IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND …alpha.uhasselt.be/frank.neven/papers/miningmdl.pdf · 2013-07-05 · Mining minimal motif pair sets maximally covering interactions

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, VOL. ???, NO. ???, ??? 2010 1

Mining minimal motif pair setsmaximally covering interactions

in a protein-protein interaction networkPeter Boyen, Frank Neven, Dries Van Dyck, Felipe L. Valentim, and Aalt D.J. van Dijk

Abstract—Correlated motif covering (CMC) is the problem of finding a set of motif pairs, i.e., pairs of patterns, in the sequences ofproteins from a protein-protein interaction network which describe the interactions in the network as concisely as possible. In otherwords, a perfect solution for CMC would be a minimal set of motif pairs which describes the interaction behavior perfectly in the sensethat two proteins from the network interact if and only if their sequences match a motif pair in the minimal set. In this paper, we introduceand formally define CMC and show that it is closely related to the Red-Blue Set Cover (RBSC) problem and its weighted version(WRBSC) — both well-known NP-hard problems for which there exist several algorithms with known approximation factor guarantees.We prove the hardness of approximation of CMC by providing an approximation factor preserving reduction from RBSC to CMC. Weshow the existence of a theoretical approximation algorithm for CMC by providing an approximation factor preserving reduction fromCMC to WRBSC. We adapt the latter algorithm into a functional heuristic for CMC, called CMC-approx, and experimentally assess itsperformance and biological relevance.The implementation in Java can be found at http://bioinformatics.uhasselt.be

Index Terms—Graphs and networks; Biology and genetics; Correlated motifs; PPI networks; Local search;

F

1 INTRODUCTION

P ROTEINS form the basic building blocks of all livingorganisms. They perform all basic functions to which end

they physically bind together, forming different complexes.These interactions occur via connections between parts ofthe respective proteins. Protein residues involved in suchconnections are called binding residues or interface residues.Knowledge of binding sites is critical for the prediction ofnovel protein-protein interactions, understanding of the evo-lution of protein interactions, or for the creation of drugs totarget a specific protein. As large-scale biological networksare available, incorporating an ever increasing number ofinteractions within the same organism [1], several computa-tional approaches have been proposed to locate binding sitesby mining overrepresented patterns in the interacting proteinpairs [2], [3], [4]. In that setting, patterns are instantiated asmotif pairs X,Y which select the subnetwork induced bythe proteins matching either X or Y and several successfulcomputational methods have been devised to search for highscoring motif pairs.

Unfortunately, the best scoring motif pairs often refer tohighly similar selected networks. For instance, among the best

• P. Boyen and F. Neven are with Hasselt University and TransnationalUniversity of Limburg.E-mail: peter.boyen, [email protected]

• D. Van Dyck is with ICR, Advanced Nuclear Systems of Belgian NuclearResearch Centre (SCK-CEN).E-mail: [email protected]

• F. Valentim and A. van Dijk are with Applied Bioinformatics - PlantResearch International, Wageningen UR.E-mail: felipe.lealvalentim, [email protected]

1 000 motif pairs found using a brute force approach in thePPI-network of Human (a network consisting of 8 872 nodes)only 598 refer to dissimilar subnetworks [2] (SupplementalMaterial). This effect becomes even worse when consideringmore specific motif pairs (i.e., restricting the number of al-lowed wildcards). For example, in Yeast (a network consistingof 1 620 nodes) out of the 1 000 best motif pairs only 382refer to distinct networks. Every found motif pair selects asubnetwork identical to one of these, even to the point ofhaving the same motif positions in the proteins. Moreover,using all 1 000 motif pairs, only 5% of all nodes and 8% ofall edges of the Yeast network are described. As a result onlybinding sites of proteins in a small, dense part of the networkcan be found.

In the present paper, we therefore take a radically differentapproach where instead of scoring the power of a patternto explain interactions on an individual basis as in [2], [3],[4], [5], [6], [7], we score the explicative power of a set ofmotif pairs as a whole. In essence, we target a minimal set ofmotif pairs M which covers a maximal part of the network.To balance the minimality of M with the ‘goodness’ of Mrepresenting the network, we employ a minimum descriptionlength measure to score sets of motif pairs. We formalize thelatter as the Correlated Motif Covering (CMC) problem.

We first analyze CMC from a computational viewpoint.We prove that CMC is NP-hard and that it belongs to aclass of problems for which it is practically impossible tofind “good” approximation algorithms. Here, “good” meansapproximations with a sub-square root approximation rate.More accurately, we show that, unless P = NP, it is impossibleto achieve a polynomial-time algorithm for CMC with anO(2log1−δ

√n) approximation ratio, with δ = 1 / log logcn,

0000–0000/00/$00.00 c© 2010 IEEE Published by the IEEE Computer Society


for any constant 0 < c < 1/2, with n the size of theinput. Specifically, we reduce the Red-Blue Set Cover (RBSC)problem, a well-known combinatorial problem, to CMC usingpolynomial time reductions that maintain constant factors ofapproximation. On the other hand, we exhibit a reductionfrom CMC to the weighted Red-Blue Set Cover (WRBSC)problem allowing to transfer known approximation algorithmsfor the latter to CMC. Unfortunately, due to the large sizesof biological networks none of the available algorithms forWRBSC remain feasible when directly adapted for CMC. Wetherefore introduce the novel heuristic CMC-approx for CMCwhich is based on an approximation algorithm by Peleg [8]for WRBSC. Although the adaptation no longer guaranteesthe same approximation rate, we experimentally assess itsmerits by comparing it to two alternative algorithms for CMC.The first alternative algorithm is SEQ-SLIDER [2] which isconsidered as a baseline as it simply mines for the k best motifpairs in a network which are then considered to form a cover.The second alternative is the expected greedy algorithmicsolution to CMC. We refer to the latter algorithm as CMC-greedy.

In an experimental validation on biological networks, weconfirm that the network coverage of CMC-approx is indeedmuch larger than that of SEQ-SLIDER and CMC-greedy atthe expense of a slightly increased running time. To furtherevaluate the biological relevance of CMC-approx, we tested theeffectiveness of the derived motif pairs in several scenarios.In a first scenario, we determine the overlap of our foundmotif pairs with actual interaction sites in the human andyeast network for which there is 3D-structure informationavailable. We obtain that the coverage (the number of proteinsand interactions covered) drastically increases (at the expenseof a slightly lower precision) compared to the baseline, therebygreatly improving the utility for experimental biologists whowant to predict binding sites in order to perform further exper-imental studies and for whom it obviously is very important toobtain predictions for as much proteins as possible. In a secondscenario, we investigate how cross-species motif pair miningcan be used as a filtering step to increase accuracy. That is,we provide evidence for the hypothesis that motif positionswhich are found in more than one species through mining ofmotif pairs are more likely to be interface residues. This meansthat a cross-species comparison could be used to filter noisefrom the predictions. Finally, we consider the prediction ofprotein interactions from two-dimensional sequences, whereCMC-approx is shown to slightly outperform the two otheralgorithms.

Outline. We formally introduce the Correlated Motif Cov-ering (CMC) problem in Section 2 and show its complexityin Section 3. Our algorithms are presented in Section 4. InSection 5 we describe the used data sets, followed by avalidation that the algorithms achieve their intended purpose(increased coverage) in Section 6 and a validation of theirbiological meaningfulness in Section 7. Finally, we discussrelated work in Section 8 and conclude in Section 9.

2 CORRELATED MOTIF COVERING (CMC)PROBLEMIn this section, we define the Correlated Motif Covering (CMC)problem. We use notation from Boyen et al. [2].

2.1 Basic definitionsTable 1 in the supplementary material explains the meaningof the most frequently used symbols.

A protein-protein interaction network (PPI-network) can berepresented as a labeled graph G = (V,E, λ), with protein setV , interaction set E ⊆ V ×V , and a labeling λ : V → L, withL a finite alphabet of labels. As we label every protein with itsamino acid sequence, the label function λ maps each vertexv ∈ V to a string λ(v) over the alphabet Σ = A, . . . , Z \B, J, O, U, X, Z.

An (`, d)-motif is a string of length ` over the alphabetΣ ∪ x containing exactly d x-characters. The character x

is interpreted as a wildcard-symbol, i.e., it matches with anycharacter of Σ. For instance, GAQPRNMY matches the (8, 3)-motif GAxPxNxY. We do not allow motifs to start with awildcard-symbol, eliminating a lot of redundant motifs. In thesequel, we sometimes refer to (`, d)-motifs simply as motifs,when parameters ` and d are clear from the context.

A protein contains an (`, d)-motif X if its amino acidsequence contains a substring of length ` that matches X . Aprotein pair contains a motif pair, if one protein contains onemotif, and the other protein contains the other. Conversely, wesay a motif (pair) selects a protein (pair) if the protein (pair)contains it.

A motif pair M = X,Y now selects a subnetwork of aPPI-network G = (V,E, λ) as follows. Let VX = v ∈ V |v contains X, be the set of proteins in the network containingthe motif X , and EM = EX,Y = u, v ∈ E | u ∈ VX ∧v ∈VY be the set of interactions between proteins containingX and proteins containing Y . Similarly, we define the setof anti-edges AM = AX,Y = u, v 6∈ E | u ∈ VX ∧v ∈ VY , the set of protein pairs selected by X,Y withoutinteractions (the false postive interactions for X,Y ). Thesubgraph GM = GX,Y selected by X,Y is then

GX,Y := (VX ∪ VY , EX,Y , λ|VX∪VY )

with λ|VX∪VY the restriction of λ to VX ∪ VY .The amount of (`, d)-motif pairs is given by

MP(`, d) =n(n− 1)

2+ n =

n(n+ 1)

2,

where n =(`−1d

)|Σ|(`−d) is the number of different (`, d)-

motifs. Here, n(n−1)2 is the number of motifs of the form

X,Y with X 6= Y and n is the number of motifs X,X.By |G| we denote the size of G, defined as |V | + |E| +∑

v∈V|λ(v)|, where |S| denotes the number of elements in a set

S and |s| denotes the length of a string s.In Fig. 1, we see an example PPI-network G = (V,E, λ),

where λ is not represented.1 The letters A, B, C, and D refer

1Although the graph in Fig. 1 is kept bipartite for illustration purposes,we do not restrict the form of considered PPI-networks.


