The generalised quasispecies
Raphaël Cerf, Joseba Dalmau
(Submitted on 22 Apr 2015)
We study Eigen’s quasispecies model in the asymptotic regime where the length of the genotypes goes to infinity and the mutation probability goes to 0. We give several explicit formulas for the stationary solutions of the limiting system of differential equations.
When the mean is not enough: Calculating fixation time distributions in birth-death processes
Peter Ashcroft, Arne Traulsen, Tobias Galla
(Submitted on 16 Apr 2015)
Studies of fixation dynamics in Markov processes predominantly focus on the mean time to absorption. This may be inadequate if the distribution is broad and skewed. We compute the distribution of fixation times in one-step birth-death processes with two absorbing states. These are expressed in terms of the spectrum of the process, and we provide different representations as forward-only processes in eigenspace. These allow efficient sampling of fixation time distributions. As an application we study evolutionary game dynamics, where invading mutants can reach fixation or go extinct. We also highlight the median fixation time as a possible analog of mixing times in systems with small mutation rates and no absorbing states, whereas the mean fixation time has no such interpretation.
Rate of Adaptive Evolution under Blending Inheritance
Alan R. Rogers
(Submitted on 1 Apr 2015)
In a population of size N, adaptive evolution is 2N times faster under Mendelian inheritance than under the 19th-century theory of blending inheritance.
Genetic variability under the seed bank coalescent
Jochen Blath , Bjarki Eldon , Adrian Casanova , Noemi Kurt , Maite Wilke-Berenguer
We analyse patterns of genetic variability of populations in the presence of a large seed bank with the help of a new coalescent structure called the seed bank coalescent. This ancestral process appears naturally as scaling limit of the genealogy of large populations that sustain seed banks, if the seed bank size and individual dormancy times are of the same order as the active population. Mutations appear as Poisson processes on the active lineages, and potentially at reduced rate also on the dormant lineages. The presence of `dormant’ lineages leads to qualitatively altered times to the most recent common ancestor and non-classical patterns of genetic diversity. To illustrate this we provide a Wright-Fisher model with seed bank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seed bank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Estimates (obtained by simulations) of the distributions of commonly employed distance statistics, in the presence and absence of a seed bank, are compared. The effect of a seed bank on the expected site-frequency spectrum is also investigated using simulations. Our results indicate that the presence of a large seed bank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect the presence of a large seed bank in genetic data.
Analysis of adaptive walks on NK fitness landscapes with different interaction schemes
Stefan Nowak, Joachim Krug
Comments: 29 pages, 9 figures
Subjects: Populations and Evolution (q-bio.PE); Disordered Systems and Neural Networks (cond-mat.dis-nn)
Fitness landscapes are genotype to fitness mappings commonly used in evolutionary biology and computer science which are closely related to spin glass models. In this paper, we study the NK model for fitness landscapes where the interaction scheme between genes can be explicitly defined. The focus is on how this scheme influences the overall shape of the landscape. Our main tool for the analysis are adaptive walks, an idealized dynamics by which the population moves uphill in fitness and terminates at a local fitness maximum. We use three different types of walks and investigate how their length (the number of steps required to reach a local peak) and height (the fitness at the endpoint of the walk) depend on the dimensionality and structure of the landscape. We find that the distribution of local maxima over the landscape is particularly sensitive to the choice of interaction pattern. Most quantities that we measure are simply correlated to the rank of the scheme, which is equal to the number of nonzero coefficients in the expansion of the fitness landscape in terms of Walsh functions.
Selective strolls: fixation and extinction in diploids are slower for weakly selected mutations than for neutral ones
fabrizio mafessoni , Michael Lachmann
In finite populations, an allele disappears or reaches fixation due to two main forces, selection and drift. Selec- tion is generally thought to accelerate the process: a selected mutation will reach fixation faster than a neutral one, and a disadvantageous one will quickly disappear from the population. We show that even in simple diploid populations, this is often not true. Dominance and recessivity unexpectedly slow down the evolutionary process for weakly selected alleles. In particular, slightly advantageous dominant and mildly deleterious recessive mu- tations reach fixation more slowly than neutral ones. This phenomenon determines genetic signatures opposite to those expected under strong selection, such as increased instead of decreased genetic diversity around the selected site. Furthermore, we characterize a new phenomenon: mildly deleterious recessive alleles, thought to represent the vast majority of newly arising mutations, survive in a population longer than neutral ones, before getting lost. Hence, natural selection is less effective than previously thought in getting rid rapidly of slightly negative mutations, contributing their observed persistence in present populations. Consequently, low frequency slightly deleterious mutations are on average older than neutral ones.
The fate of a mutation in a fluctuating environment
Ivana Cvijovic , Benjamin H. Good , Elizabeth R. Jerison , Michael M. Desai
Natural environments are never truly constant, but the evolutionary implications of temporally varying selection pressures remain poorly understood. Here we investigate how the fate of a new mutation in a variable environment depends on the dynamics of environmental fluctuations and on the selective pressures in each condition. We find that even when a mutation experiences many environmental epochs before fixing or going extinct, its fate is not necessarily determined by its time-averaged selective effect. Instead, environmental variability reduces the efficiency of selection across a broad parameter regime, rendering selection unable to distinguish between mutations that are substantially beneficial and substantially deleterious on average. Temporal fluctuations can also dramatically increase fixation probabilities, often making the details of these fluctuations more important than the average selection pressures acting on each new mutation. For example, mutations that result in a tradeoff between conditions but are strongly deleterious on average can nevertheless be more likely to fix than mutations that are always neutral or beneficial. These effects can have important implications for patterns of molecular evolution in variable environments, and they suggest that it may often be difficult for populations to maintain specialist traits, even when their loss leads to a decline in time-averaged fitness.