Fixation Probability Research Articles

Of variants and vaccines

In this issue of Cell, Bushman et al. show how more transmissible variants, even if they do not escape immunity, can be strongly selected during the early pandemic. This explains the dynamics of past SARS-CoV-2 variants, but as immunity increases, it is difficult to predict what will emerge next.

Haldane\u2019s formula in Cannings models: the case of moderately strong selection

For a class of Cannings models we prove Haldane’s formula, pi (s_N) sim frac{2s_N}{rho ^2}, for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for s_N sim N^{-b} and 0< b<1/2. Here, s_N is the selective advantage of an individual carrying the beneficial type, and rho ^2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

Fixation and effective size in a haploid–diploid population with asexual reproduction

The majority of population genetic theory assumes fully haploid or diploid organisms with obligate sexuality, despite complex life cycles with alternating generations being commonly observed. To reveal how natural selection and genetic drift shape the evolution of haploid–diploid populations, we analyze a stochastic genetic model for populations that consist of a mixture of haploid and diploid individuals, allowing for asexual reproduction and niche separation between haploid and diploid stages. Applying a diffusion approximation, we derive the fixation probability and describe its dependence on the reproductive values of haploid and diploid stages, which depend strongly on the extent of asexual reproduction in each phase and on the ecological differences between them.

Toward a Universal Model for Spatially Structured Populations.

A key question in evolution is how likely a mutant is to take over. This depends on natural selection and on stochastic fluctuations. Population spatial structure can impact mutant fixation probabilities. We introduce a model for structured populations on graphs that generalizes previous ones by making migrations independent of birth and death. We demonstrate that by tuning migration asymmetry, the star graph transitions from amplifying to suppressing natural selection. The results from our model are universal in the sense that they do not hinge on a modeling choice of microscopic dynamics or update rules. Instead, they depend on migration asymmetry, which can be experimentally tuned and measured.

Bernoulli and binomial proliferation on evolutionary graphs

In this paper we introduce random proliferation models on graphs. We consider two types of particles: type-1/mutant/invader/red particles proliferates on a population of type-2/wild-type/resident/blue particles. Unlike the well-known Moran model on graphs –as introduced in Lieberman et al. (2005)–, type-1 particles can occupy in a single iteration several neighbouring sites previously occupied by type-2 particles. Two variants are considered, depending on the random distribution involving the proliferation mechanism: Bernoulli and binomial proliferation. By comparison with fixation probability of type-1 particles in the Moran process, critical parameters are introduced. Properties of proliferation are studied and some particular cases are analytically solved. Finally, by updating the parameters that drive the processes through a density-dependent mechanism, it is possible to capture additional relevant features as fluctuating waves of type-1 particles over long periods of time. In fact, the models can be adapted to tackle more general, complex and realistic situations.

The network structure affects the fixation probability when it couples to the birth-death dynamics in finite population.

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.

Impact of the baseline payoff on evolutionary outcomes.

Do individuals enjoying a higher baseline payoff behave similarly in competitive scenarios compared to their counterparts? The classical replicator equation does not answer such a question since it is invariant to the background or baseline payoff of individuals. In reality, however, if one's baseline payoff is higher than the possible payoffs of an interaction (or game), the individual may respond generously or indifferently if s(he) is satisfied with the prevailing benchmark payoff. This work intends to explore such a phenomenon within the realm of pairwise interactions-taking the prisoner's dilemma as a metaphor-in well-mixed finite and infinite populations. In this framework, a player uses the payoff (comprising baseline and game payoffs) -expectation difference to estimate a degree of eagerness and, with that degree of eagerness, revises his or her strategy with a certain probability. We adopt two approaches to explore such a context, naming them as the Fermi and imitation processes, in which the former uses a pairwise Femi function and the latter considers the relative fitness to estimate probabilities for strategy revision. In a finite population, we examine the effect of intensities to payoff-expectation and strategic payoff differences (denoted by k_{1} and k_{2}, respectively) as well as the level of contentment (ω) on the fixation probability and fixation time (for a single defector). We observe that the fixation probability surges with the increase of intensity parameters. Nevertheless, the maximum fixation probability may require a substantially larger time to fixate, especially when the expectation is lower than the baseline payoff. This means that cooperators can persist for a longer period of time. A higher expectation or greed, however, considerably reduces the fixation time. Interestingly, our numerical simulation reveals that both approaches are equivalent under weak k_{2}(≪1) in the Fermi process. We further derive mean-field equations for both approaches in the context of an infinite population, where we observe two possible evolutionary consequences: either full-scale defection or the persistence of the initial frequency of cooperators. The latter scenario indicates players' uninterested or neutral behavior in relation to the interaction due to their satisfaction on the baseline payoff.