A B

B

A

C

D

C D

1

2

3

4

5

6

Fig. 1. An example protein-protein interaction network

to motif matches. For instance, proteins 1 and 3 contain themotif A, and the motif A therefore selects those proteins. Ifyou pair either protein 1 or 2, with protein 4, both proteinpairs contain the motif pair B,C, though only one hasan interaction. The subgraph GB,C consists of proteins 1,2 (containing B), 4, 6 (containing C) and the three edgesbetween them.

2.2 Scoring an individual motif pair

A support measure f is simply a function mapping a motifpair X,Y and a PPI-network G to a positive real numberf(X,Y , G). We refer to f(X,Y , G) as the support ofX,Y in G. Support measures should reflect the powerof a motif pair to describe interactions. That is, we areinterested in those motif pairs for which the selected networkresembles large bicliques (i.e., complete bipartite graphs).Several support measures to this end have been investigatedincluding the χ2-score [3] and the p-score [4].

The Correlated Motif Mining (CMM) problem is then de-fined as follows:

The Correlated Motif Mining problem (CMM)• Input: A PPI-network G, three numbers`, d, k ∈ N+, with d < ` and a supportmeasure f

• Output: the k (`, d)-motif pairs X1, Y1,. . . ,Xk, Yk with highest support in G with respectto f

Boyen et al. [2], Leung et al. [4], and Tan et al. [3] proposedmethods for CMM which succeed in finding the best scoringmotif pairs. Unfortunately, as explained in the introduction, thebest scoring motif pairs often refer to highly similar selectednetworks, a shortcoming we address in the next section.

2.3 Covering a graph with a set of motif pairs

Rather than scoring motif pairs on an individual basis as inCMM, we now formalize how to score the explicative power ofa set of motif pairs as a whole. To this end, let M be a set ofmotif pairs. Then the graph GM = (VM, EM, λM) induced

by M on G is defined as

GM :=⋃

X,Y ∈M

(VX ∪ VY , VX × VY , λ|VX∪VY ),

where the union of two consistently labeled graphs G1 =(V1, E1, λ1) and G2 = (V2, E2, λ2) is simply G1,2 = (V1 ∪V2, E1 ∪E2, λ1 ∪λ2). Note that GM might contain edges notpresent in G, contrary to the case for GM for a motif pairM . For example, if we were to take GB,C in Fig. 1, weget the network containing proteins 1, 2, 4, 6, with four edgesconnecting those proteins, even though in G and GB,C onlythree of those edges are present.

For a result set M to have little redundancy and highcoverage, we wantM to be small while maximizing the simi-larity between GM and G. Therefore, we adopt the minimumdescription length (MDL) principle [9] which embraces theslogan of Induction by Compression. More specifically, wecan compress G by GM encoded as M. We then only needto list the false positive and false negative edges. That is, theedges present in EM but not in EG, and the edges missingin EM but present in EG. The size of M, denoted |M| issimply counted as the number of motif pairs occurring in it.More formally, we calculate the size of the latter compressionrelative to three nonnegative numbers α, β, and γ as follows

costα,β,γ(M, G) := α|M|+ β|EM \ EG|+ γ|EG \ EM|.

The table below lists some example solutions for the networkin Fig. 1:

M costα,β,γ(M, G)∅ 7γ

B,C α+ β + 4γB,C, A,D 2α+ 2β + 2γA,C, B,D 2α

The last line contains a solution that describes the interactionsperfectly, and, given reasonable values for α, β, and γ, shouldbe the optimal solution.

In the experimental section, following the MDL principle,we will set α to the number of bits to represent a motif pairand β = γ to the number of bits to represent an interaction.

We are now ready to define the problem central to this paper:

The Correlated Motif Covering problem (CMC)• Input: A PPI-network G, numbers `, d ∈ N+ withd < ` and α, β, γ ∈ Q+

• Output: a set M of (`, d)-motif pairs minimizingcostα,β,γ(M, G).

3 COMPLEXITY AND APPROXIMATION OF CMC

We show that CMC is NP-hard and that there are limits tohow well it can be approximated within polynomial time.On the positive side, we show that there are algorithms thatprovide approximation guarantees that are within those limitsby establishing an approximation factor preserving reductionto the weighted Red-Blue Set Cover (WRBSC) problem.


3.1 Lower boundsIn this section, we show that CMC is NP-hard and that itis hard to approximate. Specifically, we show that, unlessP = NP, it is impossible to achieve a polynomial-time algo-rithm for CMC with an O(2log1−δ

√n) approximation ratio, with

δ = 1 / log logcn, for any constant 0 < c < 1/2, with nthe size of the input.

We start by reducing the Red-Blue Set Cover (RBSC) prob-lem to CMC using approximation factor preserving reductions.

First, we define RBSC [10]. The input consists of a universeU of elements which are either red or blue. In addition, a setS of subsets of U is given. The objective is to choose setsfrom S covering all blue elements while selecting as few redelements as possible.

Formally, we define the Red-Blue Set Cover (RBSC) prob-lem [10] as follows:• Input: Finite sets U,B,R,S, with U = B∪R, B∩R = ∅,

and S ⊆ 2U , with⋃S∈S

S = U.2

• Output: a set S∗ ⊆ S such that B ⊆⋃S∈S∗

S minimizing

the cost function

costRBSC(S∗, B,R) := |⋃S∈S∗

S ∩R|.

To obtain the lower bounds for CMC, we make use of areduction from RBSC to CMC that preserves not only constantfactor approximability but also the constant itself. Therefore,we will need the following notion:

Definition 1: [11] Let Π1 and Π2 be two minimiza-tion problems. An approximation factor preserving reduction(AFP-reduction) from Π1 to Π2 is a pair of PTIME functions(f, g) such that:• for any instance I1 of Π1, I2 = f(I1) is an instance of

Π2 such that OPT2(I2) ≤ OPT1(I1), where OPT1 (resp.OPT2) is the quality of an optimal solution to I1 (resp.I2), and

• for any solution s2 to I2, s1 = g(s2) is a solution toI1 such that obj1(s1) ≤ obj2(s2), where obj1() (resp.obj2()) is a function measuring the quality of a solutionto I1 (resp. I2).

By Π1 ≤AFP Π2, we denote that there exists an AFP-reduction from Π1 to Π2.

Throughout the remainder of this section, objΠ is the costfunction of problem Π and OPTΠ(I) equals the cost of anoptimal solution for instance I of problem Π.

The proofs in the rest of this section are based on thefollowing lemma which is an immediate consequence of thedefinition of an AFP-reduction (and is left as an exercise in[11]):

Lemma 1: If there exists an approximation algorithm withapproximation factor α for problem Π1, and Π2 ≤AFP Π1,

2We note that the standard definition of RBSC does not require⋃S = U .

The latter restriction ensures a solution to always exist and is equivalent tostandard RBSC in that case. Indeed, when blue balls are missing from

⋃S then

there is no solution. Missing red balls can always be added as an additionalset without affecting the cost of an optimal solution (a set containing onlyred elements can never belong to an optimal solution).

then there exists an approximation algorithm with approxima-tion factor α, for problem Π2.

In the remainder of this section, we will prove the followingtheorem.

Theorem 1: CMC is NP-hard and unless P = NP, there isno polynomial-time algorithm for CMC with an O(2log1−δ

√n)

approximation ratio, with δ = 1 / log logcn, for any constantc < 1/2, with n the size of the input.

We need the following lemma:Lemma 2: RBSC ≤AFP CMC.

Proof: (of Lemma 2) Let IRBSC = (U,B,R,S) be aninstance of RBSC. Let S = S1, . . . , S|S|. We start bydefining the function f mapping IRBSC to f(IRBSC) = ICMC =(G, `, d, α, β, γ). An example, graphically illustrating the func-tion f , is given in the supplementary material. Here, ` = |S|and d = |S| − 1. This means that motifs only have one non-wildcard character. Furthermore, set α = 0, β = 1, and γ toone more than the maximal number of red elements occurringin a set in S. That is, γ = max

S∈S|S ∩ R| + 1. It remains to

define G = (V,E, λ). Define V = U ∪ c ∪ O where c is anew element and O consists of 2γ|B|+1 new elements. Here,c is the center of the graph connected to all elements in B.Furthermore, O, and R consist of isolated vertices. That is, setE = b, c | b ∈ B. Although Σ contains the 20 charactersused to specify amino acids, G uses only three characters. Forease of exposition, we shall refer to three characters in Σ as0, 1, and 2. Then, λ(c) is a sequence of length |S| consistingonly of 2-characters; for each u ∈ U , λ(u) = c1 . . . c|S|, withci = 1 if u ∈ Si and 0 otherwise; and; for each o ∈ O, λ(o)is a sequence of length |S| consisting only of 0-characters.Clearly, f is PTIME-computable.

We next argue that OPTCMC(ICMC) ≤ OPTRBSC(IRBSC).First, we need some terminology concerning motif pairs. As` = d + 1, every motif contains precisely one non-wildcardcharacter σ. We refer to such a motif as a σ-motif. We saythat a motif pair X,Y where X is a σ-motif and Y is aσ′-motif, is a motif pair of type Jσ, σ′K.

Let sCMC = M be a solution to ICMC. Then objCMC(M) =α|M|+β|EM\EG|+γ|EG\EM| = |EM\EG|+γ|EG\EM|by choice of α and β. Assume M is optimal. Then M hasto cover all edges. That is, EG \ EM = ∅. Indeed, assumetowards a contradiction that b, c ∈ EG \EM and let b ∈ Si(b is always in at least one set, according to the definition ofRBSC). Then, we can extend M with a motif pair selectingb, c thereby decreasing its cost and contradicting the fact thatM is optimal. Indeed, denote by Mi the motif pair Xi, Yiwhere Xi is the 1-motif with 1 on the i-th position and Yi isthe 2-motif with 2 on the i-th position. Then

objCMC(M∪ Mi) =

= |EM∪Mi \ EG|+ γ|EG \ EM∪Mi|≤ (|EM \ EG|+ |Si ∩R|) + (γ|EG \ EM| − γ) (†)< |EM \ EG|+ γ|EG \ EM| (‡)= objCMC(M).

Here, (†) is an equality if Si contains only one blue elementnot yet covered, and (‡) follows as γ = maxS∈S |S ∩ R| +


1 > |Si ∩ R|. So, it follows that EG \ EM = ∅ and thatobjCMC(M) = |EM \ EG| when M is optimal.

