Appeared in Proc. Roy. Soc. Lond. A, 445, p.529-541.(1994)

On the absolute meaning of
the energy scale ~kT
in the 'thermal interference'
involved in enzyme coupled reactions
Ken Mogi
Laboratory for Neural Networks
The Institute of Physical and Chemical Research (RIKEN)
Hirosawa 2-1, Wako-shi, Saitama 351-01 JAPANNow atSony Computer Science Laboratory
Takanawa Muse Bldg. 3-14-13, Higashigotanda
Shinagawa-ku, Tokyo, 141-0022 Japan
Tel +81-3-5448-4380 Fax +81-3-5448-4273
kenmogi@csl.sony.co.jp
http://www.csl.sony.co.jp/person/kenmogi.html
Summary
We propose a novel mechanism involved in enzyme-catalyzed
coupled reactions in biological systems. It is assumed that an enzyme setsa constraint on the coupling ratio between the degrees of freedom coupledin the reaction. We show that this assumption leads to a mixing of freeenergy values between the degrees of freedom coupled in the reaction.
This is an interference effect of thermal origin, which shares someproperties with the quantum interference effect. We discuss the possibility
that the energy scale ~kT has an absolute meaning in determining themagnitude of the thermal interference effect.

Through the action of enzyme E , thesetwo degrees of freedom become interrelated.
Coupling ratio C is defined as the ratio of the net
occurrence of these two reaction steps (Mogi
1993a).

1. Introduction
Coupling is a ubiquitous and importantmechanism employed in many crucial aspects ofthe biological processes. The exact nature of thisvery important biological process, however, is not
yet known. Recently, through the observation of
the properties of mechanochemical coupling inmuscle contraction (Yanagida et al. 1985, Haradaet al. 1990, Higuchi & Goldman 1991, Ohno &Kodama 1991, Yasunaga & Wakabayashi 1991,
Lombardi et al. 1992, Irving et al. 1992), a newquestion as to the nature of coupling in biologicalsystems has been brought up. Through theanalysis of this question, we present, in this paper,
a novel mechanism of coupling in biologicalsystems. We postulate that a 'thermal interference
effect', namely the interference between mutuallyinconsistent events coupled through the action of
the enzyme, is involved in the coupled reactions.
This is an effect similar to the quantum interference
effect (see for example, Dirac 1958 or Bohm
1951), but of different physical origin. We discuss
the possibility that, as in the quantum interference
effect, there is an absolute meaning to the scale ofenergy ~kT (where k is Boltzmann's constant,and T is the absolute temperature) in themechanism leading to the thermal interference
effect.
The new question about coupling that has
been brought up through the studies of muscle
contraction can be summarized as follows.
Under constant temperature and constantpressure conditions of biological systems, theGibbs free energy determines the direction inwhich reaction proceeds. Namely, a reaction must
involve a negative free energy change in order toproceed spontaneously. In coupling, enzyme E
drives a reaction step involving an unfavorable(positive) free energy change by coupling it toanother step involving a favorable (negative) freeenergy change. Let us denote the two coupled
reaction steps as A 1→A 2(energeticallyC=occurrence of A1 → A2
occurrence of B1 → B2

In the conventional understanding of thecoupled reaction (see, for example, Stryer 1988),
it has been assumed that coupling ratio C is one.
In this situation, it was assumed that when one unitof reaction A 1→A 2 occurs, one unit of reaction
B 1→B 2also occurs. Namely, the conventional

scheme is
In coupling, EA1 B1 → EA2B2 always occurs(A)
However, it has been suggested in the
study of muscle contraction and other biologicalsystems (e.g., Shimizu et al. 1991) that the
coupling ratio is not necessarily one. In particular,
in the study of muscle contraction, coupling ratio
C has been shown to be much larger than one
under certain conditions. This means that the tworeactions A 1→A 2 and B 1→B 2 do not always
occur simultaneously. Namely, in view of therecent experimental evidence coupling should
rather be expressed in the scheme
In coupling, sometimes EA1 B1 → EA1 B2 occurs
and sometimes EA1 B1 → EA2 B1 occurs .
(B)
In the new scheme, the conventional
coupling scheme (A) can be considered as a special
case in which the two reactions in scheme (B)occur simultaneously.
To illustrate the theoretical problems thatunfavorable reaction) and B 1→B 2 (energeticallyfavorable reaction).
2

