1. SNP-pair Tetrahedron: Geometric Presentation of Haplotype Space of Pairwise SNPs
Yamada R(1)(2), Okada Y(1)
(1)Laboratory of Functional Genomics, Human Genome Center, IMS, U of Tokyo, Tokyo, Japan
(2)Unit of Human Disease Genomics, Center for Genomic Medicine, Kyoto University, Kyoto, Japan
September 7-10, 2007 York, England (IGES 2007)
Three parameters to express frequency of four haplotypes (ref 1)
Fifteen statuses of SNP pairs
SUMMARY AB Ab aB ab
(I)
(V)
(II)
(III)
(IV)
Drift-out
Mutation
Recombination
We recently published a new method to express frequency of haplotypes for a locus with multiple SNPs, which gave a generalized definition of linkage disequilibrium (LD) for multiple SNPs (ref 1). Assume a locus with n SNPs. This new method gave a set of haplotype frequencies a linear expression with 2n-1 variables, that were mutually independent. In case of SNP pairs, a frequency vector of four haplotypes is expressed with 22-1=3 variables. Two of the three variables correspond to frequency of two SNPs and the last one represents LD. The three variables are defined so that they are mutually independent; therefore the corresponding three vectors are orthogonal. We do not place the three orthogonal vectors as base vectors for three dimensional Euclidean space but propose to let them construct a regular tetrahedron so that components of the tetrahedron represent conditions of pairwise SNPs: Tetrahedron is consisted of four vertices, six edges and four faces. Four vertices stand for clonal conditions. Four out of six edges correspond to conditions where only one of two sites is polymorphic. The two other edges are conditions where a LD index r2=1. Four faces are conditions where another LD index, D'=1. The space inside of the tetrahedron represents conditions where four haplotypes exist. The tetrahedron contains a curved surface of linkage equilibrium (LE). This presentation characterizes the SNP-pair tetrahedron and introduces its applications to evaluations of genetic heterogeneity and linkage disequilibrium.
F={f(AB),f(Ab),f(aB),f(ab)} f(AB)=(x1+x2+x3+1)/4, f(Ab)=(x1-x2-x3+1)/4, f(aB)=(-x1+x2-x3+1)/4, f(ab)=(-x1-x2+x3+1)/4; x1=f(A)-f(a) x2=f(B)-f(b) x3=f(AB)-f(Ab)-f(aB)+f(ab) AB Ab ab aB x1 x2 x3
Fifteen statuses are grouped into five; (I) no SNP, (II) one SNP, (III) two SNPs with three haplotypes, (IV) two SNPs with four haplotypes and (V) two SNPs with two haplotypes.
Tetrahedron Tetrahedron is 3-simplex, a polytope in three dimensional space that has 4 vertices, 6 edges, 4 facets and 1 inner-space; =15
V1:{AB}
V2:{Ab}
V3:{aB}
V4:{ab}
E1:{AB,Ab;A=1}
E2:{aB,ab:a=1}
E3:{AB,aB:B=1}
E4:{Ab,ab:b=1}
E5:{AB,ab}
E6:{Ab,aB}
F1:{AB,Ab,aB}:Translucent
F2:{AB,Ab,ab}:Orange
F3:{AB,aB,ab}:Pink
F4:{Ab,aB,ab}:Green
IS:{AB,Ab,aB,ab}:Inner space
LE-surface in the tetrahedron V1 V2 V3 V4 E1 E2 E3 E4 E5 E6
Three axes in the tetrahedron ((x1)Axis for SNP A, (x2)Axis for SNP B and (x3)Axis for LD for the pair)
Concluding remarks
Statuses of SNP pairs are expressed in a tetrahedron. This spatial expression of conditions will contribute to analyze and quantitate polymorphic conditions.
Potential Usage of expression of SNPs in polytopes.
The most popular method to describe relation between two SNPs is linkage disequilibrium index. The tetrahedron gives completely different expression of relation between SNP pairs. This may give a novel index to quantitate of haplotype status including LD.
It can characterize each SNP in relation with many other SNPs. That may allow to distinguish SNPs with same allele frequency by their relation with other SNPs.
E2:{a=1}
E1:{A=1}
x1-axis
E4:{b=1}
E3:{B=1}
x2-axis
E5:{AB,ab}
E6:{Ab,aB}
x3-axis
Life (Birth, Drift and Death) of a SNP pair is traced in the tetrahedron AB Ab aB ab
(I)
(V)
(II)
(III)
(IV)
References (1) A novel method to express SNP-based genetic heterogeneity, , and its use to measure linkage disequilibrium for multiple SNPs, Dg, and to estimate absolute maximum of haplotype frequency (p n/a) Ryo Yamada, Fumihiko Matsuda Gen. Epidemiol.　Published Online: 16 May 2007 DOI: /gepi
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