Transcript Bayesian Hierarchical Model for QTLs

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Bayesian Hierarchical Model
for QTLs
Susan Simmons
University of North Carolina
Wilmington
Collaborators
Dr. Edward Boone
Dr. Ann Stapleton
Mr. Haikun Bao
DNA
Chromosome
Genes
Genetic Map
Chromosome 1 of Protozoa
Cryptosporidium parvum
Chromosome 1 of Homo
sapiens
Alleles
Genetic Maps

Many more maps available at
www.ncbi.nih.gov
 Knowing information about genes now
allows us to find associations between
genes and outcomes (phenotypes)
Some examples

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In 1989 a breakthrough was made for the disease
of cystic fibrosis.
Location (or locus) is 7q31.2 - The CFTR gene is
found in region q31.2 on the long (q) arm of
human chromosome 7 (single gene responsible for
this disease).
The disease arises when an individual has two
recessive copies at this location.
An individual with one dominant and one
recessive is said to be a carrier of the disease.
Genetic screening to determine disease.
Green revolution

The Green Revolution is the increase in food
production stemming from the improved strains of
wheat, rice, maize and other cereals in the 1960s
developed by Dr Norman Borlaug in Mexico and
others under the sponsorship of the Rockefeller
Foundation
 Created new species of wheat and rice that
produced higher yield.
QTL

Better medical treatments and increased
agriculture are only two examples in which
identifying the location on the genome can have
an impact.
 Identifying the region on the genome (or on the
chromosome) responsible for a quantitative trait
(as opposed to qualitative as disease) is known as
Quantitative Trait Locus (QTL).
Existing software
Zhao-Bang Zeng’s group at NC State has
QTL Cartographer
 Karl Broman (John Hopkins) has an R
program that performs a number of
algorithms for QTLs
 To use these algorithms (and a number of
other published algorithms) only one
observation per genotype can be used

World of plants
Why plants?
 Increase
yield to feed our increasing
population
 Make plants resistant to UV-B
exposure
Plants, continued

Control
– Design and Environment
– Reproduction
– Design (RIL is one of the best designs for
detecting QTLs)… Alleles are homozygous

Cost
 Time
Plant QTL experiments

In most experiments, a number of replicates or
clones are observed within each line
 A number of plant biologist use some summary
measure to use conventional methods
 Information is lost (and can be
misleading…example in Conte et al
(unpublished))
 Hierarchical model to incorporate replicates
within each line
Data

Trait or phenotype, yij , i = 1,..,L where L is
the number of lines and j = 1, …, ni
(number of replicates within each line)
 Design matrix, X is L x M where M is the
number of markers on the genetic map
Hierarchical Model
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Hierarchical Model
yij ~ N(li,si2)
li ~ N(XiTb,t 2)
 Priors
t 2 ~ Inverse c 2 (1)
bk ~ N(0,100)
si2 ~ Inverse c 2 (1)
Posterior Model Probability

Let  denote the set of all possible models.
Given data D, the posterior probability of
model ki is given by Bayes Rule
P ( ki | D ) 
P ( D | ki ) P ( k i )

 P( D | k ) P(k )
j 1
i
i
(These probabilities are implicitly conditioned
on the set )
Posterior Model continued
To compute probability of the model given the
) need to
data in previous slide ( P(ki),| Dwe
compute P(D|ki), where
P( D | ki )   P ( D | qi , ki ) P(qi | ki )dqi
qi is the vector of unknown parameters for
model ki
Integration

This integration can become difficult since the
length of the unknown parameters is 2*L + M +2.
Use Monte Carlo estimate of the integral
1 t
( j)
P
(
D
|
q
,
k
)
P
(
q
|
k
)
d
q

P
(
D
|
q

i
i
i
i
i
i , ki )

t j 1
Where qi( j ) , j = 1,…,t are samples from the posterior
distribution
Search strategy

The activation probability, P(bj 0|D) is
defined as
P( b j  0 | D)   P( b j  0 | ki , D) P(ki | D)

There are 2M number of potential
models,which can make the calculation of
P(bj 0|D) computationally intensive
 Instead, we define a conditional probability
search approach

C1
C2
C21
C211
C3
C22
C4
C41
C212
C5
C42
C421
C4211
C422
C4212
Simulated data

Using the line information from the Bay x
Sha RIL population, a single QTL was
simulated on the fourth marker of the first
chromosome.
 The Bay x Sha population has 5
chromosomes.
C1
C2
C3
C4
C5
1
0.4
0.6
0.4
0.0029
C11
C12
C31
C32
1
0.9362
0.063
0.063
C111
C112
C121
C122
0.818
0.927
0.114
0.108
C1111
C1112
C1121
C1122
0.041 (M1)
0.014(M2)
0.083(M3)
1(M4)
Comments

Need to run model on more simulations
 Would like to compare this search strategy
to a stochastic search
 Would like to include epistasis in the model
Thank you