arise from the above picture of coupling, let us
assume, for example, that the coupling ratio is N,
where N is some integer larger than 1. Theenergetically favorable reaction B 1→B 2 occurs,

assume the alternating conformations of the motor
molecules.
To conclude, there is a very severe
conceptual difficulty when we try to understand acoupling mechanism in which the coupling ratio is
not one. In view of the recent experimental
evidence, we must abandon the conventional
scheme, and endeavor to find a coupling principle
which can incorporate the newly revealed nature ofcoupling. In particular, we need a 'credit'mechanism in which the energetically unfavorable
reaction is driven on the 'credit' that theenergetically favorable reaction would occur with acertain probability.
In the next section, we propose a coupling
principle which can solve the problem formulated
above.

on average, only in one case out of N occurrencesof the energetically unfavorable reaction A 1→A 2 .
Namely, in N-1 out of N cases, the energeticallyunfavorable reaction A 1→A 2
should occur
without the accompanying occurrence of thedriving reaction B 1→B 2. We then ask ourselves
how the unfavorable reaction A 1→A 2 is driven in
the absence of the favorable reaction B 1→B 2 . To
understand such a mode of coupled reactions, weneed a coupling mechanism in which theenergetically unfavorable reaction can be driven on
the 'credit' that the energetically favorable reaction
would occur in one out of N cases.
Many hypotheses have been proposed to
explain a 'credit' mechanism such as formulated
above. Storage of the free energy liberated by thedriving reaction B 1→B 2 in some mode of the

2. Coupling ratio constraint
mechanism
In this section, we propose a novel
coupling mechanism (the 'coupling ratio
constraint' mechanism).
Let us denote the two coupled degrees of
freedom as x ( A 1→A 2 , the energeticallyprotein structure (thermal ratchet model) has beensuggested as a possible solution to this question(Feynman et al. 1963, Oosawa 1989, Vale &Oosawa 1990, Cordova et al. 1992). However,
although this scheme can reproduce the basic
properties of the experimental data, in view of therapid decay time of any fluctuation in protein
structure (e.g., Brooks et al. 1988) it is very
difficult to incorporate this model in the realistic
dynamics of protein in water. It may appear thatinertia can drive the unfavorable reaction,
especially when the driven reaction A 1→A 2 is theunfavorable reaction) and y (B 1→B 2 , theenergetically favorable reaction). We formulate thecoupled reaction system as a random walk in a
two-dimensional plane (x, y). First we consider thesituation in the absence of coupling enzyme E. At
each transition time, transition can occur in one of
the 4 directions (1,0), (-1,0), (0,1), (0,-1), withprobabilities v 1, v 2, u1, u2, respectively.
The normalization condition istranslocation of macromolecules. However, inertia
cannot function as an energy-storing mechanism
because of the low value of Reynolds number for
macromolecules in water (Purcell 1977, Shapere& Wilczek 1987). The existence of multiple
intermediate states is also unlikely to explain theobserved large values of the coupling ratio inmuscle contraction, as it is unlikely that themultiple intermediate states can sequentiallyv1 + v2 + u1 + u2 = 1 .
(1)

In the absence of coupling enzyme E,
there is no correlation between the two degrees of
freedom. Let us assume that transition A 1→A 2
involves a free energy change of ∆E x >0,3

andtransition B 1→B 2

involves a free energy change

determining the kinetic rate constants). Since an
enzyme cannot change the free energy balance of asystem, the actual free energy gradients ∆E x andof ∆E y <0.
The transition probabilities v 1, v 2, u1, u2∆E y are expected to be invariant. Therefore, there

and free energy gradients ∆E x , ∆E y characterize

would be, in general, a discrepancy between theeffective and actual energy gradients.
Note that for reasons that will be clarified
later, the arguments developed in this section are
considered to be applicable only to the cases where
the absolute values of the energy gradients ∆E x

the uncoupled system. We have the Arrhenius
relationsv1 ∆E 
= exp − x  kT 
v2
 ∆E 
u1= exp  − y  kT  .
u2

and ∆E ysections 3 & 4). When the absolute values of ∆E x

(2)and ∆E y are larger than ~kT, a modification of the

If we take an appropriate origin, the state of
the system represented by the coordinate (x,y) has
the free energy value of E(x,y), wherescheme is needed, as discussed in section 4.
The novel coupling mechanism that we
propose in this paper is based on the following two
fundamental assumptions.

