Introduction

Many episodes in evolution and plant breeding are initiated by hybridizing between parent populations or individuals. Meiosis and syngamy within the hybrid (F₁) lead to a population with a different frequency structure (the F₂), i.e. the F₁ is in disequilibrium and has an ephemeral population structure. The quantitative genetic properties of the F₁ population have been examined (Gordon, 1999), but how do the properties of the emergent F₂ relate to these? Intuitively, because of meiosis, we would expect more variance: but we need to explore the quantitative relationships of this matter. Intuitively, also, we would expect to have to account for effects of inbreeding because of shared ancestral origins (F₁ individuals) amongst members of the F₂ population. This needs to be reconciled with the assumption that an allogamous F₂ can be considered as a single panmictic random fertilizing gamodeme with zero inbreeding. It appears that these, and related, matters have not been explored rigorously nor quantitatively.

These omissions became strongly apparent when the author sought to estimate the genetic advance arising from selection from an F₂. This is a fundamental phase in important protocols of plant breeding, such as Line Breeding and Pedigree Breeding (Allard, 1960; Poehlmann, 1979; Moore and Janik, 1983). Such selection from the F₂ is essential also to the ongoing contribution of a hybrid in natural selection. In order to evaluate genetic advance from such selection, we need more knowledge on the properties of the F₂.

The present paper examines the quantitative genetic properties of the bulk F₂ population generated by random fertilization amongst F₁ individuals. Some earlier focus has examined the mean of such an F₂ (Falconer, 1981), in order to demonstrate the drop in hybrid vigour compared with the F₁. Here, we derive as well the allogamous F₂ genotypic variance (²_G), genic variance (²_A), dominance variance (²_D), broad-sense and narrow-sense heritabilities (h²_B and h²_N, respectively), genotypic and allelic frequencies, and levels of inbreeding. Following this, we will be able to estimate selection genetic advance from the F₂ and we will examine the relationship of the allogamous F₂ to random fertilizing (RF) populations.

General method

A population consists of genotypes arising from alleles (or allele Expectations) A₁ and A₂ present with frequencies p and q respectively, with no omissions (i.e. p + q=1). Its individual phenotypic Expectations are its genotype effects defined as deviates from the homozygote midpoint, namely g'=[a, d, - a] for {A₁A₁, A₁A₂, A₂A₂}, respectively. This is the familiar single-factor gene model used to define classical RF populations (Falconer, 1981). It has been used also to define hybrids (Gordon, 1999), where further discussion is presented on its utility. In examples given in this paper, we will use a=10, and d=7.5 (partial dominance). The population properties also depend on the genotype frequencies; it is one of the tasks of this paper to discover these for the allogamous F₂.

The original parent populations set the parameters not only for the hybrid (F₁), but also for this F₂. The focal parent (P₁) is a population of individuals, with allele frequencies p₁ and q₁. As p₁ 1 (or 0), we are dealing with a pure-line population (or even a homozygous individual); but a focal-parent population may have any p₁ within the range 0 p₁ 1. The other parent (P₂) has frequencies defined as offsets from the P₁ frequencies, following Falconer (1981). This leads to p₂=p₁ - y, and q₂=q₁ + y, where y=p₁ - p₂. These are the same definitions as used for hybrids (Falconer, 1981; Gordon, 1999).

The RF population analogous to the allogamous F₂ is of special relevance, and is based on p_F1=p₁ - ½y and q_F1=q₁ + ½y (Gordon, 1999). It has the classical properties (Falconer, 1981):

A major task of this paper is to redefine these in terms of the F₂ 'offset' parameters, and to compare the two sets of results.

All other Methods form an integral part of the derivations which constitute the Results of this paper, and will be presented there.

Results

Genotype frequencies

Allogamous F₂

We adopt a fundamental means of deriving the genotype frequencies: we use the F₁ genotype frequencies (Gordon, 1999) as parental frequencies, and the cross-products of these form crossing (mating) frequencies. Next, we note the segregation ratios of each family of cross which, when multiplied by the crossing frequencies, gives the family frequencies of segregating progeny genotypes. When these terms are accumulated across all crossing families, we obtain the allogamous F₂ genotype frequencies. The F₁ parent frequencies and the crossing frequencies are presented in Table 1.

