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On the molecular dynamics of the chemo-mechanical conversion in muscle contraction
J. theor. Biol. (1977) 69, 415-428
On the Molecular Dynamics of the Chemo-Mechanical Conversion in Muscle Contraction K. KOMETANI AND H. SHIMIZU Division of Biophysical Dynamics, Faculty of Pharmaceutical Sciences, The University of Tokyo, Tokyo 113, Japan (Received 14 July 1976, and in revised form 14 March 1977) The molecular dynamics of energy conversion by the actomyosin system in muscle contraction is studied by comparing two different types of model on the motion of crossbridge on thin filament. The motion is associated with a transition between two stable states in Huxley and Simmons’ model while in Shimizu et al.‘s model with a transition from an unstable to a stable state. The rate of the transition, which is proportional to the velocity of shortening of muscle in steady state, is calculated by representing the motion of crossbridge by that of a Brownian particle moving on a one-dimensional linear potential. In the case of the Huxley-Simmons model the energy conversion process is essentially a thermal one and the velocity of shortening depends sharply on the number of crossbridges on muscular filament, which is proportional to the overlapping length between thin and thick filaments. On the other hand, in the case of the Shimizu model the energy conversion process is a deterministic one which means that muscle is able to shorten smoothly and that the velocity of shortening is almost independent of the overlapping length. Experimental observations by Gordon et al. are consistent with the latter model.
1. Introduction Muscle contraction is caused from mutual sliding of thick and thin filaments caused by the motive force which is generated in ATP-decomposition by crossbridges attaching to actin molecules on the thin filament. The mechanical energy of muscle contraction is obtained from the chemical energy of ATP binding to the head part of the crossbridge. However, the molecular mechanism of this chemo-mechanical
conversion
has not yet been
elucidated. In 1957, A. F. Huxley proposed an oscillator model for the molecular dynamics of the crossbridge and succeeded in deriving Hill’s equations on 415
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tension-velocity relationship and on the dependence of energy liberation from muscle upon tension (Hill, 1938). However, this model fails to account for the very early transient responses of length-step and tension-step experiments. To explain their length-step experiments, Huxley & Simmons (1971) revised the Huxley model as follows: The head part of myosin has two combining sites (M,, M2) which are respectively capable of combining reversibly with corresponding actin sites (A,, A2) on a thin filament. The affinity between these myosin and actin sites is larger for M,A, than for M, A,. The combined state M,A, will be named state I and the state M,A, state I/. The transition from I to II induces extension of the elastic element of the crossbridge and a force is exerted on the thin filament. The two states I and ZZ are stable ones with a potential barrier between them, and the motion ot the crossbridge is associated with the transition between two stable states I and Ii. To make a transition Z--f II, activation energy is required even when external load is absent not only for escaping from the state I but a&o for stretching the elastic element. Shimizu et al. (Shimizu, 1972; Shimizu & Yamada, 1975; Shimizu, Yamada, Nishiyama & Yano, 1976) have proposed a different mechanism, they suggest that the interaction between actin and myosin is unstabilized just before the motion of crossbridge starts. Qualitatively, we may denote the elementary cycle of their model in the decreasing direction of free energy as : A-M*PI
AM*P~(AM)fP~(AM)“P: 2’
0
I
M+P+A, 3’
II.4
(1)
11,
where M*P represents an intermediate complex (or complexes if the two heads of myosin work co-operatively in an elementary cycle) storing free energy (Yamada, Shimizu & Suga, 1973), and A an actin molecule (molecules). The actomyosin complex in state I is formed in process 1 and the transition from state I + II occurs thermally in process 2 where the excess energy on AM*P is transferred internally, inducing an unstable or “activated” configuration in the actomyosin complex, (AM)‘, and having a high potential energy. The motion of the crossbridge is coupled with process 3, where the state of the actomyosin changes from the substate ZZ,d to the substate II,, leading to a configuration change from (AM)’ to (AM)” which has less potential energy than (AM)’ does. II, and II, indicate two representative substates with high and low potential energy. There may exist a number of substates in state ZZ in some continuous way. It is noted that this model is essentially consistent with T. L. Hill’s model (1974), after proper modifications; for instance, his free energy in the attached states is compared with our potential energy between actin and crossbridge. Mechanical properties
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of the states Z and ZZ in Huxley-Simmons’ model are respectively different from those of the substates II, and II, of the model by Shimizu et al., e.g. Shimizu’s model. The force-generating state of Huxley-Simmons’ model. state Z, is stable with respect to the movement of the crossbridge on thin filament, while the force-generating state II, is unstable in Shimizu’s model. Barany (1967) found the existence of a close correlation between the rate of ATP splitting by myosin molecules activated by actin molecules (actomyosin ATPase) from various kinds of muscle and the maximum velocity of shortening of the muscle. In muscle the ATPase activity depends on load (Hill, 1964). In addition, the load varies the velocity of the motion of crossbridges on thin filament as is described by the characteristic equation of Hill (1938). Hence, it may be considered that the motion of the crossbridge on thin filament is one of the major rate determining steps in the actomyosin ATPase activity such that the latter is proportional to the former in steady state in the first order approximation. Now, we like to ask whether the step is associated with a transition between stable states as in HuxleySimmons’ model or with that from an unstable to a stable state as assumed in Shimizu’ model. According to Huxley-Simmon’s model the sliding motion occurs when a hypothetical particle travels in a mechanical potential from the first potential valley to the second one, going up to a potential barrier. To go up the uphill potential, energy must be supplied from outside to the particle, namely from molecular surroundings through thermal agitation. As a result, the particle is driven randomly on the potential as a Brownian particle in onedimensional co-ordinate system. If each crossbridge shows such a Brownian motion on thin filament, the random nature of the motion will influence to the sliding motion between thick and thin filaments. Particularly, when the overlap between these two kinds of filaments is so reduced that only a few actomyosin pairs can be present per thin filament, the thin filament will move randomly in the forward and backward directions, reflecting the random motion of crossbridges. It is, however, observed (Gordon, Huxley & Julian. 1966) under very light load that muscle shortens smoothly with constant velocity, which is substantially the same with the maximum velocity of shortening with the full overlap condition, even when the overlap is scarcely present between thick and thin filaments. If the sliding motion is yielded by a transition of actin and crossbridge from an unstable to a stable state (as Shimizu’s model) the particle on the downhill potential (unstable potential) may go down to the stable state in a rather mechanical and deterministic way, provided that certain conditions are satisfied for mechanical and thermal forces. That is, there is a possibility that the particle moves in the forward direction of the sliding motion without
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significant agitation by molecular surroundings. Quantitative treatments will be given in the next two sections. It will be also shown that the (excess) potential energy between actin and crossbridge in the force-generating state with unstable potential is converted directly to the mechanical energy of the crossbridge on the thin filament, because the molecular motion in an unstable potential is deterministic or mechanical rather than stochastic or thermal. On the other hand, the motion of the crossbridge becomes stochastic or thermal in the case of the stable force-generating state, leading to the conversion of the thermal energy to the mechanical energy of muscle contraction.
