Haldane relationships Ordered

Alberty first proposed the use of Haldane relations to distinguish among the ordered Bi Bi, the ordered Bi Bi Theorell-Chance, and the rapid equilibrium random Bi Bi mechanisms. Nordlie and Fromm used Haldane relationships to rule out certain mechanisms for ribitol dehydrogenase. 

Kinetic Haldane relations use a ratio of apparent rate constants in the forward and reverse directions, if the substrate concentrations are very low. For an ordered Bi Bi reaction, the apparent rate constant for the second step is Emax,f/ b (where K, is the Michaelis constant for B) and, in the reverse reaction, V ax,v/Kp. Each of these is multiplied by the reciprocal of the dissociation constant of A and Q, respectively. The forward product is then divided by the reverse product. Hence, the kinetic Haldane relationship for the ordered Bi Bi reaction is Keq = KiO V eJKp)l Kiq V eJKp) = y ,ax.f pKiq/ (yranx,rKmKif). For Completely random mechanisms, thermodynamic and kinetic Haldane relationships are equivalent. 

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. 

Haldane relationships are relationships between the equilibrium constant and the various kinetic constants defined for a given mechanism. They exist because we define for each mechanism more kinetic parameters than there are independently determinable parameters. They are of two types, kinetic and thermodynamic, and every mechanism has at least one of each. Thermodynamic Haldanes consist of the cross product of reciprocal dissociation constants for the substrates and dissociation constants for the products (i.e., the product of equilibrium constants for each step in the mechanism). For mechanisms with at least three substrates, the Cleland notation defines dissociation constants as K, values (i.e., /fia, ib, K c, etc.), but for Ordered Uni Bi and Bi Bi mechanisms, Cleland defined the dissociation constants of the inner substrates differently (4). The dissociation constants for A and Q were and /(jq, but that for B was... 

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 ... 

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 rate Eq. (12.83) can be written down directly from the general rate equation for the Ordered Ter Ter mechanism (Table 4), simply by omitting all the concentration terms that contain P and R and eliminating the Keq with the aid of Haldane relationships. Equation (12.83) is identical with the rate Eq. (12.26) for the Ordered Ter Ter mechanism, in the absence of products, except that a new term, 4>ABCQ, is added in the denominator ... 

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