E(x, y) = x∆Ex + y∆Ey .
(3)(1) Through the action of the enzyme, the ratios of
the forward and backward transition probabilities
in the coupled degrees of freedom are fixed at aconstant value. As a result, the coupling reaction iseffectively constrained to proceed in a single(average) degree of freedom (the coupling ratio
constraint hypothesis, Fig.1).
(2) Although there is, in general, a discrepancy
between the effective and actual free energy
values, they take the same value along the average
degree of freedom specified by the coupling ratio
constraint.

When we introduce enzyme E, the twodegrees of freedom are correlated through theaction of the enzyme. As a result, it is expected thatthe transition probabilities v 1, v 2, u1, u2 and thefree energy gradients ∆E x , ∆E y are transformed.
Let us write the transformed transition rate
constants and free energy gradients as v 1', v 2',u1', u2' and ∆E x ,' ∆E y '. The essence of thecoupling mechanism is reflected in thetransformation(v 1,v 2 ,u1 ,u2 ,∆ Ex , ∆Ey ) uncoupled → (v1′, v ′2, u1′, u 2′ ,∆E ′x ,∆Ey′ )coupled.

are smaller or comparable to ~kT (see(4)

We will refer to the transformed energy
gradients ∆E x ' and ∆E y ' as the effective energy
gradients. The effective energy gradients are
expected to be relevant only as far as the dynamical
behavior of the system is concerned (e.g., in4

actual degree of freedom

C=average degree of freedomyE

We express the average degree of freedom
specified by the coupling ratio constraint by the'coupling ratio vector' (p,q), wherexSv1′ − v′2
u1′ − u2′ .

actual degree of freedom

p2 + q2 = 1 .
transition in the x- degree of freedom

In the following discussion, we assume,
without a loss of generality, that p, q >0.
The coupling ratio constraint hypothesis
can be written astransition in the y- degree of freedom

Fig.1 The coupling ratio constraint
mechanismv1′ v2′ p==
u1′ u2′ q .

In the coupling ratio constraint mechanism, it isassumed that the coupling ratio (the ratio of thetransition rate constants in the coupled degrees of
freedom) is fixed at a constant value by the
enzyme. The actual transitions, however, continue
to occur in two dimensions. In this figure, theaverage degree of freedom is shown by the boldline. The transitions of the system are restricted, on
average, to this single degree of freedom. An
example of a sequence of the actual transitions isrepresented by the directed edges. When an edge
has two arrowheads, the occurrence of a reversible
transition is suggested.

(6-A)
Under condition (6-A), the coupling ratio
takes a constant value of
C=

(6-B)
Note that condition (6-A) is a stronger
requirement than condition (6-B). This stronger
requirement is necessary and appropriate for two
reasons. The first reason is that if we require only
condition (6-B), we would need one more equation
to determine the transformed rate constants andfree energy gradients. In other words, we are left
with one free parameter, and there is no natural
physical or biological requirement that would
provide the necessary equation. The second reason
is that requirement (6-B) is a generalization of theprinciple of detailed balancing (Bridgman 1928) inthat both the forward and backward reactions
proceed in the same average degree of freedom.
The random walk is now on average
restricted in the direction of the coupling ratio
vector (p,q). In other words, there is effectively
only one degree of freedom. The requirement thatNow let us obtain the transformed
transition probabilities and energy gradients basedon the above assumptions.
The normalization condition is again
v1′ + v2′ + u1′ + u2′ = 1

v1′ − v′2 p=
u1′ − u2′ q .

.
(5)

The coupling ratio C defined in theprevious section can be written as

5

∆ E′

the effective free energy and the actual free energy
take the same value along the average degree of
freedom can be written aspe− kTv1′ =∆E ′( p + q )1 + e − kT 
p
v2′ =
∆E ′
( p + q)1+ e − kT p∆Ex + q∆Ey = p∆ Ex′ + q∆ Ey′ .
(7)

∆E ′

qe − kTu1′ =∆E′( p + q)1+ e − kT 

The Arrhenius relations for the transformed
transition probabilities and free energy gradientsare ∆E ′ v1′= exp  − x v2′
 kT 
 ∆E ′ u1′= exp − y 
u2′
 kT 

u2′ =

(10)

Equations (9) and (10) give transformation(4), realized through the coupling ratio constraint
mechanism.
It is interesting to consider the implicationsof formula (9). Recall that we have assumed that∆E x >0 and ∆E y <0. The transition in the x.
(8)

