121E olu iona y Bioin o ma ics 2015:11
In oduc ion
Mic oa ay echnology is highly used in biological esea ch
en i onmen s due o i s abili y o moni o , o a g ea gene
collec ion, he RNA concen a ion le els, hus enabling he
s udy o gene ic unc ions o species.1 Bioin o ma ics and da a
mining ha e de eloped a as numbe o compu a ional ools
ha allow us o analyze da a ob ained using his echnology
and o ind new knowledge ha is hidden om human eye-
sigh .2,3 One o he mos s udied app oaches is pa e n sea ch
in gene exp ession da a. The genes exhibi ing high co ela ion
among hei exp ession le els could be in ol ed in simila eg-
ula o y p ocesses.4 The ela ionship be ween co ela ion and
unc ionali y has been p o ed in se e al s udies as in he s udy
by D’haeselee e al.5
Clus e ing echniques a e sui able o pe o ming pa e n
sea ch by c ea ing g oups o genes ha exhibi simila exp es-
sion pa e ns.6 T adi ional clus e ing algo i hms analyze he
whole mic oa ay dimensional space g ouping genes ak-
ing in o accoun all expe imen al condi ions.7 Howe e , he
ac i i y o genes could only appea unde a pa icula se o
expe imen al condi ions, exhibi ing local pa e ns. Disco e -
ing hese local pa e ns can be key o disco e gene pa hways,
which could be ha d o disco e in o he ways. Fo his eason,
he pa adigm o clus e ing echniques mus be modi ied o
me hods ha allow local pa e n disco e y in gene exp ession
da a.8 Biclus e ing9 add esses his p oblem by elaxing he
condi ions and by allowing assessmen only unde a subse o
he condi ions o he expe imen , and i has p o ed o be suc-
cess ul in inding gene pa e ns.10,11
I a hi d dimension is added o he da ase besides
genes and condi ions, such as ime, clus e ing and biclus-
e ing esul insu icien . The e is a lo o in e es in empo-
al expe imen s because hey allow an in-dep h analysis o
molecula p ocesses in which he ime e olu ion is impo -
an , o example, cell cycles, de elopmen a he molecula
le el, o e olu ion o diseases.12 In his sense, iclus e ing
appea s as a echnique going one s ep u he by g ouping
genes unde pa icula condi ions and unde pa icula ime
poin s,13 hus being capable o managing h ee-dimensional
(3D) da a. The e o e, iclus e ing is sui able o he analysis
o mic oa ay expe imen s whe e se e al samples a e aken a
di e en ime poin s.14 This is o g ea in e es since i allows
o a deep analysis o biological p ocesses whe e empo a y
de elopmen is impo an .
Bo h biclus e ing and iclus e ing a ack NP-ha d p o-
blems.15 The e o e, algo i hms based on heu is ics a e well
sui ed o manage his kind o p oblem. In his sense, de ining
an app op ia e quali y measu e o iclus e s is an impo an
and essen ial challenge.16
In his wo k, we p opose a quali y measu e called
Mul i Slope Measu e (MSL), which measu es he quali y o
a iclus e based on he simila i y among he angles o he
slopes o med by each p o ile o med by he genes, condi ions,
and imes o he iclus e .
MSL: A Measu e o E alua e Th ee-dimensional Pa e ns
in Gene Exp ession Da a
Da id Gu ié ez-a ilés and c is ina ubio-Escude o
Depa men o Compu e Science, Uni e si y o Se ille, Se ille, Spain.
Abs Ac : Mic oa ay echnology is highly used in biological esea ch en i onmen s due o i s abili y o moni o he RNA concen a ion le els. The
analysis o he da a gene a ed ep esen s a compu a ional challenge due o he cha ac e is ics o hese da a. Clus e ing echniques a e widely applied o
c ea e g oups o genes ha exhibi a simila beha io . Biclus e ing elaxes he cons ain s o g ouping, allowing genes o be e alua ed only unde a subse o
he condi ions. T iclus e ing appea s o he analysis o longi udinal expe imen s in which he genes a e e alua ed unde ce ain condi ions a se e al ime
poin s. These iclus e s p o ide hidden in o ma ion in he o m o beha io pa e ns om empo al expe imen s wi h mic oa ays ela ing subse s o genes,
expe imen al condi ions, and ime poin s. We p esen an e alua ion measu e o iclus e s called Mul i Slope Measu e, based on he simila i y among he
angles o he slopes o med by each p o ile o med by he genes, condi ions, and imes o he iclus e .
Keywo ds: iclus e ing, angula compa ison, gene ic algo i hms, i ness unc ion, mic oa ays, ime se ies
CiTATion: Gu ié ez-a ilés and ubio-Escude o. msl: a measu e o E alua e h ee-
dimensional Pa e ns in Gene Exp ession Da a. E olu iona y Bioin o ma ics 2015:11
121–135 doi: 10.4137/EBo.s25822.
RECEi ED: ma ch 11, 2015. RESubMiTTED: may 13, 2015. ACCEPTED oR
PubLiCATion: may 21, 2015.
ACADEMiC EDiToR: Jike cui, associa e Edi o
TYPE: o iginal esea ch
unDinG: The au ho s wan o acknowledge he inancial suppo gi en by he Spanish
minis y o science and echnology wi h p ojec in2011-28956-c02-02 and Jun a de
Andalucía wi h p ojec TIC-7528. The au ho s con i m ha he unde had no in luence
o e he s udy design, con en o he a icle, o selec ion o his jou nal.
CoMPETinG inTERESTS: Au ho s disclose no po en ial con lic s o in e es .
CoRRESPonDEnCE: [email p o ec ed]s, c ubioescude [email protected]
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an open-access a icle dis ibu ed unde he e ms o he c ea i e commons cc-By-nc
3.0 license.
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was subjec o an i-plagia ism scanning. P io o publica ion all au ho s ha e gi en signed
con i ma ion o ag eemen o a icle publica ion and compliance wi h all applicable e hical and
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o compe ing in e es s and unding sou ces, compliance wi h e hical equi emen s ela ing o
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Published by libe as academica. lea n mo e abou his jou nal.
