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objective function c that would result in v as the optimal solution of (5).
An optimization approach developed for this inverse task [40] has been
applied to analyze objective functions based on experimentally meas-
ured aerobic and anaerobic flux distributions in E. coli . The objective
functions defined through this approach were found to be similar to
the assumed biomass composition and were not found to vary signifi-
cantly between the aerobic and anaerobic conditions. These results
indicate that, at least for the E. coli model, the objective function
assumed previously was consistent with the flux measurements.
Analyzing Allowable Phenotypes
While FBA and its variations described above are the most common
approaches utilized in constraint-based analysis of metabolic net-
works, there are many other mathematical methods that can be applied
to studying the properties of genome-scale stoichiometric matrices
and making experimentally verifiable predictions based on these
matrices [8,41]. The linear constraints (2)-(4) define a closed convex
subspace representing all the allowed flux distributions, optimal or
not, under particular conditions, and a number of methods have been
developed to analyze this whole space of flux distributions. It should
be emphasized that, in contrast to the optimization-based approaches
described above, the methods discussed here do not require specifying
a particular objective function and do not predict a particular flux
distribution. In this sense the methods described here are more
objective than the optimization-based approaches and can be used
to analyze the relationship of metabolic network topology to its
function.
Network-based pathway analysis . Extreme pathway and elementary
mode analysis are based on enumerating basis vectors for the convex
subspace defined by constraints (2) and (3) [42-44]. Biologically, these
basis vectors correspond to network-based definitions of pathways
whose combinations can be used to represent an arbitrary flux distri-
bution within the allowed space of flux distributions. Extreme pathways
correspond directly to the edges of the open convex solution space
defined by constraints (2) and (3), and hence are the smallest set of
pathways that represent the overall functionality of the metabolic
network. Elementary flux modes are a superset of extreme pathways,
where each elementary mode corresponds to a flux distribution through
a minimal set of reactions that can operate at steady state. The pathway
enumeration methods are computationally extremely intensive because
the number of basis vectors scales exponentially with the number of
reactions in the network. For this reason, these enumeration methods
have been limited to smaller metabolic networks or to cases where the
numbers of inputs and outputs are limited.
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