Post on 13-Jan-2016
Cédric Notredame (21/04/23)
Using Dynamic Programming To Align Sequences
Cédric Notredame
Cédric Notredame (21/04/23)
Our Scope
Coding a Global and a Local Algorithm
Understanding the DP concept
Aligning with Affine gap penalties
Sophisticated variants…
Saving memory
Cédric Notredame (21/04/23)
Outline
-Coding Dynamic Programming with Non-affine Penalties
-Adding affine penalties
-Turning a global algorithm into a local Algorithm
-Using A Divide and conquer Strategy
-The repeated Matches Algorithm
-Double Dynamic Programming
-Tailoring DP to your needs:
Cédric Notredame (21/04/23)
Global Alignments Without Affine Gap
penalties
Dynamic Programming
Cédric Notredame (21/04/23)
How To align Two Sequences With a Gap Penalty, A Substitution
matrix and Not too Much Time
Dynamic Programming
Cédric Notredame (21/04/23)
A bit of History…
-DP invented in the 50s by Bellman
-Programming Tabulation
-Re-invented in 1970 by Needlman and Wunsch
-It took 10 year to find out…
Cédric Notredame (21/04/23)
The Foolish Assumption
The score of each column of the alignment is independent from the rest of the alignment
It is possible to model the relationship between two sequences with:
-A substitution matrix-A simple gap penalty
Cédric Notredame (21/04/23)
The Principal of DP
If you extend optimally an optimal alignment of two sub-sequences, the result remains an optimal alignment
X-XXXXXX
X-
XX
-X
Deletion
Alignment
Insertion
??+
Cédric Notredame (21/04/23)
Finding the score of i,j
-Sequence 1: [1-i]-Sequence 2: [1-j]
-The optimal alignment of [1-i] vs [1-j] can finish in three different manners:
X-
XX
-X
Cédric Notredame (21/04/23)
Finding the score of i,j
i-
ij
-j
1…i1…j-1
1…i-11…j-1
1…i-11…j
+
+
+
Three ways to buildthe alignment
1…i1…j
Cédric Notredame (21/04/23)
Finding the score of i,j
1…i-11…j-1
1…i1…j-1
1…i-11…j
In order to Compute the score of
1…i1…j
All we need are the scores of:
Cédric Notredame (21/04/23)
Formalizing the algorithm
F(i,j)= best
F(i,j-1) + Gep
F(i-1,j-1) + Mat[i,j]
F(i-1,j) + Gep X-
XX
-X
1…i1…j-1
1…i-11…j-1
1…i-11…j
+
+
+
Cédric Notredame (21/04/23)
Arranging Everything in a Table
- F A
-
F
A
S
T
T
1…I-11…J-1
1…I1…J-1
1…I-11…J
1…I 1…J
Cédric Notredame (21/04/23)
Taking Care of the Limits
In a Dynamic Programming strategy, the most delicate part is to take care of the limits:
-what happens when you start-what happens when you finish
The DP strategy relies on the idea that ALL the cells in your table have the same environment…
This is NOT true of ALL the cells!!!!
Cédric Notredame (21/04/23)
Taking Care of the Limits
- F A-FAS
T
T -4Match=2MisMatch=-1Gap=-1
-3
FAT---
-1
F-
-2
FA--
-1F-
-2FA--
-3FAS---
0
Cédric Notredame (21/04/23)
Filing Up The Matrix
Cédric Notredame (21/04/23)
- F A
-
F
A
S -3
-2
-1
-1 -2
T
-3
T -4
-2+2
-2 +2-3
-2
+1 +1-4
-3
0 0+1
-2
-3 +10
+4
0 +4-1
0
+3 +30
-3
-4 0+3
0
-1 +3+2
+3
+2 +3-1
-4
-5 -1+2
-1
-2 +2+2
+5
+1 +5
0
Cédric Notredame (21/04/23)
Delivering the alignment: Trace-back
Score of 1…3 Vs 1…4
Optimal Aln Score
TT
S-
AAFF
Cédric Notredame (21/04/23)
Trace-back: possible implementation
while (!($i==0 && $j==0)) { if ($tb[$i][$j]==$sub) #SUBSTITUTION
{ $alnI[$aln_len]=$seqI[--$i]; $alnJ[$aln_len]=$seqJ[--$j]; }
elsif ($tb[$i][$j]==$del) #DELETION{ $alnI[$aln_len]='-'; $alnJ[$aln_len]=$seqJ[--$j]; }
elsif ($tb[$i][$j]==$ins) #INSERTION{ $alnI[$aln_len]=$seqI[0][--$i]; $alnJ[$aln_len]='-'; }
$aln_len++; }
Cédric Notredame (21/04/23)