Gu ié ez-A ilés and Rubio-Escude o
122 E olu iona y Bioin o ma ics 2015:11
We show he esul s ob ained applying he MSL mea-
su e embedded in he T iGen Algo i hm,17 an algo i hm
based on an e olu iona y heu is ic, gene ic algo i hms. The
da ase s used a e a syn he ic da ase and h ee eal expe i-
men da ase s: he yeas cell cycle– egula ed genes,18 mouse
degene a ion o e inal cells,14 and human ansc ip ion ac o
oncogene OTX2 silencing e ec on D425 medulloblas oma
cell line.19
The esul s ha e been alida ed by h ee di e en me h-
ods. Fi s , by analyzing he co ela ion among he genes,
condi ions, and imes in each iclus e using wo di e en
co ela ion measu es (Pea son20 and Spea man21). Second,
by a g aphic alida ion o he pa e ns ex ac ed based on he
g aphic ep esen a ion (see G aphic Rep esen a ion subsec-
ion), and hi d we ha e p o ided unc ional anno a ions o
he genes ex ac ed om he Gene On ology (GO) p ojec .22
The esul s ob ained ha e been compa ed o wo p e iously
de ined quali y measu es, MSR3D23 and LSL,24 showing
imp o emen in he pe o mance o he measu e (see Resul s
and Discussion sec ion).
The es o he a icle is s uc u ed as ollows. A e iew o
he la es ela ed wo ks can be ound in S a e o he A sec-
ion. Me hods sec ion desc ibes he MSL measu e as well as
a b ie desc ip ion o T iclus e ing, he g aphic ep esen a ion
applied, and he T iGen algo i hm. In he Resul s and Dis-
cussion sec ion, we show he esul s and discussion o apply-
ing T iGen o he syn he ic and eal da ase s. The las sec ion
shows he conclusions.
s a e o he A
This sec ion is o p o ide a gene al o e iew o ecen wo ks
in he ield o gene exp ession empo al da a. In pa icula , o
hose wo ks ela ed o he applica ion o iclus e ing, we ocus
on he measu es applied o e alua e he iclus e s.
We i s p esen he au ho s’ p e ious con ibu ions o
his ield. In ou s udy,23 we desc ibed MSR3D, an adap a-
ion o he Mean Squa e Residue (MSR)9 o he 3D space,
so ha a hi d ac o , ime in his case, can be aken in o
accoun . MSR3D measu es he homogenei y o a iclus e in
he ela ion o each alue o he iclus e , wi h he a e age
o all genes, a e age o all condi ions, a e age o all imes,
a e age o all genes and condi ions, a e age o all genes and
imes, a e age o all condi ions and imes, and a e age o
all genes, condi ions, and imes in he iclus e . We also
ha e p esen ed LSL in ou ecen s udy,24 which measu es
he quali y o a iclus e based on he simila i y among he
slopes o he angles o med by he leas squa e lines om
each o he p o iles o med by he genes, condi ions, and
imes o he iclus e . LSL has ob ained be e esul s han
MSR3D applied o he same da ase s along wi h he T iGen
algo i hm.24
Rega ding o he au ho s’ con ibu ions, in 2005,
Zhao and Zaki25 in oduced he iClus e algo i hm o
ex ac pa e ns in 3D gene exp ession da a. They p esen ed
a measu e o assess iclus e s’s quali y based on he sym-
me y p ope y. This allows o e y e icien clus e mining
since clus e s a e sea ched o e he dimensions wi h he leas
ca dinali y.
g- iClus e , an ex ended and gene alized e sion o
Zhao and Zaki’s p oposal, was published one yea la e .26 The
au ho s claimed ha he symme y p ope y is no sui able
o all pa e ns p esen in biological da a and p oposed he
Spea man ank co ela ion21 as a mo e app op ia e iclus e
e alua ion measu e.
An e olu iona y compu a ion p oposal was made by Liu
e al.27 The i ness unc ion de ined is a mul iobjec i e measu e
ha ies o op imize h ee con lic ing objec i es: clus e s size,
homogenei y, and gene-dimension a iance o he 3D clus e .
LagMine was in oduced by Xu e al.28 o ind ime-
lagged 3D clus e s, wha allows in u n o ind egula o y
ela ionships among genes. I is based on a no el 3D clus e
model called S2 D3 Clus e . They e alua ed hei iclus e s
on homogenei y, egula ion, minimum gene numbe , sample
subspace size, and ime pe iods leng h.
Wang e al.29 p oposed a new algo i hm called s-clus e
basing hei de ini ion o cohe en iclus e s also on inding
egula o y ela ionships among genes. Fo ha pu pose, ime
shi ing is also conside ed among ime poin s in he e alua ed
iclus e s.
A new s a egy o mine 3D clus e s in eal- alued da a was
in oduced by Sim e al.30 The au ho s de ined he Co ela ed
3D Subspace Clus e s (CSCs), whe e he alues in each clus e
mus ha e high co-occu ences and hose co-occu ences a e
no by chance. They measu e he clus e s based on he co -
ela ion in o ma ion measu e, which akes in o accoun bo h
p e equisi es.
Hu and Bha naga p esen ed an app oach ocusing on
he concep o Low-Va iance 3-Clus e ,31 which obeys he
cons ain o a low- a iance dis ibu ion o cell alues.
The wo k by Liu e al.32 was ocused on inding Tem-
po al Dependency Associa ion Rules, which ela e pa e ns
o beha io among genes. The ules ob ained a e o ep esen
egula ed ela ions among genes.
Finally, a b ie su ey on iclus e ing applied o gene
exp ession ime se ies was published in 2011.13 The e a e h ee
main ea u es ha a iclus e ing algo i hm can pe o m.
