The advantages of dense marker sets for linkage analysis with very large families

Abstract

Dense sets of hundreds of thousands of markers have been developed for genome-wide association studies. These marker sets are also beneficial for linkage analysis of large, deep pedigrees containing distantly related cases. It is impossible to analyse jointly all genotypes in large pedigrees using the Lander–Green Algorithm, however, as marker density increases it becomes less crucial to analyse all individuals’ genotypes simultaneously. In this report, an approximate multipoint non-parametric technique is described, where large pedigrees are split into many small pedigrees, each containing just two cases. This technique is demonstrated, using phased data from the International Hapmap Project to simulate sets of 10,000, 50,000 and 250,000 markers, showing that it becomes increasingly accurate as more markers are genotyped. This method allows routine linkage analysis of large families with dense marker sets and represents a more easily applied alternative to Monte Carlo Markov Chain methods.

Appendix

Further details on the simulation process

Four sets of simulations were performed (Figs. 2, 3, 4 and 8) to examine the accuracy of this splitting method at various marker densities. The simulation process involved randomly choosing a position on the autosomal genome for a disease locus. For the bulk of the simulations, the inheritance vector at this locus was specified. Traditionally, simulations to estimate power are created by fixing the disease model and allowing the inheritance vector to vary according to the model chosen. With our simulation scheme we were not able to determine power for a given disease model, however, the scheme facilitated comparisons between linkage methods and marker sets. For the set of simulations that assumed no disease locus was present, the inheritance vector was allowed to vary.

To simulate genotypic data, for each founder, two haplotypes extending 20 cM either side of the disease locus (or to the end of the chromosome) were sampled without replacement from the phased haplotypes in phase I of the International HapMap project (Utah CEPH population). Recombinations between the markers in the 40 cM region were then simulated for every meiosis in the pedigree, using the fine-scale genetic map estimated from HapMap data available for download at http://www.hapmap.org/downloads/recombination/latest/ (McVean et al. 2004). These recombinations, in conjunction with the chosen inheritance vector, were used to generate genotypes for non-founders.

The S _pairs statistic was estimated for the simulated genotypic data, using the pedigree-splitting method described above. IBD-sharing probabilities were calculated using Merlin 1.0 (Abecasis et al. 2002; Abecasis and Wigginton 2005). LD between the markers was taken into account by defining haplotype blocks using HaploView 3.2 (Barrett et al. 2005) (spine-of-LD rule with D′ = 0.8). HaploView was also used to estimate haplotype frequencies. These were provided as inputs in Merlin 1.0 and the SNPs within haplotype blocks were then treated as single multi-allelic markers. The NPL_pairs statistic was obtained from the S _pairs statistic by subtracting the mean and dividing by the standard deviation of the S _pairs statistic under the null hypothesis of no linkage for a given pedigree. The mean and standard deviation were estimated for a fully-informative marker by 10⁷ gene-dropping simulations (Sobel and Lange 1996). In each simulation founder alleles are transmitted randomly down the complete, unsplit pedigree, and S _pairs is equal to the number of pairs of alleles from distinct cases that are copies of the same founder allele.

Estimation of the number of markers required to provide sufficient power to detect linkage

Figure 6a gives the cumulative distribution function F(x) of the number of markers, x, inherited IBD by a pair of related cases on either side of a disease locus (i.e. in the region of no recombination). To obtain this, it was assumed that both markers and recombinations are both distributed randomly across the genome with Poisson distributions relative to genetic distance. Then for a pair of cases who only have one line of common ancestry through a pair of full siblings, the expected distance in morgans to the first recombination on either side of the disease locus is ${\text{1/}}\lambda _{{\text{1}}}={\text{1/(}}k{\text{ + 1)}}$, where k is the degree of relationship between the cases (k = 3 for first cousins, k = 5 for second cousins, etc.). Letting n denote the number of markers genotyped along the genome (of approximate length 35 morgans), the expected distance to the first marker on either side of the disease locus is ${\text{1/}}\lambda _{{\text{2}}} {\text{ = 35/}}n$.

