# Quantitative genetics

Quantitative genetics is that branch of population genetics which deals with phenotypes which vary continuously (such as height or weight), rather than with phenotypes and gene-products which are discretely identifiable (such as eye-colour, or the presence of a particular biochemical). Both employ the frequencies of different alleles of a gene in breeding populations (gamodemes), and combine them with concepts arising from simple Mendelian inheritance in order to analyze inheritance patterns across generations and descendant lines. While population genetics can focus on particular genes and their subsequent metabolic products, quantitative genetics focuses more on the outward phenotypes, and makes summaries only of the underlying genetics. This, however, can be viewed as its strength, because it facilitates an interface with the biological macrocosm, including micro-evolution and artificial selection in plant and animal breeding. Both branches share some common history; and some mathematics: for example, they use expansion of the quadratic equation to represent the fertilization of gametes to form the zygote. However, because of the continuous distribution of phenotypic values, quantitative genetics needs also to employ many other statistical methods (such as the effect, the mean and the variance) in order to link the phenotype to underlying genetics principles. Some phenotypes (attributes) may be analyzed either as discrete categories or as continuous phenotypes, depending on the definition of cut-off points, or on the metric[1] used to quantify them. Mendel himself had to discuss this matter in his famous paper, especially with respect to his peas attribute dwarf/normal, which actually was "length of stem".[2] Analysis of quantitative trait loci, or QTL,[3] is a more recent addition to quantitative genetics, linking it more directly to molecular genetics.

The field was founded by the originators of the modern synthesis, Ronald Fisher and Sewall Wright.

## Basic principles

### Genotypic variance

The principles of Mendelian inheritance form the genetic basis of interpreting the statistics. In diploid organisms, the average genotypic outcome of a gene (locus) may be defined by the allele "effect" [4] and dominance,[5] and by how genes interact with genes at other loci (epistasis). The founder of quantitative genetics - Sir Ronald Fisher - perceived much of this when he proposed the first mathematics of this branch of genetics.[6] He sought to define a single statistical summary of all the variance arising from phenotypic change during the course of genetic assortment and segregation, which he called "genetic" variance. His residual genotypic variance (called simply "residual" by Fisher) represented assortment which did not lead to phenotypic change. Subsequently, these partitions became known respectively as the "additive"(σ²A) and "dominance" (σ²D) variances, which titles do not appear to convey Fisher's partitions at first glance. However, the assortment/substitution partition of Fisher can be viewed as the average effects of genes after inheritance from parents (remember they pass through meiosis and fertilization to be inherited - the very mechanisms effecting assortment/substitution).[7] So, "additive" actually means "average inherited effect after assortment" and is equivalent to Fisher's assortative disequilibrium. Recently it has been shown to contain the homozygote variance, a portion of the heterozygote variance, and a covariance between homozygote and heterozygote effects.[8] Falconer and Mackay show that Fisher's "residual" (depicted in their Fig.7.2, p. 117) is due to heterozygosis, ie "dominance"; but not all of it, as some is embedded in the "additive" component, as we noted earlier (Gordon (2003)). It actually is that part of the genotypic variance reflecting assortative equilibrium.

This is not the only approach to defining and partitioning genotypic variances. An alternative was advanced by Mather and Jinks.[9] Their notation was entirely different from that of Fisher and his method predominates. However, when their approach is translated into Fisher's notation, the relationships between the two approaches are clear. The Mather and Jinks approach is more "genetical" than Fisher's, being based on variances arising straightforwardly from homozygotes and heterozygotes. The inter-conversions between the Fisher and Mather & Jinks methods are given in Gordon (2003).

### The mating-system

The mating-system assumed in deriving these genotypic variances is panmixia: which implies random fertilization with uniform distribution of gametes in a very large population (theoretically, infinite). This rarely occurs in nature, as gamete distribution may be limited, for example by dispersal restrictions, or preferential matings, or chance sampling in small populations of gametes (gamodemes). Each gamete restriction leads to a descendant small-population (line). Individuals within a line will not all be the same, but they will be more similar than individuals from panmictic populations. In any source breeding group, many separate gamete restrictions will occur during a mating cycle, each one leading to a line. These lines also will vary with respect to their mean phenotypes, and the process is called dispersion. The inbreeding coefficient quantifies the increase in homozygosity which results. The values of this coefficient for a wide variety of situations (e.g. islands, "onion-skin" aggregates, linear strips, matings of related parents) are available.[10] As well as a general rise in homozygosity, the dispersed lines vary in their allele frequencies because of gamete sampling. However, the mean of the frequencies across all lines from the one source will be the same as the original frequencies in the source population. The phenotypic mean of all of these lines is less than that of the original source, this being inbreeding depression. The genetic variances also change relative to those of panmixia. Variance-within-lines decreases, but the variance-amongst-lines and the total-variance-in-the-system both increase (Mackay et al.; Gordon 2003). The first of these facts is common knowledge, but the latter two are not. Many of these lines will be inferior in phenotype: but, some lines will be superior, and some will be about average (Chapter 13 in Falconer et al.). Selection assisted by dispersion leads to maximum genetic advance (see previous references). Plant and animal breeders utilize these properties routinely, and have devised breeding methods especially to do so (e.g. line breeding, pure-line breeding, backcrossing).[11] The role of dispersion in natural selection has not received much attention.

