Temporal and Multiple Quantitative Trait Loci Analyses of Resistance to Bacterial Wilt in Tomato Permit the Resolution of Linked Loci

Corresponding author: N. H. Grimsley, Laboratoire de Biologie Moléculaire des Relations Plantes-Microorganismes, CNRS-INRA, B.P. 27 Auzeville, 31326 Castanet-Tolosan, France., grimsley{at}toulouse.inra.fr (E-mail)

Communicating editor: C. HALEY

	ABSTRACT

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Ralstonia solanacearum is a soil-borne bacterium that causes the serious disease known as bacterial wilt in many plant species. In tomato, several QTL controlling resistance have been found, but in different studies, markers spanning a large region of chromosome 6 showed strong association with the resistance. By using two different approaches to analyze the data from a field test F₃ population, we show that at least two separate loci ~30 cM apart on this chromosome are most likely involved in the resistance. First, a temporal analysis of the progression of symptoms reveals a distal locus early in the development of the disease. As the disease progresses, the maximum LOD peak observed shifts toward the proximal end of the chromosome, obscuring the distal locus. Second, although classical interval mapping could only detect the presence of one locus, a statistical "two-QTL model" test, specifically adapted for the resolution of linked QTL, strongly supported the hypothesis for the presence of two loci. These results are discussed in the context of current molecular knowledge about disease resistance genes on chromosome 6 and observations made by tomato breeders during the production of bacterial wilt-resistant varieties.

BACTERIAL wilt (BW) caused by Ralstonia solanacearum is a very important disease worldwide, attacking many different species, including many agriculturally important crops. As the bacterium is soilborne and enters the plant via the roots, subsequently spreading in the vascular system, chemical control of the disease is impractical and environmentally unacceptable. Polygenic resistance has been described in a number of species (KELMAN 1953 and references therein), but little is known about the molecular mechanisms of this type of resistance. Plant resistance to pathogens in various interactions is sometimes associated with a hypersensitive response (HR), a phenomenon often controlled by single dominant loci, and some of the genes controlling this type of response (R-genes) have been cloned and characterized (see BAKER et al. 1997 ; DE WIT 1997 ; GEBHARDT 1997 ; HAMMOND-KOSACK and JONES 1997 for examples of recent reviews). However, there is no clear genetic evidence that HR is necessary for resistance to R. solanacearum, although a kind of HR has been observed in some interactions and may be found in some plant accessions that also show resistance (CARNEY and DENNY 1990 ; ARLAT et al. 1994 ). In the genus Lycopersicon, several accessions of tomato (Lycopersicon esculentum) and its close relatives show resistance to BW (WANG et al. 1996 ). Although resistant plants do not show symptoms, their vascular systems are nevertheless invaded to varying extents by the bacterium (GRIMAULT and PRIOR 1993 ). The use of molecular markers and interval mapping is a powerful approach that permits the identification and genetic mapping of loci controlling a trait of interest. Several quantitative trait loci (QTL) have been shown to play a role in resistance to BW in different studies using different populations and at different geographic locations (DANESH et al. 1994 ; GRIMAULT et al. 1995 ; THOQUET et al. 1996A , THOQUET et al. 1996B ). In studies using molecular markers, one QTL with a broad LOD peak on chromosome 6 was of overriding importance. However, the map position of the maximum LOD score observed on chromosome 6 differed between populations and between environments. Here, by temporal analysis of disease development and use of two different kinds of statistical analyses, we show that there are most probably at least two QTL present on chromosome 6.

	MATERIALS AND METHODS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX 1 APPENDIX 1 LITERATURE CITED

Plant growth and inoculation with R. solanacearum:
The F₃ population of ~3500 individuals was obtained from a cross between Hawaii7996 (L. esculentum, resistant) and WVa700 (L. pimpinellifolium, susceptible) and was planted in the field in randomized blocks (THOQUET et al. 1996B ). Bacteria (strain GMI8217; A. TRIGALET, personal communication) resistant to streptomycin and rifampicin (a derivative of GT1; PRIOR and STEVA 1990 ) were grown as previously described (THOQUET et al. 1996A ), and the plants were inoculated under shade cover, using 2 ml of bacterial suspension per plant. Plants were kept under shade for a further 2 days before transfer to an infested field.

