Preponderance of generalized chain functions in reconstructed Boolean models of biological networks

Abstract

Boolean networks (BNs) have been extensively used to model the dynamics of gene regulatory networks (GRNs) that underlie cellular decisions. The dynamics of BNs depend on the network architecture and regulatory logic rules (or Boolean functions (BFs)) associated with nodes, both of which have been shown to be far from random in large-scale studies of reconstructed Boolean models. At the level of the BFs, nested canalyzing functions (NCFs) have been shown to be strongly enriched in such GRN models. The central question we address here is whether that enrichment is due to certain sub-types of NCFs. To answer this, we build on one sub-type of NCFs, the chain functions (or chain-0 functions) proposed by Gat-Viks and Shamir. First, we propose 2 other sub-types of NCFs, namely, the class of chain-1 functions which is the dual of the class of chain-0 functions, and generalized chain functions, the union of the chain-0 and chain-1 types. Next, we find that the fraction of NCFs that are chain-0 functions decays exponentially with the number of inputs, and exhibits a fractal-like behaviour as a function of the bias for a fixed number of inputs. Moreover, we explain several of these observations analytically. Then, by analyzing 5990 BFs extracted from a large dataset of reconstructed Boolean models, and 2 other datasets, we find that generalized chain functions are significantly enriched within the NCFs. Lastly, we illustrate the severe restriction imposed by generalized chain functions compared to NCFs for 3 biological models and perform model selection on them using known relative stability constraints.