Affine Gap Alignment CS 181 Fall 2020 Definitions

Definitions: Inputs and Outputs Inputs: 〈X, Y, �� 〉 ● ● ● X, Y

Definitions: Auxiliary Data Structures Matrices: V, G, E, F ● ● V = the

The Algorithm: 1) Initialize the matrices 2) Apply the recurrence relations to fill each

An Example: ● ● ● X = ATCGGC Y = AGC �� = 2

And so on. . . The algorithm continues like so until all the matrices

Traceback We use the backpointers in our matrices to reconstruct our alignment. At each

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Affine Gap Alignment CS 181 Fall 2020

Definitions: Inputs and Outputs Inputs: 〈X, Y, �� 〉 ● ● ● X, Y = strings of length m, n with characters indexed by i, j, respectively �� = match score �� = mismatch penalty �� = gap opening penalty �� = gap extension penalty (single-letter gap penalty) Output: An alignment which maximizes the following score: �� (# matches) - �� (# mismatches) - �� (# gap clusters) - �� (# single-letter gaps)

Definitions: Auxiliary Data Structures Matrices: V, G, E, F ● ● V = the best-score matrix G = the match-mismatch matrix E = the X-gap matrix F = the Y-gap matrix

The Algorithm: 1) Initialize the matrices 2) Apply the recurrence relations to fill each matrix 3) Traceback through V (not shown)

An Example: ● ● ● X = ATCGGC Y = AGC �� = 2 (score = +2) �� = 1 (penalty = -1) �� = 2 (penalty = -2) �� = 1 (penalty = -1)

G E G A - G - C - F V A T C G G C - - - *0 th row, 0 th column in G is unused → initialize with error values �� = 2 �� = 1 �� =1

G E E -∞ A -3 G -4 C -5 F V A T C G G C -∞ -∞ -∞ *0 th row in E is unspecified → initialize with negative infinity to favor opening gaps from V once we start calculating down �� = 2 �� = 1 �� =1

G E E -∞ A -3 V G -4 E C -5 E F V A T C G G C -∞ -∞ -∞ *Also keep backpointers to let you know which matrix was used to compute the score in each cell �� = 2 �� = 1 �� =1

G E F -∞ A -∞ G -∞ C -∞ F V A T C G G C -3 -4 -5 -6 -7 -8 *0 th column in F is unspecified → initialize with negative infinity to favor opening gaps from V once we start calculating across �� = 2 �� = 1 �� =1

G E F -∞ A -∞ G -∞ C -∞ F V A T C G G C -3 V -4 F -5 F -6 F -7 F -8 F *Also keep backpointers to let you know which matrix was used to compute the score in each cell �� = 2 �� = 1 �� =1

G E V 0 A -3 E G -4 E C -5 E F V A T C G G C -3 F -4 F -5 F -6 F -7 F -8 F *In V, keep track of which matrix gave you each score �� = 2 �� = 1 �� =1

G E F V 0 2 G A T C G G C - - - - A - 2 G - C - *We don’t need backpointers in G because all entries are computed from the same cell in V �� = 2 �� = 1 �� =1

G E F 2 -6 -6 V 0 A -3 E G -4 E C -5 E V A T C G G C -3 F -4 F -5 F -6 F -7 F -8 F �� = 2 �� = 1 �� =1

G E F 2 -6 -6 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G G -4 E C -5 E �� = 2 �� = 1 �� =1

G E F -4 -7 -1 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G G -4 E C -5 E �� = 2 �� = 1 �� =1

G E F -4 -7 -1 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F G -4 E C -5 E �� = 2 �� = 1 �� =1

G E F -5 -8 -2 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F G -4 E C -5 E �� = 2 �� = 1 �� =1

G E F -5 -8 -2 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F G -4 E C -5 E �� = 2 �� = 1 �� =1

And so on. . . The algorithm continues like so until all the matrices are filled. We’ll skip ahead, stopping at some interesting intermediate states which cover new branches in the algorithm. As an exercise, try filling out these matrices on your own and checking the values against our final solution!

