Page 110 - Thermodynamics of Biochemical Reactions

106 Chapter 6 Systems of Biochemical Reactions

6.1 CALCULATION OF NET REACTIONS USING
MATRIX MULTIPLICATION

In dealing with large systems of biochemical reactions it is important to find ways
to obtain a more global view, and one way is to use net reactions because they
show what is accomplished by a set of reactions. In calculating a net reaction, the
intermediates are eliminated because they are produced and consumed in equal
amounts. In order to prevent accumulation of intermediates, some of the reactions
in the set have to run at 2, 3,. .. times the rates of other reactions. These integers
are referred to as the stoichiometric numbers of steps, which are represented by s:
for step i. The pathway matrix (column vector) for a set of biochemical reactions
is represented by s'. The pathway matrix is R'x 1. The advantage of using
matrices is that linear algebra and computers can be used.
The relation between a stoichiometric number matrix v' for a set of R'
reactions involving N' reactants and the stoichiometric number matrix vket for a
net reaction is a system of linear equations that is represented by the following
matrix multiplication (Alberty, 1996):

............... = ... (6.1 - 1)
............ sk ...
...... '
v;2 "NR 'ne1N
Equation 6.1-1 can be written in the form

(6.1-2)

& 1 44 2 "NR "netN
This shows that the solution s' to the system of linear equations represented by
equation 6.1-1 is made up of the stoichiometric numbers sI that give the number
of times the various biochemical reactions have to occur to accomplish the net
reaction. Equation 6.1-1 is conveniently written in matrix notation as
VI * = VI net (6.1-3)
The stoichiometric number matrix v' for the system is N' x R', the pathway matrix
s' is R' x 1, and the stoichiometric number matrix vie, for the net reaction is N' x 1.
When the pH (and perhaps the free concentrations of cations that are bound
reversibly) is specified, a prime is used on the symbols in equation 6.1-3 to
distinguish the stoichiometric numbers of the biochemical reactions from those of
the underlying chemical reactions. Since it is easy to make errors in typing a
stoichiometric number matrix into a computer, it is useful to check the matrix by
using it to print out the reactions. This can be done using the programs mkeqn
(Alberty, 1996a) and nameMatrix (Alberty, 2000c), which are given in Problem
6.1. The use of these programs is illustrated in Problem 6.1.
The net reaction for the 10 steps of glycolysis is
glucose + 2Pi + 2ADP + 2NAD,, = 2pyruvate + 2ATP + 2NAD,,, + 2H,O
(6.1-4)

This net reaction is obtained by multiplying the first five reactions of glycolysis
by 1, the second five reactions by 2, and adding. This causes the intermediates to
cancel. Alternatively, this net reaction can be calculated by multiplying the
stoichiometric number matrix v' for the 10 reactions of glycolysis by the pathway
matrix s', where (s')~ = { 1, 1,1, 1,1,2,2,2,2,2), according to equation 6.1-3.