ISBRA 2007 Tutorial A:

Scalable Algorithms for Genotype

and Haplotype Analysis

Ion Mandoiu (University of Connecticut)

Alexander Zelikovsky (Georgia State University)

Outline

Background on genetic variationGenotype phasingError detectionDisease association searchDisease susceptibility prediction

3

Main form of variation between individual genomes: single nucleotide polymorphisms (SNPs)

High density in the human genome: 1 107 SNPs out of total 3 109 base pairs

Single Nucleotide Polymorphisms

… ataggtccCtatttcgcgcCgtatacacgggActata …… ataggtccGtatttcgcgcCgtatacacgggTctata …… ataggtccCtatttcgcgcCgtatacacgggTctata …

Haplotypes and Genotypes

Diploids: two homologous copies of each autosomal chromosome

One inherited from mother and one from father

Haplotype: description of SNP alleles on a chromosome 0/1 vector: 0 for major allele, 1 for minor

Genotype: description of alleles on both chromosomes 0/1/2 vector: 0 (1) - both chromosomes contain the major (minor)

allele; 2 - the chromosomes contain different alleles

011100110001000010021200210

+two haplotypes per individual

genotype

5

Identification and fine mapping of disease-related genes

Methods: Linkage analysis, allele-sharing, association studies Genotype data: large pedigrees, sibling pairs, trios, unrelated

Why SNPs?

6

Latest technologies deliver 1M SNP genotypes per sample, at low cost

Major challenges Efficiency Reproducibility Need simple methods!

Challenges in SNP Data Analysis

Genotype Phasing

Genotype Phasing

For a genotype with k 2’s there are 2k-1 possible pairs of haplotypes explaining it

g: 0010212 ?

h1:0010111

h2:0010010

h3:0010011

h4:0010110

Computational approaches to genotype phasing Statistical methods: PHASE, Phamily, PL, GERBIL … Combinatorial methods: Parsimony, HAP, 2SNP, ENT …

Minimum Entropy Genotype Phasing

Phasing – function f that assigns to each genotype g a pair of haplotypes (h,h’) that explains g

Coverage of h in f – number of times h appears in the image of f

Entropy of a phasing:

)||2

),cov(log(

||2

),cov()(

0),cov(: G

fh

G

fhfEntropy

fhh

Minimum Entropy Genotype Phasing [HalperinKarp 04]: Given a set of genotypes, find a phasing with minimum entropy

Connection with Likelihood Maximization

Iterative Improvement Algorithm[Gusev et al. 07]

InitializationStart with random phasing

Iterative improvement stepWhile there exists a genotype whose re-phasing decreases the entropy, find the genotype that yields the highest decrease in entropy and re-phase it

Overlapping Window approach

Entropy is computed over short windows of size l+f l “locked” SNPs previously phased f “free” SNPs are currently phased

locked free

…4321

g1

gn

…

…

• Only phasings consistent with the l locked SNPs are considered

Effect of Window Size

Time Complexity

n unrelated genotypes over k SNPs k/f windows n*2f candidate haplotype pairs evaluated per

window O(1) time per pair to compute the entropy gain Empirically, the number of iterations is linear in

n, but is reduced to O(log3n) by re-explaining multiple genotypes per iteration (batching)

Total runtime O(n log3n 2f k/f)

Empirical Runtime

Extension to general pedigrees

Parent-child relationships can be exploited to infer haplotype phase for a substantial fraction of the SNPs

Phasing related genotypes based on the no recombination assumption

Algorithm modifications: At each step re-explain an entire family

Cache inheritance pattern given by first window to speed-up computations for subsequent windows

Entropy computation based on founder haplotypes only

Enumeration No-Recombination Phasings for a Pedigree

• Gaussian elimination [Jiang et al.]• [Gusev et al. 07] implementation based on simple backtracking

Empirical Evaluation

International HapMap Project, Phase I & II datasets 3.7 million SNP loci Trio and unrelated genotypes from 4 different populations Reference haplotypes obtained using PHASE

Accuracy measures Relative Genotype Error (RGE): percentage of missing

genotypes inferred differently from the reference method Relative Switching Error (RSE): number of switches

needed to convert inferred haplotype pairs into the reference haplotype pairs

Empirical Evaluation (cont.)

