Introduction

The majority of quantitative genetics is based on the properties of the disomic randomly fertilized population, either as a single panmictic gamodeme in equilibrium or as a dispersed inbred derivative of it (e.g. Wright, 1951, 1952; Crow & Kimura, 1970; Mather & Jinks, 1971; Falconer, 1981). The fundamental quantitative genetic concepts and details are defined with respect to such populations, and include most of the common facts: allele and genotype frequencies, population mean, genotypic variances, heritabilities, selection genetic advance and effects of inbreeding. However, there are other types of populations which are common in nature and in plant breeding for which these properties may not apply. These populations include hybrids (F1 ‘hybrid swarms’ and simple crosses), and segregating F2 bulks of either sib-crossed or selfed origin. These are, intrinsically, in disequilibrium by virtue of their very origin. The basic properties arising from the randomly fertilized population (RF) generally may not apply to them. In order to evaluate this, and to account for the effects of such populations in evolution and in plant breeding, it is necessary to reveal their quantitative genetic properties.

The present paper examines the quantitative genetic properties of the bulk hybrid produced by unhindered crossing between two parent populations.Footnote 1 Previous work has explored the mean of such an F1, having been stimulated by interest in hybrid vigour (Falconer, 1981). Also, some knowledge on the genotype frequencies within hybrids is widely extant, but only for the simple case of a hybrid between two complementary pure lines (homozygotes) (Wright, 1952; Kempthorne, 1956). Here, all of the properties mentioned earlier are derived from first principles, including an outbreeding coefficient. The main novel information from this paper is the definition of the hybrid’s genotypic variance (σ2G), and broad-sense heritability (h2B). We will discover that the genic variance (σ2A) and dominance variance (σ2D) cannot be defined, and are inappropriate for hybrids.

General method

The classical two-allele population gene-model is used (e.g. Falconer, 1981), where p=frequency of A1 (the better allele-expectation, or allele), and q=frequency of A2 (the lesser allele-expectation, or allele), without any omission (i.e. p+ q=1).

The genotype effects are deviates from the homozygote midpoint (MP), where a is the A1A1 homozygote effect (an expectation), −a is the A2A2 effect, and d is the heterozygote effect. The row vector of these effects is g′=[a, d, −a]. Notice that this is a biometrical model and not a single-gene model. In this form, it describes either the net results (expectations) of polygenic systems (Mather & Jinks, 1971; Jana, 1972), or the sensu strictu single-gene case. It is a single main-effect model only, and does not attempt to define epistasis as interactions amongst factorial main effects. Some forms of epistasis may be accommodated by overdominance (d > a). Although this is a simplification, it is a standard approach which has served quantitative genetics well for many decades (Mather & Jinks, 1971; Falconer, 1981).

Alternative models, with factorial main effects and interactions, fall into two categories: biometrical expectations over many loci (Cockerham, 1954; Kempthorne, 1956; Hayman, 1958), or sensu strictu polygene specifications (Seyffert, 1966; Jana, 1971). The latter refer to specified loci situations, such as transgenics, mutants, or simply inherited specific phenotypes; they are not considered in this paper. For this initial examination of the properties of the hybrid, we will use the classical, robust, biometrical simple main-effects model which is the foundation of quantitative genetics.

In most of the examples used in this paper, a=10 and d=7.5 (a case of partial dominance with d < a). When overdominance (d > a) is being considered, the following values of d are substituted: 15, 22.5 and 30. (In my experience, biometrical overdominance is extremely rare in practice, and allowing for a case of d=3a probably exceeds any reality.)

One random-fertilizing parent population (P1) is regarded as focal, with frequencies p1 and q1. The other random-fertilizing parent population (P2) is defined as an offset from this where p2=p1 − y, and q2=q1 + y, with y=p1 − p2. This follows the procedure of Falconer (1981), and defines the two parental populations within a single system. This is intrinsically better than Kempthorne’s approach (Kempthorne, 1956) which is incapable of revealing the interconnectedness arising from hybridization, i.e. the disequilibrium (y). As p1 → 1 and p2 → 0 (or vice versa) with y → ±1, a simple hybrid between two complementary pure lines is defined.

The methods used actually to derive the hybrid properties form an integral part of the Results of this paper, and are presented in that section.

