Genome-Scale Models of Metabolic and Regulatory Networks - Systems Biology: Networks, Models, and Applications - page 240

Flux coupling analysis . Recently, other more scalable methods have
been developed to analyze flux subspaces defined by constraints (2)
and (3) based on optimization techniques [45] or random sampling of
the solution space [46-48]. The flux coupling finder (FCF) [45] is an
optimization-based approach for finding general dependences
between pairs of fluxes in the metabolic network under steady-state
conditions without assuming a particular objective for the network.
The FCF approach enumerates coupled reaction subsets and directional
couplings between reactions. Coupled subsets are sets of reactions
that always have to be co-utilized because of stoichiometric constraints,
and in most cases the fluxes through the reactions are also constrained
to fixed ratios with respect to each other. Directional couplings are uni-
directional relationships between pairs of fluxes in the network that
represent the requirement for a nonzero flux through one reaction for
the functioning of another reaction, but not vice versa. The FCF method
is scalable to genome-scale metabolic networks, because it only requires
solving a fixed number of linear optimization problems.
Random sampling . Random sampling approaches represent perhaps
the most unbiased way to analyze the overall properties of the flux
spaces defined by constraints (2)-(4) as well as the effect of imposing
further constraints on the network [46-48]. While uniform sampling of
points (each corresponding to an allowed flux distribution) within the
closed convex set defined by (2)-(4) is in principle straightforward, in
practice efficient algorithms are required to ensure that the sampling
converges to a uniform distribution in a reasonable time. The standard
approach for sampling convex spaces, the hit-and-run algorithm [49],
is based on uniformly randomly selecting a direction from a random
starting point within the space and moving a random distance (drawn
from a uniform distribution) along this direction so that the resulting
point is still within the constraints. This process is then repeated until
the sampling distribution has converged to a stationary uniform distri-
bution. Extensions to this basic method have been proposed that
potentially allow faster convergence of the sampler [50]. The output of
the sampling methods is a large set of flux distributions whose charac-
teristics can be analyzed using statistical multivariate analysis tools.
The methods described above are all based on the stoichiometric
matrix as opposed to using a graph representation of the metabolic
network. It should be emphasized that representing the connectivity of
the metabolic network in the form of a stoichiometric matrix allows
automatically accounting for the mass balancing of the entire meta-
bolic reaction network. This is in contrast to most graph-based
methods [51,52], which usually fail to account for the correct cofactor
connectivity and balancing in these networks. Since metabolism is
fundamentally driven by these cofactors such as ATP, NADH, and
NADPH, the failure to account for the balancing of these factors