Constrained evolution of a quantitative character by pleiotropic mutation

ARTICLE IN PRESS

Theoretical Population Biology 68 (2005) 243–251 www.elsevier.com/locate/ytpbi

Constrained evolution of a quantitative character by pleiotropic mutation Y. Tanaka Faculty of Economics, Chuo University, Higashinakano 742-1, Hachioji, Tokyo 192-0393, Japan Received 15 November 2004 Available online 27 July 2005

Abstract The long-term response to directional selection and its selection limit are derived for a quantitative character that is controlled by pleiotropic mutations with direct deleterious effect on fitness. Directional selection is assumed to be weaker than the selection acting directly on mutations via deleterious effects (purging selection), which renders all mutations to eventual elimination. The analysis embedding this restrictive assumption indicates that the evolutionary response of the character starting from an equilibrium state, in which mutation and purging selection balance but no directional selection is operating, decreases monotonically with time at an exponential rate. And the fading rate of responses is mostly determined by the direct deleterious effect. Contrary to the expectation by the standard selection limit theory based on fixation of extant genetic variation, the present model predicts that the selection limit depends on the intensity of directional selection, the limit being proportional to the ratio of the directional selection intensity to the direct deleterious effect. A slightly larger genetic variance is maintained at the selection limit than would be without directional selection. r 2005 Elsevier Inc. All rights reserved. Keywords: Deleterious mutation; Directional selection; Long-term responses; Pleiotropy; Quantitative traits

1. Introduction If deleterious mutations have additive pleiotropic effect on a quantitative trait, and if such pleiotropic effects explain the most genetic variance of the trait, the genetic response to directional selection is greatly affected by the direct deleterious effect (Keightley and Hill, 1990). Genes that are selected for by the directional phenotypic selection will be lost due to the deleterious effect on fitness and replaced by new mutations. Some recent theoretical studies have investigated the effect of pleiotropic mutations on the maintenance of genetic variance of a quantitative trait (Barton, 1990; Kondrashov and Turelli, 1992; Tanaka, 1996; Zhang and Hill, 2002; Zhang et al., 2002), and have predicted the equilibrium genetic variance to be considerably smaller with pleiotropic mutations than without pleioFax: +81 426 74 3444.

E-mail address: [email protected]. 0040-5809/$ - see front matter r 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2005.06.005

tropic effect of mutations (Turelli, 1988; Barton, 1990; Keightley and Hill, 1990; Kondrashov and Turelli, 1992; Tanaka, 1996). Pleiotropic mutations influence the short-term evolutionary rate of change in fitness as well. Crow and Nagylaki (1976) presented a modified version of the Fundamental Theorem, pointing out that the evolutionary rate of change in fitness depends on the covariance between deleterious effects and phenotypic effects of mutations. They concluded that the rate of change in fitness was not exactly equal to additive genetic variance of fitness if the additive variance of fitness was partly attributed by the covariance between fitness and phenotype. An important question yet explored is how and what extent to which pleiotropic mutations affect directional selection (but see Barton and Turelli, 1987; Keightley and Hill, 1989; Tanaka, 1998). Some long-term selection experiments using mainly Drosophila bristle traits observed that the genetic responses partly reversed when the directional selection

ARTICLE IN PRESS 244

Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251

was relaxed after a great deal of genetic responses (Robertson, 1955; Roberts, 1966; Dudley, 1977; Yoo, 1980a), indicating that some of the genes responsible for the long-term responses had adverse fitness effect. In addition, segregation of rare recessive pleiotropic mutations explained why the genetic variance increased during the long-term responses (Yoo, 1980b). More recent long-term selection experiments on isogenic lines and experimental subline divergences by accumulation of spontaneous mutations have revealed widespread pleiotropy of new mutations among quantitative traits and fitness components (Santiago et al., 1992; Lopez and Lopez-Fanjul, 1993a,b; Houle et al., 1994; review in Hill and Caballero (1992) and Garcia-Dorado et al. (1999)). Numerous theoretical studies have formulated shortterm or long-term dynamics of quantitative characters by directional selection (Barton and Turelli, 1987; Turelli and Barton, 1990; Burger, 1993). Nonetheless, most studies do not examine the effect of pleiotropic gene action on the dynamics of quantitative characters. The present analysis examines impacts of pleiotropic mutations on responses to directional selection and the selection limit. Mutations are subject to two kinds of selection, the selection against direct deleterious effect (purging selection) and the selection via additive effects on the quantitative character (phenotypic directional selection). On the contrary to the traditional explanation of the selection limit that the alleles that are favored by directional selection are fixed at all loci, damping genetic variation of the trait (Falconer and Mackay, 1996), the explanation by the pleiotropic model is that all mutations including those favored by directional selection are eventually lost by the direct deleterious effect. One of the important issues yet explored is the evolutionary rate or the asymptotic cumulative response (ACR) (selection limit). Analytical approximations rely on restrictive assumptions, a large population size and weak directional selection. As a consequence, the major scope of the present analysis is the evolutionary changes in nature where the above assumptions are more likely than animal or plant breeding programs.

