Abstract
This article presents a method for QTL interval mapping in autotetraploid species for a fullsib family derived by crossing two parents. For each offspring, the marker information on each chromosome is used to identify possible configurations of chromosomes inherited from the two parents and the locations of crossovers on these chromosomes. A branch and bound algorithm is used to identify configurations with the minimum number of crossovers. From these configurations, the conditional probability of each possible QTL genotype for a series of positions along the chromosome can be estimated. An iterative weighted regression is then used to relate the trait values to the QTL genotype probabilities. A simulation study is performed to assess this approach and to investigate the effects of the proportion of codominant to dominant markers, the heritability, and the population size. We conclude that the method successfully locates QTL and estimates their parameters accurately, and we discuss different modes of action of the QTL that may be modeled.
LINKAGE analysis and quantitative trait loci (QTL) mapping methods are now well established and widely used for diploid plant species, and there is an increasing interest in extending these methods to autopolyploid species, despite the complications of polysomic inheritance. Linkage maps have been calculated for autotetraploid potato (Meyeret al. 1998), autotetraploid alfalfa (Brouwer and Osborn 1999; Diwanet al. 2000), autohexaploid sweet potato (Ukosit and Thompson 1997), and autooctaploid sugarcane (AlJanabiet al. 1993; Da Silva et al. 1993, 1995; Ripolet al. 1999). Most of these studies used dominant markers or individual alleles from codominant markers that are present in one parent and segregate in a 1:1 ratio in the mapping population (simplex markers). Methods for calculating recombination frequencies between simplex markers were discussed by Wu et al. (1992).
Unless a mapping population is very large, it is difficult to detect repulsion linkages between simplex markers in polyploids. Yu and Pauls (1993) and Hackett et al. (1998) examined the use of doubledose (duplex) markers as well as simplex markers to identify homologous chromosomes in autotetraploid populations, and Meyer et al. (1998) used this approach in their map of potato. Da Silva (1993) and Ripol et al. (1999) gave similar theoretical results for octaploid species and used this to obtain a map of sugarcane. Recently Luo et al. (2001) showed how to calculate recombination frequencies and LOD scores for all possible configurations of codominant multiallelic markers in autotetraploid species and how this information may be used to construct a linkage map. They showed that codominant markers are, in general, more powerful for detecting linkage and give more precise estimates of the recombination fraction.
An important use of linkage maps is to locate major genes and QTL for important traits. Early studies of diploid species compared trait means for different phenotypes at a single marker using regression models, and some authors used the same approach in polyploid species. Sills et al. (1995) used regression models to establish associations between four stalk traits in sugarcane and a set of simplex markers. A similar approach was used by Meyer et al. (1998) and Bradshaw et al. (1998) to locate QTL in potato for quantitative resistance to late blight [Phytophthora infestans (Mont.) De Bary] and the white potato cyst nematode [Globodera pallida (Stone)], respectively. These authors used both simplex and duplex markers and used permutation tests (Churchill and Doerge 1994) to establish an appropriate significance level for testing multiple markers. In an autotetraploid cross, however, there could be up to eight different QTL alleles and the regression approach will give little insight into their individual effects and interactions. We also want to test for the presence of a QTL at locations between markers.
In this article we present an approach for QTL interval mapping in autotetraploid species in a fullsib family derived by crossing two parents. As for similar populations derived from outbreeding diploid parents, we use all the markers on a chromosome to estimate conditional probabilities for QTL genotypes. The trait values are then related to the QTL genotypes, using a mixture model. We present a simulation study to look at effects of the proportion of codominant to dominant markers, the heritability, and the population size on the ability of the model to locate QTL and to estimate their effects.
METHODS
The mapping population: The QTL mapping approach is developed for an F_{1} population derived by crossing two parents, P_{1} and P_{2}. The phenotypes of m molecular markers are assumed to be known for the parents and n offspring, and trait data are available for the parents and offspring. The parents can have up to eight distinct alleles at each marker or quantitative trait locus: These are represented by A–H or O for a “null” allele (Callenet al. 1993). In practice, most loci will have less than eight alleles, and the parental phenotypes will generally be compatible with more than one genotype. Luo et al. (2000) have shown how the probabilities of possible parental marker genotypes may be inferred from the parental and offspring phenotypes using Bayes' theorem.
