A general model for likelihood computations of genetic marker data accounting for linkage, linkage disequilibrium, and mutations

Abstract

Several applications necessitate an unbiased determination of relatedness, be it in linkage or association studies or in a forensic setting. An appropriate model to compute the joint probability of some genetic data for a set of persons given some hypothesis about the pedigree structure is then required. The increasing number of markers available through high-density SNP microarray typing and NGS technologies intensifies the demand, where using a large number of markers may lead to biased results due to strong dependencies between closely located loci, both within pedigrees (linkage) and in the population (allelic association or linkage disequilibrium (LD)). We present a new general model, based on a Markov chain for inheritance patterns and another Markov chain for founder allele patterns, the latter allowing us to account for LD. We also demonstrate a specific implementation for X chromosomal markers that allows for computation of likelihoods based on hypotheses of alleged relationships and genetic marker data. The algorithm can simultaneously account for linkage, LD, and mutations. We demonstrate its feasibility using simulated examples. The algorithm is implemented in the software FamLinkX, providing a user-friendly GUI for Windows systems (FamLinkX, as well as further usage instructions, is freely available at www.famlink.se). Our software provides the necessary means to solve cases where no previous implementation exists. In addition, the software has the possibility to perform simulations in order to further study the impact of linkage and LD on computed likelihoods for an arbitrary set of markers.

Appendix

The following section includes a more detailed description of the notation used in the paper. First, we assume locus i, (i = 1, … , I) has A _i possible alleles, and let p _i be a vector specifying the probabilities of a haplotype’s alleles at locus i given the haplotype’s alleles at lower indexes. We let r ₂, … , r _I denote the recombination rates between the loci, which are assumed known. For a locus i, let t be a transmission, specifying a start allele in the parent, a resulting allele in the child, and whether the parent is a mother or a father. We then denote with m _i (t) the probability that the child obtains the resulting allele, given that the parent has the start allele. This function specifies the mutation model at locus i. The parameters of our model are p = (p ₁, … , p _I ), r = (r ₂, … , r _I ), and m = (m ₁, … , m _I ).

If parents’ alleles follow the population frequencies, the probabilities for a child to have various alleles are not given by the population frequencies, unless the process represented by the mutation model happens to have the population frequencies as stationary distribution. This means that adding the untyped father or mother to a person in the pedigree may change the probability results we are computing. To avoid this nuisance, we recommend that all untyped founders with only one child in the pedigree are (recursively) removed prior to computations. In our pedigree, a person may have specified no parents, only a mother, only a father, or both parents. Founders are those who have no parents in the pedigree. We also assume the pedigree does not contain untyped children with no descendants as such children cannot affect the result.

Our observed data is divided into data s for S typed founders and data d for M typed non-founders: Let s _{i j} for i = 1, … , I, j = 1, … , S denote the observed allele or alleles of typed founder j at locus i. For males and X- chromosomal data, s _{i j} specifies only one allele, otherwise s _{i j} specifies the two observed alleles in no particular order. For the typed non-founders, let d _{i j} specify the similar data. We write s _i = (s _i1, … , s _{i S} ), s = (s ₁, … , s _I ), d _i = (d _i1, … , d _{i M} ), and d = (d ₁, … , d _I ).

We also need a number of ancillary variables: The inheritance pattern at locus i can be described as a vector v _i of length N, with one component for each parent-child relationship in the pedigree when the locus is autosomal, and one for each mother-child relationship for X- chromosomal loci. Each component is 0 or 1 depending on whether the paternal or maternal allele is inherited, we write v = (v ₁, … , v _I ). We also need to describe the founder alleles of the pedigree: These are maternal or paternal alleles whose relevant parent is not in the pedigree. First, there are founder alleles belonging to typed founders: Let g _{i j} be the allele or alleles of typed founder j at locus i listed with the paternal allele first. Write g _i = (g _i1, … , g _{i S} ) and g = (g ₁, … , g _I ). For the remaining F founder alleles, let f _{i j} denote the j ^′ t h founder allele at locus i. Finally, we write f _i = (f _i1, … , f _{i F} ) and f = (f ₁, … , f _I ).