# Inverse Alignment - PowerPoint PPT Presentation

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Inverse Alignment. CS 374 Bahman Bahmani Fall 2006. The Papers To Be Presented. Sequence Comparison - Alignment. Alignments can be thought of as two sequences differing due to mutations happened during the evolution. AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC.

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Inverse Alignment

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## Inverse Alignment

CS 374

Bahman Bahmani

Fall 2006

### Sequence Comparison - Alignment

• Alignments can be thought of as two sequences differing due to mutations happened during the evolution

AGGCTATCACCTGACCTCCAGGCCGATGCCC

TAGCTATCACGACCGCGGTCGATTTGCCCGAC

-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC---

| | | | | | | | | | | | | x | | | | | | | | | | |

TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC

### Scoring Alignments

• Alignments are based on three basic operations:

• Substitutions

• Insertions

• Deletions

• A score is assigned to each single operation (resulting in a scoring matrix and also in gap penalties). Alignments are then scored by adding the scores of their operations.

• Standard formulations of string alignment optimize the above score of the alignment.

AKRANR

KAAANK

-1 + (-1) + (-2) + 5 + 7 + 3 = 11

### Scoring Matrices in Practice

• Some choices for substitution scores are now common, largely due to convention

• Most commonly used Amino-Acid substitution matrices:

• PAM (Percent Accepted Mutation)

• BLOSUM (Blocks Amino Acid Substitution Matrix)

BLOSUM50 Scoring Matrix

### Gap Penalties

• Inclusion of gaps and gap penalties is necessary to obtain the best alignment

• If gap penalty is too high, gaps will never appear in the alignment

AATGCTGC

ATGCTGCA

• If gap penalty is too low, gaps will appear everywhere in the alignment

AATGCTGC----

A----TGCTGCA

### Gap Penalties (Cont’d)

Separate penalties for gap opening and gap extension

Opening: The cost to introduce a gap

Extension: The cost to elongate a gap

Opening a gap is costly, while extending a gap is cheap

Despite scoring matrices, no gap penalties are commonly agreed upon

LETVGY

W----L

-5 -1 -1 -1

### Parametric Sequence Alignment

• For a given pair of strings, the alignment problem is solved for effectively all possible choices of the scoring parameters and penalties (exhaustive search).

• A correct alignmentis then used to find the best parameter values.

• However, this method is very inefficient if the number of parameters is large.

### Inverse Parametric Alignment

• INPUT: an alignment of a pair of strings.

• OUTPUT: a choice of parameters that makes the input alignment be an optimal-scoring alignment of its strings.

• From Machine Learning point of view, this learns the parameters for optimal alignment from training examples of correct alignments.

### Inverse Optimal Alignment

Definition (Inverse Optimal Alignment):

INPUT: alignments A1, A2, …, Ak of strings,

an alignment scoring function fw with parameters w = (w1, w2, …, wp).

OUTPUT: values x = (x1, x2, …, xp) for w

GOAL: each input alignment be an optimal alignment of its strings under fx .

ATTENTION: This problem may have no solution!

### Inverse Near-Optimal Alignment

• When minimizing the scoring function f, we say an alignment A of a set of strings S is –optimal, for some if:

where is the optimal alignment of S under f.

### Inverse Near-Optimal Alignment (Cont’d)

• Definition (Inverse Near-Optimal Alignment):

INPUT: alignments Ai

scoring function f

real number

OUTPUT: find parameter values x

GOAL: each alignment Ai be -optimal under fx .

The smallest possible can be found within accuracy using calls to the algorithm.

### Inverse Unique-Optimal Alignment

• When minimizing the scoring function f, we say an alignment A of a set of strings S is -uniquefor some if:

for every alignment B of S other than A.

### Inverse Unique-Optimal Alignment (Cont’d)

• Definition (Inverse Unique-Optimal Alignment):

INPUT: alignments Ai

scoring function f

real number

OUTPUT: parameter values x

GOAL: each alignment Ai be -unique under fx

The largest possible can be found within accuracy using calls to the algorithm.

### Let There Be Linear Functions …

• For most standard forms of alignment, the alignment scoring function f is a linear function of its parameters:

where each fi measures one of the features of A.

### Let There Be Linear Functions … (Example I)

• With fixed substitution scores, and two parameters gap open ( ) and gap extension ( ) penalties, p=2 and:

where:

g(A) = number of gaps

l(A) = total length of gaps

s(A) = total score of all substitutions

### Let There Be Linear Functions … (Example II)

• With no parameters fixed, the substitution scores are also in our parameters and:

where:

a and b range over all letters in the alphabet

hab(A) = # of substitutions in A replacing a by b

### Linear Programming Problem

• INPUT: variables x = (x1, x2, …, xn)

a system of linear inequalities in x

a linear objective function in x

OUTPUT: assignment of real values to x

GOAL: satisfy all the inequalities and minimize the objective

In general, the program can be infeasible, bounded, or unbounded.