We distinguish the following kinds of motif pairs:• We say that a motif pair is good when it is of type J1, 2K.

Note that every such motif pairs selects the subgraphconsisting of the center and all elements in some set ofS.

• We say that a motif pair is bad when it is of type J0, 2KNote that every such motif pairs selects the subgraph con-sisting of the center and all elements in the complementof some set of S, together with all the elements in O.

• Motif pairs which only contain σ-motifs with σ ∈0, 1, 2 but which are not good nor bad are called super-fluous. Specifically, they are of type J0, 0K, J1, 1K, J2, 2K,and J0, 1K. They are called superfluous as EM ∩EG = ∅for every superfluous motif pair M .

• A motif pair M = X,Y which is not good, bad, orsuperfluous is called empty. Specifically, it is of typeJσ, σ′K where σ 6∈ 0, 1, 2 or σ′ 6∈ 0, 1, 2. They arecalled empty because at least one of VX or VY is empty.

Next, we argue that the optimal solution M cannot containbad motif pairs. Furthermore, we will show that for everyoptimal solution M there is a reduced optimal solution M′which does not contain empty or superfluous motif pairs andonly a minimal number of good motif pairs which can bemapped in a one-to-one fashion on sets in S.

Indeed, assumeMbad to be the non-empty set of bad motifpairs in any solution M. Then

objCMC(M\Mbad)

= |EM\Mbad\ EG|+ γ|EG \ EM\Mbad

|≤ |EM\Mbad

\ EG|+ γ|EG|≤ |EM \ EG| − |O|+ 2γ|EG| (†)< |EM \ EG| (by def. of O)

≤ |EM \ EG|+ γ|EG \ EM|= objCMC(M).

Here, (†) follows as the motif pairs in Mbad select at leastall elements in O, and at most all the elements in |EG|.

Also, assume Md to be the non-empty set of superfluousmotif pairs in any solution M. Then

objCMC(M\Msup)

= |EM\Msup\ EG|+ γ|EG \ EM\Msup

|= |EM \ EG| − |EMsup |+ γ|EG \ EM| (†)≤ |EM \ EG|+ γ|EG \ EM|= objCMC(M).

Here, (†) follows from the fact that EMsup ∩ EG = ∅.Finally, assume Mempty to be the non-empty set of empty

motif pairs in any solution M. Then

objCMC(M\Mempty)

= |EM\Mempty\ EG|+ γ|EG \ EM\Mempty

|= |EM \ EG|+ γ|EG \ EM|= objCMC(M).

So, given an optimal set of motif pairsM, it cannot containany bad motif pairs. Furthermore, letM′ be obtained fromMby removing all empty and superfluous motif pairs, then M′is also optimal and contains only good motif pairs. Therefore,ifM is optimal objCMC(M) = objCMC(M′) = |EM′ ∩Red| =|EM ∩ Red| with Red = c, r | r ∈ R. Thus, each motifpair in M′ is of the form Xi, Yj where Xi is the 1-motifwith 1 on the i-th position and Yj is the 2-motif with 2 on thej-th position. We replace each motif pair Xi, Yj, with themotif pair Mi = Xi, Yi and remove any duplicates. Sinceany 2-motif selects only the central node, the solution stillselects the same node pairs, so its cost and thus its optimalityare maintained.

Each Mi selects exactly those u, c with u ∈ Si. Sincewe have shown that EG \ EM = ∅, we can create a solutionfor IRBSC by taking Si for each Mi ∈ M′. Also, becauseOPTCMC(ICMC) = |EM ∩ Red|, this solution for IRBSC has thesame cost as the optimal solution to ICMC. This solution is alsooptimal, because if there was a S∗ with a lower cost we couldcreate a set of motif pairs with that same cost (by taking Mi

for each Si ∈ S∗), which would be in contradiction with Mbeing optimal. This means OPTCMC(ICMC) = OPTRBSC(IRBSC),and so definitely OPTCMC(ICMC) ≤ OPTRBSC(IRBSC).

Let sCMC = M. We next define the function g mappingsCMC to g(sCMC) = sRBSC = S∗. Here, let Mgood be the setof good motif pairs in M. Define Sgood as the set of sets Sifor which there is a motif pair X,Y in Mgood, with X the1-motif with 1 on its ith position. For every b ∈ B, let Sb bean arbitrary set in S containing b. Then, define

Sextra :=⋃

b∈B,b6∈ ⋃S∈Sgood

S

Sb.

Finally, set

S∗ := Sgood ∪ Sextra.

Clearly, g is PTIME-computable.We next argue that objRBSC(sRBSC) ≤ objCMC(sCMC). Indeed,

objRBSC(sRBSC)

= |⋃S∈S∗

S ∩R|

≤ |⋃

S∈Sgood

S ∩R|+ (maxS∈S|S ∩R|)|Sextra|

≤ |EMgood∩ Red|+ γ|EG \ EMgood

|= objCMC(Mgood)

≤ objCMC(sCMC).

Note that the above reduction makes use of very specificmotifs, those with only one non-wildcard character, and spe-cific values for α and β. Although the reduction can be adaptedto allow for an arbitrary number of non-wildcard characters,


we can not exclude the existence of classes of parameters forwhich CMC does become tractable.

Since an AFP-reduction is a polynomial-time reduction andRBSC is an NP-hard problem, this lemma already shows thatCMC is NP-hard.

Lemma 3: If there is a constant δ, with 0 < δ < 1, forwhich there exists a polynomial-time algorithm for CMC withan O(2log1−δ

√n) approximation ratio, with n the size of the

input, then there exists a polynomial-time algorithm for RBSC

with an O(2log1−δn) approximation ratio, with n the size ofthe input.

Proof: (of Lemma 3) A polynomial-time algorithm forRBSC with an O(2log1−δn) approximation ratio is achieved bycomposing the AFP-reduction from the proof of Lemma 2 andthe polynomial-time algorithm for CMC.

It remains to discuss the approximation rate. First, we dis-cuss the size of the obtained CMC instance, which is |ICMC| =|(G, `, d, α, β, γ)| = O(|S||V |+ log(|S|) + log(|S|) + 1 + 1 +log(|R|)) = O(|S||V |), where |G| = O(|V | + |E| + |λ|) =O(|V |+ |λ|), because it is a sparse graph by construction and|λ| = O(|S||V |). So, O(|S||V |) = O(|S||U |2), because ofthe size of O.

The size of the RBSC instance is n = |IRBSC| =|(U,B,R,S)| = O(|U |+ |U |+ |U |+ |S||U |) = O(|S||U |).

Thus, the size of the CMC-instance created by the reductionis O(|S||U |2) = O(n2).

Since OPTCMC(ICMC) = OPTRBSC(IRBSC) andobjRBSC(sRBSC) ≤ objCMC(sCMC) (as shown by the proofof Lemma 2), it follows that our composed algorithm hasan approximation rate of O(2log((

√n)2)1−δ) = O(2log(n)1−δ).

Take a solution sCMC to an instance ICMC from the theorizedpolynomial-time algorithm for CMC. The algorithm has aO(2log1−δn) approximation ratio which means

objRBSC(sRBSC)

≤ objCMC(sCMC)

≤ O(2log1−δ√nCMC ) OPTCMC(ICMC)

= O(2log1−δ√nCMC ) OPTRBSC(IRBSC)

= O(2log1−δ√

(nRBSC)2) OPTRBSC(IRBSC)

= O(2log1−δnRBSC ) OPTRBSC(IRBSC).

Hence, the lemma follows.

RBSC was shown to be Ω(2log1−εn)-inapproximable byDinur et al. [12], Carr et al. [10], and Elkin et al. [13]. Dinuret al. [12] prove this inapproximability under the weakestassumption (P 6= NP).3 Specifically, they proved unless P =NP, it is impossible to achieve a polynomial-time algorithmfor RBSC with an O(2log1−δn) approximation ratio, withδ = 1 / log logcn, for any constant c < 1/2, withn the size of the input. So, since we know a polynomial-time algorithm for RBSC with an O(2log1−δn) approximation

3To be precise, Dinur and Safra [12] showed the mentioned inapprox-imability result for the MINIMUM-MONOTONE-SATISFYING-ASSIGNMENTproblem for formulae of depth 3, denoted by MMSA3, which is equivalentto RBSC as argued by Carr et al. ( [10], Section 3).

ratio is impossible unless P = NP, this means a polynomial-time algorithm with an O(2log1−δ

√n) approximation ratio is

impossible for CMC unless P = NP, which proves Theorem 1.

3.2 Upper bounds

In this section, we show that we can use existing approxi-mation algorithms to solve CMC. Specifically, we show thatCMC is polynomial time reducible to the weighted versionof the Red-Blue Set Cover problem using AFP-reductions.This allows to transfer known approximation algorithms (suchas the one given by Peleg [8]) from WRBSC to CMC, whilepreserving any constant factor approximations. In section 3.1,we have shown that such algorithms are unlikely to exist, butin Section 4.3 we will show that we can still use this AFP-reduction to obtain an approximation algorithm with a non-constant approximation guarantee.

We start by introducing WRBSC which is the weightedversion of RBSC (defined in Section 3.1), in which the red ele-ments have a nonnegative cost assigned to them. By choosingelements from S, the objective is to select all the blue elements(to which no cost is assigned), while incurring as little costby selecting red elements as possible.

Formally, we define the Weighted Red-Blue Set Cover(WRBSC) problem [10] as follows:• Input: Finite sets U , B, R, S with U = B ∪ R andB ∩ R = ∅, S ⊆ 2U , with

⋃S∈S

S = U , and a function

c : R→ Q+.• Output: a set S∗ ⊆ S such that B ⊆

⋃S∈S∗

S and that

minimizes the cost function

costWRBSC(S∗, B,R, c) :=∑

r∈(

⋃S∈S∗

S ∩R)

c(r).

By restricting the length of motifs to be at most logarithmicin the size of the network, we can reduce CMC to WRBSC usingapproximation factor preserving reductions.

