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In this work we propose a model for gene expression based on the theory of random dynamical systems (RDS) and show that it has a "modularity property" in the following sense: given any collection of genes that are linked in a transcriptional network, if each of them is individually described by a certain class of RDS then there is a natural, and essentially unique, prescription for coupling them together, respecting the network topology, in such a way that the collective system formed by all genes is a RDS as well. Moreover, the class of RDS used to describe the individual genes is flexible enough to account for a wide range of stochastic behaviors within the realm of stationary processes., Comment: 18 pages, 2 figures. Withdrawn for review and rewriting

Since the foundations of Population Genetics the notion of genetic equilibrium (in close analogy to Classical Mechanics) has been associated to the Hardy-Weinberg (HW) Principle and the identification of equilibrium is currently assumed by stating that the HW axioms are valid if appropriate values of Chi-Square (p<0.05) are observed in experiments. Here we show by numerical experiments with the genetic system of one locus/two alleles that considering large ensembles of populations the Chi-Square test is not decisive and may lead to false negatives in random mating populations and false positives in nonrandom mating populations. As a result we confirm the logical statement that statistical tests can not be used to deduce if the genetic population is under the HW conditions. Furthermore, we show that under the HW conditions populations of any finite size evolve in time according to what can be identified as neutral dynamics to which the very notion of equilibrium is unattainable for any practical purpose. Therefore, under the HW conditions equilibrium properties are not observable. We also show that by relaxing the condition of random mating the dynamics acquires all the characteristics of asymptotic stable equilibrium. As a consequence our results show that the question of equilibrium in genetic systems should be approached in close analogy to non-equilibrium statistical physics and its observability should be focused on dynamical quantities like the typical decay properties of the allelic auto correlation function in time. In this perspective one should abandon the classical notion of genetic equilibrium and its relation to the HW proportions and open investigations in the direction of searching for unifying general principles of population genetic transformations capable to take in consideration these systems in their full complexity., Comment: 14 pages, 6 figures

In this paper we revisit and adapt to viral evolution an approach based on the theory of branching process advanced by Demetrius, Schuster and Sigmund ("Polynucleotide evolution and branching processes", Bull. Math. Biol. 46 (1985) 239-262), in their study of polynucleotide evolution. By taking into account beneficial effects we obtain a non-trivial multivariate generalization of their single-type branching process model. Perturbative techniques allows us to obtain analytical asymptotic expressions for the main global parameters of the model which lead to the following rigorous results: (i) a new criterion for "no sure extinction", (ii) a generalization and proof, for this particular class of models, of the lethal mutagenesis criterion proposed by Bull, Sanju\'an and Wilke ("Theory of lethal mutagenesis for viruses", J. Virology 18 (2007) 2930-2939), (iii) a new proposal for the notion of relaxation time with a quantitative prescription for its evaluation, (iv) the quantitative description of the evolution of the expected values in in four distinct "stages": extinction threshold, lethal mutagenesis, stationary "equilibrium" and transient. Finally, based on these quantitative results we are able to draw some qualitative conclusions., Comment: 23 pages, 1 figure, 2 tables. arXiv admin note: substantial text overlap with arXiv:1110.3368

In the present work we analyze the problem of adaptation and evolution of RNA virus populations, by defining the basic stochastic model as a multivariate branching process in close relation with the branching process advanced by Demetrius, Schuster and Sigmund ("Polynucleotide evolution and branching processes", Bull. Math. Biol. 46 (1985) 239-262), in their study of polynucleotide evolution. We show that in the absence of beneficial forces the model is exactly solvable. As a result it is possible to prove several key results directly related to known typical properties of these systems like (i) proof, in the context of the theory of branching processes, of the lethal mutagenesis criterion proposed by Bull, Sanju\'an and Wilke ("Theory of lethal mutagenesis for viruses", J. Virology 18 (2007) 2930-2939); (ii) a new proposal for the notion of relaxation time with a quantitative prescription for its evaluation and (iii) the quantitative description of the evolution of the expected values in four distinct regimes: transient, "stationary" equilibrium, extinction threshold and lethal mutagenesis. Moreover, new insights on the dynamics of evolving virus populations can be foreseen., Comment: 39 pages, 3 figures. International Symposium on Mathematical and Computational Biology, Tempe, Arizona, USA, 6 - 10 November 2012. Fernando Antoneli, Francisco Bosco, Diogo Castro, And Luiz Mario Janini (2013) Viral Evolution and Adaptation as a Multivariate Branching Process. Biomat 2012: pp. 217-243. Ed.: R. P. Mondaini. World Scientific, Singapore