The figure illustrates at least part of the reason that we need models of sequence evolution. It comes from a very nice book by Page & Holmes.

What I want to do here is to follow the derivation of the equations for P

_{XX}and P

_{XY}as a function of time, as developed in Higgs & Atwood. This isn't really necessary from a mathematical viewpoint, since we have already guessed the equations, but it's a fun argument.

Consider the following path: we start with an A at some position at time-zero, and after time t + Δt we observe that it is still A, but realize that at a short time prior to the second observation it might have been any nucleotide (since we weren't looking then):

There are four possible paths to get from A to A, through each of the possible intermediates. We sum over the probabilities... We have:

We can expand the last term to:

A wee bit o'calculus. We want to know

`P`_{AA}(t + Δt)

. Since `Δt`

is small, we can take the value of the function at t and correct it by adding the slope of the function (at t) times `Δt`

. That is:So we substitute this expression for the left-hand side of the first equation and then notice that we can subtract P

_{AA}(t) from both sides, leaving:

Since

`Δt`

occurs in each term on both sides it cancels (which is really the whole point of this). Also the sum of the three P_{AX}(t) terms is equal to

`1 - P`_{AA}(t)

, and so we have:The rate of change of

`P`_{AA}(t)

is proportional to `P`_{AA}(t)

, which is pretty obvious when you think about it, and so the form of the equation is an exponential:We need the -4α in the exponent, so that it will come out front when we take the derivative (see here).

We evaluate the constants A and B by considering the boundary conditions, namely, at long times

`P`_{AA}(t) = 1/4

, so `B = 1/4`

; and `P`_{AA}(0) = 1

, so `A + B = 1`

and `A = 3/4`

.Since the other three

`P`_{AX}(t)

are all equal and also equal to `1 - P`_{AA}(t)

, we have:which is just what we said the other day!

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