Acco ding o Mahan a e al.13, hese ea u es a e empo al
cohe ence ha makes e e ence o he abili y o he algo i hm
o cap u e he cohe ence o di e en genes in a single ime
poin ac oss samples while gene a ing he inal iclus e s
and he abili y o ind iclus e s wi h nonconsecu i e ime
poin s and iclus e wi h a speci ic ype o pa e n (shi -
ing, scaling, delayed). g- iClus e ,26 Moga3c,27 LagMine ,28
s-clus e ,29 and Tempo al Dependency Associa ion Rules32
pe o m he empo al cohe ence ea u e and only T iclus e 25
and Moga3c27 pe o m inding iclus e s con aining non-
consecu i e ime poin s. T iclus e 25 inds scaling pa e ns,
LagMine 28 inds shi ing and scaling pa e ns and s-clus e 29
MSL: 3D pa e ns om gene exp ession.
123E olu iona y Bioin o ma ics 2015:11
ocuses on ime-delayed pa e ns; he es do no ocus on
inding a speci ic ype o pa e n. Ano he ea u e examined
is he algo i hm ype dis inguishing be ween de e minis ic
(T iclus e 25, g- iClus e 26) and nonde e minis ic (Moga3c27)
app oaches.
Me hods
In his sec ion, we desc ibe ou p oposal, he iclus e quali y
measu e called MSL ha is based on iclus e ’s angula ea-
u es. We will analyze all MSL p inciples and undamen als
and how i has been de eloped.
This sec ion is s uc u ed as ollows: T iclus e ing sub-
sec ion desc ibes he iclus e ing p ocedu e as an e olu ion
o i s well-known p edecesso biclus e ing. In he subsec-
ion G aphic Rep esen a ion, we in oduce he g aphic
ep esen a ion, which is key o unde s anding he MSL
measu e. Then, in he subsec ion MSL Measu e, we ana-
lyze he co e o ou wo k, he MSL measu e. Finally, in he
T iGen Algo i h subsec ion, we b ie ly desc ibe he T iGen
algo i hm.
iclus e ing. Clus e ing echniques a e applied o ana-
lyze gene exp ession da a om mic oa ay expe imen s. The
da ase ob ained om he expe imen , D, con ains genes and
expe imen al condi ions and clus e ing aims a inding sub-
g oups o genes ha sha e a beha io pa e n acco ding o hei
exp ession le el. Biclus e ing appea s as an e olu ion o clus e -
ing due o i s abili y o mine subg oups o genes and condi ions
om he da a se D, whe e he genes exhibi highly co ela ed
pa e ns o beha io unde ce ain expe imen al condi ions.9
T iclus e ing eme ges as an e olu ion o biclus e ing,
aking in o accoun he empo a y e olu ion o genes unde
pa icula expe imen al condi ions. In his way, om a da ase
D ob ained om a mic oa ay expe imen , which con ains
genes GD, condi ions CD, and ime poin s TD, we de ine i-
clus e ing as a echnique ha inds iclus e s TRI1,…,TRIn
om D, whe e a iclus e TRI is o mally de ined as TRI =
G × C × T, whe e G ⊆ GD, C ⊆ CD, and T ⊆ TD,17 ie, a subse
o genes ha con ains in o ma ion ela ed o he beha io o
some genes om da ase G unde condi ions C a imes T.
Figu e 1 shows a iclus e wi h genes as ows, condi ions as
columns, and ime as dep h.
G aphic ep esen a ion. In o de o explain he MSL
measu e, we de ine he g aphic ep esen a ion o a iclus e
TRIxop, wi h x, o, and p being ei he genes G, expe imen-
al condi ions C, o ime poin s T, so ha he x elemen s in
TRIxop will be on X axis and o elemen s in TRIxop will be he
ou lines ep esen ed in as many panels as p elemen s in TRIxop
indica es, as can be seen in Figu e 2.
To isually analyze he beha io pa e ns o a iclus e
TRI, we always conside h ee g aphical iews:
• TRIgc (x = G, o = C, p = T): one panel o each ime,
genes on he X axis, he exp ession le els on he Y axis,
and he lines o condi ions as he ou line.
• TRIg c (x = G, o = T, p = C): one panel o each condi ion,
genes on he X axis, he exp ession le els on he Y axis,
and he ime lines as he ou line.
• TRI gc (x = T, o = G, p = C): one panel o each condi ion,
imes on he X axis, he exp ession le els on he Y axis,
and he genes as he ou line.
Wi h TRIgc and TRIg c, we can analyze how each gene
exp ession le el a ies h oughou condi ions and imes,
espec i ely. TRI gc ep esen s how each gene a ies h ough-
ou ime o each condi ion.
MsL measu e. A e analyzing he g aphic ep e-
sen a ion o a iclus e , we desc ibe ou p oposal: he
Mul i Slope Measu e (MSL). MSL measu es he di e -
ences among he angles o med by e e y se ies aced on
each o h ee g aphic ep esen a ions aking in o accoun
TRIgc , TRIg c, and TRI gc (subsec ion G aphic Rep esen-
a ion). MSL akes in o accoun he in luence o neigh-
bo ing ime poin s. We can obse e an example o TRI gc
iew o TRI = G {g1, g4, g7, g10}, C {c2, c5, c8}, T { 0, 2, 11}
in Figu e 3. We can see how each ou line o gene o ms a
se o angles ( wo o his pa icula example) de ined by
each ime poin in he X axis o e e y panel o expe i-
men al condi ion.
To calcula e he MSL measu e o a iclus e , we i s
pe o m h e m ul ian g ula co m p a is o n e m ca l cula i o n . Th e
mul iangula compa ison ope a ion o a g aphic ep esen-
a ion xop om a iclus e TRI is de ined in Equa ion 1a.
We de ine ACmul i o a iclus e ’s g aphic ep esen a ion
TRIxop as he a e age o he di e ences ∆ o angles ec o s
a opε ang se (Equa ion 1b) o all ou lines o o each panel
p (Vmc in Equa ion 1c) and i s equi alen o he es o he
panels (Hmc in Equa ion 1d), wi h Nmc being he numbe
Condi ions
Times
Genes
igu e 1. iclus e ep esen a ion.
