incremental inference of relational motifs with a degenerate alphabet nadia pisanti, lipn paris 13...

Incremental Inference of Relational Motifs with a Degenerate Alphabet

Nadia Pisanti, LIPN Paris 13 & ABI Paris 6

joint work with:

H.Soldano, LIPN Paris 13 & ABI Paris 6M.Carpentier, ABI Paris 6

CPM 2005

Summary

Relational Motifs: The model. A few motivations.

Previous work: KMR: the idea of the paradigm. KMRC: using a degenerate alphabet &

maximality. KMRELAT=KMRC + “relations”. Problems.

The algorithm: The idea. Properties that guarantee correctness and

efficiency. Preliminary tests on 3D Proteins.

Relational Motifs: the idea

Motif TCAGTCTCA

occurrences

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

Alphabet: {A,C,G,T}Length: 9Quorum: 5Sequence

IN

OUT

“normal” motifs:

Relational Motifs: the idea

Motif TCAGTCTCA

occurrences

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

TCAGTCTCA

Alphabet: {A,C,G,T}Length: 9Quorum: 5Sequence

IN

OUT

Alphabet: natural numbersLength: 3Quorum: 4Relations alphabet: {<,>,=}.Sequence and pairwise relations

IN

OUT

relational motifs (trivial case: only relations):

“normal” motifs:

252

161 161

151

151Relations: (p1,p2) <, (p2,p3) >, and (p1,p3) =

Music: detecting scales and tunes in different keys.

3D structures of proteins: amino acids in 3D space and pairwise distance as relation.

Motif inference in structured texts: data structures, source codes.

Numbers and arithmetical relations. Events and temporal relations.

Some possible applications

KMR (1972): concatenation of short motifs into long ones

Ex: Length k with quorum 3

Length k/2:

occurs at b1,b2,b3,...

occurs at r1,r2,r3,...

occurs at y1,y2,y3,...

y1

y2

b1

b8 r7

r2b2 b3r1 r3

y7 r8

distance k/2 = | | = | |



Length k/2:

occurs at b1,b2,b3,...



y1

y2

b1

b8 r7

r2b2 b3r1 r3

y7 r8

distance k/2 = | | = | |, hence = occurs at y7and | | = k



Length k:

occurs at y2,y7,...

y1

y2

b1

b8 r7

r2b2 b3r1 r3

y7



Length k:

occurs at y2,y7,...

occurs at y1,...

y1

y2

b1

b8 r7

r2b2r1 r3

y7



Length k:

occurs at y2,y7,...

occurs at y1,...

y1

y2

b1

b8

r2b2 r3

y7

occurs at b1,b8.



Length k:

occurs at y2,y7,...

occurs at y1,...

y1

y2

b1

b8 y7

occurs at b1,b8.



Length k:

occurs at y2,y7,...

occurs at y1,...

y1

y2

b1

b8 y7

occurs at b1,b8.no quorum



Length k:

occurs at y2,y7,...

occurs at y1,...

y1

y2

y7

OUTPUT:

With O(log k) steps all k-motifs are generated



Length k:

occurs at y2,y7,y5

occurs at y1,...

y1

y2

y7

OUTPUT:

extent of y5

KMRC: Degenerate AlphabetFinding exact motifs is not enough for certain applications...

(nor so challenging!)

Motif {C,T}{C,G}A{C,T}A

CCATA

TCACA

TGATA

TCACACGACA

TCATATGATA

Length: 5Quorum: 5Input Sequence (hence implicitely {A,C,G,T})

&Motif’s alphabet (cover of ): {A},{C,G},{C,T}

IN

OUT

KMRC: Degenerate AlphabetFinding exact motifs is not enough for certain applications...

(nor so challenging!)

Motif {C,T}{C,G}A{C,T}A

CCATA

TCACA

TGATA

TCACACGACA

TCATATGATA

Length: 5Quorum: 5Input Sequence (hence implicitely {A,C,G,T}).

&Motif’s alphabet (cover of ): {A},{C,G},{C,T}

IN

OUT

Alphabet with degeneracy 2

Sequence alphabet: amino acids of primary structure.

Motifs degenerate alphabeth: grouping amino acids with similar chemical properties.

Relations: distance of -carbons in 3D tertiary structure of the protein.

Relations degenerate alphabet: e.g. discretizing the distance.

Repeated structures in 3D proteins

Relational motifs with degenerate alphabet(one for symbols and one for relations)

1

2

3

4

8

76

955

3

4

3

3

44

3

2

44 6

r(1,4) = 5r(1,3) = 4 r(2,4) = 3r(1,2) = 3 r(2,3) = 4 r(3,4) =3

r(6,9) = 6r(6,8) = 3 r(7,9) = 4r(6,7) = 4 r(7,8) = 4 r(8,9) =2

Motifs & Extents

In AAAAAAAAAAAAAAAAAAAAA ... AAAAAAAAAAAAAAAAAAAAAAAAAAAA = An

• Every k-long word on {C1,C2,C3} is a different k-motif!• Each one of them has extents {1,2,3,…,n-k+1} (indeed, it’s always the same…)

Hence {1,2,3,…,n-k+1} represents gk motifs!

