Hi Sergei,

I am interested in using likelihood to compare substitution rates between specific nucleotides.  The problem I am running into is in interpretation of the various rates that are calculated, because it seems there are at least three different categories of 'rates'.

The first category is simply the parameters used to define a particular model.  For instance, we use the parameters a, b, c, d, e, and f to define the general time-reversible model.  I think these are called rate ratios.

The second category is defined in Q, the instantaneous rate matrix, where each rate ratio is multiplied by the appropriate nt frequency.  So each element, qXY, of the matrix represents the instantaneous rate from nt X to nt Y.  For example,
qAC = a * freqC
qAG = b * freqG

A third category is also commonly calculated, where the instantaneous rate is multiplied by another nt frequency parameter.  So you get a rate defined by qXY * freqX.  For instance, this rate is used when testing for reversibility of a Q matrix (Does qXY * freqX equal qYX * freqY?).

Here's what I'd like to know: 1) what are the proper names for each of these three different categories of rates, 2) what is the precise interpretation of these different categories (if there is one), and 3) do you know of any references that provide a good explanation of these various rates?

Thanks!
Jeff

Hi Jeff/Sergei,

Was looking through the forum, and found your post on interpretation of Q that is relevant to some problems I have understanding it.

Sergei - you mention that
Quote:

But, when I read something like Huelsenbeck et al. 2002 (Syst Biol 51(1):32-43 "Inferring the Root of a Phylogenetic Tree" PMID:11943091), or Swofford et al. 1996 (in Molecular Systematics book by Hillis et al.), a reversible matrix is shown of the form where q_ij = r_ij * pi_j   (for both articles, this is equation 3).

Not sure how to interpret the difference between these representations of these reversible matrices, and your comment...? Is it that, as these general non-reversible models are stationary, there are indeed pi_i for each state, and that there are r_ij for each possible transition, but that these are perhaps non-unique or something like that?

Illumination much appreciated

Thanks

Aidan

Hi Sergei,

Firstly - many many thanks for taking the time to help me with these questions.

One reason my question was confusing is that I mis-typed something - I meant to say that when looking at other descriptions of the 12-model matrix (e.g. in Swofford et al.), the non-reversible matrix has q_ij = r_ij * pi_j   even if r_ji =/= r_ij - and I understood from your message that
Quote:

(which I thought meant that the decomposition of q_ij into s_ij * pi_j was not possible for these matrix).

The reason why I'm asking along these lines is that I'm keen to get a reasonable idea of the implications (and interpretation) of non-reversible models. In particular, what I don't understand is how the equilibrium state of these models is maintained. For the reversible models,  pi_i * q_ij = pi_j * q_ji for all i,j - so at the steady-state/equilibrium, in a given time interval, the same number of transitions occur from all i to all j [e.g. with a two-state model, the same number of 1->0 transitions occur as 0->1 transitions).

Do I understand right when I read that the above model is reversible, and that for a non-reversible model then, at equilibrium (and taking the binary state model), the number of 1->0 transitions is DIFFERENT from the number of 0->1 transitions? i.e. if for example in a binary model q_01 is greater than q_10 (i.e. r_10 > r_01), then at equilibrium the number of changes of state from 0->1 is greater than the number of changes from state 1->0.

But if that's the case, I'd expect the frequency of state 1 to be increasing with time.... but we're at equilibrium, so the frequencies of the different states should be constant...? This is the contradiction I can't understand - I think that my problem lies somewhere with my interpretation of what it means to be in equilibrium - perhaps you can see where I'm making a mistake in my understanding?

Thanks

Aidan