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In Class 12 genetics you learn that the percentage of recombinant offspring between two genes can be used as a distance on a chromosome map. That works well for genes close together. For genes far apart it breaks down, because recombination never goes above 50 per cent however far apart the genes are. The fix is a mapping function, and one of the two in everyday use was published in 1944 by an Indian mathematician better known to many readers as a historian, D. D. Kosambi.
Who was D. D. Kosambi?
The MacTutor biography gives his dates as 31 July 1907 to 29 June 1966. He was born in Kosben, Goa, studied at Harvard, where he graduated summa cum laude in 1929, and then taught at Banaras Hindu University, Aligarh Muslim University and Fergusson College, Pune. In 1945 he joined the Tata Institute of Fundamental Research in Bombay. He published on mathematics, statistics, genetics, the weights of ancient coins and Indian history. MacTutor credits him with a 1943 paper on decomposing a signal into components, now known as the Kosambi-Karhunen-Loeve theorem.
His genetics paper was The estimation of map distance from recombination values, in the Annals of Eugenics, volume 12, pages 172 to 175.
Why do recombination percentages not add up?
Our post on Alfred Sturtevant and the first genetic map shows the problem in his 1913 data: for two distant genes the map predicted 57.6 units but he observed only 37.6 per cent recombinants. When two crossovers fall between a pair of genes, the outer genes end up back in their original combination, so the offspring look like non-recombinants. The longer the stretch, the more crossovers are hidden.
A mapping function converts the observed recombination frequency r into a map distance d that does add up. The two classic ones, as given by Kivikoski and colleagues in Heredity (2022), are:
Haldane (1919): r = ½(1 − e^(−2d))
Kosambi (1944): r = ½ tanh(2d)
Here d is in Morgans (1 Morgan = 100 centimorgans, cM). The same paper points out that Haldane's function assumes there is no crossover interference, while Kosambi's builds interference in: one crossover makes another one nearby less likely.
Worked example: from recombination to distance
Turn each formula round to get d from r:
Haldane: d = −½ ln(1 − 2r)
Kosambi: d = ¼ ln[(1 + 2r) ÷ (1 − 2r)]
Now try three values of r.
r = 0.05 (5 per cent). Haldane: −½ ln 0.9 = 0.0527 M = 5.27 cM. Kosambi: ¼ ln(1.1 ÷ 0.9) = ¼ × 0.2007 = 0.0502 M = 5.02 cM. Both are close to the simple answer of 5 cM.
r = 0.20. Haldane: −½ ln 0.6 = 0.2554 M = 25.5 cM. Kosambi: ¼ ln(1.4 ÷ 0.6) = ¼ × 0.8473 = 0.2118 M = 21.2 cM.
r = 0.40. Haldane: −½ ln 0.2 = 0.8047 M = 80.5 cM. Kosambi: ¼ ln(1.8 ÷ 0.2) = ¼ ln 9 = 0.5493 M = 54.9 cM.
For short distances everything agrees. For long ones the corrections are large, and Haldane's is larger because, without interference, it assumes more hidden double crossovers.
How do the formulas combine two intervals?
Suppose genes A–B and B–C each show r = 0.20, with B in the middle. What is r for A–C?
Naive addition: 0.20 + 0.20 = 0.40.
Kosambi: distances add, so d = 0.2118 + 0.2118 = 0.4236 M. Then r = ½ tanh(2 × 0.4236) = ½ tanh(0.847) = 0.345. Using the addition rule for tanh, the same answer comes directly from r = (r₁ + r₂) ÷ (1 + 4r₁r₂) = 0.40 ÷ 1.16 = 0.345.
Haldane: (1 − 2r) multiplies, so r = r₁ + r₂ − 2r₁r₂ = 0.40 − 0.08 = 0.32.
The observed A–C recombination is always less than the sum of the parts, and the size of the shortfall depends on how much interference you assume. Joseph Felsenstein showed in Genetics (1979) that Kosambi's argument can be extended into a whole family of mapping functions, with a parameter for the strength of interference, that includes Haldane's and Kosambi's as special cases.
Are mapping functions still used?
Yes, though their role has shifted. Kivikoski and colleagues describe Haldane's and Kosambi's as probably the two most widely recognised mapping functions. With modern dense marker maps, neighbouring markers are so close that multiple crossovers between them are rare and every mapping function gives nearly the same answer. Their data from humans and sticklebacks showed that turning long map distances back into recombination frequencies with either formula systematically underpredicts the real values. A 1944 formula is still the reference point that new work is measured against.
Common mistakes:
Treating r above 0.5 as possible. It is not; 0.5 is what unlinked genes give.
Mixing Morgans and centimorgans. 0.2118 M is 21.2 cM.
Adding recombination fractions over long distances. Add map distances, then convert back.
What can a student take from Kosambi?
He was a mathematician who read a biology problem closely enough to see that a measurement did not behave like a distance, and wrote down the function that would. The same move, asking whether a quantity really adds, comes up in physics and chemistry too. For the Class 12 side of the topic, our guide to linkage and crossing over covers the basics.
If logarithms or tanh are the sticking point, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revisit the maths before the genetics.
Try it now: for r = 0.30, work out the Haldane and Kosambi distances and see which is larger.