Compared algorithms ENT [Gusev et al. 07] 2SNP [Brinza&Zelikovsky 05] Pure Parsimony Trio Phasing (PPTP) [Brinza et al. 05] PHASE [Stephens et al 01] HAP [Halperin&Eskin 04] FastPhase [Scheet & Stephens 06]

Results on Hapmap Phase II Trio Populations

ENT needs only few hours on a regular workstation to phase the entire HapMap Phase II dataset, compared to PHASE which required months of CPU time on two clusters with a total of 238 nodes

Complex Pedigree Phasing

Exploiting pedigree info significantly improves accuracy!

Application of Phasing: Missing data recovery

Genotype Error Detection

Genotyping Errors

A real problem despite advances in technology & typing algorithms

1.1% of 20 million dbSNP genotypes typed multiple times are inconsistent [Zaitlen et al. 2005]

Systematic errors (e.g., assay failure) typically detected by departure from HWE [Hosking et al. 2004]

In pedigrees, some errors detected as Mendelian Inconsistencies (MIs)

Many errors remain undetected As much as 70% of errors are Mendelian consistent for

mother/father/child trios [Gordon et al. 1999]

Effects of Undetected Genotyping Errors

Even low error levels can have large effects for some study designs (e.g. rare alleles, haplotype-based)

Errors as low as .1% can increase Type I error rates in haplotype sharing transmission disequilibrium test (HS-TDT) [Knapp&Becker04]

1% errors decrease power by 10-50% for linkage, and by 5-20% for association [Douglas et al. 00, Abecasis et al. 01]

Related Work

Improved genotype calling algorithms [Di et al. 05, Rabbee&Speed 06, Nicolae et al. 06]

Explicit modeling in analysis methods [Sieberts et al. 01, Sobel et al. 02, Abecasis et al. 02,Cheng 06] Computationally complex

Separate error detection step [Douglas et al. 00, Abecasis et al. 02, Becker et al. 06] Detected errors can be retyped, imputed, or ignored in

downstream analyses

Likelihood Sensitivity Approach to Error Detection [Becker et al. 06]

0 1 2 1 0 2

0 2 2 1 0 2

0 2 2 1 0 2

Mother Father

Child

Likelihood of best phasing for original trio T

0 1 1 1 0 0 h1

0 0 0 1 0 1 h3

0 1 1 1 0 0 h1

0 1 0 1 0 1 h2

0 0 0 1 0 1 h3

0 1 1 1 0 0 h4

)()()()( MAX)( 4321 hphphphpTL

Likelihood Sensitivity Approach to Error Detection [Becker et al. 06]

0 1 2 1 0 2

0 2 2 1 0 2

0 2 2 1 0 2

Mother Father

Child

Likelihood of best phasing for original trio T

)()()()( MAX)( 4321 hphphphpTL

? 0 1 0 1 0 1 h’ 1 0 0 0 1 0 0 h’ 3

0 1 0 1 0 1 h’1

0 1 1 1 0 0 h’2

0 0 0 1 0 0 h’ 3

0 1 1 1 0 1 h’ 4

Likelihood of best phasing for modified trio T’

)'()'()'()'( MAX)'( 4321 hphphphpTL

Likelihood Sensitivity Approach to Error Detection [Becker et al. 06]

0 1 2 1 0 2

0 2 2 1 0 2

0 2 2 1 0 2

Mother Father

Child

?

Large change in likelihood suggests likely error Flag genotype as an error if L(T’)/L(T) > R, where R is the detection threshold (e.g., R=104)

Implementation in FAMHAP[Becker et al. 06]

Window-based algorithm For each window including the SNP

under test, generate list of H most frequent haplotypes (default H=50)

Find most likely trio phasings by pruned search over the H4 quadruples of frequent haplotypes

Flag genotype as an error if L(T’)/L(T) > R for at least one window

Mother …201012 1 02210...Father …201202 2 10211...Child …000120 2 21021...

Limitations of FAMHAP Implementation

Truncating the list of haplotypes to size H may lead to sub-optimal phasings and inaccurate L(T) values

False positives caused by nearby errors (due to the use of multiple short windows)

[Kennedy et al.] HMM model of haplotype diversity all haplotypes are

represented + no need for short windows Alternate likelihood functions scalable runtime

HMM Model

Similar to models proposed by [Schwartz 04, Rastas et al. 05, Kimmel&Shamir 05, Scheet&Stephens 06]

Block-free model, paths with high transition probability correspond to “founder” haplotypes

(Figure from Rastas et al. 07)

HMM Training

Previous works use EM training of HMM based on unrelated genotype data

2-step algo exploiting pedigree info [Kennedy et al. 07]

Step 1: Infer haplotypes using pedigree-aware algorithm based on entropy-minimization

Step 2: train HMM based on inferred haplotypes, using Baum-Welch

Complexity of Computing Maximum Phasing Probability

How hard is to compute the likelihood function of Becker et al.?

