A Hyb id Me aheu is ic o Biclus e ing
Based on Sca e Sea ch and Gene ic Algo i hms
Juan A. Nepomuceno1, Alicia T oncoso2, and Jesús S. Aguila –Ruiz2
1Depa men o Compu e Science, Uni e si y o Se illa, Spain
[email p o ec ed]
2A ea o Compu e Science, Pablo de Ola ide Uni e si y o Se illa, Spain
{ali,aguila }@upo.es
Abs ac . In his pape a hyb id me aheu is ic o biclus e ing based
on Sca e Sea ch and Gene ic Algo i hms is p esen ed. A gene al scheme
o Sca e Sea ch has been used o ob ain high–quali y biclus e s, bu a
way o gene a ing he ini ial popula ion and a me hod o combina ion
based on Gene ic Algo i hms ha e been chosen. Expe imen al esul s
om yeas cell cycle and human B-cell lymphoma a e epo ed. Finally,
he pe o mance o he p oposed hyb id algo i hm is compa ed wi h a
gene ic algo i hm ecen ly published.
Keywo ds: Biclus e ing, Gene Exp ession Da a, Sca e Sea ch, E olu-
iona y Compu a ion.
1 In oduc ion
Recen ly, da a mining echniques a e being applied o mic oa ay da a analysis
in o de o ex ac use ul in o ma ion [1]. Clus e ing echniques find g oups o
genes wi h simila pa e ns om a mic oa ay. Howe e , genes a e no necessa y
ela ed o e e y condi ion. Thus, he goal o he biclus e ing is o iden i y genes
wi h he same beha io only unde a specific g oup o condi ions.
In he con ex o mic oa ay analysis, many app oaches ha e been p oposed o
biclus e ing [2]. Biclus e ing echniques ha e wo impo an aspec s: he sea ch
algo i hm and he measu e o e alua e he quali y o biclus e s.
Mos o p oposed app oaches in he li e a u e a e ocussed on diffe en sea ch
me hods. Thus, in [3] an i e a i e hie a chical clus e ing is applied o each di-
mension sepa a ely and biclus e s a e buil by he combina ion o he ob ained
esul s o each dimension. In [4] an i e a i e sea ch me hod which buil biclus-
e s adding o emo ing genes o condi ions in o de o imp o e he measu e
o quali y called Mean Squa ed Residue (MSR) was p esen ed. An exhaus i e
biclus e s enume a ion by means a bipa i e g aph-based model ha nodes we e
added o emo ed in o de o find maximum weigh subg aphs was gene a ed in
[5]. The FLOC algo i hm [6] imp o ed he me hod p esen ed in [4] ob aining a
se o biclus e s simul aneously and adding missing alues echniques. In [7], a
simple linea model o gene exp ession was used assuming no mally dis ibu ed
exp ession le el o each gene o condi ion. Also, geome ical cha ac e iza ions
such as hype planes in a high dimensional da a space ha e been used o find
biclus e s [8]. In he las ew yea s, global op imiza ion echniques, such as Sim-
ula ed Annealing [9] o E olu iona y Compu a ion [10,11], ha e been applied o
ob ain biclus e s due o hei good pe o mance in se e al en i onmen s.
Recen ly, se e al pape s we e ocussed on he measu e p oposed o e alua e
he quali y o biclus e s. In [12] an analysis o he MSR was made, showing ha
his measu e is good o find biclus e s wi h shi ing pa e ns bu no scaling
pa e ns. A new measu e based on uncons ained op imiza ion echniques was
p oposed in [13] as al e na i e o he MSR in o de o find biclus e s wi h ce ain
pa e ns.
In his pape a hyb id me aheu is ic o biclus e ing based on Sca e Sea ch
and Gene ic Algo i hms (SS&GA) is p esen ed. A gene al scheme o Sca e
Sea ch has been used o ob ain high–quali y biclus e s, bu a way o gene a ing
he ini ial popula ion and a me hod o combina ion based on Gene ic Algo i hms
ha e been chosen. Finally, he pe o mance o he p oposed hyb id algo i hm is
compa ed wi h a gene ic algo i hm ecen ly published [10]. A Sca e Sea ch
has been selec ed due o he ecen success ob ained o sol e diffe en ha d
op imiza ion p oblems and o e e ences abou he applica ion o Sca e Sea ch
o biclus e ing ha e no been ound in he li e a u e.
