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# MAIZE RESEARCH RPTS BSC 5932

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Genet Rest Camb 2000 75 pp 83794 With S gures Printed in the United Kingdom 2000 Cambridge University Press 83 Predicting longterm response to selection JEFF P REEVE Department of Biology Concordia University Montreal Quebec Canada Received 26 January 1999 and in revisedform 30 March 1999 Summary Lande s equation for predicting the response of trait means to a shift in optimal trait values is tested using a stochastic simulation model The simulated population is nite and each individual has a nite number of loci Therefore selection may cause allele frequencies and distributions to change over time Since the equation assumes constant genetic parameters the degree to which such allelic changes affect predictions can be examined Predictions are based only on information available at generation zero of directional selection The quality of the predictions depends on the nature of allelic distributions in the original population lf allelic effects are approximately normally distributed as assumed in Lande s Gaussian approximation to the continuumofalleles model the predictions are very accurate despite small changes in the G matrix lf allelic effects have a leptokurtic distribution as is likely in Turelli s house of cards approximation the equation underestimates the rate of response and correlated response and overestimates the time required for the trait means to reach their equilibrium values Models with biallelic loci have limits as to the amount of trait divergence possible since only two allelic values are available at each of a nite set of loci If the new optimal trait values lie within these limits predictions are good if not singularity in the G matrix results in suboptimal equilibria despite the presence of genetic var1ance for each individual trait 1 Introduction Understanding the dynamics of phenotypic evolution is important not only for predicting how traits should respond to selection but for knowing how much can be assumed about past selective forces given present day trait distributions Selection experiments have added greatly to our understanding of shortterm response and the results have been for the most part consistent with theoretical expectations Falconer 1989 Ro 1997 Patterns oflongterm evolutionary change must be studied primarily using nonexper imental methods given the dif culties associated with collecting suitable data There is a large body of theoretical work on long term selection but most of this concerns mutation stabilizing selection balance eg Lande 1975 Turelli 1984 Barton 1986 Keightley amp Hill 1988 Burger et Room 1219 Hall Building Department of Biology Concordia University 1455 de Maisonneuve Blvd West Montreal Quebec Canada H3G 1M8 Tel 1 514 848 3397 email jreeve vax2concordiaca al 1989 These models assume that the population s mean phenotype is already at or near the optimum and are used primarily for predicting how much genetic variance can be maintained at equilibrium given various assumptions concerning genetic details Most directional selection theory is concerned with truncation selection as used in laboratory experiments eg Robertson 1970 Bulmer 1980 Hill 1982 Keightley amp Hill 1987 and has therefore focussed mainly on shortterm responses in small populations Here I use a stochastic model to simulate longterm response in nite populations undergoing directional selection to a new set of optimal trait values The basic equation for predicting response to a single generation of directional selection is R ms 1 where R is the response in the trait mean It2 is the narrowsense heritability and S 1s the selection differential The multivariate version of 1 is AZ G 2 J P Reeve Lande 1979 where AZ is a vector of changes in trait means G is the genetic variance4zovariance matrix and is the selection gradient often written as the product of the inverse of the phenotypic variancei covariance matrix P l and the vector of selection differentials s Extending 2 to more than one generation of selection presents two distinct problems The rst is that G must be assumed to remain constant over time How likely this is remains controversial and empirical ndings are equivocal Shaw et al 1995 The second problem is that directional selection is unlikely to continue at constant intensity for long periods of time in natural populations Even in experiments using truncation selection the force of arti cial directional selection is likely to be opposed by natural selection acting either on the selected or on correlated traits Lande amp Arnold 1983 Zeng amp Hill 1986 Hill amp Keightley 1988 There is strong evidence that stabilizing selection for intermediate trait values is common in nature Endler 1986 Therefore it is of some interest to investigate the predictive ability of equations that model directional selection as a shift in the optimal values of a set of traits under multivariate Gaussian selection The standard equation for shifted optima Lande 1980a is AZ GWP 1672 3 where W is a symmetrical matrix the diagonal elements being the strength of stabilizing selection acting on each trait large values weak selection and the offdiagonal elements a measure of the strength of correlational selection The superscript 71 indicates matrix inversion and 8 is a column vector of trait optima In modelling evolution with this equation it is assumed that an environmental change has brought about a change in 8 causing directional selection until the traits have evolved to their joint optima Therefore the strength of directional selection decrease as Z approaches 8 but the strength of stabilizing selection W the curvature of the tness surface remains constant Hereafter I will refer to 3 as peakshift selection since the tness optimum has been shifted to a new location This should not be confused with the use of the term to describe the shift of a population s genotype from one tness peak to another in speciation through genetic drift Equation 3 still requires a number of assumptions the most important of which are multivariate nor mality of genotypic and phenotypic trait values in both current and descendant populations and con stant G and P matrices These assumptions will be violated to some extent in nite populations with nite numbers of loci The consequences of such violations are studied here using stochastic simula tions In