachment plus Pi release is much larger if Ca2+ is bound to troponin. (b) End-to-end tropomyosin interactions are responsible for positive cooperativity. Although these ideas seem to be sufficient, this of course does not mean that they are necessary. These same ingredients were used in two previous, more elaborate models for the cooperative equilibrium binding of myosin subfragment-1 on actin-tropomyosin-troponin, with and without Ca2+, and for a study of the steady state ATPase activity of the same system. Essentially as an appendix, the above-mentioned simple treatment is extended to a somewhat more realistic and complicated model of isometric contraction.

Tension and ATPase rate in steady-state contractions of rabbit soleus fiber segments

Steady-state isometric tension and ATPase were studied in hyperpermeable segments of single muscle fibers from rabbit soleus muscle at 22 degrees C. The ATPase activity was due to actomyosin. The ratio of fiber ATPase to tension was used as an index of steady-state cross-bridge kinetics. Increasing the calcium ion concentration from pCa 8 to pCa 5 activated both tension and ATPase. The maximal tension was 1.35 +/- 0.07 kg/cm2. The maximal ATPase was 1.05 +/- 0.13 mumol X g-1. s-1 at pCa 5.2. ATPase activity increased with tension, such that the ratio of ATPase to tension remained constant at all calcium concentrations. In the absence of calcium, increasing the concentration of MgATP from 1 to 7 X 10(-7) M increased tension from zero to a maximum of 0.46 +/- 0.03 kg/cm2. Increasing MgATP concentration further to 1 X 10(-6) M inhibited tension. In the phase of rising tension, ATPase increased proportionally to tension, to 0.11 +/- 0.01 mumol X g-1 X s-1 at maximum tension. However, the ratio of ATPase to tension on the rising phase had a value only one-third of that seen with calcium-activated tension. Thus, low substrate concentrations, but not low calcium ion concentrations, influence cross-bridge kinetics under steady-state isometric conditions, possibly by an increase in the tension-time product during a cross-bridge cycle.

The role of magnesium adenosine triphosphate in the contractile kinetics of insect fibrillar flight muscle

The changes in tension produced by small step or sinusoidal changes of length have been measured for chemically skinned flight muscle fibres of the giant tropical water bug Lethocerus at MgATP concentrations in the range 0.1-10 mM. In the presence of calcium ion concentrations of about 20 microM, the rates of the rapid mechanical processes observed were found to increase with increasing MgATP, exhibiting saturation with an apparent half-saturation constant lying between 0.1 and 1 mM MgATP, depending upon the conditions used. Under the same conditions, an increase in MgATP concentration was found to lead to a slight decrease in the isometric tension generated by the preparations. The results are discussed with reference to some current crossbridge models of muscle contraction.

Metabolic cost of the stimulated beating of isolated adult rat heart cells in suspension

Heart cells from adult rats were induced to beat in suspension by electric field stimulation. We have gained evidence that all the rod-shaped cells in suspension were indeed beating, and that the beat had dynamic characteristics similar to those of intact heart muscle contracting under zero load. The cells were undamaged in the process, as judged by maintenance of ATP levels, morphology, and ability to beat. In gaining such evidence, we also measured the metabolic cost to the cells of beating under zero load. In cells with oxidative phosphorylation inhibited by rotenone plus oligomycin (termed anaerobic), the rate of beat-dependent lactate production suggested an equivalent rate of ATP utilization of 0.126 +/- 0.013 nmol ATP/beat per mg protein (plus isoproterenol), and 0.058 +/- 0.005 nmol ATP/beat per mg protein (minus isoproterenol). In respiring cells, the rate of beat-dependent oligomycin-sensitive oxygen consumption gave an equivalent rate of ATP utilization of 0.198 +/- 0.009 nmol ATP/beat per mg protein (plus isoproterenol), and 0.126 +/- 0.013 nmol ATP/beat per mg protein (minus isoproterenol). The cells beat with the same approximate maximum velocity whether isoproterenol was present or not. We calculate that--in the case of anaerobic cells without isoproterenol--this rate of ATP utilization can account for only about a 15% degree of activation of the contractile proteins. In addition, we have found an oligomycin-insensitive beat-dependent mitochondrial respiration of 0.023 +/- 0.006 nanoatom O/beat per mg. The cause of this respiration is not known. The total rate of oxygen consumption of cells and also the rate per beat was comparable to that measured in nonworking whole hearts.

