Prove that L >= G for Local and Global alignments of a specific function

Question

I'm taking a bioinformatics class this semester and I'm having trouble with a specific question from the book.

*Given two DNA sequences, S and T, of the same length n and let the scoring function be defined as follows: match = 1, mismatch = -1, indel(gap) = -2. Suppose that G and L are the scores of an optimal global alignment and an optimal local alignment between S and T, respectively.

Prove that L >= G.

I understand how to find the respective alignments of two random sequences, but I'm having trouble proving this. As far as I can tell this is true. G will never be able to be greater than L because of the indel penalty being so high and the match not being able to make up for it. I also had to generate an example to prove that they can be equal, so I know that is true.

So yeah, any hints on how to go about this would be great.

Answer 1

Well this site isn't supposed to be us doing your homework but it's a simple question so let's have a crack at it:

We'll assume the points you make originally are valid (about the scoring).

Suppose the contrary, that there exists some local alignment which is less than G. If this were true, then it means your best local alignment (meaning you started somewhere away from G's starting or end points) is actually less efficient than your global alignment. But we know this can't be the case because the local alignment is a subset of your global alignment (worst case scenario, your local alignment IS your global alignment).

Therefore we prove that there are no counterexamples so this statement must hold.

Hope that makes sense!