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BAG. Journal of basic and applied genetics

On-line version ISSN 1852-6233

BAG, J. basic appl. genet. vol.18 no.1 Ciudad Autónoma de Buenos Aires Jan./June 2007



Genetics and statistical association between lethal alleles and quantitative yield factors in maize (Zea mays l.)

Salerno, Juan Carlos¹; Kandus, Mariana¹; Boggio Ronceros, R....²; Sorarrain, O....²; González, Cecilia²; Almorza, David³

(1) Instituto de Genética “Ewald A. Favret”; INTA-Castelar. C.C. 25-1712, Castelar. Argentina.
(2) Facultad de Ciencias Agrarias y Forestales. Universidad Nacional de La Plata; Argentina.
(3) Departamento de Matemáticas. Universidad de Cádiz, España.

Corresponding author: Ing. Agr. Juan Carlos Salerno
Instituto de Genética “IGEAF”. C.C. 25-1712; Castelar, Pcia. de Buenos Aires, Argentina.


The objective of this work is to study the genetic process related to the hybrid vigour (heterosis) and the identification of quantitative trait loci (QTL) in maize.
A biometric model was built using the mathematical formalism relating to the discrete absorbent Markov chain in canonical form, in order to analyse the evolution of chromosome segments, with recessive lethal genes linked with grain yield factors through generations. The grain yield obtained from an inbred maize line regulated by a balanced lethal system, and other lines without this system as control, were evaluated during five inbreeding generations. The use of the Markov chain theory for the study of evolution during these generations was a different approach of the classical mathematical ones.

Key words: Maize; Lethal alleles; Heterosis; Markov Chains; QTL.


El objetivo de este trabajo es estudiar los procesos genéticos relacionados con el vigor híbrido (heterosis) y la identificación de QTL (Quantitative Trait Locus) en el maíz. Para obtener los resultados, se construyó un modelo biométrico usando los formalismos relacionados a las cadenas absorbentes discretas de Markov en forma canónica, con el fin de analizar la evolución de segmentos cromosómicos, con genes recesivos letales ligados con factores de rendimiento en grano a través de las sucesivas generaciones. El rendimiento en grano obtenido de una línea de maíz regulada por un sistema de letales balanceados y otras líneas sin este sistema como testigo, se evaluó durante 5 generaciones de endocría. El uso de la teoría de las cadenas de Markov con el fin de estudiar la evolución durante estas generaciones fue una aproximación diferente a los métodos matemáticos clásicos.

Palabras claves: Maíz; alelos letales; Heterosis; Cadenas de Markov; QTL.


In recent research work, maize has been used as a model for the study of molecular markers for the identification, mapping and manipulation of major genes affecting quantitative yield factors in plants. Experimental results from numerous mapping studies have shown that quantitative traits are controlled by some factors with major effects (Stuber, 1995). These results are in agreement with previous conclusions obtained long time ago in our experimental station at Castelar (Argentina) by Favret and Godeck (1959), and Favret and Ryan (1966), in barley. In 1978, we began the present study in maize; some of our findings have been partly reported by Salerno (1981), Salerno and Favret (1984), Salerno and Gardner (1985), and Salerno et al. (1986), with similar conclusions to those reported by Stuber (1995).
The present study focuses on the existence of some linkage groups in which lethal infertile genes associated with yield factors are present. These groups should be considered as special groups of major genes affecting quantitative traits in plants or animals. This hypothesis is supported by the finding of natural balanced lethal systems (Crumpacker, 1967). These systems seem to be useful to maintain the linkage because of the high frequency of lethal genes in the population. They are also useful to show the maximum genetic load (Muller, 1950) that plants or animals can tolerate (Carson, 1967). Concerning genetic load, Wallace (1970, 1981) has published a complete review on this subject. Crumpacker (1967), Gustafsson (1946), and Burnam (1993), among other authors, have revised the genetic load showing that the number of recessive mutations is quite high in different populations, either wild or domesticated, including also human beings.
These considerations moved us to study the maize balanced lethal system as a system itself in order to identify, isolate and keep up through the balanced lethal systems.
It is also useful to predict the number of generations during which a lethal system maintains the linkage, and the distance between genes with two or more lethal alleles at the repulsion phase, so as to get permanent heterozygosis without recombination. Thus, the theory of discrete Markov chains in space and time was used for the analyses (Barucha and Reid, 1960; Bosso et al., 1969; Sorarrain et al., 1980; Boggio et al 1997; Salerno et al., 1999).

