On The RNA Structure Prediction Problems: Structural Inference Technique and Other Recent Algorithms

1
On The RNA Structure Prediction Problems
Structural Inference Technique and Other Recent
Algorithms

• Hugo Willy
• HT030031E

2
Content of Presentation

• Introduction
• RNA and Its Functions
• RNA Structures
• Brief Review on Computational Methods on RNA
  Structure Prediction
• Ab Initio Predictive Methods
• Comparison Methods
• Inference Methods
• Current Work
• Introduction
• Preliminaries and Problem Definition
• Previous Approach
• Sparsification
• Recursive Dynamic Programming
• Hirschberg-like Recursive Traceback Technique
• Conclusion and Future Direction
• References

3
RNA

• Stands for Ribonucleic Acid
• A biological polymer consisting monomers called
  nucleotides
• Each nucleotide consists of a (ribose) sugar, a
  phosphate group and a base.
• There are mainly 4 types of base in RNA
  sequences.

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RNA
Watson-Crick Base Pairing
Wobble Base Pairing
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RNA Functions

• DNA transcription and translation
• Transcription Messenger RNA (mRNA),
• Translation Transfer RNA(tRNA), Ribosomal RNA
  (rRNA)
• Catalyst and regulator in nucleic acid processing
  and gene expression
• Messenger RNA splicing Small Nuclear RNA
  (snRNA)
• rRNA processing in the nucleus Small Nucleolar
  RNA (snoRNA)
• Regulators Micro RNA (miRNA) which has two
  types,
• 1)Small Interfering RNA (siRNA) and
• 2) Small Temporally Regulated RNA (stRNA)

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RNA Primary Structure

• The view of RNA from its nucleotides base
• Commonly represented by a string S over the
  alphabet SA,C,G,U
• Can be found using similar techniques for DNA
  sequencing, such as Gel Electrophoresis, etc.

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RNA Secondary Structures
Helices
Bulge Loop
Hairpin Loop
Internal Loop
Multi Loop
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RNA Tertiary Structures
Pseudoknot
Base Triple
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RNA Structure

• To a preserved function there corresponds a
  preserved molecular conformation.
• Secondary and tertiary structures are being
  solved much slower than new RNA sequences being
  discovered. Existing experimental methods are
  relatively expensive and slow.
• Denatured RNAs deterministically fold back to
  their original folding in vitro. Thus, RNA
  structure depends solely on its nucleotide
  content. Computational method should exist!

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Existing Computational Methods for RNA Structure
Prediction

• Ab-Initio Predictive Methods
• Try to compute the RNA structure solely based on
  its nucleotide contents by minimizing the free
  energy of the predicted structure.
• Comparative Methods using sequence homology
• By examining a set of homologous sequence along
  with their covarying position, we can predict
  interactions between non adjacent positions in
  the sequence, such as base pairs, triples, etc.
• Structural Inference Methods
• Given a sequence with a known structure, we infer
  the structure of another sequence known to be
  similar to the first one by maximizing some
  similarity function

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Ab Initio Predictive Methods

• Minimizing the sum of Free Energy of the
  predicted structure
• Uses experimentally determined local structures
• free energy.
• Best result without Pseudoknot O(n³) time 1
• Best result with Restricted Pseudoknot
• Simple Pseudoknot 2 O(n4) time
• Recursive Pseudoknot 2 O(n5) time
• General Pseudoknot 2 NP-Hard

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Ab Initio Predictive Methods

• Equilibrium Partition Function
• The weighted sum of probabilities over all
  possible structure, where the weight is computed
  from the free enrgy of the structure.
• Best result without Pseudoknot O(n³) time 1
• Best result with Restricted Pseudoknot O(n5)
  time 3 (Restricted to class of pseudoknots that
  are physically most likely to occur)

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Comparative Methods

• Simultaneous Sequence Structure Alignment
• First by D. Sankoff 4, with O(n6) time
  complexity
• Best result without pseudoknot O(M3n3) 5 where
  M is the maximum distance between the 2 sequences
• Stochastic Context Free Grammars (SCFG)
• Use context free grammar to produce the base
  pairing with some distribution (hence the term
  stochastic)
• Can only handle non-pseudoknotted structure
• Most recent result in 6
• A new model called Parallel Communicating Grammar
  System (PCGS) is designed in 7, to handle
  pseudoknots

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Comparative Methods

• Maximum Weighted Matching
• Based on Gabows Maximal Weighted Matching
  algorithm. Tries to find the base pairs in the
  structure given the likelihood score of all
  possible pairs.
• Computing the likelihood score might require
  multiple sequence alignment (slow)
• The most recent publications are 8 and 9. 8
  only considers bi-secondary RNA structures. While
  9 tries to find helices of some minimum length
  in the sequences and try to align them.

