Crop Science 40:659-665 (2000)
© 2000 Crop Science Society of America
CROP BREEDING, GENETICS & CYTOLOGY
Variance of Marker Estimates of Parental Contribution to F2 and BC1-Derived Inbreds
Jiankang Wanga and
Rex Bernardob
a Laboratory Center, Henan Academy of Agricultural Sciences, Zhengzhou, Henan 450002, P.R. People's Republic of China
b Dep. of Agronomy, Purdue Univ., 1150 Lilly Hall of Life Sciences, West Lafayette, IN 47905-1150 USA
bernardo{at}purdue.edu
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ABSTRACT
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An Essentially Derived Variety is a cultivar or inbred that largely retains the characteristics of an ancestral cultivar or inbred. The parental contribution to F2-derived inbreds (pF2) and BC1-derived inbreds (pBC1) can be estimated with molecular markers. A recombinant inbred (RI) with pF2 or pBC1 greater than a specified threshold is then considered essentially derived. Our objectives were (i) to derive the variance of pF2 and pBC1, and (ii) to determine the probability of obtaining an essentially derived RI for different numbers of marker loci in different species. The variances of pF2 and pBC1 are a function of the number of chromosomes, length of each chromosome, and number of marker loci on each chromosome. The standard errors (SE) of pF2 and pBC1 were smallest when the two marker loci closest to the ends of each chromosome were included. The minimum values of SE(pF2) and SE(pBC1) are useful for setting minimum values of thresholds for declaring essential derivation. Suppose selfing from the BC1 is permissible and the maximum error rate for falsely declaring an RI is essentially derived is set at 2.5%. The minimum value of the threshold for these conditions is 0.881 in maize (Zea mays L.). For a threshold of 0.90, the probabilities of an essentially derived RI from the BC1 generation were >6% in rye (Secale cereale L.), >3% in barley (Hordeum vulgare L.), <3% in tomato (Lycopersicon spp.), rice (Oryza spp.), and maize, and <1% wheat (Triticum aestivum L.). These results suggest that the thresholds used to declare essential derivation should differ among species.
Abbreviations: cM, centimorgans • RI, recombinant inbred
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INTRODUCTION
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PARENTAL CONTRIBUTION is the proportion of the genome contributed by a parent to its inbred progeny. Expected parental contributions with Mendelian inheritance are 0.5 for either parent of an F2-derived inbred, 0.75 for the recurrent parent of a BC1-derived inbred, and 0.25 for the donor parent of a BC1-derived inbred. Selection and genetic drift during selfing may cause differences between observed and expected parental contributions to inbred progeny (Lorenzen et al., 1995; Bernardo et al., 1997). Parental contribution determined from pedigree records may therefore be inaccurate.
In 1991, the Union Internationale pour la Protection des Obtentions Végétales established the concept of an Essentially Derived Variety, i.e., when "the essential part of the genome of an initial variety has been included in the new variety" (Smith et al., 1995). Molecular markers are useful for estimating parental contribution, and the use of molecular markers for assessing essential derivation has gained widespread acceptance (Dillmann et al., 1995). Implicit in the concept of an Essentially Derived Variety is that thresholds will be established beyond which an inbred will be declared essentially derived from an initial (i.e., parental or ancestral) inbred. A consensus has not been reached regarding appropriate thresholds, although a threshold of 0.90 has been proposed for maize (Smith et al., 1995).
The variance of marker estimates of parental contribution, among a set of RIs, is crucial in determining appropriate thresholds for essential derivation in different crop species. Such information is needed to calculate the probability of obtaining an essentially derived RI from an F2 or BC1 population in the absence of selection. The variance of marker estimates of parental contribution would vary among species because of differences in the number of chromosomes and length of each chromosome. Because of linkage among marker loci, a simple binomial distribution is not applicable for the variance of marker estimates of parental contribution.