Orientation of spin labels attached to cross-bridges in contracting muscle fibres

Electron micrographs showing different cross-bridge orientations in different states of muscle fibres, and X-ray diffraction patterns indicating axial cross-bridge disorder in contracting muscle first suggested that force generation in the contracting muscle involved a change in orientation of the myosin heads that form cross-bridges between thick and thin filaments. This has been supported by subsequent work; the myosin molecule has the required flexibility for changes in orientation. The orientation of muscle tryptophans and of probes attached to the myosin heads of permeable muscle fibres depends on the state of the muscle. Recently, fluorescence polarization fluctuations and time-resolved X-ray diffraction patterns have suggested that cross-bridges of a contracting muscle can rotate. We have used electron paramagnetic resonance (EPR) spectroscopy to monitor the orientation of spin labels attached specifically to a reactive sulphydryl on the myosin heads in glycerinated rabbit psoas skeletal muscle. Previously, it has been shown that the paramagnetic probes are highly ordered in rigor muscle, with a nearly random angular distribution in relaxed muscle. We show here that during the generation of isometric tension, approximately 80% of the probes display a random angular distribution as in relaxed muscle while the remaining 20% are highly oriented at the same angle as found in rigor muscle. These findings indicate that a domain of the myosin head does not change orientation during the power stroke of the contractile interaction.

Kinetic and thermodynamic properties of the ternary complex between F-actin, myosin subfragment 1 and adenosine 5'-[beta, gamma-imido]triphosphate

Equilibrium constants for the formation of a ternary complex between actin, myosin subfragment 1 (S1) and the non-hydrolyzable ATP analog adenosine 5'-[beta, gamma-imido]triphosphate (Ado PP[NH]P) were determined from light-scattering titrations under a variety of conditions. The affinities of S1 (binding constant K1) and acto . S1 (K4) for AdoPP[NH]P have relatively low dependencies on temperature (delta H degrees approximately equal to - 15 - 30 kJ mol-1) and ionic strength, in contrast to the affinities of S1 (K2) and S1 . AdoPP[NH]P (K3) for actin which are influenced quite strongly by temperature (delta H degrees approximately equal to 50 - 65 kJ mol-1) and ionic strength, K2 decreasing by a factor of 10 - 15 between I = 0.05 M and I = 0.2 M and K3 decreasing by a factor of 5.K1, and by detailed balance K2 as well, were found to be about 10-times higher than hitherto reported values (K1 = 3.4 X 10(7) M-1, K2 = 6 X 10(8) M-1, at 24 degrees C,I = 0.09 M, pH 8.0). The binding of ADP to S1 is about 10-fold weaker than that of AdoPP[NH]P, being however much more exothermic (delta H degrees = - 70 kJ mol-1 at I = 0.1 M) and having a negative standard entropy change (delta S = - 125 J mol-1 K-1), in contrast to AdoPP[NH]P binding for which the calculated delta S had positive values. The observed rate constant of dissociation of acto . S1 by AdoPP[NH]P showed an almost hyperbolic dependence on the nucleotide concentration, reaching a maximum of 15 s-1 at I = 0.055 M and 5 s-1 at I = 0.275 M, pH 8.0, 23 degrees C; at 5 degrees C this value was somewhat higher. The rate constant of dissociation of AdoPP[NH]P from its complex with acto . S1 was estimated to exceed 400 s-1 at 23 degrees C, and to be of the order of 150 s-1 at 4 degrees C. The observed rate constant for the association of the S1 . nucleotide complex and actin was proportional to actin concentrations up to 60 microM, thus defining an apparent second-order rate constant of 2 X 10(4) M-1 s-1 at I = 0.125 M and 23 degrees C. A reaction scheme is proposed in which isomerizations of the acto . S1 and acto . S1 . nucleotide complexes can occur.