It is often instructive to compare the F1 with a random-fertilizing population (RF) based on pF1. The properties of such a population are obtained in the usual way (see, e.g. Falconer, 1981) after incorporating the new allele frequency into the equations:

Results

The frequencies of progeny genotypes following hybridization are obtained as the products between the two parental arrays of allele frequencies [p1, q1] and [p1− y, q1 + y], assuming that each parent is contributing equally to the cross. Thus, we are considering the biparental cross (BiP) and hybrid-cultivar in plant breeding; and, in evolution, the mean bidirectional hybrid between two populations. The gamete union is presented in Table 1.

Table 1 Gamete fertilization outcomes and frequencies between two hybridizing populations
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These results lead readily to the following progeny genotype frequencies (row vector f ′):

As discussions of hybrids often focus on their heterozygosity, we examine f12 in, Fig. 1 where complementary opposite crosses (i.e. p2=1 – p1) are shown, together with a range of fixed other parents (i.e. p2=0, 0.15, 0.35, 0.5, 0.65, 0.85 or 1). Notice that a fully heterozygous hybrid occurs only when opposite pure lines (p1=1 and p2=0, or vice versa) are crossed.

Figure 1
figure 1

Frequencies of heterozygotes in the F1 (f12) from crosses between the focus parent (A1 frequency p1) and the other parent with various fixed p2. The complementary-opposite other parent is compared also.

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F1 allele frequency

The hybrid’s A1 ‘allele’ (allele-expectation or actual allele) frequency (pF1) is found as the weighted mean of the progeny (F1) genotype frequencies f ′. The weights reflect the proportions of the focus ‘allele’ in each genotype in the F1, i.e. for A1 the weights are w1′=[1, ½, 0] for the F1 genotypes {A1A1, A1A2 and A2A2}, respectively. Similarly, for qF1, the frequency of A2, the weights are w2′=[0, ½, 1] instead. Thus,

Similarly, the new A2 frequency is:

The frequency pF1 is examined in Fig. 2 for the complementary-opposite crosses mentioned earlier, and the previous series of fixed other-parents. Notice that complementary-opposite crosses always result in a hybrid allele frequency of 0.5, as do several other combinations of p1 and p2. In general, by rearranging the previous equations, the parental frequency required to obtain any nominated pF1 can be estimated from either p1=2pF1 − p2, or p2=2pF1 − p1, depending on which values are provided and which are sought. For the special case of pF1=½ these reduce to p1=1 − p2 and p2=1 − p1, respectively.

Figure 2
figure 2

Frequencies of the A1 allele in the F1 (pF1) of crosses between the focus parent (A1 frequency p1), and the other parent either with complementary-opposites p2 or with various fixed p2.

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Hybrid mean

The F1 mean (F¯1) is μ+G¯F1, where μ is the attribute background mean, and G¯F1 (=γ also) is the gene-model mean. This is obtained as follows, using the vectors defined earlier.

Here, α1 is the well known average-allele substitution effect for the focal parent population (see, e.g. Falconer, 1981), being equal to a + (q1 − p1) d.

This is the same result as obtained by Falconer (1981), but the derivation here is much more direct. The value of this F¯1 is explored in Fig. 3 for complementary-opposite crosses and moderate partial dominance (a=10, d=7.5 and μ=10). It is useful to compare it in Fig. 3 with the equivalent means of both P1 and P2, the parental mean (PM), and with the RF with pF1=0.5. (In order to save space, we will cease illustrating with fixed p2 crosses from here on.)

Figure 3
figure 3

Population mean of the F1 of crosses between the focus parent (A1 frequency p1) and the other parent with complementary-opposite p2. The two parent means (P1 and P2) are shown for comparison, along with the bi-parental mean (PM) and RF mean for pF1=½. (a=10, d=7.5, μ=10.)

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Notice that the F1 mean increases as p1 → 0 or p1 → 1, but it is never greater than the mean of the better parent (which is P2 when p1 < ½ and P1 when p1 > ½) in this example showing partial dominance.

Changes in dominance (particularly overdominance, in which d > a) elevate the mean considerably, especially as p1 → 0 or 1.

