Genetic Drift in Populations of Distorting Gene Complexes

Transcription

1 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 University of California, Berkeley, CA Genomics Collaborative, Cambridge, MA Computational Biology Group, University of Pennsylvania, Philadelphia, PA Department of Molecular Biology, Princeton University Princeton, NJ

2 Keywords: meiotic drive, segregation distortion, genetic drift, selfish chromosomes. Running Head: Evolution of Segration Distortion Contact Author: Dannie Durand Department of Molecular Biology Princeton University Princeton, NJ tel: or 6874 fax:

3 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 bearing the distorting complex are sterile. Although distorters promote themselves at the expense of other genes in the same genome, they have been successful in evolution: distorting gene complexes have been observed in animal, plant and fungal species. While the molecular drive mechanisms differ, distorting gene complexes exhibit similar genetic features, suggesting that the 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 of distorting gene complexes. Genetic drift, the stochastic behavior of allele frequencies in small populations, is a fundamental force in the process of evolution, yet the role of genetic drift in the evolution of distorters is not well understood. Our analysis shows how distorters evolve different characteristics in species typified by small and large populations, respectively. We apply our results to two well studied distoring gene complexes: the t-haplotype in Mus musculus and Segregation Distorter in Drosophila melanogaster. 1 Introduction Distorting gene complexes, or distorters, are chromosomal regions that enhance their own transmission relative to the rest of an individual s genome [Werren et al., 1988] and actively interfere with the functions of other genes in the same nucleus [Wu and Hammer, 1991]. These complexes are characterized by segregation distortion, also referred to as transmission ratio distortion or TRD 1. 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 been discovered 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 many species of Drosophila [Atlan et al., 1997, Hartl and Hiraizuma, 1976, Jaenike, 1996], ALSO WOOD LEMMINGS, 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- 1 We prefer the term transmission ratio distortion to segregation distortion because the underlying molecular and cell biological processes responsible for this phenomena are not well understood and may be acting after segration. 3

4 torting gene complexes are the t-haplotype in Mus musculus and Segregation Distorter or SD in Drosophila melanogaster. The t-haplotype [Silver, 1985, Silver, 1993] is a variant form of a 30 megabasepair region on chromosome 17. A series of inversions in the region results in a localized suppression of recombination and thus allows a complete t-haplotype to be transmitted as a unit from one generation to the next. Chromosomes bearing the t-haplotype are preferentially transmitted 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 the same genes involved in TRD are also responsible for the homozygous sterility [Lyon, 1986]. According to the phylogenetic evidence, the t-haplotype evolved in a stepwise manner through the sequential 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 increased with each additional inversion, increasing slowly from Mendelian transmission to its present level. These inversions were acquired over the period from three million years ago to about 100,000 years ago. Despite this antiquity, sequence comparisons among modern t-haplotypes show almost no nucleotide polymorphism in comparison to wild type chromosomes, suggesting that all contemporary t-haplotypes share a much more recent common ancestor that lived between 10,000 and 100,000 years ago. In wild populations today, the t-haplotype persists as a polymorphism with t-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 naturallyoccurring t-haplotypes are linked to mutations conferring recessive embryonic lethality. In stark contrast to the lack of polymorphism among independent t-haplotypes, there are a large number of independent recessive lethal mutations, which suggests both that the lethals are a recent accumulation 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, 198 Hiraizumi 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 homozygous males. Unlike the t-haplotype, SD is rarely linked to recessive lethal mutations. Such mutations are observed occasionally, but do not appear to persist. SD and t also differ in their transmission ratio: in SD the transmission ratio approaches 100%. In fact, the transmission ratio of the t-haplotype is lower than most known distorting gene complexes. Spore Killer in Neurospora CITATION, Sex Ratio in Drosophila CITATION and the W chromosome in butterflies CITATION are 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, in nature the effective sex ratio is lower due to a range of MD chromosones with varying sensitivities to 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 genome and the individual that carries them. Distorting gene complexes have been observed in a number of animal, plant and fungal species. Although the underlying physiological mechanisms of transmission ratio distortion differ from species to species, on the genetic level several common features have been found in all systems that have been well studied: diploidy, multiple linked genes, transmission ratio distortion and homozygous male infertility [Lyttle, 1991, Lyttle, 1993]. This suggests that the population genetic forces that allow such gene complexes to evolve and persist may be shared in 4