Notation of disease symptoms:
The development of disease symptoms was noted about every 2 days after the first signs of symptom development, from 6 days after inoculation (d.a.i.) onward. A scale of 1 to 9 was used (THOQUET et al. 1996B ).

Molecular and genetic analyses:
Molecular RFLP analysis was done as previously described (THOQUET et al. 1996A ), and marker maps were established both with MAP-MAKER (LANDER et al. 1987 ) and with JoinMap (STAM 1993 ), using the Haldane mapping function. Small differences in linkage distances were found between these maps, so further QTL analyses were performed using MAPMAKER distances because these are estimated by maximum likelihood.

For the purposes of this study, plants were scored either as healthy (stage 1 or 2) or wilted (stages 3 to 9). The proportion of plants wilted for each family, x, was then used, after transformation using y = arcsin to improve the normality of the distribution, as the statistic for QTL analysis. A few F₃ families were poorly represented due to poor seed set or germination, or to bad weather conditions during the test (tropical thunderstorms). Arbitrarily, families with <10 representatives were not used, leaving 187 families for the analysis (the average family size was 19.0 individuals). Interval mapping was used to locate QTL in the F₃ population, using both MAPMAKER and MapQTL (VAN OOIJEN and MALIEPAARD 1996 ). Multiple-QTL model (MQM) was performed with the MapQTL software.

Test for two linked QTL:
Using backcross progenies, GOFFINET and MANGIN 1998 studied tests that permit discrimination between the hypotheses of one or more than one QTL in a linkage group. Among all the tests compared, the so-called "two-QTL model" (2QM) was the most powerful test for detection of closely linked QTL.

The 2QM test is the minimum value of two statistics, denoted T(₁) and T(₂), that are obtained by comparing the likelihood maximized under the two-QTL model with the likelihood maximized under the one-QTL model with QTL position fixed at ₁ and ₂, where ₁ and ₂ are the maximum-likelihood estimators of the parameter positions t₁ and t₂ from the two-QTL model. T(₁) and T(₂) are calculated here as LOD tests (difference of the log₁₀-likelihood), like maximum-likelihood ratio tests (two times the difference of the Napierian logarithm of the likelihood). The mixture likelihood is approximated by a linear-likelihood approximation that is well-known to be effective when the QTL is not a major gene (HALEY and KNOTT 1992 ; REBAI et al. 1994 ), and the maximization under the two-QTL model is limited to QTL belonging to nonadjacent intervals.

Let T be the 2QM test; evidence for the presence of more than one QTL in the linkage group is obtained if T is greater than a threshold chosen such that Pr(T > ) for all possible parameters included in the null hypothesis that there is only one QTL in the linkage group. In F₂ progenies, five nuisance parameters are involved in the null hypothesis: µ (the global mean), ² (the variance), and parameters a, d, t (the additive effect, dominance effect, and position of the QTL).

In fact, there is no problem with the global mean and the variance because T is invariant for these two parameters, but this is not the case for parameters concerning the QTL; moreover, the distribution of T depends on the values of the other parameters (GOFFINET and MANGIN 1998 ).

Threshold by parametric bootstrap:
The simplest way to get a threshold is to estimate the nuisance parameters in a one-QTL model and to perform a Monte Carlo simulation with these estimates. This sampling procedure is called a parametric bootstrap.

