Page 101 - Thermodynamics of Biochemical Reactions

96 Chapter 5 Matrices in Chemical and Biochemical Thermodynamics

(Alberty, 1992b):

ATP4- ADP3- HP0;- H,O HATP3- HADP2- H,PO;
c 10 10 0 0 10 10 0
A' =
0 13 10 4 1 13 10 4
P 3 2 1 0 3 2 1
(5.2-1)
where the prime on A' indicates that the pH has been specified so that hydrogen
atoms are not conserved. This matrix has three pairs of redundant columns. Since
the columns for ATP4- and HATP3- are the same, we can delete one and label
the remaining column as ATP, where this abbreviation refers to the sum of
ATP4- and HATP3-. The abbreviations ADP and Pi are introduced in the same
way to obtain
ATP H,O ADP Pi

(5.2-2)
P 3 0 2 1

This is referred to as an apparent conservation matrix to distinguish it from the
conservation matrix in equation 5.1-32. Thus specifying the pH has the effect of
simplifying the conservation matrix of the system by reducing the number of rows
by one and the number of columns by four. The matrix in equation 5.2-2 is not
unique. An equivalent apparent conservation matrix can be obtained more simply
by conserving adenosine groups; this leads to
ATP H,O ADP Pi
aden 1 0 0 1
A' = (5.2-3)
0 9 1 6 4
P 3 0 2 1

The row-reduced forms of matrices 5.2-2 and 5.2-3 are the same, and so they are
equivalent.

ATP H,O ADP Pi

(5.2-4)

ADP 0 0 1 -1
Apparent conservation matrices A' and apparent stoichiometric number
matrices v' at specified pH have the properties indicated by equations 5.1-10 and
5.1-22 so that

A'v' = 0 (5.2-5)
(v')~(A')~ 0 (5.2-6)
=
The rank of the A' matrix is the number C' of apparent components, and the rank
of the apparent stoichiometric number matrix is the number R' of independent
biochemical reactions.
C' = rank A' (5.2-7)
R' = rank v' (5.2-8)

N' = C' -+ R' (5.2-9)