Evolutionary rescue by beneficial mutations in environments that change in space and time

A factor that may limit the ability of many populations to adapt to changing conditions is the rate at which beneficial mutations can become established. We study the probability that mutations become established in changing environments by extending the classic theory for branching processes. When environments change in time, under quite general conditions, the establishment probability is approximately twice the ‘effective selection coefficient’, whose value is an average that gives most weight to a mutant''s fitness in the generations immediately after it appears. When fitness varies along a gradient in a continuous habitat, increased dispersal generally decreases the chance a mutation establishes because mutations move out of areas where they are most adapted. When there is a patch of favourable habitat that moves in time, there is a maximum speed of movement above which mutations cannot become established, regardless of when and where they first appear. This critical speed limit, which is proportional to the mutation''s maximum selective advantage, represents an absolute constraint on the potential of locally adapted mutations to contribute to evolutionary rescue.     

The fixation probability of a beneficial allele in a population dividing by binary fission

We derive formulae for the fixation probability, P, of a rare benefical allele segregating in a population of fixed size which reproduces by binary fission, in terms of the selection coefficient for the beneficial allele, s. We find that an earlier result P 4s does not depend on the assumption of binary fission, but depends on an assumption about the ordering of events in the life cycle. We find that P 2s for mutations occurring during chromosome replication and P 2.8s for mutations occurring at random times between replication events.     

The rate of beneficial mutations surfing on the wave of a range expansion

Many theoretical and experimental studies suggest that range expansions can have severe consequences for the gene pool of the expanding population. Due to strongly enhanced genetic drift at the advancing frontier, neutral and weakly deleterious mutations can reach large frequencies in the newly colonized regions, as if they were surfing the front of the range expansion. These findings raise the question of how frequently beneficial mutations successfully surf at shifting range margins, thereby promoting adaptation towards a range-expansion phenotype. Here, we use individual-based simulations to study the surfing statistics of recurrent beneficial mutations on wave-like range expansions in linear habitats. We show that the rate of surfing depends on two strongly antagonistic factors, the probability of surfing given the spatial location of a novel mutation and the rate of occurrence of mutations at that location. The surfing probability strongly increases towards the tip of the wave. Novel mutations are unlikely to surf unless they enjoy a spatial head start compared to the bulk of the population. The needed head start is shown to be proportional to the inverse fitness of the mutant type, and only weakly dependent on the carrying capacity. The precise location dependence of surfing probabilities is derived from the non-extinction probability of a branching process within a moving field of growth rates. The second factor is the mutation occurrence which strongly decreases towards the tip of the wave. Thus, most successful mutations arise at an intermediate position in the front of the wave. We present an analytic theory for the tradeoff between these factors that allows to predict how frequently substitutions by beneficial mutations occur at invasion fronts. We find that small amounts of genetic drift increase the fixation rate of beneficial mutations at the advancing front, and thus could be important for adaptation during species invasions.     

Establishment of new mutations in changing environments

Understanding adaptation in changing environments is an important topic in evolutionary genetics, especially in the light of climatic and environmental change. In this work, we study one of the most fundamental aspects of the genetics of adaptation in changing environments: the establishment of new beneficial mutations. We use the framework of time-dependent branching processes to derive simple approximations for the establishment probability of new mutations assuming that temporal changes in the offspring distribution are small. This approach allows us to generalize Haldane's classic result for the fixation probability in a constant environment to arbitrary patterns of temporal change in selection coefficients. Under weak selection, the only aspect of temporal variation that enters the probability of establishment is a weighted average of selection coefficients. These weights quantify how much earlier generations contribute to determining the establishment probability compared to later generations. We apply our results to several biologically interesting cases such as selection coefficients that change in consistent, periodic, and random ways and to changing population sizes. Comparison with exact results shows that the approximation is very accurate.     

The fixation probability of two competing beneficial mutations

Suppose that a beneficial mutation is undergoing a selective sweep when another beneficial mutation arises at a linked locus. We study the fixation probability of the double mutant, i.e., one (produced by recombination) that carries both mutations. Previous analysis works well for the case where the earlier beneficial mutation confers a greater selective advantage than the later mutation, but not so well in the opposite case. We present an approach to approximating the fixation probability in the case where the later mutation confers a greater selective advantage.     