This procedure can be shown, by the use of central limit theorems, to give an asymptotically correct threshold if the observations follow the distribution chosen for the Monte Carlo simulation, and if the nuisance parameters are consistently estimated (EFRON and TIBSHIRANI 1993 , p. 307). In our case, the distribution chosen for the Monte Carlo simulation is a mixture of three normal distributions. Deviation from normality has been studied by DOERGE and REBAI 1995 for the detection of one QTL. Apparently even when the observations follow a skewed distribution, the assumption of normality does not materially change the threshold. But, because the information matrix is singular, the QTL parameters are not consistently estimated when the QTL have medium or small effect (MANGIN et al. 1994 ), so the threshold obtained by this procedure cannot be proven to lead to a correct type I error.

Threshold by intensive Monte Carlo simulations:
GOFFINET and MANGIN 1998 propose that the threshold be computed by simulating all of the possible values of the nuisance parameters. The nuisance parameters are varying in a continuous space, and even if simulations involve only a limited number of points, this procedure needs intensive simulations. However, the threshold obtained leads to a correct type I error.

Threshold by stratified shuffling:
Stratified shuffling as proposed by CHURCHILL and DOERGE 1996 is another possible way for computing a threshold for the 2QM test. The main step of this procedure is to split up the data with respect to the assigned classes given by an informative marker and to shuffle the data in each class. The shuffling is repeated N times and the 2QM test is computed for each shuffled data set. The N test values are used to produce an empirical threshold value at the marker (see Appendix 1 for more details).

The stratified shuffling procedure produces a new data set that behaves asymptotically like a random sample drawn out of a population where only one QTL located at the marker segregates (see Appendix 1).

Let us denote by _i the empirical threshold obtained at marker i. The stratified shuffling procedure is repeated for each fully informative marker, and an empirical threshold for the whole linkage group, noted , is obtained by

Theoretically, the use of this supremum is not sufficient to guarantee a correct type I error for the test over the whole linkage group because only positions corresponding to fully informative markers are investigated, but a slight modification can be proposed to handle this problem. Before shuffling, each individual that is not assigned to a class with certainty is assigned to a class by a random draw. The class probabilities for this random draw are inferred for each individual using all of its linked marker information, as are inferred the QTL genotype probabilities in the interval mapping method. However, because only chromosomal locations with markers are investigated, there is no guarantee that the whole procedure provides a correct type I error between widely spaced markers.

	RESULTS

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A molecular map of chromosome 6 was developed previously using an F₂ population of 200 individuals from the cross Hawaii7996 x WVa700 (THOQUET et al. 1996A ). For the purposes of this study, only codominant RFLP probes were used, and two further probes, Cf-2 and TG406, were placed on the map (Figure 1, left). Cf-2, a cDNA clone containing the coding sequence of the Cf-2 disease resistance gene (DIXON et al. 1996 ), showed a remarkably high level of polymorphism between the parental lines (10/14 enzyme/probe combinations polymorphic for at least one of the multiple bands present), considering the low overall level of polymorphism in the population (THOQUET et al. 1996A ). Four different RFLP bands were mapped to the same genomic location, as expected for a multi-gene family (DIXON et al. 1996 ). Alleles detected by the probe TG406 were previously shown to map to chromosomes 3 and 6 (TANKSLEY et al. 1992 ), but in our material we did not observe any polymorphism for the chromosome 3 allele. We chose to include the chromosome 6 locus in the current study because it improves the map density in a region close to one of the LOD curve peaks observed (see below).

View larger version (70K): In this window In a new window Download PPT slide   Figure 1. Analysis of resistance by interval and MQM scans of chromosome 6. LOD score curves (horizontal axes) of resistance to bacterial wilt are displayed. Markers are shown on the left (map distances on vertical axes in centimorgans). The LOD score curves are shown at different times after inoculation (d.a.i.) with the pathogen: 6 d.a.i., fine dashed lines; 14 d.a.i., thick dashed lines; 21 d.a.i., fine solid lines; 28 d.a.i., thick solid lines. The three panels represent different treatments of the data: (A) interval mapping; (B) MQM cofactor on TG118; (C) MQM cofactor on TG240.

Temporal analysis of disease resistance:
When wilting symptoms were fully developed at 28 d.a.i., a broad QTL LOD peak associated with many of the markers on chromosome 6 was observed (THOQUET et al. 1996B ; Figure 1A, 28 d.a.i.).

