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Variation of fixation probability over time in Wright-Fisher model


In Wright-Fisher model of population size N and initial mutation frequency of 1/N, how does the fixation probability vary over generations. So, mathematically, what is the function that maps the generation to its fixation probability?


The function probably look like $r(t) = A + B e^{-C t}$, where $r(t)$ is the rate of fixation at time $t$ (in generation). The reason for this exponential is that the probability that a mutation happens after $t$ generation is given by the exponential distribution.

As the rate of fixation at equilibrium must be at $2Nmu frac{1}{2N} = mu$, $A=mu$, that is $r(t) = mu + B e^{-C t}$

As at $t=0$, the rate must be 0. As $e^{0} = 1$, you should have $B=-mu$. Hence,

$$r(t) = mu - mu e^{-C t}$$

$C$ here represents the steepness of the curve. I don't know how you could calculate it a priori but it must depends upon your population size $N$. As the expected time to fixation starting at frequency $p_0$ is (from Kimura and Ohta 1968)

$$ar t(p_0)=-4Nleft(frac{1-p_0}{p_0} ight)ln(1-p_0)$$

For $p_0 = frac{1}{2N}$

$$ar tleft(frac{1}{2N} ight)=-4Nleft(2N-1 ight)lnleft(1-frac{1}{2N} ight)$$

or

$$ar tleft(frac{1}{2N} ight)=left(4N-8N^2 ight)lnleft(frac{2N-1}{2N} ight)$$

if you prefer. Maybe $C = ar tleft(frac{1}{2N} ight)$ leading to

$$r(t) = mu left(1 - e^{-left( left(4N-8N^2 ight)lnleft(frac{2N-1}{2N} ight) ight) t} ight)$$

Let me know if it matches (I'd pretty amazed)!


EVOLUTION OF EVOLVABILITY IN A DEVELOPMENTAL MODEL

Evolvability, the ability of populations to adapt, can evolve through changes in the mechanisms determining genetic variation and in the processes of development. Here we construct and evolve a simple developmental model in which the pleiotropic effects of genes can evolve. We demonstrate that selection in a changing environment favors a specific pattern of variability, and that this favored pattern maximizes evolvability. Our analysis shows that mutant genotypes with higher evolvability are more likely to increase to fixation. We also show that populations of highly evolvable genotypes are much less likely to be invaded by mutants with lower evolvability, and that this dynamic primarily shapes evolvability. We examine several theoretical objections to the evolution of evolvability in light of this result. We also show that this result is robust to the presence or absence of recombination, and explore how nonrandom environmental change can select for a modular pattern of variability.

Evolvability is the ability of populations to adapt through natural selection. This property is succinctly expressed by the observation that, through mutation, recombination and development, organisms can produce offspring that are more fit than themselves ( Altenberg 1994 ): the study of evolvability posits that this observation is surprising and that it demands an explanation. The concept of evolvability promises to integrate ideas about constraint, phenotypic correlations, and mutational biases into a systematic theory of the variational properties that underlie evolution by natural selection. Evolvability may also provide an essential framework for understanding the success of invasive species ( Gilchrist and Lee 2007 ) and the response of populations to anthropogenic change. Central to the idea of evolvability is the genotype–phenotype map: a set of rules that relate genotypes to the range of phenotypes they can produce ( Alberch 1991 Wagner and Altenberg 1996 ). This mapping emphasizes how developmental systems create phenotypic variation from underlying genetic variation, and suggests two levels of processes contributing to evolvability.

Mutation and recombination contribute to evolvability by creating genetic differences between parents and offspring. Several investigations of the evolution of mutation reference the concept of evolvability ( Radman et al. 1999 Tenaillon et al. 2001 Bedau and Packard 2003 Earl and Deem 2004 André and Godelle 2006 ), as do decades of studies on the evolution of sex and recombination (reviewed in Bell 1982 Otto and Barton 1997 Pepper 2003 Goddard et al. 2005 ). Although these studies have inspired much debate and deepened our understanding of the evolution of variability, they cover only a small fraction of the biological traits that shape evolvability.

A more diverse, and much less-explored, level of influences on evolvability includes the developmental processes that make genetic variation visible to natural selection. Evolvability is increasingly popular as a framework for interpreting a wide range of developmental traits at diverse scales: codon usage in genes ( Plotkin and Dushoff 2003 Meyers et al. 2005 ), RNA structural evolution ( Cowperthwaite and Meyers 2007 ), protein folding and stability ( Wagner et al. 1999 Bloom et al. 2006 ), gene regulatory interactions ( Wagner 1996 Tanay et al. 2005 Quayle and Bullock 2006 Tirosh et al. 2006 ), and angiogenesis and neural outgrowth in animal development ( Kirschner and Gerhart 1998 ). These two levels of influences on evolvability imply that many of an organism's traits might contribute to its evolvability. Insights from the experimental evolution of proteins ( Aharoni et al. 2005 Khersonsky et al. 2006 O'Loughlin et al. 2006 Poelwijk et al. 2007 ), microorganisms ( Burch and Chao 2000 ), and computer programs (reviewed in Adami 2006 Magg and Philippides 2006 ) are increasingly described in terms of evolvability, suggesting the exciting unifying potential of this idea. However, we still lack the theoretical tools to make rigorous measurements and comparisons of organismal evolvability, and the literature contains much confusion over the definition and utility of evolvability ( Sniegowski and Murphy 2006 Lynch 2007 ). Central to this ambiguity is the question of whether evolvability can itself evolve by natural selection.

The evolution of evolvability through direct selection is currently a controversial hypothesis for the adaptability of organisms. One problem with this idea is that selection for evolvability seems to conflict with the apparent myopia of natural selection: the benefits to evolvability lie in an unknown future, perhaps beyond the ken of selection acting on contemporary phenotypes (e.g., Kirschner and Gerhart 1998 Poole et al. 2003 Earl and Deem 2004 Sniegowski and Murphy 2006 ). However, evolutionary biology contains several frameworks for understanding adaptation, such as geometric mean fitness ( Stearns 2000 ) and lifetime reproductive success, in which selection, by integrating information about the past, appears to anticipate the future. Seen in this context, this objection to the evolution of evolvability is simply an empirical question about how well past environments predict future ones, and not a logical paradox.

Another objection is that an evolvable genotype does not survive its own success: a genotype which produced a mutant that fixed in a population may have been evolvable, but it is now also extinct ( Plotkin and Dushoff 2003 ). This difficulty may also be more apparent than real: just as offspring must merely resemble their parents for selection to cause evolution, evolvability must only be partially heritable to evolve. Again, this is an empirical question about organisms and other evolvable systems (see our discussion below and Plotkin and Dushoff (2003) for examples of how it can be answered).

The last major argument against the efficacy of selection favoring evolvability is that recombination will quickly dissociate an allele that improves variability from any positively selected variants it helps to create ( Sniegowski and Murphy 2006 ). This argument may be damning for alleles that increase the mutation rate, or “mutator” alleles, in populations with any recombination (Tenaillon et al. 2000), implying that the evolutionary relevance of mutators is small at best ( Sniegowski et al. 2000 de Visser 2002 ). Sniegowski and Murphy (2006) suggest that this result argues against all but special cases of evolvability loci with local effects, such as contingency loci in certain bacteria or transposons ( de Visser 2002 ). Because these loci cannot be readily decoupled from any adaptive variants they create, recombination does not directly limit their successful fixation through indirect selection. Although it is unknown whether local mutation-modifying loci are the exception or the rule, it is worth noting that loci that affect development can influence variability through epistatic interactions with other loci. Such epistasis binds an evolvability-modifying allele to the beneficial alleles it facilitates, setting selection in opposition to recombination. Therefore, recombination may restrict, but not preclude, the evolution of evolvability through changes in epistasis.