Effects of word predictability on eye movements during Arabic reading

Contextual predictability influences both the probability and duration of eye fixations on words when reading Latinate alphabetic scripts like English and German. However, it is unknown whether word predictability influences eye movements in reading similarly for Semitic languages like Arabic, which are alphabetic languages with very different visual and linguistic characteristics. Such knowledge is nevertheless important for establishing the generality of mechanisms of eye-movement control across different alphabetic writing systems. Accordingly, we investigated word predictability effects in Arabic in two eye-movement experiments. Both produced shorter fixation times for words with high compared to low predictability, consistent with previous findings. Predictability did not influence skipping probabilities for (four- to eight-letter) words of varying length and morphological complexity (Experiment 1). However, it did for short (three- to four-letter) words with simpler structures (Experiment 2). We suggest that word-skipping is reduced, and affected less by contextual predictability, in Arabic compared to Latinate alphabetic reading, because of specific orthographic and morphological characteristics of the Arabic script.

A Bayesian approach to dynamical modeling of eye-movement control in reading of normal, mirrored, and scrambled texts.

In eye-movement control during reading, advanced process-oriented models have been developed to reproduce behavioral data. So far, model complexity and large numbers of model parameters prevented rigorous statistical inference and modeling of interindividual differences. Here we propose a Bayesian approach to both problems for one representative computational model of sentence reading (SWIFT; Engbert et al., Psychological Review, 112, 2005, pp. 777-813). We used experimental data from 36 subjects who read the text in a normal and one of four manipulated text layouts (e.g., mirrored and scrambled letters). The SWIFT model was fitted to subjects and experimental conditions individually to investigate between-subject variability. Based on posterior distributions of model parameters, fixation probabilities and durations are reliably recovered from simulated data and reproduced for withheld empirical data, at both the experimental condition and subject levels. A subsequent statistical analysis of model parameters across reading conditions generates model-driven explanations for observable effects between conditions. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Martingales and the characteristic functions of absorption time on bipartite graphs.

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald’s martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.

Strong noise limit for population dynamics in incompressible advection.

Genetic diversity is at the basis of the evolution process of populations and it is responsible for the populations' degree of fitness to a particular ecosystem. In marine environments many factors play a role in determining the dynamics of a population, including the amount of nutrients, the temperature, and many other stressing factors. An important and yet rather unexplored challenge is to figure out the role of individuals' dispersion, due to flow advection, on population genetics. In this paper we focus on two populations, one of which has a slight selective advantage, advanced by an incompressible two-dimensional flow. In particular, we want to understand how this advective flow can modify the dynamics of the advantageous allele. We generalize, through a theoretical analysis, previous evidence according to which the fixation probability is independent of diffusivity, showing that this is also independent of fluid advection. These findings may have important implications in the understanding of the dynamics of a population of microorganism, such as plankton or bacteria, in marine environments under the influence of (turbulent) currents.

An extension of the Moran process using type-specific connection graphs

The Moran process, as studied by Lieberman, Hauert and Nowak (2005) [1], is a birth-death process that models the spread of mutations in two-type populations (residents-mutants) whose structure is defined by a digraph. The process' central notion is the probability that a randomly placed mutant will occupy the whole vertex set (fixation probability). We extend this model by considering type-specific graphs, and consequently present results on the fundamental problems related to the fixation probability and its computation. Finally, we view the resident-mutant competing forces as players that choose digraphs and indicate that the mutant's complete graph is a dominant strategy.

Evolution of resistance under alternative models of selective interference.

