Introduction

In Mexico, the use of synthetic varieties (SVs) of maize (Zea mays L.) formed with pure high yield lines is not common, due to the scarcity of this type of parents (^{Espinosa et al., 2002}; ^{Luna et al., 2012}) and the limited availability of inexpensive seed of outstanding single cross hybrids. ^{Pérez-López et al.(2014)} mention this as the reason for recurring to the production of three-way (THs) or double-cross hybrids, but they are seldom sown recurrently by the same farmer. Some farmers exploit their advanced generations instead of buying original seed again. The resulting populations could be considered like the SVs that would form with the parents of the cultivated hybrids, but this may not be the case (^{Márquez-Sánchez, 2010}; ^{Sahagún-Castellanos and Villanueva-Verduzco, 2012}). Of the differences between these two types of populations, that due to the balanced participation of the genetic material of the lines in the SV that would be formed stands out, given that this does not occur when the parents are THs formed with the same lines, for example. In this case, each line of a single cross contributes half of the genetic material with respect to what is supplied by the third line of the three-way cross. This important difference is manifested in the inbreeding coefficient reported by ^{Márquez-Sánchez (2010)} for an SV formed with THs constructed with pure lines. The differences may also appear when there is genetic erosion. ^{Escalante-González et al. (2013}) consider that the randomness of the genetic mechanism, the finite number of individuals represented by each parent of an SV, and the heterozygous condition of its genotypes enable the loss of genes during its development. These authors studied the occurrence of genetic erosion during the formation of SVs whose parents are single crosses, and found that the loss of genes that are non-identical by descent (NIBD) is inversely related to the inbreeding coefficient of their parents. Furthermore, due to differences in the gene frequencies with which the lines participate in the SVs, there must be a loss of genes NIBD in different quantities among those formed only by lines and those generated by THs whose parents are the same lines, regardless of their inbreeding coefficient.

With respect to the formation of SVs from THs, the objective of the present investigation was to determine the magnitude of genetic erosion that would occur during the formation of the sample of individuals that represent each parent.

Materials and methods

The model of a locus in a diploid species reproduced by random mating was considered in this investigation. To this respect, it was assumed that the initial L lines with which the THs are formed are not related, that their inbreeding coefficient is F (0≤F≤1), and that L is a multiple of three. According to the characteristics of the lines, independently of the gene frequencies, it was assumed that each one of these contains 2 - F NIBD genes.

The number of NIBD genes that on the average are lost by a TH [(NIBD)_pt] was expressed as the initial number of the genes that have their three parents [3(2 - F)], minus the average number of NIBD genes that reach their m representatives. The variable represented by this number was defined as Y _m , and its average as E(Y _m ), or that is, (NIBD) _pt = 3(2-F) E(Y _m ).

It was assumed that the parents of a TH are A ₁ A ₂, B ₁ B ₂ and C ₁ C ₂, and without loss of generality, this is represented as (A ₁ A ₂ x B ₁ B ₂) x C ₁ C ₂, with genotypic array [(AGE) _CT ]:

Furthermore, their m representatives are the elements of a random sample taken with replacement of the population formed by its eight genotypes (Equation 1).

The possible results of the sampling with replacement of the offspring of a TH (Equation 1) include the extreme cases in which their m elements are the same genotype m times, and when the sample contains the eight genotypes of the progeny. For this latter case it is required that m≥8.

When m is large, there are a great number of possibilities of frequencies and orders of occurrence of the genotypes that integrate each sample, and in this case the number of different NIBD genes was determined (regardless of their frequency), and the number of forms (permutations) in which this number of genes may occur. When calculating the number of events that are equally possible and mutually exclusive that can be produced by the sampling (8^m), calculations were made of E(Y _m ) and the number of NIBD genes that are lost in the passing of the virtual population of a TH (genotypic array of Equation 1) to the sample that represents it.