Despite these apparent theoretical difficulties, several studies have made strong arguments for the evolution of evolvability by the evolution of developmental processes, in both biological systems and models ( Ancel and Fontana 2000 Masel and Bergman 2003 Plotkin and Dushoff 2003 Masel 2005 Meyers et al. 2005 ). A common feature of many of these successful investigations is a tractable and explicit model of the relevant aspects of development: RNA folding in Ancel and Fontana (2000) , mRNA translation in Masel and Bergman (2003) , and the genetic code in Plotkin and Dushoff (2003) and Meyers et al. (2005) . Although the genotype–phenotype maps of even simple organisms are largely unknown, analyzing the evolution of evolvability in a diverse array of simple, well-defined models of development may reveal the mechanisms of selection that shape the variabilities and developmental systems of organisms.

To dissect the influence of natural selection on evolvability, we constructed a model of development focused on the pleiotropic effects of two genes on two quantitative characters. Pleiotropy, the effects of a single locus on multiple traits, is a ubiquitous feature of real developmental systems, and an elemental component of genotype–phenotype maps ( Baatz and Wagner 1997 Hansen 2006 ). The effects of pleiotropy on adaptation have been studied since Fisher and are central to ideas about complexity and modularity in biology ( Wagner and Altenberg 1996 Baatz and Wagner 1997 Orr 2000 Hansen 2003 Welch and Waxman 2003 Griswold 2006 ). Our model allows the evolution of both phenotypes and pleiotropic effects and therefore enables us to address the important question of how selection on the phenotype alters pleiotropy ( Hansen 2006 ). We find that subjecting simulated populations of these model organisms to fluctuating selection favors pleiotropic relationships that minimize constraints on variability. We demonstrate that this outcome is the result of selection favoring evolvability, and that our results suggest a novel perspective on the influence of selection on the evolvability of developmental systems.


Introduction

Helping behaviors, by which individuals in a population provide fitness benefits to others, are more likely to evolve in the presence of mechanisms that allow discrimination against defectors (Hamilton, 1964, 1971 Axelrod and Hamilton, 1981 Eshel and Cavalli-Sforza, 1982). Discrimination may occur if helpers can recognize each other by using conspicuous phenotypic cues, tags, or markers, and provide benefits to other individuals carrying the gene(s) underlying helping, instead of providing benefits to defectors. Both genetic kin-recognition and the so-called green-beard mechanism may involve the expression of helping conditional based on actor and recipient bearing identical recognition markers (Hamilton, 1964 Dawkins, 1982 Grafen, 1990). Such conditional helping relies on a tight linkage between the genes underlying helping and those producing the conspicuous markers. If genes for helping and markers were loosely coupled, defectors could acquire the markers expressed by helpers and then receive the benefits of helping without paying the cost. This would ultimately prevent the evolution of marker-based conditional helping.

Genetic kin-recognition based on actor and recipient bearing identical marker alleles might evolve in spatially-structured populations (Axelrod et al., 2004 Jansen and van Baalen, 2006 Rousset and Roze, 2007). Two individuals from the same group (or spatial location) are more likely to have inherited both helping and recognition marker alleles from the same recent common ancestor than are two individuals from different groups. Common ancestry can then lead to the build up of genetic associations between helping and recognition alleles between individuals, which descend from the same group even in the presence of recombination, provided migration is limited and group size is not too large. Because individuals within groups might interact with resident and immigrant individuals, recognition markers that are identical-by-descent may allow actors to discriminate among categories of recipients defined by markers within groups. Such marker-based discrimination then sustains the evolution of conditional helping under strong population structure (Axelrod et al., 2004 Jansen and van Baalen, 2006 Rousset and Roze, 2007).

An increase in group size or migration rate erodes population structure and weakens the genetic associations between individuals from the same group, within and across loci. With frequent migration, population structure is likely to vanish. In this case, marker-based helping is no longer expected to evolve. Nevertheless, if the population becomes panmictic but is of finite size, some variation may remain in the propensity of interacting individuals to share alleles identical-by-descent at many loci. Indeed, two offspring of the same parent are always more likely to have inherited identical helping and recognition marker alleles than are two individuals sampled at random from the population. The variation in the ancestry of pairs of interacting individuals within a panmictic population might then still allow actors to discriminate among categories of recipients. This variance supports the evolution of marker-based conditional helping when population size is very small (Traulsen and Nowak, 2007) because the probability that individuals descending from the same parent interact in a panmictic population is likely to be small, approximately equal to the inverse of population size in the absence of searching. Thus, whether marker-based conditional helping evolves in panmictic or structured populations, finite population size is a crucial demographic requirement for the evolution of the behavior.

With finite local group (or total population) size individuals that help each other are also more likely to compete for the same local (or global) resources. Competition between interacting individuals has been shown to partially offset the benefits of helping in both finite panmictic (Hamilton, 1971) and infinite structured populations (Taylor, 1992a,b). Finite population size thus creates ecological conditions where actors may actually benefit from expressing behaviors that reduce the fecundity of neighbours by harming them, instead of increasing it by helping them, even when the actor suffers a fecundity cost. This follows from the fact that in a finite population a single individual may have a marked effect on population productivity and decrease the intensity of competition experienced by its offspring.

From the green-beard in the red fire ant Solenopsis invicta, which kills individuals that do not have it, to bacteria releasing antagonistic compounds in their environment, to maternally transmitted symbionts generating cytoplasmic incompatibility, several examples have been documented where genotypes spread through natural populations by hampering the reproduction of those that do not carry them, thereby reducing the intensity of competition experienced by their carriers (e.g., Werren, 1997 Keller and Ross, 1998 Riley and Gordon, 1999 Brown et al., 2006). It is thus useful to understand not only what are the ecological and demographic conditions leading to the evolution of helping behaviors (cooperation and altruism), but also those conducive to the evolution of harming (exploitation and spite). This might lead to a better understanding of the type of social interactions expected to occur in natural populations.

In this paper, we try to understand the conditions under which marker-based conditional harming, whereby an actor decreases the fecundity of recipients conditional on them bearing a different phenotypic cue than the actor, is selected for. To that aim, we analyze the joint evolution of neutral recognition markers and marker-based conditional harming behaviors in a two-locus population genetic framework. The first locus controls the expression of neutral conspicuous markers. The second locus determines the expression of harming, conditional on actor and recipient bearing different conspicuous markers at the first locus. We show that under a Wright-Fisher scheme of reproduction, marker-based conditional harming can evolve for a large range of recombination rates and group sizes, in both finite panmictic and infinite structured populations. Direct comparison with results for the evolution of marker-based conditional helping reveals conditions under which, everything else being equal, the selective pressure favoring marker-based conditional harming is stronger than that on conditional helping. In particular, this is the case when only two conspicuous marker alleles at the recognition locus segregate in the population.


