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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 500

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

In summary, the time
required for a truely exhaustive multiple alignment, using the
most straightforward approach, is **O(k ^{2}2^{k}n^{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(n^{x }) steps, where x is
proportional to the size of the dataset. So, in the
equation above, the real culprits are the exponential
terms 2and^{k}
n.^{k} |

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.

Algorithm: 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 http://journal-ci.csse.monash.edu.au/ci/vol04/mulali/mulali.html.*

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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