this study I use simulated populations subject to 84 the laws of Mendelian inheritance to investigate the accuracy of 3 given various assumptions about the genetic details The trajectories of the simulated populations trait means are compared with predic tions from 3 that are based solely on information available at generation zero of directional selection Changes in the variance skew and kurtosis of the distribution of genotypic values are compared with those found or expected in previous models It is well established that the level of genetic variance that can be maintained by mutationstabi lizing selection balance with or without genetic drift depends on assumptions made about the distribution of mutational effects at each locus Turelli 1984 These assumptions have also been shown to be important in terms of the response expected when an equilibrium population is subjected to exponential directional selection Burger 1993 of the form wz e1 4 where wz is the mean tness of individuals with phenotype z and s is the strength of directional selection Therefore it follows that the accuracy of 3 should also depend on the genetic details of the starting population This section brie y describes the three genetic models that will be simulated in this paper For a thorough review see Bulmer 1989 Models of mutationiselection balance can be classi ed into two groups those that assume the mutational and therefore allelic effects are con tinuously distributed and those that assume effects are discrete and nite Models of the rst type are generally based on the continuumofalleles model of Crow amp Kimura 1964 This assumes an effectively in nite number of alleles at each locus producing a continuous distribution of effects Lande 1975 extended this model to multiple loci and developed a formula for the equilibrium variance now known as the Gaussian approximation to the continuumof alleles model This assumes that mutational effects a are normally distributed at each locus and that the variance of these effects is small compared with the standing per locus allelic variance cc2 lt 7 The Gaussian approximation requires Burger et al 1989 2 lt 4m lt5 where u is the haploid per locus mutation rate and V 42 the strength of stabilizing selection on each character equivalent to the diagonal elements of W in equation 3 02E the environmental variance Lande argued that under these conditions muta tioniselection balance could maintain levels of genetic variance consistent with those seen in natural popu lations Turelli 1984 showed that maintaining observed heritabilities under Lande s assumptions would require per locus mutation rates far in excess of Predicting response to selection what is usually thought to be realistic He proposed an alternative formula for the equilibrium genetic vari ance called the houseofcards HC or rare allele approximation This applies when the variance of mutational effects at each locus is large compared with the standing allelic variance cc2 gt 0 and requires Turelli 1984 a2 2 20p 6 This causes mutational effects to swamp the existing variance at each locus The net effect is that most genetic variance is maintained by small numbers of mutant alleles at each locus each of large effect This tends to produce highly leptokurtic allelic distri butions In the second type of model it is assumed that only a small number of allelic values are possible at each locus with mutational effects limited to moving from one value to another The rst such models assumed two possible alleles Latter 1960 Bulmer 1972 1980 These have since been extended to include three Turelli 1984 Houle 1989 and ve Slatkin 1987a alleles Since the allelic values are xed traits are restricted to a nite range of genotypic values if the number of loci is nite When multiple traits are considered there are also limits on the divergence between traits With three or more traits these limits are determined by the eigenstructure of the G matrix rather than by the correlations between pairs of traits In this paper three main types of initial population are simulated The rst two are continuumofalleles models that have either normal Gaussian or leptokurtic HC allelic distributions at equilibrium Both of these assume normally distributed mutational effects and will be referred to as continuous effects populations The third population type has two discrete values 705 and 05 per allele with equal forward and backward mutation rates and will be referred to as discrete effects populations The response in populations with continuous leptokurtic mutational effects is also compared with the main continuous effects results 2 The model The main simulations consist of 4000 diploid indi viduals with three genetically correlated traits Sexes are separate but identical and all data are averaged over the two sexes Mating is random and gener ations are nonoverlapping Populations are given 20000 generations to reach stabilizing selectioni mutationidrift equilibrium hereafter simply equi librium before the start of directional selection The trait means Z start and remain near their optimal values 8 throughout this initial phase Under most initial conditions the genetic variances decline steadily 85 for the rst few thousand generations before reaching their equilibrium levels generally before generation 10000 To simulate directional selection the optimal value for trait 3 is shifted upwards by 10 phenotypic standard deviation units All other conditions are identical in both the equilibrating and directional phases for a given population type For the directional phase of a given population type ve replicates of 1500 generations are run Although all the graphs shown are from only three initial populations many others with different par ameter values were simulated to check the generality of the results in terms of the effect of population size N 4000 or 400 magnitude of peakshift and stabilizing selection intensity W i Creating the population Each individual has L 100 unlinked loci Popu lations with continuous allelic effects are initialized by assigning a random normal variate with mean zero and standard deviation 1 to each allele in each individual Discrete effects populations are randomly assigned a 7 05 or 05 at each allele Each of the three traits is controlled by n 50 loci randomly assigned from the 100 available per individual The pleiotropic relationship between traits was produced by randomly assigning 50 1 s to each row of a 