Potential automata. Application to the genetic code III

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.

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.

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.

The rules of variation: amino acid exchange according to the rotating circular genetic code

General guidelines for the molecular basis of functional variation are presented while focused on the rotating circular genetic code and allowable exchanges that make it resistant to genetic diseases under normal conditions. The rules of variation, bioinformatics aids for preventative medicine, are: (1) same position in the four quadrants for hydrophobic codons, (2) same or contiguous position in two quadrants for synonymous or related codons, and (3) same quadrant for equivalent codons. To preserve protein function, amino acid exchange according to the first rule takes into account the positional homology of essential hydrophobic amino acids with every codon with a central uracil in the four quadrants, the second rule includes codons for identical, acidic, or their amidic amino acids present in two quadrants, and the third rule, the smaller, aromatic, stop codons, and basic amino acids, each in proximity within a 90 degree angle. I also define codifying genes and palindromati, CTCGTGCCGAATTCGGCACGAG.

Micro-RNAs: viral genome and robustness of gene expression in the host

For comparing RNA rings or hairpins with reference or random ring sequences, circular versions of distances and distributions like those of Hamming and Gumbel are needed. We define these circular versions and we apply these new tools to the comparison of RNA relics (such as micro-RNAs and tRNAs) with viral genomes that have coevolved with them. Then we show how robust are the regulation networks incorporating in their boundary micro-RNAs as sources or new feedback loops involving ubiquitous proteins like p53 (which is a micro-RNA transcription factor) or oligopeptides regulating protein translation. Eventually, we propose a new coevolution game between viral and host genomes.

Biological boundaries and biological age

The chronologic age classically used in demography is often unable to give useful information about which exact stage in development or aging processes has reached an organism. Hence, we propose here to explain in some applications for what reason the chronologic age fails in explaining totally the observed state of an organism, which leads to propose a new notion, the biological age. This biological age is essentially determined by the number of divisions before the Hayflick's limit the tissue or mitochondrion in a critical organ (in the sense where its loss causes the death of the whole organism) has already used for its development and adult phases. We give a precise definition of the biological age of an organ based on the Hayflick's limit of its cells and we introduce a desynchronization index (the cell entropy) for some critical tissues or membranes, which are mainly skin, intestinal endothelium, alveoli epithelium and mitochondrial inner membrane. In these actively metabolising interface tissues or membranes, there is a rapid turnover of cells, of their cytoplasmic constituents such as proteins, and of membrane lipids. The boundaries corresponding to these tissues, cells or membranes have vital functions of interface with the environment (protection, homeothermy, nutrition and respiration) and have a rapid turnover (the total cell renewal time is in mice equal to 3 weeks for the skin, 1.5 day for the intestine, 4 months for the alveolae and 11 days for mitochondrial inner membrane) conditioning their biological age. The biological age of a tissue is made of two major components: (1) first, its embryonic age based on the distance (in number of divisions) between the birth date of its first differentiated cell and the time until it reaches its final boundary at the end of its development and (2) second, its adult age whose complement until its death is just the lapse of time made of the sum of remaining cell cycle durations authorized by its Hayflick's limit. From this definition, we calculate the global biological lifespan of an organism and revisit notions like demographic survival curves, duration and synchrony of cell cycles, living boundaries from proto-cells to organs, and embryonic and adult phases duration.

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

Regulatory interaction networks are often studied on their dynamical side (existence of attractors, study of their stability). We focus here also on their robustness, that is their ability to offer the same spatiotemporal patterns and to resist to external perturbations such as losses of nodes or edges in the networks interactions architecture, changes in their environmental boundary conditions as well as changes in the update schedule (or updating mode) of the states of their elements (e.g., if these elements are genes, their synchronous coexpression mode versus their sequential expression). We define the generic notions of boundary, core, and critical vertex or edge of the underlying interaction graph of the regulatory network, whose disappearance causes dramatic changes in the number and nature of attractors (e.g., passage from a bistable behaviour to a unique periodic regime) or in the range of their basins of stability. The dynamic transition of states will be presented in the framework of threshold Boolean automata rules. A panorama of applications at different levels will be given: brain and plant morphogenesis, bulbar cardio-respiratory regulation, glycolytic/oxidative metabolic coupling, and eventually cell cycle and feather morphogenesis genetic control.

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.

Evolution and RNA relics. a systems biology view

The genetic code has evolved from its initial non-degenerate wobble version until reaching its present state of degeneracy. By using the stereochemical hypothesis, we revisit the problem of codon assignations to the synonymy classes of amino-acids. We obtain these classes with a simple classifier based on physico-chemical properties of nucleic bases, like hydrophobicity and molecular weight. Then we propose simple RNA (or more generally XNA, with X for D, P or R) ring structures that present, overlap included, one and only one codon by synonymy class as solutions of a combinatory variational problem. We compare these solutions to sequences of present RNAs considered as relics, with a high interspecific invariance, like invariant parts of (t)RNAs and micro-RNAs. We conclude by emphasizing some optimal properties of the genetic code.