2. Models 2.1. Cohort-of-mutations model Assume that there are n diploid loci that contribute to a quantitative character that is not itself a fitness component, and that major allelic effects on the trait are additive within and among loci. The total genotypic value G is the simple summation Pof the genotypic values of all contributing loci, G ¼ ni¼1 xi , where xi is the genotypic effect of the ith locus. For analytical tractability, all parameters are assumed to be identical

among loci. Linkage disequilibrium and dominance are also disregarded. The analysis focuses on heterozygous effect of mutations because mutations are assumed rare. Heterozygous effects of new mutations on the trait, denoted as a, follow a normal distribution with 0 mean and s2a variance, which is denoted as g(a). Regardless of phenotypic effects, all mutations have an identical deleterious effect by pleiotropy. Due to the deleterious effect the relative fitness (or survivorship) of mutations (in heterozygote) in comparison to the wild-type allele (in homozygote), which is free from deleterious effect, is less than unity, and is denoted l (0olo1). Mutations that appeared in a certain generation are lost at a rate of 1l every generation. At a hypothetical initial generation all loci are occupied by a wild-type allele that has a null (0) phenotypic and deleterious effect. Assuming an infinite population, genotypic values at a locus after mutation of one generation have the following distribution: 2mgðxÞ þ ð1  2mÞdðxÞ, where m is the mutation rate per locus per gamete per generation, and dðxÞ is the R 1 Dirac delta function (dð0Þ ! 1 otherwise 0, and 1 dðxÞ dx ¼ 1). All mutations, regardless of phenotypic effects, are deleterious on fitness. In addition to the direct effect on fitness, mutations have pleiotropic effects on a quantitative character that is subject to true directional selection. The former effect is referred to as ‘‘direct deleterious effect’’ and the latter as ‘‘phenotypic effect’’. Selective force acting on mutations is generated in two ways, i.e. selection purging deleterious mutations and directional selection picking up mutations with favorable phenotypic effects. It is assumed that for all mutations the reduction of fitness by the direct effect is larger than the fitness gain by the phenotypic directional selection, rendering all mutations to have lower fitness than the wild-type allele. Owing to the superior fitness, the wild-type allele dominates in frequency at all loci. At the same time, this assumption inevitably requires the range of phenotypic effects to be limited, the upper and the lower limit being, respectively, denoted a+limit and alimit. The above scheme of pleiotropy causes apparent stabilizing selection on the trait because individuals containing more mutations have lower fitness and higher genetic variance of the trait (Barton, 1990; Kondrashov and Turelli, 1992). The apparent selection and the genetic variance are both influenced by variable deleterious effects and their correlation with phenotypic effects (Caballero and Keightley, 1994; Zhang et al., 2002). The present analysis, however, assumes an identical deleterious effect among mutations, and disregards any association between deleterious effects and phenotypic effects, in order to derive analytical solutions in an interpretable form.

ARTICLE IN PRESS Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251

245

At mutation-selection balance, all mutations (genes identical by descent originated from the same mutation) are classified according to their age (the number of generations after their emergence as a mutation by a mutational event). Hereafter, a group of mutations that appeared at the same generation (i.e. mutations of the same age) is referred to as a cohort of mutations, and the cohort that passed t generations after its origin by mutation is called as ‘‘tth cohort’’ (Tanaka, 1996). A cohort decreases in frequency by the purging selection. ‘‘Propagation rate’’ of the tth cohort, R(t), is defined as the frequency of the tth cohort relative to that when they are new mutations (0th cohort), and is less than unity by assumption in the present analysis. The propagation rate is mostly determined by the direct deleterious effect and modified by the phenotypic selection, which changes the distribution of phenotypic effects within cohorts: Q RðtÞ ¼ lt ty wðyÞ, where wðyÞ is the mean fitness of the ¯ ¯ yth cohort due to phenotypic selection relative to the wild-type allele. Denoting the relative fitness of mutations with phenotypic effect x to that with null effect (x ¼ 0) as w(x), the probability of mutations with phenotypic effect x to survive t generations is ltw(x)t. And the frequency of mutations in the tth cohort with phenotypic effect x is 2mg(x)ltw(x)t, which is equivalent to

N½xjm; v refers to a normal distribution of mean m and variance v. At a stationary state, if it exists, each locus is composed of mutations of all cohorts and the wild-type allele. Denoting the total gene frequency of all mutations as P, and the distribution of phenotypic effects among all (cohorts of) mutations as f(x), the genotypic distribution at a locus is

2mRðtÞgc ðx; tÞ,

gation rate specified as lt expð12s2a b2 t2 Þ is not integrative from 0 to infinity. This is, however, a mathematical artifact of R(x) diverging with time with unbounded phenotypic effects. Taking account that phenotypic effects of mutations are limited and that even the most advantageous mutation under the directional selection loses its frequency by the direct effect (1  l4aþlimit b40), the propagation rate R(t) is approximated by Z aþlimit RðtÞ ¼ gðxÞðlebx Þt dx ffi est ð1 þ 12s2a b2 t2 Þ.

