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Lecture 6, part 2 of 2
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3. Constructing  multiple alignments

a. Dynamic Programming

An optimal multiple alignment depends on all possible pairwise comparisons of all amino acids in each protein at each position.  It is possible to extend the dynamic programming algorithm for pairwise comparisons from the special case of k=2 sequences to the more general case of k=n sequences.

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


 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 []

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(nk). 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  2k-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 truely exhaustive multiple alignment, using the most straightforward approach, is O(k22knk).

Figure from Zucker []

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

It is often the case that good approximations to an otherwise intractable problem can be found by heuristic methods. Heuristic methods take a "learn as you go" strategy, in which the algorithm solves small parts of the problem at a time, and gradually converges on a solution.

Heuristic methods are not guaranteed to find the optimal solution, but can sometimes be shown to produce a solution that is close to optimal. Heuristics are often very sensitive to the starting conditions, which are often arbitrary. That is, different starting conditions may yeild different solutions.

1. CLUSTAL, TCOFFEE - global alignment using Neighbor-Joining Trees

(Note: TCOFFEE should be considered the successor to the CLUSTAL family of programs. It is generally advisable to use TCOFFEE, rather than the older CLUSTAL.)

Hierarchical clustering methods work by aligning small sets of closely-related sequences into sub-alignments, and then aligning the alignments, until all sequences are represented in an alignment.

Hierarchical methods work on the hypothesis that sequences in an alignment will reflect their evolutionary history. That is, if one were to go from one sequence to the next most-closely related one, one would visit all nodes on the phylogenetic tree describing how these sequences evolved.

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

Calculate distances between all possible pairs of sequences
Construct a Neighbor-Joining tree from pairwise distances
while not all nodes on the tree have been visited
align each pair of sequences or profiles at the terminal nodes
replace aligned sequences with a profile representing the alignment
     of all sequences in below that node

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

 Remember: Once a gap always a gap.

WARNING - The guide trees produced by programs such as clustal in step A are not legitimate phylogenetic trees! They should never be used for any purpose other than creating the alignment.

Reason: A good phylogeny is calculated from all of the information in the multiple alignment. Because of the "once a gap, always a gap" rule, pairwise distances calculated from the final alignment may contain gaps not present in the original pairwise alignments. Thus, the original pairwise alignments would underestimate the distances across the whole tree.

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