No. 8]

43.

Proc. Japan

Acad.,

69, Ser. B (1993)

227

Involvement of "Thermal Interference" in the Multiple Working
Strokes Per Hydrolyzed ATP Observed in Muscle Contraction
By Kenichiro
MOGIof Physics, Faculty of Science, University of Tokyo,
7-3-1 Hongo, Bunkyo-ku 113and*) Laboratory
for Neural Networks , The Institute of Physical andChemical Research (RIKEN), 2-1 Hirosawa,Wako-shi, Saitama 351-01
Department(Communicatedby Setsuro

EBASHI, M. J. A., Oct. 12, 1993)

Abstract:
Recently, observed large values of `step size' in in vivo muscle contractionl),2) and in vitro actomyosin motility assays3)-5) have raised new questions about thenature of coupling in biological systems. It has been experimentallydemonstrated'thatmultiple working strokes occur during the hydrolysis of one ATP molecule. It is difficult toexplain these observations by conventional enzyme kinetics. In this paper, we propose anovel enzymatic mechanism (the `coupling ratio constraint' mechanism). By `free energy
mixing' between the two coupled degrees of freedom, we can explain the mechanism
underlying the large values of step size and the multiple working strokes per ATPhydrolyzed observed in muscle. We predict that myosin would behave in an unconventionalmanner during the multiple working strokes. This prediction can be tested experimentally.
Finally, we propose that `thermal interference', similar to quantum interference20~ (but ofdifferent physical origin) is involved in the observed phenomenon.
Key words:

Muscle contraction;

multipleworkingstrokes;

thermalinterference.

In many biologically important reactions, an otherwise unfavorable reaction is drivenby `coupling' it to anotherenergeticallyfavorable reaction.
The exact nature of thisimportant "coupling" mechanism, however, is not yet known. In this paper, we propose anovel principle of enzyme coupled reactions, which involves what we define in this paper as`thermal interference' .
In muscle contraction,a relative sliding between the actin and the myosin filament iscaused by the hydrolysis of ATP. The `step size' is defined as the distance of sliding causedby the hydrolysis of one ATP molecule. The `working stroke' is defined as the unit event ofsliding conducted by myosin. It has been estimated that the distance of sliding caused by
one working stroke is about 12 nm.12) Since it has been assumed that the hydrolysis of one
ATP molecule causes one workingstroke, 11)43) the sliding distance caused by thehydrolysis of one ATP molecule has been estimated to be about 12 nm. Yanagida et al.,1)
however, observed that the step size in crab muscle under free load condition is > 60 nm.
The large values of `step size' observed
by Yanagida
and othersl),2),4),5) is very
puzzling, although a smaller value of step size has not been completely excluded.6) Since atthe scale of actin filament and myosin filament we can virtually neglect the inertia, 14),15)
actin filament can slide only during the working stroke. At the end of working stroke, actin
filament is supposed to stop. As from structurallimitations the sliding distance of actin
filament caused by one working stroke cannot be larger than 40 nm,18) it has beenconcluded that observed large values of the step size must be the result of more than one*)

Corresponding

address.228

K. Mop;i[Vol. 69(B),working strokes per ATP hydrolyed.4~ Recently, it has been experimentally shown that incontracting muscle several elementary working strokes occur per ATP moleculehydrolyzed.7~ It is therefore interesting to consider how multiple working strokes canoccur during the hydrolysis of one ATP molecule.
An obvious explanation of the occurrence of multiple working strokes per ATP
molecule hydrolyzed is to assume the existence of multiple intermediate states (multipleintermediate states hypothesis). Namely, we assume that the ATP hydrolysis cycle iscomposed of intermediate states as S1-S2---~ ..
ESN-~S1, with gradually decreasing
free energy levels. If we then assume that one working stroke is caused by the transition
from one intermediate state to the next state, N working strokes can in principle occur
during the hydrolysis of one ATP molecule. However, it is difficult to make this scheme
compatible with the repeated conformation change accompanying the multiple working
strokes. 17)It is unlikely that the intermediate states can sequentially take the alternating
conformations of myosin during one ATP hydrolysis cycle.
A second explanation is that the energy liberated by the hydrolysis of ATP is stored insome form in myosin molecule, and then used little by little in the multiple working
strokes. The thermal ratchet models8~10~postulate that the free energy liberated by thehydrolysis of ATP is stored as a temperature difference between the actin filament andmyosin head. However, since the relaxation of the excitation of any mode of protein inwater is very rapid,19~ this kind of temperature difference is unlikely to sustain sliding overtime lasting for milliseconds unless `temperature' in these models represent somepreviously unknown sustainable mode of excitation.
In summary, we are encountering a fundamental difficulty in understanding themultiple working strokes per ATP hydrolyzed observed in muscle contraction.
In what follows, we propose a novel coupling principle which can explain thisinteresting phenomenon.
Let us write the two degrees of freedom that are coupled as x (sliding of actin
filament) and y (hydrolysis of ATP). We write the rates of transition in these degrees offreedom as Vx (number of working strokes per second) and Vy (number of ATP molecules
hydrolyzed per second). In the mathematical model developed here, we approximate theATP hydrolysis cycle with one step transition, which can be interpreted as the phosphaterelease step (A. M. ADPPi-~A. M.ADP). This approximation does not affect the essential
feature of our model. The free energy change accompanying the working stroke and theATP hydrolysis are written as Ex and Ey, respectively. We define `coupling ratio' C as16)
C = vVx = vl-v2
yul-u2(1)

