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Local adaptation and maintenance of variation in heterogeneous environments Yeaman, Samuel 2009

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Local adaptation and maintenance of variation in heterogeneous environments by SAMUEL YEAMAN  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Zoology) The University of British Columbia (Vancouver)  October 2009 © Samuel Yeaman, 2009  Abstract Most species inhabit environments that are spatially heterogeneous at some scale. If dispersal is low enough relative to spatial variations in the effect of natural selection, then local adaptations may emerge. On the other hand, if dispersal is high enough to prevent isolation by distance, then gene flow among populations will influence both the amount of standing genetic variation maintained within populations and the architecture of this variation. Here, I explore various genetic consequences of evolution in heterogeneous environments. I begin by reviewing two empirical studies exploring how heterogeneous selection and gene flow affect the maintenance of variation within populations. The first of these is an observational study of patterns in natural populations of Pinus contorta (lodgepole pine; Chapter 2), while the second is a manipulative laboratory evolution experiment using Drosophila melanogaster (Chapter 3). I then discuss three theoretical studies on the evolution of locally adaptive trait divergence between populations under migration-selection balance. The first of these develops analytical approximations to predict the invasion probability and persistence time of beneficial mutations in finite populations (Chapter 4). The second of these studies explores the effect of migrationselection balance on the evolution of the genetic architecture underlying a quantitative trait (Chapter 5). The final theoretical study presents an exploration of the discrepancies between quantitative genetic models of mutation-selection balance and observations based on individual-based simulations (Chapter 6). Taken together, this research contributes to our understanding of how gene flow and heterogeneous selection influence the genetics of adaptation and the maintenance of genetic variation.  ii  Table of contents ABSTRACT ..........................................................................................................................................................ii TABLE OF CONTENTS ........................................................................................................................................iii LIST OF TABLES ..................................................................................................................................................v LIST OF FIGURES................................................................................................................................................vi ACKNOWLEDGMENTS ......................................................................................................................................vii DEDICATION ................................................................................................................................................... viii CO-AUTHORSHIP STATMENT .............................................................................................................................ix 1.  INTRODUCTION........................................................................................................................................1 REFERENCES ......................................................................................................................................................9  2. REGIONAL HETEROGENEITY AND GENE FLOW MAINTAIN VARIANCE IN A QUANTITATIVE TRAIT WITHIN POPULATIONS OF LODGEPOLE PINE ..................................15 INTRODUCTION ................................................................................................................................................15 METHODS .........................................................................................................................................................19 Genetic variance ........................................................................................................................................19 Environmental heterogeneity ....................................................................................................................21 RESULTS ...........................................................................................................................................................24 DISCUSSION ......................................................................................................................................................26 REFERENCES ....................................................................................................................................................34 3. NO EFFECT OF ENVIRONMENTAL HETEROGENEITY ON THE MAINTENANCE OF GENETIC VARIATION IN WING SHAPE IN DROSOPHILA MELANOGASTER ............................40 INTRODUCTION ................................................................................................................................................40 METHODS .........................................................................................................................................................43 Establishment and maintenance of experimental lines............................................................................43 Experimental crosses .................................................................................................................................44 Wing traits and statistical analysis ...........................................................................................................45 RESULTS ...........................................................................................................................................................48 Fitness differences between homogeneous lines ......................................................................................48 Trait divergence between homogeneous lines..........................................................................................49 Trait divergence under limited migration ................................................................................................50 Differences in additive genetic variance among treatments ...................................................................51 DISCUSSION ......................................................................................................................................................53 REFERENCES ....................................................................................................................................................67 4. ESTABLISHMENT AND MAINTENANCE OF ADAPTIVE GENETIC DIVERGENCE UNDER MIGRATION, SELECTION, AND DRIFT ..................................................................................72 INTRODUCTION ................................................................................................................................................72 ANALYTICAL MODEL .......................................................................................................................................74 INDIVIDUAL-BASED SIMULATIONS ..................................................................................................................80 Probability of a new mutation rising to high frequency:.........................................................................81 Persistence time of a pair of divergent alleles:........................................................................................83 Maintenance of divergence with recurrent mutation...............................................................................84 DISCUSSION ......................................................................................................................................................85 5. THE GENETIC ARCHITECTURE OF ADAPTATION UNDER MIGRATION-SELECTION BALANCE...........................................................................................................................................................95 INTRODUCTION ................................................................................................................................................95 ANALYTICAL APPROXIMATIONS......................................................................................................................98 Evolution in a quantitative trait determined by a single locus ...............................................................98 Linkage between loci and the net strength of divergent selection ........................................................101  iii  INDIVIDUAL-BASED SIMULATIONS ................................................................................................................104 RESULTS .........................................................................................................................................................107 Adaptive divergence for a single-locus quantitative trait .....................................................................107 Critical migration threshold for multi-locus traits ................................................................................108 Evolution of genetic architecture – Migration rate ...............................................................................109 Evolution of genetic architecture – Mutation rate and effect size ........................................................110 DISCUSSION ....................................................................................................................................................112 REFERENCES ..................................................................................................................................................129 6. PREDICTING ADAPTATION UNDER MIGRATION LOAD: THE ROLE OF GENETIC SKEW ................................................................................................................................................................133 INTRODUCTION ..............................................................................................................................................133 THE GAUSSIAN APPROXIMATION MODEL......................................................................................................136 THE DISCRETE GAM MODEL .........................................................................................................................137 THE SIMULATION MODEL ...............................................................................................................................140 RESULTS .........................................................................................................................................................143 GAM vs. discrete GAM ............................................................................................................................146 Effect of genetic architecture ..................................................................................................................147 DISCUSSION ....................................................................................................................................................149 REFERENCES ..................................................................................................................................................167 7.  CONCLUSIONS ......................................................................................................................................172 REFERENCES ..................................................................................................................................................185  8.  APPENDICES ..........................................................................................................................................187 APPENDIX 1 – DETAILS FOR EXPERIMENTAL METHODS IN CHAPTER THREE ..............................................187 Culture materials and medium ................................................................................................................187 Checking for contamination by Drosophila simulans ...........................................................................187 Resolving a minor mite infestation..........................................................................................................188 Schedule for experimental crosses..........................................................................................................189 Wing measurement...................................................................................................................................190 APPENDIX 2 –DIFFERENCES IN VA AMONG TREATMENTS FOR CHAPTER THREE ........................................192 APPENDIX 3 – SUPPLEMENTARY MATERIALS FOR CHAPTER FOUR .............................................................194 Invasion probability for dominant or recessive mutations....................................................................194 APPENDIX 4 –RESULTS FOR VARIABLES OF MINOR EFFECT FROM CHAPTER SIX........................................196 Genetic variance maintained at mutation-selection balance ................................................................196 Heritability and environmental variance ...............................................................................................196 Recombination..........................................................................................................................................197 Number of loci..........................................................................................................................................198 Effect size ..................................................................................................................................................198 APPENDIX 5 – FULL EXPRESSION AND ACCURATE APPROXIMATIONS FOR THE DISCRETE GAM MODEL FROM CHAPTER SIX .......................................................................................................................................204  iv  List of tables TABLE 2.1 – CLIMATIC VARIABLES. ....................................................................................................................31 TABLE 3.1 – DESCRIPTION OF EXPERIMENTAL TREATMENTS AND ABBREVIATIONS .........................................60 TABLE 3.2 – DIVERGENCE IN TWENTY WING TRAITS. .........................................................................................61 TABLE 3.3 – TRAIT HERITABILITIES (H2), ADDITIVE GENETIC VARIATION (V A), AND 95% CONFIDENCE INTERVALS (95% CI) FOR EACH TREATMENT, AVERAGED ACROSS REPLICATE AND ASSAY...................62 TABLE 3.4 – TEST OF THE DIFFERENCES IN PHENOTYPIC VARIANCE BETWEEN MALES AND FEMALES FROM THE PARENTAL GENERATION USING LEVENE’ S TEST (DF = 1)..........................................................................63 TABLE 6.1 – MIGRANT FITNESS UNDER SELECTION REGIMES ..........................................................................166 TABLE A2.8.1 – GENETIC VARIANCE MAINTAINED BY THE SIX TREATMENTS. ...............................................193  v  List of figures FIGURE 2.1 – RELATIONSHIP BETWEEN VARIANCE WITHIN POPULATIONS AND HETEROGENEITY....................32 FIGURE 2.2 – SENSITIVITY OF RESULTS TO VARIATION IN MODEL REPRESENTING GENE FLOW ........................33 FIGURE 3.1 – LANDMARKS PRODUCED BY FINDWING SPLINE FITTING. ........................................................64 FIGURE 3.2 – DIFFERENCE IN MEAN LOG-TRANSFORMED PRODUCTIVITY BETWEEN WARM ASSAY AND COLD ASSAY FOR EACH REPLICATE OF THE CONTROL POPULATIONS. ................................................................65 FIGURE 3.3 – REDUCED DIVERGENCE BETWEEN CAGES IN THE LIMITED MIGRATION TREATMENT ..................66 FIGURE 4.1– MAGNITUDE OF ! L AS A FUNCTION OF THE RATIO BETWEEN THE ALLELIC SELECTION COEFFICIENTS (S, T) AND THE MIGRATION RATE. ......................................................................................88 FIGURE 4.2 – INVASION PROBABILITY OF A NEW MUTATION ..............................................................................90 FIGURE 4.3 – PERSISTENCE TIME AND MAINTENANCE OF POLYMORPHISM .......................................................92 FIGURE 5.1 – CRITICAL MIGRATION RATE MAINTAINING POLYMORPHISM ......................................................120 FIGURE 5.2 – INVASION OF A SECOND ALLELE FOLLOWING INITIAL DIVERGENCE AT A LINKED LOCUS .........121 FIGURE 5.3 – MAINTENANCE OF DIVERGENCE FOR A SINGLE- LOCUS QUANTITATIVE TRAIT ..........................123 FIGURE 5.4 – MAINTENANCE OF DIVERGENCE FOR A MULTI- LOCUS QUANTITATIVE TRAIT ...........................124 FIGURE 5.5 – EVOLUTION OF GENETIC ARCHITECTURE ....................................................................................125 FIGURE 5.6 – INFLUENCE OF MUTATION RATE AND EFFECT SIZE ON GENETIC ARCHITECTURE ......................127 FIGURE 5.7 – INFLUENCE OF SELECTION AND RECOMBINATION ON GENETIC ARCHITECTURE........................128 FIGURE 6.1 – SAMPLE POPULATION UNDER SELECTION AND MIGRATION ........................................................157 FIGURE 6.2 – COMPARISON OF DIVERGENCE UNDER GAM VS. SIMULATIONS ...............................................158 FIGURE 6.3 – EVOLUTION OF GENETIC SKEW AND INFLUENCE ON ACCURACY OF THE GAM.........................160 FIGURE 6.4 – G ENETIC ARCHITECTURE UNDER CONTINUUM-OF-ALLELES MODEL .........................................162 FIGURE 6.5 – EVOLUTION OF GENETIC SKEW AND DIVERGENCE AS A FUNCTION OF ARCHITECTURE.............163 FIGURE 6.6 – COMPARISON OF SIMULATIONS, GAM, AND DISCRETE GAM ...................................................164 FIGURE A3.1 – INVASION PROBABILITY FOR DOMINANT AND RECESSIVE MUTATIONS...................................195 FIGURE A4.2 – EFFECT OF ENVIRONMENTAL VARIANCE AND MIGRATION ON GAM AND SIMULATIONS ......200 FIGURE A4.3 – EFFECT OF ENVIRONMENTAL VARIANCE AND MUTATION ON GAM AND SIMULATIONS .......201 FIGURE A4.4 – EFFECT OF RECOMBINATION ON GAM AND SIMULATIONS.....................................................202 FIGURE A4.5 – EFFECT OF MUTATION EFFECT SIZE ON GAM AND SIMULATIONS ..........................................203 FIGURE A5.1 – NUMERICAL SOLUTIONS FOR EQUILIBRIUM DIVERGENCE. ......................................................206  vi  Acknowledgments First and foremost, I would like to thank my supervisor, Mike Whitlock. Looking back at where I began over five years ago, I cannot imagine getting to this point without the stimulating discussions, constructive criticism, encouragement, and guidance that surrounded all of our interactions. The experience of working closely with Mike has had a formative and long-lasting impact on the way I observe, reflect, and synthesize. Andy Jarvis deserves special thanks for starting me off on the research path and giving me the freedom to explore, which contributed greatly to my decision to pursue a doctoral degree. I would also like to thank my committee: Sally Aitken, Sally Otto, Dolph Schluter, and Mark Collard for providing lots helpful of discussion, guidance, feedback, and support. I would like to thank Sally Otto especially for helping me express my intuitions mathematically. Fred Guillaume and Alistair Blachford provided lots of help with programming in C++ and general messing about with computers, without which everything I did would have taken at least twice as long. I also had a lot of help with the fly evolution experiment from Iris Chen, Brock Glover, Lisa Correia, Michelle Wood, and especially Yukon Chen, who digitized all of the fly wings. Throughout the development of my thesis, many others have helped with discussions, assistance, and support: Jason Holliday, Rebecca Best, Brad Davis, Alana Schick, Aleeza Gerstein, Jen Guevara, J.S. Moore, Kelly Jewell, Leithen M’Gonigle, Jabus Tyerman, Dilara Ally, Jess Hill, Crispin Jordan, Alan Brelsford, Rowan Barrett, Cort Griswold, Rich Fitzjohn, Loren Rieseberg, Michael Doebeli, Mark Vellend, Troy Day, Gerry Rehfeldt, Alvin Yanchuk, Simon Cook, and the SOWD/DeltaTea lab group. Lastly, I would like to thank Michael Berrill for introducing me to the study of evolution as a science.  vii  Dedication  To my parents, all three of them. My unconventional family tree led me to reflect upon the nature of inheritance from an early age. I am here because of the unique blend of nature and nurture that you provided.  viii  Co-authorship Statement Chapter One: Written by Sam Yeaman Chapter Two: Conceived by Sam Yeaman and Andy Jarvis; research, data analysis, and manuscript preparation by Sam Yeaman. Chapter Three: Conceived by Sam Yeaman and Michael Whitlock. Experimental evolution conducted by Sam Yeaman; genetic variance assays conducted by Sam Yeaman and Yukon Chen; fly wing data capture by Yukon Chen. Data analysis and manuscript preparation by Sam Yeaman. Chapter Four: Conceived by Sam Yeaman with input from Sarah Otto. Individual-based simulations conducted and analysed by Sam Yeaman. Analytical theory developed by Sarah Otto and Sam Yeaman. Manuscript prepared by Sam Yeaman. Chapter Five: Conceived by Sam Yeaman with input from Michael Whitlock. Individualbased simulations conducted and analysed by Sam Yeaman. Analytical theory developed by Mike Whitlock and Sam Yeaman. Manuscript prepared by Sam Yeaman. Chapter Six: Conceived by Sam Yeaman and Fred Guillaume. Individual-based simulations conducted and analysed by Sam Yeaman. Analytical theory developed by Fred Guillaume. Synthetic discussion and analysis of results by Sam Yeaman and Fred Guillaume. Manuscript prepared by Sam Yeaman and Fred Guillaume. Chapter 7: Written by Sam Yeaman.  ix  1. Introduction A fundamental problem in biology is to understand how simple evolutionary processes generate and maintain the enormous diversity seen in nature. While all heritable genetic variation must ultimately arise from mutation, it is sorted, shaped, and maintained by many interrelated ecological and evolutionary processes. Genetic variation is often partitioned into components within- and among-populations, describing the level of polymorphism within populations and the amount of divergence among them, respectively. As evolvability is directly proportional to the standing genetic variation within a population (Fisher 1930), studying how this variation is maintained is critical to understanding how populations and species vary in their capacities to respond to environmental change. On the other hand, as differences in trait means between populations may lead to stable local adaptation and the eventual emergence of new species, studying how divergence arises and is affected by the various evolutionary processes is critical to understanding how biodiversity is maintained at and above the species level. Migration and heterogeneous selection can affect the evolution of variation at both of these levels, depending on the spatial scale and intensity. While heterogeneous selection will tend to increase variation among populations, migration can have the opposite effect, homogenizing differences and reducing variation between populations while simultaneously increasing variation within populations. Although migration and selection tend to affect the maintenance of variation both within and between populations, research has often focused on the effect on variation at one level or the other, as the ecological and evolutionary consequences of variation at these levels are quite different (and analytical treatments often require restrictive assumptions about  1  constant variance within populations (e.g., Hendry et al. 2001) or maximal divergence between populations (e.g. Lythgoe 1997)). The effect of the interplay between migration and selection on the maintenance of divergence between populations is one of the oldest areas of theoretical evolutionary research, with some of the earliest models in population genetics deriving the critical immigration rate above which a locally adapted allele would be lost (Haldane 1930; Wright 1931). Subsequent developments extended this allele-based approach to twopatch models (Levene 1953; Moran 1968; Bulmer 1972), habitat selection and sympatric speciation (Maynard Smith 1966), environmental gradients (Slatkin 1973), hard selection (Christiansen 1975), and a range of other increasingly realistic modifications (reviewed in Felsenstein 1976; Lenormand 2002). More recently, Tachida and Iizuka (1991), Gavrilets and Gibson (2002), and Whitlock and Gomulkiewicz (2005) have examined how migration-selection balance in heterogeneous environments affects the fixation probability of new mutations in finite populations. Quantitative genetics models have also been developed to address similar questions on a phenotypic rather than allelic level. Hendry and colleagues (2001) derived a model predicting divergence in a quantitative trait for two populations under migration-selection balance, which was later extended to include the effects of seed vs. pollen migration (Lopez et al. 2008). Other models have examined adaptation in source-sink environments (Holt and Gomulkiewicz 1997; Ronce and Kirkpatrick 2001) and the evolution of species range limits (Holt and Gaines 1992; Garcia-Ramos and Kirkpatrick 1997; Kirkpatrick and Barton 1997; Case and Taper 2000).  2  Numerous examples of local adaptation in heterogeneous environments have now been identified at the phenotypic level (Hedrick et al. 1976; Linhart and Grant 1996), and the genetic basis of these differences has been characterized in an increasing number of examples (e.g., armour plating in sticklebacks (Colosimo et al. 2004); and colouration in mice (Hoekstra et al. 2006; Steiner et al. 2007)). While the genetic basis of some traits seems relatively simple, in others it is much more complex. Genetic mapping of the loci underlying a range of quantitative traits (QTL) has revealed considerable variation in the effect size distribution and number of loci contributing to local adaptation and species differentiation (Orr 2001). It is unclear how the various factors that shape or constrain genetic architecture contribute to this variation. On the one hand, genetic architecture must be constrained at some level by the mapping from genome to phenotype and the capacity of different genes and pathways to shape phenotypic variation. On the other hand, as mutations of different size have different probabilities of contributing to adaptation, population genetic processes of selection, migration, and drift will also influence the distribution of mutations fixed during adaptive divergence (Orr 1998; Griswold 2006). From an empirical standpoint, this problem of the importance of genomic vs. evolutionary factors shaping the architecture of local adaptation may be addressed by comparing the genetic basis of adaptation among replicate populations exposed to similar environments, as has been shown with the ectodysplasin gene in sticklebacks (Colosimo et al. 2005). Further theoretical work is also required to expand existing models to incorporate the role of drift and generate more refined predictions. The maintenance of genetic variation within populations poses a related yet distinct problem. The ability of a population to respond to natural selection depends upon  3  its standing pool of genetic variation; however, this variation is itself depleted by the process of natural selection. As strong selection is commonly observed in nature (Kingsolver et al. 2001; Hereford 2009), low levels of genetic variation would be expected in most populations. Natural populations, however, often have high levels of heritable genetic variation (h2 > 0.3), while laboratory populations have shown sustained responses to selection over many generations (Falconer and Mackay 1996). In his review of 842 estimates of heritability, Houle (1992) found that levels of additive genetic variance (VA) were typically higher in traits that were more closely connected to fitness than in traits that experience weak selection. Understanding the processes that generate these seemingly counter-intuitive observations remains an unsolved problem in evolutionary biology. Two general classes of model have been proposed to resolve this problem: those that emphasize the role of mutation and those that emphasize the role of some form of balancing selection. Mutation-based models posit that most of the variation in VA among traits, populations, and species can be explained by differences in the mutation rate or strength of selection. Originally developed by Kimura (1965) and Lande (1976), theories of mutation-selection balance have been refined to consider the impacts of alternate models of mutation (Turelli 1984), finite population size (Houle 1989), apparent stabilizing selection (Kondrashov and Turelli 1992), population structure with limited migration (Goldstein and Holsinger 1991; Phillips 1996), and pleiotropy (Keightley and Hill 1988). Proponents of the mutation-selection balance hypothesis suggest that the high levels of variation in fitness-related traits are the product of larger mutational targets, greater environmental variation, and reduced heritabilities (Houle et al. 1996). Despite  4  much productive research (or because of it?), mutation-selection balance remains a controversial subject. While some authors have suggested that the modified theories described above can account for empirically observed heritabilities (Houle et al. 1996; Zhang et al. 2004), this has not reached scientific consensus. A recent review by Johnson and Barton (2005) notes that most theoretical studies have used unrealistically low strengths of selection (compared to estimates by Kingsolver et al. (2001)), and result in dynamical behaviour that is inconsistent with empirically observed mutation rates. Thus, while mutation-selection balance must account for some portion of the variation in VA, the current synthesis cannot provide a complete explanation. The second class of models that have been used to explain the maintenance of variation are those based on some form of balancing selection. The general idea underlying these theories is that certain ecological and genetic factors weaken the net strength of stabilizing selection, resulting in reduced rates of elimination of mutant traits and a higher equilibrium level of variance. The simplest of these theories suggests that a general heterozygote advantage can protect a polymorphism that would be eliminated if only one homozygote was most fit. This does not seem to be a general mechanism for the maintenance of variation however, because self-pollinating plants typically have levels of variation similar to outcrossing plants (Charlesworth and Charlesworth 1995). Positive frequency-dependent selection can also protect polymorphism through increased fitness of rare alleles (Bürger and Gimelfarb 2004) but it is not clear whether the intraspecific competition and resource distributions required to drive frequency dependence commonly occur in nature for most traits. The effects of pleiotropy and alleles with sexdependent expression have also been investigated; however, the conditions for  5  maintenance of variation are very restrictive and unlikely to have general importance (Turelli and Barton 2004). A final sub-class of balancing selection models examines the effects of heterogeneous environments on the maintenance of genetic variation. There are three principal ways that heterogeneous environments could maintain variation: through selection towards a spatially or temporally fluctuating optimum within the range of a panmictic population (Bürger and Gimelfarb 2002; Spichtig and Kawecki 2004); through genotype-by-environment interactions (G x E) and decreased sensitivity of heterozygotes (Via and Lande 1987; Gillespie and Turelli 1989; Turelli and Barton 2004); and by gene flow between differently adapted populations inhabiting regionally heterogeneous but locally homogeneous environments (Slatkin 1978; Barton 1999; Spichtig and Kawecki 2004). Models studying each of these effects suggest that increased levels of variance can be maintained over some regions of parameter space. At the present time, however, it is unclear whether these conditions are biologically plausible, and if so, whether they commonly occur (Johnson and Barton 2005). The aim of my thesis is to explore how gene flow in heterogeneous environments affects both the establishment of adaptive divergence among populations and the maintenance of genetic variation within populations. The research first chapter examines the relative potential of heterogeneous vs. homogeneous environments to maintain genetic variation within populations of lodgepole pine (Pinus contorta). This study develops a novel method to measure environmental heterogeneity in the region surrounding a population and tests how well this approach predicts levels of genetic variance within populations, estimated from a previously collected dataset from a large-  6  scale common garden study (Rehfeldt et al. 1999). The second chapter discusses the results of a laboratory study exploring the effects of different patterns of environmental heterogeneity on genetic variance for wing shape in Drosophila melanogaster. A series of twenty-five replicate populations were established from wild-caught individuals and reared under the experimental treatments for two years before assaying their effect on the maintenance of quantitative genetic variation using offspring-parent regression. Where the study in the first chapter was observational, the second chapter uses experimental manipulations to explicitly compare how spatial vs. temporal heterogeneity affect the maintenance of variation. The third chapter presents an analytical approach to studying migration-selection balance in finite populations and tests some of the resulting predictions using individual-based simulations. This approach yields accurate predictions of the probability of new mutations contributing to local adaptation as well as the critical migration rate above which divergence between populations tends to be lost due to the homogenizing effect of gene flow. The fourth chapter uses the approximations developed in the third chapter in combination with individual-based simulations to examine how genetic architecture evolves under migration-selection balance in finite populations. This work shows that stabilizing selection and migration results in qualitatively different patterns of genetic architecture, compared to previous studies of evolution under stabilizing selection without migration (Orr 1998) or a short bout of divergent selection with migration (Griswold 2006). The fifth chapter examines how quantitative genetic models of divergence under migration-selection balance can yield inaccurate predictions when the assumption of normally distributed breeding values is violated. Predictions from a previously developed quantitative genetics model (Hendry et al. 2001) are  7  compared to results from individual-based simulations, showing that migration can generate sufficient skew to significantly compromise the accuracy of these models. 