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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.

KAAANK

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

An Example Of Scoring an Alignment Using a Scoring MatrixScoring 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 Programming

- 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) Programming

- 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) Programming

- 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 Programming

- 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 Programming

- 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 Programming

- 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) Programming

- 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) Programming

- 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.

Computational Results Programming

Computational Results (Cont’d) Programming

Computational Results (Cont’d) Programming

CONTRAlign Programming

- What: extensible and fully automatic parameter learning framework for protein pair-wise sequence alignment
- How: pair conditional random fields (pair CRF s)
- Who:

Pair-HMMs for Sequence Alignment Programming

Pair-HMMs … (Cont’d) Programming

- If then:
where:

Training Pair-HMMs Programming

- 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 Programming

- 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 Programming

- Directly model the conditional probabilities:
where w is a real-valued parameter vector not necessarily corresponding to log-probabilities

Training Pair-CRFs Programming

- 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 Programming

- 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 Programming

- Some possible model topologies:
CONTRAlignDouble-Affine :

CONTRAlignLocal :

Choice of Feature Sets in CONTRAlign Programming

- 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” Programming

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? Programming

Thank You! Programming

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