5 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 small populations, is a fundamental force in the process of evolution. The mathematics of genetic drift and the corresponding evolutionary implications have been studied in great detail since the early decades of this century (see, for example, [Ewens, 1979]). However, there have been few studies of genetic drift in populations of distorters and the role of genetic drift in their evolution is not well understood. Here we present an analysis of the stochastic behavior of autosomal distorters in 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 to other autosonal distorters, notably SD. We use our results to propose novel solutions to two longstanding problems that have stymied biologists studying the t-haplotype for over forty years: why the 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 analysis also 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 are presented in Section 3. These show how the persistence of a distorter depends on distortion levels and population size. Competition between sterile and lethal distorters in a single population are examined in Section 4. Previous modeling efforts are reviewed in Section 5 and the results of the current paper are discussed in light of this earlier work. In conclusion, the evolutionary implications of 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 diploid individuals: a wild type (+) and a distorter (t). Although distorters are known to contain multiple loci, we will treat t as an indivisible unit since those loci are linked by inversions. The transmission ratio is represented as a parameter, τ. Heterozygous males pass t to their offspring with probability τ 0.5. Heterozygous females transmit t and + to their offspring with equal probability. Our model has two variants. In the sterile model, t/t males are sterile and t/t females have no reduction in fertility. This model applies equally well to SD or to any other autosomal distorting gene complex characterized by TRD, recombination suppression and homozygous male sterility. In the lethal model, 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 genetic drift 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 with two possible allelic types (A and a) is considered. The random variable W (i) refers to the number of A s in generation i. The variation in allele frequency over time is modeled by a Markov chain 5

6 with 2N +1 states, where state j refers to a population with j genes of allelic type A and 2N j genes of type a. Two states (j = 0 and j =2N) are absorbing, corresponding to the fixation of the a and A allelic types, respectively. The transition probability of this Markov chain is: P (W (i+1) = k W(i)) = ( 2N k ) (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 offspring generation are obtained by random sampling, with replacement, from the genes in the parental generation. Note that a single random variable is sufficient to specify the state under this assumption and that it is not necessary to know the distribution of genotypes in the population, so that various properties of interest can be determined using allele frequencies alone. In particular, the mean time to fixation of either a or A denoted E[T ], can be approximated accurately from Equation 1 using a one-dimensional diffusion process. This procedure requires calculation of the mean m( q) and the variance 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) d 2 E(T ) 2 dq 2 = 1, (2) subject to the boundary conditions E(T ) = 0 when q =0,q= 1. The value of E(T ) is then found by replacing q in this solution by, q(0), the initial value of the frequency of the A allele. For the model 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 main conclusion to be drawn from this equation is that the mean time to fixation is only linear in the population size, so that a small change in population size leads to a small change in the mean fixation time. We will find that this conclusion no longer holds in the models of distorting complexes we consider next. 2.2 Sterile distorters in finite populations We now construct a Markov chain model similar in principle to that described in Equation 1, but with further features necessary to describe the behavior of a population of sterile distorters. As above, we assume a population of N diploid organisms. We further assume that the number of males in each generation is fixed at N m and the number of females at N f = N N m. Variation over time in the frequency of the t allele can be modeled using a Markov chain generalizing the simple Wright-Fisher model. However, in contrast to Equation 1, which is based on allele frequencies, in the sterile model we need to keep track of genotype frequencies, because t has a different phenotype in t/t males (sterility) than in +/t males (TRD). As a result, more than one independent variable is required for the complete specification of the population in generation i. We denote the number of males and females of each genotype in generation i by the vectors {X m (i),y m (i),z m (i)} and {X f (i),y f (i),z f (i)}, as described in Table 1. For notational convenience, we define the total number of +/+ individuals of either sex in generation i to be X(i) =X m (i)+x f (i), with similar 6

