genetic drift in populations of distorting gene complexesdurand/papers/durandgenetics97.pdfthe...

Genetic Drift in Populations of Distorting Gene Complexes

Dannie Durand,∗ Eric Bendix,† Kristin Ardlie,‡ Warren Ewens.§ Lee Silver,¶

August 7, 2003

∗Department of Biological Sciences, Carnegie Mellon University, Pittsburgh, PA 15213†University of California, Berkeley, CA‡Genomics Collaborative, Cambridge, MA 02139§Computational Biology Group, University of Pennsylvania, Philadelphia, PA 19104-6228¶Department of Molecular Biology, Princeton University Princeton, NJ 08544

1

Keywords: meiotic drive, segregation distortion, genetic drift, selfish chromosomes.Running Head: Evolution of Segration Distortion

Contact Author:

Dannie DurandDepartment of Molecular BiologyPrinceton UniversityPrinceton, NJ 08544tel: 609-258-6484 or 6874fax: [email protected]

2

Abstract

We use stochastic population models to study the evolution of distorting gene complexes.Distorting gene complexes, or distorters, are chromosomal regions characterized by meiotic drive:a heterozygote bearing the distorter passes it to more than 50% of offspring. Homozygotes bear-ing the distorting complex are sterile. Although distorters promote themselves at the expenseof other genes in the same genome, they have been successful in evolution: distorting genecomplexes have been observed in animal, plant and fungal species. While the molecular drivemechanisms differ, distorting gene complexes exhibit similar genetic features, suggesting thatthe population genetic mechanisms that allow such gene complexes to evolve may be shared.We investigate such possible mechanisms in the current work.

We present Markov models and Monte Carlo simulations of genetic drift in populations ofdistorting gene complexes. Genetic drift, the stochastic behavior of allele frequencies in smallpopulations, is a fundamental force in the process of evolution, yet the role of genetic drift in theevolution of distorters is not well understood. Our analysis shows how distorters evolve differentcharacteristics in species typified by small and large populations, respectively. We apply ourresults to two well studied distoring gene complexes: the t-haplotype in Mus musculus andSegregation Distorter in Drosophila melanogaster.

1 Introduction

Distorting gene complexes, or distorters, are chromosomal regions that “enhance their own trans-mission relative to the rest of an individual’s genome” [Werren et al., 1988] and “actively interferewith the functions of other genes in the same nucleus” [Wu and Hammer, 1991]. These com-plexes are characterized by segregation distortion, also referred to as transmission ratio distortionor TRD1. TRD occurs when a heterozygote bearing the distorter passes it to more than 50%of offspring at the expense of the wild type chromosome. Distorting gene complexes have beendiscovered in mice [Silver, 1985, Silver, 1993], Neurospora [Turner and Perkins, 1979], the tomato[Rick, 1971], wheat [Loegering and Sears, 1963], Podospora [Padieu and Bernet, 1967] and manyspecies of Drosophila [Atlan et al., 1997, Hartl and Hiraizuma, 1976, Jaenike, 1996], ALSO WOODLEMMINGS, BUTTERFLIES AND MOSQUITOES.

WE NEED TO DISCUSS AND CITE THE RECENT NATURE ARTICLE ON THE T-RESPONDER SOMEWHERE IN THIS PARAGRAPH. The most well studied autosomal dis-

1We prefer the term transmission ratio distortion to segregation distortion because the underly-ing molecular and cell biological processes responsible for this phenomena are not well understoodand may be acting after segration.

3

torting gene complexes are the t-haplotype in Mus musculus and Segregation Distorter or SD inDrosophila melanogaster. The t-haplotype [Silver, 1985, Silver, 1993] is a variant form of a 30megabasepair region on chromosome 17. A series of inversions in the region results in a localizedsuppression of recombination and thus allows a complete t-haplotype to be transmitted as a unitfrom one generation to the next. Chromosomes bearing the t-haplotype are preferentially transmit-ted from heterozygous (+/t) males to 90% of their offspring, on average [Ardlie and Silver, 1996,Dunn, 1957]. Homozygous t/t males are unconditionally sterile. The genetic data suggest that thesame genes involved in TRD are also responsible for the homozygous sterility [Lyon, 1986]. Accord-ing to the phylogenetic evidence, the t-haplotype evolved in a stepwise manner through the sequen-tial accumulation of four inversions [Hammer and Silver, 1993, Morita et al., 1992, Silver, 1993].Because each inversion is associated with loci involved in the TRD phenotype, TRD probably in-creased with each additional inversion, increasing slowly from Mendelian transmission to its presentlevel. These inversions were acquired over the period from three million years ago to about 100,000years ago. Despite this antiquity, sequence comparisons among modern t-haplotypes show almostno nucleotide polymorphism in comparison to wild type chromosomes, suggesting that all contem-porary t-haplotypes share a much more recent common ancestor that lived between 10,000 and100,000 years ago. In wild populations today, the t-haplotype persists as a polymorphism witht-allele frequencies of 10% to 15% [Ardlie, 1995, Ardlie and Silver, 1998, Lenington et al., 1988,Ruvinsky et al., 1991].

In addition to the negative selection conferred by homozygous male sterility, most naturally-occurring t-haplotypes are linked to mutations conferring recessive embryonic lethality. In starkcontrast to the lack of polymorphism among independent t-haplotypes, there are a large num-ber of independent recessive lethal mutations, which suggests both that the lethals are a recentaccumulation and that they confer a selective advantage on the haplotypes that carry them.

The distorting gene complex SD in Drosophila melanogaster [Hiraizumi et al., 1960, Hiraizumi and Thomas, 198Hiraizumi et al., 1994, Powers and Ganetsky, 1991, Sandler and Golic, 1985, Wu and Hammer, 1991]is also characterized by transmission ratio distortion, chromosonal inversions and infertility in ho-mozygous males. Unlike the t-haplotype, SD is rarely linked to recessive lethal mutations. Suchmutations are observed occasionally, but do not appear to persist. SD and t also differ in theirtransmission ratio: in SD the transmission ratio approaches 100%. In fact, the transmission ratioof the t-haplotype is lower than most known distorting gene complexes. Spore Killer in NeurosporaCITATION, Sex Ratio in Drosophila CITATION and the W chromosome in butterflies CITATIONare also characterized by near 100% transmission ratios. A transmission ratio greater than 99%has been observed in Male Drive (MD) in mosquito in laboratory studies CITATION. However, innature the effective sex ratio is lower due to a range of MD chromosones with varying sensitivitiesto MD in the wild CITATION.

Distorting gene complexes are an intriguing example of selection occurring at multiple levels.They represent both competition between genes within the same genome, and between the genomeand the individual that carries them. Distorting gene complexes have been observed in a number ofanimal, plant and fungal species. Although the underlying physiological mechanisms of transmissionratio distortion differ from species to species, on the genetic level several common features havebeen found in all systems that have been well studied: diploidy, multiple linked genes, transmissionratio distortion and homozygous male infertility [Lyttle, 1991, Lyttle, 1993]. This suggests that thepopulation genetic forces that allow such gene complexes to evolve and persist may be shared in

4

common.In the current paper, we study the role of genetic drift in the evolution and persistence of

distorting gene complexes. Genetic drift, the stochastic behavior of allele frequencies in smallpopulations, is a fundamental force in the process of evolution. The mathematics of genetic driftand the corresponding evolutionary implications have been studied in great detail since the earlydecades of this century (see, for example, [Ewens, 1979]). However, there have been few studiesof genetic drift in populations of distorters and the role of genetic drift in their evolution is notwell understood. Here we present an analysis of the stochastic behavior of autosomal distortersin finite, unstructured populations as a function of population size and the degree of distortion.Although our models focus on the t-haplotype, the results are quite general and can be applied toother autosonal distorters, notably SD. We use our results to propose novel solutions to two long-standing problems that have stymied biologists studying the t-haplotype for over forty years: whythe transmission ratio seen in wild mouse populations is closer to 0.9 than 1.0 and why recessive,lethal mutations are a recent, but widespread innovation in t-haplotype evolution. Our analysisalso illustrates ways in which distorters that evolved in species with small effective population sizes(e.g., the t-haplotype in mouse) differ from distorters found in large, panmictic populations (e.g.,SD in Drosophila).

In the next section, we introduce Markov models of genetic drift in populations of distorters,treating lethals and steriles independently. Monte Carlo simulations of the Markov model arepresented in Section 3. These show how the persistence of a distorter depends on distortion levelsand population size. Competition between sterile and lethal distorters in a single population areexamined in Section 4. Previous modeling efforts are reviewed in Section 5 and the results of thecurrent paper are discussed in light of this earlier work. In conclusion, the evolutionary implicationsof our work are discussed in Section 6.

2 Population Models for Distorting Gene Complexes

In our Markov models of genetic drift, two alleles are represented in a population of N diploidindividuals: a wild type (+) and a distorter (t). Although distorters are known to contain multipleloci, we will treat t as an indivisible unit since those loci are linked by inversions. The transmissionratio is represented as a parameter, τ . Heterozygous males pass t to their offspring with probabilityτ ≥ 0.5. Heterozygous females transmit t and + to their offspring with equal probability. Our modelhas two variants. In the sterile model, t/t males are sterile and t/t females have no reduction infertility. This model applies equally well to SD or to any other autosomal distorting gene complexcharacterized by TRD, recombination suppression and homozygous male sterility. In the lethalmodel, t confers recessive lethality; no t/t animals of either sex are found in the population.

