A general reduction principle for genetic modifiers of recombination

Evolutionarily stable strategy analysis and its links to demography and genetics through invasion fitness

Evolutionarily stable strategy (ESS) analysis pioneered by Maynard Smith and Price took off in part because it often does not require explicit assumptions about the genetics and demography of a population in contrast to population genetic models. Though this simplicity is useful, it obscures the degree to which ESS analysis applies to populations with more realistic genetics and demography: for example, how does ESS analysis handle complexities such as kin selection, group selection and variable environments when phenotypes are affected by multiple genes? In this paper, I review the history of the ESS concept and show how early uncertainty about the method lead to important mathematical theory linking ESS analysis to general population genetic models. I use this theory to emphasize the link between ESS analysis and the concept of invasion fitness. I give examples of how invasion fitness can measure kin selection, group selection and the evolution of linked modifier genes in response to variable environments. The ESSs in these examples depend crucially on demographic and genetic parameters, which highlights how ESS analysis will continue to be an important tool in understanding evolutionary patterns as new models address the increasing abundance of genetic and long-term demographic data in natural populations. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

The gossip paradox: Why do bacteria share genes?

Bacteria, in contrast to eukaryotic cells, contain two types of genes: chromosomal genes that are fixed to the cell, and plasmids, smaller loops of DNA capable of being passed from one cell to another. The sharing of plasmid genes between individual bacteria and between bacterial lineages has contributed vastly to bacterial evolution, allowing specialized traits to 'jump ship' between one lineage or species and the next. The benefits of this generosity from the point of view of both recipient cell and plasmid are generally understood: plasmids receive new hosts and ride out selective sweeps across the population, recipient cells gain new traits (such as antibiotic resistance). Explaining this behavior from the point of view of donor cells is substantially more difficult. Donor cells pay a fitness cost in order to share plasmids, and run the risk of sharing advantageous genes with their competition and rendering their own lineage redundant, while seemingly receiving no benefit in return. Using both compartment based models and agent based simulations we demonstrate that 'secretive' genes which restrict horizontal gene transfer are favored over a wide range of models and parameter values, even when sharing carries no direct cost. 'Generous' chromosomal genes which are more permissive of plasmid transfer are found to have neutral fitness at best, and are generally disfavored by selection. Our findings lead to a peculiar paradox: given the obvious benefits of keeping secrets, why do bacteria share information so freely?

Non-vertical cultural transmission, assortment and the evolution of cooperation

Cultural evolution of cooperation under vertical and non-vertical cultural transmission is studied, and conditions are found for fixation and coexistence of cooperation and defection. The evolution of cooperation is facilitated by its horizontal transmission and by an association between social interactions and horizontal transmission. The effect of oblique transmission depends on the horizontal transmission bias. Stable polymorphism of cooperation and defection can occur, and when it does, reduced association between social interactions and horizontal transmission evolves, which leads to a decreased frequency of cooperation and lower population mean fitness. The deterministic conditions are compared to outcomes of stochastic simulations of structured populations. Parallels are drawn with Hamilton’s rule incorporating relatedness and assortment.

Evolution of transmission modifiers under frequency-dependent selection and transmission in constant or fluctuating environments

Although the Reduction Principle for rates of mutation, migration, and recombination has been proved for large populations under constant selection, the fate of modifiers of these evolutionary forces under frequency-dependent or fluctuating selection is, in general, less well understood. Here we study modifiers of transmission, which include modifiers of mutation and oblique cultural transmission, under frequency-dependent and cyclically fluctuating selection, and develop models for which the Reduction Principle fails. We show that whether increased rates of transmission can evolve from an equilibrium at which there is zero transmission (for example, no mutation) depends on the number of alleles among which transmission is occurring. In addition, properties of the null-transmission state are clarified.

Unified reduction principle for the evolution of mutation, migration, and recombination

Modifier-gene models for the evolution of genetic information transmission between generations of organisms exhibit the reduction principle: Selection favors reduction in the rate of variation production in populations near equilibrium under a balance of constant viability selection and variation production. Whereas this outcome has been proven for a variety of genetic models, it has not been proven in general for multiallelic genetic models of mutation, migration, and recombination modification with arbitrary linkage between the modifier and major genes under viability selection. We show that the reduction principle holds for all of these cases by developing a unifying mathematical framework that characterizes all of these evolutionary models.