(1)

where gc(x, t) is the distribution of phenotypic effect among mutations in the tth cohort. Recurrent mutation is assumed negligible since any cohort of mutations is not as frequent as the wild-type allele (house-of-cards approximation, c.f. Latter, 1960; Bulmer, 1973; Turelli, 1984). The directional selection acting on the trait is described by an exponential fitness function, W ðZÞ ¼ ebZ , where Z is the trait (phenotypic) value and b is the selection gradient. Since phenotypic values are sums of genotypicP values and environmental deviations, E, i.e. Z ¼ ni¼1 xi þ E, the marginal (or expected) fitness as regards the environmental deviation can each locus, E½W ðZÞ ¼ Qn be decomposed into bxi wðx Þ and wðx Þ ¼ e . i i i¼1 Mutations with the upper limit of mutational effect, a+limit, gain fitness by ba+limit while lose their fitness by s due to the direct deleterious effect. Therefore, it is necessary that ba+limit is smaller than s in order that most mutations have lower fitness than the wild-type allele and the above assumption is met. Provided that the relative fitness of mutations with phenotypic effect x is ebx, the propagation rate of the tth R1 cohort is calculated from RðtÞ ¼ 1 gðxÞðlebx Þt dx as RðtÞ ¼ lt expð12s2a b2 t2 Þ. From gc ðx; tÞ ¼ gðxÞðlebx Þt =RðtÞ the phenotypic distribution within the cohorts is specified as gc ðx; tÞ ¼ N½xjs2a bt; s2a , where the notation

Pf ðxÞ þ ð1  PÞdðxÞ. Because all moments of d(x) are 0, the moments of the genotypic distribution is equivalent to the moments of Pf(x). With the definition of R(x) and gc(x,t), we get Pf ðxÞ ¼ 2m lim

t!1

Z

t X

RðyÞgc ðx; yÞ;

y¼0

1

RðtÞgc ðx; tÞ dt:

’ 2m

ð2Þ

0

All genotypic moments (Mk: kth moment) including the mean and the genetic variance can be calculated R1 from the moments of (2), i.e. M k ¼ 2m 0 RðtÞmk ðtÞ dt, where mk(t) is the kth moment of the tth cohort of mutations, which is derived from the characteristic function of the tth cohort and its derivatives: ft ðyÞ ¼ R 1 yx @k 1 e gc ðx; tÞ dx and mk ðtÞ ¼ @k y ft ðyÞjy¼0 . The propa-

alimit

The equilibrium genetic variance without directional selection is derived from nM2 with b ¼ 0 as 2

Usa V~ g ¼ s

(3)

where U ¼ 2nm, n is the number of loci, and s is the selection coefficient via direct deleterious effects, s ¼ lnl (Tanaka, 1996). 2.2. Asymptotic cumulative response If the directional selection sustains indefinitely, the mean trait value reaches an equilibrium, which is determined by the balance of the directional selection, which increases the mean genotype, versus new mutations and the purging selection, both of which restore the mean genotype to the original value. The difference

ARTICLE IN PRESS 246

Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251

in the mean phenotypes between the two extremes, respectively, achieved without any directional selection and with continuous directional selection is referred to as ‘‘ACR’’. The ACR is interpreted as the total response or the selection limit of a population to long-term directional selection, starting from an equilibrium state which had never experienced directional selection, and in the present model, is equivalent to the asymptotic mean phenotype under the continuous directional selection since the mean phenotype is assumed 0 if b ¼ 0. The asymptotic mean phenotype under continuous directional selection, z¯ L , is calculated from the summation of the mean (the first moment, m1 ¼ bs2a t) of all cohorts weighted with the propagation rate, R(t): (   ) b bsa 2 2 z¯ L ¼ 2 Usa 1 þ 3 s s (   ) b ~ bsa 2 ¼ Vg 1 þ 3 . ð4Þ s s Thus, ACR is proportional to the directional selection gradient and the mutational input of genetic variance, and inversely proportional to the squared net selection coefficient against mutations. If the long-term responses are attributed to fixation of favorable mutations at all loci, and recurrent mutations are negligible, the selection limit by fixation, z¯ F , would be 2nsa (c.f. Falconer and Mackay, 1996). Using the above expression, the ratio of ACR with pleiotropy to the selection limit by fixation is (  ) z¯ L mbsa bsa 2 ¼ 2 1þ3 . z¯ F s s

pleiotropic mutations and the purging selection are balanced to generate an equilibrium genetic variance, while directional selection does not operate. After the onset of the directional selection the mean genotype is composed of two parts, i.e. the pre-existing mutations, which existed before the onset of directional selection, and new mutations, which arose after the onset of directional selection. The relative importance of the new mutations increases with time. The pre-existing mutations suffer selection via the direct effect from the mutational event (the starting point of a cohort) to the present and the directional selection after the onset of directional selection. The new mutations are subject to both selections from the mutational event to the present. Denoting the number of generations of directional selection as t, the number of generations from the start (by mutation) of a cohort of pre-existing mutations to the onset of directional selection as t, and the number of generations after the start of new mutations to the present as t (Fig. 1), the distribution of phenotypic effects within a cohort of pre-existing mutations, gc ðx; tÞ, is gc ðx; tÞ ¼ N½xja2 bt; a2 , and the distribution of phenotypic effects within a cohort of new mutations, gnc ðx; tn Þ, is gnc ðx; tn Þ ¼ N½xja2 btn ; a2 . The mean trait value at the tth generation after the onset of directional selection is derived from a calculation parallel to (4) as

