Abstract
Network motifs have been studied extensively over the past decade, and certain motifs, such as the feed-forward loop, play an important role in regulatory networks. Recent studies have used Boolean network motifs to explore the link between form and function in gene regulatory networks and have found that the structure of a motif does not strongly determine its function, if this is defined in terms of the gene expression patterns the motif can produce. Here, we offer a different, higher-level definition of the ‘function’ of a motif, in terms of two fundamental properties of its dynamical state space as a Boolean network. One is the basin entropy, which is a complexity measure of the dynamics of Boolean networks. The other is the diversity of cyclic attractor lengths that a given motif can produce. Using these two measures, we examine all 104 topologically distinct three-node motifs and show that the structural properties of a motif, such as the presence of feedback loops and feed-forward loops, predict fundamental characteristics of its dynamical state space, which in turn determine aspects of its functional versatility. We also show that these higher-level properties have a direct bearing on real regulatory networks, as both basin entropy and cycle length diversity show a close correspondence with the prevalence, in neural and genetic regulatory networks, of the 13 connected motifs without self-interactions that have been studied extensively in the literature.
Highlights
Network motifs, which are small subgraphs of directed networks, have been the subject of much research over the past decade [1,2], and certain motifs, such as the feed-forward loop, are known to play an important role in gene regulatory networks [3]
Motifs that contain a feed-forward loop and no three-node feedback loop have low basin entropy and low cycle length diversity. We connect these results to the frequencies of motifs in real-world gene regulatory networks and signal-transduction networks [2]. We show that both basin entropy and cycle length diversity are inversely correlated with the relative enrichment of network motifs, meaning that motifs containing feedback loops are suppressed in real-world regulatory networks
A number of publications have addressed the link between the structure and dynamics or, more broadly speaking, form and function of network motifs [12,13,14,15]
Summary
Network motifs, which are small subgraphs of directed networks, have been the subject of much research over the past decade [1,2], and certain motifs, such as the feed-forward loop, are known to play an important role in gene regulatory networks [3]. We examine the link between the form and function of network motifs differently, by defining their functional versatility in terms of two structural properties of their dynamical attraction basins. The second is the number of different attractor cycle lengths that can be realized in state space for a given network motif We show that both of these measures strongly depend on the presence of feedback loops and feed-forward loops in the motif. Motifs that contain a feed-forward loop and no three-node feedback loop have low basin entropy and low cycle length diversity. We connect these results to the frequencies of motifs in real-world gene regulatory networks and signal-transduction networks [2]. The structural properties of network motifs strongly determine their functional versatility as regulatory circuits, if we define this functional versatility in terms of attractor properties
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