Minimum complexity drives regulatory logic in Boolean models of living systems

Abstract

The properties of random Boolean networks have been investigated extensively as models of regulation in biological systems. However, the Boolean functions (BFs) specifying the associated logical update rules should not be expected to be random. In this contribution, we focus on biologically meaningful types of BFs, and perform a systematic study of their preponderance in a compilation of 2,687 functions extracted from published models. A surprising feature is that most of these BFs have odd “bias”, that is they produce “on” outputs for a total number of input combinations that is odd. Upon further analysis, we are able to explain this observation, along with the enrichment of read-once functions (RoFs) and its nested canalyzing functions (NCFs) subset, in terms of 2 complexity measures: Boolean complexity based on string lengths in formal logic, which is yet unexplored in biological contexts, and the so-called average sensitivity. RoFs minimize Boolean complexity and all such functions have odd bias. Furthermore, NCFs minimize not only the Boolean complexity but also the average sensitivity. These results reveal the importance of minimum complexity in the regulatory logic of biological networks.

Composition structures and biologically meaningful logics: plausibility and relevance in bipartite models of gene regulation

“…In this section, we present the results of our analyses of the abundances of composed BFs in a compiled reference biological dataset of 2687 BFs from 88 published Boolean network models of biological systems [23]. In Appendix C, we provide details about the compilation and curation of this empirical dataset.…”

“…We also computed the number and fraction of composed BFs for different composition structures which have odd bias. Recently, some of us showed that BFs with odd bias are preponderant among BFs in reconstructed Boolean network models of biological systems [23]. Furthermore, it was shown that NCFs [69] and RoFs [23] have odd bias .…”

“…We first describe the procedure that we employed to test the significance of an enrichment factor E for the set T of composed BFs in the composition structure { t 1 , t 2 ,…, t r }. A similar statistical test was presented and implemented by Subbaroyan et al [23]. First, we introduce the null hypothesis, denoted by H 0 , in which all the k -input BFs in the reference biological dataset are drawn from a random ensemble comprised of uniformly distributed k -input BFs.…”

“…Random Boolean networks are typically defined by placing interactions (directed edges) between randomly chosen genes (nodes) and assigning random logical update rules to those nodes. However, mounting evidence over the past two decades, obtained via biological network reconstruction using large-scale data from high-throughput experiments [3,18], has shown that the architecture of real gene networks is far from random, both for their network structure [3,5,13,[18][19][20][21][22][23] and for their logical update rules [11,13,[24][25][26][27][28][29][30][31], i.e., the Boolean functions (BFs) assigned to each associated gene.…”

“…This effort has led to the addition of a new type of BF previously un-explored in Boolean models of living systems, the Readonce function (RoF), to the list of biologically meaningful BFs. Remarkably, the 'simplest' logics, NCFs and RoFs also form the most restrictive subset of BFs among other biologically meaningful BFs in the space of all BFs [23].…”