Robustness in regulatory interaction networks. A generic approach with applications at different levels: physiologic, metabolic and genetic

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.

Emergence of a "Cyclosome" in a Primitive Network Capable of Building "Infinite" Proteins

We argue for the existence of an RNA sequence, called the AL (for ALpha) sequence, which may have played a role at the origin of life; this role entailed the AL sequence helping generate the first peptide assemblies via a primitive network. These peptide assemblies included "infinite" proteins. The AL sequence was constructed on an economy principle as the smallest RNA ring having one representative of each codon's synonymy class and capable of adopting a non-functional but nevertheless evolutionarily stable hairpin form that resisted denaturation due to environmental changes in pH, hydration, temperature, etc. Long subsequences from the AL ring resemble sequences from tRNAs and 5S rRNAs of numerous species like the proteobacterium, Rhodobacter sphaeroides. Pentameric subsequences from the AL are present more frequently than expected in current genomes, in particular, in genes encoding some of the proteins associated with ribosomes like tRNA synthetases. Such relics may help explain the existence of universal sequences like exon/intron frontier regions, Shine-Dalgarno sequence (present in bacterial and archaeal mRNAs), CRISPR and mitochondrial loop sequences.

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.

Discrete dynamics of contagious social diseases: Example of obesity

Modeling contagious diseases needs to incorporate information about social networks through which the disease spreads as well as data about demographic and genetic changes in the susceptible population. In this paper, we propose a theoretical framework (conceptualization and formalization) which seeks to model obesity as a process of transformation of one's own body determined by individual (physical and psychological), inter-individual (relational, i.e., relative to the relationship between the individual and others) and socio-cultural (environmental, i.e., relative to the relationship between the individual and his milieu) factors. Individual and inter-individual factors are tied to each other in a socio-cultural context whose impact is notably related to the visibility of anybody being exposed on the public stage in a non-contingent way. The question we are dealing with in this article is whether such kind of social diseases, i.e., depending upon socio-environmental exposure, can be considered as "contagious". In other words, can obesity be propagated from individual to individual or from environmental sources throughout an entire population?

Evolution of social networks: the example of obesity

The present paper deals with the effect of the social transmission of nutrition habits in a social and biological age-dependent context on obesity, and accordingly on type II diabetes and among its complications, the neurodegenerative diseases. The evolution of social networks and inside a network the healthy weight of a person are depending on the context in which this person has contacts and exchanges concerning his alimentation, physical activity and sedentary habits, inside the dominant social network in which the person lives (e.g., scholar for young, professional for adult, home or institution for elderly people). Three successive steps of evolution will be considered for social networks (like for neural one's): initial random connectivity, destruction and consolidation of links following a new transition rule called homophilic until an asymptotic architectural organization and configuration of states. The application of such a network dynamics concerns the sequence overweight/obesity/type II diabetes and neurodegenerative diseases.

Stability, complexity and robustness in population dynamics

The problem of stability in population dynamics concerns many domains of application in demography, biology, mechanics and mathematics. The problem is highly generic and independent of the population considered (human, animals, molecules,…). We give in this paper some examples of population dynamics concerning nucleic acids interacting through direct nucleic binding with small or cyclic RNAs acting on mRNAs or tRNAs as translation factors or through protein complexes expressed by genes and linked to DNA as transcription factors. The networks made of these interactions between nucleic acids (considered respectively as edges and nodes of their interaction graph) are complex, but exhibit simple emergent asymptotic behaviours, when time tends to infinity, called attractors. We show that the quantity called attractor entropy plays a crucial role in the study of the stability and robustness of such genetic networks.

Analysis of the robustness of network-based disease-gene prioritization methods reveals redundancy in the human interactome and functional diversity of disease-genes

Complex biological systems usually pose a trade-off between robustness and fragility where a small number of perturbations can substantially disrupt the system. Although biological systems are robust against changes in many external and internal conditions, even a single mutation can perturb the system substantially, giving rise to a pathophenotype. Recent advances in identifying and analyzing the sequential variations beneath human disorders help to comprehend a systemic view of the mechanisms underlying various disease phenotypes. Network-based disease-gene prioritization methods rank the relevance of genes in a disease under the hypothesis that genes whose proteins interact with each other tend to exhibit similar phenotypes. In this study, we have tested the robustness of several network-based disease-gene prioritization methods with respect to the perturbations of the system using various disease phenotypes from the Online Mendelian Inheritance in Man database. These perturbations have been introduced either in the protein-protein interaction network or in the set of known disease-gene associations. As the network-based disease-gene prioritization methods are based on the connectivity between known disease-gene associations, we have further used these methods to categorize the pathophenotypes with respect to the recoverability of hidden disease-genes. Our results have suggested that, in general, disease-genes are connected through multiple paths in the human interactome. Moreover, even when these paths are disturbed, network-based prioritization can reveal hidden disease-gene associations in some pathophenotypes such as breast cancer, cardiomyopathy, diabetes, leukemia, parkinson disease and obesity to a greater extend compared to the rest of the pathophenotypes tested in this study. Gene Ontology (GO) analysis highlighted the role of functional diversity for such diseases.