Temperature dependence of mammalian muscle contractions and ATPase activities

Isolated rat and mouse extensor digitorum longus (EDL) and soleus muscles were studied under isometric and isotonic conditions at temperatures from approximately 8 degrees -38 degrees C. The rate constant for the exponential rise of tension during an isometric tetanus had a Q10 of approximately 2.5 for all muscles (corresponding to an enthalpy of activation, delta H = 66 kJ/mol, if the rate was determined by a single chemical reaction). The half-contraction time, contraction time, and maximum rate of rise for tension in an isometric twitch and the maximum shortening velocity in an isotonic contraction all had a similar temperature dependence (i.e., delta H approximately 66 kJ/mol). The Mg++ ATPase rates of myofibrils prepared from rat EDL and soleus muscles had a steeper temperature dependence (delta H = 130 kJ/mol), but absolute rates at 20 degrees C were lower than the rate of rise of tension. This suggests that the Mg++ ATPase cycle rate is not limiting for force generation. A substantial fraction of cross-bridges may exist in a resting state that converts to the force-producing state at a rate faster than required to complete the cycle and repopulate the resting state. The temperature dependence for the rate constant of the exponential decay of tension during an isometric twitch or short tetanus (and the half-fall time of a twitch) had a break point at approximately 20 degrees C, with apparent enthalpy values of delta H = 117 kJ/mol below 20 degrees C and delta H = 70 kJ/mol above 20 degrees C. The break point and the values of delta H at high and low temperatures agree closely with published values for the delta H of the sarcoplasmic reticulum (SR) Ca++ ATPase. Thus, the temperature dependence for the relaxation rate of a twitch or a short tetanus is consistent with that for the reabsorption rate of Ca++ into the SR.

Factors modulating the sensitivity of the relaxation to the loading conditions in rat cardiac muscle

The load sensitivity of the relaxation phase was studied in rat papillary muscle, with isotonic afterloaded contractions and stretches applied after the peak of isometric twitches. The tension decay occurred earlier in isotonic than in isometric contractions. When a central region of the preparation was marked with small stainless steel pins, a lengthening of this region could be shown during relaxation of isometric (fixed end) contractions. This lengthening was earlier and faster in isotonic afterloaded contractions. Therefore the sensitivity of relaxation to load or length changes could be described in the context of the general mechanism of relaxation which takes into account the non uniform behaviour of the muscle and the internal movement during contractions. Interventions which decelerate the activation decay rate had different effects on the load dependence of relaxation. Caffeine addition and substitution of strontium for calcium abolished the load sensitivity while a temperature reduction had no influence on it.

Subunit treadmilling of microtubules or actin in the presence of cellular barriers: possible conversion of chemical free energy into mechanical work

Free microtubule or actin filaments, along with the monomeric forms of the protein, hydrolyze GTP or ATP to produce a flux of subunits through the polymer. This flux, called treadmilling, produces no useful work. In the cell, however, these filaments are likely to be constrained between nucleating sites and other barriers that will limit polymer growth. We study here the effects of a small compression of the filaments resulting from polymerization against such barriers. If subunits can still exchange at the two ends, treadmilling will take place here as well. Under these conditions, the filament system can do useful work. The free energy of NTP hydrolysis can be used to transport materials, attached to the filament, against a resisting force. This process can in principle take place at high efficiency and bears a resemblance in a bioenergetic sense to the utilization of ATP free energy in muscle contraction. The same general principles apply to a polymer in which one end is anchored and one end is free.

An EMG-level muscle model for a fast arm movement to target

A model of human muscle action is presented for a maximally fast, large-amplitude forearm movement to target. The inputs to the model are approximately the biceps and triceps EMG envelopes over a single movement. The model's output gives the corresponding displacement angle of the forearm about a fixed elbow position as a function of time. The idea of the model is to conceive of both EMG input drives as successions of millisecond input pulses, with each pulse resulting in a muscle tension twitch. Every twitch is amplitude-scaled, parametrically-shaped, and duration-limited as a function of the muscle's contractile history thus far in the movement. The muscle tension at any time t is the sum of the residual tension levels of all twitches begun before t. The model was developed and tested with special reference to two subjects: one, according to the model dynamics, was a comparatively slow-twitch type and the other modelled as a fast-twitch type. Good agreement was found between model output and subject response data whenever the subject's EMG's were "synchronous". The model can be used to characterize each subject's responses by a suite of twitch characteristics. This will enable us to check the accepted but now suspect correlation between muscle biopsy- and performance-determined muscle twitch type.