Z z¯ ðtÞ ¼ 2nm

Z

1

Rðt þ t; tÞm1 ðtÞ dt þ 0

t n

n

Rðt Þm1 ðt Þ dt

n



,

0

(6a) where R(t1,t2) is the propagation rate of a cohort that has been subject to the purging selection for t1 generations and the directional selection for t2 generations, Rðt þ t; tÞ ¼ ltþt ð1 þ 12a2 b2 t2 Þ. The mean trait value at the tth generation is " (   ! bs2a bsa 2 z¯ ðtÞ ¼ 2nm 2 1þ3 ð1  est Þ s s    3 2 2 2 2 st  b sa t þ t e . ð6bÞ 2 s

Provided that m=s51 and bss a o1, much smaller total responses are predicted with pleiotropic mutations than with fixation of extant mutations. The genetic variance at the equilibrium, V~ gL , is approximately derived from V~ gL ¼ nM 2  z¯ 2L and m2 ðtÞ ¼ s2a þ ðbs2a tÞ2 as (   ) bsa 2 ~ ~ V gL ’ V g 1 þ 3 . (5) s

The right-hand side converges to ACR if t is infinitely large ð lim z¯ ðtÞ ! z¯ L Þ, and is simplified as follows,

The genetic variance somewhat increases with directional selection since bsa must be considerably smaller than s by assumption (baþlimit os).

Onset of directional selection

t!1

time t

2.3. Long-term responses Here are formulated long-term responses subsequent to introduction of continuous directional selection upon an equilibrium population that had not experienced any directional selection. The initial state of the population is assumed to be a stationary equilibrium, at which

τ

present

t*

Pre-existing mutation New mutation

Fig. 1. Schematic drawing on generation numbers for pre-existing and new mutations.

ARTICLE IN PRESS Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251

provided that (bsa =sp0:1), bs2 z¯ ðtÞ ffi 2nm 2a s

the (

directional

selection

  ) bsa 2 1þ3 ð1  est Þ s

is

weak

(6c)

From Eqs. (4) and (6c), the cumulative response relative to ACR, which is the shape of evolutionary trajectory, is approximately z¯ ðtÞ ffi 1  est . z¯ L

(7)

The directional selection gradient and the genetic variance before the onset of directional selection cancel and do not influence the shape of evolutionary trajectory, which depends only on the selection coefficient against direct deleterious effects. The half-life of long-term response, Thalf, in which the cumulative response reaches the midpoint to the total response, is T half ¼ 1s ln 12. Thus, the half-life is inversely proportional to the selection coefficient, but is independent of the strength of directional selection and the genetic variance. If s ¼ 0:02 (Simmons and Crow, 1977; Crow and Simmons, 1983), the half-life is T half ’ 35, which is compatible with typical values of half-life of responses in long-term selection experiments (Yoo, 1980a). The selection limit scaled by the first response is approximately the inverse of the selection coefficient on L direct deleterious effects: z¯z¯ð1Þ ¼ 1e1 s ’ 1s . Since the average persistence time ðT¯ p Þ of mutations in an infinite population is T¯ p ¼ 1=s (Tanaka, 1996), the selection limit scaled by the first response is approximately equivalent to the mean persistence time of mutations, z¯ L ¯ z¯ ð1Þ ’ T p . The persistence time of typical deleterious mutations ranges from 30 to 50 generations (T¯ p ¼ 3050, Crow, 1979), suggesting that pleiotropic effects of typical deleterious mutations may explain the observed amount of total responses, which is usually less than 100 times a short-term response per generation. st Putting (5) into (6c), we get z¯ ðtÞ ffi bV~ gL 1es , from the differential of which the evolutionary response per generation is obtained, D¯zðtÞ ffi bV~ gL est .

(8)

The rate of evolution decreases monotonically with time. The fading rate of evolutionary changes is determined by the selection coefficient against deleterious effects. Heavy genetic load (large s) shortens life span of all mutations including those picked up by the directional selection. The accelerated turnover rate of mutations reduces the genetic response to directional selection.

247

3. Simulations Numerical simulations were practiced to examine the robustness of the analytical approximations. The critical assumptions are the changes in the distribution of phenotypic effects between generations within a cohort of mutations described by gc(x, t), and the total cohort frequency decreasing by 1R(t). The total range of genotypic space was defined from 0 to 80, within which 81 discrete classes of genotypes were equally decomposed (each has the same interval of 1). The value (heterozygous effect of mutations) of the kth genotype is assumed the integer k. The simulation assumed, for simplicity, only one locus, and traced the population densities of the respective 81 genotypic classes. The total population density including the non-mutant wild type genotype was kept constant. Effects of new mutations were subject to a normal distribution with mean 40 and standard deviation sa. The number of new mutations to a specific genotypic class was deterministically set by the mutation rate, the population density and the frequency distribution of mutations. The per-generation flux of new mutations R kþ0:5 into the kth class is 2mN T k0:5 N½xj40; s2a dx, where NT is the total population density. After new mutations were allocated into each genotypic class, selection was operated by multiplying the frequency of each genotypic class by the relative fitness of mutations due to the direct and the phenotypic effects, lw(k). This process was iterated for 500 generations starting from the 2 initial genotypic distribution set as 2m s N½xj40; sa þ ð1  2m s Þ dðx  40Þ, where dðx  40Þ ¼ 1 if x ¼ 40 otherwise 0. Parameter values for n, m and sa were globally set as 100, 104 and 15, while l and b varied for a wide range so that impacts of l and b on ACR and V~ gL can be examined. The relative magnitude of the directional selection to the direct deleterious effect determined whether or not the simulation and the analytical approximation coincided. Fig. 2 depicts two typical results, one with the assumption met and the other violated. As long as directional selection is enough weak, the cohort-ofmutations model approximates well the simulated genotypic distribution of mutations, while it fails to draw the simulated distribution when the directional selection is strong. Most cases of obvious discrepancies entailed an inflated peak of the simulated genotypic distribution shifted in the direction of the favored genotype from the peak by the analytical approximation (Fig. 2). The inflation of the favored genotype is the consequence of that the fitness gain by the directional selection compensates the reduction in fitness due to the direct deleterious effect. Proliferation of the initially very rare mutations that had extremely favorable effects on the trait violates the heuristic assumption that the