Gu ié ez-A ilés and Rubio-Escude o
124 E olu iona y Bioin o ma ics 2015:11
o di e ences made (Equa ion 1e). An angle ec o o an
ou line o in a panel p is de ined as a se o angles ha
a e o med by he ou line o aking in o accoun e e y da a
poin in he X axis (Equa ion 1 ), each ou line will ha e
Numbe o X axis icks – 1 angles as you can see in Figu e 3.
The di e ence ∆ be ween wo angles ec o a A and a B
is de ined as he a e age o he MAX – MIN (MAX being
he maximum and MIN he minimum o wo angles a A(i)
and a B(i)) o each componen (o angle) i o a A and a B
(Equa ion 1g).
AC TRIVH
N
mul ixop
mc mc
mc
()
=+
(1a)
angse a a a a a a
op op op op op op
UT AN
=
{}
11 21 31 12 12
,,,,,,,……
( 1 b )
Va a
mc op nex op
angse
=∆
()
()
∑,
(1c)
Ha a
mc op onex p
angse
=∆
()
()
∑,
(1d)
N
op op
mc =+−
()
|||| || ||2
2
**
(1e)
a aa
op ix xx
iAX
=∀ −
ε
,,…
1
(1 )
∆
()
=
∑
() ()
()
−
() ()
()
a a
MAXa ia iMIN a ia i
AB
ia a ABAB
AB
,
,,
,
ε
|||a AB,
(1g)
The ACmul i e m is based on se e al ope a ions wi h a op
angle ec o s. These elemen s ha e been ob ained based on
concep o se ies (Equa ion 2a) so ha a se ies Sop o a ou line
o o a panel p is a se o pai o alues om he x axis (xi) and
exp ession le els (elj) ha o m he ou line. Fo each se ies Sop,
he alpha angle
α
xi
is calcula ed as he spin o he a c angen
o he slope o he line o med by (xi, eli) and (xnex (i), elnex (i))
poin s (Equa ion 2b). The spin ope a ion o an angle showed
in Equa ion 2c is he posi i e equi alen o his angle i i is
nega i e.
Sxel xel
op AX L
=< >< >
{}
00
,,,,…
(2a)
α
x
nex ii
nex ii
ispin xx
el el
=−
−


















()
()
a c an
(2b)
elL
el0
el1el0
el1
el0
el1
elj
elL
elj
elL
elj
elL
elj
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
el0
el1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... ...
.
oUT
o0
o1
o
oUT
o0
o1
o
oUT
o0
o1
o
oUT
o0
o1
o
x0x1x1xAX ......
x0x1x1xAX ... ...
x0x1x1xAX ......
x0x1x1xAX
p0p1pspAN
igu e 2. G aphic ep esen a ion o a iclus e .
c2
g4
g1
g7
g10
g4
g1
g7
g10
g4
g1
g7
g10
αg
αc2
g7
αc2
g4
c5c8
0 2 11 0 2 11 0 2 11
βgαg
αc5
g7
αc5
g4
αc5
g1
βgαc8
g10
βc8
g10
βc5
g7αc8
g7
βc8
g7
βc5
g4
αc8
g4βc8
g4
βc5
g1αc8
g1βc8
g1
βc2
g7
βc2
g4
c2
10 c2
10 c5
10 c5
10
igu e 3. angles o TRI gc g aphic iew.
MSL: 3D pa e ns om gene exp ession.
125E olu iona y Bioin o ma ics 2015:11
spin i
xx xx
ii ii
() *
αα αα
=<⇒=+02π
(2c)
To conclude, he MSL measu e o a iclus e TRI (Equa-
ion 3) is he a e age o he angula compa ison o he h ee
g aphic ep esen a ions o he iclus e .
Following Figu e 3, we show an example o ACmul i(TRI gc )
calcula ion in Figu e 4. Fi s , we a ange he example iclus-
e TRI so ha o each condi ion (panel) we ob ain a able
wi h one ow pe gene (ou line) and one column pe ime poin
(X axis). Second, in o de o ge each a gc (Equa ion 1 ) o
angse (Equa ion 1b), we use Equa ion 2b om Sgc se ies and
ob ain all angles
α
xi
o a gc ec o s, and hi d, we use Equa-
ions 1c and 1d o Vmc and Hmc calcula ions, espec i ely,
going h ough he a se in ho izon al and e ical di ec ion.
Finally, we use Equa ion 1a o ob ain he ACmul i(TRI gc) alue.
We will ha e o epea his p ocess wice mo e, once o each
g aphic ep esen a ion TRIgc and TRIg c, o ob ain he MSL
measu e acco ding o Equa ion 3.
MSLTRI AC TRIACTRI
AC TRI
mul igc mul ig c
mul i
()
=
()


+
()
+
1
3
ggc
()
(3)
iGen algo i hm. In his sec ion, we p esen he T iGen
(T iclus e ing-Gene ic based) algo i hm,17 whe e he MSL
measu e has been embedded in o de o es i s e ec i eness.