This pape is o ganized as ollows. Sec ion 2 p esen s basic concep s abou
Sca e Sea ch. The desc ip ion o he p oposed me aheu is ic is desc ibed in
Sec ion 3. Some expe imen al esul s om wo eal da ase s and a compa ison
be ween he p oposed me hod and a gene ic algo i hm a e epo ed in Sec ion 4.
Finally, Sec ion 5 ou lines he main conclusions o he pape and u u e wo ks.
2 Sca e Sea ch
Sca e Sea ch [14] is an op imiza ion algo i hm based on popula ions in o-
duced in he se en ies. Recen ly, Sca e Sea ch algo i hms ha e been applied o
many nonlinea and combina o ial op imiza ion p oblems p o iding ema kable
ou comes due o i s flexibili y o adop diffe en sea ch s a egies mainly.
Basically, a s anda d Sca e Sea ch can be summa ized by he ollowing s eps:
1. Gene a e an ini ial popula ion in a de e minis ic manne o assu e he di-
e si y o he popula ion ega ding a dis ance.
2. A se , called se o e e ence, is buil wi h he bes indi iduals om his
popula ion. The bes indi iduals is no limi ed o a measu e o indi iduals
p o ided by a fi ness unc ion bu a indi idual ha imp o es he di e si y
can be added o he e e ence se .
3. New indi iduals a e c ea ed by he de e minis ic combina ion o indi iduals
o he e e ence se and all indi iduals o he e e ence se a e selec ed o be
combined.
4. The e e ence se is upda ed using he new indi iduals and he combina ion
is epea ed un il he e e ence se does no change.
5. The e e ence se is ebuil and i he maximum numbe o i e a ions is no
eachedgo os ep3.
The e o e, he sea ch s a egies o a Sca e Sea ch depend on a di e sifica ion
me hod o gene a e he ini ial popula ion, a me hod o buil he e e ence se ,
a me hod o combine indi iduals and a me hod o ebuil he e e ence se .
The main diffe ences be ween a Gene ic Algo i hm and a Sca e Sea ch a e
he way o gene a ing he ini ial popula ion, as he ini ial popula ion is gen-
e a ed andomly and de e minis ic, espec i ely; he selec ion o indi idual o
c ea e offsp ing, as a p obabilis ic p ocedu e is applied o selec pa en s in Ge-
ne ic Algo i hms and all indi iduals o he e e ence se a e combined in Sca e
Sea ch; he e olu ion o he popula ion, based on he su i al o he bes depend-
ing on he fi ness unc ion in Gene ic Algo i hms and he ebuilding me hod o
e e ence se used in Sca e Sea ch. Finally, he size o he popula ion in Gene ic
Algo i hms is bigge han ha o he e e ence se in Sca e Sea ch. A ypical
size in Gene ic Algo i hms is 100 and 10 in Sca e Sea ch, due o ha he com-
bina ion me hod in Sca e Sea ch akes in o accoun all pai s o indi iduals o
c ea e new indi iduals. In sho , he unde lying idea o Sca e Sea ch is o em-
phasize sys ema ic p ocesses agains andom p ocedu es o gene a e popula ions,
o c ea e new indi iduals and o injec di e si y o he popula ion.
3 Desc ip ion o he Algo i hm
In his sec ion he p oposed SS&GA algo i hm o ob ain biclus e s is desc ibed,
de ailing he s eps a o emen ioned in he p e ious sec ion such as combina ion,
gene a ion, upda ing and ebuilding me hods.
The pseudocode o he p oposed SS&GA algo i hm is p esen ed in
algo i hm 1.
3.1 Biclus e s Codi ica ion and Gene a ion
Fo mally, a mic oa ay is a eal ma ix composed by Ngenes and Mcondi ions.
The elemen (i, j)o he ma ix means he le el o exp ession o gene iunde
he condi ion j. A biclus e is a subma ix o he ma ix Mcomposed by n≤N
ows o genes and m≤Mcolumns o condi ions.
Biclus e s a e encoded by bina y s ings o leng h N+M[10]. Each o he
fi s N bi s o he bina y s ing is ela ed o he genes and he emaining M
bi s o he condi ions om mic oa ay M. Fo ins ance, he biclus e shown in
Fig. 1 is encoding by he ollowing s ing,
0010110000|01100 (1)
Thus, his s ing codifies he biclus e composed by genes numbe 3,5and 6
and condi ions 2and 3 om a mic oa ay comp ising 10 genes and 5condi ions.