Results and discussion

To derive a formula for E(Y _m ), it was considered that the sample of m size with replacement extracts g genotypes of those indicated in Equation 1 (g=1, 2,..., 8) and was defined as:

1) gkfim: frequency of genotype i (i= 1, 2, ...,g) in the k-th (k = 1, 2, ..., Ug) possible set of g frequencies of occurrence, each for one of the g genotypes that may be in the sample.

2) gkplm: frequency of the l-th smallest frequency of the k-th possible set of the frequencies of occurrence (l = 1, 2, ..., l _g ) each for each one of the g genotypes that may be in the sample.

Two quotients were constructed with both terms with which the following can be calculated:

1) The number of forms in which any g genotypes can occur in the sample with the k-th set of g frequencies of the type gkfimgk(NF)m, defined in a multinomial distribution as:

The number of permutations to assign the g elements of the k-th set of frequencies to two separate genotypes of the sample gk(NF)m:

Whatever the form of association between frequencies and genotypes, the number of NIBD genes they supply is the same.

The product gk(NF)mgk(NP)m (Ecuations 2 and 3) is the total of forms in which g genotypes can occur, regardless of what they are, in a random sample of m size. The sample with replacement is taken from the eight genotypes indicated in Equation 1 with the k-th set of frequencies.

If from the r-th group of gksrm different sets of g genotypes each set contains gkNrmNIBD genes (r = 1, 2, ..., n _g ), the sampling that captures g genotypes with the k-th set of frequencies should contribute with a total of NIBD genes NIPDgk[CG]m expressed as:

In a sample of m size the number of different genotypes (g) may be 1, 2, 3, ..., 8 (when ≥38). Furthermore, the number of sets of frequencies depends on g(k = 1, 2, 3, ..., Ug). With both considerations and by Equation 4, the expected number (average) of NIBD genes in the m representatives of a TH [E(Y _m )] is:

As the number of NIBD genes that contain the genotypic arrangement of a TH is 3(2-F), the average of NIBD lost by each three-way hybrid [(NIBD)_pt] was expressed as follows:

As an example, when m = 12 and g = 6, let’s say G ₁, G ₂, ..., G ₆, there are 11 sets of possible frequencies in which the six genotypes may occur (k = 1, 2, ..., 11; Table 1). If k = 1(set of frequencies for any six genotypes in which each one of five of these occurs once and the sixth one occurs seven times), the frequencies can be associated to the six genotypes in six different forms [6!/(5! 1!) = 6]. Furthermore, the number of forms in which these six genotypes can occur with m = 12 is 12!/(1!1!1!1!1!7!) = 95,040 [Table 1].

In this example, for each one of the 28 different sets of six genotypes the possible sets of frequencies are 11. Complementarily, the numbers of forms of assigning the frequencies of each set are shown 6k(NP)12, the corresponding numbers of the forms of occurrence of each set of six genotypes 6k(NP)12 and their product (Table 1).

If the six genotypes that appear in the sample were A ₁ C ₁, A ₁ C ₂, A ₂ C ₁, A ₂ C ₂, B ₁ C ₁ and B ₁ C ₂ (Equation 1), they would supply 5-2F genes NIBD. Based on the above, the information in Table 1 and applying Equations 1 to 4, their contribution when k=161[CG]12aE(Ym) is as follows:

To determine the total contribution when m = 12 and g = 6, we should consider the contribution of the remaining 27 sets of six genotypes which may be produced by the sampling and the amount of NIBD genes that each one contains (Table 2), as well as all of the data that permit the determination of E(Y _m ) for each one of the 11 values of m (Table 3).

