J. theor. Biol. ( 1977) 69, 187-230

The Sliding Filament Model of Muscle Contraction 11. The Energetic and Dynamical Predictions of a Quantum Mechanical Transducer Model B. F. GRAY? AND I. GONDA~ Department of Physical Chemistry, School of Chemistry, The University of Leedr, Lee& LS2 9JT, England (Received 22 September 1976, and in revised form 29 June 1977) A theoretical model of a molecular energy transducing unit designed for the production of mechanical work is constructed and its consequences examined and compared with the experimentally determined myothermal and dynamic properties of vertebrate striated muscle. The model rests on a number of independent assumptions which include: the almost instantaneous generation of mechanical force by the occurrence of a radiationless transition between vibronic states of the transducer (crossbridge) at a point of potential energy surface crossing; transmission of this force to the load via the active sites on the thin filament by means of non-bonding repulsive forces, no energy being required for detachment; “detachment” consists of a second radiationless transition at a lower energy point than the first force generating transition, the energy difference appearing largely as work. The method of force generation completely avoids problems such as the “force-rate dilemma” which occur repeatedly in any discussion where state populations are near-Boltzmann and also leads without further arbitrary assumptions to such concepts as “attached but non forceproducing states” and strongly position dependent “attachment” and “detachment” rate constants since these can only be appreciable near potential energy surface crossings. The kinetics and energetics of a transducer of this type operating cyclically and converting ATP --, ADP $- Pi are considered and shown to lead to length-tension and energetic behaviour very similar to that exhibited by vertebrate striated muscle, both for contraction and stretching. The existence of a limiting tension for stretching is predicted by the model as is the decrease of the rate of enthalpy release rate below the isometric value. At the limiting tension the rate of enthalpy release by the transducers is virtually zero, as observed. However, the stretching only inhibits the ATP hydrolysis, the cyclic synthesis from ADP and work being impossible with this model. The response to rapid t Presentaddress: Biomolecular Science Unit and School of Chemistry, Macquarie University, North Ryde 2113, New South Wales, Australia. $ Present address: Department of Pharmacy, University of Aston, Gosta Green, Birmingham, B4 7ET, England. 187

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length step changes automatically contains the asymmetry observed experimentally (with respect to lengthening and shortening) and arbitrary assumptions over and above those giving adequate explanation of the steady-state properties are not required. The asymmetry arises mainly as a consequence of the non-bonded pushing action of the crossbridges. This same assumption predicts the occurrence of an asymmetric thermoelastic ratio for active muscle with respect to stretching and contraction. The quantitative aspects of the model are satisfactory as it simultaneously reconciles the numerical magnitudes of macroscopic quantities such as isometric tension, maximum contraction velocity, limiting tension sustainable on stretching, isometric heat rate and resting heat rate with molecular parameters such as the filament and crossbridge periodicities, molecular vibrational relaxation rates, recurrence times for the radiationless transitions occurring, etc. This is achieved without any parameter optimization and only a very much smaller number of unknown parameters than the number of observed results accounted for. Many of the entities occurring in the model cycle (vibronic states of crossbridges, ATP, etc.) appear to be in one-to-one correspondence with many of the kinetic entities postulated to account for the biochemical kinetic results obtained for the actomyosin ATPase system in vitro. Finally, the rigor state has to be viewed in a different way from the conventional one; on the basis that the present model states which are part of the contraction cycle but sparsely populated during the latter (and hence are of chemical kinetic but not dynamical importance) are heavily populated during the rigor state. The mechanical properties of the rigor state would then be determined by these molecular states which would be very short-lived during the contraction cycle. If this is correct the rigor state could yield much more information about inaccessible parts of the contraction cycle than is presently supposed. The model leads one to expect a rather different response to quick length step changes in the rigor state from that of the active state, in contrast to current interpretations in terms of a large number of attached crossbridges, unable to detach due to the absence of ATP.

