SNP-pair Tetrahedron: Geometric Presentation of Haplotype Space of Pairwise SNPs

1. AB Ab aB ab AB Ab aB ab AB aB Ab ab AB ab Ab aB AB Ab aB AB Ab ab AB aB ab Ab aB ab AB Ab aB ab (I) (II) (V) (III) Drift-out (IV) Mutation Recombination AB aB x3 x2 x1 V1 V1 V1 V1 E1 E5 ab V1 Ab V2 E5 V4 E1 E5 E3 E4 E4 V2 E1 V4 E3 V4 E6 E5 E3 E5 E3 E1 V2 E4 E6 E2 E3 E2 E6 E2 E4 E2 V3 V4 V3 V3 E2 V2 E6 V3 V3 x2-axis (I) x3-axis x1-axis AB Ab aB ab (II) (V) E3:{B=1} AB ab Ab aB AB Ab aB ab AB aB Ab ab E3:{B=1} E1:{A=1} E5:{AB,ab} (III) AB Ab aB AB Ab ab AB aB ab Ab aB ab E4:{b=1} (IV) E2:{a=1} E6:{Ab,aB} AB Ab aB ab 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 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. SNP-pair Tetrahedron: Geometric Presentation of Haplotype Space of Pairwise SNPs 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) 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. TetrahedronTetrahedron is 3-simplex, a polytope in three dimensional space that has 4 vertices, 6 edges, 4 facets and 1 inner-space; 4+6+4+1=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 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. Life (Birth, Drift and Death) of a SNP pair is traced in the tetrahedron 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: 10.1002/gepi .202355 Figures were drawn with Cinderella http://cinderella.de/tiki-index.php we appreciate any comments or questions on this poster : ryamada@src.riken.go.jp