I describe an analytical approximation for calculating the short-term probability of loss of a chromosome under the neutral Wright-Fisher model with recombination. I also present an upper and lower bound for this probability. Exact analytical calculation of this quantity is difficult and computationally expensive because the number of different ways in which a chromosome can be lost, grows very large in the presence of recombination. Simulations indicate that the probabilities obtained using my approximate formula are always comparable to the true expectations provided that the number of generations remains small. These results are useful in the context of an algorithm that we recently developed for simulating Wright-Fisher populations forward in time. C++ programs that can efficiently calculate these formulas are available on request.

Genome-wide association studies (GWAS) have identified >300 loci associated with measures of adiposity including body mass index (BMI) and waist-to-hip ratio (adjusted for BMI, WHRadjBMI), but few have been identified through screening of the African ancestry genomes. We performed large scale meta-analyses and replications in up to 52,895 individuals for BMI and up to 23,095 individuals for WHRadjBMI from the African Ancestry Anthropometry Genetics Consortium (AAAGC) using 1000 Genomes phase 1 imputed GWAS to improve coverage of both common and low frequency variants in the low linkage disequilibrium African ancestry genomes. In the sex-combined analyses, we identified one novel locus (TCF7L2/HABP2) for WHRadjBMI and eight previously established loci at P < 5×10-8: seven for BMI, and one for WHRadjBMI in African ancestry individuals. An additional novel locus (SPRYD7/DLEU2) was identified for WHRadjBMI when combined with European GWAS. In the sex-stratified analyses, we identified three novel loci for BMI (INTS10/LPL and MLC1 in men, IRX4/IRX2 in women) and four for WHRadjBMI (SSX2IP, CASC8, PDE3B and ZDHHC1/HSD11B2 in women) in individuals of African ancestry or both African and European ancestry. For four of the novel variants, the minor allele frequency was low (<5%). In the trans-ethnic fine mapping of 47 BMI loci and 27 WHRadjBMI loci that were locus-wide significant (P < 0.05 adjusted for effective number of variants per locus) from the African ancestry sex-combined and sex-stratified analyses, 26 BMI loci and 17 WHRadjBMI loci contained ≤ 20 variants in the credible sets that jointly account for 99% posterior probability of driving the associations. The lead variants in 13 of these loci had a high probability of being causal. As compared to our previous HapMap imputed GWAS for BMI and WHRadjBMI including up to 71,412 and 27,350 African ancestry individuals, respectively, our results suggest that 1000 Genomes imputation showed modest improvement in identifying GWAS loci including low frequency variants. Trans-ethnic meta-analyses further improved fine mapping of putative causal variants in loci shared between the African and European ancestry populations.