Single Coupling
Hacket, C.A. et al. 1998. Linkage analysis in tetraploid
species: a simulation study. Genet. Res. Comb. 71:143-154.
Let X be the dominant allele at one locus and Y be
the dominant allele at a second locus. The four homologous
chromosomes have the following genotypes:
XY
xy
xy
xy
The XY chromosome must pair with an xy chromosome.
Double reduction gametes are not considered. Only bivalent
pairing is assumed. The probability of a crossover event
for one bivalent is r.
The only observable recombination is between the xy
chromosome that pairs with the XY chromosome. If we
consider recombination for the bivalent in coupling
phase linkage, we have the following gametic series.
(1-r)/2 XY; (1-r)/2 xy; r/2Xy:r/2XY
Let a, b, c, d, be the observed numbers of progeny
in the XY, Xy, xY, and xy classes, respectively. The
solution to the proportion of observed recombination,
'r' is solved by maximum likelihood. The probability
of observing a, b, c, and d of each class is:
| P |
( |
a,b,c,d; |
1-r |
, |
r |
, |
r |
, |
1-r |
) |
| 2 |
|
2 |
|
2 |
|
2 |
| = |
n! |
( |
1-r |
) |
a |
( |
r |
) |
b |
( |
r |
) |
c |
( |
1-r |
) |
d |
| a!b!c!d! |
2 |
|
2 |
|
2 |
|
2 |
|
We can solve for r by taking the derivative of the
log of this function with respect to r and setting the
derivative equal to zero.
The log - likelihood, L, is given by:
| L(r)=log |
[ |
n! |
] |
+a log |
(1-r) |
+b log |
( |
r |
) |
+ c log |
( |
r |
) |
+ d log |
(l-r) |
| a!b!c!d! |
2 |
2 |
2 |
2 |
| d[L(r)] |
= 0 + |
-a |
+ |
b |
+ |
c |
+ |
-d |
|
| dr |
(1-r)/2 |
r/2 |
r/2 |
(1-r)/2 |
|
| 0= |
b + c |
- |
(a + d) |
| r/2 |
(1 - r)/2 |
|
|
| (b + c)(1 - r) |
= |
(a + d)r |
| 2 |
2 |
|
| a + d |
= |
b + c |
| (1 - r)/2 |
r/2 |
| (b +c)(1 - r) |
= |
(a + d)r |
| 2 |
2 |
b + c - br - cr = ar + dr
b + c = ar + br + cr + dr
b + c = r(a+b+c+d)
b + c = rn, because a+b+c+d = n
| b + c |
= r; |
which is the solution. |
| n |
Copyright
2000©, Ted Helms |
