The purpose of global alignment (aka optimal matching algorithm) is to align two sequences from start to end, and make as many matches as possible. Algorithms do this by inserting gaps within the letters of each sequence to maximize the number of matches. This technique is often used when comparing sequences of similar length.

The algorithm for global alignment was first published by Saul Needleman and Christian Wunsch (1970), then modified over the next few years by other scientists. It aligns sequences from beginning to end and finds the best alignment that maximizes the overall score. The score is calculated based on matches, mismatches, and gaps.

The algorithm is based on an algorithmic technique known as dynamic programming, where the optimal alignment is reduced into a series of smaller problem. The solution to these smaller problems are then used to calculate the solution to the larger (or overall) problem.

The best way to go through the algorithm is to show an example, so let's compare two sequences - IDENTITY and SIMILARITY.

The first step is to set up a 2d matrix with the first sequence on the y-axis, and the second sequence on the x-axis.

Now we need to fill this grid with numerical values, which we will later use to score the matches. Let's come up with a formula to find the score per cell, which we can do with pre-determined values.

- In the first option, if a match is present (cell will be colored green),
`s(x`will be given a score of +1. If it is a mismatch, then the score will be -2._{i}, y_{i}) - In the second option, a gap will be inserted in the first sequence. Thus, we take the cell left of the current cell and -2 from it.
- In the second option, a gap will be inserted in the second sequence. We take the cell on top of the current cell and -2 from it.

The maximum of these values will be taken and placed as the current cell's score.

Start with filling a 0 at the top-left corner. Each cell along the row will be -2 since the only value it looks at is the cell to the left. Each cell along the first column will be -2 from its top, since that's the only value it has to look.

After coloring each matching letter green, you will have a table that looks like this:

We can see that for green cells (match), we take the maximum value of the left, top-left (diagonal), or top, and +1 to it. For any mismatches or gap insertions, we -2 from the largest value of the left, top-left (diagonal) and top cells.

A completed matrix gives the final score of the alignment on the bottom-right most cell. In this case, it's -7.

Before we find trace the optimal path, let's go over some conventions.

Our purpose is to find the optimal path from the top left going to the bottom right. A diagonal line denotes that the letter is aligned (can be match or mismatch). A perfect match will have a diagonal going from top left to bottom right. Mismatches can still generate this diagonal, but their scores will be different.

Horizontal and vertical lines denote the presence of a gap in the first and second sequence, respectively. Gaps may be of any length, and may occur within (internal) or at the ends (terminal) of sequences.

For our example, we start from the bottom-right cell, then move our way up to the top-left.

With this, we get the alignment:

There are other alignments that give the same score. Can you trace them out?

Depending on what kind of alignment you're looking for, you can vary the parameters. For example, some sequence alignment algorithms vary gap penalties in two subcategories. An opening gap would be given a more severe penalty than each continuing gap. In our example case, all gaps were treated equally with a penalty of -2.

Adding a harsher mismatch penalty can force more gaps to be created. However, this is non-ideal as it's good to have near-identical sequences with few deletions.

Additionally, global alignments usually use an end gap penalty. This is the distinguishing factor between global and local alignment, which we'll see next.

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