RNA Secondary Structure Prediction

1
RNA Secondary Structure Prediction

• Introduction

• RNA is a single-stranded chain of the nucleotides
  A, C, G, and U. The string of nucleotides
  specifies the linear structure of the RNA strand.
• When RNA folds, complementary nucleotides form
  base pairs (CG and AU).
• The tertiary (3 dimensional) structure is too
  complicated for us to calculate.
• We calculate only secondary structures, lists of
  base pairs.
• Knowing the base pairs tells a lot about the 3
  dimensional structure.

2
Chemical Structure of RNA

• Four base types.
• Distinguishable ends.

3
Partial Tertiary Structure

• One illustration

4
Yet Another Tertiary Structure

• Found via google

5
Our Final Tertiary Picture

• Very complex

6
A Partial RNA Secondary Structure
7
Pure Secondary Structure
8
Our Basic Model

• RNA linear structure Rr1 r2 . . . rn from
  A,C,G,U
• RNA secondary structure pairs (ri,rj) such that
  0ltiltjltn1.
• Goal secondary structures with minimum free
  energy.

9
Implementing Model Restrictions

• No knots pairs (ri,rj) and (rk,rl) such that
  iltkltjltl. RNA does contain knots.
• Program loop structure.
• No close base pairs j-igtt for some tgt0.
• High free energy.
• Complementary base pairs A-U, C-G.
• High free energy.

10
Our Two Algorithms

• Independent base pairs quite easy, but
  inaccurate.
• Calculate loops free energy best we can do for
  todays class.

11
Independent Base Pair Algorithm

• Assumption Independent base pairs.
• Advantage 1 Simpler calculations.
• Advantage 2 Illustrates ideas for a much
• more accurate
  algorithm.
• Disadvantage Unrealistic answers.

12
Independent Base PairsWhat Makes It Easy?

• Assumption The energy of each base pair is
  independent of all of the other pairs and the
  loop structure.
• Consequence Total free energy is the sum of all
  of the base pair free energies.

13
Independent Base PairsBasic Approach

• Use solutions for smaller strings to determine
  solutions for larger strings.
• This is precisely the kind of decoupling required
  for dynamic programming algorithms to work.

14
Independent Base Pairs Notation

• a(ri,rj) the free energy of a base pair joining
  ri and rj.
• Si,j The secondary structure of the RNA strand
  from base ri to base rj. Ie, the set of base
  pairs between ri and rj inclusive.
• E(Si,j) The free energy associated with the
  secondary structure Si,j.
• We define a(ri,rj) large when constraints are
  violated.

15
Independent Base PairsCalculating Free Energy

• Consider the RNA strand from position i to j.
• Consider whether rj is paired
• If rj is paired, E(Si,j)E(Si,k-1)a(k,j)E(Sk1,j
  -1) for some i-1ltkltj
• If rj isnt paired, then E(Si,j)E(Si,j-1)

16
Independent Base Pairs - Algorithm

• We search for intervals with minimum free energy.
• For each interval, the free energy is given by
  this formula
• E(Si,j) min(
• 
  E(Si1,j-1)a(ri,rj),
• 
  E(Si,k-1a(ri,rk)Sk1,j-1), i -1ltkltj1
• )
• The free energy of the RNA strand is E(S1,n).

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Independent Base PairsQuestion 1

• How does this formula deal with the case where rj
  isnt paired with any base?
• A special case of
• E(Si,k-1a(ri,rk)Sk1,j-1), i -1ltkltj1
• The special case with kj.

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Independent Base PairsQuestion 2

• What is the high level algorithm flow?
• Advance from smaller to larger intervals,
  calculating free energy costs.
• Trace back the path that corresponds to the
  maximum free energy cost.

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Independent Base PairsQuestion 3

• In what orders can the intervals free energy
  costs be evaluated?
• Major lower, minor upper bound
• Major upper, minor lower bound
• Diagonally
• Any order (eg, random) that respects the partial
  order induced by inclusion

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Independent Base PairsQuestion 4

• What are the time and storage requirements of
  this algorithm?
• Express your answer in terms of the number of
  bases in the RNA strand.
• Since the number of intervals is quadratic, the
  storage requirements are quadratic.
• Since the time requirement for each interval is
  linear, total time is cubic.

