equen ly ea anged in ad anced T-cell lymphomas FRAT
egula o o WNT signaling pa hway 1 (FRAT1) : Time
beha iou al s udy o 3 d o de combina ions in WNT3A
s imula ed HEK 293 cells
sh ip akash sinha
Independen Resea che ; O cid ID : o cid.o g/0000-0001-7027-5788
Add ess : 104-Madhu isha Heigh s Phase 1, Risali, Bhilai-490006, India
Co esponding au ho email : sinha.sh ip [email protected]
Abs ac
FRAT1 is homologous o GSK3 binding p o ein (GBP), ha is known o ac as a pos-
i i e egula o o WNT signaling by s abilizing β-ca enin. Guj al and MacBea h [1]
p o ides a quan i a i e, and dynamic s udy o WNT3A-media ed s imula ion o HEK
293 cells, whe e hey eco d ime based exp ession p o iles o se e al esponse genes
which co ela ed signi ican ly wi h p oli e a ion and mig a ion. By moni o ing he dy-
namics o gene exp ession using sel -o ganizing maps, hey iden i ied clus e s o genes
ha exhibi simila exp ession dynamics and unco e ed p e iously un ecognized posi-
i e and nega i e eedback loops. Howe e , hei s udy depic s/uses singula measu e-
men s o indi idual gene exp ession a di e en ime snapsho s/poin s o in e he sys-
em wide analysis o he pa hway. A any pa icula ime poin , i is o en he case ha
genes a e wo king syne gis ically in combina ions, e en hough hei exp ession mea-
su emen s a e singula in na u e. He e, I •enume a e and ank all 2415 FRAT1 ela ed
3 d o de combina ions in a o es o 71C3combina ions using ou di e en sensi i i y
me hods; •show he conse ed ankings o FRAT1-X-X combina ions, which poin
o exis ence o biological syne gy o some o hese combina ions ac oss he di e en
sensi i i y me hods; and •s udy he beha iou o some o hese combina ions ela ed o
WNT3A esponse genes ha a e anked by he machine lea ning sea ch engine (Sinha
[2]) in ime. Pa e ns o combina ions eme ge, some o which ha e been es ed in we
lab, while o he s equi e u he we lab analysis.
Keywo ds: Sensi i i y analysis, Suppo ec o anking, Hilbe Schmid
Independence C i e ion indices (HSIC) and Sobol indicies, WNT3A
ITime beha iou al s udy o 3-od FRAT1 comb. in WNT3A s imula ed cells
1Aspec s o unpublished wo k we e p esen ed in a pos e session a Cell Symposia: Technology. Biology.
Da a Science, 9-11 Oc obe 2016, Be keley, Cali o nia, USA.
P ep in submi ed o P ep in Ma ch 16, 2025
1. Signi icance
Sinha [2] ecen ly demons a ed he use o machine lea ning based sea ch engine o
ank/ e eal gene combina ions a 2nd o de o he ime se ies da a by Guj al and
MacBea h [1] and showed how i is possible o loca e combina ions o p io i y ha
migh be wo king syne gis ically, using sensi i i y me hods and powe ul suppo ec-
o anking algo i hm. Howe e , he p oblem explodes combina o ially wi h e en a
small se o 71 eco ded genes in he s udy by Guj al and MacBea h [1], when one
s eps o explo e 3 d o de combina ions. Wi h he o al numbe o 71C3(= 57155) com-
bina ions, i becomes nea ly impossible o any biologis o s udy he sys em wide dy-
namics o any pa hway. Also, he amoun o ime usually needed o sea ch o and es
a combina ion is a mo e han he sea ch down by he machine lea ning based sea ch
engine. He e, I ex end he esea ch wo k by Sinha [2] o conduc a beha io al s udy o
3 d o de FRAT1 ela ed combina ions using indi idual gene exp essions measu ed in
ime, in WNT3A s imula ed HEK 293 cells.
2. In oduc ion
The de ails o he machine lea ning based sea ch engine has been ecen ly published in
Sinha [2] and deployed o explo e he 2nd o de combina ions o genes in he da a se
p o ided by Guj al and MacBea h [1]. Ne e heless, he e, I poin o he undamen als
o he published wo k o comple eness.
