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ijcrb.webs.com INTERDISCIPLINARY JOURNAL OF CONTEMPORARY RESEARCH IN BUSINESS Figure 1 In Figure 1 Statistical approach to phylogenetics and modeling recombination. For a given column yt in the alignment, a probability P(yt|St, w,θ) can be computed, which depends on the tree topology, St, the vector of branch lengths, w, and the parameters of the nucleotide substitution model, θ. In the presence of recombination, the tree topology can change and thus becomes a random variable that depends on the site label t. For four taxa, there are three different tree topologies. The vectors w and θ are accumulated vectors, as defined in the paragraph above equation (6) Figure 2 COPY RIGHT © 2011 Institute of Interdisciplinary Business Research JANUARY 2011 VOL 2, NO 9 532
ijcrb.webs.com INTERDISCIPLINARY JOURNAL OF CONTEMPORARY RESEARCH IN BUSINESS Figure 2. Modeling recombination with hidden Markov models. Positions in the model, labeled by the subscript t, correspond to sites in the DNA sequence alignment. Black nodes represent observed random variables; these are the columns in the DNA sequence alignment. White nodes represent hidden states; these are the different tree topologies, shown (for four sequences) in figure 3. Arcs represent conditional dependencies. Squares represent parameters of the model. The probability for observing a column vector yt at position t in the DNA sequence alignment depends on the tree topology St, the vector of branch lengths w, and the parameters of the nucleotide substitution model θ. The tree topology at position t depends on the topologies at the adjacent sites, St-1 and St+1, and the recombination parameter v . Left: In the older approaches of McGuire, Wright, and Prentice (2000)and Husmeier and Wright (2001), v, w, and θ are parameters that have to be estimated. Right: In the Bayesian approach, v, w, and θ are random variables. The prior distribution for v is a beta distribution with hyperparameters α and ß. The prior distributions for the remaining parameters are discussed under Method: A Bayesian Approach and depend on some hyperparameters Ώ. The parameters v, w, and θ are sampled from the posterior distribution with Markov chain Monte Carlo While, in general, this problem would be intractable because of the exponential increase in the number of state sequences (see above), the reduction to nearest-neighbor interactions between hidden states and the resulting factorization (3) allows the application of a dynamic programing technique, the so-called Viterbi algorithm (Rabiner 1989, to find the mode Ŝ with computational complexity O(N). The factorization (3) contains three terms: P(yt|St), P(St|St-1), and P(S1). The transition probabilities P(St|St-1) correspond to recombination events (if St ≠ St-1). Let v denote the probability that the tree topology remains unchanged as we move from a given site in the alignment, t, to an adjacent site, t + 1 or t - 1. We then obtain for the state transition probabilities: COPY RIGHT © 2011 Institute of Interdisciplinary Business Research JANUARY 2011 VOL 2, NO 9 533
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