2. The Dependence of the Sliding Velocity on the Overlapping Length of Muscle Filaments
As discussed in the preceding section, the sliding motion between thick and thin filaments may be proportional to the transition rate from I to II for Huxley-Simmons’ model and from lIA to II, for Shimizu’s model. Such initial and final states in the both models will be tentatively named that A and B states. In muscle contraction a number of crossbridges from surrounding thick filaments, interact with a thin filament and make the thin filament slide relatively. Let the number of such crossbridges by N per thin filament. The number of crossbridges per half thick filament has been reported to be 200 by Morimoto & Harrington (1974). Haselgrove & Huxley (1973) give upper bounds of 50% of the crossbridges for attached ones. Hence, the maximum value of N is about 50 under isometric contraction. Reduction in the overlapping length between thick and thin filament decreases N. As for the Huxley-Simmon’s the motion of the ith crossbridge may be discussed in terms the ith hypothetical particle in a one-dimensional potential V (x) between the two states A and B. The equation of motion for the ith particle may be given by a Langevin equation, mdvi/dt = - myvi-dv(xi)/dxi
+Fi +fi,
i=l-N,
(2)
where nz, vi and xi refer to the mass, the velocity and the position of the ith particle, respectively; Fi denotes the interaction force between the ith particle and the other ones. The frictional coefficient y is related to the random forcefi (t) by: Cfi(t)fj(t’))
= 2myk*T6ij6(t
according to the fluctuation-dissipation
- t’),
theorem.
In equation
(3) (3) aij and
MOLECULAR
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s(t) are Kronecker’s and Dirac’s b-symbols, respectively, and ( ) denotes an average over all possible values of the random force fi(t). Quantities k, and T are Boltzmann’s constant and temperature, respectively. The co-ordinates {xi} are relative ones to a proper position on the interacting actin molecule. For the sake of simplicity we may assume a linear form : V(x) = &Kx, fior the uphill and downhill potentials, assumed among the particles; Fi = a ~
(4)
respectively. Elastic interaction
is
(Xi-Xi)
i
with force constant cc.The expression (5) is a reasonable assumption, since reactive sites are connected in series in muscle with some elastic components. From equations (2), (4) and (5), one finds m dUi/dt = -myv, T K-t-a $J (Xj-xi)+fi(t). (6) j In steady state the motion of crossbridges on a thin filament makes the filament slided with a certain velocity which is determined by the value of load. In the first order approximation, the sliding velocity may be proportional to the x-component of the mean velocity of such motions of crossbridges in the force-generating state. Taking the ensemble average of (6), one obtains mdv/dr = - myv T K +g, (7) where ZJdenotes the mean velocity and
From (3) and (8) (g(t)g(t’))
= 2myk,TG(t-
t’)/N.
(9)
We will confine ourselves to the sliding motion in steady state, which allows us to omit the inertial term from (7). Putting du/dt = 0 and u = dx/dt, one reduces equation (7) to mydx/dt = f K +g(t), (10) where x stands for the ensemble average of {Xi> with respect to N particles in the system.
The Fokker-Planck equation is derived from equation (lo), and equation (11) gives the conditional probability density P(x,lx, t), the probability
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density of finding the average co-ordinate of the system at x at t = t when the co-ordinate was present at so at t = 0.
where diffusion constant D is given as: D = k,TINmy.
(12)
Let the co-ordinates for the two states A and B be xA and xB (xA < su), respectively. The rate constant k of the transition from A to B is inversely proportional to the mean first passage time M, (B/A) of a “particle” representing the motion of the center of mass of the system travelling from X, to xB. The sliding velocity in steady state is proportional to k in the first order approximation. The time M, (BIA) and the mean square first passage time M2 (BIA) are given from equations (1 l), (A2) and (A3) as: hf,(BIA)
= T
[l f (k,T/NKL)(e*NKL’k”T-
t)]?
(13)
and,
x
e+NKL/keT
+
2(myk,T)2
e’
N2K4
ZNKL/keT
(14)
where L = xB--xA and f KL is the difference of the potential energy at xA and xP A reflecting boundary is assumed at sA in the calculation of equations (13) and (14). Under optimum conditions the chemical energy of ATP, which is about 15k,T, is converted into the mechanical energy of muscle contraction with an efficiency >50%. On the other hand, the mechanical energy results from the motion of the “particle” on the potential. Hence, it is reasonable to assume that KL is significantly larger than k,T. By putting KL s k,T, equations (13) and (14) for the case of the uphill potential t result in :
M,&E(A) N_ s:
eNKLIksT,
(15)
and 2NKLIkBT
,
(16)
MOLECULAR
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421
respectively. Hence, the variance a, around M, , becomes at = Mzt-MIT2 (17) The rate constant k, (= l/M,)
is derived from equation (IS) as: (18)
which is a sharply decreasing function of N. In the case of the downhill potential L, equations (13) and (14) lead to: (I!)) and (20) The variance and the rate constant are given as :
(21) and K k, = my(1 - k,T/NKL)’ N K/my.
