Multiple Sequence Alignment

Vasileios Hatzivassiloglou

University of Texas at Dallas

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Center star algorithm for multiple sequence global alignment

• T is the set of strings that we want to align• Pick ST that minimizes

• The initial alignment starts with S (≡S1)

• Suppose we have already aligned S1, S2, ..., Si as S′1, S′2, ..., S′i. Then we add the remaining strings one at a time by aligning Si+1 with S′1, obtaining S′i+1 and S′′1. We replace S′1 with S′′1 and add spaces to S′2, ..., S′i wherever spaces were added to S′1.

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Finding S

• S is the best representative of the set T in terms of the distance metric d

• If T is considered as a cluster of strings, then S is the centroid of the cluster

• To find S, align each string with every other ( pairs) and calculate the sum for each candidate. Pick the choice that minimizes this sum

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Example

• Three strings: GTA, CGT, CAG• Step 1: Calculate all three pairwise

similarities and pick the string that minimizes total distance; let’s say it’s CGT

• Step 2-1: Align CGT with GTACGT--GTA

• Step 2-2: Extend uninvolved, processed strings with spaces (not needed now)

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Example (continued)

• Step 3-1: Align CGT- with CAGC-GT-CAG--

• Step 3-2: Extend uninvolved, processed strings with spaces (-GTA)C-GT---GTACAG--

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Algorithm complexity – Finding S

• To find S, we consider k candidates

• For each candidate, we calculate the sum of k-1 terms – O(k2) such terms total

• If the maximum string length is n, then each term can be calculated in O(n2) time

• Total for finding S is O(k2n2)

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Algorithm complexity – Subsequent alignments

• Each subsequent alignment at step i+1 aligns a string S′1 of length at most in with a string Si+1 of length at most n

• Each alignment can be found in time O(in∙n)

• Total time for these alignments is

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Algorithm complexity – Extensions with spaces

• At step i+1 there is an extension of i-1 strings each of length at most in

• For each such string, we need to consider a total of n new space positions

• Time required is

• Overall total time for the algorithm is O(k2n2)

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Error bounds

• It is useful to know how far the solution found by an approximate algorithm is from the true optimal solution

• Sometimes (but not always) it is possible to provide error bounds, that is give upper and lower bounds for the quantity

• Bounds may depend on n and ksolution) malscore(opti

solution) oximatescore(appr

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Error analysis assumptions

• Sometimes we need additional assumptions in order to derive useful bounds

• For the approximate algorithm for multiple string alignment, we assume the triangle inequality for measure d:

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Background on distances

• A distance or metric d is formally defined as a function A×A→ℜ on a set A (called a metric space) with the following properties:– d(x,y)≥0 (non-negativity)

– d(x,y)=0 iff x=y (identity of indiscernibles)

– d(x,y)=d(y,x) (symmetry)

– d(x,y)≤d(x,z)+d(z,y) (triangle inequality)

• Metric spaces include ℜ (with d(x,y)=|x-y|), all Euclidean spaces, the Lp spaces, and inner product spaces.

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Background on distances

• A distance or metric d is formally defined as a function A×A→ℜ on a set A (called a metric space) with the following properties:– d(x,y)≥0

– d(x,y)=0 iff x=y

– d(x,y)=d(y,x)

– d(x,y)≤d(x,z)+d(z,y)

• Metric spaces include ℜ (with d(x,y)=|x-y|), all Euclidean spaces, the Lp spaces, and inner product spaces.

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Deriving an error bound

• Let v0 be the score for the optimal alignment and v* the score for the alignment produced by the center star algorithm

• Let d0(i,j) (d*(i,j)) be the corresponding induced distances on strings Si and Sj

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Lower bound for v0

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Upper bound for v*

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Triangle inequality

Symmetry

Each string is aligned with S1 optimally (there may be additional spaces in matching positions, which do not change the distance)

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Combining the bounds

• Better bound for low k

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Motif data notation• A motif is denoted by three parameters

– Its length l– The number of allowed spaces g– The number of allowed changes d– (l, d, g) notation

• Changes and gaps allowed because of mutations across organisms

• In a “good” motif, g and d are small compared to l• Most work assumes g = 0

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Finding the motif consensus

• Assume known motif instance positions and length (e.g., via multiple alignment)

• Also known as the known site problem• Input: A set of motif instances• Output: What is the motif consensus?• Further, is the consensus a valid motif, or is

it statistically indistinguishable from what we would expect from other randomly chosen regions?

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Statistical estimation

• An important approach to many data mining and machine learning tasks

• Requirement: The problem must be expressed as a probability function that depends on a number of modeled parameters whose value is unknown

• The estimation task: Find the optimal values for these parameters

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Estimation example

• Can be performed without an explicit probabilistic model

• Example: Future markets are exchanges where contracts are traded for future execution

• Contract price reflects probabilities of events

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