7/22/2006

Sequence comparison

(Bioinformatics)

OK, so I got word through from the guy in charge of the computer course about my little ethical crisis. He suggested that, instead of posting actual code, I instead merely include pseudocode for the bits covered by the course. Now why the heck didn't I think of that?

On with the show. A major part of bioinformatics involves the comparison of different genetic sequences. Say you have a working gene for vitamin C production, plus something that you think might be a broken copy of same. How do you figure out whether they are actually related?

The basic technique is to figure out how many mutations (insertions, deletions, substitutions) it takes to get from one to the other. There are many many variants on this basic principle - tailor-made modifications designed to account for a variety of changeable factors such as:

1) are certain substitutions more common than others?
2) how likely are insertions and deletions of various sizes? (coughZipf distributioncough)
3) do certain subsections match better than others?

For this post, I'll only cover the most simple case: all three types of mutation are equally likely, and indels (insertions and deletions) are always of length 1.

Cutting it down to size

OK, so we have a couple of strings (referred to as S and T) to align - we'll think of this as finding a way to mutate S so it ends up looking like T. There's no immediately-obvious way of solving this, so let's try a standard approach: see if we can reduce it to a simpler problem.

I'm going to look at a few cases and show how they reduce to various easier problems, with examples of situations where each reductive approach can be ideal. Hope people can follow! I will be using the Python slice notation extensively, so you may want to familiarise yourself with that first.

Matching

OK, so say our two strings are "CAAATAGCAA" and "CCGCGAGCAA". There's something you should notice here: we don't actually need to do anything with the last 5 digits! We can just say "hey, they match", and stop worrying about them. The problem then reduces itself to trying to align "CAAAT" with "CCGCG". Much simpler.

Formally: if S[-1] and T[-1] are identical, they may be replaced by S[:-1] and T[:-1] at no cost.

Substitution

Let's try a different set of strings. Say we have "AAATCCAGCA" and "AAATCCAGCT". Now, if you look at these, you'll see they differ by only the last element. Time for a substitution! If we transform the first string by substituting a T for the A, we'll then just be able to match the rest. Home and dry! This does "cost" us one mutation, but in this case that's a small price to pay.

Formally: S and T may be replaced by S[:-1] and T[:-1] at a cost of 1 mutation

Deletion

Our next two example strings are "GGGTCATCTTA" and "GGGTCATCTT". Now, it should be fairly obvious what you need to do to the first string to get it to match the second: just delete the "A".

Formally: S may be replaced with S[:-1] at a cost of 1 mutation

Insertion

Finally, consider the strings "CCGGAGACCT" and "CCGGAGACCTG". Our instinctive response here is to stick a G on the end of the first string. However, that presents us with a problem: if we're going to start adding stuff to strings, we're going to end up with longer strings. That ruins the whole idea of trying to simplify the problem.

There's a simple workaround - we just need to combine two operations. If we add a "G" and then immediately match the two "G"s away, we end up with the same result as if we'd simply dropped the "G" off the end of the second string. That's a lot more tractable, and incidentally is pleasingly symmetrical with the aforementioned operation of deletion.

Formally: T may be replaced by T[:-1] at a cost of 1 mutation

These are the only options for this model!

Shake it all about

OK, so let's combine these four approaches into a single algorithm. What we're going to do is take our pair of strings and apply each of the four abovementioned alterations to them (using matching or substitution as applicable). This will provide us with three pairs of strings, at least one of which will be shorter, which we can then apply the same method to. Eventually both strings will sink to size zero in every single case. We'll then select the sequence of operations that gets us to this point via the fewest expensive mutations.

We can write this more formally. Refer to the "distance" (number of mutations) between S and T as D(S,T). Then we have the following:

D(S,T) = min{
D(S[:-1],T[:-1]) + s(S[-1],T[-1])
D(S[:-1],T) + 1
D(S,T[:-1]) + 1
}

where s(A,B) = 1 if A and B are different letters and 0 if A and B are the same (this represents the cost of either substituting B for A or matching them)

If we implement this as a function that takes two strings and returns the distance between them, we can simply design it so that it recurses through all possible substrings. Puzzle solved!

A problem

We have, however, missed something. What happens if, say, we're comparing "ATGGCTA" with ""? The string T[:-1] will now no longer exist, and the program will crash horribly when it tries to evaluate D(S,T[:-1]). Doh.

Fortunately, there's an easy solution. Since the only way of getting from "ATGGCTA" to "" is by deleting all the letters, we can just say that this has a cost of 7 mutations. A similar approach applies if the first string is empty. We'll need to write this into the code:

if S=="": D(S,T) = len(T)

elif T=="": D(S,T) = len(S)

else:
D(S,T) = min{
D(S[:-1],T[:-1]) + s(S[-1],T[-1])
D(S[:-1],T) + 1
D(S,T[:-1]) + 1
}

A computational note

If you decide to implement this alignment algorithm for yourself - and I strongly recommend you try it - you'll quickly notice a major problem with the standard recursive implementation of the algorithm. In short, it's damn slow for long strings. More precisely, since each distance calculation spawns three other distance calculations, the time required to run the thing will be somewhere on the order of 3 to the power of the length of the strings[1]. This sucks.

Fortunately, there's a much nicer approach, albeit one that requires a modicum more memory (try saying that three time fast...). The problem with the recursion is that it's wasting time by repeatedly calculating the same distance. Why not simply precalculate these values, and refer to them as necessary? One way of implementing this is by creating a grid such that element (i,j) of the grid is equal to D(S[:i],T[:j]). This can be populated by filling in the sides adjacent to D("","") and then working outwards from that corner.

A rather fuzzy mathematical note

It turns out, if I understand the book I'm currently reading correctly, that the problem solved here is part of a wider class of mathematical problems, which also includes things like determining RNA secondary structure. In particular, they're ones that in some rather abstract sense involve the concept of context-free grammars, and it turns out that you can solve them all in a similar fashion.

At present, I have sod-all idea how this works. However, as an incurable maths student, I'm very interested in exploring the theoretical underpinnings involved, and will probably rant on about this in some detail at a later date. Consider yourselves warned :)

The next step

In this post I've provided a fairly solid discussion of the basic alignment algorithm. The next post on this topic will be fairly brief, and will cover a phenomenon known as BLOSUM matrices that are used to fine-tune the alignment of two protein (not DNA) sequences.

[1] At some point it will actually be beneficial to have a detailed discussion of calculations of this sort. They come up in the real world a heck of a lot - for example, in cryptography, mathematicians can spend years of their lives trying to reduce an algorithm's runtime by a comparatively tiny-looking amount. The reason they do this is because that "tiny amount" can often be disproportionately important as the codes get harder to crack.

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