Crossing over and map distance

Crossing over and map distance

• Replicated chromosomes during meiosis are comprised of two sister chromatids.

• Crossovers occur between non-sister chromatids from homologous chromosomes, thereby producing recombinant haplotypes.

• Recombination frequencies between widely spaced genes tend towards 50%.

Crossing over and map distance

• We observe the frequency of recombinants, not the frequency of crossing over

• An odd number of crossovers between two loci produces recombinant haplotypes, whereas an even number of crossovers between two loci produces non-recombinant haplotypes.

– recombination frequency ≠ crossover frequency

• The presence of one crossover often suppresses crossover in the immediate vicinity, a phenomenon known as interference.

– recombination frequencies are not additive

Map distance• The genetic map distance between two genes, measured in centi-

Morgans (cM), is the expected number of crossovers that arise on a single chromatid.

• The mean number of crossovers per chromatid between A and C in the diagram shown below is one. (0 + 1 + 2 + 1)/4 = 1

A

B

C

A

B

C

12

a

b

c

a

b

c

34

A

b

c

A

B

C

12

a

b

C

a

B

c

34

A

b

c

A

B

C

21

a

b

C

a

B

c

43

Meiotic Post-Meiotic

Mapping functions

• Recombination frequency and map distance (r = d) are equal in the absence of multiple crossovers

• Mapping functions are used to transform recombination frequencies into additive map distances to better estimate map distance by counting single crossover events once and double crossover events twice.

Morgan mapping function (1928)

• With complete interference (C = 0) and absence of multiple crossovers

rd d = map distance in Mr = recombination frequency

• In most cases, this assumption only applies for very tighly linked loci (r< 0.1)

Haldane mapping function (1919)

• With no interference (C = 1), the distribution of crossovers is Poisson

)ˆ21ln(2

1rd

x = number of crossoversd = map distance in Mr = recombination frequency

Haldane map distance

Distances will be additive only when there is no interference

dx

ex

ddx

!);Pr(

BCABAC ddd

BCABBCABAC rrrrr ˆˆ2ˆˆ

Haldane mapping function (1919)

3.0ˆ

1.0ˆ

22.0ˆ

BC

AC

AB

r

r

r

2899.0)]22.0(21ln[2

1)ˆ21ln(

2

1 ABAB rd

4015.0)]276.0(21ln[2

1)ˆ21ln(

2

1 BCBC rd

Order BAC

1116.0)]10.0(21ln[2

1)ˆ21ln(

2

1 ACAC rd

276.0)1.0)(22.0(21.022.0ˆˆ2ˆˆ ACABACABBC rrrrr

276.032.010.022.0ˆˆ ACABBC rrr

4015.01116.02899.0 ACABBC ddd

Kosambi mapping function (1944)

• Condition: C = 2r

• Make sense biologically– C tends to 1 as r approaches 0.5– C tends to 0 as r approaches 0.0.

• Widely used

• For three linked loci ordered ABC

Kosambi’s addition formula for the recombination fractions of the loci

r

rd

ˆ21

ˆ21ln4

1

2ˆˆ41

ˆˆ C

rr

rrr

BCAB

BCABAC

Kosambi mapping function (1944)

3.0ˆ

1.0ˆ

22.0ˆ

BC

AC

AB

r

r

r

2361.0)22.0(21

)22.0(21ln4

1ˆ21

ˆ21ln4

1

AB

ABAB r

rd

3375.0)2941.0(21

)2941.0(21ln4

1ˆ21

ˆ21ln4

1

BC

BCBC r

rd

Order BAC

2941.0)1.0)(22.0)(5882.0(21.022.0ˆˆ2ˆˆ ACABACABBC rrCrrr

2941.032.010.022.0ˆˆ ACABBC rrr

3375.01014.02361.0 ACABBC ddd

5882.0)1.0)(22.0(41

)1.022.0(2ˆˆ41

)ˆˆ(22

ACAB

ACABBC rr

rrrC

1014.0)1.0(21

)1.0(21ln4

1

21

ˆ21ln4

1

AC

ACAC r

rd

2ˆˆ41

ˆˆ C

rr

rrr

ACAB

ACABBC

Felsenstein Mapping Function (1979)

• equal to the Haldane mapping function when k = 1

• equal to the Kosambi mapping function when k = 0

where 0 ≤ k ≤ 2

r

kr

kd

21

)2(21ln

)2(2

1

Binomial mapping function

• proposed by Karlin (1984)• When a maximum of N crossovers arise in an interval of length

d with binomial probability, map distance is estimated by the inverse of the binomial mapping function

2

])21(1[ /1 NrNd

if d < N/2, and r = 1/2 otherwise

• incorporates interference• multilocus feasible• recommended by Ott (1991)

])2

1(1[2

1 N

N

rr

Recombination fraction as a function of the map distance

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Map distance (M)