A common understanding of ‘hybrid vigour’ is that the ‘hybrid is more vigorous than the parents’ — i.e. both parents. The formal definition does not say that, however, as it defines vigour with respect to the mid-parent (i.e. the parental mean, PM). The common hybrid vigour definition is hv=F¯1-PM (Falconer, 1981), who also shows that hv=y2d in terms of the gene model. This is a very optimistic view of hybrid vigour, because midparent is never more than the mean of the better parent. An alternative view (hybrid vigour (b): Mather & Jinks, 1971) is more in line with the common concept of hybrid vigour, and is defined as follows:

When p1 > p2, parent P1 will be the better parent. We can then derive hybrid vigour (b) (hb) in terms of the gene model as follows:

With P1 as the better parent, y is positive: so hb will be positive (i.e. real) only when α1 is negative. The definition of α1 shows that this will occur when p1d > (a + q1d). As y becomes negative, it means that P2 is now the better parent, and we simply swap our definition base to hb=−yα2. In Fig. 4, the levels of hb from several degrees of dominance are examined. Hybrids with partial dominance do not express hybrid vigour (b); nor do any hybrids with central values of p1. Vigour maximizes as p1 → 0 or 1, and as overdominance increases.

Figure 4
figure 4

Heterosis from better parent (hb) in hybrids between a focus parent (A1 frequency p1) and the other parent with complementary p2. Using a=10, partial dominance has d=7.5; while overdominance increases as d=15, 22.5 and 30.

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Hybrid genotypic variance

The genotypic variance of a hybrid population has the usual expectation:

where the first right-hand-side term is an unadjusted sum of squares (USS) and the second term is a correction factor (CF)Footnote 2. The expanded components are as follows.

After subtracting the CF from the USS and effecting several gathering and factoring of termsFootnote 3, the sum of squares becomes:

Recalling the classical definitions of σ2A and of σ2D (Falconer, 1981), and defining cov(a,d)=4pqad, we can write:

The overall genotypic variance is visualized for our partial dominance and complementary-opposites cross in Fig. 5. There, it is compared also with the genotypic variance of the focal parent (P1), and of the RF population based on pF1=½ as before.

Figure 5
figure 5

Genotypic variance (Var G) of hybrids between a focus parent (A1 frequency p1) and the other parent with complementary-opposite p2. Partial dominance (a=10, d=7.5) is shown. Variance of the focal parent (P1) and of the RF for pF1=½ are compared.

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Notice that simple hybrids between opposite pure lines have no genotypic variance for the gene effect in question, as one would expect. For p1 < ½, the F1 has smaller genotypic variance than both the focal parent and the equivalent RF. For p1 > ½, its variance lies between that of the other two populations. This pattern is true even for overdominance: which greatly inflates the total amount of genotypic variance in all three population types.

The genic and dominance variances of hybrid populations present a problem of definition, and even of existence. Therefore they will be addressed in the Discussion rather than here in the Results.

Heritability

Having defined a genotypic variance for the F1, we have a corresponding ratio of σ2G to σ2P: that is, a broad-sense heritability. As genic variance is undefined, there is no narrow-sense heritability. The broad-sense heritability (h2B) is a biometrical parameter, it being the genotypic determination of the phenotype (Wright, 1951). This broad-sense heritability would therefore be appropriate for obtaining the genetic advance from forward selection from hybrids. It is shown in Fig. 6, using the various parameter values given elsewhere, and an example environmental variance of σ2E=25.

Figure 6
figure 6

Broad-sense heritability (h2B) under four levels of dominance (d=7.5, 15, 22.5, 30) for hybrids between a focus parent (A1 frequency p1) and the other parent with complementary-opposite p2. (a=10, σ2E=25).

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Note that the broad-sense heritability attains maximum values for a wide range of hybrids with p1 of central values, and increases with increasing dominance (d). We would expect therefore to see a maximal result from forward selection in such hybrids, with a minimal response in hybrids between extreme parents (i.e. complementary-opposite parents, and p1 → 0 or 1). This matter raises interesting discussion (later) about the relative merits of hybrid vigour and selection.

Outbreeding coefficient

The definition is based on the level of heterozygosity in the hybrid as contrasted to a relevant reference population: in this case a RF population with allele frequency pF1. It is as follows:

where all symbols have been defined previously.

This is also one approach to defining an inbreeding coefficient: but in this case the higher levels of heterozygosity in the F1 make it negative. This means it indicates ‘new heterozygosity relative to a RF population’, i.e. outbreeding. Its values for complementary-opposites crosses, and for fixed P2 crosses (p2=0, 0.5, 1) are shown in Fig. 7. For the complementary-opposites, outbreeding maximizes (that is φ → −1) for extreme values of y (p1 → 0 or p1 → 1): in other words, the proportion of heterozygotes approaches twice that of the comparable RF population. When p1=p2=½, the hybrid has the same heterozygosity as the corresponding RF population, and the outbreeding coefficient is zero.

