Genetic recombination and mapping functions

Genetic mapping is based on the idea that recombination is more likely between genes that are far apart on chromosomes than between genes that are close. If we have three genes $A$, $B$, and $C$ arranged in that order on a chromosome, then

$$r_{AC} = r_{AB}(1-r_{BC}) + (1-r_{AB})r_{BC} \qquad ,$$

where $r_{AB}$, $r_{AC}$, and $r_{BC}$ are the recombination rates between $A$ and $B$, $A$ and $C$, and $B$ and $C$, respectively.⁷

Haldane pointed out that this relationship implies another, namely that the probability that there are $k$ recombination events between two loci $m$ map units apart is given by the Poisson distribution:

$$p(m,k) = {e^{-m} m^k \over k!} \qquad .$$

Now to observe a recombination event between $A$ and $C$ requires that there be an odd number of recombination events between them (1, 3, 5, $\ldots$), i.e.,

$$\begin{eqnarray*} r_{AC} &=& \sum_{k=0}^\infty \frac{e^{-m} m^{(2k+1)}}{(2k+1)!} \\ &=& \frac{1 - e^{-2m}}{2} \qquad . \end{eqnarray*}$$

This leads to a natural definition of map units as

$$m = -\ln (1-2r) /2 \qquad .$$

$m$ calculated in this way gives the map distance in Morgans ($1M$). Map distances are more commonly expressed as centiMorgans, where $100cM = 1M$. Notice that when $r$ is small, $r \approx m$, so the map distance in centiMorgans is approximately equal to the recombination frequency expressed as a percent. There are several other mapping functions that can be chosen for an analysis. In particular, for analysis of real data investigators typically choose a mapping function that allows for interference in recombination. We don't have time to worry about those complications, so we'll use only the Haldane mapping function in our further discussions.