Analysis of attractor distances in Random Boolean Networks

We study the properties of the distance between attractors in Random Boolean Networks, a prominent model of genetic regulatory networks. We define three distance measures, upon which attractor distance matrices are constructed and their main statistic parameters are computed. The experimental analysis shows that ordered networks have a very clustered set of attractors, while chaotic networks' attractors are scattered; critical networks show, instead, a pattern with characteristics of both ordered and chaotic networks.Comment: 9 pages, 6 figures. Presented at WIRN 2010 - Italian workshop on neural networks, May 2010. To appear in a volume published by IOS Pres

Topology of biological networks and reliability of information processing

Biological systems rely on robust internal information processing: Survival depends on highly reproducible dynamics of regulatory processes. Biological information processing elements, however, are intrinsically noisy (genetic switches, neurons, etc.). Such noise poses severe stability problems to system behavior as it tends to desynchronize system dynamics (e.g. via fluctuating response or transmission time of the elements). Synchronicity in parallel information processing is not readily sustained in the absence of a central clock. Here we analyze the influence of topology on synchronicity in networks of autonomous noisy elements. In numerical and analytical studies we find a clear distinction between non-reliable and reliable dynamical attractors, depending on the topology of the circuit. In the reliable cases, synchronicity is sustained, while in the unreliable scenario, fluctuating responses of single elements can gradually desynchronize the system, leading to non-reproducible behavior. We find that the fraction of reliable dynamical attractors strongly correlates with the underlying circuitry. Our model suggests that the observed motif structure of biological signaling networks is shaped by the biological requirement for reproducibility of attractors.Comment: 7 pages, 7 figure

Genetic networks with canalyzing Boolean rules are always stable

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.Comment: Final version available through PNAS open access at http://www.pnas.org/cgi/content/abstract/0407783101v

Canalizing Kauffman networks: non-ergodicity and its effect on their critical behavior

Boolean Networks have been used to study numerous phenomena, including gene regulation, neural networks, social interactions, and biological evolution. Here, we propose a general method for determining the critical behavior of Boolean systems built from arbitrary ensembles of Boolean functions. In particular, we solve the critical condition for systems of units operating according to canalizing functions and present strong numerical evidence that our approach correctly predicts the phase transition from order to chaos in such systems.Comment: to be published in PR

The effect of scale-free topology on the robustness and evolvability of genetic regulatory networks

We investigate how scale-free (SF) and Erdos-Renyi (ER) topologies affect the interplay between evolvability and robustness of model gene regulatory networks with Boolean threshold dynamics. In agreement with Oikonomou and Cluzel (2006) we find that networks with SFin topologies, that is SF topology for incoming nodes and ER topology for outgoing nodes, are significantly more evolvable towards specific oscillatory targets than networks with ER topology for both incoming and outgoing nodes. Similar results are found for networks with SFboth and SFout topologies. The functionality of the SFout topology, which most closely resembles the structure of biological gene networks (Babu et al., 2004), is compared to the ER topology in further detail through an extension to multiple target outputs, with either an oscillatory or a non-oscillatory nature. For multiple oscillatory targets of the same length, the differences between SFout and ER networks are enhanced, but for non-oscillatory targets both types of networks show fairly similar evolvability. We find that SF networks generate oscillations much more easily than ER networks do, and this may explain why SF networks are more evolvable than ER networks are for oscillatory phenotypes. In spite of their greater evolvability, we find that networks with SFout topologies are also more robust to mutations than ER networks. Furthermore, the SFout topologies are more robust to changes in initial conditions (environmental robustness). For both topologies, we find that once a population of networks has reached the target state, further neutral evolution can lead to an increase in both the mutational robustness and the environmental robustness to changes in initial conditions.Comment: 16 pages, 15 figure