However, the current analysis revealed that as early as 6 d.a.i. several markers showed significant association with resistance (Figure 1A). The disease then progressed rapidly in the population, and at 12 d.a.i. two regions of the chromosome, between Cf-2 and TG153, and between CP18 and TG406, showed clear association with resistance, with LOD score peaks ranging from 4.3 to 5.9, respectively. Figure 2 shows the temporal progression of the LOD score at these two chromosomal locations. At 14 d.a.i. (Figure 1 and Figure 2), the interval TG73 to TG406 showed the highest LOD score (8.1 at 36 cM), and markers on the upper part of the chromosome also showed increased LOD scores. Thus, at three different temporal observation points the LOD score of the distal peak exceeded that of the proximal peak (Figure 2). Subsequently, however, the upper part of the chromosome showed the strongest association with resistance, with a plateau exceeding LOD 7 (maximum LOD 10.3) extending over most of the chromosome.

View larger version (13K): In this window In a new window Download PPT slide   Figure 2. Temporal development of the association of two genomic regions of chromosome 6 with resistance. The peak close to TG118 is represented by the solid line, while the dashed line represents the peak close to TG240.

In summary, the position of the maximum LOD score varies considerably over the time course of the experiment, and suggests that two separate loci might be affecting the resistance. Cofactor analysis (JANSEN and STAM 1994 ; VAN OOIJEN and MALIEPAARD 1996 ) was therefore used in conjunction with the above temporal analysis to investigate the possible presence of more than one QTL on the chromosome. Placing a cofactor on TG118 at 12 cM, corresponding to the maximum of the peak observed at the end of the resistance test, does not support a hypothesis for the presence of two QTL because the LOD score increase on the distal part of the chromosome is insufficient (Figure 1B), in agreement with the previous results of interval mapping (THOQUET et al. 1996B ). However, the preceding temporal analysis indicates that this interpretation might be misleading, because at 14 d.a.i., the maximum LOD score of 8.1 is observed at 36 cM close to marker TG240. The chromosome was therefore scanned using MQM by placing cofactors on each marker along the length of the chromosome in a series of analyses. Use of TG240 as a cofactor confirmed the likely presence of two QTL on this chromosome, because in this case at both 21 and 28 d.a.i. the LOD score rises by >4 LOD units toward the top end of the chromosome (Figure 1C).

A statistical test for the presence of two QTL on chromosome 6:
Statistical tests that tackle the question of whether one or two QTL might be involved in the segregation of a particular character have been compared (GOFFINET and MANGIN 1998 ). Among those tests compared, 2QM was found to be the most powerful in many situations, and this was thus applied using the data collected 28 d.a.i. At this stage, standard interval mapping could not resolve the presence of two QTL on chromosome 6. The 2QM gave a LOD score of 4.36 for the presence of two loci.

To confirm that the segregation of resistance observed is most likely due to the presence of two loci, various simulations using those constraints imposed by our dataset (map distances, population size; see below) were done. These simulations gave the distribution of the values of the LOD scores that could be obtained in the 2QM by chance, using the given constraints and assuming the presence of only one QTL. For each simulation, 1000 replications were performed, and the empirical threshold value for the test of at least two QTL vs. one QTL, with a type I error of 1%, was obtained.

The nuisance parameter estimates, needed for the parametric bootstrap procedure, were computed using MAPMAKER/QTL ( = 0.037, = 0.108, â = 0.157, = 0.012). An empirical threshold value of 3.52 was obtained.

Intensive Monte Carlo simulations for a QTL position located every 10 cM on the chromosome and with the variance explained by the QTL (a²/2 + d²/4) ranging from 10 to 30% of the residual variance were done (Figure 3). The empirical threshold value obtained by this procedure (the maximum of all of the threshold values) was 4.04.

