Page 111 - Thermodynamics of Biochemical Reactions

6.3 Use of a Legendre Transform for Reactions Involving Water as a Reactant 107

A matrix multiplication can be used to calculate the A,G” values for a series
of reactions from a vector of AfG” values for the reactants involved.

[Af G;’, Af G:, . . . , Af G:] . V’ = [Af G;’, Arc:,. . . , Ar G:] (6.1 - 5)
(Note that (1 x N’)(N’x R’) = 1 x R’ and see Problem 6.2.) The for a
particular path s’ is obtained by multiplying both sides of this equation by s’:
[Af G‘,’, Af G:, . . . , A, GG] . V’ . S’ = Ar G:t (6.1 -6)
Note the first matrix is 1 x N‘, the second is N‘ x R‘, and the third is R‘ x 1, and
so the result is 1 x 1.

H 6.2 CALCULATION OF PATHWAYS BY SOLVING
LINEAR EQUATIONS

Since equation 6.1-2 represents a set of linear equations, the path can be
calculated from the stoichiometric number matrix and a particular net reaction
by solving the set of linear equations (Alberty,l996a). In Mathematica this can be
done with Linearsolve:
s’ = LinearSolveCv‘, vhe1] (6.2-1)
This calculation can be made for chemical reactions, biochemical reactions at
specified pH, or at steady state concentrations of reactants like ATP and ADP,
as is discussed in Section 6.6. The advantage of the matrix formulation of this
calculation is that very large matrices can be handled.
Problem 6.2 illustrates the use of equation 6.2-1 by applying it to four net
reactions that represent the oxidation of glucose to carbon dioxide and water: (1)
the net reaction for glycolysis, (2) the net reaction catalyzed by the pyruvate
dehydrogenase complex, (3) the net reaction for the citric acid cycle, and (4) the
net reaction for oxidative phosphorylation. The v’ in equation 6.2-1 is the
apparent stoichiometric number matrix for these four reactions. The net reaction
is

glucose + 60, + 40ATP + 40Pi = 46H,O + 6C0, + 40ATP (6.2-2)
The nunet in equation 6.1-1 is the list of stoichiometric coefficients for this
reaction for the order of the reactants in v‘. The use of equation 6.2-1 yields the
following path: {1,2,2,12}. This means that reaction 1 has to occur once, reaction
2 has to occur twice, reaction 3 has to occur twice, and reaction 4 has to occur
12 times in order to oxidize a mole of glucose to carbon dioxide and water.

6.3 USE OF A LEGENDRE TRANSFORM FOR
REACTIONS INVOLVING WATER AS A REACTANT

When water is a reactant, the calculation of K‘ using A,G“ = - RTln K‘ is based
on the convention that ArG”(H,O) is involved in calculating Arc”, but that the
activity of H,O in the expression for K‘ is taken to be unity. This is a practical
convention in treating a single reaction, but in treating systems of reactions, it is
almost a necessity to use matrices, linear algebra, and a computer. Linear algebra
can be used to convert sets of stoichiometric equations to sets of conservation
equations, and vice versa, but these operations are incompatible when H,O is a
reactant and the convention that a(H,O) is equal to unity is used (Alberty, 2001a,
2002b). In considering systems of reactions, it is advantageous to use apparent
conservation matrices and apparent stoichiometric matrices that are interconvert-
ible using the operations of linear algebra. In dilute aqueous solutions, the solvent
provides an essentially infinite reservoir for H,O, and so a Legendre transform
can be used to define a further transformed Gibbs energy G” that provides the