The distribution of fitness effects of new beneficial mutations in Pseudomonas fluorescens

Theoretical studies of adaptation emphasize the importance of understanding the distribution of fitness effects (DFE) of new mutations. We report the isolation of 100 adaptive mutants—without the biasing influence of natural selection—from an ancestral genotype whose fitness in the niche occupied by the derived type is extremely low. The fitness of each derived genotype was determined relative to a single reference type and the fitness effects found to conform to a normal distribution. When fitness was measured in a different environment, the rank order changed, but not the shape of the distribution. We argue that, even with detailed knowledge of the genetic architecture underpinning the adaptive types (as is the case here), the DFEs remain unpredictable, and we discuss the possibility that general explanations for the shape of the DFE might not be possible in the absence of organism-specific biological details.     

A branching process, its application in biology: influence of demographic parameters on the social structure in mammal groups

Branching processes are widely used in biology. This theoretical tool is used in cell dynamics, epidemics and population dynamics. In population dynamics, branching processes are mainly used to access extinction probabilities of populations, groups or families, with the Galton-Watson branching process. Many mammal species live in socially-structured groups, and the smallest units of these groups are lineages (or families) of kin-related individuals. In many primate species, these lineages are matrilines, as females remain in their natal groups most of the time, whereas males generally disperse. Lineage parameters, such as numbers of matrilines, size of each matriline and average degree of relatedness, could strongly influence the genetic composition of groups. Evidence indicates that division along matrilines could induce substantial differentiation among fission groups. Here, we develop a novel mathematical model based on the branching process theory describing demographic dynamics of groups. The main result of this model is an explicit analytical expression of the joint distribution of numbers of lineages and sizes of socially-structured groups. We investigated the influence of parameters such as natality and mortality on the outcome of the process, including extinction probability. Finally, we discuss this theoretical result with respect to biological significance.     

Rates of decay for the survival probability of a mutant gene II The multitype case

This paper extends the results of [1] to the multitype case. For a multitype branching process that is slightly supercritical, approximations for the survival probability in terms of the maximal eigenvalue of the mean matrix and a generalized variance ² are developed. Our results improve upon those of Hoppe [5] and Eshel [3] that seek to validate a conjecture of Ewens [4].Research supported in part by NSF grant DMS 9007182     

Keeping up with early springs: rapid range expansion in an avian herbivore incurs a mismatch between reproductive timing and food supply

Within three decades, the barnacle goose population wintering on the European mainland has dramatically increased in numbers and extended its breeding range. The expansion has occurred both within the Arctic as well as by the colonization of temperate areas. Studies of performance of individuals in expanding populations provide information on how well species can adapt to novel environments and global warming. We, therefore, studied the availability of high quality food as well as timing of reproduction, wing moult, fledgling production and postfledging survival of individually marked geese in three recently established populations: one Arctic (Barents Sea) and two temperate (Baltic, North Sea). In the Barents Sea population, timing of hatching was synchronized with the peak in food availability and there was strong stabilizing selection. Although birds in the Baltic and North Sea populations bred 6–7 weeks earlier than Arctic birds, timing of hatching was late in relation to the peak in food availability, and there was moderate to strong directional selection for early breeding. In the Baltic, absolute timing of egg laying advanced considerably over the 20‐year study period, but advanced little relative to spring phenology, and directional selection on lay date increased over time. Wing moult of adults started only 2–4 weeks earlier in the temperate populations than in the Arctic. Synchronization between fledging of young and end of wing moult decreased in the temperate populations. Arctic‐breeding geese may gradually accumulate body stores from the food they encounter during spring migration, which allows them to breed relatively early and their young to use the peak of the Arctic food resources. By contrast, temperate‐breeding birds are not able to acquire adequate body stores from local resources early enough, that is before the quality of food for their young starts to decrease. When global temperatures continue to rise, Arctic‐breeding barnacle geese might encounter similar problems.     

Model and test in a fungus of the probability that beneficial mutations survive drift

Determining the probability of fixation of beneficial mutations is critically important for building predictive models of adaptive evolution. Despite considerable theoretical work, models of fixation probability have stood untested for nearly a century. However, recent advances in experimental and theoretical techniques permit the development of models with testable predictions. We developed a new model for the probability of surviving genetic drift, a major component of fixation probability, for novel beneficial mutations in the fungus Aspergillus nidulans, based on the life-history characteristics of its colony growth on a solid surface. We tested the model by measuring the probability of surviving drift in 11 adapted strains introduced into wild-type populations of different densities. We found that the probability of surviving drift increased with mutant invasion fitness, and decreased with wild-type density, as expected. The model accurately predicted the survival probability for the majority of mutants, yielding one of the first direct tests of the extinction probability of beneficial mutations.     