View larger version (21K): In this window In a new window Download PPT slide   Figure 3. Empirical threshold values, for a type I error of 1%, by Monte Carlo simulations over 1000 replicates. Points for each LOD (vertical axis) threshold curve were calculated every 10 cM (horizontal axis) using different parameters: fine dashed lines, percentage of residual variance = 0.3; fine solid lines, percentage of residual variance = 0.2; thick solid lines, percentage of residual variance = 0.1. The symbols show the results of assuming d = 0 (), a = 0 (), or d = a (). The straight line indicates the 2QM LOD value for the original data (arrow).

The stratified shuffling procedure gave an empirical threshold value of 3.61 (Table 1). Thus, using the test value and the thresholds obtained above, we accept with >99% confidence that at least two QTL are involved in chromosome 6. The maximum-likelihood estimators of the positions were found near Cf-2 (at precisely 0.7 cM to the right of Cf-2) and on TG73. These loci explained 12 and 13% of the phenotypic variation, respectively, and their effects were essentially additive.

	DISCUSSION

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Using data collected at the end of the resistance test, THOQUET et al. 1996A , THOQUET et al. 1996B concluded that there was an important QTL on chromosome 6 carried by the Hawaii7996 parent; their analysis showed clearly that the LOD curve peak was very broad, but only one locus was identified. Indeed, we confirm here that if only the end of test results is taken into account, even use of the MapQTL cofactor approach fails to support a hypothesis for the presence of a second QTL. However, we now present several lines of evidence that indicate that at least two loci controlling BW resistance are present on this chromosome.

Temporal analyses of data collected from F₃ families tested in the field show changes in the maximum LOD score peaks during the course of the resistance test. A maximum around 40 cM is clearly observed and maintained over three observation dates (Figure 2), but at the end of the test the maximum LOD score is associated with markers on the top part of the chromosome. Although individual QTL may contribute different amounts to explaining the variance in different tests or at different times during a test, the map positions of real QTL are not expected to move along the chromosome. Two possible hypotheses can be proposed to account for this paradoxical observation of the apparent movement of QTL: (1) Because we can only estimate the most likely position of a QTL from the available data with a certain level of confidence for a particular map interval, it is possible that this position varies between datasets, and the observations can be explained by one QTL; (2) two linked QTL could be involved to different extents at different stages in the development of disease. We strongly favor the latter hypothesis because (i) the apparent movement of the maximum LOD score is not random but progresses mainly in one direction, and (ii) this shift occurs over a relatively large map distance (~25 cM; Figure 2). Taken together, the observed data thus constitute a strong argument in favor of the presence of at least two different loci on chromosome 6. Any biological explanation of the shift in the importance of these loci is speculative. However, the presence of a second locus might possibly be dependent on environmental conditions such as the presence of natural light, or on the physiological state of the plant, because no evidence for a second locus could be found by a similar temporal and statistical analysis (unpublished data) of an F₂ population of cuttings testing in a growth chamber, where the markers TG118 and CP18 showed the strongest associations with resistance (THOQUET et al. 1996A ).

Two different kinds of statistical analysis, cofactor analysis (VAN OOIJEN and MALIEPAARD 1996 ) and a 2QM test (GOFFINET and MANGIN 1998 ), show that at least two chromosomal regions are probably involved in resistance. A cofactor placed on or near TG240 reveals the presence of a second QTL in the upper region of chromosome 6 in all of the analyses of the F₃ population done from 12 to 30 d.a.i., whereas a cofactor placed on a proximal locus (in the uppermost 12 cM of the linkage group) reveals the presence of a second locus around TG240, albeit only in some analyses (Figure 2B, 14 d.a.i.). The 2QM test indicates with >99% confidence that two loci exist near markers Cf-2 and TG73 at the end of the test done on the F₃ population.