1. Introduction Our aim in this paper is to use the concepts presented in Part I (Gray & Gonda, 1977) in an attempt to discuss how they can be applied to real systems, such as the contractile machinery in striated muscle. To do this we shall make a number of assumptions some of which are common to a large class of muscle contraction theories, some of which are peculiar to the present work. Most of our exposition will use “molecular language”, as set out in Part I (Gray & Gonda, 1977) and it will, as far as possible, attempt to provide a molecular dynamic picture of a number of the experimental facts of muscle energetics and dynamics. As such, it is not to be regarded as an alternative to the theory of A. F. Huxley (1957) in general but as providing

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a molecular interpretation of the parts of it which we retain in terms of concepts well-founded and familiar in molecular quantum mechanics. Huxley & Simmons (1973) point out that the theory of A. F. Huxley (1957) contains three main propositions which are independent. These are stated as follows. (1) “Each zone of overlap between!thick and thin filaments contains uniformly distributed independent sites each of which generates a relative force between the filaments, and the total force on a thin filament is the sum of the forces contributed by the sites between that filament and the neighbouring thick filaments.” (2) “Each site consists of a movable side piece elastically connected to the thick filament which undergoes cycles of(i) sttachment to a thin filament site with a moderate rate constant; (ii) exerting a force on the thin filament; and (iii) detachment, the rate constant for the latter becoming large when the sliding motion of the filaments has brought the side piece to a position where the force it exerts is near to zero.” (3) “The force arises because the location required for attachment is such that attachment occurs only when a crosspiece is displaced in the direction which causes the restoring force due to its elastic element to be in the pulling direction.” In this work we accept proposition (1) in rota as axiomatic and proposition (2) in part as axiomatic, i.e. that each site consists of a movable crossbridge which undergoes cyclic attachment, force generation and detachment, with suitable interpretation of “attachment” and “detachment”. We do not take the statements about the rate constants as axiomatic, but in fact show that these statements follow in a general way from axioms made at the molecular level with regard to crossing of potential energy surfaces of the M-ATP and MADP.Pi states involved in the cycle. Thus we show that it is easy to produce a firm quantum molecular basis for rate constants which vary sharply with spatial position, in the context of an ordered array of molecules such as exists in a muscle fibril. We discard proposition (3) entirely, although Brownian motion still plays a part in our scheme as a mechanism for returning detached crossbridges back to their starting point (where they first attach, and will subsequently reattach) by a diffusion mechanism. It is easily seen that this mechanism could return crossbridges sufficiently quickly to their starting points to support a steady state in the fastest contraction, provided we are not endowing them with sufficient potential energy in the process to generate force on attachment. This latter idea leads to the “force-rate dilemma” (Hill, 1974) inevitably faced by models based on equilibrium thermodynamic considerations. A particle undergoing diffusive motion due

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to Brownian buffeting, and known to be at the origin along the x-axis at time zero (the point of detachment), will suffer displacement given by (Ax)’ = 2Dt. Taking D N 10-l’ m* s-l and x N 20 nm, we find the time required for diffusive spreading N IO-’ s and this seems short enough even for the fastest muscles with periods of N 10m3 s. In the present theory, “attachment” and generation of force are distinct processes as an inherent feature of the model, not an extra axiom necessary in order to account for the mechanical transients measured by Huxley & Simmons (1971) as in the model of Julian, Sollins & Sollins (1973). After “attachment”, which we discuss more fully later, the crossbridgeATP entity undergoes a transition to a state exerting a relatively large force on the actin filament as a result of perturbation by the appropriate actin site. This transition is not a separate axiom, nor is it to be likened to the “conformational changes” beloved of biochemists, it is a necessary consequence of our basic axioms, is well defined and similar phenomena are well understood everyday occurrences in photochemical kinetics for example. This stimulated transition (stimulated by the actin site, which also acts as recipient of the work as well as the trigger for the transition) prevents wastage of energy which would occur if the corresponding spontaneous transition were highly probable. Further discussion of this will be deferred until the details have been presented and the nature of the “attached but non forcegenerating” state further clarified. However, it is possible to point out here that at no time in the cycle does the present model require “bonding”, in the general chemical sense, between the crossbridge head and the thin filament. The force can be transmitted perfectly adequately via non-bonding interactions (ultimately due to the Pauli Exclusion principle allied to Coulomb’s Law). The attached (but non-force-generating) state will be seen as a state of the M-ATP moiety whose equilibrium position is near the thin filament, “upstream” from the position of detachment, and this is the only sense in which it can be regarded as attached, it is not “bonded” to the thin filament and longitudinal relative motion is freely allowed. The existence of this state, with a different equilibrium configuration, e.g. angle of tilt, from that of an unattached crossbridge not interacting with ATP implies motion of crossbridges on activation by interaction (possibly with Ca*+) even in the non-overlap zone. This question has been discussed by H. E. Huxley (1973) and White & Thorson (1973). The basic axiom, which is at the heart of the present model, concerns our suggestion for the ultimate transduction mechanism in muscle, and from it the rest of the model follows more or less inevitably. This mechanism can