(22) (The rate constant does not depend on T and N.) Hence, the motion of the Brownian particles in the downhill potential is almost “deterministic”. For the case of the flat potential-+, K = 0, equations (13) and (14) lead to : M&+4)
= g2,
(23) B
and M,,(BjA)
= %(%;;I. B The variance and the rate constant of transition are calculated as:
(24)
(25) and (26)
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FIG. 1. The dependence of the velocity of shortening on N and on the overlapping length for unstable (-•-) and stable (-O-) force-generating states. The numbers beside the lines in the figure stand for the value KL/ksT.
The rate constant k, decreases as the number of particles increases. Huxley and Simmons have estimated the height of their uphill potential as 3*8k,T under maximum external load (Huxley & Simmons, 1971), and lower values as load is reduced. It is, therefore, better to discuss a case where KL is not very different in size from k,T, particularly, for the case of the uphill potential. When KL = 3k,T, k, decreases of the order of 10 as one particle is added to the system. On the other hand, k, is almost unchanged. Namely, k, (N = oo)/k, (N = 1) N 2/3. The N-dependence of the rate constants is shown in Fig. 1 for KL = 2kBT and 3k,T. Even in the case of the flat potential, k, is a decreasing function of A? The rate constant k, scarcely depends on N only in the case of the downhill potential. 3. Fluctuations of the Motion of Crossbridgesaround the Mean Velocity
Subtracting equation (7) from equation (6), and omitting of the resulting equation, one obtains:
the inertial term
(27) w(WW = - aN& +fi - 9, where ti = xi-x. This equation has a closed form with respect to the suIi?x i, which will be omitted in the following discussion. The Fokker-
MOLECULAR
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Planck equation for the conditional
MUSCLE
probability
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423
W with respect to i is given
as:
(28) where
D’ = (N- l)k,T/myN = k,T,lmy,
(29)
the effective temperature T, = (N- 1) T/N. The mean first passage time M’, (r10) for the the change of < from < = 0 .-. r is obtained from equations (28), (A2) and (A3) as:
with
M;(rlO) = j; j dq j! exp [crN(q’-
0
i_‘2)/kBTJ d<.
0
Tlhe rate of fluctuation, k, is given by r/M, (rl0). When the force constant o! is as small as aNr’/k,T,< 1, the linear approximation of equation (30)
with respect to a leads to:
which may be rewritten [using equation (29)] as:
Let us consider a special case r = L in equation (32) and compare it with the rate of the averaged motion obtained in the preceding section. If the motion of the crossbridge is random, the expectation value for the rate of the displacement of a crossbridge by distance L becomes larger than that of the averaged motion, which means h-,- > k. The order of the magnitude of the ratio k/k, is obtained as: k-/k,
- l/N.
k, k, h, exp (- NKL/kBT)
1o-7
k,lk, -
KLIkBT 4
for KL 9 k,T for KL = 2k,T, N = 10,
for KL 3 k,T for KL = 2k,T. N = 10.
(33) (34)
(35)
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The ratio k,/k, decreases, while k,lk, increases, as KL/k,T becomes large. As N increases, k,/k, decreases sharply while k,/k, depends only weakly on N as N/(N- 1). Though we have no information about the potential energy in the force generating state, k, is much smaller than k, even KL - 0, because N is of the order of 10. Hence, in the case of the uphill potential the motion of crossbridges is of random nature and it induces no smooth shortening in muscle. On the other hand, crossbridges move rather coherently in the case of the downhill potential and muscle is able to shorten smoothly.
4. Comparison with Experiments Gordon, Huxley & Julian (1966) have shown that the velocity of shortening of muscle under a very light load is substantially independent of the overlapping length of thin and thick filaments. They studied shortening, for example, from 3.32 pm-l.95 pm striation spacing. For 3.32 pm striation spacing the overlapping length of thin and thick filaments is about one third of the full overlapping length. Julian & Sollins (1975) reported that the number of attached crossbridges is non-zero even at the maximum velocity of shortening. From their result we deduce the number of attached crossbridges under no load shortening is, tentatively, one fifth of that under isometric contraction. Hence, the number N of crossbridges interacting with a thin filament varies from 3-10 when the muscle shortens from 3.23 pm to 1.95 pm striation spacing under no load. The dependence of the velocity of shortening, which is proportional to the rate constant of the transition A -+ B, on N is given in Fig. 1. If the state A is stable, the velocity will be sharply decreased as N is increased. When N increases from 3 to 10, the velocity reduces to about 3 x 10m6 for KL = 2k,T. According to Huxley & Simmons (1971) the potential barrier is flat under no load shortening. Even in this case the veiocity at N = 10 will be about one third of that at N = 3, which is seen from equation (26). On the other hand, Fig. 1 shows that the velocity is almost independent of N if the state A is unstable. When KL $ k,T, which is plausible as written before, the velocity is independent of the overlapping length between thin and thick filaments. Observations by Gordon et al. are consistent with the result obtained in the case of the downhill potential. It means that the chemomechanical conversion in muscle contraction occurs in such a way that interaction of actin and myosin molecules is unstabilized by transferring the chemical energy of ATP and then gives rise to the sliding motion of muscle filaments.
