Download Global Sequence Alignment by Dynamic Programming

Global Sequence Alignment
by Dynamic Programming
Needleman-Wunsch Algorithm
• General Algorithm for sequence
comparison
• Maximize a similarity score to give the
maximum match
• Maximum match=largest number of
amino acids or nucleotides of one
sequence that can be matched with
another, allowing for all possible
deletions.
Needleman-Wunsch Algorithm
• Finds the best GLOBAL alignment of
any two sequences.
• N-M involves an iterative matrix method
of calculation
• All possible pairs (nucleotides or amino
acids) are represented in a twodimensional array.
• All possible alignments are represented
by pathways through this array.
Needleman-Wunsch Algorithm
• Sequence alignment methods predate dotmatrix searches, and all of the alignment
methods in use today are related to the
original method of Needleman and Wunsch
(1970).
• Needleman and Wunsch wanted to quantify
the similarity between two sequences.
Needleman-Wunsch Algorithm
• Over the course of evolution, some positions
undergo base or amino acid substitutions, and
bases or amino acids can be inserted or
deleted.
• Any measurement of similarity must
therefore be done with respect to the best
possible alignment between two sequences.
• Because insertion/deletion events are rare
compared to base substitutions, it makes
sense to penalize gaps more heavily than
mismatches when calculating a similarity
score.
Dynamic Programming
• Finding the best alignment of 2
sequences is a hard problem solved by a
computational method called dynamic
programming
• Multiple sequence alignment of 3 or
more sequences can be solved by
dynamic programming and statistical
methods
How Do We Generate the
Correct Alignment ?
• We can't.
• We can never guarantee that a particular
alignment is correct except for the
simplest, unambiguous alignments!
• Such an alignment would require aligning
each sequence in turn to the ancestral
sequence first.
• Since there is no possibility to know the
ancestral sequence and the evolutionary
steps, the evolutionary correctness of any
alignment cannot be determined.
The Optimal Alignment.
• If the optimal alignment does not
support homology, then the correct
alignment (which has a smaller or equal
score) will not support homology either.
• But again: there is no guarantee that
the optimal alignment is the correct
alignment, even though it may be the
best guess.
Dynamic Programming
• Dynamic programming is a term from
operations research, where it was first
used to describe a class of algorithms
for the optimization of dynamic
systems.
Dynamic Programming
• In dynamic programming the principle of
divide-and-conquer is used extensively:
subdivide a problem that is to large to be
computed, into smaller problems that may be
efficiently computed.
• Then assemble the answers to give a solution
for the large problem.
• When you do not know which smaller problem
to solve, simply solve all smaller problems,
store the answers and assembled them later
to a solution for the large problem.
Dynamic Programming
• Global optimal alignment is a difficult
problem.
• The major difficulty comes from the fact,
that one cannot simply slide one sequence
along another and sum over the similarity
scores looked up in the appropriate mutation
data matrix.
• This will not work, because biological
sequences may have gaps or insertions of
sequences relative to each other.
Three steps in Dynamic
Programming
1. Initialization
2 Matrix fill or scoring
3. Traceback and alignment
Sample Matrix
• To align with a cell in the diagonal
means an alignment in the next
position.
• An increasing diagonal line means a
stretch of sequence identity.
• To align with an off-diagonal cell
requires the insertion of a
corresponding number of gaps.
Needleman-Wunsch Algorithm
• We can compute for every cell the
highest possible score that can be
obtained for a path originating from
that cell.
• That is done by looking in all the
elements permissible for extending the
path, and adding the highest value found
to the contents of the cell.
Needleman-Wunsch Algorithm
• If this is done in an orderly way, the
highest score found is the global
maximum alignment score.
• Then the optimal path consists of all
those cells that contributed to the
global maximum alignment score.
• There can be several equivalent optimal
paths.
Sample Matrix
• Create a matrix with M + 1 columns and
N + 1 rows where M and N correspond
to the size of the sequences to be
aligned.
• Since this example assumes there is no
gap opening or gap extension penalty,
the first row and first column of the
matrix can be initially filled with 0.
Matrix Fill Step
• One possible solution of the matrix fill step
finds the maximum global alignment score by
starting in the upper left hand corner in the
matrix and finding the maximal score Mi,j for
each position in the matrix.
• In order to find Mi,j for any i,j it is minimal
to know the score for the matrix positions to
the left, above and diagonal to i, j.
• In terms of matrix positions, it is necessary
to know Mi-1,j, Mi,j-1 and Mi-1, j-1.