Consider the combined Poisson process of recombinations and markers with rate λ ₁ + λ ₂, where each event has probability, $p_{m} = \lambda _{{{\text{ }}2}} /(\lambda _{{{\text{ }}1}} + \lambda _{{{\text{ 2}}}} )=(n/35), (k + 1 + n/35)$ of being a marker. Due to the reversibility of Poisson processes, the distribution of x is the same as the distribution of the number of markers occurring before two recombination events occur in the (k + 1) meioses connecting the cases. This is given by

$$ \begin{aligned}{} f(x) & = P(x{\text{ markers before 2nd recomb}}{\text{.}}) \\ & = {\sum\limits_{i = 0}^x {P(i{\text{ markers}}){\text{ }}P({\text{recomb}}{\text{.}}){\text{ }}P{\text{(}}x - i{\text{ markers) }}P{\text{(recomb}}{\text{.)}}} } \\ & = (x + 1)p^{x}_{m} (1 - p_{m} )^{2} \\ \end{aligned} $$

(1)

The cumulative distribution function is then given by

$$ \begin{aligned}{} F(x) & \; = \;P(X \le x)\; = \;{\sum\limits_{j = 0}^x {(j + 1)p_{m} ^{j} {\left( {1 - p_{m} } \right)}^{2} } } \\ & = {\left( {1 - p_{m} } \right)}^{2} {\sum\limits_{j = 0}^x {\frac{d} {{dp_{m} }}{\left[ {p_{m} ^{{j + 1}} } \right]}} } \\ & = {\left( {1 - p_{m} } \right)}^{2} \frac{d} {{dp_{m} }}{\left[ {{\sum\limits_{j = 0}^x {p_{m} ^{{j + 1}} } }} \right]} \\ & = {\left( {1 - p_{m} } \right)}^{2} \frac{d} {{dp_{m} }}{\left[ {{\left( {\frac{{p_{m} - p_{m} ^{{x + 2}} }} {{1 - p_{m} }}} \right)}} \right]} \\ & = 1 - {\left[ {\frac{{n/35}} {{k + 1 + n/35}}} \right]}^{{x + 1}} {\left( {1 + (x + 1){\left[ {\frac{{k + 1}} {{k + 1 + n/35}}} \right]}} \right)} \\ \end{aligned} $$

(2)

Figure 6b presents an estimate of the minimum number, y of identical-by-state (IBS) markers required to be reasonably confident that a haplotype has been inherited identical-by-descent (IBD) by two individuals. An idealized situation is assumed, where there is no LD between the markers, but they are close enough to ensure that the relatives have either inherited a haplotype IBD over the entire region, or have not inherited a haplotype IBD over any of the region (that is there is no recombination in the region and thus complete separation between the scales on which LD and linkage operate). If two individuals have inherited an allele identical-by-state at y consecutive markers, the probability that these alleles have been inherited IBD is

$$ \begin{aligned}{} P({\text{IBD}}\left| {y{\text{ alleles IBS}},k} \right.) & = \frac{{P({\text{IBD}}\left| k \right.)}} {{P({\text{IBD}}\left| k \right.)\; + \;P(y{\text{ alleles IBS}}\left| {{\text{not IBD}},k} \right.)P({\text{not IBD}}\left| k \right.)}} \\ & = \frac{{\frac{1} {{2^{k-1} }}}} {{\frac{1} {{2^{k-1} }}\; + \;{\left[ {1 - 2p^{2} (1 - p)^{2} } \right]}^{y} {\left( {1 - \frac{1} {{2^{k-1} }}} \right)}}} \\ \end{aligned} $$

(3)

where k is the degree of relationship and p is the minor allele frequency. For this probability to be greater than P, y is given by

$$ y > \frac{{\log {\left[ {\frac{{1/P - 1}} {{2^{k-1} - 1}}} \right]}}} {{\log {\left( {1 - 2p^{2} (1 - p)^{2} } \right)}}}. $$

(4)

Figure 6b was generated from the above formula, with P = 0.8.