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The genomes of recombinant inbred lines

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First generation: Wi = Xi = A, Yi = Zi = B. Then Pr(AA fixed) = 2(q1 q2) Pr(AB fixed) = 4 q3 ... Markov property. 39. Markov property. 40. Markov property. 41 ... – PowerPoint PPT presentation

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Title: The genomes of recombinant inbred lines


1
The genomes ofrecombinant inbred lines
2
Inbred mice
3
C57BL/6
4
The intercross
5
Recombinant inbred lines
(by sibling mating)
6
The RIX design
7
The Collaborative Cross
Complex Trait Consortium (2004) Nat Genet
361133-1137
8
Genome of an 8-way RI
9
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.

10
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.

11
Haldane Waddington 1931
Genetics 16357-374
12
Recombinant inbred lines
(by selfing)
13
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)

14
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.

15
Equations for selfing
16
Recombinant inbred lines
(by sibling mating)
17
Equations for sib-mating
18
Result for sib-mating
19
The Collaborative Cross
20
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)

21
The X chromosome
22
Computer simulations
23
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.

24
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

25
Coincidence
No interference
26
Coincidence
27
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.

28
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.

29
Coincidence
30
But there is an easier way...
31
Equations for sib-mating
32
The simpler method
  • Consider the cross W1W2X1X2 ? Y1Y2Z1Z2
  • Let q1 Pr(W1W2 fixed)
  • q2 Pr(W1X2 fixed)
  • q3 Pr(W1Y2 fixed)
  • Then 4 q1 4 q2 8 q3 1
  • First generation Wi Xi A, Yi Zi B
  • Then Pr(AA fixed) 2(q1 q2)
  • Pr(AB fixed) 4 q3

33
The simpler method
  • W1W2X1X2 ? Y1Y2Z1Z2
  • q1 Pr(W1W2 fixed) q2 Pr(W1X2 fixed)
    q3 Pr(W1Y2 fixed)
  • Second generation Wi Yi A, Xi Zi B
  • Then Pr(AA fixed) 2(q1 q3)
  • Thus q2 q3

34
The simpler method
  • W1W2X1X2 ? Y1Y2Z1Z2
  • q1 Pr(W1W2 fixed) q2 Pr(W1X2 fixed)
    q3 Pr(W1Y2 fixed)
  • Now we use the usual trick, condition on the
    first step
  • q1 (1 r)/2 ? q1 ? 4 1/2 ? 1/2 ? q2 ?
    12
  • Combined with the previous results, we get
  • q2 r/2(16r)
  • And so Pr(AB fixed) 4q3 4r/(16r)

35
The formula
36
3-point symmetry
37
Markov property
38
Markov property
39
Markov property
40
Markov property
41
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

42
No. generations to fixation
43
No. gens to 99 fixation
44
Percent genome not fixed
45
Number of breakpoints
46
Segment lengths
47
Probability a segmentis inherited intact
48
Length of smallest segment
49
No. segments lt 1 cM
50
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

51
Acknowledgement
Friedrich Teuscher Research Institute for the
Biology of Farm Animals Dummerstorf, Germany
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