T iGen applies a bio-inspi ed pa adigm o an e olu iona y
heu is ic, gene ic algo i hms, in such a way ha inds a se
o iclus e s om gene exp ession da ase s whe e he ime is
also a componen aken in o accoun in he expe imen . This
me hod mimics he p ocess o na u al selec ion by c ea ing
an ini ial popula ion o indi iduals ep esen ing solu ions ha
p1 = c2
x1 = 0
1 s
10 515
30
42
100
40
45
85
30
50
20 10 15
35
45
80
20
20 10 10
40
45
70
35
55
80
30
50
70
50
70
30
40
70
80
2 s3 s 1 s2 s3 s 1 s2 s3 s
o1 = g1
o2 = g4
o3 = g7
o4 = g10
Hmc = 17.01
ACmul i (TRI gc) = 1.97
Vmc = 42.37
a g
a g
a g
a g
o1 = g1
o2 = g4
o3 = g7
o4 = g10
o1 = g1
o2 = g4
o3 = g7
o4 = g10
x2 = 2x3 = 11 x1 = 0x2 = 2x3 = 11 x1 = 0x2 = 2x3 = 11
p2 = c5p3 = c8
= {1.37, 1.5}
= {4.9, 5.03}
= {1.47, 4.81}
= {4.9, 1.47}
= {0, 1.47}
= {1.47, 4.9}
= {4.81, 1.5}
= {4.81, 1.37}
= {1.47, 4.81}
= {1.37, 4.81}
= {1.37, 1.37}
= {4.81, 0}
a g c21 c81
a g c2
4
a gc5
4c8
4
c8
7
c810
a g c2
10
a g c2
7
= 1.78
∆ (a g , a g )
c21 c2
7
= 3.38
∆ (a g , a g )
c21 c2
4
= 1.83
∆ (a g , a g )
c24 c2
7
= 3.53
∆ (a g , a g )
c27 c2
10
= 1.7
∆ (a g , a g )
c24 c2
10
= 1.78
∆ (a g , a g )
c21 c2
10
= 0.06
∆ (a g , a g )
c51 c5
4
= 3.43
∆ (a g , a g )
c51 c5
7
= 3.37
∆ (a g , a g )
c54 c5
7
= 2.45
∆ (a g , a g )
c51 c5
10
= 2.42
∆ (a g , a g )
c54 c5
10
= 2.45
∆ (a g , a g )
c57 c5
10
= 2.4
∆ (a g , a g )
c81 c8
4
= 0.09
∆ (a g , a g )
c21 c2
1= 3.32
∆ (a g , a g )
c24 c5
4
= 1.76
∆ (a g , a g )
c24 c8
4
= 1.78
∆ (a g , a g )
c54 c8
4
= 2.4
∆ (a g , a g )
c510 c8
10
= 1.7
∆ (a g , a g )
c210 c8
10
= 0.7
∆ (a g , a g )
c210 c5
10
= 0.78
∆ (a g , a g )
c21 c8
1
= 0.68
∆ (a g , a g )
c51 c8
1
= 1.87
∆ (a g , a g )
c27 c8
7
= 0.09
∆ (a g , a g )
c57 c8
7
= 1.78
∆ (a g , a g )
c27 c8
7
= 4.12
∆ (a g , a g )
c81 c8
7
= 1.71
∆ (a g , a g )
c84 c8
7
= 1.76
∆ (a g , a g )
c84 c8
10
= 4.07
∆ (a g , a g )
c81 c8
10
= 0.04
∆ (a g , a g )
c87 c8
10
a g c51
a g c5
7
a g c5
10
igu e 4. ACmul i(TRI gc) example.

Gu ié ez-A ilés and Rubio-Escude o
126 E olu iona y Bioin o ma ics 2015:11
1. Inpu : The T iGen algo i hm has wo inpu a gumen s:
• D: A da ase con aining he gene exp ession alues om a
mic oa ay expe imen con aining genes DG, expe imen-
al condi ions DC, and imes DT . The e o e, each cell [i, j,
k] om D whe e i ∈ DG, j ∈ DC, and k ∈ DT , ep esen s
he exp ession le el o he gene i unde he expe imen al
condi ion j a ime k.
• P: Se o pa ame e s o execu e he algo i hm as desc ibed
in Table 1. These pa ame e s con ol he numbe o solu-
ions o iclus e s o ind (N), he numbe o gene a ions
o execu e (G), he numbe o indi iduals in he popula-
ion (I), and he andomness ac o hey a e gene a ed
wi h he ini ial popula ion (Ale) as well as weigh s o he
selec ion and mu a ion ope a o s (Sel and Mu ), weigh s
o con ol he e ec o he MSL measu e (w ), he size
o he iclus e s (wg, wc, w ), and weigh s o con ol he
o e lap among solu ions (wog, woc, wo ).
The pa ame e s ha e been chosen a e an exhaus i e
expe imen a ion wi h all possible anks o alues. Each o he
pa ame e s has e ec on he iclus e s ound: size, o e lapping,
explo a ion e sus exploi a ion o he algo i hm, e c. We now
desc ibe he e ec s o each o he pa ame e s. In he execu ion
o T iGen, each pa ame e is associa ed o a gene ic ope a o ,
ha is, G con ols he whole e olu iona y p ocess so an inc ease
in he numbe o gene a ions implies a g ea e numbe o
ecombina ion o indi iduals. The e o e, an excessi e inc ease in
G may a o exploi a ion e sus explo a ion in excess and he
algo i hm may e u n solu ions ha all in o a local minimum.
I and Ale con ol he ini ial popula ion c ea ion, and when he
numbe o indi iduals I is inc eased, a la ge sea ch space o
he solu ions is c ea ed so ha an excessi e inc ease can c ea e
a sca e sea ch e ec , and he e o e, no e u n good quali y
a e c ossed and mu a ed o a numbe o gene a ions, wi h
he bes indi iduals in he popula ion being inally selec ed.
The MSL measu e has been applied as he i ness unc ion o
assess he quali y o he iclus e s o solu ions in he popula-
ion. The lowcha o he T iGen algo i hm can be seen in
Figu e 5. We now de ine he mos impo an elemen s o he
algo i hm such as inpu s, ou pu s, codi ica ion o indi iduals,
and gene ic ope a o s.
Table 1. T iGen algo i hm pa ame e s.
PARAMETER DESCRiPTion
Nnumbe o iclus e s ex ac ed
Gnumbe o gene a ions
Inumbe o indi iduals in he popula ion
Ale andomness a e
Sel selec ion a e
Mu mu a ion p obabili y
w Weigh o MSL measu e
wgWeigh o he numbe o genes
wcWeigh o he numbe o condi ions
w Weigh o he numbe o imes
wogWeigh o he o e lap among genes
wocWeigh o he o e lap among
condi ions
wo Weigh o he o e lap among imes
D, P
Ini ial popula ion
E alua ion
Selec ion
C osso e
Mu a ion
P ocessed
gene a ions < G
Upda e SOL
Solu ions ound < N
Ge bes TRI
SOL
igu e 5. T iGen algo i hm lowcha .