The ini ial popula ion o biclus e s is s ic ly andomly gene a ed ( ypical in
Gene ic algo i hms) wi hou aking in o accoun he di e si y ( ypical in Sca e
Sea ch). Random s ings composed by 0and 1a e gene a ed un il nB biclus e s
a e buil , whe e nB is he size o he s a ing popula ion, i.e. he numbe o
biclus e s.
Algo i hm 1. SS&GA o Biclus e ing
INPUT Mic oa ay M, penaliza ion ac o s M1and M2, size o popula ion nB,size
o e e ence se , S, and maximum numbe o i e a ions, MaxI e .
OUTPUT The e e ence se , Re Se
begin
Ini ialize P andomly wi h nB biclus e s
//Building e e ence se
R1←S/2bes biclus e s om P(acco ding o hei fi ness unc ion)
R2←S/2mos sca e biclus e s, ega ding R1, omPR1(acco ding o a dis-
ance).
Re Se =R1∪R2
P=PRe Se
//Ini ializa ion
s able ←FALSE
i e =0
while (i e ≤MaxI e )do
//Upda ing e e ence se
while (NOT s able) do
A←Re Se
B←Combina ionMe hod(Re Se )
Re Se ←Sbes biclus e s om Re Se ∪B
i (A=Re Se ) hen
s able ←TRUE
end i
end while
//Rebuilding e e ence se
R1←S/2bes biclus e s om Re Se (acco ding o hei fi ness unc ion)
R2←S/2mos sca e biclus e s, ega ding R1, omPR1.
Re Se =R1∪R2
P←PRe Se
i e =i e +1
end while
end
3.2 Building Re e ence Se
The e e ence se comp ises he bes Sbiclus e s o he ini ial popula ion, P,
whe e Sis he numbe o biclus e s ha belong o his se . The e e ence se is
buil aken in o accoun bo h quali y and sca e ing o biclus e s. The quali y o
biclus e s is measu ed e alua ing he fi ness unc ion conside ed in he e olu i e
p ocess. Thus, a biclus e is be e han ano he i he fi ness unc ion alue
is lowe han ha o he second one. On he o he hand, a dis ance mus be
defined in o de o show how he sca e ing is in oduced in he sea ch space.
In he p oposed SS&GA app oach he dis ance used is he Hamming dis ance.
The Hamming dis ance be ween wo bina y s ings is defined by he numbe o
posi ions o which hei co esponding 0/1 alues a e diffe en . Fo example,
he Hamming dis ance o s ings 001001001|001 and 001011001|101 is 2.
1C 3C 2C 5C 4C
3.0 6.2 - 3.5 6.3 2.2 1G
2 - 2 1.2 - 1.3 - 5.1 3.1 2G
3G 7.4 0.1 0.1 4.0 9.7
4.1 1.3 2.2 3.0 - 8.3 - 4G
5G 5.7 0.1 0.1 3.2 - 1.2
6G 4.0 0.1 0.1 3.0 4.0
1.3 5.2 - 5.2 - 3.8 2.3 7G
1.0 3.0 1.4 1.3 5.2 8G
2.0 2.9 9.6 4.0 1.3 9G
1.0 - 3.0 3.0 5.0 3.0 01G
1.0 1.0
1.0 1.0
1.0 1.0
biclus e
00101100000|1100
codi ica ion
mic oa ay
Fig. 1. Mic oa ay and biclus e along wi h i s codifica ion
The e o e, he e e ence se is o med by he S/2bes biclus e s om P(se
R1) acco ding o hei fi ness unc ion and he S/2biclus e s om PR1
(se R2) wi h he highes dis ances o he se R1acco ding o he Hamming
dis ance.
3.3 Combina ion Me hod and Upda ing Re e ence Se
Combina ion me hod is he mechanism o c ea e new biclus e s in Sca e Sea ch.
All pai s o biclus e s a e combined gene a ing S∗(S−1)/2new biclus e s. In he
SS&GA algo i hm he ypical uni o m c osso e ope a o used in Gene ic Algo-
i hms is he p oposed combina ion me hod. This c osso e ope a o is shown in
Fig. 2. A bina y mask is andomly gene a ed and a child is composed by alues
om he fi s pa en when he e is a 1 in he mask, and om he second pa en
when he e is a 0.
