The Breeder's Equation

Most of the traits that interest biologists, such as behavior, morphology, physiology, and the susceptibility to disease, are quantitative. This means that variation between individuals is due to the combined effect of differences in genotype and in environmental experience. The genetic component of this variation typically involves many genes. Because the phenotype emerges from the complex interplay of genes and environment, biologists are compelled to take a statistical approach to study the evolution of quantitative traits. Perhaps the most basic form of this approach is the breeder's equation(Lush 1937). The breeder's equation predicts evolutionary change in a trait:

Here, Z is the mean of this trait in the population and ΔΖ refers to the change in the mean over one complete generation. The process of selection, either natural or artificial, comes into the prediction through S, the selection differential, which is a measure of association between trait values and fitness (Falconer &Mackay 1996). S will be negative if lower values of a trait confer increased survival or higher reproductive success; positive if selection favors higher trait values. With truncation selection, where a fixed proportion of the population is chosen to reproduce and found the next generation, S is equal to the difference in mean trait values between the selected individuals and the entire population.

The elegance of the breeder's equation is that all of the complexities of multi-locus inheritance are distilled into h^2.. A single statistic known as the narrow sense heritability, h² is the proportion of trait variation statistically attributed to additive genetic effects. If we knew all the genes that influencing a trait and how these genes affect phenotype and the frequencies of alternative alleles within the population, then we could calculate h² directly. However, we never have all of this information. The broad relevance of the breeder's equation owes to the fact that h² can be estimated from the resemblance of relatives. In other words, h² can be inferred by measuring the phenotypic similarity of parents to their offspring or of siblings to each other without knowing the number of genes that affect the trait or where there are located in the genome or how they interact with the environment. Here, it is worth emphasizing that the narrow sense heritability is a population parameter, not a feature of individuals. The failure to fully appreciate this has caused substantial confusion in the public interpretation of h² estimates of human traits (see Freeman & Herron1998 and Visscher et al. 2008 for recent commentaries).

The breeder's equation is used for a diversity of purposes in evolutionary biology. The most direct application is to predict evolutionary change. Figure 1 provides an example for flower size evolution in the yellow monkeyflower (Mimulus guttatus). This experiment started with two populations that were genetically equivalent and medium in flower size. In each, flower size was measured on 1,000 plants. The 200 plants with the smallest flowers were selected in the ‘low' population and the 200 largest in the 'high' population. Selected plants were randomly mated within populations to generate seed for the next generation. This process was repeated distinctly in each population for six generations. As is typical of artificial selection experiments, the populations diverge through time.

The breeder's equation can be applied here to predict the divergence in mean flower size. In total, there were twelve generations of selection (six in each direction). The heritability of flower size was estimated to be about 0.3 by comparing flower size in other plants of this population. Finally, selecting the top (or bottom) 20% yields a selection differential of S = 1.4 σ, where σ is the phenotypic standard deviation of flower size in the initial population. The discrepancy between observed and predicted response is only about 4% (Figure 1), which is small considering that h² is estimated with error. Greater deviations are observed in many selection experiments (Hill & Caballero 1992), an outcome that likely reflects small population sizes. The breeder's equation is deterministic and does not directly accommodate genetic drift, the random changes in allele frequency due to sampling in a finite population.

An important consequence of directional selection on quantitative traits is that it can rapidly produce novel phenotypes. After only six generations of selection, we see flowers that are larger and smaller than any observed in the initial population (Figure 2). Here, the production of novel phenotypes is not caused by new mutations. Instead, it is caused by the sorting of multi-locus variation into previously unrealized genotypes. At any one point in time, a population will normally exhibit only a small fraction of what is possible for a quantitative trait. For example, with only ten genes contributing to trait variation (a modest number), there are over 59,000 possible multi-locus genotypes. Whether particular genotypes exist, say those producing extreme phenotypes, depends on population size and allele frequencies. Selection changes allele frequencies and thus increases the probability of aggregating many alleles with similar phenotypic effects into the same multi-locus genotype.

Evolutionary biologists routinely employ the breeder's equation with only partial information. If divergence (ΔΖ) has been estimated and the accumulated effects of selection are known, an estimate for the trait heritability can be calculated as ΔΖ/S. This is known as the realized heritability. In contrast, the history of selection is often unknown when divergence is estimated by comparing the mean trait values of different natural populations. Here, S can be estimated as ΔΖ/h², given an estimate for the heritability. This is known as a retrospective selection analysis. Both realized heritability and retrospective selection analyses are encumbered with notable assumptions. In particular, each requires that ΔΖ is determined by selection on the trait and that selection is accurately characterized by the breeder's equation.

Figures 1-2 were derived from an artificial selection experiment. The breeder's equation has been successfully applied to predict ΔΖ in natural populations (e.g. (Grant & Grant 1995)), but there are few convincing examples. In part, this reflects the practical difficulties of accurately estimating h² and S in natural populations. However, application is also hindered by the strong assumption that the measured trait directly generates the differences in fitness characterized by S (Morrissey et al. 2010). In nature, survival and reproduction will depend on many distinct features of the organism. For this reason, evolutionary biologists routinely apply the multivariate form of the breeder's equation (Lande 1979; Lande & Arnold 1983).

It is noteworthy that while the breeder's equation is named for prediction in agricultural genetics(Lush 1937), its most frequent use in evolutionary biology is as a conceptual device. The complicated process by which selection causes genetic change within a population is decomposed into two components, phenotypic selection (S) and response to selection (h²). The former determines the relationship between phenotype (trait values) and fitness (survival and reproductive success). The latter is determined by the mapping from genotype to phenotype. Phenotypic selection can be measured in nature for many organisms for which heritability estimation is difficult or impossible. For organisms that cannot be monitored for survival and reproduction in the wild, heritability studies provide information about the evolutionary potential of populations. Admittedly, the partitioning of evolution into phenotypic selection and response to selection formally depends on assumptions that are frequently violated in nature (Hadfield 2008; Rauscher 1992). For example, environmental effects can simultaneously affect trait values and fitness measurements producing spurious estimates for S. It can also be very difficult to get an accurate estimate for selection when many individuals die before expressing a trait. If this "invisible fraction" (Graffen 1988) is genetically distinct from the individuals that are included in the calculation of S (in terms of loci affecting the trait of interest), then the selection differential will again be biased. Despite these difficulties however, the promise of estimating statistics that are directly related to genetic change has motivated a great deal of empirical research in evolutionary biology (Kingsolver et al. 2001; Mousseau & Roff 1987).