Theorem 2: If ` ≤ log(|G|), CMC ≤AFP WRBSC.The rest of this section is dedicated to prove this theorem.First, we reformulate CMC on a more abstract level which

facilitates the reduction to WRBSC. Let U be a finite set ofelements divided into two disjoint subsets U = C ∪ I . Here,C contains the correct elements while I contains the incorrectones. Let S be a set of subsets of U . The relationship withCMC is as follows: Given a PPI-network G = (VG, EG, λG)and numbers ` and d, U corresponds to the set of all proteinpairs, C corresponds to the edges in G, while I are the non-edges (anti-edges). Finally, S contains precisely those sets ofprotein pairs defined by (`, d)-motif pairs X,Y . That is,

S =⋃X,Y

VX × VY .

We will show that if we restrict ` to its typical small values,CMC reduces to the following problem which we refer to asthe MDL-cover (MDL-COVER) problem:


• Input: Finite sets U , C, I with U = C ∪ I , C ∩ I = ∅,S ⊆ 2U and nonnegative numbers α, β, and γ

• Output: a set S∗ ⊆ S that minimizes the cost function

α|S∗|+ β|I ∩⋃S∈S∗

S|+ γ|C \⋃S∈S∗

S|.

Lemma 4: If ` ≤ log(|G|), CMC ≤AFP MDL-COVER.Proof: (of Lemma 4) Let ICMC = (G, l, d, α, β, γ) be an

instance of CMC. Let G = (V,E, λ). We start by definingthe function f mapping ICMC to f(ICMC) = IMDL-COVER =(U,C, I,S, α, β, γ). An example, graphically illustrating thefunction f , is given in the supplementary material. Here, α, β,and γ remain the same. Let C = E, and I = (V × V ) \ E,then U = C ∪ I . Lastly, define S =

⋃X,Y

VX × VY .

We argue that f is PTIME-computable. To construct S,we need to generate all possible motif pairs. The amount ofpossible (`, d)-motifs is given by the following equation

(`− 1

d

)|Σ|(`−d) ≤

`−1∑i=0

(`− 1

i

)|Σ|` = 2`−1|Σ|`.

If ` ≤ log(|G|), then this amount is polynomial in |G|.Since the number of motif pairs is polynomial in the numberof motifs, we can enumerate all motif pairs in time polynomialin the size of G. For each motif pair X,Y , we then needto find VX × VY . This can be done in polynomial time, bypassing over all the sequences.

Let sMDL-COVER = S∗. We next define the function g mappingsMDL-COVER to g(sMDL-COVER) = sCMC = (M). By construction,for any set Si ∈ S∗, there is at least one motif pair X,Y with VX × VY = Si. Call one of those motif pairs Mi. Then,define

M =⋃

Si∈S∗Mi.

By the same reasoning as above, it can be argued that g isPTIME-computable.

We next argue that objCMC(sCMC) ≤objMDL-COVER(sMDL-COVER). By construction, the sets insMDL-COVER contain the same correct and incorrect elementsthat are selected by the motif pairs in sCMC. That is, EG = C

and V × V \ EG = I . Further, EM =⋃S∈S∗

S. It is however

possible that multiple sets get represented by the same motifpair, i.e., |M| ≤ |S∗|. So,

objCMC(sCMC)

= α|M|+ β|EM \ EG|+ γ|EG \ EM|≤ α|S∗|+ β|EM \ EG|+ γ|EG \ EM|= α|S∗|+ β|

⋃S∈S∗

S \ C|+ γ|C \⋃S∈S∗

S|

= α|S∗|+ β|I ∩⋃S∈S∗

S|+ γ|C \⋃S∈S∗

S|

= objMDL-COVER(sMDL-COVER).

We next show that OPTMDL-COVER(IMDL-COVER) ≤OPTCMC(ICMC).

We first argue that if an optimal set of motif pairs Mcontains two motif pairs which select the same node pairs,we can remove one of them without removing optimality. Werefer to such motif pairs as duplicates. Indeed, assume M to bea motif pair in M that selects the same node pairs as anothermotif pair in M. Then,

objCMC(M\ M)= α|M \ M|+ β|EM\M \ EG|+ γ|EG \ EM\M|= α(|M| − 1) + β|EM \ EG|+ γ|EG \ EM|≤ α|M|+ β|EM \ EG|+ γ|EG \ EM|= objCMC(M).

So, for an optimal solution M, we can create a solutionM′ which contains no duplicates and is still optimal.

For convenience, we now construct the function g−1 thatmaps M′ to g−1(M′) = S∗, as

g−1(M′) = S∗ =⋃

Mi∈M∗Si.

Since |M′| = |S∗|,

objCMC(M′)= α|M′|+ β|EM′ \ EG|+ γ|EG \ EM′ |= α|S∗|+ β|EM′ \ EG|+ γ|EG \ EM′ |= α|S∗|+ β|

⋃S∈S∗

S \ C|+ γ|C \⋃S∈S∗

S|

= α|S∗|+ β|I ∩⋃S∈S∗

S|+ γ|C \⋃S∈S∗

S|

= objMDL-COVER(S∗).

This solution S∗ to IMDL-COVER is also optimal. Indeed,assume towards a contradiction that there exists S∗2 withobjMDL-COVER(S∗2 ) < objMDL-COVER(S∗). We have already shownobjCMC(g(sMDL-COVER)) ≤ objMDL-COVER(sMDL-COVER), so

objCMC(g(S∗2 ))

≤ objMDL-COVER(S∗2 )

< objMDL-COVER(S∗)= objCMC(M′),

which is in contradiction with M′, and thus M, being op-timal. Therefore, OPTMDL-COVER(IMDL-COVER) ≤ OPTCMC(ICMC).The lemma now follows.

Lemma 5: MDL-COVER ≤AFP WRBSC.Proof: (of Lemma 5) Let IMDL-COVER =

(UMDL-COVER, C, I,SMDL-COVER, α, β, γ) be an instanceof MDL-COVER. We start by defining the functionf mapping IMDL-COVER to f(IMDL-COVER) = IWRBSC =(UWRBSC, B,R,SWRBSC, c). An example, graphically illustratingthe function f , is given in the supplementary material. Here,


B = C, R = I ∪ rT | T ∈ SMDL-COVER ∪ rb | b ∈ C,where all rt and rb are new objects, and UWRBSC = B ∪ R.Define

c(r) =

α r ∈ rT | T ∈ SMDL-COVER;γ r ∈ rb | b ∈ C; and,β r ∈ I.

For every T ∈ SMDL-COVER, we define ST = rT ∪ T . Forevery b ∈ B, we define Sb = rb ∪ b. Finally, SWRBSC isthe following disjoint union:

SWRBSC =

( ⋃T∈SMDL-COVER

ST

)∪⋃b∈B

Sb. (?)

Clearly, f is PTIME-computable.We next define the function g mapping sWRBSC to

g(sWRBSC) = sMDL-COVER. Simply set

sMDL-COVER = T ∈ SMDL-COVER | ST ∈ sWRBSC. (??)

Clearly, g is PTIME-computable.We next argue that objMDL-COVER(sMDL-COVER) ≤

objWRBSC(sWRBSC). Indeed,

objWRBSC(sWRBSC)

=∑c(r) | r ∈ (

⋃S∈sWRBSC

S ∩R)

=∑c(r) | T ∈ sMDL-COVER, ST ∈ sWRBSC, r ∈ (ST ∩R)

+∑c(r) | b ∈ C, Sb ∈ sWRBSC, r ∈ (Sb ∩R)

=∑c(r) | T ∈ sMDL-COVER, r ∈ ((T ∪ rT ) ∩R)

+∑c(r) | b ∈ C, Sb ∈ sWRBSC, r ∈ (rb ∩R)

=∑c(r) | T ∈ sMDL-COVER, r ∈ (rT ∩R)

+∑c(r) | T ∈ sMDL-COVER, r ∈ (T ∩R)

+∑c(r) | b ∈ C, Sb ∈ sWRBSC, r ∈ (rb ∩R)

= α|sMDL-COVER|+ β|⋃

T∈sMDL-COVER

T ∩ I|

+γ|⋃

Sb∈sWRBSC

rb|

≥ α|sMDL-COVER|+ β|⋃

T∈sMDL-COVER

T ∩ I|

+γ|C \⋃

S∈sMDL-COVER

S| (†)

= objMDL-COVER(sMDL-COVER).

Here, (†) follows as a set Sb can be selected in sWRBSC, whileanother set already selects b.

We argue in the supplementary material thatOPTWRBSC(IWRBSC) ≤ OPTMDL-COVER(IMDL-COVER), whichcompletes the proof of the lemma.

Theorem 2 now follows from Lemma 4 and Lemma 5and the fact that AFP-reductions are closed under composi-tion [11].

Input: PPI-network G = (V,E, λ), `, d ∈ N+, d < `, and α,β, γ ∈ Q+, TIME ∈ N+

Output: best cover M according to costα,β,γ found in G1: M← ∅2: cost ← costα,β,γ(M, G)3: oldCost ←∞4: while cost < oldCost do5: oldCost ← cost6: bestMotifPair = undef7: startTime = getTime()8: while getTime() < startTime + TIME do9: M ← localSearch(G, `, d,M, costα,β,γ)

10: if costα,β,γ(M∪ M, G) < cost then11: bestMotifPair ←M12: cost ← costα,β,γ(M∪ M, G)13: if cost < oldCost then14: M←M∪ bestMotifPair

Fig. 2. The algorithm CMC-greedy.

4 ALGORITHMS

In the next three subsections, we present three algorithms forCMC; (i) SEQ-SLIDER, (ii) CMC-greedy, and (iii) CMC-approx.

4.1 SEQ-SLIDER

We use the SEQ-SLIDER algorithm [2] as a baseline. Usinglocal search, the latter returns the set of the k best motif pairs(according to its χ2-score) it can find, which is then interpretedas a cover.

4.2 CMC-greedyWe devise a naive greedy algorithm, called CMC-greedy, forCMC which repeatedly looks for a motif pair to add to Mwhich reduces costα,β,γ(M, G) the most. Since the space ofpossible motif pairs is far too large to consider every motifpair (see Section 2.1), we replace this step with a local searchapproach, similar to that of SEQ-SLIDER, that looks for thebest motif pair to add to M during a set time period. Thealgorithm keeps adding motif pairs until it no longer finds onethat reduces costα,β,γ(M, G). The full algorithm is shown inFig. 2; the local search step is shown in Fig. 3, with neighborsthe neighbor function of SEQ-SLIDER [2]. When used by CMC-greedy costf (M,M,G) = costα,β,γ(M,M,G), which issimply defined as costα,β,γ(M∪ M, G).

