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The evolution of the genetic load caused by recurrent mutation in small populations : genetic context… Poon, Arthur F. 2000

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T H E E V O L U T I O N OF T H E G E N E T I C L O A D C A U S E D B Y R E C U R R E N T M U T A T I O N I N S M A L L P O P U L A T I O N S : G E N E T I C C O N T E X T A N D D E M O G R A P H I C H I S T O R Y by A R T P O O N Hon. B . Sc., The University of Toronto, 1998. A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y OF G R A D U A T E S T U D I E S (Department of Zoology) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A July 2000 © Art Poon, 2000 U B C Special Collections - Thesis Authorisation Form 6/7/2000 10:13 A M In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date Page 1 of 1 ABSTRACT I have conducted two separate theoretical investigations on the consequences o f recurrent mutation for the extinction risk of small populations. The continual generation of deleterious mutations and their subsequent rise in frequency in the population cause genetic load: a reduction in the mean fitness of the population. This evolutionary process is a function of natural selection, mutation, and genetic drift. Predictions made by theoretical studies on the extinction risk depend on what assumptions are made about selection, mutation, and drift. The following work modifies standard assumptions that are made in models that study genetic loads in finite populations such that the sensitivity of the results can be evaluated. First of all , many models assume that most mutations are unconditionally deleterious — i.e. they have the same effect in any genome. There may be, however, a large fraction of mutations that conceal the expression of another mutation when placed in the same genome. These are compensatory mutations. With compensatory mutation, qualitatively different conclusions are reached about the extinction risk of a small population by genetic causes. Secondly, the models that study the effect of decreasing population size on the genetic load simplify the demographic change to an instantaneous drop from infinite size. Declining populations often gradually decline in size over several generations, however. I have studied the effect of gradual demographic change by numerical methods. The results indicate that transient changes in the mean load caused by recessive alleles can be delayed by a gradual change in population size. i i T A B L E O F C O N T E N T S Abstract i i List of Figures i v Acknowledgements v C H A P T E R I The genetic load of small populations 1 Selection, mutation, and extinction 1 Evolutionary consequences of small population size 4 Genetic loads 7 Statement of authorship 11 C H A P T E R II The effect of compensatory mutation 12 2.1 Preface 12 2.2 Overview 12 2.3 Evidence for compensatory mutations 14 2.4 Fisher's geometrical model of adaptation 17 2.5 Moment analysis of the master equation 21 2.6 Distribution functions and numerical methods 28 2.7 Results in the one-dimensional model 34 2.8 Results in the ^-dimensional model 39 2.9 Sensitivity of the model to its assumptions 43 2.10 Implications for the extinction risk of small populations 49 2.11 Implications for pleiotropy 51 C H A P T E R III The effect of demographic history 54 3.1 Overview 54 3.2 Steady-state expectations of the genetic load 54 3.3 Marginal fitness and purging 57 3.4 Numerical methods 59 3.5 Numerical results 63 3.6 Concluding remarks 71 C H A P T E R IV Conclusion and recommendations for further work 77 Bibliography 79 i i i LIST OF FIGURES Chapter II: 2-1 Fisher's geometrical model of adaptation 18 2-2 Approximation of the fixation probability for new mutant alleles 25 2-3 Mutant vectors in two- and three-dimensional space 31 2-4 Mean fixed drift load against effective population size for n - 1 36 2-5 Mean fixed drift load against A, for n = 1 38 2-6 Mean fixed drift load against effective population size for n = 10 41 2- 7 Mean fixed drift load against n 42 Chapter III: 3- 1 The glide and shift trajectories 61 3-2 Mean genetic load ( x l O - 6 ) of glide trajectories plotted against time in generations 65 3-3 Mean genetic load ( x l O - 6 ) of glide trajectories at generation 10, plotted against s 66 3-4 Mean genetic load ( x l O - 6 ) of shift trajectories against time in generations. . . 69 3-5 Mean genetic load ( x l O - 6 ) o f shift trajectories at generation 10, plotted against s 70 'iv A C K N O W L E D G E M E N T S I thank my advisor, S. Otto, for her patience through the development of my thesis and for making possible my transition to graduate-level mathematics; and the rest of my committee, M . Whitlock and R. Redfield, for their guidance and insight. I also thank the numerous professors that I have approached with theoretical problems including M . Doebeli, P. Greenwood, S. Marion, M . Barlow, B . Bergerson, and H . A . Orr; and doctoral students T. Duty and T. Johnson. For their help in programming and other computational issues, I thank A . Blachford and J. Hauser of the Zoology Computing Unit, and M . Choptuik for use of the Physics Department Linux Cluster. This thesis was supported by N S E R C P G S - A scholarship number 208606-1998 and an N S E R C grant to S. Otto. CHAPTER I: OVERVIEW 1.1 Selection, mutation, and extinction The mandate of conservation biology is the preservation of biological diversity, in part, by reducing the extinction risk of natural populations. Although every population wi l l eventually go extinct, there are intrinsic (e.g. genetic) and extrinsic (e.g. environmental) factors that decrease the expected time to extinction. Evolutionary biology can contribute to this effort by characterizing the genetic factors that contribute to the extinction risk. Before discussing the relevance of evolutionary processes to the extinction risk, it is necessary to first explain the demographic process of population growth and decline. The expected time to extinction is determined by the population's ability to maintain its numbers. To model the growth and decline o f a population, it is useful to assume that its generations are discrete such that there is no mating between generations because the equations become much simpler (Hastings 1997). Consider a population of size N. Let the expected (i.e. average) number of offspring of the z t h individual that survive to adulthood be represented by <W,>. It follows that the expected number of individuals comprising the next generation, <N'>, is: {N') = i(Wt). (1.1.1) A useful quantity for describing the growth of populations is derived from the ratio of population sizes between generations: 1 AT N , v — r N 1=1 where I have used (1.1.1) to show that this ratio is equal to the expected number of surviving offspring averaged over all individuals in the parental generation. A population in which R is less than one, such that the number of offspring that w i l l survive to adulthood is less than the number of parents, wi l l soon go extinct. Furthermore, i f R only slightly exceeds one, then a substantial extinction risk remains because the actual (or realized) population size, TV, may by chance be less than expected over a number of generations. Such variation in R reflects the cumulative effect of fluctuations in the realized number of offspring, Wt, over all individuals in the population. The fluctuations around each average, <W,>, are either caused by chance (i.e. demographic stochasticity) or by environmental changes (Lande 1988). The mean time to extinction is maximized when R is considerably greater than one, when the carrying capacity o f the habitat is very large, and when there is little variation over time in R. The summary variable R retains no information about individual variation in Wh nor about the variation of the mean values, <W,>, among individuals. A s we shall see, the latter is necessary for studying the evolutionary dynamics o f R. In fact, W is considered to be synonymous with Darwinian fitness (Crow and Kimura 1970) and reflects the performance of an individual in its environment. Similarly, R is referred to as the "mean absolute fitness" of a population. The number of descendants that an individual w i l l produce on average, <Wj>, is determined by how well-adapted it is to its environment. Variation among individuals in the degree of adaptation, and hence in the expected number of descendants left, is the sine qua non of natural selection. Yet there is no response to natural selection, and thus 2 evolution, unless fitness variation in the population is inherited by subsequent generations. The principal mechanism of biological inheritance is genetic. Therefore, i f variation in <W,> among individuals has a heritable genetic basis, then there w i l l be evolution by natural selection. Indeed, genetic material that, when expressed, tends to produce a phenotype (i.e. characteristic or form) that outperforms any other can be expected to be found in a progressively greater proportion of the population over time. A s a result, the mean fitness, R, of the population wi l l often increase except when it is at the carrying capacity of its habitat. It follows that selection on variation in fitness can reduce the risk of extinction. Genetic variance is constantly being introduced into a population by recurrent mutation. A mutation is a change in the genetic sequence that is brought about by the inherent inaccuracy of our genomic replication machinery, as well as other genetic processes including transposable genetic elements (Nuzhdin and Mackay 1995). Every biological organism exhibits some measurable rate of mutation (Drake et al. 1999). Although any mutation in a gene sequence in a strict sense creates a different version (i. e. allele) of that gene, I am mainly concerned with those that modify the performance of the subsequent gene product. Because of the inherent complexity of biological systems, predicting the effect that a particular mutation wi l l have on fitness is a difficult task. However, both the results of experimental studies on mutation (Mukai 1964; Fry et al. 1999) and our understanding of how genes and proteins work imply that most mutations that affect fitness reduce individual fitness (i.e. are deleterious). It follows that an individual carrying a large number of mutations, relative to the mean number in the population, w i l l on average contribute fewer offspring to the next generation, reducing the transmission probability of those mutant 3 alleles. This variance among individuals in <W,> introduced by mutation causes a net reduction in R. The magnitude of this reduction is determined by the number of deleterious mutant alleles in the population (Haldane 1937). This number reflects a balance between the generation o f mutations and their elimination by selection. This dynamic equilibrium is often referred to as the mutation-selection balance. Ih a small population, the mean sojourn time of a new deleterious mutant allele (i.e. the mean number of generations until its descendants are eliminated from the population) is also affected by genetic drift. The details of genetic drift w i l l be clarified in the next section. This phenomenon is particularly relevant to problems in conservation biology, as the populations that are studied often have been reduced to small numbers (Caughley 1994). 1.2 Evolutionary consequences of small population size If the offspring are equivalent with respect to juvenile mortality, then the subset that survive to become the next adult generation w i l l be a random sample of those offspring. There is some chance that any particular parent w i l l be over- or under-represented in this sample of offspring. Therefore the expected number of offspring, <W>, is a different quantity than the realized number, W. Random fluctuations in the relative success of genetically distinct groups in a population vary the expected pattern of inheritance and thus cause genetic drift. When the size of this sample between generations is small, these fluctuations increase in magnitude. Because the sample is by definition independent of the biological character of each individual, genetic drift is a process that is independent of natural selection. Put another way, an offspring that is genetically well-adapted to its 4 environment has no guarantee of surviving to reproductive maturity because of accidents that occur over time. The variance of allele frequencies in the sample of offspring caused by genetic drift is a decreasing function of the population size, N. However, this relationship makes several assumptions about the reproductive character of the population, such as equal sex ratio and random mating. I f these assumptions are violated, then drift in the population w i l l behave as it would in an ideal population of another size in which the assumptions hold true. The size of this ideal population is the effective population size. Whereas the original quantity, called the census population size, is the countable number of individuals, the latter is an evolutionary quantity that describes the effect of drift on the genetic state of that population (Crow and Kimura 1970). When the sex ratio is unequal, for example, the mean number of gametes contributed to the next generation is no longer equal among individuals. The effective population size consequently becomes smaller than the census size (Crow and Kimura 1970). Indeed, empirical estimates of the effective size of populations is roughly one-tenth the census size on average (Frankham 1995). Genetic drift can interfere with natural selection by causing individuals with many deleterious alleles to become over-represented in subsequent generations and as a result cause a net reduction in R. There is, in fact, some probability that sampling error can eventually cause every individual in the population to have descended from a common ancestor that carried the most deleterious mutant alleles in its generation. For a haploid asexual population, this process is analogous to Muller's ratchet (Muller 1964; Felsenstein 5 1974), in which the class of individuals with the least number of mutations goes extinct by chance. It is now necessary for our purposes, however, to concentrate on the number of gene copies in the population rather than the number of individuals. When the population has more than one allele at a locus (i.e. a polymorphic locus), it is more convenient to collect gene copies together according to their allelic form (i. e. identity by state) and characterize the genetic state of a population by the frequencies of each allele. There is a chance that drift w i l l eventually cause a deleterious allele to become the only allele in the population. This state is called fixation. The average number of generations required for a new mutant allele to be fixed in a population is determined by the intensity of genetic drift, which in turn is dependent on the effective population size over time. Kondrashov (1995) inferred from the available empirical data on the rate and severity of deleterious mutation that the accumulation of slightly deleterious mutant alleles in the genome should be sufficient to cause the extinction of most populations. This, of course, assumes that the accumulation of mutant alleles corresponds to an accumulation of unconditionally deleterious effects on fitness. On the contrary, it is possible that new mutant alleles that cancel out the deleterious effects of previous mutations are frequently generated. Mutations that are not required to be in combination with other genes to increase fitness are called beneficial. If a mutation w i l l only increase fitness in the presence of certain deleterious alleles, it is called a compensatory mutation. In the second chapter, I wi l l investigate the effect of compensatory mutation on the extinction risk of small populations that accumulate many deleterious mutations. Moreover, the effects of genetic drift are often studied in a population held at a constant size (e.g. Kimura 1955; Lande 1994). Endangered 6 populations, however, are often in decline or have only recently declined to their current level because of human-mediated disturbance (Ehrlich and Ehrlich 1981; Wilson 1988). Very few studies have investigated how the demographic history of a population affects its genetic state and subsequently its extinction risk (Kirkpatrick and Jarne 2000). This is the subject of the third chapter. Hence, the following two chapters represent further contributions to an existing literature on the application of evolutionary theory to the problem of extinction risk (e.g. Lynch and Gabriel 1990; Lande 1988, 1994; Burger and Lynch 1995; Frankham 1995; Hedrick and Mi l l e r 1992). A common feature o f these studies is the genetic load, which acts as an index describing the genetic component of a population's extinction risk. The next section of this chapter wi l l be devoted to identifying the various types of genetic load that are directly caused by recurrent mutation in finite populations. 