2.1 The Wright-Fisher Model: non-distorting alleles in finite populations

Since our models are generalizations of the simple Wright-Fisher model, a Markov model of geneticdrift in populations of non-distorting alleles, we briefly review that model here. The simple Wright-Fisher model assumes a finite population of N diploid, monoecious individuals. A single locus withtwo possible allelic types (A and a) is considered. The random variable W (i) refers to the numberof A’s in generation i. The variation in allele frequency over time is modeled by a Markov chain

5

with 2N+1 states, where state j refers to a population with j genes of allelic type A and 2N−jgenes of type a. Two states (j = 0 and j = 2N) are absorbing, corresponding to the fixation of thea and A allelic types, respectively. The transition probability of this Markov chain is:

P (W (i+1) = k|W (i)) =(

2Nk

)(q(i))k (1− q(i))2N−k , (1)

where q(i) = (W (i)/2N) and p(i) = 1 − q(i) are the A- and a-allele frequencies in generation i,respectively. These transition probabilities are derived by assuming that the genes in the offspringgeneration are obtained by random sampling, with replacement, from the genes in the parental gen-eration. Note that a single random variable is sufficient to specify the state under this assumptionand that it is not necessary to know the distribution of genotypes in the population, so that variousproperties of interest can be determined using allele frequencies alone. In particular, the mean timeto fixation of either a or A denoted E[T ], can be approximated accurately from Equation 1 using aone-dimensional diffusion process. This procedure requires calculation of the mean m(∆q) and thevariance v(∆q) of the change, ∆q, in the frequency of the A allele from one generation to the next.The first step in the calculation of E(T ) is to solve the so-called backward Kolmogorov equation

m(∆q)dE(T )dq

+v(∆q)

2d2E(T )dq2

= −1, (2)

subject to the boundary conditions E(T ) = 0 when q = 0, q = 1. The value of E(T ) is then foundby replacing q in this solution by, q(0), the initial value of the frequency of the A allele. For themodel 1, m(∆q) = 0, v(∆q) = q(1− q)/2N , and solution of (2) yields

E[T ] ≈ −4N · (q(0) log q(0) + (1−q(0)) log (1−q(0))), (3)

where q(0) and (1−q(0)) are the initial allele frequencies of A and a, respectively. The mainconclusion to be drawn from this equation is that the mean time to fixation is only linear in thepopulation size, so that a small change in population size leads to a small change in the mean fixationtime. We will find that this conclusion no longer holds in the models of distorting complexes weconsider next.

2.2 Sterile distorters in finite populations

We now construct a Markov chain model similar in principle to that described in Equation 1, butwith further features necessary to describe the behavior of a population of sterile distorters. Asabove, we assume a population of N diploid organisms. We further assume that the number ofmales in each generation is fixed at Nm and the number of females at Nf = N−Nm. Variationover time in the frequency of the t allele can be modeled using a Markov chain generalizing thesimple Wright-Fisher model. However, in contrast to Equation 1, which is based on allele frequen-cies, in the sterile model we need to keep track of genotype frequencies, because t has a differentphenotype in t/t males (sterility) than in +/t males (TRD). As a result, more than one independentvariable is required for the complete specification of the population in generation i. We denote thenumber of males and females of each genotype in generation i by the vectors {Xm(i), Ym(i), Zm(i)}and {Xf (i), Yf (i), Zf (i)}, as described in Table 1. For notational convenience, we define the totalnumber of +/+ individuals of either sex in generation i to be X(i) = Xm(i)+Xf (i), with similar

6

definitions for Y (i) and Z(i). Since the sex ratio is fixed, the number of +/+ individuals is con-strained to be Xm(i) = Nm−Ym(i)−Zm(i) for males and Xf (i) = Nf−Yf (i)−Zf (i) for females.Thus, four independent variables {Ym(i), Zm(i), Yf (i), Zf (i)} are required to describe the state of

i Current generation.N The number of diploid individuals in the population.Nm, Nf The number of males (resp. females) in the population.X(i), Xm(i), Xf (i) The number of +/+ individuals (total, male, female) in generation i.Y (i), Ym(i), Yf (i) The number of +/t individuals (total, male, female) in generation i.Z(i), Zm(i), Zf (i) The number of t/t individuals (total, male, female) in generation i.q(i) = (Y (i) + 2Z(i))/2N t-allele frequency in generation irm(i), rf (i) Probability that a male (resp. female) in generation i will transmit t to offspring.p(t/t, i) Probability that an embryo will have genotype t/t.p(+/t, i) Probability that an embryo will have genotype +/t.p(+/+, i) Probability that an embryo will have genotype +/+.

Table 1: Notation for the sterile model.

the Markov chain in generation i.The population will either eventually consist entirely of +/+ individuals (both male and female)

or will become extinct (because males in some generation are all t/t). We wish to consider theprobabilities of these two outcomes, as well as the mean time until one outcome or the other occurs.In the Wright-Fisher model (1), the probability that a parent will transmit a particular allele to theoffspring generation is simply the allele frequency, q(·). However, in a population of distorters theallele transmission frequency is influenced by TRD and male sterility. The respective probabilitiesrm(i) (rf (i)) that a gene transmitted to generation i+1 from a male (female) of generation i is a tallele are

rm(i) =Ym(i) · τ

Xm(i) + Ym(i),

rf (i) =Yf (i)2Nf

+Zf (i)Nf

.

Thus the probabilities p(+/+), p(+/t) and p(t/t) that an embryo (male or female) in the nextgeneration will have genotype +/+, +/t and t/t, respectively, are given by

p(+/+) = (1− rm(i)) · (1− rf (i)),p(+/t) = (1− rm(i)) · rf (i)+(1− rf (i)) · rm(i),p(t/t) = rm(i) · rf (i).

The genotypes in the offspring generation are obtained by sampling from these genotype frequencies,yielding the transition probabilities

P (Ym(i+1) = ym, Zm(i+1) = zm, Yf (i+1) = yf , Zf (i+1) = zf | Ym(i), Zm(i), Yf (i), Zf (i)) =Nm!Nf !

(Nm−ym−zm)!ym!zm!(Nf−yf−zf )!yf !zf !· p(+/+)xp(+/t)yp(t/t)z (4)

7

for the sterile Markov model, where x = xm + xf , y = ym + yf and z = zm + zf . In contrast to thebinomial form of the simple Wright-Fisher model (1), this model has a multinomial form, becausemore than one random variable is required to specify the state. For the same reason, the onedimensional diffusion approximation cannot be used to obtain a tractable, analytical expression forthe mean time to loss of t, analogous to the expression for the estimated mean fixation time givenin (3). Thus, it appears difficult, if not impossible, to obtain explicit analytic results describing theoutcomes of a population of sterile distorters from the Markov chain model (4). Some simplificationto the model is obtained if the frequencies of the three respective genotypes are assumed to be thesame within males and females. However this simplification reduces the Markov chain to onedescribed by two variables, and analytic results are still difficult, if not impossible, to obtain. Wetherefore do not pursue this amended chain further. Instead, all our results obtained for this modelare found by simulation of the Markov chain (4). These are presented in Section 3. By contrast, wewill find below that the assumption of equal genotype frequencies betwen males and females doesmake the chain for the lethal distorter case amenable to theoretical calculations.

2.3 Lethal distorters in finite populations

In this section we derive Markov models for populations of lethal distorters in which both maleand female t/t embryos die before birth. A consequence of lethality is that the highest t-haplotypefrequency is 0.5 and occurs in the state where all individuals are heterozygotes. Thus, a populationof lethal distorters can be modeled by a Markov chain with only one absorbing state (where allalleles are +). Since there are no t-bearing homozygotes, two degrees of freedom are sufficient tospecify the state of a population of lethals, namely the number of male and female +/t heterozgotes.While this allows us to derive a Markov model for lethal distorters with transition probabilities thatare simpler than the expression for the sterile model given in Equation 4, it is still not possible toapply the one-dimensional diffusion approximation to obtain tractable estimates of mean fixationtimes. However, by making the simplifying assumption that the heterozygote frequency is the sameamong both males and females, we can obtain more tractable Markov models that are amenable toanalysis. We present two such models. The first, which is analogous to the Wright-Fisher model,describes the change in the genetic distribution of the population from one generation to the next.In the second model, based on the Moran model of haploid populations, the replacement of a singleindividual is the basic unit of change.

The parameters used in these models are defined in Table 2. As in the sterile case, we supposethat the number of males and females in the population is fixed over all generations, with Nf

females and Nm = N−Nf males. The number of female heterozygotes in generation i is Yf (i) andthe number of male heterozygotes is Ym(i). In the lethal case, Z(·) = 0 and the total number ofwild type homozygotes is constrained to be X(i) = N − Y (i). A further consequence of lethality isthat, unlike the sterile model, genotype frequencies do not give us more information about the statethan allele frequencies; in fact, they are interchangeable because the t-allele frequency always differsfrom the heterozygote frequency by a factor of two. We denote the frequency of +/t individuals ingeneration i by Q(i), where Q(i) = 2q(i) = Y (i)/N .

The composition of generation i+1 is found by sampling from the distribution of viable embryogenotypes obtained from the male and female gene pools in generation i. These frequencies dependon the probability of transmitting t from parent to offspring. As in the sterile case, this transmissionfrequency depends not only on the allele frequency but also on the forces of TRD and negative

8

i Current time unit.N The number of diploid individuals in the population.Nm, Nf The number of males (resp. females) in the population.X(i), Xm(i), Xf (i) The number of +/+ individuals (total, male, female) at time i.Y (i), Ym(i), Yf (i) The number of +/t individuals (total, male, female) at time i.Z(·) = 0 No t/t adults in population.q(i) = Y (i)/2N t-allele frequency at time iQ(i) = Y (i)/N Heterozygote frequency at time irm(i), rf (i) Probability that a male (resp. female) will transmit t to offspring.Q′(i) Probability after selection that an offspring embryo will be +/t.

