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Thermodynamic coupling in chemical reactions
J. theor. Biol. (1975) 49, 323-335
Thermodynamic Coupling in Chemical Reactions JOEL KEIZER
ChemistlT Department, University of California, Davis, California 95616, U.S.A. (Receioed 14 January 1974, and in revised form 10 June 1974) The significance of thermodynamic coupling in chemical reactions--which recently has been questioned on thermodynamic grounds--is examined from the point of view of kinetics. It is shown that considerations of stoichiometry lead to a meaningful definition of velocity for elementary and certain elementary-complex reactions, whereas the non-equilibrium thermodynamic definition of reaction velocity is ambiguous in multireaction systems. Based on rate laws which include the effects of nonideality, it is proven that elementary-type reactions are not coupled thermodynamically and concluded that thermodynamic coupling has no kinetic significance. A discussion is given to show that this result is compatible with coupling in both biochemical systems and oscillating reactions. 1. Introduction in his pioneering work on non-equilibrium thermodynamics De Donder (1936) formulated the concepts of the rate and driving force o f a chemical reaction. When a single reaction occurs in a uniform system at constant temperature and pressure, De Donder showed that
-dG/dt = AV, where G is the Gibbs free energy, A is the affinity of the reaction, and V is the reaction velocity. Because the Second Law asserts that under these conditions the free energy is a monotone decreasing function of time, one has A V >_ 0,
(i)
and so the affinity and velocity in such a system always have the same sign. This result will be called De Donder's Theorem and means that the direction in which the reaction proceeds is governed by the sign o f the affinity. F o r example, if A > 0, then V > 0 so the reaction goes from left to right. On 323
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the other hand, if A < 0, then Y < 0 and the reaction proceeds in the backward direction. Also the equilibrium condition is determined by ,4 - 0, and so deviations of ,4 from this null point will cause the reaction to proceed spontaneously in the prescribed direction. When several reactions can occur in a uniform system at constant temperature and pressure, the local equilibrium theory used by De Donder to obtain equation (l) gives only that (Prigogine, 1967) -dG/dt = ~, Ap Vp (2) P
where p labels the reactions. Thus it would seem that it is possible for individual terms Ap Vp in this sum to be negative--as long as the sum ~ Ap Vp p
is itself positive (Katchalsky & Curran, 1965; Prigogine, 1967, p. 25). As a consequence a reaction could spontaneously convert reactants into products (Vp > 0) even though products were in excess (Ap < 0). This has been called "thermodynamic coupling" and is reputed to be of "great importance in biological processes" (Prigogine, 1967, p. 25). The first indication that this might not be the case appears in an interesting paper by Koenig, H o r n e & Mohilner (1961). They consider an arbitrary collection of independent reactions and show that there always exists a thermodynamically equivalent set for which ,4p Vp >_ 0. Since thermodynamics by itself provides no criterion for choosing among equivalent descriptions, they conclude that thermodynamic coupling has no thermodynamic significance. They leave open, however, the possibility that the study of reaction mechanisms may provide non-thermodynamic criteria for picking reactions which show thermodynamic coupling and, thereby, give meaning to the concept. It is the purpose of this note to present some considerations about thermodynamic coupling from the perspective of chemical kinetics. The definition of velocities for elementary reactions is examined, and it is shown that it can be extended only to a small class of complex reactions. This is contrasted with the usual non-equilibrium thermodynamics definition which is shown to lead to both ambiguous and physically unintuitive velocities. This ambiguity is due to the fact that the theory makes no distinction between complex reactions and elementary-type ones. However, the reactions in a system can always be analyzed in terms of the elementary-type ones and it is shown from the properties of the rate laws for these reactions that their velocities and affinities always have the same sign. Thus elementary-type reactions do not show thermodynamic coupling. From the view of kinetics, then, it is not possible to attribute any significance to thermodynamic coupling. The compatibility of this result with established ideas about chemical coupling is discussed.
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2. Velocities for Elementary Reactions As is weU-known an elementary chemical reaction is a reaction which proceeds according to microscopic events involving only the species represented in the reaction stoichiometry (Frost & Pearson, 1965). For example, the gas phase reaction between H and Br2 is an elementary reaction and so the stoichiometric equation H+Br2 = HBr+Br accurately represents that the forward and reverse rate processes result from encounters involving only H and Br2, and HBr and Br respectively. On the other hand, the stoichiometric equation for a complex reaction does not correspond to a reaction mechanism. Such is the case with the enzymatic conversion of a substrate S into a product P with the stoichiometric equation S=P. Since the enzyme E is a catalyst, it does not enter into the stoichiometry. Many enzyme catalyzed reactions, however, proceed through an intermediate complex with a mechanism represented by two elementary reactions (Gardiner, 1962) E + S = ES ES = E + P . Thus in addition to the reactants and products a new species, the enzymesubstrate complex ES, is present in the system. The difference between elementary and complex reactions is important in non-equilibrium thermodynamics because a complex reaction may not have a well-defined velocity. For example, consider the enzyme catalyzed reaction above. If the reaction stoichiometry told the whole story of the mechanism, then conservation of mass would imply that changes in the number of molecules of S and P due to reaction would be related by - d N s = dare = d~, which would define a progress variable 4. However, because the reaction is complex, the conservation relationship is actually that Ns + N~s + Np is constant, so that - d N s = dNp + dN~. Thus, in general, - d N s ~ dNp and so the loss of reactant S is not the same as the increase in product. This is clearly a property of all complex reactions and means that complex reactions will not, in general, have a weN-defined velocity. In fact in the reaction above only when the enzyme-substrate complex is a transient intermediate, so that d N ~ ~ 0, does the stoichiometric equation permit a reaction velocity to be assigned. If the species ES is present in experimentally negligible amounts then it is negligible in the thermodynamic treatment. In addition, if its rate of change is experimentally negligible, then the time dependence of the intermediate can be neglected in the rate equations. It seems reasonable to call complex reactions in which all inter-