ARTICLE IN PRESS Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251

total frequency of a cohort decreases at a constant rate mostly determined by the direct deleterious effect. The selection coefficient, s, must be larger than ba+lim (aþlim ¼ 20 in this simulation) in order to meet the assumption. Hereafter, all simulations were undertaken with this restriction.

0.03

400

Simulation (0.98) Analytic (0.98) Simulation (0.97) Analytic (0.97) Simulation (0.96) Analytic (0.96) Simulation (0.95) Analytic (0.95)

350 Genetic Variance

248

300 250 200 150 100

Simulation Frequency

0.0225

50

Analytic

0 0

0.015 0.0075

0

10

20

30 40 50 60 Phenotypic effect

(a)

70

80

0.1 Simulation Analytic

Frequency

0.075 0.05 0.025

0 (b)

10

20

30 40 50 60 Phenotypic effect

70

80

Fig. 2. Simulated and analytic frequency distribution of mutations within a cohort. The curve represents the analytic distribution and the dots represents the simulated distribution. Results are two typical cases where (a) the basic assumption, baþlim os, is met, and (b) it is not met. Parameter values were set l ¼ 0:98, and b ¼ 0:00025 for (a) or b ¼ 0:001 for (b).

50

Simulation (0.98) Analytic (0.98) Simulation (0.97) Analytic (0.97) Simulation (0.96) Analytic (0.96) Simulation (0.95) Analytic (0.95)

49

Selection Lim

48 47 46 45

0.0005 Selection Intensity

0.001

Fig. 4. Genetic variances with various selection coefficient (s) for direct effect and selection intensity (b, gradient) on the trait. Lines represent the simulation results while the broken lines represent the analytical results (Eq. (5) in the text). Parameter values followed those in Fig. 3. See text for explanation.

For ACR and V~ gL the simulations were in good accord with the analytical predictions. The ACR is quasi-linearly related to the directional selection intensity. Beyond the relevant range of b, ACRs in simulations departed from the analytical predictions. Essentially the same trend was observed for the equilibrium genetic variance, and the analytical predictions were constantly overestimated to the simulation results. The series of lines in Figs. 3 and 4 indicates that the directional selection and the direct effect interact in affecting ACR and V~ gL . The quasi-linear relationship between b and ACR is largely dependent of the selection coefficient against direct deleterious effects, which substantially reduce the cumulative responses if the directional selection is held constant. The limited range of b and s, in which the analytic and the numerical results coincided, implies the limited applicability of the present model. In any case, ACR qffiffiffiffiffiffi larger than one genetic standard deviation, V~ g , of the initial equilibrium population without directional selection is hardly explained unless the number of loci is very large.

44 43

4. Discussion

42 41 40 0

0.0005 Selection Intensity

0.001

Fig. 3. Selection limits with various selection coefficient (s) for direct effect and selection intensity (b, gradient) on the trait. Lines represent the simulation results while the broken lines represent the analytical results (Eq. (4) in the text). Parameter values were set n ¼ 100, m ¼ 104 , and sa ¼ 15. See text for explanation.

The predictions of the present pleiotropic model are qualitatively different from what are predicted by fixation of beneficial mutations. The non-pleiotropic model without recurrent mutations predicts that the selection limit does not depend on the directional selection intensity if population is infinite, while the half-life of long-term responses does (Falconer and Mackay, 1996). The differences in conclusions between