Assessing the amount of variation among random RIs, in terms of their marker estimate of parental contribution, is conceptually different from determining the minimum number of marker loci for estimating the parental contribution for a specific RI. We do not make any recommendations regarding the number of marker loci needed for assessing essential derivation. Specifically, our first objective was to derive the variance of marker estimates of parental contribution for different lengths of chromosomes and numbers of marker loci on each chromosome. Our second objective was to determine the probability of obtaining a random F2- or BC1-derived RI that is essentially derived, given different thresholds and numbers of marker loci in different species.
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Theory
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Genetic Model and Notation
We considered an RI derived from a (P1 x P2)F2 or [(P1 x P2) x P1]BC1 population. The species has n pairs of chromosomes. On the kth chromosome, P1 and P2 are polymorphic at lk mapped, single-locus markers. Across all chromosomes, P1 and P2 are polymorphic at
marker loci. The marker genotypes are M1M1M2M2 ... MlMl in P1, and m1m1m2m2...mlml in P2.
The lk marker loci are evenly distributed along a chromosome that is D centimorgans (cM) long. The distance (d) between adjacent loci depends on the placement of markers on the chromosome. We considered two models for the placement of marker loci (Fig. 1) . In the Terminal Marker Model, the distance between adjacent markers is d = D/(lk – 1), and the marker loci are at the 0, d, 2d, ..., D cM positions on the chromosome. The two marker loci that map closest to the ends of each chromosome are chosen first, then marker loci located between these two terminal markers are chosen next. In the Nonterminal Marker Model, the distance between adjacent marker loci is d = D/lk, and the marker loci are at the (0.5)d, (1.5)d, (2.5)d, ..., (lk – 0.5)d cM positions.
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Fig. 1 Locations of five marker loci on an 80 centimorgan (cM) chromosome with the Terminal Marker Model and Nonterminal Marker Model. The kd values are the distances between adjacent markers on the kth chromosome
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For the kth chromosome, marker estimates of parental contribution are denoted as kpF2 for an F2-derived inbred and kpBC1 for a BC1-derived inbred. Across all chromosomes, marker estimates of parental contribution are denoted as pF2 for an F2-derived inbred and pBC1 for a BC1-derived inbred.
Individual Chromosome
F2-Derived RI
At the kth chromosome,
is an indicator variable that is equal to 1 if the RI has the same marker genotype as P1, and 0 if the RI has the same marker genotype as P2. The number of marker loci at which the RI and P1 are homozygous for the same allele is
. The parental contribution at Chromosome k of P1 to the F2-derived RI is estimated as
, with a mean and variance of
The mean and variance of kX are
Consider two linked marker loci, i and j (i < j). With a single meiosis, the recombination frequency between i and j is rij. The frequency of recombinant genotypes among RIs is
(Haldane and Waddington, 1931). On the basis of the expected frequencies of marker genotypes among RIs (Table 1)
, the means, variances, and covariances of kXi and kXj are
where
. Let dij be the distance in cM between i and j. We used the Kosambi (1944) mapping function because, unlike the Haldane mapping function, it allows for modest crossover interference in adjacent marker intervals. With the Kosambi mapping function,
. Consequently,
.
