A Comparative Study on Genetic Heterogeneity in Tuberous Sclerosis: Evidence for One Gene on 9q34 and a Second Gene on 11q22–23 a

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In contrast with these reports, a substantial number of families were described with free recombination between TSC and the chromosome 9 markers.s.h Recently Kandt et al. excluded TSC from an area of 20 cM encompassing the ABO locus. ' In 1988 Clark et a/. described a TSC patient with trisomy for I lq23.3-qter.8 The mother was reported to have a balanced t( I lq23.3; 22ql I .2) rearrangement, but there was no evidence of TSC in either parent. The gene for neural cell adhesion molecule, which is located on llq23, was postulated as a candidate locus. Linkage studies in IS North American families using four chromosome 9q markers and six chromosome I Iq markers revealed evidence for linkage to chromosome I 1 .y A maximum two-point lod score of 3.26 at 0 = 0.08 was obtained with the marker MCT128.1.
The present and other" collaborative efforts aim at ending the controversy by combining data from several studies, including those in which linkage to either chromosome 9 or I I has been established. Data were combined without selection for any type of families. We compared different statistical strategies to investigate the possibility of locus heterogeneity.

LOCUS HETEROGENEITY
In the last decade reverse genetics has become an important tool to localize and identify disease genes. This approach is based on the search for co-segregation o f a disease trait and genetic markers of known location. The growing number of mapped informative polymorphic markers enhanced the success of this approach. Several genes causing diseases have been mapped and identified this way.
This approach is based, however, on the assumption that data from informative meioses in different families can be combined. This assumption would be invalid if the disease-causing locus is not identical in all families, that is, if locus heterogeneity exists. Given the complexity of biochemical pathways, it is expected that locus heterogeneity will prove to be a feature of a significant proportion of genetic diseases.
Of interest, then, is the analysis of diseases that have been mapped in some families to a locus, although in other phenotypically similar families a mutation at the mapped locus cannot account for the observed data.
Tuberous sclerosis may very well be a member of this small group of dis-eases4.10.1 1.1 la showing locus heterogeneity. Other examples of diseases belonging to this group are elliptocytosis, X-linked retinitis pigmentosa, Charcot-Marie-Tooth disease type I , adult polycystic kidney disease, and manic depression. Only for Charcot-Marie-Tooth and X-linked retinitis pigmentosa has more than one locus been mapped.l?.I3.l4 In some diseases the assignment of families to a certain locus is facilitated by the type of segregation. This is the case when one or more loci are X-linked, while other loci are autosomal. Diseases like limb girdle muscular dystrophy and manic depression are phenotypically indistinguishable from their X-linked counterparts, Becker dystrophy and X-linked manic depression.
Most inherited diseases show autosomal inheritance, however. For autosomal diseases with locus heterogeneity, arguments for classification apart from linkage data are hard to obtain. Lack of linkage data may explain why the diseases have not yet been recognized as being heterogeneous. Even when linkage data are available, a locus may be overlooked, if the statistical analysis does not allow for locus heterogeneity. Lander and Botstein15 estimated that a trait-causing locus that accounts for 60% of all cases may be missed completely in a linkage study under the false assumption of homogeneity. If the tested marker lies within 1 cM of the disease locus, linkage may still be excluded from a region of about 20 cM encompassing the marker. This implies that the recent exclusion data of Kandt ef u1.' do not undermine the validity of the previous assignment of a TSC locus to 9q34.
The power of reverse genetics is clearly reduced in the case of locus heterogeneity. This complicated problem requires the development and use of powerful statistical approaches.