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5. Discussions and Conclusion The sliding velocity of thin filaments is not only determined by the rate of the change A --f B. It may be a function of the overall rate of the elementary cycle. However, we are only concerned in the present paper with the dependence of the sliding velocity on the overlapping length between thick and thin filaments, which is proportional to N. The N-dependence of the sliding velocity comes only from the change A -+ B, because motions of crossbridges on a thin filament are closely correlated with each other due to the feedback interaction through the sliding motion of the thin filament (Shimizu ef al., 1976). The other processes in the elementary cycle are independent from each other and give no significant effect on the N-dependence of the sliding velocity. Therefore we may take the rate of the change A --+ .B in the force-generating state to be proportional to the velocity of shortening of muscle as long as discussions are confined to the N-dependence of the of the velocity. As is seen from equation (22), the velocity o1 is independent of temperature, provided that KL 9 k,T. However, this should not be compared directly with experimental observations, because the actual velocity of shortening is determined by the overall rate of the elementary c:ycle. Processes other than the change A -+ B are driven thermally and may give rise to the temperature-dependence of the velocity of shortening of muscle. If the two states A and B are stable ones, the transition A -+ B requires a supply of thermal energy to the motion of crossbridges. Since the strength of thermal force acting on the center of mass of the system is inversely proportional to the number of particles effectively, the rate of the co-operative change A -+ B is reduced as N increases. Hence, the velocity of shortening is reduced N increases, which is inconsistent with results observed by Gordon et al. (1966). In the case of the transition from an unstable to a stable state, the motive force of the crossbridge is given in the forward direction both by potential force and by thermal force. As discussed before the condition k,T < KL i:j realized in muscle contraction, which means the potential force is appreciably larger than the thermal one. Therefore, in the case of the downhill potential, muscle is able to shorten smoothly and the velocity of shortening is independent of N, which is consistent with experimental results by Gordon et al. This is also consistent with McClare’s picture (1971) in a sense that the motion of crossbridges is “deterministic”. Moreover, a consistency is further found with Huxley’s model (1957), because the motion of the site IM in his model changes from a stochastic to a determinstic one after binding with the site A on the thin filament. In the same spirit our scheme is con-
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sistent with Huxley’s picture (1973) which introduces two attaching states in his model of 1957. Recently, Nishiyama, Shin&u, Kometani & Chaen ( 1977) proposed a three-state model for the elementary cycle of ATP-splitting by the actomyosin molecule. They succeeded in describing consistently mechanical properties of steady shortening and lengthening processes, as well as those of isometric and isotonic transients, in a quantitative way. In their model the motion of crossbridges on thin filament, causing the sliding motion of muscle filaments, starts on a potential which is unstable with respect to the displacement. This model is consistent with our results obtained above that the forcegenerating state must be an unstable one. T. L. Hill (1974) gave general discussions about the chemo-mechanical coupling in terms of free energy. In his discussions the sliding motion is associated with the change from unstable states to stable state, which is also consistent with our three-state model. The mechanical energy for muscle contraction comes directly from the (excess) energy of the inter-molecular potential between actin and myosin unstabilized by utilizing the chemical energy of ATP splitting, beccause the motion of crossbridges proceeds in a mechanical manner. However, this is not the case for the uphill potential; the motion of crossbridges occurs thermally and the mechanical energy of muscle contraction is converted mainly from thermal energy of heat reservoir. The free energy of ATP is utilized for giving a definite direction to the motion of the crossbridge such as A -+ B. It may be compared with the role of Maxwell’s demon who is able to transform the thermal energy of heat reservoir to macroscopic work in a definite direction. These two types of energy conversion, the unstable energy conversion and the thermal energy conversion, are both possible from the principle of thermodynamics and probably may be found in biological systems. In order to develop large tension in muscle contraction, the number N of interacting sites must be large, since tension is proportional to N. On the other hand, skeletal muscles must shorten rapidly with large velocity. In general, large tension is compatible with such a large velocity only when the force-generating state A is unstable, namely, only when the energy conversion is of unstable energy conversion. (Thermal energy conversions may be found in the active transport of substances across biomembrane, partly because the reactions there may be independent and partly because no mechanical energy is not always necessary.) Our conclusion in the present paper is that the chemomechanical conversion in muscle contraction is an unstable energy conversion. Strictly speaking, the effect of the Z-band must be considered for the N-dependence
MOLECULAR
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of the velocity of shortening. such a modification.
OF
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421
is not altered by
The authors wish to thank Drs K. Nishiyama, H. Yamamoto and T. Yamada for their valuable discussions and suggestions. They acknowledge financial support from the Ministry of Education, Science and Culture of Japan and from the Kurata Research Grant. REFERENCES BA.RANY, M. (1967). J. gen. Physiol. 50, 197. GOEL, N. S. & RICHTER-DYN, N. (1974). Stochastic
Models
in Biology.
Press.
New York: Academic
GORDON, A. M., HUXLEY, A. F. & JULIAN, F. J. (1966). J. Physiol. 184, 170. HASELGROVE, J. C. & HUXLEY, H. E. (1973). J. Moiec. &of. 77, 549. HILL, A. V. (1938). Proc. R. Sot. B126, 136. HILL, A. V. (1964). Proc. R. Sot. B159, 297. HILL, T. L. (1974). Prog. Biophys. molec. Biol. 28, 267. HIJXLEY, A. F. (1957). Prog. Biophys. biophys. Chem. 7, 255. HIJXLEY, A. F. (1973). Proc. R. Sot. Land. Blt33, 83. HIJXLEY, A. F. & SIMMONS, R. M. (1971).Nature, Lond. 233, 533. JULIAN, F. J. & SOLLINS, M. R. (1975).J. gen. Physiol. 66, 287. MCCLARE, C. W. F. (1971). J. theor. Biol. 30, 1. MORIMOTO, K. & HARRINGTON, W. F. (1974). J. mol. Biol. 83, 83. N:ISHIYAMA, K., SHIMIZU, H., KOMETANI, K. & CHAEN, S. (1977).Biochim. biophys. Acto. 460, 523. SI-IIMIZU, H. (1972). J. Phys. Sot. Japn. 32, 1323. SI-IIMIZU, H. & YAMADA, T. (1975). J. theor. Biol. 49, 89. SIIIMIZ~, H., YAMADA, T., NISHIYAMA, K. & YANO, M. (1976). J. theor. Biol. 63, 189. YWADA. T., SHIMIZU, H. & SUGA, H. (1973).Biochim. bi0ph.w. Acta. 305, 642.
APPENDIX Let us consider a Brownian particle moving in a domain (x,, xa) following to a Fokker-Planck equation:
ap - =-- a a(x)P at ax
+; a;2 b(x)P.
The time, the first passagetime, which is needed for a Brownian particle found initially at y to reach the position z such that (xA 5 y < z I xB) will be considered here. The first passage time is a random variable, because the motion of the particle is essentially of randomness. By assuminga reflecting boundary at xA, the mean first passage time M, (z/y) and the mean square first passagetime M, (zl~~)are given as follows (Goel & Richter-Dyn, 1074).
