Poisson Distribution
We have previously shown that a Poisson distribution has the form:
e-m(1,
m, m2/2!,
m3/3!,
m4/4!...
mi/i!)
| Events(X) |
P(x) |
| 0 |
e-m |
| 1 |
e-m(m) |
| 2 |
e-mm2/2 |
| 3 |
e-mm3/3! |
| 4 |
e-mm4/4! |
Let l = mean number of crossover events.
The probability of zero crossover events is e-l.
The probability of at least one crossover event is:
1 - e-l = 1 - p(zero crossover
events)
p(at least one crossover event) = 1 - e-l
2r = 1 - e-l
r = 1/2(1-e-l)
2r = 1-e-l
e-l
= 1-2r
ln(e-l)
= ln(1-2r)
-l = ln(1-2r)
l = -ln(1-2r)
The distance = l/2, because l
is the probability of a crossover. Distance is measured in recombinantion
units.
p = 1/2(1 - e-2d)
p = 1/2(1 - e-l)
l=-ln(1-2p)
d = 1/2(-ln(1-2p))
The actual map distance is on the horizontal axis. The observed recombination
is on the vertical axis. p is the observed relative frequency of recombination
in % (verticle axis). d is the actual physical distance between two
loci in Morgans.
* Map distances are given in Morgans and the recombination
fraction is given in percent.
|
Let
p = 0.5 = 50%
d
= -1/2ln(1 - 2p)
d = -1/2ln(0)
d = infinity
|
Let
d = 50cM = 0.5M
d
= -1/2ln(1 - 2r)
0.50 = -1/2ln(1 - 2r)
-1 = ln(1 - 2r)
e-1=
1 - 2r
1 - e-1=
2r
1/2(1 - e-1)
= r
1/2(1 - 1/e1)
= r
r = 1/2(1 - 0.368)
r = 0.316
|
|
Let
p = 0.49 = 49%
d
= -1/2 ln(1 - 0.98)
d = -1/2(-3.912)
d = 1.956M
= 195.6cM
|
The recombination frequency = 1/2(1 - e-2d) = r.
d = actual recombination, r = observed recombination
because only 50% of crossover events result in a non-parental
type. 2d is the mean number of crossover events. Each
crossover event results in 50% recombination. d = l/2,
l = mean # of crossover events.
l = 2d.