Deconstruction and dynamical robustness of regulatory networks: application to the yeast cell cycle networks

Analyzing all the deterministic dynamics of a Boolean regulatory network is a difficult problem since it grows exponentially with the number of nodes. In this paper, we present mathematical and computational tools for analyzing the complete deterministic dynamics of Boolean regulatory networks. For this, the notion of alliance is introduced, which is a subconfiguration of states that remains fixed regardless of the values of the other nodes. Also, equivalent classes are considered, which are sets of updating schedules which have the same dynamics. Using these techniques, we analyze two yeast cell cycle models. Results show the effectiveness of the proposed tools for analyzing update robustness as well as the discovery of new information related to the attractors of the yeast cell cycle models considering all the possible deterministic dynamics, which previously have only been studied considering the parallel updating scheme.

Predictive power of "a minima" models in biology

Many apparently complex mechanisms in biology, especially in embryology and molecular biology, can be explained easily by reasoning at the level of the "efficient cause" of the observed phenomenology: the mechanism can then be explained by a simple geometrical argument or a variational principle, leading to the solution of an optimization problem, for example, via the co-existence of a minimization and a maximization problem (a min-max principle). Passing from a microscopic (or cellular) level (optimal min-max solution of the simple mechanistic system) to the macroscopic level often involves an averaging effect (linked to the repetition of a large number of such microscopic systems with possible random choice of the parameters of each of them) that gives birth to a global functional feature (e.g. at the tissue level). We will illustrate these general principles by building in four different domains of application "a minima" models and showing the main properties of their solutions: (1) extraction of a minimal RNA structure functioning as the first "peptidic machine," a kind of ancestral ribosome; (2) study of a genetic regulatory network of Drosophila centred on Engrailed gene and expressing successively two genes inside a limit cycle; (3) study of a genetic network regulating neural activity and proliferation in mammals; and (4) study of a simple geometric model of epiboly in zebrafish.

"Immunetworks", intersecting circuits and dynamics

This paper proposes a study of biological regulation networks based on a multi-level strategy. Given a network, the first structural level of this strategy consists in analysing the architecture of the network interactions in order to describe it. The second dynamical level consists in relating the patterns found in the architecture to the possible dynamical behaviours of the network. It is known that circuits are the patterns that play the most important part in the dynamics of a network in the sense that they are responsible for the diversity of its asymptotic behaviours. Here, we pursue further this idea and argue that beyond the influence of underlying circuits, intersections of circuits also impact significantly on the dynamics of a network and thus need to be payed special attention to. For some genetic regulation networks involved in the control of the immune system ("immunetworks"), we show that the small number of attractors can be explained by the presence, in the underlying structures of these networks, of intersecting circuits that "inter-lock".

The isochronal fibration: characterization and implication in biology

Limit cycles, because they are constituted of a periodic succession of states (discrete or continuous) constitute a good manner to store information. From any points of the state space reached after a perturbation or stimulation of the cognitive system storing this information, one can aim to join through a more or less long return trajectory a precise neighbourhood of the asymptotic trajectory at a specific moment (or a specific place) on the limit cycle, i.e. where the information of interest stands. We propose that the isochronal fibration, initially imagined and described by A. T. Winfree may be an excellent way to connect directly those two locations. Each isochron is indeed the set of points in temporal phase with one single point of the attractor. The characterisation of the isochronal fibration of various dynamical systems is not easy and until now has principally only been done numerically but not analytically. By integrating the homogeneous solutions of the dynamical system we can solve this fibration in the case of the well known anharmonic pendulum. Other isochronal fibration on classical examples such as the van der Pol system and the non-symmetrical PFK limit cycle are obtained numerically and we also provide the first numerical study on 3-dimentional systems like the anharmonic pendulum with a linear relaxation on its third variable and the Lorenz attractor. The empirical approach seems us useful for dealing with the isochronal fibration which could constitute a powerful tool for understanding and controlling the dynamics of biological or biological-inspired systems.

RNA relics and origin of life

A number of small RNA sequences, located in different non-coding sequences and highly preserved across the tree of life, have been suggested to be molecular fossils, of ancient (and possibly primordial) origin. On the other hand, recent years have revealed the existence of ubiquitous roles for small RNA sequences in modern organisms, in functions ranging from cell regulation to antiviral activity. We propose that a single thread can be followed from the beginning of life in RNA structures selected only for stability reasons through the RNA relics and up to the current coevolution of RNA sequences; such an understanding would shed light both on the history and on the present development of the RNA machinery and interactions. After presenting the evidence (by comparing their sequences) that points toward a common thread, we discuss a scenario of genome coevolution (with emphasis on viral infectious processes) and finally propose a plan for the reevaluation of the stereochemical theory of the genetic code; we claim that it may still be relevant, and not only for understanding the origin of life, but also for a comprehensive picture of regulation in present-day cells.