High-energy phosphate metabolism and energy liberation associated with rapid shortening in frog skeletal muscle

1. High-energy phosphate metabolism and energy liberated as heat and work were measured in 3 sec tetani of frog sartorius muscles at 0 degrees C.2. Three contraction periods were studied: (a) shortening at near-maximum velocity for 0.3 sec from sarcomere length 2.6 to 1.8 mum, beginning after 2 sec of isometric stimulation, (b) the 0.7 sec isometric period immediately following such rapid shortening, (c) the period from 2 to 3 sec in an isometric tetanus at sarcomere length 1.8 mum.3. There were no significant changes in levels of ATP, ADP or AMP in any contraction period. The observed changes in inorganic phosphate and creatine levels indicated that the only significant reaction occurring was phosphocreatine splitting.4. The mean rate of high-energy phosphate splitting during rapid shortening, 0.48 +/- 0.24 mumole/g.sec (mean +/- s.e. of mean, n = 29; ;g' refers to blotted muscle weight), was not significantly different from that in the 1 sec period in the isometric tetanus, 0.32 +/- 0.11 mumole/g.sec (n = 17). The mean rate in the post-shortening period, 0.71 +/- 0.10 mumole/g.sec (n = 22), was greater than that in the 1 sec period in the isometric tetanus, and this difference is significant (P < 0.02, t test).5. A large quantity of heat plus work was produced during the rapid shortening period, but less than half of this could be accounted for by simultaneous chemical reactions. The unexplained enthalpy production was 6.5 +/- 2.6 mJ/g (mean +/- s.e. of mean). No significant unexplained enthalpy was produced in the 1 sec period in the isometric tetanus.6. In the post-shortening period the observed enthalpy was less, by 6.2 +/- 2.6 mJ/g, than that expected from the simultaneous chemical reactions.7. The results are interpreted in terms of an exothermic shift in the population of cross-bridge states during rapid shortening. It is suggested that a relatively slow subsequent step prevents many of these cross-bridges from completing the cycle and splitting ATP until after the end of shortening.

Can free energy transduction be localized at some crucial part of the enzymatic cycle?

Free energy transfer from one small molecule (ligand, substrate, etc.) to another can in general be comprehended only in terms of complete kinetic cycles, not in terms of an individual transition in the cycle, or of a single 'energized' state in the cycle, or of binding strengths of the small molecules on the enzyme, or of their standard chemical potentials when bound. The reason these latter approaches fail is that the enzyme molecule is as much a specific participant in the step by step proceedings as are the small molecules; small molecule free energies cannot be separated from enzyme contributions or, in general, from each other except at the complete cycle level. It is possible to follow the 'flow' of the total free energy of enzyme + small molecules among various subdivisions or categories, as the system proceeds through the states of the transducing cycle. These categories can be understood in molecular terms but several of them involve the enzyme in a way that is inseparable from the small molecules. Hence this procedure also, does not allow localization within the cycle of the supposed point of transfer of free energy from one small molecule to another.

Influence of osmotic compression on calcium activation and tension in skinned muscle fibers of the rabbit

Single skinned muscle fibers were osmotically compressed back to and below their in situ size by addition of a large, random-coil polymer (Deytran T500; MN = 180,000; MW = 461,000) to the bathing medium. Maximal Ca2+-activated tension in fibers swollen (zero Dextran, fiber width 21% above in situ) or near in situ size (5% Dextran, in g/100 ml final solution) was similar, but compression to 86% of in situ width with 10% Dextran decreased maximal force by 15% relative to polymer-free control. While the relative tension-pCa relation in 0 and 10% Dextran was similar, with a pCa of 6.37 required for 50% activation, that in 5% Dextran was more sensitive to Ca2+, with a pCa50 of 6.66. We feel these effects are most likely due to changes in interfilament spacing with compression and that alterations in Ca2+-sensitivity might be explained by changes in cross-bridge angle or in the concomitant attachment-detachment rate constants which would be expected to influence the troponin-Ca2+ binding equilibrium, as has been proposed by others.

