Biology Reference
In-Depth Information
In our model, all reactants (places) are modeled in non-limited amounts with a capacity of K ( p ) →∞
and an initial marking of one token per place to enable each transition.
The model has been validated using PN analysis techniques. We determine the static and dynamic
properties, using the programs I NA [Starke, 1998] and T INA N ET [Thormann et al. , 2009]. With increasing
network size and complexity, the number of T-invariants can exponentially grow. Two approaches
were employed to further decompose the network and thus, to facilitate the validation of the model: i )
decomposition into disjunctive subnetworks (MCTS) and ii ) decomposition into overlapping subnetworks
(T-clusters).
Partitioning of T-invariants into MCTS
MCTS [Sackmann et al. , 2006] are based on a matrix D in which rows and columns correspond to T
and Y , respectively, with T
defining the set of transitions and Y
defining the set of T-invariants. Each
row constitutes a subset I
Y of T-invariants that share a considered transition t . Hence, I ( t i ) denotes
the subset of T-Invariants, which share the transition t i . Biologically, this means that a specific reaction is
part of a certain number of all possible and minimal steady state signaling pathways within the network.
All transitions, which in this way are exclusively shared by the same set of T-invariants, form an MCTS
A
T for which holds:
t i ,t j
T
: t i ,t j
A
I ( t i )= I ( t j ):
(5)
This can be also expressed by:
y
Y
: A
supp ( y )
A
supp ( y )=
(6)
All reactions of an MCTS occur always together, i.e. , reactants and enzymes have to follow a similar
scheme of regulation. The transitions of an MCTS and the places in between describe a subnetwork.
Clustering of T-invariants
We performed a clustering of T-invariants to find similarities imposed by transitions that are shared
between different T-invariants. T-clusters define subnetworks that can overlap or contain each other
[Grafahrend-Belau et al. , 2008]. They facilitate the identification of traversed routes, which are formed
by common subsets of reactions and highlight more important structures within the net.
The clustering was computed using the Tanimoto coefficient [Grafahrend-Belau et al. , 2008] as distance
measure, which is also known as binary distance or Jaccard index [Cormack, 1971]. The corresponding
distance tree of related T-invariants was constructed using the UPGMA-algorithm [The R Development
Core Team, 2005]. A threshold of 80% was chosen to merge T-invariants with less than 20% difference
into the same subtree.
The same distance measure and clustering algorithm was used to create a color map . A color map is
a graphical way of displaying matrices by using colors to represent the numerical values. Due to the
binary (on/off = present / non-present) representation of transitions within the support of T-invariants, a
simplified two color mode was chosen, where dark (light) blue tones indicate the presence (absence) of a
reaction. The color map also re-arranges rows and columns of the distance matrix such that similar rows,
and similar columns, are grouped together, according to the distance tree. This representation facilitates
the visualization of block patterns of transitions, shared by different T-invariants.
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