A note on Makeev's conjectures

A counterexample is given for the Knaster-like conjecture of Makeev for functions on $S^2$. Some particular cases of another conjecture of Makeev, on inscribing a quadrangle into a smooth simple closed curve, are solved positively

In search of causal variants: refining disease association signals using cross-population contrasts

<p>Abstract</p> <p>Background</p> <p>Genome-wide association (GWA) using large numbers of single nucleotide polymorphisms (SNPs) is now a powerful, state-of-the-art approach to mapping human disease genes. When a GWA study detects association between a SNP and the disease, this signal usually represents association with a set of several highly correlated SNPs in strong linkage disequilibrium. The challenge we address is to distinguish among these correlated loci to highlight potential functional variants and prioritize them for follow-up.</p> <p>Results</p> <p>We implemented a systematic method for testing association across diverse population samples having differing histories and LD patterns, using a logistic regression framework. The hypothesis is that important underlying biological mechanisms are shared across human populations, and we can filter correlated variants by testing for heterogeneity of genetic effects in different population samples. This approach formalizes the descriptive comparison of p-values that has typified similar cross-population fine-mapping studies to date. We applied this method to correlated SNPs in the cholinergic nicotinic receptor gene cluster <it>CHRNA5-CHRNA3-CHRNB4</it>, in a case-control study of cocaine dependence composed of 504 European-American and 583 African-American samples. Of the 10 SNPs genotyped in the r²≥ 0.8 bin for <it>rs16969968</it>, three demonstrated significant cross-population heterogeneity and are filtered from priority follow-up; the remaining SNPs include <it>rs16969968 </it>(heterogeneity p = 0.75). Though the power to filter out rs16969968 is reduced due to the difference in allele frequency in the two groups, the results nevertheless focus attention on a smaller group of SNPs that includes the non-synonymous SNP rs16969968, which retains a similar effect size (odds ratio) across both population samples.</p> <p>Conclusion</p> <p>Filtering out SNPs that demonstrate cross-population heterogeneity enriches for variants more likely to be important and causative. Our approach provides an important and effective tool to help interpret results from the many GWA studies now underway.</p

Joint effects of known type 2 diabetes susceptibility loci in genome-wide association study of Singapore Chinese: The Singapore Chinese health study

Background: Genome-wide association studies (GWAS) have identified genetic factors in type 2 diabetes (T2D), mostly among individuals of European ancestry. We tested whether previously identified T2D-associated single nucleotide polymorphisms (SNPs) replicate and whether SNPs in regions near known T2D SNPs were associated with T2D within the Singapore Chinese Health Study. Methods: 2338 cases and 2339 T2D controls from the Singapore Chinese Health Study were genotyped for 507,509 SNPs. Imputation extended the genotyped SNPs to 7,514,461 with high estimated certainty (r2>0.8). Replication of known index SNP associations in T2D was attempted. Risk scores were computed as the sum of index risk alleles. SNPs in regions ±100 kb around each index were tested for associations with T2D in conditional fine-mapping analysis. Results: Of 69 index SNPs, 20 were genotyped directly and genotypes at 35 others were well imputed. Among the 55 SNPs with data, disease associations were replicated (at p<0.05) for 15 SNPs, while 32 more were directionally consistent with previous reports. Risk score was a significant predictor with a 2.03 fold higher risk CI (1.69-2.44) of T2D comparing the highest to lowest quintile of risk allele burden (p = 5.72×10-14). Two improved SNPs around index rs10923931 and 5 new candidate SNPs around indices rs10965250 and rs1111875 passed simple Bonferroni corrections for significance in conditional analysis. Nonetheless, only a small fraction (2.3% on the disease liability scale) of T2D burden in Singapore is explained by these SNPs. Conclusions: While diabetes risk in Singapore Chinese involves genetic variants, most disease risk remains unexplained. Further genetic work is ongoing in the Singapore Chinese population to identify unique common variants not already seen in earlier studies. However rapid increases in T2D risk have occurred in recent decades in this population, indicating that dynamic environmental influences and possibly gene by environment interactions complicate the genetic architecture of this disease. © 2014 Chen et al