Emergence and Growth of Complex Networks in Adaptive Systems

We consider the population dynamics of a set of species whose network of catalytic interactions is described by a directed graph. The relationship between the attractors of this dynamics and the underlying graph theoretic structures like cycles and autocatalytic sets is discussed. It is shown that when the population dynamics is suitably coupled to a slow dynamics of the graph itself, the network evolves towards increasing complexity driven by autocatalytic sets. Some quantitative measures of network complexity are described.Comment: 10 pages (including figures), 3 Postscript figure

Autocatalytic Sets and the Growth of Complexity in an Evolutionary Model

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 (ACS) 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 ACS 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 non-random, being localized on an exponentially small region of graph space for large $s$.Comment: 13 pages REVTEX (including figures), 4 Postscript figure

Inducing phase-locking and chaos in cellular oscillators by modulating the driving stimuli

Inflammatory responses in eucaryotic cells are often associated with oscillations in the nuclear-cytoplasmic translocation of the transcription factor NF-kB. In most laboratory realizations, the oscillations are triggered by a cytokine stimulus, like the tumor necrosis factor alpha, applied as a step change to a steady level. Here we use a mathematical model to show that an oscillatory external stimulus can synchronize the NF-kB oscillations into states where the ratios of the internal to external frequency are close to rational numbers. We predict a specific response diagram of the TNF-driven NF-kB system which exhibits bands of synchronization known as "Arnold tongues". Our model also suggests that when the amplitude of the external stimulus exceeds a certain threshold there is the possibility of coexistence of multiple different synchronized states and eventually chaotic dynamics of the nuclear NF-kB concentration. This could be used as a way of externally controlling immune response, DNA repair and apoptotic pathways.Comment: 12 pages, 3 figure

Symbolic dynamics of biological feedback networks

We formulate general rules for a coarse-graining of the dynamics, which we term `symbolic dynamics', of feedback networks with monotone interactions, such as most biological modules. Networks which are more complex than simple cyclic structures can exhibit multiple different symbolic dynamics. Nevertheless, we show several examples where the symbolic dynamics is dominated by a single pattern that is very robust to changes in parameters and is consistent with the dynamics being dictated by a single feedback loop. Our analysis provides a method for extracting these dominant loops from short time series, even if they only show transient trajectories.Comment: 4 pages, 4 figure

Network Models of Phage-Bacteria Coevolution

Bacteria and their bacteriophages are the most abundant, widespread and diverse groups of biological entities on the planet. In an attempt to understand how the interactions between bacteria, virulent phages and temperate phages might affect the diversity of these groups, we developed a novel stochastic network model for examining the co-evolution of these ecologies. In our approach, nodes represent whole species or strains of bacteria or phages, rather than individuals, with "speciation" and extinction modelled by duplication and removal of nodes. Phage-bacteria links represent host-parasite relationships and temperate-virulent phage links denote prophage-encoded resistance. The effect of horizontal transfer of genetic information between strains was also included in the dynamical rules. The observed networks evolved in a highly dynamic fashion but the ecosystems were prone to collapse (one or more entire groups going extinct). Diversity could be stably maintained in the model only if the probability of speciation was independent of the diversity. Such an effect could be achieved in real ecosystems if the speciation rate is primarily set by the availability of ecological niches.Comment: 8 pages, 6 figure