Biomedical Engineering Reference
In-Depth Information
The inequality constraints in Eq. (
2
) refer to the subset j of irreversible reactions
with nonnegative flux values. Equation (
2
) expresses an undetermined system of
algebraic equations because q [[ m, and thus it has no unique solution. The
universe of solutions of Eq. (
2
) forms a polyhedral cone in the fluxome solution
space whose edges correspond to independent elementary modes (elementary
modes are discussed in more detailed in the next section).
Equation (
2
) applies only to balanced intracellular metabolites. For extracel-
lular metabolites the net accumulation is nonzero and the following equation
applies:
b
¼
A
0
v
v
j
0
ð
3
Þ
with b the vector of fluxes of extracellular metabolites across the cellular mem-
brane and A
0
the stoichiometric matrix of such extracellular metabolites.
2.3 Elementary Modes
Elementary mode analysis has become a widespread technique for systems-level
metabolic pathway analysis [
28
,
29
]. An elementary mode can be defined as a
minimal set of enzymes able to operate at steady state, with the enzymes weighted
by the relative flux they need to carry for the mode to function [
10
]. The universe
of elementary modes of a given metabolic network defines the full set of nonde-
composable steady-state flux distributions that the network can support. Any
particular steady-state flux distribution can be expressed as a nonnegative linear
combination of elementary modes.
As such, the phenotype of a cell, as defined by its fluxome, v, can be expressed
as a weighted sum of the contribution of each elementary mode
v
¼
k
1
e
1
þ
k
2
e
2
þ
...
þ
k
k
e
k
¼
X
K
k
i
e
i
;
ð
4
Þ
i
¼
1
where e
i
is an elementary mode vector with dim(e
i
) = q, k
i
is the weighting factor
of e
i
, K is the number of elementary modes, and dim(v) = dim(e
i
) = q is the
number of metabolic reactions of the metabolic network. Geometrically the ele-
mentary modes correspond to the edges of the polyhedral cone in the fluxome
solution space (Fig.
1
).
The elementary mode matrix, EM, is obtained by concatenating all the e
i
vectors into a q 9 K matrix
EM
¼
e
1
½
e
2
e
K
:
ð
5
Þ
Multiplying the EM matrix by the stoichiometric matrix of the extracellular
metabolites, A
0
, one obtains the elementary mode stoichiometric matrix
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