Table 2: Notation for the lethal models.

selection. The (unconditional) probabilities rf (i) and rm(i) of drawing a t gene from the femaleand male gene pools in generation i are given respectively by

rf (i) =Yf (i)2Nf

, rm(i) =τYm(i)Nm

. (5)

From these, we wish to determine the heterozygote frequency in state i+1. However, the genotypefrequencies for embryos and adults differ in the lethal case due to recessive embryonic lethality.The probability, s(i), that an offspring will be heterozygous is the probability that an embryo willbe heterozygous normalized for the loss of t-homozygous embryos and is given by

s(i) =rm(i)

(1− rf (i)

)+ rf (i)

(1− rm(i)

)1− rf (i)rm(i)

. (6)

This probability is the same for both for males and females. The transition probability in theMarkov chain model for the evolution of the pair {X(·), Y (·)} is thus

P(Yf (i+1) = yf , Ym(i+1)) = ym|(Yf (i), Ym(i)

)=

Nf !Nm!xf !(Nf−xf )!ym!(Nm−ym)!

s(i)y(1− s(i)

)N−y,(7)

where y = yf+ym. Although the model defined by this equation has fewer independent variablesthan the sterile model (Equation 4), it is still not amenable to the diffusion approximation giventhe two-dimensional nature of the variable in the Markov chain. We have therefore obtainedour information about its behavior from simulations, discussed below in Section 3. However, bymaking some additional simplifying assumptions, it is possible to derive an approximate Markovchain model that yields tractable expressions for quantities of interest in the lethal case.

We now derive a simpler lethal distorter model by assuming that the male and female frequenciesof heterozygous +/t individuals in any generation are equal. This allows us to reduce the numberof independent variables required to describe the state to one. We further assume that the number

9

of males and females in each generation is the same (Nm = Nf = N/2)2. Under these assumptions,Yf (i) = Ym(i) = Y (i)/2 and the population composition in generation i can be described by a singlevariable, namely Y (i), the number of heterozygous individuals in that generation. The possiblevalues of this number are 0, 1, 2, . . . , N .

The probability Q′(i) that an individual in generation i+1 is a heterozygote can now be ex-pressed simply in terms of Q(i) by observing that Equation 5 simplifies to rf = Y (i)/2N = Q(i)/2and rm = τY (i)/N = τQ(i). Insertion of these values in Equation 6 gives

Q′(i) =Q(i)(1 + 2τ)− 2τ(Q(i))2

2− τ(Q(i))2(8)

The Markov chain describing the evolution of the number of heterozygous individuals in successivegenerations is then given by

P(Y (i+1) = y|Y (i)) =

(N

y

)(Q′(i)

)y(1−Q′(i))N−y. (9)

Note that because we have assumed that Yf (i) = Ym(i), the transition probability of this Markovchain has a binomial coefficient in contrast to the multinomial form of the more general lethal model(Equation 7). Equation 9 is analogous in form to the classic Wright-Fisher model (1). The numberof heterozygotes in generation i+1 is obtained by random sampling of heterozygote embryos afterselection, just as the allele frequency in the simple Wright-Fisher model is obtained by randomsampling of genes from the previous generation.

The only absorbing state in this Markov chain is Y (·) = 0 and the main quantity of interest isE(T ), the mean number of generations until this state is reached, given some initial frequency Q(0)of heterozygotes. The value of E(T ) can be approximated by a diffusion process using Equation 2.Given the number Y (i) of heterozygotes in generation i, Equation 9 and standard properties of thebinomial distribution show that the mean value of the heterozygote frequency in generation i + 1is Q′(i). The value of the mean change in the heterozygote frequency from one generation to the

2While this second assumption is not necessary to obtain a one-dimensional Markov model, itsimplifies the exposition considerably. The extension of the analysis given here to any fixed sexratio requires a more complex algebraic formulation but is otherwise straightforward. We requireonly that the sex ratio be sufficiently balanced to allow the maintenance of a fixed population sizeof N individuals.

10

next is thus Q′(i)−Q(i), yielding

m(∆Q) =Q(−1 + 2τ − 2τQ+ τQ2)

2− τQ2. (10)

Similarly, using standard properties of the binomial distribution, the value of v(∆Q) is

v(∆Q) =Q′(i)

(1−Q′(i)

)N

=Q(1 + 2τ − 2τQ)(2−Q− 2τQ+ τQ2)

N(2− τQ2)2. (11)

These expressions for m(∆Q) and v(∆Q) can now be inserted into Equation 2, with Q replacingq, to yield a differential equation for E(T ). Because of the forms of the expressions for m(∆Q)and v(∆Q), the integrations required to solve that equation cannot be carried out analytically,although an approximate solution can be obtained through numerical integration. Because of this,and because this model involves the assumption that the frequency of heterozygotes is the same inmales and females, we have nor pursued this model further from a theoretical point of view.

Instead we introduce a third model, in which the assumption that the frequency of heterozygotesis the same in males and females allows for straightforward theoretical analysis, yielding explicitand exact expressions for all quantities of interest. In this model, a birth-death model analogousto the Moran haploid model [Ewens, 1979], we define the basic unit of change in the population tobe the replacement of one individual. More specifically, at time unit i, (i = 1, 2, 3, . . . ), a randomlychosen individual in the population is chosen to die, and is immediately replaced by a newborn3.Thus, the number of heterozygotes either increases by one, remains the same, or decreases by oneat each time unit. For example, the number of heterozygotes increases by one if a homozygote+/+ is chosen to die and a heterozygote is born. The probability that a homozygote is chosen todie is (1−Q(i)), since in the lethal model, the frequency of heterozygotes is just twice the t-allelefrequency. The probability that the newborn is a heterozygote is Q′(i), as defined in Equation 8above, but where the time parameter i now refers to the number of birth-death events ratherthan the number of generations that have elapsed. ??? From these probabilities, we obtain theprobability that the number of heterozygotes will increase by one at time i:

λi = (1−Q(i)) ·Q′(i) =2τ(Q(i))3 − (1 + 4τ)Q(i))2 + (1 + 2τ)Q(i)

2− τ(Q(i))2. (12)

3In order to maintain a fixed sex ratio, Nf = Nm, we will assume that, at any given time unit,the randomly chosen individual is replaced by an individual of the same sex.

11

Similarly, the probability µi that the number of heterozygotes decreases by one at this birth-deathevent is given by the probability that a heterozygote dies and a wild type is born, or

µi = Q(i) · (1−Q′(i)) =τ(Q(i))3 − (1 + 2τ)(Q(i))2 + 2Q(i)

2− τ(q(i))2. (13)

If an individual is replaced by one with the same genotype, the number of heterozygotes does notchange at time unit i. This occurs with probability 1− λi − µi.

Unlike the Wright-Fisher model, in which any state can be reached from any other state, in thebirth-death model the set of accessible states is severely restricted. Given a population in stateY (i) = j, only three states, Y (i+1) = j−1, Y (i+1) = j and Y (i+1) = j+1, are possible after onetime step. Stated formally, the transition probability from state j to state k is

P (Y (i+1) = k|Y (i) = j) =

λ(i) k = j+1,1− λ(i)− µ(i) k = j,µ(i) k = j − 1,0 otherwise.

(14)

Note that this expression is a continuant, a matrix in which only the three central diagonalsare non-zero. The advantage of a model having a continuant transition matrix is that explicit,exact expressions for all quantities of interest can be obtained (see, for example, [Ewens, 1979]),in contrast to the various generation-based Markov models described above. In particular, themean time T (j) for loss of the heterozygous genotype, given an initial number Y (0) = j of suchgenotypes, can be expressed in the form

T (j) =N−1∑`=1

tj`. (15)

In this expression, tj` is defined for ` = 1, 2, . . . j by

tj` = µ−1`

(1 +

λ`−1

µ`−1+ · · ·+ λ`−1λ`−2 · · ·λ1

µ`−1µ`−2 · · ·µ1

), (16)

and for ` = j + 1, j + 2, . . . N by

tj` = tjj

(λjλj+1 · · ·λ`−1

µj+1µj+2 · · ·µ`

). (17)

In equation (16) the value of tjj is found from the ` = j case of equation (16). In these equations,tj` is the mean number of time units for which there are exactly ` heterozgotes in the populationbefore the (certain) eventual loss of heterozygotes.

CONCLUDING PARA

3 Impact of Drift on the Survival of Distorters

A comparison of the sterile and lethal models with the Wright-Fisher model for non-distortingalleles, gives a qualitative understanding of the behavior of genetic drift in populations of distorting

12

loci. The Wright-Fisher model tells us that there are two possible outcomes for a population of non-distorting alleles: loss of a and loss of A. In the absence of selection, a and A are equally likely toprevail given an initial allele frequency of 50% and these two outcomes are symmetric. In the sterilemodel, there are also two possible outcomes: wild type fixation and extinction. However, these twooutcomes are not symmetric in that the fixation of t results in extinction for the population and fort itself. In the lethal models, only one outcome is possible: wild type fixation. The t-allele frequencycan never rise above 50%. Thus, in both lethal and sterile models, the eventual outcome is alwaysthe loss of t, either through wild type fixation or extinction of the population. The expected timeuntil loss occurs is a measure of the negative selective pressure on t due to sterility or lethality,and is thus an important characteristic of the two models. In this section, we present quantitativeestimates of the characteristic features of these models; in particular, the expected time to reachan absorbing state. In the simple Wright-Fisher model, fixation of one or the other allele tendsto occur fairly quickly in small populations: as noted above, the mean fixation time grows onlylinearly with population size. We will see that this is not the case for a population of distortinggene complexes.

