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I just started reading about recombination frequencies of genes and related stuff, and can't find an answer to this (possibly naive) question. Can recombination possibly "kill off" a gene ?

To be more specific, say a gene is located at position $x$ to $y$ on some chromosome. Then, we are so unlucky that recombination decides to cut off the chromosome at location $x + (y - x)/2$, right in the middle of the gene. So both halves will get recombined with the other halves of the gene on the homolog chromosome. If the allele on this chromosome was different, could it result in destroying both alleles of the gene ? Is there something that prevents this ?

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Yes, you could imagine a situation where two functional alleles could produce a recombinant which was non-functional.

chromosome 1 has A and B at positions x and y - functional

chromosome 2 has C and D at positions x and y - functional

Recombination between positions x and y produces A/D and C/B versions of the gene. Either or both of these could be non-functional.

Nothing prevents this from happening, but natural selection will act on the recombinants.

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