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". Analysis of quantitative trait loci, or QTLs, 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. 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. 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. 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. 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. 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: ( p + q ) 2 = p 2 + 2 p q + q 2 = 1 {\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.) Note 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. 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. 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. 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 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 world'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. 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 [ p 2 ( 1 − f ) + p f ] {\textstyle [p^{2}(1-f)+pf]} for AA and 2 p q ( 1 − f ) {\textstyle 2pq(1-f)} for Aa and [ q 2 ( 1 − f ) + q f ] {\textstyle [q^{2}(1-f)+qf]} for aa. Notice that the frequency of the heterozygote declines in proportion to f. When f = 1, these three frequencies become respectively p, 0 and q Conversely, when f = 0, they reduce to the random-fertilization quadratic expansion shown previously.
Population mean The population mean shifts the central reference point from the homozygote midpoint (mp) to the mean of a sexually reproduced population. This is important not only to relocate the focus into the natural world, but also to use a measure of central tendency used in statistics/biometry. In particular, the square of this mean is the Correction Factor, which is used to obtain the genotypic variances later. For each genotype in turn, its allele effect is multiplied by its genotype frequency; and the products are accumulated across all genotypes in the model. Some algebraic simplification usually follows to reach a succinct result.
The mean after random fertilization The contribution of AA is p 2 ( + ) a {\textstyle p^{2}(+)a} , that of Aa is 2 p q d {\textstyle 2pqd} , and that of aa is q 2 ( − ) a {\textstyle q^{2}(-)a} . Gathering together the two a terms and accumulating over all, the result is: a ( p 2 − q 2 ) + 2 p q d {\textstyle a(p^{2}-q^{2})+2pqd} . Simplification is achieved by noting that ( p 2 − q 2 ) = ( p − q ) ( p + q ) {\textstyle (p^{2}-q^{2})=(p-q)(p+q)} , and by recalling that ( p + q ) = 1 {\textstyle (p+q)=1} , thereby reducing the right-hand term to ( p − q ) {\textstyle (p-q)} . The succinct result is therefore G = a ( p − q ) + 2 p q d {\textstyle G=a(p-q)+2pqd} . This defines the population mean as an "offset" from the homozygote midpoint (recall a and d are defined as deviations from that midpoint). The Figure depicts G across all values of p for several values of d, including one case of slight over-dominance. Notice that G is often negative, thereby emphasizing that it is itself a deviation (from mp). Finally, to obtain the actual Population Mean in "phenotypic space", the midpoint value is added to this offset: P = G + m p {\textstyle P=G+mp} . An example arises from data on ear length in maize. Assuming for now that one gene only is represented, a = 5.45 cm, d = 0.12 cm [virtually "0", really], mp = 12.05 cm. Further assuming that p = 0.6 and q = 0.4 in this example population, then: G = 5.45 (0.6 − 0.4) + (0.48)0.12 = 1.15 cm (rounded); and P = 1.15 + 12.05 = 13.20 cm (rounded).
The mean after long-term self-fertilization The contribution of AA is p ( + a ) {\textstyle p(+a)} , while that of aa is q ( − a ) {\textstyle q(-a)} . [See above for the frequencies.] Gathering these two a terms together leads to an immediately very simple final result:
G ( f = 1 ) = a ( p − q ) {\textstyle G_{(f=1)}=a(p-q)} . As before, P = G + m p {\textstyle P=G+mp} . Often, "G(f=1)" is abbreviated to "G1". Mendel's peas can provide us with the allele effects and midpoint (see previously); and a mixed self-pollinated population with p = 0.6 and q = 0.4 provides example frequencies. Thus: G(f=1) = 82 (0.6 − .04) = 59.6 cm (rounded); and P(f=1) = 59.6 + 116 = 175.6 cm (rounded).
The mean – generalized fertilization A general formula incorporates the inbreeding coefficient f, and can then accommodate any situation. The procedure is exactly the same as before, using the weighted genotype frequencies given earlier. After translation into our symbols, and further rearrangement:
G f = a ( q − p ) + [ 2 p q d − f ( 2 p q d ) ] = a ( p − q ) + ( 1 − f ) 2 p q d = G 0 − f 2 p q d {\displaystyle {\begin{aligned}G_{f}&=a(q-p)+[2pqd-f(2pqd)]\\&=a(p-q)+(1-f)2pqd\\&=G_{0}-f\ 2pqd\end{aligned}}}
Here, G0 is G, which was given earlier. (Often, when dealing with inbreeding, "G0" is preferred to "G".) Supposing that the maize example [given earlier] had been constrained on a holme (a narrow riparian meadow), and had partial inbreeding to the extent of f = 0.25, then, using the third version (above) of Gf: G0.25 = 1.15 − 0.25 (0.48) 0.12 = 1.136 cm (rounded), with P0.25 = 13.194 cm (rounded). There is hardly any effect from inbreeding in this example, which arises because there was virtually no dominance in this attribute (d → 0). Examination of all three versions of Gf reveals that this would lead to trivial change in the Population mean. Where dominance was notable, however, there would be considerable change.
