The recently measured yeast transcriptional network is analyzed in terms of simplified Boolean network models, with the aim of determining feasible rule structures, given the requirement of stable solutions of the generated Boolean networks. We find that, for ensembles of generated models, those with canalyzing Boolean rules are remarkably stable, whereas those with random Boolean rules are only marginally stable. Furthermore, substantial parts of the generated networks are frozen, in the sense that they reach the same state, regardless of initial state. Thus, our ensemble approach suggests that the yeast network shows highly ordered dynamics.genetic networks ͉ dynamical systems T he regulatory network for Saccharomyces cerevisiae was recently measured (1) for 106 of the 141 known transcription factors by determining the bindings of transcription factor proteins to promoter regions on the DNA. Associating the promoter regions with genes yields a network of directed genegene interactions. As described in refs. 1 and 2, the significance of measured bindings with regard to inferring putative interactions are quantified in terms of P values. Lee et al. (1) did not infer interactions having P values above a threshold value, P th ϭ 0.001, for most of their analysis. Small threshold values, P th , correspond to a small number of inferred interactions with high quality, whereas larger values correspond to more inferred connections, but of lower quality. It was found that for the P th ϭ 0.001 network, the fan-out from each transcription factor to its regulated targets is substantial, on the average 38 (1). From the underlying data (http:͞͞web.wi.mit.edu͞young͞regulatory network), one finds that fairly few signals feed into each of them; on the average 1.9. The experiments yield the regulatory network architecture but yield neither the interaction rules at the nodes, nor the dynamics of the system, nor its final states.With no direct experimental results on the states of the system, there is, of course, no systematic method to pin down the interaction rules, not even within the framework of simplified and coarse-grained genetic network models; e.g., ones where the rules are Boolean. One can nevertheless attempt to investigate to what extent the measured architecture can, based on criteria of stability, select between classes of Boolean models (3).We generate ensembles of different model networks on the given architecture and analyze their behavior with respect to stability. In a stable system, small initial perturbations should not grow in time. This aspect is investigated by monitoring how the Hamming distances between different initial states evolve in a Derrida plot (4). If small Hamming distances diverge in time, the system is unstable and vice versa. Based on this criterion, we find that synchronously updated random Boolean networks (with a flat rule distribution) are marginally stable on the transcriptional network of yeast.By using a subset of Boolean rules, nested canalyzing functions (see Methods and Models), th...

Circadian rhythms are ubiquitous in eukaryotes, and co-ordinate numerous aspects of behaviour, physiology and metabolism, from sleep/wake cycles in mammals to growth and photosynthesis in plants1,2. This daily timekeeping is thought to be driven by transcriptional/translational feedback loops, whereby rhythmic expression of clock gene products regulates expression of associated genes in approximately 24-hour cycles. The specific transcriptional components differ between phylogenetic kingdoms3. The unicellular pico-eukaryotic alga, Ostreococcus tauri, possesses a naturally minimised clock, which includes many features that are shared with higher eukaryotes (plants), such as a central negative feedback loop that involves the morning-expressed CCA1 and evening-expressed TOC1 genes4. Given that recent observations in animals and plants have revealed prominent post-translational contributions to timekeeping5, a reappraisal of the transcriptional contribution to oscillator function is overdue. Here we show that non-transcriptional mechanisms are sufficient to sustain circadian timekeeping in the eukaryotic lineage, though they normally function in conjunction with transcriptional components. We identify oxidation of peroxiredoxin proteins as a transcription-independent rhythmic biomarker, which is also rhythmic in mammals6. Moreover we show that pharmacological modulators of the mammalian clockwork have the same effects on rhythms in Ostreococcus. Post-translational mechanisms, and at least one rhythmic marker, appear to be better conserved than transcriptional clock regulators. It is plausible that the oldest oscillator components are non-transcriptional in nature, as in cyanobacteria7, and are conserved across kingdoms.

We determine stability and attractor properties of random Boolean genetic network models with canalyzing rules for a variety of architectures. For all power law, exponential, and flat in-degree distributions, we find that the networks are dynamically stable. Furthermore, for architectures with few inputs per node, the dynamics of the networks is close to critical. In addition, the fraction of genes that are active decreases with the number of inputs per node. These results are based upon investigating ensembles of networks using analytical methods. Also, for different in-degree distributions, the numbers of fixed points and cycles are calculated, with results intuitively consistent with stability analysis; fewer inputs per node implies more cycles, and vice versa. There are hints that genetic networks acquire broader degree distributions with evolution, and hence our results indicate that for single cells, the dynamics should become more stable with evolution. However, such an effect is very likely compensated for by multicellular dynamics, because one expects less stability when interactions among cells are included. We verify this by simulations of a simple model for interactions among cells.