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mediates are thermodynamically and kinetically negligible "elementarycomplex" reactions. As seen from the example such reactions will have an experimental reaction velocity defined by De Donder's procedure from the reaction stoichiometry. An important point to note about elementarycomplex reactions is that since the rate of change of all intermediates is negligible, the intermediates may be eliminated from the rate equations. Thus the reaction velocity of an elementary--complex reaction will not depend explicitly on the intermediate compounds. Many elementarycomplex reactions are known, the archetype in the gas phase being the unimolecular decomposition of cyclopropane. In the case of the enzyme catalyzed reaction discussed above the well-known Michaelis-Menten steady state treatment is based on the neglect of the enzyme-substrate complex. This condition is dNEs = 0, and so the reaction does have a velocity. Situations are known, however, in which the enzyme-substrate complex is not a transient intermediate and in these cases the complex reaction must be treated as two separate, but interlocking, elementary reactions--each with its own velocity (Keizer & Bernhard, 1966). The important differences between elementary, elementary--complex, and complex reactions do not seem to have been recognized by De Donder (1931) and others (Katchalsky & Curran, 1965). Indeed, De Donder introduces the idea of a reaction velocity using the Haber reaction 3Hz + N2 = 2NH3 without commenting about the complexity of its mechanism. As shown here a truly complex reaction does not possess a reaction velocity and it will be shown below that the failure to recognize this is, in a sense, the real origin of thermodynamic coupling. 3. Thermodynamic Reaction Velocities The usual non-equilibrium thermodynamic approach to chemical reactions involves adopting a maximal set of m linearly independent reactions. If there are n species--denoted by C ~ , . . . , C,--involved in these reactions, then the chemical equations are written symbolically as Z vp~Ct = 0, i
where the vp~ is the stoichiometric coefficient for species i in the pth reaction. The affinity is a property of a symbolic reaction and is definedby /lp = - Z vpi#~, t
where/q is the molecular chemical potential. The usual theory then defines the reaction velocities, Vp,by the equations (Koenig, Horne & Mohilner, 1961) dNffdt =-"E vp, Vp, (3) P
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~vhere N~ is the number of molecules of species i. This definition makes the theory macroscopic and empirical since once the set of reactions is fixed, i.e. once the matrix vpi is delineated, the velocities may be determined from experiment by inversion of this equation. The inversion is always possible since the reactions are independent. The definitions lead in the usual way to the expression for the free energy production given in equation (2). Although this procedure is consistent, it has the ambiguous feature of allowing the set of reactions to be chosen as any linear combination from any linearly independent set of reactions. This is the origin of the so-called transformation theory of chemical reactions (Prigogine, 1967, p. 41). It is also the origin of the remark that one can "confer or revoke thermodynamic coupling ... by the stroke of a pen" (Koenig et al., 1961) since certain linear combinations are thermodynamically coupled and others are not. Although this ambiguity does not produce mathematical errors, it makes the "rate of reaction" impossible to interpret because a given reaction can be assigned a number of velocities by this procedure. Two examples illustrate the problem quite clearly. First consider two linearly independent elementarytype reactions: I A=B 2 B=C. Because these are elementary-type reactions, they have well-defined velocities Vt and I/2 as described in the preceeding section. From the so-called thermodynamic point of view, these reactions are "equivalent" to the complex reactions, I' A = B 2'
A + C = 2B,
which using the definition in equation (3) have velocities V~ = VI + V2 and V~ = - V2. Reaction I', however, is identical to reaction I (it involves identical stoichiometry and has the same affinity), yet in this "equivalent" scheme the reaction has a different velocity. The new velocity has also lost its intuitive meaning since the reaction A = B can now go from left to right (V~ > 0) or from right to left (V~ < 0) even when no A is being converted to B! This will happen when VI = 0 with V2 ~ 0 and certainly is an unsatisfactory property of a reaction velocity. The same problem occurs when the elementary reactions in a system are linearly dependent, as in the set I
A=B
2
B=C
3
C=A,
with velocities V1, V2 and V3. If the linearly independent set is chosen to be r.B.
22
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reactions 1 and 2, then definition (3) confers the velocities V; = 1/1- 113 and V~ = V2 - / / 3 ; whereas if reactions 2 and 3 are chosen, then they have velocities V~' = V2- Vt and V~' = 173- Vt. Thus the velocity of the reaction B = C can be V2, 112- V3, or 1/'2- V, depending on the choice of independent reactions. Since the velocity is the time derivative of the extent of reaction, there is the further problem that the extent of a reaction also depends on this choice. Thus using equation (3) neither the velocity or progress of a reaction are properties of the reaction, itself, but rather reflect only the arbitrary choice made for the whole collection of reactions. This, of course, is not the case with the affinity which is a property of a given reaction. From the point of view of kinetics, this multivalueness of the velocity of a reaction is an unacceptable shortcoming of definition (3). It should be noticed that this problem does not exist at full equilibrium since then all velocities vanish and so this arbitrariness will have no special consequences there. Away from equilibrium its drawbacks are typified by the two examples given above. In the first example, it might be said that the problem arises because the reaction mechanism has been ignored by considering the kinetically meaningless net reaction A + C = 2B, whereas in the second example important mechanistic steps have been deleted by not considering all the elementary processes. In case the stoichiometric coefficients of the elementary-type reactions are independent, it is clear from a mechanistic point of view that these reactions should be used in the thermodynamic treatment. In this case the velocities defined in the previous section will agree with those obtained from equation (3). On the other hand, even when the stoichiometric coefficients are dependent, the use of the elementary-type reactions is dictated by kinetics. Actually the compelling reason for choosing linearly independent reactions is to assure that definition (3) can be inverted for the velocities. Indeed independent variables need be chosen in dynamical problems only out of convenience, and in some cases dependent variables may be a better choice. This certainly seems the case for coupled reactions---especially in view of the arbitrary way in which a velocity is associated with a reaction when independent reactions are used. 4. De Donder's Theorem for Elementary-type Reactions
In this section the possibility of thermodynamic coupling among elementary-type reactions is examined. If n such reactions occur in a system at constant temperature T and pressure p, then the local equilibrium theory (Katchalsky & Curran, 1965, chap. 7) gives
dG/dt + S d T / d t - V dp/dt = ~ ~li d N / d t
(4)
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with S the entropy and V the volume. The method for defining the velocity of the pth elementary-type reaction has been described earlier and permits the rate of change of the amount of the ith species to be calculated as a sum of contributions from each reaction:
dNl/dt = ~. vpi Vp. P
This then gives equation (4) the form
dG/dt+S d T / d t - V dp/dt = - • At, Vp
(5)
P
where the sum is over all the elementary-type reactions. By introducing activities, the chemical potentials may be written tti =/~o + kB T In as, where kB is Bottzmann's constant and #~' is the chemical potential of some standard state and depends only on the temperature. With this definition the affinities become
Ap = - ~ Vp,l ~ - k , TIn [I-I (a?')l. Ll
I
Finally recalling the definition of the equilibrium constant (Wall, 1958)
- ~ vpi#~ = kBT In Kp, t
and denoting the bracketed activity product above by Qp, gives Ap = kB T In (KflQp).
