--Fidel Márquez-Sánchez

With mathematical formulae based upon gene frequencies, the theoretical selection limit in allogamous plants is asymptotically reached at generation infinity. There, frequencies of the favourable alleles of the involved loci are supposed to be one. However, because linkage fixation occurs for both favourable and unfavourable alleles, there is more on the former than in the latter. This is why breeders try to avoid inbreeding by recombining selected plants at the end of each selection cycle. As in most autogamous plants recombining is not possible, inbreeding can not be avoided, thus the result of selection are superior inbred lines chosen among large numbers of them.

On the other hand, as massive crossing between populations in maize is quite possible, inbreeding is used as a means of utilizing heterosis by crossing inbred lines. Such utilization may be through either different types of F1 hybrids or through multi-line F2-synthetics.

If an F1-synthetic is subjected to random mating inbreeding depression occurs depending mainly on the degree of inbreeding of parental lines and on their number (S. Wright, USDA Bull. 1121, 1929; Gilmore, Crop Sci. 9:102, 1969; Busbice, Crop Sci. 10:265, 1970). For diploid plants the smaller the inbreeding of the parental lines and their number the higher the inbreeding depression (Busbice, 1970).

If selection of desirable plants could be made before pollination (called individual selection by D. S. Falconer, Intro. Quant. Genet., 1961, as opposed to mass selection), a recurrent selection process may be visualized as a cyclic F2-synthetic methodology. In each cycle the F1-synthetic is the composite of selected plants (conceptual single-cross hybrids) while the result of its random mating recombination is the F2-synthetic, called by maize breeders the ith cycle of selection (Ci). According to Wright's formula the prediction of an F2-synthetic is

F2 = F1 - F1 - P
                    n

On the other hand, any population can be considered as n/2 conceptual single-cross hybrids (the individual plants) resulting from crossing n conceptual homozygous lines (the male and female gametes); therefore if the base population, in which selection is going to be practiced is large, i.e., n is large, random mating does not cause any inbreeding depression (Márquez-Sánchez, Crop Sci. 19:439, 1979). However, as soon as selection takes place, the saved portion (selection pressure) consists of a relatively small number of plants (n is small), and inbreeding depression occurs. As selection advances, even if the selection pressure is kept constant, the number n of involved conceptual selected lines gets progressively smaller gradually increasing the inbreeding depression.

A mathematical function that describes the F1-synthetic behaviour through selection cycles is of increasing nature, for instance, an exponential function. Its higher limit is the conceptual single cross (n = 2) of highest theoretical yield, and reaches asymptotically its lower limit, the base population, at n=*.

In order to find out a corresponding function for the F2-synthetic Márquez-Sánchez (1979) used Wright's formula, demonstrating that both F1 and F2 functions approach asymptotically the base population (V). However, while the F1 function is of increasing nature, the F2 curve has a minimum at n - 2 and maximum at n = k. That is to say, with k specific lines the F2-synthetic of maximum yield is obtained. In Fig. 1 the graphical description of this model is shown (n: 2, 3, ..., k,... *).

Figure 1. General model for maize breeding through selection (V), synthesis (S), and hybridization (H). P, average of homozygous lines.

The model proposed for the selection process is shown in Fig. 2; it is the same as Fig. 1 but now the involved curves are drawn from right to left (n:*, ..., k, ..., 3, 2). If the selection limit is defined as the point where inbreeding causes the yield of the selected sample to decline, then the limit is reached when the number of conceptual selected lines is k and the number of selected plants is k/2.

Figure 2. Proposed model for selection limit in maize.

Empirical studies on maize use numbers of selected plants around 200 (out of 4000 or 5000), which for the first cycle of selection would mean an F2-synthetic of 400 conceptual lines, a number which causes a negligible inbreeding depression. However, as phenotypic similarity of selected plants tends also to cause gene and genotypic resemblance, sooner or later the number (n) of conceptual selected lines will have to decrease gradually down to k, where the selection limit for practical purposes has to be reached.

If the number of conceptual selected lines can be associated to the number of cycles or generations of selection through a mathematical function, the generation at which the selection limit so described is reached may be predicted. A simulation study with such a purpose is being carried out at our institution.

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