E.g. Motif’s alphabet C1={A,C},C2={A,G},C3={A,T} with degeneracy g=3

A k-motif is a k-long word on {C1,C2,C3} (occurring q times)

Different motifs can occur at the same position...

Even worse: two different motifs may have the very same extents

Maximal motifs

Motif C2 C2 C1 C2 C1 occurs at p3,p4,p5,p6,p7 non maximal

CCACA

CCACA

CGATA

CCACACGACA

CCACACGATA

e.g. motif’s alphabet: C1={A},C2={C,G},C3={C,T}

p1 p2p3

p4p5

p6 p7

Motif C2 C2 C1 C3 C1 occurs at p3,p4,p5,p6,p7 and in p1,p2 maximal

Maximal motifs: good and bad news

Good news: Each maximal k-motif can be built from two

maximal (k/2)-motifs. Bad news:

Two maximal (k/2)-motifs can generate a non-maximal motif.

Non-maximal motifs have to be detected and discarded at each step.

Very bad news: There can be an exponential number of

maximal motifs (theoretically).

KMRelat (2003): introducing relations


Length k/2:





Length k/2:

occurs at r1,r2,r3,... and relations are conserved

occurs at y1,y2,y3,... and relations are conserved



Length k:

occurs at y1,y2,y3,... and SOME relations are conserved



Length k:

occurs at y1,y2,y3,... and SOME relations are conserved

There are still O(k2) relations to be checked.. per each occurrence... and at each step...

Overlap steps

Length k-d

d

d

Overlap steps

Length k-d

d

d

Length k

Overlap steps

Length k-d

d

d

Length k

Only O(d2) relations to check where d is a constant

Why KMR+overlap

It takes O(k) steps, each one taking O(n), hence O(kn) [regardless the degenerate alphabet].

Possible alternatives: KMR would take O(log k) steps with step i

concatenating two 2i-motifs and checking (2i)2 relations, that is

i=1 n * 22i = ... = O(k2n). With an in depth approach (not KMR-like) it would

take O(n) steps where at step i an i-motif is extended of one position and i relations are checked, that is

i=1 n * i = O(k2n).

(log2 k)-1

k-1

KMRoverlap and relations

Inferring relational k-motifs with degenerate alphabets performing overlap steps:

Maximal motifs still suffice. No need to explicitely store relations: the

extents suffices still. Relations refine the query and thus reduce

the search space and the output size. More sensitive motifs inference.

KMRoverlap: sketch of the algorithm

l:=1;REPEATOverlap two relational l-motifs;Check relations and generate as many relational

(l+d)-motifs as conserved ones;Check quorum;Eliminate non maximal;l:=l+d;UNTIL l=k.

KMRoverlap: sketch of the algorithm

l:=1;REPEATOverlap two relational l-motifs;Check relations and generate as many relational

(l+d)-motifs as conserved ones;Check quorum;Eliminate non maximal;l:=l+d;UNTIL l=k.

... but there is a problem...

Pseudomotifs

Motif’s alphabet: C1={a,b}, C2={b,c}, and C3={x}. Input sequence =xbxcxaxbxc. Quorum q=2 and length k=3.

1 2 3 4 5 6 7 8 9 0

An example:

xbxcxaxbxcx

b c

aC1

C2

C3

Pseudomotifs


1 2 3 4 5 6 7 8 9 0

An example:

xbxcxaxbxcx

b c

aC1

C2

C3

The extent {1,7} corresponds to xbx occurring twice (so far so good...)and corresponding to the motif C3 C1C2 C3 (strange..)

Pseudomotifs


1 2 3 4 5 6 7 8 9 0

An example:

xbxcxaxbxcx

b c

aC1

C2

C3

The extent {1,7} corresponds to xbx occurring twice (so far so good...)and corresponding to the motif C3 C1C2 C3 (strange..)

