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- Models and methods
The evolution of segregation distortion is governed by the interplay of selection at different levels. Despite their systematic advantage at the gamete level, none of the well-known segregation distorters spreads to fixation since they induce severe negative fitness effects at the individual level. In a deme-structured population, selection at the population level also plays a role. By means of a population genetical model, we analyse the various factors that determine the success of a segregation distorter in a metapopulation. Our focus is on the question of how the success of a distorter allele is affected by its segregation ratio and its fitness effects at the individual level. The analysis reveals that distorter alleles with high segregation ratios are the best invaders and reach the highest frequencies within single demes. However, the productivity of a deme harbouring a distorter with a high segregation ratio may be significantly reduced. As a consequence, an efficient distorter will be underrepresented in the migrant pool and, moreover, it may increase the probability of deme extinction. In other words, efficient distorters with high segregation ratios may well succumb to their own success. Therefore, distorters with intermediate segregation ratios may reach the highest frequency in the metapopulation as a result of the opposing forces of gamete, individual and group selection. We discuss the implications of this conclusion for the t complex of the house mouse.
- Top of page
- Models and methods
The phenomenon of segregation distortion has long fascinated evolutionary biologists because it exemplifies that selection at a lower level can lead to maladaptive features at a higher level. By disturbing Mendelian segregation in their favour, segregation distorters obtain a systematic advantage at the gamete level. In all examples of segregation distortion (reviewed by Lyttle, 1991), this advantage is counterbalanced by negative fitness effects at the individual level. In a deme-structured population, selection at the population level also plays a role ( Lewontin, 1962). In fact, demes that contain a high proportion of individuals of low fitness will typically perform less well than demes without such individuals.
The opposing forces of gamete, individual and interdeme selection will often lead to a stable polymorphism. Such a polymorphism is the result of an intricate interplay between selection at different levels, and it is difficult to ascertain the relative importance of each level. To gain a better understanding of the various forces and their interaction, we analyse, by means of a population genetical metapopulation model, the most important determinants for the success of a segregation distorter. In particular, we consider (1) the ability of a distorter to invade a wildtype deme, (2) the frequency that it reaches in a deme once established, (3) the time it is able to persist in a deme, (4) its effect on the productivity of a deme and (5) its representation in the migrant pool. The analysis of these factors will shed some light on the question of how the overall success of a segregation distorter depends on its basic characteristics.
Our model is motivated by the t complex of the house mouse, the standard example for segregation distortion in a deme-structured population ( Williams, 1966; see Silver, 1993, for a review). Certain variants at this gene complex, the so-called t haplotypes, strongly distort segregation in their favour in males heterozygous for the wildtype and a t haplotype. Segregation ratios as high as 0.95 are not uncommon. However, there is considerable variation in the degree of distortion ( Petras, 1967; Bennett et al., 1983 ; Gummere et al., 1986 ; Lenington et al., 1988 ; Ardlie & Silver, 1996). At the individual level, t haplotypes have severe negative fitness effects. In homozygous condition, all t haplotypes induce male sterility (at least all ‘complete’t haplotypes; Lyon, 1991; Johnson et al., 1995 ). Moreover, many t haplotypes carry recessive lethals, leading to the death and abortion of both female and male embryos. Up to now no less than 16 different recessive lethals, located at different positions within the t complex, have been found ( Klein et al., 1984 ). Evolution at the t complex is probably also affected by interdeme selection ( Lewontin, 1962). In fact, house mouse populations are generally thought to be subdivided into small, relatively isolated breeding clusters (e.g. Singleton & Hay, 1983; Lidicker & Patton, 1987), and t-bearing demes may have a much higher probability of extinction.
