Lecture 10. Sequence alignments

Transcription

1 Lecture 10 Sequence alignments

2 Alignment algorithms: Overview Given a scoring system, we need to have an algorithm for finding an optimal alignment for a pair of sequences. We want to maximize the score (represented by log-odds scores) to find the optimal alignment. The algorithm for finding optimal alignment given an additive alignment score is called dynamic programming. Dynamic programming algorithms are guaranteed to find the optimal scoring alignment or set of alignments.

3 Global alignment The first problem is that of obtaining the optimal global alignment between two sequences, allowing gaps. The dynamic programming algorithm for solving this problem is known as the Needleman-Wunsch algorithm. The idea is to build up an optimal alignment using previous solutions for optimal alignments of smaller subsequences.

4 Needleman-Wunsch algorithm

5 Dynamic programming To set about developing an algorithm based on dynamic programming, one needs a collection of subproblems derived from the original problem that satisfies a few basic properties: 1) There are only a polynomial number of subproblems 2) The solution to the original problem can be easily computed from the solution to the subproblems. 3) There is a natural ordering on subproblems from smallest to largest together with an easy-to-compute recurrence that allows one to determine the solution to a subproblem from the solutions to some number of smaller subproblems.

6 Alignment Suppose we are given two sequences x and y, where x consists of the sequence of symbols x 1 x 2 x m and y consists of the sequence of symbols y 1 y 2 y n. Consider the sets {1,2,, m} and {1,2,, n} as representing the different positions in the sequences x and y, and consider a matching of these sets. A matching is a set of ordered pairs with the property that each item occurs in at most one pair. A matching M of these two sets is an alignment if there are no crossing pairs: if i, j, (i, j ) M and i < i, then j < j

7 Alignment An alignment gives a way of lining up the two sequences, by telling us which pairs of positions will be lined up with one another. For example, stop- -tops corresponds to the alignment { 2,1, 3,2, 4,3 }.

8 Optimal alignment Suppose M is a given alignment between x and y. First, there is a parameter d that defines a gap penalty. For each position of x and y that is not matched in M, we incur a cost of d. Second, for each pair of letters a b in our alphabet, there is a mismatch score of s a, b < 0 for lining up a with b. Thus, for each i, j M, we pay the appropriate mismatch cost s(a, b). One generally assumes that s a, a > 0 for each letter a. The score of M is the sum of its gap penalties, mismatch scores, and match scores. We seek an alignment of maximum score.

9 Optimal alignment The process of maximizing this score is referred to as sequence alignment in the biology literature. The quantities d and s(a, b) are external parameters that must be plugged into software for sequence alignment. The higher the cost, the more similar we declare the sequences to be.

10 Designing the algorithm Let M be any alignment of x and y. Then m, n M or m, n M. If m, n M, there are three possible cases: x m is not matched. y n is not matched. Both x m and y n are matched. But I will show that this case is impossible.

11 Designing the algorithm [Theorem] Let M be any alignment of x and y. If m, n M, then either the m th position (last) of x or the n th position (last) of y is not matched in M. Proof. Suppose by way of contradiction that m, n M, and there are numbers i < m and j < n so that m, j M and i, n M. But this contradicts our definition of alignment: we have i, n, m, j M with i < m, but j < n so the pairs i, n and m, j cross.

12 Designing the algorithm There is an equivalent way to write the theorem that exposes three alternative possibilities, and leads directly to the formulation of a recurrence. In an optimal alignment M, at least one of the following is true: 1) m, n M; or 2) the m th position of x is not matched; or 3) the n th position of y is not matched.

13 Designing the algorithm Let F(i, j) denote the maximum score of an alignment between x 1 i and y 1 j. If case 1) holds, we pay s(x m, y n ) and we get F m, n = F m 1, n 1 + s(x m, y n ) If case 2) holds, we pay a gap penalty of d since the m th position of x is not matched and we get F m, n = F m 1, n d If case 3) holds, we pay a gap penalty of d since the n th position of y is not matched and we get F m, n = F m, n 1 d

14 Designing the algorithm Using the same argument for the subproblem of finding the maximum-score alignment between x 1 i and y 1 j, we get the following fact: The maximum alignment scores satisfy the following recurrence for i 1 and j 1: Moreover, (i, j) is in an optimal alignment M for this subproblem if and only if the maximum is achieved by the first of these values.

15 Designing the algorithm We build up the values of F(i, j) using the recurrence. There are only O(mn) subproblems, and F(m, n) is the value we are seeking. We now specify the algorithm to compute the value of the optimal alignment. For purpose of initialization, we note that F i, 0 = id F 0, j = jd for all i and j, since the only way to line up the i-letter word with 0-letter word is to use i gaps.

16 Designing the algorithm Alignmnet(x,y) Array F[0 m,0 n] Initialize F[i,0]=-id for each i Initialize F[0,j]=-jd for each j For j=1,,n For i=1,,m Use the recurrence to compute F(i,j) Endfor Endfor Return F[m,n]

17 Designing the algorithm To find the alignment itself, we must find the path of choices that led to this final value. This procedure is known as a traceback.

18 Example: BLOSUM50, gap penalty=-8

19 BLOSUM50

20 Running time The algorithm takes O(mn) time and O(mn) memory. O(mn) is a standard notation, called big-o notation, meaning of order mn. The computation time or memory storage required to solve the problem scales as the product of the sequence lengths mn, up to a constant factor.

21 Local alignment A much more common situation is where we are looking for the best alignment between subsequences of x and y. This arises for example when it is suspected that two protein sequences may share a common domain, or when comparing two very highly diverged sequences. The highest scoring alignment of subsequences of x and y is called the best local alignment.

22 Smith-Waterman algorithm The algorithm for finding optimal local alignments is closely related to that for global alignments. There are two differences. First,

23 Smith-Waterman algorithm Taking the option 0 corresponds to starting a new alignment. If the best alignment up to some point has a negative score, it is better to start a new one. Note that F i, 0 = 0 F 0, j = 0

24 Smith-Waterman algorithm Second, an alignment can end anywhere in the matrix. Instead of taking the value at F(m, n) for the best score, we look for the highest value of F(i, j) over the whole matrix, and start the traceback from there. The traceback ends when we meet a cell with value 0, which corresponds to the start of the alignment.

25 Smith-Waterman algorithm

26 Smith-Waterman algorithm The local version of the dynamic programming sequence alignment algorithm is known as the Smith-Waterman algorithm.

27 Smith-Waterman algorithm