Genetic drift
Genetic drift was introduced when discussing the likelihood of panmixia being widely extant as a natural fertilization pattern. (See section on Allele and genotype frequencies). Here the sampling of gametes from the potential gamodeme is discussed in more detail. The sampling involves random fertilization between pairs of random gametes, each of which may contain either an A or an a allele. The sampling is therefore binomial sampling. Each sampling "packet" involves 2N alleles, and produces N zygotes (a "progeny" or a "line") as a result. During the course of the reproductive period, this sampling is repeated over and over, so that the final result is a mixture of sample progenies. The result is dispersed random fertilization ( ⨀ ) {\displaystyle \left(\bigodot \right)} These events, and the overall end-result, are examined here with an illustrative example. The "base" allele frequencies of the example are those of the potential gamodeme: the frequency of A is pg = 0.75, while the frequency of a is qg = 0.25. [White label "1" in the diagram.] Five example actual gamodemes are binomially sampled out of this base (s = the number of samples = 5), and each sample is designated with an "index" k: with k = 1 .... s sequentially. (These are the sampling "packets" referred to in the previous paragraph.) The number of gametes involved in fertilization varies from sample to sample, and is given as 2Nk [at white label "2" in the diagram]. The total (Σ) number of gametes sampled overall is 52 [white label "3" in the diagram]. Because each sample has its own size, weights are needed to obtain averages (and other statistics) when obtaining the overall results. These are ω k = 2 N k / ( ∑ k s 2 N k ) {\textstyle \omega _{k}=2N_{k}/(\sum _{k}^{s}2N_{k})} , and are given at white label "4" in the diagram.
The sample gamodemes – genetic drift Following completion of these five binomial sampling events, the resultant actual gamodemes each contained different allele frequencies—(pk and qk). [These are given at white label "5" in the diagram.] This outcome is actually the genetic drift itself. Notice that two samples (k = 1 and 5) happen to have the same frequencies as the base (potential) gamodeme. Another (k = 3) happens to have the p and q "reversed". Sample (k = 2) happens to be an "extreme" case, with pk = 0.9 and qk = 0.1; while the remaining sample (k = 4) is "middle of the range" in its allele frequencies. All of these results have arisen only by "chance", through binomial sampling. Having occurred, however, they set in place all the downstream properties of the progenies. Because sampling involves chance, the probabilities ( ∫k ) of obtaining each of these samples become of interest. These binomial probabilities depend on the starting frequencies (pg and qg) and the sample size (2Nk). They are tedious to obtain, but are of considerable interest. [See white label "6" in the diagram.] The two samples (k = 1, 5), with the allele frequencies the same as in the potential gamodeme, had higher "chances" of occurring than the other samples. Their binomial probabilities did differ, however, because of their different sample sizes (2Nk). The "reversal" sample (k = 3) had a very low Probability of occurring, confirming perhaps what might be expected. The "extreme" allele frequency gamodeme (k = 2) was not "rare", however; and the "middle of the range" sample (k=4) was rare. These same Probabilities apply also to the progeny of these fertilizations. Here, some summarizing can begin. The overall allele frequencies in the progenies bulk are supplied by weighted averages of the appropriate frequencies of the individual samples. That is: p ⋅ = ∑ k s ω k p k {\textstyle p_{\centerdot }=\sum _{k}^{s}\omega _{k}\ p_{k}} and q ⋅ = ∑ k s ω k q k {\textstyle q_{\centerdot }=\sum _{k}^{s}\omega _{k}\ q_{k}} . (Notice that k is replaced by • for the overall result—a common practice.) The results for the example are p• = 0.631 and q• = 0.369 [black label "5" in the diagram]. These values are quite different to the starting ones (pg and qg) [white label "1"]. The sample allele frequencies also have variance as well as an average. This has been obtained using the sum of squares (SS) method [See to the right of black label "5" in the diagram]. [Further discussion on this variance occurs in the section below on Extensive genetic drift.]