2.1. A combina o ial p oblem
Sensi i i y analysis plays a majo ole in compu ing he s eng h o he in luence o
in ol ed ac o s in any phenomena unde in es iga ion. When applied o exp ession
p o iles o a ious in a/ex acellula ac o s ha o m an in eg al pa o a signaling
pa hway, he a iance and densi y based analysis yields a ange o sensi i i y indices
o indi idual as well as a ious combina ions o ac o s. These combina ions deno e
he highe o de in e ac ions among he in ol ed ac o s. Compu a ion o highe o -
de in e ac ions is o en ime consuming bu i gi es a chance o explo e he a ious
combina ions ha migh be o in e es in he wo king mechanism o he pa hway. Fo
example, in a ange o ou h o de combina ions among he a ious ac o s o he Wn
pa hway, i would be easy o assess he in luence o he des uc ion complex o med by
APC, AXIN, CSKI and GSK3 in e ac ion. Bu he e ec o hese combina ions a y
o e ime as measu emen s o old changes and de ia ions in old changes a y. So
i is impe a i e o know how an in e ac ion o a combina ion o he in ol ed ac o s
beha e in ime and Sinha [2] de elops a p ocedu e o ack he beha iou by exploi ing
he in luences o hese in ol ed ac o s.
2.2. A possible solu ion
In his wo k, a e es ima ing he indi idual e ec s o ac o s o a highe o de combi-
na ion, he indi idual indices a e conside ed as disc imina i e ea u es. A combina ion,
2
hen, is a ea u e se in highe o de (≥2 ,i.e mul i a ia e). Wi h an excessi ely la ge
numbe o ac o s in ol ed in he pa hway, i is di icul o sea ch o impo an com-
bina ions in a wide sea ch space o e di e en o de s. Exploi ing he analogy wi h
he issues o p io i izing webpages using anking algo i hms, o a pa icula o de , a
ull se o combina ions o in e ac ions can hen be p io i ized based on hese ea u es
using a powe ul anking algo i hm ia suppo ec o s Joachims [3]. Reco ding he
changing ankings o he combina ions o e ime e eals how highe o de in e ac ions
beha e wi hin he pa hway and when an in e en ion migh be necessa y o in luence
he in e ac ion wi hin he pa hway.
2.3. equen ly ea anged in ad anced T-cell lymphomas FRAT eg-
ula o o WNT signaling pa hway 1 (FRAT1)
Embi e al. [4] pu i ied a glycogen syn hase kinase (GSK) om abbi skele al muscle
by p ecipi a ion wi h ammonium sulpha e, ch oma og aphy on DEAE-cellulose and
ch oma og aphy on hyd oxy-apa i e. This enzyme was highly speci ic o glycogen
syn hase and was sepa a ed om i ually all phos i in kinase and casein kinase ac-
i i y by hech oma og aphy on DEAE-cellulose. Fu he , he ac i i y o his enzyme
was una ec ed by cyclic AMP, cyclic GMP, calcium ions, calcium ions plus calmod-
ulin, and he speci ic p o ein inhibi o o cyclic-AMP-dependen p o ein kinase. They
obse ed ha he p ope ies o he enzyme demons a ed ha i was dis inc om bo h
cyclic-AMP-dependen p o ein kinase and phospho ylase kinase, he wo well cha -
ac e ized glycogen syn hasekinases in skele al muscle. This enzyme was he e o e
e med GSK3. The phospho yla ion o glycogen syn hase by GSK3 eached a plea eau
nea 1.5 molecule phospha e inco po a ed pe subuni unde op imal condi ions. Fi-
nally, glycogen syn hase was phospho yla ed by cyclic-AMP-dependen p o ein kinase,
phospho -ylase kinase and GSK3, using condi ions whe e he phospho yla ion by any-
one p o ein kinase eached a pla eau nea one molecule o phospha e inco po a ed pe
subuni . These esul s implied ha each p o ein kinase p e e en ially phospho yla ed a
di e en si e(s) on glycogen syn hase, which was con i med by he aminoacid sequence
analysis la e on.
Accele a ion o lymphomagenesis in oncogene-bea ing ansgenic mice by slow-
ans o ming e o i uses has p o en a aluable ool in iden i ying coope a ing onco-
genes. Jonke s e al. [5] modi ied his p o ocol o sea ch o genes ha could collab-
o a e wi h he ansgene in la e s ages o umo de elopmen . P opaga ion o umo s
induced by Moloney mu ine leukemia i us (M-MuLV) in Eµ-PIM1 o H2-K-MYC
ansgenic mice by ansplan a ion o syngeneic hos s pe mi ed p o i al agging o
’p og ession’ genes. They iden i ied a no el gene, designa ed F a 1 (in his a icle
mFRAT1), upon molecula cloning o common p o i al inse ion si es ha we e de-
ec ed p e e en ially in ansplan ed umo s. They u he cloned and sequenced bo h
he mouse F a 1 (mFRAT1) gene and i s human coun e pa (hFRAT1), he encoded
p o iens o which we e highly homologous. Upon in ec ion wi h a mFRAT1-IRES-
lacZ e o i us, hey obse ed ha umo cell lines wi h high exp ession o MYC and
PIM1 acqui ed an addi ional selec i e ad an age in i o, which unde sco ed he ole o
mFRAT1 in umo p og ession, and he abili y o mFRAT1 o collabo a e wi h PIM1
3
and MYC in lymphomagenesis.
