I. that-. Biol. (1977) 69, 167-l 85

The Sliding Filament Model of Muscle Contraction I. Quantum Mechanical Formalism B. F. GRAY?

AND I. GONDA~

Department of Physical Chemistry, School of Chemistry, The University of Leeds, Leeds LS2 9JT, England (Received 22 September 1976, and in revised form 29 June 1977) The quantum mechanical analog of work is defined and discussed by using a simple hypothetical molecular machine, thus enabling the introduction of clearly defined ideas which are necessary for a molecular discussion of biological

machines

such as the contractile

machinery

in striated

muscle. The problem of control of such quantum machines is discussedand shown to be possible using the concept of a stimulated transition. The problem of “reversibility” is also discussed and shown to have a satisfactory solution for the orders of magnitude of the forces and velocities involved in muscular contractile machinery.

1. Introduction All textbooks on thermodynamics are concerned with the idea of “work”, its definition, performance and calculation, and so to a certain extent overlap with the corresponding concepts of classical mechanics. On the other hand. we have been unable to find a quantum mechanical text where the word “work” appears in the index at all, and it certainly appears in none with a quantum-chemical leaning, although it was discussed briefly by Feynman (1939). This is an unfortunate state of affairs since, particularly in biological systems, work is now being discussed at the molecular level where a quantummechanical treatment is essential, e.g. in the case of muscle contraction, Nihei, Mendelson & Botts (1974) state: “It is now generally accepted that the contraction of vertebrate skeletal muscle-the relative translation of thick (myosin) and thin (actinfcontrollers) filaments-results from repetitive j’ Correspondence: Biomolecular Science Unit and School of Chemistry, Macquarie University, North Ryde 2113, New South Wales, Australia. 1 Correspondence: Department of Pharmacy, University of Aston in Birmingham, Gosta Green, Birmingham, B4 7ET, U.K. 167

168

B.

F.

GRAY

AND

I.

GONDA

mechanical impulses that projections or ‘crossbridges’ on thick filaments deliver to thin filaments. Accordingly research is now focussed on the general outline of how a single impeller works”. We might add that an impeller is a part of a single (albeit large) molecule. Similarly in active transport the various “pumps” transport ions from regions of low to high concentration, the opposite direction to that expected on the basis of a purely dissipative process such as diffusion. The phenomenological approach of thermodynamics allows the feasibility of such processes provided they are “coupled” with a process which may take place spontaneously, i.e. with a decrease in Gibbs free energy, and this is often the hydrolysis of ATP. However, thermodynamics, from its very nature, can tell us nothing about two very interesting questions which immediately arise: (i) What is the nature of the “coupling”?, and (ii) What determines the rates of these processes? With regard to (i) saying that two processes are “coupled” in no way throws light on, for example, how an exothermic chemical reaction can “drive” an endothermic reaction in a constant temperature medium. This coupling is easy if we allow the temperature to rise as a result of the exothermic reaction, thus supplying the activation energy of the endothermic reaction via the usual energy relaxation channels, but this would be disastrous in a biological system, therefore energy must be transferred directly without becoming thermalized as a Boltzmann distribution among the translational energy levels of the system. With regard to (ii) it is well known that classical thermodynamics has absolutely nothing to say about the rates of processes, either purely chemical or mechanochemical or electrochemical. If we require information about rates, we start to study kinetics, either phenomenologically or at the molecular level or preferably both. It might be objected that “irreversible thermodynamics” does have something to say about rates, and this is true up to a point sufficiently close to an equilibrium situation, but even here it is being wise after the event. For example the “flow and force” treatment of a chemical reaction (Prigogine, 1967) rewrites the kinetic expression for reaction rate in terms of affinity rather than concentrations, useful though this may be in exhibiting features common to chemical, thermal and mechanical rate processes. A muscle in the relaxed steady state is extremely far from equilibrium in every sense, e.g. a nerve impulse of some 10e7 J initiates a twitch releasing some lo-’ J g-’ in the muscle. It is on the brink of very rapid transition to an entirely different steady state, e.g. isometric or isotonic contraction, and it is certain that neither of the latter steady states is on the equilibrium or thermodynamic branch for muscle, in fact it is rather unlikely that the relaxed living state itself is on this branch, as even the latter appears to be more sensibly

MUSCLE

CONTRACTION.

