Grouping loci Criteria • Maximum two-point recombination fraction – Example -r ij ≤ 0.40 • Minimum LOD score - Z ij – For n loci, there are n(n-1)/2 possible combinations that will be tested – Expect probability of false positives • Significant probability value - p ij – Example p ij ≤ 0.00001
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Grouping loci Criteria Maximum two-point recombination fraction –Example -r ij ≤ 0.40 Minimum LOD score - Z ij –For n loci, there are n(n-1)/2 possible.
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Grouping loci
Criteria• Maximum two-point recombination fraction
– Example -rij ≤ 0.40
• Minimum LOD score - Zij
– For n loci, there are n(n-1)/2 possible combinations that will be tested
– Expect probability of false positives
• Significant probability value - pij
– Example pij ≤ 0.00001
Locus ordering• Ideally, we would estimate the likelihoods for all
possible orders and take the one that is most probable by comparing log likelihoods
• That is computationally inefficient when there are more than ~10 loci
• Several methods have been proposed for producing a preliminary order
Locus ordering
6
)2)(1(
kkkntriplets
No. of locik
Possible orders
No. of triplets
2 1 0
3 3 1
5 60 10
10 1,814,400 120
20 1.22 X 1018 1,140
40 4.08 X 1047 9,880
Number of orders among k loci
Number of triplets among k loci
2
)1)(2)(3)...(2)(1(
2
!
kkkknk
Three-point Analysis
32
)1)(2)(3(
2
!33 n
Number of unique orders among k loci2
!knk
Order Mirror Order
ABC CBA
ACB BCA
BAC CAB
For three loci (k = 3 )
Three-point analysis
Non-Additivity of recombination frequencies
A B C
rABrBC
rAC
The recombination frequency over the interval A – C (rAC) is less than the sum of rAB and rBC : rAC < rAB + rBC.This is because (rare) double recombination events (a recombination in both A - B and B - C) do not contribute to recombination between A and C.
Non-Additivity of recombination frequenciesA B C
A B C
A B C
A B C
P00=(1-rAB)(1-rBC)
P10=rAB(1-rBC)
P01=(1-rAB)rBC
P11=rABrBC
rAC=rAB(1-rBC)+(1-rAB)rBC
rAC=rAB+rBC-2rABrBC
• Interference means that recombination events in adjacent intervals interfere. The occurrence of an event in a given interval may reduce or enhance the occurrence of an event in its neighbourhood.
• Positive interference refers to the ‘suppression’ of recombination events in the neighbourhood of a given one.
• Negative interference refers to the opposite: enhancement of clusters of recombination events.
• Positive interference results in less double recombinants (over adjacent intervals) than expected on the basis of independence of recombination events.
Interference
rAC=rAB+rBC-2CrABrBC
Interference
C = coefficient of coincidence
A B C
a b c
Interference I = 1 - C
overs sdoublecors ofnumber Expected
overs cross double ofnumber ObservedC
Coefficient of coincidence
Expected number of double crossovers = rABrBCN
Observed Count: 22 24 16 14 8 10 2 4
24.0100
)42108(ˆ
36.0100
)421416(ˆ
BC
AB
r
r
DH population N=100, locus order ABC
69.064.8
6
10024.036.0
42
DCs ofnumber Expected
DCs ofnumber Observed
C
Interference
• No interference– C = 1 and Interference = 1-C = 0
• Complete interference– C = 0 and Interference = 1-C = 1
• Negative interference– C > 1 and Interference = 1-C < 0
• Positive interference– C < 1 and Interference = 1-C > 0
Minimum Sum of Adjacent Recombination Frequencies (SARF) (Falk 1989)
1
1
ˆl
iaa jirSARF
Order SARF
ABC 0.22 + 0.30 = 0.52
BAC 0.22 + 0.10 = 0.32
ACB 0.10 + 0.30 = 0.40
r = recombination frequency between adjacent loci ai and ajfor a given order: 1, 2, 3, …, l -1, l
The B-A-C order gives MIN[SARF] and the minimum distance (MD) map
3.0100
)451011(ˆ
1.0100
)1045(ˆ
22.0100
)101011(ˆ
BC
AC
AB
r
r
r
Simulations have shown that SARF is a reliable method to obtain markers orders for large datasets
Minimum Product of Adjacent Recombination Frequencies (PARF) (Wilson 1988)
Order PARF
ABC 0.22 x 0.30 = 0.066
BAC 0.22 x 0.10 = 0.022
ACB 0.10 x 0.30 = 0.0303.0ˆ
1.0ˆ
22.0ˆ
BC
AC
AB
r
r
r
r = recombination frequency between adjacent loci ai and ajfor a given order: 1, 2, 3, …, l -1, l
The B-A-C order gives MIN[PARF] and the minimum distance (MD) map
SARF and PARF are equivalent methods to obtain markers orders for large datasets
1
1
ˆl
iaa jirPARF
Maximum Sum of Adjacent LOD Scores(SALOD)
1
1
l
iaa ji
zSALOD
Order SALOD
ABC 3.135 + 1.551 = 4.686
BAC 3.135 + 6.942 = 10.077
ACB 6.942 + 1.551 = 8.493551.1;3.0ˆ
942.6;1.0ˆ
135.3;22.0ˆ
BCBC
ACAC
ABAB
Zr
Zr
Zr
Z = LOD score for recombination frequency between adjacent loci ai and aj
for a given order: 1, 2, 3, …, l -1, l
The B-A-C order gives MAX[SALOD]
SALOD is sensitive to locus informativeness
Minimum Count of Crossover Events (COUNT) (Van Os et al. 2005)
1
1
l
iaa ji
XCOUNT
Order COUNT
ABC 22 + 30 = 52
BAC 22 + 10 = 32
ACB 10 + 30 = 40
X = simple count of recombination events between adjacent loci ai and aj
for a given sequence: 1, 2, 3, …, l -1, l
The B-A-C order gives MIN[COUNT]
3.0100
)451011(ˆ
1.0100
)1045(ˆ
22.0100
)101011(ˆ
BC
AC
AB
r
r
r
COUNT is equivalent to SARF and PARF with perfect data. COUNT is superior to SARF with incomplete data
Locus Order- Likelihood Approach
k
iii pfCrrZ
121 )4log(),,(
r1 = Recombination fraction in interval 1r2 = Recombination fraction in interval 2C = Coefficient of coincidencepi = fi / nfi = Expected frequency of the ith pooled phenotypic classI = 1, 2, …, kk = No. of pooled phenotypic classes