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Fig. 13.2 Flux Balance Analysis. The
axes
represent metabolic fluxes. In this example, only three
fluxes are considered. The possible fluxes can adopt any value of
v
1
,
v
2
, and
v
3
. Stoichiometry
constraints and capacity constraints restrict the fluxes. The stoichiometry constraints are imposed
by the reactions of the network, and capacity constraints derive from known limits of the reactions.
For example, we may know that the flux
v
1
has to lie between two known values:
a
b
.
Since these types of constraints are linear, possible fluxes now lie within the straight line
boundaries of a polygonal cone starting at the origin. FBA then maximizes (or minimizes) an
objective function (biomass, ATP production,
<
v
1
<
) that can be any linear combination of the fluxes.
The optimal solution (
dark blue point
) lies on the edge of the allowable solution space
...
illustrated in Fig.
13.2
. Like in MFA, the basis of FBA is the stoichiometry matrix,
i.e., the complete topological description of the metabolic network. The stoichiom-
etry matrix already imposes a first set of constraints on possible flux distributions.
Additional constraints can be added, for example, in the form of maximal and
minimal fluxes for a particular reaction. The second, and crucial, step of FBA
consists on defining an “objective function” that will be maximized. A commonly
used objective function is biomass, which amounts to adding a flux called
“biomass” to the metabolic network. Different reactions can contribute to the
biomass function, and the relative contributions can change according to growth
conditions (Meadows et al.
2010
). Mathematically, the constraints and the
objective function form a system of linear equations that can be solved, using linear
programming, to maximize the objective function. Similar to MFA, FBA does not
require any knowledge of kinetic parameters and can rapidly be calculated even for
large networks. Furthermore, prior knowledge of, e.g., reaction constants can be
incorporated into the algorithm in the form of additional constraints.
The calculated optimal flux distribution depends on the choice of the objective
function (Feist and Palsson
2010
). Evolutionary arguments favor the biomass
objective function for
E. coli
and other bacteria, at least for laboratory strains that
have been grown for a long time on commonly used growth substrates. The basic
operation for calculating biomass involves defining the macromolecular composi-
tion of the cell and thus the metabolites necessary for assembling these cellular
constituents. This objective function can be further improved by considering the
energy needed for macromolecular assembly, for example the number of ATP
molecules needed for incorporating amino acids into proteins. Even more detailed
formulations of the biomass include secondary metabolites such as vitamins and
cofactors. Even though the biomass objective function can be calculated with high
precision, different bacteria under different conditions may well be optimized for
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