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Table 1 Frequency of marker genotypes, at Loci i and j on Chromosome k, among recombinant inbreds (RI) derived from a (P1 x P2)F2 population
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Consider three linked marker loci, i, j, and h. If Locus h is between i and j, then dij = dih + dhj, and
. Therefore, the mean and variance of kX are
 | (1) |
If the lk loci are evenly distributed along the chromosome, the distance between any two adjacent markers is dij = d. For both the Terminal Marker and Nonterminal Marker Models, the frequency of recombinants between any two adjacent loci is
and therefore,
From Eq. [1], the variance of kX is
We found that the last part of this previous equation reduces to
Hence, the variance of kX is equal to
For the kth chromosome, the mean and variance of the parental contribution of P1 to an F2-derived RI is
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For a chromosome that is D cM long, the limit of V
as lk approaches infinity is
BC1-Derived RI
At the kth chromosome,
is an indicator variable that is equal to 1 if the BC1-derived RI has the same marker genotype as P1, and 0 if the RI has the same marker genotype as P2. The number of marker loci at which the RI and P1 are homozygous for the same allele is
. On the basis of the expected frequencies of marker genotypes in the BC1 population (Table 2)
, the means, variances, and covariances of kXBC1i and kXBC1j are
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Table 2 Frequency of marker genotypes, at Loci i and j on Chromosome k, among recombinant inbreds (RI) derived from a [(P1 x P2) x P1]BC1 population
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Therefore, the means and variances of kXBC1 and kpBC1 are
Several Chromosomes
Across all n chromosomes, the total number of marker loci at which an F2-derived RI and P1 are homozygous for the same allele is
. The kX values are independently distributed, and the mean and variance of XT are
where l is the total number of marker loci across all chromosomes. Therefore, the mean and variance of pF2 are
For a BC1-derived RI, the mean and variance of pBC1 are
 | (3) |
The estimates of parental contribution are expected to approach a normal distribution as the number of marker loci increases. The standard errors (SE) of pF2 and pBC1 are equal to the square root of their respective variances. We transformed pF2 and pBC1 into z-scores, i.e.,
for F2-derived RIs and
for BC1-derived RIs. On the basis of z-scores, we determined the probability of obtaining an essentially derived RI for thresholds of 0.65, 0.70, and 0.75 for F2-derived RIs, and thresholds of 0.85, 0.90, and 0.95 for BC1-derived inbreds.
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Results and discussion
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Terminal versus Nonterminal Marker Models for a Single Chromosome
The location of marker loci on a chromosome affected the nature of the relationship between the number of marker loci and SE. In the Terminal Marker Model (Fig. 1), there was a specific number of marker loci that minimized SE for each chromosome length (Fig. 2)
. In contrast, SE in the Nonterminal Marker Model, decreased asymptotically as the number of marker loci increased. These results indicated that V and V
cannot be infinitely small. The difference in SE between the two models was greater with shorter chromosomes than with longer chromosomes. But regardless of chromosome length, the Terminal Marker Model always had a lower minimum SE than the Nonterminal Marker Model. For example, the minimum SE
in the Terminal Marker Model was 0.3624 for a 50 cM chromosome and 0.2101 for a 250 cM chromosome. In contrast, the SE in the Nonterminal Marker Model as the number of marker loci approached infinity was 0.3767 for a 50 cM chromosome and 0.2121 for a 250 cM chromosome.
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Fig. 2 Standard error (SE) of the parental contribution to F2-derived recombinant inbreds for a single chromosome for (i) different numbers of marker loci on the chromosome, (ii) different lengths (in centimorgans, cM) of the chromosome, and (iii) terminal versus nonterminal locations of markers
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As indicated in Eq. [2], V has a variance component (i.e., 1/4lk) and a covariance component. We are unable to provide a simple, intuitive explanation for the existence of a specific number of markers that minimized V
in the Terminal Marker Model but not in the Nonterminal Marker Model. Instead, we can only describe the behavior of the variance and covariance components. As the number of marker loci on a chromosome of fixed length increases, the variance component decreases whereas the covariance component increases. The variance component is equal for both models. But due to a smaller distance between adjacent marker loci, the Nonterminal Marker Model has a larger covariance component than the Terminal Marker Model. Lower thresholds will help minimize unjustified disputes regarding essential derivation. The remainder of our discussion will focus only on the results for the Terminal Marker Model because its standard error, being always smaller than that for the Nonterminal Marker Model, leads to lower thresholds for declaring essential derivation.