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Regional heterogeneity and gene flow maintain variance in a quantitative trait within populations of lodgepole pine1 Introduction The maintenance of genetic variation within populations remains a central question in the study of evolutionary biology (Johnson and Barton 2005). Strongly selected traits are expected to have low levels of genetic variance and low heritabilities, due to the purging of non-optimal alleles (Fisher 1930). Empirical evidence, however, shows that variation is ubiquitous. Both natural and experimental populations often have high heritabilities for many traits (Mousseau & Roff 1987; Roff & Mousseau 1987), and traits that are closely associated with fitness tend to have higher levels of genetic variance (but lower heritability) than weakly selected traits (Houle 1992). Understanding the population genetic processes that maintain such high levels of variation is central to the study of evolution. Numerous theoretical approaches have been taken to address this problem, including models of mutation-selection balance (Lande 1975; Turelli 1984; Bulmer 1989), genotype-environment interactions (Via & Lande 1987; Gillespie & Turelli 1989; Turelli & Barton 2004), population structure and stabilizing selection (Goldstein & Holsinger 1992), temporally fluctuating selection pressures (Waxman & Peck 1999; Bürger & Gimelfarb 2002), and pleiotropic overdominance (Bulmer 1973; Gillespie 1984; Turelli & Barton 2004). Despite extensive theoretical research, however, there is still no consensus about which processes are most important and only limited evidence to evaluate this question empirically (Johnson and Barton 2005). 1  A version of this chapter has been published. Yeaman, S. and A. Jarvis. 2006. Regional heterogeneity and gene flow maintain variance in a quantitative trait within populations of lodgepole pine. Proceedings of the Royal Society, Series B. 273:1587-1593. 15  One of the more intuitive mechanisms proposed to maintain genetic variation is through spatially varying selection and gene flow. When populations inhabit environments with different local optima, the reduction in variation caused by selection within each population can be opposed by gene flow between them. As long as gene flow is insufficient to overwhelm local adaptations and prevent divergence among populations, it will increase the standing genetic variance within populations. While numerous models have examined this migration-selection balance in single locus traits (reviewed in Felsenstein 1976), there have been comparatively few treatments of polygenic traits. Slatkin (1978) showed how gene flow through an environmental gradient increases the additive genetic variance within populations inhabiting intermediate regions of the cline, due to the mixing of alleles from populations with different means. Although Slatkin suggested that this effect would only be important under strong selection, Barton (1999) has shown that gene flow through a cline can result in substantial increases in variance under weak selection as well. Other models based on single populations experiencing stabilizing selection and immigration of non-locally adapted alleles (Tufto 2000), or two populations inhabiting different environments connected by gene flow (Spichtig & Kawecki 2004) have made similar qualitative conclusions, showing that gene flow can maintain genetic variation within populations, but that this effect will only occur under limited migration and strong selection. Although these theoretical treatments have demonstrated the possibility that gene flow and spatially heterogeneous selection can maintain variation within populations, it is unclear whether this effect makes a substantial contribution to levels of variation in nature. Several studies have shown that gene flow in clines can maintain heterozygosity  16  in single locus traits (Barton & Hewitt 1985; Lenormand & Raymond 2000), but there have been fewer demonstrations of this effect in quantitative traits. Sgro and Blows (2003) examined genetic variance in a cline for development time in wild populations of Drosophila serrata and found that non-additive genetic variance was highest at the region with the sharpest change in trait mean. While this evidence conforms to the predictions of Barton’s analysis of polygenic clines (1999), it is unclear why patterns of additive genetic variance did not also increase in this transition zone, as predicted by theory. Other studies of quantitative genetic clines have reported ambiguous patterns in genetic variance (van t’Land et al. 1999; Magiafoglou et al. 2002) or have focused on changes in mean rather than variance (e.g. James et al. 1995; Hoffmann et al. 2002; Palo et al. 2003). Part of the reason for this lack of evidence may stem from the difficulty of testing predictions based on genetic variance; the theoretical expectations for patterns of genetic structure in polygenic traits change depending upon the strength of selection and the shape of the change in optimum. For example, under a gentle linear gradient, genetic variance should be relatively constant across the cline, whereas for a steep linear gradient, genetic variance should peak at the centre of the cline (Barton 1999). Because the shape of the environmental gradient is often unknown, and because similar patterns could be expected under alternative processes such as localized mutation-selection balance, it is difficult to test hypotheses based on clines in polygenic traits. In a slightly different context, empirical studies of maladaptation caused by gene flow (Riechert 1993; Bossart & Scriber 1995; King & Lawson 1995; Hendry & Taylor 2004) suggest that gene flow can be a persistent source of variance within populations, yet there is still considerable disagreement about the strength of this effect (Hendry & Taylor 2004). Thus, while  17  theoretical research suggests that gene flow through heterogeneous environments may maintain genetic variance, there is little empirical evidence to evaluate the importance of this process. A general prediction about the effect of gene flow that is more easily tested is that genetic variance within populations should be correlated to regional environmental heterogeneity. Assuming homogenous migration rates and dispersal distances across the range of a species, the variance in trait values carried by alleles flowing into any given population should be proportional to the regional variance in trait means, which should in turn be positively related to the regional variance in environment. This prediction is a direct extrapolation from cline theory, as variance is expected to be higher in populations inhabiting steep clines with more heterogeneity within the region defined by dispersal distance (Slatkin 1978; Barton 1999). Importantly, this prediction will only hold when there is sufficient regional variation in environment to maintain local adaptations among populations (analogous to Slatkin’s ‘characteristic length’ threshold in a linear cline). To test this prediction, we studied patterns of genetic variation within populations of lodgepole pine (Pinus contorta) using previously collected data from a long-term common garden experiment (Rehfeldt et al. 1999) and compared these to quantitative estimates of environmental heterogeneity. Lodgepole pine is a long-lived, highly outcrossed conifer (Yeh & Layton 1979) with an extensive range spanning climatically heterogeneous mountain environments and relatively homogenous valleys and plateaus from Baja California to the Yukon Territories and Alaska. Adaptation to the climatic variations across this range requires tradeoffs between maximizing growth rate and tailoring phenology to avoid exposing sensitive growing tissues to extreme environmental  18  conditions (Howe et al. 2003). Like many tree species, neutral genetic variation in lodgepole pine is mostly partitioned within populations (Yang et al. 1996; Hamrick 2004). Studies of growth response, however, have found significant patterns of local adaptation and divergence in trait means among populations (Yang et al. 1996; Rehfeldt et al. 1999; Wu & Ying 2004). Because the environment is sufficiently heterogeneous to maintain divergence between populations in the presence of gene flow, lodgepole pine conforms to the assumptions of the gene flow/environmental heterogeneity hypothesis described above. Thus, if gene flow through heterogeneous environments is an important process maintaining variation, we should see significant correlations between genetic variance for growth response within populations and environmental heterogeneity in the surrounding region. We tested this prediction by comparing levels of genetic variance within populations to quantitative measures of heterogeneity in climatic conditions in their surrounding region.  Methods Genetic variance Genetic variance was estimated using previously collected data from a common garden experiment testing growth response to climate in 142 populations of lodgepole pine. This experiment was established by Keith Illingworth and maintained by the British Columbia Forest Service; the details are described more thoroughly in other papers (Rehfeldt et al.1999; Wu & Ying 2004). Basic experimental design was as follows: seeds were collected from approximately 15 different trees in each population (over an area of approximately 1 km2) and bulked according to their population of origin (Alvin Yanchuk,  19  pers. comm.). Seeds from each population were then planted in several common gardens across a range of climatic conditions, with two randomized blocks of nine individuals in each site (Rehfeldt et al. 1999). Due to the limitations of such an extensive study, populations were not planted in the full set of 60 common gardens used for testing, but were planted in approximately 33 different sites on average. Following 20 years of growth, the heights of all surviving individuals were recorded. For the analysis in this study, we excluded any trials with fewer than 10 surviving individuals (out of 18) to minimize the error in estimating variance in small samples. Similarly, we also excluded populations that had been planted in fewer than 5 locations. Following these restrictions, 103 populations remained with an average of 28 planting sites per population (sd = 13.9). For each population, we calculated the mean and variance in 20-year growth response using LSMEANS and VARCOMP in SAS with a restricted maximum likelihood model to control for block effects in the planting design. These measures of variance showed a weak correlation with the population means (r = 0.29, p < 0.01, df = 97). In attempts correct for this correlation, we tried three different transformations: log-transforming the raw height data, standardizing the final variances by the population means, and standardizing the standard deviations by the population means. Each of these transformations, however, yielded estimates of variance that had stronger, negative correlations with the population means (rlog = -0.68; r"2/µ = -0.61; r"/µ = -0.87), so for all subsequent analysis we used the original untransformed measures of variance. A major limitation facing studies investigating the maintenance of variation is the difficulty of estimating genetic variance for a large number of populations. Here, we use  20  phenotypic variance (VP) measured from multiple common garden trials as a proxy for genetic variance (VG) within each population. While imperfect, this approach may provide a reasonable estimation of genetic variance, assuming environmental variance (VE) is either positively correlated with VG or is roughly constant among populations planted within the same test environment. Randomized planting design and the large number of replicate test sites across a broad range of environmental conditions (mean = 33) should reduce the likelihood of consistent differences in VE due to experimental design and/or genotype x environment interactions.  Environmental heterogeneity To represent patterns of spatial environmental variation, we constructed GIS-based maps of precipitation and temperature (Table 2.1). Each of these datasets consists of a square lattice with a resolution of approximately 1 km2 per cell, covering a total area of roughly 4000 km x 4000 km. This spans the entire range of lodgepole pine, providing an approximate picture of the spatial variation in climatic conditions experienced by the species. All variables were created from 1 km GIS-based rasters in the WorldClim global climatic datasets (Hijmans et al. 2004). The original WorldClim dataset consists of maps showing mean monthly estimates of precipitation and minimum and maximum temperature, created by using thin plate splines to interpolate from a global network of weather stations based on distance and altitude. Weather station data typically represent 30–year averages from the period of 1960-1990. The WorldClim data does not account for rainshadow effects or other complicated meteorological phenomena, however it should be sufficiently accurate to capture medium- and large-scale climatic patterns and  21  approximate regional heterogeneity. As these climatic variables are continuous representations of the abiotic conditions rather than biologically-based models of stress (e.g. incorporating the importance of the transition from freezing to non-freezing temperatures), they are only approximate representations of selection pressure. To increase the biological relevance of these maps and avoid the inclusion of climatic data from uninhabited alpine areas and lowland valleys, we eliminated all areas that are currently non-forested. We identified non-forested areas using maps of coniferous forest from 1992 (DeFries et al. 2000), excluding any area with less than 10% cover. For any given cell in the climatic maps, we calculated heterogeneity in the surrounding region using a weighted measure of variance:  "2  $ ( x # x) = $m ij  ij  ij  !  2  % mij  (2.1)  ij  where i,j are the geographic coordinates of the cells being considered, xij is the environmental condition at cell [i,j], x is the weighted mean for the region, and mij is the weighting at cell [i,j]. In this study, the region of analysis is a 201 km x 201 km square ! surrounding the focal cell and the weights represent the relative probability of gene flow  from the surrounding cells to the focal cell. In lodgepole pine, gene flow is mainly the product of wind-mediated pollen dispersal, with seeds typically moving only short distances (Ennos 1994). While we could find no quantitative data on pollen dispersal curves in lodgepole pine, studies of other wind-pollinated conifers have generally found that pollen flow decreases with distance and is typically leptokurtic (Schuster & Mitton  22  2000; Robledo-Arnuncio and Gil 2005). Following Austerlitz and colleagues (2004), we used a Weibull distribution to calculate the weights for each cell, as this was the bestfitting of all distributions they tested and is also flexible in its shape:  mij =  " *1 " $ d ' *( d # ) " & ) e # %# (  (2.2)  where d is the distance between the focal cell and cell [i,j], # is the scale parameter and $  !  is the shape parameter. For this distribution, # corresponds to the mean distance of gene flow, and $ describes the degree of leptokurtosis (decreasing values of $ result in increasing leptokurtosis). It is important to note that in this application, the mean distance of gene flow does not correspond exactly to the dispersal distance of pollen, due to the possibility of multi-generational stepping stone migration. As an initial parameter set, we measured heterogeneity under # = 0.5 km and $ = 1 (which corresponds to an exponential distribution with mean = 0.5). To examine the effect of different parameters describing the dispersal curve, we applied this method to calculate heterogeneity under a range of values of # (200 m – 10 km) and $ (0.2 – 2), using an analysis window of 201 x 201 km. This simple representation of gene flow will fail to account for the influence of physical and phenological barriers to pollen flow and may be less accurate in mountainous areas where differences in flowering time can present effective barriers to gene flow (Schuster et al. 1989); it should, however, provide a coarse approximation. In all applications of the above methods, we calculated heterogeneity for the cell corresponding to the geo-referenced point of origin for each provenance. Four of the provenance geo-reference points corresponded to cells that lay more than 2 km outside of currently forested areas, and these were excluded from the analysis. Following these restrictions, 99 populations had sufficient data for estimates of both genetic diversity and 23  environmental heterogeneity. All variables except DROUGHT were log-transformed prior to the calculation of weighted variance in order to reduce the effect of large ranges in magnitude. To facilitate comparisons among variables, all measurements of heterogeneity were standardized to an index between 0 and 1 by subtracting the minimum value from all scores and dividing by the range. To evaluate the relationship between withinpopulation variance and regional heterogeneity, we performed multiple linear regressions and tested Pearson correlations using ‘lm’ and ‘cor.test’ in the statistical package R. As a test of an alternate hypothesis, we calculated correlations between genetic variance within populations and both latitude and distance from the center of the range (defining the center as the mean of all latitudinal and longitudinal coordinates).  Results Genetic variance in 20-year growth responses within populations was correlated with regional heterogeneity in all four environmental variables (Figure 2.1). Surprisingly, these correlations were relatively insensitive to variations in the parameters defining the dispersal distance under the Weibull distribution (Figure 2.2). Pearson correlations (r) calculated under all parameter combinations were significantly different from 0 (p < 0.05, df = 97), however 95 % confidence limits overlapped for all dispersal functions considered (not shown). Thus, although correlations tended to be somewhat lower under certain parameter combinations (e.g. # < 1 and $ > 1 for DROUGHT, COLD, and TEMP), these were not statistically different from correlations under other combinations. By calculation of r2, regional heterogeneity in DROUGHT explained 8% - 20% of the variation in genetic diversity, depending upon the values of # and $ (Figure 2.2).  24  Heterogeneity in the other variables explained 7% - 14% (COLD), 7% - 13% (TEMP), and 7% - 19% (PRECIP) of the variation in genetic diversity within populations. Although some variables explained more of the variation than others, and all were significantly different from zero (p << 0.05, df = 97), differences between them were not statistically significant (all estimates lie within 95% confidence limits of other estimates). Thus, it was not possible to evaluate the relative biological importance of the different climatic variables used here. Multiple linear regression of genetic variance on heterogeneity in all four variables showed that they could collectively explain between 17% and 28% of the variation in genetic diversity within populations, depending upon the values of # and $. To examine other possible explanations for patterns in genetic variance, we used linear regression to test whether variance within populations was correlated to either their latitude or their distance from the geographic centre of their range. Both variables showed significant inverse relationships, with latitude explaining 5% of the variation (p < 0.05, df = 97) and distance explaining 10% of the variation (p < 0.01, df = 97) in genetic variance. To test whether these two terms covaried with any of the heterogeneity measurements, we calculated the residuals of the single factor regressions of diversity on latitude and distance, and then calculated the correlations between these residuals and the measurements of heterogeneity. In almost all cases, the correlations between genetic variance and heterogeneity were still significant, with heterogeneity explaining 9% - 18% (DROUGHT), 4% - 12% (COLD), 6% - 10% (TEMP), and 6% - 19% (PRECIP) of the variation in residuals of the distance regression and 7% - 16% (DROUGHT), 6% - 12% (COLD), 7% 11% (TEMP), and 4% - 18% (PRECIP) of the variation in residuals of the latitude regression, depending upon the values of # and $. Of these, all correlations were  25  significant (p < 0.05, df = 97) except for 6 of the 120 parameter combinations in COLD vs. distance residuals and 3 of the 120 parameter combinations in PRECIP vs. latitude residuals. We then calculated the residuals from the multiple linear regression of diversity on both latitude and distance. Multiple linear regression of these combined residuals on heterogeneity in all four variables showed that they could collectively explain approximately 20% of the variation in genetic diversity within populations (p < 0.001, df = 97; for # = 500 m and $ = 1). Thus, while latitude and distance from the centre of the range are significantly correlated to genetic variance, for the most part, they explain different portions of the variance than regional environmental heterogeneity.  Discussion Evolutionary theory suggests that quantitative genetic variation within populations can be maintained by spatially varying selection and gene flow (Slatkin 1987; Barton 1999; Tufto 2000; Spichtig & Kawecki 2004). Although local adaptation is pervasive (Hedrick et al. 1976; Linhart & Grant 1996), it is unclear whether gene flow is strong enough to maintain the high levels of genetic variation found in nature. Other processes such as mutation-selection balance or population bottlenecks could have a greater effect on levels of genetic variation within populations, obscuring any contribution by gene flow. Evidence from this study shows strong correlation between regional heterogeneity and genetic variance in lodgepole pine (r2 ~ 20%), suggesting that gene flow and heterogeneous selection are making significant contributions to levels of genetic variation within populations.  26  As this inference is based on correlation, however, there are two possible alternate explanations that should be addressed. Firstly, it is possible that temporal heterogeneity is responsible for maintaining diversity within populations (as per Bürger & Gimelfarb 2002). If areas that are more spatially heterogeneous are also more temporally heterogeneous, then it would be impossible to evaluate which of these factors is maintaining diversity with the correlation-based approach used here. There is good reason to suspect that temporal and spatial heterogeneity in climate would not be well correlated, because other factors such as continentality and oceanic currents can influence temporal variations in climate without correlation to spatial influences such as altitude. At the present time, however, there are no long-term climatic records available at a sufficiently fine spatial scale, so this alternate hypothesis cannot be conclusively rejected. As a second explanation, it is possible that environments within populations are also heterogeneous and that variance within populations is the product of microenvironmental adaptation in their immediate environment. If areas that are regionally heterogeneous also tend to be more heterogeneous at this small scale within populations, then correlations with regional heterogeneity could be an artifact, as described above with respect to temporal heterogeneity. Because the datasets we used are interpolated on a coarse 1 km x 1 km scale, they are inappropriate for estimating fine-scale heterogeneity and testing this alternate hypothesis. While local adaptation in Douglas fir (Pseudotsuga menziesii) has been found over changes in altitude of only a few hundred meters (Campbell 1979), sampling for the establishment of common gardens in this experiment was conducted over small areas (~1 km2) and care was taken to avoid sampling over obvious sources of environmental variation within a population. Thus, although it was not  27  possible to conclusively test that variance was not being maintained by microenvironmental adaptation within populations, this seems an unlikely explanation for the correlations found in this study. While theoretically possible, these alternate explanations are less likely than the suggestion that gene flow and regional heterogeneity maintain diversity within populations. Local adaptation has been demonstrated (Yang et al. 1996; Rehfeldt et al. 1999; Wu & Ying 2004), and gene flow in lodgepole pine is extensive (Perry 1978), so some effect of gene flow on levels of variance is expected. Assuming that our interpretation is correct, gene flow and regional heterogeneity explain approximately 20% of the variation in diversity within populations of lodgepole pine. In fact, due to the potential for errors in the heterogeneity modeling and estimation of genetic variance, the true correlation may be considerably stronger. This evidence suggests that gene flow and heterogeneity play an important role in maintaining genetic variance, but how general is this effect? Under what conditions would we expect to see maintenance of variation by environmental heterogeneity and gene flow in other species? Firstly, natural selection must be strong enough to maintain variation in trait means between populations in spite of gene flow. Secondly, for the increase in variance to be significant, there must be considerable spatial variation in the optimum trait within the effective range of gene flow. Linear cline models show how the environment must change over a ‘characteristic length’ defined by the strength of selection and distance of gene flow in order for selection to maintain localized adaptations (Slatkin 1978; Barton 1999). Assuming there is some analogous threshold in more complex heterogeneous environments, there will be some scale of change in  28  environment below which local adaptations are not maintained and genetic variance is unaffected by migration-selection balance. Lodgepole pine is an ideal species in which to detect such an effect, as it has extensive gene flow and inhabits both heterogeneous mountain environments and homogenous plateaus. It remains to be seen whether migration-selection balance plays a significant role in other species, where gene flow is limited and/or environments are more uniform. Unfortunately, the method used here is only applicable when there is substantial variation in the regional heterogeneity; tests based on correlation are powerless to detect an effect when all populations experience equal conditions. It is worth noting that if migration-selection balance plays a significant role in driving levels of diversity, the common statistical assumption of homogeneity of variances across populations may not be warranted when they inhabit environments with different levels of heterogeneity. Most studies of the impact of gene flow on genetic structure within populations have focused on its maladaptive consequences. Empirical studies have described the impacts of migration load (eg. Storfer & Sih 1998), gene swamping (eg. Raymond & Marquine 1994), and outbreeding depression (eg. Price & Waser 1979), while theoretical studies have noted the limits to local adaptation (eg. Kirkpatrick & Barton 1997). Here, we have found evidence suggesting that variance within populations can be maintained by gene flow without homogenizing local adaptations and eliminating diversity between populations. While populations of lodgepole pine are not always perfectly locally adapted (Wu and Ying 2004), they are able to persist under this genetic load and even maintain ecological dominance. Although genetic load is always maladaptive in environments that do not change over time, genetic variance is necessary for any response to selection  29  (Fisher 1930); thus, genetic load may be beneficial in times of rapid environmental change. While both mutation and migration can increase variance, migrant alleles from neighbouring populations with slightly different environments are more likely to be adaptive to slight temporal variations in environment than random mutations generated de novo. For example, if two populations inhabit wet and dry environments respectively, and climatic change causes these habitats to reverse their characteristics, allelic variation maintained by gene flow would be inherently adaptive to the novel change, whereas alleles generated by mutation would often be maladaptive. If migration-selection balance proves to be a widespread and significant factor maintaining variation in quantitative traits, it will be important to consider environmental heterogeneity and gene flow when evaluating conservation and management options, especially when considering adaptation to climatic change. Generally speaking, conserving both heterogeneous landscapes and historical levels of gene flow should maintain diversity within and between populations.  30  Table 2.1 – Climatic variables. Variable Name DROUGHT PRECIP  Description Number of ! month periods with less than 25mm of precipitation Mean annual precipitation  COLD  Coldest mean monthly temperature  TEMP  Mean annual temperature  31  Figure 2.1 – Relationship between variance within populations and heterogeneity. Scatterplots shown for four climatic variables, calculated using # = 500m, $ = 1 for the Weibull distribution representing gene flow.  32  Figure 2.2 – Sensitivity of results to variation in model representing gene flow The z-axis shows the strength of the Pearson correlation between variance within populations and heterogeneity in the four climatic variables, calculated under a range of values of " and #$for the Weibull distribution representing gene flow.  33  References Austerlitz, F., C.W. Dick, C. 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C. and C. Layton. 1979. Organization of genetic variability in central and marginal populations of lodgepole pine (Pinus contorta Spp Latifolia). Can. J. Genet. Cytol. 21:487-503.  39  3. No effect of environmental heterogeneity on the maintenance of genetic variation in wing shape in Drosophila melanogaster1 Introduction In 1981, Mackay published the results of a landmark experiment examining the effect of heterogeneous environments on the maintenance of quantitative genetic variation in bristle traits and body size. Interestingly, she found that both long- and short-period temporally heterogeneous environments (ethanol vs. standard fly medium) maintained more additive genetic variance (VA) in sternopleural bristles than spatially varying environments, which is not predicted from classical single-locus theory (Felsenstein 1976). While her study presented strong evidence for an effect of heterogeneity on variation, finding up to 3.5 times more VA in the heterogeneous treatments, other similar studies have yielded inconsistent results. Of two studies with Drosophila spp. that preceded Mackay’s, one found either an increase in genetic variance for sternopleural bristles of ~50% under diurnal temperature fluctuations (Beardmore 1961), while the other found a small and non-significant relationship between VG and heterogeneity on a one-month or three-month cycle (Long 1970). A third study that varied the type of medium given to successive generations of Tribolium castaneum found no increase in genetic variance relative to constant controls (Riddle et al. 1986). Two subsequent studies that inferred genetic variation from heritability or response to selection also did not find consistent trends (Verdonck 1987; Garcia-Dorado et al. 1991). By contrast, studies of the maintenance of allozyme heterozygosity found comparatively consistent evidence of  1  A version of this chapter will be submitted for publication. Yeaman, S., Y. Chen, and M.C. Whitlock. No effect of environmental heterogeneity on the maintenance of genetic variation in wing shape in Drosophila melanogaster. 40  modest increases (usually under 50%) under heterogeneous environments (Powell 1971; McDonald and Ayala 1974; Powell and Wistrand 1978; but see Haley and Birley 1983), which may be due to the relatively less stringent conditions for maintenance of polymorphism in single-locus traits (Spichtig and Kawecki 2004). Given the variation in the results of these studies and the range of study designs they have employed (often with limited replication), it is unclear how broadly Mackay’s results would generalize to other traits and environmental stresses. Over a quarter of a century after Mackay’s paper, we are still uncertain about the relative importance of the various evolutionary processes for the maintenance of genetic variation. Predictions from models of mutation-selection balance are still thought to be inconsistent with empirical measurements of the strength of selection and rate of mutation (Johnson and Barton 2005; but see Zhang and Hill 2004, 2005), and while a range of models have shown that heterogeneous environments have the potential to maintain more variation (Via and Lande 1987; Gillespie and Turelli 1989; Bürger and Gimelfarb 2002; Spichtig and Kawecki 2004; Turelli and Barton 2004), there is little formal quantification of the likely magnitude of this effect under biologically reasonable parameters. There is some evidence that gene flow in heterogeneous environments can cause a detectable increase in genetic variance within populations (Yeaman and Jarvis 2006), but the generality of this effect is unclear. Given the difficulties in parameterizing even the simplest models of mutation-selection balance, further experimental studies may yield the most productive evidence to illuminate our understanding of the maintenance of variation. Here, we test the effect of various patterns of environmental heterogeneity on the  41  maintenance of genetic variation in wing traits in Drosophila melanogaster, replicating and extending Mackay’s classic study in a different trait and environment, and including a treatment examining the effect of spatial heterogeneity with limited migration (treatments described in Table 3.1). As differences in temperature are known to cause adaptive responses in Drosophila for wing size (Partridge et al. 1994) and shape (Santos et al. 2004), we established 25 replicate populations from a sample of wild-collected Drosophila melanogaster and exposed each them to one of five different heterogeneous or homogeneous temperature treatments for 116 weeks (five replicate populations per treatment). To simulate a spatially heterogeneous environment, lines were divided into two cages maintained at different temperatures (16oC and 25oC) with either panmictic (S; m = 0.5) or limited migration (M; m ~ 0.001). To simulate a temporally heterogeneous environment, lines were moved every four weeks from one of the experimental temperatures to the other (T). These experimental treatments were compared to two homogeneous treatments, where lines were maintained at either the cold or hot temperature for the duration of the experiment (C & H). At the end of the period of adaptation, we assayed genetic variance in 20 wing morphology traits at both experimental temperatures using an offspring-parent breeding design. This design allows us to test whether more variance is maintained under temporal than spatial heterogeneity, as found unexpectedly by Mackay (1981), and to compare the variance maintained by these treatments to a pattern of spatial heterogeneity with limited migration.  