Quantitative genetics

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.

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. 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.

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.^[1]^: 27–69 Mendel himself had to discuss this matter in his famous paper,^[2] especially with respect to his peas' attribute tall/dwarf, which actually was derived by adding a cut-off point to "length of stem".^[3]^[4] Analysis of quantitative trait loci, or QTLs,^[5]^[6]^[7] 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.^[8]

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.^[9] 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.^[10] 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.^[11]^[12] Epistasis has been approached statistically as interaction (i.e., inconsistencies),^[13] 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".^[14]

Mendel's pea attribute "length of stem" provides us with a good example.^[3] 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 = 82 cm = -. The dominance effect (d) is = 90 cm.^[15] This historical example illustrates clearly how phenotype values and gene effects are linked.

Allele and genotype frequencies

To obtain means, variances and other statistics, 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 p_P = p_g = p; and similarly for q.^[14] This mating system, dependent upon these assumptions, became known as "panmixia".

Panmixia rarely actually occurs in nature,^[16]^{: 152–180}^[17] 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.^[18]^[19] 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.^[13]^: 34–39^[20]^[21]^[22]^[23] 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.^[13]^: 32 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: ${\textstyle (p+q)^{2}=p^{2}+2pq+q^{2}=1}$ . (The "=1" states that the frequencies are in fraction form, not percentages; and that there are no omissions within the framework proposed.)

Notice that "random fertilization" and "panmixia" are not synonyms.

Mendel's research cross – a contrast

Mendel's pea experiments were constructed by establishing true-breeding parents with "opposite" phenotypes for each attribute.^[3] 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 = 1/2. However, the frequency of this heterozygote = 1, because this is the F1 of an artificial cross: it has not arisen through random fertilization.^[24] The F2 generation was produced by natural self-pollination of the F1 (with monitoring against insect contamination), resulting in p = q = 1/2 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.^[25] We can notice that, because of the dominance of T- over tt , 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.^[24] The general properties of intra-species hybrids (F1) and F2 (both "autogamous" and "allogamous") are considered in a later section.

Self fertilization – an alternative

Having noticed that the pea is naturally self-pollinated, we cannot continue to use it as an example for illustrating random fertilization properties. Self-fertilization ("selfing") is a major alternative to random fertilization, especially within Plants. Most of the Earth's cereals are naturally self-pollinated (rice, wheat, barley, for example), as well as the pulses. Considering the millions of individuals of each of these on Earth at any time, it is obvious that self-fertilization is at least as significant as random fertilization. Self-fertilization is the most intensive form of inbreeding, which arises whenever there is restricted independence in the genetical origins of gametes. Such reduction in independence arises if parents are already related, and/or from genetic drift or other spatial restrictions on gamete dispersal. Path analysis demonstrates that these are tantamount to the same thing.^[26]^[27] Arising from this background, the inbreeding coefficient (often symbolized as F or f) quantifies the effect of inbreeding from whatever cause. There are several formal definitions of f, and some of these are considered in later sections. For the present, note that for a long-term self-fertilized species f = 1. Natural self-fertilized populations are not single " pure lines ", however, but mixtures of such lines. This becomes particularly obvious when considering more than one gene at a time. Therefore, allele frequencies (p and q) other than 1 or 0 are still relevant in these cases (refer back to the Mendel Cross section). The genotype frequencies take a different form, however.

In general, the genotype frequencies become ${\textstyle }$

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