Mathematics and the Genome: Near and Far (Strings)
5. Near and Far (Strings)
One dramatic way that mathematics has come in handy for the study of the genome involves the concept of distance. What is the distance between New York and Boston? is a very familiar use of the word distance. Yet the word distance is very rich in linguistic connotations. We speak of distant relatives and distant memories. Distance suggests not only the ability to tell how close or far apart things are but also issues of similarity and dissimilarity. It is because of the varied contexts in which distance arises that a mathematical framework has been helpful. Given a collection of objects (cities, bird songs, genes, languages, manuscripts), one can assign a number to pairs of these objects which tell one how distant or how similar or how close they are. What properties do we want the number associated with the pair of objects to have?
However, the context here illustrates an important point: often there is more than one way to define the distance between the same objects. Here, in getting between two locations in an urban setting, we can not get between the opera house and the commuter rail terminal as the crow flies. We will probably want to take a taxi. The taxicab distance from O to C will not be 5 but 7, because the taxi might take the route going from O to B (3 units) and then from B to C (4 units). In general, taxicab distance between two points (x1, x2) and (y1, y2) is given by:
This illustrates that often one gets a new and different insight into the same objects by changing perspective on them.
would be 6 because the strings, when lined up one on top of the other differ in six columns. This distance, which is a distance because it obeys the 4 rules described earlier, is today called Hamming Distance in honor of Richard Hamming.
The operations we will allow to transform one string into another, which also permits the addition of dashes as noted above, are:
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