MSL: 3D pa e ns om gene exp ession.
127E olu iona y Bioin o ma ics 2015:11
solu ions; an inc ease o he andomness a e Ale in he ini ial
popula ion has o be combined wi h he o e lap con ol o make
su e ha a wide a ea o he space o solu ions is ini ially co -
e ed. Sel con ols he selec ion mechanism, and as a esul , he
c osso e and a high Sel c ea es indi iduals wi h a low le el o
gene ic ecombina ion, a o ing exploi a ion e sus explo a ion,
and i he pa ame e is inc eased in excess, he algo i hm may
all in o a local minimum. On he con a y, a high p obabili y
o mu a ion Mu a o s explo a ion e sus exploi a ion, and i
inc eased in excess, we will end up wi h solu ions in many a eas
o he sea ch space bu wi h low quali y le els. w and wo com-
bined wi h w con ol he i ness unc ion; w weigh s con ol he
numbe o i ems in he solu ions, an inc ease o hese weigh s
in ol es a o ing solu ions wi h mo e olume; he inc ease o wo
weigh s leads o li le o nono e lapped solu ions, an excessi e
inc ease can lead o he loss o in e es ing solu ions. The con en
unde Gene ic Ope a o s in he subsec ion T iGen Algo i hm
explains all ope a o s and how a pa ame e a ia ion a ec s he
execu ion o T iGen.
2. Ou pu : The T iGen algo i hm’s ou pu will be a se o N
iclus e s, o mally SOL = {TRI1, TRI2,…, TRIN}. Each
TRIi ε SOL is composed o a subse o genes TRIG, condi-
ions TRIC, and imes TRIT om he inpu da ase D and
has he bes sco e in i s popula ion when e alua ed unde
he MSL measu e.
3. Codi ica ion o indi iduals: Each indi idual in he e olu ion-
a y p ocess o he T iGen algo i hm ep esen s a iclus e ,
which is a po en ial solu ion. The e o e, an indi idual is
ep esen ed as a subse o genes
Gggg
ii iF
=< >
1,,,,
2
…
a
subse o condi ions
Cccc
ii iQ
=< >
12
,,,…
, and a subse o
ime poin s
T
ii iw
=< >
12
,,,.…
The genes, condi ions,
and imes subse s a e ex ac ed om he inpu da ase D.
The algo i hm selec s ime poin s as pa o he iclus e
bu always keeping hei o de . All gene ic ope a o s a e
applied o each indi idual in he popula ion, in each o
hese h ee subse s.
4. O e lapping con ol: We ha e designed an o e lapping
con ol mechanism o a oid o e lapping among he i-
clus e solu ions ob ained. I is called Da a Hie a chy and
consis s in main aining he numbe o occu ences o
genes, condi ions, and ime poin s o da ase D in each
iclus e solu ion om less o mos isi ed in such a way
ha , as we will explain in Ini ial Popula ion in he sub-
sec ion T iGen Algo i hm, he ini ial popula ion c e-
a ion uses his s uc u e o ini ialize popula ion wi h he
minimum o e lapping. This Da a Hie a chy is upda ed a
e e y gene a ion a new iclus e solu ion is selec ed.
5. Gene ic ope a o s:
a. Ini ial popula ion: Wi h he ini ial popula ion me hod,
I indi iduals a e gene a ed a ending o he Ale an-
domness pa ame e . An Ale pe cen o indi iduals a e
c ea ed a andom by wo me hods: hal o he indi-
iduals a e pu ely andomly gene a ed, his is, a an-
dom subse o genes TRIG, condi ions TRIC, and imes
TRIT a e chosen om D and he o he hal is also
andomly c ea ed bu con olling ha he alues o
he genes TRIG a e con iguous, he alues o he con-
di ions TRIC a e con iguous and he imes TRIT a e
con iguous as well. The es o he indi iduals a e an-
domly c ea ed, bu aking in o accoun he p e iously
c ea ed indi iduals o con ol o e lapping o solu ions
acco ding o Da a Hie a chy s uc u e (O e lapping
Con ol in he subsec ion T iGen Algo i hm).
b. Fi ness unc ion: Ou p oposed measu e has been
included as he gene ic algo i hm’s i ness unc ion
FF(TRI) along wi h he size and he o e lapping
con ol. As can be seen in Equa ion 4, MSL has been
combined wi h six o he ac o s as a weigh ed a e -
age. Th ee o hese ac o s
1||
||
1||
||
−−
TRI
D
TRI
D
G
G
C
C
,
,
and
1||
||
−TRI
D
T
T
measu e he numbe o genes, con-
di ions, and imes o TRI (|TRIG, C, T|) ela i e o he
da ase ’s size (|DG, C, T|). MSL is a minimizing i ness
unc ion, and we ha e o se 1 minus each amoun
p opo ion in o de o a o TRI wi h g ea e sizes
when wg, wc, o w a e inc eased. The o he h ee
membe s
RTRI SOL
TRISOL
RTRI SOL
TRISOL
G
G
C
C
,
*,,
*
()
()
||||||||
, and
RTRI SOL
TRISOL
T
T
,
*
()
||||
measu e he epea ed genes, condi-
ions, o imes elemen s o TRI in he se o al eady
ound solu ions SOL (RG, C, T(TRI, SOL)) p opo -
ionally calcula ed supposing all o genes, condi ions,
o imes a e epea ed in SOL (|TRIG, C, T|*|SOL|) in
o de o a o TRI wi h less o e lapping when wog,
woc, o wo a e inc eased. Finally, he main membe
MSLTRI
()
2π
measu es MSL (TRI) p opo ionally
calcula ed o i s maximum alue 2π in o de o
a o TRI wi h smalle MSL when w is inc eased.
A de aul con igu a ion o w , wg, wc, w , wog, woc,
and wo consis s in ixing w o 0.8 and dis ibu ing
0.2 among wg, wc, w , wog, woc, and wo .
