The Haldane relationship

All reactions are to some degree reversible, and many enzyme-catalysed reactions can take place in either direction inside a cell. It is therefore interesting to compare the forward and back reactions, especially when the reaction approaches equilibrium, as in an enzyme reactor. Haldane derived a relationship between the kinetic and equilibrium constants. The derivation of the relationship is shown in Appendix 5.4. 

If the equilibrium constant is known, this relationship can be used to check the validity of the kinetic constants which have been determined. In general, the equilibrium will favour the metabolically important direction. However, it should be noted that the direction of metabolic flow in cell systems will also be dependent of the concentrations of substrate and products. 

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For reversible enzymatic reactions, the Haldane relationship relates the equilibrium constant KeqsNith the kinetic parameters of a reaction. The equilibrium constant Keq for the reversible Michaelis Menten scheme shown above is given as... 

The importance of the Haldane relationship Eq. (42) relates to the fact that the kinetic parameters of a reversible enzymatic reaction are not independent but are constraint by the equilibrium constant of the overall reaction [157]. 

The use of Haldane relationships to verify the magnitude of the equilibrium constant or, conversely, to determine (or verify) one of the kinetic parameters requires that aU constants be measured under the same experimental conditions (eg., temperature, pH, buffer species, ionic strength, free metal ion concentrations, etc) If not, the Haldane relationship has no meaning. In addition, kinetic data are often limited in precision, unlike equilibrium measurements. For multisubstrate reactions, there are at least two different Haldane relationships. Thus,... 

Haldane relationships can also be useful in characterizing isozymes or the same enzyme isolated from a different source. Reactions catalyzed by isozymes must have identical equilibrium constants, but the magnitudes of their kinetic parameters are usually different (e.g., the case of yeast and mammalian brain hexokinase ). Note that the Haldane relationship for the ordered Bi Bi mechanism is = Hmax,f p i iq/(f max.r ia b)- This same... 

Haldane is also valid for the ordered Bi Bi Theorell-Chance mechanism and the rapid equilibrium random Bi Bi mechanism. The reverse reaction of the yeast enzyme is easily studied an observation not true for the brain enzyme, even though both enzymes catalyze the exact same reaction. A crucial difference between the two enzymes is the dissociation constant (i iq) for Q (in this case, glucose 6-phosphate). For the yeast enzyme, this value is about 5 mM whereas for the brain enzyme the value is 1 tM. Hence, in order for Keq to remain constant (and assuming Kp, and are all approximately the same for both enzymes) the Hmax,f/f max,r ratio for the brain enzyme must be considerably larger than the corresponding ratio for the yeast enzyme. In fact, the differences between the two ratios is more than a thousandfold. Hence, the Haldane relationship helps to explain how one enzyme appears to be more kmeticaUy reversible than another catalyzing the same reaction. 

In Scheme 1, the rate parameters Vmax,f and Emax,r are the maximum velocities in the forward and reverse direction, respectively (such that Emax,f = [Etotai] and Emax,r = ki [Etotai]), a is the Michaelis constant for substrate A (Xa = (/c2 + ksykb), and Xeq is the equihbrium constant (equal to kikslk2k, and having the Haldane relationships... 

As in any other chemical reaction, there is a relationship between the rate constants for forward and reverse enzyme-catalyzed reactions and the equilibrium constant. This relationship, first derived by the British kineticist J. B. S. Haldane and proposed in his book Enzymes41 in 1930, is known as the Haldane relationship. It is obtained by setting v( = vr for the condition that product and substrate concentrations are those at equilibrium. For a single substrate-single product system it is given by Eq. 9-42. 

A , is the rate constant for combination of A with the free enzyme E. However, = ( 4 6 + 4 7 + k k-j)/k k kj. This involves 5 rate constants Aj, A4, Aj, Ag and kj all of which involve both A and B or, in the case of Ag and k-, both corresponding products. Thus is unlikely to be independent of the nature of A for this mechanism. Clearly this criterion will distinguish between two mechanisms shown in Schemes 5 and 8, and in fact the prediction is unique to the ping-pong mechanism. Another distinctive test of the ping-pong mechanism is the Haldane relationship. Haldane pointed out that from the initial rate kinetic parameters for the forward and reverse directions of a reversible enzyme-catalysed-reaction it was possible to obtain an expression for the overall equilibrium constant [66]. 