42  Methods Establishment and maintenance of experimental lines All lines used in this experiment were established from a large sample Drosophila melanogaster captured on September 25th, 2005, using a sweep net over a permanent compost pile from a certified organic orchard near Cawston in the Similkameen Valley, British Columbia, Canada. From an initial capture of ~2000 adult males and females, 400 mated females were placed individually into vials and allowed to lay eggs. Virgin females were collected from these vials and used to establish lines by making reciprocal crosses between the numbered isofemale lines. All isofemale lines were test-crossed with males from a known Drosophila simulans strain from the Tucson Stock Centre to confirm species identity (see Supplementary Materials for details). The large cage population was allowed to expand in size over the next nine generations, with bottles randomly reassigned among cages to ensure random mating (see Supplementary Materials for culture medium and cage design). On February 26th, 2006, 400 bottles were added to the cages of the breeding population, then removed two days later and redistributed among 50 new cages. Two cages were then randomly assigned to each replicate population of each treatment and transferred to either 16oC or 25oC according to their experimental treatment (see Table 3.1). In cages maintained at 25oC (16oC), a new generation began every two (four) weeks, with fresh bottles added on Wednesday evening (Tuesday morning) and transferred to new cages on Friday morning. Cages in the cold room thus had approximately twice the laying time of warm room flies to account for their approximately two-fold slower metabolism. Migration occurred every four weeks, such that cages in the cold room migrated every generation, while cages in  43  the warm room migrated every second generation. Migration for the ‘M’ treatment was performed on Tuesday morning before feeding; a random sample of ~15-30 flies was aspirated from the top of each cage, lightly anaesthetized using CO2, and sorted by sex. Two migrant females from each sample were retained and moved to the corresponding cage in the opposite temperature (excess flies from these samples were not returned to the cages). Population size at the end of the generation typically ranged from 2000-4000 adults per cage but on rare occasions was observed as low as ~800 in some cages, due to natural variability of the populations.  Experimental crosses Experimental crosses to assay genetic variance in wing traits were begun in early May, 2009, following 116 weeks of the experiment, which corresponds to 29 generations in the cold room or 58 generations in the warm room. Assays were conducted under both experimental temperatures by initiating two assay lines from each experimental line. Each assay line was established with 10 vials of 30 randomly-picked eggs taken from laying plates left overnight in both cages of each experimental line. To allow for comparisons between the two sub-populations of each replicate of the limited migration treatment (MC/MH), we treated the sub-populations from the hot and cold chambers as separate replicate lines, establishing hot and cold assay lines from each cage. This yielded a total of 30 replicate lines in each assay temperature. Because the procedure for maintaining the assay lines was labour-intensive, we staggered the establishment of assay lines by replicate number, starting one line per treatment every working day over a one week period. Following this initial staggering, a strict schedule was observed such that  44  each replicate block was transferred for mating and laying at the same time relative to its initiation (see Supplementary Materials for a details). Except where specifically stated, all activities associated with the establishment and maintenance of assay lines were conducted at the assay temperature. To control for maternal effects, we reared three generations of flies from each replicate line at controlled densities under both experimental temperatures, using the final two generations as parents and offspring to assay genetic variation. Productivity trials were performed with the C and H treatment parents at each assay temperature by allowing them to lay for either two (warm assay) or four (cold assay) days and counting the number of offspring that emerge from each vial (see Supplementary Materials for details).  Wing traits and statistical analysis Images of the left wings of flies were captured using the approach of Houle et al. (2003), with B-splines fit to the wing veins using FINDWING (see Supplementary Materials for details), yielding 12 landmarks which were used for all subsequent analysis (Figure 3.1). Wing data were included in the analysis for all families with suitable images of both parents and at least one offspring of each sex, with typical families having either two males and two females or three males and three females (after digitizing several thousand wings from families of six offspring, it was determined that little power would be lost by reducing family size). To explore the effect of temperature on the genetic variation, we chose 20 wing traits a priori: centroid size, length to centroid ratio, four allometries (from Weber et al. 2005), five angles (from Whitlock and Fowler 1999), and an additional seven line segments and two angles (described in Table 2). Generalized Procrustes  45  analysis in the statistical package R (procGPA, ‘shapes’ library; Dryden 2007) was used to calculate the centroid size of each wing and to scale and rotate the landmarks to a common orientation, such that the rotated values represent deviations from the average location of the landmark across all individuals. Following Mezey and Houle (2005), we ran procGPA on eight groups of wings (sires, dams, male offspring, female offspring in the cold vs. warm assay) to avoid introducing error into the Procrustes analysis due to differences in trait variance among these groups. The scaled and rotated landmarks were used to calculate the length of all of the line segments described in Table 3.2, while the allometries and angles were calculated from the raw landmark data. By choosing a wide range of traits, we maximize the chances of detecting divergence between the homogeneous populations (C and H) for any biologically relevant dimensions of the wing. To identify traits that had likely been under divergent selection between the conditions in the cold and hot chambers, we fit linear mixed effects models (lme) to the full dataset using the nlme library in R (Pinheiro et al. 2008): lme (trait_value ~ treatment * assay * sex * gen, random = ~1 | replicate), where ‘treatment’ is a fixed effect referring to the C/H experimental treatment, ‘replicate’ is a random effect representing the replicate populations in each treatment, ‘assay’ is a fixed effect representing the two assay conditions, and ‘sex’ and ‘gen’ are fixed effects representing the possible combinations of male and female parents and offspring. We used ‘stepAIC’ in the MASS library in R (CRAN) to simplify the model for each trait, removing nonsignificant effects and interaction terms by comparing the AIC values with and without the term of least significance to yield a most-parsimonious model. The null hypothesis of no difference between trait means of C and H was then evaluated using planned contrasts.  46  The average magnitude of trait divergence was calculated as the mean difference between C  and H treatments standardized by the mean trait standard deviation, both of which were  averaged across the eight possible combinations of the levels of ‘sex’, ‘gen’, and ‘assay’. Phenotypic correlations were analyzed between all pairwise combinations of the traits with evidence for significant divergence by calculating the average correlation across sons, daughters, sires, and dams. For any pairs of traits that yielded r2 > 0.1, we excluded the trait that had a lower average divergence between the C and the H treatments from further analyses. The analysis to compare genetic variation among the full set of experimental treatments was then undertaken using this subset of divergent and weaklycorrelated traits. When males and females have equal phenotypic variance, narrow sense heritability (h2) can be calculated by regressing the mid-offspring values on the midparent values for each trait, with the slope of this relation = h2 (Falconer and Mackay 1996). When males and females have unequal variance, separate heritabilities must be calculated for each sex, with male heritability equal to twice the average of the slopes yielded by regressing sons on sires and daughters on sires, after multiplying the daughtersire regression by "m / "f (with the reverse procedure for female h2, multiplying the sondam coefficient by "f / "m; Falconer and Mackay 1996). We used Levene’s test (‘car’ library in R) to compare phenotypic variance between sires and dams for both the warm and cold assay, bulked across all populations. We implemented both of these methods of estimating heritability using ‘lm’ in R, calculating the phenotypic standard deviations across all individuals of a given sex for each line assay. Additive genetic variance (VA) can then be estimated from: VA = h2 (VP,sires + VP,dams) / 2 (for "m = "f ), or  47  VA,m = hm2 " VP,sires and VA, f = h 2f " VP,dams (for "m % "f), where VP is the phenotypic variance. Additive genetic variance maintained within each line was then compared among  !  ! a linear mixed effects model (VA ~ treatment * assay, random = ~ 1 | treatments using  replicate / assay), while 95% confidence intervals for these estimates were calculated using ‘ci’ in the ‘gmodels’ library in R.  Results Fitness differences between homogeneous lines The warm and cold homogeneous populations (C and H) produced more offspring that survived to adulthood when assayed in the temperature at which they had evolved over the course of the experiment (relative to the opposite-temperature homogeneous populations). In the cold assay, the flies from the C and H treatments produced 67.0 and 59.4 offspring on average, respectively, over a four day laying period, while in the warm assay the C and H treatment flies produced an average of 84.3 and 86.4 offspring, respectively, over a two day laying period. Variance in productivity among replicate populations was lower in the cold assay than the warm assay, so to statistically analyze this pattern we log-transformed all productivity measures. To test the significance of the interaction between treatment, assay, and productivity, we subtracted the log-transformed mean productivity of each replicate in the cold assay from the same quantity in the warm assay (Figure 3.2), and compared the differences in the resulting values between the C and H populations using a Wilcoxon rank sum test (W = 1, p = 0.016) and Welch’s test for unequal variances (t = 2.37; df = 4.13, p = 0.075). The differences in productivity are statistically significant by the former test and marginally non-significant by the latter,  48  suggesting that these lines have adapted to their respective environments. A more accurate measure of fitness accounting for the effects of density and development time would likely reveal more pronounced differences between the homogeneous populations. We also examined selection differentials for the 20 traits by regressing productivity on dam phenotype but found no consistent significant associations (either when grouped by replicate or bulked across all replicates; results not shown).  Trait divergence between homogeneous lines Of the twenty traits that we included in our initial survey, six were found to have p-values < 0.05 for the test of different trait means between the C and H treatments (Table 3.2). Of these six traits, only ANGLE2-4-9 and ANGLE2-4-8 were highly correlated with each other (r2 > 0.1). Based on the average trait divergence between the homogeneous populations (Table 3.2), we excluded ANGLE2-4-9 from further study, as it was less divergent than ANGLE2-4-8.  Of the remaining five traits, only ANGLE7-8-9 and LINE9-10 had p-values  that were significant following Bonferroni correction for multiple comparisons (#/n = 0.0025), and are thus the only traits for which we have strong evidence of divergence. Nevertheless, we used all five remaining five traits that were divergent and weaklycorrelated to explore the effect of the experimental heterogeneity treatments on the maintenance of genetic variation (referred to hereafter as ‘divergent traits’; shaded grey in Table 3.2). We also examined the effect of the treatments on VA for CENTROID, as size is commonly reported to diverge with temperature, although we did not see significant differences in size between the homogeneous treatments. We refer to these six traits as ‘biologically motivated’, as we used criteria defined before analyzing changes in variance  49  to identify them as biologically interesting to avoid the loss of power associated with post-hoc corrections for multiple comparisons.  Trait divergence under limited migration The populations in the ‘M’ treatment consisted of one cage in each chamber, with two randomly selected gravid females moved reciprocally between the cages every four weeks to simulate limited migration rate of approximately m = 0.001. To ask whether this low rate of migration was sufficient to constrain the adaptation of the sub-populations in each chamber, we compared the average divergence between the hot and cold cages within each replicate of the M treatment (DMC-MH) to the average divergence between all possible pairs of replicate lines between the C and H treatments (DC-H) for each of the divergent traits. For all five traits, the absolute value of the average difference between MC  and MH cages within a replicate was less than the absolute value of the average  difference between the C and the H cages (Table 3.2). For three out of the five traits, the mean trait values averaged over all 5 replicates for MC and MH fell between the values for the C and H traits and diverge in the expected direction, such that H > MH > MC > C or the reverse (e.g., Figure 3.3). The two exceptions to this pattern nearly conformed with the expectation, but differed in the sign of one comparison (ANGLE3-10-4 had MH > H > MC > C  while LINE9-10 had H > MC > MH > C). A problem with drawing inference from this evidence, however, is due to  ascertainment bias, as these five traits were selected because they were known to be divergent between populations in the C and H treatments. As such, the finding of less divergence between MC and MH could simply be due to the trait selection criteria. If  50  ascertainment bias is an issue, we should find the reverse pattern (more divergence between MC and MH than C and H) if we select traits based on divergence between MC and MH.  To check for ascertainment bias, we used the above methodology to examine  divergence between MC and MH, and found that only one of the 20 traits showed evidence of divergence (ANGLE7-8-9), even with the raw p-values unadjusted for multiple comparisons (Table 3.2). Of the traits with the five lowest p-values for this comparison, four of them were the same traits identified above as divergent for C and H, but had less divergence between MC and MH. This general pattern therefore suggests that migration is constraining adaptation and divergence between MC and MH in at least some of the traits, although we do not have strong support for any individual comparisons (we did not have enough replication to detect significant differences between MH and H or MC and C for the five divergent traits).  Differences in additive genetic variance among treatments Narrow-sense heritabilities for the six traits were high and significantly different from zero for most replicate populations and traits using the equal variance heritability estimate (Table 3.3). Despite these high heritabilities, we did not find any statistically significant differences in either the additive genetic variance or heritability among any of the experimental treatments for any of the biologically motivated traits identified above, using the equal variance method for estimating heritability (p > 0.3 in all cases; df = 24). While Levene’s test suggests that phenotypic variance differed between the sexes for at least two of the six biologically motivated traits, the ratio of "P,m / "P,f was not very different from one in all cases except LINE3-10 (Table 3.4). In any case, we still found no  51  effect of treatment on VA when calculated using the unequal variance method for estimating heritability. Statistical non-significance aside, we also did not observe any obvious trends in the rank-order of VA by treatment across the six traits (Table 3.3). This lack of effect of experimental treatment on standing genetic variation was also insensitive to whether we excluded populations with non-significant slopes for the offspring-parent regression from the analysis, analyzed variation in VA for the males and females assays separately, or averaged measures of VA for each replicate across both assays. Approaching the limited migration treatment as a pair of connected subpopulations and assaying the genetic variance within each cage (i.e., MC and MH) focuses on the effect of migration on the maintenance of variation within populations. If we instead pool the families from both cages within each replicate of the limited migration treatment (MPOOLED), we can ask whether this treatment maintains more genetic variance across the two environments. Taking this approach, we see much the same results as when we consider the sub-populations individually (Table 3.3), with no significant increase in VA in any trait relative to the lines in either the homogeneous or the other heterogeneous treatments. This further suggests that this type of heterogeneity does not substantially influence the maintenance of standing genetic variation. Despite using the most powerful analysis available by testing the null hypothesis of equal VA across all treatments, there was no evidence of any effect of any of the heterogeneous treatments on genetic variation. We then expanded the above analysis to include all traits, not just those that displayed evidence of divergence, but still found no evidence for an effect of the heterogeneous treatments on genetic variance (even without Bonferroni corrections; see Supplementary Materials). In any analysis that fails to detect  52  an effect when one was expected, it is important to consider the power of the methods employed. Namely, if there was a true effect of heterogeneity on genetic variance that we failed to detect, can we place an upper bound on the maximum effect that would be likely to have gone undetected by the study design we employed? The 95% confidence intervals around the estimates of VA are reasonably small; in almost all cases the upper bounds of intervals for the heterogeneous treatments do not exceed the lower bounds of the intervals of the homogeneous treatment by more than a factor of two (with the exception of LINE310; Table 3.3). Thus, although it is impossible to rule out an effect of the heterogeneous treatments on genetic variance, we may be reasonably confident that they could not have caused an increase of genetic variance of more than approximately two-fold, relative to the homogeneous treatments. If our experiment had resulted in a real effect of heterogeneity on variance of the magnitude observed by Mackay (up to 3.5 fold;1981), we should have had the statistical power to detect it.  Discussion Population genetic theory generally suggests that heterogeneous environments have the potential to maintain more genetic variation than homogeneous environments, especially when there are genotype-by-environment interactions for fitness (Felsenstein 1976; Spichtig and Kawecki 2004). Despite high trait heritabilities (Table 3.3), divergence in trait values (Table 3.2) and evidence for environment-dependent differences in productivity between homogeneous lines (Figure 3.2), we found no evidence that any of the environmentally heterogeneous treatments maintained more variation than the homogeneous treatments (Table 3.3). Perhaps most surprising is that while limited  53  migration in the M treatment seems to have constrained local adaptation and divergence between the warm and cold cages (Table 3.2), it does not seem to have had an effect on the maintenance of variance within the sub-populations (Table 3.3). Thus, although our study has replicated the conditions under which one would might expect to see an effect of heterogeneity on VA, we witnessed no such effect. One possible explanation for the lack of effect of heterogeneity on VA is that selection simply wasn’t strong enough to significantly increase the variance in the heterogeneous treatments relative to the effects of drift and background variation maintained by mutation or within-patch balancing selection. While we found significant differences between C and H populations in the mean values of five traits, the magnitudes of these differences were not extreme (Table 3.2), suggesting that selection during the course of the experiment was weak in strength or involved relatively small differences in optimum. If we assume constant directional selection operating at equal strength but opposite sign in each treatment, the trait with the largest difference between C and H (ANGLE7-8-9; Table 3.2) would have had to change by only 0.0053 "p per generation (Haldanes) to result in total divergence of 0.461 "p by the end of the experiment (through either direct or correlated selection). Given a mean heritability of ~0.635, this would correspond to selection differentials of approximately 0.0083"p, which is over an order of magnitude weaker than median estimates of ! ~ |0.16| from single-generation studies of natural populations (Kingsolver et al. 2001). By contrast, if most trait divergence occurred in the first few generations followed by a period of stabilizing selection around the different optima, selection could have been much closer to typical estimates in natural populations.  54  Comparing the amount of divergence we observed between the populations in the homogeneous treatments to divergence observed in other similar studies that did find a positive effect of heterogeneity on genetic variance could help interpret our results. Unfortunately, no study that has documented an increase in variance or heterozygosity under heterogeneous conditions has also explored whether the environmental variable used in the experiment caused divergence in phenotype or allele frequency between populations exposed to constant conditions of one extreme or the other (see Introduction). As such, it is unclear whether the lack of response to heterogeneity that we observed was due to relatively weaker selection than previous experiments that detected a positive effect of heterogeneity on VA or heterozygosity. The rates of divergence observed in our study are roughly similar to others that have studied response to laboratory natural selection in Drosophila. Santos et al. (2004) found that an overall wing shape metric diverged at a rate of ~0.01 "p per generation (Haldanes) in response to divergent temperature treatments, which is only twice as fast as the rate for ANGLE7-8-9 found here. Cavicchi and colleagues (1989) found that wing length diverged in response to temperature by approximately 2.25 phenotypic standard deviations after 208 generations of divergent laboratory natural selection, which yields a rate of divergence similar to that observed in our experiment if constant gradual change is assumed (two populations were founded from a long-term stock that had been maintained at 18oC, and were subsequently maintained at 28oC). As such, it is reasonable to assume that the strength of selection resulting from our experimental design was on the same order of magnitude as the other heterogeneity-genetic variance studies.  55  Reconciling our results with those of previous studies is further complicated by considerable variation in experimental design, choice of traits, heterogeneity treatments, and limited replication and testing of controls (see Introduction). The biggest difference between the design of our study and that of Mackay (1981) was the potential for habitat choice in all of her heterogeneous treatments, as regular and alcohol-containing medium were simultaneously available for some period of the life cycle in all three of her heterogeneous treatments. In another study that used a design very similar to that of Mackay, Garcia Dorado and colleagues (1991) found evidence for the evolution of habitat choice in response to heterogeneous alcohol/regular medium treatments, but they did not explicitly measure genetic variance and found no consistent effects of treatment on heritability. As habitat choice is potentially a strong mechanism maintaining variation (Edelaar et al. 2008; Ravigné et al. 2009), this seems the most plausible explanation for the increased variance in Mackay’s study and the lack of effect in the S and T treatments in our study, where habitat choice was prevented by enforced random mating. Mackay detected up to ~3.5 times more VA in the heterogeneous treatments than in the homogeneous treatment, which is well above the ~2 fold upper limit to the difference in VA that could have gone undetected in our experimental assays. The lack of effect on VA in the M treatment of our study is more puzzling, as theoretical models suggest that significant increases in variance are typically seen when migration constrains adaptation in heterogeneous environments (Spichtig and Kawecki 2004; their Figure 3.4). If mutation is a much stronger driver of variation in wing traits than migration, it would likely have affected all treatments in a similar manner and could have overwhelmed the relative contribution of the treatment effects to differences in  56  variance (as shown in Figure 5 in Spichtig and Kawcki 2004). Frequency-dependent selection could also maintain considerable variation irrespective of the environmental heterogeneity treatment (Bürger and Gimelfarb 2004), and strongly-selected polymorphisms such as rover-sitter have been identified in D. melanogaster (Fitzpatrick et al. 2007). If wing shape polymorphisms are maintained directly by frequencydependent selection or are linked to genes that experience frequency-dependent selection, this could also explain the lack of effect of heterogeneity that we observed in the M treatment. While differences in Drosophila spp. wing size are commonly reported to evolve in response to temperature in both natural (Huey et al. 2000; Gilchrist et al. 1994; Wayne et al. 2005; Hoffmann and Weeks 2007) and laboratory populations (Anderson 1966; Powell 1974; Cavicchi et al. 1989; Partridge et al. 1994), we did not observe any statistically significant differences in centroid size between the C and H populations. Unlike the other studies reported above, Santos et al. (2004; 2005) also found evidence of divergence in wing shape but no divergence in wing size, which they attributed to the fact that they controlled larval densities during the maintenance of the populations. As populations maintained under warm laboratory conditions tend to equilibrate at higher densities than cold populations when density is uncontrolled, Santos et al. (2004, 2005) interpreted the lack of response in body size as suggesting that perhaps population density is more critical than temperature for body size evolution in a laboratory setting. Our experiment did not control larval density but did impose non-overlapping generations, which could have also resulted in very different selection pressures than previous studies. Anderson (1966), Powell (1974), and Partridge et al. (1999) all used  57  overlapping generations, while Cavicchi et al. (1989) used discrete generations of the same length used here, but with 18oC vs. 25oC (instead of 16oC vs. 25oC). Flies in our cold treatment tended to mature much closer to the day of the cycle where food was added than those in the warm treatment, suggesting that development time might be under greater selection in the cold room. If this was the case, selection for rapid development time in the cold room could be expected to oppose selection for large body size, on account of the typical positive genetic correlation between these traits (Partridge and Fowler 1993). If mutation provides a sufficient explanation for the maintenance of variation, then evolvability is largely determined by processes that operate within a single population, irrespective of the environment. If instead, the maintenance of variation also depends upon the spatial and temporal qualities of the environment that a species inhabits, then evolvability may depend upon processes that operate at the metapopulation level. If migration between populations maintains a considerable fraction of the standing variation within populations, then habitat loss and fragmentation may have long term impacts on evolvability and species survival, especially in the face of climatic change. While it is clear that mutation is the ultimate source of all variation, empirical evidence suggests that migration and heterogeneity may cause substantial increases in genetic variation within natural populations (Yeaman and Jarvis 2006). The lack of effect of heterogeneity on VA observed in this study should not be taken as evidence against an effect of heterogeneity in general, but does suggest that random panmictic migration does not facilitate the maintenance of much genetic variation. As the constraint on habitat choice was the principal difference in design between our study and that of Mackay  58  (1981), who found the strongest evidence of an effect of heterogeneity on genetic variance, we suggest that this may be a feature of evolutionary ecology worthy of further consideration.  59  Table 3.1 – Description of experimental treatments and abbreviations Code  Name  Description  C  Cold control  H  Hot control  S  Spatial heterogeneity with panmixia Temporal heterogeneity  Two cages at 16oC, 8 bottles per cage, 4 bottles migrate reciprocally Two cages at 25oC, 8 bottles per cage, 4 bottles migrate reciprocally One cage at 25oC, one at 16oC, 8 bottles per cage, 4 bottles migrate reciprocally Two cages maintained together, 8 bottles per cage, 4 bottles migrate reciprocally, cages move between 25oC and 16oC every 4 weeks One cage at 25oC (MH), one at 16oC (MC), 8 bottles per cage, 2 mated females migrate reciprocally  T  MC/MH  Spatial heterogeneity with limited migration  60  Table 3.2 – Divergence in twenty wing traits.  Trait centroid line7-8 line9-10 line2-12 line1-4 line1-5 line2-8 line3-10 angle1-2-4 angle1-5-4 angle7-8-9 angle3-10-4 angle1-8-2 angle2-4-9 angle2-4-8 allometry1 allometry2 allometry3 allometry4 centroid / length  Description centroid line between 7 and 8 line between 9 and 10 line between 2 and 12 line between 1 and 4 line between 1 and 5 line between 2 and 8 line between 3 and 10 angle between 1,2,4 angle between 1,5,4 angle between 7,8,9 angle between 3,10,4 angle between 1,8,2 angle between 2,4,9 angle between 2,4,8 line1-2 / line4-5 line2-4 / line1-5 line1-4 / line5-12 line4-5 / line1-12 centroid / line2-12  [ZC - ZH] / "p 0.19 -0.16 -0.34 0.09 -0.16 -0.11 0.07 0.20 -0.10 -0.12 -0.46 -0.43 0.08 0.34 0.41 0.07 -0.19 -0.07 0.004  DC-H DMC-MH 0.14 -0.08 0.17 -0.01 0.05 0.08 -0.05 0.07 -0.03 -0.08 0.21 0.09 -0.04 0.30 0.38 -0.10 -0.06 0.06 -0.051  p [C vs. H] 0.2950 0.2941 * 0.0017 0.1476 0.1312 0.3308 0.1022 0.0081 0.1347 0.1229 * 0.0003 0.0041 0.8114 0.0103 0.0060 0.5860 0.5605 0.7730 0.5341  p [MC vs. MH] 0.5456 0.1641 0.1045 0.8915 0.9100 0.7089 0.6785 0.0624 0.8021 0.3997 0.0317 0.0565 0.4088 0.5336 0.9479 0.5891 0.4066 0.5006 0.5306  -0.09  -0.01  0.1455  0.9522  Description of the wing traits used in the preliminary analysis and evidence for divergence in trait means in C vs. H and MC vs. MH. p-values for the comparisons between MC  vs. MH and C vs. H are shown for the test of the null hypothesis of no divergence (df =  24). Rows shaded grey indicate the five divergent and minimally correlated traits used in the analysis of VA; (*) indicates significance following Bonferroni correction.  61  Table 3.3 – Trait heritabilities (h2), additive genetic variation (VA), and 95% confidence intervals (95% CI) for each treatment, averaged across replicate and assay.  LINE9-10  LINE3-10  ANGLE7-8-9  ANGLE3-10-4  ANGLE2-4-8  CENTROID  h2 VA (x10-5) 95% CI h2 VA (x10-5) 95% CI h2 VA 95% CI h2 VA 95% CI h2 VA 95% CI h2 VA 95% CI  Treatment C 0.54 (10) 0.173 (0.116-0.204) 0.25 (5) 2.179 (1.187-3.279) 0.63 (10) 19.11 (17.35-25.5) 0.62 (10) 0.333 (0.273-0.397) 0.67 (10) 1.645 (1.579-2.159) 0.43 (8) 26.82 (23.68-38.77)  H  MC  MH  0.56 (10) 0.190 (0.133-0.221) 0.19 (4) 1.745 (0.753-2.844) 0.64 (10) 17.98 (16.23-24.38) 0.61 (10) 0.362 (0.302-0.426) 0.67 (10) 1.442 (1.375-1.956) 0.43 (7) 24.81 (21.66-36.75)  0.49 (9) 0.170 (0.114-0.202) 0.23 (5) 2.335 (1.343-3.434) 0.55 (10) 16.27 (14.51-22.66) 0.66 (10) 0.328 (0.268-0.392) 0.69 (10) 1.526 (1.459-2.04) 0.59 (9) 30.63 (27.48-42.57)  0.54 (10) 0.184 (0.128-0.216) 0.26 (4) 2.572 (1.58-3.671) 0.61 (9) 19.19 (17.43-25.58) 0.65 (10) 0.358 (0.298-0.422) 0.64 (10) 1.591 (1.525-2.105) 0.41 (9) 25.66 (22.51-37.6)  MPOOLED 0.53 (10) 0.192 (0.140-0.223) 0.26 (9) 2.540 (1.73-3.61) 0.59 (10) 18.54 (16.8-24.9) 0.69 (10) 0.330 (0.320-0.436) 0.67 (10) 1.650 (1.519-2.150) 0.50 (10) 26.82 (26.97-40.27)  S  T  0.46 (8) 0.142 (0.085-0.173) 0.21 (3) 1.870 (0.877-2.969) 0.67 (10) 17.92 (16.16-24.32) 0.62 (10) 0.313 (0.253-0.377) 0.73 (10) 1.717 (1.651-2.231) 0.54 (10) 28.15 (25.01-40.09)  0.57 (10) 0.203 (0.146-0.235) 0.32 (7) 3.164 (2.172-4.263) 0.6 (10) 18.29 (16.53-24.68) 0.7 (10) 0.350 (0.29-0.414) 0.65 (10) 1.574 (1.507-2.088) 0.46 (9) 26.37 (23.23-38.31)  Numbers in brackets adjacent to each heritability estimate represent the number of replicate/assay combinations with h2 significantly different from 0 (out of 10). Treatment symbols indicate: cold homogeneous (C); hot homogeneous (H); spatial heterogeneity with limited migration, cold and hot subpopulations (MC and MH) and from samples pooled from both subpopulations (POOLED); spatial heterogeneity with panmictic migration (S); temporal heterogeneity (T).  62  Table 3.4 – Test of the differences in phenotypic variance between males and females from the parental generation using Levene’s test (df = 1).  Trait LINE9-10 LINE3-10 ANGLE7-8-9 ANGLE3-10-4 ANGLE2-4-8 CENTROID  !P,m/!P,f Warm assay Cold assay 0.96 1.06 1.38 1.21 1.03 0.96 1.05 1.05 0.95 0.94 1.05 1.03  Levene’s test p-value Warm assay Cold assay 0.0672 0.0681 0.0000 0.0000 0.5809 0.0535 0.1506 0.3977 0.0439 0.0052 0.2364 0.7443  63  Figure 3.1 – Landmarks produced by FINDWING spline fitting.  64  Figure 3.2 – Difference in mean log-transformed productivity between warm assay and cold assay for each replicate of the control populations.  65  Figure 3.3 – Reduced divergence between cages in the limited migration treatment Mean trait values for ANGLE7-8-9 averaged across both assay temperatures for the five replicates in each treatment for the two control treatments and the warm and cold cages of the limited migration treatment. Grand means across all replicates shown as red bars.  66  References Anderson, W. W. 1966. Genetic divergence in M. Vetukhiv’s experimental populations of Drosophila pseudoobscura. 