RNA Folding Algorithms. Michal Ziv-Ukelson Ben Gurion University of the Negev
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1 RNA Folding Algorithms Michal Ziv-Ukelson Ben Gurion University of the Negev
2 The RNA Folding Problem: Given an RNA sequence, predict its energetically most stable structure (minimal free energy). AUCCCCGUAUCGAUC AAAAUCCAUGGGUAC CCUAGUGAAAGUGUA UAUACGUGCUCUGAU UCUUUACUGAGGAGU CAGUGAACGAACUGA
3 RNA structure prediction by basepair maximization Input: Output: a string over A,C,G,U A pairs with U, C pairs with G a subset of possible base-pairs of maximal size such that no two base-pairs intersect.
4 Sequence Alignment as a method to determine structure Bases pair in order to form backbones and determine the secondary structure Aligning bases based on their ability to pair with each other gives an algorithmic approach to determining the optimal structure Problem: number of potential structures grows exponentially with the number, n, of bases
5 RNA secondary structure prediction algorithms cont d We must distinguish the biologically correct structure from all the incorrect structures. We need both a function that assigns the highest score to the correct structure, and an algorithm for evaluating the scores of all possible structures דצמבר 15
6 RNA secondary structure prediction algorithms cont d One approach might be to find the structure with the most base pairs. Nussinov introduced an efficient dynamic programming algorithm for this problem in Although this criterion is too simplistic, the mechanics of this algorithm are the same as those of more sophisticated energy minimization folding algorithms
7 Ruth Nussinov Professor in the Department of Human Genetics, School of Medicine, Tel Aviv University. Proposed the first dynamic programming algorithm for RNA secondary structure prediction, by maximizing the number of base pairs (1978).
8 Folding an RNA Sequence of length n. 1. Classical, O(n 3 ), [Nussinov et. Al. 1978, 1980] [Waterman and Smith 1978] [Zuker and Stiegler 1981] MFOLD: Vienna RNA Package: 2. Complex worst case speedup based on Fast Matrix Multiplication: O(n 3 * log 3 logn /(log 2 n) [Akutsu 1999] Based on Four Russians: O(n 3 /(logn) [Frid&Gusfield] O(n 3 /(log 2 n) [Pinhas*, Zakov*, Tsur and Ziv-Ukelson 2013] 3. Sparsification: Practical speed up: O(nZ) where Z is in [n, n 2 ] [Wexler, Zilberstein, Ziv-Ukelson 2007] [Backofen, Tsur, Zakov, Ziv-Ukelson 2009] 9
9 Assumptions of the RNA secondary structure prediction algorithm, based on MFE: 1. The most likely structure of the RNA molecule is identical or similar to the energetically most stable structure. 2. The energy associated with any position in the structure is only influenced by local sequence structure. 3. The structure is assumed to be formed by folding of the chain back on itself in a manner that does not produce any pseudoknots. i i j j i Legal structural elements Illegal structural elements
10 (Simplified) Problem Definition An RNA molecule is a sequence of length n over the alphabet {A, C, G, U}. Each base (= letter) in the sequence may form a bond with at most one other base, where A can pair only with U, and C only with G. The base-pairs are nested: Why can this problem be solved by dynamic programming? AAAGUUUCGUCCGGG (((.)))((.)(.)) A set of nested base-pairs is called a secondary structure, or a folding of the sequence. Goal: for a given RNA sequence, compute a folding with a maximum number of base-pairs 12
11 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 13
12 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 14
13 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 15
14 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 16
15 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 17
16 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 18
17 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 19
18 Co-terminus foldings: A U C A U G G C A U Partitionable foldings: 20
19 A recursive solution Co-terminus foldings: Partitionable foldings: L c (i,j) the maximum cardinality of a co-terminus folding of S i,j A U C A U G G C A U A U C A U G G C A U L p (i,j) the maximum cardinality of a partitionable folding of S i,j L(i,j) the maximum cardinality of a folding of S i,j (the objective function) - i<q j q-1 q 21
20 A recursive solution
21 The Nussinov-Jacobson Algorithm A DP algorithm which performs a bottom-up computation of the recurrence. Uses a table M which stores solutions for subsequences: M[i,j] = L(i,j). Upon reaching M[i,j], all entries which are needed for the computation of L(i,j) have already been computed and stored in M. 23 i 1 A C A G U U G C A 0 0 j
22 The Nussinov-Jacobson Algorithm 1 A ? 2 C A G U U G C A
23 The Nussinov-Jacobson Algorithm - 1 A C A G U U G C A
24 The Nussinov-Jacobson Algorithm i < q j q-1 q 1 A C A G U U G C A
25 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 2 1 A C A G U U G C A
26 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 3 1 A C A G U U G C A
27 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 4 1 A C A G U U G C A
28 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 5 1 A C A G U U G C A
29 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 6 1 A C A G U U G C A
30 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 7 1 A C A G U U G C A
31 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 8 1 A C A G U U G C A
32 The Nussinov-Jacobson Algorithm i < q j q-1 q q = 9 1 A C A G U U G C A
33 The Nussinov-Jacobson Algorithm i < q j q-1 q Space complexity: O(n 2 ) Time complexity: O(n 3 ) 1 A C A G U U G C A
34 The Nussinov-Jacobson Algorithm i < q j q-1 q Sparsification: Do we really need to consider all O(n) sums of pairs? Space complexity: O(n 2 ) Time complexity: O(n 3 ) 1 A C A G U U G C A
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