Tetrasomic inheritance: The model for QTL mapping is developed by assuming random chromosomal segregation. The four homologous chromosomes are assumed to pair at random to give two bivalents, and crossing over is assumed to be restricted to within each bivalent. All bivalent pairings are assumed to be equally likely. The possibilities of nonrandom chromosomal pairing, or multivalent formation, are not considered here. We assume that there is no chromatid and no crossover interference. Luo et al. (2001) have shown how recombination frequencies and LOD scores between pairs of markers can be estimated for all possible phases, assuming random chromosomal segregation, and how the most likely phase may be identified. Using this information, markers can be partitioned into linkage groups, ordered, the distances between them calculated, and the phases of the markers in the parents can be deduced. We assume that this analysis has been carried out, so that the map of molecular markers is known.
Model for a quantitative trait: Kempthorne (1957) discussed the partitioning of the genetical variance of polyploid individuals in a random mating population at equilibrium. He expressed the genotypic value Y_{G} of a tetraploid individual with genotype A_{i}A_{j}A_{k}A_{l} as
There are further constraints on the higherorder terms of Equation 2, for example,
When fitting models in the simulation study, we concentrate on the main effects model (4), with cornerpoint constraints on the parameter estimates. However, there is no theoretical difficulty in including higherorder terms.
In practice, the offspring QTL genotypes are unknown, and the conditional probabilities of QTL genotypes must be estimated from the marker information.
A mixture model for QTL mapping: Here we develop a maximumlikelihood approach for fitting a single QTL model, considering one chromosome at a time. The analysis is an extension of that used by Jansen (1992). We assume that we have a population of n offspring and that for offspring i we observe trait value y_{i} and marker phenotype data o_{i} for the chromosome. Let q_{i} ∈ Q_{i} be the set of possible QTL genotypes, and let g_{i} ∈ G_{i} be the set of chromosome configurations that are compatible with phenotype o_{i}. By “chromosome configurations” we mean the marker genotypes and the parental chromosomes from which the marker alleles come, so that it is clear how the bivalents occurred to form that offspring and where recombinations occurred. We adopt an interval mapping approach, fitting a QTL at a set of locations along the chromosome and maximizing the likelihood for each location as a function of the QTL parameters θ = (μ, α_{2}, α_{3}, α_{4}, α_{6}, α_{7}, α_{8}, σ^{2}), where μ and α_{i} are as in Equation 4 and σ^{2} is the residual variance.
The likelihood of the trait and marker data is
We can maximize the loglikelihood by
Estimation of QTL genotype probabilities: In populations such as doubled haploids from inbred diploid lines, every marker is fully informative, and the conditional probability of a QTL genotype depends only on the genotypes of the markers flanking the possible QTL location. When markers are not fully informative, we need to consider the information from all markers on a chromosome to calculate the conditional probabilities of QTL genotypes. The QTL genotype probabilities P(q_{i}g_{i})P(g_{i}o_{i}) factor into two terms, and we consider them separately.
The conditional probability of the chromosome configuration, given the marker phenotypes, P(g_{i}o_{i}): One way to calculate these probabilities would be to use a hidden Markov model (HMM). These have been widely used for multipoint mapping in diploid species (e.g., Lander and Green 1987; Jiang and Zeng 1997) and Xie and Xu (2000) tried to apply them to mapping in tetraploid species under random chromosomal segregation. There are theoretical problems with the latter's approach, however, due to the form used for the matrix of transition probabilities between neighboring markers, and as a consequence their HMM represents multivalent formations (Hackett 2001).
We preferred to use an alternative method via a “branch and bound” algorithm, where we search for chromosome configurations for each offspring that are compatible with the marker phenotype information, arise from a possible bivalent pairing in each parent, and have the minimum number of crossovers. appendix b describes the process of reconstruction for one individual from a cross between the two parents shown in Table 1 and Figure 1 illustrates the eight possible chromosome configurations for this individual that are compatible with the phenotypic information and have the minimum number of crossovers (six).