From relations (5), (6-A) and (7)-(8), theeffective free energy gradients are obtained as asingle value ∆E'.
∆ Ex′ = ∆Ey′ = ∆E ′ =q∆E′( p + q)1 + e− kT .

degree of freedom cannot proceed by itself.
Through the 'coupling ratio constraint' of (6-A),
the free energy gradients in the two coupleddegrees of freedom are 'mixed' into a single valueof ∆E'. As long as the conditionp∆Ex + q∆E y
p+ q
(9)

∆ E′ =The transformed transition probabilities are
obtained asp∆Ex + q∆Ey<0p+ qis satisfied (i.e., if the coupling ratio is not verylarge), the transition in the x- degree of freedom
would be driven by an effective free energy
gradient of ∆E'. In this way, the negative freeenergy change ∆E y accompanying the favorable
transition B 1→B 2 is used to drive the unfavorable
reaction A 1→A 2. This is a 'free energy mixing'
effect, which can reproduce the essential featuresof a coupling reaction for which coupling ratio C
is not one.
The effective energy value of the state (x,y)
is now given as6

E ′(x, y) = x∆Ex ′ + y∆Ey ′= (x + y)temperature (~300K). If we assume, for example,
that the 'load' ∆E x is 3 kT, and the coupling ratiop∆Ex + q∆Eyp+q.

is 4, the effective free energy gradient ∆E' will be 1.6 kT. Therefore, each 'working stroke' would
be driven by a constant effective free energy
gradient, regardless of the particular time atwhich the hydrolysis of ATP is completed. This
picture is compatible with the smooth sliding
movement of actin filament observed in in vitro
motility assays (Ishijima et al. 1991, Harada et al.
1990).
Let us verify that the energy balance (i.e.,the change in the Gibbs free energy) is always
negative, as is required by the second law ofthermodynamics. We assume that after n steps ofrandom walks, the system has undergone atransition of (x(n), y(n)). The energy balance U(n)(change in the Gibbs free energy) for the coupled
reaction is defined as(11)
We see that the question formulated in theprevious section of how the enzyme can establish a'credit' mechanism is now answered. Namely, inthe above scheme, the unfavorable reaction A 1→
A2

is driven on the 'credit' that the favorable

reaction B 1→B 2 would, on average, occur in
one out of C cases. We also see that there is a'thermal interference effect' involved in ourscheme. The 'thermal interference effect' isformulated as the interference between two
mutually inconsistent events that are coupledthrough the action of enzyme E. In our scheme,
the two events

U(n) = x(n)∆E x + y(n)∆E y .

EA1 B1 → EA1B2

(12)

and
EA1 B1 → EA2B1

The average value of the energy balance isthen calculated as

are mutually inconsistent. However, through theaction of the enzyme, these two inconsistent eventsare coupled, and the free energy gradients
accompanying these events 'interfere' with eachother. As a result, we obtain a single effective freeenergy gradient (11) which is common for these
two mutually inconsistent events. This is a novel
feature of the 'coupling ratio constraint'
mechanism.
As a concrete example, let us consider thesliding movement of actin filament driven by thehydrolysis of ATP observed in muscle contraction.
It has been experimentally demonstrated thatseveral elementary 'working strokes' areconducted during the hydrolysis of one ATP
molecule (Lombardi et al. 1992). The free energy
liberated by the hydrolysis of ATP underphysiological conditions is about 12 kcal/mol(Shriver 1984) or 20 kT per molecule at room

U(n) = x(n) ∆Ex + y(n) ∆Eyp
= n(u1′ − u2′ )  ∆Ex + ∆E y 
q

∆E ′

= −n

1 − e − kT∆E ′1 + e − kT

∆ E′
.
(13)

We see that the average value of the energybalance is always zero or negative, regardless ofthe value of the effective free energy gradient ∆E'.
Therefore, in our scheme, the second law ofthermodynamics is not violated, and there is no
Maxwell's demon (Maxwell 1871) at work.
7

then it would have to satisfy certain conditions inaccordance with the statistical properties of thethermal fluctuation. Let us therefore check if thestatistical behavior of the difference between the'effective' and 'actual' free energy values iscompatible with the assumption that its origin is thethermal fluctuation.
Let us again begin with the assumption thatafter n steps of random walks, the system has
undergone a transition of (x(n), y(n)). The
transition of the system (x(n), y(n)) can be
decomposed into a superposition of the coupling
ratio vector (p,q) and a unit vector (-q,p)perpendicular to it.