I

169

regarded as a dissipative structure beyond an instability of the equilibrium branch, but is close to a further instability stimulatable by the nerve impulse. Such dissipative structures (often characterized by co-operative behaviour) do not obey the “laws” of linear irreversible thermodynamics (Nicolis & Prigonine, 1974) and the conditions under which they become unstable depend on the intensities of the external constraints, diffusion coefficients and other non-thermodynamic properties. For such reasons we feel that treatments of the muscle contraction problem such as those of Caplan (1966) and Hill (1974) are inappropriate. Living systems are unique in that they contain devices of molecular dimensions which can produce mechanical work. It is highly unfortunate that in the biological area the word “work” immediately evokes the reflex response “thermodynamics”. Work performed by a muscle is essentially a mechanical entity (McClare, 1971), and if performed at the molecular level it must be treated as a quantum mechanical entity. We do not, however, regard the impotence of classical thermodynamics in this area as legitimate grounds for its criticism as a discipline. but nevertheless wish to stress its inability with respect to the study of rates and mechanisms of molecular energy couplings. Clearly then if we wish to obtain information on the molecular dynamics of work-performing processes, we must study the definition and production of work in quantum mechanics. Theoretical chemistry, the area of quantum mechanics which one would expect to influence our considerations more than any other, is largely concerned with isolated systems (a similar concept to a thermodynamically isolated system) where both energy and matter are constrained by boundary conditions applied on a fixed surface definining the limit of the system under consideration. This then leads to the concept of discrete energy levels and stationary states arise, but, not surprisingly, the concept of work is clearly absent whenever the “boundaries” of the system remain fixed. By “boundary” we mean any point in space where a boundar) condition is applied, and it could be “internal” in the case of a molecule whcrc we consider the motion of electrons and apply boundary conditions lo the electronic wave function at the nuclei, as well as to other places possibly. The notion of work involves the movement of the boundary of a system (in the generalized sense used above) against an external force applied at the boundary. This implies the existence of an internally generated force: at the nuclei. The Hellmann-Feynman theorem and its special case the Electrostatic Theorem (Levine, 1970) have useful statements to make about the forces on nuclei within a molecule and in the next two sections we present a very simple example to illustrate these ideas before proceeding to a more realistic quantum mechanical model for muscle contraction.

170

B.

F.

GRAY

AND

1.

GONDA

2. The Quantum Mechanical “Particle in a Box Machine”

(P.I.B.M.)

The one-dimensional particle in a box model with fixed impenetrable walls is a much used illustrative example in quantum chemistry for the application of boundary conditions and the appearance of a discrete energy level spectrum as a result. If we consider a box of length Z, with impenetrable walls, the Schriidinger equation for the system is (in atomic units, taking the mass of the particle as unity):

(1)

HO> = w = 0,

(2)

where the boundary condition (2) shows that the walls are placed at x = 0 and x = I, respectively. The energy levels and wave functions of this system are : E,, = ~n27C2 212 ’

n = 1,2,. . .

(3)

and the ground state is 12= 1. Defining force as we would classically, i.e.

then the quantum mechanical expectation value of the force in stationary state n is: (F,), = s Wx$, dx (4) and this is easily shown to vanish. However at the two walls of the box, equal and opposite forces are found to exist, i.e. at x = 1,

and at the other wall at x = 0 VA

n27c2 = - 13)

(6)

so there is no net force acting on the centre of mass of the system. Nevertheless there is an outward force on the walls of the box, and as the latter are assumed stationary, even after transitions from state n to state n’ (when the internal force as a rule changes abruptly), we are forced to either assume that (a) the walls are infinitely heavy, or (b) the internal force is opposed by an equal and opposite external force applied to each wall. (a) does not lead to any new useful conclusions and is not physically realistic, but (b) leads us to the idea

MUSCLE

CONTRACTION.

I

171

of a molecular machine. Suppose the particle in the box is initially in the ground state (n = 1) represented by a wave function with no nodes. The force on each wall will be numerically n’/Ii where l,, is the initial length of the box. These forces (akin to the pressure of a gas) will be balanced by an equal and opposite constant external force if the system is initially in equilibrium, i.e. F,, = rt ~~11;.

(7)

If we induce a transition in the particle-in-the-box system to an excited state (the exact mechanism is not important here) with quantum number n, then the internal force will instantaneously increase to nzz2/I~, but the restoring

[I

FIG.

5(J1:‘.

1.

force remains constant. This system will no longer be in mechanical equilibrium (point B in Fig. 1) and the walls will experience a net force outwards of [(n2rr2/@ - F,,] which could be used to perform work if coupled to a load at the right time. For the time being we will assume that this is possible, but this is an important assumption. The motion increasing the size of the box will tend to decrease the force since 1 > I, and eventually a new equilibrium position is found, depending upon n, the level of excitation produced and the external constraint F. Provided F+F,, is less than the maximum force produced after excitation, useful work will be performed until a new equilibrium is attained when: n27t2 --jr= l

F+F.,=F+lf:.

