Hi HyPhyers

Got a question about interpretation of non-reversible substitution models (as it happens, the application I'm working on is for binary characters i.e. with only two possible states, 1 and 0, rather than nucleotides or amino-acids)

An example of non-reversible models and their interpretation can be seen in PMID: 17940211 where Mower (using HyPhy!) models edited RNA as an additional state added onto a ACGU RNA substitution model.

As I understand Markov chains, it is possible for them to be

• stationary
• homgeneous
• **nonreversible**

in which case they have a stationary equilibrium distribution of states, even though they are not reversible - in a general non-reversible model of DNA substitution, for example, one would see this if the relative rate parameters in Q are such that rel_rate_param_ij =/= relative_rate_param_ji

In the application I'm looking at, we have a non-reversible chain when pi₁ * q₁₀ =/= pi₁ * q₀₁

Taking the situation, for example, where q₀₁ < q₁₀, then as time progresses with this non-reversible model, the frequency of 1 looks like it will decrease.

Here's what I don't understand - I thought this was a stationary homogeneous model? In a stationary homogeneous model the frequencies of the different states should be at equilibrium i.e. not tend to change with time....?

Mower (PMID: 17940211) finds a better fit with a nonreversible model for edited bases - with rates of base loss being much greater than gain. This is then interpreted as demonstrating that for the angiosperm dataset being examined, there was an ancestrally a higher proportion of edited bases, and that this proportion has been falling through time.    This doesn' t sound stationary to me?! But I thought the chain was being assumed to be stationary...?

You get (hopefully) the idea of my confusion - my guess is that I've missed/don't understand something about the conditions/requirements for some important properties for chains to be homogeneous/stationary/reversible. If someone can see where I'm making my mistake/misunderstanding - well, I'd love to hear from them!

Thanks

Aidan