7 definitions for Y (i) and Z(i). Since the sex ratio is fixed, the number of +/+ individuals is constrained to be X m (i) =N m Y m (i) Z m (i) for males and X f (i) =N f Y f (i) Z f (i) for females. Thus, four independent variables {Y m (i),z m (i),y f (i),z f (i)} are required to describe the state of i Current generation. N The number of diploid individuals in the population. N m, N f The number of males (resp. females) in the population. X(i),X m (i),x f (i) The number of +/+ individuals (total, male, female) in generation i. Y (i),y m (i),y f (i) The number of +/t individuals (total, male, female) in generation i. Z(i),Z m (i),z f (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 i r m (i),r f (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 the probabilities 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 the offspring generation is simply the allele frequency, q( ). However, in a population of distorters the allele transmission frequency is influenced by TRD and male sterility. The respective probabilities r m (i) (r f (i)) that a gene transmitted to generation i+1 from a male (female) of generation i is a t allele are Y m (i) τ r m (i) = X m (i)+y m (i), r f (i) = Y f(i) + Z f(i). 2N f N f Thus the probabilities p(+/+), p(+/t) and p(t/t) that an embryo (male or female) in the next generation will have genotype +/+, +/t and t/t, respectively, are given by p(+/+) = (1 r m (i)) (1 r f (i)), p(+/t) = (1 r m (i)) r f (i)+(1 r f (i)) r m (i), p(t/t) = r m (i) r f (i). The genotypes in the offspring generation are obtained by sampling from these genotype frequencies, yielding the transition probabilities P (Y m (i+1) = y m,z m (i+1) = z m,y f (i+1) = y f,z f (i+1) = z f Y m (i),z m (i),y f (i),z f (i)) = N m!n f! (N m y m z m )!y m!z m!(n f y f z f )!y f!z f! p(+/+) x p(+/t) y p(t/t) z (4) 7

8 for the sterile Markov model, where x = x m + x f, y = y m + y f and z = z m + z f. In contrast to the binomial form of the simple Wright-Fisher model (1), this model has a multinomial form, because more than one random variable is required to specify the state. For the same reason, the one dimensional diffusion approximation cannot be used to obtain a tractable, analytical expression for the mean time to loss of t, analogous to the expression for the estimated mean fixation time given in (3). Thus, it appears difficult, if not impossible, to obtain explicit analytic results describing the outcomes of a population of sterile distorters from the Markov chain model (4). Some simplification to the model is obtained if the frequencies of the three respective genotypes are assumed to be the same within males and females. However this simplification reduces the Markov chain to one described by two variables, and analytic results are still difficult, if not impossible, to obtain. We therefore do not pursue this amended chain further. Instead, all our results obtained for this model are found by simulation of the Markov chain (4). These are presented in Section 3. By contrast, we will find below that the assumption of equal genotype frequencies betwen males and females does make 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 male and female t/t embryos die before birth. A consequence of lethality is that the highest t-haplotype frequency is 0.5 and occurs in the state where all individuals are heterozygotes. Thus, a population of lethal distorters can be modeled by a Markov chain with only one absorbing state (where all alleles are +). Since there are no t-bearing homozygotes, two degrees of freedom are sufficient to specify 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 that are simpler than the expression for the sterile model given in Equation 4, it is still not possible to apply the one-dimensional diffusion approximation to obtain tractable estimates of mean fixation times. However, by making the simplifying assumption that the heterozygote frequency is the same among both males and females, we can obtain more tractable Markov models that are amenable to analysis. 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 single individual is the basic unit of change. The parameters used in these models are defined in Table 2. As in the sterile case, we suppose that the number of males and females in the population is fixed over all generations, with N f females and N m = N N f males. The number of female heterozygotes in generation i is Y f (i) and the number of male heterozygotes is Y m (i). In the lethal case, Z( ) = 0 and the total number of wild type homozygotes is constrained to be X(i) = N Y(i). A further consequence of lethality is that, unlike the sterile model, genotype frequencies do not give us more information about the state than allele frequencies; in fact, they are interchangeable because the t-allele frequency always differs from the heterozygote frequency by a factor of two. We denote the frequency of +/t individuals in generation 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 embryo genotypes obtained from the male and female gene pools in generation i. These frequencies depend on the probability of transmitting t from parent to offspring. As in the sterile case, this transmission frequency depends not only on the allele frequency but also on the forces of TRD and negative 8