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be likened to neither a heat engine (or equivalent devices, i.e. Carnot limited) nor an electrochemical cell. In the former type of machine chemical energy (which is electronic energy) is degraded first to vibrational energy, then very quickly to translational energy, and the latter is used to push a piston, dropping its temperature in the process. The heat is released by chemical reactions, which are normally accelerated by a rise in temperature, in turn releasing more heat etc., i.e. we have a positive feedback process resulting in a point of instability where the system is on the brink of ignition, or the brink of extinction if it is already ignited, clearly a desirable and useful feature if we require rapid and easy control, and can supply selective perturbations. On the other hand, electrochemical devices, which are not subject to the Carnot limitation of efficiency are negative feedback systems. In a typical open-circuited cell the chemical reaction proceeds and generates a potential difference which inhibits the reaction and results in the setting up of an electrochemical equilibrium. These devices lend themselves to reversible operation and the steady state set up is always stable, hence they are highly efficient but inflexible, as is well known. They are not efficient at high power outputs, or under conditions of rapidly changing demand. Our transduction mechanism has the advantages of both and the disadvantages of neither. It involves conversion of electronic (chemical) energy directly into mechanical work and so is not subject to the Carnot limitation. On the other hand, it does not have a self-inhibiting characteristic like the cell, and can easily be arranged to have a positive feedback characteristic. What is the catch? The mechanism requires an accurate and characteristic arrangement of potential energy surfaces allowing perfusive access to fuel molecules and exit to waste molecules which it would be beyond present day technology to build, but this of course does not preclude its occurrence as a result of natural selection. A special form of the transduction mechanism we propose has already been discussed (Gray & Gonda, 1975) involving an electronically excited state of one of the participants. This is not necessary and all that is required is a molecule, or part of a molecule, far from its equilibrium configuration, as the starting-point of a cycle. For simple molecules the obvious way to achieve this is by absorption of radiation, which by the Franck-Condon Principle will produce an electronically excited state momentarily at the equilibrium configuration of the ground state (see Fig. I). This would be point Y. In a normal gas phase reaction dissociation would now take place along YU, the atoms A and B having relative kinetic energy AE,, which would quickly be thermalized. If the point Y happens to lie below the dissociation limit of curve 2 a vibrationally and electronically excited state would be produced. The vibrational energy would quickly be dissipated as heat and the electronically excited AB

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molecule would sit near Z until it emitted a photon (assuming the states 1 and 2 have different multiplicities thus giving an appreciable radiative lifetime). It would drop to point W in the ground state and then relax vibrationally back to X. The most likely vibrational quantum number determining the equilibrium configurations of states 1 and 2 would of course depend on temperature. The potential for extracting work from this device lies in the segment YZ of the excited curve; we disregard here the possibility of eccentric coupling for the WX segment (see Part I). Normally the outward force at Y is large and results either in rapid acceleration of the atoms apart or an energetic vibration. However, if we could prevent this acceleration by suitable application of a constraint ultimately coupled to a weight to be lifted, we could obtain work directly. The energy of the molecule would drop (down to point Z) and the potential energy of the weight would be raised by a similar amount. Of course the constraint would have to be another molecule and both molecules would have to be part of a regular spatial array. At point Y it is crucial to understand the physical origin of the force--it is simply that the electron cloud of the molecule is appropriate to state 2 as a result of rapid response to the absorption of a quantum of energy whilst the nuclei, due to their much greater inertia, are still in the position of equilibrium for state 1. The system is somewhat akin to a deformable but responsive jelly containing heavy ball-bearings embedded therein. A rapid deformation of the jelly will result in a state with potential for doing work as the bearings move towards their new equilibrium positions. In complex polyatomic molecules there is even more scope for imagining this type of device, and excited states are not, strictly speaking, necessary. We must be careful to avoid semantic problems here, as isomers with