The use of multiple pesticides or drugs can lead to a simultaneous selection pressure for resistance alleles at different loci. Models of resistance evolution focus on how this can delay the spread of resistance through a population, but often neglect how this can also reduce the probability that a resistance allele spreads. This neglected factor has been studied in a parallel literature as selective interference. Models of interference use alternative constructions of fitness, where selection coefficients from different loci either add or multiply. Although these are equivalent under weak selection, the two constructions make alternative predictions under the strong selection that characterizes resistance evolution. Here, simulations are used to examine the effects of interference on the probability of fixation and time to fixation of a new and strongly beneficial mutation in the presence of another strongly beneficial allele with variable starting frequency. The results from simulations show a complicated pattern of effects. The key result is that, under multiplicativity, the presence of the strongly beneficial allele leads to a small reduction in the probability of fixation for the new beneficial mutation up to ~10%, and a negligible increase in the average time to fixation up to ~2%, whereas under additivity, the effect is more substantial at up to ~50% for the probability of fixation and ~100% for the average time to fixation. Consequently, the effect of interference is only an important feature of resistance evolution under additivity. Current evidence from studies of experimental evolution provides widespread support for the basic features of additivity, which suggests that interference may afford resistance a different pattern of evolution than other adaptations: rather than the gradual and simultaneous selection of many alleles with small effects, the rapid evolution of resistance may involve the sequential selection of alleles with large effects.

Fixation probabilities in network structured meta-populations

The effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two patch structure with unequal population size suppresses selection for low migration probabilities.

Dynamic sampling bias and overdispersion induced by skewed offspring distributions.

Natural populations often show enhanced genetic drift consistent with a strong skew in their offspring number distribution. The skew arises because the variability of family sizes is either inherently strong or amplified by population expansions. The resulting allele-frequency fluctuations are large and, therefore, challenge standard models of population genetics, which assume sufficiently narrow offspring distributions. While the neutral dynamics backward in time can be readily analyzed using coalescent approaches, we still know little about the effect of broad offspring distributions on the forward-in-time dynamics, especially with selection. Here, we employ an asymptotic analysis combined with a scaling hypothesis to demonstrate that over-dispersed frequency trajectories emerge from the competition of conventional forces, such as selection or mutations, with an emerging time-dependent sampling bias against the minor allele. The sampling bias arises from the characteristic time-dependence of the largest sampled family size within each allelic type. Using this insight, we establish simple scaling relations for allele-frequency fluctuations, fixation probabilities, extinction times, and the site frequency spectra that arise when offspring numbers are distributed according to a power law.

Recombination Facilitates Adaptive Evolution in Rhizobial Soil Bacteria.

Homologous recombination is expected to increase natural selection efficacy by decoupling the fate of beneficial and deleterious mutations and by readily creating new combinations of beneficial alleles. Here, we investigate how the proportion of amino acid substitutions fixed by adaptive evolution () depends on the recombination rate in bacteria. We analyze 3,086 core protein-coding sequences from 196 genomes belonging to five closely related species of the genus Rhizobium. These genes are found in all species and do not display any signs of introgression between species. We estimate using the site frequency spectrum (SFS) and divergence data for all pairs of species. We evaluate the impact of recombination within each species by dividing genes into three equally sized recombination classes based on their average level of intragenic linkage disequilibrium. We find that varies from 0.07 to 0.39 across species and is positively correlated with the level of recombination. This is both due to a higher estimated rate of adaptive evolution and a lower estimated rate of nonadaptive evolution, suggesting that recombination both increases the fixation probability of advantageous variants and decreases the probability of fixation of deleterious variants. Our results demonstrate that homologous recombination facilitates adaptive evolution measured by in the core genome of prokaryote species in agreement with studies in eukaryotes.

Fast and strong amplifiers of natural selection

Selection and random drift determine the probability that novel mutations fixate in a population. Population structure is known to affect the dynamics of the evolutionary process. Amplifiers of selection are population structures that increase the fixation probability of beneficial mutants compared to well-mixed populations. Over the past 15 years, extensive research has produced remarkable structures called strong amplifiers which guarantee that every beneficial mutation fixates with high probability. But strong amplification has come at the cost of considerably delaying the fixation event, which can slow down the overall rate of evolution. However, the precise relationship between fixation probability and time has remained elusive. Here we characterize the slowdown effect of strong amplification. First, we prove that all strong amplifiers must delay the fixation event at least to some extent. Second, we construct strong amplifiers that delay the fixation event only marginally as compared to the well-mixed populations. Our results thus establish a tight relationship between fixation probability and time: Strong amplification always comes at a cost of a slowdown, but more than a marginal slowdown is not needed.