• C3 C1 C3 has extent {1,5,7}{1,7}

• C3 C2 C3 has extent {1,3,7}{1,7}

Pseudomotifs


1 2 3 4 5 6 7 8 9 0

An example:

xbxcxaxbxcx

b c

aC1

C2

C3

The extent {1,7} corresponds to xbx occurring twice (so far so good...)and corresponding to the motif C3 C1C2 C3 (strange...)• C3 C1 C3 has extent {1,5,7}{1,7}

• C3 C2 C3 has extent {1,3,7}{1,7}

C3 C1C2 C3 is a pseudomotif... and {1,7} a pseudoextent

Pseudomotifs


1 2 3 4 5 6 7 8 9 0

An example:

xbxcxaxbxcx

b c

aC1

C2

C3

The extent {1,7} corresponds to xbx occurring twice (so far so good...)and corresponding to the motif C3 C1C2 C3 (strange...)• C3 C1 C3 has extent {1,5,7}{1,7}

• C3 C2 C3 has extent {1,3,7}{1,7}

C3 C1C2 C3 is a pseudomotif... and {1,7} a pseudoextent

The extent {2,8} corresponds to C1C2 C3 C2, but it is also the extent of the 3-motif C2 C3 C2 {2,8} is not a pseudoextent.

Pseudomotifs

Why are pseudomotifs dangerous? The overlap of two maximal (k-d)-motifs can

generate a pseudomotif. There can be O(2|G|k) distinct pseudomotifs of length

k. Pseudomotifs can never be maximal. Thus:

They will never have to be output. They will never be useful to generate longer motifs.

We need to find a way to avoid generating them...

Storing prefixes and suffixes

Length k-d

d

d

Length k-d

d

d

Length k


prefix

suffix

Length k-dLength k


prefix

suffix

also of inherited motifs

prefix

suffix

Length k-dLength k


prefix

suffix


prefix

suffix

The extent of is included in that of

Length k-dLength k


prefix

suffix


prefix

suffix


Hence is eliminated and inheritates it.

Length k-dLength k


prefixes

suffixes



Hence is eliminated and inheritates it.

Avoiding pseudomotifs

Length k-d

d

d

with prefixes in the set P


with suffixes in the set S


Avoiding pseudomotifs

Length k-d

d

d





Length kis generated iff S P

Some interesting properties

The prefix-suffix condition avoids generating exactly pseudomotifs!

It is enough that ONE maximal motif inheritates the prefix and suffix of a discarded one:

Only (or ) inheritates .

The KMRoverlap algorithm

endwhile;Output all left motifs.

l := 1;while l < k do

for each l-motif occurring at x and l-motif occurring at x+d do:

If S P then generate a per each different set of conserved relations;

Eliminate extents that are < q;Eliminate nonmaximal extents;

l := l + d;

KMRoverlapEx: Length k with quorum 2

Length k-d:

d


Length k-d:

d

Overlap and


Length k-d:

d...

...

...

check relations and generate asmany as relations sets.

Overlap and :


Length k-d:

d...

...

...

Length k:

occurs twice

occurs once

check relations and generate asmany as relations sets.

Overlap and :


Length k-d: Length k:

occurs twice

occurs once

Overlap and

...

...

occurs twice


Length k-d: Length k:

occurs twice

occurs once

Overlap and

occurs twice

occurs twice

occurs twice

......

...

...


Length k:

occurs twice

occurs once Check quorumoccurs twice

occurs twice

occurs twice


Length k:

occurs twice

They are all maximal(just a coincidence to simplify!)

occurs twice

occurs twice

occurs twice


Output:

occurs twiceoccurs twice

occurs twice

occurs twice

Complexity

O(k) steps. At each step i there are O(ngl) motifs of length

l. Generating new motifs takes O(n gl). Detecting possible inclusions takes O(n g2l).

Overall complexity in O(k n g2k), [linear w.r.t. input size but still looks bad]

but it is a very rough approximation...

To be precise...

There are two degeneracies: g and g’ At each step i there are O(ngl) motifs of length l.

Generating new motifs takes O(n gl + (g’)2l). Detecting possible inclusions takes O(n (g+(g’)2l)2).

Overall complexity in O(k n (g+(g’)2k)2),[linear wrt input size but exponential in k] but it even a more rough approximation...

KMRoverlap: correctness and completeness

The algorithm is correct (it generates ONLY maximal k-motifs) because: Non maximal are discarded. It stops when k-motifs are generated.

The algorithm is complete (it generates ALL maximal k-motifs) because: Overlapping two maximal (k-d)-motifs is enough to generate all maximal k-motifs. The prefix-suffix condition only discards pseudomotifs.

Preliminary tests

8973 0 124

81757 76953 1110

550911 1881241 936

165727 1673186 167

12502 15668 27

4 5

3 4

6 7

5 6

7 8

generatedmotifs

avoidedpseudo-motifs

maximalmotifsoverlap step

k = 8d = 1q = 5n ~103

• As expected there are many pseudo-motifs.

• Alhought they have the same theoretical upper bound, the number of maximal k-motifs is sensibly smaller than that of the k-motifs.

THANK YOU

incremental inference of relational motifs with a degenerate alphabet nadia pisanti, lipn paris 13...

Documents

quorum slide

length k2

concatenation of short

y1 y2 y7 output

y1 y2 b1 b8y7

normal motifs

y1 y2 b1 b8 r2b2 r3

exact motifs