Entropy as a Robustness Marker in Genetic Regulatory Networks.

Genetic regulatory networks have evolved by complexifying their control systems with numerous effectors (inhibitors and activators). That is, for example, the case for the double inhibition by microRNAs and circular RNAs, which introduce a ubiquitous double brake control reducing in general the number of attractors of the complex genetic networks (e.g., by destroying positive regulation circuits), in which complexity indices are the number of nodes, their connectivity, the number of strong connected components and the size of their interaction graph. The stability and robustness of the networks correspond to their ability to respectively recover from dynamical and structural disturbances the same asymptotic trajectories, and hence the same number and nature of their attractors. The complexity of the dynamics is quantified here using the notion of attractor entropy: it describes the way the invariant measure of the dynamics is spread over the state space. The stability (robustness) is characterized by the rate at which the system returns to its equilibrium trajectories (invariant measure) after a dynamical (structural) perturbation. The mathematical relationships between the indices of complexity, stability and robustness are presented in case of Markov chains related to threshold Boolean random regulatory networks updated with a Hopfield-like rule. The entropy of the invariant measure of a network as well as the Kolmogorov-Sinaï entropy of the Markov transition matrix ruling its random dynamics can be considered complexity, stability and robustness indices; and it is possible to exploit the links between these notions to characterize the resilience of a biological system with respect to endogenous or exogenous perturbations. The example of the genetic network controlling the kinin-kallikrein system involved in a pathology called angioedema shows the practical interest of the present approach of the complexity and robustness in two cases, its physiological normal and pathological, abnormal, dynamical behaviors.

Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks.

Networks used in biological applications at different scales (molecule, cell and population) are of different types: neuronal, genetic, and social, but they share the same dynamical concepts, in their continuous differential versions (e.g., non-linear Wilson-Cowan system) as well as in their discrete Boolean versions (e.g., non-linear Hopfield system); in both cases, the notion of interaction graph G(J) associated to its Jacobian matrix J, and also the concepts of frustrated nodes, positive or negative circuits of G(J), kinetic energy, entropy, attractors, structural stability, etc., are relevant and useful for studying the dynamics and the robustness of these systems. We will give some general results available for both continuous and discrete biological networks, and then study some specific applications of three new notions of entropy: (i) attractor entropy, (ii) isochronal entropy and (iii) entropy centrality; in three domains: a neural network involved in the memory evocation, a genetic network responsible of the iron control and a social network accounting for the obesity spread in high school environment.

Ten-year follow-up of cluster-based asthma phenotypes in adults. A pooled analysis of three cohorts.

RATIONALE: The temporal stability of adult asthma phenotypes identified using clustering methods has never been addressed. Longitudinal cluster-based methods may provide novel insights in the study of the natural history of asthma. OBJECTIVES: To compare the stability of cluster-based asthma phenotype structures a decade apart in adults and to address the individuals' phenotypic transition across these asthma phenotypes. METHODS: The latent transition analysis was applied on longitudinal data (twice, 10 yr apart) from 3,320 adults with asthma who took part in the European Community Respiratory Health Survey, the Swiss Cohort Study on Air Pollution and Lung and Heart Diseases in Adults, or the Epidemiological Study on Genetics and Environment of Asthma. Nine variables covering personal and phenotypic characteristics measured twice, 10 years apart, were simultaneously considered. MEASUREMENTS AND MAIN RESULTS: Latent transition analysis identifies seven asthma phenotypes (prevalence range, 8.4-20.8%), mainly characterized by the level of asthma symptoms (low, moderate, high), the allergic status, and pulmonary function. Phenotypes observed 10 years apart showed strong similarities. The probability of membership in the same asthma phenotype at both times varied across phenotypes from 54 to 88%. Different transition patterns were observed across phenotypes. Transitions toward increased asthma symptoms were more frequently observed among nonallergic phenotypes as compared with allergic phenotypes. Results showed a strong stability of the allergic status over time. CONCLUSIONS: Adult asthma phenotypes identified by a clustering approach, 10 years apart, were highly consistent. This study is the first to model the probabilities of transitioning over time between comprehensive asthma phenotypes.