Table 1 - Crossing frequencies of F₁ genotypes ( and ) to produce allogamous F₂, together with segregations (A₁A₁: A₁A₂: A₂A₂) in the resultant progeny lines. Allele frequencies p₁ and q₁ refer to original focal parent, and y = p₁ - p₂, the offset between the two parent populations.

Full table (22K)

Gathering the outcomes for A₁A₁ progenies leads to:

Similarly, for A₁A₂ progenies:

And lastly, for A₂A₂ progenies:

We define the row-vector containing these results as f'.

These results reveal that for any allogamous F₂ from crosses with complementary opposite parents (i.e. p₂=1 - p₁), the heterozygote frequency is constant (f₁₂=0.5) across all values of p₁. Whenever the other parent has a fixed p₂, the general levels of heterozygosity are often lower than for the complementary opposites cases. The heterozygosity of any F₂ is generally less than that in its corresponding F₁; but the two are equal when p₁=p₂.

Equivalent RF and panmictic

Here, we use the classical RF genotype frequencies, and substitute (p₁ - ½y) for p_F1 and (q₁ + ½y) for q_F1 (Gordon, 1999). Thus:

We notice that these are the same results as obtained for the allogamous F₂, proving that the genotype frequencies at least are the same for the two situations. Note that these two results were derived in different but relevant ways, and obtaining the same outcome establishes their equivalence.

Allele frequencies

We need to check the allele frequencies of the F₂ using the genotype frequencies f'. This can be done as weighted means using the allele structures of genotypes as weights. Thus, for the A₁ allele, the weights are w^'₁=[1, 1/2, 0] for {A₁A₁, A₁A₂, A₂A₂}, respectively; and for the A₂ allele, the respective weights are w^'₂=[0, 1/2, 1]. The allele frequencies are then found as follows:

and

Note that these are the same as the F₁ frequencies (Gordon, 1999), which we would expect. (The RF frequencies were defined to these values at the outset).

Inbreeding coefficient

The natural inbreeding reference population for the F₂ is the F₁, as the former is sexually derived from the latter. The F₂ bulk population is a mixture of many different full-sib lines and half-sib lines, with varying degrees of coancestry amongst them. The simplest approach to defining inbreeding in this bulk mixture is the net drop in heterozygosity of the F₂ relative to the F₁. This approach precludes any need to unravel the breeding system mixture. Furthermore, it automatically reflects the dynamics of the bulk which depend on the parental inputs (p₁). Therefore, the inbreeding coefficient is defined as:

where f₁₂ of the F₂ is defined in this paper, and f₁₂ of the F₁ is defined in Gordon (1999). The degree of inbreeding in the allogamous F₂, for all possible focus-parent inputs (p₁), is visualized in Fig. 1, where the other-parent is complementary opposite or fixed p₂=0.01, 0.5 or 0.8 as examples.

Figure 1.

Inbreeding coefficient (phi ()) for the allogamous F₂ from crosses of the focus parent (A₁ frequency p₁), and the other parent either with complementary-opposite p₂, or with fixed p₂=0.5 or 0.8.

Full figure and legend (21K)

We observe that the level of inbreeding depends on the relatives-mixture present in the F₁ bulk, which changes as does p₁. Where p₁ 0 (or 1), for complementary opposite crosses, the hybrid is virtually entirely heterozygous, and the F₂ is equivalent to selfing a heterozygote; hence the inbreeding is ½. Again, for this same crossing system, when p₁ p₂ ½, the 'hybrid' is virtually a RF population, and the F₂ is equivalent to a second-generation RF in equilibrium with the 'hybrid'. Therefore, there is no change in heterozygosity, and 0.

Now, we will compare this F₂ heterozygosity against that of the equivalent RF, as an alternative reference. This is achieved simply by replacing f₁₂ of the F₁ with f₁₂ of the RF in the equation for . We recall that we have shown that both the allogamous F₂ and the equivalent RF (based on p_F1) have the same level of heterozygosity. Therefore, the inbreeding of the allogamous F₂ is zero when referenced against the equivalent RF population. This conforms with widespread notions already extant. But it is of doubtful relevance, because the natural reference population is the parental origin, viz. the F₁. The whole issue becomes more intriguing still if we use the mean heterozygote frequency of the two parents as the reference basis. We then find that nearly all allogamous F₂ are outbred (negative ) with respect to the parents, except at p₁ ½. This dependence of the value of inbreeding on an appropriate reference base was noted by its pioneer, Sewall Wright (Wright, 1921).

