Sequence Alignment .

Sequences Much of bioinformatics involves sequences DNA sequences RNA sequences Protein sequences We can think of these sequences as strings of letters DNA & RNA: alphabet of 4 letters Protein: alphabet of 20 letters

20 Amino Acids Glycine (G, GLY) Alanine (A, ALA) Valine (V, VAL) Leucine (L, LEU) Isoleucine (I, ILE) Phenylalanine (F, PHE) Proline (P, PRO) Serine (S, SER) Threonine (T, THR) Cysteine (C, CYS) Methionine (M, MET) Tryptophan (W, TRP) Tyrosine (T, TYR) Asparagine (N, ASN) Glutamine (Q, GLN) Aspartic acid (D, ASP) Glutamic Acid (E, GLU) Lysine (K, LYS) Arginine (R, ARG) Histidine (H, HIS) START: AUG STOP: UAA, UAG, UGA

Sequence Comparison Finding similarity between sequences is important for many biological questions For example: Find similar proteins Allows to predict function & structure Locate similar subsequences in DNA Allows to identify (e.g) regulatory elements Locate DNA sequences that might overlap Helps in sequence assembly g1 g2

Sequence Alignment Input: two sequences over the same alphabet Output: an alignment of the two sequences Example: GCGCATGGATTGAGCGA TGCGCCATTGATGACCA A possible alignment: -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A

Alignments -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A Three elements: Perfect matches Mismatches Insertions & deletions (indel)

Choosing Alignments There are many possible alignments For example, compare: -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A to ------GCGCATGGATTGAGCGA TGCGCC----ATTGATGACCA-- Which one is better?

Scoring Alignments Rough intuition: Similar sequences evolved from a common ancestor Evolution changed the sequences from this ancestral sequence by mutations: Replacements: one letter replaced by another Deletion: deletion of a letter Insertion: insertion of a letter Scoring of sequence similarity should examine how many operations took place

Simple Scoring Rule Score each position independently: Match: +1 Mismatch : -1 Indel -2 Score of an alignment is sum of positional scores

Example Example: -GCGC-ATGGATTGAGCGA TGCGCCATTGAT-GACC-A Score: (+1x13) + (-1x2) + (-2x4) = 3 ------GCGCATGGATTGAGCGA TGCGCC----ATTGATGACCA-- Score: (+1x5) + (-1x6) + (-2x11) = -23

More General Scores The choice of +1,-1, and -2 scores was quite arbitrary Depending on the context, some changes are more plausible than others Exchange of an amino-acid by one with similar properties (size, charge, etc.) vs. Exchange of an amino-acid by one with opposite properties

Additive Scoring Rules We define a scoring function by specifying a function (x,y) is the score of replacing x by y (x,-) is the score of deleting x (-,x) is the score of inserting x The score of an alignment is the sum of position scores

Edit Distance The edit distance between two sequences is the “cost” of the “cheapest” set of edit operations needed to transform one sequence into the other Computing edit distance between two sequences almost equivalent to finding the alignment that minimizes the distance

Computing Edit Distance How can we compute the edit distance?? If |s| = n and |t| = m, there are more than alignments The additive form of the score allows to perform dynamic programming to compute edit distance efficiently

Recursive Argument Suppose we have two sequences: s[1..n+1] and t[1..m+1] The best alignment must be in one of three cases: 1. Last position is (s[n+1],t[m +1] ) 2. Last position is (s[n +1],-) 3. Last position is (-, t[m +1] )

Recursive Argument Suppose we have two sequences: s[1..n+1] and t[1..m+1] The best alignment must be in one of three cases: 1. Last position is (s[n+1],t[m +1] ) 2. Last position is (s[n +1],-) 3. Last position is (-, t[m +1] )

Recursive Argument Suppose we have two sequences: s[1..n+1] and t[1..m+1] The best alignment must be in one of three cases: 1. Last position is (s[n+1],t[m +1] ) 2. Last position is (s[n +1],-) 3. Last position is (-, t[m +1] )

Recursive Argument Define the notation: Using our recursive argument, we get the following recurrence for V: V[i,j] V[i+1,j] V[i,j+1] V[i+1,j+1]

