Bayesian Hierarchical Model for QTLs

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www. geocities.com/ResearchTriangle/Forum/4463/ani
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Bayesian Hierarchical Model for QTLs

• Susan Simmons
• University of North Carolina Wilmington

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CollaboratorsDr. Edward BooneDr. Ann
StapletonMr. Haikun Bao
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DNA
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Chromosome
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Genes
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Genetic Map
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Chromosome 1 of ProtozoaCryptosporidium parvum
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Chromosome 1 of Homo sapiens
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Alleles
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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)

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Some examples

• 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.

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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.

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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).

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Existing software

• Zhao-Bang Zengs 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

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World of plants
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Why plants?

• Increase yield to feed our increasing population
• Make plants resistant to UV-B exposure

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Plants, continued

• Control
• Design and Environment
• Reproduction
• Design (RIL is one of the best designs for
  detecting QTLs) Alleles are homozygous
• Cost
• Time

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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
  misleadingexample in Conte et al (unpublished))
• Hierarchical model to incorporate replicates
  within each line

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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

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Hierarchical Model

• 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)

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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
• (These probabilities are implicitly conditioned
  on the set ?)

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Posterior Model continued

• To compute probability of the model given the
  data in previous slide ( ), we need to
  compute P(Dki), where
• qi is the vector of unknown parameters for model
  ki

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Integration

• This integration can become difficult since the
  length of the unknown parameters is 2L M 2.
  Use Monte Carlo estimate of the integral
• Where , j 1,,t are samples from the
  posterior distribution

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Search strategy

• The activation probability, P(bj ?0D) is defined
  as
• There are 2M number of potential models,which can
  make the calculation of P(bj ?0D)
  computationally intensive
• Instead, we define a conditional probability
  search approach

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C2
C3
C4
C5
C1
C41
C42
C21
C22
C211
C212
C422
C421
C4212
C4211
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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.

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C2 0.4
C3 0.6
C4 0.4
C5 0.0029
C1 1
C31 0.063
C32 0.063
C12 0.9362
C11 1
C111 0.818
C112 0.927
C122 0.108
C121 0.114
C1112 0.014(M2)
C1111 0.041 (M1)
C1121 0.083(M3)
C1122 1(M4)
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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

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Thank you