A large number of models have been analysed to study the evolutionary dynamics of t haplotypes, both in the context of a large well-mixed population (e.g. Bruck, 1957; Dunn & Levene, 1961) and in the context of a deme-structured metapopulation (e.g. Lewontin & Dunn, 1960; Lewontin, 1962; Levin et al., 1969 ; Petras & Topping, 1983; Nunney & Baker, 1993; Durand et al., 1997 ). The deterministic models for large, well-mixed populations generally predict unrealistically high distorter frequencies (but see Petras, 1967; Lewontin, 1968). Most metapopulation models have focused on this problem: they consider one specific distorter allele and address the question of whether a satisfying fit between empirical estimates and theoretical expectation is obtained if population structure is taken into account. We are less interested in such realistic predictions for a specific distorter allele. Instead, we consider a broad spectrum of distorter alleles which we compare with respect to their frequency in the metapopulation. This allows us, in the spirit of Lewontin’s (1962) seminal paper, to investigate how the success of a distorter is affected by its segregation ratio and its fitness effects.
We present a metapopulation model which captures the most important features of sex-specific segregation distortion in a deme-structured population. First, we show how the frequency of a distorter allele depends on its basic characteristics (i.e. segregation ratio, fitness effects), and on structural aspects of the population (e.g. deme size, migration rate, sex ratio). We then try to gain a better understanding of the factors that determine the success of a distorter (invasion efficiency, typical frequency, persistence ability, deme productivity, migrant production). To this end, we analyse a simple Markov model, which describes the dynamics of segregation distortion in a single deme. Compared to the individual-based simulation model, the state-based Markov model has the advantage of analytical tractability.
Let us stress from the beginning that we strive more for conceptual clarification than for the most adequate representation of a particular system. Although the model structure and parameters resemble the situation at the t complex, we have not tried to capture all aspects of house mouse populations in the most realistic way. In our opinion, a realistic model is hardly achievable since population structure is not known in sufficient detail, and probably varies considerably. We show, however, that our conclusions are rather robust and that they also apply to Nunney & Baker’s (1993) model that was specifically tailored to the t complex.
Models and methods
- Top of page
- Models and methods
Motivated by the t complex of the house mouse and other prominent examples of segregation distortion, we consider segregation distorters which have strong negative fitness effects in homozygous condition, and which distort Mendelian segregation in heterozygous males. More specifically, we focus on segregation distorters which lead to sterility of males homozygous for the distorter allele (‘sterile’ distorters) and on distorter alleles that induce lethality of both female and male zygotes when homozygous (‘lethal’ distorters). A single autosomal gene locus is considered with a wildtype allele and a distorter allele t. The segregation ratio or fraction of distorter alleles contributed by heterozygous +t males is denoted by σ. Hence, σ = 0.50 means that segregation is Mendelian. We will first present our basic simulation model which describes the dynamics of segregation distortion in a deme-structured population. To analyse the dynamics of segregation distortion in small, isolated populations (i.e. in the absence of migration), we also consider a Markov model. This model closely resembles the simulation model for the case of no migration.
The simulation model
Consider a metapopulation which consists of a large number of demes that are connected by migration. Within each deme there is a fixed maximum number of N♀ adult females and N♂ adult males. Generations are nonoverlapping, mating occurs at random, and each female is able to produce a maximum of z zygotes. As a result, a deme may produce N♀z offspring per generation. However, the actual number of offspring produced will be lower if a deme contains sterile males, or if zygotes happen to be homozygous for a lethal distorter allele. Per female gamete, a male gamete is chosen from a randomly chosen male. In case of a sterile distorter, we assume that no zygote is produced if the male is sterile. Hence, the reduction in the actual number of zygotes produced is proportional to the number of sterile males. In case of a lethal distorter, no zygote is produced if both the female and the male contribute a gamete carrying a lethal distorter allele. The sex of the offspring is determined at random: with probability ½ it is a female, and with probability ½ it is a male. From the offspring a number of N♀ females and N♂ males is chosen at random to make up the next generation of adults. The supernumerary offspring that do not succeed in acquiring a position in their local deme enter a common pool of migrants. In case that less than N♀ female offspring or less than N male offspring are produced, deme size is reduced accordingly, and such a deme does not produce female or male migrants.
Migration operates via the replacement of deme members by randomly chosen individuals from the migrant pool. With probability m, a resident individual is replaced by a migrant. Moreover, if a deme does not contain any female or male, an individual from the missing sex is added from the migrant pool. In this way, recolonization by a founding individual from the missing sex prevents extinction of a deme.