Probability of Obtaining Essentially Derived Inbreds
Crop species differ in genome size (in centimorgans), number of chromosomes, and length of each chromosome. Estimated genome sizes and the numbers of linkage groups, corresponding to the haploid number of chromosomes (n), are 724 cM and n = 7 in rye (Wanous et al., 1998); 1088 cM and n = 7 in barley (Langridge et al., 1995); 1294 cM and n = 12 in tomato (Tanksley et al., 1992); 1491 cM and n = 12 in rice (Causse et al., 1994); 1749 cM and n = 10 in maize (Senior et al., 1996); and 2828 cM and n = 21 in wheat (Gale et al., 1995). Rye, which has the fewest chromosomes and smallest genome, had the largest values of SE (Fig. 3)
. Wheat, which has the most chromosomes and largest genome, had the smallest values of SE. The numbers of polymorphic marker loci that minimized SE were 56 in rye, 76 in barley, 97 in tomato, 111 in rice, 128 in maize, and 206 in wheat. The minimum SE with these numbers of marker loci were 0.1117 in rye, 0.0965 in barley, 0.0842 in tomato, 0.0801 in rice, 0.0771 in maize, and 0.0588 in wheat. These numbers of marker loci corresponded to distances between adjacent marker loci of 15 cM in rye, tomato, rice, maize, and wheat, and 16 cM in barley. As indicated by Eq. [3], the minimum SE
was equal to (3/4)1/2 SE.
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Fig. 3 Standard error (SE) of the parental contribution to F2-derived recombinant inbreds with different numbers of marker loci in different species
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In a study of essential derivation in maize, the American Seed Trade Association has proposed estimating SE and SE empirically with at least 80 marker loci (A.R. Hallauer, 1997, personal communication; Bernardo, 1999, unpublished). The observed variation in pF2 and pBC1 in a diverse set of populations would provide benchmarks for setting thresholds. An alternative approach is to use the theoretical values of SE and SE we have derived. The SE and SE indicate the variation in pF2 and pBC1 among a series of random RIs, but do not indicate the error of the estimate of pF2 and pBC1 for a single RI. In other words, our results are consistent with the American Seed Trade Association approach for determining appropriate thresholds, but not for determining whether a specific RI is essentially derived or not. Further research is needed on the variance of the error of the estimate of pF2 and pBC1 for an individual RI.
We propose that the minimum values of SE and SE be used to establish minimum values of thresholds for declaring essential derivation. Suppose selfing is permissible from a BC1 population with a third party inbred as the recurrent parent. Assume the maximum error rate for falsely declaring an RI is essentially derived is set at 2.5%. The upper limit of a 95% confidence interval on pBC1, equal to 0.75 + z0.975 x SE(pBC1), would then serve as the minimum value of the threshold for declaring essential derivation. This upper limit is equal to 0.881 in maize. This result implies that, given the specified rate of 2.5% for false positives, any BC1-derived RI with pBC1 
0.881 should not be in any danger of being declared essentially derived. A BC1-derived RI with pBC1 > 0.881 may or may not be declared essentially derived, depending on the final threshold used for assessing essential derivation.
Averaged across the six crop species we considered, the probability of obtaining an essentially derived RI was greatest when the threshold was 0.85 among BC1-derived RIs (Fig. 4)
. For the threshold of 0.75 among F2-derived RIs, the probabilities of an essentially derived RI were <2% in rye, <1% in barley, and <0.3% in tomato, rice, maize, and wheat (results not shown in Fig. 4). For the extreme threshold of 0.95 among BC1-derived RIs, the probabilities were <3% in rye, <2% in barley, and <0.6% in tomato, rice, maize, and wheat (results not shown in Fig. 4). Rye, which has the smallest genome, had the highest probability of an essentially derived RI across all thresholds. The probability of an essentially derived RI from the BC1 was >15% when the threshold was 0.85, and >6% when the threshold was 0.90. The probabilities were lowest in wheat, which has the largest genome. When the threshold was 0.85, the probability of an essentially derived RI from the BC1 was about five times lower in wheat than in rye. The probabilities in wheat were <0.1% when the threshold was 0.70 among F2-derived RIs, and <0.4% when the threshold was 0.90 among BC1-derived RIs (results not shown in Fig. 4). The probabilities of essentially derived RIs in barley, tomato, rice, and maize were intermediate to those in rye and wheat. In maize, the probability in the BC1 was 6 to 9% when the threshold was 0.85, and 1 to 2% when the threshold was 0.90. These results suggest that the thresholds used to declare essential derivation should differ among species.
