last  page

PLNT4610/PLNT7690 Bioinformatics Lecture 6, part 2 of 2

next page

3. Constructing  multiple alignments

a. Dynamic Programming

Scoring must account for those comparisons. Consider the alignment of k sequences of length n

MQPILLL
MLR-LL-
MK-ILLL
MPPVLIL

 For every position (column) in the alignment, we calculate the sum of scores, based on the amino acid substitution matrix (eg. PAM250) using the function

where i and j refer to all unique pairs of sequences, ignoring self comparisons, and eliminating duplicate comparisons.

The sum-of-pairs (SP) function scores each position in the protein, that is, each column, as the sum of the pairwise scores. For k sequences, there are k(k-1)/2 unique pairwise comparisons, excluding self comparisons.

For example, in column 4, the score would be

SP-score(I,-,I,V) =  p(I,-) + p(I,I) + p(I,V) + p(-,I) + p(-,V) + p(I,V)

where p(a,b) is the pairwise score of two amino acids. This function is valid regardless of the order of sequences in the alignment.

Unlike pairwise alignments, it is legitimate in multiple alignments to have a match between two gap characters. By definition , p(-,-) = 0. Although one might be tempted to score a gap as highly negative, when two sequences match at a gap, it indicates that both sequences have a deletion at that position, that is not shared by other sequences.  Gaps that are conserved between two sequences represent the conservation of a unique evolutionary event. Put another way, if two sequences have a gap at the same position, they haven't changed. In contrast, when two amino acids are drastically different, that indicates a significant mutational event. Matching gaps therefore should get a higher score than, for example, a mismatch between two very different amino acids, such as Proline and Lysine.

Generalizing the pairwise dynamic programming algorithm to k sequences is, in essence, doing an alignment in k dimensions. For example, an optimal alignment between k=3 sequences could be visualized in 3 dimensions:

Figure 2 from Zucker [http://www.genetics.wustl.edu/bio5495/1999-course/lecture.7/]

The problem with the dynamic programming algorithm is size and speed. To start with, where k sequences of length n are compared, the storage required for the alignment matrix is O(n^k). For 20 sequences of 500 amino acids each, it would require 500²⁰ units of storage (eg. bytes).

Recall that in a pairwise  comparison, for each cell in the 2-dimensional array the score was dependent on the 3 preceeding adjacent cells. In a k-dimensional array, there are  2^k-1 preceeding cells. (This makes sense: for k=2 ie. a pairwise comparison, 2²-1 = 3). Finally, for each column, there are k(k-1)/2 pairwise comparisons.

In summary, the time required for a truly exhaustive multiple alignment, using the most straightforward approach, is O(k²2^kn^k).

Figure from Zucker [http://www.genetics.wustl.edu/bio5495/1999-course/lecture.7/]

Although some methods have introduced great efficiencies that bring global dynamic programming to a level that is practical for a handful of sequences, more efficient approximate methods are needed for typical alignment problems. Probably the single most important speedup method for dynamic programming methods [Carillo and  Lipman (1988) SIAM J. Appl. Math.48:1073-1082, Altschul and Lipman (1989) SIAM J. Appl. Math. 49:197-209]. This method only scales to about 8 sequences, so it is not usually practical. Optimal multiple alignments are an example of NP complete problems. NP complete problems are problems whose solutions can not be done in polynomial time. Polynomial time requires O(n c ) steps, where c is a constant. In contrast, an NP complete problem requires O(nx ) steps, where x is proportional to the size of the dataset. So, in the equation above, the real culprits are the exponential terms 2k and nk.

b. Heuristic alignment methods

1. CLUSTAL Omega - global alignment using Neighbor-Joining Trees

As illustrated, the sequences to be aligned are the end nodes, the "leaves" of the tree.

CLUSTAL ALGORITHM

In effect, this algorithm keeps going deeper and deeper into the tree, clustering larger and larger groups of sequences. Clusters (profiles) are merged, until the root of the tree is reached.

 
from G. Fuellen, Multiple Alignment. Complexity International 4, 1997. URL http://journal-ci.csse.monash.edu.au/ci/vol04/mulali/mulali.html.