MI(+) = 2 iY dvh)
j Cb(5)n(5)1-1 dt, XA
(A2)
428
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and &(zf~)
= 4 j dv(v) Y
with
j ~,@XWM)l-
’ d5,
x.4
4y1) = exp - j! WW43
.
d5 I
(A3)
644)
On the Molecular Dynamics of the Chemo-Mechanical Conversion in Muscle Contraction K. KOMETANI AND H. SHIMIZU Division of Biophysical Dynamics, Faculty of Pharmaceutical Sciences, The University of Tokyo, Tokyo 113, Japan (Received 14 July 1976, and in revised form 14 March 1977) The molecular dynamics of energy conversion by the actomyosin system in muscle contraction is studied by comparing two different types of model on the motion of crossbridge on thin filament. The motion is associated with a transition between two stable states in Huxley and Simmons’ model while in Shimizu et al.‘s model with a transition from an unstable to a stable state. The rate of the transition, which is proportional to the velocity of shortening of muscle in steady state, is calculated by representing the motion of crossbridge by that of a Brownian particle moving on a one-dimensional linear potential. In the case of the Huxley-Simmons model the energy conversion process is essentially a thermal one and the velocity of shortening depends sharply on the number of crossbridges on muscular filament, which is proportional to the overlapping length between thin and thick filaments. On the other hand, in the case of the Shimizu model the energy conversion process is a deterministic one which means that muscle is able to shorten smoothly and that the velocity of shortening is almost independent of the overlapping length. Experimental observations by Gordon et al. are consistent with the latter model.
1. Introduction Muscle contraction is caused from mutual sliding of thick and thin filaments caused by the motive force which is generated in ATP-decomposition by crossbridges attaching to actin molecules on the thin filament. The mechanical energy of muscle contraction is obtained from the chemical energy of ATP binding to the head part of the crossbridge. However, the molecular mechanism of this chemo-mechanical
conversion
has not yet been
elucidated. In 1957, A. F. Huxley proposed an oscillator model for the molecular dynamics of the crossbridge and succeeded in deriving Hill’s equations on 415
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tension-velocity relationship and on the dependence of energy liberation from muscle upon tension (Hill, 1938). However, this model fails to account for the very early transient responses of length-step and tension-step experiments. To explain their length-step experiments, Huxley & Simmons (1971) revised the Huxley model as follows: The head part of myosin has two combining sites (M,, M2) which are respectively capable of combining reversibly with corresponding actin sites (A,, A2) on a thin filament. The affinity between these myosin and actin sites is larger for M,A, than for M, A,. The combined state M,A, will be named state I and the state M,A, state I/. The transition from I to II induces extension of the elastic element of the crossbridge and a force is exerted on the thin filament. The two states I and ZZ are stable ones with a potential barrier between them, and the motion ot the crossbridge is associated with the transition between two stable states I and Ii. To make a transition Z--f II, activation energy is required even when external load is absent not only for escaping from the state I but a&o for stretching the elastic element. Shimizu et al. (Shimizu, 1972; Shimizu & Yamada, 1975; Shimizu, Yamada, Nishiyama & Yano, 1976) have proposed a different mechanism, they suggest that the interaction between actin and myosin is unstabilized just before the motion of crossbridge starts. Qualitatively, we may denote the elementary cycle of their model in the decreasing direction of free energy as : A-M*PI
AM*P~(AM)fP~(AM)“P: 2’
0
I
M+P+A, 3’
II.4
(1)
11,
where M*P represents an intermediate complex (or complexes if the two heads of myosin work co-operatively in an elementary cycle) storing free energy (Yamada, Shimizu & Suga, 1973), and A an actin molecule (molecules). The actomyosin complex in state I is formed in process 1 and the transition from state I + II occurs thermally in process 2 where the excess energy on AM*P is transferred internally, inducing an unstable or “activated” configuration in the actomyosin complex, (AM)‘, and having a high potential energy. The motion of the crossbridge is coupled with process 3, where the state of the actomyosin changes from the substate ZZ,d to the substate II,, leading to a configuration change from (AM)’ to (AM)” which has less potential energy than (AM)’ does. II, and II, indicate two representative substates with high and low potential energy. There may exist a number of substates in state ZZ in some continuous way. It is noted that this model is essentially consistent with T. L. Hill’s model (1974), after proper modifications; for instance, his free energy in the attached states is compared with our potential energy between actin and crossbridge. Mechanical properties
MOLECULAR
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of the states Z and ZZ in Huxley-Simmons’ model are respectively different from those of the substates II, and II, of the model by Shimizu et al., e.g. Shimizu’s model. The force-generating state of Huxley-Simmons’ model. state Z, is stable with respect to the movement of the crossbridge on thin filament, while the force-generating state II, is unstable in Shimizu’s model. Barany (1967) found the existence of a close correlation between the rate of ATP splitting by myosin molecules activated by actin molecules (actomyosin ATPase) from various kinds of muscle and the maximum velocity of shortening of the muscle. In muscle the ATPase activity depends on load (Hill, 1964). In addition, the load varies the velocity of the motion of crossbridges on thin filament as is described by the characteristic equation of Hill (1938). Hence, it may be considered that the motion of the crossbridge on thin filament is one of the major rate determining steps in the actomyosin ATPase activity such that the latter is proportional to the former in steady state in the first order approximation. Now, we like to ask whether the step is associated with a transition between stable states as in HuxleySimmons’ model or with that from an unstable to a stable state as assumed in Shimizu’ model. According to Huxley-Simmon’s model the sliding motion occurs when a hypothetical particle travels in a mechanical potential from the first potential valley to the second one, going up to a potential barrier. To go up the uphill potential, energy must be supplied from outside to the particle, namely from molecular surroundings through thermal agitation. As a result, the particle is driven randomly on the potential as a Brownian particle in onedimensional co-ordinate system. If each crossbridge shows such a Brownian motion on thin filament, the random nature of the motion will influence to the sliding motion between thick and thin filaments. Particularly, when the overlap between these two kinds of filaments is so reduced that only a few actomyosin pairs can be present per thin filament, the thin filament will move randomly in the forward and backward directions, reflecting the random motion of crossbridges. It is, however, observed (Gordon, Huxley & Julian. 1966) under very light load that muscle shortens smoothly with constant velocity, which is substantially the same with the maximum velocity of shortening with the full overlap condition, even when the overlap is scarcely present between thick and thin filaments. If the sliding motion is yielded by a transition of actin and crossbridge from an unstable to a stable state (as Shimizu’s model) the particle on the downhill potential (unstable potential) may go down to the stable state in a rather mechanical and deterministic way, provided that certain conditions are satisfied for mechanical and thermal forces. That is, there is a possibility that the particle moves in the forward direction of the sliding motion without
418
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significant agitation by molecular surroundings. Quantitative treatments will be given in the next two sections. It will be also shown that the (excess) potential energy between actin and crossbridge in the force-generating state with unstable potential is converted directly to the mechanical energy of the crossbridge on the thin filament, because the molecular motion in an unstable potential is deterministic or mechanical rather than stochastic or thermal. On the other hand, the motion of the crossbridge becomes stochastic or thermal in the case of the stable force-generating state, leading to the conversion of the thermal energy to the mechanical energy of muscle contraction.