We used use Monte Carlo simulation to study the stochastic behavior of both the sterile andlethal generation-based models. For the birth-death model of lethals, the expected time to wildtype fixation as a function of population size and transmission ratio was conputed directly fromEquation 15 using Mathematica CITATION. The simulation procedure is the same in principle as,although clearly different in the details from, the sampling process leading to the theoretical models(Equations 1 through 9). Our simulator models a population of N individuals, each representedby sex and genotype (+/+, +/t and, in the case of sterile alleles, t/t). The population size, N ,is fixed from generation to generation and the sex ratio is fixed at 50:50. Each generation iscalculated from the previous generation. For each offspring in the new generation, two parents areselected at random from the current generation and the offspring genotype is calculated from theparental genotypes following the genetics of transmission ratio distortion. If the offspring is notviable (because the father is sterile or because it carries two lethal t’s), the program tries again.The simulation of new generations is repeated until t is lost, either through wild type fixation orthrough extinction. The variation of t allele frequency as a function of generation and the time toloss of t were measured as a function of τ and N . In the case of steriles, the manner in which t waslost (wild type fixation or extinction) was also recorded. Preliminary experiments indicated thatinitial t-allele frequency has very little impact on the final outcome, so an initial allele frequency of50% was always used.

The expected time to fixation as a function of transmission ratio and population size for thesterile model is shown in Figures 1 – 3. Each point is the average of 20 runs. Figure 1 showsthe impact of τ on expected time to loss of t when N = 30. For low and high values of τ , thepopulation reaches an extreme state within tens of generations. However, for intermediate valuesof τ (0.70 < τ < 0.85) the expected time to loss of t rises quickly to thousands of generations.The degree of fluctuation in allele frequencies was measured by computing 90% confidence intervalsfor each experiment, shown as bars extending from each data point in the figure. These barsdemonstrate that stochastic variation also increases dramatically in the intermediate region.

This effect becomes more pronounced as population size increases as can be seen in Figures 2and 3. For populations of size ten, fixation occurs quickly for all values of τ , which is consistentwith the observations of [Lewontin, 1962]. However, the time to fixation grows exponentially as

13

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

50 55 60 65 70 75 80 85 90 95 100

Ge

ne

ratio

ns

Transmission Ratio (%)

N = 30

Figure 1: Expected time to loss of sterile distorters in a population of 30 individuals as a functionof transmission ratio. Each point is averaged over twenty experiments. Error bars represent 90%confidence intervals.

N increases, resulting in a peaked bell curve. As population size increases, this peak also movesslowly to the right.

The short absorption times at low values of τ occur when the system is rapidly captured bythe absorbing state corresponding to wild type fixation. Short absorption times associated withhigh values of τ occur when t quickly spreads through the population, leading to extinction. Thisis illustrated in Figure 4, which shows the fraction of experiments which terminated through wildtype fixation or extinction, respectively. For low values of τ , wild type fixation occurs in mostexperiments, while extinction is the prevalent outcome for high values of τ . A transition zonewhere both outcomes are possible is seen in the middle. This transition moves to the right andbecomes more abrupt as the population size increases. Notice that for each value of N , the crossoverpoint is located at roughly the same value of τ as the peak of the corresponding curve in Figure 2.

Figures 5 and 6 show the impact of N and τ , respectively, on expected time to wild type fixationfor the generation-based model, while Figures 7 and 8 show the same quantities for the birth-deathlethal model. Since Figures 5 and 6 and Figures 7 and 8 are based on different models, theycannot be compare quantitatively. Qualitatively, however, both figures show the same behavior:the expected time to fixation grows exponentially with both N and τ in populations of lethal t-haplotypes. This is due to the absence of a second absorbing state at high t-allele frequencies. The

14

1

10

100

1000

10000

100000

1e+06

1e+07

1e+08

50 55 60 65 70 75 80 85 90 95 100

Ge

ne

ratio

ns


N = 10N = 20N = 30N = 40N = 50N = 60

Figure 2: Expected time to loss of sterile distorters as a function of transmission ratio for varyingpopulation sizes. Each point is the average over twenty experiments.

lethal case differs from the sterile case in that the greater the transmission distortion, the longer thedistorter survives. The agreement of the analytical and simulation results validates the simulationapproach.

The simulation results show that populations of distorters exhibit “quasi-stable” behavior, inwhich the time to absorption is larger by at least one order of magnitude than standard Wright-Fisher selectively neutral case, resulting in a long-term polymorphism whose allele frequency ischaracterized by a well-defined mean and variance. This quasi-stable behavior is seen at highvalues of τ for lethals and occurs at intermediate values of τ for steriles. We refer to the set ofvalues of τ and N for which quasi-stable behavior occurs as the quasi-stable region of the parameterspace. In the sterile model, the quasi stable region is delineated by the peaked bell in the curvesin Figure 2. The values of τ at which the bell rises and falls give the lower and upper limits of thequasi-stable region for a given population size.

Quasi-stable behavior can be understood by looking at Figure 9, which shows t-allele frequencyas a function of generation in a population of size 40 for τ = 0.70, τ = 0.78 and τ = 0.85, thelower limit, peak and upper limit of the quasi-stable region, respectively. Each point is the averageallele frequency, over all twenty experiments, seen at the current generation. The bars represent theminimum and maximum t-allele frequency seen at that generation in any of the twenty experiments.

15

1

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40 45 50

Ge

ne

ratio

ns

Population Size

tr = 0.80tr = 0.85tr = 0.90tr = 0.95tr = 0.99

Figure 3: Expected time to loss of sterile distorters as a function of population size for varyingtransmission ratios. Each point is the average over twenty experiments.

As Figure 9 shows, the average allele frequency increases with τ , consistent with previousmodels CITATION. At the lower limit of the quasi-stable region (τ = 0.70), the average allelefrequency is low enough so that the bottom of the range is close to q = 0, resulting in wild typefixation. Similarly, for the upper limit of the quasi-stable region (τ = 0.85), the top of the rangeapproaches q = 1.0, which would lead to extinction. However, for τ = 0.78, near the center of thepeak, fluctuations in t-allele frequency rarely approach either absorbing state, enabling a long-termt-polymorphism

The limits of the quasi-stable region (that is, the minimum and maximum values of τ associatedwith long-term polymorphism) vary with population size. For the sterile case, we used our simulatorto determine the dependence of the upper limit on the population size. Figure 10 shows the expectedtime to loss at high transmission ratios for population sizes ranging from N = 50 to N = 500.For each curve, the transmission ratio at which the expected fixation time drops by an order ofmagnitude and the curve flattens out indicates the upper limit of the quasi-stable region. As shownin the figure, the maximum transmission ratio for which a long-term t-polymorphism is possible isτ = 0.87 for N = 50, τ = 0.93 for N = 100 and τ > 0.99 for N = 500. For lethals, the maximumapproaches τ = 1.0 for all population sizes (Figure 6).

16

0

0.2

0.4

0.6

0.8

1

50 55 60 65 70 75 80 85 90 95 100

Fre

quen

cy


N = 10

+-fixationExtinction

0

0.2

0.4

0.6

0.8

1

50 55 60 65 70 75 80 85 90 95 100

Fre

quen

cy


N = 20


0

0.2

0.4

0.6

0.8

1

50 55 60 65 70 75 80 85 90 95 100

Fre

quen

cy


N = 30


0

0.2

0.4

0.6

0.8

1

50 55 60 65 70 75 80 85 90 95 100

Fre

quen

cy


N = 40


Figure 4: Frequency of expected outcome (wild type fixation or extinction due to sterility) of fixationexperiments as a function of transmission ratio for populations of 10, 20, 30 and 40 individuals,respectively. Each point is the average over twenty experiments.

4 Competition between Lethals and Sterile Distorters

The experiments described in the previous section investigate the dependence of t-haplotype persis-tence on transmission ratio and population size for either sterile distorters alone or lethal distortersalone. The results show that there are regions of the (τ ,N) parameter space that allow the long-term persistence of a distorter and that these regions differ for sterile and lethal distorters. Inparticular, in small populations sterile distorters are rapidly lost to extinction at high transmissionratios. In contrast, for lethal distorters expected persistence time increases with transmission ratiofor all population sizes. This suggests that lethal distorters may have a competitive advantage oversteriles at high transmission ratios in small populations.

To test this hypothesis, we simulated competition between sterile and lethal t-haplotypes ina single population. Recessive lethal mutations linked to the t-haplotype observed in nature aretypically point mutations; that is, changes that arise suddenly CITATION?. To investigate thelikelihood that a new lethal mutation will persist, we simulated the introduction of a single lethal

17

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40 45 50

Ge

ne

ratio

ns

Population Size

tr = 0.5tr = 0.6tr = 0.7tr = 0.9tr = 0.8

Figure 5: Expected time to loss of lethal distorters in the generational model as a function ofpopulation size, for varying transmission ratios. Each point is the average over twenty experiments.

distorter into a population of sterile distorters and wild type alleles, for population sizes N = 10,50, 100 and 500 and transmission ratios τ = 0.75, 0.85, 0.90 and 0.99. The initial population of 2Nalleles contained one lethal distorter and 0.3N sterile distorters. The remaining alleles were wildtype. These initial conditions are consistent with the 15% t-allele frequency observed in natureCITE KRISTIN.

