An Alternative to the Breeder’s and Lande’s Equations

The breeder’s equation is a cornerstone of quantitative genetics, widely used in evolutionary modeling. Noting the mean phenotype in parental, selected parents, and the progeny by E(Z0), E(ZW), and E(Z1), this equation relates response to selection R = E(Z1) − E(Z0) to the selection differential S = E(ZW) − E(Z0) through a simple proportionality relation R = h2S, where the heritability coefficient h2 is a simple function of genotype and environment factors variance. The validity of this relation relies strongly on the normal (Gaussian) distribution of the parent genotype, which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian with mean μ, an alternative, exact linear equation of the form R′ = j2S′ can be derived, regardless of the parental genotype distribution. Here R′ = E(Z1) − μ and S′ = E(ZW) − μ stand for the mean phenotypic lag with respect to the mean of the fitness function in the offspring and selected populations. The proportionality coefficient j2 is a simple function of selection function and environment factors variance, but does not contain the genotype variance. To demonstrate this, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between them. These results generalize naturally to the concept of G matrix and the multivariate Lande’s equation Δz¯=GP−1S. The linearity coefficient of the alternative equation are not changed by Gaussian selection.

below). Another important case is when the genotype is a cross between different breeds due to external gene flow or the breeder's scheme. In many cases, the phenotype can have a bell shape and thus is assumed to be Gaussian, when the genotype is indeed far from it (see, for example, Figure 2A). It is sometimes argued that even if the breeding value does not follow a normal distribution, a scale can be used to restore it to a normal distribution. Such a scale, however, will also distort the distribution of environment factors and the assumptions of breeder's equation are violated even in this case.
For additive genetic effects and in the absence of epistasis and dominance, I derive here a precise functional relation between the mean of the trait in the selected subpopulation and in their progeny for the general case. The mathematical formulation is close to the framework used by many authors such as Slatkin, Lande and Karlin (Slatkin 1970;Karlin 1979;Lande 1979). I then use a standard tool of functional analysis, the Fourier transform (FT), to deduce all the cases that lead to a linear relation between the response R and the selection differential S, regardless of the selection function. These cases imply a precise form of the distributions of genotype and environment factors, and I show that the proportionality factor between R and S is the heritability coefficient h 2 only if these distributions are normal.
The genotype, however, is not observable or controllable, and its normal distribution cannot be assumed a priori. I show that if instead of the genotype, the fitness function and environment factors are Gaussian, then a new proportionality relation can be obtained in the form of (1) regardless of the genotype distribution. Noting the mean of the Gaussian selection function by m, R9 = E(Z 1 ) 2 m and S9 = E (Z W ) 2 m are the mean phenotypic lag with respect to the mean of the fitness function of the progeny and the selected population ( Figure 1). As E(Z 1 ), E(Z W ), and m are all measurable, R9 and S9 are both measurable in the same way as R and S are. The j 2 coefficient contains only the width of the fitness function and environment factors. The use of a Gaussian selection function, both in artificial and natural selection (as an approximation of stabilizing selection), is widespread (Lewontin 1964;Lande 1976;Kimura and Crow 1978;Zhang and Hill 2010) and the aforementioned relationship is potentially as useful as the standard breeder's equation. The advantage is more critical when the breeder's or Lande's equations are used in long-term evolution, where the variance of the genotype (or the G matrix) also varies and h 2 cannot be assumed to remain constant (Gavrilets and Hastings 1995;Pigliucci and Schlichting 1997;Roff 2000) ; in contrast, the relation (1) remains valid if each round of selection uses a Gaussian fitness function.
The aforementioned results generalize naturally to multivariate trait selection where the alternative Lande's equation is where R9 and S9 are the vectorial phenotype lag, and V and E are the covariance matrices of the fitness function and the environment respectively. This article is organized as follows: in the Results section, I first derive the general functional relationship between R and S; the second subsection is devoted to all the cases where these two quantities can be linearly related, including the special case of the breeder's equation. The alternative breeder's equation is derived in the third subsection, and all the results are generalized to selection on multiple traits in the fourth subsection. The aforementioned results are put into perspective in the Discussion section. Technical details, such as the use of FTs, are treated in the Appendix.