4.3 CMC-approxBy Theorem 2, we can apply to CMC any constant factorapproximation algorithm for WRBSC while preserving theapproximation factor. However, no such algorithms are known,and by Theorem 1 are very unlikely to exist. We can, however,also apply the reduction to algorithms with known (but non-constant) approximation ratios and then deduce the approx-imation ratio of this algorithm for CMC. Therefore, we firstgive an overview of such algorithms.

WRBSC admits naive approximation algorithms with ratios|B|, |R| or |S|log(|B|) [8]. Carr et al. [10] suggest an


Input: PPI-network G = (V,E, λ), `, d ∈ N, d < `, set ofmotif pairs M, a cost function costf

Output: best correlated motif pair M∗ to add toM accordingto costf found in G

1: M∗ ← randomMotifPair()2: cost ← costf (M,M∗, G)3: oldCost ←∞4: while cost < oldCost do5: oldCost ← cost6: M ←M∗

7: for all M ′ ∈ neighbors(M) do8: if costf (M,M ′, G) < cost then9: M∗ ←M ′

10: cost ← costf (M,M ′, G)

Fig. 3. The local search algorithm for CMC, with costf asthe cost function.

algorithm that gives an approximation ratio of 2√KB |S| or

O(|S|1−1/KR log(|S|)) (with KB (KR) the maximum amountof blue (red) elements in a single set). These ratios are lowwhen KB or KR are small, but may go up to Ω(

√|S||B|)

or Ω(|S|log(|S|)). Peleg [8] suggests a greedy algorithm withapproximation factor ∆(S)log(|B|) (with ∆(S) the maximumamount of sets that contain the same red element) and anotheralgorithm with factor 2

√|S|log(|B|).

Theoretically, the naive approximation algorithm with ratio|B| is the most appealing as B in our setting corresponds tothe number of edges and is far smaller than the number ofpossible motif pairs |S|. The naive algorithm selects for eachblue element a set containing it with the least cost worth of redelements. For a blue element b, there is the option to take a setST or the set Sb (cf. equation (?) in the proof of Lemma 5).Here, ST has a cost of at least α and corresponds to a motifpair, while the set Sb contains b, has associated cost γ and doesnot correspond to a motif pair. As in our setting α > γ, thenaive algorithm will always select the sets Sb which results inreturning the empty set of motif pairs as a solution for CMC (cf.equation (??) in the proof of Lemma 5). So, although this naivealgorithm has the best theoretical approximation guarantee, itcan not be meaningfully applied towards CMC.

Therefore, we turn to Peleg’s algorithm with approxi-mation factor 2

√|S|log(|B|) [8] which by composing the

reductions in the proofs of Lemmas 4 and 5 results in atheoretical algorithm for CMC with an approximation factorof 2

√(MP(`, d) + |E|)log(|E|). We next explain how the

algorithm works. Peleg’s approximation algorithm [8], trans-forms an instance of WRBSC into an instance of the weightedSet Cover (WSC) problem by assigning to each set the cost ofthe red elements in it and then removing the red elements fromeach set. He applies the greedy algorithm for WSC, which isknown to have an approximation ratio of log(|B|) [14], to thisinstance. The greedy algorithm works by selecting the mostcost-efficient set at each time point. He proves this algorithmhas approximation guarantee 2

√|S|log(|B|), when applied

to an instance of WRBSC adapted from the original instanceusing Z, the maximum cost of red elements in a single set inthe optimal solution to that instance. As Z is unknown, the

Input: PPI-network G = (V,E, λ), `, d ∈ N+, d < `, and α,β, γ ∈ Q+, TIME ∈ N+

Output: best cover M according to costα,β,γ found in G1: M = ∅2: for Z = 1→ α+ β|V × V | do3: M′ ← ∅4: while E 6⊂ EM′ do5: bestMotifPair = undef6: cost ←∞7: startTime = getTime()8: while getTime() < startTime + TIME do9: M ← localSearch(G, `, d,M′, costWSC,α,β,γ,Z)

10: if costWSC,α,β,γ,Z(M′,M,G) < cost then11: bestMotifPair ←M12: cost ← costWSC,α,β,γ,Z(M,M,G)13: M′ ←M′ ∪ M14: if costα,β,γ(M′, G) < costα,β,γ(M, G) then15: M←M′

Fig. 4. The adapted approximation algorithm for WRBSCapplied to CMM.

algorithm is run repeatedly with every possible value and thebest result is saved. The adaptation of the original instancestarts by removing any set with total cost greater than Z.Then it removes from the remaining set all red elements stilloccurring in more than

√|S|/log(|B|) sets. Unfortunately, the

latter algorithm is infeasible as it iterates over all elements inS corresponding to all possible motif pairs.

Therefore, we consider an adaption, called CMC-approx,which is shown in Fig. 4. Rather than iterating over allpossible motif pairs, CMC-approx employs local search to findthe most cost-efficient motif pair to be added. Specifically,CMC-approx iterates over the possible values for Z, themaximum amount of cost for a single motif pair, and then useslocal search utilizing a cost function costWSC,α,β,γ,Z . Here,costWSC,α,β,γ,Z(M,M,G) calculates the cost (as the inverseof cost efficiency) of adding M to M as follows:

costWSC,α,β,γ,Z(M,M,G) := α+ β|AM ||EM \ EM|

α+ β|AM | <= Z;

∞ otherwise.

where AM is AM with the anti-edges selected by√(MP(`, d) + |E|)/log(|E|) or more motif pairs removed. 4

We cannot determine the amount of motif pairs that selectan anti-edge without knowledge of all the sets, but we cancompute an upper bound. Indeed, for every protein p, wecompute the number Np of motifs selecting p (note that thenumber of motifs is significantly lower than the number ofmotif pairs). The number of motif pairs selecting an anti-edgep, q is then bounded above by Np ×Nq .

4Recall that AM stands for the set of anti-edges and MP(`, d) for thenumber of (`, d)-motif pairs as defined in Section 2.1.


5 DATA

To assess the performance of our methods in covering net-works, we analyzed various networks. First, we predicted inter-action motifs in the high-confidence PPI-network of yeast [15].This network consists of 1 620 proteins and 9 060 interactions.In addition, we used the dataset presented by Yu et al. [16],which contains known positive and negative protein-proteininteraction cases created by balanced sampling. We took theirfiles for yeast containing 4 972 physical interactions and 4 972non-interactions and created from those a network with allthe proteins contained in these files and all the physicalinteractions present. We ran our methods on cross-validationnetworks created from this network.

For these networks, in addition to analyzing the coverageobtained by the predicted interaction motifs, we also used theresulting covers to predict protein-protein interactions by usingmotif occurrences as binary attributes in an SVM approach,where each case consisted of a protein pair with associatedclass label describing interaction vs. non-interaction. Protein-protein interaction prediction can be used in two differentways: (i) Predicting interaction/non-interaction between pro-teins for which no previous observation exists at all, and (ii)predicting interaction/non-interaction for proteins for whichsome knowledge of their interactions has been obtained al-ready. Both scenarios are relevant in biology and are testedhere. For each of these setups, we made ten training and testsets for cross-validation purposes: (i) In the first case, thefocus lies on separating test-proteins from training-proteins.We divided the list of all proteins into ten sets randomly.For each of these sets, we took the original (non-)interactiondata set and put each (non-)interaction where one or moreproteins are contained in the set of selected proteins, intothe test set. Remaining (non-)interactions became the trainingset. (ii) In the second case, the focus lies on separatingtest-(non-)interactions from training-(non-)interactions. In thiscase, the positive and negative data sets were divided randomlyinto ten parts. Each part became a test set, with the other(non-)interactions forming the training set. For every trainingset, a network was created with all the proteins remainingin the positive and negative training set and as interactionsthose in the positive training set. Practically, our algorithms didnot have substantially different results depending on the case.Therefore, we treat this as a homogeneous set of interactionnetworks for prediction purposes.

In addition to these analyses of network coverage andperformance in predicting protein-protein interactions basedon networks covers, we analyzed whether our methods locateactual binding sites. To do so, we compared our resultsto known protein structures. We took the high-confidencenetworks for human and yeast presented by Yu et al. [16]and mapped the protein sequences (retrieved from uniprot.organd yeastgenome.org respectively) to structures in the PDBdatabase. In order to perform the mapping of the proteinsequences to PDB structures, we used a strategy similar to theone used by Yabuki at al. [17], where each protein sequencein the input dataset is searched against the PDB databaseby gapped BLAST. The protein structure was mapped to the

protein sequence if the following conditions were met: thealigned regions present sequence identity of at least 40%,coverage of aligned region on query sequence of at least30%, coverage of aligned region on PDB sequence of at least30%, and bit score of at least 70, respectively. By doingso, it is guaranteed that at least 30% of the length of theprotein sequence is mapped to the PDB structure, and the PDBstructure is covered at least 30% by the alignment with identityof at least 40% and with bit score of at least 70. After mapping,we obtained a human network containing 578 proteins and 547interactions, and a yeast network containing 526 proteins and263 interactions.

6 EVALUATION

All experiments were run on a 3GHz Mac Pro using 2GB ofRAM and 8 cores. Our Java-implementations of SEQ-SLIDER,CMC-greedy and CMC-approx can use as many processors asare available.

We set out to check if the algorithms we present, fulfill therequirement we first laid out: the result set should be smalland have a high coverage of the network. We performed arun of SEQ-SLIDER on each of the cross-validation networksdescribed in Section 5 taking about 75 minutes requesting10 000 result motif pairs. A run of CMC-greedy with 30seconds for each iteration was also performed, as well as arun of CMC-approx with 15 seconds for each iteration. Therun of CMC-greedy took on average 14 minutes while CMC-approx took 120 minutes. While in theory the results of SEQ-SLIDER can keep improving over time until they reach thesame results as a brute force search, in practice, the gain ofallowing additional time over 75 minutes of computation isnegligible (data not shown).

