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08 - Computing RNA secondary structure

# 08 - Computing RNA secondary structure - β 1 β n ∈{A C...

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2/13/08 - Computing RNA secondary str... RNA: a single-stranded molecule made up of {A, C, G, U} Secondary structure: certain base pairs on same molecule match up Constraints: (i) A pairs only with U C pairs only with G U pairs only with A G pairs only with C (ii) A base pairs 1 other base. (iii) [No hairpin turns] If i pairs with j then |i - j| >= 5 (iv) [No knotting] If (i, j) and (k, l) are paired then NOT i < k < j < l. Sequence β 1 , β 2 , ... , β n {A, C, G, U} (iii) If " β i pairs with β j } ... (iv) "If ( β i , β j ) and ( β k , β l ) are paired" ... 13-1 13

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The molecule maximizes the number of matched pairs. Problem: Given a string
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Unformatted text preview: β 1 , . .., β n ∈ {A, C, G, U} n , Fnd a matching satisfying (i)-(iv) with max # of matched pairs. Strategy: Compute OPT(i, j) for every subsequence β i , β i+1 , . .., β j in order of increasing j - i = k. for k = 1, 2, . .., n - 1 for i = 1, . .., n - k if k ≤ 4 OPT(i, i + k) = 0 else j = i + k 13-2 OPT(i, j) = max { OPT(i, j - 1), max i ≤ t ≤ j - 5, ( β t , β j ) ∈ { (A,U), (C, G), (G,C), (U,A)} {1 + OPT(i + 1, t - 1) + OPT(t + 1, j - 1) } Proof by induction over k. Running time: O(n 2 ) iterations of loop ≤ O(n) word per iteration Runtime: O(n 3 ) 13-3...
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