Attractor-Specific and Common Expression Values in Random Boolean Network Models (with a Preliminary Look at Single-Cell Data)

Abstract

Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the “common sea” (CS) as the set of nodes that take the same value in all the attractors of a given network realization, and the “specific part” (SP) as the set of all the other nodes, and we study their properties in different ensembles, generated with different parameter values. Both the CS and of the SP can be composed of one or more weakly connected components, which are emergent intermediate-level structures. We show that the study of these sets provides very important information about the behavior of the model. The distribution of distances between attractors is also examined. Moreover, we show how the notion of a “common sea” of genes can be used to analyze data from single-cell experiments.

Pseudo-attractors in Random Boolean Network Models and Single-Cell Data

“…The action of a gene on the activation of other genes takes place through the action of its corresponding protein; therefore, the notion of a single time step corresponds to assuming a common decay time of the different regulatory proteins, which is not supported by biological data. By following [6] we therefore define, for each N-dimensional attractor, a corresponding constant N-dimensional "pseudoattractor", in which each component assumes the value 1 if its time average in the dynamic attractor is ≥θ (in the following we suppose θ=0.5) and take the value 0 otherwise. As a consequence, the relationship between dynamical attractors and pseudo-attractors is not injective, and it qualitatively corresponds to a kind of coarse graining in phase space.…”

“…We studied the properties of the CS and the SP by simulating RBNs which belong to different ensembles, generated with different parameter values. The following results have been presented in [6], so we will avoid to continuously refer to it. Most simulations concern dynamically critical networks (i.e.…”

Review and assessment of Boolean approaches for inference of gene regulatory networks

“…This final step is encoded as a satisfiability (SAT) problem. For more information see [42] , [44] , [45] . One limitation of SCNS is its computational complexity.…”