(6)
This form for the affinity will be used below. As discussed in the Introduction, even when the temperature and pressure are held constant, the monotonicity of G cannot be used to determine the relative signs of Ap and Vp. This depends on the form of the rate law, and the rate laws for elementary-type reactions have properties which make Ap Vp > 0. In order to see this more clearly, the reaction velocity is broken up into a positive forward velocity V+ and a positive reverse velocity VZ, as is usual in chemical kinetics. Because of the sign convention for the stoichiometric coefficients, these two velocities are positive and their difference is the reaction velocity, so Vp = V+ - V ; . (7) The following two properties of the rate laws for elementary-type reactions imply that ApVp >_ 0: (i) The ratio of the forward velocity to the reverse velocity depends on the number of molecules only through the ratio Kp/Qp, i.e.
V~/V;- = fp(Kp/Qp),
(8)
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where the dependence of fp on other local thermodynamic variables has been made implicit. (ii) In the absence of other processes the rate law given by equation (7) corresponds to a unique stable equilibrium at KflQp = 1. With the possible exception of the singular points KflQp = 0 and oo, all points must asymptotically proceed to the equilibrium point.t Neither of these properties is arbitrary and both actually hold for the form of rate laws which describe elementary reactions in the gas phase and in solutions. For example, for the bimolecular elementary reaction in solution, k_._~+ CI-~ C 2 ~ - C3-~-C4, transition state theory gives (Frost & Pearson, 1965) V ; = k+ala2 V;- = k - a 3 a 4
(9)
where the a's are activities. So that
v; /v;
=
(k +/k-)/(a3 a ,/a, a2).
But the equilibrium condition Vv+ = Vv- implies that the equilibrium constant for this reaction is K = k+/k -, so condition (i) is clearly met using f ( X ) = X. In the gas phase a similar result holds using fugacities or--in the case of an ideal gas, partial pressures-in the place of activities. Hence the well-known mass action laws guarantee that condition (i) will be met. For non-ideal systems the form of the mass action laws as written above is less well-established, but is implied by the transition state theory. Since transition state theory successfully describes many of the effects of non-ideality in solution reactions, these expressions probably represent the dominant kinetic effects of non-ideality. Thus to the extent that mass action laws involving the affinities are valid, condition (i) will be met even in non-ideal systems such as encountered in solution. An example of an elementary complex reaction satisfying condition (i) is the generalized Michaelis-Menten scheme E+S ~
k-I
ES ~
k-2
ES
E'+P.
If dNEs[dt ,.~ O, this will be an elementary--complex reaction. Using this steady state assumption it is shown in the Appendix that it is possible to write the velocity for this reaction as V = V + - V- and again V + / V - = K/Q. t An example of a singularity at K/Q = 0 comes from the reaction C~+C2 = 2C2 which has a metastable state at N~ = O.
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It is possible to construct other examples of elementary-complex reactions satisfying condition (i), and following Hollingsworth (1952) it is believed that the rate laws for all elementary--complex reactions may be written so that V + / V - = f ( K / Q ) . The validity of the second condition, i.e. existence of a unique, stable equilibrium, is commonly assumed by chemists and has actually been proven (Shear, 1968) for polynominal rate laws. PROOF THAT Ap Va > 0
From equation (6) the sign of the affinity Ap is determined solely by the size of the dimensionless ratio X = Kp/Qp. In fact, as is shown in Fig. 1, A p > 0 when X > 1, A p = 0 at X = 1, and A p < 0 when 0 < X < 1.
/ 0
/ /../.I I
I 2
X=K# Op FIG. 1. A schematic representation of the functions In X and forward to reverse reaction rates.
fD(X),
the ratio of the
Using conditions (i) and (ii) above it will be shown that Vp has the same sign as Ap in these regions. Equivalently equation (7) shows that Vp = v ; ( v ; / v ; - l ) = V~-(fp(X) - 1).
Thus since Vp- is strictly positive, Vp is positive whenfp(X) > 1 and negative when fp(X) < 1. Hence it suttees to show that on 1 < X < o% f p ( X ) > 1 and on 0 < X < 1, fp(X) < I. To see this, recall that X = 1 is the equilibrium point, so fp(1) -= 1 by condition (ii). But since the equilibrium is stable, it follows for small positive 5 thatfp(1 +6) > 1. This is so because if X = K f l Q p > 1, then the activity quotient Qp must increase as time
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proceeds in order that equilibrium be reattained. Thus the activity of the products must be increased, i.e. the reaction must go in the forward direction. Thus Vp > I or fp(1 +6) > 1. Similarly it follows from the stability of the equilibrium that for small positive 6, fp(1 - ~ ) < 1. Furthermore there can be no X on (0, co) other than X = 1 for whichfp(X) = 1. Otherwise there would be another point with Vp = 0, implying the existence of a collection of states which would not proceed to equilibrium. This would violate condition (ii). Thus fp(X) is greater than 1 on 1 < X < oo and less than 1 on 0 < X < 1 and so Ap and Vo have the same sign.