General results
Consider a continuous phenotype Z, which is the result of additive genetic effect Y and the environment j (Fisher 1918;Lynch and Walsh 1998;Visscher et al. 2008) The term environment encompasses here any source of noise that causes the observed phenotype z to deviate from the (unobserved) breeding value y (Wright 1920;Lynch and Walsh 1998;Raj and van Oudenaarden 2008). In the following, the population distribution of the breeding value (genotype) and its variance in the parental generation are denoted p 0 (y) and s 2 A . The environment effect is captured by the distribution law f(z|y), the probability density of observing phenotype z with the given genotype y. We will suppose that f is a symmetric function of its argument of the form f(z|y) = f(z 2 y) and denote its width by s 2 E . A subpopulation among the parental generation is selected according to a fitness or selection function W(z), the proportion of phenotypes in [z, z + dz] to be selected for the production of the next generation. The selected individuals produce offspring which will constitute the next generation. As we will show herein, the response R (the mean of the phenotype trait in the offspring) and the selection differential S (the mean of the phenotype trait in the selected parents) are given by where W is the mean fitness of parental generation. Equations (3) and (4) are used, for example, by Lande (1979), although their derivation there depended on the normal distribution of the genotype. I derive these equations here for the more general case.
Before going into the details of calculations, note that the genotype distribution p 0 (y) and the selection function W(z) play a symmetric role in the aformentioned expressions. In the following sections, we will explore specific functional forms of p 0 (y) and W(z), which lead to a linear relationship between R and S. Because of the symmetric role of these two functions however, once a particular relation is obtained for a specific form of p 0 (y) regardless of W(z), an analogous relationship can be obtained for a similar form of W(z) regardless of p 0 (y). This is what leads us to an alternative form of the breeder's equation.
Let us now derive the equations (3,4). We note that the distribution of the phenotype Z in the parental generation is given by We will denote its variance by s 2 P . The distribution of the phenotype z in the parental population selected according to the fitness function W(z) is The genotype distribution of the selected population is (Turelli and Barton 1994) where is the genotype fitness function, i.e., the convolution of the phenotype fitness function by the environment factors. W y is the mean genotype fitness: Note that W ¼ W y as both these quantities are defined by the same double integration over the domains of y and z. For a large, randomly mating population, reproduction gives for the distribution of breeding values in the next generation (Slatkin 1970;Karlin 1979;Bulmer 1985;Turelli and Barton 1994) The exact form of the probability density L(y) that captures the inheritance process (recombination, segregation, . . .) is not important here; Turelli and Barton (1994), for example, use a normal distribution for L(y) in the framework of the infinitesimal model. For our purpose, it is enough to suppose that the mean of the distribution L(y) is zero, i.e., R y yLðyÞdy ¼ 0 which is valid in the absence of dominance and epistasis effects (Turelli and Barton 1990) (see also Appendix/Segregation density function).
The phenotype distribution of the progeny is We now make the further assumption that (1) the environment and genotype are independent random variables, so that f ðzjyÞ ¼ f ðz 2 yÞ and therefore the variances are additive: ). An environmental noise with such a distribution law does not change the mean of the random variable: . Therefore, the mean phenotype of the offspring is which is equation (3). Note that the first lines of the above equations merely state that the expectations of the breeding's value of parent and offspring are equal for purely additive traits.
On the other hand, the mean phenotype of the selected parents is which is equation (4).
For an asexually reproducing organism, or for a sexually reproducing population which remains at Hardy-Weinberg equilibrium after selection-reproduction, we would have p 1 (y) = p w (y) ; this would again lead to the same equation (10) and the same response (11). The conditions for the existence of multilocus Hardy-Weinberg equilibrium were analyzed by Karlin and Liberman (1979a,b), who concluded that for additive traits, the equilibrium is stable for a wide range of recombination distributions. The general relation between R and S can also be studied in the context of the Price equation. A detailed study of this relation has been performed by Heywood (2005).
Conditions for proportionality of R and S The relations (3) and (4) show that the selection differential S and the response R to it are related through a functional equation involving three factors: genotype distribution, the selection function and the environmental noise. It is far from obvious that R and S could be proportional, a question we will investigate by using FTs.
FTs in functional analysis play a role analogous to logarithms in algebra. They are useful for clarifying the R 2 S relation, where we can Figure 1 Schematic representation of the selection lag S9, the response lag R9, and their relation to the selection differential S and the response R. The mean phenotype of parental generation z 0 , selected population z w , the progeny z 1 , and the peak of selection function m are represented on the phenotype axis z. Dashed curves represent a sketch of the distributions of parental phenotype q 0 (z), selected parents q w (z), the progeny q 1 (z), and the selection function W(z).
transform the double integrations into simple ones. The FT of the function u(x) is the functionũðkÞ defined as (see Appendix/Fourier Transforms)ũ For example, the FT of the function u(x) = exp(2a|x|) is uðkÞ ¼ 2a=ða 2 þ k 2 Þ. Part of the usefulness of FT is due to the fact that they transform convolution products into simple products: given two functions u(x) and v(x) and their convolution product h(z): the relation between their FT is a simple product: hðkÞ ¼ũðkÞṽðkÞ As the general relations (11) and (12) involve convolutions, FT proves to be very useful in their handling. Using the various properties of FT (see Appendix/Fourier Transforms), the relation between R and S in the Fourier space reads: and where the mean fitness W is itself defined in Fourier space as Here a Ã designate the complex conjugate of a, i 2 = 21 and we have set the origin of the breeding values at its mean in the parental population, i.e., R ℝ yp 0 ðyÞdy ¼ 0. In general, the FT of a function is complex. However, as the function in direct space here are real, it can be shown that expressions (13) and (14) are indeed real; the fact that i appears in these expression insures this fact (see Appendix/Fourier Transforms). It is worthwhile to consider a particular case to clarify the above expressions. The detailed computations for a truncation selection in which breeding value and environmental factors are normally distributed are provided in Appendix/Truncation selection.
We see from equations (13) and (14) that S and R can be proportional if the second term of the r.h.s. of equation (14) is proportional to R; this will be true, regardless of the selection function W, if where a is an arbitrary constant. Equation (15) is the necessary and sufficient condition that defines the functional shape of the genotype distribution and the environment noise compatible with the proportionality of R and S regardless of the selection function. If condition (15) is fulfilled, then On the other hand, equation (15) can be seen as a differential equation whose solution is given bỹ where b is another arbitrary constant. Let us consider some particular case where the aforementioned relation is obeyed.