9 i Current time unit. N The number of diploid individuals in the population. N m, N f The number of males (resp. females) in the population. X(i),X m (i),x f (i) The number of +/+ individuals (total, male, female) at time i. Y (i),y m (i),y f (i) The number of +/t individuals (total, male, female) at time i. Z( ) = 0 Not/tadults in population. q(i) = Y(i)/2N t-allele frequency at time i Q(i) = Y(i)/N Heterozygote frequency at time i r m (i),r f (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 r f (i) and r m (i) of drawing a t gene from the female and male gene pools in generation i are given respectively by r f (i) = Y f(i) 2N f, r m (i)= τy m(i) N m. (5) From these, we wish to determine the heterozygote frequency in state i+1. However, the genotype frequencies 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 will be heterozygous normalized for the loss of t-homozygous embryos and is given by s(i) = r m(i) ( 1 r f (i) ) +r f (i) ( 1 r m (i) ). (6) 1 r f (i)r m (i) This probability is the same for both for males and females. The transition probability in the Markov chain model for the evolution of the pair {X( ),Y( )} is thus P ( Y f (i+1) = y f,y m (i+1)) = y m (Y f (i),y m (i) ) = N f!n m! x f!(n f x f )!y m!(n m y m )! s(i)y( 1 s(i) ) N y, where y = y f +y m. Although the model defined by this equation has fewer independent variables than the sterile model (Equation 4), it is still not amenable to the diffusion approximation given the two-dimensional nature of the variable in the Markov chain. We have therefore obtained our information about its behavior from simulations, discussed below in Section 3. However, by making some additional simplifying assumptions, it is possible to derive an approximate Markov chain 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 frequencies of heterozygous +/t individuals in any generation are equal. This allows us to reduce the number of independent variables required to describe the state to one. We further assume that the number (7) 9

10 of males and females in each generation is the same (N m = N f = N/2) 2. Under these assumptions, Y f (i) =Y m (i)=y(i)/2 and the population composition in generation i can be described by a single variable, namely Y (i), the number of heterozygous individuals in that generation. The possible values 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 expressed simply in terms of Q(i) by observing that Equation 5 simplifies to r f = Y (i)/2n = Q(i)/2 and r m = 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 successive generations is then given by P ( ( ) N (Q Y (i+1) = y Y (i)) = (i) ) y( 1 Q (i) ) N y. (9) y Note that because we have assumed that Y f (i) =Y m (i), the transition probability of this Markov chain 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 number of heterozygotes in generation i+1 is obtained by random sampling of heterozygote embryos after selection, just as the allele frequency in the simple Wright-Fisher model is obtained by random sampling of genes from the previous generation. The only absorbing state in this Markov chain is Y ( ) = 0 and the main quantity of interest is E(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 the binomial distribution show that the mean value of the heterozygote frequency in generation i + 1 is Q (i). The value of the mean change in the heterozygote frequency from one generation to the 2 While this second assumption is not necessary to obtain a one-dimensional Markov model, it simplifies the exposition considerably. The extension of the analysis given here to any fixed sex ratio requires a more complex algebraic formulation but is otherwise straightforward. We require only that the sex ratio be sufficiently balanced to allow the maintenance of a fixed population size of N individuals. 10