The progeny lines – dispersion The genotype frequencies of the five sample progenies are obtained from the usual quadratic expansion of their respective allele frequencies (random fertilization). The results are given at the diagram's white label "7" for the homozygotes, and at white label "8" for the heterozygotes. Re-arrangement in this manner prepares the way for monitoring inbreeding levels. This can be done either by examining the level of total homozygosis [(p2k + q2k) = (1 − 2pkqk)], or by examining the level of heterozygosis (2pkqk), as they are complementary. Notice that samples k= 1, 3, 5 all had the same level of heterozygosis, despite one being the "mirror image" of the others with respect to allele frequencies. The "extreme" allele-frequency case (k= 2) had the most homozygosis (least heterozygosis) of any sample. The "middle of the range" case (k= 4) had the least homozygosity (most heterozygosity): they were each equal at 0.50, in fact. The overall summary can continue by obtaining the weighted average of the respective genotype frequencies for the progeny bulk. Thus, for AA, it is p ⋅ 2 = ∑ k s ω k p k 2 {\textstyle p_{\centerdot }^{2}=\sum _{k}^{s}\omega _{k}\ p_{k}^{2}} , for Aa, it is 2 p ⋅ q ⋅ = ∑ k s ω k 2 p k q k {\textstyle 2p_{\centerdot }q_{\centerdot }=\sum _{k}^{s}\omega _{k}\ 2p_{k}q_{k}} and for aa, it is q ⋅ 2 = ∑ k s ω k q k 2 {\textstyle q_{\centerdot }^{2}=\sum _{k}^{s}\omega _{k}\ q_{k}^{2}} . The example results are given at black label "7" for the homozygotes, and at black label "8" for the heterozygote. Note that the heterozygosity mean is 0.3588, which the next section uses to examine inbreeding resulting from this genetic drift. The next focus of interest is the dispersion itself, which refers to the "spreading apart" of the progenies' population means. These are obtained as G k = a ( p k − q k ) + 2 p k q k d {\textstyle G_{k}=a(p_{k}-q_{k})+2p_{k}q_{k}d} [see section on the Population mean], for each sample progeny in turn, using the example gene effects given at white label "9" in the diagram. Then, each P k = G k + m p {\textstyle P_{k}=G_{k}+mp} is obtained also [at white label "10" in the diagram]. Notice that the "best" line (k = 2) had the highest allele frequency for the "more" allele (A) (it also had the highest level of homozygosity). The worst progeny (k = 3) had the highest frequency for the "less" allele (a), which accounted for its poor performance. This "poor" line was less homozygous than the "best" line; and it shared the same level of homozygosity, in fact, as the two second-best lines (k = 1, 5). The progeny line with both the "more" and the "less" alleles present in equal frequency (k = 4) had a mean below the overall average (see next paragraph), and had the lowest level of homozygosity. These results reveal the fact that the alleles most prevalent in the "gene-pool" (also called the "germplasm") determine performance, not the level of homozygosity per se. Binomial sampling alone effects this dispersion. The overall summary can now be concluded by obtaining G ⋅ = ∑ k s ω k G k {\textstyle G_{\centerdot }=\sum _{k}^{s}\omega _{k}\ G_{k}} and P ⋅ = ∑ k s ω k P k {\textstyle P_{\centerdot }=\sum _{k}^{s}\omega _{k}\ P_{k}} . The example result for P• is 36.94 (black label "10" in the diagram). This later is used to quantify inbreeding depression overall, from the gamete sampling. [See the next section.] However, recall that some "non-depressed" progeny means have been identified already (k = 1, 2, 5). This is an enigma of inbreeding—while there may be "depression" overall, there are usually superior lines among the gamodeme samplings.
