![GTR model of sequence evolution GTR model of sequence evolution](https://s3-eu-west-1.amazonaws.com/gb.awsbucket.1/0047860_0.jpeg)
Ĭorollary The 12 off-diagonal enteries of the rate matrix, (note the off-diagonal enteries determine the diagonal enteries, since the rows of sum to zero) can be completely determined by 9 numbers these are: 6 exchangeability terms and 3 stationary frequencies, (since the stationary frequencies sum to 1). In other words, is the fraction of the frequency of state that results as a result of transitions from state to state. Under the time reversibility assumption, let, then it is easy to see that:ĭefinition The symmetric term is called the exchangeability between states and. Not all stationary processes are reversible, however, almost all DNA evolution models assume time reversibility, which is considered to be a reasonable assumption. Thus, by using the differential equation above,ĭefinition: A stationary Markov process is time reversible if (in the steady state) the amount of change from state to is equal to the amount of change from to, (although the two states may occur with different frequencies). In DNA evolution, under the assumption of a common process for each site, the stationary frequencies, correspond to equilibrium base compositions.ĭefinition A Markov process is stationary if its current distribution is the stationary distribution, i.e. if all states communicate, then the Markov chain has a stationary distribution where each is the proportion of time spent in state after the Markov chain has run for infinite time, and this probability does not depend upon the initial state of the process. If all the transition probabilities, are positive, i.e. Note that by definition, the rows of sum to zero. In other words (in frequentist language), the frequency of 's at time is equal to the frequency at time minus the frequency of the lost 's plus the frequency of the newly created 's. The changes in the probability distribution for small increments of time are given by: Letīe the transition rate from state to state. For a fixed site, letīe the column vector of probabilities of states and at time. Assume that the processes followed by the m sites are Markovian independent, identically distributed and constant in time. Theorem: Continuous-time transition matrices satisfy:Ĭonsider a DNA sequence of fixed length m evolving in time by base replacement. This then allows us to write that probability as. Where the top-left and bottom-right blocks correspond to transition probabilities and the top-right and bottom-left blocks corresponds to transversion probabilities.Īssumption: If at some time, the Markov chain is in state, then the probability that at time, it will be in state depends only upon, and. The corresponding transition matrices will look like:
![GTR model of sequence evolution GTR model of sequence evolution](https://www.frontiersin.org/files/Articles/303410/fpls-08-01927-HTML/image_m/fpls-08-01927-g001.jpg)
Jukes-Cantor, Kimura, etc.) in a continuous time fashion. Įxample: We would like to model the substitution process in DNA sequences ( i.e. Where each individual entry, refers to the probability that state will change to state in time. Specifically, if are the states, then the transition matrix Which are, in addition, parameterized by time. 2.4 HKY85 model (Hasegawa, Kishino and Yano 1985)ĭNA Evolution as a Continuous Time Markov Chain Continuous Time Markov ChainsĬontinuous-time Markov chains have the usual transition matrices.2.1 JC69 model (Jukes and Cantor, 1969).1.2 Deriving the Dynamics of Substitution.1 DNA Evolution as a Continuous Time Markov Chain.