21
Independent Base Pairs Question 5

• Why not simply calculate free energies as they
  are needed? Why store them at all?
• Because the recursive calls would turn our
  polynomial algorithm into an exponential
  algorithm.

22
Independent Base PairsQuestion 6

• How does traceback work for this algorithm?
• Recalculate which subinterval yields the
  maximum free energy.
• Save traceback paths.

23
Loop Free Energy Algorithm

• An RNA molecules free energy is not independent
  of all other base pairs.
• An RNA molecules free energy actually depends on
  its loop structure.
• What do we mean by loops?

24
Types of Loops

• Each base pair (ri,rj) encloses a loop
• Hairpin loop
• Bulge on i or j
• Interior loop
• Helical region

25
Hairpin Loop

• There are no base pairs (rk,rl) for iltkltlltj.

26
Bulge on i and j

• Bulge on i
• (ri,rj) and (rk,rj-1) are base pairs with kgti1.
• ri1 is not paired.
• The bulge on j is symmetric.

27
Interior loop

• (ri,rj) and (rk,rl) are base pairs with
  i1ltk1ltk2ltj-1.
• ri1 and rj-1 are not in base pairs

28
Helical region

• (ri,rj) and (ri1,rj-1) are base pairs.

29
Free energy analysis

• E(Si,j) E(Si1,j) when ri isnt paired.
• E(Si,j) E(Si,j-1) when rj isnt paired.
• E(Si,j) min(E(Si,k)E(Sk1,j)) for iltkltl,
• k between is and js pairs
• when i and j are paired
• but not to each other
• E(Si,j) E(Li,j) where Li,j is loop energy
• when I and j are paired
• to each other

30
Free Energy Functions

• a(ri,rj) Free energy of base pair (ri,rj)
• H(k) Destabilizing free energy of a hairpin
  loop with size k.
• R Stabilizing free energy of adjacent base
  pairs (helical region).
• B(k) Destabilizing free energy of a bulge of
  size k.
• I(k) Destabilizing free energy of an interior
  loop of size k.

31
Loop Energy Formulas

• H(j-i-1) for a hairpin loop
• R E(Si1,j-1) for a helical region
• B(k) E(Sik1,j-1) for a bulge on i
• B(k) E(Si1,j-k-1) for a bulge on j
• I(k1k2) E(Sik11,j-k2-1)
• for an interior
  loop

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Free Energy Calculationfor interval (i,j)

• Minimize over
• Case where (ri,rj) is not a pair.
• Case where (ri,rj) is a pair.
• Add a(ri,rj) to the formulas.
• Minimize over k, k1, and k2.

33
What is the Apparent Complexity?

• The interior loop calculations are given by
  I(k1k2) E(Sik11,j-k2-1)
• The number of inner loop possibilities is
  quadratic in the interval size.
• The number of intervals is quadratic in the size
  of the problem.
• The complexity appears to grow as n4.

34
What is the Actual Complexity?

• Overall reduction from n4 to n3 is possible.
• Interval reduction from n2 to linear.
• Store the minimum free energy Vi,j,k where the
  interval (i,j) contains an interior loop of size
  k.

35
Multiple Solutions

• Care must be taken to define the issues.
• Multiple solutions can be obtained by adding
  flexibility to the traceback logic.
• The number of solutions can grow exponentially.

36
References

• M. Zuker, The Use of dynamic programming in RNA
  secondary structure prdiction. In M. S.
  Waterman, editor, Mathematical Methods for DNS
  Sequences. Boca Raton, FL CRC Press, 1989
• J, Setubal and J. Meidanis,Ch 8.1, Introduction
  to Computational Molecular Biology, Pacific
  Grove, CA Brooks/Cole Publishing Co., 1997