Dynamic Programming
(cont’d)
CS 466
Saurabh Sinha
Spliced Alignment
• Begins by selecting either all putative exons
between potential acceptor and donor sites or by
finding all substrings similar to the target protein
(as in the Exon Chaining Problem).
• This set is further filtered in a such a way that
attempt to retain all true exons, with some false
ones.
• Then find the chain of exons such that the
sequence similarity to the target protein
sequence is maximized
Spliced Alignment Problem: Formulation
• Input: Genomic sequences G, target
sequence T, and a set of candidate
exons (blocks) B.
• Output: A chain of exons Γ such that
the global alignment score between Γ*
and T is maximized
Γ* - concatenation of all exons from chain Γ
The DAG
•
•
•
•
Vertices: One vertex for each block in B
Directed edge connecting non-overlapping blocks
Label of vertex = string of block it represents
A path through the DAG spells out the string
obtained by concatenating that particular chain of
blocks
• Weight of a path is the score of the optimal
alignment between the string it spells out and the
target sequence
Dynamic programming
• Genomic sequence G = g1g2…gn
• Target sequence T = t1t2…tm
• As usual, we want to find the optimal
alignment score of the i-prefix of G and
the j-prefix of T
• Problem is, there are many i-prefixes
possible (since multiple blocks may
include position i)
Idea
• Find the optimal alignment score of the
i-prefix of G and the j-prefix of T
assuming that this alignment uses a
particular block B at position i
• S(i, j, B)
• … for every block B that includes i
Recurrence
If i is not the starting vertex of block B:
• S(i, j, B) =
max { S(i – 1, j, B) – indel penalty
S(i, j – 1, B) – indel penalty
S(i – 1, j – 1, B) + δ(gi, tj) }
If i is the starting vertex of block B:
• S(i, j, B) =
max { S(i, j – 1, B) – indel penalty
maxall blocks B’ preceding block B S(end(B’), j, B’) – indel penalty
maxall blocks B’ preceding block B S(end(B’), j – 1, B’) + δ(gi, tj)
}
RNA secondary structure
prediction
RNA
• RNA is similar to DNA chemically. It is usually only a
single strand. T(hyamine) is replaced by U(racil)
• Some forms of RNA can form secondary structures by
“pairing up” with itself. This can change its properties
dramatically.
tRNA linear and 3D view:
http://www.cgl.ucsf.edu/home/glasfeld/tutorial/trna/trna.gif
RNA
• There’s more to RNA than mRNA
• RNA can adopt interesting non-linear
structures, and catalyze reactions
• tRNAs (transfer RNAs) are the
“adapters” that implement translation
Secondary structure
• Several interesting RNAs have a conserved
secondary structure (resulting from basepairing interactions)
• Sometimes, the sequence itself may not be
conserved for the function to be retained
• It is important to tell what the secondary
structure is going to be, for homology
detection
Conserved secondary
structure
N-Y
A
A
N-N’
N-N’
R
N-N’
N-N’
N-N’
N-N’
N-N’
/
N
Consensus binding
site for R17 phage coat
protein. N = A/C/G/U,
N’ is a complementary
base pairing to N,
Y is C/U, R is A/G
Source: DEKM
Basics of secondary structure
• G-C pairing: three bonds (strong)
• A-U pairing: two bonds (weaker)
• Base pairs are approximately coplanar
Basics of secondary structure
Basics of secondary structure
•
•
•
•
G-C pairing: three bonds (strong)
A-U pairing: two bonds (weaker)
Base pairs are approximately coplanar
Base pairs are stacked onto other base
pairs (arranged side by side): “stems”
Secondary loop
structure
elements
at the end
of a stem
single stranded
bases within
a stem
stem loop
… on both
sides of stem
… only on one
side of stem
Loop: single stranded subsequences bounded by base pairs
Non-canonical base pairs
• G-C and A-U are the canonical base pairs
• G-U is also possible, almost as stable
Nesting
• Base pairs almost always occur in a
nested fashion
• If positions i and j are paired, and
positions i’ and j’ are paired, then these
two base-pairings are said to be nested if:
• i < i’ < j’ < j
OR
i’ < i < j < j’
• Non-nested base pairing: pseudoknot
Pseudoknot
9
2
18
11
(9, 18)
(2, 11)
NOT NESTED
Pseudoknot problems
• Pseudoknots are not handled by the
algorithms we shall see
• Pseudoknots do occur in many
important RNAs
• But the total number of pseudoknotted
base pairs is typically relatively small
Secondary structure prediction
Approach 1.