Evolution of reduced mutation under frequency-dependent selection

Most models for the evolution of mutation under frequency-dependent selection involve some form of host-parasite interaction. These generally involve cyclic dynamics under which mutation may increase. Here we show that the reduction principle for the evolution of mutation, which is generally true for frequency-independent selection, also holds under frequency-dependent selection on haploids and diploids that does not involve cyclic dynamics.

Sexual antagonism and meiotic drive cause stable linkage disequilibrium and favour reduced recombination on the X chromosome

Sexual antagonism and meiotic drive are sex-specific evolutionary forces with the potential to shape genomic architecture. Previous theory has found that pairing two sexually antagonistic loci or combining sexual antagonism with meiotic drive at linked autosomal loci augments genetic variation, produces stable linkage disequilibrium (LD) and favours reduced recombination. However, the influence of these two forces has not been examined on the X chromosome, which is thought to be enriched for sexual antagonism and meiotic drive. We investigate the evolution of the X chromosome under both sexual antagonism and meiotic drive with two models: in one, both loci experience sexual antagonism; in the other, we pair a meiotic drive locus with a sexually antagonistic locus. We find that LD arises between the two loci in both models, even when the two loci freely recombine in females and that driving haplotypes will be enriched for male-beneficial alleles, further skewing sex ratios in these populations. We introduce a new measure of LD, Dz', which accounts for population allele frequencies and is appropriate for instances where these are sex specific. Both models demonstrate that natural selection favours modifiers that reduce the recombination rate. These results inform observed patterns of congealment found on driving X chromosomes and have implications for patterns of natural variation and the evolution of recombination rates on the X chromosome.

Evolution of recombination rates in a multi-locus, haploid-selection, symmetric-viability model

A fast algorithm for computing multi-locus recombination is extended to include a recombination-modifier locus. This algorithm and a linear stability analysis is used to investigate the evolution of recombination rates in a multi-locus, haploid-selection, symmetric-viability model for which stable equilibria have recently been determined. When the starting equilibrium is symmetric with two selected loci, we show analytically that modifier alleles that reduce recombination always invade. When the starting equilibrium is monomorphic, and there is a fixed nonzero recombination rate between the modifier locus and the selected loci, we determine analytical conditions for which a modifier allele can invade. In particular, we show that a gap exists between the recombination rates of modifiers that can invade and the recombination rate that specifies the lower stability boundary of the monomorphic equilibrium. A numerical investigation shows that a similar gap exists in a weakened form when the starting equilibrium is fully polymorphic but asymmetric.

The evolutionary reduction principle for linear variation in genetic transmission

The evolution of genetic systems has been analyzed through the use of modifier gene models, in which a neutral gene is posited to control the transmission of other genes under selection. Analysis of modifier gene models has found the manifestations of an "evolutionary reduction principle": in a population near equilibrium, a new modifier allele that scales equally all transition probabilities between different genotypes under selection can invade if and only if it reduces the transition probabilities. Analytical results on the reduction principle have always required some set of constraints for tractability: limitations to one or two selected loci, two alleles per locus, specific selection regimes or weak selection, specific genetic processes being modified, extreme or infinitesimal effects of the modifier allele, or tight linkage between modifier and selected loci. Here, I prove the reduction principle in the absence of any of these constraints, confirming a twenty-year-old conjecture. The proof is obtained by a wider application of Karlin's Theorem 5.2 (Karlin in Evolutionary biology, vol. 14, pp. 61-204, Plenum, New York, 1982) and its extension to ML-matrices, substochastic matrices, and reducible matrices.

Exact phase diagram of a quasispecies model with a mutation rate modifier

We consider an infinite asexual population with a mutator allele which can elevate mutation rates. With probability f, a transition from nonmutator to mutator state occurs but the reverse transition is forbidden. We find that at f=0, the population is in the state with minimum mutation rate, and at f=fc, a phase transition occurs between a mixed phase with both nonmutators and mutators and a pure mutator phase. We calculate the critical probability fc and the total mutator fraction Q in the mixed phase exactly. Our predictions for Q are in agreement with those seen in microbial populations in static environments.

