A simple scheme for the analysis of HLA linkages in pedigrees

In the maximum-likelihood analysis of genetic linkage in human pedigrees as carried out, for example, by the Elston-Stewart (1971) algorithm, one has to specify, for each individual, an array of values P(x1g) denoting the probability of observing the individual’s phenotype x given the individual has genotype g. The number of elements in this array is determined by the total number ala2 (a,a, + 1)/2 of genotypes, where a, and a, denote the respective number of alleles a t the first and second locus under consideration. For large numbers ( > 4) of alleles, this array will contain so many elements that even for moderately sized pedigrees the necessary calculations become very lengthy. This is the case for the HLA locus especially when the HLA-A/B haplotypes are treated as elleles of a single locus. However, under certain conditions that are often met in practice, there is a simple scheme that allows a proper maximum-likelihood (lod score) linkage analysis requiring the specification of only four HLA alleles. It is based on the fact verified below that when the genotypes a t a locus are completely known for all individuals marrying into the pedigree (originals), then the choice of gene frequencies at this locus has no effect on the lod score, i.e. the gene frequencies can arbitrarily be set equal to a fixed constant such as 1. This allows for the ‘recycling ’ of some HLA alleles (haplotypes) as described below so that only four HLA alleles have to be carried along which will be denoted by A, B, C and D. They determine ten genotypes, AA, AB, AC, A D , BB, . . ., DD, where, for instance, the phenotype BC reflects the genotypes B/C and CIB. The conditions under which the simple scheme leads to correct answers are the following two: (1) The genotypes at the HLAlocus are known for all originals, either by typing or by deduction from other pedigree members. Note that there is no restriction on the phenotypes or genotypes at the other locus (trait locus). (2) There is at most one mate per individual. Relaxation of these conditions and corresponding extensions of the basic scheme will be