11 next is thus Q (i) Q(i), yielding m( Q) = Q( 1+2τ 2τQ+τQ2 ) 2 τq 2. (10) Similarly, using standard properties of the binomial distribution, the value of v( Q) is v( Q) = Q (i) ( 1 Q (i) ) = Q(1 + 2τ 2τQ)(2 Q 2τQ+τQ2 ) N N(2 τq 2 ) 2. (11) These expressions for m( Q) and v( Q) can now be inserted into Equation 2, with Q replacing q, 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 in males 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 heterozygotes is the same in males and females allows for straightforward theoretical analysis, yielding explicit and exact expressions for all quantities of interest. In this model, a birth-death model analogous to the Moran haploid model [Ewens, 1979], we define the basic unit of change in the population to be the replacement of one individual. More specifically, at time unit i, (i =1,2,3,...), a randomly chosen individual in the population is chosen to die, and is immediately replaced by a newborn 3. Thus, the number of heterozygotes either increases by one, remains the same, or decreases by one at 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 to die is (1 Q(i)), since in the lethal model, the frequency of heterozygotes is just twice the t-allele frequency. The probability that the newborn is a heterozygote is Q (i), as defined in Equation 8 above, but where the time parameter i now refers to the number of birth-death events rather than the number of generations that have elapsed.??? From these probabilities, we obtain the probability that the number of heterozygotes will increase by one at time i: λ i =(1 Q(i)) Q (i) = 2τ(Q(i))3 (1 + 4τ)Q(i)) 2 +(1+2τ)Q(i) 2 τ(q(i)) 2. (12) 3 In order to maintain a fixed sex ratio, N f = N m, we will assume that, at any given time unit, the randomly chosen individual is replaced by an individual of the same sex. 11

12 Similarly, the probability µ i that the number of heterozygotes decreases by one at this birth-death event is given by the probability that a heterozygote dies and a wild type is born, or µ i = 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 not change at time unit i. This occurs with probability 1 λ i µ i. Unlike the Wright-Fisher model, in which any state can be reached from any other state, in the birth-death model the set of accessible states is severely restricted. Given a population in state Y (i) =j, only three states, Y (i+1) = j 1, Y (i+1) = j and Y (i+1) = j+1, are possible after one time step. Stated formally, the transition probability from state j to state k is λ(i) k=j+1, 1 λ(i) µ(i) k = j, P (Y (i+1) = k Y (i) =j)= (14) µ(i) k = j 1, 0 otherwise. Note that this expression is a continuant, a matrix in which only the three central diagonals are 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, the mean time T (j) for loss of the heterozygous genotype, given an initial number Y (0) = j of such genotypes, can be expressed in the form T (j) = N 1 l=1 In this expression, t jl is defined for l =1,2,...j by ( t jl = µ 1 l 1+ λ l 1 µ l λ l 1λ l 2 λ 1 µ l 1 µ l 2 µ 1 t jl. (15) ), (16) and for l = j +1,j+2,...N by ( ) λj λ j+1 λ l 1 t jl = t jj. (17) µ j+1 µ j+2 µ l In equation (16) the value of t jj is found from the l = j case of equation (16). In these equations, t jl is the mean number of time units for which there are exactly l heterozgotes in the population before 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-distorting alleles, gives a qualitative understanding of the behavior of genetic drift in populations of distorting 12

