Page 103 - Thermodynamics of Biochemical Reactions

98 Chapter 5 Matrices in Chemical and Biochemical Thermodynamics

The use of Nullspace yields the following row reduced conservation matrix:
Glutamate ATP Amm ADP Pi Glutamine
1 0 0 0 0 1
0 1 0 0 0 1
A' =z (5.3-5)
0 0 1 0 0 1
0 0 0 1 0 -1
0 0 0 0 1 -1

This shows that five apparent components are in agreement with C' = N' - R'
= 6 - 1 = 5, but only four elements (C, 0, N, and P) are to be conserved. The
fifth conservation equation is needed to tie reactions 5.3-1 and 5.3-2 one to one.
This conservation equation can be written in a number of ways, but one way is

n(ATP) + n(g1utamine) = const. (5.3-6)
Thus the apparent conservation matrix is given by

Glutamate ATP Amm ADP P, Glutamine
C 5 10 0 10 0 5
0 4 13 0 10 4 3
A' = (5.3-7)
N 1 5 1 5 0 2
P 0 3 0 2 1 0
con 1 0 1 0 0 0 1
Where con1 is the component represented by equation 5.3-6. Row reduction
yields equation 5.3-5, which shows that the stoichiometric number matrix and
conservation matrix are equivalent. The last column of equation 5.3-5 shows that
there is a single reaction and that it agrees with equation 5.3-3. When coupling
introduces additional conservation equations, components can be chosen in such
a way that the conservation relations are all expressed in terms of conservations
of reactants that are chosen as components. Thus equation 5.3-5 utilizes the five
components glutamate, ATP, ammonia, ADP, and P,.
Coupling does not necessarily involve constraints in addition to element
balances. For example, glucokinase (EC 2.7.1.2) couples the hydrolysis of ATP to
ADP with the phosphorylation of glucose to G6P. The reaction catalyzed is

ATP + Glc = ADP + G6P (5.3-8)
The apparent conservation matrix is

ATP Glc ADP G6P
aden 1 0 1 0
A' = (5.3-9)
glc 0 1 0 1
P 3 0 2 2

Note that here the reactions that are coupled share a reactant that is not a
reactant in reaction 5.3-8. It is usually more convenient to count groups rather
than atoms. Row reduction and use of the analogue of equation 5.1-19 show that
reaction 5.3-8 is obtained.
An extreme example of additional constraints introduced by the enzymatic
mechanism of a biochemical reaction is the NAD synthase (glutamine-hydrolyz-
ing) reaction (EC 6.3.5.1) (Alberty, 1994b):
ATP + deamido-NAD,, + L-glutamine + H,O

= AMP + PPi + NAD,, + L-glutamate (5.3-10)