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unequal energies are certainly involved and one could in principle refer to a high energy isomer as an excited state, but this is not normal practice. Simply stated, a segment of a polyatomic molecule (the part which is going to do the pushing) can be brought to the equivalent of the point Y as a result of reaction of another part of the molecule with a substrate (ATP for example). Of course we have to deal with potential energy surfaces in polyatomic molecules, and these can touch and intersect each other. At a point of intersection we have what can be very crudely termed an “electronic isomer”. For a given position of the nuclei the electron distributions are different (giving rise to different forces unless the contact of the curves is tangential) but the energies of the two states are equal. This latter circumstance, as discussed fully in part I (Gray & Gonda, 1976) allows the possibility of transitions from one state to the other in this region without significant energy input. The transitions can be induced by very small perturbations of the right type and they are thus a suitable point for the imposition of control of some sort. In a mechanism of this type, it is essential that the molecule acting as the thrust receiver be very rigid, otherwise it would simply become vibrationally excited, thus dissipating the energy. There is evidence (Huxley, 1974) that the filaments themselves have virtually zero compliance and there is also evidence (van Eerd & Kawasaki, 1972) for an increase of rigidity in the region of the binding sites of the thin filament in the presence of Ca2+ ions, just what would be required to accept thrust from the transducing unit. Another feature of this transduction mechanism which follows automatically from the model is that up to the work performing step, the reactions (in particular binding of ATP to myosin) are reversible, the irreversibility appearing the moment the transition into the pushing state occurs and some work is performed. This initial performance of work reduces the energy of the “machine” below the level of the potential surface crossing thus preventing the reverse reaction as discussed in part I. After performance of all the work other reversible reactions can take place, e.g. dissociation and binding of ADP and Pi, but these yield states different from the complex existing before the transition into the work performing state (M-ATP) (Bagshaw & Trentham, 1975), and this latter will not be accessible by the reverse route, without doing work 011 the complex and pushing it uphill. The present transduction model contains the new feature such that a single mechanism ensures that catalysis of the reaction and production of tension or mechanical work are synonymous, and this depends on the structural integrity of the filament-crossbridge system. A further axiom adopted in this paper which is independent of the assumptions about the ultimate transduction mechanism concerns the nature I .“.

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of the interaction between the crossbridge head in the pushing state and the thrust-receiving site on the thin filament. No force of attraction between these is assumed, i.e. no chemical bonding. However, there is in the model an apparent force of attraction between them, arising from the force on the crossbridge head in its pushing state centred on its equilibrium position which is downstream behind the actin site which is postulated to sterically hinder the motion of the crossbridge towards its equilibrium position, thus preventing acceleration and dissipation of the energy.7 Clearly the present model of the transduction mechanism has a built in asymmetry with respect to stretching and contraction and this is relevant in discussion of the response of activated muscle to rapid length and tension steps as performed by Huxley & Simmons (1971), and also the asymmetry of the thermoelastic heat with respect to stretching and release (Woledge, 1971; Woledge, 1961). In the next section the detailed basis of the model will be set out, making full use of the development in Part I, the appropriate dynamical and kinetic equations formulated and the steady state solutions presented. In the following sections these results will be used for comparison with experimental results such as the tension-velocity relation for contraction, the tensionvelocity relation for stretching, the total energy release rate for contraction and stretching, work rates, efficiency, energy release in the isometric state compared with energy release in the resting condition, level of control by CaZf ions required for consistency (this turns out to be necessary at at least two points in the cycle). A further section contains discussion of furthei developments required to treat quick release results properly and difficulties and puzzles remaining from the present point of view. A further section attempts to relate biochemical studies on actomyosin ATPase to this model. Finally conclusions and predictions are summarized.

2. The Transduction Model Formulation The various states in the crossbridge cycle can be identified by referring to the energy level diagram in Fig. 2 and this energy includes that of a substrate (ATP) molecule, considered as part of the system. The effect of the t This apparent force of attraction appears to have very strange properties if regarded as a function of the distance between the two bodies concerned (the actin site and the “binding” site on the crossbridge head). It would appear to be non-local, i.e. dependent on variables other than the distance between the sites. For example, if this distance were increased by rapidly moving the actin site downstream (in the direction of contraction) before the myosin site could respond, the “force” would appear to be absent, but if in some way one attempted to move the myosin site upstream it would be present. This can be seen more clearly by referring to Fig. 4 in Part I which is essentially a potential energy diagram for this sort of system.