At each step of the simulation, the genetic composition of the population was determined fromthe previous generation following the genetics of sterile and lethal t-haplotypes as shown in Table 4.There are five possible genotypes in this population: +/+, +/l, +/s, s/l and s/s, where s and lrefer to sterile and lethal t-haplotypes, respectively. There are no l/l individuals of either sex in thepopulation and s/l and s/s males are sterile. Females bearing either the s/l or the s/s genotypesare fertile.

The evolution of allele frequencies from one generation to the next was simulated until anabsorbing state was reached. A population containing both sterile and lethal distorters can be inone of five possible states are:

3: all three alleles present,

s: steriles and wild types present,

18

FatherMother +/+ +/l +/s

+/+ +/+: 1 +/+: (1−τ) +/+: (1−τ)+/l: τ +/s: τ

+/l +/+: 0.5 +/+: 0.5(1−τ) +/+: 0.5(1−τ)+/l: 0.5 +/l: 0.5 +/l: 0.5(1−τ)

l/l:0.5τ +/s: 0.5τs/l: 0.5τ

+/s +/+: 0.5 +/+: 0.5(1−τ) +/+: 0.5(1−τ)+/s: 0.5 +/s: 0.5(1−τ) +/s: 0.5

+/l: 0.5τ s/s: 0.5τs/l: 0.5τ

s/l +/s: 0.5 +/s: 0.5(1−τ) +/l: 0.5(1−τ)+/s

+/l: 0.5 +/l: 0.5(1−τ) +/s: 0.5(1−τ)s/l: 0.5τ s/l: 0.5τl/l: 0.5τ s/s: 0.5τ

s/s +/s:1 +/s: (1−τ) +/s: (1−τ)+/s

s/l: τ s/s:τ

Table 3: Frequencies of embryonic genotypes before selection resulting from matings between indi-viduals bearing both lethal and sterile distorters. l/l embryos will die before birth. NEEDS MOREDETAIL.

19

10

100

1000

10000

100000

1e+06

1e+07

1e+08

50 55 60 65 70 75 80 85 90 95 100

Ge

ne

ratio

ns


N = 10N = 20N = 30N = 40

Figure 6: Expected time to loss of lethal distorters as a function of transmission ratio for varyingpopulation sizes. Each point is the average over twenty experiments.

l: lethals and wild types present,

wt: wild types fixation

x: extinction.

The first three states are transient, the last two absorbing states. The t-allele frequency and thestate of the population at each generation was noted. Each experiment was repeated 10,000 timesto determine the probability of finding the population in a given state as a function of generation,calculated by dividing the number of runs in which the population was in that state in generationi by the total number of runs. CHECK DETAILS OF EXPERIMENTAL PARAMETERS.

We are interested in what happens immediately after a lethal allele is introduced into thepopulation. How long can lethals, steriles and wild types coexist in the same population? Once oneof these three alleles is eliminated, with what probability will the population inhabit each of theother four states? Since, in the absence of mutation, the three-allele state can never be regained,we focus on the short term behavior of the system. The long-term behavior of the other stateshas already been discussed in the previous sections. The results of these simulations are showngraphically in Figures 11 – 15 for parameter values of particular interest. As the figures show,these experiments initially exhibit transient behavior in which the t-allele frequency and the state

20

10

100

1000

10000

100000

1e+06

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Tim

e (

birth

-de

ath

eve

nts

)

Transmission Ratio

N = 10N = 20N = 30N = 40N = 50

Figure 7: Expected time to loss of lethal distorters under the birth-death model as a functionof population size, for varying transmission ratios, given an initial heterozygote frequency of 0.5.Values were computed from Equation refabsorptionTime using Mathematica.

probabilities change rapidly over time. Next a quasi-stable phase is reached in which the stateprobabilities change very slowly. This quasi-stable phase ends when the probability of finding adistorter in the population approaches zero and wild type fixation and extinction are the only stateswith non-zero probabilities.

Consider, for example, the progression of a population of ten individuals with a transmissionratio of 0.90, shown in Figure 12. Initially, all three alleles coexist in the population. During thefirst few generations, all five states can be observed. However, the three-allele state is not seenafter generation 20 and, although initially a sterile distorter polymorphism is quite prevalent, thesterile allele is never able to persist longer than 40 generations. Forty generations after the lethalmutation is introduced, the system has reached the quasi-stable phase and only three states areobserved: extinction, wild type fixation and a quasi-stable lethal t-polymorphism. The probabilityof wild type fixation slowly increases until the probability of lethal t-persistence reaches zero ingeneration ??? (data not shown.)

In all of these experiments, the population typically reached the quasi-stable phase within 40to 60 generations. Therefore, we used the probability of finding t-haplotypes in the population atgeneration 100 as a measure of the success of t-haplotype persistence for a given set of parameters.

21

10

100

1000

10000

100000

1e+06

0 10 20 30 40 50 60

Tim

e (

birth

-de

ath

eve

nts

)

Population size

t = 0.5t = 0.6t = 0.7t = 0.8t = 0.9

Figure 8: Expected time to loss of lethal distorters under the birth-death model as a function oftransmission ratio distortion, , for varying transmission ratios, for an initial heterozygote frequencyof 0.5. Values were computed from Equation refabsorptionTime using Mathematica.

The state probabilities for these experiments are summarized in Table 4. These show that forthe range of parameters we considered, a quasi-stable polymorphism of all three alleles was neverobserved. The probability of the three-allele state approached zero within a few tens of generations.Furthermore, once the quasi-stable phase is reached, only one of the two distorting alleles survives;we never observe both sterile persistence (state s) and lethal persistence (state l) in the same regionof the parameter space.

For small populations (N = 10), lethal t never survived the transient phase for low transmissionratios (τ = 0.70, shown in Figure 12). At generation 100, the population had reached wild typefixation in approximately 80% and extinction in approximately 15% of simulations, while sterilesand wild types continued to coexist in the remaining 6% of the experiments. For larger transmissionratios, however, the lethal alleles survived to reach quasi-stability while the steriles did not. Lethalsand wild types persisted in 4% of the simulations for τ = 0.85, in 10% of the simulations for τ = 0.90and in 11% of the simulations for τ = 0.99. Notice, that while the probability of lethal persistenceis not very high, steriles were not able to persist in this regime at all.

For larger populations (N ≥ 50), a different picture emerges (Figures 13 - 15). Only two statesare seen at generation 100 for τ ≥ 0.85: extinction and coexistence of steriles and wild types.

22

0

20

40

60

80

100

0 100 200 300 400 500

Alle

le fr

eque

ncy

(%)

Generation

N = 40

tr = 0.70tr = 0.78tr = 0.85

Figure 9: Variation in average frequency of sterile distorters over time. Each point represents thedistorter allele frequency seen at the current generation averaged over twenty experiments. Thebars represent the minimum and maximum allele frequency seen at the current generation in anyof the twenty experiments.

The probability of sterile t-haplotype persistence increases with population size, reaching 100% forN = 500 (Figure 15). The lethal mutation introduced in the first generation never persists afterthe transient regime for N ≥ 50.

In summary, in larger populations, the lethal mutation is quickly lost, but sterile t-haplotypeswill persist for certain values of τ and N . In contrast, in populations of ten individuals with hightransmission ratio (0.85 or greater), only t-haplotypes linked to a lethal mutation survived thetransient regime.

5 Related Work

Most previous models of the t-haplotype have focussed on questions of population dynamics: whatforces in modern day mouse populations account for the t-haplotype frequency seen in nature?The earliest population models assumed random mating in infinite populations. Based on theseassumptions, Bruck [Bruck, 1957] predicted allele frequency as a function of transmission ratiodistortion in populations of lethal t-haplotypes:

q = .5−√τ(1− τ)/2τ . (18)

This panmictic model predicts a t-haplotype frequency of 0.333 when τ = 0.9, which is much higherthan the empirical range of 0.1 - 0.15 [Ardlie, 1995, Ardlie and Silver, 1998, Lenington et al., 1988,

23

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

84 86 88 90 92 94 96 98 100

Gen

erat

ions


N = 50 N = 75 N = 100N = 250N = 500

Figure 10: Expected time to loss of sterile distorters for large populations in the high τ region.Populations of 500 or more individuals see no negative selective pressure due to male sterility. Inpopulations of less than 100 individuals, a value of τ ≥ 0.95 will lead to extinction.

Ruvinsky et al., 1991]. A similar model for sterile t-haplotypes

q = 2τ−1 (19)

was presented by [Dunn and Levene, 1961]. This model also predicts t-haplotype frequencies thatare much greater than those seen in the wild, suggesting that infinite population analyses are notgood models for distorters in mouse populations.

In the early sixties, Lewontin and Dunn studied the behavior of t in finite populations usingMonte Carlo simulation [Lewontin and Dunn, 1960, Lewontin, 1962]. They did not perform a com-prehensive study of the stochastic behavior of t-haplotype frequencies as a function of transmissionratio and population size but the cases that they studied provided good intuition concerning the

τ N = 10 N = 50 N = 100 N = 5003 s l 3 s l 3 s l 3 s l

0.70 0.00 0.06 0.00 0.01 0.94 0.01 0.01 0.98 0.00 0.03 0.97 0.000.85 0.00 0.00 0.04 0.00 0.99 0.01 0.00 1.00 0.00 0.02 0.98 0.000.90 0.00 0.00 0.10 0.00 0.84 0.00 0.00 1.00 0.00 0.01 0.99 0.000.99 0.00 0.00 0.11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.87 0.00

Table 4: State probabilities 100 generations after the introduction of a lethal distorter into apopulation of 15% sterile alleles as a function of transmission ratio and population size.

