Moreover, the circular logic


 Reginald Rich
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1 Moreover, the circular logic How do we know what is the right distance without a good alignment? And how do we construct a good alignment without knowing what substitutions were made previously? ATGCGTGCAAGT GCGGCGGCAGTT ATGCGTGCAAGT GCGGCGGCAGTT Which is better? Less gaps but 7matches More gaps but 8 matches
2 Substitution matrices for proteins We need to compare, DNA or protein sequences, estimate their distances etc (e.g. Helpful for inferring molecular function by finding similarity to a sequence with known function). For that purpose we need: Good alignment of sequences. For that purpose we need: A measure for judging the quality of an alignment in relation to other possible alignments, a scoring system.
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5 A practical aspect as well, since we don t study very diverged DNA sequences
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25 Commonly multiplied by a power of 10 to deal with decimals
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28 A sample substitution matrix 134 LQQGELDLVMTSDILPRSELHYSPMFDFEVRLVLAPDHPLASKTQITPEDLASETLLI 137 LDSNSVDLVLMGVPPRNVEVEAEAFMDNPLVVIAPPDHPLAGERAISLARLAEETFVM BLOSUM62 D:D = +6 C 9 S 1 4 T P A G D:R = 2 N D E Q H R K M I L V F Y W C S T P A G N D E Q H R K M I L V F Y W
29 BLOSUM Brief Overview Based purely on counts and alignment BLOSUM65 < BLOSUM45 in evolutionary distance a set of sequences S = S...S, where S = s...s, ij 20 j i j i i= 1 j= 1 i= 1 1 k i i1 in s is the jth amino acid in sequence S. The probability of observing each amino acid X =p(x) p(x) cx (, S)/ cx (, S) where cx (, S k = j i k ) is the count of amino acid X in sequence S PX X random= px px orpx ifl= m 2 ( l, m ) 2 ( l). ( m) ( l), i j i
30 In general logodds for alignment PX (, X data) = # of X, X combinations/all possible pairwise combinations l m l m finally take the log2odds likelihood ratio and multiply by 2 for easy representation { } Note: all possible pairwise combinations=k. nn ( 1)/2 Blosum does use pseudo count to accomadate for combinations not observed: add 1 in numerator, add 210 (20*19+20) in denominator ie P(X,X data) = l m count(x l,x m) k. ( 1) / { nn } P(X l,x m data) λ.log = subs. matrix P (X,X random) l m λ being a scaling factor for representation
31 BLOSUM vs PAM vs others? BLOSUM: Based on a range of evolutionary periods  Each matrix constructed separately Indirectly accounts for interdependence of residues Range of sequences, range of replacements PAM Based on extrapolation from a short evolutionary period  Errors in PAM1 are magnified through PAM250 Assumes Markov process too much extrapolation Rare replacements too infrequent to be represented accurately Others Incorporation of secondary structure/structural alignments Use of structural alignments Transmemberane proteinspecific matrices
32 How do we use the matrices for sequence alignment? AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC Definition Given two strings x = x 1 x 2...x M, y = y 1 y 2 y N, An alignment of two sequences x and y is an arrangement of x and y by position, where a and b can be padded with gap symbols to achieve the same length.
33 Why do we bother aligning? Finding important molecular regions that are conserved across species Given a new sequence, infer its function based on similarity to another Seq. Determining the evolutionary constraints at work Mutations in a population or a family of genes Find similar looking sequences in a database Inferring the secondary or tertiary structure of a sequence of interest Molecular modeling using a template (homology modeling)
34 Alignment is a path in the alignment matrix X: ACACACTA Y: AGCACACA X: AGCACACA Y: ACACACTA m+ n! mn!! m+ n 2 Alignments possible!
35 Dynamic programming for global alignment simple case Consider two sequences: x[1.. M], and y[1.. N] for a new position to be aligned we have ONLY three choices 1. x i aligns to y j x 1 x i1 x i Align x[1.. i] with y[1.. j] y 1 y j1 y j 2. x i aligns to a gap x 1 x i1 x i y 1 y j  3. y j aligns to a gap x 1 x i  y 1 y j1 y j x[1..(i1)] is already aligned with y[1..(j)], so align x[i] with a gap in y x[1.. i] is already aligned with y[1.. (j1)], so align a gap in x to y[j]
36 1. x i aligns to y j x 1 x i1 y 1 y j1 x i y j F (i,j) = F (i1, j1) + s(i,j) 2. x i aligns to a gap x 1 x i1 x i y 1 y j  3. y j aligns to a gap x 1 x i  y 1 y j1 y j F (i,j) = F (i1, j) gap_open_penalty (go) F (i,j) = F (i, j1) gap_open_penalty (go)
37 If we could make F (i, j1), F (i1, j), F (i1, j1) optimal, then we can make the next ones optimal as well F (i1, j1) + s (x i, y j ) F (i, j) = max F (i1, j) go F (i, j1) go Where s (x i, y j ) = Score for a match, if x i = y j ; score for a mismatch, if x i y j ;