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Establishment and maintenance of adaptive genetic divergence under migration, selection, and drift1 Introduction Understanding how evolutionary processes affect the establishment and maintenance of genetic divergence between populations is critical to interpreting empirical evidence for local adaptation and speciation. Most theoretical research on population divergence under migration-selection balance has focused on deterministic models (reviewed in Felsenstein 1976; Lenormand 2002). The earliest studies by Haldane (1930) and Wright (1931) considered the maintenance of a locally adapted allele in a single patch subject to immigration of a maladapted allele from a source population (a continent-island model). These studies showed that the locally adapted allele would be deterministically maintained as long as the rate of immigration was lower than the selection coefficient favouring the local allele (m < s). Moran (1962), Maynard Smith (1966), and Bulmer (1972) explored a similar model where selection favored different alleles in two neighbouring patches connected by reciprocal migration (m1, m2). A rare allele will spread deterministically when  m1 m2 + < 1, where the fitnesses of the common s1 s2  homozygotes in the two patches are defined as 1–s1 and 1–s2 relative to the rare ! 1972). While the two-patch threshold simplifies to the continentheterozygotes (Bulmer  island threshold when one of the migration rates tends towards zero, it predicts the maintenance of polymorphism across a much wider region of parameter space when selection coefficients and migration rates are more closely balanced. In the most extreme 1  A version of this chapter will be submitted for publication. Yeaman, S. and S.P. Otto. Establishment and maintenance of adaptive genetic divergence under migration, selection, and drift 72  case, if s1 = –s2 and m1 = m2, any migration rate will satisfy the inequality, implying that even the most weakly selected loci can diverge between populations, regardless of how much migration connects them. It should be noted, however, that although this extreme case predicts the maintenance of divergence for any migration rate, the actual difference between the frequencies of the locally adapted and migrant alleles in each patch would tend towards zero with decreasing magnitudes of selection or increasing migration rates. As a consequence of the near balance between migration and selection in these cases, we might expect polymorphism to be readily lost in finite populations. In contrast to the predictions from infinite population models, the maintenance of polymorphism in a two-patch system with finite populations inevitably requires the input of novel mutations, as global fixation of one allele or another will eventually occur. Thus, polymorphism will effectively be maintained when the rate at which locally beneficial mutations occur and rise to high frequency outpaces the rate at which polymorphism is lost due to drift. That said, as migration rates decrease and the strength of divergent selection increases, the persistence time of a pair of divergent alleles will tend towards infinity, such that in some cases polymorphism can be maintained over biologically relevant time scales without recurring mutation. While several studies have explored the effects of finite population size on the probability of global fixation in a two-patch model (i.e., the probability of a single mutation fixing in both patches; Tachida and Iizuka 1991; Gavrilets and Gibson 2002; Whitlock and Gomulkiewicz 2005), previous work has not yet explored the probability of a locally beneficial mutation rising to high frequency under recurrent migration. Billiard and Lenormand (2005) derived an approximation predicting equilibrium allele frequencies under selection, drift, and low rates of  73  migration, but they found that this approach would not extend to high migration rates. The aim of this paper is to understand the effect of drift on the maintenance of polymorphism by approximating the conditions under which a protected polymorphism occurs in a finite two-patch model. Specifically, approximations from the deterministic analytical framework are applied to make predictions about the effect of migration, selection, and drift on the probability of an allele rising when rare, the persistence time of a pair of divergent alleles, and the extent of adaptive divergence among patches. These predictions are then tested using individual-based simulations.  Analytical model We begin by rederiving the conditions for a protected polymorphism for the deterministic two-allele model (A, a) with two patches connected by migration (Bulmer 1972), which is the foundation for our results with finite patch sizes. We use the following notation for parameters in patch i: the allele frequencies of the two competing alleles (A and a) are pi and qi, Wi represents the fitness of each genotype, mij represents the proportion of gametes in patch i that have arrived from patch j, and W i represents the mean fitness (with i and j subscripts reversed for patch j). Beginning a generation immediately before mating and assuming random mating within!each patch, selection alters the allele frequency in patch i by: pis = pi ( piW i,AA + qiW i,Aa ) /W i qis = qi (qiW i,aa + piW i,Aa ) /W i  (4.1)  We assume throughout that selection favors allele a in patch 1 and allele A in patch 2; to !  emphasize this assumption, we will use the following alternative fitness definitions where  74  convenient: W1,AA = 1" s , W1,Aa = 1" hs , W1,aa = 1, W 2,AA = 1, W 2,Aa = 1" kt , and  W 2,aa = 1" t . After selection, surviving adults produce gametes that migrate, leading to ! ! ! allele frequencies in patch i: !  pim = (1" mij ) pis + mij p sj qim = (1" mij )qis + mij q sj  !  !  .  (4.2)  At this point, the next generation begins.  !  The standard approach to finding the regions of parameter space that allow the maintenance of polymorphism is to consider the stability of the equilibria when either allele is fixed (e.g., Moran 1962; Maynard Smith 1966; Bulmer 1972). If both of these equilibria are unstable, it follows that there is some stable equilibrium where both alleles are present in the system, even if its solution is not easily derived. The stability of the equilibrium where A is fixed can be found by evaluating the Jacobian matrix for qi = 0:  # "q1m % "q1 % m % "q2 $ "q1  W "q1m & # 1) m12 ) 1,Aa ( % W1,AA "q2 ( =% m( W "q2 ( % m21 1,Aa W1,AA "q2 ' q =q =0 $ 1 2  W 2,Aa & W 2,AA ( W ( (1) m21) W 2,Aa ( 2,AA ' m12  (4.3)  The leading eigenvalue of this matrix (!l,a) represents the rate of change in frequency of  !  the a allele when rare: W W 1 1 " l,a = # + # 2 $ 4(1$ m12 $ m21 ) 1,Aa 2,Aa 2 2 W1,AA W 2,AA  (4.4)  where !  " = (1# m12 )  W1,Aa W + (1# m21 ) 2,Aa . W1,AA W 2,AA  A similar result holds for the invasion of A when rare (!l,A, interchanging A and a as well  ! as m and m ). In an infinite population, we expect the population to remain 12 21 75  polymorphic as long as both !l,a and !l,A are greater than unity, because both alleles increase in frequency when rare (i.e., balancing selection).  To simplify the presentation, we now assume that migration is symmetric between the two patches (mij = mji); analogous results can be obtained using the following methods for unequal migration. To begin, we find the critical migration rates below which each allele will invade by solving for when !l,a > 1 and !l,A > 1. For the invasion of the a allele when rare, the critical migration threshold is: " mcrit,a =  1 W1,Aa W 2,Aa # W1,Aa # W1,AA W 2,AA # W 2,Aa  (4.5)  where the " superscript is added to differentiate this critical threshold from the one that is  !  derived below for finite populations. Equation 4.5 is equivalent to the threshold found by Bulmer (1972) using his fitness definitions ( W1,AA /W1,Aa = 1" s1 and W 2,AA /W 2,Aa = 1" s2 ). We have written (5) so that the two fractions in the denominator are positive under the  ! patch 2. If the difference assumption that allele a is favored!in patch 1 and allele A in " between these two fractions is negative ( mcrit,a < 0), then allele a can invade for any  migration rate. If there is intermediate dominance with h = 0.5 and k = 0.5, equation (4.5) reduces to: " mcrit,a =  st 2t # 2s  ! (4.6)  Similarly, m"crit,A can be obtained by interchanging the allele designations, A and a, in  !  equation (4.5).  !  76  The critical threshold for the invasion of the a allele, given by equation (4.5), is more restrictive than the equivalent threshold for the invasion of A whenever their selection and dominance coefficients satisfy:  (1" k)kt (1" h)hs . > 1" kt 1" hs  (4.7)  Because we are free to define which allele is which, we assume that the alleles are  !  labeled such that equation (4.7) is satisfied, keeping the patch labels such that allele a is favored in patch 1. For example, in the additive case, equation (7) reduces to t > s, indicating that we should label the a allele as the one that is more strongly selected against in the patch where it is locally maladapted. Defining the alleles such that equation (4.7) is satisfied, m"crit,a is then sufficient to determine whether a polymorphism will be maintained. We follow these allelic definitions throughout the remainder of this  !the deterministic critical threshold as m" for simplicity. paper and refer to crit For some combinations of parameters, the critical threshold rises above one,  ! implying that both alleles will be maintained deterministically for any migration rate. It can be shown that for m"crit to be greater than one, the geometric mean fitness of the heterozygotes across patches must be higher than that of the resident homozygotes: W1,Aa W 2,Aa! > 1, W1,AA W 2,AA  (4.8)  Equation (4.8) is most likely to be satisfied when the dominance coefficients (h and k) are !  low, causing the heterozygotes to have a high relative fitness. Figure 4.1 shows how the magnitude of !l (from equation 4.4) varies with the relation between s, t, and m for the simple case of intermediate dominance (h = 0.5; k = 0.5) with symmetric migration (mij = mji = m), plotting the smaller of the two values of  77  !l. Values above one (above the white plane at !l = 1) indicate regions where a balanced polymorphism is expected in the deterministic model. In the most asymmetric cases (t >> s, or vice versa), the system tends towards the threshold for the continent-island model found by Haldane (1930) and Wright (1931), where the continent is assumed to have little polymorphism (the patch with strong selection), in which case selection in the other patch must be stronger than the thresholds at m = s/2 or m = t/2 (black crossing lines in Figure 4.1A) for polymorphism to be maintained (s and t are divided by two to account for the homozygous fitness definitions used here). Similar behaviour is seen in the continuous-time model of Gavrilets and Gibson (2002; their Figure 1). Note that in the typical deterministic analysis, the maintenance of polymorphism is inferred from the sign of (!l – 1) rather than its magnitude, which indicates the net strength of selection maintaining polymorphism. With finite population size, the magnitude of (!l – 1) is important, not just its sign, as polymorphisms experiencing very weak diversifying selection could be lost due to stochastic fixation of one allele or another. Billiard and Lenormand (2005) used numerical methods to solve for equilibrium allele frequencies under migration, selection, mutation, and drift, but they suggested that a closed form solution to this problem was unattainable. Here, we are less interested in deriving an exact solution for the migrationselection-drift interaction, focusing instead on approximations that can be applied to predict regions of parameter space that are more or less likely to maintain allelic divergence. While the critical migration threshold for infinite populations ( m"crit ) is solved by finding the migration rate at which the deterministic effect of heterogeneous selection is exactly opposed by that of migration, an approximation for the!critical migration  78  threshold in finite populations can be derived by solving for the migration rate at which the probability that a rare allele becomes established due to the net effect of heterogeneous selection and migration is equal to the probability that it would establish due to drift alone. Because the invasion probability of an allele is dominated by selection only when its selective advantage, S, is sufficiently large relative to drift that 2S > 1/(2N) (Crow and Kimura 1970), the critical migration rate defined above (equation 4.5) can be adjusted to account for the role of drift by solving for the migration rate that results in 2(!l – 1) > 1/(2N), which uses the fact that !l – 1 is the asymptotic rate of increase of a rare allele in a deterministic analysis, which would equal the strength of selection acting on that allele, S, in a simple one-patch model. This approach basically finds the boundary where the net evolutionary force acting deterministically on the allele frequencies (migration and selection) has the same probability of explaining the spread of the rare allele as the stochastic noise generated by sampling error in finite populations. Throughout the following, we assume that the dominance coefficients (h and k) are not so high (or the selection coefficients so low) that drift would overwhelm selection and hamper the spread of the alleles where they are locally favored, even without the swamping effects of migration. Specifically, we assume that: #W & # W 2,Aa & 1 1 2% 1,Aa " 1( >> 2 " 1( >> % and 2N 2N . $ W1,AA ' $ W 2,aa '  (4.9)  Assuming the above, we obtain a new threshold for the migration rate below which the !system is expected to maintain ! polymorphism: f mcrit =  1 W1,Aa  (  W1,Aa " W1,AA 1 +  !  1 4N  )  "  .  W 2,Aa  (  W 2,AA 1 +  1 4N  )"W  (4.10)  2,Aa  79  f When m < mcrit , the evolutionary forces driving rare alleles to rise in frequency are  stronger than the tendency for alleles to be lost by random genetic drift. Decreasing the  ! population size makes m f smaller whenever m f is positive and makes m f more crit crit crit likely to be positive, both of which reduce the parameter range under which  ! can be maintained. For ! the special case of additive ! selection, equation polymorphism (4.10) reduces to: f mcrit =  (2Ns + 1)(2Ns + s "1) (2N)(2 " s) 2  (4.11)  when s = t and h = k = 0.5.  !  Again, the critical threshold can rise above one for some parameters, implying f that both alleles will be maintained deterministically for any migration rate. For mcrit to  be greater than one, the geometric mean fitness of the heterozygotes across patches must be sufficiently higher than that of the resident homozygotes that drift is!overwhelmed: W1,Aa W 2,Aa " 1 $ . > 1+ # W1,AA W 2,AA 4N %  (4.12)  Drift in smaller populations makes equation (4.12) harder to satisfy unless the dominance !  coefficients are sufficiently low.  Individual-based simulations The main approximation used in this paper substitutes the net result of the deterministic dynamics (!l – 1) as a proxy for selection in the classic one-locus one-patch model of selection with drift (Crow and Kimura 1970). This deterministic-stochastic splicing f approximation was used above to derive mcrit and will be used below to obtain the  probability of establishment of a locally favored allele and the waiting time until loss of  ! 80  the polymorphism. To assess the accuracy of this approximation, we used a modification of the Nemo platform (Guillaume and Rougemont 2006) to run individual-based simulations to test the accuracy of the splicing approximation. Individuals inhabit an environment consisting of two patches with different selection regimes connected by gametic migration at rate m. At the beginning of each generation, diploid offspring with single locus genotypes were created by drawing parental alleles from either patch (Pr[local] = 1 – m; Pr[non-local] = m), which then survived with a probability equal to their fitness (W), as defined in the previous section. Offspring were created until filling the local patch to carrying capacity (N = 1000). Thus, the entire system effectively experienced soft selection followed by migration of the gametes. These simulations were used to test the splicing approximation by finding: i) the probability of a novel mutation rising to high frequency in the patch where it is favoured, ii) the persistence time of a pair of divergent alleles, and iii) the extent of adaptive population divergence. For case iii), we wanted to focus on the level of divergence, rather than whether divergence was possible, and so we included bidirectional mutation at rate between µ = 10–4 to facilitate the maintenance of a polymorphism. Populations were considered to be adaptively differentiated when the frequency of the locally favoured allele in each patch was greater than 0.5.  Probability of a new mutation rising to high frequency: For a given mutation experiencing selection of strength S in a single population of size N, Kimura (1962) showed that the probability of rising from a single copy to fixation is:  1" e"2S Pr[ fix] = 1" e"4 NS  !  (4.13)  81  Several papers have used Kimura’s diffusion approach to solve for the probability of global fixation in the two-patch model (Tachida and Iizuka 1991; Gavrilets and Gibson 2002; Whitlock and Gomulkiewicz 2005). Here, we are interested in the probability of a novel mutation rising to high frequency in the patch where it is favoured (hereafter, local invasion probability), which can be approximated by substituting !l – 1 for S in Kimura’s equation, as this represents the average fitness of the new mutation when rare, with N assumed to equal the local patch size. To test this approximation using simulations and to examine the relationship between mutation effect size and local invasion probability, all individuals in both patches were initialized with homozygous A genotypes and a single a mutant was introduced in one randomly chosen individual in the patch where it experiences positive selection. No mutations occurred thereafter and simulations were run for 5000 generations, with the invasion probability calculated from the fraction of simulations where q rose to a frequency above 0.5 where it was locally favoured or persisted for 5000 generations without reaching high frequency. Kimura’s approximation (equation 4.13) provided an accurate prediction for the invasion probability for a range of selection coefficients when mutations were codominant (h = k = 0.5; Figure 4.2A). Alleles introduced where they were locally favored had a much higher invasion probability when selection was stronger, and in all cases, invasion probability increased with decreasing migration. While it is intuitive that strongly selected mutations should have a higher probability of contributing to local adaptation, this is not evident from the threshold migration rates derived by Bulmer (1972) and Spichtig and Kawecki (2004), which are undefined when selection coefficients and migration rates are symmetrical. When the invading locally favoured  82  allele was partially dominant to the locally disfavoured allele (h = k = 0.25), the analytical approximation tended to slightly overestimate the invasion probability, whereas in the opposite case (h = k = 0.75), the analytical approximation tended to slightly underestimate the invasion probability (with the differences more pronounced for weak selection and high migration). These slight inaccuracies are mainly due to using the closed form of Kimura’s equation for fixation probability (equation 4.13), which assumes additivity. If we instead use Kimura’s general solution for arbitrary dominance, performing numerical integrations to predict invasion probability for specific cases, we find better agreement between the observed and expected values (see Supplementary Materials). Nevertheless, the closed form approximation (equation 4.13) provides reasonably accurate predictions as long as dominance levels are intermediate and inequality (4.9) is satisfied.  Persistence time of a pair of divergent alleles: f To explore the utility of the mcrit threshold for predicting regions of parameter space that  would maintain adaptive divergence for long periods of time, individuals were initialized with homozygous ! genotypes of the allele favoured in their local patch and allowed to evolve without any subsequent mutation. It was not possible to follow populations until f loss of polymorphism for m << mcrit because time to fixation tends towards infinity with  decreasing m. As such, all simulations were run for 100,000 generations, with the  ! as the elapsed time before loss of either allele or the time at persistence time measured the end of the simulation if fixation did not occur, averaged over all simulation runs.  83  f Persistence time tended to increase with decreasing migration for m < mcrit , and as this  threshold rate was considerably higher for alleles with larger selection coefficients, they  ! maintained polymorphism over a much larger region of parameter space (Figure 4.3A). In all cases, the deterministic threshold was infinite, yet the transition to longer persistence f times happened at migration rates that were accurately described by mcrit . For  codominant mutations of equal effect size and N = 1000, the waiting time before the loss of either one of the alleles was approximately described by!the neutral expectation f (equation 8.9.3, Crow and Kimura 1970) when m > mcrit , but increased considerably as m  fell below this threshold (Figure 4.3A).  ! Maintenance of divergence with recurrent mutation f To test the accuracy of the mcrit threshold for predicting maintenance of polymorphism  under recurrent mutation and drift, simulations were initialized with all individuals heterozygous for ! the A/a genotype. Simulations were run for 20,000 generations, with a census taken every 50 generations for the final 5000 generations. At each census, populations were considered to exhibit adaptive divergence if the locally favoured allele in each patch was at a frequency greater than 0.5. The critical threshold provided an accurate prediction for the migration rate below which local adaptation increased from zero across a wide range of selection coefficients for codominant mutations (Figure 4.3B) and for partly recessive (h = k = 0.25) and partly dominant (h = k = 0.75) mutations (not shown).  84  Discussion If two alleles experience opposing selection pressures in different parts of the range of a species, divergent selection and/or limited migration may favour the maintenance of both alleles. Because alleles can be lost stochastically, however, drift can reduce the potential for the maintenance of polymorphism, especially in small populations. The results of this study show that the rate of invasion of a locally beneficial allele (!l) from a deterministic model can be spliced together with classic results describing the interplay between selection and drift to predict the probability of an allele rising to high frequency in the patch where it is favoured (Figure 4.2) and the critical migration rate below which polymorphism will tend to be maintained under migration, selection, and drift (Figure 4.3). While Rousset has suggested that predicting the equilibrium allele frequencies under migration, selection, drift, and mutation cannot be solved analytically (pers. comm. in Billiard and Lenormand 2005), the results of this study show that other aspects of evolutionary dynamics under these processes can be approximated, yielding useful descriptions of the parameter space that is conducive to the maintenance of polymorphism. The accuracy of the deterministic-stochastic splicing approximation is somewhat surprising, given the potentially restrictive assumptions involved in the derivation. The equation used here (equation 4.13) for fixation probability from Kimura (1962) was derived for genic rather than zygotic selection, yet it still provides a reasonably accurate prediction of invasion probability when mutations are not codominant (especially when f selection is strong and m < mcrit ). Similarly, the stochastic critical migration threshold  was derived by solving for 2(!l – 1) = 1/(2N), based on the logic that this is the  ! 85  approximate region of parameter space where selection and drift have equal effect on average. Despite the coarseness of this approach, the stochastic migration threshold functions well as long as selection is not too weak and/or the population size is not too f small (1/N < mcrit ), because when the number of migrants drops below one per  generation, populations will tend to diverge due to drift alone (Crow and Kimura 1970). f ! Rearranging equation (4.11) to find when mcrit = 1/N in the additive and symmetric case  (s = t; h = k = 0.5), this approximation for the critical threshold should work as long as s >> 3 / (2N). We note that for this!case of additive alleles with equal and opposite effects in the two patches, the deterministic analysis ( m"crit ) would predict that polymorphism can be maintained regardless of the strength of selection (Figure 4.1), but that this result does  ! critical migration thresholds are more restrictive not extend to finite populations, where when selection is weaker. The splicing approximation may also be usefully applied to more complex cases where selection operates on phenotypic traits determined by many loci. The values for s, t, h, and k could then be calculated from the relative fitnesses of the phenotypes produced by the various homozygous and heterozygous genotypes. If mutations are small relative to the curvature of the fitness function, the fitness of heterozygotes may tend to fall close to half way between the two homozygotes, which would increase the accuracy of simple approximations such as equation (4.13), as h and k would fall more close to 0.5. Future papers will explore the utility of these approximations for understanding the evolution of multi-locus architecture under migration-selection balance. Based on the results of the current study, prolonged periods of stabilizing selection with migration would be expected to select for genetic architectures with fewer large mutations, as these would  86  have longer persistence times than numerous small mutations with the same net additive effect. This general prediction was observed in a model by Griswold (2006), who found that in populations evolving under divergent directional selection and migration, the distribution of mutation effect sizes was shifted towards larger effect alleles, relative to the expectation for adaptation without migration. While Griswold also noted that mutations of size s < m were often found to have diverged between the simulated populations, our results suggest that this is not altogether surprising, as the critical threshold may often exceed s (or t) when selection and migration are symmetrical (equation 4.11). The results of this study illustrate how the effect of the interplay between migration and selection in finite populations can be approximated using the rate of change in frequency of an invading allele within an infinitely large populations (!l; equation 4.4) as an indication of the net effect of the deterministic evolutionary processes (here, migration and selection). When the diversifying effect of these processes is strong relative to drift (i.e., 2(1 – !l) >> 1/(2N)), locally adapted alleles will have a high probability of invading when rare and a long persistence time before fixation of one or the other. As migration rates increase, the net diversifying effect decreases, which results in decreased persistence times and invasion probabilities. We show that the interplay between migration, selection, and drift tends to result in threshold behavior of the system, f with rapid transitions at the critical migration rate ( mcrit ; equation 4.10) between regions  of parameter space that have a high vs. low probability of maintaining polymorphism.  ! Generally speaking, the potential for the maintenance of polymorphism decreases with population size, as the role of drift increases relative to selection.  87  Figure 4.1– Magnitude of !l as a function of the ratio between the allelic selection coefficients (s, t) and the migration rate. Panel (A) shows the smaller of the two leading eigenvalues (!l,a or !l,A) for additive alleles with h = k = 0.5 and m = 1/100. The white plane at of !l = 1 separates regions where deterministic selection favors a balanced polymorphism (above plane) and where loss of one of the alleles is expected (below plane). Panel (B) shows the critical migration rate, below which polymorphism will be maintained using equation (5) for infinitely large populations and (10) for finite populations, assuming s = 0.8 t and h = k = 0.5. The four points correspond to the same parameter sets in both panels and illustrate the fact that even though !l > 1 in a deterministic analysis (points “!,” “",” and “+”), the evolutionary forces can be too weak to maintain a polymorphism in the face of drift in finite populations (when N < 2643 for “!” and when N < 220 for “"”).  88  89  Figure 4.2 – Invasion probability of a new mutation Relation between migration rate and the probability of invasion of a new locally adaptive mutant for three different dominance regimes and N = 1000. Symbols correspond to selection coefficients (s = t) of: 0.0025 (stars); 0.005 (filled circles); 0.01 (triangles); 0.025 (open circles). Dashed lines correspond to predictions from Kimura’s fixation probability using !l (Equation 4.12). Results for s = t = 0.025 and 0.01 and h = k = 0.75 not shown for high migration rates, as the low frequency of invasion precluded an accurate estimate of its probability.  90  91  Figure 4.3 – Persistence time and maintenance of polymorphism Relation between migration rate and the persistence of a pair of divergent alleles (A), and maintenance of divergence under recurrent mutation (B), for h = k = 0.5, and N = 1000 and the same selection coefficients and symbols as Figure 4.2. Dashed lines correspond f to mcrit for the four selection regimes while the solid line in (A) indicates the expected  persistence time under neutrality. The deterministic critical threshold predicts the  !  maintenance of both alleles for any migration rate when h = k = 0.5, because m"crit ! " as t ! s. Simulations run for 105 generations, so points below the critical thresholds  ! during this period. underestimate the persistence time in cases where neither allele fixed 92  References Billiard, S. and T. Lenormand. 2005. Evolution of migration under kin selection and local adaptation. 59:13-23. Bulmer, M.G. 1972. Multiple niche polymorphism. American Naturalist. 106:254-257. Crow, J.F. and Kimura, M. 1970. An introduction to population genetics theory. Harper and Row. New York. Felsenstein, J. 1976. The theoretical population genetics of variable selection and migration. Annual Review of Genetics. 10:253-280. Gavrilets, S. and N. Gibson. 2002. Fixation probabilities in a spatially heterogeneous environment. Population Ecology. 44:51-58. Griswold, C.K. 2006. Gene flow’s effect on the genetic architecture of a local adaptation and its consequences for QTL analyses. Heredity. 96:445-453. Guillaume, F., and J. Rougemont. 2006. Nemo: an evolutionary and population genetics programming framework. Bioinformatics. 22:2556-2557. Haldane, J.B.S. 1930. A mathematical theory of natural and artificial selection. Part VI. Isolation. Proceedings of the Cambridge Philosophical Society. 26:220-230. Kimura, M. 1962. On the probability of fixation of mutant genes in a population. Genetics. 47:713-719. Lenormand, T. 2002. Gene flow and the limits to natural selection. Trends in Ecology and Evolution. 17:183-189. Maynard Smith, J. 1966. Sympatric speciation. American Naturalist. 100:637-650. Moran, P.A.P. 1962. The statistical processes of evolutionary theory. Clarendon Press. Oxford.  93  Spichtig, M. and T.J. Kawecki. 2004. The maintenance (or not) of polygenic variation by soft selection in heterogeneous environments. American Naturalist. 164:70-84. Tachida, H. and M. Iizuka. 1991. Fixation probability in spatially changing environments. Genetical Research. 58:243-251. Whitlock, M.C. and R. Gomulkiewicz. 2005. Probability of fixation in a heterogeneous environment. Genetics. 171:1407-1417. Wright, S. 1931. Evolution in Mendelian populations. Genetics. 16:97-159.  94  5. The genetic architecture of adaptation under migrationselection balance1 Introduction Many species inhabit coarse-grained environments where selection pressures vary across their distribution and dispersal distances are low enough to permit local adaptations to emerge (Hedrick et al. 1976; Linhart and Grant 1996). Mounting evidence from studies of the loci underlying adaptation in quantitative traits (QTL) suggests that there is considerable heterogeneity in architecture among species and among traits (Orr 2001; Slate 2005), with some traits defined by only a few QTL of large effect (e.g. pelvic girdle (Shapiro et al. 2004) and armour plating (Colosimo et al. 2004) in sticklebacks; flower architecture in Mimulus spp. (Bradshaw et al. 1995, 1998)), others defined by many QTL of small effect (e.g., cold hardiness in conifers (Howe et al. 2003); flowering time in maize (Buckler et al. 2009)), and others defined by QTL with a range of effect sizes (e.g., body shape in sticklebacks (Albert et al. 2006)). Furthermore, fine scale dissection of large effect QTL has sometimes revealed that they contain several tightly linked genes each contributing small individual effects (e.g. skeletal traits in mice (Christians and Senger 2007)). What factors shape this variation in genetic architecture? If there are only a limited number of genes that control the expression of a phenotypic trait under selection, then the genetic architecture of local adaptation will necessarily be constrained by the rate and size of beneficial mutations at these genes. The observation of tight linkage of small effect alleles in a single large effect QTL might also be expected if 1  A version of this chapter will be submitted for publication. Yeaman, S. and M.C. Whitlock. The genetic architecture of adaptation under migration-selection balance. 