Biochemical interpretation of tension transients produced by a four-state mechanical model

We have simulated transient tension responses seen in rabbit psoas and insect flight muscle using a four-state cycle. A wide range of all the rate constants in the cycle and a minimum number of simple mechanical assumptions were used. The rate constants were noted for those cases in which the transient resembled a response seen in real muscle and in which the values of ATP turnover were realistic. The rate constants fell within narrow ranges. With a suitable correspondence of mechanical and biochemical state the actual rate constants were compatible with those determined in biochemical experiments. It was possible to model the effects of temperature, calcium ion concentration, length change and the removal of inorganic phosphate.

Local movement in stimulated frog sartorius muscle

Local movement was recorded in tetanically contracting frog sartorius muscle to estimate the nonuniformity in the distribution of compliance in the muscle preparation and the compliance that resides in the attachments of the preparation to the measuring apparatus. The stimulated muscle was also subjected to rapid length changes, and the local movements and tension responses were recorded. The results indicate that during tension development at resting length the central region of the muscle shortens at the expense of the ends. After stimulation the "shoulder" in the tension, which divided the relaxation into a slow decline and a subsequent, rather exponential decay toward zero, was accompanied by an abrupt increase in local movement. We also examined the temperature sensitivity of the two phases of relaxation. The results are consistent with the view that the decrease in tension during relaxation depends on mechanical conditions. The local movement brought about by the imposed length changes indicates that the peak value of the relative length change of the uniformly acting part was approximately 20% less than the relative length change of the whole preparation. From these observations, corrections were obtained for the compliance data derived from the tension responses. These corrections allow a comparison with data in the literature obtained from single fiber preparations. The implications for the stiffness measured during the tension responses are discussed.

Theoretical aspects of translocation on DNA: adenosine triphosphatases and treadmilling binding proteins

The basic kinetic and bioenergetic theory is outlined for two kinds of translocation on DNA: (i) helicases that use ATP to move along single-stranded DNA or to move on and invade double-stranded DNA at a replication fork; and (ii) DNA-binding proteins (not ATPases) that form bound aggregates on single-stranded DNA and facilitate replication by steady-state treadmilling of molecules between the ends of the aggregate. The respective resemblances to myosin--actin in muscle and to steady-state treadmilling in solution of actin or tubulin are pointed out.

Theoretical models for cooperative steady-state ATPase activity of myosin subfragment-1 on regulated actin

Recent theoretical work on the cooperative equilibrium binding of myosin subfragment-1-ADP to regulated actin, as influenced by Ca2+, is extended here to the cooperative steady-state ATPase activity of myosin subfragment-1 on regulated actin. Exact solution of the general steady-state problem will require Monte Carlo calculations. Three interrelated special cases are discussed in some detail and sample computer (not Monte Carlo) solutions are given. The eventual objective is to apply these considerations to in vitro experimental data and to in vivo muscle models.

The kinetics of post-vibration tension recovery of the isolated rat portal vein

1. The kinetics of post-vibration tension recovery have been examined during electrical, noradrenaline or KCl stimulation of the isolated rat portal vein. 2. Inhibition of isometric contractions produced by a combination of noradrenaline (20 microM) and KCl (53 mM) by longitudinal, 100 Hz sinusoidal vibration increased with increasing vibration amplitude up to a maximum of 78.7% of the active tension. This inhibition was little affected by a decrease in temperature from 37 to 25 degrees C. Recovery of tension after the end of vibration was complete and took place exponentially. The time constant for this recovery was little affected by changes in vibration amplitude, but increased from 1.72 +/- 0.09 to 4.35 +/- 0.33 sec, for large amplitude vibrations, when the temperature was lowered from 37 to 25 degrees C. 3. The increase in isometric tension during 50 Hz a.c. electrical field stimulation was exponential, apart from a minor initial activation component, and took place with a time constant of 1.25 +/- 0.17 sec. Neither delaying nor interrupting development of this contraction with inhibitory vibration altered the time constant for this exponential increase in tension. There was no correlation between the time constant and the maximum active tension achieved after vibration was stopped. 4. Post-vibration tension recovery during electrical, noradrenaline (20 microM) or KCl (120-130 mM) stimulation was independent of the nature of the stimulus at comparable times of stimulation, but the time constant increased during exposures of more than 10 sec to either noradrenaline or KCl. With noradrenaline, the increase was from 1.45 +/- 0.10 sec after 50 sec of stimulation to 2.24 +/- 0.16 sec after 336 sec of stimulation (P less than 0.0005). Such an increase in the time constant may reflect slower cycling of cross-bridges with an improvement in the efficiency by which contraction is maintained. 5. The kinetics of post-vibration tension recovery were those of a monomolecular or, as is more likely, a pseudo-monomolecular chemical reaction. A cross-bridge attachment model based on such a reaction has been used to interpret the observations.