5. Scope and Consequence of the Theorem In order to emphasize the range of validity of this extension of De Donder's theorem, we note that the proof given above in no way depends on the constancy of the temperature and pressure. In fact what is required beyond conditions (i) and (ii) is that the system be uniform and in local equilibrium, so that equation (6) can be used to define the affinities. Furthermore, if the reactions occur in an open system in which the number of molecules are being uniformly changed by an external source, the result will also hold. Indeed, in this case
dNJdt = ~ vp~Vp+ Ri, P
where R~ is the external rate and Vp are the reaction velocities. Thus even in an open system as long as the chemical rate law satisfies conditions (i) and (ii), it remains true that Ap Vp > 0. In this regard it should be noted that conditions (i) and (ii) require detailed balance to hold only at the equilibrium state implied by each elementary reaction when decoupled from the remaining reactions. Since detailed balance need not occur at the steady states of the open system, the result applies even to far from equilibrium steady states. Thus the result that the direction of an elementarytype reaction is determined by its al~nity is seen to hold for a broad class of uniform coupled chemical systems throughout the entire domain of local equilibrium. The proof of De Donder's theorem for elementary-type reactions is based on the form of the chemical rate laws and so is a kinetic proof. As pointed out above, a purely thermodynamic proof of the theorem is not possible. On the other hand, thermodynamic considerations do show that thermodynamically coupled reactions can always be uncoupled by a simple transformation (Koenig et al., 1961). The results obtained here make an even stronger point, namely, that the only cases in which thermodynamic coupling is found are those in which inherently uncoupled elementary processes are
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333
naively added together or deleted yielding a description in terms of complex reactions. In order to reinforce this conclusion, two examples of the effect of chemical coupling are discussed to show that they are not compromised by this result. Consider first the condensation of glucose and fructose to form sucrose in aqueous solution (Lehninger, t965, chap. 4) glucose + fructose = sucrose + H20. This reaction is thermodynamically unfavorable under standard conditions because the equilibrium constant is less than one. In fact, AG ° = 5-5 kcal. However, when the hydrolysis of adenosine triphosphase (ATP) is mechanistically coupled to this reaction one has ATP+glucose+fructose = sucrose+ADP+phosphoric acid (10) which now is a favorable reaction since AG° = - 1 . 5 kcal. Thus the hydrolysis of ATP is said to "drive" the condensation of the disaccharide toward completion. This sort of "driving" through coupled reactions clearly has nothing to do with the affinity, since it is merely a change in the equilibrium due to coupling and the affinities all vanish at equilibrium. Moreover this is a complex reaction and so does not even have a reaction velocity. Indeed its mechanism may be written as (White, Handler & Smith, 1964, p. 409) ATP + glucose = glucose 6-phosphate + ADP glucose 6-phosphate = glucose I-phosphate glucose 1-phosphate + fructose = sucrose + phosphoric acid. These are all elementary-complex enzyme reactions catalyzed by hexokinase, phosphoglucomutase, and sucrose phosphorylase respectively and each has a well-defined velocity and a AG°. The sum of the standard free energies for the three component reactions is just the - 1.5 kcal required for the net reaction written in equation (t0). Hence this well-known sort of biochemical coupling has nothing to do with kinetics, and its validity is not endangered by a lack of thermodynamic coupling. Purely kinetic-type coupling does exist in complex homogeneous chemical systems--a striking example occurring in the oscillating Belousov reaction (Field, KOrrs & Noyes, 1972). While the details of the mechanism of this process are quite complicated, Field & Noyes (1974) have summarized the mechanism in terms of the symbolic elementary reactions A+Y~eX X+Y~P B+X~2X+Z 2Z~ Q z~eY
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where X - HBrO2, Y - Br-, Z - Ce(IV), and the other symbols represent kinetically unimportant products or quantities held in excess. The kinetic equations for the rate of change of HBrO2, Br-, and Ce(IV) are easily written down and have been solved numerically to give a good representation of the concentration oscillations in this system. The kinetic equations used by Field & Noyes (1974) for these elementary reactions are compatible with the conditions (i) and (ii) used to prove that reaction velocities have the same sign as their affinity. Thus even chemical oscillations do not require a reaction to be driven against its affinity by other reactions. Indeed the generalization of De Donder's theorem implies through equation (5) that at constant temperature and pressure each elementary reaction will decrease the free energy. This means that chemical reactions are still inherently dissipative even though the concentrations of particular compounds may rise and fall almost periodically for long periods of time. In this sense the origin of the so-called "order" due to the chemical oscillations is kinetic and not thermodynamic. It should be clear now that the reason De Donder's theorem is valid is that it focuses on reactions rather than compounds. In fact, if attention is shifted back to individual chemical species, then at constant temperature and pressure equation (4) may be written dG/dt = ~ p~ dNJdt. i
Each term in this expression is not inherently negative and in the case of chemical oscillations can even change sign during the course of reaction. This, however, can be adequately explained by kinetic coupling alone. It is only when the terms are grouped together for each elementary reaction that the inherently dissipative nature of the chemical reactions--and their thermodynamic independence--becomes clear. The author gratefully acknowledges several useful conversations with Professors Peter Rock, Nobuhiko Saito, and Ronald Fox. REFERENCES DE DONDER, T. & VAN RYSSELBERGHE, P. (1936). Theromdynamic Theory of Affinity. Stanford: Stanford UniversityPress. FIELD,R., K6R6S,E. & NoY~, R. (1972). J. Am. chem. Soc. 94, 8349. FIELD,R. & NOYES,R. (1974). J. chem. Phys. 60, 1877. FROST,A. A. & PEARSON,R. G. (1965). Kinetics and Mechanism, 2nd edn. New York: Wiley. GARDINER,W. C. JR (1972). Rates and Mechanism of Chemical Reactions. New York: Benjamin. HOLLINGSWORTH,A. (1952). J. chem. Phys. 20, 921. KATCHALSKY,A. & CrdRRAN,P. F. (1965). Non-Equilibrium Thermodynamics in Biophysics. Cambridge: Harvard UniversityPress.
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KEIZER,J. & BERNHARD,S. A. (1966). Biochemistry, N.Y. 12, 4127. KOENIG,F. O., HORNE,F. H. & MOHILNER,D. M. (1961). J. Am. chem. Soc. 83, 1029. LEttNINGER,A. L. (1965). Bioenergetics. New York: Benjamin. PRIGOGINE,I. (1967). Introduction to Th&modynamics of Irreversible Processes. 3rd edn, pp. 23-25. New York: Interscience. SHEAR,D. B. (1968). J. chem. Phys. 48, 4144. WALL, F. T. (1958). Chemical Thermodynamics. San Francisco: Freeman. WHITE, A., HANDLER, P. & SMrm, E. (1964). Principles o f Biochemistry. New York: McGraw Hill.
Appendix For the reaction kl
E + S ~ - ES k-i
k2
ES ~
k-2
E'+P
the stoichiometry is E+S =E'+P and the equilibrium constant is K = k 2 k l / k _ 2 k _ 1. Since N s + N ~ + N p is constant and d N B [ d t ,,~ O, the reaction velocity V = - d N s / d t = dNp[dt. Using the rate laws for the c o m p o n e n t reactions gives the equations V = -dNs/dt
= k t aEas - k _ I a~:s
= d N p / d t = - k _ 2 aE" av + k 2 aEs
(A1)
and 0 = dNEs/dt = - ( k _ 1 + k2)aEs + k l aE as + k _ 2 aE" av
where the as are activities. The steady state condition implies klaEas
k - 2 a E , ap
aES -- k _ l + k ~2 + k _ l + k 2 .
(-A2)
Using equation (A2) and the first equation for V in (A1) forward and reverse velocities m a y be defined, for example, b y t and
V + =ktk2aEas/(k_t+k2)
V- =k_tk_2aE, ap/(k_l+k2)
and (A1) implies V = V + - V - . F r o m this result and the definition of the equilibrium constant it follows immediately that V*lV
-
=KIQ.
t This simple decomposition was suggested by a thoughtful referee who also conjectured that a result like equation (A3) might hold in general.