Of the 28 different sets of six genotypes, four contain 5 2F NIBD genes each, and each one of the 24 remaining sets supplies 6 - 3F (Table 2). With the contributions of the 11 sets of possible genotypic frequencies (k = 1, 2, 3, ..., 11; Table 1), the total contribution can be calculated when m = 12 and g = 6 to E(Y _m ). That is (Tables 1 to 3):

The obtainment of this result involved the consideration already evidenced that the 28 combinations of six of the eight genotypes of the offspring of a TH (Equation 1) have the same probability of occurrence in the sample. Furthermore, with m≥8, the results with respect to genotypes obtained in the random sample with replacement are all of the possible combinations that can be formed with g of the eight genotypes for each one of the possible values of g (1, 2, 3, ..., 8).

The information of Tables 2 and 3 make it possible to calculate the numerator of the quotient that permits the calculation of the average of NIBD genes that reach the m representatives of the threeway cross. The denominator (8 ^m ) is the number of results equally possible and mutually exclusive that can be produced by the sample; to define the numerator, it is necessary to calculate the number of NIBD genes supplied by each one of the 8 ^m possible cases (with 8≤m). For example, according to the information of Table 3, for m = 12, which allows the obtainment of samples that may contain 1, 2, 3, 4, ..., 7 or 8 genotypes:

This result, E(Y ₁₂) = 5.8728 - 2.8733F, is important for breeders and specialists in genetic resources and is an indicator that one is close to what should be expected when m is “big”: no NIBD gene must be lost; that is, their number in the sample should be 6-3 F. From Table 4, various relevant facts are revealed: for a fixed value of F when m increases, E(Y _m ) increases, and more rapidly when F is smaller. Furthermore, with pure lines (F=1), a sample of 10 individuals of a three-way cross is enough to reach the equilibrium in three NIBD genes, and therefore, zero genes are lost. On the other hand, with F=0.5, this condition begins to be ostensible with m=12. It is also inferred that when F=0.75, starting from m=15 there is no loss of NIBD genes.

With an increasingly larger sample its number of NIBD genes tends to increase because each additional individual included is a new opportunity for it to have a genotype that supplies genes that were not yet part of that sample. This can have repercussions on the inbreeding coefficient; ^{Márquez-Sánchez (2010)} found that an SV formed with t THs (Sin _t ) constructed with pure non-related lines has an inbreeding coefficient (FSin _t ) equal to (3m + 1)/ (8tm). According to this result m is related inversely to FSin _t and the number of NIBD genes lost is reduced, which would predictable when FSin _t is larger.

For ^{Busbice (1970)}, the coefficient of endogamy of an SV is linearly and inversely related to its genotypic mean. In this context, the inbreeding coefficients of a synthetic variety formed with an even number of pure L lines (Sin _L ) and another formed with L/2 simple crosses (Sin _CS ) made with the same pure L lines are not dependent on m, are equal to 1/L, and the genotypic means of the two synthetic varieties should be equal. However, if L is an even number and multiple of three, the inbreeding coefficient of a Sin _t made with L/3 three-way crosses would be greater than 1/L according to the result reported by ^{Márquez Sánchez (2010)}. Furthermore, in consistence with the equality of its inbreeding coefficients, if F=1 the Sin _CS and the Sin _L never lose NIBD genes when the sample is formed from the m representatives of each one parent; also, the m representatives of each parent have the same genotype. On the other hand, the Sin _t loses NIBD genes (Table 4). ^{Sahagún-Castellanos and Villanueva-Verduzco (2012)} report losses of NIBD genes in synthetics formed only with double or single crosses (F<1).

Conclusions

Starting at the formation of the m individuals of each one of the parent three-way crosses, loss of NIBD genes may occur; this is higher when m is smaller and its parent lines have a lower inbreeding coefficient (F). In contrast, if F=1 a sample size of eight (m=8) is enough for this loss to be reduced to zero. If F=0.5 and m=12, the expected number of NIBD genes lost is 0.06. In addition, the imbalance of the genetic frequencies of the parents of each three-way cross produces a higher inbreeding coefficient in the Sin _t than in the Sin _L , which has a negative effect on the genotypic mean of the former.