What I want to do here is to wrap up something from the first post. There we had two differential equations for the rate of change of a particular nucleotide position:

And we'd like to express these results in terms of

`P`_{XX}(t) and P_{XY}(t)

:Taking the first one, we have

And for the second

So the slopes are proportional to the probabilities, with an extra term. But the most interesting thing is that the form is the same for both

`P`_{XX}

and `P`_{XY}

!I wasn't expecting this but it makes sense, because at long times we come to equilibrium (the

*stationary distribution*of the Markov chain), and all rates are the same. At time-zero we have

`P`_{XX} = 1

and the rate is `-3*α`

, while `P`_{XY} = 0

and the rate is `α`

. I think it's OK.
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