Statistical methods for quantitative trait loci qtl mapping ii
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Statistical Methods for Quantitative Trait Loci (QTL) Mapping II. Lectures 5 – Oct 12, 2011 CSE 527 Computational Biology, Fall 2011 Instructor: Su-In Lee TA: Christopher Miles Monday & Wednesday 12:00-1:20 Johnson Hall (JHN) 022. Course Announcements. HW #1 is out Project proposal

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Statistical Methods for Quantitative Trait Loci (QTL) Mapping II

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Statistical methods for quantitative trait loci qtl mapping ii

Statistical Methods for Quantitative Trait Loci (QTL) Mapping II

Lectures 5 – Oct 12, 2011

CSE 527 Computational Biology, Fall 2011

Instructor: Su-In Lee

TA: Christopher Miles

Monday & Wednesday 12:00-1:20

Johnson Hall (JHN) 022


Course announcements

Course Announcements

  • HW #1 is out

  • Project proposal

    • Due next Wed

    • 1 paragraph describing what you’d like to work on for the class project.


Why are we so different

Why are we so different?

Any observable characteristic or trait

  • Human genetic diversity

  • Different “phenotype”

    • Appearance

    • Disease susceptibility

    • Drug responses

      :

  • Different “genotype”

    • Individual-specific DNA

    • 3 billion-long string

TGATCGAAGCTAAATGCATCAGCTGATGATCCTAGC…

TGATCGTAGCTAAATGCATCAGCTGATGATCGTAGC…

……ACTGTTAGGCTGAGCTAGCCCAAAATTTATAGCGTCGACTGCAGGGTCCACCAAAGCTCGACTGCAGTCGACGACCTAAAATTTAACCGACTACGAGATGGGCACGTCACTTTTACGCAGCTTGATGATGCTAGCTGATCGTAGCTAAATGCATCAGCTGATGATCGTAGCTAAATGCATCAGCTGATGATCGTAGCTAAATGCATCAGCTGATGATCGTAGCTAAATGCATCAGCTGATTCACTTTTACGCAGCTTGATGACGACTACGAGATGGGCACGTTCACCATCTACTACTACTCATCTACTCATCAACCAAAAACACTACTCATCATCATCATCTACATCTATCATCATCACATCTACTGGGGGTGGGATAGATAGTGTGCTCGATCGATCGATCGTCAGCTGATCGACGGCAG……

TGATCGCAGCTAAATGCAGCAGCTGATGATCGTAGC…


Motivation

AG

GTC

Instruction

Different instruction

XX

XXX

DNA – 3 billion long!

ACTTCGGAACATATCAAATCCAACGC

cell

Motivation

  • Appearance, Personality, Disease susceptibility, Drug responses, …

  • Which sequence variation affects a trait?

    • Better understanding disease mechanisms

    • Personalized medicine

Sequence variations

Obese?15%

Bold?30%

Diabetes?6.2%

Parkinson’s disease?0.3%

Heart disease?20.1%

Colon cancer?6.5%

:

cell

A different person

A person


Qtl mapping

3000 markers

mouse

individuals

1

1

0

:

0

0

1

0

:

0

1

0

0

:

0

QTL mapping

Genotype data

Phenotype data

1 2 3 4 5 … 3,000

0101100100…011

1011110100…001

0010110000…010

:

0000010100…101

0010000000…100

:

  • Data

    • Phenotypes: yi = trait value for mouse i

    • Genotypes: xik = 1/0 (i.e. AB/AA) of mouse i at marker k

    • Genetic map: Locations of genetic markers

  • Goals: Identify the genomic regions (QTLs) contributing to variation in the phenotype.


Outline

mouse

individuals

1

1

0

:

0

0

1

0

:

0

1

0

0

:

0

Outline

  • Statistical methods for mapping QTL

    • What is QTL?

    • Experimental animals

    • Analysis of variance (marker regression)

    • Interval mapping (EM)

QTL?

1 2 3 4 5 … 3,000

:


Interval mapping lander and botstein 1989

Interval mapping [Lander and Botstein, 1989]

  • Consider any one position in the genome as the location for a putative QTL.

  • For a particular mouse, let z = 1/0 if (unobserved) genotype at QTL is AB/AA.

  • Calculate P(z = 1 | marker data).