Population mean

Allogamous F₂

The F₂ mean (F¯₂) is +G¯_F2, where is the attribute background mean, and G¯_F2 (= also) is the gene-model mean. The latter is obtained as follows:

The ₁ is the usual average allele substitution effect for the focal parent population (e.g. Falconer, 1981), and equals a + (q₁ - p₁)d. This is the same result as presented by Falconer (1981) using another method of derivation, and the - ½y²d represents the drop in hybrid vigour, which was his focus. Several examples of this mean are shown in Fig. 2, all for partial dominance (a=10, d=7.5, =10). Figure. 2 shows both complementary opposite parents and fixed other-parent p₂=0.5, and compares both the F₂ and F₁ means. The loss of vigour is obvious, especially for p₁ 0 (or 1). Two other F₂-mean examples are shown to illustrate extremes, namely fixed p₂=0.01 and fixed p₂=0.8.

Figure 2.

Population mean of the allogamous F₂ from crosses between the focus parent (A₁ frequency p₁), and the other parent either with complementary-opposite p₂, or with fixed p₂=0.5, 0.8, or 0.01. The corresponding F₁ means are given also in the first two cases, for comparison.

Full figure and legend (24K)

It is interesting to note the patterns of increase in the mean with increasing overdominance. The effect of parental complementarity (i.e. p_F1 = ½ always) leads to the mean being constant across all p₁, but rising in steps with increasing overdominance. The patterns with other forms of parental relationship (with respect to p₁ and p₂) are more varied, but there is still a general increase in the mean with increasing overdominance.

Equivalent RF and panmictic

We use at once the classical RF definition of the gene-model mean (see General Method), and substitute into it the values for p_F1 in offset form. Thus:

Through this appropriate independent derivation, we find that the RF mean is the same as the F₂ mean. We now have proved that in two properties the allogamous F₂ and the RF population are identical, viz. genotype frequencies and gene-model mean.

Genotypic variance

Allogamous F₂

The allogamous F₂ genotypic variance has the Expectation

where the right-hand terms are an unadjusted sum-of-squares (USS) and a correction factor (CF), respectively. Using the previously defined f' and g' vectors and the scalar , and after subtracting the CF from the USS followed by gathering and factoring several terms ^*, the SS (i.e. ²_G the genotypic variance ^†), becomes:

Now, recalling the classical definitions of ²_A, and of ²_D (Falconer, 1981), and of cov(ad) and ²_d, (Gordon, 1999), this whole expression resolves into:

The previous equation (eqn 2) is simpler, but this one provides some link with the traditional nomenclature. However, there is some danger in it, for these components are not the genic and dominance variances of the F₂ ²_G. We will explore that issue in due course.

The variance can be visualized for our partial dominance and complementary opposites and fixed other-parent p₂=½ crosses in Fig. 3. In each case, it is compared also with the ²_G of its F₁.

Figure 3.

Genotypic variance (VG) of the allogamous F₂ from crosses between the focus parent (A₁ frequency p₁), and the other parent either with complementary-opposite p₂, or with fixed p₂=0.5. Partial dominance is depicted (a=10, d=7.5). The corresponding F₁ variances are given also for comparison.

Full figure and legend (25K)

Notice from Fig. 3 that this F₂ variance is generally greater than that of its F₁, except for the region p₁ ½ where they are virtually equal. This was one of the questions we posed at the beginning. For the complementary opposites cross, the F₂ variance greatly exceeds that of its F₁ as p₁ 0 (or 1). Another conspicuous result for the complementary opposites is that the variance is constant across all p₁; this results from the facts that p_F1 is always ½, and F₂ genotype frequencies are constant for such crosses. The results for the fixed p₂=½ crosses are conspicuously different. These general differences between the F₂ and F₁ variances have interesting consequences regarding ongoing selection efficiency, and the balances between selection and hybrid vigour, which we will discuss later.

