Stochastic Evolution of a Single Locus in a Virus Population: An Analytic Review

I.M. Rouzine and J.M. Coffin.

Tufts University, Boston MA

A broad range of problems of population genetics is analyzed with a view to virological applications, such as growth competition assay and fixation of advantageous alleles in vitro, and evolution of HIV within and between infected patients. We start from a one-locus, two-allele model of haploid populations including the following factors: random drift (as represented by a finite population size, N), purifying selection (selection coefficient, s) and mutations (mutation rate, m). Evolutionary behavior, at any population size, is analyzed by a unified method using a stochastic equation of diffusion type, and by the Monte-Carlo simulation. We calculate quantitative parameters suitable for comparison with experiment such as the probability density of the mutant frequency, as well as the expectation values and variances of the mutant frequency and the intra-population genetic distance. The analysis shows that that evolution becomes almost deterministic at population sizes much larger than the inverse mutation rate, N >> 1/m. The smallest populations, N << 1/s, exhibit behavior described by "neutral model". The two limits are joined by a broad interval of population sizes, 1/s << N << 1/m, termed the "selection-drift" regime. In this regime, weakly polymorphous populations behave essentially randomly, while the dynamics of highly polymorphous populations is almost deterministic and controlled by selection pressure. We discuss the biological impact of these findings. For general evolution, we suggest that most biological species may evolve while in the selection-drift regime, in which case both selection and random drift are equally important for the fixation speed of new advantageous mutations. For HIV in vivo, we applied a specially designed (and almost model-independent) linkage disequilibrium test to show that an effective HIV population, in a typical untreated patient, is larger than 1/m ~ 10^{5} infected cells, i.e., either at the border or within the deterministic regime (1). We also studied different models of random sampling during transmission between patients to show that the observed high level of genetic diversity in the pro gene is, most likely, due to strong differences in the best-fit sequence between individuals, 16%. Such a strong difference is consistent with cytotoxic immune response combined with co-selection (2).

- Rouzine, I. M. & Coffin, J. M. (1999) Proc. Natl. Acad. Sci. U. S. A. 96, 10758-63.
- Rouzine, I. M. & Coffin, J. M. (1999) J. Virol. 73, 8167-78.

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