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Fig. 4 Probability of obtaining an essentially derived recombinant inbred (RI) when parental contribution is estimated with different numbers of marker loci in different crop species. Thresholds for essential derivation are 0.65 and 0.70 for F2-derived RIs, and 0.85 and 0.90 for BC1-derived RIs
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Robustness of the Model
The probabilities we calculated depended on four assumptions: (i) the linkage maps we used are representative of the crop species; (ii) the marker loci are evenly spaced along a chromosome; (iii) pF2 and pBC1 among RIs follow a normal distribution, and (iv) selection is absent. The probabilities of obtaining essentially derived RIs are only as good as the linkage maps from which they were determined. Different linkage maps for the same species vary in the estimated size of each chromosome. If a linkage map differs substantially from the map we used, then the probabilities of obtaining essentially derived RIs may need to be recalculated.
The assumption of evenly spaced markers may not be met in practice. The number of marker loci that minimized SE was 11 for a 150 cM chromosome, with a corresponding distance of 15 cM between adjacent marker loci and SE of 0.2586. Suppose the distance between marker loci is 10, 15, or 20 cM instead of a constant 15 cM. If the marker loci are at the 0, 10, 25, 45, 55, 70, 90, 100, 115, 135, and 150 cM positions, the SE increases only slightly from 0.2586 to 0.2587. When the distance between marker loci is 5, 15, or 25 cM, SE increases to 0.2609. Hence, the formulas for SE and SE
are robust with regards to variation in the distance between marker loci.
In simulation studies, we found that pF2 among 100000 F2-derived RIs closely followed a normal distribution (results not shown). For BC1-derived RIs, the pBC1 values approached a normal distribution as the number of chromosomes and marker loci increased. Suppose the genome comprises only n = 4 chromosomes, each 80 cM long and each with five marker loci. When the threshold was 0.85, the probability of obtaining an essentially derived RI in the simulation study, compared with the probability for a normal distribution (in parentheses), was 0.213 (0.166). The corresponding probability was 0.114 (0.085) when the threshold was 0.90. The probabilities in this paper may therefore underestimate the actual probability of obtaining an essentially derived RI if n and the number of marker loci are small. This discrepancy largely disappeared when the genome comprised n = 8 chromosomes, each 140 cM long and each with eight marker loci. The probability of an essentially derived RI was 0.127 (0.126) when the threshold was 0.85, and 0.037 (0.047) when the threshold was 0.90.
The assumption of no selection implies that the mean parental contribution among a set of RIs is 0.50 for F2-derived RIs and 0.75 for BC1-derived RIs. We speculate that selection could change the probability of obtaining an RI with pF2 or pBC1 exceeding the threshold. A comparison of our theoretical results with empirical data on pF2 and pBC1 among sets of RIs, developed with selection during inbreeding, would be useful.Wanous Heredia-Diaz Ma Goicoechea Wricke Ferrer Gustafson 1997
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ACKNOWLEDGMENTS
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The China Scholarship Council supported Dr. Jiankang Wang as a visiting scholar at Purdue University. We thank an anonymous reviewer for pointing out a conceptual error in a previous version of this manuscript.
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NOTES
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Purdue Agric. Res. Programs Journal Paper 16015.
Received for publication June 4, 1999.
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ABSTRACT
INTRODUCTION