3 traits x 100 loci matrix B equivalent to Wagner s 1989 B matrix All other elements of B are assigned to 0 A 1 at element Bu indicates that locus j contributes its allelic values to trait i Columns with no 1 s represent loci that are not assigned to any trait and are therefore selectively neutral All individuals use the same B which is assumed to be constant The same matrix is used for all simulations discussed in this paper ii Assigning trait values The genotypic value of each trait in an individual is de ned as the sum of all allelic values at all loci that code via B for that trait and is therefore additive between and within loci The expected average genotypic value for all traits is zero before directional selection Phenotypic values equal the genotypic values plus a random normal deviate with a mean of zero and a standard deviation set so as to produce an initial heritability of 05 for all traits This heritability is in general higher than that present after the quot has 1 quotquot The 39 39 vari ance for each trait remains constant throughout selection The environmental deviates added to each genotypic value within an individual are independent thus the expected environmental covariance between traits is zero J P Reeve iii Assigning survival probabilities and selecting parents Each offspring is assigned a survival probability according to wZ exp 7 05Z 7 6T W 1Z 7 8 7 Lande 1980 a where superscript T indicates matrix transposition For directional selection 8 for trait 3 83 is set to 100 phenotypic standard deviation units 8 for all other traits remains at zero throughout the simulation Equation 7 gives values between 00 and 10 and can be interpreted as the probability of survival Therefore selection is frequencyindepen dent since the tness of each individual and the population as a whole is determined solely by its proximity to the optimal vector of phenotypes From those individuals that survive viability selection in the previous generation males and females are randomly assigned to monogamous pairs Pairs are then randomly sampled with replacement each time pro ducing one offspring of each sex Offspring consist of a random haploid complement of genes from each parent Offspring phenotypes and tnesses are assigned as above This procedure is repeated until there are enough surviving offspring to replenish the original population The number of offspring that have to be sampled in order to reestablish the initial population size is therefore a measure of the mean tness of the population This method of viability selection as used in Baatz amp Wagner 1997 produced results virtually identical to the alternative whereby parents were sampled with replacement each gen eration in proportion to their tness w and produced offspring that automatically survived to stock the next generation results not shown All statistics and data are collected only from the surviving offspring Mutations are applied after selection and do not affect that individual s phenotype Mutational effects are added to the value of preexisting alleles The formula for the houseofcards HC approximation for the equilibrium genetic variance assumes that mutational effects are essentially independent Turelli 1984 of preexisting allelic values However this assumption is required in order to simplify the mathematics and is not intended as a statement concerning the actual effect of mutations in real populations Therefore HC populations in this paper have mutational effects of relatively high variance and low mutation rates compared with the Gaussian populations but do not implement the simpli cation required for Turelli s approximation iv Constants and parameter estimates G and P are estimated at generation zero of directional selection from the genotypic and phenotypic values 86 of all individuals The diagonal elements of W are set to 15 times the environmental variance of each corresponding trait This is a value within the range of experimental estimates Johnson 1976 Turelli 1984 The offdiagonal elements of W are set to zero For the continuous effects models genetic variance VG 2na 2 100 assuming global linkage equilibrium Since the heritability of each trait is set at 05 V the initial VG Mutational heritability 11 de ned as the mutational variance VM 2n a2 VE is set to 0001 a value consistent with empirical ndings Lynch 1988 Houle et al 1996 Given VE 100 and Zn 100 1062 must equal 0001 to produce this value note that VE is not set to the conventional 10 The Gaussian simulations uset 0001 and a2 10 While this violates a2 lt 02 it does create Gaussian allelic distributions at equilibrium con rmed by simulation To simultaneously satisfy hi1 0001 a2 lt 7 and n 50 would require mutation rates on the order of 10 The HC simulations uset 00001 and a2 100 For populations with discrete allelic effects a It of 00001 is used as the rate at which each allele changes from 05 to 705 or vice versa In all populations the number of mutations per generation is drawn from a Poisson distribution with a mean of 2LiN For populations with leptokurtic mutational effects the re ected gamma distribution is used where the density function of mutational effects a randomly assigned either a positive or negative sign is given by kbe k lab lHb 8 where F is the gamma function Z is a shape parameter and k is a scaling parameter adjusted so as to produce a mutational variance of 0001 VE as in the above simulations with normally distributed mutational effects The value of b is set to 05 to produce a highly leptokurtic distribution as in several previous simu lation studies eg Keightley amp Hill 1989 Burger amp Lande 1994 3 Results Fig 1 shows the observed and predicted trajectories of the trait means for the different models The prediction for all three traits is very accurate for the Gaussian population Fig 1A The discrepancy between the average observed and predicted values is never greater than 030F for any generation With a population size of 400 results not shown the predictions were nearly as good maximum discrepancy 060 The predic tions for the HC population Fig 1B are much less accurate with discrepancies as large as 370 In this population 3 underestimates the rate of response and correlated response while overestimating the time required to reach equilibrium Predictions for N 400 HC population underestimated the true re Predicting response to selection A O 100 200 300 400 500 C Response GP units O 100 200 300 400 500 87 B O 