# Gap Penalties

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Gap PenaltiesCMSC 423

General Gap CAThese have the same score, but the second one is often moreplausible.A single insertion of “GAAT” into the first string could changeit into the second. Now, the cost of a run of k gaps is gap k It might be more realistic to support general gap penalty, so thatthe score of a run of k gaps is gap(k) gap k. Then, the optimization will prefer to group gaps together.

General Gap CAPrevious DP no longer works with general gap penalties becausethe score of the last character depends on details of the -Instead, we need to “know” how long a final run of gaps is inorder to give a score to the last subproblem.

Three MatricesWe now keep 3 different matrices:M[i,j] score of best alignment of x[1.i] and y[1.j] ending with a charactercharacter match or mismatch.X[i,j] score of best alignment of x[1.i] and y[1.j] ending with a space in X.Y[i,j] score of best alignment of x[1.i] and y[1.j] ending with a space in Y.8 M [i 1, jM [i, j] match(i, j) max X[i 1, j :Y [i 1, j(M [i, jX[i, j] maxY [i, j(M [iY [i, j] maxX[i1]1]1]k] gap(k)k] gap(k)for 1 k jfor 1 k jgap(k)gap(k)for 1 k ifor 1 k ik, j]k, j]

The M MatrixWe now keep 3 different matrices:M[i,j] score of best alignment of x[1.i] and y[1.j] ending with a charactercharacter match or mismatch.X[i,j] score of best alignment of x[1.i] and y[1.j] ending with a space in X.Y[i,j] score of best alignment of x[1.i] and y[1.j] ending with a space in Y.By definition, alignmentends in a match.8 M [i 1, jM [i, j] match(i, j) max X[i 1, j :Y [i 1, jAny kind of alignment isallowed before the match.AG1]1]1]

The X (and Y) matriceskixG--- ACGT Gyj-k(M [i, jX[i, j] maxY [i, jixyjk] gap(k)k] gap(k)kG--- -CGT Gj-kjk decides how long tomake the gap.We have to make thewhole gap at once in orderto know how to score it.for 1 k jfor 1 k j

The X (and Y) matriceskixG--- ACGT Gyj-k(M [i, jX[i, j] maxY [i, jixyjk] gap(k)k] gap(k)kG--- -CGT Gj-kjk decides how long tomake the gap.We have to make thewhole gap at once in orderto know how to score it.for 1 k jfor 1 k jThis case is automaticallyhandled.kixy---- GCGT Gj-kj

Running Time for Gap Penalties8 M [i 1, jM [i, j] match(i, j) max X[i 1, j :Y [i 1, j(M [i, jX[i, j] maxY [i, j(M [iY [i, j] maxX[i1]1]1]k] gap(k)k] gap(k)for 1 k jfor 1 k jgap(k)gap(k)for 1 k ifor 1 k ik, j]k, j]Final score is max {M[n,m], X[n,m],Y[n,m]}.How do you do the traceback?Runtime: Assume X Y n for simplicity: 3n2 subproblems2n2 subproblems take O(n) time to solve (because we have to try all k)O(n3) total time

Affine Gap Penalties O(n3) for general gap penalties is usually too slow. We can still encourage spaces to group together using a special caseof general penalties called affine gap penalties:gap start the cost of starting a gapgap extend the cost of extending a gap by one more space Same idea of using 3 matrices, but now we don’t need to search overall gap lengths, we just have to know whether we are starting a newgap or not.

Affine Gap Penalties8 M [i 1, j 1]M [i, j] match(i, j) max X[i 1, j 1] match:Y [i 1, j 1]betweenx and y8 gap start gap extend M [i, jX[i, j] max gap extend X[i, j 1] :gap in xgap start gap extend Y [i, j8 gap start gap extend M [iY [i, j] max gap start gap extend X[i :gap in ygap extend Y [i 1, j]If previousalignment ends inmatch, this is anew gap1]1]1, j]1, j]

Affine Gap as Finite State Machinematch(i,j)Mgs gegs gematch(i,j)match(i,j)geYgs gegs geXge

Affine Base Cases (Global) M[0, i] “score of best alignment between 0 characters of x and icharacters of y that ends in a match” - because no such alignmentcan exist. X[0, i] “score of best alignment between 0 characters of x and icharacters of y that ends in a gap in x” gap start i gap extendbecause this alignment looks like:--------yyyyyyyyy X[i, 0] “score of best alignment between i characters of x and 0characters of y that ends in a gap in X” - xxxxxxxxx---------- not allowedM[i, 0] M[0, i] and Y[0, i] and Y[i,0] are computed using the same logicas X[i,0] and X[0,i]

Affine Gap Runtime 3mn subproblems Each one takes constant time Total runtime O(mn): back to the run time of the basic running time.Traceback Arrows now can point between matrices. The possible arrows are given, as usual, by the recurrence. E.g. What arrows are possible leaving a cell in the M matrix?

Why do you “need” 3 matrices? Alternative WRONG algorithm:M[i][j] max(M[i-1][j-1] cost(x[i], y[i]),M[i-1][j] gap (gap start if Arrow[i-1][j] ! ),M[j][i-1] gap (gap start if Arrow[i][j-1] ! ))WRONG Intuition: we only need to know whether we are starting a gap orextending a gap.The arrows coming out of each subproblem tell us how the best alignment ends, so wecan use them to decide if we are starting a new gap.The best alignmentup to this cell endsin a gap.The best alignmentup to this cell endsin a match.PROBLEM: The best alignment for stringsx[1.i] and y[1.j] doesn’t have to be usedin the best alignment betweenx[1.i 1] and y[1.j 1]

Why 3 Matrices: Examplematch 10, mismatch -2, gap -7, gap start -15CARTCA-TOPT(4, 3) optimal score 30 - 15 - 7 8CARTSCA-T-WRONG(5, 3) 30 - 15 - 7 - 15 - 7 -14CARTSCAT--OPT(5, 3) 20 - 2 - 15 - 14 -11this is why we need to keep the X and Y matrices around.they tell us the score of ending with a gap in one of the sequences.

Side Note: Lower Bounds Suppose the lengths of x and y are n. Clearly, need at least Ω(n) time to find their global alignment(have to read the strings!) The DP algorithms show global alignment can be done in O(n2) time.

Side Note: Lower Bounds Suppose the lengths of x and y are n. Clearly, need at least Ω(n) time to find their global alignment(have to read the strings!) The DP algorithms show global alignment can be done in O(n2) time. A trick called the “Four Russians Speedup” can make a similar dynamicprogramming algorithm run in O(n2 / log n) time. We probably won’t talk about the Four Russians Speedup.The important thing to remember is that only one of the four authors is Russian.(Alrazarov, Dinic, Kronrod, Faradzev, 1970) Open questions: Can we do better? Can we prove that we can’t dobetter? No one knows.

Recap Local alignment: extra “0” case. General gap penalties require 3 matrices and O(n3) time. Affine gap penalties require 3 matrices, but only O(n2) time.

What you should know by now. Dynamic programming framework Global & local sequence alignment algorithms with basic gappenalties Alignment with general gap penalties Alignment with affine gap penalties Longest common subsequence (board lecture) Subset Sum (board lecture)

General Gap Penalties Now, the cost of a run of k gaps is gap k It might be more realistic to support general gap penalty, so that the score of a run of k gaps is gap(k) gap k. Then, the optimization will prefer to group gaps together. AAAGAATTCA A-A-A-T-CA AAAGAATTCA AAA----TCA vs. Th

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