3. Statistical properties ofdifference between effective andactual free energy values
In the previous section, we presented the'coupling ratio constraint' mechanism which can
explain the basic properties of a coupled reaction
where the coupling ratio is not one. We formulated
the model in terms of two different concepts of freeenergy values, namely, the effective and actual freeenergy values. We postulated that the effectiveenergy values are relevant when we consider the
dynamical behavior (the kinetic rate constants, etc.)of the system. The actual free energy values, on
the other hand, are assumed to be relevant when
we consider the energy balance. In the 'coupling
ratio constraint' mechanism, there is, in general, adiscrepancy between the effective and actual freeenergy values. The difference between the effective
and actual free energy values is a novel feature of
the 'coupling ratio constraint' mechanism. In theconventional scheme of coupled reactions, there isno discrepancy between the effective and actual
free energy values (see, for example, Mogi
1993a).
The question then arises as to the origin of
the difference between the effective and actual freeenergy values. Since the difference between thetwo free energy values can ultimately be only
accounted for by the heat bath, it is reasonable toassume that the difference between the effective
and actual free energy values is provided
temporarily by the thermal fluctuation. Namely,
we assume that the coupling ratio constraint
mechanism works by utilizing a temporal energy
difference (i.e., the difference between theeffective and actual free energy gradients) provided
on loan from the heat bath in the form of thethermal fluctuation.
In consolidating the physical validity of the'coupling ratio constraint' mechanism, thestatistical properties of the scheme are expected to
be important. In particular, if the difference
between the 'effective' and 'actual' free energy
values is to be provided by the thermal fluctuation,( x( n), y(n)) = a(n)(p,q) + b(n)(−q, p)(14)
We havea(n) = px(n) + qy(n)b(n) = −qx(n) + py(n).
(15)

The average values of a(n) and b(n) are
obtained asa(n) = p(v1′ − v2′ )n + q( u1′ − u2′ )n =1(v ′ − v′ )np 1 2

b(n) = 0.
(16)

The standard deviation of b(n) is related tothe standard deviations of x(n) and y (n) as
σ b(2 n) = q 2σ x2( n) + p2σ y2( n),(17)where the standard deviations on the right side are
given as8

σ x2(n ) = n(v1′ + v2′ ) (u1′ + u2′ ) + 4nv1′v2′
σy2( n) = n (u1′ + u2′ )( v1′ + v2′ ) + 4nu1′u2′

gradients is the thermal fluctuation. Note also that
this result is guaranteed by the condition of (7).
Using (19), we obtain the standard
deviation for δE(n) as.
(18)

Using these relations, and making use of(10), the standard deviation of b(n) is obtained as

2

∆E ′
∆ E′
pq  1+ e − kT  + 8 pqe− kTσ δE ( n) = n( ∆Ex − ∆E y )∆E ′2( p + q) 1 + e − kT 

∆E ′
∆E ′
pq 1 + e − kT  + 8pqe − kT∆E ′( p + q )1 + e − kT 
.
(19)
2

σ b( n) = n(22)
A problem arises when we examine theproperty of formula (22). From the nature ofthermal fluctuation, the probability of theoccurrence of a fluctuation of the scale ~∆E is
expected to scale as

Let us now examine the statisticalproperties of the difference between the effectiveenergy value E' (11) and the actual energy value
E (3). If the system has undergone a transition of(x(n), y(n)), the difference between the effectiveenergy value E'(x(n), y(n)) and the actual energy
value E(x(n),y(n)) is given asδE(n ) = E ′ (x( n), y(n)) − E(x (n), y( n))= x (n)∆E ′ + y(n) ∆E ′ − x (n)∆E + y (n) ∆E
x

y

(

x

y

∆E

~ e− kT(23)
However, the fluctuation in the difference
between the effective and actual energy values
δE(n) (22) for the coupled reaction does not scale
as in condition (23). It is seen, for example, thatthe fluctuation given in (22) increases linearly withthe absolute values of ∆E x and ∆E y when the)( py(n) − qx( n))(∆Ex − ∆Ey )
p +q
b (n)
=
( ∆Ex − ∆E y )p +q=

effective energy gradient ∆E' is ~0. This situation
is clearly incompatible with the assumption that theorigin of the difference between the actual andeffective energy values is the thermal fluctuation.
This is the 'fluctuation problem'.
We therefore postulate that there is somemechanism that restricts the 'thermal interference'
effect so that the 'fluctuation problem' disappears.
One possibility would be that the coupling
constraint mechanism as developed in the previous
section is valid only when the conditions(20)
The average value of δE(n) is thereforeb(n)( ∆Ex − ∆E y )p+q=0

