791_cb_lecture6

791_cb_lecture6 - 7.91 7.36 BE.490 Lecture#6 Mar 11 2004...

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7.91 / 7.36 / BE.490 Lecture #6 Mar. 11, 2004 Predicting RNA Secondary Structure Chris Burge
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Review of Markov Models & DNA Evolution Ch. 4 of Mount • CpG Island HMM • The Viterbi Algorithm • Real World HMMs • Markov Models for DNA Evolution
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DNA Sequence Evolution Generation n-1 (grandparent) 5’ TGGCATGCACCCTGTAAGTCAATATAAATGGCTACGCCTAGCCCATGCGA 3’ |||||||||||||||||||||||||||||||||||||||||||||||||| 3’ ACCGTACGTGGGACATTCAGTTATATTTACCGATGCGGATCGGGTACGCT 5’ 5’ TGGCATGCACCCTGTAAGTCAATATAAATGGCTA T GCCTAGCCCATGCGA 3’ |||||||||||||||||||||||||||||||||||||||||||||||||| 3’ ACCGTACGTGGGACATTCAGTTATATTTACCGAT A CGGATCGGGTACGCT 5’ Generation n (parent) Generation n+1 (child) 5’ TGGCATGCACCCTGTAAGTCAATATAAATGGCTA T GCCTAGCCC G TGCGA 3’± ||||||||||||||||||||||||||||||||||||||||||||||||||± 3’ ACCGTACGTGGGACATTCAGTTATATTTACCGAT A CGGATCGGG C ACGCT 5’±
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What is a Markov Model (aka Markov Chain)? Classical Definition A discrete stochastic process X 1 , X 2 , X 3 , … which has the Markov property : P(X n+1 = j | X 1 =x 1 , X 2 =x 2 , … X n =x n ) = P(X n+1 = j | X n =x ) n (for all x, all j, all n) i In words: A random process which has the property that the future (next state) is conditionally independent of the past given the present (current state) Markov - a Russian mathematician, ca. 1922
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DNA Sequence Evolution is a Markov Process No selection case P AA P AC P AG P AT P CC P CG P CT S n = base at generation n P = P CA P GA P GC P GG P GT P ij = P ( S = j | S n = i ) P TA P TC P TG P TT n + 1 G q n = ( q A , q C , q , q T ) = vector of prob’s of bases at gen. n G Handy relations: G q n + 1 G P q n = G q n + k = G q n P k
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Limit Theorem for Markov Chains S n = base at generation n P ij = P ( S n + 1 = j | S n = i ) If P ij > 0 for all i,j (and P ij = 1 for all i ) j G then there is a unique vector P n G P r G r r such that G q G r G lim = q = and (for any prob. vector ) n →∞ G r is called the “stationary” or “limiting” distribution of P See Ch. 4, Taylor & Karlin, An Introduction to Stochastic Modeling, 1984 for details
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Stationary Distribution Examples 2-letter alphabet: R = purine, Y = pyrimidine Stationary distributions for: 10 01 I = Q = 1 p p P = p 1 p 0 < p < 1 1 p p 0 < p < 1, 0 < q < 1 P ′ = q 1 q
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How are mutation rates measured?
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This note was uploaded on 11/11/2011 for the course BIO 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.

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791_cb_lecture6 - 7.91 7.36 BE.490 Lecture#6 Mar 11 2004...

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