The solitary wave of asexual evolution

Abstract

Using a previously undescribed approach, we develop an analytic model that predicts whether an asexual population accumulates advantageous or deleterious mutations over time and the rate at which either process occurs. The model considers a large number of linked identical loci, or nucleotide sites; assumes that the selection coefficient per site is much less than the mutation rate per genome; and includes back and compensating mutations. Using analysis and Monte Carlo simulations, we demonstrate the accuracy of our results over almost the entire range of population sizes. Two limiting cases of our results, when either deleterious or advantageous mutations can be neglected, correspond to the Fisher-Muller effect and Muller's ratchet, respectively. By comparing predictions of our model (no recombination) to those of simple single-locus models (strong recombination), we show that the accumulation of advantageous mutations is slowed by linkage over a broad, finite range of population size. This supports the view of Fisher and Muller, who argued in the 1930s that progressive evolution of organisms is slowed because loci at which beneficial mutations can occur are often linked together on the same chromosome. These results follow from our main finding, that distribution of sequences over the mutation number evolves as a traveling wave whose speed and width depend on population size and other parameters. The model explains a logarithmic dependence of steady-state fitness on the population size reported recently for an RNA virus. T he scope of evolutionary biology ranges from understanding the origin and extinction of species to predicting the accumulation of drug-or antibody-resistant mutations in a population of microbes during an infection of an individual. Viruses like HIV in which persistent infection of individuals lasts for large numbers of viral generations provide a valuable opportunity to test evolutionary theory by comparing model predictions to a wealth of readily obtained data. In addition, evolutionary models can be used to infer important properties of viral populations.The forces that produce and maintain genetic variation in a population are thought to be known. These include the ''systematic pressures'' (1) of mutation, natural selection, and migration. If these were the only forces operating, the fate of the population could be modeled deterministically. However, all evolution occurs in finite-size populations, which is the source of the other major evolutionary factor: random genetic drift. Drift adds a stochastic element to evolution, resulting from the chance sampling of individuals from one generation to the next and from the fact that not all possible genetic variants can be present in a finite population. For given levels of systematic pressure, evolution will be mostly deterministic if the population size is large enough but will be mostly neutral and dominated by drift (2) when the population size is small. Between these two limits there exists a large intermediate region, in which both selec...

Adaptation in Simple and Complex Fitness Landscapes

“…Under this assumption a deterministic description suffices and we can write down the time evolution equation for the average population fraction X(σ, t) of sequence σ at time t. Although this is often unrealistic, the analysis is simpler in this limit which in many cases can be adapted to the finite population case to provide quantitative agreement with experiments [25,26,27]. The infinite population limit can be justified if the population is known to be localized in a small region of sequence space around a fitness peak, if one is interested in a short piece of the genome such as a single regulatory binding site [16] (see also Sect.…”

“…Note that the model is well defined only for N µ < 1. At large times, Z(σ, t + 1) ≈ ΛZ(σ, t) where Λ is the largest eigenvalue of the evolution matrix on the right hand side of (27). In the delocalised phase, the population is spread over the entire sequence space with mean fitness W = 1, so that Λ = 1 whereas in the localised phase, a finite fraction has fitness W 0 > 1 and hence Λ > 1.…”

“…In addition, the continuum limit of (75) should be carried out on the level of ln Y rather than for Y itself, which leads to a nonlinear drift-diffusion equation replacing (76) [27]. Recent applications of fitness space models that go beyond the present discussion include studies of the in vitro evolution of DNA sequences selected for protein binding [46], viral populations undergoing serial population transfers [91], and the effects of recombination in asexual populations [92].…”

“…Finally, we note that under certain conditions populations of RNA viruses display a linear increase or decrease of fitness with time [27,110], which can be analyzed within the framework of the fitness space models discussed in Sect. 5.3.…”

Analysis of Evolution through Competitive Selection

“…For asexual models of this type, the dynamics are fairly easily solved for very small populations or for infinite populations. However, even with very simple assumptions regarding the fitness landscape, the evolution dynamics of a large but finite population far from equilibrium turned out to be a difficult problem, as the evolution rate diverges for large populations [9]; only recently have general analytical results emerged [10].…”