13 loci. The Wright-Fisher model tells us that there are two possible outcomes for a population of nondistorting alleles: loss of a and loss of A. In the absence of selection, a and A are equally likely to prevail given an initial allele frequency of 50% and these two outcomes are symmetric. In the sterile model, there are also two possible outcomes: wild type fixation and extinction. However, these two outcomes are not symmetric in that the fixation of t results in extinction for the population and for t itself. In the lethal models, only one outcome is possible: wild type fixation. The t-allele frequency can never rise above 50%. Thus, in both lethal and sterile models, the eventual outcome is always the loss of t, either through wild type fixation or extinction of the population. The expected time until 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 quantitative estimates of the characteristic features of these models; in particular, the expected time to reach an absorbing state. In the simple Wright-Fisher model, fixation of one or the other allele tends to occur fairly quickly in small populations: as noted above, the mean fixation time grows only linearly with population size. We will see that this is not the case for a population of distorting gene complexes. We used use Monte Carlo simulation to study the stochastic behavior of both the sterile and lethal generation-based models. For the birth-death model of lethals, the expected time to wild type fixation as a function of population size and transmission ratio was conputed directly from Equation 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 represented by 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 is calculated from the previous generation. For each offspring in the new generation, two parents are selected at random from the current generation and the offspring genotype is calculated from the parental genotypes following the genetics of transmission ratio distortion. If the offspring is not viable (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 or through extinction. The variation of t allele frequency as a function of generation and the time to loss of t were measured as a function of τ and N. In the case of steriles, the manner in which t was lost (wild type fixation or extinction) was also recorded. Preliminary experiments indicated that initial t-allele frequency has very little impact on the final outcome, so an initial allele frequency of 50% was always used. The expected time to fixation as a function of transmission ratio and population size for the sterile model is shown in Figures 1 3. Each point is the average of 20 runs. Figure 1 shows the impact of τ on expected time to loss of t when N = 30. For low and high values of τ, the population reaches an extreme state within tens of generations. However, for intermediate values of 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 intervals for each experiment, shown as bars extending from each data point in the figure. These bars demonstrate 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 2 and 3. For populations of size ten, fixation occurs quickly for all values of τ, which is consistent with the observations of [Lewontin, 1962]. However, the time to fixation grows exponentially as 13

14 9000 N = Generations Transmission Ratio (%) Figure 1: Expected time to loss of sterile distorters in a population of 30 individuals as a function of 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 moves slowly to the right. The short absorption times at low values of τ occur when the system is rapidly captured by the absorbing state corresponding to wild type fixation. Short absorption times associated with high values of τ occur when t quickly spreads through the population, leading to extinction. This is illustrated in Figure 4, which shows the fraction of experiments which terminated through wild type fixation or extinction, respectively. For low values of τ, wild type fixation occurs in most experiments, while extinction is the prevalent outcome for high values of τ. A transition zone where both outcomes are possible is seen in the middle. This transition moves to the right and becomes more abrupt as the population size increases. Notice that for each value of N, the crossover point 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 fixation for the generation-based model, while Figures 7 and 8 show the same quantities for the birth-death lethal model. Since Figures 5 and 6 and Figures 7 and 8 are based on different models, they cannot 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

15 1e+08 1e+07 1e+06 N = 10 N = 20 N = 30 N = 40 N = 50 N = Generations Transmission Ratio (%) Figure 2: Expected time to loss of sterile distorters as a function of transmission ratio for varying population 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 the distorter survives. The agreement of the analytical and simulation results validates the simulation approach. The simulation results show that populations of distorters exhibit quasi-stable behavior, in which 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 is characterized by a well-defined mean and variance. This quasi-stable behavior is seen at high values of τ for lethals and occurs at intermediate values of τ for steriles. We refer to the set of values of τ and N for which quasi-stable behavior occurs as the quasi-stable region of the parameter space. In the sterile model, the quasi stable region is delineated by the peaked bell in the curves in Figure 2. The values of τ at which the bell rises and falls give the lower and upper limits of the quasi-stable region for a given population size. Quasi-stable behavior can be understood by looking at Figure 9, which shows t-allele frequency as a function of generation in a population of size 40 for τ =0.70, τ =0.78 and τ =0.85, the lower limit, peak and upper limit of the quasi-stable region, respectively. Each point is the average allele frequency, over all twenty experiments, seen at the current generation. The bars represent the minimum and maximum t-allele frequency seen at that generation in any of the twenty experiments. 15