### Environmental variance

The environmental variance is phenotypic variability which cannot be ascribed to genetics. This sounds simple, but the experimental design needed to separate the two needs very careful planning. Even the "external" environment can be divided into geographical and temporal components, as well as partitions such as "litter" or "family" and "culture" or "history". Where does epigenetic variance get placed? Is it embedded within epistasis: or is it "internal environment"? These components are very dependent upon the actual experimental model used to do the research. Such issues are very important when doing the research itself, but in this article on quantitative genetics this may suffice.

It is an appropriate place, however, for a summary:

Phenotypic variance = genotypic variances + environmental variances + genotype-environment interaction + experimental "error" variance

ie σ²P = σ²G + σ²E + σ²GE + σ²e

or σ²P = σ²A + σ²D + σ²e + σ²E + σ²GE + σ²e

after partitioning the genotypic variance (G) into the components of "additive" (A), "dominance" (D), and "epistasic" (I) variance mentioned above ("σ²" is a statistics symbol meaning "variance".)

### Heritability and repeatability

The heritability of a trait is the proportion of the total (phenotypic) variance (σ²P) that is explained by the total genotypic variance (σ²G). This is known as the "broad sense" heritability (H2).[12] If only additive genetic variance (σ²A) is used in the numerator, the heritability is called "narrow sense" (h2).

The broad sense heritability indicates the proportion of the phenotypic variance due to the whole genotypical variance. In colloquial terms, it indicates the extent of "nature" while (1-H2) indicates the extent of "nurture". Narrow sense heritability indicates the proportion of the phenotypic variance attributable to the "additive" genetic variance, discussed above. It was pointed out there that this variance arises from assortative disequilibrium. Fisher proposed that this narrow-sense heritability might be appropriate in considering the results of natural selection, focusing as it does on disequilibrium and hence adaptation. It has been used also for predicting generally the results of artificial selection. In the latter case, however, the broad sense heritability may be more appropriate, as the whole attribute is being altered: not just adaptive capacity. Generally, advance from selection is more rapid with higher heritability. In animals, heritability of reproductive traits is typically low, while heritability of disease resistance and production are moderately-low to moderate, and heritability of body conformation is high.

Repeatability (r2) is the proportion of phenotypic variance attributable to differences in repeated measures of the same subject, arising from later records. It is used particularly for long-lived species. This value can only be determined for traits that manifest multiple times in the organism's lifetime, such as adult body mass, metabolic rate or litter size. Individual birth mass, for example, would not have a repeatability value: but it would have a heritability value. Generally, but not always, repeatability indicates the upper level of the heritability.

r2 = (σ²G + σ²PE)/σ²P

where σ²PE = phenotype-environment interaction ≡ repeatability.

The above concept of repeatability is, however, problematic for traits that necessarily change greatly between measurements. For example, body mass increases greatly in many organisms between birth and adult-hood. Nonetheless, within a given age range (or life-cycle stage), repeated measures could be done, and repeatability would be meaningful within that stage.

## Relationship

### Resemblance between relatives

Central in estimating the variances for the various components is the principle of relatedness. A child has a father and a mother. Consequently, the child and father share 50% of their alleles, as do the child and the mother. However, the mother and father normally do not share alleles as a result of shared ancestors. Similarly, two full siblings share also on average 50% of the alleles with each other, while half siblings share only 25% of their alleles. This variation in relatedness can be used to estimate which proportion of the total phenotypic variance (σ²P) is explained by the above-mentioned components.