24

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Pro

babi

lity

Generation

steriles and wildtypeslethals and wildtypes

Wildtypes persistExtinctionall alleles

Figure 11: State probabilities as a function of generation after a single lethal allele is introducedinto a population of N = 10 individuals with three sterile alleles. τ = 0.7.

population dynamics of t and was an inspiration for the current work. FOOTNOTE ABOUTCOMPUTER TIME. In particular, they observed that for small demes (approximately ten mice),wild type fixation occurs relatively rapidly whereas for larger demes (20 – 50 mice) lethal t hap-lotypes will persist over many generations. From this observation, they hypothesized that the lowlevel t polymorphism observed in nature is due to a balance between the loss of t in small demesdue to genetic drift and the reintroduction of t through interdemic migration.

This hypothesis was widely accepted for thirty years. In separate studies, [Levin et al., 1969]and [Nunney and Baker, 1993] presented Monte Carlo simulations of structured populations inwhich they demonstrated that a deme size and mutation rate could be found that, taken together,predict the t-haplotype frequency seen in nature. However, in 1997, [Durand et al., 1997] showedthat although it is possible to find such values, they are not stable with respect to small pertur-bations in the interdemic migration rate. If the interdemic migration rate is decreased slightly, twill be lost from the population. If it is raised slightly, the allele frequency will rapidly increaseto the high frequencies predicted by the panmictic model. This instability in allele frequencies isnot consistent with natural observations and suggests that a balance between drift and migrationis not a sufficiently powerful model to explain the stable t-polymorphism seen in nature.

While the focus of the current work is on evolutionary questions, the results reported in theprevious section have some relevance to the population dynamics work surveyed here. They showthat genetic drift is fundamentally different in population of distorting gene complexes in thatexpected absorption time varies nonlinearly with N and τ and in that, for certain regions ofthe parameter space, a quasi-stable state can exist. This explains Lewontin’s observation that t

25

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Pro

babi

lity

Generation



Figure 12: State probabilities as a function of generation after a single lethal allele is introducedinto a population of N = 10 individuals with three sterile alleles. τ = 0.90.

disappears in small populations but persists in larger ones. Our results also suggest why a balancebetween the loss of t-alleles due to genetic drift and reintroduction of t-alleles through interdemicmigration does not result in a stable, low-level t-haplotype polymorphism. A change in interdemicmigration rate is analogous to a change in effective population size. The results in Figure 5 showthat small changes in population size can result in large changes in persistence time. Thus, a smallchange in migration rate could move the population from a regime where the t allele is lost in 10to 100 generations to a quasi-stable state where it persists for a long time.

In addition to work on the population genetics of t-haplotypes, there is a substantial bodyof theoretical analyses of ultraselfish genes in other organisms (see [Feldman and Otto, 1991] fora survey.) This work, which deals entirely with infinite populations, includes one-locus mod-els of the sex-ratio (SR) chromosome in Drosophila pseudoobscura [Curtsinger and Feldman, 1980,Thomson and Feldman, 1976, Edwards, 1961] and the Segregation Distorter (SD) complex in Drosophilamelanogaster [Hiraizumi et al., 1960] and two-locus models of SD [Prout et al., 1973, Charlesworth and Hartl, 1978Hartl, 1975, Liberman, 1976, Thomson and Feldman, 1975] and SR distortion in Aedes aegypti[Maffi and Jayakar, 1981, Lessard, 1987, Feldman and Otto, 1989]. The role of recombination andselection in modifiers of SR has also been studied [Lessard, 1987, Feldman and Otto, 1989]. Ananalysis of the evolutionary dynamics of spore killer genes has been presented recently [Nauta and Hoekstra, 1993].

The observation that recessive lethality is linked to t-haplotypes in most populations has stimu-lated a number of theoretical investigations of the evolution of lethal t-haplotypes. [Dunn and Levene, 1961]suggested that recessive lethality confers a selective advantage on t-haplotypes by eliminating ster-ile t/t males, who cannot transmit t to future generations and yet compete for reproductive and

26

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Pro

babi

lity

Generation



Figure 13: State probabilities as a function of generation after a single lethal allele is introducedinto a population of N = 50 individuals with 15 sterile alleles. τ = 0.90.

material resources. While many authors find this argument compelling, a specific mechanism bywhich this advantage is attained has not been limned. [Charlesworth, 1994] has suggested repro-ductive compensation as a mechanism that would facilitate the success of lethal t-haplotypes. Thishypothesis stems from the argument that litter sizes resulting from heterozygote crosses wouldnot be greatly reduced if the death of t/t embryos occurs early in embryonic development, sincethere are more implanted embryos than live births. In Charlesworth’s study, one or more lethalt-haplotypes are introduced into an infinite, panmictic population of wild types and steriles andthe persistence of those alleles over time as a function of transmission ratio and the degree of repro-ductive compensation is determined. The results show that at high transmission ratios a recessivelethal will survive in a polymorphic equilibrium in which all three alleles are present, if the degreeof compensation is sufficiently high. While Charlesworth’s results are provocative, experimentalevidence suggests that reproductive compensation is not an important factor in determining littersize in t bearing mice. In a study of litter sizes from the long term records of Princeton University’smouse colony, Ardlie and Silver observed a reduction of roughly 40% in average litter sizes resultingfrom matings of +/tw5 heterozygotes when compared with average wild type litters (unpublisheddata). Dunn and Bennett 1960 CITATION analyzed the ratio of normal to abnormal embryos inlitters of tw5/tw5 crosses and found approximately 44% abnormal embryos in utero. Both of theseobservations are consistent with the theoretical prediction of a 45% reduction in litter size whenτ = 0.9, suggesting that reproductive compensation does not occur.

[van Boven et al., 1996] present two models of competition between a lethal and a sterile dis-torter as part of a larger study of the role of allele frequency and transmission ratio in the coexistence

27

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Pro

babi

lity

Generation




of different t-haplotypes. They consider a sterile distorter t1 and lethal distorter t2, such that al-though t1/t1 males are sterile, t1/t2 males are fully fertile (there are no t2/t2 males). The firstmodel considers an infinite, panmictic population of +, t1 and t2 alleles. This population reachesa stable equilibrium within 100 generations in which all three alleles coexist. This is consistentwith the results of Charlesworth. The second model considers a subdivided population involvingmigration between 100 demes of either 10 or 20 individuals. Normally the genotypes of offspringin the next generation are determined by parental genotypes drawn from the same deme, but withprobability m = 0.05 a resident is replaced by a migrant from another deme. In the subdividedpopulation, wild type fixation occurs for low transmission ratios ( τ ≤ 0.75). All three alleles coexistat intermediate transmission ratios while at very high transmission ratios the wild type allele is lostand t1 and t2 coexist. Note that the model of van Boven et al. is fundamentally different from ours.We investigate the fate of a new lethal mutation linked to an existing sterile t-haplotype. Theyconsider the introduction of a new, unrelated lethal t-haplotype into a population of steriles. Whilethere are many known complementing lethal mutations (i.e., an embryo bearing two different lethalmutations does not die), there is no evidence of two or more “complementing” sterile segregationdistorters in mice. Thus, while van Boven et al. may describe some as yet undiscovered segregationdistortion system, their models do not give insight into the evolution of lethal segregation distortersin mice.

There has been very little work on analytical models of distorting gene complexes in finitepopulations. van Boven [van Boven, 1997, ?] used a Markov analysis to compare the persistenceof distorters to that of non-distorting alleles using Robertson’s concept of a retardation factor

28

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Pro

babi

lity

Generation




[Robertson, 1962]. This factor is the ratio of the largest non-unit eigenvalue in the ultraselfish caseto the corresponding eigenvalue of the neutral Wright-Fisher model (Equation 1). A disadvantageof this approach is that it only gives a good approximation if the other non-unit eigenvalues aremuch smaller than the leading eigenvalue and hence higher powers of these eigenvalues approachzero rapidly as time increases. van Boven has not demonstrated that the largest non-unit eigenvector dominates in this context. Whereas his predictions for sterile alleles are similar to ours, hispredictions for lethal t-haplotypes differ. Unlike the results shown in Figures 5 and 6, van Boven’spredictions of time to fixation for lethals do not show exponential increase as τ increases. This is atodds with the results presented in Figures 5 – 8, which were obtained using two different approaches.Intuitively, we would expect a significant increase in mean absorption time with increasing τ . Thisis because the Markov chain describing populations of lethal t-haplotypes has only one absorbingstate, corresponding to wild type fixation, and the probability of reaching this state in a shorttime drops dramatically as τ increases. This discrepancy between our predictions and those of vanBoven’s approach could be due to the limitations of using only the largest non-unit eigenvalue toanalyze the convergence of t-allele frequencies.

6 Discussion

Our results show that the population genetics of distorting gene complexes differs from Mendelianpopulation genetics in several important ways. There are substantive differences between geneticdrift in populations of non-distorting alleles. When genetic drift acts on a Mendelian locus with

29

two possible alleles, both alleles have a non-zero probability of fixation and the expected timeto fixation is linearly related to the population size in the absence of selection. In contrast, theultimate outcome for any distorter under genetic drift is always loss, either through the fixation ofa competing wild type allele or through extinction. The expected time to loss varies non-linearlywith both transmission ratio, τ , and population size, N , so that a small change in population size ortransmission ratio can result in a dramatic change in the dynamics of the population. Furthermore,for certain values of the N and τ , distorters will reach a quasi-stable state in which the expectedtime to loss grows exponentially with population size, resulting in a long term polymorphism. Thisquasi-stable state occurs at intermediate transmission ratios for steriles and at high transmissionratios for lethals. In both cases, distorting alleles with transmission ratios that place them in thequasi-stable region have a selective advantage over those that are rapidly lost. What do thesemathematical results suggest about the evolution of distorters seen in nature?

