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

Dear Aidan,

The property of stationarity actually says something about the distribution of character frequencies, not just the rate matrix itself.

EVERY (ergodic, i.e. where any state can be reached from any other state) Markov process will have SOME stationary distribution. Once the process arrives at that distribution, it will preserve it, but if you start the process in another frequency distribution, then the frequencies will change over time, eventually converging to the stationary distribution.

I would imagine the Mower paper simply assumes that the current state of the system is not stationary, i.e. base frequencies are still drifting towards an equilibrium.

Cheers,
Sergei