where (Vi, v2) and (u1, u2) are the forward and backward rate constants for the working
stroke and ATP hydrolysis, respectively. In other words, coupling ratio is the number ofworking strokes conducted per ATP molecule hydrolyzed.
We assume that through the interaction between actin filament and myosin coupling
ratio C is determined as a function of the load (E,). Namely, we assume that the essence of
coupling through actomyosin interaction is the constraint imposed on coupling ratio C(`coupling ratio constraint' hypothesis) (Fig. 1).
Our hypothesis takes the form ofLet

In order to satisfy relation
us write the transformed

Vi -V2 - Culu2(2)(2), free energy changes Ex and Ey must be transformed.
free energy changes as Ex' and Es'.

"ThermalNo. 8]

Interference"in Muscle

Contraction

229

Fig. 1. Coupling ratio constraint mechanism. In the coupling ratio constraint mechanism,
it is assumed that the coupling ratio (the ratio of the transition rate constants in thecoupled degrees of freedom) is fixed at a constant value by the enzyme. The actualtransitions, however, continue to occur in two dimensions. The discrepancy between theaverage and actual degrees of freedom leads to the thermal interference effect. In thisfigure, the average degree of freedom is shown by the bold line. The transitions of thesystem are restricted,
on average, to this single degree of freedom. An example of asequence of the actual transitions is represented
by the directed edges.

We make a furtherassumptionthat the free energy change is invariant
in thedirection of vector (C,1), which is the direction of the actual change under condition (2).
This assumption
is necessary
for the conservationof energy. Namely,
CEx+EInordertoderive

Ex'andEy',

we

(3)

y = CE x' +Ey'
makev 1 = exp
v2

use-of theArrhenius

relationsxkT(4)

ui =exp(--b-)(5)

u2
Fromequations(2)-(5),

we can derive
E x' = E'=ylevels

In particular,
undercan be given as

thethetransformed1+CC

E x+ 1+C
1free load (Ex=O) condition,

Ex' = E y' =1+C
1

Ey

free energy
Ey

the transformedchanges

as(6)

free energy(7)

We see that the free energy released by the hydrolysis of ATP (Ey) is used to drive
the working stroke by a free energy gradient of Ex'. Multiple working strokes per ATPhydrolyzed is thus possible in this scheme. Moreover, since every working stroke is driven

230

K.

MOGI

[Vol. 69(B),by a free energy gradient of Ex', the sliding of actin filament is smooth.5)
It is interesting to consider implications of transformation (6). In our scheme, the twodegrees of freedom are related through coupling ratio constraint of (2) and are no longer
independent. As a result, the free energy changes Ex and Ey are `mixed' into a singlevalue. The hydrolysis of ATP drives the working strokes through this `free energy mixing'.
Note that mathematically, `coupling ratio constraint' effectively reduces the dimension of
the space in which transition occurs from two to one. The actual transitions of the state,
however, continue to occur in two dimensions. It is this discrepancy that is the origin of the`free energy mixing' .
From the analysis of the model we presented in this paper, we can make the following
predictions about the nature of coupling between the hydrolysis of ATP and the sliding ofactin filament.
(i) ATP is not necessarily hydrolyzed when one working stroke occurs.
(ii) When ATP is not hydrolyzed, the chemical state of myosin is the same before andafter the working stroke.
(iii) The sliding of actin filament is uniform regardless of the particular time at which thehydrolysis of ATP is completed.
(iv) The multiple working strokes during the hydrolysis of ATP is driven by exactly thesame amount of free energy change.
(v) The number of intermediate states involved in the coupling can be much smaller than
the number of working strokes conducted per ATP hydrolyzed.
(vi) One myosin molecule is sufficient to sustain multiple working strokes per ATPhydrolyzed.
These predictions can be tested in future experiments to verify the validity of ourmodel. In particular, it would be interesting to test the above predictions under conditions
where large values of coupling ratio is realized.21)
The `coupling ratio constraint' hypothesis that we introduced in this paper has several
novel features.
Firstly, our model predicts an unconventional behavior of myosin during the multiple
working strokes. If the coupling ratio is C, C working strokes are conducted during thehydrolysis of one ATP molecule. In C-i out of C cases, one working stroke would be
conducted without the completion of the hydrolysis of ATP. Therefore, in our model thestate of myosin before and after the working strokes remains the same in C-i out of C cases(Fig. 2).
Secondly, a `thermal interference effect' is involved in our model. The `coupling ratio
constraint' is imposed between two events, where the first event is the event `working
stroke is not done and ATP is hydrolyzed', and the second event is the event `working
stroke is done and ATP is not hydrolyzed'. These two events are mutually inconsistent.
However, the free energy changes accompanying these two inconsistent events are `mixed'
into a single value Ex' = Ey' through the `thermal interference' effect. It should be noted
here that one myosin molecule is sufficient to induce the `thermal interference' effect.
Interaction between multiple myosin molecules is not necessary.
We can summarize the situation in which a thermal interference effect occurs asfollows.
(i) There is a stochastic process, which occurs in two degrees of freedom.
(ii) Through a constraint imposed by the enzyme (myosin), the stochastic transition isrestricted on average to one degree of freedom.
(iii) The actual transitions, however, continue to occur in two degrees of freedom.
(iv) The discrepancy between the average and actual behaviors of the system leads to the`thermal interference' effect .

