Computing maximal and minimal trap spaces of Boolean networks

Abstract

Asymptotic behaviors are often of particular interest when analyzing Boolean networks that represent biological systems such as signal transduction or gene regulatory networks. Methods based on a generalization of the steady state notion, the so-called trap spaces, can be exploited to investigate attractor properties as well as for model reduction techniques. In this paper, we propose a novel optimization-based method for computing all minimal and maximal trap spaces and motivate their use. In particular, we add a new result yielding a lower bound for the number of cyclic attractors and illustrate the methods with a study of a MAPK pathway model. To test the efficiency and scalability of the method, we compare the performance of the ILP solver gurobi with the ASP solver potassco in a benchmark of random networks.

Computing Signal Transduction in Signaling Networks modeled as Boolean Networks, Petri Nets, and Hypergraphs

“…Different update strategies yield different dynamics, with subtleties: steady-state attractors are the same both for synchronous and asynchronous updates of a Boolean Network [19,3]. Stable motifs [100] and trap spaces [46,47], that is, trap sets with particularly simple dynamics, are also independent of the particular updating schedule strategy.…”

“…In particular, stable motifs are independent of the update scheduling strategy. Stable motifs are closely related to the concepts of trap spaces [47]. Using the MAPK pathway modeling cell fate decisions [33], the minimal trap spaces are computed and used to give a lower bound on the number of cyclic attractors [47].…”

“…Stable motifs are closely related to the concepts of trap spaces [47]. Using the MAPK pathway modeling cell fate decisions [33], the minimal trap spaces are computed and used to give a lower bound on the number of cyclic attractors [47]. To compute trap spaces, a graph expansion method is used; the prime implicant graph which is a unique hypergraph expansion of a Boolean Network not depending on any particular normal form.…”

“…This expansion method is closely related to the translation from [17], although the latter heavily relies on either read-arcs or inhibitory arcs to preserve the dynamics. Similarly, the unique prime implicant graph of a Boolean Network is a B-hypergraph [46,47], and can easily be translated into a Petri Net structure or into a graph with composite nodes. It should be further investigated how this translation preserves the dynamics.…”

General method to find the attractors of discrete dynamic models of biological systems

“…Boolean models, which characterize each node with two states and describe regulation in a parameter-free manner, are most strongly based on the network structure. Many methods exist for finding attractors of Boolean networks [19][20][21][22]. Although for some systems Boolean modeling is appropriate, often at least a subset of the nodes needs to be characterized by multiple levels, in order to accurately describe experimentally observed relative outcomes in case of combinations of inputs [23,24].…”