Matrix Fill Step
• In the example, Mi-1,j-1 will be red,
Mi,j-1 will be green and Mi-1,j will be
blue.
• Since the gap penalty (w) is 0, the rest
of row 1 and column 1 can be filled in
with the value 1.
Matrix Fill Step
• Now let's look at column 2. The location
at row 2 will be assigned the value of
the maximum of 1(mismatch),
1(horizontal gap) or 1 (vertical gap). So
its value is 1.
• At the position column 2 row 3, there is
an A in both sequences. Thus, its value
will be the maximum of 2(match), 1
(horizontal gap), 1 (vertical gap) so its
value is 2.
Traceback Step
• After the matrix fill step, the maximum
alignment score for the two test sequences is
6.
• The traceback step determines the actual
alignment(s) that result in the maximum score.
• With a simple scoring algorithm like this one,
there are likely to be multiple maximal
alignments.
• The traceback step begins in the M,J position
in the matrix- the position that leads to the
maximal score.
• In this case, there is a 6 in that location.
Traceback Step
• Traceback takes the current cell and looks to
the neighbor cells that could be direct
predecessors.
• This means it looks to the neighbor to the left
(gap in sequence #2), the diagonal neighbor
(match/mismatch), and the neighbor above it
(gap in sequence #1).
• The algorithm for traceback chooses as the
next cell in the sequence one of the possible
predecessors. The neighbors are marked in red
and are also equal to 5.
Traceback Step
• Since the current cell has a value of 6
and the scores are 1 for a match and 0
for anything else, the only possible
predecessor is the diagonal
match/mismatch neighbor.
• If more than one possible predecessor
exists, any can be chosen.
Traceback Step
• This gives us a current alignment of
(Seq #1)
A
|
(Seq #2)
A
• So now we look at the current cell and
determine which cell is its direct
predecessor.
• In this case, it is the cell with the red
5.
• This is One Traceback Giving an
alignment of :
G A A T T C A G T T A
|
|
| |
|
|
G G A _ T C _ G _ _ A
• This is an alternative Traceback giving
an alignment of :
G _ A A T T C A G T T A
|
|
| |
|
|
G G _ A _ T C _ G _ _ A
What Is the Best Alignment
Between These Two Amino
Acid Sequences?
• A zinc-finger core sequence:
CKHVFCRVCI
• A sequence fragment from a viral
protein:
CKKCFCKCV
Practice Matrix
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------|
|
|
|
|
|
|
|
|
Place a 1 Where There Is a
Match in the Matrix
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
1
1
|
1
|
1
| 1
1
1
|
1
| 1
1
1
|
1
| 1
1
1
|
1
1
Place Zeroes on the Ends
Since They Can’t Align
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
1
1 0
|
1
0
|
1
0
| 1
1
1 0
|
1
0
| 1
1
1 0
|
1
0
| 1
1
1 0
| 0 0 0 1 0 0 0 1 0 0
Loading the Matrix
• Then proceed to the next row and
column, adding to each matrix cell the
maximal value of any other cell that
could be the next step on a path to the
matrix.
• For instance the value 1 at (C6,C8) now
becomes a 2, since it could be extended
from the 1 at (V8,V9) through a 1.