• Find the secondary structure with most
base pairs.
• Nussinov’s algorithm
• Recursive: finds best structure for small
subsequences, and works its way
outwards to larger subsequences
Nussinov’s algorithm: idea
• There are only four possible ways of
getting the best structure for
subsequence (i,j) from the best
structures of the smaller subsequences
(1) Add unpaired position i onto best
structure for subsequence (i+1,j)
i+1
i
j
Nussinov’s algorithm: idea
• There are only four possible ways of
getting the best structure for
subsequence (i,j) from the best
structures of the smaller subsequences
(2) Add unpaired position j onto best
structure for subsequence (i,j-1)
i
j-1
j
Nussinov’s algorithm: idea
• There are only four possible ways of
getting the best structure for
subsequence (i,j) from the best
structures of the smaller subsequences
(3) Add (i,j) pair onto best structure for
subsequence (i+1,j-1)
i
i+1
j-1
j
Nussinov’s algorithm: idea
• There are only four possible ways of
getting the best structure for
subsequence (i,j) from the best
structures of the smaller subsequences
(4)Combine two optimal substructures (i,k)
and (k+1,j)
i
k
k+1 j
Nussinov RNA folding
algorithm
• Given a sequence s of length L with symbols
s1 … sL. Let (i,j) = 1 if si and sj are a
complementary base pair, and 0 otherwise.
• We recursively calculate scores g(i,j) which
are the maximal number of base pairs that
can be formed for subsequence si…sj.
• Dynamic programming
Recursion
• Starting with all subsequences of length
2, to length L
• g(i,j) = max of
•
•
•
•
g(i+1, j)
g(i,j-1)
g(i+1,j-1) + (i,j)
maxi < k < j [g(i,k) + g(k+1,j)]
• Initialization
• g(i,i-1) = 0
• g(i,i) = 0
O(n2) ?
No. O(n3)
Traceback
• As usual in sequence alignment ?
• Optimal sequence alignment is a linear
path in the dynamic programming table
• Optimal secondary structure can have
“bifurcations”
• Traceback uses a pushdown stack
Traceback
Push (1,L) onto stack
Repeat until stack is empty:
pop (i,j)
if i >= j continue
else if g(i+1,j) = g(i,j) push (i+1,j)
else if g(i,j-1) = g(i,j) push (i,j-1)
else if g(i+1,j-1) + (i,j) = g(i,j)
record (i,j) base pair
push (i+1,j-1)
else for k = i+1 to j-1, if g(i,k)+g(k+1,j) g(i,j)
push (k+1,j)
push (i,k)
break (for loop)
Secondary structure prediction
Approach 2
• Based on minimization of ∆G, the
equilibrium free energy, rather than
maximization of number of base pairs
• Better fit to real (experimental) ∆G
• Energy of stem is sum of “stacking”
contributions from the interface between
neighboring base pairs
U U
A
A
G-C
G-C
A
G-C
U-A
A-U
C-G
A-U
A
4nt loop +5.9
1nt bulge +3.3
dangle -0.3
A
Source: DEKM
-1.1 terminal mismatch of hairpin
-2.9 stack
-2.9 stack (special case)
-1.8 stack
-0.9 stack
-1.8 stack
-2.1 stack
• Neighboring base pairs: stack
• Single bulges OK in stacking
• Longer bulges: no stacking term
• Loop destabilisation energy
• Loop terminal mismatch energy
hairpin loop: exactly one base pair
internal loop: exactly two base pairs
bulge: internal loop with one base from
each base pair being adjacent
multibranched loop: > 2 base pairs
in a loop, one base pair is closest to ends of RNA.