Local Alignments Without Affine Gap
penalties
Smith and Waterman
Cédric Notredame (21/04/23)
Getting rid of the pieces of Junk between the
interesting bits
Smith and Waterman
Cédric Notredame (21/04/23)
Cédric Notredame (21/04/23)
The Smith and Waterman Algorithm
F(i,j)= best
F(i-1,j) + Gep
F(i-1,j-1) + Mat[i,j]
F(i,j-1) + Gep X-
XX
-X
1…i1…j-1
1…i-11…j-1
1…i-11…j
+
+
+
0
Cédric Notredame (21/04/23)
The Smith and Waterman Algorithm
F(i,j)= best
F(i-1,j) + Gep
F(i-1,j-1) + Mat[i,j]
F(i,j-1) + Gep
0
Cédric Notredame (21/04/23)
The Smith and Waterman Algorithm
0
Ignore The rest of the Matrix
Terminate a local Aln
Cédric Notredame (21/04/23)
Filing Up a SW Matrix
0
Cédric Notredame (21/04/23)
Filling up a SW matrix: borders
* - A N I C E C A T - 0 0 0 0 0 0 0 0 0C 0A 0T 0A 0N 0 D 0O 0G 0
Easy:Local alignments
NEVER start/end with a gap…
Cédric Notredame (21/04/23)
Filling up a SW matrix
* - A N I C E C A T - 0 0 0 0 0 0 0 0 0C 0 0 0 0 2 0 2 0 0 A 0 2 0 0 0 0 0 4 0T 0 0 0 0 0 0 0 2 6A 0 2 0 0 0 0 0 0 4N 0 0 4 2 0 0 0 0 2D 0 0 2 2 0 0 0 0 0O 0 0 0 0 0 0 0 0 0G 0 0 0 0 0 0 0 0 0
Best Local score
Beginning of the trace-back
Cédric Notredame (21/04/23)
for ($i=1; $i<=$len0; $i++) { for ($j=1; $j<=$len1; $j++)
{ if ($res0[0][$i-1] eq $res1[0][$j-1]){$s=2;}
else {$s=-1;} $sub=$mat[$i-1][$j-1]+$s; $del=$mat[$i ][$j-1]+$gep; $ins=$mat[$i-1][$j ]+$gep; if ($sub>$del && $sub>$ins && $sub>0)
{$smat[$i][$j]=$sub;$tb[$i][$j]=$subcode;} elsif($del>$ins && $del>0 )
{$smat[$i][$j]=$del;$tb[$i][$j]=$delcode;} elsif( $ins>0 )
{$smat[$i][$j]=$ins;$tb[$i][$j]=$inscode;} else {$smat[$i][$j]=$zero;$tb[$i][$j]=$stopcode;}
if ($smat[$i][$j]> $best_score) { $best_score=$smat[$i][$j]; $best_i=$i; $best_j=$j; }
} }
PrepareTraceback
Turning
NW
into
SW
Cédric Notredame (21/04/23)
A few things to remember
SW only works if the substitution matrix has been normalized to give a Negative score to a random alignment.
Chance should not pay when it comes to local alignments !
Cédric Notredame (21/04/23)
More than One match…
-SW delivers only the best scoring Match
-If you need more than one match:-SIM (Huang and Millers)Or-Waterman and Eggert (Durbin, p91)
Cédric Notredame (21/04/23)
Waterman and Eggert
-Iterative algorithm:
-1-identify the best match-2-redo SW with used pairs forbidden
-Delivers a collection of non-overlapping local alignments
-Avoid trivial variations of the optimal.
-3-finish when the last interesting local extracted
Cédric Notredame (21/04/23)
Adding Affine Gap Penalties
The Gotoh Algorithm
Cédric Notredame (21/04/23)
Forcing a bit of Biology into your alignment
The Gotoh Formulation
Cédric Notredame (21/04/23)
Why Affine gap Penalties are Biologically better
Cost
L
Afine Gap Penalty
GOP
GEP
GOP GOP
GOP
Parsimony: Evolution takes the simplest path
(So We Think…)
Cost=gop+L*gep
Or Cost=gop+(L-1)*gep
Cédric Notredame (21/04/23)
But Harder To compute…
More Than 3 Ways to extend an Alignment
X-XXXXXX
X-
XX
-X
Deletion
Alignment
Insertion
??+
Opening
Extension
Opening
Extension
Cédric Notredame (21/04/23)
More Questions Need to be asked
For instance, what is the cost of an insertion ?
1…I-1 ??X1…J-1 ??X
1…I ??- 1…J ??X
1…I ??-1…J-1 ??X
GOP GEP
Cédric Notredame (21/04/23)
Solution:Maintain 3 Tables
Ix: Table that contains the score of every optimal alignment 1…i vs 1…j that
finishes with an Insertion in sequence X.
Iy: Table that contains the score of every optimal alignment 1…I vs 1…J that
finishes with an Insertion in sequence Y.