LOCUS HETEROGENEITY IN TUBEROUS SCLEROSIS
For most heterogeneous traits only one candidate region is known. Since there are two candidate regions for TSC, TSCl (9q) and TSC2 (I Iq), our problem is quite exceptional. We compared various approaches that allow for locus heterogeneity. These methods may be divided into two groups: The first group contains approaches that start with a separation of families, followed by conventional twopoint analyses. Results obtained with these methods are presented in detail elsewhere.20 Approaches from the second group start with conventional two-point or multipoint analyses, followed by weighted family assignment to the putative disease loci. Results obtained this way are presented in this paper.
A well-known test for heterogeneity is the "classic" admixture test.I6 It has been used in the analysis of almost all genetically heterogeneous diseases mentioned above. For each map location (x) and for each proportion of linked families ( a ) a lod score under heterogeneity [ Z i ( a , x ) The combination of a and x for which Z i ( a , x ) is maximal is evaluated in a Chi square test, using the maximum value of Z , ( a , x ) , assuming homogeneity ( a = 1) as a reference. The admixture tests were performed by using the HOMOG computer program." An alternative approach is also based on the admixture test. Instead of one position (x), two positions ( x l and x2) are evaluated. The equation Z,(a,x, ,x2) = log,,[a . 10"'.'1) + ( 1 -a ) . 1 0 L q is similar to the equation given above, where x? is at 0 = 0.5. This alternative approach can be taken to test for two distinct loci on one chromosome. Such a test has been performed successfully for X-linked retinitis pigmentosa. l 4 If both loci are not on the same chromosome, a so-called "imaginary chromosome" has to be constructed, as described previously. We composed the imaginary chromosome by combining the results of two multipoint analyses (one for each candidate region) in a head-to-tail orientation. The most distal marker on chromosome 9 and the most proximal marker on chromosome 1 I were flanking the junction. The recombination fraction between these markers and the junction was set at 0.5. By the use of this imaginary construct, the TSC linkage problem to markers on separate chromosomes was reduced to a linkage problem with two regions of interest on a single (imaginary) chromosome. The imaginary chromosome was analyzed with the programs HOMOGZ and POINT.1." Both programs are part of the HOMOG package. This method is only valid in the absence of further heterogeneity in the data set. Therefore a final check using the HOMOG3 program was performed. This program allows for a third locus, which may be unlinked to either candidate region.
Seventy-three families were analyzed, the majority of which have been included in previous publication^.^^^^'" For two-point and multipoint analyses the various programs of the LINKAGE package were utilized."

RESULTS OBTAINED WITH THE "CLASSIC" ADMIXTURE TESTS
The simplest application of the "classic" admixture test is the analysis of twopoint linkage data using HOMOG. Two of these tests were performed, one for the ABO locus on chromosome 9 and one for the anonymous chromosome I 1 marker MCT128. I . The results of these tests are shown in Table I . Significant support for heterogeneity could only be obtained with the ABO linkage data. A similar test using MCT128.1 revealed only insignificant support for heterogeneity.
A more advanced application of the admixture test uses a map of markers. The markers MCOA12 (D9S28, M s p l ) , AKI ( T q I or protein polymorphism), abl (Psrl or Trcql), and ABO were used as chromosome 9 markers in a five-point analysis. Another five-point analysis was carried out using the chromosome I I markers  The results obtained with HOMOG are presented in TABLE 1 . The use of a map of four chromosome 9 markers instead of one resulted in an increased significance when heterogeneity was tested versus homogeneity. However, a similar study on the chromosome I I region showed no increased significance.

RESULTS OBTAINED WITH THE IMAGINARY CHROMOSOME APPROACH
By combining the results of both five-point analyses into one larger structure an imaginary chromosome was constructed. 'The results obtained with this approach are shown in TABLES I , 2, and 3. The outcome of the analysis showed strong support for locus heterogeneity, with one putative TSC locus 6 cM proximal to ABO (TSCI) and another putative TSC locus 35 cM distal to PBGD    NorE: ICA: imaginary chromosome approach; MLSM: family assignment by maximum lod score method; NLSM: possible family assignments by negative lod score method. (Since families can be selected twice by using the NLSM, all possible assignments are indicated instead of a definitive assignment.) Next to the family assignment by the ICA, the posterior probability of the assignment of each family (by ICA) is given in parentheses. The last two columns show the lod scores at the putative TSCl and TSC2 loci, as calculated using the imaginary chromosome approach. Rot = Rotterdam, Car = Cardiff, Irv = Irvine, Lon = London and Bos = Boston.  Given at the top of each list. NOTE: The odds are calculated using the imaginary chromosome approach (TSC2). Additional tests revealed that other possible orders could not always be significantly excluded (TABLE 3). The most likely location of both putative TSC loci was confirmed by a HOMOG3 analysis. These results revealed no evidence for a third locus in our data set.