95  tandem gene duplications have played an important role in the elaboration of a given phenotype. On the other hand, genetic architecture can also be shaped by evolutionary processes of natural selection and migration. Orr (1998) has shown that the distribution of allele effect sizes fixed during an adaptive walk (without migration) will be approximately exponential, even if the underlying distribution of mutation sizes is much different, while Griswold (2006) has shown that migration constraining the process of adaptation will cause this distribution to be skewed towards larger allele effect sizes. If there are many different ways to build a given adaptive phenotype, different populations may fix different combinations of alleles. As mutations with larger selection coefficients have a higher probability of contributing to adaptation and a longer persistence time under migration-selection balance in finite populations (Chapter 4), prolonged bouts of adaptation under gene flow could result in the gradual replacement of many small effect alleles by fewer large effect alleles (provided their effects on the phenotype are approximately interchangeable). Furthermore, if translocations, inversions, or transposable elements rearrange the physical distribution of locally adaptive alleles within and among chromosomes or if a modifier alters the rate of recombination between them, the creation of tight linkage groups can allow several small effect alleles to function effectively as a single large linkage group. Understanding how the interplay between migration, selection, and drift affects the evolution of genetic architecture and favours or disfavours these types of patterns is thus critical to interpreting the variation in QTL effect size, number, and composition seen in the natural world. In the absence of epistasis, the interplay between divergent natural selection and migration is expected to favour architectures that minimize recombination between  96  locally adapted alleles, as recombination breaks up the positive disequilbrium generated by migration, yielding maladaptive intermediate genotypes (Maynard Smith 1977; Pylkov et al. 1998; Lenormand and Otto 2000). Considering two populations evolving under divergent selection with migration, Kirkpatrick and Barton (2006) showed that a chromosomal inversion eliminating recombination between previously unlinked loci will experience positive selection in proportion to the number of loci involved in local adaptation and the migration rate. Interestingly, they found that the net strength of selection favouring the inversion was independent of the selection coefficients on the alleles involved. While not explicitly focusing on the evolution of genetic architecture underlying quantitative traits, these studies indicate that prolonged bouts of stabilizing selection with migration should tend to favour architectures characterized by fewer, larger, more tightly linked alleles. Most theory that explicitly explores adaptation under migration-selection balance has focused on deriving the conditions that maintain genetic polymorphism at a single locus (e.g., Haldane 1930; Wright 1931; Bulmer 1972), multiple loci (Lythgoe 1997; Spichtig and Kawecki 2004), or divergence at a quantitative trait (Hendry et al. 2001; Lopez et al. 2008), rather than examining the evolution of genetic architecture per se. To arrive at analytically tractable solutions to the migration threshold problem, the multilocus studies have typically made restrictive assumptions about the genetic architecture underlying the phenotype (e.g., diallelic loci of equal effect size), limiting the scope of inferences about how migration-selection balance shapes the distributions of effect size and number of loci. While Spichtig and Kawecki (2004) assumed diallelic loci of equal effect size for much of their analysis of the conditions maintaining multi-locus  97  polymorphism, they also showed that when they relaxed this assumption and allowed unequal effect sizes among loci there was less polymorphism maintained, which is consistent with the expectation of fewer loci contributing to divergence under migrationstabilizing selection balance. While this work is suggestive, they did not explicitly compare the relative fitness of genotypes with different architectures or examine the effect of linkage between loci. The aim of this paper is to examine how prolonged bouts of divergent stabilizing selection with migration and recurrent mutation and recombination affect the evolution of the genetic architecture underlying a locally adapted trait. We use a combination of individual-based simulations and analytical approximations based on work by Bengtsson (1985) and Yeaman and Otto (Chapter 4) to explore how the interplay of these evolutionary processes affects the establishment of adaptive divergence between populations and the subsequent elaboration of genetic architecture. Specifically, we seek to understand how the interplay between migration, selection, and drift affects the number of loci contributing to divergence, the size of alleles that diverge between populations, and their propensity to cluster together in tight linkage on a chromosome.  Analytical approximations Evolution in a quantitative trait determined by a single locus Chapter 4 examined a two-patch, two-allele model with migration and divergent selection, deriving several approximations to identify the conditions favouring the maintenance of polymorphism in finite populations. They showed that the rate of change in allele frequency when one allele is fixed and the other is a new mutant can provide a  98  useful approximation for the net deterministic effect of the tension between migration and selection, represented by the leading eigenvalue of the stability analysis (!l). Because !l represents the same sort of systematic change in allele frequency that is caused by selection in a simplified model of evolution in a finite population under directional selection, it can be spliced into existing population genetic approximations to predict the probability of a new locally-beneficial mutation contributing adaptive divergence (hereafter, ‘invasion probability’; Equation 4.13) and the critical migration rate above which migration and drift overwhelm the diversifying effect of selection (‘critical migration threshold’, or mcrit ; Equation 4.10). Using this approach, Yeaman and Otto (Chapter 4) found that mutations with larger selection coefficients had a higher invasion ! persistence time once established, and a higher critical migration probability, longer  threshold, suggesting that they might make a proportionally greater contribution to locally adapted phenotypes under migration-selection-drift balance. Here, we apply the general approach used by Yeaman and Otto (Chapter 4) to examine the evolution of adaptive divergence and genetic architecture in quantitative traits. Where Chapter 4 modeled the fate of mutations with set fitness effects, here we specify a function defining the absolute fitness (W) of a phenotype (Z) under stabilizing selection of strength " towards a local optimum (#) that is positive in one patch and negative in the other, such that W = 1" #[ $ " Z /(2$ )]% (modified from Spichtig and Kawecki 2004). By this function, an individual that is perfectly adapted in one patch  ! experiences a disadvantage of exactly " in the other patch, regardless of the value of $, which defines the curvature of the fitness function. When $ > 1 the fitness function has a convex shape around the optimum, when $ = 1 it is linear, and when $ < 1 it has a  99  concave shape. While the effect of individual mutations on the phenotype is assumed to be additive, dominance for fitness emerges as a result of the curvature of the fitness function (!). We can calculate the relative fitness of heterozygous and homozygous genotypes with mutations of any given size (!) in either patch by comparing their absolute fitness to that of the most fit genotype in the patch, yielding the relative fitness values used by Yeaman and Otto (wij, where i is the patch and j is the genotype). The approximations developed Chapter 4 can then be applied using these relative fitness values to approximate the net effect of migration and selection on an invading mutation ("l; their Equation 4.4): w w 1 1 "l = # + # 2 $ 4(1$ m12 $ m21 ) 1,Aa 2,Aa 2 2 w1,AA w 2,AA  (5.1)  where: !  " = (1# m12 )  w1,Aa w + (1# m21 ) 2,Aa w1,AA w 2,AA  ! and m12 and m21 are the probabilities of immigration for each patch (here m12 = m21). The net effect of the deterministic interaction between selection and migration is increasingly diversifying as ("l – 1) increases away from zero, and increasingly homogenizing as ("l – 1) decreases away from zero (see Chapter 4 for a full derivation). Using ("l – 1), we can predict the invasion probability of the mutation using Kimura’s equation for fixation probability (Equation 4.13), while we can identify the critical migration rate below which both locally adapted alleles will tend to be maintained in the system by solving for when 2("l – 1) = 1/2N ( mcrit ;Equation 4.10):  ! 100  mcrit =  1 w1,Aa  (  w1,Aa " w1,AA 1+  1 4N  )  "  (5.2)  w 2,Aa  (  w 2,AA 1+  1 4N  )"w  2,Aa  This equation can also be solved to predict the minimum mutation effect size necessary  !  for divergence to have a high probability of being maintained for a given migration rate, strength of selection, and population size. Figure 5.1 shows that larger effect alleles have a higher critical migration threshold, indicating that polymorphism can be maintained more readily under higher migration rates. This suggests that divergence at the genotypic level in finite populations will be constrained to alleles with sufficiently large effect sizes, assuming there is no linkage between the loci involved in local adaptation (we will relax this assumption below). If the migration rate is so high that even a single perfectly adapted allele (2! = ") does not result in mcrit > m, selection is insufficiently strong for any adaptive phenotypic divergence to be maintained between populations, which is seen in Figure 5.1 as the curves asymptote. This critical migration threshold for the phenotype evaluated at (2! = Z ") will be referred to as mcrit to distinguish it from the lower critical migration thresholds  derived for alleles with specific effect size, m"crit .  ! Linkage between loci and the net ! strength of divergent selection If the strength of selection is too weak for a small locally adaptive mutation to overcome the homogenizing effect of migration, it may still invade and contribute to adaptation if it is linked to a larger allele at a second locus that has already diverged. Also, over time, selection might favour the replacement of weakly linked pairs of alleles by more tightly linked pairs. To explore the interplay between recombination, selection, and migration,  101  we combine the approximations for net diversifying effect of selection developed above (!l – 1) with a model by Bengtsson (1985) for the decrease in effective migration rate at a neutral focal locus linked to a background locus under selection. Bengtsson showed that a fixed difference at a background locus that is maintained by selection of strength s against immigrant types would decrease the effective migration rate at an adjacent focal locus linked at recombination rate r by a factor of:  me = m  r(1" s) 1" (1" r)(1" s)  (5.3)  This effective migration rate can be used to incorporate the effect of linkage into the  !  calculation of the net diversifying effect of selection on an invading mutation at the focal locus by substituting me into equation 5.1, and substituting wAa for 1 – s (as Bengtsson followed selection on heterozygotes). While this yields only an approximate solution for !l – 1, due to Bengtsson’s assumption that the background alleles are fixed in their local patches, it illustrates how the interaction between the strength of divergent selection, migration, and recombination rate influences the net strength of selection on the invasion of a second allele at a linked locus (Figure 5.2A). When migration rates are very low (Figure 5.2A, upper curves), there is comparatively little difference in the magnitude of !l – 1 at low vs. high recombination and thus little difference in the invasion probability and persistence time of a tightly vs. loosely linked allele. This is because migration rates are already low enough that the deterministic force of selection is strong relative to migration, such that the increase in the efficacy of selection due to linkage to a locally adapted gene is small by comparison. By contrast, under sub-critical migration rates (Figure 5.2A, middle curves), !l – 1 depends much more strongly on the rate of recombination. This general pattern 102  reproduces the qualitative results of Kirkpatrick and Barton (2006), derived by considering dynamics of the invasion of secondary divergent allele linked to an alreadydiverged background locus (as opposed to the spread of an inverted chromosome). The transition from low to high values of !l – 1 happens over different ranges in recombination rate under different rates of migration, with the transition occurring over lower rates of recombination for higher migration rates, as shown by the curve through !l50  (the midpoint between the lowest and highest values of !l – 1 for a given migration  rate). For example, in the case illustrated in Figure 5.2A, there is little change in !l – 1 with a decrease in recombination from 10-1 to 10-3 at the highest migration rate. Figure 5.2B shows a similar relationship, viewed instead from the perspective of the relationship between the strength of phenotypic selection (") and the recombination rate at which the value of !l – 1 passes through its midpoint (!l-50). Again, the increase in the net diversifying effect of selection occurs over different ranges in the rate of recombination for different strengths of phenotypic selection. For strong selection, the region over which a drop in recombination yields a substantial change in !l occurs at higher rates of recombination than under weak selection (Figure 5.2B). Figure 5.2 suggests that strong phenotypic selection will cause clusters of locally adapted mutations to be favoured over much wider map distances than under weak selection. We stress that due to the coarseness of these approximations, this approach is intended more as a useful heuristic than a means of formulating quantitative estimates about the effective strength of selection on a secondary invading allele.  103  Individual-based simulations We used individual-based simulations to explore the utility of the above approximations for understanding multi-locus evolution under migration-selection-drift balance, using a modification of the Nemo platform (Guillaume and Rougemont 2006). Individuals inhabit an environment consisting of two patches with different selection regimes connected by migration at rate m. In all cases, the absolute fitness of an individual in patch i is defined by the function described above, where W i = 1" #[ $ i " Z /(2$ i )]% . For the multi-locus cases, the phenotype is defined by n additive loci arranged on a single  ! chromosome, with recombination occurring between adjacent loci at rate r. At the beginning of each generation, diploid offspring are created by drawing parental gametes from either patch (Pr[local] = 1 – m; Pr[non-local] = m), which go on to survive with a probability equal to their absolute fitness (W). Offspring are created until filling the local patch to carrying capacity (N), such that the entire system effectively experiences soft selection followed by gamete migration. For simplicity, all cases discussed below use an optimum phenotype (!) of +/-1 in each patch and a local patch size of N = 1000. As mutation effect size and rate are expected to strongly influence the dynamics, we employ a flexible approach to modeling mutations, drawing their values from a Gaussian distribution with standard deviation of " and a mean of zero. Mutations occur at a per locus rate of µ with the value of the new mutation added onto the previous allelic value at the locus. Single-locus simulations with a continuum-of-alleles model of recurrent mutation were used to examine the accuracy of the critical migration thresholds derived in Chapter 4 when applied to mutations building a phenotype under selection. Multi-locus simulations were then used to explore how migration, selection, drift, and recombination  104  combine to influence the genetic architecture of local adaptation. In most cases, simulations were run for 500,000 generations with summary statistics calculated every 500 generations; for simulations with low mutation rates (µ = 10-5), we ran simulations until there was no consistent change in mean phenotype (usually >1,000,000 generations). Except where otherwise noted, statistics reported in the figures were averaged over at least 100 census points during the final portion of the simulations, with 20 to 40 replicates per parameter set. All multi-locus simulations were initialized with mutations at 5% of the loci to provide some initial variability. Individual ‘loci’ in this continuum-of-alleles mutation model can be thought of either as groups of tightly linked genes or a single gene subject to recurrent mutation. The term ‘allele’ and ‘locus’ will be used throughout this paper to simplify discussion, however it should be kept in mind that these terms could represent several tightly linked genes. An ideal simulation would include tens of thousands of loci with heterogeneous recombination rates across the genome and explore whether locally adapted mutations tended to cluster around areas of reduced recombination. As this would be computationally prohibitive, we focus instead on a single chromosome with 50 loci and examine the effect of the rate of recombination between adjacent loci to represent what would occur in areas of high vs. low recombination. Under the continuum-of-alleles model, genetic architecture can evolve to mimic the genotype created by a single large mutation through two different mechanisms. If individual mutation effect sizes are small, several mutations can stack up on the same simulated locus over time, effectively building a single large effect allele (hereafter ‘stacking’). This is roughly analogous to either recurrent mutations at a single gene (e.g.,  105  the Drosophila shavenbaby gene (McGregor et al. 2007)) or to several mutations at different genes that are tightly linked (r ~ 0). Alternatively, if two mutations have occurred at different parts of the chromosome (such that r >> 0 between them), a second mutation closer to one of them can invade and replace the more distant allele, reducing the recombination rate between the locally adapted alleles (hereafter ‘clustering’). While clustering can produce a genotype that is identical to a stacked genotype as r ! 0 between the locally adapted alleles, the two terms are useful to differentiate the intermediate cases where r > 0 between linked alleles from cases where several successive mutations build a large allele at a single locus. Genetic architectures that tend to have fewer loci of larger effects and/or tighter linkage between loci contributing to divergence will be referred to as ‘concentrated’, while those with more loci of smaller effects and looser linkage will be referred to as ‘diffuse’. As multiple alleles may segregate at a single locus within each population under the continuum-of-alleles model, the most common allele at the ith locus in the jth patch is referred to as the ‘leading allele’: a"ij , with the difference between leading alleles in the two patches represented by d = a"i,1 # a"i,2 . For the single-locus case, the mean value of d  ! averaged over time and replicate ( d ) provides an average measure of adaptive divergence  ! the multi-locus simulations, dmax is used to refer to the size of the under gene flow. In  ! census point, averaged across all replicates. Individual-based largest value of d at a given simulations involve considerable stochasticity in values of d; while a large number of loci may be differentiated at any given census point, only a few of these typically make a long-term contribution to divergence. We use the term ‘transient’ to differentiate these short-lived polymorphisms from loci with longer-lasting ‘stable’ allelic divergence,  106  which are given the symbol d " (defined as those with d ! 0 at a single locus for at least 18 out of 20 census points during the 20,000 generations preceding the focal census  ! the extent of clustering in the genetic architecture, we calculated the point). To represent ‘clustering distance’ as the average physical distance between dmax and all other stable differentiated loci (i.e., absolute difference in position on the chromosome), setting this to zero if only a single locus was stably differentiated. To estimate the average size of alleles contributing to long-term divergence, we calculated the average size of d at all stable diverged loci (dmean), averaged over all replicates. Throughout the following analysis, the magnitudes of dmax and dmean will be used both as indicators of the size of alleles contributing to divergence and the number of loci involved in local adaptation, as the number of loci increases as dmax and dmean decrease.  Results Adaptive divergence for a single-locus quantitative trait Reproducing the approach of Chapter 4 with alleles contributing to a phenotype under selection (as opposed to mutations with explicitly defined selection coefficients), we generally find very good agreement between results from individual-based simulations and the predicted invasion probability and persistence time of individual mutations (results not shown). As expected, large mutations have much longer persistence times and higher invasion probabilities than small mutations, especially at sub-critical migration rates. When populations evolve with recurring mutations of variable size, however, an Z interesting and unpredicted pattern emerges. As expected, when m << mcrit , populations  spend most of their time differentiated with allelic values that fall close to the optimum  ! 107  (Figure 5.3). As migration rate increases, however, the extent of divergence ( d ) increases at sub-critical migration rates, such that the leading alleles are of an extreme size,  ! generating phenotypes that surpass the local optimum in the homozygous state (Figure Z 5.3). As m " mcrit , d then decreases again as populations spend an increasing proportion  of time undifferentiated, reaching zero at approximately the point where 2(!l – 1) = 1/2N.  !While the!analytical theory does not predict the extreme alleles, these results show that the critical migration threshold (from Equation 5.2) provides an accurate prediction of the migration rate below which there is a high probability of divergence being maintained between populations for a quantitative trait with a single locus.  Critical migration threshold for multi-locus traits As discussed by Yeaman and Guillaume (2009), variation in the way that mutations affect a phenotypic trait (i.e., mutation effect size (!), mutation rate (µ), number of loci, and the rate of recombination between adjacent loci (r)) has very little effect on the phenotypic divergence between populations at equilibrium under the continuum-ofalleles mutation model. Rather, patterns at the phenotypic level are largely defined by the balance between migration, selection, and drift, with the critical migration threshold (Equation 5.2) providing a good approximation of the point where the system transitions from an undiverged state to a diverged one (Figure 5.4). That said, genetic architecture is itself greatly influenced by the interaction between these processes and the architecture of mutations. The following sections explore the effect of these interactions on the distribution of allele effect sizes, the clustering of small alleles into tight linkage groups on a chromosome, and the persistence times of alleles and linkage groups.  108  Evolution of genetic architecture – Migration rate As migration rate increases, the system passes through two thresholds where the genetic architecture of adaptive divergence changes dramatically. When fewer than one individual migrates per generation (drift threshold, ~Nm < 1), the two populations evolve largely independently of each other, rapidly approaching their local optima and then gradually cycling through different combinations of alleles with opposing effects that sum to yield a locally optimal phenotype (Figure 5.5A). While both populations are well adapted to their local optima from very early in the simulations (as indicated by the black line in Figure 5.5D), as time progresses, the genetic architectures of the two populations drift further and further apart from each other with increasingly large effect size alleles at the locus with the largest value of d (dmax; Figure 5.5D; red). Eventually, the size of dmax surpasses the value conferring optimal adaptation in each patch and is compensated by alleles with opposing effects at other loci (Figure 5.5D; blue). Although the recombinant F2 offspring of a mating between individuals from the different patches would have very low fitness, such matings are infrequent enough at low migration that selection is too weak to prevent the architectures from drifting apart, ultimately leading to speciation. Z When m is above the drift threshold but below the selection threshold (m < mcrit ),  migration is frequent enough to entrain the architectures of the two populations, such that  ! (Figures 5B; they differ at only a few loci necessary to build a locally-adaptive phenotype 5E). In the example shown in Figure 5.5B, the small mutation effect size and high mutation rate result in the gradual stacking of mutations (indicated by the gradual increase in dmax and decrease in d at all other loci), yielding an architecture with a single  109  locus conferring most of the adaptive divergence between the populations, and transient divergence contributed by small effect alleles at other loci. Z Above mcrit , migration is too strong to permit local adaptation and transient  differences in the genetic architectures of the two populations are quickly homogenized  ! 5C; 5F). The interaction between selection and drift determines the threshold (Figures migration rates at which the system transitions through these three phases, from effective allopatry (m < 1/N), to migration-selection balance where architectures are entrained but Z locally adapted (1/N < m < mcrit ), to homogenization of differences between populations Z ( mcrit < m). The remainder of this paper will focus on evolution in the middle of these  ! phases, examining the impacts of mutation rate and effect size, selection, and !  recombination on the architecture of local adaptation.  Evolution of genetic architecture – Mutation rate and effect size Somewhat surprisingly, when individual mutations are smaller on average (! = 0.05 vs. 0.5), the architecture underlying local adaptation tends to be characterized by divergence at fewer loci with larger effect alleles that persist for longer periods of time than when individual mutations are large (Figure 5.6). Generally speaking, mutations with large selection coefficients tend to be favoured in finite populations because they have a higher probability of rising to high frequency and a longer persistence time than small-effect mutations, even if migration is low enough that both could be maintained deterministically (Chapter 4). The large-effect alleles built by stacking when individual mutations are smaller (! = 0.05) tend to have very long persistence times, whereas when individual mutations are larger (! = 0.5), large alleles occur commonly and are more  110  rapidly replaced by other alleles of similar effect size at different loci (Figure 5.6B). Higher per-locus mutation rates have little effect when mutation effect sizes are small, but result in even faster cycling of the alleles contributing to divergence when effect sizes are large (Figure 5.6B), along with a lower average size of dmax and therefore an increased number of loci involved in divergence (Figure 5.6A). As migration decreases, genetic architectures generally become less concentrated, with more loci contributing to adaptation with smaller average effect sizes. Even so, very few loci are typically involved in local adaptation, despite the high availability of beneficial mutations. As seen with the single-locus models, a single allele with an effect size that surpasses the local optimum in its homozygous state evolves in cases that are sufficiently mutation-limited (Figure 5.6A), but when mutations are more common (µ = 10-4), these extreme alleles are not observed nearly as commonly as when they are more rare (µ = 10-5), due to the lower persistence time of leading alleles under rapid mutation. Taken together, these results show that the availability of mutations of large effect has considerable influence on the shape and temporal stability of the genetic architecture of local adaptation. While small Z effect mutations may constrain local adaptation when m"crit < m < mcrit (especially if the  mutation rate is also low; Figure 5.6), if the genome is flexible enough to permit  ! be overcome ! by the creation of large effect clustering or stacking, then this constraint can genes or linkage groups.  Evolution of genetic architecture – Strength of selection and recombination rate Generally speaking, clusters of linked locally adaptive alleles persist over much larger regions of the chromosome when selection is strong, recombination is low, and migration  111  is low (Figure 5.7). As migration rates increase, the average size of alleles contributing to divergence increases and the clustering distance decreases, indicating that more concentrated architectures are more strongly favoured, which is consistent with Kirkpatrick and Barton (2006) and Figure 5.2A. This occurs for two reasons: the disadvantage of recombination increases with migration load and the relative advantage Z of larger alleles over smaller beneficial alleles increases as m " mcrit . Selection thus  affects genetic architecture in two ways: through its primary effect defining the critical  ! migration threshold and therefore the range in migration rate over which clustering gives way to stacking, and through a secondary effect on the physical distance on the chromosome across which stable clusters of linked alleles can persist and contribute to adaptation. This secondary effect is not predicted by the model of Kirkpatrick and Barton, but is predicted by the approach based on Bengtsson’s model (Figure 5.2B).  Discussion Migration-selection balance in finite populations tends to favour genetic architectures that minimize the number of loci or linkage groups involved in local adaptation. If the number of genes that can potentially mutate to shape a locally adaptive phenotype is much larger than the number of mutations necessary to actually yield an optimal phenotype, then evolution over long periods of time will eventually favour architectures with fewer, larger, and more tightly linked alleles. In the opposite case, where mutations at most or all of the genes that could potentially shape a phenotype are required to build an optimum phenotype, evolution will be more constrained by the underlying genomic and developmental biology of the organism. In such cases, more concentrated architectures  112  could be created by chromosomal rearrangements (i.e., inversions, translocations), transposon mediated gene movement, gene duplication and degeneration of the less clustered copy, or through some modifier reducing the rate of recombination between these genes. In any case, we would expect traits under migration-selection balance to have fewer QTL with higher percent variance explained (PVE) by each locus than traits under purifying or directional selection without migration, all else being equal. As such, while the relatively common finding of large effect QTL underlying locally adaptive traits may simply be a product of their increased probability of discovery, it might also be a direct consequence of their relative advantage under migration-selection balance in finite populations. These results thus have implications for predicting the number of undetected small effect QTL underlying a locally adaptive trait (e.g., Otto and Jones 2000; Hayes and Goddard 2001). The distributions of allele effect size observed in this study have much fewer alleles of small effect than expected from an exponential distribution (e.g., Orr 1998), suggesting caution in making assumptions about the expected distribution of QTL effect sizes for traits evolving under migration-selection balance. Interestingly, we also found much more concentrated architectures than reported by Griswold (2006), who used a very similar individual-based model. Where Griswold suggested that ‘it is unexpected to find alleles of large effect when populations have diverged for a long period of time and their phenotypic optima are far apart’ (2006; p. 452), we find that given sufficient time, very concentrated architectures will evolve even when mutations are very small relative to the difference in optimum (Figure 5.6). Our study examined changes in architecture over much longer periods of time than that of Griswold (>500,000  113  vs. 10,000 generations), during which we observed a steady increase in dmax and a concomitant decrease in the number of loci involved in adaptive divergence (Figure 5.5), which would seem to account for the more concentrated architectures observed in our study. Also, Griwsold did not allow recurrent mutations at a single locus and did not examine whether locally adapted alleles were tending to cluster in tight linkage groups, which might also explain some of the difference in our results. We note that Van Doorn and Dieckman (2005) also found concentrated architectures evolving under frequency dependent disruptive selection. It would be interesting to compare the architecture of adaptation under other selection regimes (e.g., temporal variation in optimum) to explore the robustness of these patterns. On a slightly different track, if migration-selection balance actually plays a strong role in the evolution of genetic architecture, we might expect some very coarse patterns at the genomic level, with genes for traits often involved in local adaptation (e.g., phenology, water use efficiency, or temperature tolerance) clustering more closely around each other than observed for a random sample of genes. The results of this study suggest that lineages that have undergone these sorts of genomic reorganizations would be better able to adapt to the challenges posed by heterogeneous environments, potentially increasing their long-term survival or their propensity to speciate. This work also has implications for modeling quantitative traits under migrationselection balance, as Yeaman and Guillaume (2009) showed that although simulations using an equal-effect diallelic model with free recombination would agree well with the predictions of a quantitative genetics model, considerable genetic skew could be generated by either decreased recombination or heterogeneity in allele effect sizes,  114  causing the quantitative genetic model to underestimate divergence under migration load. Here, we show that the very architectures that facilitate the generation of skew are those that are most strongly favoured under migration-selection balance, suggesting that quantitative genetic models assuming a Gaussian distribution of genotypes may be inherently unsuitable for modeling these types of evolutionary processes. The expectation of selection for reduced recombination between locally adapted loci has been previously been discussed by Otto and Lenormand (2000) as a mechanism affecting the genetic architecture of traits under domestication. The concentrated architectures expectation is also implicit in other studies on the evolution of recombination, which have concluded that recombination is disfavoured under divergent selection and migration when there is no epistasis within patches, as this minimizes the load created when locally adapted genotypes recombine with maladapted immigrant genotypes (Maynard Smith 1977; Pylkov et al. 1998; Martin et al. 2006). Clustering of reproductive isolating mechanisms into ‘islands of genomic divergence’ is also predicted in models of chromosomal speciation (Rieseberg 2001; Noor et al. 2001; Navarro and Barton 2003; Gavrilets 2004; Nosil et al. 2009). More recently, Kirkpatrick and Barton (2006) showed that the net strength of selection favouring the elimination of recombination between previously unlinked loci (e.g., by an inversion) is proportional to the migration rate and number of loci involved in local adaptation, but independent of the strength of selection on the