### Reducing The Inverse Alignment Problems To Linear Programming

• Inverse Optimal Alignment: For each Ai and every alignment B of the set Si, we have an inequality:

or equivalently:

The number of alignments of a pair of strings of length n is hence a total of inequalities in p variables. Also, no specific objective function.

### Separation Theorem

• Some definitions:

• Polyhedron: intersection of half-spaces

• Rational polyhedron: described by inequalities with only rational coefficients

• Bounded polyhedron: no infinite rays

### Separation Theorem (Cont’d)

• Optimization Problem for a rational polyhedron P in :

INPUT: rational coefficients c specifying the objective

OUTPUT: a point x in P minimizing cx, or determining that P is empty.

• Separation Problem for P is:

INPUT: a point y in

OUTPU: rational coefficients w and b such that for all points x in P, but (a violated inequality) or determining that y is in P.

### Separation Theorem (Cont’d)

• Theorem (Equivalence of Separation and Optimization): The optimization problem on a bounded rational polyhedron can be solved in polynomial time if and only if the separation problem can be solved in polynomial time.

That is, for bounded rational polyhedrons:

OptimizationSeparation

### Cutting-Plane Algorithm

• Start with a small subset S of the set L of all inequalities

• Compute an optimal solution x under constraints in S

• Call the separation algorithm for L on x

• If x is determined to satisfy L output it and halt; otherwise,

add the violated inequality to S and loop back to step (2).

### Complexity of Inverse Alignment

• Theorem:Inverse Optimal and Near-Optimal Alignment can be solved in polynomial time for any form of alignment in which:

1. the alignment scoring function is linear

2. the parameters values can be bounded

3. for any fixed parameter choice, an optimal alignment can be found in polynomial time.

Inverse Unique-Optimal Alignment can be solved in polynomial time if in addition:

3’. for any fixed parameter choice, a next-best alignment can be found in polynomial time.

### Application to Global Alignment

• Initializing the Cutting-Plane Algorithm: We consider the problem in two cases:

• All scores and penalties varying: Then the parameter space can be made bounded.

• Substitution costs are fixed: Then either (1) a bounding inequality, or (2) two inequalities one of which is a downward half-space, the other one is an upward half-space, and the slope of the former is less than the slope of the latter can be found in O(1) time, if they exist.

### Application to Global Alignment (Cont’d)

• Choosing an Objective Function: Again we consider two different cases:

• Fixed substitution scores: in this case we choose the following objective:

• Varying substitution scores: In this case we choose the following objective:

where s is the minimum of all non-identity substitution scores and i is the maximum of all identity scores.

### Application to Global Alignment (Cont’d)

• For every objective, two extreme solutions exist: xlarge and xsmall. Then for every we have a corresponding solution:

x1/2 is expected to better generalize to alignments outside the training set.

### CONTRAlign

• What: extensible and fully automatic parameter learning framework for protein pair-wise sequence alignment

• How: pair conditional random fields (pair CRF s)

• Who:

• If then:

where:

### Training Pair-HMMs

• INPUT: a set of training examples

• OUTPUT: the feature vector w

• METHOD: maximizing the joint log-likelihood of the data and alignments under constraints on w:

### Generating Alignments Using Pair-HMMs

• Viterbi Algorithm on a Pair-HMM:

INPUT: two sequences x and y

OUTPUT: the alignment a of x and y that maximizes P(a|x,y;w)

RUNNING TIME: O(|x|.|y|)

### Pair-CRFs

• Directly model the conditional probabilities:

where w is a real-valued parameter vector not necessarily corresponding to log-probabilities

### Training Pair-CRFs

• INPUT: a set of training examples

• OUTPUT: real-valued feature vector w

• METHOD: maximizing theconditional log-likelihood of the data (discriminative/conditional learning)

where is a Gaussian prior on w, to prevent over-fitting.

### Properties of Pair-CRFs

• Far weaker independence assumptions than Pair-HMMs

• Capable of utilizing complex non-independent feature sets

• Directly optimizing the predictive ability, ignoring P(x,y); the model to generate the input sequences

### Choice of Model Topology in CONTRAlign

• Some possible model topologies:

CONTRAlignDouble-Affine :

CONTRAlignLocal :

### Choice of Feature Sets in CONTRAlign

• Some possible feature sets to utilize:

1. Hydropathy-based gap context features (CONTRAlignHYDROPATHY)

2. External Information:

2.1. Secondary structure (CONTRAlignDSSP)

2.2. Solvent accessibility (CONTRAlignACCESSIBILITY)

### Results: Alignment Accuracy in the “Twilight Zone”

For each conservation range, the uncolored bars give accuracies for MAFFT(L-INS-i), T-Coffee, CLUSTALW, MUSCLE, and PROBCONS (Bali) in that order, and the colored bar indicated the accuracy for CONTRAlign.

Questions?

Thank You!