1. Structure Prediction
dmitra
11/18/2018

2. Methods Ab initio Heuristics Machine learning Homology modeling
Threading
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3. RNA Structure Prediction: Ab-initio
Sequence over {A, C, G, U}
Complementary pairs attract, form base-pairs or minimizes energy
We are not interested in overall energy of the sequence, just the process of minimization
Just the linear sequence, zero base pairs, energy=0
Physics is embedded within “free-energy” parameter/function
Minimization of energy is objective
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4. RNA Structure Prediction: Knot-free
Knot-free assumption
Knot: base pairs (I, j) and (k, l) where I<j<k<l
Knot-free causes planar graph, and makes DP algorithm feasible
Base pairs are disjoint or embed in each other
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5. RNA Structure Prediction: Principle of optimality
Assumption 1: Base-pairing do not affect each other’s energy
Now one can add energy minimization by all base pairs in a string and check which configuration produces lowest energy
Combinatorics is exponential
Need further assumption
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6. RNA Structure Prediction: DP Algorithm
Assume energy for each component can be calculated independently
a(r,k): free energy for base pair (r,k), where r, k from ACGU
a is zero for self-pairing (impossible)
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7. RNA Structure Prediction: DP Algorithm
E(Sij)= min{
E(SI+1,j-1 ) + a(ri,rj), when i,j pairs,
Min{E(SI,k-1) + E(Sk+1,j )}, when j pairs with k, I<k=<j}
Compute (n x n) matrix for I and j, bottom up, for I-j=0, I-j=1, I-j=2,…
Complexity: O(n^3)
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8. RNA Structure Prediction: relax assumptions
Consider some special energy functions, other than just the base pairing ones a(r,k)
This means: different “types” of base pairings
Some more practical topology
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9. RNA Structure Prediction: Loops
Say, base pair at (I,j) and I<u<v<w<j
v is accessible from base pair (I,j) if there is no base pair at (u,v)
Loop is the bases accessible from base pair (I,j)
Note, still no knot
Some loops: p249
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10. RNA Structure Prediction: Energy over loops
Say, (I,j) base pair closes a loop
Si+1,j-1 may not have the minimum energy configuration
Because energy of Si+1,j-1 plus free energy of a(ri,rj) may be less than min-energy configuration of string (I+1 to j-1) without base pairing at (I,j)
This interactive-ness was ignored at the previous assumption level
Dynamic Programming can still be done, if we explicitly specify energy parameters
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11. RNA Structure Prediction: Energy over loops
E(Sij)= min{
E(SI+1,j ), I is not paired
E(SI+1,j-1 ), j is not paired
min{E(S,i,k-1) + E(Sk+1,j )}, when i or j pairs with k, i<k<j},
E(LI,j ), when (I,j) base pairs and all special structures may appear within [embeds first formula of previous assumption] }
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12. RNA Structure Prediction: More assumptions
Disregard free energies that do not belong to any loops
Added energy of only components is the final energy of the string: no interaction between components
Only 4 types of loops’ as in p249 for E(LI,j ), (can add more, if you know their energy parameterization)
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13. RNA Structure Prediction: free energies for 4 loops
Hairpin loop of size k: Zi(k)
Additional stabilizing energy for two adjacent base pairs(in addition to a(r,k)): eta, constant
Destabilizing energy for bulge of size k: beta(k)
Destabilizing energy for interior loop of size k: gamma(k)
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14. RNA Structure Prediction: E(LI,j )
Hairpin: a(ri,rj) + zi(j-I+1)
Stacked-pair: a(ri,rj)+eta+E(Si+1,j-1)
Bulge on i: min{a(ri,rj)+beta(k)+ E(Si+k+1,j-1), k>=1
Bulge on j: min{a(ri,rj)+beta(k)+ E(Si+1,j-k-1), k>=1
Interior loop: min{a(ri,rj)+gamma(k1+k2)+ E(Si+k1+1,j-k2-1), k1,k2>=1
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15. RNA Structure Prediction: complexity
O(n^2) table entries
On each entry:
First 2 formulae: O(1) leading to O(n^2)
Third formula: O(n) :: O(n^3)
4.1 (E(L) hairpin): O(1) :: O(n^2)
4.2: O(1) :: O(n^2)
4.3: O(n), run on k :: O(n^3)
4.4: O(n), run on k :: O(n^3)
4.5: O(n^2), run on k1, k2 :: O(n^4)
Final complexity from 4.4: O(n^4)
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16. Protein Threading
Interactions in proteins are between 20x20 residues, as opposed to 4x4 NA’a at most in RNA’s
Residue interactions are quite non-local, causing much more structural complexity
Proteins have frequent loops (helices are loops)
So, prediction by Ab initio is extremely difficult
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17. Protein Threading
Number of protein folds are few (~1,000 for 20,000+ proteins)
Threading: map the target sequence over a template fold
Threading is an alignment problem, Torda, Fig1
Find the fold to which target “aligns” optimally (minimum “energy” function)
Needs basic scoring functions as in sequence alignment
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18. Protein Threading: number of folds
More the number of folds in database: more time to find correct template
Scoring function for threading is quite imperfect: need more available templates (contradictory requirements)
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19. Protein Threading: Scoring functions
Full force field is not necessarily ideal:
it involves dynamics between molecules, stretch, torsion, etc.
Unimportant for a static alignment
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20. Protein Threading: Scoring functions
Scoring function could be between residues from the same sequence: for coming close to each other on the alignment
Torda, Fig 5
Example scoring function (free energy):
For pair of residues A and B to be at distance r (Torda, p7):
G(AB) = kT ln(rho-rAB / rho-0-rAB),
rho-rAB is probability of AB to be at distance r,
rho-0 is probability of random occurrence of that (k,T usual)
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21. Protein Threading: Scoring functions
Probabilities are collected from PDB proteins with known structure
Different threading scheme uses different scoring functions, but mostly they are derived from PDB
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22. Protein Threading: Scoring functions
Example (Setubal-Meidanis, p257):
G1(I, ti) for placing i-th residue in sequence to the ti position in the fold
G2(I, j, ti, tj) simultaneous placements of i, j, for I<j
Constrained to be within a range, say bi<ti<ei
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23. Protein Threading
Optimization is not only on placement, but also on multiple folds in database
Accuracy is very sensitive to alignment errors
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24. Protein Threading: Dynamic programming
Advantage/disadvantage of DP is that it is deterministic
Problem: “adjacency” is hard to define in 3D
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25. Protein Threading: Dynamic programming
DP: try out different combination of “adjacent” residues on different parts of a template (Torda, Fig 5c: adjacent comes from template sequence)
Start with smaller number of elements and build up to the full sequence
Alternative approach: start with placing each residue to one of its “possible” positions and see where next residue should go: continue residue by residue
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26. Protein Threading: Probabilistic algorithm
Monte Carlo simulation: randomly throw residues at positions on fold and check aggregate scoring function
Simulated annealing: gradually move residues to optimize, stochastically making random shifts to avoid local optimum
Time consuming, & the result is non-deterministic
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27. Protein Threading: Branch and bound
In the worst case try all possible alignments, but prune the search space for non-useful branches using some bounding function
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28. Protein Threading: Search on folds
Divide and conquer over the space of folds
Assumption: folds can be ordered for their “goodness” for the target protein
Example: Setubal-Meidanis, p258
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29. Protein Threading: Future
Slow
Subsumed by Ab intio of IBM Blue Gene™ type projects
De Novo technique using linear programming (Xu and Li, 2003)
Threading techniques are not only useful for structure prediction but for fold recognition problem also: no alignment, just find the template (fold suggests function)
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