How to Measure Genetic Heterogeneity

International Workshop on Statistical-Mechanical Informatics

2009/09/13-2009/09/16

Unit of Statistical Genetics

Center for Genomic Medicine

Kyoto University

Ryo Yamada

What is genetic heterogeneity?

Biological Strategies

Landcover map by Environmental Research and Teaching at the University of Toronto

Lives cover land.

Slime moldchanges its shape and moves

around but uses spores to reproduce.

Wikipedia

Slime mold keeps looking for new

(better?) conditions.Space is too big to be covered completely.Therefore, multiple places are selected

and they are bridged without break.

Each part seems to act independently.

Food Slime mold is clever enough to find the shortest

route in the labyrinth.

Its strategy is being

investigated as a new model of

parallel computing

system.

Phylogenic tree

LIFE keeps looking for new (better?)

conditions.

Space is too big to be covered completely.

Therefore, multiple places are selected

and they are bridged without break.

Each part seems to act independently.

Phylogenic tree

They are bridged without break.

WE are here because WE are all offspring of “No-break” family sharing the features of continuous LIFE.

?? Features of LIFE ??

• Keeps looking for something.

• Accepts multiple conditions as good ones.

• Stays contiguous each other.

• Acts independently.

Slime mold distributes in physical space.

LIFE distributes in genetic space.

Distributions ～ Heterogeneity

LIFE distributes in genetic space.

What is genetic space?

DNA molecules4 letters, {A,T,G,C}L=3 x 109 in length (Homo sapience)

Sequence variations

4L; L=1,2,…

Biological space is a part of physico-chemical space.

Biological space is far much smaller than chemical space, But still enormously big.

Environmental fluctuations change width of pathways in biological space

Nature Reviews Genetics 3, 380-390 (2002); doi:10.1038/nrg795GENEALOGICAL TREES, COALESCENT THEORY AND THE ANALYSIS OF GENETIC POLYMORPHISMS

Nature Reviews Genetics 3, 380-390 (2002); doi:10.1038/nrg795GENEALOGICAL TREES, COALESCENT THEORY AND THE ANALYSIS OF GENETIC POLYMORPHISMS

Inter-species

Phylogeny

Intra-species

Recombination Graph

Inter-species heterogeneityIntra-species heterogeneity

MutantLetters are changed

(Mutation)

Combination of letters are changed

(Recombination)

Letters are changed : MutationCombination of letters are changed : Recombination

4L → 2L

Space is too big to be covered completely.

L=3 x 109

Variable sites ～ 10 x 106

Population size of homo sapience ～ 6x109

210,000,000>>>>> 6x109

k sites → 2k sequence variations

00…000 p(1)

00…001 p(2)

00…010 p(3)

00…011 p(4)

…

11…111 p (2k) =1-(p(1)+…+p(2k-1))

• 2k -1 parameters

• Flat and equal

Genetic heterogeneity

Dependency or association among variable sites

How to summarize the heterogeneity

with how many parameters?

Pairwise relation : r2

Variance-covariance matrix• describes the

heterogeneity with k(k-1)/2 parameters for individual pairs.

• predicts test statistics of associated markers for association study.

Ψ• Power set of {1,2,

…,k} is consisted of 2k subsets.

φ{1},{2},…,{k}{1,2},{1,3},…,{2,3},{2,4},…,{k-1,k}→Pairwise

{1,2,3},{1,2,4},…,{2,3,4},…,{k-2,k-1,k}…{1,2,…,k}

• Hierarchic parameters in full.

Hyper-cubes or lattice

Subsets with tandem elements

in Ψ

Pairwise relation : r2

Tandem pairs are elements of

both.

{1,2,…,k}

{1,2}{1,2,3}

{1,2,3,4}

{1,3}

{1,4}

{1,k}

One parameter for heterogeneity (1)

• EntropyH=-Σ p(i) ln(p(i)).

Effective No. sites to describe heterogeneity.

H=k ln(2) when all sites are independent.

H=0 when a clone (no variation).

• Entropy-based standardized measure of allelic association : ε ε=0 when all sites are independent.

ε=1 when only 2 types of sequence exist.

One parameter for heterogeneity (2)

• Entropy-based measure of allelic association : ε When k=2,

ε=r2=Σ((obs-exp)2/exp)=Σ(obs2/exp)-1

• rk keeps the shape of the equation of r2 and fits the value from 0 to 1 for any k:rk=Σ(obs(1+1/(k-1))/exp (1/(k-1))) -1

Space is too big to be covered completely.

210,000,000>>>>> 6x109

Every sequence is unique.Frequency is not useful.

Sparse graph

• Sequences can be plotted at nodes in k-dimensional hyper cube.

• Graph distance between sequences is No. mutations.

Graph distance between sequences is No. mutations.

Recombination’s distance?

Recombination is three-term relation.But graph is for two-

term relation.A more informational

tool is necessary.

• Biological meaning of heterogeneity:

• It does not want to lose variations even when a significant part of it can not survive because they might be useful sometime.

Survival curve of variable sites when a fraction of population extinct

• Each sequence set draws different survival curve.

• A measure to represent curves :

• The area upper the curve.

Various ways to measure

• Pairwise relation r2 k(k+1)/2• Power set Ψ 2k-1 Hierarchic• Entropy H 1• Entropy-based ε 1• r2-generalization rk 1• Graph Mutation distance• Graph+α +Recombination

distance• Survival curve 1 Simulation,

Mutation and Recombination

distance

Unit of Statistical Genetic, Center for Genomic MedicineGraduate School of Medicine, Kyoto University

http://www.genome.med.kyoto-u.ac.jp/wiki_tokyo/index.php/