Effects of AMPPNP and vanadate on the mechanochemical crossbridge cycle in flagella

The non-hydrolysable ATP analogue, beta,gamma-imido-adenosine-5'-triphosphate (AMPPNP, adenylilimidodiphosphate) does not reduce the stiffness of demembranated sea urchin sperm flagella measured in the rigor state in the absence of MgATP2- and does not cause relaxation of rigor bends. The competitive inhibitions provide further evidence that the dynein-tubulin crossbridge cycle in flagella is similar to the actin-myosin crossbridge cycle.

Steady-state head-to-tail polymerization of actin or microtubules. II. Two-state and three-state kinetic cycles

In a previous paper, bioenergetic aspects of head-to-tail polymerization for a two-state actin ATPase cycle were discussed. In section 2, here, the steady-state polymer length distribution for this case is derived. The distribution has the same mathematical form as at equilibrium, but the parameters are different. In section 3, both bioenergetic topics and the polymer length distribution are considered for the more general and realistic case of a three-state actin ATPase cycle. Again, the mathematical form of the steady-state distribution is the same as at equilibrium, but the parameters are more complicated. In section 4, the question is examined of how much the mean and variance of a polymer length distribution, obtained from a finite experimental sample of polymer (aggregate) molecules, would be expected to deviate from the true mean and variance (from an infinite sample). Also considered briefly in section 4 is the effect of hard polymer-polymer interactions on the equilibrium polymer length distribution, at finite polymer concentrations.

Bioenergetic aspects and polymer length distribution in steady-state head-to-tail polymerization of actin or microtubules

Wegner's theory of steady-state head-to-tail polymerization of actin (or microtubules) is extended somewhat in order to show the explicit role of the ATP (or GTP) free energy of hydrolysis (X) in the steady-state kinetics. The monomer flux and the ATP flux can both be expressed in terms of X and rate constants of the model. Both fluxes approach zero as X leads to 0 (by variation of the concentrations of ATP, ADP, and Pi); this limit corresponds to ATP equilibrium. The dependence of rate constants on these concentrations is examined. Free energy levels of the monomer kinetic cycle and the rate of free energy dissipation are discussed. The steady-state polymer length distribution is derived for a special case.

Theoretical considerations in the equilibrium binding of myosin fragments on F-actin

In a previous paper, equilibrium ocnstants for the binding of myosin fragments onto F-actin were assumed known and the statistical problems encountered when the actin sites are occupied to an arbitrary fractional extent were analyzed. The object of the present paper is to attempt to understand the observed order of magnitude of these equilibrium constants in terms of the statistical mechanical degrees of freedom involved. That is, we examine here the equilibrium constants themselves rather than the statistical consequences of the equilibrium constants. The treatment given amounts to a semi-quantitative sketch or outline of the problem. Structural details are much too uncertain to warrant a careful and rigorous treatment at this time. But the discussion suffices to establish the essential qualitative features of the problem. The procedure used is to examine the important equilibrium constants, one at a time, in terms of the factors (partition functions) that contribute to each constant, together with numerical estimates for these factors.

Cross-bridge behavior in rigor muscle

Properties of the rigor state in muscle can be explained by a simple cross-bridge model, of the type which has been suggested for active muscle, in which detachment of cross-bridges by ATP is excluded. Two attached cross-bridge states, with distinct force vs. distortion relationships, are required, in addition to a detached state, but the attached cross-bridge states in rigor muscle appear to differ significantly from the attached cross-bridge states in active muscle. The stability of the rigor force maintained in muscle under isometric conditions does not require exceptional stability of the attached cross-bridges, if the positions in which attachment of cross-bridges is allowed are limited so that the attachment of cross-bridges in positions which have minimum free energy is excluded. This explanation of the stability of the rigor state may also be applicable to the maintenance of stable rigor waves on flagella.