THE VALIDITY OF THE MODEL
Although the quality of the model we tested permitted locus heterogeneity to be proven, we may still question whether we tested the optimal model. The model can be regarded to be less optimal if we used false inter-marker distances or orders, if we did not test markers from the real TSC regions, if further locus heterogeneity is the case, or if the penetrance (IN%) is not correct. Testing a less optimal model might result in reduced significance and incorrect putative locations. The putative TSC2 locus is not flanked by a distal marker. The additional tests on the chromosome 1 1 region revealed an alternative TSC2 locus 30 cM proximal to MCT128.1. This alternative position is almost as likely as the position 35 cM distal to PBGD (TABI.~: 3). The validity of the tested region on chromosome I I seems questionable.
Exclusion of relatives without symptoms in additional analyses did not alter the outcome. Therefore the assumed full penetrance did not influence the model to a large extent (results not shown). Furthermore the maps of chromosome 9 and chromosome I I markers are rigorously tested by several groups.?',?? Therefore analysis of the data under the assumption of further locus heterogeneity or the analysis of more distant chromosome 1 I markers are the best candidate strategies for improving the model.

COMPARISON WITH RESULTS OBTAINED USING ALTERNATIVE METHODS
Although locus heterogeneity may not be an uncommon feature, only a few examples are known. So far it has not been possible to test methods for heterogeneity analysis on a large scale. Therefore it is wise to use multiple methods in each study, in order to avoid biases. Povey and coworkers used two alternative methods in the analysis of the same family material.?() Both methods start with a separation of families, followed by conventional two-point analysis.
The first approach is called the Negative Lod Score Method (NLSM).?" The method only uses linkage information from a candidate region if the alternative region is associated with negative lod scores.?3 This analysis is followed by conventional two-point analysis on the selected families. This way the problem is simplified into two segments for which homogeneity is assumed. No linkage data are used twice.
The second method is called the Maximum Lod Score Method (MLSM).?" It uses the maximum lod scores from both candidate regions to classify the families (N. Morton, personal communication).
Part of the results obtained by these methods are included in TABLE 2. The results on the chromosome I I localization support a TSC2 locus proximal of MCT128.1 A position between 2-7-ID6 and L424 seems likely. There seems to be an inconsistency within the results on the localization of the TSCl locus. The NLSM supports a locus between ABO and EFD126.3, which is in contrast with the position near abl as supported by the MLSMZa and ICA. This discrepancy is not more than a seeming discrepancy since none of the resulting locus orders are inconsistent with the odds against these orders as given in TABLE 3.
All approaches used have shown their value. The imaginary chromosome approach has increased power in heterogeneity analysis when compared to the "classic" admixture tests. The imaginary chromosome approach seems to be more refined and less biased, since data from all of the families are used to identify the two most likely locations, without prior selection. However, misleading results could be generated by using an incorrect model. For instance, if one of the putative locations was inaccurate, then the probabilities of linkage assigned to each family would be incorrect, and the associated lod scores would be low. Because of these potential problems, the use of an alternative method such as the NLSM or the MLSM to verify the results is also recommended.

CONCLUDING REMARKS
Our studies revealed evidence for a model with two different loci independently causing TSC. The first locus (TSCI) maps on chromosome 9 between the abl oncogene and the ABO blood group locus. Several methods confirmed a position about 6 cM proximal to ABO. The second locus (TSC2) maps on chromosome 11q22-23. The exact position of the TSCZ locus remains unclear.
Our current studies aim at finding closer linked markers to both TSCI and TSCZ loci. On chromosome I I a set of informative markers that span a wider range is under investigation. On chromosome 9 our studies focus on new markers, closely linked to abl and ABO.