Genetic map function

Mapping functions in genetics are functions used to model the relationship between genetic map distance (measured in map units or centimorgans) and recombination frequency.

The simplest mapping function was the Morgan Mapping Function, eponymously devised by Thomas Hunt Morgan. Other well-known mapping functions include the Haldane Mapping Function, introduced by J. B. S. Haldane in a 1919 paper titled "The combination of linkage values, and the calculation of distances between the loci of linked factors", and the Kosambi Mapping Function introduced by Damodar Dharmananda Kosambi in 1944.

Morgan Mapping Function

Where d is the distance in map units, the Morgan Mapping Function states that the recombination frequency r can be expressed as $\ r=d$ . This assumes that one crossover occurs, at most, in an interval between two loci, and that the probability of the occurrence of this crossover is proportional to the map length of the interval.

Where d is the distance in map units, the recombination frequency r can be expressed as:

$\ r={\frac {1}{2}}[1-(1-2d)]=d$

The equation only holds when ${\frac {1}{2}}\geq d\geq 0$ as, otherwise, recombination frequency would exceed 50%. Therefore, the function cannot approximate recombination frequencies beyond short distances.

Haldane Mapping Function

Overview

Two properties of the Haldane Mapping Function is that it limits recombination frequency up to, but not beyond 50%, and that it represents a linear relationship between the frequency of recombination and map distance up to recombination frequencies of 10%. It also assumes that crossovers occur at random positions and that they do so independent of one another. This assumption therefore also assumes no crossover interference takes place; but using this assumption allows Haldane to model the mapping function using a Poisson distribution.

Definitions

• r = recombination frequency
• d = mean number of crossovers on a chromosomal interval
• 2d = mean number of crossovers for a tetrad
• e^-2d = probability of no genetic exchange in a chromosomal interval

Formula

Based on the definitions, Mather's formula can be used to derive the Haldane mapping function as:

$\ r={\frac {1}{2}}(1-e^{-2d})$

Inverse

$\ d=-{\frac {1}{2}}ln(1-2r)$

Kosambi Mapping Function

Overview

The Kosambi mapping function was introduced to account for the impact played by crossover interference on recombination frequency. It introduces a parameter C, representing the coefficient of coincidence, and sets it equal to 2r. For loci which are strongly linked, interference is strong; otherwise, interference decreases towards zero.

Formula

$\ d=-{\frac {1}{4}}ln({\frac {1+2r}{1-2r}})$