The metapopulation model was analysed by means of computer simulations. In all simulations the number of demes was n = 1000, so that the effects of population-wide genetic drift are minimized. We focused on metapopulations with small deme size (N = 6), intermediate deme size (N = 10) and large deme size (N = 20), where N=N♀+N♂. Throughout the sex ratio is 1:1, unless otherwise stated. The number of offspring per female in the absence of segregation distortion is set at z = 6. This corresponds, roughly, to empirical estimates for the house mouse ( Pelikan, 1981; Sage, 1981). Immigration rates varied from m = 0.025 to m = 0.1. All simulations shown here were started with one randomly assigned copy of the distorter allele per deme. However, the initial composition of the metapopulation did not seem to affect the results, since regardless of the initial composition populations quickly (within 200 generations, say) reached a characteristic composition. All simulations were run for 1000 generations. The frequency of the distorter allele in the migrants, and the distorter frequency in the metapopulation were determined by taking the metapopulation average over the last 500 generations.
The Markov model
In the Markov model we consider a single deme with a fixed number of N♀ adult females and N♂ adult males. The unordered genotype frequencies at the adult stage are denoted by P♀++, P♀+t and Ptt♀ in the females, and P++♂, P+t♂ and Ptt♂ in the males. Let us first consider gt♀ and gt♂, the relative frequencies of t-bearing gametes produced by females and males, respectively. In case of a lethal distorter, tt adults are absent (Ptt♀=Ptt♂ =0). As a consequence, the gamete frequencies are given by
In case of a sterile distorter, homozygous tt individuals are viable, but the males are sterile. Hence a fraction Ptt♂ of males does not contribute gametes to the next generation. Therefore, the relative gamete frequencies are given by
To avoid unnecessary complexity, we assume random union of gametes rather than random mating. Hence, the probabilities of forming a ++, +t, or tt zygote are given by
where g+♀ = 1−gt♀ and g+♂ =1−gt♂ are the relative frequencies of wildtype gametes. To make up the new generation of adults, N♀females and N♂males are formed by randomly assigning genotypes according to the probability distribution specified by eqs (1) and (2).
In the Markov model, extinction of a population is modelled as follows. The females of a deme are able to produce a number of N♀z potential zygotes, where z is a fixed fertility parameter. If ϕ denotes the probability that a female gamete is fertilized and viable, Nz = N♀zϕ viable zygotes will be produced on average. We assume that a population will go extinct if the number of viable zygotes is smaller than a certain critical number N0. The probability of population extinction is therefore
Typically, we have taken N0 = N♀ + N♂, which means that extinction occurs if not enough viable zygotes are formed to make up a next generation of N♀ + N♂ individuals.
The probability ϕ that a female gamete is fertilized and that the resulting zygote is viable depends on the availability of fertile males in case of a sterile distorter and on the production of viable zygotes in case of a lethal distorter. For a sterile distorter we assume, in accordance with the metapopulation model, that each female mates with a single, randomly chosen male. Hence, ϕ should be proportional to the fraction of fertile males:
The consequences of this assumption will be discussed below. In case of a lethal distorter, all zygotes are fertilized and ϕ is given by the probability that a zygote is viable, i.e. that it is not of the lethal genotype tt:
For any choice of ϕ, the Markov model can now easily be specified: the states of the model are given by all possible distributions of the three genotypes over N♀ females and N♂ males plus one state corresponding to extinction. The number of states may be quite large, and is given by 1/4(N♀+1)(N♂+1)(N♂+2)+1. For a given state, the corresponding genotype distributions P♀.. and P♂.. are readily calculated. The probability distribution over the states in the next generation is then given by eqs (1) and (2) in case of deme survival, while the probability of deme extinction is given by eqs (3) and (4). This process can be characterized by a matrix M of transition probabilities, where mij denotes the probability that a population in state j enters state i in the next generation (e.g. Kemeny & Snell, 1960).