Material and Methods

We have analysed grain yield corresponding to different kinds of selfing cross lines including lethal factors in three maize populations: a) Colosal original population, flint (type of endosperm grain), variety population of open pollination; b) Colosal with genetic load, same population with presence of lethal factors like deficient chlorophyll mutations; c) Colosal without genetic load, same population without chlorophyll mutations.
From these populations, BLS1 Argentine flint line was obtained for this study. It is regulated by a balanced lethal system located at the pericentric region of chromosome six, and present two varieties: BLS1 1:1, with two lethal alleles; and BLS1 3:1, with only one lethal allele.
The results obtained about the grain yield are compared with that corresponding to different kinds of public lines: N28; B73; MO17 and B14. These are inbred lines used as control during five years at the same location.
In order to check our findings, we considered another Argentine flint line regulated by a balanced lethal system, BLS14, and it was then compared through a grain yield experiment using the same line without lethal alleles.
In all these lines a randomised block was designed with three replications in a row of 7 x 1.4 meters. Two kernels per gap were planted, leaving one plant per gap at harvest in order to have 60 plants per plot. The experiments were performed at the same station in Castelar.
Finally, we evaluated the combining ability between this balanced lethal lines with public lines with commercial hybrids during two years, at two locations during the first year (Junín and Castelar), and at three locations, the second year (Junín, Castelar and Los Hornos). A randomised block was then designed with two replications in a row of 7 x 1.4 meters. Two kernels per gap were planted, leaving one plant per gap at harvest in order to have 60 plants per plot.
The model for the analysis of the states comprises different kinds of selfing crosses, and it has been performed assuming each state as defined by self fertilisation of any genotype, as shown in figure 1. These genotypes belong to some isolated populations located at a limited closed field. The genotypes entail the action of two genes, and in both loci, lethal alleles l1 and l2 have been found in natural form in the population.

Figure 1. Means separated chromosomes (actually in the same chromosome with negligible linkage). Cases 1-3. Fully linked chromosomes.


A) About the model

Since the model involves (we have) a self fertilisation, the lethal effect at the homozygous state of alleles l1 and l2 will be considered as recessive, since any of them appears in both chromosomes for each locus, in crosses CC, FF and GG. Obviously, the same occurs in II cross, where recessive alleles were found in both loci. These genotypes must be also considered as infertile because they will disappear from the field for the following generation.
According with the absorbent Markov chains theory, genotypes A, C, F, G, H and I act as absorbent states. Also, during self fertilisation of genotype A, the same cross is obtained in the following generation; therefore AA is also an absorbent non-lethal state. Transient states are only BB, DD, EE and also KK (figures 2 and 3).

Figure 2. Non balanced system (lethals in the same chromosome)

Figure 3. Balanced system (lethal in opposite chromosomes). Fully linked loci

Afterwards, beginning with selfing crosses for the independent segregation case (loci with very weak linkage, practically independent for the segregation), selfing crosses AA, CC, FF, GG, HH, and II can be recognised as absorbent states, while BB, DD, EE are equivalent to KK.
Under these conditions, following Mendel's laws for the calculation of the resulting probabilities for the selfing in the following generation, results are given by the transition matrix P written in canonical form, in the way shown in table 1 and figure 4.

Table 1. Matrix P in canonical form

Figure 4. Matrix P, in canonical form; r = 9, s=3.

For an absorbing chain, only transient and absorbent states must be considered. The transition matrix P can be written as four sub-matrixes: sub-matrix O, a zero matrix of s·(r-s) order; sub-matrix Q, providing the probabilities for the transition between transient states; sub-matrix R, denoting the probabilities between transient and absorbent states; and sub-matrix S, (rs)·(r-s) identity matrix. From this matrix P it is possible to calculate many characteristic parameters of the evolution process (Boggio et al. 1997). The relative frequencies of the genotypes from the starting to the n-th generation are given by the relation (1) or (2) (Bosso et al. 1969),

where Pn is a row vector with k-th component Pn,k, and P0, another row vector with j-th component P0,j. P0 denotes the relative frequencies of the starting generation. We shall now consider the following situations:

a) Independence between both loci.

When both loci are independent, starting from vector P0=(0,0,0,0,0,0,0,0,1), only genotype E is at the beginning in the field population. The relative frequencies for the non-lethal genotypes are shown in table 2, from the first to the sixth generation.

Table 2. Generations for independence between both loci (lethal states have been excluded from the table, and the relative frequencies normalized)

In these calculations, non-selectivity between both loci has been assumed; this is, non-selective advantages in frequencies between mono- and double heterozygous genotypes have been considered. In this case, the only absorbent non-lethal genotype increases in the way shown in figure 5.

Figure 5. Relative frequencies in the population (independent segregation) for AA (1) genotype.

b) The fully linked genotype E

The fully linked genotype E (figure 2) shows the lethal alleles in the same chromosome. In this case, starting at P0 with only genotype E in the population, this is, for fully linked loci (negligible crossing-over), and applying Mendel's rules, the elements of the last row of matrix P must be changed in the way shown in table 3.

