# a linear-algebra problem in genetic inheritance

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.

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## 2 Answers

I think this is merely a subtle linguistic misunderstanding. You're using the verb "cross" with the genotypes as the subject ("For AA to cross with AA") and speak of "the cross with AA". From this and your general introduction, it seems you're thinking of a situation in which the animals freely cross. This is not what that web page is talking about. It uses "cross" with the subject "we" and speaks of "the crossing", which refers to a human act of intentionally crossing specific animals. So these are simply two different processes; they choose to consider what happens when animals are intentionally crossed in a certain way, and you're quite right in thinking that this is not what would happen if the animals freely crossed.

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yeah, I get your point. My previous thought confused a hell out of me. Thanks. –  xiaohan2012 Sep 13 '11 at 15:55

It is because, as the web site you are referring to explains, "Let us consider a series of experiments in which we keep crossing offspring with dominant animals only. Thus we keep crossing $AA$, $Aa$, and $aa$ with $AA$". In other words, the reason they chose $AA$ is simply a biological one.

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Yes, that is the point I missed. You are right.Thanks! –  xiaohan2012 Sep 13 '11 at 15:56