(8) 0

172

B.

F.

GRAY

AND

I.

GONDA

We define the work done in this process W by: W = f F dx,

(9) lo in general, but if the load is constant, e.g. raising a weight, we can write [cf. equation (S)]: W,, = F(l,-I,)

= (7

- ;;)

(1,-l,).

The energy originally supplied to the machine AE,,-the -is given by:

(10)

source of this work

AE,, = gZ (n’-- l),

(11)

0

so an obvious definition of efficiency for our machine working quantum levels 1 and n is

between

(13) where r = lo/l, < 1. In general r, the “compression ratio” will be a function of both n and F. If we regard n as a constant, which would be the case if a fixed quantity of energy were available for resonant excitation of the machine, then we can find an optimum compression ratio r which will maximize the efficiency for a given pair of quantum levels. By differentiating equation (13) with respect to r and setting the derivative equal to zero we obtain the optimum ratio as a solution of: 1+ n2(2r;ta, - 3r&J

= 0

(14) and for n = 2, rmax N 0.82 giving czl = 0.224. tnl increases with n, tending to a limit of 0.29 as n becomes very large. For n = 2, the total energy of the machine at t = 0 after excitation, but before performing work, is 27r2/1$ After the configuration change to the new equilibrium position, the internal energy of the machine is 2n2/1,2, i.e. 1*345n2/1i at this state. The work done against the external force in the configuration change is given by equation (10) and is O~336rc2/Z$The work performed against the restoring force, Feq is easily calculated as O*219%r2/1& giving in total 1 *9005n2/Zg compared with the original total energy immediately after excitation of 2x2/1$ We have thus O*1rc2/1t or 5 % of the total energy unaccounted for; what has happened to this energy? It is easily traced by considering the situation at t = 0, just

MUSCLE

CON

rRACTI0I-i.

I

173

as the configuration change commences. The barrier experiences a large net force of 47?/1: - F- F,, = 3rc2/1g- F,, and this will cause acceleration of the barrier giving it kinetic energy. For the moment we will assume that this is dissipated eventually as heat and so we are correct in not including it in the efficiently used energy. If we complete the cycle by dropping down from level 2 to level 1 at f, (C -+ D in Fig. 1) we lose a quantum of energy of magnitude 1 *O097c2/1& compared with the input quantum of 1 *57c2/14$We assume also that on deexcitation the machine is uncoupled from the load, completing its recovery autonomously. Conditions for the validity of these assumptions about coupling and uncoupling will be discussed later. In the ground state at I, (point D) the force generated by the system is less than Feq the restoring force, which therefore moves the system back to its original configuration at /,, using some of the energy imparted to it during excitation. The low value of the efficiency, 22x, is not important here for an illustrative machine of this type. The inefficiency is not due to “irreversibility” it is due to the large size of the quantum of energy emitted by the machine moving along CD in Fig. 1. We can devise a machine where the paths BC and AD cross, i.e. au accidental degeneracy occurs. This is easily contrived with a more realistic three-dimensional particle in a box with only one side movable, e.g. a piston in a cylinder. This gives actual efficiencies in the SO-90% region if the external force is such that Z,is near the degenerate conformation. There is then no necessity for an emitted quantum, instead a radiationless transition can take place. However in a case of this type, the lower energy state of the cycle cannot be the ground state of the system if it becomes strictly degenerate due to Kramer’s Theorem. For a finite barrier mass the efficiency would be less than calculated above due to purely quantum effects arising from perturbation of the excited state by the movement of the barrier but this effect can usually be neglected.

3. Conditions for Adiabacityt

The relationship between the motion of the barrier in this case and the particle in the box is completely analogous to the electron-nucleus relationship in the case of molecules, and is similarly amenable to treatment by the Born-Oppenheimer approximation. In zeroth approximation (the fixed nucleus approximation) the motion of the barrier is ignored. In the first approximation the motion of the barrier (calculated on the basis of the zeroth approximation wave function) is treated as a perturbation causing t May be omitted by non-mathematical

readers.

174

B.

F.

GRAY

AND

I.

