Quantitative genetics is the study of quantitative traits, which are phenotypes that vary continuously—such as height or mass—as opposed to phenotypes and gene-products that are discretely identifiable—such as eye-colour, or the presence of a particular biochemical.
While population genetics can focus on particular genes and their subsequent metabolic products, quantitative genetics focuses more on the outward phenotypes, and makes only summaries of the underlying genetics. Both of these branches of genetics use the frequencies of different alleles of a gene in breeding populations (gamodemes), and combine them with concepts from simple Mendelian inheritance to analyze inheritance patterns across generations and descendant lines.
Due to the continuous distribution of phenotypic values, quantitative genetics must employ many other statistical methods (such as the effect size, the mean and the variance) to link phenotypes (attributes) to genotypes. Some phenotypes may be analyzed either as discrete categories or as continuous phenotypes, depending on the definition of cut-off points, or on the metric used to quantify them. Mendel himself had to discuss this matter in his famous paper, especially with respect to his peas' attribute tall/dwarf, which actually was derived by adding a cut-off point to "length of stem".The Mendel G.; Bateson W. (1891) paper, with additional comments by Bateson, is reprinted in: Sinnott E.W.; Dunn L.C.; Dobzhansky T. (1958). "Principles of genetics"; New York, McGraw-Hill: 419-443. Footnote 3, page 422 identifies Bateson as the original translator, and provides the reference for that translation. Analysis of quantitative trait loci, or QTLs,A QTL is a region in the DNA genome that effects, or is associated with, quantitative phenotypic traits. is a more recent addition to quantitative genetics, linking it more directly to molecular genetics.
Gene effects
In diploid organisms, the average genotypic "value" (locus value) may be defined by the allele "effect" together with a dominance effect, and also 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.
Being a statistician, he defined the gene effects as deviations from a central value—enabling the use of statistical concepts such as mean and variance, which use this idea. The central value he chose for the gene was the midpoint between the two opposing homozygotes at the one locus. The deviation from there to the "greater" homozygous genotype can be named "+a"; and therefore it is "-a" from that same midpoint to the "lesser" homozygote genotype. This is the "allele" effect mentioned above. The heterozygote deviation from the same midpoint can be named "d", this being the "dominance" effect referred to above.Other symbols are sometimes used, but these are common. The diagram depicts the idea. However, in reality we measure phenotypes, and the figure also shows how observed phenotypes relate to the gene effects. Formal definitions of these effects recognize this phenotypic focus.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.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. Epistasis has been approached statistically as interaction (i.e., inconsistencies), but epigenetics suggests a new approach may be needed.
If 0<d<a, the dominance is regarded as partial or incomplete—while d=a indicates full or classical dominance. Previously, d>a was known as "over-dominance".
Mendel's pea attribute "length of stem" provides us with a good example. Mendel stated that the tall true-breeding parents ranged from 6–7 feet in stem length (183 – 213 cm), giving a median of 198 cm (= P1). The short parents ranged from 0.75 to 1.25 feet in stem length (23 – 46 cm), with a rounded median of 34 cm (= P2). Their hybrid ranged from 6–7.5 feet in length (183–229 cm), with a median of 206 cm (= F1). The mean of P1 and P2 is 116 cm, this being the phenotypic value of the homozygotes midpoint (mp). The allele affect (a) is [P1-mp] = 82 cm = -[P2-mp]. The dominance effect (d) is [F1-mp] = 90 cm.Mendel commented on this particular tendency for F1 > P1, i.e., evidence of hybrid vigour in stem length. However, the difference may not be significant. (The relationship between the range and the standard deviation is known [Steel and Torrie (1980): 576], permitting an approximate significance test to be made for this present difference.) This historical example illustrates clearly how phenotype values and gene effects are linked.
Allele and genotype frequencies
To obtain means, variances and other statistical values, both quantities and their occurrences are required. The gene effects (above) provide the framework for quantities: and the frequencies of the contrasting alleles in the fertilization gamete-pool provide the information on occurrences.
Commonly, the frequency of the allele causing "more" in the phenotype (including dominance) is given the symbol p, while the frequency of the contrasting allele is q. An initial assumption made when establishing the algebra was that the parental population was infinite and random mating, which was made simply to facilitate the derivation. The subsequent mathematical development also implied that the frequency distribution within the effective gamete-pool was uniform: there were no local perturbations where p and q varied. Looking at the diagrammatic analysis of sexual reproduction, this is the same as declaring that pP = pg = p; and similarly for q. This mating system, dependent upon these assumptions, became known as panmixia.