100 200 300 400 500 D 439 039 rld 9 DLD DBDBDD u 0 100 200 300 400 500 Generation Fig 1 Response of trait means to shiftedoptima selection on trait 3 Units are phenotypic standard deviations A Gaussian conditions B Houseof cards conditions C D Biallelic loci All peakshifts are for 100P except C which is for 5 Filled symbols continuous lines simulation results Open symbols dashed lines predictions from 3 Trait 1 triangles trait 2 circles trait 3 squares Heritabilities averaged over traits of the starting populations were 044 014 016 and 016 for A D respectively All graphs in this paper are based on the average of ve replicate runs sponse by 270 P When the starting population from Fig 1B was given a 93 shift of l 50 instead of 10 the results were qualitatively similar with average discrepancies as large as 14 The quality of predictions from biallelic models depends on the relationship between the peakshift and the selection limit see Section 4 Fig 1 C shows that when the peakshift 50 P in this case is within the limit predictions are good and the average dis crepancy was never larger than O 20 P Average discrepancies in the 400 population size results not shown approached O3930 P In Fig 1D a peakshift of 100 as in Fig 1A and B exceeds the limit resulting in suboptimal evolutionary equilibria Popu lation size and mutation rate have no effect on this limit For all three types of population running the directional phase of selection without mutation has virtually no effect on the trajectory of the means Therefore for the continuous effects populations there is enough standing variance to easily move 100 For the biallelic population mutation rate has little effect on anything but the equilibrium variance see Section 4 Fig 2 shows the changes in several genetic parameters caused by directional selection for the continuous effects models of Fig 1A and B In the Gaussian population genotypic variances increase by 15 25 peaking at generation 80 90 Fig 2A The variances of traits 1 and 2 change more than that of trait 3 despite the means being displaced far less In the HC population Fig 2b the variance peaks at generations 30 80 with a 6fold increase in trait 3 and a 4fold increase in traits 1 and 2 When the HC population was run at N 400 variance still increased by up to 4 times With N 400 and VS 2 60 rather than 16 as in the main simulations variance increased by a factor of 18 although there was not a noticeable increase until about generation 20 The skew and kurtosis of the Gaussian population s trait genotypic values remain near the values of normal distributions 00 and 30 respectively In the HC population the skew for all traits is initially near 00 The skew in traits 1 and 2 remains near 0 except for the rst 15 25 generations when there is a positive skew of up to x 03 The skew for trait 3 reaches a higher peak 04 but continues to decline for hundreds of generations Kurtosis was high in the starting population 3394 but was quickly driven to normal levels and then steadily increased from generation 300 onward It eventually returned to pre directional selection levels These gures show how the genetic characteristics of the populations continue to evolve long after the trait means have reached an apparent equilibrium Fig 3 shows the genetic changes in the biallelic populations from Fig 1C and D As with the continuous effects models both populations initially show an increase in genetic variance Notice that in Fig 3B the population maintains variance for all J P Reeve A 74 88 B 1393 Genetic variance Genetic skew Genetic kurtosis 2398 2 8 0 Generation 500 1000 1500 500 1000 1500 Fig 2 Genetic variances skews and kurtoses for A Gaussian and B houseofcards populations corresponding to Fig 1A and B respectively Genetic variances are standardized to the level in generation zero Note that the scale is di erent for the two variance graphs Trait 1 thin dotted line trait 2 thin continuous line trait 3 thick continuous line traits despite the trait means being at a suboptimal equilibrium The normality of the starting population is a consequence of the symmetry of the allelic effects 05 and 05 about the optimum 00 This guarantees that directional selection will produce skew and positive kurtosis proportional to the peak shift given the restrictions on mutational effects In Fig 4 the effect of directional selection on genetic correlations in the four populations is shown Only in the Gaussian population do the correlations remain relatively constant Fig 4A In HC populations Fig 4B the patterns of change are irregular and are apparently very dependent on the exact starting conditions and the nature of the peakshift In the biallelic population Figs 4 C D correlations increase as loci unique to trait 3 increase the frequency of their high alleles Fig 4D shows that pairwise correlations of less than 10 do not guarantee that traits will reach their optima in biallelic models if there are more than two traits in the system This effect has previously been noted in algebraic models of multitrait systems Slatkin 1987b Charlesworth 1990 Textbook descriptions of changes in the genetic correlation during directional selection are usually based on the biallelic model as is appropriate for small laboratory populations that are often derived from crosses between lines The changes expected in large continuumofalleles populations Fig 4AB Predicting response to selection A 89 B Genetic variance I 3M wh hh39 b Genetic skew Genetic kurtosis 28 28 V 0 500 1000 1500 0 Generation 1000 1500 Fig 3 Genetic variances skews and kurtoses for biallelic populations where the selective optima 9 is either A within or B beyond the selection limit Genetic variances are standardized to the level in generation zero A and B correspond to the populations from Figs 1 C and D respectively Symbols as in Fig 2 are far less intuitive due to the presence of con tinuously distributed allelic mutational effects The median allelic skew and kurtosis from the HC population are shown in Fig 5 where loci have been classi ed according to the combination of traits they control There are eight classes 0 1 2 3 1 2 1 I 3 231 23 containing 19 711141310 6 and 20 loci respectively The 19 neutral loci of class 0 are not shown in this gure as they do not respond in any directed manner to the directional selection encoun tered here As directional selection starts and rare alleles with large positive effect on trait 3 increase in frequency the