38 The NeedlemanWunsch Algorithm pioneering application of DP to biological sequences 1. Initialization. a. F(0, 0) = 0 b. F(0, j) =  j go c. F(i, 0) =  i go 2. Main Iteration. Fillingin partial alignments For each i = 1 M For each j = 1 N F(i1,j1) + s(x i, y j ) [case 1] F(i, j) = max F(i1, j) go [case 2] F(i, j1) go [case 3] DIAG, if [case 1] Ptr(i,j) = LEFT, if [case 2] UP, if [case 3] 3. Termination. F(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment
39 An example T C G C A F (i1, j1) + s (x i, y j ) F (i, j) = max F (i1, j) go F (i, j1) go T s (x i, y j )=1 for match, 1 for mismatch go=2 C C A Trace back arrows from lower right to upper left and Maximize the scores Alignment score = 2 T C G C A T C  C A
40 Are gaps that bad? Current model: Gap of length n incurs penalty n.go γ(n) Bunch of gaps are better than individual gaps Convex (saturating) gap penalty function: γ(n): for all n, γ(n + 1)  γ(n) γ(n)  γ(n 1) A common function is Affine gap penalty γ(n) = go + (n 1) ge gap gap open extend γ(n)
41 Convex gap dynamic programming Initialization: same as before Iteration: F(i1, j1) + s(x i, y j ) F(i, j) = max max k=0 i1 F(k,j) γ(ik) max k=0 j1 F(i,k) γ(jk) Termination: same Running Time: O(N 2 M) Space: O(NM) (assume N>M)
42 Fast implementation of affine gap penalty We need three matrices for tracking scores Matrix1: a[i,j]= to store maximum score of an alignment that ends in x[i] matched to y[j] Matrix2: b[i,j]= to store maximum score of an alignment that ends in gap matched to y[j] Matrix3: c[i,j]= to store maximum score of an alignment that ends in gap matched to x[i] x i y j  y j.x i 
43 Implementation Cont d ai [ 1, j 1] ai [, j] = max bi [ 1, j 1] + si (, j) ci [ 1, j 1] ai [, j 1] + go bi [, j] = max bi [, j 1] + ge ci [, j 1] + go ai [ 1, j] + go ci [, j] = max bi [ 1, j] + go ci [ 1, j] + ge Pointermatrices: Three matrices to figure out which state within each score matrix maximization was used to obtain the optimal alignment of position i,j. Of course you need to know which matrix yielded the best score in the end (m,n) as well Covered by assignment!
44 For checking results of your code
45 Pair HMMs for alignment HMM for sequence alignment, which incorporates affine gap scores. Match (M) Insertion in x (X) insertion in y (Y) Hidden States Observation Symbols Match (M): {(a,b) a,b in }. Insertion in x (X): {(a,) a in }. Insertion in y (Y): {(,a) a in }.
46 Simple representation 1δ M p xiyj 1ε 1ε X q xi δ δ Y q yj ε ε M X Y M 12δ 1 ε 1 ε X δ ε 0 Y δ 0 ε
47 Full representation δ ε Begin 12δτ M p xiyj δ X q xi 1ετ τ τ End 12δτ δ δ 1ετ Y q yj ε τ τ
48 Initialization: v M (0, 0) = 1. v X (0, 0) = v Y (0, 0) = 0 v * (1, j) = v * (i, 1) = 0. Recurrence: i = 0,,n, j = 0,,m, except for(0,0); Termination: = = = 1), ( 1), ( max ), ( ) 1, ( ) 1, ( max ), ( 1) 1, ( ) (1 1) 1, ( ) (1 1) 1, ( ) 2 (1 max ), ( X M Y X M X Y X M M j i v j i v q j i v j i v j i v q j i v j i v j i v j i v p j i v j i j i y x y x ε δ ε δ τ ε τ ε τ δ )), ( ),, ( ),, ( max( Y X M E m n v m n v m n v v τ = Algorithm: Viterbi algorithm for pair HMMs
49 Viterbi Pair HMM is equivalent to global dynamic programming with affine gap! Proof (sketch) Add the following missing details to DEKM s summary Take the viterbi matrices and divide by the terms for random sequence probability, and take the log of resulting transformation i.e, (1 2 δ τ) v ( i 1, j 1). p q q (1 η) v i j v i j p q q M 2 xiyj xi yi M X 2 (, ) = max (1 ε τ) ( 1, 1). xiyj xi yi (1 η) Y 2 (1 ε τ) v ( i 1, j 1). pxiyj qxiqyi (1 η) M δv ( i 1, j). qxi qxi (1 η) (, ) = max i X εv ( i 1, j). qxi qxi (1 η) X v i j qx
50 Multiple Sequence Alignment (MSA) Si () = sx ( i, yi) all possible pairs How to score? and How to align? Scoring, common approach: Sum of pairs or its variants for each column Consider L aligned at all N positions, LL score =5 total score, Si ( ) = 5 NN ( 1) 2 Now consider one position being G, rest being L GL score=4 ΔSi ( ) 5( N 1) + 4( N 1) 9( N 1) 9 = = = Si () Si () 2.5 NN ( 1) 2.5N Relative evidence decreases with N, opposed to having increased confidence in L being the "correct" residue at that position
51 Clustal W Pairwise alignment: calculation of distance matrix Rooted nj tree (guide tree) and calculation of sequence weights Progressive alignment following the guide tree
52 Step 1Calculation of Distance Matrix using pairwise alignment Use the Distance Matrix to create a Guide Tree to determine the order of the sequences. HbbHu 1  HbbHo HbaHu HbaHo MygPh GibPe LgbLu D = 1 (I) D = Difference score I = # of identical aa s in pairwise global alignment total number of aa s in shortest sequence
53 Step 2Create Rooted Tree and calculate weights Neighbor joining algorithm simple, to be discussed later Weight Alignment Order of alignment: 1 HbaHu vs HbaHo 2 HbbHu vs HbbHo 3 A vs B 4 MygPh vs C 5 GibPe vs D 6 LghLu vs E
54 Step 3Progressive alignment
55 Step 3Progressive alignment Scoring during progressive alignment
56 Recommended MSA Programs MUSCLE (fast and accurate) MAVID (genomescale alignment) SAM ( hidden markov, powerful and wide range of options)
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