2. The Dependence of the Sliding Velocity on the Overlapping Length of Muscle Filaments
As discussed in the preceding section, the sliding motion between thick and thin filaments may be proportional to the transition rate from I to II for Huxley-Simmons’ model and from lIA to II, for Shimizu’s model. Such initial and final states in the both models will be tentatively named that A and B states. In muscle contraction a number of crossbridges from surrounding thick filaments, interact with a thin filament and make the thin filament slide relatively. Let the number of such crossbridges by N per thin filament. The number of crossbridges per half thick filament has been reported to be 200 by Morimoto & Harrington (1974). Haselgrove & Huxley (1973) give upper bounds of 50% of the crossbridges for attached ones. Hence, the maximum value of N is about 50 under isometric contraction. Reduction in the overlapping length between thick and thin filament decreases N. As for the Huxley-Simmon’s the motion of the ith crossbridge may be discussed in terms the ith hypothetical particle in a one-dimensional potential V (x) between the two states A and B. The equation of motion for the ith particle may be given by a Langevin equation, mdvi/dt = - myvi-dv(xi)/dxi
+Fi +fi,
i=l-N,
(2)
where nz, vi and xi refer to the mass, the velocity and the position of the ith particle, respectively; Fi denotes the interaction force between the ith particle and the other ones. The frictional coefficient y is related to the random forcefi (t) by: Cfi(t)fj(t’))
= 2myk*T6ij6(t
according to the fluctuation-dissipation
- t’),
theorem.
In equation
(3) (3) aij and
MOLECULAR
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419
s(t) are Kronecker’s and Dirac’s b-symbols, respectively, and ( ) denotes an average over all possible values of the random force fi(t). Quantities k, and T are Boltzmann’s constant and temperature, respectively. The co-ordinates {xi} are relative ones to a proper position on the interacting actin molecule. For the sake of simplicity we may assume a linear form : V(x) = &Kx, fior the uphill and downhill potentials, assumed among the particles; Fi = a ~
(4)
respectively. Elastic interaction
is
(Xi-Xi)
i
with force constant cc.The expression (5) is a reasonable assumption, since reactive sites are connected in series in muscle with some elastic components. From equations (2), (4) and (5), one finds m dUi/dt = -myv, T K-t-a $J (Xj-xi)+fi(t). (6) j In steady state the motion of crossbridges on a thin filament makes the filament slided with a certain velocity which is determined by the value of load. In the first order approximation, the sliding velocity may be proportional to the x-component of the mean velocity of such motions of crossbridges in the force-generating state. Taking the ensemble average of (6), one obtains mdv/dr = - myv T K +g, (7) where ZJdenotes the mean velocity and
From (3) and (8) (g(t)g(t’))
= 2myk,TG(t-
t’)/N.
(9)
We will confine ourselves to the sliding motion in steady state, which allows us to omit the inertial term from (7). Putting du/dt = 0 and u = dx/dt, one reduces equation (7) to mydx/dt = f K +g(t), (10) where x stands for the ensemble average of {Xi> with respect to N particles in the system.
The Fokker-Planck equation is derived from equation (lo), and equation (11) gives the conditional probability density P(x,lx, t), the probability
420
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density of finding the average co-ordinate of the system at x at t = t when the co-ordinate was present at so at t = 0.
where diffusion constant D is given as: D = k,TINmy.
(12)
Let the co-ordinates for the two states A and B be xA and xB (xA < su), respectively. The rate constant k of the transition from A to B is inversely proportional to the mean first passage time M, (B/A) of a “particle” representing the motion of the center of mass of the system travelling from X, to xB. The sliding velocity in steady state is proportional to k in the first order approximation. The time M, (BIA) and the mean square first passage time M2 (BIA) are given from equations (1 l), (A2) and (A3) as: hf,(BIA)
= T
[l f (k,T/NKL)(e*NKL’k”T-
t)]?
(13)
and,
x
e+NKL/keT
+
2(myk,T)2
e’
N2K4
ZNKL/keT
(14)
where L = xB--xA and f KL is the difference of the potential energy at xA and xP A reflecting boundary is assumed at sA in the calculation of equations (13) and (14). Under optimum conditions the chemical energy of ATP, which is about 15k,T, is converted into the mechanical energy of muscle contraction with an efficiency >50%. On the other hand, the mechanical energy results from the motion of the “particle” on the potential. Hence, it is reasonable to assume that KL is significantly larger than k,T. By putting KL s k,T, equations (13) and (14) for the case of the uphill potential t result in :
M,&E(A) N_ s:
eNKLIksT,
(15)
and 2NKLIkBT
,
(16)
MOLECULAR
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421
respectively. Hence, the variance a, around M, , becomes at = Mzt-MIT2 (17) The rate constant k, (= l/M,)
is derived from equation (IS) as: (18)
which is a sharply decreasing function of N. In the case of the downhill potential L, equations (13) and (14) lead to: (I!)) and (20) The variance and the rate constant are given as :
(21) and K k, = my(1 - k,T/NKL)’ N K/my.
(22) (The rate constant does not depend on T and N.) Hence, the motion of the Brownian particles in the downhill potential is almost “deterministic”. For the case of the flat potential-+, K = 0, equations (13) and (14) lead to : M&+4)
= g2,
(23) B
and M,,(BjA)
= %(%;;I. B The variance and the rate constant of transition are calculated as:
(24)
(25) and (26)
422
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FIG. 1. The dependence of the velocity of shortening on N and on the overlapping length for unstable (-•-) and stable (-O-) force-generating states. The numbers beside the lines in the figure stand for the value KL/ksT.
