Suppose there is a pair of genes A and a. And there are three types of genotype, AA,*Aa* and aa.
Now, study the problem of genetic transition. That's, given an initial genotype fraction, say AA for 1/2, Aa for 1/2 and aa for 0 in an population.
After n generation, what will the fraction for each genotype be?
It is reasonable to construct a transition matrix.
And have the following equation.
Xn=A*Xn-1, where Xn-1 is the (n-1)th generation genotype fraction vector, Xn the similar meaning and A is the transition matrix mentioned above.
Now the problem is how to construct the transition matrix A.
I referenced a link on web, which follows the idea like the following:
For each genotype(AA,*Aa* and aa) to cross with AA( This is where I am confused with), the fraction or probability for getting each genotype is:
For AA to cross with AA
- 1 probability to get AA
- 0 probability to get Aa
- 0 probability to get aa
For Aa to cross with AA
- 1/2 probability to get AA
- 1/2 probability to get Aa
- 0 probability to get aa
For aa to cross with AA
- 0 probability to get AA
- 1 probability to get Aa
- 0 probability to get aa
And combining the three column vectors yields the transition matrix A, where is
1 1/2 0 0 1/2 1 0 0 0
So, why is it like this? The reference on the link is over. More details see here
And what I had in mind is like this:
Why consider the cross with AA (the dominant gene type,(in converse, the recessive gene type)) every time when constructing the transition matrix's columns. This is where I marked above.
Why not set Aa or aa as the basis?
For example, when setting Aa as the basis, instead of AA.
The transition matrix will be
1/2 1/2 0 1/4 1/2 1/4 0 1/2 1/2
And the result(the next generation's gene fraction) will be different.
In summary, my question is:
Is it necessary to set AA as the basis for each crossing when building the transition matrix?
PS: It may be more of a biological problem on selecting the crossing basis. I will try to seek help from that biology field.