Filling the Matrix
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
1
1 0
|
1
0 0
|
1
0 0
| 1
1
1 0
|
1
0 0
| 1
1
1 0
|
1
0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
Repeat This Procedure for the
Next Row and Column:
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
1
1 1 0
|
1
1 0 0
|
1
1 0 0
| 1
1
1 1 0
|
1
1 0 0
| 1
1
1 1 0
| 2 3 2 2 2 1 1 1 0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
And the Next …
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
1 1 1 1 0
|
1
1 1 0 0
|
1
1 1 0 0
| 1
1 1 1 1 0
|
1
1 1 0 0
| 4 2 2 2 2 2 1 1 1 0
| 2 3 2 2 2 1 1 1 0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
And the Next Row…
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
2 1 1 1 0
|
1
1 1 1 0 0
|
1
1 1 1 0 0
| 1
2 1 1 1 0
| 3 2 2 2 3 1 1 1 0 0
| 4 2 2 2 2 2 1 1 1 0
| 2 3 2 2 2 1 1 1 0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
And the Next …
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
2 2 1 1 1 0
|
1
2 1 1 1 0 0
|
1
2 1 1 1 0 0
| 4 3 3 3 2 2 1 1 1 0
| 3 2 2 2 3 1 1 1 0 0
| 4 2 2 2 2 2 1 1 1 0
| 2 3 2 2 2 1 1 1 0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
And the Next …
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
3 2 2 1 1 1 0
|
1
3 2 1 1 1 0 0
| 3 4 3 3 2 1 1 1 0 0
| 4 3 3 3 2 2 1 1 1 0
| 3 2 2 2 3 1 1 1 0 0
| 4 2 2 2 2 2 1 1 1 0
| 2 3 2 2 2 1 1 1 0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
And the Next …
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 1
3 3 2 2 1 1 1 0
| 4 4 3 3 2 1 1 1 0 0
| 3 4 3 3 2 1 1 1 0 0
| 4 3 3 3 2 2 1 1 1 0
| 3 2 2 2 3 1 1 1 0 0
| 4 2 2 2 2 2 1 1 1 0
| 2 3 2 2 2 1 1 1 0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
Another Path Marked
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 5 3 3 3 2 2 1 1 1 0
| 4 4 3 3 2 1 1 1 0 0
| 3 4 3 3 2 1 1 1 0 0
| 4 3 3 3 2 2 1 1 1 0
| 3 2 2 2 3 1 1 1 0 0
| 4 2 2 2 2 2 1 1 1 0
| 2 3 2 2 2 1 1 1 0 0
| 2 1 1 1 1 2 1 0 1 0
| 0 0 0 1 0 0 0 1 0 0
Filling the Matrix
• The globally optimal score is the highest
score in the first row or column - for the Nterminus of either protein sequence.
• In our example it is 5. Therefore, the globally
optimal alignment will have 5 matches. Let us
now back trace our path ending at the cell
with value 5 to see how we arrived at the
value.
• Strip away all the cells that could not have
contributed to the sum.
There are several equally good
paths to a score of 5
C
K
K
C
F
C
K
C
V
C K H V F C R V C I
+-------------------| 5
|
4
|
4 3 3
|
3 3
|
3
|
2
|
2 1 1
|
2 1
1
|
1
A Number of Different
Alignments Are Possible:
•C
C
•C
C
•C
C
•C
C
K
K
K
K
K
K
K
H
K
H
K
K
K
H
K
V
C
V
C
H
C
C
F
F
F
F
V
V
-
C
C
C
C
F
F
F
F
R
R
K
C
C
C
C
V
K
V
R
R
-
C
C
C
C
V
K
V
K
I
V
I
V
C
C
C
C
I
V
I
V
Advanced Dynamic
Programming
• An advanced scoring scheme is assumed
where
– Si,j = 2 if the residue at position i of
sequence #1 is the same as the residue at
position j of sequence #2 (match score);
otherwise
– Si,j = -1 (mismatch score)
– w = -2 (gap penalty)
Adding Gaps and Mismatches
• The second A-G pair is a mismatch.
G A A T T C A G T T A
|
| |
|
|
|
G G A T _ C _ G _ _ A
• The fifth T has no pair so this is a gap.
• Or it could be an insertion or gain of
sequence.
Creating the Matrix
• The first step in the global alignment
dynamic programming approach is to
create a matrix with M + 1 columns and
N + 1 rows where M and N correspond
to the size of the sequences to be
aligned.
• The first row and first column of the
matrix can be initially filled with 0.
Creating the Matrix
• For each position, Mi,j is defined to be the
maximum score at position i,j; i.e.
• Mi,j = MAXIMUM[
•
Mi-1, j-1 + Si,j (match/mismatch in the
diagonal),
•
Mi,j-1 + w (gap in sequence #1),
•
Mi-1,j + w (gap in sequence #2)]
• Note that in the example, Mi-1,j-1 will be red,
Mi,j-1 will be green and Mi-1,j will be blue
• Moving down the first column to row 2, we can
see that there is once again a match in both
sequences. Thus, S1,2 = 2. So M1,2 =
MAX[M0,1 + 2, M1,1 - 2, M0,2 -2] = MAX[0 +
2, 2 - 2, 0 - 2] = MAX[2, 0, -2].
• A value of 2 is then placed in position 1,2 of
the scoring matrix and an arrow is placed to
point back to M[0,1] which led to the
maximum score.
Filling the Matrix
• Looking at column 1 row 3, there is not a
match in the sequences, so S 1,3 = -1.
M1,3 = MAX[M0,2 - 1, M1,2 - 2, M0,3 2] = MAX[0 - 1, 2 - 2, 0 - 2] = MAX[-1,
0, -2].
• A value of 0 is then placed in position
1,3 of the scoring matrix and an arrow is
placed to point back to M[1,2] which led
to the maximum score.