If there are m marker loci, the recombination frequency between loci i and i + 1 is r_{i}, and there are x_{i} recombinations between them (0 ≤ x_{i} ≤ 4), then the probability of a configuration can be calculated as
The probability of the QTL genotype, given the chromosome configuration P(q_{i}g_{i}): Once we have calculated a set of possible chromosome configurations for each individual, we can identify possible QTL genotypes and calculate their probability for putative QTL locations at a set of positions along the chromosome. We assume that there are no double crossovers between markers. The individual illustrated in Figure 1, for example, has inherited chromosomes 1, 2, 6, and 8 at marker loci L_{5} and L_{6} for all configurations. Therefore, for QTL locations between L_{5} and L_{6}, we assume that the QTL genotype is Q_{1}Q_{2}Q_{6}Q_{8}, with probability 1. This individual has also inherited chromosomes 1, 2, 6, and 5 at locus L_{7}, with a crossover between chromosomes 5 and 8. For QTL locations between L_{6} and L_{7}, there are two possible QTL genotypes, Q_{1}Q_{2}Q_{6}Q_{8} and Q_{1}Q_{2}Q_{6}Q_{5}. The probability of the former genotype will decrease, and the probability of the latter will increase, as we consider locations at an increasing distance from L_{6} and closer to L_{7}. To calculate these probabilities, we assume that crossovers follow a Poisson process with the probability of no crossovers in an interval of M morgans equal to e^{−}^{M} and the probability of one crossover as Me^{−}^{M}. If the positions of L_{6} and L_{7} are m_{6} and m_{7} and we want to calculate the probability associated with a QTL at position m_{Q} between them,
Simulation study: A simulation study was carried out to investigate this approach for QTL mapping and to quantify the effects of the marker type, the trait heritability, and the population size. Two parents were simulated initially and these were crossed to give a population of 200 offspring by random chromosomal segregation. The first simulation consisted of one chromosome with 10 codominant markers, spaced at 10cM intervals. There were five possible alleles (A–E) and a null allele (O) at each marker locus, with equal probability. The parental marker genotypes were simulated by sampling four alleles for each parent from the set of possible alleles, with replacement. One such parental configuration is shown in Table 1. A QTL was assumed to be situated halfway between markers L_{2} and L_{3} and to have eight different alleles. The QTL alleles were assumed to have additive effects, of sizes 0, 1, 1, 2, 0, −1, −1, and −2 for alleles Q_{1}–Q_{8}, respectively. The trait values for each offspring were calculated as an overall mean of 10.0, plus the sum of the effects of their alleles, plus an environmental effect distributed as N(0, σ^{2}). The value of σ^{2} was chosen to give the desired trait heritability
The above simulation has three random stages: simulation of the parents, simulation of the offspring given the parents, and simulation of the environmental error to add to the genotype values; and the study can be replicated at each stage. To see the effect of each level of replication, 10 pairs of parents were simulated, 20 sets of 200 offspring were simulated for each set of parents, and 20 sets of environmental error were simulated for each set of offspring, giving a total of 4000 sets of marker and trait data for analysis. Simulations A and B had heritabilities of 25 and 10%, respectively.
Most experimental data sets will be a mixture of codominant and dominant marker types, and in general the dominant markers are less informative. A further set of simulations (C) was generated to investigate this. The dominant markers were simulated as a mixture of simplex markers (AOOO × OOOO), duplex markers (AAOO × OOOO), and doublesimplex markers (AOOO × AOOO), in the proportion found in potato by Meyer et al. (1998). Other configurations of dominant markers were excluded. Dominant and codominant markers were ordered so that there were two loci with dominant alleles present in each parent between the codominant loci. A typical parental configuration is shown in Table 2. For this configuration, loci L_{1}, L_{5}, L_{9}, and L_{14} are codominant. Between L_{1} and L_{5} there are three dominant markers (one double simplex, one duplex originating from parent 1, and one simplex originating from parent 2). Between L_{5} and L_{9} there are another three dominant markers (one double simplex and two duplex, one from each parent) and between L_{9} and L_{14} there are four simplex markers, two from each parent. A QTL was positioned halfway between the first and second codominant markers for each configuration. A heritability of 10% was used for set C. A further set D was run to investigate the effect of reducing the population size from 200 to 100 offspring. Set D was otherwise the same as C.
We need a threshold for
Computing: All routines were written in Fortran 90 (Digital 1997). The branch and bound algorithm was adapted from a branch and bound algorithm for best subset selection by Roberts (1984).
RESULTS
Reconstruction of the chromosome configurations: The reconstruction of the chromosome configurations depends only on the marker data and is not affected by the heritability of the trait. The computer program to calculate the possible configurations had an upper limit of 5000 for each individual so that individuals with >5000 possible configurations were excluded from further analysis. No individuals were excluded, using the codominant markers for simulations A and B. The mean numbers of individuals excluded from simulations C and D were 8.8 (SE 0.93) out of 200 individuals and 4.8 (SE 0.96) out of 100 individuals, respectively.