1.3 Genetic load Genetic variation is the raw material from which all evolutionary adaptations are derived. A lack of genetic variation may even contribute to the extinction risk of a population, as adaptation to a changing environment is required for survival (Burger and Lynch 1995). In addition, there is some evidence that populations with a lack of genetic variation are less resistant to disease and parasitism (Frankham 1995), although we do not know whether the level of resistance depends on genetic variation itself or reflects the fixation of deleterious alleles. Nevertheless, many features of our genetic system that maintain genetic variation are unavoidably associated with a cost, paid as a reduction in the mean fitness of the population (Crow 1970). We refer to each of these fitness reductions as a "genetic load," following Muller 's (1950) use of the term "load" to describe the deleterious 7 impact of mutation on populations. Load is measured relative to an ideal population that lacks the genetic feature of interest. Mathematically, this may be written: where Ro is the mean absolute fitness of the ideal population (Crow 1970). Although mutation generates genetic variation, for example, most new mutant alleles that affect fitness have deleterious effects on fitness (Mukai 1964). Thus, recurrent mutation causes, on average, a reduction in fitness that is called the mutation load (Muller 1950; Morton et al. 1956) and is measured relative to the mean fitness of a population that does not experience mutation. If the effects of drift are negligible, as is the case in an infinitely-large population, then selection wi l l act against mutation to establish an equilibrium frequency for deleterious alleles (Haldane 1937), such that mutant alleles are being removed from the population as quickly as they are being generated. In this thesis, I w i l l limit the term "mutation load" to refer to the expected load in a large population where drift is absent. Mutation load, then, is assumed to be present in a population of any given size, in addition to which other genetic loads w i l l cause further changes in the mean fitness. In finite populations, genetic drift w i l l move the frequencies of mutant alleles away from the equilibria determined by mutation and selection, by causing an allele to be over- or under-represented in subsequent generations. This in turn can create an additional change in fitness called the drift load. The drift load can be apportioned into two components, each associated with a different feature of the probability distribution of allele frequencies. First 8 of all , there is a portion of the drift load that is caused by asymmetries in the spread of the frequencies of deleterious unfixed alleles about their deterministic equilibria, due to genetic drift. We refer to the difference between the expected mutation load and the total reduction in fitness caused by variation in the frequency of polymorphic genes as the segregating drift load. The segregating drift load can contribute the greater fraction of the drift load in moderately large populations, in which the rate of fixation is low and allelic polymorphism is maintained. A s population size decreases, however, allele frequency distributions become more U-shaped (Wright 1937) because new mutant alleles are rapidly eliminated or fixed in the population so that few loci are polymorphic with alleles that affect fitness. Deleterious alleles that are fixed in a population contribute to the second component of the drift load, which I w i l l refer to as the fixed drift load. Because the fixation probability o f deleterious mutations increases exponentially with decreasing population size (Crow and Kimura 1970), fixed mutations make a larger contribution to the drift load in smaller populations. Furthermore, fixed mutations accumulate over time, which can contribute to an incremental reduction in fitness and population size known as the mutational meltdown process (Gabriel et al. 1993; Lynch and Gabriel 1990). The accumulation of fixed drift load may eventually exceed both the mutation and segregating drift loads in small populations (Lande 1994). Therefore, it is reasonable and useful simplification to focus on the fixed drift load to assess the genetic contribution to the extinction risk of a small population. The model presented in Chapter II takes advantage of this simplification to model the effect of compensatory mutations on the genetic load. Because the mutation load does not change much as a function of N, it acts as a baseline from which further reductions in mean fitness occur. The segregating drift load is assumed to be small relative to the fixed drift load when the 9 population is held at a small size. On the other hand, chapter III presents numerical results that do not require such simplifications so the genetic load is treated as a single quantity including mutation load and drift load, both segregating and fixed. Both chapters demonstrate how the expected genetic load of a small population is affected by changes in standard assumptions of previous evolutionary models. 10 STATEMENT OF AUTHORSHIP The following chapter is adapted from a co-authored manuscript to appear in the journal Evolution. I appear as main author and my supervisor, S. Otto, is senior (second) author. I was responsible for the majority of the analysis and for preparing the manuscript. S. Otto was responsible for the initial solution of the one-dimensional analysis. We are in agreement that the contributions of the thesis author are as stated above: Senior Author Thesis Author 11 Chapter II: T H E E F F E C T S O F C O M P E N S A T O R Y M U T A T I O N 2.1 Preface This chapter appears in the journal Evolution under the title "Compensating for our load of mutations: freezing the meltdown of small populations" (Poon and Otto, in press), as a manuscript co-authored with my advisor, S. Otto. The contributions of each author to the manuscript are described on a separate document that precedes this chapter, as per the requirements of the Special Collections and University Archives Division. I was responsible for the majority of the analysis and writing. Redundancies in the manuscript in the context of this thesis have been removed. 2.2 Overview Do small populations, in which there is a substantial probability that slightly deleterious mutations w i l l fix, have a high risk of going extinct as a result? The majority of theoretical studies that have addressed this question affirm that many asexual and some sexual populations with low effective population sizes (see Section 1.2) w i l l inevitably go extinct through mutation accumulation (Butcher 1995; Gabriel et al. 1993; Lande 1994, 1998; Lynch et al. 1993, 1995a, 1995b; Lynch and Gabriel 1990; Schultz and Lynch 1997). Even populations as large as our own species, given what we have measured of mutation, may be in genetic peril (Crow 1997; Eyre-Walker and Keightley 1999; Kondrashov 1995). Thus it appears that deleterious mutation presents a genuine hazard to the conservation of species and the well-being of human populations. However, this threat may have been 12 overestimated as an artifact of how the effect of mutation on the mean fitness of populations has been modeled. The earliest work, such as Haldane's (1937) pioneering investigations into the effect of variation in fitness in an infinite population, simplified the biology of mutations by ignoring the phenotypic origin of their effects on fitness (Hartl and Taubes 1998). Subsequent investigators have done much to improve the biological realism within this classical framework by introducing rates of reversal, epistasis, and effect variation (Butcher 1995; Kondrashov 1994; Lande 1998; Schultz and Lynch 1997). Nevertheless, most previous investigations have assumed that mutations can be classified as either beneficial or deleterious regardless of the current phenotype. Such mutations are said to be unconditionally beneficial or deleterious (Wagner and Gabriel 1990). The effect of a mutation at biochemical, physiological, morphological and behavioral levels may, however, depend on its genomic context (Wright 1968). In other words, a mutation may be deleterious in some genomes and advantageous in others. Interactions within and among gene products, furthermore, provide opportunities for one mutation to compensate for the effects of another at some level of the phenotype. Hence, new mutations may restore fitness losses incurred by previous mutations without requiring true reversals in the gene sequence (Burch and Chao 1999; Hartl and Taubes 1996; Kimura 1990; Ohta 1992; Wagner and Gabriel 1990). These are known as compensatory or suppressor mutations. There is an abundance of experimental evidence, reviewed below, for mutations that conceal maladaptive mutant phenotypes at molecular, biochemical, and organismal levels. With few exceptions (e.g. Hartl and Taubes 1998), we do not yet know how compensatory mutations might influence predictions made by evolutionary models because context-dependence is not 13 a part of the classical framework. Therefore it would be useful to determine how changing the way we model mutations affects the outcome of models addressing biological problems, such as the extinction risk of small populations. To do so requires a framework that defines a mutation by its effect on the phenotype, subsequently treating the fitness effect as a result of the phenotypic alteration. Fisher's geometrical model of adaptation (Fisher 1930), which I w i l l describe in Section 2.4, is a good candidate that has been applied in a handful of studies to evolutionary models of mutation (Haiti and Taubes 1998; Orr 1998; Wagner and Gabriel 1990). One study in particular (Hartl and Taubes 1998) has derived an analytical estimate for the long-term mean fitness reduction caused by fixed mutations, which is called the fixed drift load (see Section 1.3). For reasons discussed previously, this is an important quantity regarding the extinction risk of small populations. I have conducted an analysis of a model similar to Hartl and Taubes' (1998), representing the evolution o f a finite population under mutation, selection, and drift. In this analysis, a probability distribution describing the amount by which a population needs to adapt is summarized by its moments (e.g. its central tendency and spread; Kendall and Stuart 1963). Furthermore, I have supplemented the results of the analysis with computer simulations to evaluate the precision of approximations made in the analysis. Differences between Hartl and Taubes' (1998) analysis and the model presented here have substantial effects on fixed drift load predictions, which I w i l l investigate at the conclusion of this chapter. 