skew for the four classes that include trait 3 moves to a high level Note that the exact value of the skew in the equilibrium population is highly variable between generations so these classes may start with almost any value The other nonneutral classes generate skew in the opposite direction at the point where the net selective forces start favouring smaller values of traits 1 and 2 compare with Fig 18 This effect of having both directions of skew in different subsets of genes will tend to produce genetic distributions that are less skewed than their underlying allelic effects This is important because standard Gaussian predictions such as 3 assume that genetic variance remains constant on directional selection Barton amp Turelli 1987 have shown that this is a consequence of assuming that there is no allelic skew At the postdirectional selection equilibrium nega tive skew was highly signi cant in classes 3 16 and J P Reeve 90 B Genetic correlation A M A A 06 quot W o oVV VW WJ39quotwl z W O o 6 a 3quot Myquot 0 039 AA via I b O 04 W 04 I 02 I I 0392 I 0 500 1000 1500 0 Generation 500 1000 1500 Fig 4 Changes in genetic correlations caused by peakshift selection A Gaussian conditions B Houseofcards conditions C Biallelic loci Peakshift within limit D Biallelic loci Peakshift beyond the selection limit A D correspond to the trait responses with the same letter in Fig 1 Correlation rm between traits i and j r12 thin dotted line r13 thin continuous line r23 thick continuous line Allelic skew 100 l U Allelic kurtosis 11 0 J 11 I r III III 39quot IIIIIIIIIII39 unnuuuuuu O 39 I I 0 10 20 30 40 50 Generation Fig 5 Changes in median allelic skew and kurtosis for the HC population of Fig 1B Loci are classi ed according to the combinations of traits they control as determined by the B matrix Loci controlling traits 1 1 2 2 thin continuous line 1 2 l 3 thin dotted line 1 3 2 3 thick continuous line 3 thick dotted line 1 3 13903 based on the median skew per class per generation averaged over generations 2000 5000 measured every 50 generations to reduce autocor relation Keightley amp Hill 1988 Burger et al 1989 Thus these two classes account for most of the negative skew seen in the genotypic distribution of trait 3 in the same simulations Fig 213 Although the variability amongst replicates is very small for the predicted and observed trait means it increases rapidly with increasing moments of the genotypic distribution As an example two other sets of ve replicates under the same conditions produced a a longterm de pression in the skew for trait 2 and b no longterm depressions It is likely that the populations can move between different equilibrium states as in Barton 1986 which can have a large in uence on the behaviour of the higher moments but less on the variance and especially the means However on the introduction of directional selection all replicate sets behaved qualitatively as described above for Fig 5 The kurtosis of the trait 3 classes declines rapidly on directional selection again due to selection for rare alleles The decline in kurtosis for the other classes is much weaker and is generated by the same processes that produce negative skew The continuumofalleles simulations with lepto kurtic mutational effects produced the expected results greater allelic leptokurtosis and therefore a greater increase in variance in response to directional selection than populations with normally distributed mutational effects results not shown A mutation rate of 0001 produced a maximum average deviation between simulation and equation that was approxi mately 5 times as large and an increase in variance that was twice as large as in the population of Fig 1A Lowering the mutation rate to 00001 at a xed mutational heritability resulted in 4fold increase in Predicting response to selection genetic variance over that of the population in Fig 1B but little difference in the accuracy of trait mean predictions 4 Discussion This paper attempts to answer a relatively straight forward question If a change in environmental conditions causes selection for a new value of a single trait can Lande s shifted optima equation be expected to accurately predict the trajectory by which this trait and any others correlated with it will evolve The answer like that to the question of how much variation can be maintained through mutationiselection bal ance depends crucially on the nature of mutational effects As almost nothing is known about the frequency magnitude or distribution of mutations at typical polygenic trait loci it is not possible to assess the predictive accuracy of the equation All that can be done at least until more empirical data are available is to show what conditions are required for making accurate predictions and to describe quali tatively the nature of the errors produced when these conditions are not met Like other models this simulation makes numerous simplifying and un realistic assumptions For instance dominance trait value epistasis and physical linkage are absent and tness is determined completely by multivariate Gaussian selection on the trait values However the simulation is not primarily intended as a model of how evolution works Rather it makes assumptions consistent with those from standard quantitative genetic theory and asks how the unavoidable com plications associated with nite populations and nite numbers of loci are likely to affect the predictions of a speci c equation The main results from this paper can be summarized as follows 1 If allelic distributions are approximately Gaussian 3 can produce very accurate predictions based only on information gathered at generation zero of directional selection These predictions were accurate despite the fact that the genetic variances changed by more than 20 during the directional selection phase Such populations start with little genetic skew or kurtosis and this changes little with directional selection Thus under these conditions evolutionary trajectories may be understood in terms of the simple parameters of 3 2 Under HC conditions the predictions can be very inaccurate Equation 3 underestimates the rate of response and correlated response and overestimates the time it takes for the traits to get to their equilibrium values In most of the populations tested including that in Fig 1B correlated responses were also of greater magnitude than predicted sometimes mark 91 edly so Directional selection can result in very large