The equivalent post-dispersion panmictic – inbreeding Included in the overall summary were the average allele frequencies in the mixture of progeny lines (p• and q•). These can now be used to construct a hypothetical panmictic equivalent. This can be regarded as a "reference" to assess the changes wrought by the gamete sampling. The example appends such a panmictic to the right of the Diagram. The frequency of AA is therefore (p•)2 = 0.3979. This is less than that found in the dispersed bulk (0.4513 at black label "7"). Similarly, for aa, (q•)2 = 0.1303—again less than the equivalent in the progenies bulk (0.1898). Clearly, genetic drift has increased the overall level of homozygosis by the amount (0.6411 − 0.5342) = 0.1069. In a complementary approach, the heterozygosity could be used instead. The panmictic equivalent for Aa is 2 p• q• = 0.4658, which is higher than that in the sampled bulk (0.3588) [black label "8"]. The sampling has caused the heterozygosity to decrease by 0.1070, which differs trivially from the earlier estimate because of rounding errors. The inbreeding coefficient (f) was introduced in the early section on self fertilization. Here, a formal definition of it is considered: f is the probability that two "same" alleles (that is A and A, or a and a), which fertilize together are of common ancestral origin—or (more formally) f is the probability that two homologous alleles are autozygous. Consider any random gamete in the potential gamodeme that has its syngamy partner restricted by binomial sampling. The probability that that second gamete is homologous autozygous to the first is 1/(2N), the reciprocal of the gamodeme size. For the five example progenies, these quantities are 0.1, 0.0833, 0.1, 0.0833 and 0.125 respectively, and their weighted average is 0.0961. This is the inbreeding coefficient of the example progenies bulk, provided it is unbiased with respect to the full binomial distribution. An example based upon s = 5 is likely to be biased, however, when compared to an appropriate entire binomial distribution based upon the sample number (s) approaching infinity (s → ∞). Another derived definition of f for the full distribution is that f also equals the rise in homozygosity, which equals the fall in heterozygosity. For the example, these frequency changes are 0.1069 and 0.1070, respectively. This result is different to the above, indicating that bias with respect to the full underlying distribution is present in the example. For the example itself, these latter values are the better ones to use, namely f• = 0.10695. The population mean of the equivalent panmictic is found as [a (p•-q•) + 2 p•q• d] + mp. Using the example gene effects (white label "9" in the diagram), this mean is P ⋅ = {\textstyle P_{\centerdot }=} 37.87. The equivalent mean in the dispersed bulk is 36.94 (black label "10"), which is depressed by the amount 0.93. This is the inbreeding depression from this Genetic Drift. However, as noted previously, three progenies were not depressed (k = 1, 2, 5), and had means even greater than that of the panmictic equivalent. These are the lines a plant breeder looks for in a line selection programme.
Extensive binomial sampling – is panmixia restored? If the number of binomial samples is large (s → ∞ ), then p• → pg and q• → qg. It might be queried whether panmixia would effectively re-appear under these circumstances. However, the sampling of allele frequencies has still occurred, with the result that σ2p, q ≠ 0. In fact, as s → ∞, the σ p , q 2 → p g q g 2 N {\textstyle \sigma _{p,\ q}^{2}\to {\tfrac {p_{g}q_{g}}{2N}}} , which is the variance of the whole binomial distribution. Furthermore, the "Wahlund equations" show that the progeny-bulk homozygote frequencies can be obtained as the sums of their respective average values (p2• or q2•) plus σ2p, q. Likewise, the bulk heterozygote frequency is (2 p• q•) minus twice the σ2p, q. The variance arising from the binomial sampling is conspicuously present. Thus, even when s → ∞, the progeny-bulk genotype frequencies still reveal increased homozygosis, and decreased heterozygosis, there is still dispersion of progeny means, and still inbreeding and inbreeding depression. That is, panmixia is not re-attained once lost because of genetic drift (binomial sampling). However, a new potential panmixia can be initiated via an allogamous F2 following hybridization.
Continued genetic drift – increased dispersion and inbreeding Previous discussion on genetic drift examined just one cycle (generation) of the process. When the sampling continues over successive generations, conspicuous changes occur in σ2p, q and f. Furthermore, another "index" is needed to keep track of "time": t = 1 .... y where y = the number of "years" (generations) considered. The methodology often is to add the current binomial increment (Δ = "de novo") to what has occurred previously. The entire binomial distribution is examined here. (There is no further benefit to be had from an abbreviated example.)
Dispersion via σ2p,q Earlier this variance (σ 2p,q) was seen to be:-
σ p , q 2 = p g q g / 2 N = p g q g ( 1 2 N ) = p g q g f = p g q g Δ f when used in recursive equations {\displaystyle {\begin{aligned}\sigma _{p,q}^{2}&=p_{g}q_{g}\ /\ 2N\\&=p_{g}q_{g}\left({\frac {1}{2N}}\right)\\&=p_{g}q_{g}\ f\\&=p_{g}q_{g}\ \Delta f\ \scriptstyle {\text{when used in recursive equations}}\end{aligned}}}
With the extension over time, this is also the result of the first cycle, and so is σ 1 2 {\textstyle \sigma _{1}^{2}} (for brevity). At cycle 2, this variance is generated yet again—this time becoming the de novo variance ( Δ σ 2 {\textstyle \Delta \sigma ^{2}} )—and a