2.3 Evidence for Compensatory Mutations There are two types o f empirical evidence that support the postulate that compensatory mutations are a common biological phenomenon. First, there is an extensive 14 body of experimental work identifying "suppressor" mutations — an older term synonymous with compensatory mutation — that conceal the expression o f a mutant gene (Hartman and Roth 1973; Jarvik and Botstein 1975). These mutations may occur within the mutant gene itself or in other genes. Compensation by additional mutations in the mutant gene (i.e. intragenic compensation) is often accomplished by completely or partially restoring the original structural or functional conformation of the gene product (e.g. Hou and Schimmel 1992; Mateu and Fersht 1999; K i m et al. 1994; Hanson et al. 1993). A recent study o f human hemoglobin ( K i m et al. 1994) provides a good example of this process. The authors modeled the energetic folding of a mutant hemoglobin protein and identified a second base-pair mutation that best recovered normal quaternary structure. That these mutations were compensatory was then verified empirically by site-directed mutagenesis. Compensatory intragenic mutations are particularly well understood in the case o f R N A structures ( rRNA, Clark et al. 1984, Hancock et al. 1988, Springer et al. 1995; m R N A , Stephan and Kirby 1993; t R N A , Hou and Schimmel 1992). For example, mutations in stem structures of tRNAs can be compensated for by a matching mutation in the opposite strand (Cedergren et al. 1981; Steinberg and Cedergren 1994). Compensation by mutations in genes other than that containing the first mutation — which w i l l herein be referred to as the target gene — (i.e. extragenic or intergenic compensation) is often mediated by some biochemical relationship between gene products. For example, a compensatory mutation in a gene involved in translation can conceal the expression of the target gene by causing occasional errors in the translation of the target gene's m R N A transcript, such that the subsequent gene product is wildtype (Hartman and Roth 1973; Waterston and Brenner 1978; Murgola 1985; Y u and Spreitzer 1992; E l Mezaine et al. 1998); frequently, these compensatory mutations turn out 15 to be point-mutations in certain t R N A genes. Alternatively, compensation can be mediated through biochemical pathways (Hartman and Roth 1973). When a mutant enzyme does not function sufficiently, the build-up of unmetabolized substrate or the lack of metabolized product may have deleterious effects. Subsequent mutations that shunt flux towards an alternate pathway (Maringanti and Imlay 1999; Dickinson et al. 1995; Gachotte et al. 1997), block flux into the impaired pathway (Manning et al. 1999), or deactivate negative extragenic regulators (Matsuno and Sonenshein 1999) may compensate for this blockage. A second source of evidence for compensatory mutation comes from studies in which direct measures of fitness compensation are made. Many classical studies find that deleterious mutations kept in laboratory populations for extended periods have decreasing fitness effects over time (Haldane 1957; Muller 1938). Furthermore, there are more recent experimental studies in which populations with reduced fitness due to the effects of a particular allele recover through compensatory effects at other loci (Burch and Chao 1999; Elena et al. 1998; Liang et al. 1998). It is worth discussing the results from Burch and Chao (1999) because they are particularly relevant to our model. Burch and Chao (1999) fixed a highly deleterious mutation in a strain of the bacterial virus 06 and then propagated several descendant lineages, varying the effective population size among the lineages by allowing only a small sample to propagate (i.e. bottlenecking). The mean fitness of these bottlenecked lineages recovered over time at a rate inversely proportional to their effective population size. Lineages that recovered the fitness loss in one step were presumed to carry the original allele that was regenerated by a reverse mutation in the deleterious allele. Other lineages that recovered fitness in a gradual step-wise fashion were inferred to have fixed compensatory 16 mutations at other sites in the genome. Burch and Chao's (1999) experiment provides evidence that compensatory mutations are more common than a reverse mutation, and therefore that the reverse mutations are relatively rare in small populations that have fewer genomes in which to accumulate mutations. It follows that small populations may depend more heavily on compensatory mutations for fitness recovery. 2.4 Fisher's Geometrical Model of Adaptation Fisher's (1930) geometrical model of adaptation provides a framework in which unconditionally deleterious and compensatory mutations can be studied together. This geometrical model was originally devised by R. A . Fisher to promote a micromutational view of adaptation, in which adaptation is achieved by successive substitutions of mutations with small effects on the phenotype (Leigh 1987). Fisher's model has made several recent appearances in the literature (Burch and Chao 1999; Hartl and Taubes 1998; Orr 1998; Peck et al. 1997). Its original premise is that an organism's phenotype can be broken down into some number o f components, n, that evolve independently o f one another. I f each component is represented by an axis, then an ^-dimensional rectangular coordinate space (i.e. «-space) can be constructed that represents the full range of possible phenotypes. We let the scale of each axis be equal to the fitness reduction caused by a given change in phenotype from the optimum. Suppose that a given population is genetically monomorphic and that the fixation of mutant alleles occurs instantaneously. Let a point in the «-space labeled A represent the current average phenotype of the population. If there exists a single optimally-adapted phenotype, represented by a point in the «-space labeled O (see F ig . 2-1), then the 17 Figure 2-1. - Fisher's geometrical model of adaptation. The model for two dimensions (n = 2) is shown. The population state is represented by point A that resides at some distance \AO\ from the optimum O, which is set at the origin for convenience. The fitness of a population is a function of \AO\ with a maximum dX\AO\-Q. Mutations are represented by vectors in the n-dimensional character space. (/) Mutations that increase \AO\ are deleterious. (//') Conversely, those decreasing \AO\ are advantageous, (iii) Mutations between points on the circle So do not change \AO\ and are neutral, (iv) The probability that a mutation w i l l be advantageous is a decreasing function of its length. Mutation vectors longer than the diameter of So cannot be advantageous. The effect of a mutation on fitness depends on its context. A mutation has an advantageous effect on one side of the optimum (ii) and a deleterious effect on the other (v). 18 population's degree of adaptiveness is some function of the Euclidean distance between these two points: where a, and o, are the coordinate projections along the z'th axis such that A = (a\, ai, ctn) and 0 = (o\,02, on). For convenience, we place the optimum at the origin such that o, = 0 for all /. It is useful to use a spherical coordinate system instead of the Cartesian coordinate system, such that \AO\ becomes the radius of a hypersphere, So, that is centered at O. Because every point on the surface of So is equidistant from the origin, each corresponds to a phenotype with the same fitness value. A mutation is represented by a vector in this «-space that originates from the point A and terminates at some other point A' (Fig. 2-1). The length of a mutation vector corresponds to the amount of phenotypic change that it has caused along some combination of components. Mutations of a fixed length wi l l terminate on the surface of a hypersphere SA centered at A; this sphere w i l l be useful for explaining the derivation of the probability distribution functions that characterize the mutation process in this system. The fitness effect associated with a mutation depends on where A' is located relative to the surface of So (see Fig. 2-1, i-iii). Without this contextual information, one cannot determine the fitness effects of a particular mutation even i f one knows its size and orientation in Fisher's model. Mutations between points on the surface of So w i l l have a neutral effect on fitness because there is no change in distance to the origin, although each brings about a different change in phenotype. A mutation that brings a population inside So reduces the distance to the origin 19 and thus has an advantageous effect on fitness. Conversely, a mutation that moves the population outside So has a deleterious effect. It is useful to note that a correspondence exists between the size of mutations and their expected effect on fitness. Small mutations are more likely to be favorable because the local curvature of So becomes flatter, such that the probability of being advantageous approaches 1 /2 (Fisher 1930). Although the probability that a mutation is favorable decreases with increasing mutation size (see F ig . 2-1, iv), advantageous mutations of larger effect are more likely to be incorporated into an adaptive walk because they confer a greater fitness advantage than their slightly advantageous counterparts and are therefore less likely to be lost by drift (Kimura 1983). When mutations become too large, however, this effect disappears because large mutations tend to overshoot the optimum and the mean fitness advantage decreases. Mutations larger than the diameter of So are never advantageous. Unlike most classical population genetic models, the genetic context of a mutation defines its effect on fitness in Fisher's model. For example, we can visualize a vector with some fixed direction which is adaptive on one side of the origin and maladaptive on another (see Fig. 2-1, ii and v). Thus, in Fisher's model there are no mutations that are intrinsically and unconditionally deleterious or advantageous. It is this quality of the model that makes it useful to our purposes. The number of orthogonal axes, n, in Fisher's model corresponds to the number of phenotypic aspects by which an organism must adapt (Fisher 1930) assuming that only one optimal phenotype exists. Consequently, the model makes certain postulates about the number of ways that an organism can have a suboptimal fitness. Specifically, there are 2"~l more ways to have a fitness of l - 2 e than l-e (e > 0). This becomes evident i f one considers 20 that the surface area of an ^-dimensional hypersphere increases at a rate proportional to the (fl-l)-th power of its radius. Therefore n, to some extent, quantifies the rarity of optimal versus nearly optimal states. To better understand this we make use of Fisher's (1930) original analogy of a microscope, except that it is now focused with n dials. The accumulation of error in the tuning of each dial results in a large number of alternate settings of the dials that could all lead to the same degree of poor focus. 2.5 Moment analysis of the master equation A s a population fixes mutations, it w i l l wander in «-space at varying distances to the optimum. We use z' to denote the Euclidean distance of the population from the optimum. Throughout this analysis, we w i l l assume that the mean environmental effect is zero and that the population size is small enough that the segregating drift load is negligibly small relative to the fixed drift load. The state of a population can be represented by a probability distribution p such that p(z', t) is the probability that the population state is within the interval [z, z'+dz') at time t. We can describe the dynamics of this probability distribution over time with the following differential equation: (Kimura 1965), where z is a state variable similar to z' that is used to integrate over the entire range of previous states, <U/is the genomic rate of fixation for new alleles, and g(z', z) is the transition probability from state z to z'. The first term represents the rate that the population dt 21 state leaves the interval [z', z'+dz') upon mutation, and the second term integrates the cumulative probability that the population state wi l l enter the interval [z', z'+dz') from another interval [z, z+dz). We assume that the occurrence of mutant alleles is rare enough and/or that recombination is frequent enough that the fixation of a mutation is not interfered with by the simultaneous segregation of other alleles. If p(z', t) converges on the same distribution in the long term regardless of its initial state, then a steady state exists for this differential equation. The steady state is very useful because it describes the probability distribution in the long term regardless of the initial conditions. Because we are primarily interested in finding a steady state we can ignore the details of the approach to a steady state which allows us to redefine the time scale through a change of variables: where T measures time in units based on the average waiting time between fixation events. The differential equation then becomes: which implies that a new mutation fixes with every time step dr. To use this equation requires that we define the transition probability density function g(z', z). Because each transition represents the fixation of a new allele, g{z', z) depends on the probability density function of mutations from z to z' — which we w i l l denote as m(z', z) — weighted by the T — t • fAf (2.5.1) 22 fixation probability u(s) of the new allele, where s is the selection coefficient that depends on z and z: The denominator of (2.5.2) is a normalizing constant such that J g(z',z)dz' = 1. Note that the probability that a mutation moves the population from z ' to z, m{z', z), w i l l depend on the number of orthogonal axes, n. For example, a mutation w i l l have a greater chance of bringing a population closer to the origin of a plane (n = 2) than the origin of a three-dimensional volume (n - 3), because a larger proportion o f mutations w i l l l ie within the hypersphere So- This dependence on n w i l l be elaborated upon below, where simulation methods are presented. , The resulting phenotypic change, once mapped to a change in the Euclidean distance from the origin, can be translated via the fitness function into a change in fitness. Because we are only concerned with changes of Euclidean distance, all calculations from this point are independent of the number of dimensions, n, in the model. Given a fitness function w, we can calculate the selection coefficient: g(z',z) = m(z',z)u(s) (2.5.2) s = w(z') - w(z) w(z) (2.5.3) 23 Note that s is a relative quantity, such that the same change in fitness causes far stronger selection when fitness is low (w(z) « 1) than when it is high (w(z) ~ 1). The fixation probability of a mutation in a diploid population with an effective size Ne is a function of s: l-e~2s u ( s ) = : =4177 1 — e (2.5.4) (Crow and Kimura 1970). This function is well approximated in the neighborhood of s = 0 (see Fig. 2-2) by: 1 u(s;s<0) = — e~2NM 2N u(s;s>0) = 2s + —e~2N's 2N To simplify the analysis, I choose a linear function to describe absolute fitness: w ( z ' ) = l - -B (2.5.5) where z' = B defines the point beyond which fitness is zero and where the optimal fitness was set, without loss of generality, to one. Because dw(z') dz' B (2.5.6) 24 Figure 2-2. - Approximation of the fixation probability for new mutant alleles. The exact and approximate fixation probabilities are plotted against the selection coefficient, s. The dashed line corresponds to the exact formula (Eq. 2.5.4; Crow and Kimura 1970) for effective population size JV= 10. Our approximation, represented by the solid line, provides a good fit in the vicinity o f s = 0, and overestimates the probability of fixation in the region \s\ > 0.05. 25 the intensity of selection equals the reciprocal of the upper bound. We are interested in finding a steady state of the differential equation (2.5.1), so we set the time-derivative on the left-hand side to zero, giving: 0 = -peq(z') + [j(z,z)Peq(z)dz (2.5.7) wherep^z') represents the probability that the population state w i l l be in the range [z' z'+dz') at steady state. Although it is difficult to solve (2.5.7) forpeq(z^) directly, we can characterize the distribution, Peqiz*), by its moments. The nth moment of a distribution h(x) is given by the integral: E[xn] = j:jx-a)nf(x)dx (2.5.8) where a is usually set to 0 to produce the non-central moments (Kendall and Stuart 1963). I proceed by using the method of moment generating functions (Burger 1991) which takes advantage of the expansion of the exponential: 26 ex = l + — + — + — + . 1! 2! 