First, these results offer a hypothesis concerning the evolution of transmission ratios in specieswith different population sizes. It is likely that the transmission ratio in early t-haplotypes was justabove 0.5 and increased slowly over time as additional inversions containing alleles that contributeto distortion were recruited to the t-haplotype and as additional mutations in the initial distortingalleles increased their effectiveness. Under this hypothesis, the transmission ratio observed inmodern day t-haplotypes is the result of the balance between the positive and negative selectiveforces on TRD. Since lethals observed today arose only recently CITATIOIN?, we assume TRDoriginally evolved in populations of sterile t-haplotypes. Positive selection promotes an increasedtransmission ratio since variants with a greater degree of distortion will be transmitted to moreoffspring. Genetic drift combined with sterility exerts a negative selective force, since for distortingalleles, drift always results in the loss of t.

Our results suggest that variants of t within the quasi-stable region have an advantage overthose outside the region and that sterile t alleles will not evolve transmission ratios correspondingto the region where the expected absorption time is on the order of 10 to 100 generations. Whichvariants within the quasi-stable region will be most successful? Theoretically, those alleles withthe longest persistence time (i.e., variants whose value of τ corresponds to the maximum of thepeak in Figure 2) will be most successful. However, in the real world, negative selective pressureon transmission ratio will not be sensitive to variations in persistence times in the range of millionsof years. Rather, negative selective pressure is uniformly negligible in the quasi-stable region.

We have seen that the range of transmission ratios at which quasi-stable persistence can occurdepends on population size. Male infertility does not exert a negative selective force on the trans-mission ratio in species with effective population sizes above 500, while in species with effectivepopulation sizes of 50 or less, homozygous male sterility will limit the evolution of the transmissionratio. In wild mouse populations, studies of variations in gene frequency over short time scales(several generations) suggest that the effective population size is on the order of 10–40 individ-uals [Anderson, 1964, Selander, 1970, Lidicker, 1976, Dallas et al., 1995]. In fruitfly populations,estimated effective population sizes are on the order of one million individuals [Kreitman, 1983].The mean observed modern day transmission ratios of roughly τ ≈ 0.9 in mouse and τ ≈ 0.99 inDrosophila CITATION are consistent with the hypothesis that the evolution of transmission ratiois limited by genetic drift in small populations but not in populations over 500 individuals.

Second, our results offer a theoretical framework for understanding the selective forces actingon recessive lethal mutations linked to distorters. The fact that, in small populations, steriles

30

persist at intermediate transmission ratios while lethals persist at high transmission ratios suggeststhat lethality confers a selective advantage on the t-haplotype in populations where extinction isthe dominant cause of loss of t. This is confirmed by the simulations of sterile and lethal allelescompeting in a single population. When the population size is small (N = 10) and the transmissionratio is high (τ ≤ 0.85), extinction is the dominant cause of loss. Our simulations show that underthese conditions, lethal t-haplotypes can survive more than 60 generations while steriles cannot. Inlarge populations or small populations with low transmission ratios (τ ≥ 0.70), wild type fixationis the dominant outcome. Under these conditions, steriles persist while lethals are quickly lost.

These observations suggest a hypothesis for the recent but widespread appearance of lethal,recessive mutations linked to the t-haplotype and the absence of lethal mutations linked to SD.In early t-haplotype populations with low transmission ratio, wild type fixation was probably theprimary cause of loss of both lethal and sterile t-haplotypes. The frequency of t is higher inpopulations of steriles since t/t females are able to pass t on to their offspring. Thus, recessivelethality confers no selective advantage on t in this region and, in fact, reduces the t-haplotypefrequency. There may have been lethal mutations linked to early t-haplotypes; but, in the absenceof any selective advantage, there is no reason why they should have persisted. However, oncethe transmission ratio increased to the point where male sterility threatened the survival of thepopulation, lethal populations acquired a selective advantage. This hypothesis explains why mostt-haplotypes found in the wild carry recessive lethal mutations and is consistent with the recentappearance of lethals as well as the many different lethal variants found in wild populations. SinceDrosophila populations are so large that male sterility never causes extinction, our hypothesis couldexplain why lethals mutations are not generally associated with SD.

In summary, our analytical and simulation models show that genetic drift imposes a negativeselective force on the transmission of distorters in small populations. This suggests the hypothesisthat the transmission ratios of autosomal distorting gene complexes in species with small populationsizes will never approach 100% and that these complexes will tend to be linked with recessivelethal mutations. While a mathematical model cannot prove that genetic drift is responsible forthe low transmission ratio and recessive embryonic lethality associated with the t-haplotype, itis suggestive that the predictions of our model are consistent with the characteristics of the twoautosomal distorting gene complexes that are well understood. The validity of our hypotheses willbe tested further as additional distorting gene complexes come to light. Since the lack of a specificphenotypic manifestation of transmission ratio distortion makes these complexes difficult to detect,the fact that they have been observed in five of the twenty to thirty model organisms ever subjectedto detailed genetic analysis suggests that transmission ratio distortion may exist undiscovered inmany other species.

The models we have presented assume a panmictic population. An important next step toconsider is the impact of genetic drift on distorters in substructured populations. There is evidenceof substructuring on several levels in wild mice populations. Various studies have found evidence,both physical and genetic, that mouse populations are subdivided among different farms or vil-lages [Anderson, 1964, Selander, 1970, Lidicker, 1976, Dallas et al., 1995]. Gene frequencies overtime scales of many generations indicate that local populations are linked by migration yieldinglarger effective population sizes [Dallas et al., 1995]. A few studies [Anderson, 1964, Ardlie, 1995,Selander, 1970] indicate that some of the very large populations might be further subdivided intomuch smaller “demes” that are true territorial/social units. While a population of demes linked

31

by migration can be modelled as a panmictic population with a smaller effective population size, itwill be of great interest to determine if the behavior of genetic drift in substructured populationsis consistent with the results presented here.

Our models are also useful for interpreting complex modeling efforts, in which genetic drift insubpopulations has an impact on the dynamics of a larger structured population. There have beenmany simulation studies of t-haplotype population dynamics, including studies of migration in sub-structured populations, as surveyed in the previous section. Our results suggest how measurementsshould be performed in the presence of stochastic variation in simulations of populations of dis-torting gene complexes. Computer simulations of finite populations are by their nature stochastic.This leads to questions such as: (1) when is the simulator in a quasi-stable state? (2) when shouldthe allele frequency be measured? and (3) how long should the simulation be allowed to run? Ourresults delineate ranges of N and τ for which a quasi-stable state occurs. In regions of the pa-rameter space where a quasi-stable state occurs, allele frequency can be characterized by the meanand variance. Outside those regions, the allele frequency is not stable. Instead, it is appropriate tomeasure whether t fixes or is lost from the population and the mean time until this event occurs.

Acknowledgements

We thank Almut Burchard, Max Mintz and Ken Steiglitz for helpful discussions and the PrincetonUniversity Computer Science Department for welcoming DD as a visiting researcher. DD wassupported by NSF grants BIR-94-13215 A01 and DEB-9752945. WJE was supported by NIH grantGM21135. LMS was supported by NIH grant HD20275.

References

[Anderson, 1964] Anderson, P. (1964). Lethal alleles in mus musculus: Local distribution andevidence for isolation of demes. Science, 145:177–178.

[Ardlie, 1995] Ardlie, K. G. (1995). The Frequency, Distribution and Maintenance of t-haplotypesin Natural Populations of Mice. PhD thesis, Princeton University.

[Ardlie and Silver, 1996] Ardlie, K. G. and Silver, L. M. (1996). Low frequency of mouse t haplo-types in wild populations is not explained by modifiers of meiotic drive. Genetics, 144:1787–1797.

[Ardlie and Silver, 1998] Ardlie, K. G. and Silver, L. M. (1998). Low frequency of t-haplotypes innatural populations of house mice (mus musculus domesticus). Evolution, 52:1185–1196.

[Atlan et al., 1997] Atlan, A., Mercot, N., Landre, C., and Montchamp-Moreau, C. (1997). The sex-ratio trait in drosophila simulans: Geographic distribution of distortion and resistance. Evolution,51:1886–1895.

[Bruck, 1957] Bruck, D. (1957). Male segregation ratio advantage as a factor in maintaining lethalalleles in wild populations of house mice. Genetics, 43:152–58.

[Charlesworth, 1994] Charlesworth, B. (1994). The evolution of lethals in the t-haplotype systemof the mouse. Proc R Soc Lond B Biol Sci, 258:101–107.

32

[Charlesworth and Hartl, 1978] Charlesworth, B. and Hartl, D. L. (1978). Population dynamics ofthe segregation distorter polymorphism of drosophila melanogaster. Genetics, 89:171–192.

[Curtsinger and Feldman, 1980] Curtsinger, J. W. . and Feldman, M. W. (1980). Experimentaland theoretical analysis of the ”sex-ratio” polymorphism of drosophila melanogaster. Genetics,94:445–466.

[Dallas et al., 1995] Dallas, J. F., Dod, B., Boursot, P., Prager, E. M., and Bonhomme, F. (1995).Population subdivision and gene flow in Danish house mice. Molecular Ecology, 4:311–320.