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repulsion of the actin sites is represented by the vertical dotted line. Thus replacement of an ADP ligand by an ATP ligand would result in a “transition” up the diagram to a higher level, e.g. from r to p. Of course the configuration of the system cannot be represented properly in one dimension, and this must be kept in mind when attaching any geometrical interpretation to Fig. 2.T A single cycle would consist of the crossbridge in its lowest state r becoming “charged” by binding an ATP substrate molecule, the system moving into a higher energy state p. This state p would have an equilibrium position such that the crossbridge head was in the vicinity of the thin filament, but it would not interact strongly with the latter as its potential energy curve is shallow. This r $p transition would be completely reversible and would result in motion of the crossbridges from their resting position in state r to their “cocked” position in state p. In state p the bridges would appear to all intents and purposes to be “bound

but not force-producing”,

but there is

no need, and indeed disadvantage, in postulating any sort of binding in the chemical sense. The interaction with ATP is postulated to place them in an equilibrium position close to the thin filament and away from the thick filament axis. The next step in the cycle is interaction of a crossbridge in the p state with an actin site which causes a radiationless transition to the “pushing”

state n, when work is done along XY. The rate of this transition

t A discussionof the juxtaposition of the crossbridgesand actin sitesis given by Gonda (Gonda, 1974).For example,the probability that a crossbridgein thep state will be pushed by the actin rather than vice versacan be assignedby an arbitrary value. It is also possible that the two-headed nature of the myosin molecule simplifiesthis problem in so far as the pushing of one head by actin could easily be envisagedas a mechanism for producing correct alignment, in the pushing position, of the secondhead.

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will be evaluated in detail below, but will obviously depend on the rate ot presentation of actin sites to the “cocked” crossbridges and also on the rate at which the transition can occur when the site and crossbridge arc in proximity. Either of these can be rate determining under various conditions. Of course it will be impossible to prevent some p states from making spontaneous transitions, i.e. without the stimulation of an actin site. The number of such crossbridges going from X to Y without performing work will be denoted by n’. These represent a degradation of energy to heat, and as such a waste, but the rate constant for this spontaneous transition is shown late1 to be the main determinant of the acceleration of the muscle on stimulation. This heat will be a part of the resting heat and this model requires it to be larger for faster muscles than for slower muscles. On purely steady-state considerations, one would wish this spontaneous transition not to happen of course. In the region of point Y a further radiationless transition is postulated to take place out of the pushing state into the state m which has a fairly flat potential energy curve. This would be interpreted as “detachment” and at the point Y the crossbridge will still be downstream, having completed its power stroke. If the equilibrium configuration of the state m is back upstream, the crossbridge will move back under the action of a small force as indicated by the shallowness of the potential energy curve for this state. Finally, this “detached” and “spent” state makes a transition back to the lowest energy point in the cycle, the state r, whereupon a second molecule of ATP can interact and the cycle can be repeated. The correlation of these states and transitions with some current biochemical ideas will be attempted in a separate section, presently we wish to formulate the kinetics of this cycling. We begin with the state r, the lowest energy state occurring. We can immediately write down (we deal with a single half sarcomere):

where k3 is a “composite” rate constant of the form k; [ATP], (depending on the biochemical interpretation), k-, = k’-, [ADP], Similarly, for the state m:

and etc.

dm =-kk,m+k!,n’+k-,vn*+k-,r, dt where k!., is the rate constant for the spontaneous transition from the n’ to m. k-,v is the rate of presentation of the pushing states n* to the point Y and this is fixed by the periodic structure of the muscle, it cannot be taken as a variable parameter. The transition out of the pushing state into the m state will not occur in stretching, and the pushing states will get trapped to