Variation of fixation probability over time in Wright-Fisher model - Biology

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Evolution of Evolvability in a Developmental Model

Jeremy Draghi, 1,* Günter P. Wagner 1,**

1 1 Department of Ecology & Evolutionary Biology, Yale University, New Haven, CT 06511

* E-mail: [email protected]
** E-mail: [email protected]

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Evolvability, the ability of populations to adapt, can evolve through changes in the mechanisms determining genetic variation and in the processes of development. Here we construct and evolve a simple developmental model in which the pleiotropic effects of genes can evolve. We demonstrate that selection in a changing environment favors a specific pattern of variability, and that this favored pattern maximizes evolvability. Our analysis shows that mutant genotypes with higher evolvability are more likely to increase to fixation. We also show that populations of highly evolvable genotypes are much less likely to be invaded by mutants with lower evolvability, and that this dynamic primarily shapes evolvability. We examine several theoretical objections to the evolution of evolvability in light of this result. We also show that this result is robust to the presence or absence of recombination, and explore how nonrandom environmental change can select for a modular pattern of variability.

Jeremy Draghi and Günter P. Wagner "Evolution of Evolvability in a Developmental Model," Evolution 62(2), 301-315, (1 February 2008). https://doi.org/10.1111/j.1558-5646.2007.00303.x

Received: 11 June 2007 Accepted: 27 October 2007 Published: 1 February 2008

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Discussion

Natural selection favors those genotypes that confer on their carriers the highest lifetime reproductive success (fitness defined here as the expected number of offspring that reach the stage of reproduction) because these genotypes are more likely to introduce replicate copies of themselves into the next generation than alternative genotypes. There are two basic and very different means by which a mutant allele can cause its carriers to have a higher fitness than those individuals bearing an alternative, resident allele. Either the mutant confers higher vital rates on its carriers (higher fecundity or survival) or the mutant confers lower vital rates to noncarriers. The latter case can be defined as harming, and when it occurs it may decrease the intensity of competition experienced by a carrier of the harming allele, or that experienced by its offspring.

EVOLUTION OF MARKER-BASED CONDITIONAL HARMING

We have analyzed the joint evolution of neutral recognition markers and marker-based conditional harming under two different but complementary demographic scenarios: finite panmictic and infinite structured populations. Our results show that for a mutant harming allele to be selected for under these two scenarios, the costs to an actor of expressing harming and being harmed must be offset by the benefits obtained from the reduction in competition faced by the actor's offspring, which is due to all actors in the population expressing conditional harming (eqs. 4 and 8). The first cost (cost of harming) depends on the probability that an actor interacts with another individual bearing a different marker allele from itself the second cost (cost of being harmed) depends on the probability that an actor interacts with another individual bearing the harming allele and a different marker allele from the actor finally, the benefit of harming depends on the probability that in the population of a focal actor, actors (including the focal actor) interact with other individuals bearing different marker alleles from those of the actor.

The interaction probabilities that weight the costs and benefits of harming depend on population size, N, recombination rate, r, and migration rate m (for the structured population case). Total finite population size (or finite local group size with limited dispersal) results in genetic drift, which entails that an actor may interact with other individuals that have the same common ancestor as the actor (i.e., coalescence of alleles sampled in different individuals occurs). These individuals are then likely to carry the mutant harming allele and the same marker allele as the actor. Interaction with such individuals will reduce the costs of being harmed, but also the benefits, because fewer individuals are likely to be harmed in the population (or local group). In the absence of recombination, the benefits might exceed the costs because an individual bearing the harming allele will never be harmed (perfect recognition), but as long as there is genetic variation in the population, individuals bearing the resident allele are harmed, which results in a decrease in competition felt by the actor or its offspring. Recombination increases the cost of being harmed, because the descendants of an ancestor bearing the harming allele may carry different marker alleles, but it also increases the benefits of harming for the same reason, as more individuals are harmed in the population.

Our results show that the selective pressure on marker-based conditional harming is a decreasing function of the three parameters N, m, and r (eq. 6). This is qualitatively exactly what is usually found for the selective pressure on unconditional helping in a spatially subdivided population with the island model of dispersal (e.g., Eshel 1972 Aoki 1982 Rogers 1990 Taylor and Irwin 2000 Gardner and West 2006 ), and is also what was found for the invasion of marker-based conditional helping ( Rousset and Roze 2007 ). But our results also suggest that the selective pressure on conditional harming can be stronger than that on conditional helping under otherwise similar life-cycle assumptions. This result holds for both our finite panmictic and infinite structured population scenarios (compare eqs. 5 and 6, and eqs. 9 and 10). The difference between the selective pressure on conditional harming and helping is actually expected to be greatest in finite panmictic populations (see section “The relationship between conditional harming and helping”). We then observe that a single mutant harming allele can be selected for under a wide range of parameter values, even in the presence of recombination (Fig. 1).

Under our life-cycle assumptions, the selective pressure on conditional harming is often stronger than that on conditional helping because the inclusive fitness benefits obtained through conditional harming and helping (total effect through B and D, assuming they are of similar magnitude) are identical in the infinite structured population case, and larger under harming for the panmictic population case (see section “The relationship between conditional harming and helping”). But at the same time, the inclusive fitness cost of expressing harming (total effect through C, eq. 11) can be lower than that of expressing helping. In particular, when there are only two recognition markers segregating in the population, individuals tend to interact more often with others having identical marker alleles (HR≤ 1/2) and thus pay the direct cost of expressing conditionally helping more often than they would if they expressed harming conditionally. When there are more marker alleles segregating at the recognition locus (HR≤ (K− 1)/K with K marker alleles), the selective pressure on conditional helping can become stronger than that on conditional harming as individuals tend to pay the direct cost of harming more often than that of helping (e.g., eqs. 5 and 6).

Different life-cycle assumptions might also lead to different selective regimes on conditional harming. For instance, in patch-structured populations with maternally transmitted symbionts spreading through host populations by hampering the reproduction of uninfected females (cytoplasmic incompatibility), the condition for invasion of harming was found to be a nonmonotonic, dome-shaped function of group size in a model with similar basic structure to ours ( Reuter et al. 2008 ). It is also well known that introducing overlapping generations with only juvenile dispersal into the type of models considered here can increase the selective pressure on unconditional helping (S > 0 in eq. 13, Taylor and Irwin 2000 Irwin and Taylor 2001 ), which may tip the balance in favor of conditional helping instead of conditional harming, although this is likely to depend on the life-cycle parameter values ( Johnstone and Cant 2008 ).

HARMING VERSUS SPITE

The selective pressure on a mutant harming allele depends on how its expression benefits its carrier and the carrier's relatives by reducing the fecundity of individuals bearing the alternative allele. Hamilton (1970) called “spiteful” a behavior decreasing the fitness of the actor and that of the recipient of the act of harming. We mention that harming might qualify as spiteful (sensu Hamilton (1970) ) in both our finite panmictic and infinite structured population models. This can be seen by noting that the net change in the fitness of a carrier due to it expressing the harming allele (and holding everything else constant) can be obtained for the infinite structured population model by setting HR(F−φ) and HR(F−γ) equal to zero in equation (4) and for the finite panmictic population model by setting P2 and P3 equal to zero equation (8). In both cases, there is a wide range of a parameter values (D, C, m, N, and r) where the resulting change in fitness can be negative (hence expressing the mutant allele results in a net fitness cost to the carrier when everything else is held constant) but the harming allele will still be favored by selection because these direct costs are offset by the reduction in competition felt by the offspring of relatives of the focal individual. However, the conditions under which harming qualifies as spiteful in the sense of Hamilton (1970) are complicated and do not help us here to further understand the ecological and demographic conditions under which conditional harming is selected for, so that we did not present such computations.