ARTICLE IN PRESS Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251

the two types of models may be the consequence of the different genetic regimes that cause selection limits and long-term responses. The pleiotropic model assumes the limit to be balanced by antagonistic selection pressures, while the non-pleiotropic model assumes that the fixation of directionally selected mutations brings about sustainable and constant long-term responses with recurrent mutations or a selection limit without recurrent mutations. Extensive theoretical works have explored on longterm responses by fixation of mutations in finite populations, and have found that evolutionary rates and selection limits increase with directional selection intensities because positive selection increases the fixation probability (Hill, 1982a,b; Hill and Rashbash, 1986). In a large population, however, which is common in nature, the association between directional selection intensities and selection limits vanishes (Zeng and Cockerman, 1990). On the contrary, the genetic regime explored in this study is likely to make the linear association between directional selection intensities and selection limits (ACRs) independent of population sizes. The pleiotropic model may be verified if it is frequently observed that ACRs linearly depend on directional selection intensities. The pleiotropic model is unlikely to explain all evolutionary processes in nature as it does not allow permanent evolutionary changes, which are apparently the basis of enormous extant biodiversity. The two types of models, the non-pleiotropic and the pleiotropic models, are not exclusive to each other, and the pleiotropic model may explain a partial process of the total evolutionary change. More explicitly, if each locus has a specific mutation regime (distribution of mutational effect on fitness and trait, mutation rate and the degree of dominance), all loci may be classified into two parts, one meeting baþlim 4s and the other baþlim os. The real evolutionary process may be a composite of the two parts, each described by the non-pleiotropic or the pleiotropic model. Some empirical evidences suggest the importance of pleiotropic effects of loci responsible to phenotypic evolution. The discrepancy of the evolutionary rate expected by quantitative genetic models and observed evolutionary rates is one of them. Neutral expectations of the divergence rate of quantitative characters, based on the mutational variances mainly observed in Drosophila and Daphnia (Lynch, 1988a; Santiago et al., 1992), often exceed by orders of magnitudes the observed phenotypic divergences although many of the tested characters are apparently driven to adaptive phenotypic shift (Lande, 1977; Turelli et al., 1988; Martins, 1994; Schluter, 2000). The most likely explanation for the long-term stability of quantitative characters is the reduced evolutionary rate with pleiotropic mutations (Turelli et al., 1988; Schluter, 2000).

249

Another line of evidence comes from artificial selection experiments: observed half-lives of responses and selection limits are compatible with the present predictions (Clayton and Robertson, 1957; Yoo, 1980a; review in Falconer and Mackay, 1996). In addition, it is often observed that relaxation of artificial selection brought about incomplete reversion of genetic responses, which is explained by segregation of pleiotropic mutations at the selection limit (Yoo, 1980b). Such genetic regimes may cause apparent directional selection, with which genetic responses do not occur whereas genetic variances are maintained (Santos, 1996; Kruuk et al., 2002). Besides the relative strength of the directional selection and the direct deleterious effect, the present model relies on some restrictive assumptions, i.e. uniformity of the direct deleterious effect, no dominance and epistasis, linkage equilibrium or no linkage, the normal mutational effects on the trait, and an infinite population size. The uniform deleterious effect of mutations regardless of the variable (Gaussian) phenotypic effects may be the most serious oversimplification. Recent experimental studies have detected large kurtosis among genetic values in quantitative characters including fitness components contributed by accumulated mutations (Mackay et al., 1992; Santiago et al., 1992; Lopez and Lopez-Fanjul, 1993b; c.f. Hill and Caballero, 1992; Mackay et al., 1992; Keightley, 1994; Garcia-Dorado et al., 1999; Lynch et al., 1999). The genetic kurtosis may be explained by a compound effect of loci with nonequivalent mutation regimes, i.e. loci comprising either ‘‘minor’’ or ‘‘major’’ mutations, or nearly identical loci comprising both the minor mutations and a few major mutations (Lopez and Lopez-Fanjul, 1993b). Mutant effects on viability in D. melanogaster were well fit by a gamma function, and exhibited a significant kurtosis (Keightley, 1994). Nonetheless, a joint gamma function, which is composed of two factors each explaining major or minor mutations, produced a much smaller estimate of kurtosis, implying there were two discrete classes of mutations (Keightley, 1994). The non-equivalent loci each having normal mutational effects reduces the theoretical prediction of the genetic variance by the mutation-selection balance (Burger and Lande, 1994; Burger, 1998; Keightley and Hill, 1989; Welch and Waxman, 2002). In contrast, variability of mutant effects on fitness increases the equilibrium genetic variance predicted by the pleiotropic model. Having analyzed the joint effect of stabilizing selection and variable direct deleterious effects on the equilibrium genetic variance and the apparent stabilizing selection, Zhang and Hill (2002) indicated that the leptokurtic distribution of mutant effects on fitness greatly increased the predicted equilibrium genetic variance so that the pleiotropic model was compatible

ARTICLE IN PRESS 250

Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251

with the frequently observed fact that large genetic variances can be maintained by mutation and relatively strong stabilizing selection (Barton and Keightley, 2002; Lynch, 1988a; but see Kingsolver et al., 2001). It was also suggested that the association between direct deleterious effects and phenotypic effects did not noticeably influence equilibrium genetic variances (Zhang and Hill, 2002). Thus, the variability is more important than the association. The mutations that have very small direct deleterious effects and beneficial phenotypic effects may increase in frequency, bringing about large genetic variances and perpetual responses to the directional selection, which violates the assumption of the present model. Bearing such restrictions in mind, the present model may have some implications to evolutionary process in nature. Provided that a large fraction of mutations that are responsible to phenotypic evolution is loaded with pleiotropic direct deleterious effects that exceed fitness gain by the directional selection, the standard quantitative genetic model may give overestimates of true evolutionary rates and evolutionary endpoints. The present study suggests that the long-term evolution is considerably limited by pleiotropic effects. Furthermore, the long-term responses are unstable in that the phenotype recovers its original state whenever the directional selection is relaxed. These properties of evolution by pleiotropic mutations may be associated with two topics in evolution, the long-term stability of phenotypes and the fitness optimization. One of the major features of phenotypic evolution is characterized by its long-term stasis interspersed with rapid phenotypic changes or adaptive radiation (Simpson, 1944; Schluter, 2000). The most plausible explanation is continuous stabilizing selection (Charlesworth et al., 1982; Maynard Smith et al., 1985). But the relevance of stabilizing selection may be limited by selection load, the cumulative effect of which may overcome the tolerable limit (Barton, 1990), and by the random genetic drift (Lande, 1979; Lynch, 1988b). The pleiotropic model can explain the long-term stasis unless the directional selection is strong and sustainable. As long as additive genetic variance in fitness is supplied by mutation, adaptive characters and fitness are optimized at the evolutionary equilibrium (Fisher, 1930; Kimura, 1958; Charnov, 1989; Charlesworth, 1990). Nonetheless, if mutations coding for the character are pleiotropically constrained to load direct adverse effects, the phenotype may not reach its fitness optimum, keeping itself at a suboptimal state.

