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CS 451 / 558. Week 4, Tue. Scoring an alignment. Let: x k := the kth letter of x y k := the kth letter of y. Input : string x , length m string y , length n (both from alphabet S [ACGT]) scoring matrix s , s.t. s ( a,b ) := the score of aligning a to b gap penalty g

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## CS 451 / 558

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**CS 451 / 558**Week 4, Tue**Scoring an alignment**Let: xk:= the kthletter of x yk:= the kthletter of y Input: string x, length m string y, length n (both from alphabet S [ACGT]) scoring matrix s, s.t.s(a,b) := the score of aligning a to b gap penalty g S = 0 # the score for (i=0; i<m; i++) if (Si or Ti is a gap character) S -= g else S += s(xi ,yi)**Finding an optimal alignment**Dynamic Programming • Looks like a merger of the 2D dotplot array and the alignment scoring**Finding an optimal alignment**Dynamic Programming • Looks like a merger of the 2D dotplot array and the alignment scoring • But it’s actually more than that**Finding an optimal alignment**Dynamic Programming • a recursive definition of the optimal score**a recursive definition of the optimal score**the optimal solution depends on • optimal solutions to subproblems of the same form • local calculations based on those solutions • e.g. score for alignment of x and y • Can only end one of three ways: • xm aligned to yn • xm aligned to nothing (ynalready used) • yn aligned to nothing (xmalready used) S = S(m-1,n-1) + s(xm,yn) S = S(m-1,n) + g S = S(m,n-1) + g**a recursive definition of the optimal score**the optimal solution depends on • optimal solutions to subproblems of the same form • local calculations based on those solutions • e.g. score for alignment of x and y • Can only end one of three ways: • xm aligned to yn • xm aligned to nothing (ynalready used) • yn aligned to nothing (xmalready used) • generally S = S(m-1,n-1) + s(xm,yn) S = S(m-1,n) + g S = S(m,n-1) + g**Finding an optimal alignment**Dynamic Programming • a recursive definition of the optimal score**Finding an optimal alignment**Dynamic Programming • a recursive definition of the optimal score**Finding an optimal alignment**Dynamic Programming • a recursive definition of the optimal score • a dynamic programming matrix for remembering optimal scores of subproblems**a dynamic programming matrix for remembering optimal scores**of subproblems**Finding an optimal alignment**Dynamic Programming • a recursive definition of the optimal score • a dynamic programming matrix for remembering optimal scores of subproblems • a bottom-up approach of filling the matrix by solving the smallest subproblems first**a bottom-up approach of filling the matrix by solving the**smallest subproblems first move me**Scoring an optimal alignment**Input: strings x & y, lengths m & n scoring matrix s, s.t.s(a,b) := the score of aligning a to b gap penalty g S0,0 = 0 # the score for (i=0; i<m; i++) initialize Si,0 for (j=0; j<n; j++) initialize S0,j for (i=0; i<m; i++) for (j=0; j<n; j++) Si-1,j-1 +s(xi ,yi), Si,j = max Si,j-1 – g , Si-1,j– g**Finding an optimal alignment**Dynamic Programming • a recursive definition of the optimal score • a dynamic programming matrix for remembering optimal scores of subproblems • a bottom-up approach of filling the matrix by solving the smallest subproblems first • a traceback of the matrix to recover the structure of the optimal solution that gave the optimal score**0**0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 5 0 3 0 0 0 4 2 0 0 3 0 8 10 4 0 8 6 0 2 0 0 2 4 0 2 0 5 12 0 6 0 0 0 2 0 0 6 0 18 0**local alignment**Input: strings x & y, lengths m & n scoring matrix s, s.t.s(a,b) := the score of aligning a to b gap penalty g S0,0 = 0 # the score for (i=0; i<m; i++) initialize Si,0 for (j=0; j<n; j++) initialize S0,j for (i=0; i<m; i++) for (j=0; j<n; j++) Si-1,j-1 +s(xi ,yi), Si,j = max Si,j-1 – g , Si-1,j– g 0**Score matrix**P(xi , yj| model of homology)**Score matrix**P(xi , yj | model of homology) P(xi , yj | model of nonhomology)**Score matrix**f (xi , yj) * f (xi) f (yj) ** * From alignments of trusted homologs ** Observed frequencies in large database of repr. seqs**Score matrix**f (xi , yj) * s(xi , yj)= log f (xi) f (yj) ** * From alignments of trusted homologs ** Observed frequencies in large database of repr. seqs**Score matrix**BLOSUM PAM VTML … f (xi , yj) * s(xi , yj)= log f (xi) f (yj) ****Score matrix**f (xi , yj) * s(xi , yj)= log f (xi) f (yj) ****Gap penalties**Usually ad hoc What works well with the chosen score matrix Linear / affine gap penalties Affine = geometric length distribution

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