• Eventually, we get to column 3 row 2.
Since there is not a match in the
sequences at this position, S3,2 = -1.
M3,2 = MAX[ M2,1 - 1, M3,1 - 2, M2,2 2] = MAX[0 - 1, -1 - 2, 1 -2] = MAX[-1, 3, -1].
• So you begin to get loss of scores.
Filling the Matrix and Using
Pointers
• Note that in the above case, there are
two different ways to get the maximum
score. In such a case, pointers are
placed back to all of the cells that can
produce the maximum score.
Traceback
• This gives an alignment of
G A A T T C A G T T A
|
|
| |
|
|
G G A _ T C _ G _ _ A
Traceback
• This gives another alignment of
G A A T T C A G T T A
|
| |
|
|
|
G G A T _ C _ G _ _ A
Gap Penalty
• The gap penalty is used to help decide
whether on not to accept a gap or insertion in
an alignment when it is possible to achieve a
good alignment residue-to-residue at some
other neighboring point in the sequence.
• You need a penalty, because an unreasonable
'gappy' alignment would result.
• Biologically, it should in general be easier for
a protein to accept a different residue in a
position, rather than having parts of the
sequence chopped away or inserted.
Gap Penalty
• Gaps/insertions should therefore be more
rare than point mutations (substitutions).
• Some different possibilities:
– A single gap-open penalty. This will tend to stop
gaps from occurring, but once they have been
introduced, they can grow unhindered.
– A gap penalty proportional to the gap length. This
will work against larger gaps.
– A gap penalty that combines a gap-open value with
a gap-length value.
Substitution Matrix
• A substitution matrix describes the
likelihood that two residue types would
mutate to each other in evolutionary
time.
• This is used to estimate how well two
residues of given types would match if
they were aligned in a sequence
alignment.
Substitution Matrix
• An amino acid substitution matrix is a
symmetrical 20*20 matrix, where each
element contains the score for substituting a
residue of type i with a residue of type j in a
protein, where i and j are one of the 20
amino-acid residue types.
• Same residues should obviously have high
scores, but if we have different residues in a
position, how should that be scored?
Substitution Matrix Scoring
• The same residues in a position give the score
value 1, and different residues give 0.
• The same residues give a score 1, similar
residues (for example: Tyr/Phe, or Ile/Leu)
give 0.5, and all others 0.
• One may calculate, using well established
sequence alignments, the frequencies
(probabilities) that a particular residue in a
position is exchanged for another.
Substitution Matrix Scoring
• This was done originally be Margaret Dayhoff,
and her matrices are called the PAM (Point
Accepted Mutation) matrices, which describe
the exchange frequencies after having
accepted a given number of point mutations
over the sequence.
• Typical values are PAM 120 (120 mutations
per 100 residues in a protein) and PAM 250.
• There are many other substitution matrices:
BLOSUM, Gonnet, etc.
Scoring Similarity
1) Can only score aligned sequences
2) DNA is usually scored as identical or not
3) Modified scoring for gaps - single vs.
multiple base gaps (gap extension)
4) AAs have varying degrees of similarity
– a. # of mutations to convert one to another
– b. chemical similarity
– c. observed mutation frequencies
5) PAM matrix calculated from observed
mutations in protein families
The PAM 250 Scoring Matrix
GCG Wisconsin Package GAP
• GAP is the implementation of the NeedlemanWunsch algorithm in the GCG program package.
• The NW algorithm will present you with a single
globally optimal alignment, not all possible
optimal alignments - different alignments may
exist that give the same score.
• GAP presents you with one member of the family
of best alignments that align the full length of
one sequence to the full length of a second
sequence.
• There may be many members of this family, but
no other member has a higher score.
GCG Wisconsin Package GAP
• The primary use of a global alignment algorithm is
when you really want the whole of two sequences to
be aligned, without truncation.
• GAP could completely bypass a region of high local
homology, if a better (or even just as good) path can
be found in a different way.
• This is problematic if one short sequence is aligned
against a longer one with internal repeats.
• If there is weak or unknown similarity between two
sequences, a local alignment algorithm (BESTFIT) is
the better choice.
• Use GAP only when you believe the similarity is over
the whole length.
Limitations to NeedlemanWunsch
• The problem with Needleman-Wunsch is
the amount of processor memory
resources it requires.
• Because of this, it is not favored for
practical use, despite the guarantee of
an optimal alignment.
• The other difficulty is that the concept
of global alignment is not used in
pairwise sequence comparison searches.