Equivalent RF and panmictic

As before, we establish an independent derivation by using the classical RF approach, and substitute into it the offset p_F1 structures. Therefore, for the RF base ²_G=²_A+²_D, where ²_A is the genic variance and ²_D is the dominance variance. We define each of these separately, and accumulate them at the end.

(i) Genic variance

Substituting previous definitions of ²_A, , ²_d, cov(a,d) into this, it becomes:

It is convenient subsequently to leave the several ²_d terms separated rather than gathered together.

We can note immediately that this equation converts the genic variance of one RF population (the focal parent) into the genic variance of another RF population (the p_F1 equivalent), which presages things to come. Notice that this genic variance absorbs several genetical terms, involving not only the focal parent genic variance, but also cov(a,d), average allele substitution effect (), gene-model homozygote effect (a), and much of the gene-model heterozygote variance (²_d) ^*. Definitely, this is not simple in terms of the gene-model interpretation; but this difficulty extends to the widely accepted classical (RF) genic variance as well.

(ii) Dominance variance

After substituting previous definitions of ²_D and ²_d this becomes:

As previously, it is convenient to leave the ²_d terms separated.

Once again, we have a conversion equation from one RF (parental) to another RF (hybrid equivalent). Notice that this dominance variance collects together all of the gene-model heterozygote variance (²_d) not absorbed into the genic variance. It can be shown that this occurs also in the classical definition of dominance variance.

(iii) Genotypic variance

Upon accumulating eqns (5) and (7), four cancellations occur (terms 2, 3, 4 and 5 of eqn 7 with respective terms 7, 8, 9 and 10 of eqn 5) and one half-cancellation occurs (the two last terms), leading to the RF genotypic variance:

Comparison of eqn (9) with eqn (2) reveals that the genotypic variance (²_G) of this RF population is identical with that of the allogamous F₂. That is, we have now proved that these two populations are the same for yet another key property. An implication is that the allogamous F₂ is an origin of RF populations. In the next section we will also make a comparison against conventionally obtained RF variances based on p_F1, which will affirm this result.

Genic and dominance variances

Having shown that the allogamous F₂ and the equivalent RF are the same in several key properties, including the genotypic variance, it is obvious that the genic (²_A) and dominance (²_D) components of the RF population are also those of the allogamous F₂. Furthermore, these two components account for all of the genotypic variance, as expected from the classical RF variances (e.g. Falconer, 1981). Even some forms of epistasis may be included as part of the dominance/overdominance specifications. Therefore, eqn (5) or eqn (3) defines the genic variance of the allogamous F₂ (as well as of the equivalent RF), and eqn (7) or eqn (8) defines its dominance variance. These are visualized in Fig. 4 for our partial dominance example, where they are compared with the overall genotypic variance as well. Both the complementary opposite and the fixed p₂=½ parental examples are depicted.

Figure 4.

Genic variance (VA) and dominance variance (VD) components of the genotypic variance (VG) of the allogamous F₂ from crosses between a focus parent (A₁ frequency p₁), and other parent either with complementary-opposite p₂, or with fixed p₂=0.5. Partial dominance is illustrated (d=7.5 with a=10).

Full figure and legend (26K)

We noted earlier that complementary opposites genotypic variance was constant across all p₁, owing to the result that p_F1=½ in this situation. We see that the same outcome applies to the genic (²_A) and dominance (²_D) variances as well. The fixed p₂=½ case reveals that the genic and dominance variances curve-shapes are reminiscent of a segment of the curves for classical RF populations. Indeed, this is exactly the situation. If we remember that the x-axis in these graphs is p₁ and not p_F1, and that only a segment of the possible p_F1 spectrum can result from our example parentage, we then can observe the fixed p₂=½ curves as a segment of the entire RF distribution based on its own p. This is shown in Fig. 5, where classical RF variances for our examples are graphed. The x-axis is now p_RF (i.e. p_F1 in our present context). The 'window' between the two vertical lines shows the same variances as in Fig. 4 for the fixed p₂=½ case; and the vertical dashes at p=0.5 show where lie the variances for the complementary opposites case. This brings all of our present variances into line with the more familiar RF variances.

Figure 5.

