Haldane's Mapping Function
r = actual recombination; d = observed recombination.
A mapping function adjusts the proportion of observed
recombinant genotypes to better estimate the map distance
by counting single crossover events once and double
crossover events twice to improve the estimate of the
distance between loci. Haldane's mapping function assumes
that crossover interference is zero.
The number of double crossovers is distributed as a
Poisson variable. Let l equal
the number of crossover events and
l = 2m. Where m is the map
distance. For loci located less than 10 Centimorgans
apart m=p. For loci located more than 10 Centimorgans
apart, m does not equal p and m must be estimated using
a mapping function. The mapping function considers the
probability of double crossover events which causes
m to be greater than p.
| Distribution
of the number of events |
Probability
of that number of events |
| 0 |
e-2m |
| 1 |
me-2m |
| 2 |
(m2/2)(e-2m) |
| 3 |
(m3/6)(e-2m) |
If e-2m is the probability of zero crossover events,
then 1 - e-2m is the probability of at least one crossover
event. The proportion of observed recombinant genotypes
is (1/2)(1 - e-2m) because for each crossover event
only one-half of the progeny will be recombinant types
since only two of four strands participate in a crossover.
Therefore there is a coefficient of 1/2 in front of
the (1 - e-2m) term. The exponent of -2m is explained
by the fact that is the probability of a crossover =
2m.
p = (1/2)(1 - e-2m) and this can be solved for m.
Determining the map distance using Haldane's mapping
function is based on the observed proportion of recombinant
genotypes adjusted for the number of unobservable double
crossovers. Solving for m:
m = -(1/2) ln(1 -2p) and this equation converts the
observed
recombination fraction (p) to Haldane's map distance
(m). Haldane's mapping function estimates the map distance
for at least one crossover (or more) between two loci.
If the map distance is estimated based on the observed
proportion of recombinant genotypes (without using the
mapping function), then we are erroneously assumming
that no double crossover events have occurred.
Liu, B.H. 1997. Statistical Genomics: Linkage Mapping
and QTL Analysis. CRC Press. pg 18-19. pgs 328-328
Haldane, J.B.S. 1919. J. Genet. 8:299-309.
For short chromosome segments map distance = recombination
fraction d = r "i.e. 4% recombination"
= 4cM = 8% crossover.
i.e. 4% recombination = 4cM = 8% c.o. events
However, consider three loci each separated by more
than 10cm.
r
does not equal r
+ r;
r
= r
+ r
- 2r
r.
r
does not equal 40cM; r
= 0.2 + 0.2 - 2(0.04) = 0.32.
Compare this to three loci closely liked.
The combination is additive for short distance.
r
= r
+ r
= 0.06 + 0.08 = 0.14 = 14%
2m = prob. of single crossover event between A and
B loci
2n = prob. of single crossover event between B and C
loci
2(m-mn) = probability of single crossover between A
and B, excluding double crossovers between the A and
B locus with B and C locus.
2(m-mn+mn) = 2m = probability of crossover between
the A and B loci when 2-strand double crossover events
are added to the single-crossover events.
2(n-mn+mn) = 2n = probability of crossover between
B and C loci when both 2-strand double crossovers and
single crossover are considered.