For the individual illustrated in Figure 1, there are eight possible configurations. These agree, and are the same as the true genotypic configuration, for 37 of the 40 alleles. For the other 3 alleles onehalf of the configurations have the simulated allele. The mean proportions of alleles that are correct for every chromosome configuration are summarized in Table 3 for simulations A, B, C, and D. The proportions of correct reconstructions were significantly higher for simulations A and B, using codominant markers only, but the proportions for simulations C and D were still high (0.81).
Interval mapping: For each trait, we obtained a profile of
Model fitting by iterative or noniterative weighted regression: As discussed above, the QTL model can be fitted at each position by a single weighted regression or by an iterative process, updating the QTL genotype probabilities using trait information as in Equation 10. The third and fourth columns of Table 4 compare the model parameters from these two methods of model fitting for simulation set A. The estimated value for the constant term was close to the true value in both cases, and the mean position, the proportion of QTL located to the region between 5 and 25 cM, and the total weight of correct and threequarters correct QTL genotypes were very similar. For all the QTL allele effects, however, the iterative approach gave an estimate close to the true value, while the noniterative approach gave estimates whose absolute values were biased downward. The noniterative approach also underestimated the percentage variance accounted for and overestimated the residual variance. The same pattern was observed for all of the other simulation sets (results not shown). We conclude that the noniterative approach to model fitting is inadequate, and it is not considered further.
The threshold for declaring a QTL present: Simulation sets E and F (codominant markers and n = 200, and dominant and codominant markers and n = 100, respectively, and with all QTL effects equal to zero) were used to investigate the distribution of
The effect of heritability: The fourth and fifth columns of Table 4 compare simulations with heritabilities of 25 and 10%, respectively. Means and standard errors were calculated over all 4000 data sets in each case. For a heritability of 25%, 3998/4000 data sets had
The effect of marker type: The effect of the change from all codominant markers (simulation B) to a mixture of codominant and dominant markers (simulation C) can be seen by comparing columns five and seven (all 4000 data sets) and columns six and eight (significant data sets) of Table 4. In each case the heritability is 10%. For simulation C, 2848/4000 data sets had
The effect of population size: The last four columns of Table 4 compare the effect of decreasing the population size from 200 to 100 (simulation D), for the situation of 10% heritability and a mixture of codominant and dominant markers. As discussed above, the threshold for declaring a significant QTL with a population of 100 is
Comparison with regression on true QTL genotype: In a simulation study such as this, the true QTL genotype is known and we can compare the parameter estimates from a regression of each trait on the true QTL genotype to that obtained by our interval mapping procedure. An examination of the estimates produced by regression on the true QTL genotypes shows that these estimates varied substantially. To illustrate this, Figure 3 shows the relationship between the estimate of
DISCUSSION
In this article we proposed and tested a method for interval mapping of QTL in a fullsib population of an autotetraploid species. This method could also be extended for QTL mapping in plant species of higher ploidy. As with diploid species, the precision with which a QTL may be located is affected by the heritability of the trait, the size of the mapping population, and the informativeness of the markers. It is useful to have as high a proportion of codominant markers as possible, both for precision of QTL mapping and for linkage map construction (Luoet al. 2001).
The threshold at which a QTL was declared present was calculated by simulating data sets with QTL effects set to zero and examining the distribution of the adjusted coefficient of determination
We used a novel approach of reconstructing possible chromosome configurations from the observed marker phenotypes for each offspring. The branch and bound algorithm was used to identify configurations with the minimum number of crossovers consistent with the observed data. Configurations that did not come from bivalent pairings were rejected. This analysis was motivated by the need for QTL mapping studies in tetraploid potato. A recent ultrahighdensity diploid genetic linkage map of potato chromosome 1 found that chromatids had experienced 0, 1, or 2 recombination events during meiosis (E. Isidore, personal communication); that is, one or two chiasmata per chromosome pair had occurred during the meiosis. The same is likely to be true for chromosomes 2–12, given the lengths of the linkage groups (68–108 cM) found using molecular markers (Van Ecket al. 1995). Hence it is not surprising that the limited cytological evidence available suggests that bivalents predominate in potato, although low frequencies of quadrivalents, trivalents, and univalents also occur (Swaminathan and Howard 1953). As ring and chain quadrivalents can give an equal 2:2 distribution of homologues and hence balanced gametes, we must assume that departures from chromosomal segregation (multivalents and double reduction) will occur at a low frequency, which may or may not be higher for the two long chromosomes of potato (Pijnacker and Ferwerda 1984). Hence when analyzing real data under the assumption of crossing over restricted to bivalents, some anomalous progeny may occur and need to be eliminated from the analysis and this may affect the power of QTL detection and bias the estimation of QTL effects. However, our computer simulation, based on bivalent pairing, has shown that for some offspring multivalent configurations with fewer than the true number of recombinations could be constructed. It would be incorrect in these simulations to infer multivalent formation for such offspring, and the same problem may arise with real data. Clearly, more theoretical work and much more experimental data will be required to resolve these issues.
