Genetic Boolean regulatory networks have been first studied by Stuart Kauffman and Rene Thomas in the late 1960’s. Such networks are highly simplified models of genetic regulatory networks, where genes (nodes) have binary states (ON/OFF) and regulatory actions between them are described by Boolean functions. Kauffman was interested in general properties of regulatory networks, not in the detailed modeling of specific systems, motivated by the observation that
Kauffman started to study random Boolean networks (RBN) and found that depending on the topology and functions used the networks either tend to chaotic behaviour while others show ordered behaviour, i.e., most nodes can be perturbed without changing the long-term behaviour and many nodes freeze, that is they stop changing state. Based on his findings Kauffman conjectured that biological systems operate at the border between the chaotic and ordered regime. Further he concluded that order and stability are not solely the result of natural selection, and that there has to be a statistical tendency towards order and self-organization. Hence, natural selection acts on self-organizing systems, rather than creating them.
In the first part of the talk I will mainly concentrate on mathematical proofs of Kauffman’s findings given first by Lynch. We will then discuss recent results on the properties of canalyzing and nested-canalyzing functions, classes of functions that have been conjectured to be a primary source of order in RBNs. In a slightly different framework I will further discuss some results that indicate that canalyzing functions are, in a certain sense, optimal regarding there information processing capabilities.