3! (Abramowitz and Stegun 1965) together with (2.5.8) to decompose functions into their moments. A n y density function h(x) of an absolutely continuous random variable x can be transformed by integration into a moment generating function h(k) = \ e h(x)dx, where k is a real number. The moment generating function can be expanded to yield all moments of the distribution described by h(x), and it also uniquely determines h(x). Applying the method of moment generating functions to peq(z'): If we assume that the population is not too far from the optimum at steady state, we can take a Taylor series expansion of g(k;z) around z = 0 to produce a polynomial of z that we integrate to give the moments of peq: 0 = j eki'Peq(z')dz'+ J J eh'g(z',z)Peq(z)dzdz', we produce 0 = \ g(k\z)peq{z)dz 0 =-&,(*) + (• Vo W + Yx(k)mx + y2(k)m2 +...) (2.5.9) 27 where m\ is the i moment of the distribution peq(z) and yfjc) is the j coefficient of the Taylor series expansion of g(k;z) around z = 0. Taking derivatives of (2.5.9) with respect to k gives us a system of equations that we limit to terms of 0(z2) and less: -mx + (y 0 L + Y[ L™, + 72 L0m2) = 0 (2.5.10) Simulations indicate that the higher moments are small and may be safely neglected in most cases. A l l analytical results and approximations were compared to simulation results. It remains to determine what form the mutational distribution function m(z', z) assumes in this analysis. I w i l l discuss this while presenting details of the simulation methods in the next section. 2.6 Distribution functions and numerical methods The evolution of a population by the fixation of new mutations is a simple Markovian stochastic process. Hence, no further information is required to determine the probability that a random variable assumes a particular state when the state at some previous moment in time is given (Gillespie 1992). It is relatively simple to simulate a Markov process with a Monte Carlo simulation program. I developed Monte Carlo simulations of the one-dimensional and ^-dimensional models in order to evaluate their analysis. Uniformly-distributed random numbers over the interval (0, 1) were generated with shuffling to avoid low-order correlations (ran2 from Press 1992). These were in turn used to generate random deviates of other probability distributions by transformation whenever possible, or else by the 28 rejection method (Press 1992). In each cycle, a mutation vector was generated with some length r and orientation 6, each being drawn at random from probability distributions that differed between the one- and ^-dimensional models. In the one-dimensional simulations, mutation lengths were drawn from an exponential distribution with mean 1 / A: I chose (2.6.1) for its relative mathematical simplicity (Mukai et al. 1972; Ohta 1977). Although the general consensus is that the distribution of mutant effects on fitness is probably L-shaped (Keightley 1994; Ohta 1998), there is no simple correspondence between the phenotypic and fitness effects of a mutation, and we cannot be certain what distribution is most representative. With n=\, mutations were oriented towards or away from the optimum with equal probability. That is, the direction of the mutation (6>i) was set to +1 half of the time and -1 otherwise. A mutation vector with length r less than z terminates on the same side of the optimum. If a mutation vector is oriented towards the optimum but is so large that it overshoots the optimum, then the resulting z is the absolute difference between its length r andz. Hence, (2.6.1) z = Z + 9xr . (2.6.2) 29 The selection coefficient (2.5.3) corresponding to a shift from z to z ' then determined the probability of fixation for this mutation. If the mutation was fixed, z became z 'as the population changed state, and the process began another cycle. When evolution takes place in more than one dimension, each mutation can always be described by a triangle with one vertex at the origin, such that the two adjacent sides become radii z ' and z and the remaining side becomes the mutation vector (Fig. 2-3; see also Hartl and Taubes 1998). Thus, regardless of how many dimensions in) there are, the process can be summarized in two. B y this scheme, any mutant vector is completely characterized by its length r and orientation 6, defined as the angle between the mutant vector and the distance vector from the optimum (Fig. 2-3). To determine the probability distribution of r in the n-dimensional model, I made use of the fact that all mutation vectors of equal length terminated on a hypersphere SA- Recall that according to the one-dimensional model, the probability that one of these vectors is of length r is described by (2.6.1). A l l that is needed then is to extrapolate (2.6.1) to account for all mutation vectors of length r (a "bottom-up" procedure). The surface area of a hypersphere increases at a rate proportional to rn~X with respect to the radius r. Using this relation we weight (2.6.1) by r" - 1 and normalize to get: \;Xe'Xrrn-'dr Xe~lrr' T(n) (2.6.3) 30 r Figure 2-3. - Mutant vectors in two- and three-dimensional space. A n y mutation can be completely described by a triangle o f vectors regardless of the number of dimensions n. z represents the distance from the optimum before mutation. Each mutation is a vector of some length r and angle 9 relative to z. Determining the next distance to the optimum, z', is then a simple matter of trigonometric relations. A number of vectors share the same values r and 9, which can be quantified by the sphere with radius r sm9 projected onto the plane orthogonal to the distance vector z. This quantity can then be translated into a probability function of 9 (see text). 31 This is the gamma probability density function (Press 1992). For n>\, (2.6.3) produces a bell-shaped curve with a mean mutation length equal to n I A. I w i l l discuss the relative merits of bottom-up against alternative methods of modeling mutations in Section 2.9. For large n and small A, implementing (2.6.3) in a simulation becomes very inefficient because there is an excess of unrealistically large mutations with a near-zero probability of fixation. Consequently, I chose to truncate the distribution by the same upper bound as that defining the fitness function, B , assuming that mutations of length greater than B were very unlikely to fix. Rescaling (2.6.3) accordingly gives us an incomplete gamma probability density function (Press 1992) for mutation length: A s n increases and A decreases, a greater proportion of the original bell-shaped curve is truncated until only a monotonically-increasing left tail remains. Having described the distribution of mutation lengths, we turn now to a description of the probability distribution of 6. Increasing n expands the number of positions associated with mutation vectors of the same length r and angle 6, which in turn modifies the probability distribution of 6. To illustrate, consider that when n = 2 there are only two vectors associated with each value of 6 (0 < 6 <ri). When n = 3, there is a cone of vectors A V A V (2.6.4) B 32 sharing an angle 6 around radius z, tracing a circular path on a plane orthogonal to the original position vector, z (see Fig . 2-3). Similarly, with increasing n this path becomes a hypersphere of n - 1 dimensions. We can take advantage of this trend to calculate the probability distribution of 0 as a function of n, apportioning the surface area of the mutant n-dimensional hypersphere SA (see Fig . 1) into (n - l)-dimensional hyperspheres associated with values of 6. Because the radius of the hypersphere orthogonal to radius z is equal to r sin 6, the probability distribution function for 9 is: f(9) = ~7n rn (2 rsin 0) n-2 Jo ^ c ' s i n ^ f l (2.6.5) where 0 < 6 < n (Hartl and Taubes 1998; Rice 1990) and where c' is a normalizing constant equal to {* sin-2 ddd V^m(n-l)] ' Jo where T denotes the gamma function (Abramowitz and Stegun 1965). Equation (2.6.5) produces a distribution similar to a Gaussian function that becomes narrower with increasing n. I used a rejection algorithm to generate pseudo-random numbers from this distribution (Press 1992). This method becomes less efficient, however, as the distribution becomes 33 narrower, causing the simulations to take longer when n is large. For large n (n > 10), (2.6.5) is well-approximated by a transformed Gaussian function centered at n 12 (Hartl and Taubes 1996; Fisher 1930), which implies that most mutations point in a direction perpendicular to the direction of the optimum. We have used this approximation to generate random deviates of 6 when n is large to reduce computational time. Given a mutation vector (r, 6) and the current distance from the optimum z, we can calculate the next distance z' from the optimum by the classic L a w of Cosines (see Fig . 2-3): Again, the mutation is fixed with probability u(s), a function of the selection coefficient associated with a phenotypic change from z to z' (Eq. 2.5.3). This completes the derivation of the transition probability g(z', z) in the ^-dimensional model. 2.7 Results in the one-dimensional model Here I w i l l describe the one-dimensional analysis as a means of introduction to the problem and as a segue into the general n-dimensional analysis discussed below. Recall that the transition probability g(z', z) is generated from the probability density function for mutational effects on the phenotype, m(z', z), weighted by the fixation probability, u{s). The first step is to determine the probability density function, m(z', z), based on an exponential distribution of mutation lengths. Because distance to the optimum {i.e. z or z) is an absolute quantity, every transformation from z to z' is associated with two points on the real line on (2.6.6) 34 either side of the origin. Assuming exponentially distributed mutation lengths and treating advantageous and deleterious mutations separately, we have: mi (14 N; M<N) = f («" M\z\-\z'\) -M\z\+\z'\) ) [advantageous] mi (14 14 k1>N)=2 ( e~' M\z'\-\z\) -A(|z'|+|z|) ) [deleterious] Partitioning the denominator of (2.5.2) into two integrals over z> z and z' < z and substituting the above equations produces a complicated solution for g(z', z) in one dimension. First moment estimates from a second-order moment analysis of this model correspond relatively well to simulation results (see Fig . 2-4). The analytical solution for the first moment, which represents the mean distance from the optimum at steady state, can be approximated by the sum of two rational polynomials: where we have omitted the subscript from Ne for clarity. Equation (2.7.1) assumes that e2N » N3, and so gives a good approximation of the exact solution when N> 10. When compared to simulation results, we find that this second-order system makes an accurate prediction of the first moment for low values of 'X (Fig. 2-5). However, for large values of A, B(2N + BX) B2A (2.7.1) m, ~ 2N + (2N + BX)2 27V(27V + 5 A ) 35 0.05 T 0 20 40 60 80 100 120 N Figure 2-4. - Mean fixed drift load against effective population size for n = 1. The mean fixed drift load is equal to the first moment of the distribution Peq(x) when B=\. Solid squares indicate the average of five simulation runs evaluated at A = 10, each consisting of 1000 fixed mutation steps. (Throughout, the first 50 steps were ignored to minimize the influence of starting conditions.) Crosses indicate the average of five simulation runs evaluated at A = 1, each consisting of 1000 fixed mutation steps. The solid curve represents equation (2.7.1) evaluated at A = 10, and the dotted curve represents the upper limit approximation at A = 0 (2.7.2). 36 (2.7.1) predicts that the first moment is nearly independent of A, while simulation results show that the mean distance to the optimum decreases with A (Fig. 2-5). Consequently, (2.7.1) w i l l overestimate the mean drift load when A is large (i.e. when the average phenotypic effect of a mutation is small). Given that the first moment obtains a maximum value at A = 0 in simulation results, we can approximate an upper bound estimate of the mean distance to the optimum by setting A = 0 such that (2.7.1) becomes: B (2.7.2) 1 + 27V B y substituting (2.7.2) into the absolute fitness function (2.5.5), we find that the upper bound estimate of the expected drift load from fixed mutations is: L = — — (2.7.3) \ + 2N Note that this upper bound estimate is independent of the selection intensity, 1 / B (see Eq . 2.5.6). Because (2.7.3) is a reciprocal function of N, the mean drift load due to fixed mutations becomes very small for N> 10. For reasonably-sized populations, the probability of fixation (2.5.4) for deleterious mutations becomes very small and most such mutations are readily compensated. 37 •o cd o c co CD E 0 . 0 1 2 0 . 0 1 0 . 0 0 8 0 . 0 0 6 0 . 0 0 4 0 . 0 0 2 • • < • simulation equation 1 0 2 0 3 0 lambda 4 0 5 0 Figure 2-5. - Mean fixed drift load against A for n = 1. The mean fixed drift load is equal to the first moment of the distribution Peq(x) when 5 = 1 . For this plot, the effective population size is set to N = 50. Points indicate the average of 10 simulation runs, each consisting of 1000 fixed mutation steps. Standard errors are too small to appear on the graph. Simulation results indicate that the first moment decreases monotonically with A. The dashed line represents the second-order approximation (2.7.1). 