MARKER-BASED RECOGNITION, GREEN-BEARDS, AND MATE CHOICE

Marker-based recognition in panmictic populations is sometimes referred to as the green-beard mechanism: a conspicuous phenotypic effect of a gene is recognized by other individuals bearing that gene, and where that gene also has a pleiotropic effect on the behavior of individuals expressing the conspicuous phenotype ( Dawkins 1982 ). Strictly speaking our marker-based conditional harming and helping models for both finite panmictic and infinite structured populations correspond to kin-recognition mechanisms. This is so because in both cases the marker alleles are exchangeable (whether the harming or helping alleles arise on an R or r marker allele background does not affect discrimination). By contrast, under the green-beard mechanism, a particular marker allele is postulated to be associated with a specific behavioral phenotype so that conspicuous markers are not exchangeable (see below for an empirical example).

Independently of the exact nature of the marker-based recognition mechanism (e.g., kin recognition or green beard), our results suggest that the evolution of marker-based helping in panmictic populations may be selected for only under small population size whereas marker-based harming might evolve under a much larger set of parameter values. In the light of this observation it is interesting that the compelling documented examples of green-beards in natural populations are of the harming type. Indeed, the green-beard found in the red fire ant S. invicta is of this type ( Keller and Ross 1998 ), where workers homozygous for allele b at the Gp-9 locus kill those individuals that do not contain it (BB queens) while not inducing killing of individuals that do (Bb queens). Note that here allele B cannot be exchanged with allele b without affecting discrimination. Other examples of marker-based conditional harming may be found among bacterial strains. Some bacteria release into their environment intraspecific antagonistic compounds such as bacteriocins and bacteriophages, which allows them to suppress the growth of competing strains ( Riley and Gordon 1999 Gardner et al. 2004 ). Recognition in this case is molecular with the bacteriocin gene tightly linked to specific immunity genes that block the effect of the bacteriocins and molecular discrimination may even occur between carriers and noncarriers of isogenic phages ( Brown et al. 2006 ). For bacteria with recurrent cycles of colonization and population growth, the number of founding clones will probably be more relevant than the stationary population size to describe the change in genotype frequency in the population in that case our parameter N can be thought of as the number of founding clones.

Another situation in which marker-based recognition can be used to discriminate between categories of recipients is in the context of assortative mating ( Crow and Kimura 1970 Kirkpatrick 1982 Seger 1985 ). Females (or males) could prefer to mate with those individuals of the opposite sex that carry identical marker alleles to them at an arbitrary recognition locus that has no direct effects on fitness ( Castro and Toro 2006 ). There are several similarities between our models and the mate choice model of Castro and Toro (2006) . These authors also consider that individuals carry two loci: one where arbitrary recognition alleles segregate and another that codes for mating expressed conditionally on pairs of individuals bearing identical marker alleles at the recognition locus the result is that individuals are more likely to interact with relatives. Castro and Toro then show by simulations that the spread of a choice allele resulting in females mating only with males carrying identical markers is enhanced by finite population size effects (whether the population is panmictic or structured), which corroborates our own results. However, our formalization does not apply directly as it stands to mate choice. By contrast to the model of Castro and Toro, we do not consider a process by which individuals search for others carrying identical (or different) recognition markers. Such a search process could be included in our models by introducing different acceptance probabilities for individuals bearing identical or different marker alleles, so that individuals would stop searching once they have found a partner they accept to interact with (or mate with in the context of mate choice). This deserves further formalization, especially because mate choice is also likely to depend on inbreeding, an inevitable consequence of finite patch or population size.

CONCLUSIONS

Harming behaviors may not be uncommon in nature. For instance, segregation distorter alleles may produce toxins during meiosis to which they but not their alternatives are resistant the distorter thus increases in frequency by reducing competition for fertilization (e.g., Lyttle 1991 Ridley 2003 Burt and Trivers 2006 ). Maternally transmitted symbionts can spread through host populations by hampering the reproduction of uninfected females, thereby reducing competition for symbiont carriers (e.g., Werren 1997 Ridley 2003 Burt and Trivers 2006 ). In all these cases a mutant allele spreads by harming others and this functions because the interaction neighborhood is small enough that the reduction of vital rates of others due to the behavior of the actor, or that of its relatives, decreases the intensity of competition experienced by the actor or its offspring (the interaction neighborhood is actually very small for segregation distorters).

In addition to the results reported here, several models have already identified ecological and demographic conditions for the evolution of harming behaviors in structured populations, where localized migration generates small interaction neighborhoods ( Gardner et al. 2004 Lehmann et al. 2006 Gardner et al. 2007 Lehmann et al. 2007a Johnstone and Cant 2008 El Mouden and Gardner 2008 ). All these results broaden the scope of biological situations where harming may occur. They show not only that harming might evolve in both finite panmictic and structured populations, but suggest that, under certain situations, harming is actually more likely to evolve than helping. This should encourage behavioral ecologists to seek evidence for conditional harming rather than conditional helping.

Associate Editor: M. Van Baalen


Discussion

Interference between beneficial mutations with partial selfing and dominance

Multilocus models of adaptation in partial self-fertilizing species can inform on how the interplay between homozygote creation and reduction in recombination jointly affects selection acting on multiple sites. It is already known that the presence of linked deleterious variation means that mildly recessive beneficial mutations (h just less than 1/2) are more able to fix in outcrossers than in selfing organisms by recombining away from the deleterious allele, in contrast to single-locus theory (Hartfield and Glémin 2014). More generally, genome-wide background selection can substantially reduce adaptation in highly selfing species (Kamran-Disfani and Agrawal 2014). Yet the extent that linkage between beneficial mutations affects mating-system evolution remains poorly known.

Here we extended several previous models of selection interference to consider how adaptation is impeded in partially selfing organisms. We considered two possibilities. First, given that an existing sweep is progressing through the population, subsequent mutations confer a lesser selective advantage and can fix only if recombining onto the fitter genetic background (the “emergence” effect). Alternatively, a second mutant could be fitter and replace the existing sweep, unless recombination unites the two alleles (the “replacement” effect). We found that the emergence effect is generally stronger than the replacement effect and is more likely to lead to loss of beneficial mutations (Figure 4).

Furthermore, selection interference has two opposite effects on Haldane’s sieve. In mainly outcrossing populations (where it operates), Haldane’s sieve is reinforced because recessive mutations are even more likely to be lost when rare compared to dominant ones, compared to single-locus results. However, when comparing different mating systems, interference reduces or nullifies the advantage of selfing of not being affected by Haldane’s sieve. Consequently, weakly beneficial mutations are more likely to be fixed in outcrossers, irrespective of their dominance level (Figure 7). These findings thus contribute to a body of literature as to when the predictions of Haldane’s sieve should break down or otherwise be altered. Other examples include the fixation probability of mutations being independent from dominance if arising from previously deleterious variation (Orr and Betancourt 2001) more generally, outcrossers are more able to fix mutations with any dominance level compared to selfers if arising from standing variation and when multiple linked deleterious variants are present (Glémin and Ronfort 2013). Conversely, dominant mutations can be lost in metapopulations due to strong drift effects (Pannell et al. 2005).

