The genomes of recombinant inbred lines

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The genomes ofrecombinant inbred lines
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Inbred mice
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C57BL/6
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The intercross
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Recombinant inbred lines
(by sibling mating)
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The RIX design
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The Collaborative Cross
Complex Trait Consortium (2004) Nat Genet
361133-1137
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Genome of an 8-way RI
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The goal(for the rest of this talk)

• Characterize the breakpoint process along a
  chromosome in 8-way RILs.
• Understand the two-point haplotype probabilities.
• Study the clustering of the breakpoints, as a
  function of crossover interference in meiosis.

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2 points in an RIL

• r recombination fraction probability of a
  recombination in the interval in a random meiotic
  product.
• R analogous thing for the RIL probability of
  different alleles at the two loci on a random RIL
  chromosome.

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Haldane Waddington 1931
Genetics 16357-374
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Recombinant inbred lines
(by selfing)
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Markov chain

• Sequence of random variables X0, X1, X2,
  satisfying
• Pr(Xn1 X0, X1, , Xn) Pr(Xn1 Xn)
• Transition probabilities Pij Pr(Xn1j Xni)
• Here, Xn parental type at generation n
• We are interested in absorption probabilities
• Pr(Xn ? j X0)

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Equations for selfing
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Absorption probabilities

• Let Pij Pr(Xn1 j Xn i) where Xn state
  at generation n.
• Consider the case of absorption into the state
  AAAA.
• Let hi probability, starting at i, eventually
  absorbed into AAAA.
• Then hAAAA 1 and hABAB 0.
• Condition on the first step hi ?k Pik hk
• For selfing, this gives a system of 3 linear
  equations.

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Recombinant inbred lines
(by sibling mating)
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Equations for sib-mating
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Result for sib-mating
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The Collaborative Cross
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8-way RILs

• Autosomes
• Pr(G1 i) 1/8
• Pr(G2 j G1 i) r / (16r) for i ? j
• Pr(G2 ? G1) 7r / (16r)
• X chromosome
• Pr(G1A) Pr(G1B) Pr(G1E) Pr(G1F) 1/6
• Pr(G1C) 1/3
• Pr(G2B G1A) r / (14r)
• Pr(G2C G1A) 2r / (14r)
• Pr(G2A G1C) r / (14r)
• Pr(G2 ? G1) (14/3) r / (14r)

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The X chromosome
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Computer simulations
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r ? R, autosomes
Selfing Sib-mating
2-way 2r / (12r) 4r / (16r)
4-way 3r / (12r) 6r / (16r)
8-way r (4 r) / (12r) 7r / (16r)
16-way r (5 3r r2) / (12r) r (8 r) / (16r)
2n-way 1 (1 r)n-1 / (12r) 1 (1 r)n-2 / (16r)

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3-point coincidence

• rij recombination fraction for interval i,j
• assume r12 r23 r
• Coincidence c Pr(double recombinant) / r2
• Pr(recn in 23 recn in 12) / Pr(recn in
  23)
• No interference ? 1
• Positive interference ? lt 1
• Negative interference ? gt 1
• Generally c is a function of r.

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3-points in 2-way RILs

• r13 2 r (1 c r)
• R f(r) R13 f(r13)
• Pr(double recombinant in RIL) R R R13 /
  2
• Coincidence (in 2-way RIL) 2 R R13 / 2
  R2

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Coincidence
No interference
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Coincidence
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Why the clusteringof breakpoints?

• The really close breakpoints occur in different
  generations.
• Breakpoints in later generations can occur only
  in regions that are not yet fixed.
• The regions of heterozygosity are, of course,
  surrounded by breakpoints.

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Coincidence in 8-way RILs

• The trick that allowed us to get the coincidence
  for 2-way RILs doesnt work for 8-way RILs.
• Its sufficient to consider 4-way RILs.
• Calculations for 3 points in 4-way RILs is still
  astoundingly complex.
• 2 points in 2-way RILs by sib-mating
• 55 parental types ? 22 states by symmetry
• 3 points in 4-way RILs by sib-mating
• 2,164,240 parental types ? 137,488 states
• Even counting the states was difficult.

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Coincidence
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3-point symmetry
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Markov property
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Markov property
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Markov property
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Markov property
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Whole genome simulations

• 2-way selfing, 2-way sib-mating, 8-way sib-mating
• Mouse-like genome, 1665 cM
• Strong positive crossover interference
• Inbreed to complete fixation
• 10,000 simulation replicates

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No. generations to fixation
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No. gens to 99 fixation
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Percent genome not fixed
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Number of breakpoints
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Segment lengths
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Probability a segmentis inherited intact
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Length of smallest segment
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No. segments lt 1 cM
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Summary

• The Collaborative Cross could provide one-stop
  shopping for gene mapping in the mouse.
• Use of such 8-way RILs requires an understanding
  of the breakpoint process.
• Weve extended Haldane Waddingtons results to
  the case of 8-way RILs R 7 r / (1 6 r).
• Weve shown clustering of breakpoints in RILs by
  sib-mating, even in the presence of strong
  crossover interference.
• Broman KW (2005) The genomes of recombinant
  inbred lines. Genetics 1691133-1146