GONI)A

transitions of the particle between its energy levels. Fortunately this perturbation is so small in realistic cases that it can be ignored and the quantum number n can be treated as an adiabatic invariant, i.e. the system remains in the same quantum state as the barrier performs its work. On a quantitative basis the Adiabatic Theorem tells us that the quantum number will not change during the motion provided that the change in the Hamiltonian for the system during the Bohr period of the system is small compared to the spacing of the energy levels of the system. The Bohr period is given by: z = l/AE,,, (15) so the condition becomes:

in general, and for our particular problem this inequality gives: dl < (n%r~‘)~ dt -_ -nm~~~~~Iz. (17) I I The larger the mass of the barrier M, the smaller we would expect IdZ/dtl to be for a given amount of kinetic energy transfer, so the larger the ratio of the masses of the barrier and particle the more likely that inequality (17) will be satisfied. This is precisely the reason why the electronic and nuclear motions can be factorized in the Born-Oppenheimer separation where the large ratio of nuclear to electronic mass (> 103) makes the adiabatic approximation a good starting point. Assuming the fundamental biological quantum of energy to be -40 kJ mol-’ (the free energy of hydrolysis of ATP) and as this is almost universally agreed to be the source of mechanical work in muscle it is of interest to calculate the Bohr’s period for a quantum of this size and compare it with the time period over which the work in a single crossbridge cycle is performed. The Bohr period is N lo-l4 s and if the work is performed in 10e3 s then for this case (irrespective of the detailed mechanism of how the force is generated) inequality (16) becomes: lo-”

4 19 (18) so clearly non-adiabaticity is no problem at all in biological systems where numbers of this order are almost always involved. We shall henceforth assume that the quantum numbers describing the state of the machine are invariant during the work performing stroke. It has been conjectured (Kushmerick & Davies, 1968; McClare, 1972a) that biological molecular machines can approach 100% efficiency, on the grounds of evolution on the one hand and on the grounds that they are not

MUSCLE

COYTRACTJON.

I

115

conventional thermodynamic machines with speed-limited efficiency on the other. The simplest possible type of molecular machine working in a cycle such as is depicted in Fig. 1 shows that this conjecture is not necessarily true simply because the machine cycles at the molecular level, even if we assume it does so reversibly and this is not feasible as has been shown by Gray (1975) due to the cost of control. On the other hand we might expect that specialized molecular machines would have evolved which would certainly have superior efficiency to our simple example. The point must be made however, that there are purely quanta1 sources of inefficiency in the conversion of excitation energy into work. It does not seem to have been pointed out previously that this is so and the opposite has sometimes been assumed. The laws of quantum theory do not permit us to convert energy into work with loo:/, efficiency either because of the cost of the control required fol reversible operation or because of the kinetic energy imparted to the “barrier” in irreversible operation. Inefficiency due to induced transitions from the upper (work performing) state caused by the motion of the barrier can be ignored for all but the most weakly quantized systems.

4. A Two-box Machine

Before pursuing the question of efficiency further in the abstract, let us consider a slightly more complex machine consisting of two particles in boxes with a common movable barrier, to which the external load is coupled (see Fig. 2). The two boxes in general need not be identical. If we excite the left-hand box to say n = 2 we will make the transition I + II in Fig. 2 and this will be almost instantaneous on our time scale. There will now be a net force to the right on the barrier, which will therefore move to a new equilibrium position (II + III) performing work on the external load in the process. De-excitation of the left-hand side whilst in its new equilibrium position will result in the instantaneous occurrence of a net force on the barrier to the left, resulting in the transition IV -+ I and completion of the cycle. Although the barrier is moving in the opposite direction in the stroke IV -+ I to that in the stroke II -+ III it is still possible to imagine a simple way of coupling to the external load so that positive work is done in both strokes, e.g. by using an eccentric coupling, thus increasing the efficiency. The increase in complexity of the model enables us to use the right-hand box (which is never excited in the cycle) as a kind of “flywheel” since during the cycle energy is stored in it, then used to return the machine to its original state I, performing work in the process. The two-box machine can be operated cyclically, i.e. returned to its original condition after performing external

176

FIG.

2.

work (IV -+ I). The second box is in fact a representation of the restoring force Feg in the first model. This interpretation would increase the efficiency considerably since the step DA in Fig. 1 would also be useful. It is interesting to speculate on signi&ance of this sort for the two-headed myosin fragments. The above discussion can be generalized in a number of ways. Mechanically the particle in a box system can be replaced by a system with any potential function in its Hamiltonian, and provided the latter depends on certain parameters (which take the place of the barrier position), work and force can again be defined and cycles traversed, provided we now interpret the figure symbolically. The above discussion is purely for illustrative purposes using the simplest possible quantum mechanical system. 5. Control -

Coupling and Uncoupling to Load

It is convenient to pursue the simple P.I.B.M. further in order to illustrate the problem of coupling and uncoupling the load at the right time in order to extract net work from a cycle. In considering a quantum molecular machine of this type operating as in Fig. 1, two questions present themselves.