this is the “exterior” or “closing” base pair
all other base pairs are “interior”
Source: Martin Tompa’s lecture notes
Energy contributions
• eS(i,j): Free energy of stacked pair (i,j) and (i+1,j-1)
• eH(i,j): Free energy of a loop closed by (i,j): depends on
length of loop, bases at i,j, and bases adjacent to them
• eL(i,j,i’,j’): Free energy of an internal loop or bulge, with (i,j)
and (i’,j’) being the bordering base pairs. Depends on bases
at these positions, and unpaired bases adjacent to them
• eM(i,j,i1,j1,…ik,jk): Free energy of a multibranch loop with (i,j)
as the closing base pair and i1j1 etc as the internal base pairs
Zuker’s algorithm:
Dynamic programming
• W(j): FE of optimal structure of s[1..j]
• V(i,j): FE of optimal structure of s[i..j]
assuming i,j form a base pair
• VBI(i,j): FE of optimal structure of s[i..j]
assuming i,j closes a bulge or internal loop
• VM(i,j): FE of optimal structure of s[i..j]
assuming i,j closes a multibranch loop
• WM(i,j): used to compute VM
Dynamic programming
recurrences
• W(j): FE of optimal structure of s[1..j]
• W(0) = 0
s[j] is external base
(a base not in any loop)
• W(j) = min( W(j-1),
min1<=i<jV(i,j)+W(i-1))
s[j] pairs with s[i],
for some i < j
Dynamic programming
recurrences
• V(i,j): FE of optimal structure of s[i..j]
assuming i,j form a base pair
• V(i,j) = infinity if i >= j
• V(i,j) = min( eH(i,j),
eS(i,j) + V(i+1,j-1),
i,j is exterior base
VBI(i,j),
pair of a hairpin
VM(i,j)) if i < j
loop
Dynamic programming
recurrences
• V(i,j): FE of optimal structure of s[i..j]
assuming i,j form a base pair
• V(i,j) = infinity if i >= j
• V(i,j) = min( eH(i,j),
eS(i,j) + V(i+1,j-1),
i,j is exterior pair
VBI(i,j),
of a stacked pair.
if i < j
i+1,j-1 is therefore VM(i,j))
a pair too.
Dynamic programming
recurrences
• V(i,j): FE of optimal structure of s[i..j]
assuming i,j form a base pair
• V(i,j) = infinity if i >= j
• V(i,j) = min( eH(i,j),
eS(i,j) + V(i+1,j-1),
i,j is exterior pair
VBI(i,j),
of a bulge or
VM(i,j)) if i < j
interior loop
Dynamic programming
recurrences
• V(i,j): FE of optimal structure of s[i..j]
assuming i,j form a base pair
• V(i,j) = infinity if i >= j
• V(i,j) = min( eH(i,j),
eS(i,j) + V(i+1,j-1),
i,j is exterior pair
VBI(i,j),
of a multibranch
VM(i,j)) if i < j
loop
Dynamic programming
recurrences
• VBI(i,j): FE of optimal structure of s[i..j]
assuming i,j closes a bulge or internal
loop
• VBI(i,j) = min (eL(i,j,i’,j’) + V(i’,j’))
i’,j’
i<i’<j’<j
Energy of the bulge
• Slow !
Dynamic programming
recurrences
• VM(i,j): FE of optimal structure of s[i..j] assuming i,j
closes a multibranch loop
• VM(i,j) = min (eM(i,j,i1,j1,..,ik,jk) + ∑hV(ih,jh))
k>=2
i1,j1,…ik,jk
Energy of the loop itself
• Very slow !
Order of computation
• What order to fill the DP table in ?
• Increasing order of (j-i)
• VBI(i,j) and VM(i,j) before V(i,j)