The results averaged over all twenty networks are shownin the table in Fig. 5. Protein coverage (indicated in thefigure by proteins) shows the percentage of proteins in thenetwork that is covered by at least one motif in the resultset. Interaction coverage (interactions) shows the percentageof interactions that is covered by at least one motif pair. Size(size) is the average size of the result sets of the differentmethods (for SEQ-SLIDER this is constant). It is clear thateven though the amount of results returned is much larger forSEQ-SLIDER, it does not cover the interactions in the networkwell. Even CMC-greedy, with only a fraction of the resultscovers more interactions. CMC-greedy covers less proteinsthan SEQ-SLIDER, though. The overall winner on coverageis CMC-approx which, with less than a tenth of the resultsof SEQ-SLIDER, manages to cover about three quarters of theinteractions, and almost all the proteins. If we look at coverageper result, CMC-greedy and CMC-approx score about the sameon interactions (and much better than SEQ-SLIDER), but CMC-approx covers much more proteins.

We also ran all algorithms on the high-confidence yeastnetwork described in Section 5. The results are shown inTable 6. It is clear that the coverage of CMC-approx is farsuperior to that of both other algorithms. The latter comes atthe expense of a longer running time and a larger set of resultscompared to CMC-greedy.


Method SEQ-SLIDER CMC-greedy CMC-approxproteins 43.8% 23.4% 96.2%

interactions 11.8% 19.0% 73.5%size 10 000 214.95 867.5

Fig. 5. Protein and interaction coverage of results ob-tained for yeast cross-validation networks.

Method SEQ-SLIDER CMC-greedy CMC-approxproteins 21.2% 44.9% 100%

interactions 23.1% 48.0% 99.9%size 10 000 663 1 786

Fig. 6. Protein and interaction coverage of results ob-tained for yeast high-confidence network.

7 BIOLOGICAL VALIDATION

To further evaluate the biological relevance of our algorithms,we tested the effectiveness of the derived motif pairs in severalscenarios. In the first scenario, we investigate how well motifhit locations correspond to interface residues by a comparisonwith protein structure data. In a second scenario, we investigatehow cross-species motif pair mining can be used as a filteringstep to increase accuracy. Finally, in the supplementary mate-rial, we discuss prediction of protein interactions from proteinsequences.

7.1 Comparison with protein structure dataTo determine the overlap of our predicted interaction motifswith actual interaction sites, we ran all three methods on thehuman and yeast network for which we have a mapping toprotein structures as described in Section 5. The table in Fig. 7shows the results for the yeast network, and the table in Fig. 8for the human network.

We start with the amount of motif pairs returned by themethod (indicated in the figure by motif pairs) and check howmany unique motifs we have (unique motifs). We then check athow many positions those motifs occur in the proteins (motifhits), and how many positions remain if a position occurringmultiple times is only counted once (unique hits). In addition,we provide the number of hits obtained when hit locations thatoccur within seven amino acids of a hit we already found, arenot taken into account (dissimilar). In this way, hit locationsthat have at least one residue overlap are considered as onehit location. The precision (precision) is the percentage of theremaining hits that is found at the interface. The decision ofwhether an amino acid is an interface amino acid is based onits relative surface area (RSA) in the structure of the unboundprotein and in the protein-protein complex, in the same way asdescribed by Boyen et al. [2]. At last, we show the percentageof proteins (proteins) and interactions (interactions) covered.

The comparison with networks containing proteins mappedto protein structure data indicates that for the yeast as wellas the human network, coverage of the network increases forCMC-approx, as would be expected, at the expense of only aslight reduction in precision compared to SEQ-SLIDER.

Method SEQ-SLIDER CMC-greedy CMC-approxmotif pairs 10 000 6 82

unique motifs 5 963 11 156motif hits 8 547 40 340

unique hits 1 792 40 336dissimilar (7) 388 40 319

precision 49.74% 65% 44.51%proteins 21.13% 15.49% 100%

interactions 31.94% 20.91% 97.72%

Fig. 7. Comparison with protein structure data for motifhits obtained for yeast.

Method SEQ-SLIDER CMC-greedy CMC-approxmotif pairs 10 000 15 135

unique motifs 2 387 30 268motif hits 10 774 104 972

unique hits 2 246 104 941dissimilar (7) 393 100 879

precision 39.95% 53% 31.29%proteins 9.39% 15.48% 100%

interactions 9.46% 23.56% 100%

Fig. 8. Comparison with protein structure data for motifhits obtained for human.

7.2 Cross-species comparison

We investigate the hypothesis that motif positions which arefound in more than one species could be more likely to beinterface residues. In that case, a cross-species comparisoncould be used to filter noise from the predictions. For thevalues (`, d) = (8,3), (8,4) and (8,5), resulting motifs for SEQ-SLIDER (75 minutes), CMC-approx (2 minutes per iteration),and CMC-greedy (4 minutes per iteration) for the humanand yeast networks for which we have a mapping to 3Dprotein structures were mapped to the sequences. Note thatthis analysis was restricted to the networks for which wehave a mapping to 3D-structures because we need structuredata to asses the precision of the predicted residues. CMC-greedy never returned motif pairs which could be mapped toa decent amount of proteins in these networks, so its resultsare omitted. Subsequently, a cross-species comparison wasperformed between yeast and human. Proteins from thesespecies were linked to each other by finding best BLAST hitsand requiring a sequence identity of at least 50%. The yeast-human pairs were aligned to each other using Muscle [18].Subsequently, the positions of the motifs in each of the twosequences in such alignment were compared to each other tosee if they overlapped or not.

The table in Fig. 9 shows the results from this analysis. Foreach run, the table shows the number of proteins in whichmotifs are found and for which a link to a protein in the otherspecies exist (Nprot). Next, it shows how many positions inthose proteins contain motifs in both species (same), and howmany of those are interface residues (same, TP). Subsequently,it shows the amount of positions in those proteins containing


motifs in only a single species (diff), and how many of thoseare interface residues (diff, TP). Finally, the table provides thefraction of positions that are found in both species, which areindeed interface residues (Fs) and the fraction of positions thatare found in only one of the two species, which are indeedinterface residues (Fd).

The data in the table supports the hypothesis that motifpositions which are found in more than one species could bemore likely to be interface residues to some extent. In five outof six cases, positions that are found in both species have aindeed a higher chance to be interface residues than positionsthat are found in only one species. These are the rows whereFs > Fd. The latter does not hold for SEQ-SLIDER (8,3) butthe motifs from that run only select two proteins linked toeach other between the two species. Besides this anomaly,there is not a clear difference between SEQ-SLIDER and CMC-approx in this respect, confirming that the CMC-approx motifsare of comparable quality as the SEQ-SLIDER motifs. Whatis also clear is that the number of proteins which could beused in this comparison was in most cases much higher forthe CMC-approx results than for the SEQ-SLIDER results. Thisis obviously related to the fact that CMC-approx obtains ahigher network coverage; in this application, we compare twonetworks from different species which means that a methodwith a high network coverage is clearly advantageous.

To conclude, comparison of results from networks fromvarious species seems to be a promising strategy to reducenoise in the predictions. The much higher network coveragethat CMC-approx obtains compared to SEQ-SLIDER makes itwell suited for such an approach.

8 RELATED WORK

8.1 Pattern miningMany pattern mining applications suffer from pattern explo-sion. That is, when an interestingness measure is chosen toretrieve novel results, such a large amount of patterns isobtained as to eclipse the data itself by several orders ofmagnitude. This is mostly due to the locality of the minimumsupport constraint, which causes patterns to be added to theresults set independent of patterns already found, causing greatredundancy in the results. This problem is often alleviated, butnot solved, by looking for special types of patterns, such asclosed [19] or maximal [20] patterns, that encompass a groupof redundant patterns. To provide less redundant patterns, therehas been much research toward finding small sets of represen-tative macro-patterns for databases. Bringmann et al. [21] usea post-processing technique to filter item set mining results andremove redundant item sets. They show that this reduction ofthe result set can improve the quality of classification. Geertset al. [22] consider the top-k tiling problem. Here, tiles arefrequent item sets that cover a large area of the transactiondatabase. Their top-k tiles problem looks for the k (possiblyoverlapping) tiles that together cover the largest surface in thedatabase. Vreeken et al. [23] introduce KRIMP, which usesMDL to find the item sets that can best be used to compress adatabase. So far, all these techniques only select item sets thatoccur exactly within the data. Xiang et al. [24] adapt tiling

to allow for item sets with a few missing items. They focus,however, on covering every cell in the database, while we aremore interested in informative patterns which give us insightinto the data. None of these techniques focuses on biologicaldata.

8.2 Binding site prediction

The prediction of binding sites is mostly used in two fields;(i) the prediction of transcription factor binding sites. Thesebinding sites are the locations on strands of DNA whereproteins bind in order to activate or inhibit its transcription.This search is typically performed by looking for statisticallyoverrepresented motifs among DNA strands of coregulatedgenes [25]. (ii) Other tools exist for predicting protein bindingsites [2], [3], [4], [5], [6], [7], [26], [27], [28], [29], [30],[31], [32], [33], [34], [35], [36], [37], [38], [39]. They useseveral types of information, e.g. amino acid sequence, solventaccessibility, conservation, and many others. This informationis not always available, especially for newly sequenced pro-teins. Many learning methods are used to predict the bindingsites, such as SVMs or neural networks. One, in particular,named PRISM [32] should be mentioned as it also uses motifpairs. Using known pairs of binding sites, it looks for astructural match for the left and right binding sites. If found,it predicts interaction through those sites. All these worksfocus on predicting binding sites separately. To the best ofour knowledge, this is the first attempt to look for a grandexplaining set of binding sites for a protein-protein interactionnetwork.

It is promising that our algorithms expand the coverage ofproteins and interactions, at the expense of only a slight reduc-tion in precision when comparing with protein structure data(cf. Section 7.1). We recently demonstrated that the precisionof SEQ-SLIDER predictions could be further improved bycombining with additional biological datasources, in partic-ular predicted surface accessibility and sequence conservation[40]. We expect that integrating such additional sources ofinformation could also further improve the precision of theCorrelated Motif Covering approaches presented in this work.An additional way to improve precision was demonstratedin Section 7.2, where we showed that comparison of resultsfrom various species improved prediction performance. Thebiological background of this type of data integration is thatthe location of interface residues in a protein sequence is quiteconserved, even more than the interactions in which theseproteins are involved [41]. In addition, cases where interac-tion sites are not conserved between species might explainvariability of protein interactions. We previously analyzedthe relationship between sequence variation and variation ofprotein interactions within a single protein family [39], [42]but the results presented in this work indicate the possibilityof extending this to a genome-wide scale. Given the availabil-ity of experimental protein interaction networks for variousspecies in databases such as IntAct [43] it will in principle bestraightforward to analyze data from additional species.