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Comparative Methods

• Iterative Loop Matching 10
• Applies the Loop Matching algorithm by Nussinov
  et.al. The algorithm finds a non-pseudoknotted
  structure in each iteration and run the same
  algorithm on the remaining unpaired bases.
• Genetic algorithm 11
• Find a set of possible structures. The algorithm
  will
• pass these structures through several stage of
  evolution.
• Bayesian Network and other approaches

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Structural Inferring Methods

• Given two RNA sequences S1 and S2, where the
  secondary structure of S1 is known. The method of
  this class will infer the secondary structure for
  S2 by aligning S1 and S2. Let the length of S1
  be n and length of S2 be m
• Previously, Bafna et.al uses dynamic programming
  to solve the problem in O(n2m2nm3) time and
  O(n2m2) space 12. K. Zhang improves the result
  to O(nm3) time and O(nm2) space 13.
• We further improve it to O(nm2 log n) time and
  O(m2 log n) space. The algorithm will be
  described later.
• The survey of works related to this class of
  algorithm can be found here

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Current Work

• We submitted a paper to WABI 2004 under the title
  A Faster and More Space-EfficientAlgorithm for
  Inferring Arc-Annotations of RNA Sequences
  through Alignment.
• Our contributions
• Improvement in running time by sparsification and
  recursive dynamic programming
• Improvement in space requirement using score-only
  dynamic programming with a Hirschberg-like trace
  back algorithm and compression.

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Preliminaries

• Consider two RNA sequences S1 and S2 with length
  equal to n and m respectively. Only S1s
  secondary structure is provided
• To represent a base pair between the base S1i
  and S1j, we use the pair (i,j), denoted as an
  arc, where 1?iltj? n. The structure of S1 can
  then be defined by a set P1 of arcs. The pair
  (S1,P1) is called an arc-annotated sequence.
• For RNA, it is obvious that S1i and S1j must
  be complementary to each other.

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Preliminaries

• Considering secondary structures, the arc
  annotation that corresponds to such structures is
  the Nested Arc Annotation
• Any two arcs (i,j) , (k,l) in a nested arc
  annotation P1 satisfy iltkltj ? iltlltj
• For any arc u in P1 let u_l be its left endpoint
  and u_r be its right endpoint. The size of an arc
  is equal to u_r-u_l1.

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Alignment Score Function

• Unpaired base alignment score function
• ?(S1i,S2j) ß if S1i and S2j are
  complementary
• 0 otherwise

• Arc Alignment Score Function

• a1, a2, and ß are positive integers.

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Problem Formulation

• The Weighted Largest Common Substructure (WLCS)
  of 2 arc-annotated sequence (S1,P1) and (S2,P2)
  is the maximum weighted alignment between S1 and
  S2 where unpaired bases are aligned to unpaired
  bases and arcs are aligned to arcs.
• The problem we address is, given (S1,P1) and S2,
  infer the arc annotation P2 of S2 such that their
  WLCS is maximized.

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Previous Algorithm
EXTEND(DP(i,i))
MERGE(DP(i,ul-1),DP(ul,ur))
ARC-MATCH (DP(i,i))
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Previous Algorithm (2)

• All EXTEND operations take O(nm2) time and space
• All ARC-MATCH operations also take O(nm2) time
  and
• space
• The bottleneck of the computation is the
  procedure
• MERGE, each requiring O(m3) time yielding a
  total of
• O(nm3) time

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Sparsification Technique

• Based on the observation that the entries in the
  rows of table DP is monotonically increasing, we
  do not need to check all possible j in the
  MERGE equation. Instead, we check the positions
  of j where the corresponding DP entries are
  distinct.
• This way, we can reduce the cost of each MERGE
  operation to
• O(minul-ur,ul-im2)

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Recursive Dynamic Programming

• An arc u is the parent of arc v iff ulltvlltvrltur
  and there is no arc w s.t ulltwlltvlltvrltwrltur
• Conversely, v is the (one of the) child of u
• Let core-arc(u) be the child of arc u with the
  biggest arc.
• Let side-arc(u) be the set of children of u
  excluding its core-arc.
• Let core-path(u) be the transitive closure of
  core-arc(u).