Do sal accumula ion o β-ca enin in ea ly Xenopus emb yos is equi ed o body
axis o ma ion. E idence has indica ed ha β-ca enin is do sally s abilized by he lo-
calized inhibi ion o he kinase X-GSK3, u ilizing a no el WNT ligand-independen
mechanism. Using a wo-hyb id sc een, Yos e al. [6] iden i ied GBP, a ma e nal
X-GSK3-binding p o ein ha is homologous o a T cell p o ooncogene in h ee well-
conse ed domains. They obse ed ha GBP inhibi s in i o phospho yla ion by X-
GSK3, and ec opic GBP exp ession induced an axis by s abilizing β-ca enin wi hin
Xenopus emb yos. Using he simila i y o GBP o T cell p o o-oncogene FRAT1 by
Jonke s e al. [5] and a human EST, hey demons a e ha GBP con ains a highly
conse ed domain equi ed o GSK3 binding and inhibi ion. They obse ed ha
bo h he mouse and human p o eins con ained h ee egions ha a e well-conse ed
wi hin GBP. A compa ison o GBP and F a 1/FRAT1 showed ha domain I was 63%
iden ical, II was 70% iden ical, and III was 83% iden ical. Thei s udies wi h GBP
p o ide a basis o unde s anding he p oli e a ion o lymphoma cells, ha o e exp ess
F a 1. Because F a 1 con ains he conse ed domain III, i is likely o bind and inhibi
GSK3, and hey p edic ed ha i s imula ed T cell lymphoma p oli e a ion by dec eas-
ing GSK3 unc ion. In suppo o his, Beals e al. [7] ha e shown ha GSK3 ac s as a
nega i e egula o o NF-ATc (nuclea ac o ac i a ed in T cells).
GSK3 ac i i y can also be inhibi ed in esponse o Wn g ow h ac o signaling.
Cy oplasmic β-ca enin is cen al o he ansmission o Wn signals o he nucleus;
in he absence o Wn signaling, β-ca enin is phospho yla ed by GSK3, which a -
ge s β-ca enin o deg ada ion ia ubiqui in-dependen p o eolysis. In esponse o Wn
signals, howe e , GSK3 ac i i y owa d β-ca enin is inhibi ed, esul ing in he s abi-
liza ion and accumula ion o β-ca enin and he ac i a ion o TCF/LEF-1 ansc ip ion
ac o s. Fu he , gene ic s udies ha e shown ha DVL, a cy oplasmic p o ein down-
s eam o Wn ecep o s, is essen ial o ansmi ing WNT signals o GSK3. AXIN
in e ac s wi h DVL and, in addi ion, binds bo h GSK3 and β-ca enin, hus acili-
a ing GSK3-media ed phospho yla ion o β-ca enin. Consequen ly, dissocia ion o
he GSK3/AXIN/β-ca enin complex p e en s GSK3-media ed phospho yla ion o β-
ca enin, esul ing in i s s abiliza ion. Jonke s e al. [5] ha e a p oposed mechanism o
WNT-induced dissocia ion o he GSK3/AXIN/β-ca enin complex which in ol es he
mammalian p o ein FRAT1.
To unde s and he mechanism by which DVL ac s h ough GSK o egula e LEF-1,
Li e al. [8] in es iga ed he oles o AXIN and FRAT1 in WNT-media ed ac i a ion
o LEF-1 in mammalian cells. They ound ha DVL in e ac ed wi h AXIN and wi h
FRAT1, bo h o which in e ac ed wi h GSK. They also ound ha DVL, AXIN and
GSK can o m a e na y complex b idged by AXIN, and ha FRAT1 could be ec ui ed
in o his complex p obably by DVL. Thei obse a ion ha he DVL-binding domain o
ei he FRAT1 o AXIN was able o inhibi WNT1-induced LEF-1 ac i a ion sugges ed
ha he in e ac ions be ween DVL and AXIN and be ween DVL and FRAT may be
impo an o WNT signaling pa hway. Fu he mo e, WNT1 appea ed o p omo e he
disin eg a ion o he FRAT1-DVL-GSK-AXIN complex, esul ing in he dissocia ion
o GSK om AXIN. Thus, o ma ion o he qua e na y complex may be an impo an
s ep in WNT signaling, by which DVL ec ui s FRAT1, leading o FRAT1-media ed
dissocia ion o GSK om AXIN.