G E E F V -6 2 -7 -1 A T C G G C -∞ -∞ A -3 V -6 V -7 V -8 V -9 V -10 V -11 V G -4 E C -5 E �� = 2 �� = 1 �� =1

G E F V -7 E -1 A T C G G C -∞ -∞ A -3 V -6 V -7 V -8 V -9 V -10 V -11 V G -4 E -1 V C -5 E �� = 2 �� = 1 �� =1

G E F -4 -1 -7 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E C -5 E �� = 2 �� = 1 �� =1

G E F -4 -1 -7 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E C -5 E �� = 2 �� = 1 �� =1

G E E F V -4 1 -5 -2 A T C G G C -∞ -∞ A -3 V -6 V -7 V -8 V -9 V -10 V -11 V G -4 E -1 V -4 V -5 V -6 V -7 V -8 V C -5 E -2 E �� = 2 �� = 1 �� =1

G E F V -5 E -2 A T C G G C -∞ -∞ A -3 V -6 V -7 V -8 V -9 V -10 V -11 V G -4 E -1 V -4 V -5 V -6 V -7 V -8 V C -5 E -2 V �� = 2 �� = 1 �� =1

G E F F V -8 -2 -9 -5 A T C G G C -∞ -3 V -4 F -5 F -6 F -7 F -8 F A -∞ -6 V -1 V -2 F -3 F -4 F -5 F G -∞ -7 V -4 V -2 V -3 F -3 V -4 V C -∞ -8 V �� = 2 �� = 1 �� =1

G E F F V -9 -5 A T C G G C -∞ -3 V -4 F -5 F -6 F -7 F -8 F A -∞ -6 V -1 V -2 F -3 F -4 F -5 F G -∞ -7 V -4 V -2 V -3 F -3 V -4 V C -∞ -8 V -5 V �� = 2 �� = 1 �� =1

G E F -2 -2 -5 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 E �� = 2 �� = 1 �� =1

G E F -2 -2 -5 V V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 G/E �� = 2 �� = 1 �� =1

G E E F V A T C G G C -∞ -∞ A -3 V -6 V -7 V -8 V -9 V -10 V -11 V G -4 E -1 V -4 V -5 V -6 V -7 V -8 V C -5 E -2 V -5 V -3 V -4 V -7 V �� = 2 �� = 1 �� =1

G E F F V A T C G G C -∞ -3 V -4 F -5 F -6 F -7 F -8 F A -∞ -6 V -1 V -2 F -3 F -4 F -5 F G -∞ -7 V -4 V -2 V -3 F -3 V -4 V C -∞ -8 V -5 V 0 V -1 F -2 F �� = 2 �� = 1 �� =1

G E V F V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 G/E 3 G 0 F -1 G/F 1 G �� = 2 �� = 1 �� =1

Traceback We use the backpointers in our matrices to reconstruct our alignment. At each position, we can recover the single-letter alignment of the prior two characters based on which matrix produced our maximum score. Starting from V(m, n) , at every V(i, j): ● If argmax = G → recover a match/mismatch; recurse on V(i-1, j-1) ● If argmax = E → recover a gap in X; follow the backpointers of E(i, j) recursively, inserting gaps in X until we return to V; recurse ● If argmax = F → recover a gap in Y; follow the backpointers of F(i, j) recursively, inserting gaps in Y until we return to V; recurse

V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 G/E 3 G 0 F -1 G/F 1 G Score: +1

V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 G/E 3 G 0 F -1 G/F 1 G C C Score: +1

V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 G/E 3 G 0 F -1 G/F 1 G G G C C Score: +1

F A T C G G C -∞ -3 V -4 F -5 F -6 F -7 F -8 F A -∞ -6 V -1 V -2 F -3 F -4 F -5 F G -∞ -7 V -4 V -2 V -3 F -3 V -4 V C -∞ -8 V -5 V 0 V -1 F -2 F G G C C Score: +1

F A T C G G C -∞ -3 V -4 F -5 F -6 F -7 F -8 F A -∞ -6 V -1 V -2 F -3 F -4 F -5 F G -∞ -7 V -4 V -2 V -3 F -3 V -4 V C -∞ -8 V -5 V 0 V -1 F -2 F G – G G C C Score: +1

F A T C G G C -∞ -3 V -4 F -5 F -6 F -7 F -8 F A -∞ -6 V -1 V -2 F -3 F -4 F -5 F G -∞ -7 V -4 V -2 V -3 F -3 V -4 V C -∞ -8 V -5 V 0 V -1 F -2 F C – G G C C Score: +1

V A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 G/E 3 G 0 F -1 G/F 1 G T – C – G G C C Score: +1

V A A A T C G G C 0 -3 F -4 F -5 F -6 F -7 F -8 F A -3 E 2 G -1 F -2 F -3 F -4 F -5 F G -4 E -1 E 1 G -2 G/F 0 G -1 G -4 F C -5 E -2 G/E 3 G 0 F -1 G/F 1 G T – C – G G C C Score: +1

Results: A A T – C – G G C C . . . is our optimal alignment with score +1!