Biochemical Reaction Networks
171
Glc
Fig. 5.11 Metabolic fluxes in
S. cerevisiae
estimated using l
3
C-labeled glucose. For each step the flux is
shown as a/b. The yeast was grown at high and low specific glucose uptake rates, respectively, and a is
the flux at high specific glucose uptake rate while b is the flux at low specific glucose uptake rate. At
high specific glucose uptake rates there is ethanol formation due to the Crabtree effect, and respiration is
repressed resulting in almost no flux through the TCA cycle. Due to the Crabtree effect the yield of
biomass on glucose is low and there is consequently a low requirement for NADPH and precursors of the
pentose phosphate pathway. Thus, the flux through this pathway is low. [The data are taken from
Gombert
et al.
(2001)].
5.4.2 Use of Linear Programming
In a metabolic model the elements of the flux vector can in principle take any value. Balancing of
all the intracellular metabolites does, however, impose a set of constraints on the flux vector v.
Even with the constraints imposed by the metabolite balancing there are in practice several
degrees of freedom in the system, and infinitely many solutions for the flux vector v result. The
total solution space for the flux vector v that satisfies the constraints imposed by the metabolite
balances is given by the
null space
of the stoichiometric matrix T. Thus, the null space defines
the possible solution space for the flux vector. Furthermore, the shape of the null space is given
by the stoichiometry of all the reactions used in the metabolic network model, and the network
model represented by the stoichiometric matrix T defines all possible modes of operation of the
network. Above we discussed how experimentally determined rates or measurements of :’C-
enrichments can be used to further constrain the flux vector. This corresponds to the situation
where some of the fluxes are measured (or additional constraints are added and this results in a
reduction in the dimension of the null space), and hereby one may find a unique solution, that is
given as the intercept of the flux vector with one of the edges of the null space.
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