SNPpair Tetrahedron: Geometric Presentation of Haplotype Space of Pairwise SNPs

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
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 22-13 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 r21. 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.
Ff(AB),f(Ab),f(aB),f(ab)f(AB)(x1x2x31)/4,
f(Ab)(x1-x2-x31)/4,f(aB)(-x1x2-x31)/4,f(ab)
(-x1-x2x31)/4x1f(A)-f(a)x2f(B)-f(b)x3f(A
B)-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 464115
V1AB V2Ab V3aB V4ab
E1AB,AbA1 E2aB,aba1 E3AB,aBB1 E4A
b,abb1
F1AB,Ab,aBTranslucent F2AB,Ab,abOrange F3
AB,aB,abPink F4Ab,aB,abGreen
ISAB,Ab,aB,abInner space
E5AB,ab E6Ab,aB
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
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we appreciate any comments or questions on this
poster ryamada_at_src.riken.go.jp