    • Need only consider nearby genotyped markers.

    • May allow for the presence of genotypic errors.

  • Given genotype at the QTL, phenotype is distributed as N(µ+∆z, σ2).

  • Given marker data, phenotype follows a mixture of normal distributions.


Im the mixture model

IM: the mixture model

Nearest flanking markers

99% AB

M1/M2

65% AB

35% AA

M1 QTL M2

35% AB

65% AA

0 7 20

99% AA

  • Let’s say that the mice with QTL genotype AA have average phenotype µA while the mice with QTL genotype AB have average phenotype µB.

  • The QTL has effect ∆ = µB - µA.

  • What are unknowns?

    • µA and µB

    • Genotype of QTL


Im estimation and lod scores

IM: estimation and LOD scores

Use a version of the EM algorithm to obtain estimates of µA, µB, σ and expectation on z (an iterative algorithm).

Calculate the LOD score

Repeat for all other genomic positions (in practice, at 0.5 cM steps along genome).


A simulated example

A simulated example

Genetic markers

LOD score curves


Interval mapping

Interval mapping

  • Advantages

    • Make proper account of missing data

    • Can allow for the presence of genotypic errors

    • Pretty pictures

    • High power in low-density scans

    • Improved estimate of QTL location

  • Disadvantages

    • Greater computational effort (doing EM for each position)

    • Requires specialized software

    • More difficult to include covariates

    • Only considers one QTL at a time


Statistical significance

Statistical significance

Null hypothesis – assuming that there are no QTLs segregating in the population.

Null distribution of the LOD scores at a particular genomic position (solid curve)

Null distribution of the LOD scores at a particular genomic position (solid curve) and of the maximum LOD score from a genome scan (dashed curve).

Large LOD score → evidence for QTL

Question: How large is large?

Answer 1: Consider distribution of LOD score if there were no QTL.

Answer 2: Consider distribution of maximum LOD score.

Only ~3% of chance that the genomic position gets LOD score≥1.


Lod thresholds

LOD thresholds

  • To account for the genome-wide search, compare the observed LOD scores to the null distribution of the maximum LOD score, genome-wide, that would be obtained if there were no QTL anywhere.

  • LOD threshold = 95th percentile of the distribution of genome-wide max LOD, when there are no QTL anywhere.

  • Methods for obtaining thresholds

    • Analytical calculations (assuming dense map of markers) (Lander & Botstein, 1989)

    • Computer simulations

    • Permutation/ randomized test (Churchill & Doerge, 1994)


More on lod thresholds

More on LOD thresholds

  • Appropriate threshold depends on:

    • Size of genome

    • Number of typed markers

    • Pattern of missing data

    • Stringency of significance threshold

    • Type of cross (e.g. F2 intercross vs backcross)

    • Etc


An example

An example

Permutation distribution for a trait


Modeling multiple qtls

Modeling multiple QTLs

Trait variation that is not explained by a detected putative QTL.

The effect of QTL1 is the same, irrespective of the genotype of QTL 2, and vice versa

The effect of QTL1 depends on the genotype of QTL 2, and vice versa

  • Advantages

    • Reduce the residual variation and obtain greater power to detect additional QTLs.

    • Identification of (epistatic) interactions between QTLs requires the joint modeling of multiple QTLs.

  • Interactions between two loci


Multiple marker model

Multiple marker model

  • Let y = phenotype,

    x = genotype data.

  • Imagine a small number of QTL with genotypes x1,…,xp

    • 2p or 3p distinct genotypes for backcross and intercross, respectively

  • We assume that

    E(y|x) = µ(x1,…,xp), var(y|x) = σ2(x1,…,xp)


Multiple marker model1

Multiple marker model

  • Constant variance

    • σ2(x1,…,xp)=σ2

  • Assuming normality

    • y|x ~ N(µg, σ2)

  • Additivity

    • µ(x1,…,xp) = µ + ∑j ∆jxj

  • Epistasis

    • µ(x1,…,xp) = µ + ∑j ∆jxj + ∑j,k wj,kxjxk


Computational problem

Computational problem

  • N backcross individuals, M markers in all with at most a handful expected to be near QTL

  • xij = genotype (0/1) of mouse i at marker j

  • yi= phenotype (trait value) of mouse i

  • Assuming addivitity,

    yi = µ + ∑j ∆jxij + e which ∆j ≠ 0?

