Kosambi's Map Function

Haldane's mapping function assumes that there is no interference which would increase or decrease the proportion of double crossovers. Kosambi's mapping function is based on empirical data regarding the proportion of double crossovers as the physical distance varies. Kosambi's function adjusts the map distance based on interference which changes the proportion of double crossovers. Where m = average number of crossovers between A and C loci; is the probability of a crossover event; and m is the map distance.

4m = ln(1 + 2r) - ln(1 - 2r)

- see Liu text pgs. 322-324
Haldane's map function is:
r_AC = r_AB + r_BC - 2r_AB r_BC

and did not adjust for crossover interference.

Kosambi's map function is:
r_AC = r_AB + r_BC -2Cr_AB r_BC

where C is the coefficient of coincidence

and m = 1/4log [(1+2r)/(1-2r) for 0 <= r < 0.5

We can calculate m or map distance when we know r.
(recombination fraction).

How do we estimate r from observed numbers of crossover and non-crossover progeny? There are several methods which include the product method and maximum likelihood.

C =coefficient of coincidence; I = interference; I = 1 - C;

C =	obs. d.c.o.
	expected d.c.o.

Expected number of double crossovers = r_ABr_BCN.

Example:
Expected number of d.c.o. = 0.148(0.107)740 = 12.
Observed number of d.c.o. = 6.
C = 6/12 = 0.5; I = 1 - 0.5 = 0.5.

Now the expected number of double crossovers would equal the observed number of double crossovers (except for sampling) when the probability of a crossover in the G-S region was independent of the probability of a crossover in the S-L region. When the probability of a crossover in the first region is reduced due to the occurrence of a double crossover in the second region, then we have interference. The crossover in one region interfered with a crossover in the second region. If the probability of a crossover in the G-L region is independent of the probability of a crossover in the S-L region, then Haldane's mapping function would fit this biological situation.

However, in this example I = 0.5, so use Kosambi's map function. Haldane's map function assumes that the probability of a crossover in one region is independent of the probability of a crossover in the second region.
Haldane's map function r_AC = r_AB + r_BC - 2r_ABr_BC
Kosambi's map function r_AC = r_AB + r_BC - 2Cr_ABr_BC

When C=1 there is no interference. If you substitute C = 1 into Kosambi's map function, you can see that in this case the two map functions are equivalent. In Kosambi's map function C = 2r. When r_AB = r_BC = 0.5, then C = 1. This means that when the loci are located far apart, interference is absent. When the loci are close together, interference occurs and Kosambi's map function would be expected to give the best results.

Kosambi used C = 2r. When r = 0.5, then C = 2(0.5) =1 and there is no interference. In this case the two mapping functions are equivalent.