Thermodynamic Coupling in Chemical Reactions JOEL KEIZER
ChemistlT Department, University of California, Davis, California 95616, U.S.A. (Receioed 14 January 1974, and in revised form 10 June 1974) The significance of thermodynamic coupling in chemical reactions--which recently has been questioned on thermodynamic grounds--is examined from the point of view of kinetics. It is shown that considerations of stoichiometry lead to a meaningful definition of velocity for elementary and certain elementary-complex reactions, whereas the non-equilibrium thermodynamic definition of reaction velocity is ambiguous in multireaction systems. Based on rate laws which include the effects of nonideality, it is proven that elementary-type reactions are not coupled thermodynamically and concluded that thermodynamic coupling has no kinetic significance. A discussion is given to show that this result is compatible with coupling in both biochemical systems and oscillating reactions. 1. Introduction in his pioneering work on non-equilibrium thermodynamics De Donder (1936) formulated the concepts of the rate and driving force o f a chemical reaction. When a single reaction occurs in a uniform system at constant temperature and pressure, De Donder showed that
-dG/dt = AV, where G is the Gibbs free energy, A is the affinity of the reaction, and V is the reaction velocity. Because the Second Law asserts that under these conditions the free energy is a monotone decreasing function of time, one has A V >_ 0,
(i)
and so the affinity and velocity in such a system always have the same sign. This result will be called De Donder's Theorem and means that the direction in which the reaction proceeds is governed by the sign o f the affinity. F o r example, if A > 0, then V > 0 so the reaction goes from left to right. On 323
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the other hand, if A < 0, then Y < 0 and the reaction proceeds in the backward direction. Also the equilibrium condition is determined by ,4 - 0, and so deviations of ,4 from this null point will cause the reaction to proceed spontaneously in the prescribed direction. When several reactions can occur in a uniform system at constant temperature and pressure, the local equilibrium theory used by De Donder to obtain equation (l) gives only that (Prigogine, 1967) -dG/dt = ~, Ap Vp (2) P
where p labels the reactions. Thus it would seem that it is possible for individual terms Ap Vp in this sum to be negative--as long as the sum ~ Ap Vp p
is itself positive (Katchalsky & Curran, 1965; Prigogine, 1967, p. 25). As a consequence a reaction could spontaneously convert reactants into products (Vp > 0) even though products were in excess (Ap < 0). This has been called "thermodynamic coupling" and is reputed to be of "great importance in biological processes" (Prigogine, 1967, p. 25). The first indication that this might not be the case appears in an interesting paper by Koenig, H o r n e & Mohilner (1961). They consider an arbitrary collection of independent reactions and show that there always exists a thermodynamically equivalent set for which ,4p Vp >_ 0. Since thermodynamics by itself provides no criterion for choosing among equivalent descriptions, they conclude that thermodynamic coupling has no thermodynamic significance. They leave open, however, the possibility that the study of reaction mechanisms may provide non-thermodynamic criteria for picking reactions which show thermodynamic coupling and, thereby, give meaning to the concept. It is the purpose of this note to present some considerations about thermodynamic coupling from the perspective of chemical kinetics. The definition of velocities for elementary reactions is examined, and it is shown that it can be extended only to a small class of complex reactions. This is contrasted with the usual non-equilibrium thermodynamics definition which is shown to lead to both ambiguous and physically unintuitive velocities. This ambiguity is due to the fact that the theory makes no distinction between complex reactions and elementary-type ones. However, the reactions in a system can always be analyzed in terms of the elementary-type ones and it is shown from the properties of the rate laws for these reactions that their velocities and affinities always have the same sign. Thus elementary-type reactions do not show thermodynamic coupling. From the view of kinetics, then, it is not possible to attribute any significance to thermodynamic coupling. The compatibility of this result with established ideas about chemical coupling is discussed.
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2. Velocities for Elementary Reactions As is weU-known an elementary chemical reaction is a reaction which proceeds according to microscopic events involving only the species represented in the reaction stoichiometry (Frost & Pearson, 1965). For example, the gas phase reaction between H and Br2 is an elementary reaction and so the stoichiometric equation H+Br2 = HBr+Br accurately represents that the forward and reverse rate processes result from encounters involving only H and Br2, and HBr and Br respectively. On the other hand, the stoichiometric equation for a complex reaction does not correspond to a reaction mechanism. Such is the case with the enzymatic conversion of a substrate S into a product P with the stoichiometric equation S=P. Since the enzyme E is a catalyst, it does not enter into the stoichiometry. Many enzyme catalyzed reactions, however, proceed through an intermediate complex with a mechanism represented by two elementary reactions (Gardiner, 1962) E + S = ES ES = E + P . Thus in addition to the reactants and products a new species, the enzymesubstrate complex ES, is present in the system. The difference between elementary and complex reactions is important in non-equilibrium thermodynamics because a complex reaction may not have a well-defined velocity. For example, consider the enzyme catalyzed reaction above. If the reaction stoichiometry told the whole story of the mechanism, then conservation of mass would imply that changes in the number of molecules of S and P due to reaction would be related by - d N s = dare = d~, which would define a progress variable 4. However, because the reaction is complex, the conservation relationship is actually that Ns + N~s + Np is constant, so that - d N s = dNp + dN~. Thus, in general, - d N s ~ dNp and so the loss of reactant S is not the same as the increase in product. This is clearly a property of all complex reactions and means that complex reactions will not, in general, have a weN-defined velocity. In fact in the reaction above only when the enzyme-substrate complex is a transient intermediate, so that d N ~ ~ 0, does the stoichiometric equation permit a reaction velocity to be assigned. If the species ES is present in experimentally negligible amounts then it is negligible in the thermodynamic treatment. In addition, if its rate of change is experimentally negligible, then the time dependence of the intermediate can be neglected in the rate equations. It seems reasonable to call complex reactions in which all inter-
