Page 100 - Thermodynamics of Biochemical Reactions

5.2 Biochemical Equations as Matrix Equations 95

chemical reactions involved are
ATP4- + H,O = ADP3- + HPOi- + H+ (5.1-28)

HATP3- = H+ + ATP4- (5.1-29)
HADP2- = H+ + ADP3- (5.1 - 30)
H,PO, = H+ + HP0;- (5.1-3 1)
This set of chemical reactions is not unique; for example, the reference reaction
can be written with H,PO,. Additional reactions are involved if Mg2+ or other
cations are bound reversibly by these species. The conservation matrix for this
system is
ATP4- H+ H,O HPO2- ADP3- HATP3- HADP2- H,PO,
c 1 0 0 0 0 10 10 10 0
A=H 12 1 2 1 12 13 13 2
0 1 3 0 1 4 10 13 10 4
P 3 0 0 1 2 3 2 1
(5.1-32)
The rows for nitrogen and electric charges are redundant, and therefore are
omitted. The row-reduced conservation matrix is
ATP4- H+ H,O HP0;- ADP3- HATP3- HADP2- H,PO,
ATP4 t o o 0 1 1 1 0
A= H+ 0 1 0 0 -1 1 0 1
H2O 0 0 1 0 1 0 1 0
HPOi- 0 0 0 1 -1 0 -1 1
(5.1-33)

The last four species can be considered to be made up of the four components
labeling the rows.
A basis for the stoichiometric number matrix can be calculated using
M athematica.
ATP4- H+ H,O HPOi- ADP3- HATP3- HADP2- H,PO,
0 -1 0 -1 0 0 0 1
T
v = -1 0 -1 1 0 0 1 0
-1 -1 0 0 0 1 0 0
-1 1 -1 1 1 0 0 0

(5.1-34)
This does not correspond with reactions 5.1-28 to 5.1-32, but it is equivalent
because the row-reduced form of equation 5.1-34 is identical with the row-reduced
form of the stoichiometric number matrix for reactions 5.1-28 to 5.1-32 (see
Problem 5.2). The application of matrix algebra to electrochemical reactions is
described by Alberty (1993d).

5.2 BIOCHEMICAL EQUATIONS AS MATRIX
EQUATIONS

As discussed in Chapter 4, biochemists are generally more interested in reactions
at specified pH. At specified pH, hydrogen is not conserved, and so this row and
column of matrix 5.1-32 are omitted to obtain the following conservation matrix