In our model we assumed that no more than two beneficial mutations simultaneously interfere in the population. However, even if mutation does not occur frequently enough to lead to multiple mutations interfering under outcrossing, the presence of a few sweeping mutations throughout a genome can jointly interfere in highly selfing species. Obtaining a general model of multiple substitutions in a diploid partially selfing population is a difficult task, but it is likely that the rate of adaptation would be further reduced compared to the two-locus predictions (as found in haploid populations by Weissman and Barton 2012).

It is also of interest to ask whether our calculations hold with different types of inbreeding (such as sib mating). For a single unlinked mutant, Caballero and Hill (1992) showed how various inbreeding regimes determine the value of F used in calculating fixation probabilities (Equation 1). However, it is unclear how effective recombination rates will be affected. For example, Nordborg’s (2000) rescaling argument relies on the proportion of recombination events that are instantly repaired by direct self-fertilization these dynamics would surely be different under alternative inbreeding scenarios. Further work would be necessary to determine how other types of inbreeding affect net recombination rates and thus the ability for selection interference to be broken down.

Causes of limits to adaptation in selfing species

We have already shown in a previous article how adaptation can be impeded in low-recombining selfing species due to the hitchhiking of linked deleterious mutations (Hartfield and Glémin 2014), with Kamran-Disfani and Agrawal (2014) demonstrating that background selection can also greatly limit adaptation. Hence the question arises of whether deleterious mutations or multiple sweeps are more likely to impede overall adaptation rates in selfing species.

Background selection due to strongly selected deleterious alleles causes a general reduction in variation across the genome by reducing (Nordborg et al. 1996) here the overall reduction in emergence probability is proportional to where is mediated by the strength and rate of deleterious mutations (Barton 1995 Johnson and Barton 2002) and thus affects all mutations in the same way [note that this process becomes more complicated with weaker deleterious mutations (McVean and Charlesworth 2000)]. Because of background selection, selfing is thus expected to globally reduce adaptation without affecting the spectrum of fixed mutations. Similarly, adaptation from standing variation, which depends on polymorphism level, is expected to be affected by the same proportion (Glémin and Ronfort 2013). Alternatively, interference between beneficial mutations is mediated by φ, the ratio of the selection coefficients of the sweeps. For a given selective effect at locus A, weak mutations at locus B are thus more affected by interference than stronger ones, and the net effect of interference cannot be summarized by a single change in (Barton 1995 Weissman and Barton 2012). Because of selective interference, selfing is also expected to shift the spectrum of fixed mutations toward those of strong effects. Interestingly, Weissman and Barton (2012) showed that neutral polymorphism can be significantly reduced by multiple sweeps, even if they do not interfere among themselves. This suggests that in selfing species, adaptation from standing variation should be more limited than predicted by single-locus theory (Glémin and Ronfort 2013). Selective interference could thus affect both the number and the type of adaptations observed in selfing species.

Reflecting on this logic, both processes should interact and we therefore predict that background selection will have a diminishing-returns effect. As background selection lowers then the substitution rate of beneficial mutations will be reduced (since it is proportional to for μ the per-site mutation rate), and hence interference between beneficial mutations will subsequently be alleviated. No such respite will be available with a higher adaptive mutation rate on the contrary, interference will increase (Figure 6). Impediment of adaptive alleles should play a strong role in reducing the fitness of selfing species, causing them to be an evolutionary dead end. Further theoretical work teasing apart these effects would be desirable. Given the complexity of such analyses, simulation studies similar to those of Kamran-Disfani and Agrawal (2014) would be a useful approach to answering this question.

In a recent study, Lande and Porcher (2015) demonstrated that once the selfing rate became critically high, selfing organisms then purged a large amount of quantitative trait variation, limiting their ability to respond to selection in a changing environment. This mechanism provides an alternative basis as to how selfing organisms are an evolutionary dead end. However, they consider only populations at equilibrium our results suggest that directional selection should further reduce quantitative genetic variation due to selective interference among mutations. Subsequent theoretical work is needed to determine the impact of interference via sweeps on the loss of quantitative variation. Furthermore, complex organisms (i.e., those where many loci underlie phenotypic selection) are less likely to adapt to a moving optimum compared to when only a few traits are under selection (Matuszewski et al. 2014) and can also purge genetic variance for lower selfing rates (Lande and Porcher 2015). Complex selfing organisms should thus be less able to adapt to environmental changes.

Empirical implications

The models derived here lead to several testable predictions for the rate of adaptation between selfing and outcrossing sister species. These include an overall reduction in the adaptive substitution rate in selfing populations a shift in the distribution of fitness effects in selfing organisms to include only strongly selected mutations that escape interference and a difference in the dominance spectrum of adaptive mutations in outcrossers compared to selfers, as already predicted by single-locus theory (Charlesworth 1992) and observed with quantitative trait loci (QTL) for domesticated crops (Ronfort and Glémin 2013).

Few studies currently exist that directly compare adaptation rates and potential between related selfing and outcrossing species, but they are in agreement with the predictions of the model. In plants, the self-incompatible C. grandiflora exhibited much higher adaptation rates [where of nonsynonymous substations were estimated to be driven by positive selection, using the McDonald–Kreitman statistic (Slotte et al. (2010)] than in the related selfing species Arabidopsis thaliana (where α is not significantly different from zero). Similarly, the outcrossing snail Physa acuta exhibited significant adaptation rates ( ), while no evidence for adaptation in the selfing snail was obtained (Burgarella et al. 2015) in fact, evidence suggests that deleterious mutations segregate due to drift ( ). In agreement with the predicted inefficacy of selection on weak mutations, Qiu et al. (2011) also observed significantly lower selection on codon usage in the Capsella and Arabidopsis selfers than in their outcrossing sister species.

In addition, as only strong advantageous mutations are expected to escape loss through selection interference, this result can explain why selective sweeps covering large tracts of a genome are commonly observed, as with A. thaliana (Long et al. 2013) and Caenorhabditis elegans (Andersen et al. 2012). Extended sweep signatures can also be explained by reduced effective recombination rates in selfing genomes. Finally, selective interference between beneficial mutations could explain why maladaptive QTL are observed as underlying fitness components, as detected in A. thaliana (Ågren et al. 2013). Direct QTL comparisons between selfing and outcrossing sister species would therefore be desirable to determine to what extent selection interference leads to maladaptation in selfing species.


Discussion

Summary of results

When many beneficial alleles are sweeping through a population, interference among them may greatly retard adaptation. In this case, the rate of adaptation may be primarily limited by the rate at which recombination can bring beneficial alleles together in the same genome. A scaling argument shows that for a given distribution of selection coefficients, the density of successful substitutions per generation per chromosome arm, , is a function solely of the density that would be expected in the absence of interference, , and does not depend on the beneficial mutation rate , the total genetic map length , the population size , or strength of selection separately. When mutations have equal effects, we obtain an explicit approximate formula for the density of substitutions, . This implies that there is an “upper bound” to the density of sweeps, . When the population variance in log fitness, , is large, interference from unlinked loci further reduces the rate of sweeps by a factor or , depending on the mating system. However, for , most interference occurs between linked loci separated by a map distance .

