Boolean networks with biologically relevant rules show ordered behavior

It was found recently that natural gene regulatory systems are governed by hierarchically canalyzing functions (HCFs), a special subclass of Boolean functions. Here we study the HCF class in detail. We present a new minimal logical expression for all HCFs. Based on this formula, we calculate the cardinality of the HCF class. Moreover, we define HCF subclasses and calculate their cardinality as well. Using the well-known critical connectivity condition 2Kcp(1−p)=12Kcp(1−p)=1, we discuss order–chaos transitions of Boolean networks (BNs) regulated by functions of given HCF subclasses. Finally, analysing real gene regulatory rules we show that nearly all of the biologically relevant functions belong to the simplest HCF subclasses. This restriction is important for reverse engineering of transcription regulatory networks and for ensemble approach studies in systems biology. It is shown that Boolean networks with functions belonging to the biologically realized HCF subclasses show ordered behavior.