MUSCLE

CONTRACTION.

I

177

(i) How can we justify neglecting the reverse transitions such as B --) A when we know that the quantum mechanical transition probability between two states, [j$,u$g dz]’ where v is the perturbation causing the transition, is a symmetrical function with respect to initial and final states ? (ii) How can we ensure that the load is coupled to the machine at just the right time after excitation in order that the excitation energy produces work and is not dissipated uselessly ? Whereas control of a macroscopic machine offers no problems whatsoever, it has been shown by Gray (1975) that for machines sufficiently small control can be self defeating as far as efficiency is concerned. This is simply because a measurement must be made to reveal the state of the machine before the load is coupled or uncoupled. For example with a macroscopic piston and cylinder undergoing a Carnot cycle one needs to know when the piston is approaching the crossing points between the adiabatic and the isothermal in order to change the external pressure at the right time. For such a machine the energy cost of such an observation is trivial and completely negligible. hut for a machine of macromolecular dimensions or smaller this energy can be several times greater than the energy released by the hydrolysis of ;I molecule of ATP and we shall assume that this is the order of magnitude of the excitation AB. Direct control of this sort is then clearly uneconomic and there does not seem to be any experimental evidence for its existence in biological systems. This question arises in the currently accepted versions of muscle contraction theory based on the original formulation of Huxley (1957) where crossbridges bond, pull, detach and return to start another cycle. In postulating rate constants for attachment and detachment which vary with configuration in a way which successfully reproduces some of the steady state experimental data this question has been avoided. To the theoretical chemist this question is of great interest and importance and our work here and following can be seen as a natural development from phenomenological to truly molecular theories. At this point a peculiarly quantum mechanical concept comes to our aid, the idea of a stimulated or induced transition. Qualitative models of muscle contraction using similar concepts have been proposed previously by McClare (1972b) and Shimizu (1972). The whole basis of quantum-mechanical discussion of rate processes of whatever kind rests on the idea of transitions between states of the system caused by a “perturbation” of some sort. This perturbation can be within the system itself, perhaps a collision between two molecules, or an externally applied one such as the field of a light wave. It offers the possibility of synchronization without direct control. For instance it is possible to postulate that a transition into the “pushing” state of our II , .!I.

178

B.

F.

GRAY

AND

I.

GONDA

P.I.B.M. occurs as the result of perturbation by the load, i.e. a stimulated transition (Gray & Gonda, 1975). This transition can itself be uncoupled from the excitation process AB with considerable advantages by having two excited states as in Fig. 3, excitation occurring (reversibly) between A and B, the excited “pushing” state being labelled B*, with the force exertable being proportional to the slope of the energy curve. The stimulated transition would now occur at the crossover point X and would be of the type referred to as “radiationless” since at X the energies of the two states are equal and no energy needs to be emitted or absorbed in the form of a quantum for conservation. The probabilities of such transitions at crossover

Plslonce

FIG.

3.

points have been discussed extensively in connection with spectroscopy and photochemistry, and the simplest formula, which is suitable for our purposes takes the form: 2n P B-t,,* = y c2sp,

(19)

where C is an electronic coupling factor between the two states which is fixed in this discussion by the states themselves. S is the square of the socalled Franck-Condon overlap integral which is taken between the vibrational wave functions in the two states and p is the density of vibrational states in the final electronic state B*. p will be discussed in connection with irreversibility later, here we are interested in S and ways in which it (and hence P B+g*) can be drastically increased when the machine is “loaded”. This will happen when the vibrational wave function of the system in state B is made

MUSCLE

CONTRACTION.

1

179

larger in a region where the vibrational wave function in state B* is already large (or vice versa). One way (not by any means the only one) in which this could occur is when an obstruction is placed in the path of the oscillator in state B (the barrier in our P.I.B.M.) forcing it to spend more of its time in the crossover region X, thus increasing the vibrational wave function in this region (strictly speaking it would increase the positional probability density). This is equivalent to changing the shape of the potential in state B to something like that shown in Fig. 4. The obstruction (which could be another

I D1sionce

Fw.4.