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the Zeft of the point X, being unable to complete the cycle. This results, as we shall see in a later section, in a very high tension being developed at very little energy cost, reminiscent of the “catch” states in shellfish muscle. We shall formulate the appropriate differential equations in a later section on stretching. The number n’ of crossbridges traversing the segment XY without pushing, i.e. making a spontaneous transition at point X. is given by: dn’ -= dt

kyp-kO_,n’,

the superscript 0 being used for the spontaneous rate constants. ky is the rate constant for transition from p -+ n’ and k? r is the rate for the transition from n’ -+ m. The rate equations for p and n* are determined by the rate of induced transition at the crossover point X. If we denote the net rates of removal ofp and production of n* by R and R’, respectively, then we can write : dp - = -(k:+k-,)p+k,r-f?. dt dn* = R’-k-,vn*, dt where R and R’ are to be determined by detailed considerations of the next section. These five differential equations (only four of which are independent as we assume a fixed number of crossbridges) must be supplemented with an equation of motion if we are to construct a mechanochemical system. We use the quantum mechanical version, since although the classical approximation will hold for particles of the masses involved here in general, it will not hold near the turning points of the motion and indeed at the curve crossing points quantum-mechanical ideas are essential. We write therefore:

where M is the effective mass per half sacromere of thin filament and (F) is the quantum mechanical expectation value of the net force per half sarcomere acting on the thin filament. IJ is the relative velocity of the thin filament with regard to the thick filament. (F) will be the difference between the tension generated by the half sarcomere and the external tension P, i.e. (F) = -P. In the present model the bulk of (F,,,) is derived from those crossbridges in the pushing state n* and it will be proportional to n* since we are assuming the crossbridges are independent force generators. At this point it is convenient to make a further assumption that the force is constant over the distance XV, i.e. the potential energy curve is a straight line. This greatly

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simplifies the analysis and it does not appear that one loses any qualitative or semi-qualitative information. If we denote this constant force by fi, then the tension generated will be fin *. Of course other states p, m, etc., will contribute to (F,,,), as resistances, but we have already assumed from the general structure of Fig. 2 that the forces associated with other states are small compared with fi. We will take these as being represented by an average value per crossbridge regardless of its state (other than n*) f2 < 1;. On this simple picture the resistance will then be (n - n*)fi where CCis the total number of crossbridges per half sarcomere. The equation of motion will be : do M-& =f~n*-f~(a--n*)-P = (fi +f&*

- afz - P.

3. The Transition Rate at X

Here we complete the mathematical formulation of the model by evaluating the transition rate R on the basis of molecular and structural information. We wish to evaluate the rate at which crossbridges in the p state, i.e. in the neighbourhood of the thin filament but near an actin site and therefore not interacting strongly, are transformed into the n* state where the crossbridge is pushing strongly against an actin site, and hence remaining in register with it until making the radiationless transition at Y. From the laws of quantum mechanics it is known that radiationless transitions of this type only take place at an appreciable rate in configurations near the curve crossing (otherwise energy input would be needed). We shall denote by pi the number of crossbridges in the p state which are “in register” at any one time with an actin site, but have not yet made the transition. The rate of formation of these will clearly be the presentation rate of actin sites to crossbridges in the p state, and this in turn is determined by the periodicity of the thin filament and the relative velocity v. If some of these pi crossbridges do not make the transition whilst in register, they will go back to the state of p again as the actin site passes. Those which do make the transition will momentarily be in the n* state but near the crossover point and therefore able to make the reverse transition back to pi. We denote these by nf. These will be removed from the crossover region by the motion of the thin filament and the pushing crossbridge down the curve XY. It is also possible that initially formed n* states are vibrationally excited and rapidly relax down from the crossover region A’. This would be very rapid (in a time < 10Vh s) and although it would represent an efficiency loss (the vibrational energy would appear as heat) it would ensure the irreversibility of the transition.

MUSCLL



(39)

’

and a second solution for 1~1> 0 is:

~=[l+~~{k,+:f(I+~,+:::+~i)+

>I - [l+yIuI]-I.

+ k;(k;+k-,)(I--k-,/k,) -I ~_~ _~--~~ ~.. (40) k,k, The equation of motion for stretching will be slightly different from equation (7) for contraction since the non-pushing crossbridges (a-n*) will again resist the motion, so effecively the sign of f2 will be changed : Ill 4 = n*f +(a-n*)f dt .’