Normal distributions
Iff ðkÞ andp 0 ðkÞ are both Gaussians, i.e., Á then the relation (16) is satisfied by and we retrieve the usual breeder's equation Of course, iff ðkÞ andp 0 ðkÞ are of the above form, their inverse FTs represent normal distributions of width s E and s A respectively (see Appendix/Fourier Transforms).

Stretched exponentials
We see, however, that even if the strict condition (16) is fulfilled, the proportionality constant need not be h 2 . Consider, for example, the class of stretched exponential functions f(k) = exp(2|k| a ), which generalizes Gaussians (case a = 2). Setf ðkÞ ¼ fðs E kÞ,p 0 ðkÞ ¼ fðs A kÞ. The inverse FT of these functions gives the distribution of the genotype Y and environment effect E and it is straightforward to show that as for the Gaussian case, VarðEÞ=VarðYÞ ¼ s 2 E =s 2 A . Condition (16) however is satisfied this time with a ¼ s a E =s a A and therefore the realized heritability h a = R/S is The aforementioned examples were to emphasize the fact that selection-independent proportionality is achieved only for particular pairs of genotype/environment distributions. In general, as shown in Figure 2, the realized heritability is not constant and depends critically on the selection function W(z).

Alternative breeder's equation
Optimal phenotypic selection approximated by Gaussians has been considered by many authors both in artificial (as early as Lush 1943) and in natural selection (as early as Wright 1935;Haldane 1954) and it is widespread in the literature (Lewontin 1964;Lande 1976;Kimura and Crow 1978;Karlin and Liberman 1979a;Zhang and Hill 2010). If the selection function is Gaussian, a new linear relation can be extracted from the general relations (3) and (4), regardless of the (unobservable) breeding value distribution. Note that a symmetric role is played by W(z) and p 0 (y) in the general expressions (3) and (4). Hence permuting their role will lead us, following the same line of arguments, to deduce all linear cases regardless of genotype. Equations (3) and (4) are obtained by multiplying the function F(y, z) = W(z)p 0 (y)f(z 2 y) either by y or z and integrating over ℝ 2 . To obtain the breeder's equation of the previous section, we wrote the integration over the y variable as a convolution product and performed the FT on the z variable.
On the other hand, we could have proceeded by writing equations (3) and (4) first as a convolution product on z and then perform a FT on the variable y (see Appendix/Fourier Transform). In this case, we get and The arguments of the previous section can be repeated. Let us center the selection function by setting W and The quantities S9 and R9 are alternative selection differential and response and represent the lag with respect to the mean of the selection function (Figure 1). In the case in which the selection function and the environment factors are both normally distributed with width s W and s E , a repetition of the arguments of the previous sections leads to where We stress that relation (21) is obtained regardless of the unknown genotype distribution p 0 (y). The alternative breeder's equation (21) may seem unusual as it does not contain the genetic variance. Such a result may seem at first glance in contradiction with our basic understanding of the selection process. Fisher fundamental's theorem for example explicitly relates the rate of increase in fitness to the genetic variance. There is, however, no contradiction: Both R9 and S9 are dependent on the genetic variance, as can be seen in the general equations (3) and (4) ; however, their ratio, i.e., the coefficient of the linear equation (21) relating them, is free of genetic variance. A similar situation occurs for the classical breeder's equation, where both R and S depend on the selection function W(z) but their ratio contains only the heritability coefficient, independently of W(z). Equation (21) has been obtained through the tools of functional analysis and its demonstration may seem a little abstract. It is worthwhile to further illustrate this equation by considering few examples where the computations can be carried out explicitly. Let us designate the normally distributed selection function W(z) and the environment factors as:

No genetic variance
The first example we consider is the extreme case in which there is no genetic variance (sA = 0) in the parental generation. The distribution of the breeding value then becomes a Dirac's delta function p0(y) = d(y). The basic rule of Dirac's delta, i.e., R I dðyÞfðyÞdy ¼ fð0Þ reduces the double integrations of equations (11) and (12) to simple integrations which involve only Gaussian functions. Note that for the general case, reduction of double integration to simple one was achieved by the use of FTs. The value of R and S are therefore readily obtained in this case: As expected, in the absence of genetic variance, there is no response to selection. The response and selection lag R9 and S9 read: Therefore, R9 = j 2 S9, and equation (21) is verified. This example shows explicitly that there is no contradiction between the alternative breeder's equation and Fisher's fundamental theorem.

Gaussian breeding values distribution
Let us now consider a less-extreme case in which there exists a normal genetic variability The double integrations (11,12) giving S and R can again be carried out exactly, as all the integrands are Gaussian: where a = (j 2 2 1)/(j 2 2 h 2 ). Therefore, and equation (21) is verified. Note that in the aforementioned case, R and S are both proportional to the mean of the selection function m. Applying a Gaussian selection function can therefore be used as a test of the normal distribution of the breeding values.

Non-Gaussian breeding values distribution
Let us now consider a case in which parental breeding values are not normally distributed but are concentrated around two particular values: and therefore E(Y 0 ) = 0 and VarðY 0 Þ ¼ s 2 A . The computation of the expressions (11) and (12) can be again carried out exactly: We note that in this case, the ratio R/S 6 ¼ h 2 and the classical breeder's equation does not hold. The alternative breeder's equation however is again verified:  (21), as any function can be seen as a superposition of Dirac's deltas: p 0 ðyÞ ¼ R ℝ p 0 ðuÞdðu 2 yÞdu. The demonstration we provided using FT is, however, more straightforward.

Selection on multiple traits
The results of the aforementioned sections are naturally generalized to selection on multiple traits. Consider the vectors of parental breeding values y 0 = (y 1 , y 2 , . . ., y N ), environmental effects e = (e 1 , . . ., e N ) and their phenotype z 0 = y 0 + e, to which a selection function W(z) is applied. Using the same notations as in the previous sections, we find without difficulty that As before, using FT, these relations transform into where = is the gradient operator: =f = (@f/@x 1 , . . . @f/@x N ). We see again that z 1 and z w are linearly related if where A is a constant matrix. The linear relation is automatically satisfied if both p 0 and f follow a Gaussian distribution where G and E are the covariance matrices for the genotype and environmental effects. Defining P = G + E as the phenotype covariance matrix, it is straightforward to show that in this case A = EG 21 and therefore (Lande 1979) z 1 ¼ GP 21 z w which is the usual breeder's equation for multiple traits. We stress that the limitation of this relation is the same as that of the scalar version: it relies on the normal distribution of the genotype. On the other hand, if the selection function W(z) is Gaussian WðzÞ } exp 2 1 2 ðz2mÞ T V 21 ðz 2 mÞ the arguments of the previous section 2 can be repeated and lead to the generalization of the alternative vectorial breeder's equation (21) z 1 2 m ¼ ðV þ EÞV 21 À z w 2 m Á which, in analogy with equation (21) we write as

