QTL mapping in mice

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QTL mapping in mice

• Lecture 10, Statistics 246
• February 24, 2004

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The mouse as a model

• Same genes?
• The genes involved in a phenotype in the mouse
  may also be involved in similar phenotypes in the
  human.
• Similar complexity?
• The complexity of the etiology underlying a mouse
  phenotype provides some indication of the
  complexity of similar human phenotypes.
• Transfer of statistical methods.
• The statistical methods developed for gene
  mapping in the mouse serve as a basis for similar
  methods applicable in direct human studies.

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Backcross experiment
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F2 intercross experiment
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F2 intercross another view
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Quantitative traits (phenotypes)
133 females from our earlier (NOD ? B6) ? (NOD ?
B6) cross
Trait 4 is the log count of a particular white
blood cell type.
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Another representation of a trait distribution
Note the equivalent of dominance in our trait
distributions.
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A second example
Note the approximate additivity in our trait
distributions here.
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Trait distributions a classical view
In general we seek a difference in the phenotype
distributions of the parental strains before we
think seeking genes associated with a trait is
worthwhile. But even if there is little
difference, there may be many such genes. Our
trait 4 is a case like this.
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Data and goals

• Data
• Phenotypes yi trait value for mouse i
• Genotype xij 1/0 of mouse i is A/H at
  marker j (backcross) need two
  dummy variables for intercross
• Genetic map Locations of markers
• Goals
• Identify the (or at least one) genomic region,
  called quantitative trait locus QTL, that
  contributes to variation in the trait
• Form confidence intervals for the QTL location
• Estimate QTL effects

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Genetic map from our NOD B6 intercross
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Genotype data
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Models Recombination

• We assume no chromatid or crossover interference.
• ? points of exchange (crossovers) along
  chromosomes are distributed as a Poisson process,
  rate 1 in genetic distancce
• ? the marker genotypes xij form a Markov chain
  along the chromosome for a backcross
  what do they form in an F2 intercross?

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Models Genotype?Phenotype

• Let y phenotype,
  g whole genome genotype
• Imagine a small number of QTL with genotypes
  g1,., gp (2p or 3p distinct genotypes for BC, IC
  resp).
• We assume
• E(yg) ?(g1,gp ), var(yg)
  ??2(g1,gp)

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Models Genotype?Phenotype, ctd

• Homoscedacity (constant variance)
• ? ?2(g1,gp) ? ?2 ?(constant)
• Normality of residual variation
• yg N(?g ,?2 ?)
• Additivity
• ?(g1,gp ) ? ??j gj (gj 0/1 for
  BC)
• Epistasis Any deviations from additivity.

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Additivity, or non-additivity (BC)
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Additivity or non-additivity F2
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The simplest method ANOVA

• Split mice into groups
• according to genotype
• at a marker
• Do a t-test/ANOVA
• Repeat for each marker
• Adjust for multiplicity

LOD score log10 likelihood ratio, comparing
single-QTL model to the no QTL anywhere model.
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Exercise

• Explain what happens when one compares trait
  values of individuals with the A and H genotypes
  in a backcross (a standard 2-sample comparison),
  when a QTL contributing to the trait is located
  at a map distance d (and recombination fraction
  r) away from the marker.
• 2. Can the location of a QTL as in 1 be
  estimated, along with the magnitude of the
  difference of the means for the two genotypes at
  the QTL? Explain fully.

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Interval mapping (IM)

• Lander Botstein (1989)
• Take account of missing genotype data (uses the
  HMM)
• Interpolates between markers
• Maximum likelihood under a mixture model

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Interval mapping, cont

• Imagine that there is a single QTL, at position z
  between two (flanking) markers
• Let qi genotype of mouse i at the QTL, and
  assume
• yi qi Normal( ?qi , ?2 )
• We wont know qi, but we can calculate
• pig Pr(qi g marker data)
• Then, yi, given the marker data, follows a
  mixture of normal distributions, with known
  mixing proportions (the pig).
• Use an EM algorithm to get MLEs of ? (?A, ?H,
  ?B, ?).
• Measure the evidence for a QTL via the LOD score,
  which is the log10 likelihood ratio comparing the
  hypothesis of a single QTL at position z to the
  hypothesis of no QTL anywhere.

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Exercises

• Suppose that two markers Ml and Mr are separated
  by map distance d, and that the locus z is a
  distance dl from Ml and dr from Mr.
  a) Derive the relationship between
  the three recombination fractions connecting Ml ,
  Mr and z corresponding to dl dr d.
  b)
  Calculate the (conditional) probabilities pig
  defined on the previous page for a BC (two g,
  four combinations of flanking genotypes), and an
  F2 (three g, nine combinations of flanking
  genotype).
• Outline the mixture model appropriate for the BC
  distribution of a QT governed by a single QTL at
  the locus z as in 1 above.

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LOD score curves
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LOD curves for Chr 9 and 11 for trait4
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LOD thresholds

• To account for the genome-wide search, compare
  the observed LOD scores to the distribution of
  the maximum LOD score, genome-wide, that would be
  obtained if there were no QTL anywhere.
• LOD threshold 95th ile of the distribution of
  genome-wide maxLOD,, when there are no QTL
  anywhere
• Derivations
• Analytical calculations (Lander Botstein, 1989)
• Simulations
• Permutation tests (Churchill Doerge, 1994).

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Permutation distribution for trait4
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Epistasis for trait4
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Acknowledgement

• Karl Broman, Johns Hopkins