RNA Folding Algorithms

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RNA Folding Algorithms

• BioE131, Ian Holmes

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Free energy of RNA folding
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Watson-Crick base pairing
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Nussinov Algorithm
?(i,j) is the max number of basepairs in
subsequence i..j ?(i,j) is 1 iff (i,j) form a
basepair
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Example implementation

• For j 1 to L (in ascending order)
• Set ?(j,j) 0
• If jgt1 Set ?(j-1,j) ?(j-1,j)
• If jgt2 For i j-2 to 1 (in descending order)
• Set ?(i,j)
• For k i1 to j-1 (in ascending order)
• Set ?(i,j)

Rate-limiting step
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Nussinov memory complexity

• Number of stored values ?(i,j) isusing the
  result...

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Asymptotic memory complexity

• Number of stored values ?(i,j) isFor
  sufficiently large L,
• the L2 term will come to dominate
• the coefficient (1/2) will be irrelevant if we
  compare to other powers of L (e.g. L, L3, L4)

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Asymptotic memory complexity

• Number of stored values ?(i,j) isWe write
  this as O(L2)...which simply means that for some
  k, L0 and L gt L0...

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Nussinov time complexity

• Number of iterations of inner k-loop is using
  the results...

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Time complexity Big O notation

• Number of iterations of inner k-loop is...or
  O(L3)
• Much easier to estimate the asymptotic (big-O)
  expression than the exact expression!

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Example what resources needed to fold HIV genome?

• Genome is 10kb in size
• i.e. 104 bases
• Memory 108 cells
• Time 1012 operations
• Each cell is 4 bytes(8 bytes on a 64-bit
  machine)
• Each operation is 10-7 seconds(assuming 100
  cycles on a 1GHz CPU)

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Wobble non-canonical base pairs
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Base-pair stacking
p-orbital conjugation Induced polarity Van der
Waals forces
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Loop closure
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Tetraloops, etc.
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Pseudoknots
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RNA Free Energy Terms

• Sequence-dependent
• Base pairing (16 possibilities)
• Base stacking (256 possibilities)
• Stem opening, closing terms
• Tetraloops, triloops, triple-A platforms, etc.
• Length-dependent
• Loop closure
• Topology
• Stacked and nested basepairs
• Pseudoknots

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Zukers algorithm

• Let E(S) be the free energy of folding of
  structure S
• Zukers algorithm computes E0 maxS E(S)
• From this, one can find S0 argmaxS E(S)
• Sequence length LTime complexity O(L3)Memory
  complexity O(L2)

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Zukers algorithm
?(x,y) energy of pairing x-y h(n) loop b(n)
bulge i(n) interior s stacked
Max energy (i..j) with i and/or j dangling
W(i,j) max
Max energy (i..j) given that i and j are to be
paired
loop
stacked basepair
V(i,j) max
right bulge
left bulge
interior loop
multibranched loop
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McCaskills algorithm

• Let E(S) be the free energy of folding of
  structure S
• McCaskills algorithm computes the partition
  function Z ? exp(-E(S)/kT)
• From this, one can find the Boltzmann probability
  of a particular structure S P(S)
  exp(-E(S)/kT) / Z
• Can also find probabilities of individual
  basepairs
• Same complexity as Zuker algorithm

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Programs

• Without pseudoknots (Zukers algorithm)
• MFOLD
• Vienna
• With pseudoknots
• PKNOTS
• NUPACK
• Many variants (two strands of RNA, etc.)