increases in genetic variance as initially rare alleles increase in frequency 3 If mutational effects in continuumof alleles populations are leptokurtically distributed directional selection will cause larger increases in genetic variance for any given mutational heritability This causes the trait means to respond more quickly than predicted even in high mutation rate smaller mutational effect populations 4 For biallelic models 3 makes good predictions as long as the peakshift does not require trait mean equilibrium values that are more divergent than can be accommodated by the genetic architecture of the species see below Predictions become progressively worse as the optimum exceeds the selection limit In such cases suboptimal equilibria will be reached despite the presence of genetic variation in each individual trait 5 For many of the parameter combinations used to test the generality of the main results peak shifts of 100P resulted in genetic variances and covariances at the new equilibrium that were remarkably close to those seen in the population before directional selection Therefore interpretation of the role of drift versus selection in shaping the G matrix should be made with caution It may be that the changes in G brought about by selection to new optima are often temporary even in relatively small populations If this is the case the changes in G found in shortterm laboratory selection experiments may be funda mentally di erent from those expected between populations or closely related species that have been experiencing different selection regimes for long periods of time Peakshift models of the sort considered here have received relatively little theoretical attention and most of this has dealt with very different types of questions For instance conservation biology issues have motivated research on the ability of a population to keep up with an optimum that is changing either gradually or randomly Lynch amp Lande 1993 Burger amp Lande 1994 Burger amp Lynch 1995 Charlesworth 1993 considered an optimum that could also change cyclically to study the effect of directional selection on the evolution of sex and recombination Each of the above studies used a single trait and did not use the univariate form of 3 to describe the evolution of the mean phenotype Zeng 1988 used a modi cation of3 to look at the effects of correlational selection on patterns of longterm correlated response in in nite populations Other papers have considered the evol ution of two traits one under stabilizing and the other under exponential directional selection eg Burger 1986 Wagner 1988 Baatz amp Wagner 1997 Barton amp Turelli 1987 used allelic recursion equations to simulate peakshift selection in a singletrait system J P Reeve using momentgenerating functions to make pre dictions for the mean and higher moments This is the rst paper to test the predictions of 3 by simulation Although more general predictive equa tions are available Barton amp Turelli 1987 Burger 1993 they require detailed information about higher genetic moments or cumulants that are generally unobtainable In addition none of these have been extended to multivariate systems Equation 2 and its descendants including 3 are popular because they attempt to predict or at least explain evolution in terms of a small number of relatively familiar parameters that can in principle be estimated i Causes of prediction error The rapid increase in genetic variance seen in the HC population Fig ZB causes the means to respond to selection far more rapidly than predicted by 3 This increase in variance as a result of directional selection has been shown previously in singletrait simulations by Barton amp Turelli 1987 peakshift Keightley amp Hill 1989 pure directional w 2 Burger 1993 exponential and Burger amp Lande 1994 shifting optimum The increase is largely due to selection for rare alleles that initially contribute little to the variance As these alleles increase in frequency they contribute more to the variance and soon cause the mean to increase at an accelerating rate Barton amp Turelli 1987 For a xed mutational variance lowering the mutation rate will result in equilibrium populations that have less genetic variance but higher allelic skew and kurtosis The accelerated response to the mean is not simply a consequence of the lower equilibrium heritability see caption for Fig 1 in HC populations Increasing this heritability to Gaussian population levels either by decreasing the strength of stabilizing selection or increasing the total L and traitspeci c n number of loci did reduce the amount by which the variance increased during directional selection However the quality of the predictions was only slightly improved with the dis crepancy between observed and predicted response remaining roughly an order of magnitude greater than in the Gaussian populations results not shown In using 3 it is assumed that the distribution of genotypic values is and will continue to be multi variate normal and therefore that the dynamics of trait mean evolution can be described completely in terms of the mean and variance HC populations have far more evolutionary potential in terms of rate of response than Gaussian populations with the same variance Therefore heritability is not an accurate predictive statistic in such populations An increase in genetic variance is seldom seen in arti cial selection experiments which would seem to 92 be evidence against the generality of HC conditions discussed in Keightley amp Hill 1989 Burger 1993 concluded that a signi cant increase in variance is unlikely if Ne is less than about 500 This gure was based on typical parameter estimates for the HC model combined with the fact that genetic variance in his model converged to the mutationidrift equilibrium level under a particular form of exponential selection For the peakshift selection used in the presen simulation HC populations with an N of 400 Ne 300 still had a 4fold increase in variance When the intensity of stabilizing selection was decreased N 6 there was a 1 8fold increase in variance Therefore Burger s conclusion may not extend to all forms of directional selection However the Ne in selection experiments is typically much smaller than 300 In addition we know nothing about how existing univariate estimates of stabilizing selection intensity should be adjusted when considering multivariate systems Therefore failure to detect increased variance in