Biology Reference
In-Depth Information
respectively, and the matrix elements describe the change of token number on a place, when a transition
fires. For metabolic networks, the incidence matrix corresponds to the stoichiometric matrix.
The firing sequence
w
can be determined from the solution of the homogeneous equation system
C
w
=
0, which is valid if the signal or information flow within the network is assumed to be conserved.
This forms the base to compute system's invariants from the incidence matrix, which can be divided in
two different vectors
w
=
x
or
w
=
y
depending on the orientation of
C
.
The vector
x
is called
nonnegative place invariant
(P-invariant) if it solves the homogeneous equation
system
·
C
T
x
=0:
x
1
...x
n
∈
N
0
(2)
The elements of a P-invariant can be interpreted as
conservation relation
for tokens.
For an initial
marking
M
0
holds:
[
M
0
]
>
N
:=
{
M
0
w
|
w
∈
T
∗}
(3)
M
∈
[
M
0
]
>
N
:
M
·
∀
x
=
M
0
·
x
N
defines the set of reachable markings and
M
is a consecutive marking (see Box 1)
that can be reached from
M
0
by firing of
w
,
i.e.
, a subset of the reachability set.
The solution vector
y
is called non-negative transition invariant (T-invariant) if the following equation
holds:
whereat
[
M
0
]
>
C
·
y
=0:
y
1
...y
n
∈
N
(4)
0
A T-invariant is a transition sequence that after firing reproduces a marking (state) of the network Eq. (4).
A T-invariant's
Parikh-vector
indicates how often each transition has to fire in order to reach the same
state (marking) again. In the following, we write
y
as
y
for short and
Y
}
for the set of all
T-Invariants that can be computed from the incidence matrix
C
. The support of
y
, written as
supp
(
y
),
contains the nodes corresponding to the non-zero entries of
y
,
i.e.
,
=
{
y
1
,...
,
y
n
supp
(
y
) denotes all transitions that
belong to a T-invariant.
T-invariants are
minimal
if there exists no smaller positive T-invariant
y
:(
y
y
)
>
0 [Baumgarten,
1996],
i.e.
, the support of
y
does not contain the support of any other invariant
y
and the largest
common divisor of all non-zero entries of
y
is equal to 1. Hence, minimal T-invariants are not further
decomposable into smaller T-invariants. The same holds for minimal P-invariants. In the following
the term “
T-invariant
”(“
P-invariant
”) stands as abbreviation for minimal non-negative T-invariant (P-
invariant).
T-invariants can be interpreted as flux vectors. In biochemical terms, it was shown that under
steady
state
conditions a metabolic network can be decomposed into sets of minimal meaningful reaction
sequences (
elementary flux modes
), which form a variety of flux patterns when expressed as non-
negative linear combinations [Schuster and Hilgetag, 1994; Schuster
et al.
, 1999]. Elementary flux
modes correspond to T-invariants. The spliceosomal assembly net was modeled as transition-bounded
net,
i.e.
, no places without pre- or post-transitions exist, but transitions without pre- and post-places.
The reactants of sinks and sources are assumed as buffered substances at fixed concentrations [Schuster
and Hilgetag, 1994]. In biological interpretation that means, all reactants feeding input transitions or
leaving output transitions are considered to be external. All others are internal.
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