A first indication of the importance of a locus on chromosome 6 in resistance to BW came from breeders who had difficulty combining the resistance to BW with resistance to nematodes conferred by the gene Mi (GILBERT and MCGUIRE 1956 ), and this observation has subsequently been substantiated in different lines of tomato (P. DEBERDT, G. ANAÏS and P. PRIOR, personal communication). Because the map location of Mi is known (KLEIN-LANKHORST et al. 1991 ; HO et al. 1992 ; WEIDE et al. 1993 ) and lies very close (VAN WORDRAGEN et al. 1996 ) to the resistance gene Cf-2 (placed on our map), the latter being a gene conferring resistance to the fungus Cladosporium fulvum (DIXON et al. 1995 , DIXON et al. 1996 ), it seems likely that several different sources of resistance to BW carry a locus in this region. Several other genes encoding resistance to pathogens have been localized to this chromosomal region (VAN DER BEEK et al. 1994 ; WILLIAMSON et al. 1994 ; ZAMIR et al. 1994 ; KALOSHIAN et al. 1995 ; SANDBRINK et al. 1995 ; CHAGUE et al. 1997 ). Possible biological explanations for this observation are very speculative, but might arise due to common aspects of the disease response mechanism, because such resistance genes may rely on common kinds of signal transduction pathway (BONAS and VAN DEN ACKERVEKEN 1997 ; DE WIT 1997 ; GEBHARDT 1997 ; HAMMOND-KOSACK and JONES 1997 ), or on the ability of certain genomic regions to generate the variation needed for rapid evolution of resistance (PARNISKE et al. 1997 ; SONG et al. 1997 ; THOMAS et al. 1997 ). This BW resistance locus, on the upper part of chromosome 6, is therefore probably different from the locus carried by tomato line L285 (DANESH et al. 1994 ), because the latter maps close to CT184, at ~40 cM on tomato chromosome 6. On the reference tomato map produced from an interspecific cross of tomato with the wild relative L. pennellii (TANKSLEY et al. 1992 ), CT184 and TG240 are both located at the same map position, 38 cM, and it is therefore likely that the QTL from L285 and one of the QTL from Hawaii7996 map close to this position. Because this locus is >30 cM away from Mi, it is most likely to be a separate locus to that inferred previously by plant breeders to be close to Mi, because it should easily be possible to obtain recombinants over such a large map distance. However, data collected by breeders on the linkage of the locus sp (a morphological marker) to BW resistance from the line HES 5808-2, a descendant of PI 127805A (ACOSTA et al. 1964 ), also suggest the existence of a distal locus, because sp is located near the bottom end of chromosome 6, close to CT109 in our material (THOQUET et al. 1996A ), and is thus too distant (>50 cM) to detect linkage with the proximal locus. The accession PI 127805A was used as a source of resistance in Hawaii tomato breeding programs (ACOSTA et al. 1964 ) and may be a progenitor for some of the BW resistance found in Hawaii7996. All of these observations lend further weight to the notion that there are at least two QTL controlling BW on chromosome 6. In conclusion, there is evidence for one resistance locus close to Mi in Hawaii7996, and evidence for a second locus in the region of TG73/TG240/CT184 in tomato lines Hawaii 7996 and L285. Although these QTL have been observed in different tomato lines and under different experimental environments, it is not yet clear how general these resistances are. The resistance carried by L285 close to CT184 is probably specific for certain bacterial strains (DANESH and YOUNG 1994 ). Further work is required to map these loci more precisely, to define whether or not more than two QTL are present on chromosome 6, and to clarify the biological nature of these resistances. To this end, the development of recombinant inbred lines (P. HANSON and C. BALATERO, personal communication) will provide an invaluable tool for future analyses.

	ACKNOWLEDGMENTS

We thank the laboratories of Steve Tanksley, Christiane Gebhardt, and Jonathan Jones for supplying RFLP probes, and Christian Boucher for reading the manuscript. The Region Midi-Pyrénées and the European Community provided part of the financial support (grant nos. 3172A and CIP CT 840050 to N.G.).

Manuscript received May 29, 1998; Accepted for publication November 16, 1998.

