Global Alignment

Example 2.5   Given a string "ACBCDDDB" and a string "CADBDAD", one possible alignment will be:

                    A C - - B C D D D B
                      |     |   | | |
                    - C A D B - D A D -

A more useful than the general case is the following problem:

Problem 2.6   INPUT: Two strings S and T |T|=m, |S|=n (n and m are of roughly the same magnitude)
QUESTION: Establish the optimal alignment according to the alignment quality (or scoring) which will be defined next.

 

 

Lemma 2.7   V(A,B) has the following properties:

$$\begin{eqnarray*}{\em Base\; conditions:} \hspace{2cm} V(i,0) &=& \sum_{k=0}^i{... ...gma(S_i, - )\\ V(i ,j-1)+\sigma( - ,T_j) \end{array} \right. \end{eqnarray*}$$

Proof:Base condition:
The only way to align the first i elements of the string S with zero elements of the string T is to align each of the elements with a space in the string T. The score for that operation is by definition $\sigma(S_i, -)$ for each of the i elements and $V(i,0) = \sum_{k=0}^i{\sigma(S_k, -)}$ for the total sum.

Similarly, the expression $V(0,j) = \sum_{k=0}^j{\sigma(-, T_k)}$ follows from matching the first j elements of T with i blanks in string S.

Proof:Recurrence relation:
Let us consider an optimal alignment of $S_1 \ldots S_i$ and $T_1 \ldots T_j$. We shall distinguish between three cases according to the three possible scoring for the three operations are:

• Aligning S_i with T_j: The score in this case is the score of the operation $\sigma(S_i,T_j)$ plus the score of aligning i-1 elements of S with j-1 elements of T, namely, $V(i-1,j-1)+\sigma(S_i,T_j)$
• Aligning S_i with a space character in string T: The score in this case is the score of the indel operation $\sigma(S_i, -)$ plus the score of aligning the previous i-1 elements of S with j elements of T (Since the space is not an original character of T), $V(i-1,j) +\sigma(S_i, - )$
• Aligning T_j with a space character in string S: Similar to the previous case, the score will be $V(i,j- 1) +\sigma(-, T_j)$