Theorem [Kennedy et al. 07]

• Cannot approximate L(T) within O(n1/4 -), unless ZPP=NP, where n is the number of SNP loci

• For unrelated genotypes, computing maximum phasing probability is hard to approximate within a factor of O(n½-)

Open: complexity for fixed number of founder haplotypes

)()()()( MAX)( 4321 hphphphpTL

Complexity of Computing Maximum Phasing Probability

• Reductions from the clique problem

Alternate Likelihood Functions

• Viterbi probability (ViterbiProb): the maximum probability of a set of 4 HMM paths that emit 4 haplotypes compatible with the trio

• Probability of Viterbi Haplotypes (ViterbiHaps): product of total probabilities of the 4 Viterbi haplotypes

• Total Trio Probability (TotalProb): total probability P(T) that the HMM emits four haplotypes that explain trio T along all possible 4-tuples of paths

For a fixed trio, Viterbi paths can be found using a 4-path version of Viterbi’s algorithm in time

K3 speed-up by factoring common terms:

Efficient Computation of Viterbi Probability for Trios

)( 8NKO

)},'()',,,;({max),,,;1(),,,;1( 4443213'43214321 4qqqqqqjPreqqqqjEqqqqjV

jQq

• = maximum probability of emitting SNP genotypes at locus j+1 from states • = transition probability

),,,;1( 4321 qqqqjE ),,,( 4321 qqqq

Where:

Viterbi probability Likelihoods of all 3N modified trios can be computed within

time using forward-backward algorithm Overall runtime for M trios

Probability of Viterbi haplotypes Obtain haplotypes from standard traceback, then compute

haplotype probabilities using forward algorithms Overall runtime

Total trio probability Similar pre-computation speed-up & forward-backward algorithm Overall runtime

Overall Runtimes

)( 5MNKO

))(( 25 KNNKMO

)( 5MNKO

)( 5NKO

Empirical Evaluation

Real dataset [Becker et al. 2006] 35 SNP loci on chromosome 16 covering a region of 91kb 551 trios

Synthetic datasets 35 SNPs, 30-551 trios, same missing data pattern as real

dataset Haplotypes assigned to trios based on frequencies inferred

from real dataset 1% error rate, four error insertion models

Random allele Random genotype Heterozygous-to-homozygous Homozygous-to-heterozygous

Comparison of Alternative Likelihood Functions (1% Random Allele Errors)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.005 0.01 0.015

FP rate

Se

ns

itiv

ity

VitHaps-P

VitProb-P

TotalProb-P

VitHaps-C

VitProb-C

TotalProb-C

Parents vs. Children (1% Random Allele Errors)

Parents-TRIOS

1

10

100

1000

10000

100000

1000000

0

0.2

7

0.5

4

0.8

1

1.0

8

1.3

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1.6

2

1.8

9

2.1

6

2.4

3

2.7

2.9

7

3.2

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3.5

1

3.7

8

4.0

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4.3

2

4.5

9

4.8

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5.1

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5.4

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5.9

4

NO_ERR ERR

Children-TRIOS

1

10

100

1000

10000

100000

1000000

0

0.2

7

0.5

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0.8

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1.0

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1.3

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2.7

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NO_ERR ERR

FPs caused by same-locus errors in parents

“Combined” Detection Method

Compute 4 likelihood ratios

Trio Mother-child duo Father-child duo Child (unrelated)

Flag as error if all ratios are above detection threshold

Comparison with FAMHAP (Children)

0

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0 0.005 0.01 0.015

FP rate

Sen

siti

vity

TotalProb-UNO

TotalProb-DUO

TotalProb-TRIO

TotalProb-COMBINED

FAMHAP-1

FAMHAP-3

Comparison with FAMHAP (Parents)

0

0.1

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0.3

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0.5

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0 0.005 0.01 0.015

FP rate

Sen

siti

vity

TotalProb-UNO

TotalProb-DUO

TotalProb-TRIO

TotalProb-COMBINED

FAMHAP-1

FAMHAP-3

Acknowledgements

Sasha Gusev, Justin Kennedy, Bogdan Pasaniuc

NSF funding (Awards 0546457 and 0543365)

Software available at http://www.engr.uconn.edu/~ion/SOFT/