FF TRIwwwwwo wo w
wMSLTRI wTRI
gc gc
g
G
()
=++++ ++
()
+−
1
21||
*
{* *
π|||
1||
|| 1||
||
D
wTRI
DwTRI
D
wo
G
c
C
C
g
T
T




+
−




+−




+**
gg
G
G
g
C
C
RTRI SOL
TRISOL wo RTRI SOL
TRISOL
wo
*,
**,
*
()
+
()
+
|||| ||||
** ,
*}
RTRI SOL
TRISOL
T
T
()
||||
(4)
Gu ié ez-A ilés and Rubio-Escude o
128 E olu iona y Bioin o ma ics 2015:11
c. Selec ion: Th ee g oups o indi iduals a e andomly
selec ed so ed om lowes o highes acco ding o
he i ness unc ion, and hen a andom selec ion om
he h ee g oups is made. The Sel pa ame e indica es
how many o hese indi iduals will pass o he nex
gene a ion. The es o he indi iduals un il comple -
ing he nex popula ion (I – #Selec ed indi iduals) will
be c ea ed based on he c osso e ope a o .
d. C osso e : To comple e he nex gene a ion, we c e-
a e new indi iduals wi h his ope a o as ollows: wo
indi iduals (pa en s, A and B) a e combined o c ea e
wo new indi iduals (o sp ings, child1 and child2).
The pa en s a e andomly chosen. Thei gene ic
ma e ials a e combined by a andom one-poin c oss
in he genes TRIG, condi ions TRIC, and ime TRIT
and mixing he coo dina es in bo h child en.17
e. Mu a ion: An indi idual can be mu a ed acco d-
ing o a p obabili y o mu a ion, Mu . The mu a ion
p obabili y is e i ied o e e y indi idual, and i
i is sa is ac o y, one ou o nine possible ac ions is
aken. These ac ions a e add a new andom gene o
TRIG, add a new condi ion o TRIC, o add a new
ime poin o TRIT , by emo ing a andom gene,
condi ion, o ime, o by changing a andom gene o
condi ion o ano he ha is andomly chosen. The
elec ion o hese ac ions is also andom. Fo he case
o addi ion o a new gene, condi ion, o ime, he
ope a o checks whe he he new membe is al eady
in he indi idual o no .
esul s and discussion
In his sec ion, we show he esul s ob ained by applica ion o
MSL as a i ness unc ion embedded in he T iGen algo i hm17
(see he subsec ion T iGen Algo i hm).
MSL has been applied o ou di e en da ase s: one syn-
he ically gene a ed da ase and h ee eal da ase s. The eal
da ase s a e ob ained om expe imen s wi h he yeas cell
cycle (Saccha omyces ce e isiae),18 an expe imen wi h mice (Mus
musculus) called GDS451014 and da a om expe imen s wi h
humans (Homo sapiens) called GDS4472.19 The las wo da a
se s ha e been e ie ed om Gene Exp ession Omnibus,33 a
da abase eposi o y o high- h oughpu gene exp ession da a.
All biological expe imen s examine he beha io o genes
unde condi ions a ce ain imes.
Fo he analysis o he iclus e s ob ained as he esul o
he expe imen s, we ha e de eloped a h ee-s ep p ocess based
on p o iding in o ma ion ela ed o co ela ion among exp es-
sion alues, he g aphic p ope ies o he ep esen a ion o he
alues, and biological alida ion.
The co ela ion alida ion is based on he Pea son and
Filon20 and Spea man21 coe icien s. Fo e e y iclus e ,
we calcula e he a e age o he co ela ion coe icien s
be ween each combina ion o gene, condi ion, and ime
o all genes. Fo ins ance, o a iclus e wi h ou genes
{1, 4, 8, 10}, wo condi ions {3 and 7}, and h ee imes {1, 3
and 5}, we p o ide he Pea sons and Spea mans co ela ion
coe icien a e age o alues a he eigh possible combi-
na ions, each ha ing h ee ime poin s: Vg=1, c=3, Vg=1, c=7,
Vg=4, c=3, Vg=4, c=7, Vg=8, c=3, Vg=8, c=7, Vg=10, c=3, and Vg=10, c=7.
The g aphic p ope ies o he ep esen a ion a e shown as
desc ibed in he subsec ion G aphic Rep esen a ion as a way
o isually check how he gene pa e ns beha e.
Finally, o he biological alida ion, we will show he
GO e ms22 ela ed o he iclus e s. We p esen a GO anal-
ysis able in which we include he mos ep esen a i e e ms
ex ac ed by he On ologize so wa e,34 each e m associa ed
o a P- alue ha deno es he ele ance le el o he e m. In
his ype o s udies, P- alues a e conside ed as ele an below
0.05 and a e be e when close o 0. Rega ding he GO
p ojec , i is a majo bioin o ma ics ini ia i e wi h he aim
o s anda dizing he ep esen a ion o gene and gene p oduc
a ibu es ac oss species and da abases. The p ojec p o ides
an on ology o e ms o desc ibing gene p oduc cha ac e -
is ics and gene p oduc anno a ion da a. The on ology co e s
h ee domains: cellula componen , he pa s o a cell o i s
ex acellula en i onmen ; molecula unc ion, he elemen-
al ac i i ies o a gene p oduc a he molecula le el, such
as binding o ca alysis; and biological p ocess, ope a ions, o
se s o molecula e en s wi h a de ined beginning and end,
pe inen o he unc ioning o in eg a ed li ing uni s: cells,
issues, o gans, and o ganisms.
We ha e compa ed he esul s ob ained o hose om
Gu ié ez-A ilés and Rubio-Escude o,23 whe e he i ness
unc ion was he MSR3D measu e and also o he esul s in
Gu ié ez-A ilés and Rubio-Escude o,24 whe e he i ness
unc ion was he LSL measu e. The compa ison has been
made in e ms o co ela ion and GO analysis. Fo each eal
expe imen , we ha e compa ed he maximum, minimum,
and mean Pea son’s and Spea man’s co ela ion index and he
maximum, minimum, and mean P- alue o each solu ion
conside ed.