δE(n) =

(21)
Namely, there is, on average, nodiscrepancy between the effective and actual freeenergy values. This is in accordance with theassumption that the origin of the difference
between the effective and actual free energy∆Ex <~ kT and ∆Ey <~ kT(24)
are satisfied. In this picture, the 'thermal
9interference effect' would be prominent only when
the conditions in (24) are satisfied.
We conclude from the argument developedin this section that there is a mechanism which
restricts the occurrence of 'thermal interference' inaccordance with the free energy change involved.
It is to be noted that the situation is similar to thecase of quantum interference, where theinterference is restricted within the energy scale of~h. However, in the case of 'thermal interference'
the role of scale ~kT is expected to be more
complicated, as will be discussed later.

v1′ + v2′ + u1′ + u2′ = 1(v1′ − v2′ )( ∆Ex′ − ∆E x ) + (u1′ − u2′ )( ∆E′y − ∆E y ) = 0
∆E x′
v1′
= e − kT
v2′
∆E y′
u1′
= e − kT
u2′

(25)
We solve equations (25) taking the effective
free energy gradients ∆E x ,' ∆E y ' as given.
We then obtain∆E y′ ∆ E′x e− kT 1− e − kT ( ∆E y − ∆Ey′ )


v1′ =
W
∆E y′


1 − e − kT  (∆ Ey −∆ Ey′ )


v2′ =
W
∆E y′
∆E x′
e − kT 1 − e − kT  (∆E ′x −∆ Ex )


u1′ =
W
∆E x′


1− e − kT ( ∆E ′x −∆Ex )


u2′ =
W

4. Absolute meaning of scale ~kT
In the previous section, we saw that the
coupling ratio constraint mechanism introduced inthis paper does not on average violate the second
law of thermodynamics. However, we have seenthat the energy fluctuation is expected to become
too large when the energy gradients ∆E x and ∆E y
become large compared to the energy scale ~kT.
In this section, we show that if we
introduce an absolute meaning to scale ~kT and
restrict the 'thermal interference' within this energy
scale, we can solve the 'fluctuation problem'.
Let us write down again the equations that
determine the transformed rate constants and freeenergy gradients(v1′, v2′ , u1′ , u2′ ,∆Ex′ , ∆Ey′ )coupledin the 'coupling ratio constraint' mechanism.
Namely,

,
(26)

where
∆E′y 
∆E′y 
∆E ′x ∆E x′ 
W = 1 + e − kT  1 − e− kT  (∆E y −∆ Ey′ ) + 1 − e − kT  1 + e− kT  (∆ Ex′ −∆ Ex )







.
We now introduce an absolute meaning toscale ~kT and restrict the 'thermal interference'
within this energy scale. In order to achieve this,we define scale function F, which is assumed tosatisfy the following conditions.
(1)
F is a monotonously decreasing function of
the absolute values of the free energy gradients∆E x and ∆E y .
10

∂F
<0
∂ ∆E x∂F<0∂ ∆E y(2)

F tends to 1 as both |∆E x | and |∆E y |approach zero.
lim F = 1
∆E → 0x∆E y → 0(3)

F tends to zero as|∆E x | and |∆E y |(∆E −∆ E ) pq 1− exbecome large.
lim F = lim F = 0∆Ex →∞

energy gradients, and there is no thermalinterference effect. In addition, the thermalinterference effect is prominent only in the energy
scale of <~kT. Namely, scale function F
introduces an absolute meaning to scale ~kT, andrestricts the thermal interference effect within thisenergy scale.
Under the assumption of (28), the standard
deviation for δE(n) is obtained asσ δE( n) = nFy2 ∆E x′−kT2
2∆ Ey ′
 ∆E ′x 
∆E y′
∆Ey′
∆Ex ′ 2 





 1− e− kT  + 4p 2q2  e− kT  1 −e − kT  +e − kT  1−e − kT  







∆E ′x



∆Ey′





∆Ex′



∆ E′y



( p + q) p1 + e − kT 1− e− kT  + q1 −e − kT 1+ e− kT ∆E y →∞(4) The characteristic energy scale of function F
is ~kT.