Table 3. Matrix P in canonical form, starting with only genotype E in the population

Summing up, starting from the initial vector P0=(0,0,0,0,0,0,0,0,1) only two absorbing states must be considered, lethal II state and non-lethal AA, already mentioned. In this case, after a few generations, nonlethal genotypes A dominate in the population (table 4), in a similar way as shown in figure 5.

Table 4. Generations for the fully linked genotype E (relative frequencies for non-lethal genotypes)

c) The fully linked genotype K

For the fully linked genotype K (figure 3), with lethal alleles l1 and l2 in opposite chromosomes, this is, for the balanced lethal systems, we consider selfing without crossing over. A new matrix P should be defined including the row for lethal alleles l1 an l2 in the same chromosome. This new matrix P (r = 10 and s = 4) is shown in table 5.

Table 5. Matrix P in canonical form, considering fully linked genotype K.

Then, for the selfing crosses, the absorbent lethal infertile genotypes G and C and the non-lethal balanced genotype K are the ones which grow in the field. In other words, the balanced fully linked K is the only one that survives in the population
It should be interesting to include in the analysis the following theorems related with absorbent Markov chains (Kemeny and Snell, 1960):

Theorem 1: In any finite absorbing Markov chain, no matter the initial distribution P0, the probabilities that process will be finished in any one of the states belonging to the set of absorbing states tend to 1 as n tends to infinity.
Theorem 2: There are numbers b > 0 and 0 < c < 1, so that Pij n ≤ b·c n for any transient states Si and Sj, that is, Pij n → 0 as n →.

The former paragraphs and the last theorems allow us to conclude that after a small number of generations (in an interval from six to ten), only genotypes A and K remain in the population. The recombination rate results quite relevant since the crossing leads from genotype K to E in both ways, and from genotype E to the whole dominance in the population; in other words, lethal alleles tend to disappear, and this is the reason for using balanced lethal systems with negligible recombination rate.
The use of balanced lethal systems for the study of short segments is not new, but we have never found previous theoretical studies concerning their persistence in the population and their relation with other genotypes including zero, one or two alleles in a given segment.
Finally, genotypes A and K can be easily detected because genotype K is the only one able to produce infertile lethal descendents. This property is very significant for future applications.

B) Grain yield and balanced lethal system

The BLS1 Argentine line ith a balanced lethal system has a significantly higher grain yield when it is compared with the other lines (table 6). This could indicate a selective advantage of the heterozygous segment with lethal alleles in this part of the genome.

Table 6. Grain yield (kg/ha) of BLS 1 and public lines.

The BLS14 Argentine line with a balanced lethal system shows significant higher grain yield than the same line without lethal alleles (table 7). This is another proof about the association of grain yield with lethal alleles already mentioned.

Table 7. Grain yield (kg/ha) of BLS 14 line, with and without lethal.

C) Combining ability

There is a good combining ability between the lines regulated by a balanced lethal system and public lines (tables 8 and 9). In other words, the grain yield assays of balanced lethal systems (without asterisks) do not show significant differences with the single commercial hybrids obtained using the crossing of pure selected lines in every generation, as usual in commercial lines.

Table 8. Grain yield assays of maize single hybrids belonging to the Genetics Institute of INTA Castelar, in two locations (Junín and Castelar)

Table 9. Grain yield assays of maize single hybrids belonging to the Genetics Institute of INTA Castelar, in three locations (Junín, Castelar and Los Hornos)

These experiences suggest that the use of lethal alleles could increase yield grain assays. These lethal alleles can be maintained in the population in order to make the handling of hybrids cheaper in grain production, at least in some cases.


From the analysis of this work, we have concluded that for a given chromosomic segment containing recessive alleles, in few generations only two genotypes which have been denoted by A and K remain in the population. These genotypes can be easily distinguished, being genotype K an excellent marker for studies on chromosome segments and the association with yield in grain production. There are only two possibilities of getting recessive alleles in two loci defining a chromosome segment: with genotypes E and K, but only genotype K should be useful to maintain the heterozygous condition.
Lethal alleles seem to belong to the genetic load of the population. This genetic load might be useful for genetic studies and grain production in major genes related with recessive alleles since these heterozygotes can be compared with those produced with usual commercial techniques.
The theory of absorbing Markov chains can be used for predictions about the number of generations to maintain the heterozygous state as a function of the recombination rate (crossing-over) in these segments.


This research work was supported by INTA and PICT 10995/02 Grant from ANPCyT (Agencia Nacional de Promoción Científica y Técnica). Mariana Kandus has been granted an ANPCyT scholarship.


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