Transcript Rich Probabilistic Models for Genomic Data

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
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Course Announcements
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HW #1 is out
Project proposal
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Due next Wed
1 paragraph describing what you’d like to work on for
the class project.
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Why are we so different?
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Any observable
characteristic or trait
Human genetic diversity
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Different “phenotype”
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TGATCGAAGCTAAATGCATCAGCTGATGATCCTAGC…
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Different “genotype”
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TGATCGTAGCTAAATGCATCAGCTGATGATCGTAGC…
TGATCGCAGCTAAATGCAGCAGCTGATGATCGTAGC…
Appearance
Disease susceptibility
Drug responses
:
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Individual-specific DNA
3 billion-long string
……ACTGTTAGGCTGAGCTAGCCCAAAATTTATAGC
GTCGACTGCAGGGTCCACCAAAGCTCGACTGCAGTCGACGACCTA
AAATTTAACCGACTACGAGATGGGCACGTCACTTTTACGCAGCTTG
ATGATGCTAGCTGATCGTAGCTAAATGCATCAGCTGATGATCGTAG
CTAAATGCATCAGCTGATGATCGTAGCTAAATGCATCAGCTGATGA
TCGTAGCTAAATGCATCAGCTGATTCACTTTTACGCAGCTTGATGA
CGACTACGAGATGGGCACGTTCACCATCTACTACTACTCATCTACT
CATCAACCAAAAACACTACTCATCATCATCATCTACATCTATCATCA
TCACATCTACTGGGGGTGGGATAGATAGTGTGCTCGATCGATCGAT
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CGTCAGCTGATCGACGGCAG……
Appearance, Personality, Disease
susceptibility, Drug responses, …
Motivation
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Which sequence variation affects a trait?
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Better understanding disease mechanisms
Personalized medicine
Sequence variations
Instruction
Different
instruction
…
AG
GTC
XXX
ACTTCGGAACATATCAAATCCAACGC
XX
…
DNA – 3 billion long!
cell
A different A person
person
Obese?
Bold?
Diabetes?
Parkinson’s disease?
Heart disease?
Colon cancer?
:
15%
30%
6.2%
0.3%
20.1%
6.5%
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QTL mapping
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Data
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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.
1 2 3
mouse
individuals
0
1
0
:
0
1
0
0
:
0
4
5
…
:
3,000
1
1
0
:
0
Genotype data Phenotype
data
3000 markers
0101100100…011
1011110100…001
0010110000…010
:
0000010100…101
0010000000…100
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Outline
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Statistical methods for mapping QTL
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What is QTL?
Experimental animals
Analysis of variance (marker regression)
Interval mapping (EM)
QTL?
1 2 3
mouse
individuals
0
1
0
:
0
1
0
0
:
0
4
5
…
:
3,000
1
1
0
:
0
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Interval mapping
[Lander and Botstein, 1989]
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Consider any one position in the genome as the location
for a putative QTL.
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For a particular mouse, let z = 1/0 if (unobserved)
genotype at QTL is AB/AA.
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Calculate P(z = 1 | marker data).
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Need only consider nearby genotyped markers.
May allow for the presence of genotypic errors.
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Given genotype at the QTL, phenotype is distributed as
N(µ+∆z, σ2).
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Given marker data, phenotype follows a mixture of
normal distributions.
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IM: the mixture model
Nearest flanking markers
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M1
QTL
M2
0
7
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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?
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µA and µB
Genotype of QTL
M1/M2
99% AB
65% AB
35% AA
35% AB
65% AA
99% AA
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IM: estimation and LOD scores
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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).
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A simulated example
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Genetic markers
LOD score curves
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Interval mapping
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Advantages
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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
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Greater computational effort (doing EM for each
position)
Requires specialized software
More difficult to include covariates
Only considers one QTL at a time
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Statistical significance
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 P( D | QTL at the position ) 
log 10

QTL
P( D | no QTL)