DISCUSSION AND CONCLUSION
The breeder's equation is a cornerstone of quantitative genetics and appears as a fundamental equation in all the important textbooks of this field (Lynch and Walsh 1998;Falconer and Mackay 1995;Crow and Kimura 2009). It is widely used in artificial selection (Lush 1943;Hill and Kirkpatrick 2010); its usage in natural selection was popularized by Lande (1976), when he formalized the main idea of phenotypic evolution and it is now commonly used in many articles based on Lande's work (see, for example, Hansen et al. 2011;Manna et al. 2011;Svardal et al. 2011). The mathematical foundation of this equation rests upon the hypothesis that the breeding value is normally distributed. This hypothesis is plausible for a continuous trait in a population not subject to selection (see, however, Appendix/Segregation density function). The normal distribution of the breeding value is more fragile in populations subjected to selection on this trait (Turelli and Barton 1990), as the genotype of selected parents is given by (equation 7) where W y (y) is the genotype fitness function defined by equation (8). Even if p 0 (y) were Gaussian, the very act of multiplying it by an arbitrary function makes p w (y), and hence p 1 (y) non-Gaussian. Therefore after the first round of selection, the normal distribution hypothesis of parental genotype cannot be sustained. Turelli and Barton (1994) have shown that for the infinitesimal model, the non-normality may not have large effects on the predictions of the breeder's equation, but they argued that when the number of loci is limited the discrepancy can grow much larger. Of course even p 0 (y) cannot be assumed to be Gaussian if different breeds are crossed to constitute the parental generation, which happens in artificial selection and in natural selection when gene flow from nearby patches is important. The breeding value is not an observable quantity. The fitness or selection function W(z) is more quantifiable and many authors have considered a Gaussian selection function. In artificial selection, it dates back at least to the work of Lush (Lush 1943), p140). In natural selection, it is used by most authors as a model for stabilizing selection. If Gaussian selection is used to evolve a population, then the alternative breeding equation (21) we derived is more precise and predictive and rests on more robust mathematical grounds while retaining the same simplicity of the standard breeder's equation. Note that the analysis of this article is not restricted to the infinitesimal model, but applies to all inheritance processes involving purely additive genetic effects. The alternative breeder's equation generalizes to selection on multiple traits in a way similar to the standard breeder's equation and can therefore be incorporated in the "adaptive landscape" formalism (Arnold et al. 2001) with the same ease.
In conclusion, we believe that in all cases where Gaussian selection functions are used to evolve a population, the alternative breeder's equation we develop above is a useful alternative approach to the standard method. ACKNOWLEDGMENTS I thank Jarrod Hadfield for discussions and comments on earlier drafts that helped to improve the manuscript. I am also grateful to M. Vallade, E. Geissler, O. Rivoire, and A. Dawid for careful reading of the manuscript and fruitful discussions.

APPENDIX FTs and convolutions
Logarithm was invented to simplify algebraic operations: a multiplication in the direct space transforms into an addition in the logarithm space, were it can be performed easily and the result brought back to direct space via the inverse transformation. FT plays a similar role in functional analysis, where derivation/integration of functions in direct space are transformed into multiplication/division by the variable in the Fourier space.
The FT of a function f(x) is defined here as (Byron and Fuller 1992) where i 2 = 21. For example, the FT of the function f(x) = exp(2a|x|) is The main properties of FT we use here are where a Ã stands for the conjugate complex of a. Note that if f(x) is a real function, thef Ã ðkÞ ¼f ð 2 kÞ, which ensures that the right hand side of the above expression is always real if both functions f and g are real. Note that we can exchange the order of integration on y and z, write the first integral as a convolution product on functions of z and proceed to the second integral by using the FT on y. For R, we have

Derivation property
ðkÞf ðkÞ Ã dk and for S we get ðkÞ Ãf ðkÞdk The translation property was used in the derivation of the functional lags (eqs 19,20). Finally, note that the FT of a Gaussian is a Gaussian: Computation of truncation selection with FT Consider a normal breeding value and environmental factor distribution and after n rounds of reproduction,p n ðkÞ ¼p 2 n 0 ðk=2 n Þ Y n 2 1 i¼0L 2 i À k=2 i Á As both p 0 (y) and L(y) are probability distribution functions of zero mean, we havẽ p 0 ð0Þ ¼Lð0Þ ¼ 1 p 0 9ð0Þ ¼L9ð0Þ ¼ 0 and thereforep Let V ¼ R ℝ y 2 LðyÞdy . We see then that VarðY n Þ ¼ 1 2 n VarðY 0 Þ þ 2 2 1 2 n21 V So the variance of the breeding values converges fast to twice the variance of the segregation density function. The distribution function p n (y), however, converges to a normal distribution only if L(y) is normal.