individual alleles. Like Kirkpatrick and Barton (2006), our approach shows that the advantage of more concentrated architectures increases with migration up to the critical migration threshold (Figure 5.2A). But where Kirkpatrick and Barton (2006) found that the advantage of completely eliminating recombination is  115  insensitive to the strength of selection, when we allowed incremental changes in the rate of recombination, we found a strong nonlinear relation between the strength of selection and the rates of recombination across which there are significant benefits of alleles being more tightly clustered (Figure 5.2B). This suggests that clustering should be much more likely in strongly selected traits, due to the increased probability of locally adaptive mutations occurring across the larger physical distances that can be maintained in tight linkage. In all of the above models, concentrated architectures evolve because decreased recombination between locally adaptive alleles is deterministically favoured under migration-selection balance. Here, we suggest a second complementary line of reasoning for the advantage of concentrated architectures, based on the relative advantage of larger alleles and linkage groups in finite populations (as per Chapter 4). Exploring the evolution of genetic architecture in finite populations using individual-based simulations also reveals some unpredicted patterns. Firstly, there is often an inverse relationship between the size of individual mutations and the size of the stable alleles/linkage groups that contribute to long-term divergence (Figure 5.6). When mutations have small effects, their individual persistence time is low and they tend to be lost to drift more frequently (Chapter 4). Clustering into tighter linkage groups or stacking of mutations can greatly increase persistence time, especially if average-sized mutations are not sufficiently large to surpass the critical threshold, as shown in Figure 5.1. When mutations are small and rare, it is very rare for another large allele or cluster to evolve and displace the resident cluster, resulting in much more stable architectures. Alternatively, when mutations are large and occur frequently, they tend to cycle rapidly through the population because it is easy for another high fitness allele to arise by a single  116  mutation and drift to high frequency. As a result, at high mutation rates and large effect sizes, individual alleles rarely persist for long enough for clusters to be of much importance (Figure 5.6). This has implications for the extension of our results to more complicated meta-population structure, as the rate and size of mutations might greatly affect the variation among patches in evolved architecture. In his recent examination of a multi-patch migration-selection model, Bürger (2009) has shown that as many alleles can be maintained at each locus as there are patches in a meta-population, although the likelihood of this sort of variation in architecture in finite populations is unclear, and may depend heavily on the dynamics discussed above. Another unpredicted pattern observed in the simulations was the evolution of Z alleles and chromosomes with extreme effects when m was just below the mcrit threshold  and when ! > 1, resulting in genotypes that surpassed the local optimum in the  ! low (Figure 5.6). homozygous state (Figure 5.3), especially when mutation rates were The explanation for this is as follows: when locally optimal alleles are segregating in each patch (i.e., " = +/- 0.5) at sub-critical migration rates, the maladaptive immigrant alleles/genotypes reach relatively high frequencies, resulting in large numbers of hybrid individuals. When there are a large number of hybrid individuals, an extreme effect allele (i.e., " > |0.5|) can invade where it is locally favoured, because the fitness advantage in its heterozygous state can exceed the fitness disadvantage in its homozygous state when the fitness function has convex curvature (! = 2). For example, under a selection regime of # = 0.75, ! = 1, and ! = 2, when the resident allele "r = 0.5 and the invading extreme allele, "e = 0.6, the relative fitness of "e vs. "r would be 1.044 in the heterozygous state vs. 0.9925 in the homozygous state. As long as migration is high enough to result in a large  117  number of heterozygotes, !e would invade and replace !r. This would result in strong selection for an opposing extreme allele with !e = -0.6 to invade in the other patch. Once both extreme alleles are established, the heterozygous benefit would be nullified, as the phenotype produced by two opposing extreme alleles would again be zero. While an optimal allele could then reinvade, this would again generate the advantage for reinvasion by another extreme allele. Extreme genotypes are thus a robust phenomenon under finite population sizes and convex fitness functions, especially when mutation rates are low, as shown in Figure 5.6. It would be interesting to see whether these extreme genotypes are ever observed in nature. Taken together, these results suggest that migration-selection balance should typically favour concentrated genetic architectures, if they are at all biologically possible. The most significant constraint to the evolution of concentrated architectures is likely the possibility for a sufficient number of beneficial mutations to occur in tight enough linkage to build a locally adapted phenotype. Clusters of locally adapted alleles can be maintained over greater physical distances on the chromosome under reduced recombination or stronger selection, so if there are only a limited number of ways for mutations to build a phenotype, stable concentrated architectures are more likely under these conditions. As long as this constraint is satisfied, neither low rates of mutation nor small individual effect size seems to pose a significant constraint on the evolution of concentrated architectures (rather, the opposite was observed), although we did not examine the effect of very small mutations or extremely low rates of mutation. While we formulated a model with a single trait under selection, multiple traits would likely exhibit similar patterns, with alleles adapting to the same environmental conditions tending to  118  cluster together regardless of the specific trait they affect. There is still little empirical data upon which we can test these hypotheses, although a recent review of clustering of FST outlier loci assumed to experience divergent selection (Nosil et al. 2009) documented a range of patterns (from weakly to strongly clustered) in five studies that had mapped these loci. As this field of empirical research is still in its infancy, further empirical work is required to explore whether migration commonly affects the genetic architecture of local adaptation.  119  Figure 5.1 – Critical migration rate maintaining polymorphism Increase in the critical migration threshold, m"crit , with allele effect size, indicating the migration rate below which invasion of the allele tends to be favoured, for three different strengths of phenotypic selection ! (N = 1000 and ! = 2; initial population fixed for an allele with an effect size " = 0). As the local optima are set to # = +/- 1, the critical Z phenotypic threshold, mcrit , occurs for " = 0.5 (horizontal dashed lines).  !  120  Figure 5.2 – Invasion of a second allele following initial divergence at a linked locus A) Relationship between recombination rate and the net strength of diversifying selection (!l – 1) favouring the invasion of a focal allele of size ! = 0.05, given the facilitating effect of an established divergent background allele of size ! = 0.25 separated from the new mutant by a rate of recombination of r (from Equation 5.3). Single-line curves indicate a range of migration rates, from m = 10-5 to m = 10-1, with " = 0.075 and # = 1. Invasion is calculated for the focal allele where the resident allele has ! = 0. The single dashed horizontal line indicates where 2("l – 1) = 1/(2N), which is the point where the net selection favouring the invasion of the focal allele is balanced by drift (Chapter 4). The double dashed curve indicates the recombination rate at which !l reaches a value halfway between its minimum and maximum for that migration rate (!l-50). B) Relationship between the strength of phenotypic selection (") and the recombination rate that yields !l50  at this strength of selection for the same scenario described in A), but with m =10-3.  121  122  Figure 5.3 – Maintenance of divergence for a single-locus quantitative trait Average difference in effect size between the leading alleles in each population for a quantitative trait determined by a single locus ( d ). Genotypes are perfectly adapted to their local patch when d = 1, but extreme alleles that surpass the optimal value ! predominate at sub-critical migration rates. Dashed black curve indicates the magnitude ! of !l for the invasion of an optimal genotype (" = 0.5) when an intermediate genotype is  fixed (" = 0); the horizontal grey line indicates where 2(!l – 1) = 1/(2N), resulting in the Z critical threshold mcrit . Parameters for these simulations are: # = 0.075, N = 1000, µ = 10-  4  , and $ = 2.  !  123  Figure 5.4 – Maintenance of divergence for a multi-locus quantitative trait Mean genotypic divergence between populations increases as migration rates drop below Z the critical thresholds ( mcrit ), indicated by the vertical lines, for three strengths of  selection: ! = 0.75 (solid lines; open circles), ! = 0.075 (dashed lines; filled circles), ! =  ! lines; triangles). In all cases, µ = 0.0001, " = 0.05; r = 0.02, n = 50 loci, N 0.0075 (dotted = 1000.  124  Figure 5.5 – Evolution of genetic architecture Evolution of genetic architecture over time at low (m = 10-6; A and D), intermediate (m = 10-3; B and E), and high (m = 0.05; C and F) levels of migration. Panels A-C show the difference in the size of the leading alleles (d) at each of the 50 loci for a single simulation replicate; white space signifies no differentiation between patches at that locus, while colours indicate the magnitude of d, whenever d ! 0 (as shown in legend). For these simulations, the local optima are set to " = +/- 1, such that a single locus with d = 1 can cause both populations to be perfectly adapted to their local conditions. Panels DF show trends in summary statistics describing the genetic architecture, averaged over 20 simulation replicates: the average size of largest allelic difference at any locus, dmax (red); the sum of d over all other loci (blue); and sum of values of d across all loci (black). In all cases, µ = 0.0001, # = 0.05; r = 0.02, n = 50, $ = 0.075, N = 1000.  125  126  Figure 5.6 – Influence of mutation rate and effect size on genetic architecture Genetic architecture is more concentrated and persists for longer under smaller mutation effect sizes (!) and lower mutation rates (µ), as indicated by the size of allelic divergence at the locus with the largest effect (dmax; A) and the average persistence time of the divergence at this locus (B). Per-locus mutation rates are µ = 10-4 (solid lines); µ = 10-5 (dashed); effect sizes are ! = 0.5 (grey lines; filled circles); ! = 0.05 (black lines; open circles), as indicated in the legend. In all cases, r = 0.02, n = 50 loci, " = 0.075, N = 1000. Z The solid vertical line indicates the position of the mcrit threshold, evaluated for invasion  vs. a genotype of 0. For all cases, persistence time of dmax was measured during the first 500,000 generations only.  ! 127  Figure 5.7 – Influence of selection and recombination on genetic architecture. Large clustering distances (A) and smaller mean sizes of allelic divergence (dmean; B) are common under strong selection, low recombination, and low migration, indicating extensive clustering. Clustering gives way to stacking as migration increases towards the critical thresholds. Recombination rates are r = 0.02 (open circles); r = 0.0002 (filled circles); selection coefficients are ! = 0.75 (solid line); ! = 0.075 (dashed line). All other parameters are as in Figure 6, but with µ = 10-4. The vertical lines indicates the position Z of the mcrit thresholds, evaluated for invasion vs. a genotype of 0.  !  128  References Albert, A.Y.K., S. Sawaya, T.H. Vines, A.K. Knecht, C.T. Miller, B.R. Summers, et al. 2006. The genetics of adaptive shape shift in stickleback: Pleiotropy and effect size. Evolution. 62:76-85. Bengtsson, B.O. 1985. The flow of genes through a genetic barrier. In: Evolution Essays in honour of John Maynard Smith, Greenwood, J. J., P.H. Harvey, and M. Slatkin, (eds.) Cambridge University Press, Cambridge, pp. 31–42. Bradshaw Jr., H.D., S.M. Wilbert, K.G. Otto, and D.W. Schemske. 1995. Genetic mapping of floral traits associated with reproductive isolation in monkeyflowers (Mimulus). Nature. 376:762-765. Bradshaw, Jr. H.D., K.G. Otto, B.E. Frewen, J.K. Mckay, and D.W. Schemske. 1998. Quantitative trait loci affecting differences in floral morphology between two species of monkeyflower (Mimulus). Genetics. 149:367-382. Bulmer, M.G. 1972. 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Predicting adaptation under migration load: The role of genetic skew1 Introduction Populations inhabiting heterogeneous environments tend to adapt to their local conditions, provided levels of gene flow are low enough for selection to cause allele frequencies to diverge. Because gene flow generally decreases with geographic distance, it is unusual to see local adaptation emerging at very fine spatial scales. If natural selection is sufficiently strong, some differentiation may be possible, even under very high rates of gene flow. The classic studies of adaptation to contaminated soils in mine tailings by McNeilly (1968) and Antonovics and Bradshaw (1970) provide a striking example of divergent selection causing adaptive responses over spatial scales small enough to allow considerable gene flow between environments. While many studies have shown evidence of some level of local adaptation (Hedrick et al. 1976; Linhart and Grant 1996), it is unclear whether gene flow commonly constrains the amount of divergence between populations. Unfortunately, implicating gene flow as a primary causal factor limiting local adaptation is difficult. While several studies have found positive correlations between genetic distance and phenotypic divergence between pairs of populations inhabiting different environments (e.g., Smith et al. 1997; Lu and Bernatchez 1999; Nosil and Crespi 2004; Hendry and Taylor 2004), this alone does not prove that gene flow constrains adaptation because a similar pattern would be expected if divergence times  1  A version of this chapter has been published. Yeaman, S. and F. Guillaume. 2009. Predicting adaptation under migration load: The role of genetic skew. Evolution (in press). 133  differed between the population pairs. Furthermore, an opposite causal explanation is also possible, as reproductive isolation mechanisms that evolve as a byproduct of adaptation may constrain gene flow, resulting in a similar correlation (Räsänen and Hendry 2008). While reciprocal transplant experiments can help by demonstrating fitness differences between populations (or lack thereof), they do not rule out the possibility that any observed lack of divergence is simply the result of a non-equilibrium state as populations evolve towards their local optima following some perturbation (e.g., colonization or shift in environment). Other studies have used ecological factors that should be insensitive to demographic history, such as differences in patch size or connectivity to infer relative amounts of gene flow and test correlations between gene flow and trait divergence among populations (Dhondt et al. 1990; Riechert 1993; Sandoval 1994; Storfer et al. 1999). Combining these coarse estimates of contemporary gene flow with evidence from molecular divergence would strengthen the inferences drawn from these correlationbased studies. Even so, it is unclear whether a linear relationship between gene flow and trait differentiation should always be expected, especially in light of the non-linear behaviour seen in related population genetic models (e.g., Hendry et al. 2001; Spichtig and Kawecki 2004). Another approach that can be used to test hypotheses about the role of gene flow is to compare empirical observations to predictions based on models that explicitly represent the evolutionary processes involved. Where variation in the trait of interest is driven by a single locus, population genetic models can be applied to predict allele frequencies under migration-selection balance (King and Lawson 1995; Lenormand et al. 1998; Bolnick and Nosil 2007). King and Lawson (1995) also derived a simple model of  134  migration-selection balance in a quantitative trait where divergence did not depend upon the additive genetic variance. Hendry, Day, and Taylor (2001) corrected this model, providing a more general solution for predicting divergence between populations which has been applied in several empirical studies (Hendry et al. 2002; Saint-Laurent et al. 2003; Moore et al. 2007). A similar model was derived by Lopez and colleagues (2008), who explored the effect of seed vs. pollen migration on genetic load and divergence between populations. Both Hendry et al. (2001) and Lopez et al. (2008) suggested that deviations from a Gaussian distribution of breeding values might compromise the accuracy of this approach by violating the assumption of normality, but as yet, there has been no systematic exploration of this problem. Lopez et al. (2008) used simulations to test their analytical model and concluded that only the most heterogeneous environments resulted in inaccurate predictions. While they suggested that any differences between simulations and analytical predictions were caused by deviations from a Gaussian distribution of breeding values, they did not explore this explanation in detail. Furthermore, they limited their study to very high mutation rates, which might be expected to counteract asymmetries caused by migration and increase the accuracy of the analytical model across a wider range of migration rates and strengths of selection. Here, we revisit this problem, comparing the predictions of the Gaussian approximation model (hereafter GAM) developed by Hendry, Day, and Taylor (2001) to predictions from an individual-based simulation model. We show that the magnitude of the discrepancy between model predictions correlates with the amount of skew in the distribution of the breeding values, suggesting that departures from normality have important consequences for predicting evolutionary trajectories. We also explore the  135  causal basis for these differences, showing that the response to selection within a single generation is greater from a skewed distribution of genotypes than from a normal distribution with the same mean and variance. High mutation rates can reduce the discrepancies between the GAM predictions and simulations, reproducing the patterns observed by Lopez et al. (2008). Observed discrepancies between predictions of the GAM and those of the simulations tend to be most extreme at intermediate to high migration rates (10-4 < m < 10-1) and medium to strong selection, which is exactly the region of parameter space that is biologically most interesting as this is the region where adaptive constraint is predicted by the GAM.  The Gaussian approximation model Using quantitative genetics theory, the Gaussian Approximation Model solves for the equilibrium divergence in phenotype between two populations by finding when the change in mean trait value caused by the mixture of the immigrants and locals (!zm) is counterbalanced by the change caused by response to selection (!zs ; Hendry et al. 2001). When migration is symmetrical and precedes selection, this yields the equilibrium divergence: % ( VG D* = D" ' * 2 &VG (1# m) + ($ + VP )m )  (6.1)  where D! is the difference between the optimal phenotype in each patch, m is the !  migration rate, VP and VG are the phenotypic and additive genetic variance, and !2 is the width of the fitness function (Hendry et al. 2001, equation 8). This model parallels Via and Lande’s (1985) model for the evolution of phenotypic plasticity in heterogeneous environments. The basic assumption of this type of model is that the system can be 136  accurately approximated by modeling the effect of migration on the mean population phenotype and ignoring its effects on the shape of the phenotypic distribution. Mixing of immigrants and locals changes the mean trait value (!zm), and the response to selection is calculated from this new average value under the assumption of constant and normally distributed genetic values (Figure 6.1). Migration followed by introgression, however, modifies the phenotypic composition of a population by adding distinct classes of individuals (e.g. immigrants, hybrids, backcrosses, etc., see Figure 6.1), which results in a distribution of genetic values that is skewed towards the immigrant’s trait value. Each of these classes of individuals will then respond to selection differently, and by aggregating all of them into a single Gaussian distribution with a single mean, the GAM is likely to misrepresent the actual response to selection under certain circumstances. In particular, it ignores that selection is stronger on hybrids and immigrants, which may make up a large proportion of the total distribution prior to selection but be of little importance after selection, especially if selection is strong. To correct for this problem, we now present a model that partitions the total distribution of trait values into discrete phenotypic classes.  The discrete GAM model The model presented here follows closely the developments presented in Hendry et al. (2001). Our approach decomposes the total distribution of trait values in a population into three main phenotypic classes: the residents, the immigrants, and their hybrids, and then expresses the response to selection of the whole population as the sum of the responses of each phenotypic class weighted by their relative contributions based on the migration rate  137  and relative fitness. The distribution of trait values within each class is still considered a Gaussian distribution and is represented in the model by its trait mean (  ) and genetic  variance VGk . For simplicity, each class k has the same genetic variance VG in all populations. The average trait value in population i after mixing and random mating can thus be written as:  (6.2) with:  (6.3) We consider here only two populations, i and j, connected by migration at rate m. The average trait value for the different classes in population i are hybrid,  for the residents, and  j is, by symmetry: is then given by: selection gradient, and  for the  for the immigrants. The mean trait value in population . The response to selection of class k in population i , with  the  the trait optimum (substitute j for i for population j). The mean  trait value after selection is then given by the sum of the contributions of each phenotypic class weighted by their relative fitness and frequency:  ,  (6.4)  138  with  , the mean fitness of phenotypic  class k (following Via and Lande 1985), and  , the average fitness in  population i. We then proceed as Hendry et al. (2001) by finding the per generation changes in mean trait values in each population "z = z s # z and in the divergence between those populations:  . The equilibrium phenotypic divergence between  ! population i and j at migration-selection balance is then found by setting  solving for the divergence  and  . However, as we consider a perfectly  symmetrical system, for simplicity, we can substitute  for  and  for  overall  and solve  for  expression for  at equilibrium is included in Appendix 5. A first linear approximation  of  , the equilibrium trait value in population i. The complete  can be found by using the assumption that both m and zi are of order ! (small  deviation) and finding the Taylor series expansion of order ! around m = 0 and zi = "i yields:  (6.5) with  , the strength of stabilizing selection. Equation 6.5 simply shows that,  first, the effect of migration on the equilibrium trait value is strongest when the response to selection after mixing is likely to be smallest (larger VS means weaker selection), and second, that this response to selection depends on the phenotypic divergence between the two populations ("); it first increases as " increases from 0 to the critical value and then decreases. This dependence of the equilibrium mean trait value on the phenotypic optimum divergence between the two populations captures the fact that as the two 139  populations diverge, selection against the hybrid and immigrant individuals increases, which compensates for their increased phenotypic effect on the population equilibrium value. The original GAM does not take into account this balanced effect of divergence and increased selection against intermediate phenotypes and thus predicts that the amount of constraint is independent of the total divergence between habitats (see Results). Because of the assumption that m is of the order of ! around 0, Equation 6.5 gives accurate predictions for small migration rates only and numerical solutions of were used to compare predictions from this model with simulation results. The complete expression for  is given in Appendix 5 along with a more accurate solution  ignoring the simplification that m is small.  The simulation model To test the predictions of the GAM and discrete GAM, we ran individual-based simulations using a modification of the Nemo platform (Guillaume and Rougemont 2006). Individuals inhabited an environment consisting of two patches with different local optima (+/- ") that were connected by migration at some specified rate (m). Offspring were created by drawing parental gametes from either patch (Pr[local] = 1 – m; Pr[non-local] = m), with recombination occurring within the parents prior to migration at rate r between adjacent loci (n = 50 loci). Newly created offspring with phenotype z experienced stabilizing viability selection around the local optimum with survival probability equal to their fitness (W) as defined by a Gaussian function with variance #2. When some non-zero level of environmental variance (VE) was included, a random value drawn from a Gaussian distribution with a mean of zero and variance equal to VE was  140  added to the individual’s genotypic trait value to calculate its fitness. Offspring were created until filling the local patch to carrying capacity (N), such that the entire system effectively experienced soft selection following migration of the parental gametes. Individuals had diploid genotypes with n loci arranged on a single chromosome, with each locus contributing additively to the phenotype. At initialization of the simulations, all loci were given allelic values of 0 with a 5% chance of mutating to provide some initial variability. All mutations were drawn from a Gaussian distribution with a standard deviation of !, with their values added to the existing allelic value at the locus (continuum-of-alleles model). Following initialization, mutations occurred at a per-locus rate of µ, such that mutational variance was VM = 2nµ!2. Simulations were run until there were no consistent directional changes in mean, variance or skew for at least 20,000 generations (this typically required at least 100,000 generations, depending on the parameter values). We ran twenty replicates of each parameter set, taking all statistical measurements of the populations after selection on juveniles every 50 generations for at least 400 intervals during the stable period at the end of the simulation. Genetic skew was calculated as:  $[(z " z )  3  /# 3 ]/N , where " is the standard deviation of the genotypes and  z is the mean genotype. To check that the simulations were yielding accurate results, we ! compared genetic variance maintained at equilibrium under a homogenous environment  !  to the variation predicted by Burger et al. (1989) for the stochastic house-of-cards model of mutation-selection balance, and we found good agreement (see Appendix 4 for details). As currently formulated, these simulations represent only the classical effects of migration, selection, mutation, recombination, and drift and do not include any effects of  141  gene flow that have the potential to facilitate adaptation (e.g., demographic rescue or reinforcement). To compare simulation results with the predictions of the GAM (Equation 6.1) we used the parameter values from each set of simulations to parameterize the GAM. Because the GAM also requires VG and VP as parameters (but these are dynamic products of the simulations), we used the within-patch genetic variance from the simulation results for VG, and calculated VP = VG + VE, as VP was not measured directly from the simulations. We also note that m from the GAM is equal to 2m in the simulations and that the simulations measure VG following selection. We used the migration rate parameter from the simulations to parameterize the GAM, as this is congruent with the theoretical development of the analytical model. Empirical studies, however, typically use estimates of m based on the realized gene flow inferred from divergence at molecular markers (e.g., Moore et al. 2007), which represents successful migration events and will be lower than the overall migration rate, therefore predicting less constraint. We sought to examine model behaviour over a biologically reasonable range of parameter values for selection, migration, and mutation rate. We used two differences in optimum (! = +/- 1; +/- 2.50) and three strengths of stabilizing selection (!2 = 2.5, 5, 25), yielding the migrant and hybrid fitnesses shown in Table 1. Nosil and co-authors (2005) reviewed a range of estimates for the fitness of locally adapted individuals when transplanted into a different environment, finding estimates for migrant fitness ranging from 0.087 to 0.99, which roughly corresponds to the fitnesses shown in Table 1 (with the exception of ! = 2.5 and !2 = 2.5, which results in stronger selection). A recent review by Hereford (2009) found similar fitness values. For each of these combinations  142  of optimum and selection, we ran a range of migration rates from 10-6 up to 10-1 and a range of per-locus mutation rates from 10-6 up to 10-3. Except where specifically indicated otherwise, all results shown below are for simulations with a patch size of N = 1000, a mutation effect size of ! = 0.5, a recombination rate of r = 0.02 between adjacent loci, n = 50 loci, and VE = 0.5. Choosing a biologically representative level of VE is complicated by trying to scale it both to VS (VS = "2 + VE) and to VM. Reviews by Johnson and Barton (2005) and Stinchcombe et al. (2008) concluded that the strength of stabilizing selection is often stronger (ie., VS is smaller) than the values of VS/VE of 10100 typically used in theoretical literature; while estimates of VE/VM are typically in the range of 100-1000 (Houle et al. 1996). Our choice of parameters for "2 yields values of VS within this range, while our choice of mutational parameters yields VM = 25µ, resulting in VE = 2000 VM for µ = 10-5, but VE = 20 VM for µ = 10-3. In any case, these variables (N, r, n, !, and VE) had very limited effects on the results of the simulations; as such, we limit our discussion here to the major effects caused by migration, mutation, and selection and discuss the effects of these other variables in Appendix 4.  Results In all cases examined, the mean divergence in phenotypes in the simulations either equaled or exceeded the predictions of both the GAM and the discrete GAM, meaning that gene flow imposed less constraint on adaptation than predicted under the GAM. The discrete GAM generally predicted less constraint than the GAM, but still predicted more constraint than the simulations (Figure 6.2). Models agreed in the less interesting set of parameters leading to nearly complete adaptation (with m < 10-4) or constraint (with m >  143  0.1, !2 = 25). The difference between the predictions was most pronounced in the region of parameter space where the GAMs predicted some intermediate level of constraint, with the difference depending heavily upon the migration (m) rate and the combination of the strength of selection (!2) and difference in optima. In some cases, the GAM predicted the phenotype would deviate from its local optimum by as much as ~25% (~15% for the discrete GAM), where the simulations predicted nearly perfect adaptation (Figure 6.2AC). At intermediate to strong selection (!2 ! 5) and larger differences in optimum, the greatest discrepancies were seen at m > 10-3; at weaker strengths of selection (!2 = 25) the greatest discrepancies were seen at intermediate migration rates, where 10-4 < m < 10-2 (Figure 6.2A-C). The effect of mutation was less pronounced, but in all cases, higher mutation rates resulted in greater agreement between the two approaches (Figure 6.2DF). The differences between the results of the approaches can be quantified by comparing the deviation from optimum predicted under the GAM relative to the deviation from optimum found in the simulations (hereafter referred to as the ‘discrepancy score’):  (6.6) (while this measure exaggerates the degree of difference between the models, it provides a useful means to explore the causal basis for the differences). To select some illustrative examples, when µ = 10-5, m = 10-3, and ! = +/- 2.5, the GAM predicted 21.3, 17.9, and 20.0 % deviation from the optimum for !2 = 2.5, 5, and 25, respectively, whereas the simulations found deviations of 0.19, 0.65, and 2.1% for the same values (as shown in Figure 6.2). By contrast, when µ = 10-3, the same choices of parameters gave GAM 144  predictions of 0.58, 0.68, and 1.0 % vs. 0.06, 0.1, and 0.68 % in the simulations. Stated as discrepancy scores, at µ = 10-5, the GAM predicted approximately 113, 27, and 10 times more constraint than the simulations, while at µ = 10-3, it predicted ~ 10, 7, and 1.6 times more constraint (parameter values as above). We found a strong positive relationship between the genetic skew measured in the simulations and the discrepancy score (Figure 6.3A), suggesting that skew is indeed important. This relationship persisted after correcting for the differing contributions of the migrant classes to the total population response to selection using the discrete GAM, suggesting that these differences in accounting cannot explain all of the discrepancies between the GAM and the simulations. The region of parameter space with the highest discrepancy scores corresponds to the region of highest genetic skew which peaks at intermediate migration rates. Genetic skew then decreases at higher migration rates as populations divergence at equilibrium decreases (Figure 6.3B). To explore the causal basis for the differences between the predictions of the GAM and the simulations, we simulated the change in mean due to migration and response to selection within a single generation using the actual distribution of genotypes sampled at the endpoint of each simulation replicate. We then compared this to the response to selection from a randomly generated normally distributed population with the same genotypic mean and variance as the post-migration distribution of genotypes. For each parameter set with m > 0.001 (i.e., with at least one migrant per generation), we performed 30 random draws of genotype values to generate post-migration distributions and then performed 30 replicate responses to selection on each of these distributions. These simulations showed that the change in mean genotype due to selection (!zs) within  145  a single generation was greatly affected by the skew in the distribution of genotypes. Normally distributed populations had much lower variance in fitness (not shown) and response to selection (Figure 6.3C) than the simulation-result populations with large amounts of skew but the same mean and genetic variance. These results conform with the general expectation under Fisher’s fundamental theorem, wherein the rate of change in fitness of a population is equal to its genetic variance in fitness (Fisher 1930). In all of the simulations, skew was generated with the opposite sign as the direction of selection (i.e., in the direction of immigration); in principle, skew could also reduce the magnitude of the selection differential when in the same direction as selection, depending on the curvature of the fitness surface, distribution of phenotypes, and resulting variance in fitness (T. Day, personal comm., 2009). Finally, all else being equal, as mutation rate increased, genetic variance increased and skew in the distribution of genotypes decreased, due to changes in the relative contributions of mutation and migration to genetic variation. Thus, the greatest differences between the predictions of the GAM and the simulations are seen when migration causes substantial deviations from normality in the distribution of breeding values, and this is typically most pronounced at lower mutation rates. At higher mutation rates, mutation contributes relatively more to genetic variance and the skewing effect of migration is less pronounced, resulting in better agreement between the predictions of the two approaches.  