Large LOD score → evidence for
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.
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) and of the maximum LOD score
from a genome scan (dashed curve).
Only ~3% of chance that the genomic
position gets LOD score≥1.
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LOD thresholds
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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
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Analytical calculations (assuming dense map of markers)
(Lander & Botstein, 1989)
Computer simulations
Permutation/ randomized test (Churchill & Doerge, 1994)
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More on LOD thresholds
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Appropriate threshold depends on:
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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
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An example
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Permutation distribution for a trait
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Trait variation that
is not explained
Modeling multiple
QTLs
by a detected putative QTL.
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Advantages
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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
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
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Multiple marker model
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Let y = phenotype,
x = genotype data.
Imagine a small number of QTL with genotypes x1,…,xp
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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)
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Multiple marker model
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Constant variance
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Assuming normality
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y|x ~ N(µg, σ2)
Additivity
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σ2(x1,…,xp) =σ2
µ(x1,…,xp) = µ + ∑j ∆jxj
Epistasis
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µ(x1,…,xp) = µ + ∑j ∆jxj + ∑j,k wj,kxjxk
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Computational problem
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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
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Mapping QTL as model selection
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Select the class of models
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Additive models
Additive with pairwise interactions
Regression trees
x1
w1
x2
…
w2
xN
wN
Phenotype (y)
y = w1 x1+…+wN xN+ε
minimizew (w1x1 + … wNxN - y)2 ?
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Linear Regression
minimizew (w1x1 + … wNxN - y)2+model complexity
x1
w1
x2
…
w2
parameters
xN
wN
Phenotype (y)
Y = w1 x1+…+wN xN+ε
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Search model space
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Forward selection (FS)
Backward deletion (BE)
FS followed by BE
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Lasso* (L1) Regression
L1 term
minimizew (w1x1 + … wNxN - y)2+  C |wi|
x1
w1
x2
…
w2
parameters
xN
L2
L1
wN
Phenotype (y)
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Induces sparsity in the solution w (many wi‘s set to zero)
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Provably selects “right” features when many features are irrelevant
Convex optimization problem
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No combinatorial search
Unique global optimum
Efficient optimization
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* Tibshirani, 1996
Model selection
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Compare models
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Likelihood function + model complexity (eg # QTLs)
Cross validation test
Sequential permutation tests
Assess performance
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Maximize the number of QTL found
Control the false positive rate
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Outline
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Basic concepts
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Haplotype, haplotype frequency
Recombination rate
Linkage disequilibrium
Haplotype reconstruction
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Parsimony-based approach
EM-based approach
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Review: genetic variation
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Single nucleotide polymorphism (SNP)
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Hardy Weinberg equilibrium (HWE)
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Each variant is called an allele; each allele has a frequency
Relationship between allele and genotype frequencies
How about the relationship between alleles of
neighboring SNPs?
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We need to know about linkage (dis)equilibrium
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Let’s consider the history of two
neighboring alleles…
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History of two neighboring alleles
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Alleles that exist today arose through ancient
mutation events…
Before mutation
A
After mutation
A
C
Mutation
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History of two neighboring alleles
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One allele arose first, and then the other…
Before mutation
A
G
C
G
After mutation
A
G
C
G
C
C
Mutation
Haplotype: combination of alleles present in a chromosome 28
Recombination can create more haplotypes
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A
G
C
C
No recombination (or 2n recombination events)
A
G
C
C
Recombination
A
C
C
G
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Without recombination
A
G
C
G
C
C
With recombination
A
G
C
G
C
C
A
C
Recombinant haplotype
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Haplotype
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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
Consider N binary SNPs in a genomic region
There are 2N possible haplotypes
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But in fact, far fewer are seen in human population
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More on haplotype
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What determines haplotype frequencies?
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Linkage disequilibrium (LD)
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Recombination rate (r) between neighboring alleles
Depends on the population
r is different for different regions in genome
Non-random association of alleles at two or more loci,
not necessarily on the same chromosome.
Why do we care about haplotypes or LD?
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References
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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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