Combinatorial methods in Bioinformatics: the haplotyping problem

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Combinatorial methods in Bioinformatics the
haplotyping problem

• Paola Bonizzoni
• DISCo
• Università di Milano-Bicocca

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Content

• Motivation biological terms
• Combinatorial methods in haplotyping
• Haplotyping via perfect phylogeny the PPH
  problem
• Inference of incomplete perfect phylogeny
  algorithms
• Incomplete pph and missing data
• Other models open problems

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Biological terms
Diploid organism
Biallelic site i Value(?i) ? A,C,G,T ? 2
heterozygous
?i ?i1 ?i2
haplotype
homozygous
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Motivations

• Human genetic variations are related to diseases
  (cancers, diabetes, osteoporoses) most common
  variation is the Single Nucleotide Polymorphism
  (SNP) on haplotypes in chromosomes
• The human genome project produces genotype
  sequences of humans
• Computational methods to derive haplotypes from
  genotype data are demanded
• Ongoing international HapMap project find
  haplotype differences on large scale
• population data

graphs
Set-cover problems
Combinatorial methods
Optimization problems
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Haplotyping the formal model

• Haplotype m-vector hlt0, 1,, 0gt over
  0,1m
• Genotype m-sequence glt0,1, ,0,0,
  1,1gt over 0,1,

g lt, , 0,, 1 gt
Def.
Haplotypes lth, kgt solve genotype g iff
g(i) implies h(i) ? k(i)
h(i) k(i) g(i) otherwise
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Examples

• g lt0,,1,,0,1,1gt

h
g solved by ltk,hgt g k
klt0,0,1,1,0,1,1gt
hlt0,1,1,0,0,1,1gt
Clark inference rule
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Haplotype inference the general problem

• Problem HI
• Instance a set Gg1, ,g m of genotypes and a
  set Hh1, ,h n of haplotypes,
• Solution a set H of haplotypes that solves
  each genotype g in G s.t. H ? H.

H derives from an inference RULE
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Type of inference rules

• Clarks rule haplotypes solve g by an iterative
  rule
• Gusfield coalescent model haplotypes are related
  to genotypes by a tree model
• Pedigree data haplotypes are related to
  genotypes by a directed graph

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Mendelian law and Recombination
Mother
Father
Parent
B D
A C
B
A
C
D
Child
A C
B D
A D
B C
A
C
A
D
B
C
D
B
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Pedigree

• Pedigree, nuclear family, founder

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Pedigree

• Pedigree, nuclear family, founder

Mother
Father
ID Num
Founders
Genotypes
Family trio
loop
Children
Nuclear family
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Haplotyping from genotypes The problem methods

• Problem
• Input genotype data (missing).
• Output haplotypes.
• Input data
• Data with pedigree (dependent).
• Data without pedigree info (independent).
• Statistical methods
• Find the most likely haplotypes based on genotype
  data.
• Adv solid theoretical bases
• Disadv computation intensive
• Rule-based methods
• Define rules based on some plausible assumptions
  and find those haplotypes consistent with these
  rules.
• Adv usually simple thus very fast
• Disadv no numerical assessment of the
  reliability of the results

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HI by the perfect phylogeny model

• IDEA

00000
g1 0, 1,,,1
G
H
g2 , 0,0,0,1
1, 0,0,0,1
0, 0,0,0,1
0, 1,1,0,1
Genotypes are the mating of haplotypes in a tree
Given G find H and T that explain G!
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Perfect Phylogeny models

• Input data 0-1 matrix A characters, species
• Output data phylogeny for A

R
1 1 0 0 0
0 0 1 0 0
1 1 0 0 1
0 0 1 1 0
Path c3c4
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Perfect phylogeny
Def.
A pp T for a 0-1 matrix A

• each row si labels exactly one leaf of T
• each column cj labels exactly one edge of T
• each internal edge labelled by at least one
  column cj
• row si gives the 0,1 path from the root to si