The rate constant k, decreases as the number of particles increases. Huxley and Simmons have estimated the height of their uphill potential as 3*8k,T under maximum external load (Huxley & Simmons, 1971), and lower values as load is reduced. It is, therefore, better to discuss a case where KL is not very different in size from k,T, particularly, for the case of the uphill potential. When KL = 3k,T, k, decreases of the order of 10 as one particle is added to the system. On the other hand, k, is almost unchanged. Namely, k, (N = oo)/k, (N = 1) N 2/3. The N-dependence of the rate constants is shown in Fig. 1 for KL = 2kBT and 3k,T. Even in the case of the flat potential, k, is a decreasing function of A? The rate constant k, scarcely depends on N only in the case of the downhill potential. 3. Fluctuations of the Motion of Crossbridgesaround the Mean Velocity
Subtracting equation (7) from equation (6), and omitting of the resulting equation, one obtains:
the inertial term
(27) w(WW = - aN& +fi - 9, where ti = xi-x. This equation has a closed form with respect to the suIi?x i, which will be omitted in the following discussion. The Fokker-
MOLECULAR
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Planck equation for the conditional
MUSCLE
probability
CONTRACTION
423
W with respect to i is given
as:
(28) where
D’ = (N- l)k,T/myN = k,T,lmy,
(29)
the effective temperature T, = (N- 1) T/N. The mean first passage time M’, (r10) for the the change of < from < = 0 .-. r is obtained from equations (28), (A2) and (A3) as:
with
M;(rlO) = j; j dq j! exp [crN(q’-
0
i_‘2)/kBTJ d<.
0
Tlhe rate of fluctuation, k, is given by r/M, (rl0). When the force constant o! is as small as aNr’/k,T,< 1, the linear approximation of equation (30)
with respect to a leads to:
which may be rewritten [using equation (29)] as:
Let us consider a special case r = L in equation (32) and compare it with the rate of the averaged motion obtained in the preceding section. If the motion of the crossbridge is random, the expectation value for the rate of the displacement of a crossbridge by distance L becomes larger than that of the averaged motion, which means h-,- > k. The order of the magnitude of the ratio k/k, is obtained as: k-/k,
- l/N.
k, k, h, exp (- NKL/kBT)
1o-7
k,lk, -
KLIkBT 4
for KL 9 k,T for KL = 2k,T, N = 10,
for KL 3 k,T for KL = 2k,T. N = 10.
(33) (34)
(35)
424
K.
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AND
H.
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The ratio k,/k, decreases, while k,lk, increases, as KL/k,T becomes large. As N increases, k,/k, decreases sharply while k,/k, depends only weakly on N as N/(N- 1). Though we have no information about the potential energy in the force generating state, k, is much smaller than k, even KL - 0, because N is of the order of 10. Hence, in the case of the uphill potential the motion of crossbridges is of random nature and it induces no smooth shortening in muscle. On the other hand, crossbridges move rather coherently in the case of the downhill potential and muscle is able to shorten smoothly.
4. Comparison with Experiments Gordon, Huxley & Julian (1966) have shown that the velocity of shortening of muscle under a very light load is substantially independent of the overlapping length of thin and thick filaments. They studied shortening, for example, from 3.32 pm-l.95 pm striation spacing. For 3.32 pm striation spacing the overlapping length of thin and thick filaments is about one third of the full overlapping length. Julian & Sollins (1975) reported that the number of attached crossbridges is non-zero even at the maximum velocity of shortening. From their result we deduce the number of attached crossbridges under no load shortening is, tentatively, one fifth of that under isometric contraction. Hence, the number N of crossbridges interacting with a thin filament varies from 3-10 when the muscle shortens from 3.23 pm to 1.95 pm striation spacing under no load. The dependence of the velocity of shortening, which is proportional to the rate constant of the transition A -+ B, on N is given in Fig. 1. If the state A is stable, the velocity will be sharply decreased as N is increased. When N increases from 3 to 10, the velocity reduces to about 3 x 10m6 for KL = 2k,T. According to Huxley & Simmons (1971) the potential barrier is flat under no load shortening. Even in this case the veiocity at N = 10 will be about one third of that at N = 3, which is seen from equation (26). On the other hand, Fig. 1 shows that the velocity is almost independent of N if the state A is unstable. When KL $ k,T, which is plausible as written before, the velocity is independent of the overlapping length between thin and thick filaments. Observations by Gordon et al. are consistent with the result obtained in the case of the downhill potential. It means that the chemomechanical conversion in muscle contraction occurs in such a way that interaction of actin and myosin molecules is unstabilized by transferring the chemical energy of ATP and then gives rise to the sliding motion of muscle filaments.
