Adaptive gene regulatory networks

Regulatory interactions between genes show a large amount of cross-species variability, even when the underlying functions are conserved: there are many ways to achieve the same function. Here we investigate the ability of regulatory networks to reproduce given expression levels within a simple model of gene regulation. We find an exponentially large space of regulatory networks compatible with a given set of expression levels, giving rise to an extensive entropy of networks. Typical realisations of regulatory networks are found to share a bias towards symmetric interactions, in line with empirical evidence. Copyright c © EPLA, 2009 Introduction. – The expression of genes is regulated such that the right combinations of gene products are generated at the right time and place of an organism. Key regulators of gene expression are transcription factors, proteins which bind to specific sites on DNA and influence the expression of nearby genes. Typically, the expression of a gene is effected by a combination of several transcription factors, and conversely, a transcription factor regulates several genes. Expression levels can thus depend on the entire set of regulatory interaction between transcription factors and their target genes, referred to as a regulatory network. These intracellular reaction networks process extracellular information to induce specific gene expression patterns, allowing, for instance, the development of a complex body plan, or responses to external conditions. Even though regulatory networks are tuned carefully to produce specific expression patterns, there are in general many networks fulfilling a regulatory task. One example is the control of mating type in different yeast species: The same set of genes controlled in S. cerevisiae by an activator which is upregulated in a certain state is controlled by a repressor which is downregulated in that state in C. albicans [1]. A second prominent example is the development of the anterior patterning in insect embryos, leading to the formation of the insect’s head. The gene crucial to this process in the fruit fly Drosophila, called bicoid, (a)E-mail: [email protected] is absent in many other insects, where a combination of different genes take on the same task [2]. Even whole sets of genes which are co-expressed across the entire yeast family can have different regulatory interactions in different species [3]. Across bacteria, widespread ‘tinkering’ at the level of individual interactions is found, even though similar subnetwork topologies (network motifs) appear in organisms with similar lifestyle [4]. Source of these changing interactions is a rapid evolutionary turnover of transcription factor binding sites at the level of DNA sequences [5,6]. This can generate new regulatory interactions. A recent essay on the degeneracy of regulatory networks can be found in [7]. The large number of networks resulting in a viable organism (viable regulatory networks) does not imply that there are many dispensable interactions: viable networks may occupy only a small fraction of the set of all possible regulatory networks, and elaborate counter-changes may be needed to restore viability once an interaction has been altered. An analogy is the set of all RNA sequences which fold into a given secondary structure, which stretches across the entire sequence space [8]. Numerical studies in regulatory networks, based on simple models of gene regulation [9–12] found that the space of viable networks can be crossed in small steps, and that a wide range of new expression patterns can be generated by small changes to different viable networks. The large number of viable regulatory networks is particularly relevant from