38 2.8 Results in the n-dimensional model The mutation probability function m(z', z) becomes very complicated when Fisher's space consists of more than one dimension. Given a population residing at a distance z from the optimum, we must consider the cumulative probability of every mutation vector of some length r and orientation 6 that terminates at the surface of a hypersphere with radius z' in order to calculate m(z', z). It is simpler to index each mutation by the angle (f> internal to So formed by the radii extending to A and to A': m(z',z) = C / ; A e x p ( - A ^ z 1 + (z'f - 2z^'cos</>)sin' ,-2 040 which contains (2.6.6); C is a normalizing coefficient. A n exact solution of this integral appears to be mathematically intractable. However, simulation results indicate that a maximum with respect to A exists for the first moment of the distribution peq(z') at A = 0. Therefore, we set A = 0 to obtain an upper bound estimate of the fixed drift load for the case of exponentially distributed mutation lengths along each axis. This assumption allows mutations in a population at a distance z from the optimum to jump to any point within the n-space with equal probability. The rc-space can then be thought of as a "uniform probability volume" divided into a series of concentric shells around the optimum like an onion. It follows that the probability density function of mutations terminating on the surface of a hypersphere centered at the optimum with radius z is: 39 (2.8.1) where I have bounded the distribution to exclude lethal mutations (z' < B). Note that (2.8.1) is independent of the original position z. Equation (2.8.1) is equal to the limit of (2.6.4) as A approaches zero. To prove this, I w i l l use the following asymptotic expansion for the denominator of (2.6.4): r (n ) - T(n, Afl) = e - " ( A B ) " f : r 0 * } UB)J (Eq. 6.2.5 from Press 1992). Substituting this expansion into (2.6.4) and taking the limit, we have: lim em(XB)n £ 1 W (XBy -Xz' / f\n—\ A^O ,„ / 1 Afl A ex\B)n + + ... n (n + \)n n(z') / \ n - \ B" The moment analysis of an ^-dimensional model using this approximation produces results which are very close to those from the simulations when A is near zero (see Figs. 40 T J CO O 0.35 0.3 0.25 0.2 C _ 0.1 5 i 0.1 0.05 0 A O A - e q u a t i o n • s imulat ion, X=0.1 A exponent ia l , X=0.1 O g a m m a , X=0.1 20 A 40 60 80 O A 1 oo N Figure 2-6. - Mean fixed drift load against effective population size for n = 10. The dashed curve represents the upper limit equation (2.8.3), where A = 0. Solid squares indicate simulation results with A = 0.1 using the original mutational model, which assumes exponential distributions of mutational effects along each axis that accumulate to produce a bell-shaped gamma distribution for total mutation length (2.6.4). We were unable to obtain simulation runs for values of N greater than 30, because mutations that fixed were extremely rare under these conditions. Open triangles represent the mean of five simulation runs consisting of 1000 fixed mutation steps each, with an exponential distribution (A = 0.1) describing the total mutation length. Open diamonds represent the mean of five simulation runs consisting of 1000 fixed mutation steps each, with an L-shaped gamma distribution (A = 0.1, a = 0.5) describing the total mutation length. 41 T3 ro o 0.6 0.5 -I 0.4 0.3 c £ 0.2 \ 0.1 0 0 s x J l -/ A A - e q u a t i o n A • s i m u l a t i o n , =^1 A s i m u l a t i o n , X=20 X a p p r o x i m a t i o n 10 15 20 # dimensions (n) 25 Figure 2-7. - Mean fixed drift load against n. For this plot, the effective population size is set to N= 10. The dashed curve represents the upper limit approximation (2.8.3), where A = 0. Solid squares indicate simulation results for A = 1. We were unable to obtain results for n > 10, because fixation events became extremely rare under these conditions. Crosses indicate the means of five simulation runs consisting of 500 mutation steps for a truncated distribution at A = 0 that generated only values of r that were likely to fix. Truncating the distribution decreased computation time but in exchange slightly overestimated the load for larger values of n. Closed triangles indicate the average of five simulation runs for A = 20, each consisting of 1000 fixed mutation steps. 42 2-6, 2-7). The upper limit solution for the first moment can be approximated by the sum of a simple rational polynomial and a vanishing term: nB J (2N)ne-2N ^ n + 2N O r(n)-T(n,2N) (2.8.2) The neglected terms are very nearly zero in the domain N> n. Even for N<n, simulations indicate that nB I (n + 2N) is a good approximation for the mean distance to the optimum for low A and provides an upper bound for m\ as A increases. Note that the first term reduces appropriately to (2.7.2) in one dimension (n = 1). Consequently, the mean equilibrium drift load caused by the fixation of new mutations is: n + 2N when A is small, which overestimates the load when small mutations are common (higher A). Equation (2.8.3) is again a reciprocal function of N, such that the mean drift load due to fixed mutations rapidly decreases with effective population size. Also , (2.8.3) is a hyperbolic function of the number of dimensions n. 2.9 Sensitivity of the model to its assumptions The features of Fisher's geometric model of adaptation that make it mathematically useful also make it biologically oversimplified. One frequent criticism of Fisher's model is that its spherical symmetry is too idealized to apply to real organisms. A l l o f the orthogonal 43 axes are standardized to fit under the same fitness function and mutational distribution. With respect to fitness, we can change the scale for each axis such that the same displacement experiences the same selection intensity. However, this would alter the mutation probability associated with a displacement along each axis. Consequently, the spherical geometry of Fisher's model cannot perfectly capture both the fitness effects of mutations and their frequency distribution. The model also assumes that the mutational distribution is symmetrical along each axis, with an equal probability of going towards and away from the optimum. If the number of potentially compensating loci varies among traits, for example, then this assumption would not hold. Hence Fisher's model is almost surely an imperfect representation of a biological system. On the other hand, models that ignore compensatory mutations are also imperfect representations. Given that neither Fisher's model nor a model lacking compensatory mutations perfectly captures reality, modeling both is still instructive because it allows us to determine the sensitivity of evolutionary predictions to two extreme alternatives. There is good reason to believe that the results obtained from this formulation are robust enough to changes in the assumptions to be useful for understanding the fixed drift load in small populations. There is a handful of theoretical studies that have either examined a related problem in a roughly similar fashion (Lynch and Gabriel 1990; Hai t i and Taubes 1998) or addressed a different evolutionary problem in a model analogous to Fisher's geometric model (Robertson 1970). It is interesting that the results of all these studies bear a non-trivial resemblance to those presented here. Robertson (1970) studied the load caused by the drift of two alleles maintained near a polymorphic equilibrium q by heterozygote 44 superiority. The departure of allele frequencies from the equilibrium could be represented by the variance of Wright's (1937) distribution, giving an expected load of: (2.9.1) l + 4Nq(l-q) (Robertson, 1970). This model is superficially analogous to Fisher's model for n = 1, in that there is an optimum located on a real line, over which selection and drift act antagonistically to determine a steady-state distribution. Note that equation (2.9.1) and the result obtained by the one-dimensional model, (2.7.3), are similar in their inverse dependence on population size. A simulation study of the mutational meltdown of small populations by Lynch and Gabriel (1990), like Fisher's model, allows for compensatory mutations. In this model, Af individuals experience a Poisson distribution (Feller 1951) of mutations whose effects on fitness have a constant mean and variance. Although mutations are on average deleterious, there is a constant probability that a beneficial mutation wi l l appear in this model, with the restriction that individuals with a fitness greater than one are not allowed. Consequently, the model is similar to ones investigating the incorporation of unconditionally beneficial mutations (Lande 1994; Schultz and Lynch 1997), except that beneficial mutations only appear once fitness has decayed to some extent. This differs from Fisher's model, where the probability of a beneficial mutation continues to increase as the population moves further from the optimum and approaches 1/2 . Lynch and Gabriel (1990) found that allowing for beneficial mutations caused the mean extinction time of the population to increase by orders 45 of magnitude. The mutational meltdown is not frozen, however, unless the fraction of beneficial mutations is sufficiently high (Schultz and Lynch 1997). More recently, Hartl and Taubes (1998) estimated the fixed drift load in their study of adaptation in Fisher's model as: Although we have studied the same model here, different assumptions have been made about the nature of mutations. We have evaluated the effect of each difference by running simulations under various alternative assumptions. First, Hartl and Taubes (1998) used an absolute selection coefficient s = w(z') — w(z) whereas the probability of fixation for a new mutation is really a function o f the relative selection coefficient s = (w(z') - w(z)) I w(z) (Eq. 2.5.3; i.e. selective advantage, Fisher 1930). Effectively, their assumption was equivalent to relaxing selection away from the optimum. According to our simulations, changing s from an absolute to a relative measure in Hartl and Taubes' (1998) study produces results that can be approximated by: Note that whereas the absolute selection coefficient causes the load estimate to have a linear dependency on the number of dimensions, n, (2.8.3) and (2.9.3) are nonlinear functions (2.9.2) L = n (2.9.3) n + SN 46 which asymptote at 1 as n gets large. The remaining difference between (2.8.3) and (2.9.3) results from what is assumed about selection and mutation. Hartl and Taubes (1998) used a quadratic fitness curve instead of a linear one (Eq. 2.5.5). Simulations indicate that the term 8N in (2.9.3) becomes roughly AN with a linear fitness function, suggesting that the fixed drift load is lowered when the fitness function curves downward away from the optimum (i.e. with synergistic epistasis; Schultz and Lynch 1997). Furthermore, I had assembled a probability distribution for mutation length, r, from component distributions along each axis in a bottom-up fashion, whereas Hartl and Taubes (1998) assumed that the total mutation length in n dimensions was uniformly distributed (see below). This difference explains the remaining discrepancy between (2.9.3) and (2.8.3). That both estimates of the fixed drift load are inversely related to the effective population size, however, implies that this result from Fisher's model is fairly robust. Comparing the results from Hartl and Taubes (1998) and this study highlights our ignorance about how mutations affect phenotype. Recall that I have assumed that the distribution for mutation length along each axis is exponential and that the total mutation length is derived from these component effects (a bottom-up derivation). However, the conventional procedure with respect to Fisher's model is to select a specific shape to the total distribution of mutational lengths, leaving the component distributions along each axis unspecified (a top-down derivation) (Kimura 1983; Orr 1998; Hartl and Taubes 1998). Little data exist concerning the shape of the distribution of mutational effects on phenotype. There is a general consensus, however, that the distribution o f deleterious mutant effects on fitness is L-shaped (Mackay et al. 1992; Lyman et al. 1992; Keightley 1994; but see Garcia-47 Dorado 1997). For mathematical reasons, it is difficult to derive a probability density function of mutant effects on phenotype that is L-shaped using the bottom-up approach (Orr 2000). Nevertheless, the mutant effects on fitness may still have an L-shaped distribution because the phenotypic changes must first be mapped onto some fitness function. This mapping depends, however, on the current state o f the population, as well as on the number of dimensions of the system in Fisher's model. Recall that mutations become less likely to be oriented directly towards or away from the optimum as the population moves further away from the optimum and as n increases (Figs. 2-1, 2-3; Eq . 2.5.3; Hartl and Taubes 1998). In other words, mutation vectors become increasingly likely to be tangential to the surface of the hypersphere 5*0. Consequently, mutations with vanishingly small effects on fitness may increase in frequency. In this case, a roughly L-shaped distribution of fitness effects may be seen even when the distribution of phenotypic effects is bell-shaped (as in the bottom-up derivation). On the other hand, very large populations (such as the Drosophila populations used to generate the expectation of an L-shaped distribution of mutational effects on fitness) are expected to reside at or close to an optimum. If this is true, then the observed distribution of fitness effects should be similar in shape to the distribution of total mutation length for populations near an optimum. Consequently, it is worth investigating mutations whose total length follows an L-shaped distribution. We have run simulations using an exponential distribution and an L-shaped gamma distribution (that can be obtained from (2.6.3) by setting n = 1 and 0.5, respectively, with A = 0.1 in both cases) to describe the distribution of total mutation lengths in n dimensions (Fig. 2-6). In both cases, the dependency of the fixed drift 48 load on the effective population size, N, is qualitatively similar to our original result (Fig. 2-6). Therefore, results from Fisher's model remain fairly robust under varying mutational models. 2.10 Implications for the extinction risk of small populations The most important result of this model is that changing the way mutations are modeled results in qualitatively different predictions regarding the extinction risk of small populations. The inevitable extinction of small populations is a common feature of genetic models developed within the classical framework of mutations whose effects on fitness are unconditionally deleterious and cannot be compensated (Lynch et al. 1993; Lande 1994). Al lowing for compensatory mutations, however, we find that mean fitness does not suffer an inexorable decline but rather reaches a steady-state level. This "freezing" of the mutational meltdown occurs because, as the population moves further from the optimum, a higher fraction of mutations increase fitness. Although it seems unlikely that all mutations can be compensated (as in Fisher's model), it is equally unlikely that compensatory mutations can be safely ignored (as in the classic framework). Whether reality lies nearer the level of compensation assumed in Fisher's model or in the classic framework is an empirical matter that has yet to be addressed. Such data are needed to resolve whether small populations are at risk of genetic meltdown and to determine the rate of this meltdown. Some previous models of the accumulation of mutations in small populations focus on unconditionally deleterious mutations with a constant fitness effect s occurring at a constant rate /I (Lynch et al. 1993; Lynch and Gabriel 1990, but see discussion above). 