Biochemical Reaction Networks
143
Appearance o f linearly dependent columns in the stoichiometric matrix
T. We remember
that columns in the stoichiometric matrix specify the stoichiometric coefficients for the
compounds included in the model, and linearly dependent columns appear if the
stoichiometric coefficients for some of the compounds are identical or linearly dependent.
This rarely happens in practice, but if one included both co-factors in co-factor couples
like NADH/NAD’, NADPH/NADP’ or ATP/ADP then the stoichiometry for one of these
co-factors would be identical with that of the other - except for the sign. A simple
solution to this problem is that only one of the co-factors is included for each co-factor
pair. Notice that if linear dependency occurs for stoichiometric coefficients of some
intracellular compounds then the linearly dependent columns will always be transferred
to linearly dependent rows in T2, as this matrix always will contain all the stoichiometric
coefficients for all intracellular compounds.
Appearance o f linearly dependent rows in the stoichiometric matrix
T. The rows in the
stoichiometric matrix specify the stoichiometry for reactions in the model, and if for
some reason one or more reaction stoichiometries are linearly dependent this will transfer
to linearly dependent columns in the T 2 matrix, and it hereby becomes singular. Linearly
dependent reaction stoichiometries rarely appear in simple reaction networks, but they
often occur in larger reaction networks, and we will therefore return to this problem in
Section 5.4.1 (see Note 5.3).
Non-observable system with the chosen measured rates.
In principle the measured rates
can be chosen arbitrary, but in practice one may chose a set of measured rates for which
the system is not observable,
i.e.
with the chosen set of measured rates the matrix T 2 is
singular. It is often difficult to identify which set of measured rates results in an
observable system and which do not, and if singularity occurs the only practical way is to
try a different set of measured rates and check whether the matrix T2 is non-singular. This
problem is discussed further in Examples 5.4 and 5.5.
This procedure to calculate the flux vector v and the non-measured rates is independent of the
size of the reaction network, but it only applies when exactly
F
rates are measured. The
procedure is illustrated below with two examples on relatively simple reaction networks. In
Section 5.4 we are going to analyze large reaction networks and will also discuss how the fluxes
can be calculated if we measure more or fewer than
F
rates.
Example 5.5 Anaerobic growth of
Saccharomyces cerevisiae
Consider anaerobic growth of
S. cerevisiae.
Here ethanol is the major metabolic product, but some
glycerol is also formed as discussed in Section 3.3. Glycerol formation is primarily a result of redox
balancing since NADH is formed in connection with biomass synthesis, and the only way for the cell to
balance the level of NADH at anaerobic conditions is through conversion of glucose to glycerol.
Glycerol is formed from dihydroxyacetone phosphate, an intermediate in the EMP pathway (see Fig.
2.4), through the following reactions:
NAD NAD
Dihydroxyacetone.
phosphate
Glycerol-3-phosphate
Glycerol
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