16 tr = 0.80 tr = 0.85 tr = 0.90 tr = 0.95 tr = 0.99 Generations Population Size Figure 3: Expected time to loss of sterile distorters as a function of population size for varying transmission ratios. Each point is the average over twenty experiments. As Figure 9 shows, the average allele frequency increases with τ, consistent with previous models CITATION. At the lower limit of the quasi-stable region (τ = 0.70), the average allele frequency is low enough so that the bottom of the range is close to q = 0, resulting in wild type fixation. Similarly, for the upper limit of the quasi-stable region (τ = 0.85), the top of the range approaches q =1.0, which would lead to extinction. However, for τ =0.78, near the center of the peak, fluctuations in t-allele frequency rarely approach either absorbing state, enabling a long-term t-polymorphism The limits of the quasi-stable region (that is, the minimum and maximum values of τ associated with long-term polymorphism) vary with population size. For the sterile case, we used our simulator to determine the dependence of the upper limit on the population size. Figure 10 shows the expected time 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 of magnitude and the curve flattens out indicates the upper limit of the quasi-stable region. As shown in the figure, the maximum transmission ratio for which a long-term t-polymorphism is possible is τ =0.87 for N = 50, τ =0.93 for N = 100 and τ>0.99 for N = 500. For lethals, the maximum approaches τ = 1.0 for all population sizes (Figure 6). 16

17 1 N = 10 1 N = Frequency fixation Extinction Frequency fixation Extinction Transmission Ratio (%) N = Transmission Ratio (%) N = Frequency fixation Extinction Frequency fixation Extinction Transmission Ratio (%) Transmission Ratio (%) Figure 4: Frequency of expected outcome (wild type fixation or extinction due to sterility) of fixation experiments 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 persistence on transmission ratio and population size for either sterile distorters alone or lethal distorters alone. The results show that there are regions of the (τ,n) parameter space that allow the longterm persistence of a distorter and that these regions differ for sterile and lethal distorters. In particular, in small populations sterile distorters are rapidly lost to extinction at high transmission ratios. In contrast, for lethal distorters expected persistence time increases with transmission ratio for all population sizes. This suggests that lethal distorters may have a competitive advantage over steriles at high transmission ratios in small populations. To test this hypothesis, we simulated competition between sterile and lethal t-haplotypes in a single population. Recessive lethal mutations linked to the t-haplotype observed in nature are typically point mutations; that is, changes that arise suddenly CITATION?. To investigate the likelihood that a new lethal mutation will persist, we simulated the introduction of a single lethal 17

18 tr = 0.5 tr = 0.6 tr = 0.7 tr = 0.9 tr = Generations Population Size Figure 5: Expected time to loss of lethal distorters in the generational model as a function of population 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 The initial population of 2N alleles contained one lethal distorter and 0.3N sterile distorters. The remaining alleles were wild type. These initial conditions are consistent with the 15% t-allele frequency observed in nature CITE KRISTIN. At each step of the simulation, the genetic composition of the population was determined from the 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 l refer to sterile and lethal t-haplotypes, respectively. There are no l/l individuals of either sex in the population and s/l and s/s males are sterile. Females bearing either the s/l or the s/s genotypes are fertile. The evolution of allele frequencies from one generation to the next was simulated until an absorbing state was reached. A population containing both sterile and lethal distorters can be in one of five possible states are: 3: all three alleles present, s: steriles and wild types present, 18

19 Father Mother +/+ +/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: 0.5τ s/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:0.5τ s/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:0.5τ l/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 individuals bearing both lethal and sterile distorters. l/l embryos will die before birth. NEEDS MORE DETAIL. 19

20 1e+08 1e+07 N = 10 N = 20 N = 30 N = 40 1e+06 Generations Transmission Ratio (%) Figure 6: Expected time to loss of lethal distorters as a function of transmission ratio for varying population 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 the state of the population at each generation was noted. Each experiment was repeated 10,000 times to 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 generation i 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 the population. How long can lethals, steriles and wild types coexist in the same population? Once one of these three alleles is eliminated, with what probability will the population inhabit each of the other 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 states has already been discussed in the previous sections. The results of these simulations are shown graphically in Figures 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

