Crossing over and map distance

1. 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. • Recombinationfrequencies between widely spaced genes tend towards 50%.

2. 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

3. Meiotic Post-Meiotic 2 1 4 3 1 2 3 4 2 1 4 3 A A a a A A a a A A a a B b B b B b B b B B b b C c c C C c c C C C c c 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

4. 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.

5. Morgan mapping function (1928) • With complete interference (C = 0) and absence of multiple crossovers d = map distance in M r = recombination frequency • In most cases, this assumption only applies for very tighly linked loci (r< 0.1)

6. Haldane mapping function (1919) • With no interference (C = 1), the distribution of crossovers is Poisson x = number of crossovers d = map distance in M r = recombination frequency Haldane map distance Distances will be additive only when there is no interference

7. Haldane mapping function (1919) Order BAC

8. 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

9. Kosambi mapping function (1944) Order BAC

10. Felsenstein Mapping Function (1979) where 0 ≤ k≤ 2 • equal to the Haldane mapping function when k = 1 • equal to the Kosambi mapping function when k = 0

11. 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 if d < N/2, and r = 1/2 otherwise • incorporates interference • multilocus feasible • recommended by Ott (1991)

13. 1 ( ) f = 1 r – - - - 1 A B 2 NR a b 1 f = r - - - 2 2 A b R a b 1 f = r - - - 3 2 a B R a b 1 ( ) f = 1 r – - - - 4 2 a b NR a b Backcross (testcross) A B a b P1 P2 coupling linkage phase X A B a b A B a b F1 (tester) X a b a b expected frequency AB/ab Ab/ab aB/ab ab/ab

14. Genotyping Error or Incomplete Penetrance Aa misclassified as aa • r =prob. of recombinants 1-r=prob. of non-recombinants s = prob. of misclassifying recombinants as non- recombinants and vice versa Prob. of apparent recombinants is: p = r(1-s) + (1-r)s p = r + s(1-2r ) aa misclassified as Aa

15. Incomplete Penetrance

16. Incomplete Penetrance Fully penetrant Aa misclassified as aa aa misclassified as Aa

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

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

19. Segregation Distortion d = nAnB(1 - r)+ nAr+ nBr+(1 - r) • - nA= is the viability coefficient for the A locus • nB= is the viability coefficient for the B locus • The viability effects for the two loci are independent

20. Segregation Distortion d = nAnB(1 - r)+ nAr+ nBr+(1 - r) • - nA= is the viability coefficient for the A locus • nB= is the viability coefficient for the B locus The solution for recombinant fraction with differential viability for the A and B loci is NOT the same as the solution without differential viability. MLE

21. 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

22. Criteria for Evaluation • Replicates with correctly estimated orders • mean rank correlation between estimated versus true orders • mean total map length

23. Results

24. 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