The accuracy with which the chromosome configurations were reconstructed depended on the type of markers used. In this study we considered codominant markers (for example, microsatellites) and dominant markers (for example, amplified fragment length polymorphisms). For the codominant markers, the dosages and configurations of alleles were obtained by random sampling with replacement from a maximum of five alleles and a null allele. The proportion of alleles reconstructed correctly was lower for a mixture of dominant and codominant markers than for codominant markers alone. The codominant markers with most alleles, and in particular those with most alleles in simplex configurations, gave offspring phenotypes that could occur in the fewest ways. Such markers are therefore the most useful for chromosome reconstruction. Useful marker information could also be obtained by pyrosequencing single nucleotide polymorphisms to measure dosages of alleles for each offspring, which should be more informative in chromosome reconstruction than presence/absence data.
This simulation study, and in particular the chromosome reconstructions, assumed that the marker order was known without error. This may not be the situation for experimental data, and the reconstruction method could also be used to check and improve the locus ordering. The current strategy (Luoet al. 2001) is to calculate the recombination frequency and LOD score for all possible phases for each pair of markers on a linkage group and to use the recombination frequency and LOD score of the most likely phase for assembling the map of that group. The final step uses the module JMMAP of JoinMap (Stam and Van Ooijen 1995). This strategy was tested for a map of simplex and duplex dominant markers by Hackett et al. (1998) and was found to give locus orders with high rank correlations with the true order for populations of 150 or more offspring. Once the offspring genotypes have been reconstructed on the basis of the order from JoinMap, the order could be investigated in two ways:
Drop each marker in the linkage group in turn and calculate the total number of crossovers for all the offspring. If the omission of any marker reduces the number of crossovers markedly compared to the order for the full group, then try other positions for this marker and reposition it where the total number of crossovers is lowest. Repeat for other markers if necessary.
Examine the distribution of the crossovers for all the offspring and identify individuals with large numbers of crossovers. The marker data corresponding to the crossovers should be checked and corrected where necessary.
Theoretically, we could also use a computerintensive search method to search directly for the locus order and parental phases that minimize the total number of crossovers in the offspring. However, there are a very large number of possible orders for a tetraploid cross [m!/2 orders for m loci, and up to (4!)^{2} possible phases at each locus], so it is preferable to use pairwise information to reduce the search space as far as possible. There is a need for further research here.
This analysis was restricted to the case of additive effects of the QTL alleles. However, this model may be too simple. For example, many traits in potato display specific as well as general combining ability (Bradshaw and Mackay 1994). Furthermore, Barnes and Hanson (1967) postulated that downy mildew resistance in alfalfa is controlled by a gene that confers resistance only when in a triplex or quadruplex state. In theory, there is no problem in extending the weighted regression model to fit separate means for each of the 36 QTL genotypes but a large population would be required to obtain good estimates of the means. One possible strategy is to use our additive model to locate regions of the genome associated with the trait. The mean trait value for each QTL genotype could then be calculated from the conditional probabilities of the genotypes for putative QTL locations in this region, to examine how they change. A range of models for the gene action (additive effects, simple models for dominance, etc.) could be tested to identify which are compatible with the means. If the additive model is rejected in favor of an alternative model, the interval mapping process should be repeated to see if the most likely QTL region changes. This strategy will be used to explore experimental data from autotetraploid potatoes at the Scottish Crop Research Institute. Computer routines may be obtained by contacting the first author.
Acknowledgments
The computer programs for simulating tetraploid data were written by Dr. Z. W. Luo. This research was supported by a research grant from the United Kingdom Biotechnology and Biological Sciences Research Council and by the Scottish Executive Rural Affairs Department.
