Invariant Cycles in the Random Mapping of N Integers Onto Themselves. Comparison with Kauffman Binary Network

Abstract

According to Kauffman’s idea [Kauffman 1970a,b, 1979], one considers an ensemble of P genes which may be found in two possible states s., labelled as 0 and 1. An overall state S of the ensemble is the set {s₁, s₂,…, s_p}, which is an element of {0,1}^p. Given a mapping of {0,l}^p→{0,1}^p, the iteration of this mapping defines the dynamics of any initial S. In Kauffman model s_i at time (t+1) is determined by the states of k genes at time t, -possibly including s_i itself. Therefore the dynamics is defined by the set of all gene connections and, for each gene, by the data of a Boolean function, that is by an array of 2^k elements whose values are either 0 or 1 (there are ${2^{{{2^k}}}}$ possible Boolean functions). The dynamics drives any S towards a cycle of period m (1 ⩽ m ⩽ 2^p), and the problem is to find out the number and the periods of those cycles when S is varied over the various possible states. A numerical study has been performed by Kauffman for k=2 and choosing at random the set of gene connections and the P Boolean functions. It appeared that:

• i) The average number of cycles is of the order of $\sqrt {P}$

• ii) The average period of the cycles is also of the order of $\sqrt {P}$.

This remarkable result shows up some amazing simplicity in the dynamics of a large system and, in particular, helps one to understand how a so large number of interacting genes can produce only few cellular types.