The e e ence se is upda ed wi h he Sbes biclus e s om he e e ence se
and he new biclus e s gene a ed by he combina ion me hod acco ding o he
fi ness unc ion. This p ocess is epea ed i e a i ely un il he e e ence se does
no change.
pa en 1 1 0 1101 1
0 1 1100 0
1 1 1100 1child
pa en 2
mask 1 0 0100 1
Fig. 2. Uni o m c osso e ope a o o Gene ic Algo i hms

3.4 Rebuilding Re e ence Se
A e ge ing he s abili y o e e ence se in he upda ing p ocess, his se is
ebuil o in oduce di e si y in he sea ch p ocess. This ask is made by mu a ion
ope a o s in Gene ic Algo i hms. Thus, he e e ence se is composed by he S/2
bes biclus e s om he upda ed e e ence se (se R1) acco ding o he fi ness
unc ion and he S/2mos dis an om PR1acco ding o he Hamming
dis ance.
3.5 Biclus e s E alua ion
The fi ness unc ion is undamen al in o de o e alua e he quali y o biclus-
e s. Cheng and Chu ch p oposed he MSR which measu es he co ela ion o a
biclus e . Gi en a biclus e comp ising he subse o genes Iand he subse o
condi ions J, he MSR is defined as ollows,
MSR(I,J)= 1
|I||J|
i∈I,j∈J
R(i, j)2(2)
whe e
R(i, j)=eij −eIj −eiJ +eIJ (3)
eIj =1
|I|
i∈I
eij (4)
eiJ =1
|J|
j∈J
eij (5)
eIJ =1
|IJ|
i∈I,j∈J
eij (6)
In his wo k, biclus e s wi h low esidue and high olume a e p e e ed. The e-
o e, he fi ness unc ion is defined by:
(B)=MSR(B)+M11
G+M21
C(7)
whe e MSR(B)is he MSR o he biclus e B,M1and M2a e penaliza ion
ac o s o con ol he olume o he biclus e B,andGand Ca e he numbe
o genes and condi ions o he biclus e B, espec i ely.
The use o MSR in he fi ness unc ion conside ed in he p oposed SS&GA
algo i hm allows o es ablish a compa ison wi h a p e ious e olu iona y-based
biclus e ing me hod and he Cheng and Chu ch algo i hm.
4 Expe imen al Resul s
Two well known da ase s [4] ha e been used o shows he pe o mance o he
p oposed SS&GA algo i hm. The fi s da ase is he yeas Saccha omyces ce e-
isiae cell cycle exp ession and he second one is he human B-cells exp ession
da a o igina ed om [15] and [16], espec i ely. O iginal da a we e p ep ocessed
in [4] eplacing missing alues wi h andom numbe s. The Yeas da ase con ains
2884 genes and 17 expe imen al condi ions and he Human da ase consis s o
4026 genes and 96 condi ions.
The main pa ame e s o he p oposed SS&GA algo i hm a e as ollows: 200 o
he ini ial popula ion; 10 o he e e ence se and 20 o he maximum numbe o
i e a ions. The penaliza ion ac o o he numbe o condi ions has been chosen
o one o de o magni ude la ge han he ange in which he fi ness unc ion
a ies o bo h da ase s. Howe e , ha o he numbe o genes has been chosen
o same o de o magni ude o he ange o alues o he fi ness unc ion o bo h
da ase s. The main goal o his choice is o es he influence o he penaliza ion
ac o s on he olume o he biclus e s.
4.1 Yeas Da a Se
Table 1 p esen s se e al biclus e s ob ained by he applica ion o he SS& GA
app oach om Yeas da ase . Fo each biclus e is shown an iden ifie o he
biclus e , he alue o i s MSR, he numbe o genes and he numbe o condi-
ions. I can be obse ed ha high–quali y biclus e s ha e been ob ained as he
alues o he MSR a e lowe han 220. Mo eo e , he olume o he ob ained
biclus e s is sa is ac o y showing ha he SS& GA app oach find non– i ial
biclus e s. Conc e ely, biclus e s and no clus e s a e ob ained since he numbe
o condi ions is less han 17 always.
Biclus e s p esen ed in Table 1 a e shown in Fig. 3. Al hough biclus e s a e
good aking in o accoun he alues o hei MSR, in his figu e hei ends
canno be obse ed easily. This is due o he o e lapping among biclus e s as
he same gene can be ound in diffe en biclus e s.