Genotypic (VG), genic (VA) and dominance (VD) variances of a classical random fertilizing (RF) population across the Universe of allele frequency p, for partial dominance (d=7.5 and a=10). Allogamous F₂ form subsets of this Universe in which p is equivalent to p_F1 from a cross between a focal parent (with A₁ frequency p₁), and an other parent (with p₂ defined as an offset y from p₁). The values marked at p=0.5 are those of the F₂ in which the other parent is a complementary opposite to the focal parent (over all p₁); while the window between the vertical lines is the sub-Universe for all p₁ crossed with fixed p₂=½. These are actually the same variances as shown in Fig. 4, where, however, the x-axis is p₁ rather than p_F1.

Full figure and legend (27K)

Overdominance (d > a) has a marked effect on both genic and dominance variances. The dominance variance generally increases with increasing overdominance. However, the effect on the genic variance is more dynamic. For complementary opposites crosses, overdominance does not alter the genic variance, which is constant for all p₁ and d. For other kinds of parental relationships, the genic variance rises with increasing overdominance for p₁ < ½, and drops with increasing overdominance for p₁ > ½. With extreme overdominance, the genic variance is zero for some p₁.

Heritabilities

As all the genetical variances of the allogamous F₂ and the RF population are identical, we can discuss heritabilities for both. The broad-sense heritability (h²_B=²_G/²_P, where ²_P=²_G+²_E, and ²_E=25 in the examples) generally increases with increasing overdominance. For other parent fixed p₂ crosses (as opposed to complementary opposites), there is a decline in broad-sense heritability as p₁ increases.

Narrow-sense heritabilities (h²_N=²_A/²_P) are of greater interest than broad-sense because of their prominence in selection theory, and because they exist for this F₂ case. Those for the complementary opposites crosses are constant across all levels of p₁, as expected from the constancy of the genic variances. Notice, however, that the level of this heritability drops with increasing dominance, because of the rising genotypic variance against the constancy of the genic variance (see earlier paragraph). For cases involving fixed other-parent p₂, the levels of heritability have various patterns which reflect the underlying patterns of the genic variance (see earlier paragraph). Generally for these F₂s, h²_N is higher for p₁ < ½, indicating that genetic advance will be greater in such parent combinations. As noted earlier, some combinations of a, d and p₁ lead to zero genic variance, and hence zero h²_N and zero genetic advance from selection.

Discussion

Randomly fertilized populations

Quantitative genetics is built largely around the randomly fertilized (RF) population, or its inbred derivatives. Also, it is often an added implication that the population is one RF gamodeme in which the frequencies of alleles are uniform over the entire area of the population (i.e. panmixia). Research on gene-flow (Levin and Kerster, 1974) has revealed that such a uniform one-gamodeme view is often unreasonable. For instance, of the examples tabulated in Richards (1986) (Table 5.10, p. 179), five-eighths of the examples definitely were not panmictic. However, an alternative realistic approach exists, i.e. to consider the population as a bulk of small dispersing RF gamodemes (Wright, 1943, 1946). This fits well with the natural evidence cited above. Nevertheless, we have shown here that the allogamous F₂ is a RF population, and that the assumption to this effect is valid! Furthermore, we have discovered a plausible origin for RF populations in the allogamous F₂ from hybrid bulks. We should realize that this RF may not necessarily be panmictic, however, and certainly not with the passage of subsequent generations when it will almost certainly become dispersed into smaller RF gamodemes.

Also, we now have the equations for converting one RF population (the focus parent) into another RF population (the derivative allogamous F₂). Such re-construction of RF populations via hybridization would be expected to occur commonly in natural genetics; and it is a frequently planned activity in plant breeding.

Inbreeding

It is widely assumed that a RF population has zero inbreeding. This is largely because of its being the philosophical base of Quantitative Genetics for which it may be defined as the "natural" reference for a definition of inbreeding. The problem is that, until now, we had no clear view of a natural history of which RF populations were a part. However, this paper has demonstrated that an allogamous F₂ is an origin for RF populations, thereby providing an external reference by which its inbreeding can be measured. The sexual pathway from F₁ to F₂ unequivocally shows that the natural reference population for the F₂ is its own F₁; which, in turn, has its properties set by the parents which crossed to produce it. This paper has derived the inbreeding of the allogamous F₂ (and equivalent RF population) with respect to that F₁. On that basis it has considerable inbreeding (see Fig. 1), the value of which depends on the relatives mixture arising from the values of p₁ and p₂ in the original parents. We have already noted that, if we adopt the RF population as the reference base, then inbreeding is zero for the allogamous F₂; conversely, if we adopt the parental mean heterozygosity as the base, then this F₂ is outbred (with negative ).

