Duplex-duplex Linkage In Coupling Phase
Hackett, C.A. et al. 1998. Linkage analysis in tetraploid species:
a simulation study. Genet. Res. Comb. 71:143-154.
| X Y(A) |
|
X Y(C) |
|
| o |
|
o |
Case
1 |
| |
|
|
A+B |
| o (B) |
|
o (D) |
C+D |
| x y |
|
x Y |
|
| X Y(A) |
|
x y(B) |
|
| o |
|
o |
Case
2 |
| |
|
|
A+C |
| o |
|
o |
B+D |
| X Y
(C) |
|
x y(D) |
|
| X Y(A) |
|
X Y(C) |
|
| o |
|
o |
Case
3 |
| |
|
|
A+D |
| o |
|
o |
B+C |
| x y(D) |
|
x Y(B) |
|
Case 1 - gametes and Case 3-gametes.
Case 2 - gametes
| |
½
XY |
½
XY |
| ½
xy |
| ¼
XxYy |
¼
XxYy |
| ¼
XxYy |
¼
XxYy |
|
| ½
xy |
We can now summarize the probabilities of each type of gamete, using
probabilities of 1/3 for each of the three cases.
| Gametes |
Probability |
| XXYY |
 |
| XXYy |
 |
| XXyy |
 |
| XxYY |
 |
| XxYy |
 |
| Xxyy |
 |
| xxYY |
 |
| xxYy |
 |
| xxyy |
 |
Now we must collect the nine gametic classes into four classes due
to dominance gene action. With dominance gene action we can only distinguish
between four classes.
| Gametes |
Probability |
Observed
Number |
| X_Y_ |
1/12
[10 - 4r + 2r2] |
a |
| X_yy |
1/12
[4r - 2r2] |
b |
| xxY_ |
1/12
[4r - 2r2] |
c |
| xxyy |
1/12
(2 - 4r + 2r2) |
d |
Now we need to double check the results because the four probabilities
should sum to unity.
Next we find the likelihood for r.
Now we find the multinomial expression for the likelihood for the observed
data and solve to find the maximum likelihood estimate of r.
The quadratic equation is in the form
and has the solution
In this case a = bcd + abc + acd + abd
In this case b = -2(bcd + abc + acd + abd)
In this case c = 5bcd + bca
To derive the same results as Hackett et al. We substitute
Y is an intermediate variable
Y is an5-2r + r2 = 4 + y
Y is an2r - r2 = 1 - y
Y is an1 - 2r2 + r = y
so we can substitute
n = a + b + c + d; 
The quadratic equation is in the form
with the solution
In our case a = -n, b = (a - 4b - 4c - 3d), c = 4d