49 Similar models draw unconditionally deleterious fitness effects from a continuous distribution (Lande 1994) or both deleterious and advantageous mutations at different rates (Schultz and Lande 1997; Lande 1998). In these examples, a population is unable to halt or even slow its fitness decline once it has begun, because then the influx of deleterious mutations overwhelms that of beneficial mutations. I f deleterious mutations are progressively less likely to accumulate as mean fitness decreases — as occurs when there is synergistic epistasis among deleterious mutations (Kondrashov 1994; Schultz and Lynch 1997) — the population reaches a point beyond which its mean fitness no longer decreases. Schultz and Lynch (1997) argue, however, that an unrealistically high level of synergistic epistasis is required. Interestingly, the meltdown is similarly frozen in Fisher's model, in which a population at the optimum has no beneficial mutations available to it and an increasing number as it declines in fitness. B y evaluating the problem in terms of Fisher's model, we are explicitly considering the context-dependence of a mutation's effect on fitness. If there is a substantial class of mutations that are readily compensated, we expect . that the drift load caused by the fixation of new mutations in small populations is unlikely to be sufficient to cause the extinction of all but the smallest populations. I have shown that the load caused by the fixation of deleterious mutations in the range N > n is too small to affect a population with a modest reproductive excess. There may exist, however, a sub-class of mutations that cannot be compensated. Such mutations cannot be modeled using Fisher's model and w i l l contribute to a mutational meltdown (Lynch and Gabriel 1990), albeit at a slower rate. O f course, other risks of extinction including demographic stochasticity and ecological degradation are more than sufficient to place many small populations in peril 50 (Lande 1988, 1993). Furthermore, small populations may be unable to adapt quickly enough to a changing world, which would also place them at risk (Burger and Lynch 1995). The pervasiveness of anthropogenic disturbance of habitats may reduce the population size of species to a point where all these forces can act in concert to cause extinction. Therefore, although our analysis lends insight into the effects of mutation on small populations when compensatory mutations are allowed, it would be unwise to base decisions regarding the conservation of endangered populations solely on these results. 2.11 Implications for pleiotropy Hartl and Taubes (1998) have discussed the relationship between the number of dimensions, n, in Fisher's adaptive space and concrete biological properties, such as the number of loci underlying a quantitative trait. It is also possible that n reflects the extent of pleiotropy in the average mutation (Orr 1998). Because this analysis substantially alters predictions of how n affects the fixed drift load, it is worth discussing the biological significance of n. To avoid confusion, I w i l l denote the number of axes in Fisher's model as « F and use n to denote the hypothetical number of components that make up an organism's phenotype. Although there is an indefinitely large number of measurable characters in any organism, this does not necessarily imply that n is also large. This is analogous to a principal components analysis of multiple morphological measurements, where most of the variation present can be accounted for by a much smaller number of axes. We can partition n into discrete sets and apply Fisher's model to each set, effectively making different assumptions about how mutation and selection act on these different trait combinations. A t one extreme, letting «F = n implies that the average mutation wi l l nearly always have a non-zero effect on 51 every phenotypic component, which is biologically equivalent to being completely pleiotropic. A t the other extreme, n can be completely partitioned into n individual sets, each represented by a one-dimensional version of Fisher's model (i.e. n?= 1). This assumes that every mutation wi l l affect only one of the n phenotypic components at a time; there are no pleiotropic effects on any other components. The subsequent fixed drift load is predicted by the product of n and the one-dimensional result, 1 / (1 + 2N) i f loads combine additively across sets. This load without pleiotropy, n I (1 + 2N), is always greater than the load with complete pleiotropy, nl(n + 2N), for n> 1. This implies that the pleiotropy inherent in Fisher's model substantially reduces the predicted drift load caused by new mutations, because every mutation can compensate several components at once. Alternatively, let n represent how an organism's phenotype w i l l respond to mutation and selection, rather than being a description of the phenotype itself (Orr 2000). Increasing n within Fisher's model (with « F = n) w i l l decrease the chance that a new mutation bring a population closer to the optimum (Fisher 1930). If the organism's phenotype evolves as i f it were a single character (n = 1), then half of the mutations that occur point in an opposite direction to the remaining half. When n is greater than one, then mutations do not affect all characters equally. Because the chance that maladapted characters w i l l be little affected by a new mutation increases with n (see Fig . 2-3), the delay between deleterious and compensatory mutation events also increases. This explains why the fixed drift load is an increasing function of n within Fisher's model. 52 I have reviewed the experimental literature that provides good evidence that compensatory mutations occur (Section 2.3), but there is a lack of empirical measures for what the overall extent o f compensation might be. From Fisher's model, we can predict how n is related to measurable quantities, such as the fixed drift load and the average degree of compensation among mutations, that can be used to extrapolate estimates of n from empirical data. Such interpretations are only as relevant, however, as Fisher's model is an accurate representation o f biology. That predictions concerning the extinction risk o f small populations depend qualitatively on this issue emphasizes the importance of further research on compensatory mutation. 53 C H A P T E R III: T H E E F F E C T S O F D E M O G R A P H I C H I S T O R Y 3.1 Overview Most of the theoretical studies that I have discussed so far have been concerned with the evolutionary consequences of a constant small population size in relation to the genetic load (e.g. Lynch and Gabriel 1990; Lande 1994). This "small-population paradigm" (sensu Caughley 1994) is common because it is mathematically simpler to model allele frequency evolution in a population of constant size. The evolutionary consequences of population decline, however, do not readily accommodate theoretical study (Caughley 1994). It remains to be understood how changes in the size of a population (i.e. its demographic history) affect allele frequency evolution and the corresponding genetic load. Because fluctuations in the allele frequency accumulate over time, alleles in a finite population w i l l rarely occur at the equilibrium frequency determined by mutation-selection balance (see Section 1.1). The corresponding change in the genetic load is one consequence of the demographic history. In this chapter, I w i l l investigate how the demographic history of a population that has recently declined from a very large size affects the transient genetic load. In other words, consider a population that is currently at a small size, but was at a very large size some known number of generations ago. With all else being equal, how does the size of the population in the intervening generations affect the current genetic load? 3.2 Steady-state expectations of genetic load 54 To understand the effect of demographic history on the genetic load, we must place results o f the model in the context o f the steady-state load expected in populations o f constant infinite or finite size. The genetic load caused by deleterious alleles at equilibrium in an infinite population for an arbitrary degree of dominance, h, is given by Crow (1970) as being approximately: L = n + qhs(\- ju) (3.2.1) where s is the selection coefficient of the deleterious allele, ju is the forward mutation rate to the allele, and q denotes the equilibrium allele frequency under mutation and selection (Haldane 1937). When the deleterious allele is completely recessive such that h = 0, (3.2.1) simplifies to fi. If h is appreciably greater than 0, then q can be approximated by fi I (hs) and (3.2.1) can be approximated by 2^u; on the other hand, i f h is very nearly 0, then (3.2.1) lies between jl and 2^ u and the exact equilibrium solution for q is required (Crow 1970). Note that the load caused by a deleterious allele in an infinite population is independent of the strength o f selection against the homozygote, s. This is equivalent to stating that every mutation with some heterozygous expression, whether lethal or nearly neutral, contributes equally to the genetic load in an infinitely large population. Numerical results obtained by Kimura et al. (1963) indicated that the steady-state genetic load in a finite population caused by a moderately deleterious allele (s < 0.1) increases above the equilibrium (3.2.1) with smaller constant population size, regardless of 55 whether the allele was additive or recessive (h = 0.1). Moreover, the steady-state load in either case was never less than the value attained in an infinite population (Kimura et al. 1963). Their most interesting result, however, was that the steady-state load caused by a slightly deleterious mutation was greater than the load caused by a comparatively severe mutation until the population size became small enough that drift overcame selection against the severe mutation. For s > 0.5, however, the required population size was very small (TV < 10). With the exception of some special cases, the steady-state distribution of allele frequency obtained by Wright (1945) and Kimura (1955) cannot be explicitly integrated to derive the moments of the genetic load (Kimura et al. 1963). In the case of lethal alleles (e.g. h = 0,s= 1), however, the steady-state allele frequency distribution can be approximated by: - 2^\xp(-2Nx2)x4N^ (3.2.2) (Crow and Kimura 1970). Integrating the product of the load caused by a completely-recessive lethal allele, L = x2, over the distribution (3.2.2) gives the mean load: - r q + 2 ^ , 2 ^ *\ ni+2Nfi) j (3.2.3) where T(-) denotes the gamma function. The second term is vanishingly small for all u. and JV> 0, such that (3.2.3) is well-approximated by u,. Hence, the mean steady-state load caused by a completely recessive lethal allele is essentially independent of population size. Together, these results imply that mildly deleterious alleles (s < 0.1) are more important in 56 the long-term effect of a reduction in size on the genetic load (Kimura et al. 1963). Results from the steady-state distribution, however, conceal the transient evolution of the genetic load that can be revealed by iterative calculations of the distribution o f genetic load. Before I discuss the basis of such calculations, however, I w i l l discuss a process that has an important effect on the transient load. 3.3 Marginal fitness and purging A change in allele frequency does not only alter the genetic load, but it can also alter the marginal fitness o f a deleterious allele. The marginal fitness o f an allele is the fitness averaged over all possible zygotic combinations between the allele and other alleles in the population. With inbreeding, the marginal fitness of an allele is more influenced by its effect in homozygotes. During a transition from panmixis to an inbred mating system, the onset o f inbreeding may therefore be accompanied by a change in the marginal fitness of deleterious alleles. The effect that a change of an allele's marginal fitness has on its frequency in subsequent generations depends on the dominance relationship between alleles in the population. Because homozygotic combinations become more common with inbreeding, the marginal fitness of a recessive deleterious allele declines with inbreeding. The reduction in the mean fitness of a population that accompanies this decline is called inbreeding depression (Charlesworth and Charlesworth 1987). In the generations following the onset of inbreeding, the overabundance of homozygotes of the recessive deleterious allele may cause more copies of the allele to be eliminated by selection from the population than expected under random mating (i.e. before inbreeding). This phenomenon is called purging (Templeton and Read 1984). One motivation for the large amount of literature (Lacy and Bal lou 1998; Fu 1999; 57 Fu et al. 1998; Hedrick 1994; Barrett and Charlesworth 1991) devoted to studying purging is the possibility that prolonged inbreeding may be a useful technique for improving the "genetic quality" of small populations (Templeton and Read 1984; Hedrick 1994; Hedrick and Mi l le r 1992). However, because the elimination of alleles causes a temporary reduction in the mean fitness of the population, the inbreeding process may itself cause extinction (Hedrick 1994; Fu etal. 1997). Because the marginal fitness of a deleterious allele also changes with the frequency of the allele in the population, purging is not unique to changes in the mating system but can also be produced by genetic drift (Wang et al. 1999; Kirkpatrick and Jarne 2000). A recessive deleterious allele that becomes over-represented in the population after genetic drift forms more homozygotes, thus exposing the allele to increased selection (Lande and Schemske 1985; Wang et al. 1999; Kirkpatrick and Jarne 2000). This phenomenon has recently been studied in theoretical models of finite populations initially sampled from an infinitely large ancestral population, and subsequently held at a constant size (Wang et al. 1999) or allowed to return to the original population size (Kirkpatrick and Jarne 2000). Although such bottleneck models can be associated with a founding event (i.e. the colonization of an unoccupied habitat) or a sudden and very severe truncation in population size, it is less useful for studying the dynamics of load in gradually declining populations. For mathematical reasons, the analytical methods available for the study of the evolution of genetic loads due to recurrent mutation are not easily extended to declining populations. Therefore, I have chosen to conduct a numerical investigation of the transient genetic load in a declining population. 58 3.4 Numerical methods I have carried out exact calculations in a Wright-Fisher model of allele frequency evolution in a randomly-mating diploid population as described by Ewens (1979), in which the dot product of the probability distribution of allele frequencies at the t-th generation, Pt, and the matrix of transition probabilities, M , gives the distribution of allele frequencies in the next generation. Each distribution Pt is a vector of 2Nt+ 1 elements, and the matrix M i s a rectangular (2A / t+i+l)x(2A / t+l) matrix. Modifying the size of the matrix and the associated vectors permits changes in population size (Tyvand 1993). The /, k-th entry of M i s equal to the transition probability from k to i alleles in the population according to the binomial formula: P{i\k) = . 