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Recursive Dynamic Programming
Left Part
Right Part
Computed
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Running Time Analysis

• Since we compute the DP table only for side arcs,
  and since the size of any side arc will not
  exceed ½ size of its parent, the recursion will
  reach at most log n levels.
• In each level, the total time spent by MERGE is
  at most O(nm2)
• Thus the total running time is still bounded by
  the MERGE operation which is O(nm2 log n)

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Space Improvement

• In order to use standard trace back, we need to
  store all the tables corresponding to an arc in
  P1.
• This requires O(nm2) storage. For sequence of
  length 3-5K, which is commonly used in lab
  experiments, the storage requirement can reach
  tens of gigabytes.
• Solution Use the score only version of the
  dynamic programming and recursion to traceback.

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Hirschberg-like Trace Back Algorithm
1. Find two points p1 and p2 in S1 such that
p2-p1 is at least 1/3n 2. Find the positions in
S2 to which p1 and p2 is aligned 3. Divide the
problem into two subproblems, each having a
fractional size of the original 4. Summing up the
decreasing geometric series, we still have the
same running time as before
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Conclusion

• RNA structure prediction is in general has many
  yet to be done.
• We considered a problem of RNA structure
  inference where we infer the structure of an RNA
  sequence given a similar sequence with known
  structure.
• Our technique is quite general that it can also
  directly solve the LAPCS problem mentioned
  elsewhere 14
• We wish to handle pseudoknot in the future by
  applying the algorithm iteratively, following the
  idea of Iterative Loop Matching

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References

• 1 R. B. Lyngsø, M. Zuker, and C.N.S. Pedersen.
  Internal loops in RNA secondary structure
  prediction. In ICMB, pages 260267, 1999.
• 2 T. Akutsu. Dynamic programming algorithms for
  RNA secondary structure with pseudoknots. In
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• 3 R. M. Dirks and N. A. Pierce. A partition
  function algorithm for nucleic acid secondary
  structure including pseudoknots. In J. Comput.
  Chem., volume 24, pages 16641677, 2003.
• 4 D. Sanko. Simultaneous solution of the RNA
  folding alignment and protosequence problem. In
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  1985.
• 5 D. Mathews and D. Turner. Dynalign an
  algorithm for finding the secondary structure
  common to two RNA sequences. In J. Mol. Biol,
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• 6 B. Knudsen and J. Hein. RNA secondary
  structure prediction using stochastic context
  free grammars and evolutionary history. In
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• 7 L.M. Cai, R. L. Malmberg, and Y. Z. Wu.
  Stochastic modeling of RNA pseudoknotted
  structures a grammatical approach. In
  Bioinformatics (suppl. 3), volume 15, pages
  166173, 2003.

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References (2)

• 8 C. Witwer, I. L. Hofacker, and P. F. Stadler.
  Prediction of consensus RNA secondary structures
  including pseudoknots. In to appear in European
  Conference on Computational Biology, 2004.
• 9 Yongmei-Ji, Xing-Xu, and G. D. Stormo. A
  graph theoretical approach to predict common RNA
  secondary structure motifs including pseudoknots
  in unaligned sequences. In to appear in
  Bioinformatics, 2004.
• 10 Jianhua Ruan, G. D. Stormo, and Weixiong
  Zhang. An iterated loop matching approach to the
  prediction of RNA secondary structures with
  pseudoknots. In Bioinformatics (1), volume 20,
  pages 5866, 2004.
• 11 J.H. Chen, S. Y. Le, and J. V. Maizel.
  Prediction of common secondary structures of
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• 12 V. Bafna, S. Muthukrishnan, and R. Ravi.
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References (3)

• 14 T. Jiang, G. H. Lin, B. Ma, and K. Zhang.
  The longest common subsequence problem for
  arc-annotated sequences. In Proceedings of
  the11th Annual Symposium on Combinatorial Pattern
  Matching, volume 1848, pages 154165.
  Springer-Verlag, 2000.