molecule coupled directly to the load) would then increase the transition rate from B + B*, which is the pushing state. In the presence of the “barrier” the potential energy curve for state B will now be YVZ and that for the state B* will be UWZ. The oscillator in state B will now be much more likely to be found near the crossover region X, thus increasing the overlap integral and therefore the transition probability PB+B*. It is important to form a clear physical picture of the meaning of Fig. 4. The distance co-ordinate is a measure of the position of the body whose motion is being considered (e.g. the particle itself in the P.I.B.M. or in a specific muscle model discussed in detail in section 2, the swivelling head of a crossbridge). The “barrier” position ZW represents the (movable) wall of the box in the P.I.B.M. and a part of the thin filament in the following muscle model. This is here being regarded as a parameter, i.e. it varies only “slowly” in some way compared with the body itself. Its position determines part of the potential in which the body moves, e.g. UWZ. However, when work is performed in the system this “barrier” moves, i.e. the motion of the body reacts on the potential and change.s it for example to positions Z’W’,

180

B.

F.

GRAY

AND

1.

GONDA

etc. This means that the solution of the appropriate equation of motion (the Schriidinger equation in this case) will not represent a conservative system. If the body modifies the potential in which it moves only “slowly” in some sense then it can be permissible to treat the barrier position as a parameter, simply solving for the motion of the body for different fixed parameter values. This is common practice in separating the motion of electrons from that of nuclei in molecular quantum mechanics and rests largely on the disparity in kinetic energy of the two sets of particles. There a given fixed arrangement of nuclei provides a field which enables the motion of the electrons to be calculated. This calculation is performed for a number of configurations of the nuclei and then in turn the variation of the electronic energy with nuclear configuration enable us to calculate the forces on (and hence the motions of) the nuclei. We can justify a similar procedure for calculation of the motion of our body and the “barrier”, provided the “barrier” is much heavier than the body. Thinking classically this is obvious since if the “barrier” were of comparable mass to the body, it would be knocked out of the way, receiving a large transfer of momentum. The quantum mechanical treatment of this is given in the next section.

Z”W”,

6. Separation of Motion of “Particle”

and “Barrier”t

The Schrodinger equation describing the motion of the particle in a box and the barrier, with a constant external force acting on the barrier is:

i a2 _---& $ + FL ti(x, 0 = W(x, 0, 2 a2 >

(20)

where M is the ratio of the barrier mass to the particle mass ( $1). Following the Born-Oppenheimer procedure we define a complete set of particle wave functions depending parametrically on the position of the barrier:

i.e. the “instantaneous” eigenfunctions for a fixed barrier position and then try for a solution of equation (20) in the form: (22)

We find that the H’s have to satisfy:

I ;a2

{-

---+Fl+$ 2M al2

t May be omitted by non-mathematical

H;(l) = E:H;(l), readers.

(23)

MUSCLE

CONTRACTION.

1

1x1

i.e. the barrier moves in an effective potential Ff+n2n2/212 or in general for any external applied force it feels an effective potential Fl+r~~71~/21~. This depends on n, the quantum number describing the motion of the particle of course and is completely parallel to the potential energy curve of a molecule regarded as a function of nuclear position. This curve, basic in physical chemistry, also depends on a quantum number, the electronic one, and a different curve can be drawn for each electronic state. Of course the wave function pm is not the exact solution to the problem, which can only be written in terms of the complete set of functions: I&,

1) = f Of(x)HK(i). (24) K=1 In the separation of electronic from nuclear motion the necessary conditions for the validity of the approximation of taking only the first term in the series are almost always satisfied, and for our particular example the single term wave function is a good approximation provided M B 3. The potential curve determining the motion of the barrier is then given by

y(“)(l) = F 1+ n2nz 2E2 .

(7.5)

Near its minimum (the equilibrium position of the barrier) this potential can be adequately fitted to a harmonic potential, simply by expanding V”(r) as a Taylor’s series and then neglecting cubic and higher terms. We obtain :

where l$i”, the position of the minimum is given by:

The effective force constant (coefficient of the quadratic term) is then seen to be:

Using standard formulae we can now obtain the oscillation frequency of the barrier with mass M’ in terms of the force constant as c1(“) ,oG = 1 _ (29) 271 J M” The P.I.B.M. will be able to perform work in the positive sense if it generates

182

B.

F.

GRAY

AND

I.