Short- and long-term benefits and detriments to recombination under antagonistic coevolution

We explored the evolution of recombination under antagonistic coevolution, concentrating on the equilibrium frequencies of modifier alleles causing recombination in initially nonrecombining populations. We found that the equilibrium level of recombination in the host depended not only on parasite virulence, but also on the strength of the modifier allele, and on whether or not the modifier was physically linked to the parasite interaction loci. Nonetheless, the maximum level of recombination for linked loci at equilibrium was about 0.3 (60% of free recombination) for interactions with highly virulent parasites; the level decreased for unlinked modifiers, and for lower levels of parasite virulence. We conclude that recombination spreads because it provides a combination of an immediate (next-generation) fitness benefit and a delayed (two or more generations) increase in the rate of response to directional selection. The relative impact of these two mechanisms depends on the virulence of parasites early in the spread of the modifier, but a trade-off between the two dictates the equilibrium modifier frequency for all nonzero virulences that we examined. In addition, population mean fitness was higher in populations at intermediate equilibria than populations fixed for free recombination or no recombination. The difference, however, was not enough on its own to overcome the two-fold cost of producing males.

On the evolution of epistasis II: a generalized Wright-Kimura framework

The evolution of fitness interactions between genes at two major loci is studied where the alleles at a third locus modify the epistatic interaction between the two major loci. The epistasis is defined by a parameter epsilon and a matrix structure that specifies the nature of the interactions. When epsilon=0 the two major loci have additive fitnesses, and when these are symmetric the interaction matrices studied here produce symmetric viabilities of the Wright [1952. The genetics of quantitative variability. In: Reeve, E.C.R., Waddington, C.H. (Eds.), Quantitative Inheritance. Her Majesty's Stationary Office, London]-Kimura [1956. A model of a genetic system which leads to closer linkage by natural selection. Evolution 10, 278-281] form. Two such interaction matrices are studied, for one of which epistasis as measured by |epsilon| always increases, and for the other it increases when the linkage between the major loci is tight enough and there is initial linkage disequilibrium. Increase of epistasis does not necessarily coincide with increase in equilibrium mean fitness.

On the evolution of epistasis I: diploids under selection

One interpretation of recent literature on the evolution of phenotypic modularity is that evolution should act to decrease the degree of interaction between genes that contribute to different phenotypes. This issue is addressed directly here using a fitness scheme determined by two genetic loci and a third locus which modifies a measure of statistical interaction between the fitnesses due to the first two. The equilibrium structure of such an epistasis-modifying locus is studied. It is shown that under well-specified conditions a modifying allele that increases epistasis succeeds. In other words, genetic interactions tend to become stronger. It is speculated that this occurs because the mean fitness in such models is locally increasing as a function of the degree of epistasis.

The Optimum Recombination Rate That Realizes the Fastest Evolution of a Novel Functional Combination of Many Genes

The effect of genetic recombination (or crossover) by sexual reproduction on the time until a novel set of genes performing a combined function appears, spreads, and becomes fixed is studied. First, we study a haploid finite population with many binary loci, in which only one sequence (called a functional gene set) is significantly advantageous over the others. The time for evolution of the function (Td) is defined as the mean number of generations until the advantageous sequence dominates in an initially random population. When the sequence diversity is initially stored sufficiently, the evolution time Td is roughly the product of the waiting time until the appearance of the advantageous sequence (creation time Tc) and the average number of appearances of the advantageous sequence from its absence until its fixation (destruction number Nd). Mutation and crossover reduce the former but enlarge the latter. If the mutation rate is low, there is an intermediate optimal rate of crossover that achieves the minimum Td. In contrast, if the mutation rate is sufficiently high, Td is smallest without crossover. Second, the breakdown of established functions by recurrent deleterious mutation in an infinite population is examined. The number of functional genes maintained decreases monotonically with the recurrent deleterious mutation rate. Thus in higher organisms having many functional sets of genes in the genome, the mutation rate must be kept very low to preserve them, and hence a high crossover rate made possible by sexual reproduction is important in accelerating the evolution of novel functional sets of genes. Implications of this long-term advantage of recombination in the maintenance of sexual reproduction in higher organisms are discussed.

