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If irreversible reactions are taken into account, the signs of the coefficients are
positive for the incoming arcs, or negative in the opposite case. As a
consequence, in the second row of
N
there is a 1 in the fifth column because
v
v
is leading away from
enters B and there is a -1 in the fourth column because
B. At steady state we have
dS
=
Nv
=
0
dt
Nv
holds. The problem of finding
solutions to this kind of equations is well known in linear algebra where it is
called as the
kernel
(or null) of
N
and it is known that any possible solution can
be found as linear combinations of special solutions called
basis vectors
,
corresponding to basis stationary flows in the network. The question whether
such flows are biologically feasible must be addressed separately, using
experimental evidences. However, such decomposition of steady state flows in a
minimal number of elementary flows, form the basic tool for the dynamic
analysis of metabolic networks. When considering irreversible reactions, the
stoichiometric matrix remains the same while constraints on the signs of some
basis vector elements arise.
=
0
and the allowed fluxes are those for which
8.3.2.
Elementary flux modes
The stoichiometry of a metabolic network can be used to determine the so-called
flux modes
, which are sets of flow vectors
v
that lead from one external
metabolite to another external metabolite. A flux mode is an
elementary flux
mode
(EFM) if it uses a minimal number of reactions and cannot be further
decomposed.
An elementary flux mode fulfills the following three conditions: 1) steady
state condition; 2) feasibility (irreversible reactions have to proceed in the 'right'
direction, meaning that all vectors are nonnegative); 3) non-decomposability,
implying that the participating enzymes in one EFM are not a subset of the
enzymes from another EFM.
The number of elementary flux modes is no less than the number of basis
vectors in the null space of
N
. An
extreme pathway
(EP) is a particular EFM that
satisfies two additional conditions: 4) each reaction must be classified either as
an exchange flux, which allow a metabolite to enter or to exit the system, or as an
internal reaction. All reversible internal reactions must be decomposed as two
separate irreversible reactions, thus implying that no internal reaction can have a
negative value. Exchanges fluxes can be reversible but the usefulness of this
