Asymptotic expression for the fixation probability of a mutant in star graphs
Fabio A. C. C. Chalub
(Submitted on 15 Apr 2014)
We consider the Moran process in a graph called “star” and obtain the asymptotic expression for the fixation probability of a single mutant when the size of the graph is large. The expression obtained corrects previously known expression announced in reference [E Lieberman, C Hauert, and MA Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312-316, 2005] and further studied in [M. Broom and J. Rychtar. An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc. R. Soc. A-Math. Phys. Eng. Sci., 464(2098):2609-2627, 2008]. We also show that the star graph is an accelerator of evolution, if the graph is large enough.
Historical contingency and entrenchment in protein evolution under purifying selection
Premal Shah, Joshua B. Plotkin
(Submitted on 15 Apr 2014)
The fitness contribution of an allele at one genetic site may depend on the states of other sites, a phenomenon known as epistasis. Epistasis can profoundly influence the process of evolution in populations under selection, and shape the course of protein evolution across divergent species. Whereas epistasis among adaptive substitutions has been the subject of extensive study, relatively little is known about epistasis under purifying selection. Here we use mechanistic models of thermodynamic stability in a ligand-binding protein to explore computationally the structure of epistatic interactions among substitutions that fix in protein sequences under purifying selection. We find that the selection coefficients of mutations that are nearly neutral when they fix are highly conditional on the presence of preceding mutations. In addition, substitutions which are initially neutral become increasingly entrenched over time due to antagonistic epistasis with subsequent substitutions. Our evolutionary model includes insertions and deletions, as well as point mutations, which allows us to quantify epistasis between these classes of mutations, and also to study the evolution of protein length. We find that protein length remains largely constant over time, because indels are more deleterious than point mutations. Our results imply that, even under purifying selection, protein sequence evolution is highly contingent on history and it cannot be predicted by the phenotypic effects of mutations introduced into the wildtype sequence alone.
Intermediate Migration Yields Optimal Adaptation in Structured, Asexual Populations
Arthur Covert III, Claus O Wilke
Most evolving populations are subdivided into multiple subpopulations connected to each other by varying levels of gene flow. However, how population structure and gene flow (i.e., migration) affect adaptive evolution is not well understood. Here, we studied the impact of migration on asexually reproducing evolving computer programs (digital organisms). We found that digital organisms evolve the highest fitness values at intermediate migration rates, and we tested three hypotheses that could potentially explain this observation: (i) migration promotes passage through fitness valleys, (ii) migration increases genetic variation, and (iii) migration reduces clonal interference through a process called leapfrogging. We found that migration had no appreciable effect on the number of fitness valleys crossed and that genetic variation declined monotonously with increasing migration rates, instead of peaking at the optimal migration rate. However, the number of leapfrogging events, in which a superior beneficial mutation emerges on a genetic background that predates the previously best genotype in the population, did peak at the optimal migration rate. We thus conclude that in structured, asexual populations intermediate migration rates allow for optimal exploration of multiple, distinct fitness peaks, and thus yield the highest long-term adaptive success.
Inferring fitness landscapes by regression produces biased estimates of epistasis
Jakub Otwinowski, Joshua B. Plotkin
(Submitted on 3 Apr 2014)
The genotype-fitness map plays a fundamental role in shaping the dynamics of evolution. However, it is difficult to directly measure a fitness landscape in practice, because the number of possible genotypes is astronomical. One approach is to sample as many genotypes as possible, measure their fitnesses, and fit a statistical model of the landscape that includes additive and pairwise interactive effects between loci. Here we elucidate the pitfalls of using such regressions, by studying artificial but mathematically convenient fitness landscapes. We identify two sources of bias inherent in these regression procedures that each tends to under-estimate high fitnesses and over-estimate low fitnesses. We characterize these biases for random sampling of genotypes, as well as for samples drawn from a population under selection in the Wright-Fisher model of evolutionary dynamics. We show that common measures of epistasis, such as the number of monotonically increasing paths between ancestral and derived genotypes, the prevalence of sign epistasis, and the number of local fitness maxima, are distorted in the inferred landscape. As a result, the inferred landscape will provide systematically biased predictions for the dynamics of adaptation. We identify the same biases in a computational RNA-folding landscape, as well as in regulatory sequence binding data, treated with the same fitting procedure. Finally, we present a method that may ameliorate these biases in some cases.
Stability and response of polygenic traits to stabilizing selection and mutation
Harold P. de Vladar, Nick Barton
(Submitted on 3 Apr 2014)
When polygenic traits are under stabilizing selection, many different combinations of alleles allow close adaptation to the optimum. If alleles have equal effects, all combinations that result in the same deviation from the optimum are equivalent. Furthermore, the genetic variance that is maintained by mutation-selection balance is 2μ/S per locus, where μ is the mutation rate and S the strength of stabilizing selection. In reality, alleles vary in their effects, making the fitness landscape asymmetric, and complicating analysis of the equilibria. We show that that the resulting genetic variance depends on the fraction of alleles near fixation, which contribute by 2μ/S, and on the total mutational effects of alleles that are at intermediate frequency. The interplay between stabilizing selection and mutation leads to a sharp transition: alleles with effects smaller than a threshold value of 2μ/S‾‾‾‾√ remain polymorphic, whereas those with larger effects are fixed. The genetic load in equilibrium is less than for traits of equal effects, and the fitness equilibria are more similar. We find that if the optimum is displaced, alleles with effects close to the threshold value sweep first, and their rate of increase is bounded by μS‾‾‾√. Long term response leads in general to well-adapted traits, unlike the case of equal effects that often end up at a sub-optimal fitness peak. However, the particular peaks to which the populations converge are extremely sensitive to the initial states, and to the speed of the shift of the optimum trait value.
Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments
Steve Evans, Alexandru Hening, Sebastian Schreiber
We consider consider a population living in a patchy environment that varies stochastically in space and time. The population is composed of two morphs (that is, individuals of the same species with different genotypes). In terms of survival and reproductive success, the associated phenotypes differ only in their habitat selection strategies. We compute invasion rates corresponding to the rates at which the abundance of an initially rare morph increases in the presence of the other morph established at equilibrium. If both morphs have positive invasion rates when rare, then there is an equilibrium distribution such that the two morphs coexist; that is, there is a protected polymorphism for habitat selection. Alternatively, if one morph has a negative invasion rate when rare, then it is asymptotically displaced by the other morph under all initial conditions where both morphs are present. We refine the characterization of an evolutionary stable strategy for habitat selection from [Schreiber, 2012] in a mathematically rigorous manner. We provide a necessary and sufficient condition for the existence of an ESS that uses all patches and determine when using a single patch is an ESS. We also provide an explicit formula for the ESS when there are two habitat types. We show that adding environmental stochasticity results in an ESS that, when compared to the ESS for the corresponding model without stochasticity, spends less time in patches with larger carrying capacities and possibly makes use of sink patches, thereby practicing a spatial form of bet hedging.
The distribution of the quasispecies for the Wright-Fisher model on the sharp peak landscape
(Submitted on 27 Mar 2014)
We consider the classical Wright-Fisher model with mutation and selection. Mutations occur independently in each locus, and selection is performed according to the sharp peak landscape. In the asymptotic regime studied in , a quasispecies is formed. We find explicitly the distribution of this quasispecies, which turns out to be the same distribution as for the Moran model.
Multidimensional epistasis and the transitory advantage of sex
Stefan Nowak, Johannes Neidhart, Ivan G. Szendro, Joachim Krug
(Submitted on 25 Mar 2014)
Identifying and quantifying the benefits of sex and recombination is a long standing problem in evolutionary theory. In particular, contradictory claims have been made about the existence of a benefit of recombination on high dimensional fitness landscapes in the presence of sign epistasis. Here we present a comparative numerical study of sexual and asexual evolutionary dynamics on tunably rugged model landscapes, paying special attention to the temporal development of the evolutionary advantage of recombination and the link between population diversity and the rate of adaptation. We show that the adaptive advantage of recombination on static rugged landscapes is strictly transitory. At early times an advantage of recombination through the Fisher-Muller effect is generally observed, but this effect is reversed at longer times by the much more efficient trapping of recombining populations at local fitness peaks. These findings are explained by means of well established results for a setup with only two loci. In accordance with the Red Queen hypothesis the transitory advantage can be prolonged indefinitely in fluctuating environments, and it is maximal when the environment fluctuates on the same time scale on which trapping at local optima typically occurs.
Range Expansion of Heterogeneous Populations
Matthias Reiter (1), Steffen Rulands (1), Erwin Frey (1 contributed equally)
(Submitted on 25 Mar 2014)
Risk spreading in bacterial populations is generally regarded as a strategy to maximize survival. Here, we study its role during range expansion of a genetically diverse population where growth and motility are two alternative traits. We find that during the initial expansion phase fast growing cells do have a selective advantage. By contrast, asymptotically, generalists balancing motility and reproduction are evolutionarily most successful. These findings are rationalized by a set of coupled Fisher equations complemented by stochastic simulations.
Population genetics of identity by descent
Pier Francesco Palamara, Ph.D. thesis
Recent improvements in high-throughput genotyping and sequencing technologies have afforded the collection of massive, genome-wide datasets of DNA information from hundreds of thousands of individuals. These datasets, in turn, provide unprecedented opportunities to reconstruct the history of human populations and detect genotype-phenotype association. Recently developed computational methods can identify long-range chromosomal segments that are identical across samples, and have been transmitted from common ancestors that lived tens to hundreds of generations in the past. These segments reveal genealogical relationships that are typically unknown to the carrying individuals. In this work, we demonstrate that such identical-by-descent (IBD) segments are informative about a number of relevant population genetics features: they enable the inference of details about past population size fluctuations, migration events, and they carry the genomic signature of natural selection. We derive a mathematical model, based on coalescent theory, that allows for a quantitative description of IBD sharing across purportedly unrelated individuals, and develop inference procedures for the reconstruction of recent demographic events, where classical methodologies are statistically underpowered. We analyze IBD sharing in several contemporary human populations, including representative communities of the Jewish Diaspora, Kenyan Maasai samples, and individuals from several Dutch provinces, in all cases retrieving evidence of fine-scale demographic events from recent history. Finally, we expand the presented model to describe distributions for those sites in IBD shared segments that harbor mutation events, showing how these may be used for the inference of mutation rates in humans and other species.