Prediction of RNA structure

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Prediction of RNA structure

• Vineet Bafna

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The central dogma

• The DNA contains the code to all of the essential
  machinery of the cell, mainly proteins.
• Information about proteins is in genes.
• The genic information is first copied to an
  intermediate molecule (transcription)
• The transcript goes outside the nucleus and is
  translated into a protein.

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The Central Dogma

• However, not all genes are translated!

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Example tRNA
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RNA

• Could it be that, there is a large trove of
  non-coding RNA, that is not discovered yet?
• The answer seems to be YES
• Todays lecture How can we computationally
  identify ncRNA?

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Novel ncRNAs are abundant Ex miRNAs

• miRNAs were the second major story in 2001 (after
  the genome).
• Subsequently, many other non-coding genes have
  been found

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Scientific American, 2006
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Bacterial Riboswitches

• -Breaker Lab

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ncRNA gene finding

• The RNA world hypothesis
• RNA are as important as protein coding genes.
• Many undiscovered ncRNA exist
• Computational methods for discovering ncRNA are
  not mature.
• What are the clues to non-coding genes?
• Structure Given a sequence, what is the
  structure into which it can fold with minimum
  energy?

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RNA structure Basics

• Key RNA is single-stranded. Think of a string
  over 4 letters, AC,G, and U.
• The complementary bases form pairs.
• A lt-gt U, C lt-gt G, G lt-gt U
• Base-pairing defines a secondary structure. The
  base-pairing is usually non-crossing.

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RNA structure Basics

• Key RNA is single-stranded. Think of a string
  over 4 letters, AC,G, and U.
• The complementary bases form pairs.
• Base-pairing defines a secondary structure. The
  base-pairing is usually non-crossing.

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RNA structure pseudoknots

• Sometimes, unpaired bases in loops form crossing
  pairs. These are pseudoknots.

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De novo RNA structure prediction

• Any set of non-crossing base-pairs defines a
  secondary structure.
• Abstract Question
• Given an RNA string find a structure that
  maximizes the number of non-crossing base-pairs
• Incorporate the true energetics of folding
• Incorporate Pseudo-knots

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De novo RNA prediction a combinatorial
formulation
A C G A U U

• Input
• A string over A,C,G,U
• A pairs with U, C pairs with G, G with U
• Output
• A subset of possible base-pairings of maximum
  size (score) such that
• No two base-pairs intersect
• Each nucleotide pairs with at most one other
  nucleotide
• How can we compute this set efficiently?

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RNA structure

• Nussinovs algorithm
• Score B for every base-pair. No penalty for
  loops. No pesudo-knots.
• Let W(i,j) be the score of the best structure of
  the subsequence from i to j.

for i n down to 1 for j i1 to n

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Obtaining RNA structure
for i n downto 1 for j i1 to n

i
j
i
j
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Obtaining RNA Structure
Procedure print_RNA(i,j) if S(i,j) /
print (i,j)
print_RNA(i1,j-1) else if (S(i,j) -)
print_RNA(i1,j) else if (S(i,j) )
print_RNA(i,j-1) else
kS(i,j) print_RNA(i,k)
print_RNA(k1,j)
i
j
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RNA structure example
i 1 2 3 4 5 6
j
2
3
4
5
6
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RNA Structure Details
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Base-pairing Loops

• Base-pairs arise from complementary nucleotides
• Single-stranded
• Stack is when 2 base-pairs are contiguous
• Loops arise when there are unpaired bases.
• They are characterized by the number of
  base-pairs that close it.
• Hairpin closed by 1 base-pair
• Bulge/Interior Loops (2 base-pairs)
• Multiple Internal loops (k base-pairs)

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Scoring Loops, multi-loops

• Zuker-Turner Energy Rules
• http//www.bioinfo.rpi.edu/zukerm/rna/energy/node
  2.html
• Stacking Energies
• Energy for Bulges and Interior Loops
• Energy for Multi-loops

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Improving the efficiency

• Computing structure on a 2000nt sequence is about
  20s (2GHz, 2Gb)
• (Wexler et al., 2006)
• How much time would it take to search 10000
  candidate regions?
• What would be the impact of an O(n) speedup on
  this search?

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A triangle inequality

• After the computations are done,
• A triangle inequality is satisfied

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Towards more efficient algorithms

• Consider the main loop again
• We only need to branch, when i forms a base-pair.
  Therefore, w.l.o.g

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A candidate list approach

• Note that if C is the maximum size of the
  candidate list, then the running time is O(Cn2)
• How can we reduce the size of the candidate list?
• Consider only those k s.t. bk is complementary to
  bi

We will consider cases when k is NOT optimal,
using the triangle inequality
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Candidate list

• Consider i,j,k s.t.
• W(i,j) B(ri,rk)W(i1,k-1)W(k1,j) (1)
• Claim For all j gt j
• j is not a candidate in computing W(i,j)

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Proof
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The candidate list algorithm

• For all i n down to 1
• For all j i1 to n

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How large is the candidate list?

• Here, the answer depends very much on the energy
  computations.
• If we simply count the number of base-pairs, the
  candidate list should be large (probably)
• For more realistic functions, it is very small
  (experiments with n upto 1000, showed C6)

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Empirical estimates of C
A Study of Accessible Motifs and RNA Folding
Complexity Ydo Wexler, Chaya Zilberstein, and
Michal Ziv-Ukelson LNBI 3909, p. 473 ff.
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A thermodynamic argument

• It is conjectured, and empirically shown that RNA
  has the polymer-zeta property
• Probability that a string of length m folds into
  an RNA structure is b/mc for constants b,cgt0
• Note that if cgt0, the number of indices j that
  base-pair with i is O(1)
• In practice c1.5

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End of Lecture
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Incorporating pseudoknots in structure prediction

• Q Determine optimal structure, when pseudoknots
  are allowed.
• Pseudoknots are only loosely defined.
• If any interleaving is allowed, then simply
  select a structure in which a max number of
  nucleotides can be paired.
• Under some restrictive notions, the structure
  problem becomes NP-hard.

