Global Stability of Reversible Enzymatic Metabolic Chains

We consider metabolic networks with reversible enzymatic reactions. The model is written as a system of ordinary differential equations, possibly with inputs and outputs. We prove the global stability of the equilibrium (if it exists), using techniques of monotone systems and compartmental matrices. We show that the equilibrium does not always exist. Finally, we consider a metabolic system coupled with a genetic network, and we study the dependence of the metabolic equilibrium (if it exists) with respect to concentrations of enzymes. We give some conclusions concerning the dynamical behavior of coupled genetic/metabolic systems.     

Finite-size and correlation-induced effects in mean-field dynamics

The brain’s activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finite-size networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such large-scale networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of the new mean-field equations, the stability properties of limit cycles are modified by the presence of correlations, and additional non-trivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finite-size networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinite-size system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system.     

Bifurcation discovery tool

MOTIVATION: Biochemical networks often yield interesting behavior such as switching, oscillation and chaotic dynamics. This article describes a tool that is capable of searching for bifurcation points in arbitrary ODE-based reaction networks by directing the user to regions in the parameter space, where such interesting dynamical behavior can be observed. RESULTS: We have implemented a genetic algorithm that searches for Hopf bifurcations, turning points and bistable switches. The software is implemented as a Systems Biology Workbench (SBW) enabled module and accepts the standard SBML model format. The interface permits a user to choose the parameters to be searched, admissible parameter ranges, and the nature of the bifurcation to be sought. The tool will return the parameter values for the model for which the particular behavior is observed. AVAILABILITY: The software, tutorial manual and test models are available for download at the following website: http:/www.sys-bio.org/ under the bifurcation link. The software is an open source and licensed under BSD.     

On the dynamics of random Boolean networks subject to noise: Attractors, ergodic sets and cell types

The asymptotic dynamics of random Boolean networks subject to random fluctuations is investigated. Under the influence of noise, the system can escape from the attractors of the deterministic model, and a thorough study of these transitions is presented. We show that the dynamics is more properly described by sets of attractors rather than single ones. We generalize here a previous notion of ergodic sets, and we show that the Threshold Ergodic Sets so defined are robust with respect to noise and, at the same time, that they do not suffer from a major drawback of ergodic sets. The system jumps from one attractor to another of the same Threshold Ergodic Set under the influence of noise, never leaving it. By interpreting random Boolean networks as models of genetic regulatory networks, we also propose to associate cell types to Threshold Ergodic Sets rather than to deterministic attractors or to ergodic sets, as it had been previously suggested. We also propose to associate cell differentiation to the process whereby a Threshold Ergodic Set composed by several attractors gives rise to another one composed by a smaller number of attractors. We show that this approach accounts for several interesting experimental facts about cell differentiation, including the possibility to obtain an induced pluripotent stem cell from a fully differentiated one by overexpressing some of its genes.     

Balance between Noise and Information Flow Maximizes Set Complexity of Network Dynamics

Boolean networks have been used as a discrete model for several biological systems, including metabolic and genetic regulatory networks. Due to their simplicity they offer a firm foundation for generic studies of physical systems. In this work we show, using a measure of context-dependent information, set complexity, that prior to reaching an attractor, random Boolean networks pass through a transient state characterized by high complexity. We justify this finding with a use of another measure of complexity, namely, the statistical complexity. We show that the networks can be tuned to the regime of maximal complexity by adding a suitable amount of noise to the deterministic Boolean dynamics. In fact, we show that for networks with Poisson degree distributions, all networks ranging from subcritical to slightly supercritical can be tuned with noise to reach maximal set complexity in their dynamics. For networks with a fixed number of inputs this is true for near-to-critical networks. This increase in complexity is obtained at the expense of disruption in information flow. For a large ensemble of networks showing maximal complexity, there exists a balance between noise and contracting dynamics in the state space. In networks that are close to critical the intrinsic noise required for the tuning is smaller and thus also has the smallest effect in terms of the information processing in the system. Our results suggest that the maximization of complexity near to the state transition might be a more general phenomenon in physical systems, and that noise present in a system may in fact be useful in retaining the system in a state with high information content.     