Simulations show that the scaling argument is accurate over a broad range of parameters. Numerical calculations and simulations show that the explicit formula for is accurate for up to a few interacting sweeps, but substantially underestimates the rate of adaptation when there are many closely-linked, concurrent sweeps. The simulations indicate that the rate of adaptation continues to increase above the “upper bound” as and increase, perhaps logarithmically however, this increase becomes so slow that is unlikely to greatly exceed one in most populations. Simulations also indicate that the assumption that all mutations have the same effect can be relaxed without affecting the key results. Genetic draft greatly reduces neutral diversity when the density of sweeps exceeds , far lower than the density needed to cause interference however, even when sweeps are dense enough to cause extreme interference, neutral diversity is not reduced by much more.

Relation with previous work

Several authors have recently studied interference among unlinked loci [23], [24], [26], [27] . Cohen et al. [23], [24] and Rouzine et al. [26] consider models in which the total number of possible adaptive substitutions is fixed, so that sufficiently large populations reach a maximum rate of adaptation, a different situation from the one we consider. However, [26] do show that the infinitesimal model used here is a good approximation to the dynamics of unlinked loci for a broad range of parameters. Neher et al.'s model [27] includes mutations and is more similar to ours. However, [26], [27] consider only facultative sexuals and assume a small rate of outcrossing, . As mentioned above, our infinitesimal model can be straightforwardly extended to a similar case, in which individuals outcross only every generations, by scaling selective coefficients by , i.e., by replacing by . This implies that the boundary between weak and strong interference is at , consistent with [27]. [27]'s result for the weak interference regime (the second line of their Eq. 12) is the same as predicted by our Eq. (1) . For strong interference, our scaled Eq. (1) has the limit , somewhat different from the first line of their Eq. 12 ( in our notation). Both predict only a logarithmic increase in , but the dependence on the underlying parameters is different. This is because in their model rare, extremely fit genotypes can produce large clonal lineages without being broken up by recombination, whereas in ours all lineages eventually recombine. Their model is more appropriate for organisms that have a small chance of outcrossing in every generation (which is most likely for bacteria and viruses, and also some eukaryotes), while ours applies to organisms that outcross at regular intervals between rounds of asexual reproduction (as is the case with some eukaryotes).

Both [27] and [26] ignore the possibility of varying degrees of linkage among loci (i.e., there is no genetic map). This is a natural model for bacteria in which recombination typically involves the replacement of short stretches of DNA, and most loci therefore have the same recombination fraction with each other. However, in viruses and eukaryotes, recombination is primarily due to crossovers, as in our model. In this case, adjusting our Eq. (8) for facultative sexuals outcrossing at frequency gives

Eq. (15) indicates that linked loci are the primary source of interference when , which we expect to be true for many populations. Thus, we expect interference among beneficial mutations to be more prevalent than predicted by previous studies. Considering both the differences between the models of facultative sex discussed in the previous paragraph, and the differences between the models of recombination, the models of [26], [27] are generally more appropriate for bacteria, while ours is generally more appropriate for eukaryotes with an obligate outcrossing stage in their life cycle. For viruses and eukaryotes that outcross rarely and randomly, their models do a better job of capturing interference among unlinked loci, and are therefore more appropriate for organisms with , while ours is better when most interference is from tightly-linked loci ().

Neher and Shraiman [30] have recently extended [27] to consider the effect of genetic draft on neutral diversity. Although they consider different measures of diversity than we do, their results are qualitatively similar to those of our infinitesimal model ( Eq. (9) for , and scaled by the outcrossing frequency): draft is significant when the variance in log fitness exceeds the square of the outcrossing rate, , i.e., for our model of obligate sexuals. A similar result was also derived by Santiago and Caballero [74]. Note that this is the same threshold value at which interference from unlinked loci begins to affect advantageous alleles. In our model of a linear genetic map, in contrast, the rate of sweeps necessary to create significant draft is much lower than the rate needed to cause strong interference: Eq. (9) predicts that that will be much less than for , typically a much weaker condition than . If we consider the case of HIV within-host evolution addressed by [30], taking the frequency of outcrossing to be , the map length to be , and typical positive selective coefficients to be [29], [81], [82], we see that for any reasonable population size (roughly, ), the threshold value of at which draft from linked sweeps becomes important is smaller than that at which draft and interference from unlinked sweeps become important. Santiago and Caballero [83] extend [74] to allow for the effect of a genetic map their framework can be used to derive the roughly the same threshold rate of sweeps , but drastically underestimates for the draft-dominated populations described by Eq. (9).

Deleterious mutations

Because deleterious mutations are far more frequent than beneficial mutations, it is important to consider how they affect our results. The effect of unlinked deleterious mutations is easy to incorporate into the infinitesimal model by repeating the analysis using the exact expression for the rate of increase in mean log fitness, including the direct effect of new mutations, , where in the second term and the expectation over include deleterious mutations. Unlinked mutations simply increase the effective strength of drift and can be described as reducing the effective population size. The effect of linked deleterious mutations can also easily be included when deleterious mutations and sweeps are not so common that they substantially reduce the efficacy of negative selection. In this case, deleterious mutations with selective disadvantage occurring at a genomic mutation rate reduce fixation probability at linked sites by a factor , where [55]. In contrast to the effect of unlinked loci, this clearly cannot be captured by a reduction in a single effective population size, as beneficial alleles of different effects experience different amounts of interference since decreases with , strongly selected alleles experience less interference from background selection, just as they experience less interference from other sweeps ( Figure 8 ). Background selection has the largest effect when there are many linked deleterious alleles, but in this case the deleterious alleles interfere with each other and the situation becomes more complicated [76]. This case and the one in which deleterious alleles experience strong interference from sweeps remain to be investigated analytically.

Population subdivision

It is important to consider how population subdivision interacts with interference in determining the rate of adaptation. When few favorable alleles enter in each generation, so that is small, the rate of adaptation increases in proportion to population size, , while Hill-Robertson interference leads to diminishing returns for increasing population size. This appears to suggest that a subdivided population, consisting of many small demes, might adapt more efficiently. However, note that for an allele to fix in the entire population, it must fix in every deme in addition, other alleles may fix only locally before going extinct. Thus, every deme experiences at least the same rate of sweeps, , as would a single panmictic population. Thus, strong population subdivision will increase interference among sweeps, most of which enter the local deme by migration, rather than by mutation. [56], [57] showed that with conservative migration, and in which each deme contributes according to its size, the fixation probability of a favorable allele is unaffected by population structure. We believe that this result does not carry over to the effects of multiple sweeps, and that overall, the fixation probability will be reduced by subdivision. This has been found to be true for asexual populations [84], but remains an open question in sexual populations.

