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Cubic exact solutions for the estimation of pairwise haplotype frequencies: implications for linkage disequilibrium analyses and a web tool 'CubeX'.

ABSTRACT: The frequency of a haplotype comprising one allele at each of two loci can be expressed as a cubic equation (the 'Hill equation'), the solution of which gives that frequency. Most haplotype and linkage disequilibrium analysis programs use iteration-based algorithms which substitute an estimate of haplotype frequency into the equation, producing a new estimate which is repeatedly fed back into the equation until the values converge to a maximum likelihood estimate (expectation-maximisation).We present a program, "CubeX", which calculates the biologically possible exact solution(s) and provides estimated haplotype frequencies, D', r2 and chi2 values for each. CubeX provides a "complete" analysis of haplotype frequencies and linkage disequilibrium for a pair of biallelic markers under situations where sampling variation and genotyping errors distort sample Hardy-Weinberg equilibrium, potentially causing more than one biologically possible solution. We also present an analysis of simulations and real data using the algebraically exact solution, which indicates that under perfect sample Hardy-Weinberg equilibrium there is only one biologically possible solution, but that under other conditions there may be more.Our analyses demonstrate that lower allele frequencies, lower sample numbers, population stratification and a possible |D'| value of 1 are particularly susceptible to distortion of sample Hardy-Weinberg equilibrium, which has significant implications for calculation of linkage disequilibrium in small sample sizes (eg HapMap) and rarer alleles (eg paucimorphisms, q < 0.05) that may have particular disease relevance and require improved approaches for meaningful evaluation.

SUBMITTER: Gaunt TR

PROVIDER: S-EPMC2180187 | biostudies-literature | 2007 Nov

REPOSITORIES: biostudies-literature

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Cubic exact solutions for the estimation of pairwise haplotype frequencies: implications for linkage disequilibrium analyses and a web tool 'CubeX'.

Gaunt Tom R TR Rodríguez Santiago S Day Ian Nm IN

BMC bioinformatics 20071102

<h4>Background</h4>The frequency of a haplotype comprising one allele at each of two loci can be expressed as a cubic equation (the 'Hill equation'), the solution of which gives that frequency. Most haplotype and linkage disequilibrium analysis programs use iteration-based algorithms which substitute an estimate of haplotype frequency into the equation, producing a new estimate which is repeatedly fed back into the equation until the values converge to a maximum likelihood estimate (expectation- ...[more]

PMID: 17980034

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Project description:Linkage disequilibrium (LD) analysis provides information on the evolutionary aspects of populations. Recently, haplotype blocks have been used to increase the power of quantitative trait loci detection in genome-wide association studies and the prediction accuracy of genomic selection. Our objectives were as follows: to compare the degree of LD, LD decay, and LD decay extent in popcorn populations; to characterize the number and length of haplotype blocks in the populations; and to determine whether maize chromosomes also have a pattern of interspaced regions of high and low rates of recombination. We used a biparental population, a synthetic, and a breeding population, genotyped for approximately 75,000 single nucleotide polymorphisms (SNPs). The sample size ranged from 190 to 192 plants. For the whole-genome LD and haplotype block analyses, we assumed a window of 500 kb. To characterize the block and step patterns of LD in the populations, we constructed LD maps by chromosome, defining a cold spot as a chromosome segment including SNPs with the same LDU position. The LD and haplotype block analyses were also performed at the intragenic level, selecting 12 genes related to zein, starch, cellulose, and fatty acid biosynthesis. The populations with the higher and lower frequencies of |D'| values greater than 0.75 were the biparental (65-74%) and the breeding population (26-58%), respectively. There were slight differences between the populations regarding the average distance for SNPs with |D'| values greater than 0.75 (in the range of approximately 207 to 229 kb). The level of LD expressed by the r2 values was low in the populations (0.02, 0.04, and 0.04, on average) but comparable to some non-isolated human populations. The frequency of r2 values greater than 0.75 was lower in the biparental population (0.2-0.5%) and higher in the other populations (0.2-1.6%). The average distance for SNPs with r2 values greater than 0.75 was much higher in the biparental population (approximately 80 to 126 kb). In the other populations, the ranges were approximately 6 to 19 and 6 to 35 kb. The heatmaps for the regions covered by the first 100 SNPs in each chromosome, in each population (1 to 3.3 Mb, approximately), provided evidence that the comparatively few high r2 values (close to 1.0) occurred only for SNPs in close proximity, especially in the synthetic and breeding populations. Due to the reduced number of SNPs in the haplotype blocks (2 to 3) in the populations, it is not expected advantage of a haplotype-based association study as well as genomic selection along generations. The results concerning LD decay (rapid decay after 5-10 kb) and LD decay extent (along up to 300 kb) are in the range observed with maize inbred line panels. The LD maps indicate that maize chromosomes had a pattern of regions of extensive LD interspaced with regions of low LD. However, our simulated LD map provides evidence that this pattern can reflect regions with differences in allele frequencies and LD levels (expressed by |D'|) and not regions with high and low rates of recombination.

Dataset Information

Cubic exact solutions for the estimation of pairwise haplotype frequencies: implications for linkage disequilibrium analyses and a web tool 'CubeX'.

Publications

Cubic exact solutions for the estimation of pairwise haplotype frequencies: implications for linkage disequilibrium analyses and a web tool 'CubeX'.

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