[Dunn, 1957] Dunn, L. (1957). Evidence of evolutionary forces leading to the spread of lethal genesin wild populations of house mice. Proc. Nat. Acad. Sci. USA, 43:158–163.

[Dunn and Levene, 1961] Dunn, L. and Levene, H. (1961). Population dynamics of a variant t-allelein a confined population of wild house mice. Evolution, 15(4):385–93.

[Durand et al., 1997] Durand, M., Ardlie, K., Buttel, L., Levin, S., and L.M.Silver (1997). Impactof migration and fitness on the stability of lethal t-haplotype polymorphism in house mice: Acomputer study. Genetics, 145:1093 – 1108.

[Edwards, 1961] Edwards, A. W. F. (1961). The population genetics of ”sex-ratio” in drosophilapseudoobscura. Heredity, 16:291–304.

[Ewens, 1979] Ewens, W. J. (1979). Mathematical Population Genetics. Springer-Verlag.

[Feldman and Otto, 1989] Feldman, M. W. and Otto, S. P. (1989). More on recombination andselection in the modifier theory of sex-ratio distortion. Theoretical Population Biology, 35:207–225.

[Feldman and Otto, 1991] Feldman, M. W. and Otto, S. P. (1991). A comparative approach to thepopulation-genetics theory of segregation distortion. American Naturalist, 137:443–456.

[Hammer and Silver, 1993] Hammer, M. F. and Silver, L. M. (1993). Phylogenetic analysis of thealpha-globin pseudogene-4 (Hba-ps4) locus in the house mouse species complex reveals a stepwiseevolution of t haplotypes. Mol. Biol. Evol., 10:971–1001.

[Hartl and Hiraizuma, 1976] Hartl, D. and Hiraizuma, Y. (1976). Segregation distortion after fif-teen years. In Ashburner, M. and Novitski, E., editors, The Genetics and Biology of DrosophilaMelanogaster. Academic Press, New York.

[Hartl, 1975] Hartl, D. L. (1975). Modifier theory and meiotic drive. Theoretical Population Biology,7:168–174.

[Hiraizumi et al., 1994] Hiraizumi, Y., Albracht, J. M., and Albracht, B. C. (1994). X-linked ele-ments associated with negative segretation distortion in the sd system of drosophila melanogaster.Genetics, 138:145–152.

[Hiraizumi et al., 1960] Hiraizumi, Y., Sandler, L., and Crow, J. F. (1960). Meiotic drive in naturalpopulations of drosophila melanogaster III. populational implications of the segregation-distorterlocus. Evolution, 14:433–444.

33

[Hiraizumi and Thomas, 1984] Hiraizumi, Y. and Thomas, A. M. (1984). Supressor system of segre-gation distortion (sd) chromosomes in natural populations of drosophila melanogaster. Genetics,95:693–706.

[Jaenike, 1996] Jaenike, J. (1996). Sex-ratio drive in the drosophila quinaria group. AmericanNaturalist, 148:237–254.

[Kreitman, 1983] Kreitman, M. (1983). Nucleotide polymorphism at the alcohol dehydrogenaselocus of drosophila melanogaster. Nature.

[Lenington et al., 1988] Lenington, S., Franks, P., and Williams, J. (1988). Distribution of t hap-lotypes in natural populations of wild house mouse. J. Mamm., 69:489–99.

[Lessard, 1987] Lessard, S. (1987). The role of recombination and selection in the modifier theoryof sex-ratio distortion. Theoretical Population Biology, 31:339–358.

[Levin et al., 1969] Levin, B., Petras, M., and Rasmussen, D. (1969). The effect of migrationon the maintenance of a lethal polymorphism in the house mouse. The American Naturalist,103(934):647–661.

[Lewontin, 1962] Lewontin, R. (1962). Interdeme selection controlling a polymorphism in the housemouse. The American Naturalist, 96:65–78.

[Lewontin and Dunn, 1960] Lewontin, R. and Dunn, L. (1960). The evolutionary dynamics of apolymorphism in the house mouse. Genetics, 45:705–772.

[Liberman, 1976] Liberman, U. (1976). Theory of meiotic drive: is Mendelian segregation stable?Theoretical Population Biology, 10:127–132.

[Lidicker, 1976] Lidicker, W. Z. (1976). Social behavior and density regulation in house mice livingin large enclosures. Journal of Animal Ecology, 45:677–697.

[Loegering and Sears, 1963] Loegering, W. Q. and Sears, E. R. (1963). Distorted inheritance ofstem-rust resistance of Timstein wheat caused by a pollen killing gene. Can J Genet Cytol,5:65–72.

[Lyon, 1986] Lyon, M. F. (1986). Male sterility of the mouse t-complex is due to homozygosity ofthe distorter gene. Cell, 44:357–363.

[Lyttle, 1991] Lyttle, T. W. (1991). Segregation distorters. Annu. Rev. Genet., 25:511–557.

[Lyttle, 1993] Lyttle, T. W. (1993). Cheaters sometimes prosper: distortion of Mendelian segrega-tion by meiotic drive. Trends in Genetics, 9:205–210.

[Maffi and Jayakar, 1981] Maffi, G. and Jayakar, S. D. (1981). A two-locus model for polymorphismfor sex-linked meiotic drive modifiers with possible applications to aedes aegypti. TheoreticalPopulation Biology, 19:19–36.

34

[Morita et al., 1992] Morita, T., Kubota, H., Murata, K., Nozaki, M., Delabre, C., Willison, K.,Satta, Y., Sakaizumi, M., Takahata, N., Gachelin, G., and Matsushiro, A. (1992). Evolution of themouse t-haplotype: Recent and worldwide introgression to mus musculus. Proc.Natl.Acad.Sci.USA, 89:6851–6855.

[Nauta and Hoekstra, 1993] Nauta, M. J. and Hoekstra, R. F. (1993). Evolutionary dynamics ofspore killers. Genetics, 135:923–930.

[Nunney and Baker, 1993] Nunney, L. and Baker, E. (1993). The role of deme size, reproductivepatterns and dispersal in the dynamics of t-lethal haplotypes. Evolution, 47(5):1342–59.

[Padieu and Bernet, 1967] Padieu, E. and Bernet, J. (1967). Mode d’action des genes responsablesde l’avortement de certains produits de la meiose chez l’ascomycete Podospora anserina. CompRend Acad Sci Paris Ser D, 264:2300–2303.

[Powers and Ganetsky, 1991] Powers, P. A. and Ganetsky, B. (1991). On the components of Seg-regation Distortion in Drosophila melanogaster. V. Molecular analysis of the SD locus. Genetics,129:133–144.

[Prout et al., 1973] Prout, T., Bundgaard, J., and Bryant, S. (1973). Population genetics of meioticdrive I. the solution of a special case and some general implications. Theoretical PopulationBiology, 4:446–465–.

[Rick, 1971] Rick, C. M. (1971). The tomato Ge locus: Linkage relations and geographic distribu-tion of alleles. Genetics, 67:75–85.

[Robertson, 1962] Robertson, A. (1962). Selection for heterozygotes in small populations. Genetics,47:1291–1300.

[Ruvinsky et al., 1991] Ruvinsky, A., Polyakov, A., Agulnik, A., Tichy, H., Figueroa, F., and Klein,J. (1991). Low diversity of t haplotypes in the eastern form of the house mouse, mus musculus.Genetics, 127:161–168.

[Sandler and Golic, 1985] Sandler and Golic (1985). Segregation distortion in drosophila. Trendsin Genetics, 5:181–184.

[Selander, 1970] Selander, R. K. (1970). Behavior and genetic variation in natural populations.Am. Zoologist, 10:53–60.

[Silver, 1985] Silver, L. M. (1985). Mouse t haplotypes. Annual Review in Genetics, 19:179–208.

[Silver, 1993] Silver, L. M. (1993). The peculiar journey of a selfish chromosome: Mouse t haplo-types and meiotic drive. Trends in Genetics, 9:250–254.

[Thomson and Feldman, 1975] Thomson, G. J. and Feldman, M. W. (1975). Population geneticsof modifiers of meiotic drive. IV. on the evolution of sex-ratio distortion. Theoretical PopulationBiology, 8:202–211.

35

[Thomson and Feldman, 1976] Thomson, G. J. and Feldman, M. W. (1976). Population geneticsof modifiers of meiotic drive. III. equilibrium analysis of a general model for the genetic controlof segregation distortion. Theoretical Population Biology, 10:10–25.

[Turner and Perkins, 1979] Turner, B. and Perkins, D. (1979). Spore killer, a chromosomal factorin Neurospora that kills meiotic products not containing it. Genetics, 93:587–606.

[van Boven, 1997] van Boven, M. (1997). Evolutionary Dynamics of a Selfish Gene. PhD thesis,University of Groningen.

[van Boven et al., 1996] van Boven, M., Weissing, F. J., Heg, D., and Huisman, J. (1996). Compe-tition between segregation distorters: coexistence of ”superior” and ”inferior” haplotypes at thet complex. Evolution, 50(6):2488–2498.

[Werren et al., 1988] Werren, J. H., Nur, U., and Wu, C.-I. (1988). Selfish genetic elements. Trendsin Ecology and Evolution, 3:297–302.

[Wu and Hammer, 1991] Wu, C.-I. and Hammer, M. F. (1991). Molecular evolution of ultraselfishgenes of meiotic drive systems. In Selander, R. K., Clark, A. G., and Whitman, T. S., editors,Evolution at the Molecular Level, pages 177–203. Sinauer Associates.

36

genetic drift in populations of distorting gene complexesdurand/papers/durandgenetics97.pdfthe...

Documents