Scoring function: Sum of pairs (SP)

Column score:

$$S(A_i) = \sum_{j=1}^N\sum_{k=j+1}^N s(a_i^{(j)},a_i^{(k)})$$

     A-CTCAT
     A-GTC-T
     ACGTC-T

Column 3 score:

$s(C,G) + s(C,G) + s(G,G) = \text{match}+2\times\text{mismatch}$

Column 6 score:

$s(A,-) + s(A,-) + s(-,-) = -2\times\text{gap} + s(-,-) = -2\times\text{gap}$

Problems with SP scoring of MSAs

• Substitution scores derived as log-odds scores for pairwise comparisons.
• The "right thing" would be to construct log-odds scores for triples, quadruples, etc. (depending on the number of rows in the alignment). E.g. \begin{align*} s(a,b,c) & = \log\frac{p_{abc}}{f_af_bf_c}\\ & \neq \log\frac{p_{ab}}{f_af_b} + \log\frac{p_{ac}}{f_af_c} + \log\frac{p_{bc}}{f_bf_c} \end{align*}
• No probabilistic justification!

Another way to think about the problem

Tree-based scores

Thought to be the most biologically appropriate, but

• We don't know the tree.
• We need to infer the characters on internal nodes to identify most parsimonious history.
• There may be different trees for different parts of the alignment (recombination!)

Sum of pairs is almost always used in practice.

Naïve approach

Define $F(i_1,i_2,\ldots,i_N)$ to be the score of the best alignment up to the subsequences ending in $x^{(1)}_{i_1}, x^{(2)}_{i_2}, \ldots, x^{(N)}_{i_N}$.

We can then find the following recurrence relation:

Naïve approach

Computational burden of naïve DP approach

• $F$ consists of $\prod_{q=1}^{N}n_q$ terms:
  • $O(n^N)$ space complexity.
• Computing each element of $F$ requires maximizing over $2^N-1$ possibilities:
  • $O(2^Nn^N)$ time complexity.
  • (traceback negligible cost in comparison)

Even aside from the time cost, the space requirement for this algorithm is prohibitive. Storing $F$ for an alignment of 5$\times$100 character sequences requires $101^5\simeq 10^{10}$ numbers. Assuming 32 bit integers, this equates to $\sim$39 Gb of memory!

Improved DP algorithm

• Carrillo & Lipman (1988) presented an algorithm implemented by the program MSA (Lipman, Altschul & Kececioglu, 1989) with improved scaling properties.
• Implements constraints on pairwise alignment scores that can be part of the optimal multiple sequence alignment.

A big improvement, but still impractical for most alignment problems.

Multiple Alignment Software

Really need approximation methods.

Different techniques:

1. Progressive global alignment of sequence starting with an alignment of the most similar sequences and then building a full alignment by adding more sequences.
2. Iterative methods that make an initial alignment of groups of sequences and then iteratively refine the alignment to achieve a better result. (Barton-Sternberg, Simulated annealing, stochastic hill climbing, genetic algorithms.)
3. Use of probabilistic models of the indel and substitution process to do statistical inference of alignment. ("Statistical alignment.")

General idea

Decisions:

1. Order of alignments.
2. Alignment of sequence to group only, or allow group to group.
3. Method of alignment, scoring function.

Guide trees

• Progressive alignment algorithms employ "guide trees" to guide the order in which sequences are progressively aligned.
• In practice these are very rough phylogenetic trees (cheap to compute), unsuitable for serious phylogenetic inference but used instead simply to bootstrap the alignment process.

The UPGMA tree-building algorithm

• Clustering algorithm based on pairwise distances.
• Requires some distance $d(k,l)$ for every pair of items.

Distance matrix:

\begin{equation*} d = \begin{array}{cccc} & A & B & C & D \\ A & - & 4 & 8 & 8 \\ B & & - & 8 & 8 \\ C & & & - & 6 \\ D & & & & - \end{array} \end{equation*}

Distance matrix:

\begin{equation*} d = \begin{array}{cccc} & A & B & C & D \\ A & - & 4 & 8 & 8 \\ B & & - & 8 & 8 \\ C & & & - & 6 \\ D & & & & - \end{array} \end{equation*}

Distance matrix:

\begin{equation*} d = \begin{array}{cccc} & E & C & D \\ E & - & 8 & 8 \\ C & & - & 6 \\ D & & & - \end{array} \end{equation*}

Distance matrix:

\begin{equation*} d = \begin{array}{cccc} & E & C & D \\ E & - & 8 & 8 \\ C & & - & 6 \\ D & & & - \end{array} \end{equation*}

Distance matrix:

\begin{equation*} d = \begin{array}{cccc} & E & F \\ E & - & 8 \\ F & & - \end{array} \end{equation*}

Distance matrix:

\begin{equation*} d = \begin{array}{cccc} & E & F \\ E & - & 8 \\ F & & - \end{array} \end{equation*}

Feng-Doolittle: Overview

Progressive alignment algorithm published in 1987 (Feng and Doolittle, J. Mol Evol).

1. Calculate diagonal matrix of $\binom{N}{2}$ distances between all pairs of $N$ sequences by standard pairwise alignment, converting raw alignment scores to approximate pairwise "distances": \[ d(k,l) = -\log\frac{S(A^{(k)},A^{(l)})-S_{\text{rand}}}{S_{\text{max}}-S_{\text{rand}}} \]
2. Construct guide tree using UPGMA or some other clustering algorithm.
3. Starting from the first internal node added to the tree, align the child nodes (which may be two sequences, a sequence and an alignment, or two alignments). Repeat for all other nodes in the order that they were added to the tree, until all sequences have been aligned.

Feng-Doolittle: Sequence$\leftrightarrow$Group alignment

Feng-Doolittle: Group$\leftrightarrow$Group alignment

Feng-Doolittle

• After alignment is completed, gap symbols replaced by neutral "X" character.
• Feng and Doolittle call this the rule of "once a gap, always a gap".
• Encourages gaps to occur in same columns in subsequent alignments.

Progressive mis-alignment

Progressive mis-alignment

Profile alignment

CLUSTALW

• Thompson, Higgins, Gibson (1994) (55k citations!)
• Widely used implementation of profile-based progressive multiple alignment
• Essentially Feng-Doolittle, but uses profile alignment.

Overview:

1. Calculate matrix of $\binom{N}{2}$ distances between all pairs of $N$ sequences.
2. Construct a guide tree using a clustering algorithm such as UPGMA or neighbour-joining.
3. Progressively align at internal nodes in order of decreasing similarity using sequence-sequence, sequence-profile and profile-profile alignment.

Employs many other heuristics.