Method Nprot same same, TP diff diff, TP Fs FdSEQ-SLIDER (8,3) 2 8 0 157 65 0.00 0.41SEQ-SLIDER (8,4) 11 43 18 1 130 327 0.42 0.29SEQ-SLIDER (8,5) 52 4 234 1 175 8 315 1 449 0.28 0.17CMC-approx (8,3) 52 68 44 1 205 314 0.65 0.26CMC-approx (8,4) 51 96 46 1 778 360 0.48 0.20CMC-approx (8,5) 52 589 161 1 759 497 0.27 0.13

Fig. 9. Combining results from human and yeast networks for SEQ-SLIDER and CMC-approx.

9 CONCLUSION

This work sought to improve upon computational methods forderiving motif pairs that usually only find motif pairs thatselect the densest part of the network. Our new algorithmsfor CMC decrease the size of the result set while increasingthe coverage of the network, having minimal impact on theirprecision. The much higher coverage of proteins by motifsis important in many applications such as cross-species com-parison, where it increases the probability of finding interfaceresidues. In addition, for experimental biologists who want topredict binding sites in order to perform further experimentalstudies, obviously it is very important to obtain predictions foras many proteins as possible.

There is, however, ample opportunity for future research.As we have shown when applying motif pairs to predictinteractions, the prediction using (`, d)−motif pairs as binaryattributes is hardly an improvement over random. We noticedthat while the mined motif pairs cover the training set well,they hardly cover any of the proteins in the test set. In thissense, (`, d)-motifs are too selective. It could therefore beworthwhile to investigate how we can deviate from the simple(`, d)−motif model by, for instance, adding disjunctions orconsidering the addition of biological information such assimilarities between amino acids. Of course, we should takecare to balance the expressiveness, and generality, of motifswith the computational effort required to mine them. Since theβ parameter has such a natural correspondence to noise, andthe γ parameter to missing data, it might be interesting to findout if we can compensate for those factors by adjusting theirrespective parameters.

ACKNOWLEDGMENTS

Research funded by a Ph.D grant of the Institute for thePromotion of Innovation through Science and Technology inFlanders (IWT-Vlaanderen).

Aalt D.J. van Dijk is supported by an NWO (Nether-lands Organisation for Scientific Research) VENI grant(863.08.027).

Felipe L. Valentim is supported by the SYSFLO MarieCurie Initial Training Network.

REFERENCES

[1] M. P. H. Stumpf, T. Thorne, E. de Silva, R. Stewart, H. J. An,M. Lappe, and C. Wiuf, “Estimating the size of the human interactome,”Proceedings of the National Academy of Sciences, vol. 105, no. 19, pp.6959–6964, 2008.

[2] P. Boyen, D. Van Dyck, F. Neven, R. C. van Ham, and A. D. J.van Dijk, “SLIDER: A Generic Metaheuristic for the Discovery ofCorrelated Motifs in Protein-Protein Interaction Networks.” IEEE/ACMTransactions on Computational Biology and Bioinformatics, vol. 8,no. 5, pp. 1344–1357, 2011.

[3] S. H. Tan, W. Hugo, W. K. Sung, and S. K. Ng, “A correlatedmotif approach for finding short linear motifs from protein interactionnetworks.” BMC Bioinformatics, vol. 7, no. 502, 2006.

[4] H. C. Leung, M. H. Siu, S. M. Yiu, F. Y. Chin, and K. W. Sung, “Findinglinear motif pairs from protein interaction networks: a probabilisticapproach.” Computational Systems Bioinformatics Conference, vol. 6,pp. 111–119, 2007.

[5] H. Li, J. Li, and L. Wong, “Discovering motif pairs at interactionsites from protein sequences on a proteome-wide scale,” Bioinformatics,vol. 22, pp. 989–996, 2006.

[6] J. Li, G. Liu, H. Li, and L. Wong, “Maximal biclique subgraphs andclosed pattern pairs of the adjacency matrix: A one-to-one correspon-dence and mining algorithms,” IEEE Transactions on Knowledge andData Engineering, vol. 19, pp. 1625–1637, 2007.

[7] J. Li, K. Sim, G. Liu, and L. Wong, “Maximal quasi-bicliques withbalanced noise tolerance: Concepts and co-clustering applications,” inProceedings of the SIAM International Conference on Data Mining.SIAM, 2008, pp. 72–83.

[8] D. Peleg, “Approximation algorithms for the label-covermax and red-blue set cover problems,” Journal of Discrete Algorithms, vol. 5, pp.55–64, 2007.

[9] J. Rissanen, “A universal prior for integers and estimation by minimumdescription length,” Annals of Statistics, vol. 11, pp. 416–431, 1983.

[10] R. D. Carr, S. Doddi, G. Konjevod, and M. Marathe, “On the red-blueset cover problem,” in Proceedings of the ACM-SIAM Symposium onDiscrete Algorithms (SODA). SIAM, 2000, pp. 345–353.

[11] V. V. Vazirani, Approximation Algorithms. Springer, 2004.[12] I. Dinur and S. Safra, “On the hardness of approximating label-cover,”

Information Processing Letters, vol. 89, pp. 247–254, 2004.[13] M. Elkin and D. Peleg, “The hardness of approximating spanner

problems,” Theory of Computing Systems, vol. 41, pp. 691–729, 2007.[14] V. Chvatal, “A Greedy Heuristic for the Set-Covering Problem,” Math-

ematics of Operations Research, vol. 4, no. 3, pp. 233–235, 1979.[15] S. R. R. Collins, P. Kemmeren, Xue, J. F. F. Greenblatt, F. Spencer,

F. C. C. Holstege, J. S. S. Weissman, and N. J. J. Krogan, “Towardsa comprehensive atlas of the physical interactome of Saccharomycescerevisiae,” Molecular & Cellular Proteomics, vol. 6, pp. 439–450,2007.

[16] J. Yu, M. Guo, C. J. Needham, Y. Huang, L. Cai, and D. R. Westhead,“Simple sequence-based kernels do not predict protein-protein interac-tions,” Bioinformatics, vol. 26, no. 20, pp. 2610–2614, 2010.

[17] Y. Yabuki, Y. Mukai, M. B. Swindells, and M. Suwa, “Genius ii: ahigh-throughput database system for linking orfs in complete genomesto known protein three-dimensional structures,” Bioinformatics, vol. 20,pp. 596–598, 2004.

[18] R. C. Edgar, “MUSCLE: multiple sequence alignment with high accu-racy and high throughput.” Nucleic Acids Research, vol. 32, no. 5, pp.1792–1797, 2004.

[19] N. Pasquier, Y. Bastide, R. Taouil, and L. Lakhal, “Discovering frequentclosed itemsets for association rules,” in Proceedings of the InternationalConference on Database Theory (ICDT). Springer-Verlag, 1999, pp.398–416.

[20] R. J. Bayardo, “Efficiently mining long patterns from databases,” inProceedings of the International Conference on Management of Data(SIGMOD). ACM, 1998, pp. 85–93.

[21] B. Bringmann and A. Zimmermann, “One in a million: picking the rightpatterns,” Knowledge and Information Systems, vol. 18, pp. 61–81, 2009.


[22] F. Geerts, B. Goethals, and T. Mielikainen, “Tiling Databases,” inDiscovery Science, ser. Lecture Notes in Computer Science, vol. 3245.Springer, 2004, pp. 278–289.

[23] J. Vreeken, M. Leeuwen, and A. Siebes, “Krimp: mining itemsets thatcompress,” Data Mining and Knowledge Discovery, vol. 23, pp. 169–214, 2011.

[24] Y. Xiang, R. Jin, D. Fuhry, and F. F. Dragan, “Succinct summarizationof transactional databases: an overlapped hyperrectangle scheme,” inProceeding of the International Conference on Knowledge Discoveryand Data Mining (SIGKDD). ACM, 2008, pp. 758–766.

[25] M. Das and H. K. Dai, “A survey of DNA motif finding algorithms,”BMC Bioinformatics, vol. 8, no. Suppl 7, pp. S21+, 2007.

[26] J. R. Bradford and D. R. Westhead, “Improved prediction of protein-protein binding sites using a support vector machines approach.” Bioin-formatics, vol. 21, no. 8, pp. 1487–1494, 2005.

[27] S. Liang, C. Zhang, S. Liu, and Y. Zhou, “Protein binding site predictionusing an empirical scoring function,” Nucleic Acids Research, vol. 34,no. 13, pp. 3698–3707, 2006.

[28] G. Lopez, A. Valencia, and M. L. Tress, “Firestar - prediction offunctionally important residues using structural templates and alignmentreliability,” Nucleic Acids Research, vol. 35, no. Web-Server-Issue, pp.573–577, 2007.

[29] Y. Murakami and S. Jones, “Sharp2: protein–protein interaction pre-dictions using patch analysis,” Bioinformatics, vol. 22, pp. 1794–1795,2006.

[30] S. S. Negi, C. H. Schein, N. Oezguen, T. D. Power, and W. Braun,“InterProSurf: a web server for predicting interacting sites on proteinsurfaces,” Bioinformatics, vol. 23, no. 24, pp. 3397–3399, 2007.

[31] H. Neuvirth, R. Raz, and G. Schreiber, “ProMate: A Structure BasedPrediction Program to Identify the Location of ProteinProtein BindingSites,” Journal of Molecular Biology, vol. 338, no. 1, pp. 181–199, 2004.

[32] U. Ogmen, O. Keskin, A. S. Aytuna, R. Nussinov, and A. Gursoy,“PRISM: protein interactions by structural matching.” Nucleic AcidsResearch, vol. 33, no. Web Server issue, 2005.

[33] A. Porollo and J. Meller, “Prediction-based fingerprints of protein-protein interactions,” Proteins: Structure, Function, and Bioinformatics,vol. 66, no. 3, pp. 630–645, 2007.

[34] A. Shulman-Peleg, R. Nussinov, and H. J. Wolfson, “Recognition offunctional sites in protein structures.” Journal of Molecular Biology,vol. 339, no. 3, pp. 607–633, 2004.

[35] E. W. Stawiski, L. M. Gregoret, and Y. M. Gutfreund, “AnnotatingNucleic Acid-Binding Function Based on Protein Structure,” Journalof Molecular Biology, vol. 326, no. 4, pp. 1065–1079, 2003.