The fate of mutations surfing on the wave of a range expansion

Many species, including humans, have dramatically expanded their range in the past, and such range expansions had certainly an impact on their genetic diversity. For example, mutations arising in populations at the edge of a range expansion can sometimes surf on the wave of advance and thus reach a larger spatial distribution and a much higher frequency than would be expected in stationary populations. We study here this surfing phenomenon in more detail, by performing extensive computer simulations under a two-dimensional stepping-stone model. We find that the probability of survival of a new mutation depends to a large degree on its proximity to the edge of the wave. Demographic factors such as deme size, migration rate, and local growth rate also influence the fate of these new mutations. We also find that the final spatial and frequency distributions depend on the local deme size of a subdivided population. This latter result is discussed in the light of human expansions in Europe as it should allow one to distinguish between mutations having spread with Paleolithic or Neolithic expansions. By favoring the spread of new mutations, a consequence of the surfing phenomenon is to increase the rate of evolution of spatially expanding populations.     

Influence of learning on range expansion and adaptation to novel habitats

Learning has been postulated to ‘drive’ evolution, but its influence on adaptive evolution in heterogeneous environments has not been formally examined. We used a spatially explicit individual‐based model to study the effect of learning on the expansion and adaptation of a species to a novel habitat. Fitness was mediated by a behavioural trait (resource preference), which in turn was determined by both the genotype and learning. Our findings indicate that learning substantially increases the range of parameters under which the species expands and adapts to the novel habitat, particularly if the two habitats are separated by a sharp ecotone (rather than a gradient). However, for a broad range of parameters, learning reduces the degree of genetically‐based local adaptation following the expansion and facilitates maintenance of genetic variation within local populations. Thus, in heterogeneous environments learning may facilitate evolutionary range expansions and maintenance of the potential of local populations to respond to subsequent environmental changes.     

Seed origin determines the range expansion of the clonal grass Elymus athericus

The recent invasion of clonal grasses to novel habitats poses a threat to biodiversity in various habitats. Elymus athericus, a clonal grass of north-western European salt marshes, is currently increasing in abundance and invading new habitats. In this study, we analyzed controlling factors for seedling establishment of E. athericus in frequently flooded low marsh habitats. Here, biotic and abiotic conditions are very different from the conditions of the parental sites with established populations higher up in the marsh. Hence, we hypothesized that seedling establishment at the expanding low marsh edge would depend on the parental origin (either through maternal effects or heritable local adaptation). We further hypothesized that seedling origin interacts with biotic factors such as herbivory and competition as well as with abiotic factors like inundation frequency. We tested the dependence of seedling survival, growth and vegetative reproduction on these factors in a factorial transplant experiment on Schiermonnikoog. Survival was high, with 77% of the planted seedling surviving until the end of the experiment. Biotic factors had a much stronger effect on seedling growth and mortality than parental origin and were independent of inundation. However, parental origin strongly interacted with herbivory and competition, with seedlings performing better under the conditions that resembled their parental site.We conclude that seedlings of E. athericus, a species that was previously thought to occur only in mid- to high marsh elevation, can establish at a frequently inundated low-marsh sites. Long term survival and further invasion will primarily depend on biotic factors in interaction with seed origin. Our results suggest that next to herbivory, limitation of seeds adapted to colonizing conditions is likely to slow down range expansion.     

A rapid altitudinal range expansion in the pine processionary moth produced by the 2003 climatic anomaly

Climatic anomalies may produce, or accelerate, geographic range expansions of species limited by temperature or other climatic variables. Most such expansions are only temporary, before the prevailing climatic conditions drive the founder populations extinct. In contrast, here, we report a recent rapid shift of the range limit during the record hot summer of 2003 in southern Europe that has the potential to be both permanent, and to have important implications on species range dynamics in general. The winter pine processionary moth (Thaumetopoea pityocampa), an important pine defoliator whose larvae feed in colonies during the winter, is limited in its distribution by winter temperatures. In the last three decades, warmer winters have led to a gradual but substantial expansion of its range both latitudinally and altitudinally. In the summer of 2003, T. pityocampa underwent an extraordinary expansion to high elevation pine stands in the Italian Alps; its altitudinal range limit increased by one third of the total altitudinal expansion over the previous three decades. In an experiment, we found flight activity of newly emerged females to increase with temperature. By determining a threshold temperature for flight take‐offs under controlled conditions, we calculated that the nights above the threshold temperature were over five times more frequent, and considerably warmer, at the range limit in 2003 than in an average year. We therefore attribute the colonization of extreme, high‐elevation sites to increased nocturnal dispersal of females during the unusually warm night temperatures in June – August 2003. Importantly, the colonies established at extreme sites survived the winter and produced offspring in 2004, although the range did not expand further because of low night temperatures that year. We discuss several life‐history characteristics of T. pityocampa that maximize the likelihood of population persistence at the new range limit. As global warming continues and climatic anomalies are predicted to become more frequent, our results draw attention to the importance of extreme climatic events in the range formation of phytophagous insects.     

