Page 76 - Thermodynamics of Biochemical Reactions

4.7 Calculation of the Binding of Hydrogen Ions by Reactants 71

For a system of chemical reactions, the criterion for spontaneous change and
equilibrium is dG < 0 at T P, {nJ. When the pH is specified, the criterion for
spontaneous change and equilibrium becomes dC' < 0 at 7: P, {nl,,), pH, where
{&} represents the set of amounts of components other than hydrogen.

4.7 CALCULATION OF THE BINDING OF HYDROGEN
IONS BY REACTANTS AND THE CHANGES IN
BINDING OF HYDROGEN IONS IN BIOCHEMICAL
REACTIONS

When a system contains a single pseudoisomer group, equation 4.1-18 shows that
dG' = -S'dT+ VdP + ,uIdnI + RTln(lO)n,(H)dpH (4.7-1)

The Maxwell equation involving the last two terms is

(4.7-2)

Since the derivative on the right is the change in the amount of hydrogen atoms
in the system when the amount of reactant i is changed, this equation can be
written in terms of the average number N, of hydrogen atoms bound by the
reactant (Alberty, 1994~).

(4.7-3)

This equation is closely related to equation 4.2-13. Wyman (1964) was probably
the first to use this type of equation. Note that the derivative of Arcr is the same
as the derivative of Ar G'O.
In Chapter 1 it was shown that the change in binding of hydrogen ions in a
reaction can be calculated by taking the difference between the binding by
products and the binding by reactants (Section 1.5). Equation 4.2-13 shows that
the rate of change of ArG' with pH is proportional to the change in binding of
hydrogen atoms in the reaction. Since components are conserved in chemical
reactions, it may be a surprise to find that the amount of the hydrogen atoms in
the system is not conserved in biochemical reactions, but of course hydrogen
Btoms are not conserved in the system when the pH is held constant. Acid or base
may have to be added to hold the pH constant. If (d~~,(H)/dt')~,~,~~ is positive, it
means that acid has to be added to keep the pH constant. In this case the
products bind more hydrogen ions than the reactants. Since the concentration of
hydrogen ions is constant, the change in the amount of the hydrogen atoms in
the reaction system is equal to the change in the binding of hydrogen ions in the
reaction. Since the derivative of Arc' is the same as the derivative of A,G",
equation 4.2-12 can be written as

-
ArN - RTlln(l0) (%)T,p,i# (4.7-4)
Substituting Arc" = - RTln K' yields

(4.7-5)

If K' increases with pH, ArNH is negative, which indicates that less hydrogen is
bound by products than reactants. In this case H' is produced by the reaction.
The change in binding of hydrogen ions in a biochemical reaction can also be