Minimized compact automaton for Minimized compact automaton for clumps over degenerate patternsclumps over degenerate patterns

Evgeniia Furletova*, Jan Holub, Mireille Regnier

November 27, 2015

Institute of mathematical problems in biology, Russia

CollaboratorsCollaborators

Mireille RegnierEcole Polytechnique, INRIA, France

Jan HolubCzech Technical University in Prague, Czech Republic

ChallengeChallenge

Problem. Creating of an efficient algorithm of pattern occurrences P-value computation.

Functional fragments recognition in biological sequences can be reduced to finding of overrepresented occurrences of a pattern.

A measure of overrepresentation is PP-value-value of pattern occurrences

PP-value of pattern occurrences-value of pattern occurrences

P-valueP-value isis the probability to find at least one occurrence of words the probability to find at least one occurrence of words from a pattern H in a random sequence of length n generated from a pattern H in a random sequence of length n generated according to a given probability model.according to a given probability model.

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11 ,(1 ( ))

nP valueC

•C(z) – generating function of clumps; •ρ – closest to 1 root of 1 – z+C(z) = 0

Regnier M., Fang B, Iakovishina D. Clump Combinatorics, Automata, and Word Asymptotics// Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO). 2014

For a Bernoulli model P-value can be approximated by the formula* :

ClumpsClumps

ExamplesExamples of clumps for pattern ACATTACA

k-clump for a pattern H = {h1,…,hr} is a string s such that:• s consists of k overlapping occurrences of H• any two consecutive letters of s belong to an occurrence of H

• ACATTACA 1-clump

ACATTACA ACATTACA ACATTACA

• ACATTACATTACACATTACA 3-clump

Our goal is to create an efficient method for computation of probabilities of k-clumps

0 1( ) ... ,nnC z p p z p z

Clumps generating functionClumps generating function

pk – sum of probabilities of all k-clumps.

Degenerate (intermediate) patternsDegenerate (intermediate) patterns

Degenerate alphabet Σ’ – alphabet letters of which are subsets of alphabet Σ.Degenerate pattern is a string in Σ’

Example:Example: IUPAC alphabet

A = [A]C = [C]G = [G]T = [T]R = [AG]Y = [CT]S = [CG] …N = [ACGT]

Examples:Examples: IUPAC consensuses

ТАТА-box ТAТA[AТ]A[AТ] – 4 words of length 7

Consensus of transcription factor binding site Antp (Drosophila) ANNNNCATTA – 256 words of length 10

Pattern matching (Aho-Corasick) automaton for degenerate pattern H = A[CT]A

0

1

2 3

4 5

A

C T

A A

0

1

2 3

4 5

A

C T

A

Pattern matching (Aho-Corasick) automaton for degenerate pattern H = A[CT]A

Clumps: ACA, ATA, ACACA, ACATA,….

A

Overlap walking automaton* for H = A[CT]A

0

1

2 3

4 5

A

C T

A A

Clumps: ACA, ATA, ACACA, ACATA,….

0

ACA

5

ACA ATA

CA TA

CA

TA

* Regnier M., 2014

Pattern matching automaton for H = A[CT]A

Overlap walking automaton Overlap walking automaton

4

ATA

We propose a minimization of overlap walking automaton for degenerate patterns

Pattern matching automata minimization degenerate pattern H = [AT][CG][AC]

[AT]

[CG]

[A] [C]

0

1

2

3 4

Minimal pattern matching automaton degenerate pattern H = [AT][CG][AC]

This automaton can be constructed in linear time of its states

RR-equivalence-equivalence

Nodes x and y are R-equivalent (x R~ y) iff x = y or1. |x|=|y|;2. suffix_link(x) R~ suffix_link(y).

For degenerate patterns, the nodes of the same length have the same paths below

Two words are R-equivalent iff they are Nerode-equivalent

Minimal overlap walking automaton for H = [AT][CG][AC]

Clumps: [AT][CG]A, [AT][CG]C, [AT][CG]A[CG]A, [AT][CG]A[CG]C,….

0

3 4

[AT][CG]A [AT][CG]C

[CG]A

Minimal pattern matching automatonfor H = [AT][CG][AC]

[AT]

[CG]

A C

0

1

2

3 4

[CG]C

Efficiency demonstrating examplesEfficiency demonstrating examples

• H = LXDXLXD[DLE] (amino acid alphabet)PatAut: 40841 states and 81681 edgesR-minimal PatAut: 25 states and 59 edgesMinimal OWA: 6 states and 45 edges

• H = AXXXXCATTA (DNA alphabet )PatAut: 1622 states and 3243 edgesR-minimal PatAut: 64 states and 140 edgesMinimal OWA: 2 states and 3 edges

MerciMerci