Fig. 4 shows he e olu ion o he a e age MSR, fi ness unc ion alues and
olume o he e e ence se h oughou he e olu iona y p ocess o he Yeas
da ase . The alues o he MSR and he olume a e ep esen ed in he axis on
he le and ha o he fi ness unc ion in he axis on he igh . I can be no iced
ha he ini ial e e ence se imp o es he a e age MSR h oughou he i e a ions
Table 1. Resul s ob ained by SS&GA algo i hm o Yeas da ase
Biclus e MSR Genes Condi ions
bi.1 74.72 10 13
bi.2 106.25 13 13
bi.3 125.9 22 13
bi.4 216.16 25 14
bi.5 97.04 26 11
bi.6 117.25 14 14
bi.7 136.67 25 13
bi.8 159.44 39 13
bi.9 121.89 26 11
2 4 6 8 10 12
0
100
200
300
400
Condi ions
Exp ession Value
bi1
24681012
100
200
300
400
Condi ions
Exp ession Value
bi2
24681012
150
200
250
300
350
400
Condi ions
Exp ession Value
bi3
2 4 6 8 10 12 14
0
100
200
300
400
Condi ions
Exp ession Value
bi4
246810
0
100
200
300
400
Condi ions
Exp ession Value
bi5
2 4 6 8 10 12 14
150
200
250
300
350
400
Condi ions
Exp ession Value
bi6
2 4 6 8 10 12
150
200
250
300
350
Condi ions
Exp ession Value
bi7
24681012
100
200
300
400
500
Condi ions
Exp ession Value
bi8
246810
0
100
200
300
400
Condi ions
Exp ession Value
bi9
Fig. 3. Biclus e s om Yeas da ase
and he SS&GA algo i hm con e ges in 8 i e a ions app oxima ely. The a e age
olume o he e e ence se dec eases e sus he numbe o i e a ions due o he
non oo la ge penaliza ion ac o s ha e been chosen.
4.2 Lymphoma Da a Se
Table 2 p esen s in o ma ion abou se e al biclus e s ound by he SS&GA ap-
p oach o Human da ase . The alues o he MSR a e conside ably low since
all a e lowe han 1100. Thus, i can be s a ed ha ob ained biclus e s ha e
a ema kable quali y. Mo eo e , in gene al he ob ained biclus e s ha e a la ge
numbe o genes, specially he biclus e numbe 1, 2 and 4. These biclus e s a e
also ep esen ed in Fig. 5.
Figu e 6 p esen s he pe o mance o he p oposed algo i hm o Human
da ase . The e olu ion o he a e age MSR, fi ness unc ion alues and olume
100
150
200
250
300
350
1 2 3 4 5 6 7 8 9 1011121314151617181920
Numbe o i e a ions
A e age esidue
0
100
200
300
400
500
600
A e age i ness unc ion
Volume
MSR
Fi ness Func ion
Fig. 4. Pe o mance o he p oposed SS&GA algo i hm o Yeas da ase
Table 2. Resul s ob ained by SS&GA o Human da ase
Biclus e MSR Genes Condi ions
bi.1 855.17 109 13
bi.2 813.70 127 12
bi.3 642.13 85 11
bi.4 815.74 122 10
bi.5 771.69 48 12
bi.6 595.69 44 9
bi.7 1074.10 56 13
bi.8 507.17 67 8
bi.9 794.07 70 11
Table 3. Compa ison o he esul s ob ained by SS&GA, SEBI and CC algo i hms
Algo i hm-Da ase A g. Residue A g. gene num. A g. cond. num.
SS&GA–Yeas 128.37 (40.71) 22.23 (8.86) 12.78 (1.09)
SS&GA–Human 763.27 (165.73) 80.89 (36.61) 11 (1.73)
SEBI–Yeas 205.18 (4.49) 13.61 (10.38) 15.25 (1.37)
SEBI–Human 1028.84 (29.19) 14.07 (5.39) 43.57 (6.20)
CC–Yeas 204.29 (42.78) 166.71 (226.37) 12.09 (4.39)
CC–Human 850.04 (153.91) 269.22 (204.71) 24.5 (20.92)
o he e e ence se is shown. A good pe o mance o he SS&GA echnique and
a as con e gence can be app ecia ed. The alues o he fi ness unc ion dec ease
quickly and only en i e a ions app oxima ely a e needed o find high–quali y
biclus e s. In his case, he choice o penaliza ion pa ame e s o keep unde con-
ol he olume o he biclus e s p o ides a nea ly cons an olume in he las
i e a ions.
Finally, a compa ison be ween he esul s ob ained wi h he SS&GA algo-
i hm and wo ep esen a i e echniques epo ed in he li e a u e is p o ided.
Conc e ely, he SS&GA algo i hm is compa ed o SEBI [10] and Cheng and