Page 112 - Thermodynamics of Biochemical Reactions

108 Chapter 6 Systems of Biochemical Reactions

criterion for spontaneous change and equilibrium at specified pH and
a(H,O) = 1. When this is done, the apparent conservation matrix A" that does
not include the conservation of H,O becomes consistent with the apparent
stoichiometric number matrix v" that does not include the stoichiometric number
for H,O.
The Legendre transform that defines the further transformed Gibbs energy
G", which provides the criterion for spontaneous change and equilibrium in dilute
aqueous solutions, is
G" = G' - ~,(O),U'~(H,O) (6.3-1)
The amount of the oxygen component in the system is given by n,(O) =
CN,(i)ni, where N,(i) is the number of oxygen atoms in reactant i. p"(H,O) is
the standard transformed chemical potential for H,O at the specified pH and
ionic strength. The standard further transformed Gibbs energy of formation of
reactant i is given by
A,GYo = AfGio - No(i)AfG'o(H20) (6.3-2)
where A,G"(H,O) is given by equation 4.4-10. Note that AfG"'(H,O) = 0. When
this adjustment of the standard transformed Gibbs energy of formation of
reactant i is made, this reactant becomes a pseudoisomer of other reactants that
differ from it only with respect to the number of oxygen atoms they contain, and
so the standard further transformed Gibbs energy of formation of the
pseudoisomer group has to be calculated using the analogue of equation 4.5-1.
The apparent equilibrium constant K" for a biochemical reaction at specified pH
and a(H,O) = 1 is given by
ArGJr0 = -RTlnK" = Cy:(ArCYo (6.3-3)
There is no term for H,O in the summation. When the pH is specified and
u(H,O) = 1, the criterion for spontaneous change and equilibrium is dG" d 0 at
specified 7; P, pH, a(H,O) = 1, and amounts of apparent components. Note that
oxygen is no longer a component.
Thus the inconsistency between A' and v" is eliminated by using A" and v".
The number C" of apparent components can be determined by row reduction of
A" since C" = rank A". The number R" of independent reactions can be deter-
mined by row reduction of v" because R" = rankv". Note that N" = C" + R".
These two types of matrices can be interconverted by use of
A"v" = 0 and (v")~(A")~ (6.3-4)
0
=
The apparent stoichiometric number matrix v" can be obtained from the row-
reduced form of A" by use of the analogue of equation 5.1-19 or by calculating a
basis for the null space using a computer program.
Further transformed Gibbs energies of formation are especially useful in
calculating equilibrium compositions by computer programs that accept conser-
vation matrices and vectors of initial amounts, as discussed in the next section.

rn 6.4 CALCULATIONS OF EQUILIBRIUM
COMPOSITIONS FOR SYSTEMS OF
BIOCHEMICAL REACTIONS

One of the important things that thermodynamics can tell us about a system of
reactions is the composition at equilibrium for given initial amounts of reactants.
For a single reaction there is an analytic solution for this problem, but for a
system consisting of two or more reactions, an iteration using the Newton-
Raphson method is required to find the composition of the system that yields the
lowest possible transformed Gibbs energy, given the conservation equations and
equilibrium expressions. Computer programs for doing that were written by