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Here's an excerpt about Kimura's two-parameter model from Felsenstein's Inferring Phylogenies:

"The model is symmetrical, and one can immediately see that, after enough time has elapsed, it will be equally likely for the base to be a purine or a pyrimidine."

I think I understand the assumptions of the model but it's not obvious to me why the equilibrium frequencies are equal. Why is that so?

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If sufficient time has elapsed, there will be a large number of transversions present between the original base and the final state. The initial state will therefore not matter.

Say, you flip (turn over, not toss) a coin at random intervals. For short time and few flips, the initial state will matter. But if you go on flipping it for millenia, the final state will be essentially random -- you will not be able to tell what was there in the beginning.

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Good answer. In the case of the model, am I right to think that equilibrium frequencies are equal because there is an equal chance of a base being a pyrimidine or purine in both the initial and final states? With enough time the base will mutate multiple times and can also mutate back to it's original state, therefore, the initial state does not matter. ... Imaginary sequence of events for a single nucleotide: A>G>C>A>G is just as likely A>G>A>C>T and in this example there is one transition and one transversion. Am I correct in my thinking? – rg255 Nov 19 '12 at 11:44

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