Calculate Transition Probabilites In Residue Substitution Models By Matrix Exponention

Question

[Peter]

Hi all, given a rate matrix Q (instantaneous rates of change) of a nucleotide substitution model, one can calculate the transition probabilities between any pair of nucleotides by calculating a probability matrix that is derived from matrix exponentiation given a certain time interval t:

P(t) = exp(Q*t)

which can further be calculated numerically by Eigen-decomposition of Q and so on.. But I just want to know how exactly one arrives at this step with the exponentiation, or in other words, why does it work? I could not find a reference for this, everybody just takes it as granted.

cheers, Peter

Answer 1

[Lars Juhl Jensen]

The fundamental assumption behind this formula is that nucleotide substitution can be modeled as a continuous time Markov chain. This handout from Stanford gives a good introduction to the topic.

[pmenzel]

Thanks for the link, Lars.