Page 104 - Thermodynamics of Biochemical Reactions

5.4 Matrix Forms of the Fundamental Equations for Chemical Reaction Systems 99

This reaction involves eight reactants, and so C‘ = N‘ - R‘ = 8 - 1 = 7, but only
four elements are involved. So there are three additional constraints. Reaction
5.3-10 can be considered to be made up of the following two reactions:
ATP + H,O = AMP + PP, (5.3- 11)

deamido-NAD,, + L-glutamine = NAD,, + L-glutamate (5.3-12)
The constraints in the urea cycle are discussed in Alberty (1997~).
When water is a reactant in a system of reactions, there is a sense in which
oxygen atoms are not conserved because they can be brought into reactants or
expelled from reactants without altering the activity of water in the solution.
When water is not involved as a reactant in a system of reactions, this problem
does not arise, and oxygen atoms are conserved in the reactants. When water is
a reactant, there are problems with equilibrium calculations, as are discussed later
in connection with the calculation of the equilibrium composition in the next
chapter in Section 6.3. The problem encountered in using A’ and v‘ matrices can
be avoided by using a Legendre transform to define a further transformed Gibbs
energy G” that takes advantage of the fact that oxygen atoms are available to a
reaction system from an essentially infinite reservoir when dilute aqueous sol-
utions are considered at specified pH (Alberty, 2001a, 2002b).

5.4 MATRIX FORMS OF THE FUNDAMENTAL
EQUATIONS FOR CHEMICAL REACTION SYSTEMS

In treating systems of biochemical reactions it is convenient to use the fundamen-
tal equation for G’ in matrix form (Alberty, 2000b). The extent of reaction t for
a chemical reaction was discussed earlier in Section 2.1. For a system of chemical
reactions, the extent of reaction vector 5 is defined by

n = no + vk (5.4- 1)
where n is the N, x 1 column matrix of amounts of species, no is the N, x 1 column
matrix of initial amounts of species, v is the N, x R matrix of stoichiometric
numbers, and 5 is the R x 1 column vector of extents of the R independent
reactions. The differential of the matrix for amounts of species is
dn = vdk (5.4-2)
Thus the fundamental equation for the Gibbs energy of a chemical reaction
system can be written as
dG = -SdT+ VdP + pdn (5.4-3)
where p is the 1 x N, chemical potential matrix. We will see later that this
equation can also be applied to phase equilibria (Chapter 8). Substituting
equation 5.4-2 yields
dG = - SdT + VdP + pvd5 (5.4-4)
This equation is useful for setting up the fundamental equation for consideration
of a chemical reaction system described by a particular stoichiometric number
matrix.
As an example of the usefulness of equation 5.4-4, consider the fumarase
reaction (fumarate + H,O = L-malate) in the range pH 5 to 9 where the chemical
reactions are
fum2- + H,O = ma12- (5.4-5)
Hfum- = H+ + fum2- (5.4-6)

Hmal- = H+ + maI2- (5.4-7)