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mediates are thermodynamically and kinetically negligible "elementarycomplex" reactions. As seen from the example such reactions will have an experimental reaction velocity defined by De Donder's procedure from the reaction stoichiometry. An important point to note about elementarycomplex reactions is that since the rate of change of all intermediates is negligible, the intermediates may be eliminated from the rate equations. Thus the reaction velocity of an elementary--complex reaction will not depend explicitly on the intermediate compounds. Many elementarycomplex reactions are known, the archetype in the gas phase being the unimolecular decomposition of cyclopropane. In the case of the enzyme catalyzed reaction discussed above the well-known Michaelis-Menten steady state treatment is based on the neglect of the enzyme-substrate complex. This condition is dNEs = 0, and so the reaction does have a velocity. Situations are known, however, in which the enzyme-substrate complex is not a transient intermediate and in these cases the complex reaction must be treated as two separate, but interlocking, elementary reactions--each with its own velocity (Keizer & Bernhard, 1966). The important differences between elementary, elementary--complex, and complex reactions do not seem to have been recognized by De Donder (1931) and others (Katchalsky & Curran, 1965). Indeed, De Donder introduces the idea of a reaction velocity using the Haber reaction 3Hz + N2 = 2NH3 without commenting about the complexity of its mechanism. As shown here a truly complex reaction does not possess a reaction velocity and it will be shown below that the failure to recognize this is, in a sense, the real origin of thermodynamic coupling. 3. Thermodynamic Reaction Velocities The usual non-equilibrium thermodynamic approach to chemical reactions involves adopting a maximal set of m linearly independent reactions. If there are n species--denoted by C ~ , . . . , C,--involved in these reactions, then the chemical equations are written symbolically as Z vp~Ct = 0, i
where the vp~ is the stoichiometric coefficient for species i in the pth reaction. The affinity is a property of a symbolic reaction and is definedby /lp = - Z vpi#~, t
where/q is the molecular chemical potential. The usual theory then defines the reaction velocities, Vp,by the equations (Koenig, Horne & Mohilner, 1961) dNffdt =-"E vp, Vp, (3) P
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~vhere N~ is the number of molecules of species i. This definition makes the theory macroscopic and empirical since once the set of reactions is fixed, i.e. once the matrix vpi is delineated, the velocities may be determined from experiment by inversion of this equation. The inversion is always possible since the reactions are independent. The definitions lead in the usual way to the expression for the free energy production given in equation (2). Although this procedure is consistent, it has the ambiguous feature of allowing the set of reactions to be chosen as any linear combination from any linearly independent set of reactions. This is the origin of the so-called transformation theory of chemical reactions (Prigogine, 1967, p. 41). It is also the origin of the remark that one can "confer or revoke thermodynamic coupling ... by the stroke of a pen" (Koenig et al., 1961) since certain linear combinations are thermodynamically coupled and others are not. Although this ambiguity does not produce mathematical errors, it makes the "rate of reaction" impossible to interpret because a given reaction can be assigned a number of velocities by this procedure. Two examples illustrate the problem quite clearly. First consider two linearly independent elementarytype reactions: I A=B 2 B=C. Because these are elementary-type reactions, they have well-defined velocities Vt and I/2 as described in the preceeding section. From the so-called thermodynamic point of view, these reactions are "equivalent" to the complex reactions, I' A = B 2'
A + C = 2B,
which using the definition in equation (3) have velocities V~ = VI + V2 and V~ = - V2. Reaction I', however, is identical to reaction I (it involves identical stoichiometry and has the same affinity), yet in this "equivalent" scheme the reaction has a different velocity. The new velocity has also lost its intuitive meaning since the reaction A = B can now go from left to right (V~ > 0) or from right to left (V~ < 0) even when no A is being converted to B! This will happen when VI = 0 with V2 ~ 0 and certainly is an unsatisfactory property of a reaction velocity. The same problem occurs when the elementary reactions in a system are linearly dependent, as in the set I
A=B
2
B=C
3
C=A,
with velocities V1, V2 and V3. If the linearly independent set is chosen to be r.B.
22
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reactions 1 and 2, then definition (3) confers the velocities V; = 1/1- 113 and V~ = V2 - / / 3 ; whereas if reactions 2 and 3 are chosen, then they have velocities V~' = V2- Vt and V~' = 173- Vt. Thus the velocity of the reaction B = C can be V2, 112- V3, or 1/'2- V, depending on the choice of independent reactions. Since the velocity is the time derivative of the extent of reaction, there is the further problem that the extent of a reaction also depends on this choice. Thus using equation (3) neither the velocity or progress of a reaction are properties of the reaction, itself, but rather reflect only the arbitrary choice made for the whole collection of reactions. This, of course, is not the case with the affinity which is a property of a given reaction. From the point of view of kinetics, this multivalueness of the velocity of a reaction is an unacceptable shortcoming of definition (3). It should be noticed that this problem does not exist at full equilibrium since then all velocities vanish and so this arbitrariness will have no special consequences there. Away from equilibrium its drawbacks are typified by the two examples given above. In the first example, it might be said that the problem arises because the reaction mechanism has been ignored by considering the kinetically meaningless net reaction A + C = 2B, whereas in the second example important mechanistic steps have been deleted by not considering all the elementary processes. In case the stoichiometric coefficients of the elementary-type reactions are independent, it is clear from a mechanistic point of view that these reactions should be used in the thermodynamic treatment. In this case the velocities defined in the previous section will agree with those obtained from equation (3). On the other hand, even when the stoichiometric coefficients are dependent, the use of the elementary-type reactions is dictated by kinetics. Actually the compelling reason for choosing linearly independent reactions is to assure that definition (3) can be inverted for the velocities. Indeed independent variables need be chosen in dynamical problems only out of convenience, and in some cases dependent variables may be a better choice. This certainly seems the case for coupled reactions---especially in view of the arbitrary way in which a velocity is associated with a reaction when independent reactions are used. 4. De Donder's Theorem for Elementary-type Reactions
In this section the possibility of thermodynamic coupling among elementary-type reactions is examined. If n such reactions occur in a system at constant temperature T and pressure p, then the local equilibrium theory (Katchalsky & Curran, 1965, chap. 7) gives
dG/dt + S d T / d t - V dp/dt = ~ ~li d N / d t
(4)
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with S the entropy and V the volume. The method for defining the velocity of the pth elementary-type reaction has been described earlier and permits the rate of change of the amount of the ith species to be calculated as a sum of contributions from each reaction:
dNl/dt = ~. vpi Vp. P
This then gives equation (4) the form
dG/dt+S d T / d t - V dp/dt = - • At, Vp
(5)
P
where the sum is over all the elementary-type reactions. By introducing activities, the chemical potentials may be written tti =/~o + kB T In as, where kB is Bottzmann's constant and #~' is the chemical potential of some standard state and depends only on the temperature. With this definition the affinities become
Ap = - ~ Vp,l ~ - k , TIn [I-I (a?')l. Ll
I
Finally recalling the definition of the equilibrium constant (Wall, 1958)
- ~ vpi#~ = kBT In Kp, t
and denoting the bracketed activity product above by Qp, gives Ap = kB T In (KflQp).