2AT,+1 - i 1-2N. (3.4.1) tJ (Feller 1950) where the allele frequency k/(2N) may be modified by selection and mutation. I assumed equal forward and backward rates of mutation at a typical value of 10 - 6 . Thus, successive generations were calculated with evolutionary processes occurring in the following order: selection, mutation, genetic drift. A t the end of each cycle, the nth moment of the probability distribution of genetic load at that generation was calculated from Px according to the formula: (L') = _ k ,2 - - k k V 2N(t) :(1-: 2N(t) 2N(t) p(k,t). (3-4.2) 59 where p(k, t) is the k-ih. element of Pt. The mean and variance of load are easily obtained from the first and second moments obtained from (3.5.1) (Kendall and Stuart 1963). I have modeled a step-wise decline in population size. The population size is assumed to be initially large, declines to a size of N* for T generations, and then declines again to its final size of N'. I explored two different classes of trajectories (Figure 3-1). The first ("glide" trajectories) set % = 10, N' = 10, and varied the intermediate iV* from 10 to °°. The second ("shift" trajectories) set N* = 100, N' = 10, and varied x from 1 to 10. Assuming a very large initial population, the allele frequency distribution in generation 0 was the equilibrium allele frequency determined by mutation and selection. The exact numerical solution of the equilibrium frequency was used instead of Haldane's (1937) approximations because the degree of inaccuracy in the approximate solution varies with the parameters s and h and can obscure the results. I used computer algorithms written in the C programming language to perform exact calculations of the Wright-Fisher model. The number of calculations required per generation was on the order of N2 since each allele frequency class k I (2Nt), k= {0, 1,2, . . . , 2iVt}in the vector P t o f length 2JVt+l could make some contribution to every class comprising the distribution vector in the next generation, P t+i- Considering the number of calculations required to take mutation, selection, and drift into account, the total number of calculations becomes prohibitively large for large populations. Approximate methods that lower the number of calculations required for large population dynamics, such as combining adjacent 60 Figure 3-1. - The glide and shift trajectories. In each case, population size is plotted against elapsed time in generations. It is assumed that the first generation is sampled from an infinitely large population. The glide trajectories, represented by the left plot, are step-wise trajectories that summarize the effect of historical population size preceding the final generation to a single value, N*. Upon the final generation (T = 10), the population makes a second decline in numbers to If = 10. The shift trajectories, represented by the right plot, are step-wise trajectories that summarize the effect of the rate of decline to a single variable, x, that corresponds to the generation upon which the second decline from N* to i V occurs. glide shift N x=10 i i • i + + i + i » time time 61 allele frequency classes into "bins" whose behavior is determined by the mean allele frequency of its member classes, failed to achieve a sufficient degree of accuracy to be useful. Consequently, the dynamics of the allele frequency distribution were explored solely for small populations (N < 100). Recall that the effect of drift on the allele frequency distribution is inversely proportional to population size. Because the distribution changes slowly while the population is still large, a disproportionately greater part of allele frequency evolution occurs while the population is of small size. Given that the decline is recent, it is a reasonable approximation to omit the generations when the population is large so that the population immediately declines from an infinite to small size (N< 100). The calculations were not quite exact because of limits on the accuracy of computer operations with 16-bit processing. The binomial distribution (3.4.1) is prohibitively difficult to implement for large N(N> 50) because several factors become vanishingly small (e.g. high powers of x) or excessively large (i.e. the factorial AH). One method for circumventing this problem is by decomposing a function into parts so that extremely large or small numbers do not appear. For example, the binomial coefficients in (3.4.1) can each be expressed as an iterative product (e.g. ra\ a\ a a-I a-b + 1. •x x . . . x ). Such (a-b)\b\ b b-1 1 iterative calculations, however, can be time-consuming. Alternatively, performing the calculations on a logarithmic scale and subsequently back-transforming the results can also prevent the use of extremely large or small numbers. I used this second method for calculating the binomial coefficients, taking advantage of the fact that the factorial of an integer n can be written as a gamma function: 62 w! = r(w + l) = J z V ' d z 0 (Abramowitz and Stegun 1972). B y applying an approximation obtained from Press (1992, p. 213), I obtained an approximate formula for the binomial distribution that I used for all population sizes: x' (1 - x)N~' = exp{ln T(N +1) - In F(N - i +1) - In r(i +1) + i In x + (N - i) ln( l - x)} V ' J where the notation in the binomial formula (3.4.1) has been changed to avoid cluttering the right-hand side. This formula allowed fast calculations of the binomial distribution that were sufficiently accurate in comparison to exact calculations for small population size. 3.5 Numer ica l results The numerical results of the calculations from the glide trajectories are presented in Figures 3-2 and 3-3. The general qualitative result of these figures is that the mean genetic load caused by a mutant allele depends not only on the dominance and strength of selection but also on the demographic history of the declining population, represented by N*. Only the results for two members of the glide trajectories are shown for clarity: fast corresponds to the population with the smallest size before x = 10 (TV* = 10), and slow, corresponds to the population with the largest size before x = 10 (N* = 100) studied. In the additive model (Fig. 3 -2a), the load caused by a lethal (5=1) allele remains at the equilibrium value, 2jx (Eq. 3.2.1), regardless of N*. This is supported by the fact that the initial sample from the infinite 63 ancestral population (i.e. generation 1) has the same mean load irrespective of N* and s and that the mean additive load is a linear function of allele frequency when mutant homozygotes are absent (Eq. 3.4.2). In contrast, the mean load of non-lethal additive alleles increases above 2\i at a rate that is greater for smaller N* (Fig. 3-2o). Note that the load for s = 0.5 increases more rapidly but is eventually overtaken by the load for 5 = 0.1 regardless of N*. Relative to the load of recessive alleles (Figs. 3-2b, c), however, the transient mean load of additive alleles is very slow to increase from 2/1. Drift assumes a more complicated role in the transient evolution of partially recessive alleles in declining populations. This is illustrated by Figure 3-2Z>. Unlike the additive case, the initial sample of size iV* from the infinite ancestral population results in a larger mean genetic load with decreasing N*. The mean load increases at a higher rate for weakly-selected alleles (s = 0.1) than lethal alleles when partially recessive, regardless of N*. In the fast population, the purging of lethal alleles eventually overcomes this increase. The slow population also experiences a slight reduction in mean load after increase. When the slow population declines to N' - 10, drift becomes stronger and the mean load jumps upwards as a result. This is similar to the initial sample of = 10 individuals from the ancestral population (i.e. fast) except that the distribution of load has been changed slightly over nine generations of being at N* = 100. Because the decline to N' = 10 is recent in the slow population, its mean load due to partially recessive lethals is greater than that of the fast population, in which selection has had the opportunity to purge lethal alleles. 64 h=0.5 1 2 3 4 5 6 7 8 9 10 generations h=0.1 — ° — s=0.1, fast —o—s=1.0, fast — o — -s=0.1, slow — -o— -s=1.0, slow 2 3 4 5 6 7 8 9 10 generations h=0.0 120.0 100.0 •s=1.0, fast •s=0.1, fast o - - •s=1.0, slow —o - - •s=0.1, slow & - h=0.1 * - h=0.5 4 5 6 7 generations 9 10 Figure 3-2. - Mean genetic load (xlfT 6) of glide trajectories plotted against time in generations. Individual plots correspond to (a) additive, (b) partially recessive, and (c) completely recessive models of selection, respectively. Note that the scale of genetic load is much larger in (c) than for (a) and (b). The equilibrium load in an infinite population is marked by an arrow and the value corresponding to the degree of dominance. The fast trajectories represent populations that immediately declines from being infinitely large to N* = 10. The slow trajectories represent populations that decline from being infinitely large to N* = 100 and remain at that size until the final generation, whereupon they decline to Af = 10. In the additive lethal case, the fast and slow trajectories overlap. Note that in each case at generation 10 the population size was the same (AT = 10) across trajectories when load was calculated, but that all selection (and purging) occurred in previous generations (when N = A7*). 65 Figure 3-3. - Mean genetic load (xlO - 6 ) of glide trajectories at generation 10, plotted against s. Individual plots correspond to (a) additive, (b) partially recessive, and (c) completely recessive models of selection, respectively. For clarity, only the results for N* = {10, 20, 100} are plotted. Note that the scale of the mean load varies among plots. The effect of demographic history is greatest for completely recessive alleles, with the greatest divergence caused by s = 0.1 at the 10 th generation. The genetic load caused by completely recessive lethal alleles in the N* = 20 trajectory exceeds both the smallest (N* = 10) and largest (N* = 100) trajectories studied. The additive plot at s = 0.0001 goes below 2p because the corresponding equilibrium genetic load is approximately equal to 1.96. 66 Figure 3-2c shows numerical results for completely recessive alleles. The overall transient genetic load is roughly an order of magnitude greater than the load in the previous two cases of additive and partial dominance (Fig. 3-2c). The initial sample of N* individuals from the infinite ancestral population causes a larger jump in the mean load for N* = 10 (fast) than N* = \00 (slow). This jump is very large overall in comparison to the partially recessive case because the equilibrium frequency of lethal alleles (~\i/(hs); Haldane 1937) is greater in the ancestral population when completely recessive, causing the initial spread of the allele frequency distribution by drift to extend to higher values. Again, the purging of lethal alleles in the fast population eventually overcomes the increase of load due to drift so that the mean load is greater in the slow population (Fig. 3-2c). In the short-term, the mean load caused by lethal alleles is greater than that caused by non-lethal (s = 0.1) alleles. Eventually, the mean load o f a completely recessive lethal allele converges in the long-term on the steady-state value jx that is less than the steady-state load caused by non-lethal alleles (Section 3.2). This example illustrates that the transient dynamics can be substantially different than steady state expectations. Figure 3-3 again presents results from the glide trajectories except that now the abscissa is a log-transformed axis of the selection coefficient, s, and the ordinate is the mean genetic load at the last generation (when A 7 ' = 10). In the additive model (Fig. 3-3a), the mean genetic load of both lethal and weakly-selected (s < 10 ) alleles is equivalent to the corresponding equilibrium load in an infinite population, irrespective of the historical population size, N*. Moderately-selected alleles (10 - 3 < s < 0.5) cause an increase in the mean load for lower N*. Again, the mean load at generation 10 is greater overall for partially 67 recessive alleles (Fig. 3-3/3), being greatest for intermediate selection coefficients for all N*. For partially or completely recessive alleles, purging in populations with smaller N* causes the mean load plots to cross over at approximately s = 0.55, indicating that selection must be severe for the population with a historically smaller size to have the lesser mean load by the final generation. For completely recessive alleles (Fig. 3-3c), the selection coefficient causing the greatest mean load increases with N*. The plots no longer cross over at a single point as is the case for partially recessive alleles. Figures 3-4 and 3-5 illustrate the results from the shift trajectories, in which populations share the same historical population size (N* = 100) and final size JV = 10 but differ in the time at which the transition from N* to N' occurs, x. Figure 3-4 is analogous to Figure 3-2c, as only results for lethal alleles are shown. Because the mean load remains at 2p in the additive lethal case for all N*, there are no differences among shift trajectories, and I have elected to omit this plot from Figure 3-4. For partially or completely recessive alleles (Fig. 3-4a, b), the evolutionary trend of mean load in a population that declines in later generations roughly mimics the trend experienced by a population that declines early (e.g. x = 1). The mean load immediately increases upon the decline in numbers at generation x and eventually decreases due to purging. Although the mean load changes while the population is at N* = 100 (see x = 10 trajectory), this is not reflected in the maximum mean attained by different trajectories. For example, in the completely recessive case (Fig. 3-4b), the mean load while the population is at N* increases over time but populations having spent more time at N* reach lower maximum values of mean load. 68 1.8 -1.6 -i 1 1 1 1 1 1 1 1 1 — 1 2 3 4 5 6 7 8 9 10 generations b. h=0.0 140.0 1 0 J , , , , , , , , , -1 2 3 4 5 6 7 8 9 10 generations Figure 3-4. - Mean genetic load (xlO - 6 ) of shift trajectories against time in generations. Individual plots correspond to (a) partially recessive, and (b) completely recessive models of selection, respectively. Only the case for lethal alleles (s=l .0) is shown. The mean genetic load does not change from 2p for additive lethal alleles. Each shift trajectory represents a population that immediately declines from infinite size and remains at N* = 100 until the x-th generation, upon which it declines to its final population size, N* = 10. The mean load jumps upwards upon the second decline to JV by a similar amount at x = 10 as when x = 1 despite having spent a number of generations at N* = 100. Note, however, that the maximum mean load attained declines gradually with increasing x. 69 1.92 -1 1 1 — i r - i r 0 .0001 0.001 0.01 0.1 1 selection coefficient Figure 3-5 - Mean genetic load (xlO - 6 ) of shift trajectories at generation 10, plotted against s. Individual plots correspond to (a) additive, (b) partially recessive, and (c) completely recessive models of selection, respectively. The shift trajectories for x = 1 and x = 10 are equivalent to the N* = 10 and N* = 100 plots of Fig. 3-3. Trajectories for x = 9 are similar to x = 10 for all degrees of dominance. For x = 5, however, there are substantial departures of the mean load for severely deleterious and recessive alleles. This results from a delay of the increase in the load following the second decline of the population to TV = 10, as illustrated in Fig. 3-4. 