No. 8]"ThermalInterference"in Muscle

Contraction

231

Fig. 2. Change in myosin states during the multiple working strokes. In this figure,coupling ratio is taken to be 8. (a) If we assume multiple intermediate
states, every
working stroke must be accompanied by a change in the chemical state of myosin, e.g.,
A.M.ADPPi->A.M.ADP(phosphate release step). (b) In the `coupling ratio constraint'
mechanism, the change in chemical state occurs on average in 1 out of 8 working strokes.
In 7 out of 8 cases, working strokes occur without the hydrolysis of ATP. In such cases,
the working strokes are driven by the mixing of free energy changes which is the resultof the `thermal interference' effect. This unconventional behavior of the myosin moleculecan be tested in future experiments.

It is interesting to note that the above situation is similar to the situation leading to
the quantum interference effect.20~However, the underlying physical principles are quite
different. In the quantum interference effect, the scale of the system is characterized by
the Planck's constant, which is orders of magnitude smaller than the characteristic energy
scale of -kT (where k is the Boltzmann's constant, and T is the absolute temperature) for
enzyme coupled reactions.
At present, it is not clear how an enzymatic protein can impose the `coupling ratio
constraint' in the form of (2). It would be interesting to reconsider the mechanism of
enzymatic coupling from this point of view. Since myosin is expected to modulate thecoupling ratio in order to work effectively under conditions of various load, there must be acorresponding molecular mechanism involved in muscle contraction.
Mathematical details of the model we presented here will be reported elsewhere.
Acknowledgements. We are grateful to Dr. Takeyuki Wakabayashi for the introduction to the problems in muscle contraction and for many suggestions. We would like tothank Dr. Toshio Yanagida, Dr. Makio Tokunaga, and Dr. Takuo Yasunaga for helpful
discussions. We would like to thank Dr. Setsuro Ebashi, M. J. A., and Dr. Masao Ito,
M.J. A., for valuable advises and encouragements.
References

1)
2)
3)
4)
5)
6)

Yanagida, T., Arata, T., and Oosawa, F.: Nature, 316, 366-369 (1985).
Higuchi, H., and Goldman, Y. E.: ibid., 352, 352-354 (1991).
Toyoshima, C., and Spudich, J. A.: ibid., 328, 536-539 (1987).
Harada, Y. et al.: J. Mol. Biol., 216, 49-68 (1990).
Ishijima, A. et al.: Nature, 352, 301-306 (1991).
Uyeda, T. Q. P. et al.: ibid., 352, 307-311 (1991).

232
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)

K. MoGI[Vol. 69(B),Lombardi, V., Piazzesi, G., and Linari, M.: Nature, 355, 638-641 (1992).
Oosawa, F.: Jikeikai Med. J., 36, 219-231 (1989).
Vale, R. D., and Oosawa, F.: Adv. Biophys., 26, 97-134 (1990).
Cordova, N. J., Ermentrout, B., and Oster, G. F.: Proc. Natl. Acad. Sci. U.S.A., 89,339-343 (1992).
Huxley, A. F.: Progr. Biophys., 7, 255-318 (1957).
Huxley, A. F., and Simmons, R. M.: Nature, 233, 533-538 (1971).
Huxley, A. F.: Proc. R. Soc. Lond., B183, 83-86 (1973).
Purcell, E. M.: Am. J. Phys., 45, 3-11 (1977).
Shapere, A., and Wilczek, F.: Phys. Rev. Lett., 58, 2051-2054 (1987).
Mogi, K.: J. Theor. Biol., 162, 337-352 (1993).
Irving, M. et al.: Nature, 357, 156-158 (1992).
Elliot, A., and Offer, G.: J. moelc. Biol., 123, 505-519 (1978).
Brooks III, C. L., Karplus, M., and Pettitt, B. M.: Proteins. John Wiley and Sons, pp.
146-147 (1988).
Bohm, D.: Quantum theory (Dover), p. 494 (1951).
Yasunaga, T., and Wakabayashi, T.: Biophysics, 31, 195 (abstract in Japanese).