Likely strength of Hill-Robertson interference

It is unclear how important the Hill-Robertson effect due to selective sweeps is in biological populations, both because it is difficult to measure the local rate of adaptive substitutions and because the expected amount of interference had not been determined theoretically. Above, we addressed the second question, and found that interference between substitutions becomes important as the rate of adaptive substitutions approaches one per Morgan every two generations. Here we briefly discuss what is known about the first question, and what this implies for the relevance of Hill-Robertson interference from sweeps.

Artificial selection

Does Hill-Robertson interference limit the response to strong artificial selection on sexual populations? At first, the response must be due to standing variation, and may depend on alleles initially in many copies. (However, many microbial evolution experiments start with very little standing variation this situation is discussed in Text S5.) The reduction in fixation probability considered here is hardly relevant in this initial phase, though negative linkage disequilibria between favorable alleles will slow down the response. However, even completely homogeneous populations respond to selection after an initial delay, showing that there is a high rate of increase in genetic variance due to new mutations, : typically, , where is the non-genetic component of the variance in the trait [53]. Thus, after some tens of generations, new mutations will start to contribute, and ultimately, the rate of fixation of such mutations limits the selection response [85], [86]. In the absence of Hill-Robertson interference, this could in principle lead to an extremely high rate of adaptive substitution. An allele with effect on a trait with total phenotypic variance has selective advantage , where is the selection gradient, which is typically of order . (For example, if the top are selected, ). Therefore, the baseline rate of substitution due to mutations of effect , arising at net rate per genome per generation, is . Since (assuming that mutations are equally likely to increase or decrease the trait under selection), this can be rewritten as . Selection can pick up alleles with effect larger than , and so substitutions could occur at up to . Using the middle of the estimated range of from [53] and assuming gives . Thus, even moderately-sized populations could in principle sustain extremely high baseline rates of adaptive substitution, both because they generate large numbers of mutations, and because selection can be effective on alleles of small effect. It seems that populations under artificial selection could easily be in the regime in which Hill-Robertson interference is strong.

It is difficult to determine if Hill-Robertson interference has limited the response in past artificial selection experiments, largely because we still have very limited understanding of the causes of mutational heritability, and of the genetic basis of selection response [87], [88]. Sequencing of genomes from pedigrees and from mutation accumulation lines has given good estimates of the total genomic mutation rate [89], but we do not know what fraction of these mutations have significant effects on traits, or the distribution of these effects. In a classic experiment, selection for increased oil content in maize has caused a large and continuing response after 70 generations, Laurie et al. [90] identified 50 QTL responsible for of the genetic variance in a cross between selected and control lines, implying on a map of . The effective population size here is extremely small () and so much of this response must be due to new mutations [91], so the density of sweeps is . Thus, it is unclear if Hill-Robertson interference has been important, but it would likely at least be an obstacle to attempts to increase selection response further via increasing . Burke et al. [92] have recently identified many regions (“several dozens”) that show consistent changes in allele frequencies across replicate populations of Drosophila melanogaster, selected over 600 generations for accelerated development. However, these do not show the complete loss of variation expected for a classic sweep, even though most of the response over this long timespan should be due to new mutations. This may be because the causal alleles have very small effect, and have not yet fixed – implying that the long-term rate of adaptive substitution could be very high. (Similarly, there are hardly any fixed differences between human populations on different continents, despite extensive adaptive divergence [93].) Whole-genome sequencing of selection experiments may soon give us a much better understanding of the rate at which adaptive mutations are picked up by selection. At present, however, selection experiments are inherently limited to detecting at most fifty or so sweeps over some tens of generations, and so without longer-running experiments we will not know how high the long-term rate of substitution may be.

Natural populations

To see whether Hill-Robertson interference could plausibly limit adaptation or diversity in natural populations, consider the evolution of Drosophila since the divergence between simulans and melanogaster. Taking the rate of adaptive substitutions (including those in non-coding regions) to be every two years [94] and the generation time to be roughly two weeks (Table 6.11 in [95]), we find that the per-generation rate is . The total sex-averaged map length is [96], so the density of substitutions is , well below the interference threshold. Observed levels of neutral diversity [97], [98] and per-base mutation rates [99] suggest that the (long-term) effective population sizes of Drosophila melanogaster and simulans are roughly . Taking the above estimate of , and considering the effect of the of the sweeps that Sattath et al. [100] estimate to have selective coefficients , Eq. (9) tells us that this corresponds to an actual population size of about , consistent with the estimate of [75]. This suggests that Drosophila may lie in the intermediate region illustrated in Figure 6 , in which sweeps are frequent enough to suppress neutral diversity, but not frequent enough to interfere with each other. However, the estimates of the underlying parameters are very uncertain see Sella et al.'s review [101].

The above back-of-the-envelope calculation probably understates the importance of the Hill-Robertson effect in evolution for several reasons. First, our results indicate that for many populations interference occurs primarily between tightly linked sites, so that it is the local, rather than genome-wide, density of sweeps that is constrained thus, if positively selected loci are unevenly distributed across the genome, the genomic density of substitutions will underestimate the amount of interference. Similarly, regions of the genome with low recombination rates may experience increased interference. Second, we find that the interference is mainly caused by selection driving alleles from moderately low frequencies to intermediate frequencies, with relatively little interference caused by very rare alleles reaching low frequencies or common alleles going to fixation. This means that soft sweeps, partial sweeps, and polymorphic loci undergoing fluctuating selection could contribute substantially to the Hill-Robertson effect without showing up as fixed differences between species. Third, local populations may experience a substantially higher rate of selective sweeps than indicated by the species-wide molecular clock. Most importantly, organisms that have a linear genome but do not outcross every generation, such as selfers and many viruses, are more likely candidates for experiencing Hill-Robertson interference among selected alleles than are obligate out-crossers like Drosophila. For instance [29], find that interference likely reduces the rate of adaptation of HIV in the chronic stage of infection by a factor of roughly 4.

No single effective population size

The effect of selection on surrounding genetic variation is often described as a reduction in an �tive population size.” Our results show that lumping drift and interference together in a single number in this way is generally misleading. Drift and unlinked variance in fitness dominate short-term stochasticity in allele trajectories, while the effect of linked sweeps becomes important over longer time scales (see Figure S8). This means that the �tive population size” estimated from the common, old alleles that dominate heterozygosity is likely to be very different from the relevant quantity for rare, young alleles. Thus, estimates of the strength of selection against rare alleles in, e.g., Drosophila may be systematically off by orders of magnitude. This contrast between drift dominating at short time scales and draft dominating at longer ones may also be used to estimate the amount of interference in natural populations from site frequency spectra [30], [102].

Hill-Robertson interference and the evolution of recombination

If adaptation is limited by the rate of recombination, then there should be strong selection to increase it. Barton [46] outlined the results derived here, and their implications for the debate over the maintenance of sex and recombination. Our results imply that if recombination does limit adaptation, then increasing recombination would increase fitness in proportion. However, a modifier of recombination would itself gain an advantage only to the extent that it remained associated with the favorable combinations of alleles that it helped generate. With loosely linked loci, its advantage would be of the same order as the fitness gain across one generation on a linear map, a recombination modifier would gain only from tightly linked alleles, less than map units away the net effect would seem likely to be very small [19]. Yet, recombination does increase significantly in artificially selected populations [103], and simulations of populations adapting at many loci show that selection for increased recombination can be strong [28], [104]. In addition, deleterious mutations are also likely to create Hill-Robertson interference, increasing selection for recombination [18], [76]. An analytical description of the evolution of modifiers of recombination rates in populations experiencing substantial genome-wide interference remains to be found.


