A model of $s$ interacting species is considered with two types of dynamical variables. The fast variables are the populations of the species and slow variables the links of a directed graph that defines the catalytic interactions among them. The graph evolves via mutations of the least fit species. Starting from a sparse random graph, we find that an autocatalytic set inevitably appears and triggers a cascade of exponentially increasing connectivity until it spans the whole graph. The connectivity subsequently saturates in a statistical steady state. The time scales for the appearance of an autocatalytic set in the graph and its growth have a power law dependence on $s$ and the catalytic probability. At the end of the growth period the network is highly nonrandom, being localized on an exponentially small region of graph space for large $s$.

The NF-κB signaling system is involved in a variety of cellular processes including immune response, inflammation, and apoptosis. Recent experiments have found oscillations in the nuclear-cytoplasmic translocation of the NF-κB transcription factor [Hoffmann, A., et al. (2002) Science 298, 1241–1245; Nelson, D. E., et al. (2004) Science 306, 704–708.] How the cell uses the oscillations to differentiate input conditions and send specific signals to downstream genes is an open problem. We shed light on this issue by examining the small core network driving the oscillations, which we show is designed to produce periodic spikes in nuclear NF-κB concentration. The presence of oscillations is extremely robust to variation of parameters, depending mainly on the saturation of the active degradation rate of IκB, an inhibitor of NF-κB. The oscillations can be used to regulate downstream genes in a variety of ways. In particular, we show that genes to whose operator sites NF-κB binds and dissociates fast can respond very sensitively to changes in the input signal, with effective Hill coefficients of >20.

Organisms are equipped with regulatory systems that display a variety of dynamical behavior ranging from simple stable steady states, to switching and multistability, to oscillations. Earlier work has shown that oscillations in protein concentrations or gene expression levels are related to the presence of at least one negative feedback loop in the regulatory network. Here, we study the dynamics of a very general class of negative feedback loops. Our main result is that, when a single negative feedback loop dominates the dynamical behavior, the sequence of maxima and minima of the concentrations exhibit a pattern that uniquely identifies the interactions of the loop. This allows us to devise an algorithm to ( i ) test whether observed oscillating time series are consistent with a single underlying negative feedback loop, and if so, ( ii ) reconstruct the precise structure of the loop, i.e., the activating/repressing nature of each interaction. This method applies even when some variables are missing from the data set, or if the time series shows transients, like damped oscillations. We illustrate the relevance and the limits of validity of our method with three examples: p53-Mdm2 oscillations, circadian gene expression in cyanobacteria, and cyclic binding of cofactors at the estrogen-sensitive pS2 promoter.

The development of new techniques to quantitatively measure gene expression in cells has shed light on a number of systems that display oscillations in protein concentration. Here we review the different mechanisms which can produce oscillations in gene expression or protein concentration using a framework of simple mathematical models. We focus on three eukaryotic genetic regulatory networks which show 'ultradian' oscillations, with a time period of the order of hours, and involve, respectively, proteins important for development (Hes1), apoptosis (p53) and immune response (NF-kappaB). We argue that underlying all three is a common design consisting of a negative feedback loop with time delay which is responsible for the oscillatory behaviour.

The control of proteins by a transcription factor with periodically varying concentration exhibits intriguing dynamical behaviour. Even though it is accepted that transcription factors vary their dynamics in response to different situations, insight into how this affects downstream genes is lacking. Here, we investigate how oscillations and chaotic dynamics in the transcription factor NF-κB can affect downstream protein production. We describe how it is possible to control the effective dynamics of the transcription factor by stimulating it with an oscillating ligand. We find that chaotic dynamics modulates gene expression and up-regulates certain families of low-affinity genes, even in the presence of extrinsic and intrinsic noise. Furthermore, this leads to an increase in the production of protein complexes and the efficiency of their assembly. Finally, we show how chaotic dynamics creates a heterogeneous population of cell states, and describe how this can be beneficial in multi-toxic environments.

The causes of major and rapid transitions observed in biological macroevolution as well as in the evolution of social systems are a subject of much debate. Here we identify the proximate causes of crashes and recoveries that arise dynamically in a model system in which populations of (molecular) species coevolve with their network of chemical interactions. Crashes are events that involve the rapid extinction of many species, and recoveries the assimilation of new ones. These are analyzed and classified in terms of the structural properties of the network. We find that in the absence of large external perturbation, “innovation” is a major cause of large extinctions and the prime cause of recoveries. Another major cause of crashes is the extinction of a “keystone species.” Different classes of causes produce crashes of different characteristic sizes.