All expe imen s we e execu ed on a mul ip ocesso
machine wi h 64 p ocesso s, In el Xeon E7-4820 2.00 GHz
wi h 8-GB RAM memo y. We ha e used Ja a o implemen
T iGen algo i hm (and o he ad hoc de elopmen s) and an R
amewo k o c ea e g aphics and ge da ase esou ces om
GEO.33
We now analyze he esul s ob ained in each o he ou
expe imen s.
syn he ic expe imen s. Syn he ic da a a e widely used
no only o es ing he pe o mance o mic oa ay analyzing
echniques14 bu also in mo e gene al da a mining publica-
ions.35 I has he ad an age ha he p ocess ha gene a ed
he da a is well known and so one is able o judge he success
o ailu e o he algo i hm.36
In his wo k, we ha e used an applica ion designed by
ou sel es o gene a e he syn he ic da a used in his expe i-
men . We ha e execu ed he T iGen algo i hm wi h he MSL
MSL: 3D pa e ns om gene exp ession.
129E olu iona y Bioin o ma ics 2015:11
measu e o e a syn he ic da ase composed o 4,000 genes, 30
expe imen al condi ions, and 20 ime poin s whose exp es-
sion le els we e andomly gene a ed by a c yp og aphic secu e
s anda d lib a y Ma h3 p o ided by Apache Commons.37 In
his da ase , we inse ed 10 iclus e s composed o 150 genes,
6 expe imen al condi ions, and 4 ime poin s, whose exp es-
sion le els o m a cons an beha io pa e n. These iclus e s
a e loca ed in andom posi ions in he da ase .
To see he beha io o he MSL measu e applied along
wi h T iGen and also wi h he aim o analyzing he e ec
o he alue o he pa ame e s in he solu ions, we ha e
made execu ions se ing N o 200 and a ying o he con-
ol pa a me e s as ollows: G ε {100, 200}, I ε {50, 100}, Sel
ε {0.5}, Mu ε {0.2, 0.3}, Ale ε {0.3, 0.5}, wg ε {0.03, 0.05, 0},
wc ε {0, 0.01}, w ε {0,0.01}, wog ε {0.04, 0.05}, woc ε {0,0.03},
and wo ε {0,0.03} (see Inpu in he T iGen Algo i hm sub-
sec ion o a de ailed desc ip ion o hese pa ame e s). The
algo i hm has been capable o inding be ween 94% and
100% o he inse ed iclus e s. The e we e no alse posi i es
o alse nega i es ound. The applica ion o he MSR3D along
wi h he T iGen algo i hm in he s udy by Gu ié ez-A ilés
and Rubio-Escude o23 was capable o inding 91% o 95%
and LSL24 ob ained a ma ching a io be ween 93% and 97%
o he iclus e s, so we can see in sligh imp o emen when
applying he MSL measu e.
yeas elu ia ion expe imen s. Fo his expe imen , we
ha e applied he T iGen algo i hm wi h MSL measu e o he
yeas (Saccha omyces ce e isiae) cell cycle p oblem,18 speci ically
he Elu ia ion expe imen . The yeas cell cycle analysis p ojec ’s
goal is o iden i y all genes whose mRNA le els a e egula ed
by he cell cycle. The esou ces used a e public and a ailable in
h p://genome-www.s an o d.edu/cellcycle/. Da a ha e been
no malized as pa o he p ep ocessing. We ha e c ea ed a da a-
se Delu3D om he elu ia ion expe imen wi h 7,744 genes, 13
expe imen al condi ions, and 14 ime poin s. Expe imen al con-
di ions co espond o di e en s a is ical measu es o he Cy3
and Cy5 channels while ime poin s ep esen di e en momen s
o aking measu es om 0 o 390 minu es.
The pa ame e con igu a ion used o his expe imen is
shown in Table 2. We se G and I alues in o de o ob ain
a de aul explo a ion o he solu ion space and Ale, Sel, and
Mu p o ide us wi h a high andom ac o in popula ion
gene a ion, low eli ism in he nex gene a ion p omo ion,
and high mu a ion ac o , espec i ely. We a o solu ions
wi h a high numbe o genes se ing wg o 0.05 and solu ions
wi h high a iabili y in genes, condi ions, and imes hanks
o wog, woc, and wo se o 0.05. These con igu a ions ha e
been ob ained as a esul o a deep expe imen al s udy on he
Delu3D da ase .
Rega ding he co ela ion analysis, we can obse e in
Table 3 show Pea son and Spea man’s alues a y be ween
[0.95, 0.97] and [0.98, 1], espec i ely, which implies a high
co ela ion be ween genes se ies o e e y expe imen al con-
di ion h ough ime poin s. These high alues con i m us ha
he quali y o he iclus e s ob ained om hese expe imen s
is e y high in e ms o co ela ion.
We can see he g aphic ep esen a ion o iclus e TRI11
in Figu e 6. Only one ou o he 20 iclus e s ob ained has been
ep esen ed o legibili y easons. We can obse e how TRI11
Table 2. T iGen algo i hm con ol pa ame e s o yeas cell cycle
da ase .
PARAMETER ALuES
n20
G150
i200
ale 0.9
sel 0.4
mu 0.9
w 0.8
wg0.05
wc0
w 0
wog0.05
woc0.05
wo 0.05
Table 3. co ela ion esul s o iclus e s om he yeas cell cycle
da ase .
TRIsol PEARSon SPEARMAn
TRI10.97 0.98
TRI20.97 0.99
TRI30.96 0.99
TRI40.96 0.98
TRI50.97 0.99
TRI60.96 0.99
TRI70.96 0.99
TRI80.96 0.98
TRI90.96 0.98
TRI10 0.96 0.99
TRI11 0.96 0.99
TRI12 0.95 0.98
TRI13 0.96 0.98
TRI14 0.96 0.98
TRI15 0.97 1
TRI16 0.96 0.98
TRI17 0.96 0.98
TRI18 0.96 0.99
TRI19 0.96 0.99
TRI20 0.96 0.98