The principle of relationship (R) is central to understanding the resemblances within families and can be useful when calculating inbreeding. Relationship has two definitions that can be applied: -The probable portion of genes that are the same for two individuals due to common ancestry exceeding that of the base population -Additive/numerator relationship: the relationship coefficient (Rxy¬) = twice the probability of two genes at loci in different individuals being identical by descent. Rxy values can range from 0 to 1. Relationship can be calculated in several ways; from the known relationships of the individual, from bracket pedigrees, and from pedigree path diagrams.

#### Calculating relationship from known relationships

Relationship Relationship Coefficient
Individual and itself 1.00
Individual and a monozygotic twin 1.00
Individual and parent 0.50
Full siblings 0.50
Half siblings 0.25
Individual and grandparent 0.25
Son of sire and daughter of sire 0.125
Grandson and granddaughter of sire 0.0625
• Note: if the common ancestor is inbred, multiply the relationship by (1+inbreeding coefficient)

#### Calculating relationship from pathway diagrams

RXY = Σ(.5)n(1+FCA)

n = number of segregations between X and Y through their common ancestor FCA = the inbreeding coefficient of the common ancestor

Example: calculating RAE and RBE Note: valid pathways only go through ancestors (only go against the direction of the arrow). For example, to calculate the relationship of A and B, the pathway A-D-B would be acceptable, whereas the pathway A-X-B would be not. The reason behind this is that having progeny together does not make two individuals related.

RAB: there are two possible pathways from A to E. A-D-F-E = (1/2)3 = .125 A-D-E = (1/2)2 = .25 Total: .375

RBE: there are four possible pathways from B to E. B-D-E = (1/2)2 = .25 B-D-F-E = (1/2)3 = .125 B-C-D-E = (1/2)3 = .125 B-C-D-F-E = (1/2)4 = .0625 Total: .5625

The square root of h^2 equals the correlation between additive genotype and expressed phenotype, as shown through the general procedures of Path Analysis.[citation needed]

## Correlated traits

Although some genes have only an effect on a single trait, many genes have an effect on various traits, which is termed pleiotropy. Because of this, a change in a single gene will have an effect on all those traits. This is calculated using covariances, and the phenotypic covariance (CovP) between two traits can be partitioned in the same way as the variances described above. The genetic correlation is calculated by dividing the covariance between the additive genetic effects of two traits by the square root of the product of the variances for the additive genetic effects of the two traits:

$\mbox{Genetic correlation} = \frac{\mathrm{Cov}(A_{1}, A_{2})}{\sqrt{{V_{A_1}*V_{A_2}}}}$

## Footnotes and references

1. ^ A metric is the scale and its numeric properties which depend on the method of measurement used. Categorizing into two categories is a binomial metric. Scoring into five or more pre-defined "scores" is a meristic metric. Measuring on a continuous scale, such as kilograms, is a continuous metric. There are others.
2. ^ Mendel, Gregor (1865). "transl Experiments in plant hybridization". Verhandlungen naturforschender Verain iv.
3. ^ A QTL is a region in the DNA genome that affects, or is associated with, quantitative phenotypic traits.
4. ^ The allele effect is the average phenotypic deviation of the homozygote from the mid-point of the two contrasting homozygote phenotypes at one locus, when observed over the infinity of all background genotypes and environments. In practice, estimates from large unbiased samples substitute for the parameter.
5. ^ The dominance effect is the average phenotypic deviation of the heterozygote from the mid-point of the two homozygotes at one locus, when observed over the infinity of all background genotypes and environments. In practice, estimates from large unbiased samples substitute for the parameter.
6. ^ Fisher, R.A. (1930 (facsimile 1999)). The Genetic theory of natural selection. Oxford Clarendon Press. ISBN 0-19-850440-3.
7. ^ Falconer, et al (1996). Introduction to quantitative genetics. Longman. ISBN 0582-24302-5.
8. ^ Gordon, I.L. (2003). "Refinements to the partitioning of the inbred genotypic variance". Heredity 91 (1): 85–89. doi:10.1038/sj.hdy.6800284. PMID 12815457.
9. ^ Mather; Jinks, John L. (1971). Biometrical genetics. Chapman and Hall. ISBN 0-412-10220-X.
10. ^ Wright, Sewall (1951). "The genetical structure of populations". Annals of Eugenics 15: 323–354. doi:10.1111/j.1469-1809.1949.tb02451.x.
11. ^ Allard, R.W. (1960). Principles of plant breeding. Wiley.
12. ^ This type of variance-ratio is an example of a coefficient of determination. It is used particularly in regression analysis. A standardized version of regression analysis is path analysis. Standardizing here means that the data were first divided by their own experimental standard errors, in order to unify the scales for all attributes.