70 Figure 3-5 is analogous to Figure 3-3. When the mutant allele is additive, the mean genetic load of a population that historically declined from N* to A 7 ' is a decreasing function o f x (Fig. 3-5a). Thus given a population's current size, the load is least when its historic size was large and the decline in size was rapid and recent. This relationship between the load and x remains true for partially or completely recessive alleles that are mildly deleterious (s > 0.5), but changes into a non-linear relationship for severely deleterious alleles (s < 0.5) (Fig. 3-5b, c). In the case of lethal alleles, the load caused by shift trajectories with intermediate values of x can exceed the load caused by upper (x = 10) and lower (x = 1) histories, which reflects the delayed increase of the mean load revealed by Figure 3-4. 3.6 Concluding remarks Genetic drift causes the mean genetic load of a population to increase either by causing deleterious alleles to fix or by increasing the variation of the unfixed allele frequency distribution. The long-term increase in the mean load due to the fixation of alleles by drift is a decreasing function of s because the fixation probability of deleterious alleles decreases exponentially with increasing s. Thus, slightly deleterious alleles are more important in the long term with respect to the transient load of a small population. Because lethal alleles cannot be fixed in the population, changes in the mean genetic load caused by lethals must be brought about by the second process. Increasing variation in allele frequency due to drift can be converted to an increase in the mean genetic load i f the load is a non-linear function of allele frequency (i.e. i f the allele is recessive; Eq. 3.4.2). Hence, there is no change in the mean load caused by lethal alleles in the additive case (e.g. F ig 3.2a) because the alleles 71 cannot become fixed and because increasing variation does not change the mean load. Variation in the frequency of completely recessive lethal alleles, on the other hand, causes a greater increase in mean load than slightly deleterious alleles in the short term (Fig 3-2c). The amount of variation in allele frequency caused by genetic drift between generations is greater for smaller population size. If a population remains at a small size for several generations, then the cumulative allele frequency variation eventually causes the purging of severely deleterious alleles to overcome the increase of mean load due to drift. Thus, purging acts in opposition to drift. This rise and fall in the mean load due to recessive lethal alleles is a transient phenomenon that occurs at small population size (Fig. 3-4). Hence, recessive lethal alleles are more important than slightly deleterious alleles in determining the transient genetic load of a small population in the short term. This confirms numerical results obtained by previous investigators (Wang et al. 1999; Kirkpatrick and Jarne 2000). Wang et al. (1999) performed an extensive simulation study on the transient dynamics of the genetic load for a population bottlenecked from an infinite initial size to a constant size (N = 50). For 2A 7 chromosome sets containing 2900 mutable loci , new mutations were each generated with a selection coefficient, s, and degree of dominance, h, drawn from distributions chosen to reflect experimental evidence. For the case in which s was exponentially distributed and h was uniformly distributed between 0 and exp(-ks) 12, where A: is a constant (Wang et al. 1998), their results indicated that the genetic load increased at a decaying rate over time. Because the mean degree of dominance decreased with increasing strength of selection, lethal alleles were highly recessive on average and were rapidly purged from the population. A s a result, slightly deleterious mutations became more 72 important for the long-term genetic load of the small population. Given the negative correlation between s and h, however, highly deleterious mutations were more important in the short-term. Kirkpatrick and Jarne (2000) performed a similar numerical study in which a severe bottleneck from infinite size was immediately followed by rapid growth of the population. The genetic load of individual loci with specific values of s and h was calculated for different bottleneck sizes. Purging of recessive deleterious alleles caused the genetic load to temporarily drop below the equilibrium value before returning to that equilibrium as the population continued to grow. There has not been, to my knowledge, a numerical study of the evolution of the genetic load in a population whose decline has occurred gradually. Consider a population of current small size that has recently declined from a very large ancestral population, where the decline may have either been severe so that the population was historically small in size, or gradual so that it is only recently small in size. If completely recessive lethal alleles are rare, then the current genetic load is greater i f the population decline was severe so that deleterious alleles have had more opportunity to fix in the population. On the other hand, i f completely recessive lethal alleles are abundant, then the current genetic load is greater in a population that has declined gradually because the purging of recessive lethals is delayed. Therefore, estimating the genetic load in a declining population requires knowledge of the joint distribution of 5 and h for mutations in the ancestral population. The general consensus is that most mutant alleles are slightly deleterious (Mukai 1964; Keightley 1995) with lethal mutations being comparatively rare. Because mostly recessive alleles make a disproportionately large contribution to the transient genetic load, however, the cumulative 73 effect of demographic history wi l l depend more on the distribution of selection coefficients within this dominance class. In reviewing the results of several mutation accumulation experiments (e.g. Simmons and Crow 1977), Caballero and Keightley (1994) observed that mutant alleles with large effects tended to recessivity, whereas alleles with small effects exhibited no tendency. I have made several simplifying assumptions that may reduce the accuracy of genetic load estimates obtained from the numerical model. Because the model addresses a single locus in isolation, there is no direct means to represent the effects of linkage and epistasis. Linkage between deleterious alleles may have a substantial effect on the effectiveness of purging (Charlesworth et al. 1992; Wang et al. 1999). Synergistic epistasis among deleterious alleles, such that the total effect exceeds the sum of individual effects, w i l l cause purging to become more efficient (Wang et al. 1999). Furthermore, I have neglected the distinction between the census size and the effective size of a population (Section 1.2). Finally, the variance in genetic load w i l l cause empirical estimates to deviate from the mean values calculated by this model; this is the subject of further study. I know of no experimental study on the genetic load in gradually declining populations. We know from mutation accumulation experiments such as Muka i et al. (1972) that deleterious mutations are abundant, but less is known about how the genetic load caused by mutations depends on population size. Keightley et al. (1998) reviewed two contradictory experiments (Shabalina et al. 1997; Gill igan et al. 1997) in which populations of Drosophila recently caught from the wi ld were maintained as random-mating laboratory populations of 74 constant size. Although fitness assays were performed to monitor the decline of the populations relative to their controls, selection between families was minimized. Because this impedes the elimination of deleterious alleles, the genetic load estimates were greater than would be obtained in equivalent natural populations. Shabalina et al. (1997) observed a decrease in fitness over time in populations held at N - 200 when fitness was assayed under harsh conditions. Gil l igan et al. (1997), on the other hand, did not observe a lower fitness in their small population compared to their large populations or controls. There is a rich empirical literature devoted to verifying the biological importance of purging due to inbreeding. The recovery of fitness in inbred lines was initially observed by Bowman and Falconer (1960) in mice and subsequently by Templeton and Read (1984) in Speke's gazelle, although the latter has recently been criticized for containing possible biases (Will is and Wiese 1997; Bal lou 1997). Several experimental studies have since attempted to induce purging by artificially increasing the rate of inbreeding in natural populations, with both affirmative (Carr and Dudash 1997; Barrett and Charlesworth 1991), negative (Dudash et al. 1997; M c C a l l et al. 1994), or mixed results (Lacy and Ballou 1998). Others have compared the magnitude of inbreeding depression among lines derived from ancestral populations with a known history of inbreeding (Latta and Ritland 1994; Weeks et al. 1999). The evidence tends to portray purging by inbreeding as a relatively weak process. The empirical study of purging by drift, on the other hand, is less extensive. Several studies (reviewed in Hauser et al. 1994) have contrasted the inbreeding depression in large and small populations, but none of them was able to detect a significant effect of population 75 size. Brakefield and Saccheri (1994), however, observed a recovery of fitness in laboratory populations of satyrine butterflies (Bicyclus ariynand) founded from one or more pairs. The divergence of morphological and molecular markers among bottlenecked lines was consistent with strong genetic drift. Lines founded from a single pair and allowed to increase to a carrying capacity of K= 300 immediately experienced a dramatic (>50%) loss of fecundity, followed by a gradual recovery in subsequent generations (Brakefield and Saccheri 1994). Similar results have been obtained by Bryant et al. (1990) for serially-bottlenecked populations of house flies and by Fowler and Whitlock (1999) for populations of D. melanogaster each founded by a single pair. Using a different experimental design, Ehiobu et al. (1989) adjusted the number of generations in serially bottlenecked populations of four and twenty D. melanogaster such that the same inbreeding coefficient was obtained by the final generation as caused by a single full-sib mating. A full-sib mating became a third treatment. The three treatments allowed them to distinguish between the effects of drift and inbreeding on inbreeding depression. Ehiobu et al. (1989) observed a significantly greater larval viability in their N=20 lines than the full-sib treatment, suggesting purging had occured. The N = 4 lines were not significantly different than the full-sib treatment, which may reflect a lesser opportunity for purging in fewer generations or the fixation of deleterious alleles by a greater intensity of drift. These results, together with the results of the numerical model described above, suggest that population that has declined gradually in size may have a greater genetic load than a population that has declined severely. 76 C H A P T E R IV C O N C L U S I O N S A N D R E C O M M E N D A T I O N F O R F U R T H E R W O R K The evolutionary theory of genetic load has developed over several decades, from the pioneering work of Haldane (1937) and Muller (1950), into a rich literature where the influence of recombination (e.g. Otto and Feldman 1997), mating systems (e.g. Lande and Schemske 1985; Barrett and Charlesworth 1991), and mutation rate (Felsenstein 1974) on load have been considered. Application of load theory to problems in conservation biology has stimulated new interest into the effects of drift on the genetic load (Lynch and Gabriel 1990; Lande 1994), as drift is increasingly important in smaller populations. In particular, the mutational meltdown process (Lynch and Gabriel 1990), by which the mean fitness of a population is continually eroded by the fixation of deleterious mutations by drift, is predicted to cause the extinction of sexual populations as large as N= 10 3 (Lynch et al. 1995b). If mutations frequently compensate for each other's effects, however, then the erosion of fitness is halted as compensatory mutations become increasingly advantageous (Chap. II). The importance of drift also emphasizes the undeveloped state of theoretical models of drift in relation to the genetic load (Kirkpatrick and Jarne 2000), especially for the transient dynamics following a change in population size. Most of the studies on this matter have consisted of numerical simulation and have calculated load for bottlenecked populations at a constant size (Wang et al. 1999) or undergoing rapid growth (Kirkpatrick and Jarne 2000). Exact numerical evaluation of recursive formulae modeling the evolution of allele frequency distributions in a declining population indicate that processes observed in the aforementioned studies (i.e. immediate boost in load, and subsequent purging) can be delayed, such that populations that have declined gradually may have a greater genetic load than others that have undergone a severe decline in numbers (Chap. III). 77 Fisher's geometrical model of adaptation (Fisher 1930), applied in Chapter II to an analysis of compensatory mutation, is a good first step towards understanding the effects of compensatory mutations on load, but more empirically grounded models should be explored next. Although the compensatory and mutational meltdown models can produce qualitatively different predictions on the extinction risk of small populations, this divergence depends on parameters for which there is an insufficient amount o f empirical data. 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