234 R. Schwieger and H. Siebert
A similar scenario motivates the theory of qualitative differential equations
(QDE) [ 4 ]. Here, a signed interaction graph is interpreted as the sign structure
of a Jacobi matrix of an ordinary differential equation (ODE) system. All ODE-
systems, which are consistent with a given interaction graph are collected in a
so-called monotonic ensembleM(Σ). For this ensemble, a qualitative state tran-
sition graph (QSTG) can be constructed whose nodes represent derivative signs
of the system components and edges indicate possible changes in the derivative
over time. It can then be used to describe the behavior of the ensemble. Similar
ideas have been exploited successfully for piecewise linear differential equations
by de Jong and colleagues for systems biology modeling [ 2 , 3 ].
Motivated by the results in the QDE setting, we show that a similar graphG
carries meaning in the Boolean framework as well. Here, for each Boolean func-
tionfconsistent with a given interaction graph, we are interested in the asyn-
chronous state transition graph (ASTG) capturing the dynamics of the model.
We show that while the ASTG cannot be related directly toGthis becomes pos-
sible for a quotient graph derived from the ASTG by identifying system states
with the same image underf. This quotient graph needs to be a subgraph ofG.
Consequently, analysis ofGallows to infer reachability constraints valid for all
models consistent with the interaction graph. In particular, universal statements
about trap sets and attractors become possible. The close correspondence ofG
to the QSTG of a family of ODEs furthermore allows to relate discrete and con-
tinuous model ensembles, which can facilitate the preprocessing of continuous
data for Boolean models as well as prove useful in model validation.
Our paper is structured in the following way: In the first section we state def-
initions and notions about Boolean regulatory networks. Afterwards, we review
existing results for monotonic ensembles in the continuous setting and transfer
these ideas to the Boolean framework in Sect. 3. Subsequently, we exploit the
results by investigating how information about trap sets and no-return sets can
be obtained for sets of models consistent with a given interaction graph without
enumeration. Section 5 then touches upon some aspects relating Boolean and
ODE models. A short discussion concludes the paper. To allow for easy repro-
duction of our results, our Python implementation is publicly accessible in the
following git-repository:https://github.com/RSchwieger/QDE.
2 Preliminaries
Throughout the paper we consider a system of components 1,...,n,n∈N.As
general notation for different and coinciding entries or arbitrary vectors, we use
forv, w∈Sn,Sany set:
diff(v, w):={
i∈{ 1 ,...,n}|vi=wi}
,
comm(v, w):={ 1 ,...,n}\diff(v, w).