GONDA

a force greater than the load, i.e. when: (n2n2/Z;) - F,, > F

(30)

where I,, is the initial value of the barrier co-ordinate, i.e. at the time of exctiation to particle state n. Since Feq = z2/Z,3,i.e. just sufficient to balance the force generated by the machine in its ground state, this inequality can be written in terms of the force constant: a(n) < 37r2(n2-----.----1)4’3 lzn2/3

37c2n2 n2 ~ , , l40 ’

(31)

or the frequency: (32)

where &(I,) is the excitation energy of the pushing excited state. It is interesting at this point to substitute some numbers into this formula. If we take the excitation energy E,(Zo) as 40 kJ mol-‘, lo N O-1 nm and M’ > IO3 kg mol-’ we find that Y(“) < 10” s-r. M’ is chosen here to approximate the value of the mass of muscle plus load per active crossbridge, i.e. tropomyosin-troponin units, Z-line proteins, etc. This upper limit for the frequency is several orders of magnitude less than that for electronic transitions and is in fact less than the rotational level spacings for small molecules. This upper limit of 10” s- 1 for the frequency indicates that the barrier motion can be treated classically and that its energy levels are very closely spaced, i.e. ~hv N 6.10-24 J apart (U N 4x 10e2’ J). We can derive an important conclusion from this: for each adiabatic particle state {CD@)} there are a large number of virtually continuous energy levels of the barrier. The very small separation of these virtually continuous levels is an important factor in justifying the irreversible nature of some of the transitions such as B --+ B* at the point Xin Fig. 4. This will be discussed in the next section.

7. The Irreversible Transformation of Potential Energy into Work In the above discussion, which is quite general, as an example we can regard the particle as an excited bound electron in a molecule exerting a force on the nuclei, thus causing a configuration change. Alternatively we can imagine that the particle represents a vibrationally excited part of a molecule moving in a restricted space, thus forcing other parts of the molecule, or other molecules (the barrier) to leave this space. This latter interpretation must

MUSCLE

CONTRACTION.

I

183

surely be of considerable interest in biological macromolecules where catenane (chain link type) and knotted molecules occur frequently, e.g. in enzyme molecules. In a normal chaotic non-biological medium production of a system in the excited state B* at the point X would simply result in the system moving to the minimum of the curve B* (Fig. 3) by vibrational relaxation via collision. The energy difference between X and the minimum would be converted to heat and lost. Diagrams such as Fig. 1 are common in discussion of photochemical systems, where the original excitation occurs by light absorption. For our general theory of machines at the moment we can ignore detailed discussion of how the energy packet or quantum gets into the machine and excites it, it could be simply the exchange of a spent ligand molecule (ADP) for a fresh one (ATP), the energy of the “system” (protein + ligand) increasing appropriately. We wish to discuss at this point what gives us the possibility of an irreversible transition at point X, and we will show that it is precisely the possibility of performing work in the final state B* which provides this. It is necessary first to dispose of a common misconception in the biological world, i.e. that quanta1 processes involving single molecules and quanta are necessarily “reversible”. The principle of microscopic reversibility states that the transition probability for the forward process between two states is equal to that for the backward process between the states (this is to be sharply distinguished from the principle of detailed balancing). Many processes however, involve transitions from one state to a group of states, all with similar energy, and in spite of microscopic reversibility there appears to be irreversibility since one would often use a measuring apparatus which could not distinguish between the states within the group. This irreversible decay of a state within a single molecule has been studied theoretically quite extensively in recent years by Jortner et al. (1969) Freed (1970) Lefebvre (1971) Freed & Jortner (1969), Robinson (1967), Lin (1967), Fano (1961) and Kubo (1952) and it is worth outlining very briefly here the physical background. From an elementary point of view it seems strange that a firstorder radiationless exponential decay of the probability of population of an initial state should occur in isolated molecules. Since the excitation energy is trapped in the molecule, why does the reverse process not balance eventually with the decay? The key is in the word “eventually”, and the concept of recurrence time is a key factor in the discussion. In classical statistical mechanics a similar problem arose when Poincare proved his Recurrence Theorem showing that any finite classical system eventually returns arbitrarily closely to its initial state. However, in that case the relevant recurrence times are many orders of magnitude larger than current estimates for the age 01

184

t3.

I-‘.

GRAY

AND

I.

GONIl.

the universe. The quantum mechanical recurrence time is a similar concept and for transitions from a discrete energy level into a continuum of adjacent energy levels, the recurrence time rrec is in fact infinite, of course in any finite molecule we do not have a continuum of energy levels, but sometimes very densely packed discrete ones. The recurrence time is related to the spacing of the levels by: Tret = WE where 6E is the energy level spacing. In typical photochemical experiments, where irreversible decays occur, it is easily shown that the duration of the experiments is very much less than r,,,. Of course in photochemical experiments mechanical work is not performed by the excited states, the energy is lost by relaxation, eventually finishing up as heat. In an organized spatial structure such as a muscle however, it is possible for this energy to be drawn off as work, e.g. by pushing the barrier .ZW to the right after making the transition at X from B -+ B*. As ZW moves to the right the energy level of the particle in the potential XWZ will drop, thus breaking the resonance between B and B* and making a reverse radiationless transition virtually impossible. The rate of “dropping” of the energy level will approximately equal the negative of the rate of working ri/; and the transition will be virtually irreversible if during the recurrence time the energy of the particle drops well away from the region near X, of width AE, where radiationless transitions can take place, i.e. ih,,,

> AE.