Re

co

mb

ina

tio

n f

rac

tio

n

Haldane

Kosambi

C=0

Binomial N=2

Backcross (testcross)A B

BA

a b

baX

A B

ba

a b

baX

P1 P2

F1 (tester)

expected frequencyA B

ba

A b

ba

a B

ba

a b

ba

NR

R

R

NR

coupling linkage phase

AB/ab

Ab/ab

aB/ab

ab/ab

f112--- 1 r– =

f212---r=

f312---r=

f412--- 1 r– =

Genotyping Error or Incomplete Penetrance

Genotypes Observed count

Fully Penetrant

Incompletely Penetrant

ABab f1 r rp)

Abab f2 r rp

aBab f3 r r+pr

abab f4 r rpr]


Fully Penetrant

Incompletely Penetrant

ABab f1 r rpr]

Abab f2 r rpr

aBab f3 r rp

abab f4 r rp)

Prob. of apparent recombinants is:

p = r-s) + (1-r)s

p = rs(1-2r

r =prob. of recombinants

r=prob. of non-recombinants

s = prob. of misclassifying recombinants as non- recombinants and vice versa

Aa misclassified as aa

aa misclassified as Aa

Incomplete Penetrance

prrfrprfprfprfL aaAa )1(

2

1lnˆ)1(

2

1lnˆ)1(

2

1lnˆ)1)(1(

2

1lnˆ 4321

)1(

2

1lnˆ

2

1lnˆ

2

1lnˆ)1(

2

1lnˆ 4321 rfrfrfrfL

)1)(1(

2

1lnˆ)1(

2

1lnˆ)1(

2

1lnˆ)1(

2

1lnˆ 4321 prfprfrprfprrfL Aaaa

Incomplete Penetrance

4321

32

ˆˆˆˆ

ˆˆˆ

ffff

ffr

)ˆˆ(ˆ)ˆˆ(ˆ)ˆˆ(ˆ

ˆ421312

312

ffffff

fffr

)ˆˆ)(ˆˆ(

ˆˆˆˆˆ

4231

2143

ffff

ffffp

)ˆˆ(ˆ)ˆˆ(ˆ)ˆˆ(ˆ

ˆ134243

243

ffffff

fffr

)ˆˆ)(ˆˆ(

ˆˆˆˆˆ

1324

3412

ffff

ffffp

Fully penetrant

Aa misclassified as aa

aa misclassified as Aa

Segregation Distortion


Exp. freq.

No diff. viability

Exp. freq. with diff. viability

for the A locus

AaBb f1 r r) r r

r

Aabb f2 r r r r r

aaBb f3 r r r r r r

aabb f4 r r r r

r

r

-viability coefficient for the A locus- Aa genotypes are less viable than aa when 0.0 < < 1.0- Aa and aa genotypes are equally viable when = 1.0- Aa genotypes are more viable than aa when 1.0 < < ∞

Segregation distortionGenotypes Observed count Exp. freq.

No diff. viability

Exp. freq. with diff. viability for the A locus

AaBb f1 r r

Aabb f2 r r

aaBb f3 r r

aabb f4 r r

4321

32

ˆˆˆˆ

ˆˆ

ffff

ffr

rrrr

rrrr

rr

ffff

ffr

A

A

A

A

AAA

A

1

)1(

11ˆˆˆˆ

ˆˆ

4321

32

The solution for recombinant fraction with differential viability for the A locus is the same as the solution without differential viability.

43

21

ˆˆ

ˆˆ

ff

ffA

MLE



Exp. freq.

No diff. viability

Exp. freq. with diff. viability for the A and

B loci

AaBb f1 r rd

Aabb f2 r rd

aaBb f3 r rd

aabb f4 r rd

- is the viability coefficient for the A locus = is the viability coefficient for the B locus

The viability effects for the two loci are independent

d = rrrr



Exp. freq.

No diff. viability

Exp. freq. with diff. viability for the A and B loci

AaBb f1 r rd

Aabb f2 r rd

aaBb f3 r rd

aabb f4 r rd

2/1

43

21

ˆˆ

ˆˆ

ff

ffA

2/1

32

2/1

41

2/1

32

ˆˆˆˆ

ˆˆ

ffff

ffr

ABS

- is the viability coefficient for the A locus = is the viability coefficient for the B locus

d = rrrr

The solution for recombinant fraction with differential viability for the A and B loci is NOT the same as the solution without differential viability.

2/1

42

31

ˆˆ

ˆˆ

ff

ffBMLE

Effect of genotyping errors, missing values, and segregation distortion (Hackett and Broadfoot

2003)

• Locus-ordering criteria– weighted least squares– maximum likelihood– SARF

• Three linkage groups of 10 loci– 2, 6, and 10 cM spacings

• Doubled haploid population of 150 individuals

Criteria for Evaluation

• Replicates with correctly estimated orders

• mean rank correlation between estimated versus true orders

• mean total map length

Results

Summary

• Missing data and genotyping errors reduced the proportion of correctly ordered maps – this effect worsened as the distances between loci decreased

• Maximum likelihood criterion was the most successful at ordering loci correctly but gave inflated map lengths when typing errors are present

• Missing data produced shorter map lengths for more widely spaced markers under the weigthed least-squares criterion

• The presence of segregation distortion had little effect

Crossing over and map distance

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