Map Function

A map function is a mathematical relationship between recombination probabilities and map distances measured in centimorgans or Morgans. QTL Cartographer presently allows for eight map functions specified by an integer. The numbers 1, 2 or 3 correspond to the Haldane, Kosambi and Morgan (formerly Fixed) mapping functions, respectively. The default is the Haldane mapping function. If r corresponds to the recombination frequency between a pair of markers and $d_M$ is the distance between them in Morgans, then the Haldane mapping function is defined by

$$d_M = -\frac{1}{2} \ln(1 - 2r)$$ (2.1)

$$r = \frac{1}{2}[1 - \exp(-2d_M)]$$ (2.2)

The Kosambi function is

$$r = \frac{1 - \exp(-4d_M)}{2[1 + \exp(-4d_M)]}$$ (2.3)

$$d_M = \frac{1}{4} \ln[\frac{1 + 2r}{1 - 2r}]$$ (2.4)

and the Morgan function assumes $d_M=r$, which is complete interference. All eight mapping functions are discussed at length in Ben Lui's book [LiuLiu1998]: We direct the reader there for the details. Table 2.2 lists the mapping functions and their integer codes for QTL Cartographer. Some of these map functions require an extra parameter. This parameter can be set in the Rmap menu. See Section 10.3.1 of liu@98 for the details.

Table 2.2: Command Line Options for Rmap

Code	Reference	Note
1	Haldane@19	default
2	Kosambi@44
3	Morgan@28	``Fixed''
4	CarterFalconer@51
5	Raoetal@79	$0 \leq p \leq 1$
6	Sturt@76	$L$
7	Felsenstein@79	$-\infty < K < \infty, K \ne 2$
8	Karlin@84	binomial, $N > 0$