    Variable selection in linear regression models


Mapping qtl as model selection

x1

x2

Mapping QTL as model selection

xN

w2

w1

wN

Phenotype (y)

y = w1 x1+…+wN xN+ε

minimizew (w1x1 + … wNxN - y)2?

  • Select the class of models

    • Additive models

    • Additive with pairwise interactions

    • Regression trees


Linear regression

Linear Regression

x1

x2

w2

w1

wN

xN

w2

w1

wN

parameters

Phenotype (y)

Y = w1 x1+…+wN xN+ε

minimizew (w1x1 + … wNxN - y)2+model complexity

  • Search model space

    • Forward selection (FS)

    • Backward deletion (BE)

    • FS followed by BE


Lasso l 1 regression

Lasso* (L1) Regression

x1

x2

x1

x2

w2

w1

L1 term

xN

L2

L1

w2

w1

wN

parameters

Phenotype (y)

minimizew (w1x1 + … wNxN - y)2+  C |wi|

  • Induces sparsity in the solution w (many wi‘s set to zero)

    • Provably selects “right” features when many features are irrelevant

  • Convex optimization problem

    • No combinatorial search

    • Unique global optimum

    • Efficient optimization

* Tibshirani, 1996


Model selection

Model selection

  • Compare models

    • Likelihood function + model complexity (eg # QTLs)

    • Cross validation test

    • Sequential permutation tests

  • Assess performance

    • Maximize the number of QTL found

    • Control the false positive rate


Outline1

Outline

  • Basic concepts

    • Haplotype, haplotype frequency

    • Recombination rate

    • Linkage disequilibrium

  • Haplotype reconstruction

    • Parsimony-based approach

    • EM-based approach


Review genetic variation

Review: genetic variation

  • Single nucleotide polymorphism (SNP)

    • Each variant is called an allele; each allele has a frequency

  • Hardy Weinberg equilibrium (HWE)

    • Relationship between allele and genotype frequencies

  • How about the relationship between alleles of neighboring SNPs?

    • We need to know about linkage (dis)equilibrium


Let s consider the history of two neighboring alleles

Let’s consider the history of two neighboring alleles…


History of two neighboring alleles

History of two neighboring alleles

Before mutation

A

After mutation

A

Mutation

C

Alleles that exist today arose through ancient mutation events…


History of two neighboring alleles1

History of two neighboring alleles

  • One allele arose first, and then the other…

Before mutation

A

G

C

G

After mutation

G

A

C

G

Mutation

C

C

Haplotype: combination of alleles present in a chromosome


Recombination can create more haplotypes

Recombination can create more haplotypes

G

A

C

C

G

A

C

C

A

C

C

G

No recombination (or 2n recombination events)

Recombination


Statistical methods for quantitative trait loci qtl mapping ii

Without recombination

A

G

G

C

C

C

With recombination

A

G

G

C

C

C

C

A

Recombinant haplotype


Haplotype

Haplotype

  • Consider N binary SNPs in a genomic region

  • There are 2N possible haplotypes

    • But in fact, far fewer are seen in human population

A combination of alleles present in a chromosome

Each haplotype has a frequency, which is the proportion of chromosomes of that type in the population


More on haplotype

More on haplotype

  • What determines haplotype frequencies?

    • Recombination rate (r) between neighboring alleles

    • Depends on the population

    • r is different for different regions in genome

  • Linkage disequilibrium (LD)

    • Non-random association of alleles at two or more loci, not necessarily on the same chromosome.

  • Why do we care about haplotypes or LD?


References

References

  • Prof Goncalo Abecasis (Univ of Michigan)’s lecture note

  • Broman, K.W., Review of statistical methods for QTL mapping in experimental crosses

  • Doerge, R.W., et al. Statistical issues in the search for genes affecting quantitative traits in experimental populations. Stat. Sci.; 12:195-219, 1997.

  • Lynch, M. and Walsh, B. Genetics and analysis of quantitative traits. Sinauer Associates, Sunderland, MA, pp. 431-89, 1998.

  • Broman, K.W., Speed, T.P. A review of methods for identifying QTLs in experimental crosses, 1999.


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