M: Table that contains the score of every optimal alignment 1…I vs 1…J that
finishes with an alignment between sequence X and Y
Cédric Notredame (21/04/23)
The Algorithm
M(i,j)= best M(i-1,j-1) + Mat(i,j) X
X1…i-11…j-1 +Ix(i-1,j-1) + Mat(i,j)
Iy(i-1,j-1) + Mat(i,j)
X-
1…i-1 X1…j X
+
Ix(i,j)= best M(i-1,j) + gop
Ix(i-1,j) + gepX-
1…i-1 X1…j -
+
-X
1…i X1…j-1 X
+
Iy(i,j)= best M(i,j-1) + gop
Iy(i,j-1) + gep-X
1…i -1…j-1 X
+
Cédric Notredame (21/04/23)
Trace-back?
MIx Iy
Start From BEST M(i,j)Ix(i,j)Iy(i,j)
Cédric Notredame (21/04/23)
Trace-back?
M Iy
Navigate from one table to the next, knowing that a gap always finishes with an aligned column…
Ix
Cédric Notredame (21/04/23)
Going Further ?
With the affine gap penalties, we have increased the number of possibilities when building our alignment.
CS talk of states and represent this as a Finite State Automaton (FSA are HMM cousins)
Cédric Notredame (21/04/23)
Going Further ?
Cédric Notredame (21/04/23)
Going Further ?
In Theory, there is no Limit on the number of states one may consider when doing such a computation.
Cédric Notredame (21/04/23)
Cédric Notredame (21/04/23)
Going Further ?
Imagine a pairwise alignment algorithm where the gap penalty depends on the length of the gap.
Can you simplify it realistically so that it can be efficiently implemented?
Cédric Notredame (21/04/23)
Ly
Lx
Cédric Notredame (21/04/23)
A divide and Conquer Strategy
The Myers and Miller Strategy
Cédric Notredame (21/04/23)
Remember Not To Run Out of Memory
The Myers and Miller Strategy
Cédric Notredame (21/04/23)
A Score in Linear Space
You never Need More Than The Previous Row To Compute the optimal score
Cédric Notredame (21/04/23)
A Score in Linear Space
For I For J
R2[i][j]=best
For J, R1[j]=R2[j]
R1R2 R2[j-1],
+gep
R1[j-1]+mat
R1[j]+gep
Cédric Notredame (21/04/23)
A Score in Linear Space
Cédric Notredame (21/04/23)
A Score in Linear Space
You never Need More Than The Previous Row To Compute the optimal score
You only need the matrix for the Trace-Back,
Or do you ????
Cédric Notredame (21/04/23)
An Alignment in Linear Space
Forward Algorithm
F(i,j)=Optimal score of0…i Vs 0…j
Backward algorithm
B(i,j)=Optimal score ofM…i Vs N…j
B(i,j)+F(i,j)=Optimal score of the alignment that passes through pair i,j
Cédric Notredame (21/04/23)
An Alignment in Linear Space
Backward algorithm
Forward Algorithm
Optimal B(i,j)+F(i,j)
Backward algorithm
Forward Algorithm
Cédric Notredame (21/04/23)
Cédric Notredame (21/04/23)
An Alignment in Linear Space
Backward algorithm
Forward Algorithm
Recursive divide and conquer strategy: Myers and Miller (Durbin p35)
Cédric Notredame (21/04/23)
An Alignment in Linear Space
Cédric Notredame (21/04/23)
A Forward-only Strategy(Durbin, p35)
Forward Algorithm
-Keep Row M in memory
-Keep track of which Cell in RowM lead to the optimal score
-Divide on this cell
M
Cédric Notredame (21/04/23)
M
M
Cédric Notredame (21/04/23)
An interesting application: finding sub-optimal alignments
Backward algorithm
Forward Algorithm
Backward algorithm
Forward Algorithm
Sum over the Forw/Bward and identify the score of the best aln going through cell i,j
Cédric Notredame (21/04/23)
Application:Non-local models
Double Dynamic Programming
Cédric Notredame (21/04/23)
Outline
The main limitation of DP: Context independent measure
Cédric Notredame (21/04/23)
11
9
1213
8
1314
5
Double Dynamic Programming
High Level Smith and WatermanDynamic Programming
Score=MaxS(i-1, j-1)+RMSd scoreS(i, j-1)+gpS(i, j-1)+gp{
Rigid Body Superposition where i and j are forced together
RMSd Score
Cédric Notredame (21/04/23)
Double Dynamic Programming
Cédric Notredame (21/04/23)
Application:Repeats
The Durbin Algorithm
Cédric Notredame (21/04/23)
Cédric Notredame (21/04/23)
In The End:Wraping it Up
Cédric Notredame (21/04/23)
Dynamic Programming
Needleman and Wunsch: Delivers the best scoring global alignment
Smith and Waterman: NW with an extra state 0
Affine Gap Penalties: Making DP more realistic
Cédric Notredame (21/04/23)
Dynamic Programming
Linear space: Using Divide and Conquer Strategies Not to run out of memory
Double Dynamic Programming, repeat extraction: DP can easily be adapted to a special need