(6)
This form for the affinity will be used below. As discussed in the Introduction, even when the temperature and pressure are held constant, the monotonicity of G cannot be used to determine the relative signs of Ap and Vp. This depends on the form of the rate law, and the rate laws for elementary-type reactions have properties which make Ap Vp > 0. In order to see this more clearly, the reaction velocity is broken up into a positive forward velocity V+ and a positive reverse velocity VZ, as is usual in chemical kinetics. Because of the sign convention for the stoichiometric coefficients, these two velocities are positive and their difference is the reaction velocity, so Vp = V+ - V ; . (7) The following two properties of the rate laws for elementary-type reactions imply that ApVp >_ 0: (i) The ratio of the forward velocity to the reverse velocity depends on the number of molecules only through the ratio Kp/Qp, i.e.
V~/V;- = fp(Kp/Qp),
(8)
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J. KEIZER
where the dependence of fp on other local thermodynamic variables has been made implicit. (ii) In the absence of other processes the rate law given by equation (7) corresponds to a unique stable equilibrium at KflQp = 1. With the possible exception of the singular points KflQp = 0 and oo, all points must asymptotically proceed to the equilibrium point.t Neither of these properties is arbitrary and both actually hold for the form of rate laws which describe elementary reactions in the gas phase and in solutions. For example, for the bimolecular elementary reaction in solution, k_._~+ CI-~ C 2 ~ - C3-~-C4, transition state theory gives (Frost & Pearson, 1965) V ; = k+ala2 V;- = k - a 3 a 4
(9)
where the a's are activities. So that
v; /v;
=
(k +/k-)/(a3 a ,/a, a2).
But the equilibrium condition Vv+ = Vv- implies that the equilibrium constant for this reaction is K = k+/k -, so condition (i) is clearly met using f ( X ) = X. In the gas phase a similar result holds using fugacities or--in the case of an ideal gas, partial pressures-in the place of activities. Hence the well-known mass action laws guarantee that condition (i) will be met. For non-ideal systems the form of the mass action laws as written above is less well-established, but is implied by the transition state theory. Since transition state theory successfully describes many of the effects of non-ideality in solution reactions, these expressions probably represent the dominant kinetic effects of non-ideality. Thus to the extent that mass action laws involving the affinities are valid, condition (i) will be met even in non-ideal systems such as encountered in solution. An example of an elementary complex reaction satisfying condition (i) is the generalized Michaelis-Menten scheme E+S ~
k-I
ES ~
k-2
ES
E'+P.
If dNEs[dt ,.~ O, this will be an elementary--complex reaction. Using this steady state assumption it is shown in the Appendix that it is possible to write the velocity for this reaction as V = V + - V- and again V + / V - = K/Q. t An example of a singularity at K/Q = 0 comes from the reaction C~+C2 = 2C2 which has a metastable state at N~ = O.
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It is possible to construct other examples of elementary-complex reactions satisfying condition (i), and following Hollingsworth (1952) it is believed that the rate laws for all elementary--complex reactions may be written so that V + / V - = f ( K / Q ) . The validity of the second condition, i.e. existence of a unique, stable equilibrium, is commonly assumed by chemists and has actually been proven (Shear, 1968) for polynominal rate laws. PROOF THAT Ap Va > 0
From equation (6) the sign of the affinity Ap is determined solely by the size of the dimensionless ratio X = Kp/Qp. In fact, as is shown in Fig. 1, A p > 0 when X > 1, A p = 0 at X = 1, and A p < 0 when 0 < X < 1.
/ 0
/ /../.I I
I 2
X=K# Op FIG. 1. A schematic representation of the functions In X and forward to reverse reaction rates.
fD(X),
the ratio of the
Using conditions (i) and (ii) above it will be shown that Vp has the same sign as Ap in these regions. Equivalently equation (7) shows that Vp = v ; ( v ; / v ; - l ) = V~-(fp(X) - 1).
Thus since Vp- is strictly positive, Vp is positive whenfp(X) > 1 and negative when fp(X) < 1. Hence it suttees to show that on 1 < X < o% f p ( X ) > 1 and on 0 < X < 1, fp(X) < I. To see this, recall that X = 1 is the equilibrium point, so fp(1) -= 1 by condition (ii). But since the equilibrium is stable, it follows for small positive 5 thatfp(1 +6) > 1. This is so because if X = K f l Q p > 1, then the activity quotient Qp must increase as time
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J. KEIZER
proceeds in order that equilibrium be reattained. Thus the activity of the products must be increased, i.e. the reaction must go in the forward direction. Thus Vp > I or fp(1 +6) > 1. Similarly it follows from the stability of the equilibrium that for small positive 6, fp(1 - ~ ) < 1. Furthermore there can be no X on (0, co) other than X = 1 for whichfp(X) = 1. Otherwise there would be another point with Vp = 0, implying the existence of a collection of states which would not proceed to equilibrium. This would violate condition (ii). Thus fp(X) is greater than 1 on 1 < X < oo and less than 1 on 0 < X < 1 and so Ap and Vo have the same sign.
