BioSystems 73 (2004) 141–152 The circular topology of rhythm in asynchronous random Boolean networks Philipp Rohlfshagen, Ezequiel A. Di Paolo ∗ Department of Informatics, University of Sussex, Brighton BN1 9QH, UK Received 14 August 2003; received in revised form 20 November 2003; accepted 28 November 2003 Abstract The analysis of previously evolved rhythmic asynchronous random Boolean networks [Biosystems 59 (2001) 185] reveals common topological characteristics indicating that rhythm originates from a circular functional structure. The rhythm generating core of the network has the form of a closed ring which operates as a synchronisation substrate by supporting a travelling wave of state change; the size of the ring corresponds well with the period of oscillation. The remaining nodes in the network are either stationary or follow the activity of the ring without feeding back into it so as to form a coherent whole. Rings are typically formed early on in the evolutionary search process. Alternatively, long chains of nodes are favoured before they close upon themselves to stabilize. Analysis of asynchronous networks with de-correlated (non-rhythmic, non-stationary) attractors reveals no such common topological characteristics. These results have been obtained using statistical analysis and a specifically developed bottom–up pruning algorithm. This algorithm works from local interactions to global configuration by eliminating redundant links. The suitability of the algorithm has been confirmed by both numerical and single lesion analysis. The ring topology solution for the generation of rhythm implies that it will be harder to evolve rhythmic networks for big sizes and small periods and for bigger number of connections per node. These trends are confirmed empirically. Finally, the identified mechanisms are utilised to handcraft rhythmic networks of different periods showing that a low number of connections suffices for a large variety of rhythms. Random asynchronous update forces the evolved solutions to be highly robust maintaining their performance in the presence of intrinsic noise. The biological implications of such robust designs for molecular clocks are discussed. © 2003 Elsevier Ireland Ltd. All rights reserved. Keywords: Random Boolean networks; Random asynchronous updating; Ring topology; Genetic algorithms; Rhythmic phenomena 1. Introduction Intracellular molecular clocks are a necessary, but not sufficient element in the generation of circa- dian and other biological rhythms (Roennenberg and Merrow, 2001). It is generally acknowledged that regulation across networks of oscillating cells, and entrainment with external signals are necessary for regulating a 24-h cycle. However, molecular clocks in ∗ Corresponding author. Fax: +44-1273-671320. E-mail address: ezequiel@sussex.ac.uk (E.A. Di Paolo). themselves can be quite robust (Young, 1998) main- taining a rhythm even when cells divide faster than the period of oscillation, for instance in cyanobacteria (Kondo et al., 1997). Random Boolean network (RBN) models of ge- netic regulation have naturally been associated with cell cycles and molecular oscillations due to the fact that their behaviour will always fall into some sort of cyclic attractor, (Kauffman, 1969, 1993; Bagley and Glass, 1996). When these cycles are very long (much larger than the size of the network) they are often called “chaotic”, but for smaller cycles it is possible to 0303-2647/$ – see front matter © 2003 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2003.11.003