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Akutsus Simple pseudoknots

• A region i0,k0 forms a simple pseudoknot if
  there exist positions j0, j0 s.t.
• Each (i,j) ? M satisfies either
• i0 ? i lt j0 ? j lt j0 or j0 ? i lt j0 ? j ? k0
• Within one of the two groups, there is no
  interleaving.
• iltiltj0 or j0 ?ilti implies jgtj.

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Simple pseudoknots
A
B
C

• A region i0,k0 forms a simple pseudoknot if it
  can be partioined into 3 regions s.t. every
  base-pair has the following properties
• It either connects (A,B) , or (B,C)
• Two base-pairs within (A,B) (or, (B,C)) do not
  interleave

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Simple pseudoknotted structure
M
M2
M1

• Collection of simple pseudoknots (Mi) and other
  basepairs M.
• None of the simple pseudoknot regions overlap.
• M is a secondary structure without pseudoknots
  for the sequence obtained by excising all
  pseudoknotted regions.
• Can we identify a sub-structure for the
  pseudo-knots?

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Akutsus algorithm (Main idea)

• Rotate the sequence so that it forms two loops.
• Each allowed base-pair is a horizontal line in
  exactly one of the two loops. The horizontal
  lines in the two loops are non-crossing.
• All base-pairs can be ordered.

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Main idea
j0
k0
i0
j0

• The base-pairs have a total order that a d.p. can
  exploit.
• (i,j) lt (i,j) if one of the following holds
• iltiltjltj
• iltiltjltjltj0
• j0ltiltiltjltj

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Frontiers of D.P.
j0
k0
i
j
k
i0
j0

• We need to construct an increasing path of
  base-pairs.
• Consider a frontier triple (i,j,k),
  corresponding to the region (i0,i) and (j,k)
• We have the following cases
• (i,j) form a base-pair
• (j,k) form a base-pair
• Neither (i,j) nor (j,k) form a base-pair

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Ordering frontiers
j0
k0
i
j
k
i0
j0

• frontier triple (i,j,k) gt (i,j,k) if and
  only if
• i lt i
• j lt j
• k lt k

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Ordering Frontiers

• (i,j,k) gt (i,j,k) if and only if

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Recurrences

• SL(i,j,k) is the optimum score of a frontier
  (i,j,k) assuming that
• iltj0ltjltj0ltk
• (i,j) form a base-pair
• SR(i,j,k) is the optimum score of a frontier
  (i,j,k) assuming that iltj0ltjltj0ltk, and
• (j,k) form a base-pair
• SM(i,j,k) is the optimum score of a frontier
  (i,j,k) assuming that iltj0ltjltj0ltk, and
• Neither (i,j) nor (j,k) form a base-pair

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Computing SL
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Putting it all together
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Computing optimal pseudo-knotted structures

• Let S(i,j) be the opt score of a simple
  pseudoknotted structure.
• Either (i,j) is a simple pseudoknot
• S(i,j) Spseudo(i,j)
• Or, not
• S(i,j) max v(ai,aj) S(i1,j-1),
• max k S(i,k-1)S(k,j)

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Time Complexity

• We compute Spseudo(i,j) for all (i,j). Each
  computation is O(n3). Total time?
• For each i0, perform the following computation
• Spseudo(i0,k0) max i0?iltj ?k0SL(i,j,k0),
• Total time O(n4) Can you do better?

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Recursive pseudoknots

• Each loop of the RNA structure is a recursive
  pseudoknotted RNA structure.
• Optimal recursive pseudoknotted RNA structure
  problem can be solved in O(n5) time.

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Open Questions

• There should be a direct generalization of simple
  pseudoknots to the following structure.
• Rivas and Eddy do consider such a generalization,
  but a systematic treatment is missing.
• Q Given a pseudo-knotted structure, is it an
  Akutsu simple pseudoknotted structure?
• Linear time algorithm was devised recently for
  this problem.

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Structure as a proxy for ncRNA

• Any genomic region with an energetically
  favorable fold is a candidate ncRNA?
• Rivas and Eddy show otherwise.
• They use
• LOD score L Pr(sRNA)/Pr(sNull)
• Z-score Z G-?/?

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LOD scores for putative ncRNA

• A region of C. elegans is containing two tRNAs is
  chosen. The position of tRNA is indicated by
  stars.
• The true positions and the LOD-score correspond.
• Stronger results in M. janaschii (AT70)
• Weak results in E.coli (GC50)

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Correlation with GC content

• GC content detector has results similar to
  structure prediction!

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Significance of detected regions

• Test for significance was using Z-scores on
  shuffled sequences.
• Earlier tests shuffled sequences using a large
  window.
• Shuffling high scoring windows led to lower
  Z-scores.

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Testing chimeric sequence

• Test on chimeric sequence. A real tRNA was
  embedded in 2000bp random sequence with similar
  GC-composition.

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Increasing window size

• tRNA Z-score computation with a larger window
  (85nt).

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• For very strong signals, a Z-score gt 4 may still
  be significant.

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Z-score distribution

• Z-scores of 415 tRNA genes. 98 have Z-score
  lower than 4.

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Structure

• We hoped that RNA structure computation would
  give us a clue into detecting novel ncRNA
• Turns out that random DNA can fold into a low
  energy structure.