(29)

An example of scale function F thatsatisfies the above four conditions would be

Note that σδE(n) is proportional to scale

F = e−∆Ex 2 +∆ Ey 2
4( kT )2.
(27)

We assume that the effective free energy
gradients ∆E x ' and ∆E y ' depend linearly on scale
function F,∆ Ex′ = ∆Ex + F (∆E ′ − ∆Ex )∆ Ey′ = ∆Ey + F( ∆E ′ − ∆E y ),(28)

where ∆E' is the effective energy gradient (9)derived in the coupling ratio constraint mechanismof section 2.
In (28), scale function F can be considered
to be the measure of the extent to which thecoupling ratio constraint (and the thermalinterference effect) (6-A) holds. When F=1, theeffective energy gradients in the x- and y- degrees
of freedom are given by the single value of E', andthere is maximum thermal interference effect.
When scale function F=0, there is no change in thefunction F. We see that σδE(n) now tends to zero
when the absolute values of the energy gradients|∆E x | and |∆E y | become large. This is inaccordance with the assumption that the difference
between the effective and actual free energy values
is provided by the thermal fluctuation.
In conclusion, we have seen that if we
introduce an absolute meaning to scale ~kT, we canmake the coupling ratio constraint mechanism
reasonable in its fluctuation behavior. The thermalinterference effect would then be prominent only inthe energy scale of <~kT.
Let us note, however, that at the present
stage, it is premature to discuss the nature of scale
function F in a more exact manner. It is expectedthat there are some complications, as discussed in
the next section.

5. Discussions
In this paper, we proposed a couplingmechanism which can explain the basic features ofcoupling observed in biological systems.
11

We argued that in considering themechanism of enzyme coupled reactions, it isimportant to take into consideration theexperimental observation that the coupling ratio is
not necessarily one. The importance of this kind of
coupling mode (loose coupling) has already been
pointed out (Oosawa & Masai 1982, Oosawa &Hayashi 1983).
We have assumed that the action of the
enzyme can be stated in the condition that the
coupling ratio is fixed at a constant value (thecoupling ratio constraint mechanism). We haveshown that this assumption successfully explains
the transduction of free energy in a coupled
reaction where the coupling ratio is not one. It isinteresting to consider the enzymatic mechanismsthat may underlie such a constraint on the coupling
ratio. In particular, condition (6-A) should reflect
some symmetry in the reaction space of thecoupled reactions imposed by the enzyme, and it isof much interest to consider the molecularmechanisms which may be involved.
From the analysis of the scheme that we
presented in this paper, we can make the following
predictions about the nature of mechanochemical
coupling in muscle contraction (Mogi 1993b).
(1) ATP is not necessarily hydrolyzed when one
working stroke occurs.
(2) When ATP is not hydrolyzed, the chemical
state of myosin is the same before and after theworking stroke.
(3) The sliding of actin filament is uniform
regardless of the particular time at which thehydrolysis of ATP is completed.
(4) The multiple working strokes during thehydrolysis of ATP is driven by exactly the sameamount of free energy change.
(5) The number of intermediate states involved incoupling can be much smaller than the number ofworking strokes conducted per ATP hydrolyzed.
These predictions can be tested in future
experiments to verify the validity of the scheme
presented in this paper.
The coupling mechanism we introduced inthis paper shows a 'thermal interference effect', inanalogy with the quantum interference effect. The
basic features of our model that lead to the 'thermalinterference effect' can be summarized as follows.
(1) A stochastic process in two degrees of freedom
is restricted,on average, to one degree of freedom,through the 'coupling ratio constraint' imposed by
the enzyme.
(2) The actual transitions, however, continue tooccur in two degrees of freedom.
(3) The 'discrepancy' between the average single
degree of freedom and the actual two degrees of
freedom leads to an interference between the two
coupled degrees of freedom.
It is to be noted that the logical structureleading to the 'thermal interference effect' is similar
to the logical structure leading to the quantum
interference effect. Several authors have pointed
out the possible connection between the quantum
and stochastic processes (e.g. Barnes & Silverman
1934, Nelson 1966, Nelson 1967).
However, it should be stressed that the'thermal interference effect' that we proposed here
is independent of the quantum interference effect.
Limited cases of quantum mechanical effect are
observed in enzyme kinetics (e. g., electron
tunneling (de Vault 1984), and tunneling ofhydrogen atoms (Cha et al. 1989)). However,since the degrees of freedom in our model are
actually the 'degenerate' expressions for many
internal degrees of freedom, our system is too'macroscopic' for any quantum processes to takeeffect. Therefore, the 'thermal interference effect'that we proposed in this paper is of purely
stochastic character.
In quantum mechanics, an absolutemeaning is given to the size of the system (Dirac
1958). Namely, the quantum interference effectoccurs only when the scale of the system involved
is of the order of Planck's constant h. We havesuggested that there is a similar absolute meaning
to energy scale ~kT in the thermal interference
effect. Namely, if we assume that the thermalinterference effect is prominent only when theenergy scale involved in coupling is of the order of12