(33) AE is the “width” of the state B, related to its lifetime by the Uncertainty Principle for time-energy. For a typical muscle crossbridge I$’ N 5 x 10-r” J mol-’ s-l sufficiently far from isometric and from the order of magnitude estimate obtained earlier for the barrier level spacing we find ~~~~2: l/v 10 -lo s. Using the relation AEz 2 h, where z is the lifetime of the “prepared” excited state B we obtain the necessary condition for irreversibility as : T > 1o-6 (34 and this is calculated for a half sarcomere velocity of 5 x lo-’ cm s-r. This is a fairly long lifetime by the standards of chemical physics, but not unduly so. For example an electronic excited state which can make radiative transitions to the ground state has an average lifetime of lo-* s, whereas a similar state with forbidden radiative return to the ground state can have a lifetime well above the limit of 10m6 s. For example the U02+ ion has z - 10e4 s and the relatively simple organic species biacetyl, benzil and anisyl all have z > IO-’ s at room temperature in solution. As the rate of working drops

MUSCLE

(‘ON

I RACTION.

I

185

toward zero at the isometric point the lifetime requirement for irreversibility gets more stringent according to: z > lo-=/

ri/:

(35) and the breakdown of this inequality will cause the total rate of energy production (or ATP usage) to fall below that expected if irreversibility held throughout the whole range. As the total rate of energy production by the contractile machinery falls to a very small value at the isometric point this effect would hardly be noticed quantitatively, but the inclusion of this effect does affect the discontinuity in slope in the velocity-tension relation, and also the stretching behaviour. However, this point will be discussed in the second paper of this series where a detailed model for muscular contraction is presented and the steady state mechanical and energetic consequences following from it are deduced and shown to be in accord with a large body of experimental data. The authors would like to thank Dr L. Vitkovic for useful comments and corrections and LG. wishes to thank the Science Research Council for a maintenance grant for the period in which this work was carried out (1971-74). We should like to dedicate this paper to the memory of the late Cohn W. F. McClare whose pioneering work in Bioenergetics served as a great stimulus for us. REFERENCES CAPLAN, S. R. (1966). J. theor. Biof. 11, 63. FANO, U. (1961). Phys. Rev. 124, 1866. FEYNMAN, R. P. (1939). Phys. Rev. 56, 340. FREED, K. F. & JORTNER, J. (1969). J. them. Phys. 50, 2916. FREED, K. F. (1970). J. them. Phys. 52, I 345. GRAY, B. F. (1975). Nature 253, 436. GRAY, B. F. & GONDA, I. (1975). J. theor. Biol. 49,493. HILL, T. L. (1974). Prog. Biophs. molec. Biol. 28, 267. HUXLEY, A. F. (1957). Prog. Biophs. bioph. Chem. 7, 261. JORTNER, J., RICE, S. A. & HOCHSTRASSER, R. M. (1969). Advances

in Photochem. 7, 149.

KUBO, R. (1952). Phys. Rev. 86,929. KUSHMJZRICK, M. J. & DAMES, R. E. (1968). Proc. R. Sot. B174, 315. LEPEBVRE, R. (1971). Chem. Phys. L&t. 8,306. LEVINE, I. N. (1970). Quantum Chemistry I. Boston: Allyn & Bacon. LIN, S. H. (1967). J. them. Phys. 46,279. MCCLARE, C. W. F. (1971). J. theor. Biol. 30, 1. MCCLARE, C. W. F. (1972~). J. theor. Biol. 35, 233. MCCLARE, C. W. F. (19726). J. theor. Biol. 35, 569. NICOLIS, G. & PRIGOGINE, I. (1974). Faraday Sot. Symp. 9, 7. NIHEI, T., MENDELX)N, R. A. & BOTTS, JBAN (1974). Proc. natn. Acad. Sci. U.S.A. 71, 274. PRIGOGINE, I. (1967). Thermodynamics of Irreversible Processes. New York: lnterscience. ROBINSON, G. W. (1967). J. them. Phys. 47, 1967. SHIMIZU, H. (1972). J. Phys. Sot. Japan, 32, 1323.