Equation (3.41) is known as the Haldane relationship (Haldane, 1930). It is a very important relationship, because it states that the luetic parameters of every reversible enzymatic reaction are not independent of one another and are limited by the thermodynamic equihbrium constant of the overall reaction. 

The Haldane relationship can be used to ehminate partly or completely from Eq. (3.40). For example, this equation is best written for consideration of the forward reaction as... 

In order to calculate the Haldane relationship for the Ordered Bi Bi mechanism, the concentration of one pair of reactants must be set to zero (Cleland, 1982). If the concentrations of A and B is negligible, it is expected that the general rate Eq. (4.39) reduces to the rate equation for the reverse reaction in the presence of P and Q, and absence of A and B ... 

Reaction scheme (8.11) describes a reversible reaction with two maximal velocities, one in the forward and the other in the reverse direction, both being the products of catalytic constants and the total concentration of enzyme, Vi = kcenEo and Vk = jAc tEo. It is important to note that the interconversion of above two equations is achieved with the aid of the Haldane relationship (8.43). 

The general rate equations, if all four substrates and products are present, and when interconverted with the Haldane relationship (8.43) are... 

In rapid equilibrium systems, the Haldane relationship can be obtained directly from rate equations. In equilibrium, the rate equation for the Rapid Equilibrium Ordered Bi Bi system (Eq. (8.12)), becomes... 

Thus, for the Rapid Equilibrium Ordered Bi Bi system, the Haldane relationship is... 

The Haldane relationship is identical for all rapid equilibrium random systems (Haldane, 1930 Cleland, 1982). Thus, from Eq. (8.37), one also obtains... 

Note that the expression ViHYaK in the P and AP terms is replaced by an expression (lCjAKB)/(Jfp 4Q) from the Haldane relationships (Eq. (9.11)). 

Again, V,/(VaJCeq) is replaced from the Haldane relationship. When A is varied, the reciprocal form of Eq. (9.22) is... 

Note, however, that the product inhibition studies in the A P -1- Q direction do not allow Jfp and Kq to be determined, but their ratio can be calculated from the Haldane relationship (Eq. (9.45)) ... 

Equation (14.35) is written for consideration of the forward reaction and all Aeq expressions are removed from the denominator with the aid of the Haldane relationship. 

From the Haldane relationship (Section 9.2), the equilibrium concentration of reactants is equal ... 

In the treatments discussed so far, it has been assumed that the back reaction could be neglected. The reactions catalysed by many enzymes are essentially irreversible or the products are immediately subject to further reaction, so that the assumption of irreversibility is valid. However, if the reaction is reversible, the Michaelis equation must be modified. Haldane suggested a notation in which V, and V, are the maximal velocities in the forward and reverse directions, and and K ,p are the Michaelis constants for the substrate and product. The Haldane relationship for a system with a single substrate and single product is then = V,K pA, K s. 

The Haldane Relationship. Another of the properties of enzyme systems frequently measured is the equilibrium constant of the over-all reaction. This is the means for determining a fundamental thermodynamic property, the free enei (F) of a reaction. Free energy will be discussed later. At this point a relation between enz3uue kinetics and equilibrium is of interest. The equilibrium constant for a reaction... 

The Haldane relationship between the equilibrium constant, maximum velocities, and Michaelis constants.)... 

There can be many reasons why a reaction does not proceed to equilibrium. The standard free energy, then, does not define the amount of work that will be obtained from a reaction it is the maximum energy available under defined conditions. When the standard free energy is known, it can be used to determine the equilibrium constant of a reaction. This, it must be remembered, measures the extent to which a reaction may proceed, but it does not indicate the speed of a reaction or even that a reaction will occur at all. AF is related indirectly to the relative rates of enzyme-catalyzed reactions by the Haldane relationship (p. 12), but the absolute rate of reaction is determined by the amount of enzyme and substrate in a given system. 

The elaborate studies of fumarase have provided the most critical test of the Haldane relationship. For fumarase this is expressed... 

For the sequence of processes (5.8) the equilibrium constant, related to the reagents in aqueous solution, can be expressed using the Haldane relationship... 

Denoting the equilibrium constant of the reaction as Keq = Poo/Soo, we get the Haldane relationship ... 

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