One Marker In Duplex Coupling
And One Marker In Simplex Coupling
Hackett, C.A. et al. 1998. Linkage analysis in tetraploid
species: a simulation study. Genet. Res. Comb. 71:143-154.
XXxx
Yyyy
Situation A - coupling
| X Y (A) |
|
X y(C) |
| o |
|
o |
|
(B) |
|
(D) |
| o |
|
o |
| x y |
|
x y |
Situation B - repulsion
We will consider coupling phase linkage (Situation
A).
| X Y |
|
X y |
|
| o |
|
o |
Case
1 |
| |
|
|
A+B |
| o |
|
o |
C+D |
| x y |
|
x y |
|
| X Y |
|
x y |
|
| o |
|
o |
Case
2 |
| |
|
|
A+C |
| o |
|
o |
B+D |
| X y |
|
x y |
|
| X Y |
|
x y |
|
| o |
|
o |
Case
3 |
| |
|
|
A+D |
| o |
|
o |
B+C |
| x y |
|
X y |
|
There is observable recombination in cases 1 and 3,
but not in case 2.
Situation A - Coupling phase.
Case
1 Gametes
|
| |
(1/2)Xy |
(1/2)xy |
| [(1-r)/2]XY |
[(1-r)/4]XXYy |
[(1-r)/4]XxYy |
| [(1-r)/2]xy |
[(1-r)/4]Xxyy |
[(1-r)/4]xxyy |
| (r/2)Xy |
(r/4)XXyy |
(r/4)Xxyy |
| (r/2)xY |
(r/4)XxYy |
(r/4)xxYy |
Case
2 Gametes
|
| |
1xy |
| (1/2)XY |
(1/2)XxYy |
| (1/2)Xy |
(1/2)Xxyy |
Case
3 Gametes
|
| |
(1/2)xy |
(1/2)Xy |
| [(1-r)/2]XY |
[(1-r)/4]XxYy |
[(1-r)/4]XXYy |
| [(1-r)/2]xy |
[(1-r)/4]xxyy |
[(1-r)/4]Xxyy |
| (r/2)Xy |
(r/4)Xxyy |
(r/4)XXyy |
| (r/2)xY |
(r/4)xxYy |
(r/4)XxYy |
Now we will add the probabilities of each type of gamete
across cases 1, 2, and 3. We weight each probability
by a factor of 1/3 so that the total probability across
all three cases will add to unity.
| Gametes |
Probability |
|
Observed Number |
| XXYy |
1 |
[ |
1-r |
+ |
1-r |
] |
= |
1-r |
|
|
|
|
|
|
|
| 3 |
4 |
4 |
6 |
|
|
|
|
|
|
| XXyy |
1 |
[ |
r |
+ |
r |
] |
= |
r |
|
|
|
|
|
|
|
|
| 3 |
4 |
4 |
6 |
| XxYy |
1 |
[ |
1-r |
+ |
r |
+ |
1 |
+ |
1-r |
+ |
r |
] |
= |
1 |
|
| 3 |
4 |
4 |
2 |
4 |
4 |
3 |
| Xxyy |
1 |
[ |
1-r |
+ |
r |
+ |
1 |
+ |
1-r |
+ |
r |
] |
= |
1 |
|
| 3 |
4 |
4 |
2 |
4 |
4 |
3 |
|
| xxYy |
1 |
[ |
r |
+ |
r |
] |
= |
r |
|
|
|
|
|
|
|
| 3 |
4 |
4 |
6 |
| xxyy |
1 |
[ |
1-r |
+ |
1-r |
] |
= |
1-r |
|
|
|
|
|
|
|
| 3 |
4 |
4 |
6 |
We can double check our results by adding the probabilities
of each type of gamete.
| 1-r |
+ |
r |
+ |
1 |
+ |
1 |
+ |
r |
+ |
1-r |
=1 |
|
| 6 |
6 |
3 |
3 |
6 |
6 |
|
| Gametes |
Probability |
Observed
Number |
| X_Y_ |
1-r |
+ |
1 |
= |
1 |
(3-r) |
a |
| 6 |
3 |
6 |
| X_yy |
r |
+ |
1 |
= |
1 |
(2+r) |
b |
| 6 |
3 |
6 |
| xxY_ |
r |
|
|
|
|
|
c |
| 6 |
|
|
|
|
|
| xxyy |
1-r |
|
|
|
|
|
|
d |
| 6 |
Now we can set up the likelihood expression [L(r)],
take the Log L(r), take the derivative of Log L(r),
set this derivative equal to zero, and solve for r.
| L(r)= |
n! |
( |
3-r |
) |
a |
( |
2+r |
) |
b |
( |
r |
) |
c |
( |
1-r |
) |
d |
| a!b!c!d! |
6 |
|
6 |
|
6 |
|
6 |
|
| Log[L(r)]=log |
[ |
n! |
] |
+ a log |
( |
3-r |
) |
+b log |
( |
2+r |
) |
| a!b!c!d! |
6 |
6 |
| |
| |
|
|
|
+ c log |
( |
r |
) |
+ d log |
( |
1-r |
) |
| 6 |
6 |
| dLogL(r) |
=0- |
|
a |
|
+ |
|
b |
|
+ |
|
c |
|
- |
|
d |
|
| dr |
( |
3-r |
) |
( |
2+r |
) |
( |
r |
) |
( |
1-r |
) |
| 6 |
6 |
6 |
6 |
| 0= |
|
-a |
|
+ |
|
b |
|
+ |
|
c |
|
- |
|
d |
|
| ( |
3-r |
) |
( |
2+r |
) |
( |
r |
) |
( |
l-r |
) |
| 6 |
6 |
6 |
6 |
| a |
+ |
d |
= |
b |
+ |
c |
|
|
|
|
|
|
| 3-r |
1-r |
2+r |
r |
|
|
|
|
|
|
| a(1-r) + d(3-r) |
= |
rb + c(2+r) |
|
|
|
|
|
|
|
|
|
|
| (3-r)(1-r) |
r(2+r) |
|
|
|
|
|
|
|
|
|
|
| a-ar+3d-dr |
= |
rb+2c+cr |
|
|
|
|
|
|
|
|
|
|
| 3-4r+r2 |
2r+r2 |
|
|
|
|
|
|
|
|
|
|
(a - ar + 3d - dr)(2r + r2) = (3 - 4r + r2)(rb + 2c + cr)
2ar - 2ar2 + a6dr - 2dr2 + ar2 - ar3 + 3dr2 - dr3
= 3rb - 4br2 + br3 + 6c - 8cr + 2cr2 + 3cr
= r3(-a -d -b -c) + r2(-2a - 2d + a + 3d + 4b - 2c + 4c)
+ r(2a + 6d - 3b + 8c - 3c) - 6c
= -nr3 +(-a + 4b + 2c + d)rd +(2a - 3b + 5c + 6d)r - 6c = 0
0 = nr3 +(a - 4b - 2c - d)r2 +(-2a + 3b - 5c - 6d)r + 6c
Solve for r by substituting the actual numbers for
a, b, c, and d, then plot f(r) as r is varied. Where
f(r) crosses the line at Y = 0 is a solution for r.
If 0< r < 0.5, then r is for coupling.