4
Now, AXIN is a nega i e egula o o emb yonic axis o ma ion in e eb a es,
which ac s h ough a WNT signal ansduc ion pa hway in ol ing he se ine/ h eonine
kinase GSK3 and β-ca enin. I has been shown o ha e dis inc binding si es o GSK3
and β-ca enin and o p omo e he phospho yla ion o β-ca enin and i s consequen
deg ada ion hus inhibi ing signaling h ough β-ca enin. Hsu e al. [9] epo he e-
sul s o a yeas wo-hyb id sc een o p o eins ha in e ac wi h he C- e minal hi d o
AXIN, a egion in which no binding si es o o he p o eins ha e p e iously been iden-
i ied. They ound ha AXIN can bind o he ca aly ic subuni o he se ine/ h eonine
p o ein phospha ase 2A (PP2A) h ough a domain be ween amino acids 632 and 836.
Thei esul s sugges ed ha PP2A migh in e ac wi h he AXIN-APC-GSK3-β-ca enin
complex, whe e i could modula e he e ec o GSK3 on β-ca enin o o he p o eins in
he complex.
Wille e al. [10] show ha AXIN is dephospho yla ed in esponse o WNT sig-
naling and dephospho yla ed AXIN binds β-ca enin less e icien ly han he phospho-
yla ed o m, hus disengaging β-ca enin om he deg ada ion machine y. The AXIN-
dependen phospho yla ion o β-ca enin ca alysed by GSK3, is inhibi ed du ing em-
b yogenesis. This p o ec s β-ca enin agains ubiqui in-dependen p o eolysis, leading
o i s accumula ion in he nucleus, whe e i con ols he exp ession o genes impo an
o de elopmen . Thomas e al. [11] demons a ed ha FRAT ide (a pep ide co e-
sponding o esidues 188226 o FRAT1) bound o GSK3 and p e en ed GSK3 om in-
e ac ing wi h AXIN. They also obse ed ha FRAT ide blocked he GSK3-ca alysed
phospho yla ion o AXIN and β-ca enin, hus poin ing o a po en ial mechanism by
which GBP could igge axis o ma ion. In con as , hey also ound ha FRAT ide
did no supp ess GSK3 ac i i y owa ds o he subs a es, such as glycogen syn hase and
eIF2B (whose phospho yla ion is independen o AXIN bu dependen on a ’p iming’
phospho yla ion). This migh explain how he essen ial cellula unc ions o GSK3 can
con inue, despi e he supp ession o β-ca enin phospho yla ion.
Hedgepe h e al. [12] de ined a 25-amino-acid sequence in AXIN ha is comp ised
o GSK3βin e ac ion domain (GID). In con as o ull-leng h AXIN, which has been
shown o an agonize WNT signaling, hey ound ha he GID inhibi ed GSK3βin i o
and ac i a ed WNT signaling. They p oposed ha he AXIN complex migh di ec ly
egula e GSK3βenzyma ic ac i i y in i o. Al hough i is known ha GID and FRAT-
ide pep ides ha e a simila unc ionali y, he GSK3 binding domains in FRAT and
AXIN do no appea o ha e homologous amino acid sequences.
Bax e al. [13] desc ibed he c ys al s uc u es o isola ed y osine 216-phospho yla ed
GSK3 and o a complex o GSK3 wi h FRAT ide and a sul a e ion. S uc u es o un-
complexed Ty 216 phospho yla ed GSK3 and o i s complex wi h a pep ide and a sul-
a e ion bo h showed he ac i a ion loop adop ing a con o ma ion simila o ha in he
phospho yla ed and ac i e o ms o he ela ed kinases CDK2 and ERK2. The sul-
a e ion, adjacen o Val214 on he ac i a ion loop, ep esen ed he binding si e o he
phosphose ine esidue on ’p imed’ subs a es. The pep ide FRAT ide o med a helix-
u n-helix mo i in binding o he C- e minal lobe o he kinase domain, a which FRAT
and AXIN compe e o binding o GSK3. FRAT ide (and FRAT1) does no inhibi he
ac i i y o GSK3 owa d glycogen syn hase (GS). GSK3 sequen ially phospho yla es
ou se ine esidues on GS, in he sequence SxxxSxxxSxxxSxxxS(p), by ecognizing
and phospho yla ing he i s se ine in he sequence mo i SxxxS(P) (whe e S(p) ep-
5
esen s a phosphose ine). P o ein inhibi o s such as FRAT, which appea o egula e
kinase ac i i y by modula ing p o ein in e ac ions wi hin signaling complexes, migh
well inhibi ac i i y owa d a selec ed subse o subs a es and migh be he a ge o
s uc u e-based ligand design.