Figure 7
figure 7

Outbreeding coefficient (phi (φ)) for hybrids between a focus parent (A1 frequency p1) and the other parent, with p2=0, 1, 0.5 and complementary-opposite.

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The situations for fixed parents are more variable. Three special cases are shown in Fig. 7 First, for p2=0.5, the curve shape across p1 is similar to that for complementary-opposite crosses, with the exception that φmin=−0.325 rather than −1. That is, the level of heterozygosity is never more than a third higher than in the reference RF population. The other two special cases involve the two pure lines (p2=0 or 1). In each case, the level of outbreeding changes from −1 (full heterozygosity) to 0 (RF heterozygosity) as the focus parent and the other parent change from being complementary-opposite to nearly identical. At the point of both parents being the same pure lines (p1=p2=pF1=0, or 1), there is no heterozygosity in any of the three populations, and outcrossing is zero.Footnote 4 (The term ‘F1’ is, of course, somewhat forced at this juncture.)

Discussion

Genic and dominance variances in the F1

It is well-known that the classical RF genotypic variance contains the two components: genic and dominance variances. From eqn (1), it is obvious that the F1 genotypic variance has a different composition. As the hybrid population structure is sexually ephemeralFootnote 5, the very concept of average allele substitution is questionable. This means that the σ2A component (eqn 1) should not be interpreted as the F1 genic variance, because such average allele effect is the very basis of such a variance (Falconer, 1981). It is simply a function of the focal parent’s genic variance which forms part of the definition of the F1 genotypic variance. Furthermore, the F1 genotypic variance (eqn 1) also contains a third component — excess cov(a,d) — which does not appear in the classical RF caseFootnote 6. These observations remind us that a hybrid is profoundly different from a RF population. Therefore, the concepts of genic and dominance variances appear inappropriate with respect to hybrids.

Selection efficiency

Several of the properties (heritability, genotypic variance, mean) have the very important implication that some hybrids will be more efficient for subsequent selection than others. Obviously, those with h2B→0 will be of little value for forward selection (i.e. selecting F1 individuals towards an F2 nursery).Footnote 7 Likewise, those with trivial σ2G will be of little utility, because σ2P→σ2E. In the case of our illustrative complementary-opposites crosses (see Figs. 5 and 6), hybrids between ‘extremely opposite’ parents (i.e. p1 < 0.2 or p1 > 0.8) are undesirable if forward selection is the intention, especially where overdominance prevails. These properties are illustrated in Table 2, where genetic advance (ΔG) at selection pressure of 10% (P=0.1) in selecting F1 → F2 is given, for partial and overdominance (d=7.5 and 30) and complementary-opposite crosses, with parental inputs which are intermediate (p1=0.35), optimal (p1=0.45), and extreme (p1=0.97). Background properties are given also, such as hybrid σ2P and h2B.

Table 2 Genetic advance (ΔG) at selection pressure of 10% (P= 0.1), variability (σ2P), broad-sense heritability (h2B), and mean of the F1 from complementary-opposites crosses for various focus parents (allele frequency p1) and two dominance levels (d = 7.5, 30 with a = 10)
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All of the trends discussed above are borne out by these values. Notice that the greatest ΔG occurs at the p1=0.45 example within each dominance level; whereas the highest hybrid mean occurs at p1=0.97 within each dominance level. The drop in σ2P for extreme p1 is conspicuous; and reduced h2B for extreme p1 is apparent. These hybridization outcomes affect natural selection as well, i.e. adaptation is optimized out of hybrid swarms between nonextreme parental populations, rather than those arising from extreme opposites.

For backward selectionFootnote 8, matters are completely different. The relevant property of the hybrid is the mean (or more specifically, its hybrid vigour), which is higher towards extreme ends of the p1 range (Fig. 3). As we are here using the hybrid’s mean performance as a progeny test of the parents, and not selecting F1 individuals onwards into the next generation, heritability is not an issue (except insofar as it is biometrically related to the standard error of the mean). So, in this case, the ‘most extreme opposites’ crosses would be the best, just as Falconer (1981) declared when he stated that y should be maximized to maximize hybrid vigour. The last column of Table 2 shows this for the illustrative example. This aspect of using hybrids appears to be entirely anthropomorphic: it is very unlikely to apply to evolution because retrospective re-making of particular crosses requires a discerning control of the programme. Also, we have shown that hybrids which maximize hybrid vigour do not maximize forward genetic advance, indicating that natural selection does not operate to maximize hybrid vigour.