Acknowledgments This study was supported by Chuo University Grant for Special Research to Y.T.

References Barton, N.H., 1990. Pleiotropic models of quantitative variation. Genetics 124, 773–782. Barton, N.H., Keightley, P.D., 2002. The nature of quantitative genetic variation. Nat. Rev. Genet. 3, 11–21. Barton, N.H., Turelli, M., 1987. Adaptive landscapes, genetic distance, and the evolution of quantitative characters. Genet. Res. 49, 157–173. Bulmer, M.G., 1973. The maintenance of the genetic variability of quantitative characters by heterozygote advantage. Genet. Res. 22, 9–12. Burger, R., 1993. Prediction of the dynamics of a polygenic character under directional selection. J. Theor. Biol. 162, 487–513. Burger, R., 1998. Mathematical properties of mutation-selection balance models. Genetica 102/103, 279–298. Burger, R., Lande, R., 1994. On the distribution of the mean and variance of a quantitative trait under mutation-selection-drift balance. Genetics 138, 901–912. Caballero, A., Keightley, P., 1994. A pleiotropic nonadditive model of variation in quantitative traits. Genetics 138, 883–900. Charlesworth, B., 1990. Optimization models, quantitative genetics, and mutation. Evolution 44, 520–538. Charlesworth, B., Lande, R., Slatkin, M., 1982. A neo-Darwinian commentary on macroevolution. Evolution 36, 474–498. Charnov, E., 1989. Phenotypic evolution under Fisher’s Fundamental Theorem of Natural Selection. Heredity 62, 113–116. Clayton, G.A., Robertson, A., 1957. An experimental check on quantitative genetical theory. II. The long-term effects of selection. J. Genet. 55, 152–170. Crow, J.F., 1979. Minor viability mutations in Drosophila. Genetics 92 (Suppl.), s165–s172. Crow, J.F., Nagylaki, T., 1976. The rate of change of a character correlated with fitness. Am. Nat. 110, 207–213. Crow, J.F., Simmons, M.J., 1983. The mutation load in Drosophila. In: Ashburner, M., Carson, H.L., Thompson, J.N. (Eds.), The Genetics and Biology of Drosophila, vol. 3c. Academic Press, New York, pp. 2–35. Dudley, J.W., 1977. 76 generations of selection for oil and protein percentage in maize. In: Pollak, E., Kempthorne, O., Bailey, T.B. (Eds.), Proceedings of the International Conference on Quantitative Genetics. Iowa State University, Ames, Iowa, pp. 459–473. Falconer, D.S., Mackay, T.F.C., 1996. Introduction to Quantitative Genetics, fourth ed. Prentice-Hall, Edinburgh. Fisher, R.A., 1930. The Genetical Theory of Natural Selection. Claredon Press, Oxford. Garcia-Dorado, A., Lopez-Fanjul, C., Caballero, A., 1999. Properties of spontaneous mutations affecting quantitative traits. Genet. Res. 74, 341–350. Hill, W.G., 1982a. Predictions of response to artificial selection from new mutations. Genet. Res. Camb. 40, 255–278. Hill, W.G., 1982b. Rates of change in quantitative traits from fixation of new mutations. Proc. Natl. Acad. Sci. USA 79, 142–145. Hill, W.G., Caballero, A., 1992. Artificial selection experiments. Ann. Rev. Ecol. Syst. 23, 287–310. Hill, W.G., Rashbash, J., 1986. Models of long term artificial selection in finite population. Genet. Res. 48, 41–50. Houle, D., Hughes, K.A., Hoffmaster, D.K., Ihara, J., Assimacopoulos, S., Canada, D., Charlesworth, B., 1994. The effects of spontaneous mutation on quantitative traits. I. Variances and covariances of life history traits. Genetics 138, 773–785. Keightley, P.D., 1994. The distribution of mutation effects on viability in Drosophila melanogaster. Genetics 138, 1315–1322. Keightley, P.D., Hill, W.G., 1989. Quantitative genetic variability maintained by mutation-stabilizing selection balance-sampling