The survival probability of a mutant in a multidimensional population

We prove a general result about the asymptotic behaviour of the survival probability of a slightly supercritical multitype Bienaymé-Galton-Watson branching process. This is the complete analogue of a result which Ewens (1968) obtained for a Poisson branching process.Research supported by NSERC     

樟子松人工林分枝结构的分析

基于对6块樟子松(Pinus sylvestris var. mongolica)人工林固定标准地中的30株样木枝解析调查数据,通过分析不同林分、不同大小林木1级枝和2级枝的分枝概率、分枝格局和分枝角度,揭示了樟子松人工林树冠的分枝结构特点。研究结果表明：樟子松人工林1级枝和2级枝的平均分枝数量分别为3.84个和2.80个,两者分枝概率均呈正态分布;1级和2级枝条在光照条件好的几个区间（方位角46°～225°）分布较多,1级枝条的水平分布遵从均匀分布,而2级枝条则不遵从均匀分布;树冠上层枝条的分枝角度略小于树冠中、下层,上层平均分枝角度为45.6°,而中下层平均分枝角度都为49.4°。不同大小林木的1级枝分枝结构规律表明：Ⅰ级木和Ⅴ级木的每轮平均分枝数非常接近,分别为3.89和3.94个,比Ⅲ级木每轮分枝数大0.5个左右;1级枝水平分布在各区间内(45°间隔)相差在0.24%～2.81%之间,方差分析结果表明枝条水平分布与林木大小无关;不同大小林木的分枝角度有所差别,Ⅰ级木、Ⅲ级木和Ⅴ级木的平均分枝角度分别为48.5°、42.2°和50.7°。     

Host range expansion by British moths onto introduced conifers

Abstract.

• 1 Over 2% of British angiosperm-feeding moths (Lepidoptera) have been recorded feeding on conifers introduced to Britain, and may be undergoing host range expansion.
• 2 We compared some of the life-history traits and ecological characteristics of fifty such species, originally exploiting angiosperms and now recorded feeding on conifers, with those of 400 non-shifting angiosperm-feeding moths, to identify those factors linked with host shifting.
• 3 Shifting species attack a wider range of angiosperms than non-shifting species, their original hosts tend to be woody tress and shrubs, and they hatch from the egg earlier in the year.
• 4 Comparisons with the random control samples suggest that larval feeding habit and overwintering stage are also important (species with less intimate relations with the host plant, and those overwintering as eggs are more likely to shift); these trends persisted when the taxonomic distribution of the shift species was controlled for, though were no longer significant.
• 5 Moth species in habitats and on host plant families which are associated with conifer afforestation in upland Britain are more likely to shift, suggesting that ecological opportunity is an important factor in host range expansion.

     

Dynamics of escape mutants

We use multi-type Galton-Watson branching processes to model the evolution of populations that, due to a small reproductive ratio of the individuals, are doomed to extinction. Yet, mutations occurring during the reproduction process, may lead to the appearance of new types of individuals that are able to escape extinction. We provide examples of such populations in medical, biological and environmental contexts and give results on (i) the probability of escape/extinction, (ii) the distribution of the waiting time to produce the first individual whose lineage does not get extinct and (iii) the distribution of the time it takes for the number of mutants to reach a high level. Special attention is dedicated to the case where the probability of mutation is very small and approximations for (i)-(iii) are derived.     

On the probability of loss of new mutations in the presence of linkage disequilibrium

A new selectively neutral mutation occurs in a multilocus genetic background that has achieved a stable equilibrium at which there is a linkage disequilibrium. Perturbation techniques are applied to an extension of the branching process formulation of Fisher in order to address the question of extinction probabilities. We show that under appropriate conditions the probability of extinction of the new mutant is increased by the existence of linkage disequilibrium in the genetic background.Research supported in part by NIH grant GM 28016