APPENDIX A: RELATIONSHIP BETWEEN BIOMETRICAL MODELS
Bradshaw (1994) compares three notations that have been used for the genetic values of tetraploid individuals. All these notations assume that two alleles, and hence five genotypes, are at a locus affecting the trait. Table A1 shows the notation used for each genotype by Easton (1976) and Wright (1979) and compares this to the expression derived from Equation 1 of this article. From the last column, the average of the two quadruplex genotypes is
APPENDIX B: RECONSTRUCTION OF CHROMOSOME CONFIGURATIONS
Here we demonstrate the reconstruction of the possible chromosomes inherited by an offspring from the cross between parents P_{1} and P_{2}, with genotypes given in Table 1. The phenotype of this offspring is shown in Table A2.
Consider locus L_{1}, with phenotype ACD. The A allele must have come from chromosome 6 (parent P_{2}). The individual does not have a B or E allele and therefore has not inherited chromosome 2 or 5 at this locus. The possibilities are (i) chromosomes 6 and 7 from P_{2}, together with either chromosomes 1 and 3 from P_{1} (giving genotype ACDO); or (ii) chromosomes 1 and 4 from P_{1} (giving genotype ACDD); or (iii) chromosomes 6 and 8 from P_{2}, together with either chromosomes 1 and 4 (giving genotype ACCD); or (iv) chromosomes 3 and 4 (giving genotype ACDO). Table A2 shows the chromosome configurations giving rise to the phenotypes of this individual at each locus.
A branch and bound algorithm is then used to identify the chromosome configurations that give the minimum number of recombinations for the complete linkage group. An initial configuration is found by ordering the configurations according to the number of loci for which they are possible and selecting for each locus the most frequent compatible configuration. For this individual, the configurations 1257 and 2357 are jointly most frequent, each being possible for 5 of the 10 loci. Neither of these is possible for locus L_{1}, however. The initial order is with 12 recombinations. The algorithm searches for configurations with the minimum number of crossovers. It is not necessary to test every combination in Table A2: If a combination for, say, L_{1}–L_{5} has more recombinations than the current minimum, then this is rejected without considering L_{6}–L_{10}.
The minimum number of recombinations for this individual is six, and there are 20 configurations with this minimum. Up to now, the question of whether the configuration may be produced by bivalent pairing has been ignored, but now this is checked for each configuration.
One possible configuration with six recombinations is If we consider the first four loci, these suggest that the chromosomes are paired as 1 + 3 and 2 + 4 from P_{1} and 5 + 7 and 6 + 8 from P_{2}. However, L_{5} and L_{6} have chromosomes 6 and 8 from P_{2}, and L_{8}–L_{10} have chromosomes 5 and 7 together. This configuration is rejected as incompatible with bivalent pairing. However, the configuration is compatible throughout with the chromosomes pairing as 1 + 3, 2 + 4, 6 + 7, and 5 + 8. Of the 20 configurations with six recombinations for this individual, 8 were compatible with bivalent pairings. They can be summarized as This is represented as a graphical genotype in Figure 1. The 8 configurations coincide for 37 of the 40 alleles, but there is uncertainty about the other 3 alleles. For example, there is definitely a recombination between chromsomes 6 and 7 between L_{7} and L_{9}, but it is uncertain on which side of L_{8} it occurred. One of the 8 configurations is the same as the simulated genotype for this individual.
If locus L_{3} was excluded from the analysis, there would be no evidence to establish the first two crossovers between chromosomes 5 and 8. Loci L_{1}, L_{3}, and L_{5} have unique alleles on chromosome 5, and this individual carries the unique allele from L_{3} but not from L_{1} or L_{5}. The true recombinations were between L_{1} and L_{2} and between L_{4} and L_{5}, but the phenotypes observed for L_{2} and L_{4} are both compatible with inheriting chromosomes 6 and 8 from parent P_{2}. Without L_{3}, the minimum recombination configuration would have four recombinations, fewer than that simulated.
Occasionally the minimum recombination configurations have fewer than the simulated number of recombinations, but are all incompatible with bivalent pairing. In this case configurations with minimum + 1, minimum + 2, etc., recombinations are considered until compatible configurations are found.
Footnotes

Communicating editor: C. Haley
 Received May 25, 2001.
 Accepted September 17, 2001.
 Copyright © 2001 by the Genetics Society of America