An advantage of the Markov model over the metapopulation model is that the expected fate of an allele can be obtained directly and analytically from the transition matrix M (see below). On the other hand, some simplifying assumptions had to be made. For instance, extinction is modelled as an all or nothing event, rather than being the result of gradual population decline. Accordingly, the genetic composition of a population has no effect on population size until a certain threshold is crossed. Furthermore, we assumed random union of gametes instead of random mating. However, these discrepancies between the Markov model and the metapopulation model are apparently of marginal importance. With respect to all but one of the measures considered in this study, the Markov model produces virtually identical results as the metapopulation model in the absence of migration (m = 0). Only with respect to persistence ability there are some quantitative (but no qualitative) differences between both models, as will be discussed below. Regardless, our simplifying assumptions do not really pose a problem since the Markov model mainly serves a conceptual purpose.
Analysis of the Markov model
We use the Markov model to investigate various determinants of the success of a distorter allele within an isolated deme. In particular, we consider the invasion success of a rare distorter allele in a wildtype deme, the persistence ability of a distorter allele once established in a deme, the typical frequency that an established distorter reaches, and the productivity of a deme as a function of its distorter frequency.
The success of a segregation distorter in a metapopulation depends crucially on its ability to become established in a wildtype deme. Let us operationally define the invasion efficiency of a rare distorter by the probability that it ever reaches a certain critical frequency pcrit. This probability was obtained from the Markov model in the following manner: starting with a given initial genotype distribution, the Markov process is iterated, thereby killing those paths where the distorter has reached the critical frequency. An alternative method is to replace those states of the Markov model that correspond to distorter frequencies equal or above the critical frequency by a single absorbing state. The invasion efficiency then corresponds to the probability that the absorbing state is reached and can be obtained directly from the fundamental matrix of the corresponding reduced Markov model (e.g. Kemeny & Snell, 1960).
The time that a distorter is able to persist in a deme is another important determinant of its success. Quite generally, the rate at which polymorphism is lost will be higher in a small deme than in a large deme. In order to compensate for this general effect, we will compare the rate at which a distorter allele is lost with the rate at which a neutral allele is lost in a deme of size N=N♀ + N♂. To this end, we apply Robertson’s (1962) concept of a ‘retardation factor’. In our context, the retardation factor is defined as
where 1 – λn and 1 – λs represent the rate at which polymorphism is lost in case of neutral and distorter alleles, respectively.
Technically, λn and λs correspond to the largest nonunit eigenvalue of the Markov chains describing genetic drift acting on a neutral and a distorter locus, respectively (e.g. Gale, 1990). In case of segregation distortion, λs can be calculated numerically from the Markov matrix M. In case of neutral alleles, it is well known that in a haploid, asexual population of size 2N polymorphism is lost at a rate λn = 1 – [1/(2N)]. For a dioecious population consisting of N♀ females and ♂N␣males, λn is slightly larger ( Li, 1976; p.337) and given by
In an infinite population, a sterile or lethal distorter reaches a stable equilibrium where its frequency is positively related to its segregation ratio (see eqn 7 below). In a finite population, a stable polymorphic equilibrium does not exist since the ultimate fate of every deme is either extinction or fixation of the wildtype allele. We may, however, focus on the frequency that a distorter ‘typically’ reaches in a polymorphic deme. The typical frequency is of importance since it determines the productivity of a deme and the representation of the distorter allele in the migrants. This frequency is determined from the eigenvector of genotype frequencies that belongs to λs, the first nonunit eigenvalue of the Markov matrix M (e.g. Gale, 1990).
The Markov model was also used to determine the productivity of a deme as a function of the distorter frequency. A deme without any distorter allele produces exactly N♀z offspring, of which N = N♀+N♂ stay in their natal deme. Hence, such a deme produces an excess of N♀z−N migrant individuals. In case of segregation distortion, the number of viable zygotes is reduced by a fraction ϕ (see eqn 4), and the number of emigrants is reduced accordingly. To determine the number of emigrants produced by a deme with a certain distorter frequency, we calculated the expected number of migrants for all possible genotype combinations, and weighted the output of emigrants by their probability of occurrence (given by the eigenvector corresponding to the first nonunit eigenvalue λs).