selection experiments probably cannot be taken as evidence against the HC model ii Selection limits in models with discrete allelic e ects Discrete effects models have been used extensively in quantitative genetics eg Latter 1960 Bulmer 1972 1980 Barton 1986 1989 Turelli amp Barton 1990 They may be interpreted either as a realistic rep resentation of the allelic effects for at least some loci or as a method of simplifying the analysis of continuous effects models Houle 1989 In the latter case results such as those in Fig 1C may lead to unwarranted con dence in the predictive ability of standard theory 7 such as 3 7 if HC conditions are the norm The lack of rare alleles of large effect in the biallelic simulation has produced a result consistent with 3 because the behaviour of the variance is more similar to that seen in the Gaussian than in the HC population Here trying to extend the results of the simplifying model to the situation with continuous effects would bias the conclusions Alternatively if discrete effects models are taken as a realistic representation of allelic effects the limit problem deserves some consideration at least when modelling the evolution of trait means To see why the limit occurs consider a system of 20 genes with free recombination where loci 1712 and 9720 control traits 1 and 2 respectively If the two traits are selected in opposite directions the correlation between traits will approach 10 as the loci unique to each trait x in the appropriate direction Genetic variance for both traits remains however since loci 9712 will still be segregating The amount of divergence between traits is a function of the number of loci unique to each trait Predicting response to selection and their allelic effects The same situation exists for systems of three or more traits except that pairwise correlations at the limit no longer have to be 10 eg ig 4D since each trait will generally share segregating loci with more than one other trait At a suboptimal limit as in Fig 1D the genotypic tness of the population cannot increase since mutations cannot improve upon the alleles that are already present Therefore mutation rate has no effect on the limit In Fig 1D the eigenvalue corresponding to an increase in trait 3 and a decrease in the other traits is near zero so the G matrix is nearly singular for that direction of response It is not completely singular because mutation continue to produce genotypes that are slightly less t than those at the limit If the alleles at each locus are typically restricted to a nite number of values the simulations suggest that the situation found in Fig 1D might be common since all it requires is a large peakshift prolonged directional selection This situation would be char acterized by a lack of response in a population despite the presence of genetic variance for each trait genetic correlations less than 10 and nonzero values for the coef cients of the phenotypic selection gradient measured as in Lande amp Arnold 1983 These non zero coet39 cients exist because environmental variance can produce phenotypes more t than those at the genetic limit but this tness difference is not heritable There are in fact several examples of such a lack of response in natural populations eg Price et al 1988 Alatalo et al 1990 van Tienderen amp de long 1994 Weis 1996 although in most cases the authors have provided compelling evidence for simpler explana tions These include the effect of missing traits on the analysis and nonheritable traits in uencing the focal traits and tness through different pathways Price et al 1988 Rausher 1992 It should be noted that in discrete effects populations with allele frequencies near xation as when di rectional selection has driven the population to a suboptimal selection limit subsequent selection in the direction of the rare allele causes a pattern of response similar to that seen in HC populations results not shown Genetic variance increases as the rare alleles become more common and the trait means respond to selection more rapidly than pre dicted by 3 However the response is very slow compared with HC models due to much lower initial heritabilities Although this paper has examined the predictive ability of only one equation a large number of other theoretical models are based on the same underlying assumptions stemming from the use of 1 These include models for the evolution of sexual size dimorphism Lande 19805 phenotypic plasticity Via amp Lande 1985 maternal effects Kirkpatrick amp Lande 1989 and epigenetic effects Atchley amp Hall 93 1991 to name but a few If houseofcards assumptions are more realistic than those of the Gaussian model some of the conclusions of these models are likely to be at least quantitatively inaccurate For instance in Lande s 1980 5 paper on the evolution of sexual size dimorphism he models the situation where sexual selection for increased values of a trait in males causes a temporary maladaptive increase in the homologous trait in females Given typical genetic correlations between the sexes he concludes that the time for the traits in each sex to reach their equilibrium values may be on the order of millions of generations From the simulation results in this paper HC conditions might be expected to reduce that time substantially Given current estimates of mutation rates and mutational heritabilities it is likely that allelic effects are leptokurtically distributed Therefore directional selection in moderate to largesized populations is likely to cause an increase in genetic variance Because of this Gaussianbased quantitative genetic models will often underestimate the rate at which trait means respond to selection In models involving stabilizing selection this will result in overestimates of the time required for populations to reach equilibrium I would like to thank D Fairbairn for valuable advice suggestions and discussions that have helped greatly in the writing of this paper I also thank D Roff S Via and two anonymous reviewers for their many helpful comments on a previous draft of this paper This work was supported by a Natural Sciences and Engineering Research Council of Canada grant to D Fairbairn Refere nces Alatalo R V Gustafsson L amp Lundberg A 1990 Phenotypic selection on heritable size traits environ mental variance and genetic response Arnerican Naturalist 135 4644171 Atchley W R amp Hall B K 1991 A model for development and evolution of complex morphological