21 1e N = 10 N = 20 N = 30 N = 40 N = 50 Time (birth-death events) Transmission Ratio Figure 7: Expected time to loss of lethal distorters under the birth-death model as a function of 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 state probabilities change very slowly. This quasi-stable phase ends when the probability of finding a distorter in the population approaches zero and wild type fixation and extinction are the only states with non-zero probabilities. Consider, for example, the progression of a population of ten individuals with a transmission ratio of 0.90, shown in Figure 12. Initially, all three alleles coexist in the population. During the first few generations, all five states can be observed. However, the three-allele state is not seen after generation 20 and, although initially a sterile distorter polymorphism is quite prevalent, the sterile allele is never able to persist longer than 40 generations. Forty generations after the lethal mutation is introduced, the system has reached the quasi-stable phase and only three states are observed: extinction, wild type fixation and a quasi-stable lethal t-polymorphism. The probability of wild type fixation slowly increases until the probability of lethal t-persistence reaches zero in generation??? (data not shown.) In all of these experiments, the population typically reached the quasi-stable phase within 40 to 60 generations. Therefore, we used the probability of finding t-haplotypes in the population at generation 100 as a measure of the success of t-haplotype persistence for a given set of parameters. 21

22 1e t = 0.5 t = 0.6 t = 0.7 t = 0.8 t = 0.9 Time (birth-death events) Population size Figure 8: Expected time to loss of lethal distorters under the birth-death model as a function of transmission ratio distortion,, for varying transmission ratios, for an initial heterozygote frequency of 0.5. Values were computed from Equation refabsorptiontime using Mathematica. The state probabilities for these experiments are summarized in Table 4. These show that for the range of parameters we considered, a quasi-stable polymorphism of all three alleles was never observed. 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 region of the parameter space. For small populations (N = 10), lethal t never survived the transient phase for low transmission ratios (τ = 0.70, shown in Figure 12). At generation 100, the population had reached wild type fixation in approximately 80% and extinction in approximately 15% of simulations, while steriles and wild types continued to coexist in the remaining 6% of the experiments. For larger transmission ratios, however, the lethal alleles survived to reach quasi-stability while the steriles did not. Lethals and wild types persisted in 4% of the simulations for τ =0.85, in 10% of the simulations for τ =0.90 and in 11% of the simulations for τ =0.99. Notice, that while the probability of lethal persistence is 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 states are seen at generation 100 for τ 0.85: extinction and coexistence of steriles and wild types. 22

23 100 N = 40 tr = 0.70 tr = 0.78 tr = Allele frequency (%) Generation Figure 9: Variation in average frequency of sterile distorters over time. Each point represents the distorter allele frequency seen at the current generation averaged over twenty experiments. The bars represent the minimum and maximum allele frequency seen at the current generation in any of the twenty experiments. The probability of sterile t-haplotype persistence increases with population size, reaching 100% for N = 500 (Figure 15). The lethal mutation introduced in the first generation never persists after the transient regime for N 50. In summary, in larger populations, the lethal mutation is quickly lost, but sterile t-haplotypes will persist for certain values of τ and N. In contrast, in populations of ten individuals with high transmission ratio (0.85 or greater), only t-haplotypes linked to a lethal mutation survived the transient regime. 5 Related Work Most previous models of the t-haplotype have focussed on questions of population dynamics: what forces 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 these assumptions, Bruck [Bruck, 1957] predicted allele frequency as a function of transmission ratio distortion in populations of lethal t-haplotypes: q =.5 τ(1 τ)/2τ. (18) This panmictic model predicts a t-haplotype frequency of when τ = 0.9, which is much higher than the empirical range of [Ardlie, 1995, Ardlie and Silver, 1998, Lenington et al., 1988, 23