Path c3c4
0 0 1 1
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pp model another view
x
L(x) cluster of x
set of leaves of T x
A pp is associated to a tree-family (S,C) with
Ss1 ,, sn CS ? S S is a cluster s.t.
?X, Y in C , if X?Y?? then X?Y or Y ? X.
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pp another view
A tree-family (S,C) is represented by a 0-1
matrix
c i

• c i S s j ? S iff b ji1

0 1 0 0 0
0 0 1 0 0
1 1 0 0 1
0 0 1 1 0

• for each set in C at least a column

s j
Lemma A 0-1 matrix is a pp iff it represents a
tree-family
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Haplotyping by the pp

• A 0-1 matrix B represents the phylogenetic tree
  for a set H of haplotypes
• si haplotype
• ci SNPs

SNP site
00000
0-1 switch in position i only once in the tree !!
01000
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Haplotyping and the pp observations

• The root of T may not be the haplotype 000000
• 0-1 switch or 1-0 switch (directed case)

00011
00000
00011
00011
00011
0-1 switch
1-0 switch
01010
01001
01000
01000
01000
11000
11001
11010
01100
01010
01001
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HI problem in the pp model

• Input data a 0-1-matrix B n ? m of genotypes
  G
• Output data a 0-1 matrix B 2n ? m of
  haplotypes s.t.
• (1) each g ? G is solved by a pair of
  rows lth,kgt in B
• (2) B has a pp (tree family)

???
DECISION Problem
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An example
a 1 0 a 0 1
a b 0 c 1 0
b 0 1 b 0 0
c 1 0 c 1 0
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The pph problem solutions

• An undirected algorithm Gusfield Recomb 2002
• An O(nm2)- algorithm Karp et al. Recomb 2003
• A linear time O(nm) algorithm ??
• Optimal algorithm
• A related problem the incomplete directed pp
  (IDP)
• Inferring a pp from a 0-1- matrix
• O(nm klog2(n m)) algorithm Peer, T. Pupko, R.
  Shamir, R. Sharan SIAM 2004

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IDP problem
Instance A 0-1-? Matrix A Solution solve ?
Into 0 or 1 to obtain a matrix A and a pp for
A, or say no pp exists
1 2 3 4 5

• OPEN PROBLEM find an optimal algorithm ??

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Decision algorithms for incomplete pp

• Based on
• Characterization of 0-1 matrix A that has a pp
• -Tree family - - forbidden
  submatrix
• give a no certificate

Bipartite graph G(A)(S,C,E) with E(si,cj)
bij 1
Forbidden subgraph
X
Y
10
11
01
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Test a 0-1 matrix A has a pp?

• O(nm) algorithm (Gusfield 1991)
• Steps
• Given A order c1, ,cm as (decreasing) binary
  numbers A
• Let L(i,j)k , k maxl ltj Ai,l1
• Let index(j) maxL(i,j) i
• Then apply th.

TH. A has a pp iff L(i,j) index(j) for
each (i,j) s.t. Ai,j1
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Idea
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The IDP algorithm
C
c
s1
s3
s2
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Other HI problems via the pp model

• Incomplete 0-1--? matrix because of missing
  data
• haplotypes pp (Ihpp) haplotype rows
• genotype pp (Igpp) genotype rows

• Algorithms
• Ihpp IDP given a row as a root (polynomial
  time)
• NP-complete otherwise

• Igpp has polynomial solution under rich data
  hypothesis (Karp et al. Recomb 2004 Icalp 2004
  )
• NP-complete otherwise

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HI problem and other models

• Haplotype inference in pedigree data under the
  recombination model

child
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Pedigree graph
father
mather
child
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Haplotype inference in pedigree
00 01 10
10 11 00
01 11 01
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Problems

• MPT-MRHI (Pedigree tree multi-mating minimum
  recombination HI)
• SPT-MRHI (Pedigree tree single-mating minimum
  recombination HI)

Np-complete even if the graph is acyclic, but
unbounded number of children
OPEN
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Conclusions
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References