ARTICLE IN PRESS Y. Tanaka / Theoretical Population Biology 68 (2005) 243–251 variation and response to subsequent directional selection. Genet. Res. 54, 45–57. Keightley, P.D., Hill, W.G., 1990. Variation maintained in quantitative traits with mutation-selection balance: pleiotropic side-effects on fitness traits. Proc. R. Soc. London Ser. B 242, 95–100. Kimura, M., 1958. On the change of population fitness by natural selection. Heredity 12, 145–167. Kingsolver, J.G., Hoekstra, H.E., Hoekstra, J.M., Berrigan, D., Vignieri, S.N., et al., 2001. The strength of phenotypic selection in natural populations. Am. Nat. 157, 245–261. Kondrashov, A.S., Turelli, M., 1992. Deleterious mutations, apparent stabilizing selection and the maintenance of quantitative variation. Genetics 132, 603–618. Kruuk, L.E.B., Slate, J., Pemberton, J.M., Brotherstone, S., Guinness, F.E., Clutton-Brock, T., 2002. Antler size in red deer: heritability and selection, but no evolution. Evolution 56, 1683–1695. Lande, R., 1977. Statistical tests for natural selection on quantitative characters. Evolution 31, 442–444. Lande, R., 1979. Quantitative genetic analysis of multivariate evolution applied to brain:body allometry. Evolution 33, 402–416. Latter, B.D.H., 1960. Natural selection for an intermediate optimum. Aust. J. Biol. Sci. 13, 30–35. Lopez, M.A., Lopez-Fanjul, C., 1993a. Spontaneous mutation for a quantitative trait in Drosophila melanogaster. I. Response to artificial selection. Genet. Res. 61, 107–116. Lopez, M.A., Lopez-Fanjul, C., 1993b. Spontaneous mutation for a quantitative trait in Drosophila melanogaster. II. Distribution of mutant effects on the trait and fitness. Genet. Res. 61, 117–126. Lynch, M., 1988a. The rate of polygenic mutation. Genet. Res. 51, 137–148. Lynch, M., 1988b. The divergence of neutral quantitative characters among partially isolated populations. Evolution 42, 455–466. Lynch, M., Blanchard, J., Houle, D., Kibota, T., Schultz, S., et al., 1999. Perspective: spontaneous deleterious mutation. Evolution 53, 645–663. Mackay, T.F.C., Lyman, R.F., Jackson, M.S., 1992. Effects of P element insertions on quantitative traits in Drosophila melanogaster. Genetics 130, 315–332. Martins, E.P., 1994. Estimating the rate of phenotypic evolution from comparative data. Am. Nat. 144, 193–209. Maynard Smith, J., Burian, R., Kauffman, S., Alberch, P., Campbell, J., Goodwin, B., Lande, R., Raup, D., Wolpert, L., 1985. Developmental constraint and evolution. Quart. Rev. Biol. 60, 265–287. Roberts, R.C., 1966. The limits to artificial selection for body weight in the mouse. II. The genetic nature of the limits. Genet. Res. 8, 361–375. Robertson, F.W., 1955. Selection response and the properties of genetic variation. Cold Spring Harbor Symp. Quant. Biol. 20, 166–177.

251

Santiago, E., Albornoz, J., Dominguez, A., Toro, M.A., Lopez-Fanjul, C., 1992. The distribution of effects of spontaneous mutations on quantitative traits and fitness in Drosophila melanogaster. Genetics 132, 771–781. Santos, M., 1996. Apparent directional selection of body size in Drosophila buzzatii: larval crowding and male mating success. Evolution 50, 2530–2535. Schluter, D., 2000. The Ecology of Adaptive Radiation. Oxford University Press, Oxford. Simmons, M.J., Crow, J.F., 1977. Mutation affecting fitness in Drosophila populations. Ann. Rev. Genet. 11, 49–78. Simpson, G.G., 1944. Tempo and Mode in Evolution. Columbia University Press, New York. Tanaka, Y., 1996. The genetic variance maintained by pleiotropic mutation. Theor. Popul. Biol. 49, 211–231. Tanaka, Y., 1998. A pleiotropic model of phenotypic evolution. Genetica 102/103, 535–543. Turelli, M., 1984. Heritable genetic variation via mutation-selection balance: Lerch’s zeta meets the abdominal bristle. Theor. Popul. Biol. 25, 138–193. Turelli, M., 1988. Population genetic models for polygenic variation and evolution. In: Weir, B.S., Eisen, E.J., Goodman, M.M., Namkoong, G. (Eds.), Proceedings of the Second International Conference on Quantitative Genetics. Sinauer, Sunderland, MA, pp. 601–618. Turelli, M., Barton, N.H., 1990. Dynamics of polygenic characters under selection. Theor. Popul. Biol. 38, 1–57. Turelli, M., Gillespie, J.H., Lande, R., 1988. Rate tests for selection on quantitative characters during macroevolution and microevolution. Evolution 42, 1085–1089. Welch, J.J., Waxman, D., 2002. Nonequivalent loci and the distribution of mutant effects. Genetics 161, 897–904. Yoo, B.H., 1980a. Long-term selection for a quantitative character in large replicate populations of Drosophila melanogaster. I. Response to selection. Genet. Res. 35, 1–17. Yoo, B.H., 1980b. Long-term selection for a quantitative character in large replicate populations of Drosophila melanogaster. II. Lethals and visible mutants with large effects. Genet. Res. 35, 19–31. Zeng, Z., Cockerman, C.C., 1990. Long-term response to artificial selection with multiple alleles -study by simulations. Theor. Popul. Biol. 37, 254–272. Zhang, X.-S., Hill, W.G., 2002. Joint effects of pleiotropic selection and stabilizing selection on the maintenance of quantitative genetic variation at mutation-selection balance. Genetics 162, 459–471. Zhang, X.-S., Wang, J., Hill, W.G., 2002. Pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance. Genetics 161, 419–433.