Page 68 - Thermodynamics of Biochemical Reactions

4.2 Derivation of the Expression for the Apparent Equilibrium Constant 63

W 4.2 DERIVATION OF THE EXPRESSION FOR THE
APPARENT EQUILIBRIUM CONSTANT

At specified pH, the biochemical reaction that corresponds with equation 4.1-18
is represented by

N'
C vIBi = 0 (4.2-1)
i=l
where the primes on the stoichiometric numbers v: in a biochemical equation are
used to distinguish them from the stoichiometric numbers on the underlying
chemical reactions. The B, are symbols representing pseudoisomer groups, as in
ATP + H,O = ADP + Pi. Biochemical reactions can produce or consume H',
but these hydrogen ions are not shown in equation 4.2-1 because the pH is held
constant. In other words, hydrogen atoms are not conserved in the reaction vessel.
Figure 4.1 shows a thought experiment that corresponds with the interpretation
of the determination of the apparent equilibrium constant at a specified 7; P, and
pH. When hydrogen ions are produced in the reaction, they diffuse into the pH
reservoir through the membrane permeable to Hi to keep the pH constant, and
when hydrogen ions are consumed in the reaction, hydrogen ions diffuse into the
reaction chamber to hold the pH constant. This figure shows that the pH
reservoir plays the same kind of role as the heat reservoir at temperature T and
the piston exerting a constant pressure P. Therefore these three independent
variables have to be treated in the same way in thermodynamics; that is, they are
introduced by means of Legendre transforms.
It is important to have symbols, that is, names, for reactants that are different
from the symbols B, for species, like ATP4-, HATP3-, and H,ATP2-, which are
used in chemical equations. (The problems in naming are discussed later in
Section 4.11.) The N' reactants in a biochemical reaction are all pseudoisomer
groups; note that a pseudoisomer group is made up of one species over a wide
range of pH if the reactant has no pKs in the pH range considered.
Equation 4.1-18 can be used to derive the expression for the apparent
equilibrium constant K' for a biochemical reaction at a specified pH. If a single
biochemical reaction is catalyzed, the amounts n: of the pseudoisomer groups at
each stage of the reaction are given by

n: = (IZ:)~ + (5' (4.2-2)
where (i11)~ is the initial amount of reactant i (pseudoisomer group i), v: is the
stoichiometric number of reactant i in the biochemical reaction (see equation

I

Figure 4.1 Thought experiment in which a reaction is carried out in a reaction vessel
connected to a pH reservoir through a semipermeable membrane that permits H+ to go
in or out. The reaction vessel is also held at a constant temperature and pressure.