MOLECULAR
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425
5. Discussions and Conclusion The sliding velocity of thin filaments is not only determined by the rate of the change A --f B. It may be a function of the overall rate of the elementary cycle. However, we are only concerned in the present paper with the dependence of the sliding velocity on the overlapping length between thick and thin filaments, which is proportional to N. The N-dependence of the sliding velocity comes only from the change A -+ B, because motions of crossbridges on a thin filament are closely correlated with each other due to the feedback interaction through the sliding motion of the thin filament (Shimizu ef al., 1976). The other processes in the elementary cycle are independent from each other and give no significant effect on the N-dependence of the sliding velocity. Therefore we may take the rate of the change A --+ .B in the force-generating state to be proportional to the velocity of shortening of muscle as long as discussions are confined to the N-dependence of the of the velocity. As is seen from equation (22), the velocity o1 is independent of temperature, provided that KL 9 k,T. However, this should not be compared directly with experimental observations, because the actual velocity of shortening is determined by the overall rate of the elementary c:ycle. Processes other than the change A -+ B are driven thermally and may give rise to the temperature-dependence of the velocity of shortening of muscle. If the two states A and B are stable ones, the transition A -+ B requires a supply of thermal energy to the motion of crossbridges. Since the strength of thermal force acting on the center of mass of the system is inversely proportional to the number of particles effectively, the rate of the co-operative change A -+ B is reduced as N increases. Hence, the velocity of shortening is reduced N increases, which is inconsistent with results observed by Gordon et al. (1966). In the case of the transition from an unstable to a stable state, the motive force of the crossbridge is given in the forward direction both by potential force and by thermal force. As discussed before the condition k,T < KL i:j realized in muscle contraction, which means the potential force is appreciably larger than the thermal one. Therefore, in the case of the downhill potential, muscle is able to shorten smoothly and the velocity of shortening is independent of N, which is consistent with experimental results by Gordon et al. This is also consistent with McClare’s picture (1971) in a sense that the motion of crossbridges is “deterministic”. Moreover, a consistency is further found with Huxley’s model (1957), because the motion of the site IM in his model changes from a stochastic to a determinstic one after binding with the site A on the thin filament. In the same spirit our scheme is con-
426
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sistent with Huxley’s picture (1973) which introduces two attaching states in his model of 1957. Recently, Nishiyama, Shin&u, Kometani & Chaen ( 1977) proposed a three-state model for the elementary cycle of ATP-splitting by the actomyosin molecule. They succeeded in describing consistently mechanical properties of steady shortening and lengthening processes, as well as those of isometric and isotonic transients, in a quantitative way. In their model the motion of crossbridges on thin filament, causing the sliding motion of muscle filaments, starts on a potential which is unstable with respect to the displacement. This model is consistent with our results obtained above that the forcegenerating state must be an unstable one. T. L. Hill (1974) gave general discussions about the chemo-mechanical coupling in terms of free energy. In his discussions the sliding motion is associated with the change from unstable states to stable state, which is also consistent with our three-state model. The mechanical energy for muscle contraction comes directly from the (excess) energy of the inter-molecular potential between actin and myosin unstabilized by utilizing the chemical energy of ATP splitting, beccause the motion of crossbridges proceeds in a mechanical manner. However, this is not the case for the uphill potential; the motion of crossbridges occurs thermally and the mechanical energy of muscle contraction is converted mainly from thermal energy of heat reservoir. The free energy of ATP is utilized for giving a definite direction to the motion of the crossbridge such as A -+ B. It may be compared with the role of Maxwell’s demon who is able to transform the thermal energy of heat reservoir to macroscopic work in a definite direction. These two types of energy conversion, the unstable energy conversion and the thermal energy conversion, are both possible from the principle of thermodynamics and probably may be found in biological systems. In order to develop large tension in muscle contraction, the number N of interacting sites must be large, since tension is proportional to N. On the other hand, skeletal muscles must shorten rapidly with large velocity. In general, large tension is compatible with such a large velocity only when the force-generating state A is unstable, namely, only when the energy conversion is of unstable energy conversion. (Thermal energy conversions may be found in the active transport of substances across biomembrane, partly because the reactions there may be independent and partly because no mechanical energy is not always necessary.) Our conclusion in the present paper is that the chemomechanical conversion in muscle contraction is an unstable energy conversion. Strictly speaking, the effect of the Z-band must be considered for the N-dependence
MOLECULAR
DYNAMICS
of the velocity of shortening. such a modification.
OF
MUSCLE
However,
CONTRACTION
our conclusion
421
is not altered by
The authors wish to thank Drs K. Nishiyama, H. Yamamoto and T. Yamada for their valuable discussions and suggestions. They acknowledge financial support from the Ministry of Education, Science and Culture of Japan and from the Kurata Research Grant. REFERENCES BA.RANY, M. (1967). J. gen. Physiol. 50, 197. GOEL, N. S. & RICHTER-DYN, N. (1974). Stochastic
Models
in Biology.
Press.
New York: Academic
GORDON, A. M., HUXLEY, A. F. & JULIAN, F. J. (1966). J. Physiol. 184, 170. HASELGROVE, J. C. & HUXLEY, H. E. (1973). J. Moiec. &of. 77, 549. HILL, A. V. (1938). Proc. R. Sot. B126, 136. HILL, A. V. (1964). Proc. R. Sot. B159, 297. HILL, T. L. (1974). Prog. Biophys. molec. Biol. 28, 267. HIJXLEY, A. F. (1957). Prog. Biophys. biophys. Chem. 7, 255. HIJXLEY, A. F. (1973). Proc. R. Sot. Land. Blt33, 83. HIJXLEY, A. F. & SIMMONS, R. M. (1971).Nature, Lond. 233, 533. JULIAN, F. J. & SOLLINS, M. R. (1975).J. gen. Physiol. 66, 287. MCCLARE, C. W. F. (1971). J. theor. Biol. 30, 1. MORIMOTO, K. & HARRINGTON, W. F. (1974). J. mol. Biol. 83, 83. N:ISHIYAMA, K., SHIMIZU, H., KOMETANI, K. & CHAEN, S. (1977).Biochim. biophys. Acto. 460, 523. SI-IIMIZU, H. (1972). J. Phys. Sot. Japn. 32, 1323. SI-IIMIZU, H. & YAMADA, T. (1975). J. theor. Biol. 49, 89. SIIIMIZ~, H., YAMADA, T., NISHIYAMA, K. & YANO, M. (1976). J. theor. Biol. 63, 189. YWADA. T., SHIMIZU, H. & SUGA, H. (1973).Biochim. bi0ph.w. Acta. 305, 642.
APPENDIX Let us consider a Brownian particle moving in a domain (x,, xa) following to a Fokker-Planck equation:
ap - =-- a a(x)P at ax
+; a;2 b(x)P.
The time, the first passagetime, which is needed for a Brownian particle found initially at y to reach the position z such that (xA 5 y < z I xB) will be considered here. The first passage time is a random variable, because the motion of the particle is essentially of randomness. By assuminga reflecting boundary at xA, the mean first passage time M, (z/y) and the mean square first passagetime M, (zl~~)are given as follows (Goel & Richter-Dyn, 1074).
MI(+) = 2 iY dvh)
j Cb(5)n(5)1-1 dt, XA
(A2)
428
K.
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AND
H.
SHIMIZU
and &(zf~)
= 4 j dv(v) Y
with
j ~,@XWM)l-
’ d5,
x.4
4y1) = exp - j! WW43
.
d5 I
(A3)
644)