Recursive Argument Of course, we also need to handle the base cases in the recursion: S T AA - - versus We fill the matrix using the recurrence rule:

Dynamic Programming Algorithm S T We continue to fill the matrix using the recurrence rule

Dynamic Programming Algorithm S T V[0,0] V[0,1] V[1,0] V[1,1] +1 -2 -A A- -2 (A- versus -A) versus

Dynamic Programming Algorithm S T

Dynamic Programming Algorithm S T Conclusion: d(AAAC,AGC) = -1

Reconstructing the Best Alignment To reconstruct the best alignment, we record which case(s) in the recursive rule maximized the score S T

Reconstructing the Best Alignment We now trace back a path that corresponds to the best alignment AAAC AG-C S T

Reconstructing the Best Alignment Sometimes, more than one alignment has the best score S T AAAC A-GC AAAC -AGC AG-C

Complexity Space: O(mn) Time: O(mn) Filling the matrix O(mn) Backtrace O(m+n)

Space Complexity In real-life applications, n and m can be very large The space requirements of O(mn) can be too demanding If m = n = 1000 we need 1MB space If m = n = 10000, we need 100MB space We can afford to perform extra computation to save space Looping over million operations takes less than seconds on modern workstations Can we trade off space with time?

Why Do We Need So Much Space? To compute d(s,t), we only need O(n) space Need to compute V[n,m] Can fill in V, column by column, only storing the last two columns in memory Note however This “trick” fails when we need to reconstruct the sequence Trace back information “eats up” all the memory

Why Do We Need So Much Space? To find d(s,t), need O(n) space Need to compute V[n,m] Can fill in V, column by column, storing only two columns in memory A 1 G 2 C 3 4 -2 -4 -6 -8 -2 1 -1 -3 -5 -4 -1 -2 -6 -3 -2 -1 Note however This “trick” fails when we need to reconstruct the sequence Trace back information “eats up” all the memory

Space Efficient Version: Outline Input: Sequences s[1,n] and t[1,m] to be aligned. Idea: perform divide and conquer Find position (n/2, j) at which the best alignment crosses a midpoint t s Construct two alignments A= s[1,n/2] vs t[1,j] B= s[n/2+1,n] vs t[j+1,m] Return the concatenated alignment AB

Finding the Midpoint The score of the best alignment that goes through j equals: d(s[1,n/2],t[1,j]) + d(s[n/2+1,n],t[j+1,m]) Thus, we need to compute these two quantities for all values of j We need to find j that maximizes this score. Such j determines a point (n/2,j) through which the best alignment passes. t s

Finding the Midpoint (Algorithm) Define V[i,j] = d(s[1..i],t[1..j]) (As before) B[i,j] = d(s[i+1..n],t[j+1..m]) (Symmetrically) F[i,j] + B[i,j] = score of best alignment through (i,j) t s

Computing V[i,j] V[i,j] = d(s[1..i],t[1..j]) As before: V[i,j] Requires linear space complexity

Computing B[i,j] B[i,j] = d(s[i+1..n],t[j+1..m]) Symmetrically (replacing i with i+1, and j with j+1): B[i,j] B[i+1,j] B[i,j+1] B[i+1,j+1] Requires linear space complexity

Local Alignment Consider now a different question: Can we find similar substring of s and t Formally, given s[1..n] and t[1..m] find i,j,k, and l such that d(s[i..j],t[k..l]) is maximal

Local Alignment As before, we use dynamic programming We now want to setV[i,j] to record the best alignment of a suffix of s[1..i] and a suffix of t[1..j] How should we change the recurrence rule?

Local Alignment New option: We can start a new match instead of extend previous alignment Alignment of empty suffixes

Local Alignment Example s = TAATA t = ATCTAA

Local Alignment Example s = TAATA t = TACTAA

Local Alignment Example s = TAATA t = TACTAA

Local Alignment Example s = TAATA t = TACTAA

Sequence Alignment We seen two variants of sequence alignment: Global alignment Local alignment Other variants: Finding best overlap (exercise) All are based on the same basic idea of dynamic programming