Population genetic perspectives on the evolution of recombination

Optimality arguments and modifier theory are reviewed as paradigms for the study of the evolution of recombination. Optimality criteria (such as maximization of mean fitness) may agree with results from models developed in terms of the evolution of recombination at modifier loci. Modifier models demonstrate, however, that equilibrium mean fitness can decrease during the evolution of recombination rates and is not always maximized. Therefore, optimality arguments do not successfully predict the conditions under which increased or decreased recombination will evolve. The results from modifier models indicate that decreased recombination rates are usually favored when the population is initially near a polymorphic equilibrium with linkage disequilibrium. When the population is subject to directional selection or to deleterious mutations, increased recombination may be favored under certain conditions, provided that there is negative epistasis among alleles.

On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution

Eighteen different terms, currently employed to define various concepts of evolutionary stability in population dynamics are mentioned in this paper. Most of these terms are used in different connotations and even different meanings by different authors. On the other hand, different terms are often employed by different authors to define quite the same concept. Twenty-five years ago there was only one, well-defined, concept of stability, universally recognized in the field. In this paper I will try to relate the recent confusion, concerning concepts of population stability, with a more serious, though not that well-recognized, confusion in the modern analytic approach to population dynamics and quantitative evolution. Concepts of population stability will be examined in relation to each other on the one hand and, on the other hand, in relation to two dichotomies regarding the dynamic processes to which they correspond: Short-term versus long-term processes and processes concerning phenotypic changes versus process concerning genotypic changes. A hopefully more consistent use of the current terminology is suggested.

On the modification of recombination with sex-dependent fitnesses and linkage

According to the Reduction Principle, when a recombination-reducing allele is introduced near an equilibrium that depends on recombination, that allele will increase in frequency. If the allele increases the recombination rate, it will be expelled from the population. There are known cases where this principle fails. In this respect, an interesting question is what kind of two-sex viability regimes support a general Reduction Principle. In this paper, we construct a model of viabilities, due to two autosomal linked genes, which differ between the sexes, such that recombination is different in the sexes. A complete analysis is provided for the case where recombination is absent in one sex. It is proved that the Reduction Principle is still valid for recombination in the other sex.

Evolution of recombination among multiple selected loci: a generalized reduction principle

Conditions for invasion by a new allele that controls the recombination pattern among an arbitrary number of genes under viability selection are studied. The recombination pattern may include interference. The new allele increases if its appropriately averaged marginal fitness is greater than the mean fitness prior to its introduction. Under weak additive-by-additive epistatic selection, this condition involves a weighted average of the changes in pairwise recombination rates relative to those prior to the introduction of the modifier. The weights here are positive functions of the epistatic selection components. In particular, the modifier allele may succeed even if it increases recombination among some pairs of loci, provided the overall average effect is one of reduction.

More on recombination and selection in the modifier theory of sex-ratio distortion

G. Maffi and S.D. Jayakar suggested a model for the two-locus control of sex determination in the mosquito Aedes aegypti (1981, Theor. Pop. Biol. 19, 19-36). This model was extended to multiple alleles and analyzed in mathematical detail by S. Lessard (1987, Theor. Pop. Biol. 31, 339-358). The model supposes that males are "Mm" and females "mm" but the transmission from males is controlled by a second gene with alleles Ai. We show that in addition to the equilibrium in which mAi in females, MAi from males and mAi from males all have the same frequencies, a second class of polymorphic equilibria exists and can be stable. The former class was shown by Lessard to be stable for intermediate and/or loose linkage. The new class of equilibria may be stable for tight linkage under the conditions that preclude stability of the former. We also develop the theory of linkage modification from the neighborhood of the new equilibrium. Successful modifiers of recombination may either reduce or increase the recombination fraction with the outcome depending on the linkage of the modifier to the major genes.