Another issue is that it is difficult to define inbreeding in terms of full-sib, half-sib or other relationships. These terms strictly apply only to the progeny of individual parents, and not to the bulk progeny of parental populations. The proportions of full-sib, half-sib, cousin-like and backcross-like progeny outcomes within the F₂ will vary considerably according to the original parent's p₁ or p₂, making it difficult to resolve the inbreeding by this approach. The method used here (based on the net level of heterozygosity) is immediately exact and straightforward, and permits ready comparison with other kinds of populations, including all the classical situations mentioned above. For the complementary opposites, we have already noted that ½, the level for selfing (generation 1), as p₁ 1 (or 0). And for most crossing systems, there is a p₁ region for which 0. The equivalent of full-sib (generation 1) ( 0.25) and of half-sib (generation 1) ( 0.125) inbreeding can also be observed at different p₁ depending on the crossing system (i.e. the relativity of p₁ and p₂).

Selection vs. hybrid vigour

Genetic advance (_G) from forward selection increases with larger phenotypic variance and higher narrow-sense heritability (e.g. Falconer, 1981), as well as with stronger selection pressure. We now have the equations for these quantities in the allogamous F₂, which enables us to estimate the relative effects of forward selection, hybrid vigour and inbreeding depression. In an earlier paper, similar properties were presented for the F₁ (Gordon, 1999): where it was noted that maximum hybrid vigour arose from crosses with extreme p₁, whereas maximum selection advance arose from central values of p₁. With complementary opposites crosses, all F₂, irrespective of p₁, will have the same genetic advance, as discussed earlier for genic variances. With F₂ from fixed p₂ crosses, greater selection advance occurs asymmetrically, for noncentral p₁. The mean, cumulative genetic advance, and vigour for selection pressure of P=0.1 are shown in Fig. 6, using individual selection with both sexes equally selected, and with partial dominance (a=10, d=7.5). A complementary opposites cross, with p₁=0.05 is shown in Fig. 6(a), while a similar cross with p₁=0.45 is shown in Fig. 6(b). Concatenating the previous F₁ results with the present results, we would expect considerable hybrid vigour followed by some selection advance in the first case. In the second case, we expect minimum vigour followed by optimum selection advance.

Figure 6.

Generation performance from focus parent (P₁), through hybrid (F₁) and allogamous F₂ (F₂) to the allogamous F₃, with individual selection (P=0.1) at F₁ F₂ and F₂ F₃, under the partial dominance and environment variance defined previously. The mean and its components [accumulated selection advance (delta- G), and hybrid vigour from the mid-parent (vigour)] are given for two parental situations: (a) focal p₁=0.05, and (b) p₁=0.45. The other parent in each case is a complementary opposite.

Full figure and legend (34K)

In (a), hybrid vigour is observed, but it is followed by noticeable inbreeding depression from F₁ F₂. Selection begins also at this stage, but it barely offsets the effects of the depression. Finally, in selecting F₂ F₃, some real advance is made from the selection. With the case in (b), both vigour and depression are trivial, and both stages of selection yield relatively large advances. The final result at F₃ is much better in case (b) than in case (a). These results are caused solely by the different properties of F₁ and/or F₂ for different regions of the p₁ range. It appears that, in these examples, selection was more effective than hybrid vigour.

Where overdominance occurs, hybrid vigour is more conspicuous, but so is subsequent depression. Selection advance also is greater with overdominance, and always overcomes the depression to make subsequent substantial advance. In addition, if we utilize dispersion by employing selection strategies such as Combined Selection (the best individuals from the best lines), the advantage to selection becomes much greater still.

Notes

^* The following relations have been used to resolve these equations:

^† Because these SS are based on frequencies rather than counts, they equate immediately to mean-squares in fact.

^* Define the gene-model heterozygote variance as ²_d=2pqd². Also define the gene-model homozygote variance as ²_a=2pqa².

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