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I learned recently that given a pedigree it is possible to tell whether a disease is autosomal recessive by observing if the trait skips a generation. If the disease does skip a generation, then the disease is autosomal recessive. But the converse is false. That is a disease that is autosomal recessive does not necessarily mean the disease skips a generation.

Does someone know of a proof of this fact?

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Is this not just down to chance? A recessive trait may not skip a generation if the right mix of alleles are still passed to the progeny, but a dominant trait would always (by definition) be present in the offspring. I imagine there's a much better answer/way of saying this! –  Luke Jul 5 '12 at 11:29
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2 Answers

What you are saying is very similar to the statement that:

Absence of evidence is not evidence of absence

Suppose you have an autosomic gene A, which has a mutant allele a, causing an illness (or any other phenotype, for that matters). The particular nature of a makes its associated phenotype recessive.

You have three possibilities:

AA -> no illness  
Aa (or aA) -> carrier, with no illness 
aa -> ill

Now, let's take an aa individual (ill) and cross it with a sane AA individual:

aa x AA

All of the individuals of the first generation (F1) will be carriers (Aa) because the father will always give a and the mother always A.

Now if we breed them together

aA x aA

both parents have 1/2 probability of giving a and 1/2 of giving A. We will then have a proportion of:

1/4 aa
1/2 Aa (1/4 A from mother and a from father, 1/4 vice-versa) 
1/4 AA

But this is just down to statistics. You have only 1/4 chances of seeing the trait reappearing, so you are much less likely to see it if they have 1 offspring then if they have 20. Moreover, you have Aa offsprings in the progeny... this means that these could pass the mutant allele to the third or fourth generation.

Another important thing is that all we said is true if the first cross is

aa * AA

If it was

aa * aa

or

aa * Aa

The trait will (surely in the first case, and with 1/2 chance in the second) appear directly in the first generation.

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Also, note that even with 20 offspring you are not guaranteed to see an aa in the F2. –  nico Jul 10 '12 at 16:42
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A proof that autosomal recessive not necessarily skips a generation would be parents who both have the genotype and where one forth of the offspring would show the full phenotype.

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