structures Biological Review 66 1017157 Baatz M amp Wagner G P 1997 Adaptive inertia caused by hidden pleiotropic effects Theoretical Population Biology 51 496 Barton N H 1986 The maintenance of polygenic variation through a balance between mutation and stabilizing selection Genetical Research 47 2097216 Barton N H 1989 The divergence ofa polygenic system subject to stabilizing selection mutation and drift Genetical Research 54 59777 Barton N H amp Turelli M 1987 Adaptive landscapes genetic distance and the evolution of quantitative characters Genetical Research 49 1577173 Bulmer M G 1972 The genetic variability of polygenic characters under optimizing selection mutation and drift Genetical Research 19 17725 Bulmer M G 1980 The Mathematical Theory of Quan titative Genetics Oxford Oxford University Press Bulmer M G 1989 Maintenance of genetic variability by mutationiselection balance a child s guide through the jungle Genorne 31 7617767 J P Reeve Burger R 1986 Constraints for the evolution of func tionally coupled characters a nonlinear analysis of a phenotypic model Evolution 40 1827193 Burger R 199 3 Predictions ofthe dynamics ofapolygenic character under directional selection Journal of Theor etical Biology 162 4877513 Burger R amp Lande R 1994 On the distribution of the mean and variance of a quantitative trait under muta tionselectionidrift balance Genetics 138 9017912 Burger R amp Lynch M 1995 Evolution and extinction in a changing environment a quantitativegenetic analysis Evolution 49 1517163 Burger R Wagner G P amp Stettinger F 1989 How much heritable variation can be maintained in nite populations by a mutationselection balance Evolution 43 174871766 Charlesworth B 1990 Optimization models quantitative genetics and mutation Evolution 44 5207538 Charlesworth B 1993 Directional selection and the evolution of sex and recombination Genetical Research 61 2057224 Crow J F amp Kimura M 1964 The theory of genetic loads In Proceedings of the XI International Congress of Genetics pp 4957505 Endler J A 1986 NaturalSelection in the Wild Princeton NJ 2 Princeton Universit ress Falconer D S 1989 Introduction to Quantitative Genetics e York Longman Hill W G 1982 Rates of change in quantitative traits from xation of new mutations Proceedings of the National Academy of Sciences of the USA 79 1427145 Hill W G amp Keightley P D 1988 lnterrelations of mutation population size arti cial and natural selection In Proceedings of the Second International Conference on Quantitative Genetics ed E J Eisen M M Goodman G Namkoong amp B S Weir pp 57770 Sunderland MA Sinauer Houle D 1989 The maintenance ofpolygenic variation in nite populations Evolution 43 176771780 Houle D Morikawa B amp Lynch M 1996 Comparing mutational variabilities Genetics 143 146771483 Johnson C 1976 Introduction to Natural Selection Baltimore University Park Press Keightley P D amp Hill W G 1987 Directional selection ariation in nite populations Genetics 117 5737582 Keightley P D amp Hill W G 1988 Quantitative genetic variability maintained by mutationstabilizing selection balance in nite populations Genetical Research 52 33413 Keightley P D amp Hill W G 1989 Quantitative genetic variability maintained by mutationstabilizing selection balance sampling variance and response to subsequent directional selection Genetical Research 54 45417 Kirkpatrick M amp Lande R 1989 The evolution of maternal effects Evolution 43 4857503 Lande R 197 5 The maintenance of genetic variability by mutation in a polygenic character with linked loci Genetical Research 26 2217235 Lande R 1979 Quantitative genetic analysis of multi variate evolution applied to brainzbody size allometry Evolution 33 4027416 94 Lande R 1980a Genetic variation and phenotypic 1 1 11 v v 116 4637479 Lande R 19801 Sexual dimorphism sexual selection and adaptation in polygenic characters Evolution 34 2927305 Lande R amp Arnold S J 1983 The measurement of selection on correlated characters Evolution 37 121071226 Latter B D H 1960 Natural selection for an inter mediate optimum Australian Journal of Biological Sciences 13 30735 Lynch M 1988 The rate of polygenic mutation Genetical Research 51 1377148 Lynch M amp Lande R 1993 Evolution and extinction in response to environmental change In Biotic Interactions and Global Change ed P M Kareiva J G Kingsolver amp B Huey pp 2347250 Sunderland MA Sinauer Price T Kirkpatrick M amp Arnold S J 1988 Directional selection and the evolution of breeding date in birds Science 240 7987799 Rausher M D 1992 The measurement of selection on quantitative traits biases due to environmental covari ances between traits and tness Evolution 46 61626 Robertson A 1970 Some optimum problems inindividual selection Theoretical Population Biology 1 1207127 Roff D A 1997 Evolutionary Quantitative Genetics New York Chapman amp Hall Shaw F H Shaw R G Wilkinson G S amp Turelli M 1995 Changes in genetic variances and covariances G whiz Evolution 49 126071267 Slatkin M 1987 a Heritable variation and heterozygosity under a balance between mutation and stabilizing selection Genetical Research 50 532 Slatkin M 1987 17 Quantitative genetics of heterochrony Evolution 41 7997811 Turelli M 1984 Heritable genetic variation via muta tionselection balance Lerch s zeta meets the abdominal bristle Theoretical Population Biology 25 1387193 Turelli M amp Barton N H 1990 Dynamics of polygenic characters under selection Theoretical Population Biology 38 1757 van Tienderen P H amp de Jong G 1994 A general model of the relation between phenotypic selection and genetic response Journal of Evolutionary Biology 7 1712 Via S amp Lande R 1985 Genotypenvironment in teraction and the evolution of phenotypic plasticity Evolution 39 5057522 Wagner G P 1988 The in uence of variation and of developmental constraints on the rate of multivariate phenotypic evolution Journal of Evolutionary Biology 1 45766 Wagner G P 1989 Multivariate mutationiselection balance with constrained pleiotropic effects Genetics 122 2237234 Weis A E 1996 Variable selection on Eurosta s gall size 111 Can an evolutionary response to selection be detected Journal ovaolutionary Biology 9 62340 Zeng ZB 1988 Longterm correlated response inter po ulation covariation and interspeci c allometry Evolution 42 3637374 Zeng ZB amp Hill W G 1986 The selection limit due to the con ict between truncation and stabilizing selection with mutation Genetics 114 131371328

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