[36] H. Tjong, S. Qin, and H. Zhou, “PI2PE: protein interface/interiorprediction engine,” Nucleic Acids Research, vol. 35, no. Web Server,pp. W357–W362, 2007.

[37] H. X. Zhou and Y. Shan, “Prediction of protein interaction sites fromsequence profile and residue neighbor list.” Proteins, vol. 44, no. 3, pp.336–343, 2001.

[38] H.-X. Zhou and S. Qin, “Interaction-site prediction for protein com-plexes: a critical assessment,” Bioinformatics, vol. 23, no. 17, pp. 2203–2209, 2007.

[39] A. D. J. van Dijk, G. Morabito, M. Fiers, R. C. H. J. van Ham,G. C. Angenent, and R. G. H. Immink, “Sequence Motifs in MADSTranscription Factors Responsible for Specificity and Diversification ofProtein-Protein Interaction,” PLoS Computational Biology, vol. 6, no. 11,pp. e1 001 017+, 2010.

[40] F. L. Valentim, F. Neven, P. Boyen, and A. D. J. van Dijk, “Interactome-wide prediction of protein-protein binding sites reveals effects of proteinsequence variation in arabidopsis thaliana,” PLoS ONE, vol. 7, no. 10,p. e47022, 2012.

[41] Q. C. Zhang, D. Petrey, R. Norel, and B. H. Honig, “Protein inter-face conservation across structure space,” Proceedings of the NationalAcademy of Sciences, vol. 107, no. 24, pp. 10 896–10 901, 2010.

[42] E. I. Severing, A. D. J. van Dijk, G. Morabito, J. Busscher-Lange,R. G. H. Immink, and R. C. H. J. van Ham, “Predicting the impactof alternative splicing on plant MADS domain protein function,” PLoSONE, vol. 7, no. 1, p. e30524, 01 2012.

[43] S. Kerrien et al., “The intact molecular interaction database in 2012,”Nucleic Acids Research, vol. 40, no. Database-Issue, pp. 841–846, 2012.

[44] M. Hall, E. Frank, G. Holmes, B. Pfahringer, P. Reutemann, andI. H. Witten, “The weka data mining software: an update,” SIGKDDExplorations Newsletter, vol. 11, pp. 10–18, 2009.

Peter Boyen received his PhD degree in Com-puter Science from Hasselt University in 2011.His research interests include protein interactionnetworks and graph alignment.

Frank Neven received his PhD degree in Com-puter Science from Limburgs Universitair Cen-trum in 1999. Since 2001, he is a professorat Hasselt University. His main research inter-ests are databases and data mining, formal lan-guages, automata, and logic.

Dries Van Dyck received his PhD degree inComputer Science from Ghent University in2004. For his PhD, he mainly worked on graphstructural properties of cubic graphs and combi-natorial optimization. Afterwards, he held a post-doctoral position at Hasselt University where hisresearch interests were algorithmic graph the-ory, heuristics for hard graph problems, graphmining and mining of biological data. In 2010, hejoined the research effort at the Belgian NuclearResearch Centre (SCK-CEN) to develop an ul-

trasonic visualization system for liquid metal for the MYRRHA-project.

Felipe L. Valentim received the BSc degree inComputer Science from Federal University ofLavras in 2006 and the MSc degree in PlantBiotechnology from Federal University of Lavrasin 2010. He is currently a PhD student in theApplied Bioinformatics group at Plant ResearchInternational, Wageningen University and Re-search Centre. His main research interests areprotein interaction networks and modelling generegulatory networks.

Aalt D.J. van Dijk received his PhD degree inChemistry from Utrecht University in 2006. Atpresent, he is a researcher in the Bioinformat-ics group at Plant Research International, Wa-geningen University and Research Centre. Hismain research interests are protein-protein inter-actions, transcription factor - DNA interactions,computational structural biology, and modellingthe dynamics of networks of interacting proteins.


SUPPLEMENTARY MATERIAL

Notation Meaning` length of motifd number of wildcards in a motif

X,Y (`, d)−motifM motif pairM set of motif pairs

MP(`, d) amount of (`, d)-motif pairsG graphV set of vertices/nodesE set of edgesλ node labeling functionVX set of proteins containing motif XEX,Y set of edges between VX and VYAX,Y set of anti-edges between VX and VYGM subgraph of G selected by motif pair MGM subgraph of G selected by the motif pairs in Mα cost of a motif pairβ cost of a false positive edgeγ cost of a false negative edgeU universe of elementsS a setS a set of setsB set of blue elementsR set of red elementsS∗ optimal set of sets for RBSC or WRBSC

TABLE 1Explanation of the most frequently used symbols.

B RS1

S2

S3

Fig. 10. An example instance IRBSC for RBSC.Blue elements are shaded in gray.

100

110

011

001

001

110

011

001

222C

O

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

000 000 000 000 000

Fig. 11. The instance f(IRBSC) = ICMC. The newly createdelements are indicated by thin edges.

Illustrating the reduction in Lemma 2We next illustrate the reduction in the proof of Lemma 2 by an example. Consider the instance IRBSC for RBSC in Fig. 10 whichcontains three blue and five red elements and their divisions into the sets S1, S2, and S3. Now, Fig. 11 displays f(IRBSC) = ICMC.As IRBSC contains three sets, ` = 3 and d = 2. The maximum number of red elements in a set is three. Therefore, γ = 4 andO contains 2× 4× 3 + 1 = 25 nodes. The new nodes in O are all labeled with 000 and the center node c is labeled with thesequence 222. The upper most blue element is assigned the label 110 as it is an element of S1 and S2, but not of S3. Thelabels of the remaining nodes are assigned accordingly.


ABAB

AAAA

BBBB

1

2 3

Fig. 12. An example instance ICMC for CMC. Theanti-edge is indicated by a dotted line.

1,2

2,3

1,3

C IA,A

B,B

A,B

S1

S2

S3

Fig. 13. f(ICMC) = IMDL-COVER. Elements correspond to edges(thick lines) or anti-edges (dotted lines) in Fig. 12.

Illustrating the reduction in Lemma 4Consider the instance ICMC for CMC in Fig. 12 which consists of a graph with three nodes labeled by AAAA, ABAB, and BBBBrespectively. For clarity, missing edges (anti-edges) are indicated by a dotted line. Fig. 13 displays f(ICMC) = IMDL-COVER. Forease of exposition, we restrict the alphabet to Σ = A,B, and set ` = 1 and d = 0. As can be seen from the figure, Ccontains the edges, while I contains the anti-edges. Furthermore, S consists of the sets of edges S1, S2, and S3, defined bythe three motif pairs A,A, B,B, and A,B, respectively.

b1

b2

r1

C I

T1

T2

T3

Fig. 14. An example instance IMDL-COVER forMDL-COVER.

!!

!

!

!

!ST1

ST2ST3

Sb1

Sb2

Fig. 15. f(IMDL-COVER) = IWRBSC. Blue elements are shaded ingray, red elements have their weight indicated. Newly createdelements are indicated with thinner lines.

Illustrating the reduction in Lemma 5Consider the instance IMDL-COVER for MDL-COVER in Fig. 14 which consists of two correct and one incorrect element, and threesets T1, T2, and T3. Fig. 15 displays f(IMDL-COVER) = IWRBSC. Every correct element is a blue element. All incorrect elementsare red and have weight β. For every blue element b, the set Sb is created containing b as well as a newly created red elementwith weight γ. For every set T in SMDL-COVER, a set ST is created containing all blue (correct) and red (incorrect) elements inT and one additional new red element with weight α.


Proof of Lemma 5 (continued).We first argue that if an optimal solution S∗WRBSC for IWRBSC contains a set Sb and a set ST with b ∈ ST , we can remove Sbwithout removing optimality. Indeed, assume Sb to be in S∗WRBSC with another set in S∗WRBSC containing b, then

objWRBSC(S∗WRBSC \ Sb)=

∑r∈

⋃S∈S∗WRBSC\Sb

S

∩Rc(r)

=

∑

r∈

⋃S∈S∗WRBSC

S

∩Rc(r)

− γ

≤∑

r∈

⋃S∈S∗WRBSC

S

∩Rc(r)

= objWRBSC(S∗WRBSC).

So, for an optimal solution S∗WRBSC, we can create a solution S∗WRBSC′ that contains no Sb with b ∈ ST ∈ S∗WRBSC

′ and is stilloptimal.

So, for an instance IWRBSC, we can use g to transform an optimal solution S∗WRBSC into a solution sMDL-COVER for IMDL-COVER

with OPTWRBSC(IWRBSC) = objWRBSC(S∗WRBSC) = objMDL-COVER(sMDL-COVER). The solution sMDL-COVER is also an optimal solution forIMDL-COVER, because if there was a better solution, we could transform it into one for IWRBSC with the same score, by taking STfor every T and Sb, for every element of C not covered. This would be in contradiction with S∗WRBSC being optimal. Therefore,OPTWRBSC(IWRBSC) = OPTMDL-COVER(IMDL-COVER), so definitely OPTWRBSC(IWRBSC) ≤ OPTMDL-COVER(IMDL-COVER).

This completes the proof of Lemma 5.

PredictionWe used the SVM classifier available in the weka [44] data mining software to perform prediction on the twenty networksdescribed in Section 5. The training and test data for the positive and negative data sets were created with the motif pairsobtained from the runs described in Section 6 as binary attributes. Of the 10 000 results returned by SEQ-SLIDER only the1 000 with the best score were used due to limitations of the classifier. In the rare case where CMC-approx returned morethan 1 000 results, only the first 1 000 were used. CMC-greedy never returned more than 1 000 results and hence in all cases,all motif pairs returned by CMC-greedy were used. The average AUC (Area Under the ROC Curve) of both SEQ-SLIDER andCMC-greedy are 0.502, while that of CMC-approxis 0.526, which is only a very slight improvement. We performed the samepredictions with a random forest classifier, but the results were nearly identical. While slightly improving over SEQ-SLIDER, thepredictive performance remains still only marginally different from random (AUC slightly above 0.5), confirming the analysisof Yu et al. [16], showing that sequences do not contain sufficient information to be useful for predicting PPIs.

Documents

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND …alpha.uhasselt.be/frank.neven/papers/miningmdl.pdf · 2013-07-05 · Mining minimal motif pair sets maximally covering interactions