5. Scope and Consequence of the Theorem In order to emphasize the range of validity of this extension of De Donder's theorem, we note that the proof given above in no way depends on the constancy of the temperature and pressure. In fact what is required beyond conditions (i) and (ii) is that the system be uniform and in local equilibrium, so that equation (6) can be used to define the affinities. Furthermore, if the reactions occur in an open system in which the number of molecules are being uniformly changed by an external source, the result will also hold. Indeed, in this case
dNJdt = ~ vp~Vp+ Ri, P
where R~ is the external rate and Vp are the reaction velocities. Thus even in an open system as long as the chemical rate law satisfies conditions (i) and (ii), it remains true that Ap Vp > 0. In this regard it should be noted that conditions (i) and (ii) require detailed balance to hold only at the equilibrium state implied by each elementary reaction when decoupled from the remaining reactions. Since detailed balance need not occur at the steady states of the open system, the result applies even to far from equilibrium steady states. Thus the result that the direction of an elementarytype reaction is determined by its al~nity is seen to hold for a broad class of uniform coupled chemical systems throughout the entire domain of local equilibrium. The proof of De Donder's theorem for elementary-type reactions is based on the form of the chemical rate laws and so is a kinetic proof. As pointed out above, a purely thermodynamic proof of the theorem is not possible. On the other hand, thermodynamic considerations do show that thermodynamically coupled reactions can always be uncoupled by a simple transformation (Koenig et al., 1961). The results obtained here make an even stronger point, namely, that the only cases in which thermodynamic coupling is found are those in which inherently uncoupled elementary processes are
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COUPLING
333
naively added together or deleted yielding a description in terms of complex reactions. In order to reinforce this conclusion, two examples of the effect of chemical coupling are discussed to show that they are not compromised by this result. Consider first the condensation of glucose and fructose to form sucrose in aqueous solution (Lehninger, t965, chap. 4) glucose + fructose = sucrose + H20. This reaction is thermodynamically unfavorable under standard conditions because the equilibrium constant is less than one. In fact, AG ° = 5-5 kcal. However, when the hydrolysis of adenosine triphosphase (ATP) is mechanistically coupled to this reaction one has ATP+glucose+fructose = sucrose+ADP+phosphoric acid (10) which now is a favorable reaction since AG° = - 1 . 5 kcal. Thus the hydrolysis of ATP is said to "drive" the condensation of the disaccharide toward completion. This sort of "driving" through coupled reactions clearly has nothing to do with the affinity, since it is merely a change in the equilibrium due to coupling and the affinities all vanish at equilibrium. Moreover this is a complex reaction and so does not even have a reaction velocity. Indeed its mechanism may be written as (White, Handler & Smith, 1964, p. 409) ATP + glucose = glucose 6-phosphate + ADP glucose 6-phosphate = glucose I-phosphate glucose 1-phosphate + fructose = sucrose + phosphoric acid. These are all elementary-complex enzyme reactions catalyzed by hexokinase, phosphoglucomutase, and sucrose phosphorylase respectively and each has a well-defined velocity and a AG°. The sum of the standard free energies for the three component reactions is just the - 1.5 kcal required for the net reaction written in equation (t0). Hence this well-known sort of biochemical coupling has nothing to do with kinetics, and its validity is not endangered by a lack of thermodynamic coupling. Purely kinetic-type coupling does exist in complex homogeneous chemical systems--a striking example occurring in the oscillating Belousov reaction (Field, KOrrs & Noyes, 1972). While the details of the mechanism of this process are quite complicated, Field & Noyes (1974) have summarized the mechanism in terms of the symbolic elementary reactions A+Y~eX X+Y~P B+X~2X+Z 2Z~ Q z~eY
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KEIZER
where X - HBrO2, Y - Br-, Z - Ce(IV), and the other symbols represent kinetically unimportant products or quantities held in excess. The kinetic equations for the rate of change of HBrO2, Br-, and Ce(IV) are easily written down and have been solved numerically to give a good representation of the concentration oscillations in this system. The kinetic equations used by Field & Noyes (1974) for these elementary reactions are compatible with the conditions (i) and (ii) used to prove that reaction velocities have the same sign as their affinity. Thus even chemical oscillations do not require a reaction to be driven against its affinity by other reactions. Indeed the generalization of De Donder's theorem implies through equation (5) that at constant temperature and pressure each elementary reaction will decrease the free energy. This means that chemical reactions are still inherently dissipative even though the concentrations of particular compounds may rise and fall almost periodically for long periods of time. In this sense the origin of the so-called "order" due to the chemical oscillations is kinetic and not thermodynamic. It should be clear now that the reason De Donder's theorem is valid is that it focuses on reactions rather than compounds. In fact, if attention is shifted back to individual chemical species, then at constant temperature and pressure equation (4) may be written dG/dt = ~ p~ dNJdt. i
Each term in this expression is not inherently negative and in the case of chemical oscillations can even change sign during the course of reaction. This, however, can be adequately explained by kinetic coupling alone. It is only when the terms are grouped together for each elementary reaction that the inherently dissipative nature of the chemical reactions--and their thermodynamic independence--becomes clear. The author gratefully acknowledges several useful conversations with Professors Peter Rock, Nobuhiko Saito, and Ronald Fox. REFERENCES DE DONDER, T. & VAN RYSSELBERGHE, P. (1936). Theromdynamic Theory of Affinity. Stanford: Stanford UniversityPress. FIELD,R., K6R6S,E. & NoY~, R. (1972). J. Am. chem. Soc. 94, 8349. FIELD,R. & NOYES,R. (1974). J. chem. Phys. 60, 1877. FROST,A. A. & PEARSON,R. G. (1965). Kinetics and Mechanism, 2nd edn. New York: Wiley. GARDINER,W. C. JR (1972). Rates and Mechanism of Chemical Reactions. New York: Benjamin. HOLLINGSWORTH,A. (1952). J. chem. Phys. 20, 921. KATCHALSKY,A. & CrdRRAN,P. F. (1965). Non-Equilibrium Thermodynamics in Biophysics. Cambridge: Harvard UniversityPress.
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KEIZER,J. & BERNHARD,S. A. (1966). Biochemistry, N.Y. 12, 4127. KOENIG,F. O., HORNE,F. H. & MOHILNER,D. M. (1961). J. Am. chem. Soc. 83, 1029. LEttNINGER,A. L. (1965). Bioenergetics. New York: Benjamin. PRIGOGINE,I. (1967). Introduction to Th&modynamics of Irreversible Processes. 3rd edn, pp. 23-25. New York: Interscience. SHEAR,D. B. (1968). J. chem. Phys. 48, 4144. WALL, F. T. (1958). Chemical Thermodynamics. San Francisco: Freeman. WHITE, A., HANDLER, P. & SMrm, E. (1964). Principles o f Biochemistry. New York: McGraw Hill.
Appendix For the reaction kl
E + S ~ - ES k-i
k2
ES ~
k-2
E'+P
the stoichiometry is E+S =E'+P and the equilibrium constant is K = k 2 k l / k _ 2 k _ 1. Since N s + N ~ + N p is constant and d N B [ d t ,,~ O, the reaction velocity V = - d N s / d t = dNp[dt. Using the rate laws for the c o m p o n e n t reactions gives the equations V = -dNs/dt
= k t aEas - k _ I a~:s
= d N p / d t = - k _ 2 aE" av + k 2 aEs
(A1)
and 0 = dNEs/dt = - ( k _ 1 + k2)aEs + k l aE as + k _ 2 aE" av
where the as are activities. The steady state condition implies klaEas
k - 2 a E , ap
aES -- k _ l + k ~2 + k _ l + k 2 .
(-A2)
Using equation (A2) and the first equation for V in (A1) forward and reverse velocities m a y be defined, for example, b y t and
V + =ktk2aEas/(k_t+k2)
V- =k_tk_2aE, ap/(k_l+k2)
and (A1) implies V = V + - V - . F r o m this result and the definition of the equilibrium constant it follows immediately that V*lV
-
=KIQ.
t This simple decomposition was suggested by a thoughtful referee who also conjectured that a result like equation (A3) might hold in general.