	APPENDIX 1

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Note k, the indexed value of n individuals in the population, M_i,k for i = 1, ... , I, the set of marker genotypes for the kth individual, and y_k, its phenotypic observed value for the quantitative trait. Let y denote the vector of all the y_k.

Suppose that the ith marker is a fully informative one; let us assign each individual k to its corresponding genotypic class according to M_i,k. For each genotypic class, we obtain a set of n^c_i indices, where n^c_i is the number of individuals in the cth class according to the marker M_i. The data are shuffled within each class by computing a random permutation of the set of indices and by assigning to the lth individual of the class c, the trait value whose index is given by the lth element of the permutation within the class c. Let us denote by Y*ⁱ the shuffled data set. The shuffling is repeated N times, and for each shuffled data set the 2QM test is computed. The N test values are used to estimate a threshold for an type I error as the empirical quantile of the N test values.

	APPENDIX 1

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX 1 APPENDIX 1 LITERATURE CITED

The asymptotic distribution of Y*ⁱ is the distribution function of an n sample with one QTL located at the ith marker given _i = _i(y), = (y), ² = ²(y), where _i, , and ² are, respectively, the maximum-likelihood estimators for the QTL effect at the marker i (additivity + dominance in backcross, additivity and dominance in F₂), the mean, and the variance, and _i(y), (y), and ²(y) are their estimated value for the initial data set.

We focus our attention on a backcross population, but similar results could be found in an F₂ population. Working in an asymptotic and local framework for the QTL effect, as the number of observations tends to infinity, the effect of the allele substitution, noted a multiplied by , tends to a finite constant . This is the correct framework in which to study the asymptotic distributions of the likelihood ratio test for QTL that are not major genes. In this framework, we get asymptotically sufficient statistics, which are , the global mean, ², the classical estimator of the variance, and the mean class difference at each marker S_j; j = 1 ... I,

where I[M_j,k = ·] is the indicator of the events [M_j,k = ·] (MANGIN et al. 1994 ). Asymptotically sufficient statistics in an F₂ population can be found in CIERCO and MANGIN 1996 .

In the following, we assume no interference between recombination events and therefore use Haldane's mapping function. In the asymptotic framework, the maximum-likelihood estimator of the effect of a QTL located at the ith marker is equivalent to S_i divided by ², and the asymptotic distribution of the set of statistics S_j for j = 1 ... I is a multinormal with

where x_jd = (1 - 2r_jd), and r_jd denote the recombination rate between the markers j and the QTL located at d, respectively, between two markers j and j'.

To prove that the asymptotic distribution of Y*ⁱ is the distribution function of an n sample with one QTL located at the marker i, given _i = _i(y), = (y), and ² = ²(y), it is therefore sufficient to study the asymptotic distribution of the S_j for a random permuted data set Y*ⁱ and to show that E(S_j) x_ijS_i(y) and Cov(S_j, S_j') (x_jj' - x_ijx_ij') ²(y), where S_i(y) is the estimated value of S_i for the initial data set.

Proof for the expectation:
Let us note , a random stratified permutation defined by the tuple (₁, ... , _k, ... , _n); we get

(A1)

and

(A2)

where

Using (A1), (A2), and the asymptotic equivalence for the proportion in marker classes leads to

where

(respectively, for n^AB_ij, n^BA_ij, n^BB_ij).

Covariances:
Proof for the covariances needs heavy algebraic calculations and notations. As this is not an essential part of the article, we studied the covariances by simulation. Table 2 shows the theoretical and empirical covariance matrices of the S_j, j = 1 ... I statistics on 10,000 shuffled data sets at each marker, for one backcross population with n = 1000, three markers evenly spaced on a 1-M chromosome, and no QTL. For this sample, the differences between the empirical and the theoretical variances or covariances were in mean equal to -1.410^-4, with a variance equal to 1.510^-4, and a maximum in absolute value equal to 3.710^-2. For other samples with fewer individuals, more markers, and a QTL segregating in the population, we found, in certain cases, values that were not as good but that remained acceptable.

	LITERATURE CITED

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