Natural models for evolution on networks ☆

Evolutionary dynamics has been traditionally studied in the context of homogeneous populations, mainly described by the Moran process [P. Moran, Random processes in genetics, Proceedings of the Cambridge Philosophical Society 54 (1) (1958) 60–71]. Recently, this approach has been generalized in [E. Lieberman, C. Hauert, M.A. Nowak, Evolutionary dynamics on graphs, Nature 433 (2005) 312–316] by arranging individuals on the nodes of a network (in general, directed). In this setting, the existence of directed arcs enables the simulation of extreme phenomena, where the fixation probability of a randomly placed mutant (i.e., the probability that the offspring of the mutant eventually spread over the whole population) is arbitrarily small or large. On the other hand, undirected networks (i.e., undirected graphs) seem to have a smoother behavior, and thus it is more challenging to find suppressors/amplifiers of selection, that is, graphs with smaller/greater fixation probability than the complete graph (i.e., the homogeneous population). In this paper we focus on undirected graphs. We present the first class of undirected graphs which act as suppressors of selection, by achieving a fixation probability that is at most one half of that of the complete graph, as the number of vertices increases. Moreover, we provide some generic upper and lower bounds for the fixation probability of general undirected graphs. As our main contribution, we introduce the natural alternative of the model proposed in [E. Lieberman, C. Hauert, M.A. Nowak, Evolutionary dynamics on graphs, Nature 433 (2005) 312–316]. In our new evolutionary model, all individuals interact simultaneously and the result is a compromise between aggressive and non-aggressive individuals. We prove that our new model of mutual influences admits a potential function, which guarantees the convergence of the system for any graph topology and any initial fitness vector of the individuals. Furthermore, we prove fast convergence to the stable state for the case of the complete graph, as well as we provide almost tight bounds on the limit fitness of the individuals. Apart from being important on its own, this new evolutionary model appears to be useful also in the abstract modeling of control mechanisms over invading populations in networks. We demonstrate this by introducing and analyzing two alternative control approaches, for which we bound the time needed to stabilize to the “healthy” state of the system.


Abstract

Overdominance, or a fitness advantage of a heterozygote over both homozygotes, can occur commonly with adaptation to a new optimum phenotype. We model how such overdominant polymorphisms can reduce the evolvability of diploid populations, uncovering a novel form of epistatic constraint on adaptation. The fitness load caused by overdominant polymorphisms can most readily be ameliorated by evolution at tightly linked loci therefore, traits controlled by multiple loosely linked loci are predicted to be strongly constrained. The degree of constraint is also sensitive to the shape of the relationship between phenotype and fitness, and the constraint caused by overdominance can be strong enough to overcome the effects of clonal interference on the rate of adaptation for a trait. These results point to novel influences on evolvability that are specific to diploids and interact with genetic architecture, and they predict a source of stochastic variability in eukaryotic evolution experiments or cases of rapid evolution in nature.


APPENDIX

Assuming weak selection and a fluctuating population size, the average probability of fixation of a beneficial mutation is ∼ (E wens 1967 K imura and O hta 1974 O tto and W hitlock 1997). Here, the arithmetic average population size is , and the “effective” population size is Ne, whose calculation depends on the nature of the population fluctuations (O tto and W hitlock 1997 W ahl et al. 2002). Unfortunately, we lack an analytical expression for the fixation probability when selection is strong and population size varies. We conjecture that an adequate approximation for the average fixation probability under strong selection is given by , which is nearly when selection is weak but has the advantage of remaining <1 when selection is strong. This approximation is equivalent to the one used when the population size is constant i.e., (see Figure 1). This functional form is also suggested by diffusion analysis in populations of large effective size (K imura 1957, 1964), which assumes weak selection. Simulations confirm that provides a satisfactory approximation for the fixation probability over a range of parameter values in populations undergoing repeated bottlenecks (within a factor of two supplemental Figure 3 at http://www.genetics.org/supplemental/).

We next consider the time to fixation of a beneficial mutation. If the mutation arises when the population size is and fixes when the population size is , a deterministic model of selection can again be used to predict that Mutations are more likely to arise when the population size is large, but they are more likely to fix when the population size is small. Averaging the time to fixation over all possible events requires precise knowledge of the fluctuations in population size and the strength of selection. Assuming that mutations arise and fix uniformly over time, however, provides a generic approximation for the time to fixation, (A1) where is the geometric mean population size over time. In Equation A1, τ represents the period of the population size cycle if population size changes cyclically. If not, Equation A1 is evaluated by taking the limit as τ goes to infinity. Simulations indicate that provides a satisfactory approximation for the average time to fixation over a range of parameter values in populations undergoing repeated bottlenecks (within a factor of two supplemental Figure 4 at http://www.genetics.org/supplemental/).

To account for clonal interference, we should determine the expected number of mutations that compete for fixation when the focal mutation appears at time t (see Equation 5) and then average over all possible times at which the focal mutation could arise. To do so exactly requires a precise description of the manner in which the population size fluctuates. As a first-order approximation, we estimate the number of competing mutations using (A2) This approximation ignores the covariance between the number of contending mutations and the time to fixation of a focal mutation, which should be generated by the fluctuations in population size.

Using Equation A2 to rederive Equation 7, the cdf among fixed beneficial mutations becomes (A3) where κ is again given by Equation 8 and the average probability of fixation across the distribution of new mutations is now (A4) The corresponding probability density function for fixed mutations is .

We assessed the accuracy of Equation A3 against simulations of a population whose size cycles from N0 to 2 7 N0 via seven doubling events followed by a 1/2 7 serial dilution. In these simulations, the growth of the population was assumed to be deterministic (no sampling except during the dilution or “bottleneck” generation), and births occurred at a rate proportional to the fitness of an individual. Under this scenario, the size of the bottleneck, N0, and the period of the cycle, τ, determine (W ahl et al. 2002), , and for use in Equation A3. Every combination of the following parameters was explored: selection coefficients (σ = 0.01, 0.1, 1, 2, and 10), beneficial mutation rate (μ = 10 −7 and 10 −9 ), and initial population size (N0 = 10 5 , 10 6 , and 10 7 ), assuming that the fitness effects of new mutants were exponential (cv = 1).

Figure A1 indicates that Equation A3 accurately predicts the distribution of fixed selective effects across this range of parameters. Interestingly, Equation 2 of R ozen et al. (2002) provides a more accurate prediction of the distribution of fixed beneficial mutations with a fluctuating population size (with in place of N) than with a constant population size (Figure 2). The improved performance of their method is due to the fact that the fixation probability used, , remains reasonably accurate even when selection is strong ( ) because of the reduction in effective population size caused by the fluctuations ( ).

Median selection coefficient of fixed beneficial mutations estimated from numerical simulations vs. analytical results. Median estimated s is given from the results of our fluctuating population size model, given by H(s) in Equation A3, or from those of a model assuming weak selection given by Equation 2 in R ozen et al. (2002), but replacing N with Ne from W ahl et al. (2002). The horizontal axis measures the number of mutations that appear within the population over the average time to fixation, , for a new mutation with selection coefficient given by the median observed s.


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