Computational Methods in Systems Biology

Graph Representations of Monotonic Boolean

Model Pools

Robert Schwieger(B)and Heike Siebert

Freie Universit ̈at Berlin, Berlin, Germany

[email protected]

Abstract.In the face of incomplete data on a system of interest,

constraint-based Boolean modeling still allows for elucidating system

characteristics by analyzing sets of models consistent with the avail-

able information. In this setting, methods not depending on consider-

ation of every single model in the set are necessary for efficient analysis.

Drawing from ideas developed in qualitative differential equation theory,

we present an approach to analyze sets of monotonic Boolean models

consistent with given signed interactions between systems components.

We show that for each such model constraints on its behavior can be

derived from a universally constructed state transition graph essentially

capturing possible sign changes of the derivative. Reachability results of

the modeled system, e.g., concerning trap or no-return sets, can then

be derived without enumerating and analyzing all models in the set.

The close correspondence of the graph to similar objects for differential

equations furthermore opens up ways to relate Boolean and continuous

models.

1 Introduction

Mathematical modeling in systems biology is often hampered by lack of infor-
mation on mechanistic detail and parameters. Constraint-based Boolean model-
ing still allows investigations based on restricted knowledge, e.g., on component
dependencies and impact of certain interactions, by considering sets of mod-
els consistent with such constraints. However, analysis of every single model in
the set is costly. Exploiting formal verification techniques still allows to investi-
gate the behavior of large numbers of models in this context [ 10 , 14 ]. A different
approach aims at avoiding enumeration and explicit analysis of every model in
the set by deriving properties directly from the given constraints, e.g., inferring
dynamical information from coinciding structural characteristics of all models
[ 9 , 11 – 13 ]. Here, we adopt the latter approach for sets of Boolean networks con-
sistent with a given signed interaction graphΣcapturing dependencies between
system components and the type of influence exerted, activating or inhibiting.
This constitutes a scenario of particular interest in application, where interac-
tion information is usually more readily available than details on the processing
logic of multiple influences on a target component.

©cSpringer International Publishing AG 2017
J. Feret and H. Koeppl (Eds.): CMSB 2017, LNBI 10545, pp. 233–248, 2017.
DOI: 10.1007/978-3-319-67471-1 14