EXAMPLE: Progressive Alignment The alignment starts with the individual sequences and progresses up the tree until a final alignment is complete.
from https://www.ncbi.nlm.nih.gov/CBBresearch/Przytycka/download/lectures/PCB_Lect05_Multip_Align.pdf

Once a gap always a gap.

• Very fast alignment, can be done for large numbers of sequences
  

• Multiple alignment is strongly influenced by the initial pairwise-alignments
• Error from pairwise alignments propagate through subsequent steps
• Neighbor Joining, used for tree building, is one of the least reliable tree-building algorithms
  

2. MAFFT - Progressive and Iterative refinement of Alignments

FFT-NS-1 Comparable to ClustalW, but distance matrices calculated based on 6-mers shared between each pair of sequences VERY FAST (Use  for >2000 sequences) FFT-NS-2 Guide tree is reconstructed based on the first progressive alignment, and a second progressive alignment is done VERY FAST (Use  for >2000 sequences) Improves upon FFT-NS-1 by recalculating distances and the guide tree and reconstructing the alignment from the improved guide tree

FFT-NS-i (2 cycles) Similar to FFT-NS-2, but distance matrices calculated by WSP at each position for individual amino acid or nucleotide pairs, rather than using the 6-mer shortcut. still FAST, but slower than FFT-NS-2 FFT-NS-i (up to 1000 cycles) Like FFT-NS-i (2cycles), but steps are iterated (i) until no further improvement in the alignment score is seen. time proportional to the number of iterations required More accurate than FFT-NS-2

The distances between any 2 sequences in the guide tree can be used as a weight, which can be multiplied by the SP score for those two sequences. For example, the distance between sequences A and C would be wA,C = wA + wC

We can modify the SP function by multiplying each SP score between two sequences i and j at any position by the weight wij.

MAAFT implements three versions of this iterative alignment process, each optimized for different cases. As with FFT-NS-i, alignments are iterated until no further improvement is seen in the alignment score.

XXXXXXXXXXX-XXXXXXXXXXXXXXX
XX-XXXXXXXXXXXXXXX-XXXXXXXX
XXXXX----XXXXXXXX---XXXXXXX
XXXXX-XXXXXXXXXX----XXXXXXX
XXXXXXXXXXXXXXXX----XXXXXXX

ooooooooooooooooooooooooooooooooXXXXXXXXXXX-XXXXXXXXXXXXXXX------------------
--------------------------------XX-XXXXXXXXXXXXXXX-XXXXXXXXooooooooooo-------
------------------ooooooooooooooXXXXX----XXXXXXXX---XXXXXXXooooooooooo-------
--------ooooooooooooooooooooooooXXXXX-XXXXXXXXXX----XXXXXXXoooooooooooooooooo
--------------------------------XXXXXXXXXXXXXXXX----XXXXXXX------------------

oooooooooXXX------XXXXooooooooooo---------------oooooooXXXXXooooooooooooooooo--oooooooooooooooo
---------XXXXX----XXXXoooooooooooooooooooooooooooooooooXXXXX-ooooooooooooooooooooooo-----------
oooooooo-XXXXX----XXXX---------------------------------XXXXX---oooooooooo--oooooooooooo--------
---------XXXXXX---XXXX---------------------------------XXXXX-----------------------------------
---------XXXXXXXXXXXXX---------------------------------XXXXX-----------------------------------
---------XX-------XXXX---------------------------------XXXXX-----------------------------------

• Most thorough methods, most likely to give the best alignment
  
• MAFFT gives user a choice of several algorithms
• If no algorithm is selected, MAFFT will attempt to choose the most appropriate algorithm, depending on the number of sequences
  

• The more thorough algorithms such as G-NS-i and E-INS-i are only practical for small numbers of sequences
  

last  page

PLNT4610/PLNT7690 Bioinformatics Lecture 6, part 2 of 2

next page