Revisiting the notion of deleterious sweeps.

It has previously been shown that, conditional on its fixation, the time to fixation of a semi-dominant deleterious autosomal mutation in a randomly mating population is the same as that of an advantageous mutation. This result implies that deleterious mutations could generate selective sweep-like effects. Although their fixation probabilities greatly differ, the much larger input of deleterious relative to beneficial mutations suggests that this phenomenon could be important. We here examine how the fixation of mildly deleterious mutations affects levels and patterns of polymorphism at linked sites-both in the presence and absence of interference amongst deleterious mutations-and how this class of sites may contribute to divergence between-populations and species. We find that, while deleterious fixations are unlikely to represent a significant proportion of outliers in polymorphism-based genomic scans within populations, minor shifts in the frequencies of deleterious mutations can influence the proportions of private variants and the value of FST after a recent population split. As sites subject to deleterious mutations are necessarily found in functional genomic regions, interpretations in terms of recurrent positive selection may require reconsideration.

Top-down and bottom-up guidance in normal aging during scene search.

Age-related differences in visual search have been extensively studied using simple item arrays, showing an attentional decline. Little is known about how aging affects attentional guidance during search in more complex scenes. To study this issue, we analyzed eye-movement behavior in realistic scene search. We examined age-related differences in top-down guidance, manipulating target template specificity (picture vs. word cue) and target-scene semantic consistency (consistent vs. inconsistent), and in bottom-up guidance, manipulating perceptual salience (high vs. low) of targets and distractors. Compared to young adults (YA), older adults (OA) were overall slower, from the first saccade in the scene. They showed a smaller benefit of a specific target template, suggesting that precision of visual information in working memory may decrease with age. The benefit of semantic consistency did not depend on age, suggesting a preserved ability in OA to use knowledge about object occurrence in scenes. OA showed greater bottom-up search facilitation due to target's high salience, which may depend on reduced selection of low-salience stimuli. Attentional capture by distractors was greater in OA than YA, with respect to engagement (probability of distractor fixation), but only following a picture cue, and disengagement (fixation duration on distractors) in all conditions. Overall, our study shows that age-related differences in visual selection of targets and distractors depend on specific task demands in terms of top-down and bottom-up guidance. It also indicates that scene search difficulties in OA can be limited by cognitive and perceptual forms of environmental support. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Attentional processing of pain faces and other emotional faces in chronic pain-an eye-tracking study.

Altered attentional processing of pain-associated stimuli–which might take the form of either avoidance or enhanced vigilance–is thought to be implicated in the development and maintenance of chronic pain. In contrast to reaction time tasks like the dot probe, eye tracking allows for tracking the time course of visual attention and thus differentiating early and late attentional processes. Our study aimed at investigating visual attention to emotional faces in patients with chronic musculoskeletal pain (N = 20) and matched pain-free controls (N = 20). Emotional faces (pain, angry, happy) were presented in pairs with a neutral face for 2000 ms each. Three parameters were determined: First fixation probabilities, fixation durations (overall and divided in four 500 ms intervals) and a fixation bias score as the relative fixation duration of emotional faces compared to neutral faces. There were no group differences in any of the parameters. First fixation probabilities were lower for pain faces than for angry faces. Overall, we found longer fixation duration on emotional compared to neutral faces (‘emotionality bias’), which is in accord with previous research. However, significant longer fixation duration compared to the neutral face was detected only for happy and angry but not for pain faces. In addition, fixation durations as well as bias scores yielded evidence for vigilant-avoidant processing of pain faces in both groups. These results suggest that attentional bias towards pain-associated stimuli might not generally differentiate between healthy individuals and chronic pain patients. Exaggerated attentional bias in patients might occur only under specific circumstances, e.g., towards stimulus material specifically relating to the specific pain of the patients under study or under high emotional distress.