Virus replication as a phenotypic version of polynucleotide evolution.

In this paper, we revisit and adapt to viral evolution an approach based on the theory of branching process advanced by Demetrius et al. (Bull. Math. Biol. 46:239-262, 1985), 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 et al. (J. Virol. 18:2930-2939, 2007), (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 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.

Neutral and stable equilibria of genetic systems and the hardy-weinberg principle: limitations of the chi-square test and advantages of auto-correlation functions of allele frequencies.

Since the foundations of Population Genetics the notion of genetic equilibrium (in close analogy with Classical Mechanics) has been associated with 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 χ(2) (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 χ(2)-test is not decisive and may lead to false negatives in random mating populations and false positives in non-random mating populations. This result confirms the logical statement that statistical tests cannot be used to deduce if the genetic population is under the HW conditions. Furthermore, we show that under the HW conditions populations of any 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 the identification of equilibrium properties needs a different approach and the use of more appropriate concepts. 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.

Scaling properties of transcription profiles in gene networks.

Here we show that the transcriptional noise is an emergent property with scale invariance from genome level to the level of small Transcriptional Regulatory Genetic Networks (TRGN). We show that a small set of 9-12 genes reproduces the geometric mean value of transcriptional noise of the largest percolating networks and the whole 93-gene wide TRGN sub-network. Our results predict that the collapse of the standard deviation of the transcriptional noise as a function of gene sub-networks connectivity should occur for 1000 genes, the approximate size of the maximal interconnected percolating network cluster, which corresponds to the minimal genome size.