Disordered Systems and Biological Organization pp 361-365 | Cite as

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

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## 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 {s
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.

_{1}, s_{2},…, 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 \( \sqrt {P} \).

## Keywords

Boolean Function Random Mapping Average Period Successive Image Cellular Type
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## References

- Kauffman, S. 1970 a. “Behaviour of randomly constructed nets”. In
**Towards a theoretical biology**Ed. C.H. Waddington, vol.**3**, Edimburg University Press.Google Scholar - Kauffman, S. 1970 b. “The organization of cellular genetic cintrol systems”.
**Math. Life Sci. 3**, 63–116.Google Scholar - Kauffman, S. 1979 “Assessing the probable regulatory structures and dynamics of the metazoan genome. Kinetic logic”. In
**Lecture notes for Biomathematics**Ed. R. Thomas,**29**, 30–61. Berlin Springer Verlag.Google Scholar - Sherlock, R.A. 1979 “Analysis of Kauffman binary networks”
**Bull. Math. Biol. 41**, 687–724.zbMATHMathSciNetGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1986