Panmixia rarely actually occurs in nature, as gamete distribution may be limited, for example by dispersal restrictions or by behaviour, or by chance sampling (those local perturbations mentioned above). It is well known that there is a huge wastage of gametes in nature, which is why the diagram depicts a potential gamete-pool separately to the actual gamete-pool. Only the latter sets the definitive frequencies for the zygotes: this is the true "gamodeme" ("gamo" refers to the gametes, and "deme" derives from Greek for "population"). But, under Fisher's assumptions, the gamodeme can be effectively extended back to the potential gamete-pool, and even back to the parental base-population (the "source" population). The random sampling arising when small "actual" gamete-pools are sampled from a large "potential" gamete-pool is known as genetic drift, and is considered subsequently.
While panmixia may not be widely extant, the potential for it does occur, although it may be only ephemeral because of those local perturbations. It has been shown, for example, that the F2 derived from random fertilization of F1 individuals (an allogamous F2), following hybridization, is an origin of a new potentially panmictic population.An F2 derived by self fertilizing F1 individuals (an autogamous F2), however, is not an origin of a randomly fertilized population structure. See Gordon (2001). It has also been shown that if panmictic random fertilization occurred continually, it would maintain the same allele and genotype frequencies across each successive panmictic sexual generation—this being the Hardy–Weinberg equilibrium.Usually in science ethics, a discovery is named after the earliest person to propose it. Castle, however, seems to have been overlooked: and later when re-found, the title "Hardy Weinberg" was so ubiquitous it seemed too late to update it. Perhaps the "Castle Hardy Weinberg" equilibrium would be a good compromise? However, as soon as genetic drift was initiated by local random sampling of gametes, the equilibrium would cease.
Random fertilization
Male and female gametes within the actual fertilizing pool are considered usually to have the same frequencies for their corresponding alleles. (Exceptions have been considered.) This means that when p male gametes carrying the A allele randomly fertilize p female gametes carrying that same allele, the resulting zygote has genotype AA, and, under random fertilization, the combination occurs with a frequency of p x p (= p2). Similarly, the zygote aa occurs with a frequency of q2. Heterozygotes (Aa) can arise in two ways: when p male (A allele) randomly fertilize q female (a allele) gametes, and vice versa. The resulting frequency for the heterozygous zygotes is thus 2pq. Notice that such a population is never more than half heterozygous, this maximum occurring when p=q= 0.5.
In summary then, under random fertilization, the zygote (genotype) frequencies are the quadratic expansion of the gametic (allelic) frequencies: . (The "=1" states that the frequencies are in fraction form, not percentages; and that there are no omissions within the framework proposed.)
Note that "random fertilization" and "panmixia" are not synonyms.Panmixia requires random fertilization with respect to any posssible physical trait. Random fertilization resulting from random mating ignoring traits could result in bias toward particular traits.
Mendel's research cross – a contrast
Mendel's pea experiments were constructed by establishing true-breeding parents with "opposite" phenotypes for each attribute. This meant that each opposite parent was homozygous for its respective allele only. In our example, "tall vs dwarf", the tall parent would be genotype TT with p = 1 (and q = 0); while the dwarf parent would be genotype tt with q = 1 (and p = 0). After controlled crossing, their hybrid is Tt, with p = q = . However, the frequency of this heterozygote = 1, because this is the F1 of an artificial cross: it has not arisen through random fertilization. The F2 generation was produced by natural self-pollination of the F1 (with monitoring against insect contamination), resulting in p = q = being maintained. Such an F2 is said to be "autogamous". However, the genotype frequencies (0.25 TT, 0.5 Tt, 0.25 tt) have arisen through a mating system very different from random fertilization, and therefore the use of the quadratic expansion has been avoided. The numerical values obtained were the same as those for random fertilization only because this is the special case of having originally crossed homozygous opposite parents. We can notice that, because of the dominance of T- [frequency (0.25 + 0.5)] over tt [frequency 0.25], the 3:1 ratio is still obtained.
A cross such as Mendel's, where true-breeding (largely homozygous) opposite parents are crossed in a controlled way to produce an F1, is a special case of hybrid structure. The F1 is often regarded as "entirely heterozygous" for the gene under consideration. However, this is an over-simplification and does not apply generally—for example when individual parents are not homozygous, or when populations inter-hybridise to form hybrid swarms. The general properties of intra-species hybrids (F1) and F2 (both "autogamous" and "allogamous") are considered in a later section.
Self fertilization – an alternative