GAM vs. discrete GAM The multiple-class version of the GAM we derived captures a part of the effect of the departure from normality caused by intermediate migration and strong selection (Figure  146  6.2). The discrepancy between the GAM and the discrete GAM thus correlates with genetic skew (not shown) and is caused by strong selection against hybrid and immigrant genotypes that the GAM ignores. Under weaker selection and small migration, the trait difference between the different phenotypic classes is reduced, which reduces the amount of genetic skew, and the discrete GAM reaches the same predictions as the GAM, closer to the simulation results (Figure 6.2). The discrete GAM also corrects for another problem inherent in the GAM, which predicts that the equilibrium divergence (D*) should scale linearly with D! for a given value of !2; the amount of constraint is thus independent of the total divergence between habitats assuming constant variances. By contrast, the discrete GAM accounts for the fact that migrants (and to a lesser degree the hybrids) should have a fitness that tends towards zero as the local optima diverge, for a constant curvature of the selection surface. To illustrate this effect, under a fitness surface of !2= 10, VG = VP = 1, and m = 0.01, the GAM predicts a migration load of 9% of D!, regardless of whether " = ±0.1 or ±10. In the former case, a perfectly adapted individual in patch 1 would have fitness equal to 0.996 in patch 2, while in the latter case, its fitness would be nearly 0. The discrete GAM accordingly predicts a migration load of 17% and 0.002% for the first and second case, respectively.  Effect of genetic architecture All of the previous simulations draw mutations from a Gaussian distribution and thus allow mutations of a broad range of sizes to arise and contribute to adaptation (continuum-of-alleles model). Even so, in most cases, very few of the 50 loci in the model actually contributed to population differentiation at equilibrium for migration rates  147  above ~ Nm = 1 (m > 0.001; Figure 6.4). As a preliminary test of the effect of genetic architecture on our results, we reran a subset of the above simulations, constraining mutations at all loci to have the same effect size (+/- 0.1, with equal probability) and replacing the previous allelic value with the new mutation (rather than adding the new value to the old, as above). This diallelic model of genetic architecture is expected to generate predictions closer to the GAM (Tufto 2000). Our results confirm this expectation as the diallelic model generated considerably less skew than under the continuum-of-alleles model (Figure 6.5A) and resulted in less divergence between the populations, matching the results from the discrete GAM closely when loci are unlinked (Figure 6.5B). Linkage between loci in the diallelic mutation model can create groups that function effectively more like a single large locus and thus tend to behave more like the continuum-of-alleles model. As such, lower recombination rates in the diallelic mutation model resulted in more skew and a poorer agreement with the GAM predictions (Figure 6.5B). By contrast, recombination had less effect on the results for the continuum-of-alleles model, where single large alleles can occur by mutation alone (Figure 6.5B). A broad comparison across a range of strengths of selection and differences in optimum shows close agreement between the predictions of the discrete GAM and the diallelic model, in contrast to the relatively poor agreement between either GAM and the continuum-of-alleles model. This demonstrates that discrepancies are generated by ignoring both selection on hybrids (in the GAM) and departure from normality in the distribution of genetic values (in the GAM and discrete GAM). This also illustrates the importance of genetic architecture and the difficulty of accurately modeling  148  migration-selection processes using the Gaussian approximation. Biologically realistic genetic architectures will likely fall somewhere between the two extremes modeled here.  Discussion The aim of the GAM is to provide a means to quantitatively assess whether gene flow is responsible for the apparent deviations from optimal local adaptation often found in natural populations. Based on the results of our analysis, it will be difficult to have confidence in the application of models based on the Gaussian approximation to generate quantitative predictions for natural populations in many cases of empirical relevance. The predictions of the GAM were most accurate in cases that generated very little genetic skew: at very low (m << 0.0001) and very high migration rates (m >> 0.05) and at weak strengths of selection and small differences in optimum. Intermediate migration rates and stronger selection can generate more genetic skew (Figure 6.3B), which decreases the accuracy of the Gaussian approximation for predicting selection within a single generation (Figure 6.3C). This resulted in GAM predictions that were much less accurate (Figure 6.3C) and depended heavily upon the mutation rate (Figure 6.2D-F) and genetic architecture (Figure 6.5 and 6.6). These results are consistent with the hypothesis that the GAM would underestimate the response to selection most severely when the change in mean phenotype due to migration (!zm) is large relative to the standing genetic variation (VG; see Figure 1; Tufto 2000). The ratio !zm / VG, should be largest under low mutation (low genetic variance) and intermediate migration rates that cause high skew (large within-generation change in mean), which is where the response to selection is most poorly approximated by a normal distribution (Figure 6.3C), and where the discrepancy  149  scores are maximized (Figure 6.3A). The corrections made to account for the differing contributions of migrant classes (the discrete GAM) provided an improved fit to the simulation results but were unable to resolve all of the discrepancies generated under the continuum-of-alleles model. Taken together, our results suggest that the role of migration in constraining adaptation may often be less important than expected under models that employ a Gaussian approximation to represent genetic variation. The Gaussian approximation works reasonably well when there is either near perfect adaptation or near perfect constraint, but is much less reliable when dealing with the more interesting cases with intermediate constraint, where its accuracy is heavily dependent upon the mutation rate and genetic architecture of the trait. It is not clear how these problems would affect modifications of the model to include habitat choice (e.g. Bolnick et al. 2009), but qualitative conclusions about the effect of habitat choice would likely be robust to genetic architecture. While Lopez et al. (2008) used simulations to test an analytical model similar to the GAM, they concluded that departures from a Gaussian distribution of breeding values caused by migration would only result in inaccurate predictions under the most heterogeneous environments and highest migration rates. By contrast, we find considerable discrepancies between the simulations and GAM predictions across a wide range of strengths of selection and migration rates. The principal difference between the present study and that of Lopez et al. (2008) is the range of mutation rates considered; Lopez et al. examined dynamics under high mutation rates (µ = 0.005 and 0.0005; n = 10; yielding a per-trait mutation rate of U = 10-2 and 10-1) whereas our study used a much broader range of mutation rates (10-6 < µ < 10-3; n = 50; yielding per-trait mutation rates  150  of 10-4 < U < 10-1). Like Lopez and colleagues, we found good agreement between predictions generated under high mutation rates (U = 0.1; µ = 10-3), but found much poorer agreement at lower mutation rates (U ! 0.01; µ ! 10-4). This is because Gaussian mutation generates variation in both direction, opposing the directional variation introduced by skew when migration rates are low relative to mutation rates. At the present time, our understanding of biologically reasonable mutation parameters is very limited and faced with seemingly contradictory lines of evidence; per-locus mutation rates estimated from allozyme polymorphism, visible, or lethal mutations typically seem to be on the order of 10-5 or 10-6 while per-trait mutation rates are on the order of 10-2, which would seem to require an unrealistically large number of loci per trait or extensive pleiotropy (Barton and Turelli 1989). If the estimations of per-trait mutation rate are more accurate than per-locus estimates, then there are less serious problems with applying the GAM to natural populations. If, however, a substantial fraction of mutations affecting quantitative traits have unconditionally deleterious pleiotropic side-effects, the effective rate of mutations contributing to trait variance in the manner modeled by both the simulations and the analytical theory discussed here would be much lower than empirical estimates (which likely include mutations with deleterious fitness consequences). Regardless of whether per-locus or per-trait estimates of mutation are more accurate, the results of this study clearly illustrate the sensitivity of the Gaussian approximation to these variables across biologically realistic regions of parameter space, undermining the application of such models to natural populations. Taking a step back from the specific problems associated with the GAM, the comparison illustrated in this study raises important questions about how to accurately  151  model evolution in quantitative traits, given the inadequacy of the Gaussian approximation under a large region of parameter space. Turelli and Barton (1994) found that the Gaussian approximation provided a reasonably accurate description of evolutionary change under the infinitesimal assumption (i.e., infinite number of loci of small effect) for a range of selection regimes (without migration), suggesting that departures from normality produced by linkage disequilibria are unlikely to be important for traits with this genetic architecture. They performed numerical tests of their results using multi-locus diallelic models with equal effect size alleles and generally found good agreement with the analytical approximations. They also suggested, however, that departures from the infinitesimal assumption (i.e., fewer loci with heterogeneous effect sizes) could cause departures from normality to be more important, but did not test this explicitly. Tufto (2000) also compared an analytical model using the Gaussian approximation to the multi-locus diallelic model for a single population experiencing stabilizing selection and immigration of individuals with non-local genotypes. Tufto concluded that discrepancies between the analytical and the diallelic model are “a result mainly of the increased genetic variance generated by migration and only to a small extent a result of deviations from reality” (Tufto 2000, p. 291). He showed that deviations from normality were unlikely to be important except under very strong selection or large deviations in migrant genotype. Tufto also noted that large differences in allele frequency among loci could also affect the accuracy of the Gaussian approximation, but suggested that these were unlikely to be important at migration-selection equilibrium. In the present study, we found relatively good agreement between the discrete GAM and the diallelic mutation model (which is similar to the diallelic models described above) but found poor  152  agreement between the GAMs and the continuum-of-alleles model. The discrete GAM is thus a good approximation of the effect of migration on traits that are well behaved in regard to the Gaussian assumption. However, when genetic architecture was characterized by a few large effect mutations or linkage groups contributing most of the variation between populations (Figure 6.4), migration could generate considerable genetic skew (Figure 6.5), resulting in larger discrepancies between the discrete GAM and the simulations (Figure 6.6). Essentially, the discrete GAM fails because while it cannot properly approximate the shape of the distribution generated under the continuumof-alleles model. Migration-selection balance seems to favour local adaptations based on genetic architectures characterized by divergence at only a few large loci. This is likely due to the homogenizing effect of gene flow, which would be more strongly opposed by the larger selection coefficient of a single large locus (compared to several unlinked loci with small coefficients). The diallelic model was more sensitive to recombination rate (Figure 6.5) because it is limited by allele effect size; the only way that evolution can ‘build’ a large effect locus under this model is through linkage under a reduced recombination rate. The continuum-of-alleles model is not as limited by mutations and is thus able to ‘build’ single large alleles without reduced recombination and linkage; as such, it is less affected by recombination rate than the diallelic model and leads to a far smaller constraining impact of gene flow on adaptation. While many studies of migration-selection balance assume diallelic loci of equal effect size to model evolutionary processes (e.g., Phillips 1996, Lythgoe 1997, Spichtig and Kawecki 2004), nature is unlikely to be so uniform. Orr (1998) showed that the distribution of mutation effect sizes fixed during adaptive evolution (without migration)  153  would be approximately exponential under many different mutation models. Griswold (2006) showed that adaptation under migration-selection balance would result in fewer loci of small effect than predicted by Orr, but there was nevertheless considerable heterogeneity in mutation effect sizes contributing to divergence. While it is unclear how closely these models resemble reality, they suggest that the assumption of infinitely many alleles of small effect does not have strong theoretical support. Drawing on evidence from the natural environment, many QTL studies have found a large proportion of trait variation between closely related species or subspecies can be explained by a few linkage groups with large effects (Orr 2001), although we still know very little about the variation in genetic architecture across a wide range of organisms. Given the prevalence of genetic architectures that depart from the infinitesimal assumption and the predictions of the adaptation theory described above, our study suggests that accounting for genetic architecture is indeed important in modeling any evolutionary process that could generate genetic skew (not to mention the higher moments of the distributions, which were not considered here but could be equally important). As such, it may be worthwhile to revisit other related models of migration-selection processes that have used the Gaussian approximation to see if their findings are robust to different types of genetic architecture (e.g., Garcia-Ramos and Kirkpatrick (1997); Kirkpatrick and Barton (1997); Filin et al. 2008). It is important to note that even with an exact model, any predictions will only be as accurate as the parameters used in their calculation. Both the GAM and the simulations require the estimation of several parameters, each of which typically has large margins of error. Without an estimate of the compounded error from these terms, it is not possible to  154  assess the confidence we have in any estimate of the concordance between the model predictions and empirical observations. Furthermore, because considerable variation in population divergence may be expected from one generation to the next due to the stochastic nature of the processes involved, inferences based on observations from a single pair of populations over a short time scale will likely have extremely large confidence intervals. Finally, it should be emphasized that the systems under study may be far away from the equilibrium state implied by the modeling approach. In many cases, the local adaptation process may be incomplete due to recent extirpation and recolonization events or changes in the environment. Depending on the efficacy of selection and the standing genetic variation, adaptation following such disturbances could take many generations to reach equilibrium, even in the absence of migration load. Hendry et al. (2001) showed that in many cases populations will evolve to within 10% of their expected mean phenotype at equilibrium in less than 50 generations, but their analysis was based on strong selection and high migration rates, which accelerate the approach to equilibrium. These considerations have not typically been addressed in the studies that have applied the GAM to testing hypotheses in nature. For example, the parameters estimated in Moore et al. (2007; 0.0005 < m < 0.0116; VP = 1.04; VG = 0.3; 51 < !2 < 3999) result in times to approach within 10% of the expected equilibrium phenotype ranging from 132 to 4004 generations, using Equation 13 from Hendry et al. (2001). It is difficult to reject the alternative non-equilibrium hypothesis without performing manipulation experiments (e.g. Riechert 1993), observations of populations following known changes in gene flow (Nosil 2009), or measuring (lack of) change in mean phenotype over many generations.  155  In summary, we have shown that the assumption of normally distributed genetic values introduces unrealistic constraints into models of adaptation under migration load, causing them to misrepresent the distribution of phenotypes that responds to selection under realistic assumptions about the genetic architecture of adaptive traits in nature. This can cause substantial inaccuracies when predicting response to selection and adaptive divergence at equilibrium, especially at intermediate migration rates and medium to strong selection, where gene flow can generate substantial genetic skew. This is precisely the region of parameter space where evolutionary constraints are expected and where accurate predictions are critical to testing hypotheses. Nevertheless, as a general approach, the discrete GAM can give better first approximations than the classic GAM. Although it is tempting to suggest simulations as an alternative means of formulating quantitative predictions, the complications involved in representing genetic architecture (e.g., Figure 6.5 and 6.6) and our poor understanding of biologically reasonable parameters greatly limit our confidence in any such predictions. When confronting systems with many variables, we feel that experimental approaches are far superior to those based on model-fitting and observation. That said, it is clear that the modeling approach greatly helped us identify the key factors causing evolutionary constraints on adaptation under divergent selection and gene flow. In particular, given the results presented here, effects of the genetic architecture should be investigated when possible in order to explain discrepancies with theoretical results.  156  Figure 6.1 – Sample population under selection and migration Histogram shows genotype frequencies in patch 1 at equilibrium following migration (black) and then selection (grey), for !2 = 2.5 and ! = +/-2.5. Curves represent the Gaussian approximations used in the GAM, where immigration does not affect the variance, but changes the mean resident genotype by !zm, followed by response to selection (!zs) towards local optimum "1, yielding zms. Equilibrium divergence is predicted when !zm = -!zs.  157  Figure 6.2 – Comparison of divergence under GAM vs. Simulations Mean divergence as a function of migration rate (A-C) and mutation rate (D-F) for the simulations (solid lines) and the corresponding GAM (dashed lines) and discrete GAM predictions (dotted lines) . Optimum values are ! = +/-2.5 (open circles) and ! = +/-1 (filled circles); strength of stabilizing selection within each patch is: 2.5 (A & D); 5 (B & E); 25 (C & F). For panels A-C the per locus mutation rate (µ) is 10-5; for panels D-F the migration rate (m) is 10-3. Each simulated point represents the average of 20 replicates (SE bars too small to show clearly).  158  159  Figure 6.3 – Evolution of genetic skew and influence on accuracy of the GAM Plots show the relationship between: A) genetic skew and the discrepancy score for predicted constraint at equilibrium (from Equation 6.6); B) migration rate and genetic skew; C) genetic skew and the ratio of response to selection for simulated normally distributed populations vs. the distributions from the individual-based simulations. Symbols in (B) represent values of !2 of 2.5 (triangles), 5 (filled circles), and 25 (open circles); optimum values are ! = +/-1 (solid lines) and ! = +/- 2.5 (dashed lines). Triangles in A are for the discrete GAM, open circles for the GAM. Points in (A & C) are taken from the same parameter sets used in Figure 6.2A-C, excluding all points where mean genotype in the simulations was not significantly different from the optimum (A) or where m " 0.001 (C).  160  161  Figure 6.4 – Genetic architecture under continuum-of-alleles model Average number of loci contributing to divergence between populations (ie., where the most common allele in one patch differs from that in the other patch) for !2 = 2.5 (solid line) and 25 (dashed line) and optimum values of " = +/-1. All other parameters are as in Figure 6.2A-C.  162  Figure 6.5 – Evolution of genetic skew and divergence as a function of architecture Genetic skew in patch 1 (A) and the mean divergence (B) as a function of migration for the diallelic mutation model (grey) and the continuum-of-alleles mutation model (black) for r = 0.02 (open circles) and r = 0.5 (filled circles). Solid lines in B are for simulations, dotted lines are for the discrete GAM predictions; !2 = 5, " = +/- 2.5, µ = 0.00001.  163  Figure 6.6 – Comparison of simulations, GAM, and discrete GAM Mean divergence as a function of migration rate for the continuum-of-alleles model and the diallelic mutation model with r = 0.5 (grey), with the corresponding GAM (dashed) and discrete GAM (dotted) predictions. All other parameters are identical to those in figure 2, A, B, and C.  164  165  Table 6.1 – Migrant fitness under selection regimes Local optimum (+/- !) 1 2.5 2  Width of fitness function (" )  2.5  0.45 (0.82)  5 25  0.67 (0.90) 0.92 (0.98)  0.007 (0.28) 0.08 (0.53) 0.61 (0.88)  Fitness of individuals with phenotype z = ! when inhabiting a patch with local optimum of -! (fitness of hybrids with z = 0 shown in brackets) for the three values of "2 used in the simulations.  166  References Antonovics, J., and A.D. Bradshaw. 1970. Evolution in closely adjacent plant populations. VIII. 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The demographics of many species have been greatly affected by these changes, causing some species to go extinct and threatening others with extinction through decreases in population range and connectivity. While the genetic and long-term evolutionary consequences of these changes are less obvious than the immediate effects on demography, they are potentially equally problematic. If species cannot adapt to changing climates and patterns of land use, we may witness much more widespread species extinction in the coming century. There is some indication that species such as Drosophila subobscura are evolving in response to climatic change, as Balanyà et al. (2006) have found evidence of shifts in chromosome frequencies repeated on three separate continents. It is far from clear, however, if other species will be able to adapt as easily (Gienapp et al. 2008). For example, Kellerman and colleagues (2009) recently found that tropical species of Drosophila had much lower levels of standing genetic variance for desiccation and cold resistance, suggesting they would have much more limited evolutionary potential. Understanding what maintains genetic variation is critical to predicting how evolutionary potential varies between species and between populations. Furthermore, if the factors that maintain genetic variation are themselves affected by the changes in environment, a negative feedback may occur, whereby global change reduces evolvability and limits the capacity of species to adapt to further change. For example, if  172  genetic variation within populations is largely maintained by mutation-selection balance, then populations can be seen as semi-autonomous units whose evolutionary capacity depends only upon the maintenance of sufficiently large population size. If, on the other hand, the maintenance of genetic variation depends upon gene flow between differently adapted populations, then demographics at the meta-population level will strongly affect the evolutionary capacity of individual populations. Loss of connectivity between populations due to habitat fragmentation and/or loss of populations from across the range of environments inhabited by the species could then considerably compromise the maintenance of variation within populations and their long-term capacity to evolve and adapt to changing climates. Similarly, as land use change and over-harvesting can cause the extirpation of populations from some portions of the species range, it is important to understand how evolutionary processes affect the potential for the re-establishment of new locally adapted populations. While gene flow can increase variance within populations, excessive amounts of gene flow can limit the establishment of locally adaptated genotypes (Lenormand 2002). The research presented in my thesis explores various aspects of how populations evolve in heterogeneous environments and how variation is maintained both within and between populations. The first two chapters use empirical approaches to explore whether gene flow and environmental heterogeneity maintain variation within populations while the three final chapters use theoretical approaches to examine how populations diverge under migration-selection-drift balance. The first research chapter focuses directly on the problem of whether gene flow affects the maintenance of genetic variation for height growth within populations of Pinus contorta (lodgepole pine; Chapter Two). This  173  research presents a new method of analysis using GIS data, showing that populations inhabiting the most regionally heterogeneous environments also tend to have the highest genetic variance in height growth, with a model accounting for four climatic variables explaining approximately 20% of the variation in genetic variance. This suggests that gene flow plays a significant role in the maintenance of genetic variation in natural populations of lodgepole pine. Although other studies have shown that gene flow constrains adaptation to local conditions (e.g., Riechert et al. 1993; Moore et al. 2007) which should also likely affect the variance within populations, this appears to be the first study to show an effect of gene flow on the maintenance of adaptive genetic variation. Interestingly, Kellerman et al. (2009) found that tropical species of Drosophila with restricted distributions tended to have lower levels of genetic variation for cold and desiccation resistance than their more widespread temperate counterparts, and they suggested that this pattern may due to the constraining effect of limited genetic variation, effectively preventing their expansion to a larger area. The results of Chapter Two would suggest an opposite interpretation is also plausible: as temperate species inhabit a larger range with much more heterogeneous climate, they have higher genetic variance as a product of local adaptation and gene flow among populations. It may be that species range expansions are much more common in times of rapid environmental change when niche space is less crowded (e.g., following the retreat of glaciers), and that the increased variance in temperate populations has emerged following these expansions by gradual accumulation and exchange of adaptive mutations. At some level, the mechanisms in play here likely involve both directions of causality, as they would tend to feedback upon and reinforce each other.  174  While it is unclear whether similar patterns are seen in other species, it seems likely that, if anything, the effect of gene flow in lodgepole pine is more important than indicated by these results, given the limitations of the modeling approach. For example, it was not possible to measure heterogeneity in all relevant stressors that would affect divergence between populations in genes related to height growth (e.g., edaphic and biotic factors); as such, this approach inevitably underestimates the effect of gene flow. Furthermore, as this approach is based on statistical correlation and relative levels of variation among populations, it does not allow for an estimation of the absolute effect of gene flow on variance (or its effect relative to mutation), as there was no control population that was known to experience pure homogeneity without gene flow. In light of these limitations and the still significant results obtained in this study, it seems likely that gene flow maintains a substantial portion of genetic variation within populations of lodgepole pine. It would be very interesting to see whether this pattern exists in other species, but obtaining datasets of sufficient size is difficult, given the large amount of labour required to establish and maintain the common gardens needed to assess variation in quantitative traits. One potential alternative approach would be to examine correlations between the regional environmental heterogeneity index developed here and heterozygosity within populations for markers known to be tightly linked to QTL explaining variation in traits that are known to respond adaptively. As such molecular variation can be assayed easily once species-specific primers and methodologies have been established, this hypothesis could be tested in a wide range of species by collecting and genotyping new accessions from populations in heterogeneous vs. homogeneous environments. If the correlation  175  found here extends more broadly to other species, this research would indicate that conservation planners should prioritize regions of the species range that are relatively more environmentally heterogeneous, as these are likely to contain greater evolutionary potential. The second research chapter of my thesis aimed to test the hypothesis explored in the first chapter, but using manipulative experiments to see whether temporally or spatially heterogeneous selection could maintain more genetic variation than homogeneous environments (Chapter Three). This experiment was conceived to replicate and extend an earlier landmark experiment by Trudy Mackay (1981), which found that temporally heterogeneous environments maintained more variance than spatially heterogeneous environments (which was unexpected from single-locus theory; Felsenstein 1976). Where Mackay (1981) had examined variation in bristle traits following exposure to regular and/or alcohol containing medium, we examined variation in 20 wing shape traits in response to heterogeneous temperatures. Although we found significant differences in several of the mean trait values between the hot and cold control populations (indicating an adaptive response to temperature), we found no effect of treatment on genetic variance in any of the traits. Surprisingly, while we found evidence that limited gene flow between populations in different environments was constraining their adaptation (relative to the controls), we found no effect of this migration rate on genetic variation within populations. While these results would appear to suggest that gene flow and heterogeneity do not maintain much more variation than evolutionary processes operating in homogeneous environments, the magnitude of adaptive divergence between the controls was limited (although statistically significant). It seems likely that  176  we would have seen an effect of migration and possibly heterogeneity if we had used a more pronounced stressors to generate more adaptive divergence between populations. For example, a recent study by Moore et al. (2007) found that a pair of stickleback populations inhabiting lake and its outlet stream differed by approximately four phenotypic standard deviations, whereas the most pronounced difference in any trait in our experiment was on the order of half of a phenotypic standard deviation. In any case, one of the most intriguing results of this study was that gene flow constrained adaptation in the limited migration treatment (m ~ 10-3). While there was no detectable effect of this amount of migration on genetic variance, it is conceivable that if it was sufficiently large to limit divergence then it would also affect the character of the standing genetic variation, if not its magnitude. To test whether this treatment affected the evolvability of the populations we extended the study discussed here, performing a single generation of artificial selection on the most divergent trait (ANGLE7-8-9) to both increase and decrease the angle in all replicates from the two control treatments (C and H) and the cold and hot cages of the limited migration treatment (MC and MH). This supplemental study has now been completed, with the data awaiting analysis. The second part of my thesis uses a combination of analytical theory and individual-based simulations to explore how the interplay between migration, selection, and drift affects the evolution of local adaptation and the genetic architecture underlying the divergence in trait values among populations. The first part of the theoretical work (Chapter 4) derives and tests analytical