~kT, we can solve the 'fluctuation problem'.
It should be noted, however, that it is, atpresent, not clear how scale function F is exactly
given as a function of the energy gradients
involved in the coupled reaction. In muscle
contraction, the free energy change accompanying
the phosphate release step (the step considered tobe essential in mechanochemical coupling) is about
7 kcal/mol (Shriver 1984) or 12 kT per moleculeat room temperature (~300K). If scale function F
decreases rapidly in energy scale >~kT, the thermalinterference effect involved in mechanochemical
coupling under physiological conditions would besmall. However, experimental evidence in vivo
(Higuchi & Goldman 1991, Lombardi et al. 1992)
demonstrates that the coupling ratio is large underphysiological conditions, suggesting the existenceof a substantial thermal interference effect. One
possibility which we did not discuss in detail inthis paper is that scale function F may depend
critically only on the positive (unfavorable) freeenergy gradient ∆E x . Another possibility is thatprocesses. Rev. Mod. Phys. 6, 162-192.
Bohm, D. 1951 in Quantum Theory (Dover)494.
Bridgman, P.W. 1928 Note on the principle ofdetailed balancing. Phys. Rev. 31, 101-102.
Brooks III, C. L., Karplus, M. & Pettitt, B. M.
1988 in Proteins (John Wiley & Sons) 146-147.
Cha, Y., Murray, C. J. & Klinman, J. P. 1989
Hydrogen tunneling in enzyme reactions. Science
243, 1325-1330.
Cordova, N. J., Ermentrout, B. & Oster, G. F.
1992 Dynamics of single-motor molecules: The
thermal ratchet model. Proc. Natl. Acad. Sci.
U.S.A. 89, 339-343.
Dirac, P. A. M. 1958 in The Principles of
Quantum Mechanics. (Fourth Edition, Oxford
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Feynman, R., Leighton, R. & Sands, M. 1963 in
The Feynman Lectures on Physics. Vol. I,
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Harada, Y., Sakurada, K., Aoki, Y., Thomas, D.
D. & Yanagida Y. 1990 Mechanochemical couplingin actomyosin energy transduction studied by inscale function F may depend on coupling ratio C.
vitro movement assay. J. Mol. Biol. 216, 49-68.
It would be at the present stage premature,
Higuchi, H. & Goldman, Y. E. 1991 Sliding
however, to discuss the exact nature of scaledistance between actin and myosin filaments perfunction F.
ATP molecule hydrolysed in skinned muscle
fibres. Nature 352, 352-354.
Acknowledgements
Irving, M., Lombardi, V., Piazzesi, G. &Ferenczi, M. A. 1992 Myosin head movements
We would like to thank (in alphabetical are synchronous with the elementary forceorder) Dr. Horace Barlow, Dr. Takao Kodama, Dr. generating process in muscle. Nature 357, 156Vincenzo Lombardi, Dr. Gabriella Piazzesi, Dr. 158.
Ken Shiotani, Dr. Kaoru Takeuchi, Dr. Makio Ishijima, A., Doi, T., Sakurada, K. & Yanagida,
Tokunaga, Dr. Takeyuki Wakabayashi, Dr. Toshio T. 1991 Sub-piconewton force fluctuations of
Yanagida, and Dr. Takuo Yasunaga for helpful actomyosin in vitro. Nature 352, 301-306.
discussions. We are grateful to Dr. Setsuro Ebashi Lombardi, V., Piazzesi, G. & Linari, M. 1992
and Dr. Masao Ito for valuable advises and Rapid regeneration of the actin-myosin power
encouragements.
stroke in contracting muscle. Nature 355, 638641
Maxwell, J. C. 1871 in Theory of Heat 1st ed.
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13

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