= Haldane mapping function
{c}
{title2=$r(d)=\tfrac12(1-e^{-2d})$}
= Haldane's mapping function
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{synonym}
The Haldane mapping function connects additive <genetic map> distance to endpoint <recombination fraction> in a toy model with no <crossover interference>. Specify the model directly along a transmitted <chromatid>: the exchange count $N$ in the interval has a <Poisson distribution> with <expected value> $d$, where $d$ is the distance in <morgans>. Counts in disjoint intervals are <independent random variables>. This is the transmitted-<chromatid> convention; $d$ is not the mean number of exchanges across all four <chromatids> in <meiosis>.
An odd exchange count changes parental phase. Summing the odd terms of the <Poisson distribution> gives
$$
r=P(N\text{ is odd})=e^{-d}\sum_{k=0}^{\infty}\frac{d^{2k+1}}{(2k+1)!}=\frac{1-e^{-2d}}{2}.
$$
For $0\le r<1/2$, the inverse and short-interval approximation are
$$
d=-\frac12\log(1-2r),\qquad r=d-d^2+O(d^3).
$$
Here $\log$ is the <natural logarithm>. At $d=0.2$, the <recombination fraction> is $0.16483997698$; at $r=0.2$, the distance is $0.2554128119$ <morgans>, or about $25.5413$ <centimorgans>. The linear approximation $r\approx d$ is useful only near zero. The <recombination fraction> approaches $1/2$ as $d$ grows; $r=1/2$ has no finite inverse in this model.
\Image[/recombination-fraction-map-distance.png]
{title=<Haldane mapping function> versus the short-interval approximation: <genetic map> distance keeps growing while the <recombination fraction> approaches one half}
The model is a useful baseline, not a universal rule for every interval during <meiosis>. <Crossover interference> can change the relationship between distance and <recombination fraction>. For the count convention and assumptions, see https://pmc.ncbi.nlm.nih.gov/articles/PMC2704907/[Weeks, Tang and Kwon (2009): genetic map functions and exchange-count conventions]. Haldane's original paper is bibliographically recorded as \i[The combination of linkage values, and the calculation of distances between the loci of linked factors], \i[Journal of Genetics] 8 (1919), 299–309, in https://jbshaldane.org/books/1990-Selected-Genetics-Papers/index.html[Selected Genetics Papers of J. B. S. Haldane, edited by K. Dronamraju (1990), contents].
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