approximations for the probability of a locally beneficial mutation rising to high frequency where it is favoured and the critical migration rate above which locally adaptive polymorphism at a single locus tends to be  177  lost due to drift and the homogenizing effect of migration. The main finding of importance from this chapter is that larger mutations have a higher probability of contributing to local adaptation and a longer persistence time once established than smaller mutations. We show that reasonably accurate quantitative descriptions of these dynamics can be obtained using an estimate of the net effect resulting from the tension between the selection and migration (Chapter 4; Equation 4.4). As this net effect of migration-selection balance is analogous to the action of selection in a single population without migration, it can then be used as a proxy for selection within simpler models of a single finite population (e.g., Kimura’s fixation probability, Equation 4.12). This approach avoids the complex numerical integrations used by some other related studies to derive exact solutions (e.g. Billiard and Lenormand 2005). These approximations are then used in combination with individual-based simulations in Chapter Five to explore how prolonged bouts of evolution in a quantitative trait under stabilizing selection and migration affect the number of loci, size of alleles, and linkage between loci contributing to adaptive divergence. Taken together, these two studies show that evolution in quantitative traits under migration-stabilizing selection balance can result in very different patterns of genetic architecture than expected from models of evolution without migration (Orr 1998) or divergent directional selection and migration (Griswold 2006). Where these studies predict an exponential distribution of mutation effect sizes (Orr 1998) or a distribution of similar shape but with an underrepresentation of small effect mutations (Griswold 2006), the results of Chapter Five show that stabilizing selection and migration favours genetic architectures with fewer, larger, and more tightly linked alleles. This occurs because larger alleles are more  178  resistant to the homogenizing effects of migration in a finite population and because tighter linkage (or a single large allele yielding an optimal phenotype) results in less recombination load. Several studies of adaptive divergence between populations have found that much of the variation in phenotype can be explained by a few quantitative trait loci (QTL) of large effect. While it is possible that the frequency of studies observing this type of architecture is a function of the increased likelihood of detecting large effect QTL, the results of this study suggest that this type of architecture is expected when populations evolve under migration-selection balance for long periods of time. This raises the intriguing possibility that the distribution of genes among chromosomes has itself been affected by natural selection, as lineages that undergo genomic rearrangements that decrease recombination between genes involved in adaptation to heterogeneous selection would be better prepared to adapt to these types of challenging environments. While the theory developed in Chapter Five applies to a single trait under selection, the results should extend to genes underlying any number of traits under correlated selection in heterogeneous environments. As such, genomic rearrangements affecting genes shaping different traits could also be favoured by this mechanism. It would therefore be interesting to test whether genes in pathways often implicated in local adaptation (e.g., phenology, temperature tolerance, drought/desiccation resistance) tend to cluster more closely together than randomly chosen genes, after controlling for clustering due to tandem gene duplications. These results may also suggest an alternate explanation for the large islands of reduced recombination that occur along with the ‘genetic mosaic’ observed in species  179  undergoing ecological speciation. Via and West (2008) observed high concentrations of FST-outlier loci extending for up to 10 cM on either side around large-effect QTL contributing to ecological species divergence among aphid races, and suggested that this occurs because “reduced inter-race mating and negative selection decrease the opportunity for recombination between chromosomes bearing different locally adapted QTL alleles”, with this decrease in recombination thus contributing to isolation at linked marker loci and higher values of FST. This is a reasonable hypothesis, given the strength of selection affecting the aphids and the occurrence of several generations of asexuality between the single annual generation of sexual reproduction (this would increase linkage disequilibrium due to natural selection without recombination). It is possible, however, that the abnormally large map distances over which recombination is reduced and FST is increased are also partly driven by divergence at undetected QTL, a possibility noted by Via and West. While Via and West suggest that this is unlikely to be of great importance, the results of this Chapter Five suggest that QTL should tend to cluster around each other, especially in cases with strong selection and migration. If undetected QTL are clustering near the detected QTL, this could contribute to the maintenance of large FST outliers across such large distances around the QTL in their study. Further theoretical exploration of the extent of ‘islands of recombination’ around QTL with variable effect sizes under migration-selection balance would help evaluate the likelihood of seeing such extreme results as found by Via and West. An interesting extension of the study of genetic architecture under migrationselection balance is the exploration of the phase where architectures diverge substantially by the stochastic fixation of alternative combinations of locally adapted alleles when  180  migration rates drop below 1/Nm (Chapter 5, Figure 5C). In this region of parameter space, hybridization between individuals with very different genetic architectures would result in reduced fitness in the F2 generation due to recombination and mixing of maladaptive combinations of alleles. As such, this process of the accumulation of reproductive isolation would be akin to a mix between the mutation-order and ecological models of speciation described by Schluter (2009). These two modes of speciation both occur by natural selection, but differ in the way that selection generates the accumulation of genetic differences leading to reproductive isolation. Ecological speciation occurs when populations evolve towards different optima and the genes that confer this adaptation are themselves directly maladaptive when combined together, whereas mutation-order speciation occurs when populations evolve towards the same optima but fix different combinations of alleles which form reproductive incompatibilities when they hybridize (it is also possible in both cases that the reproductive incompatibilities are caused by other alleles hitchhiking at unidentified tightly linked loci). The results of Chapter Five for m < 1/N showed both divergence in genetic architecture (mutation-order based incompatibility) and divergence in alleles directly adaptive to each environment (ecological incompatibility) but did not include any loci causing explicit reproductive isolation. While Chapter Five did not examine the effects of the strength of selection or the rate of recombination on the rate of divergence in architecture and the resulting hybrid inviability, it shows that mutation-order speciation mechanisms are likely to accompany ecological speciation mechanisms when m << 1/N. Thus while Schluter (2009) suggests that mutation-order speciation can occur in both small and large populations, this result suggests that it is more likely in small populations when  181  speciation occurs in parapatry, as smaller populations would begin to drift apart at higher migration rates. This particular area of theory requires more directed study to explore the effect of the parameters involved on the rate of accumulation of differences in genetic architecture and therefore reproductive incompatibility. The final research chapter in my thesis (Chapter 6) examines the accuracy of quantitative genetics models of divergence in an adaptive trait under migration and stabilizing selection towards different optima. Quantitative genetic models are typically based on the assumption that genotypes within a population follow a Gaussian distribution, but migration can introduce skew into this distribution, violating this fundamental assumption and leading to inaccuracies in model predictions. This chapter used individual-based simulations to show that the inaccuracy of the predicted divergence under the quantitative genetic model was greatest for the parameter sets in which migration generated the most skew in the simulations. Interestingly, it also showed that the generation of skew in the simulations was dependent upon the genetic architecture; substantial amounts of skew were only generated under architectures that had very heterogeneous distributions of effect size among loci, with a few alleles of large effect and many alleles of small effect (Chapter 6; Figure 5). When mutations were constrained to have equal effect sizes and there was free recombination between all loci, little skew was generated and the simulations agreed well with the analytical quantitative genetic model. When the same equal-effect size mutation model was used with limited recombination, more skew was generated due to the build up of linkage disequilibrium, which resulted in variation in the effect size of linkage groups. As the results of Chapter 5 showed that heterogeneous distributions of allele effect size are expected under  182  migration-selection balance, this chapter indicates that models of these processes relying on the Gaussian approximation will not provide accurate quantitative predictions. Taken together, these three theoretical chapters illustrate both that evolution under migrationselection balance can significantly impact genetic architecture and that in return, genetic architecture can have considerable impact on the predictions of evolutionary models. As many studies employ the simplifying assumption of diallelic mutations of equal size across all loci (e.g., Turelli and Barton 1994; Lythgoe 1997; Tufto 2000), it may be interesting to revisit their predictions using more flexible models of mutation that allow genetic architecture itself to evolve. The five studies included in my thesis further our understanding of how heterogeneous environments affect the maintenance of variation both within and among populations. In some cases, such as the GIS-based method for predicting genetic variance based on regional environmental heterogeneity (Chapter 2), this research suggests the need to include evaluations of genetic variance and evolvability in conservation planning, although further research is required to explore the generality of these findings. In other cases, such as the analysis of the Gaussian Approximation Model (Chapter 6), this research suggests caution when attempting to apply evolutionary models to natural populations, as these models may be based on tenuous assumptions. In any case, it seems clear that while our understanding of the genetic basis of local adaptation and the evolutionary mechanisms governing it is growing considerably, there remain many unanswered questions. My research has shown that gene flow has the potential to have considerable impacts on the adaptive divergence, standing genetic variance, and genetic architecture of populations inhabiting heterogeneous environments. Many of the  183  predictions from this research, however, depend heavily on the parameters and pathways involved. How strong is selection over the long term? Are per-locus or per-trait estimates of mutation rate more reliable indicators of the frequency of occurence of beneficial mutations? How commonly do genomic reorganizations bring together genes under correlated selection? How redundant are the pathways involved in creating phenotypes and how many ways can a maladapted genotype mutate to build an adaptive phenotype? The answers to these questions will greatly influence our understanding of the importance of gene flow in shaping the evolution of populations and species. Returning to the general problems raised above, this will eventually help predict the long-term consequences of fragmentation and habitat loss on evolvability and species survival. Hopefully, we come to understand these issues before their negative consequences are more fully realized, but understanding them completely is by no means a prerequisite to taking precautionary measures.  184  References Balanyà, J., J.M. Oller, R.M. Huey, G.W. Gilchrist, and L. Serra. Global genetic change tracks global climate warning in Drosophila subobscura. Science. 313:1773-1775. Billiard, S. and T. Lenormand. 2005. Evolution of migration under kin selection and local adaptation. 59:13-23. Felsenstein, J. 1976. The theoretical population genetics of variable selection and migration. Annual Review of Genetics. 10:253-280. Gienapp, P., C. Teplitsky, J.S. Alho, J.A. Mills, and J. Merilä. 2008. Climate change and evolution: Disentangling environmental and genetic responses. Molecular Ecology. 17:167-178. Griswold, C.K. 2006. Gene flow’s effect on the genetic architecture of a local adaptation and it’s consequences for QTL analyses. Heredity 96:445-453. Kellerman, V., B. van Heerwarden, C.M. Sgrò, and A.A. Hoffmann. 2009. Fundamental evolutionary limits in ecological traits drive Drosophila species distributions. Science. 325:1244-1246. Lenormand, T. 2002. Gene flow and the limits to natural selection. Trends in Ecology and Evolution. 17:183-189. Lythgoe, K.A. 1997. Consequences of gene flow in spatially structured populations. Genetical Research 69:49-60. Orr, H.A. 1998. The population genetics of adaptation: The distribution of factors fixed during adaptive evolution. Evolution 52:935-949. Riechert, S.E. 1993. Investigation of potential gene flow limitation of behavioral adaptation in an aridlands spider. Behavioral Ecology and Sociobiology. 32:355-  185  363. Mackay, T.F.C. 1981. Genetic variation in varying environments. Genetical Research. 37:79-93. Moore, J.-S., J.L. Gow, E.B. Taylor, and A.P. Hendry. 2007. Quantifying the constraining influence of gene flow on adaptive divergence in the lake-stream threespine stickleback system. Evolution. 61:2015-2026. Schluter, D. 2009. Evidence for ecological speciation and its alternative. Science. 323:737-741. Tufto, J. 2000. Quantitative genetic models for the balance between migration and stabilizing selection. Genetical Research 76:285-293. Turelli, M. and N.H. Barton. 1994. Genetic and statistical analyses of strong selection on polygenic traits: What me normal? Genetics 138:913-941. Via, S. and J. West. 2008. The genetic mosaic suggests a new role for hitchhiking in ecological speciation. Molecular Ecology. 17:4334-4345.  186  8. Appendices Appendix 1 – Details for experimental methods in Chapter Three Culture materials and medium Throughout the experiment, flies were reared on an agar medium containing dextrose, sucrose, cornmeal, brewers yeast, propionic acid, and either tetracycline, ampicillin, or streptomycin (antibiotics alternated regularly). We used 25 mm diameter vials with approximately 10 mL of medium for the crosses and assays and 60 mm diameter plastic bottles with approximately 40-50 mL of medium for the stock maintenance. Cages were approximately 22 cm x 25 cm x 32 cm, with an open mouth covered in nylon mesh and secured with a ring clamp. A generous sprinkle of Fleishmann’s brand active dry yeast was added to both vials and bottles before use. Eggs were collected from the cages using plastic specimen cups with approximately 10 mL of medium tinted dark green using ‘Club House’ brand food colouring (McCormick Inc., London, Canada) and a small dollop of thick yeast paste.  Checking for contamination by Drosophila simulans To check for contamination by Drosophila simulans, we performed reciprocal crosses between the isofemale lines established from the wild collections. A total of 160 reciprocal crosses were performed, of which 149 produced viable progeny from both crosses. These lines were used to found a large breeding population from which we created all of the experimental lines. To check for the presence of D. simulans in the 11 crosses that did not produce viable offspring in at least one direction, the males from the crosses were mated with virgin D. simulans of known provenance (obtained from the  187  Tuscon stock center). In no case did any of these crosses produce viable offspring, suggesting that there was no D. simulans present in the population.  Resolving a minor mite infestation A mild infestation of white mites was detected in 13 of the population cages in the warm room in early October, 2006, during a census of the population size. As a precautionary measure, all experimental lines from the warm chamber were moved to a clean incubator as they eclosed and were maintained in vials (approximately 100 adults/vial, 8-10 vials/cage), transferring the lines to fresh vials once a day to attempt to ‘outrun’ the mite population, for a total of 7 days. This occurred during the middle of a cold chamber generation and was detected with enough time that it did not disrupt the rotation of the cages. To ensure that all experimental cages were treated equally, all cages in the cold chamber were run through the same protocol, although no mites were detected there. This treatment effectively rid the lines of all mites, following which the adults were returned to normal cage culture in the original disinfected chambers. We do not expect this to have had significant consequences on the outcome of the experiment, as all lines were treated similarly and mortality during the transfers was minimal, but one generation of migration was prevented by the protocol. The mites were suspected to have originated in a nonexperimental stock cage maintained in the warm chamber on a continuous generation cycle that was found to be heavily contaminated. The discrete generation cycle of the experimental lines seemed to have a short enough period to preclude the survival of mites once the continuous generation stock cage was discarded.  188  Schedule for experimental crosses Flies in the cold assay took approximately twice as long to mature but in all other respects were treated in the same manner as flies in the warm assay. The following description of the assay schedule is for the warm assay, with the timing of the cold assay transfer shown in square brackets (the schedule began with day 0 when the first eggs were laid in the medium). At day 10 [day 24] of the first assay generation, adult flies were released from their vials into single cages for mass mating within each line. Laying plates were introduced in the afternoon of day 13 [morning of day 26] and a second generation of 10 vials with 30 randomly picked eggs was initiated for each line the following morning. Adult flies from the second generation were collected and separated by sex every 8-10 hours [14-16 hours] as they eclosed to prevent mating of virgin females. At day 11 or 12 [day 26 or 27], flies were lightly anaesthetized and randomly paired into single vials to establish 60 families per assay line. Flies were allowed to mate for 24-48 hours [6-7 days] and were introduced without anesthesia into specimen cups for egg laying in the late afternoon. The following morning [the morning after the next day], 30 eggs per family were picked and transferred to fresh vials; in the case of an insufficient number of eggs, families with fewer than 20 eggs were discarded, while the full count of eggs was used for those that produced 20 or more eggs. All picks for the warm assay were done at room temperature (20oC – 23oC) by two workers, performed in groups of four families / treatment (24 families / block), ensuring that no single family was exposed to this temperature for more than 45 minutes [Cold assay picks were done at 16oC, but were similarly blocked]. Depending on the treatment, the parents were either frozen at -80oC in Eppendorf snap-cap vials after laying in the specimen cups (MH, MC, T,  189  S)  or were transferred by aspiration to fresh vials for productivity assays (C, H). Offspring  were maintained at the assay temperature until emergence and were transferred to Eppendorf vials for freezing 3-5 days after eclosion. Productivity trials were performed for the c and h treatments by transferring each parental pair by aspiration from the specimen cup to a fresh vial and allowing them to lay for 48 hours [96 hours] +/- 10 minutes, following which, the parents were frozen in Eppendorf vials at -80oC. Productivity vials were frozen 14 days [28 days] after the beginning of the laying period, and the number of adult flies present in each vial was counted. The vials were frozen on their side to prevent the flies sticking to the food and complicating the counting, but in some cases there were a few flies frozen in the food, which might have introduced some error into the final counts (likely on the order of 25%).  Wing measurement Fly wings were manipulated following the WingMachine approach of Houle et al. (2003), using a Leica dissecting scope and an ImagingSource 480 x 640 pixel firewire camera (DFK 31AF03). All images and the two orienting landmarks were captured using ImageJ on a Macintosh computer. The FINDWING program was used on a PC to fit Bsplines to the wing veins in the captured images, yielding the 12 landmarks at the intersection of the veins that were used for all subsequent analysis (Figure 3.1). An image and associated landmarks of a specific one millimeter section of a metal ruler were captured to calibrate the images at the beginning of each session, and on 35 of these sessions a single image of the same fly wing mounted on a microscope slide was also  190  captured to test the repeatability of the measurements and the usefulness of calibration. Calibration did not reduce the noise in the measurement process based on comparison of the coefficients of variation of uncalibrated vs. calibrated line segments from the 35 control images of the same wing (for the length between landmarks 2 and 12, CVcalibrated = 0.015, CVuncalibrated = 0.0025), so all subsequent analysis is based on the uncorrected landmarks produced by FINDWING (Houle et al. 2003). The default settings and Model.cp0 file included with FINDWING provided splines that fit well in almost all cases. All images were reviewed manually to check the fit of the splines, with fewer than 1% having errors or failures in the fitting. Images that did not fit a good spline with the specified parameters were discarded rather than digitizing landmarks by hand, which could introduce measurement error.  191  Appendix 2 –Differences in VA among treatments for Chapter Three We expanded our analysis of the VA maintained in the various treatments to include all traits examined in the study. We did not find significant differences in VA between treatments for any trait for which we also found significant slopes for the offspring on parent regression for most of the replicate populations (Table A2.1). The four traits shaded in grey had particularly low heritabilities (Table A2.1), which resulted in very heterogeneous estimation of their slopes, which in turn caused considerable variation in the estimation of VA. Although the p-values for these comparisons are low, we do not put much stock in these results as they cannot be considered significant due to the error in estimating the slopes with accuracy.  192  Table A2.1 – Genetic variance maintained by the six treatments. Trait  C  H  MC  MH  S  T  centroid line7-8 (x105) line9-10 (x105) line2-12 (x105) line1-4 (x105) line1-5 (x105) line2-8 (x105) line3-10 (x105) angle1-2-4 angle1-5-4 angle7-8-9 angle3-10-4 angle1-8-2 angle2-4-9 angle2-4-8 allometry1 (x104) allometry2 (x104) allometry3 (x104) allometry4 (x104) centroid:length (x105)  26.82 0.51 0.17 0.22 3.22 2.25 0.66 2.18 3.14 0.85 19.11 0.33 1.71 2.15 1.65 3.06 2.54 101.12 5.19 0.29  24.81 0.59 0.19 2.84 2.80 2.23 0.54 1.75 2.96 0.83 17.98 0.36 1.61 2.07 1.44 2.91 2.29 -51.64 8.85 10.61  30.63 0.51 0.17 0.30 3.14 2.31 0.30 2.33 3.12 0.80 16.27 0.33 1.53 2.20 1.53 2.62 2.46 5.51 3.50 1.06  25.66 0.63 0.18 2.43 3.11 2.12 0.81 2.57 2.90 0.83 19.19 0.36 1.59 2.12 1.59 2.97 2.59 14.91 7.19 8.48  28.15 0.54 0.14 1.74 2.86 2.24 0.54 1.87 3.03 0.79 17.92 0.31 1.55 2.35 1.72 2.97 2.30 35.33 7.21 7.27  26.37 0.52 0.20 0.31 2.98 2.74 0.70 3.16 3.13 0.86 18.29 0.35 1.72 2.16 1.57 3.02 2.58 28.35 3.80 0.79  Significant Slopes (/60) 52 59 57 13 48 39 18 28 60 60 59 60 59 60 60 59 60 2 14 10  p (VA) 0.88 0.56 0.35 0.02 0.98 0.83 0.29 0.32 0.97 0.99 0.90 0.80 0.91 0.88 0.76 0.92 0.69 0.37 0.45 0.02  Mean additive genetic variance for each of the six treatments (applicable multiplication factors shown in brackets beside the trait names), proportion of slopes for offspring-parent regression with significant p-values across all replicate populations and both assay conditions, and p-values for the comparison of mean VA maintained among treatments (df = 24). Grey shading indicates the traits with low heritabilities (i.e., few significant estimates of the offspring-parent regression coefficient for each replicate population).  193  Appendix 3 – Supplementary materials for Chapter Four Invasion probability for dominant or recessive mutations While Kimura’s (1962) closed form solution for the probability of fixation of a new mutation provided an accurate approximation for the invasion probability of an additive locally-advantageous allele under migration-selection (substituting !l – 1 for S), it tended to over- or underestimate the invasion probability of dominant or recessive mutations, respectively. Kimura (1962) also derived the probability of fixation for non-codominant mutations, although a closed form solution could is not available. For a mutation with a dominance coefficient of H and a selection coefficient of S, Kimura found that its probability of fixation in a diploid population would equal:  u( p) =  # #  p "2cDx(1"x )"2cx  e  dx  e  dx  0 1 "2cDx(1"x )"2cx 0  (A3.1)  where c = NS and D = 2H – 1. As !l represents the rate of increase of a mutation when  !  rare, which is equal to the rate of increase of a heterozygote, we can substitute (!l – 1) for all HS terms and (!l – 1)/(1 – h) for all S terms, to approximate the invasion probability of a mutation with arbitrary dominance by integrating numerically (h in this paper has the opposite dominance effect of H in Kimura’s formulation). Figure A3.1 shows the predicted invasion probabilities for both dominant and recessive mutations with s = t = 0.0025 over a range of migration rates. This modified approach yields very accurate predictions when h = k = 0.25, correcting the discrepancy shown in Figure 4.2C. While the modified approach is also very accurate for low migration rates when h = k = 0.75, it f fails to provide accurate predictions for higher migration rates, when m > mcrit .  ! 194  Figure A3.1 – Invasion probability for dominant and recessive mutations. Results for simulations are shown as solid lines with filled circles (h = k = 0.25) and open circles (h = k = 0.75) for s = t = 0.0025 and N = 1000; predictions based on Kimura’s closed form solution used in the main body of the paper are shown with blue dashed curves, while the predictions from the solution for arbitrary dominance are shown with f red dashed curves. The solid dashed line indicates the critical migration threshold, mcrit , f for h = k = 0.75; mcrit is undefined for h = k = 0.25, as these parameters satisfy inequality  4.10, implying that any migration rate can permit divergence.  !  !  195  Appendix 4 –Results for variables of minor effect from Chapter Six Genetic variance maintained at mutation-selection balance To check the accuracy of the simulations, we compared the genetic variance maintained at equilibrium under a homogenous environment (m = 0.5, opt = –1/+1) to that predicted by the stochastic house-of-cards model of Bürger et al. (1989): VG =  4Nnµ" 2 N" 2 1+ VS  Where Ne is the effective population size, n is the number of loci, ! is the mutation effect !  size, µ is the per-locus mutation rate, and VS = "2, the curvature of the Gaussian fitness function (when VE = 0). We found good agreement for a range of parameters (Figure A4.1), suggesting that the simulations are performing as expected.  Heritability and environmental variance To examine the model behaviour under different levels of VE, we introduced nonheritable noise around the mapping from genotype to phenotype by making a random draw from a Gaussian distribution and adding this quantity to the genotype before calculating individual fitness. We ran the full range of simulations shown in figures 1 and 2 of the manuscript using values of VE = 0, 0.5, 1.0, and 4.0. Mutational variance in all the simulations was equal to 25µ (VM = 2nµ!2; n = 50, ! = 0.5), so the non-zero values of VE used in figure S2 were 2000, 4000, and 16000 times VM (estimates from empirical investigations suggest a value of VE = 1000*VM; Barton and Turelli 1989). In almost all cases, the environmental variance had very little effect on either the results of the mean genotype evolved in simulations (Figures A4.2, A4.3). The only exceptions to this were 196  seen at the highest level of noise at strong selection and very high migration rates, where VE reduced the effectiveness of selection in the simulations (Figure A4.2A and A4.2B; open circles). There was a slight increase in the accuracy of the GAM predictions for strong selection and intermediate migration (figure A4.2A) or low mutation (figure A4.3A) due to the higher genetic variance maintained under the lower heritability at high levels of VE = 4. The heritabilities for these regions of parameter space were unreasonably low (h2 < 0.05), however, so this slight increase in accuracy can be discounted as being biologically unrealistic (h2 = VA / (VA + VE; VG in the simulations = VA). Because we found such little effect of VE on the results of the simulations and the corresponding predictions of the GAM, we conclude that it has relatively little influence on the accuracy of our conclusions described in the manuscript.  Recombination All simulations in the manuscript were run with r = 0.02. We also tested a range of recombination rates (r = 0.5, 0.001, 0.0001) under strong selection (VS = 2.5, ! = +/-2.5) and under weaker selection (VS = 25, ! = +/-1) and a range of migration rates. We found that low recombination rates resulted in slightly increased mean genotype and slightly reduced levels of genetic variation in the simulations (due to linkage and the increased efficiency of selection) causing a slight reduction in GAM predictions (Figure A4.4). Higher recombination rates (r = 0.5) resulted in the opposite effect. We saw no detectable changes in mean or variance with recombination rate under the weaker selection regime (results not shown). As a consequence of the limited response to these variations in r, we  197  concluded that our particular choice of recombination rate parameter could not explain the lack of concordance between the GAMs and the simulations.  Number of loci All simulations in the manuscript were run with 50 loci. Running the same parameter sets from Figure 1 with 500 loci and 10% of the per-locus mutation rate (effectively maintaining the per-trait mutation rate) yielded essentially the same results for all cases examined (results not shown).  Effect size All simulations in main body of the manuscript (except Figure 6.4) were run using the continuum-of-alleles model, drawing mutations from a Gaussian distribution with a standard deviation of ! = 0.5. To test the effect of smaller effect size mutations, we reran a subset of these simulations with ! = 0.1 and found only minor effects on the results, with the smaller effect size resulting in slightly more divergence in the simulations and greater differences between the results of the simulations and the GAM (figure A4.5). We note that even with smaller effect mutations (! = 0.1), genetic architecture of the trait was typically defined by divergence at only a few loci with large effect alleles.  198  Figure A4.1 – Genetic variance in a homogeneous population at equilibrium. Simulation values are shown in black; predictions from Bürger et al. (1989) are shown in red. Filled circles are for !2 = 25, open circles !2 = 5; ! = 0.05 (solid lines), 0.25 (dashed lines), 0.5 (dotted lines).  199  Figure A4.2 – Effect of environmental variance and migration on GAM and simulations Predicted values of the mean genotype in patch 1 as a function of migration rate with different amounts of environmental variance (VE). Solid lines are the simulation values; dashed lines are the GAM values; the eight lines in the upper half are for ! = 2.5, while the eight lines in the lower half are for ! = 1. Symbols correspond to different levels of VE: 0 (triangles), 0.5 (crosses), 1 (filled circles), 4 (open circles). Panels are for different strengths of stabilizing selection, !2 = 2.5 (A), 5 (B), and 25 (C). All other parameters are as in figure A4.2A.  200  Figure A4.3 – Effect of environmental variance and mutation on GAM and simulations Predicted values of the mean genotype in patch 1 as a function of mutation rate with different amounts of environmental variance (VE). Migration rate (m) is 10-3; all symbols and other parameters are as in figure A4.2.  201  Figure A4.4 – Effect of recombination on GAM and simulations. Predicted mean genotype (A) and genetic variance (B) as a function of migration rate with different rates of recombination. Recombination rates are 0.0001 (red), 0.001 (black), 0.02 (blue), 0.5 (green). Dashed lines in (A) are GAM predictions; !2 = 2.5, ! = +/-2.5, all other parameters are as in Figure 6.1.  202  Figure A4.5 – Effect of mutation effect size on GAM and simulations Mean genotype from the simulations vs. GAM predictions as a function of migration rate (left panel) and mutation rate (right panel) with different mutation effect sizes (!). Dashed lines are the GAM predictions, solid lines the simulations; ! = 0.5 (open circles);  ! = 0.1 (filled circles). "2 = 2.5 (red), 5 (black), and 25 (blue), and ! = +/- 1; all other parameters are as in Figure 6.2.  203  Appendix 5 – Full expression and accurate approximations for the discrete GAM model from Chapter Six We give here the full expression for  , the per-generation change in phenotypic  divergence between populations:  (A5.1)  with  .  An equilibrium solution for  can be obtained after setting  and solving  numerically. An accurate approximation can also be obtained after expanding as a Taylor series of first order in zi around zi = !i. This yields:  .  (A5.2)  The accuracy of this approximation is shown in Figure A5.6 by comparing its predictions with numerical solutions of Equation A5.1.  204  Figure A5.1 – Numerical solutions for equilibrium divergence. Equilibrium phenotypic divergence as given by numerically solving Equation A5.1 (dots) and by the approximation A5.2 (lines) for VE = 0.5, VS = !2 + VE = 2.5 (red) and VS = 25 (blue), and for "i = +/–1 and +/–2.5 (with VP = 1, and 2, respectively).  205  

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