2

-p

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(41)

and the steady-state equation to supplement (39) or (40) will be: n *u-i -f2> + G-2 = p* (42) Taking the 1111= 0 solution first, i.e. solving equations (39) and (42) we can obtain both IZ* and p as functions of p (not 1111) and these relations are necessary for evaluating the variation of total enthalpy release ri along the IuI = 0 curve for P > PO, discussed in a later section. For the present purposes we simply note that this solution gives the straight line branch of the curve in the tension-velocity diagram (Fig. 3). The second solution, obtained from equations (42) and (40) gives us a tension-velocity relation for stretching, easily shown to be: (43) where y is given by equation (40) and P, is given by afi, i.e. it is the maximum force that the muscle can generate when all crossbridges in the overlap region are pushing. This solution gives the branch of the curve in Fig. 3 which gives decreasing tension as Iuj increases. This type of behaviour has been observed by Curtin & Davies (1973) and Chaplain (1972). Chaplain suggested that under conditions of maximal activation with high Ca’+ level the increase of tension above the isometric value was due to actual distortion of crossbridges as the two sets of filaments are displaced contrary to the normal direction of motion. The same statement holds true in the present model which is more specific in molecular terms, as the crossbridges are moved away from the crossover point X where transition can take place. Chaplain found that the maximal tension developed was insensitive to the

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velocity for 101> O*lv,, i.e. on the steep curved branch AB in Fig. 3. The “corner” on the curve at A is a result of our simplifying assumptions, in particular the constancy of fi and fi and also our neglect of the possible transition n* -+ n’ by spontaneous detachment of Ca’+ from the thin filament, thus allowing some n* crossbridges to slip. Inclusion of these small terms would round off the corner and reduce P, below the value afi, i.e. all crossbridges trapped in the n* state. Both Chaplain and Curtin & Davies obtain P, N 1*5P,, and if we use this value in equation (43) along with n/P, = 0.25 we obtain an empirical value for y of -2 x lo4 if the point B, where the tension drops to the isometric value, occurs at a velocity of -0.50, (Curtin & Davies, 1973). In conjunction with equation (40) this determines k, from experiment as - lo6 s-r, when the numerical value assigned to the other constants later are substituted. This is entirely consistent with the physical interpretation as a vibrational relaxation process and with the numerical requirements imposed by the Hill’s equation behaviour for contraction. Furthermore, as we shall see later, the behaviour of the total enthalpy release on stretching [as measured by Curtin & Davies, 1973) is simulated accurately without the choice of any further parameters. We can also compare the expressions Pl = c@, and a = a& and if we take typical values for frog muscle of Pi = 1.5P, and a = 0.25P,, we see that .f’,/‘z = 6, again entirely consistent with our molecular assumptions as typified in Fig. 2. Finally we discuss a strange phenomenon observed after moderate speed stretching (Abbot & Aubert, 1952). After the end of the stretch tension decays to a value substantially greater than the isometric tension at the same length, i.e. stimulation after the length is set. The tension remains high throughout the tetanus but develops only the normal isometric value when the stimulus is stopped and restricted. Similarly, when a muscle shortens down to a given length, the steady tension is less than the isometric value, the discrepancy being greater the greater the velocity of shortening. These tensions reflect the immediate history of the muscle and clearly on the present model they simply reflect the number of crossbridges in the pushing state immediately before the length is held constant. Since n* < n* (isometric) for v > 0 and n* > n* (isometric) for u < 0 the observed results are explained immediately since the appropriate number of crossbridges iz* will be trapped between X and Yin Fig. 2 when the length is held constant. On removing the stimulus the Ca2+ ions will be pumped back into the sarcoplasmic reticulum leaving the crossbridges free to proceed to point Y without resistance and return to the resting state. Thus stopping stimulation wipes out the memory effect. This appears to be a simple physical interpretation of the v = 0 solution obtained from this model for P > P,,. i.rJ.

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7. Enthalpy Release Rate, Contraction In the steady state the total enthalpy production rate is proportional the rate of refuelling of the machinery with ATP, hence:

to

fi = (AHj(k,r - k-,p),

(4) where AH is the enthalpy change for hydrolysis of ATP + ADP per molecule. Ei as calculated here will be per half sarcomere of muscle. We simply substitute the steady state values of r andp into equation (44) to obtain fi as an explicit function of u (or f since we have a steady state velocity-tension relation). We choose to express fi in terms of its isometric value fi,, and obtain:

(45)

directly in terms of the number of crossbridges pushing. A more convenient form is in terms of the tension, and if we use our form for Hill’s equation we obtain after some rearrangement a linear relation with negative slope, (46)

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