Search for steady states of piecewise-linear differential equation models of genetic regulatory networks

Analysis of the attractors of a genetic regulatory network gives a good indication of the possible functional modes of the system. In this paper we are concerned with the problem of finding all steady states of genetic regulatory networks described by piecewise-linear differential equation (PLDE) models. We show that the problem is NP-hard and translate it into a propositional satisfiability (SAT) problem. This allows the use of existing, efficient SAT solvers and has enabled the development of a steady state search module of the computer tool Genetic Network Analyzer (GNA). The practical use of this module is demonstrated by means of the analysis of a number of relatively small bacterial regulatory networks as well as randomly generated networks of several hundreds of genes.     

Characterising neurological time series data using biologically motivated networks of coupled discrete maps

Artificial biochemical networks (ABNs) are a class of computational dynamical system whose architectures are motivated by the organisation of genetic and metabolic networks in biological cells. Using evolutionary algorithms to search for networks with diagnostic potential, we demonstrate how ABNs can be used to carry out classification when stimulated with time series data collected from human subjects with and without Parkinson's disease. Artificial metabolic networks, composed of coupled discrete maps, offer the best recognition of Parkinsonian behaviour, achieving accuracies in the region of 90%. This is comparable to the diagnostic accuracies found in clinical diagnosis, and is significantly higher than those found in primary and non-expert secondary care. We also illustrate how an evolved classifier is able to recognise diverse features of Parkinsonian behaviour and, using perturbation analysis, show that the evolved classifiers have interesting computational behaviours.     

Parallel evolution and inheritance of quantitative traits

Parallel phenotypic evolution, the independent evolution of the same trait in closely related lineages, is interesting because it tells us about the contribution of natural selection to phenotypic evolution. Haldane and others have proposed that parallel evolution also results from a second process, the similarly biased production of genetic variation in close relatives, an idea that has received few tests. We suggest that influence of shared genetic biases should be detectable by the disproportionate use of the same genes in independent instances of parallel phenotypic evolution. We show how progress in testing this prediction can be made through simple tests of parallel inheritance of genetic differences: similar additive, dominance, and epistasis components in analysis of line means and similar effective numbers of loci. We demonstrate parallel inheritance in two traits, lateral plate number and body shape, in two lineages of threespine stickleback that have adapted independently to freshwater streams on opposite sides of the Pacific Ocean. Notably, reduction of plate number in freshwater involves a substitution at the same major locus in both lineages. Our results represent only a first step in the study of the genetics of parallel phenotypic evolution in sticklebacks. Nevertheless, we have shown how such studies can be employed to test the genetic hypothesis of parallel evolution and how study of parallel evolution might yield insights into the roles of both selection and genetic constraint in phenotypic evolution.     

Analysis of Metabolic Subnetworks by Flux Cone Projection

Background

Analysis of elementary modes (EMs) is proven to be a powerful constraint-based method in the study of metabolic networks. However, enumeration of EMs is a hard computational task. Additionally, due to their large number, EMs cannot be simply used as an input for subsequent analysis. One possibility is to limit the analysis to a subset of interesting reactions. However, analysing an isolated subnetwork can result in finding incorrect EMs which are not part of any steady-state flux distribution of the original network. The ideal set to describe the reaction activity in a subnetwork would be the set of all EMs projected to the reactions of interest. Recently, the concept of "elementary flux patterns" (EFPs) has been proposed. Each EFP is a subset of the support (i.e., non-zero elements) of at least one EM.

Results

We introduce the concept of ProCEMs (Projected Cone Elementary Modes). The ProCEM set can be computed by projecting the flux cone onto a lower-dimensional subspace and enumerating the extreme rays of the projected cone. In contrast to EFPs, ProCEMs are not merely a set of reactions, but projected EMs. We additionally prove that the set of EFPs is included in the set of ProCEM supports. Finally, ProCEMs and EFPs are compared for studying substructures of biological networks.

Conclusions

We introduce the concept of ProCEMs and recommend its use for the analysis of substructures of metabolic networks for which the set of EMs cannot be computed.