A Complete Enumeration and Classification of Two-Locus Disease Models

There are 512 two-locus, two-allele, two-phenotype, fully-penetrant disease models. Using the permutation between two alleles, between two loci, and between being affected and unaffected, one model can be considered to be equivalent to another model under the corresponding permutation. These permutations greatly reduce the number of two-locus models in the analysis of complex diseases. This paper determines the number of non-redundant two-locus models (which can be 102, 100, 96, 51, 50, or 48, depending on which permutations are used, and depending on whether zero-locus and single-locus models are excluded). Whenever possible, these non-redundant two-locus models are classified by their property. Besides the familiar features of multiplicative models (logical AND), heterogeneity models (logical OR), and threshold models, new classifications are added or expanded: modifying-effect models, logical XOR models, interference and negative interference models (neither dominant nor recessive), conditionally dominant/recessive models, missing lethal genotype models, and highly symmetric models. The following aspects of two-locus models are studied: the marginal penetrance tables at both loci, the expected joint identity-by-descent probabilities, and the correlation between marginal identity-by-descent probabilities at the two loci. These studies are useful for linkage analyses using single-locus models while the underlying disease model is two-locus, and for correlation analyses using the linkage signals at different locations obtained by a single-locus model.Comment: LaTeX, to be published in Human Heredit

Diminishing Return for Increased Mappability with Longer Sequencing Reads: Implications of the k-mer Distributions in the Human Genome

The amount of non-unique sequence (non-singletons) in a genome directly affects the difficulty of read alignment to a reference assembly for high throughput-sequencing data. Although a greater length increases the chance for reads being uniquely mapped to the reference genome, a quantitative analysis of the influence of read lengths on mappability has been lacking. To address this question, we evaluate the k-mer distribution of the human reference genome. The k-mer frequency is determined for k ranging from 20 to 1000 basepairs. We use the proportion of non-singleton k-mers to evaluate the mappability of reads for a corresponding read length. We observe that the proportion of non-singletons decreases slowly with increasing k, and can be fitted by piecewise power-law functions with different exponents at different k ranges. A faster decay at smaller values for k indicates more limited gains for read lengths > 200 basepairs. The frequency distributions of k-mers exhibit long tails in a power-law-like trend, and rank frequency plots exhibit a concave Zipf's curve. The location of the most frequent 1000-mers comprises 172 kilobase-ranged regions, including four large stretches on chromosomes 1 and X, containing genes with biomedical implications. Even the read length 1000 would be insufficient to reliably sequence these specific regions.Comment: 5 figure