I p esen 3 d o de combina ions o FRAT1 wi h o he genes, ha he machine
lea ning based sea ch engine poin s o, as possible syne gis ic combina ions ha migh
be wo king in ime.
3. Me hods
Please e e o sec ions o Sinha [2] o me hods, design o s udy and analysis o da a
o 2nd o de combina ions. The same me hod and design o s udy is used o gene a e
esul s o 3 d o de combina ions p esen ed in his s udy.
4. Time se ies da a
Guj al and MacBea h [1] p esen a se o 71 WNT- ela ed gene exp ession alues o 6
di e en imes poin s o e a ange o 24-hou pe iod using qPCR. The changes ep e-
sen he old-change in he exp ession le els o genes in 200 ng/mL WNT3A-s imula ed
HEK 293 cells in ime ela i e o hei le els in uns imula ed, se um-s a ed cells a 0-
hou . Guj al and MacBea h [1] s a e ha qPCR da a a e he means o h ee biological
eplica es. Only genes whose mean ansc ip le els changed by mo e han wo- old a
one o mo e ime poin s du ing he 24-hou ime cou se we e conside ed signi ican .
Posi i e (nega i e) numbe s ep esen up (down) - egula ion. We ha e al eady co e ed
he issues ela ed o hese da a se s in de ail in Sinha [14]. Reade s a e eques ed o
go h ough hem in he poin ed e e ence. The ools o s udy which a e used he e ha e
been published in ano he ounda ional wo k in Sinha [14].
5. Design o expe imen
5.1. Pipeline o ime se ies da a
Fo he case o ime se ies da a, in e ac ions among he con ibu ing ac o s a e s udied
by compa ing iple s o old-changes a single ime poin s. The p odecu e begins wi h
he gene a ion o dis ibu ion a ound measu emen s a single ime poin s wi h added
noise is done o es ima e he indices. A dis ibu ion is gene a ed o he old changes
a single ime poin s. Then o e e y gene, he e is a ec o o alues ep esen ing old
changes as well as de ia ions in old changes o di e en ime poin s and du a ions
be ween ime poin s, espec i ely. Nex a lis ing o all Cn
kcombina ions o knumbe
o genes om a o al o ngenes is gene a ed. kis ≥2 and ≤(n−1). Each o he com-
bina ion o o de k ep esen s a unique se o in e ac ion be ween he in ol ed gene ic
ac o s. A e his, he da ase s a e combined in a speci ed o ma which go as inpu
as pe he equi emen o a pa icula sensi i i y analysis me hod. Thus o each p h
combina ion in Cn
kcombina ions, he da ase is p epa ed in he equi ed o ma om
6
he dis ibu ions o wo sepa a e cases which ha e been discussed abo e. (See .R code
in mainSc ip -1-1.R). A e he da a has been ans o med, ec o ized p og amming
is employed o densi y based sensi i i y analysis and looping is employed o a i-
ance based sensi i i y analysis o compu e he equi ed sensi i i y indices o each o
he pcombina ions. This p ocedu e is done o di e en kinds o sensi i i y analysis
me hods.
A e he abo e sensi i i y indices ha e been s o ed o each o he p h combina-
ion, he nex s ep in he design o expe imen is conduc ed. Since he e is only one
eco ding o sensi i i y index pe combina ion, each combina ion o ms a aining ex-
ample which is allo ed a aining index and he sensi i i y indices o he indi idual
gene ic ac o s o m he aining example. Thus he e a e Cn
k aining examples o k h
o de in e ac ion. Using his aining se SVMRank
lea n Joachims [3] is used o gene a e a
model on de aul alue C alue o 20. In he cu en expe imen on oy model C alue
has no been unned. The aining se helps in he gene a ion o he model as he di -
e en gene combina ions a e numbe ed in o de which a e used as ank indices. The
model is hen used o gene a e sco e on he obse a ions in he es ing se using he
SV MRank
classi y Joachims [3]. No e ha due o a ailabili y o only one example pe com-
bina ion, a e he model has been buil , he same aining da a is used as es da a o
gene a es he sco es. This p ocedu e is execu ed o each and e e y sensi i i y analysis
me hod. This is ollowed by so ing o hese sco es along wi h he ank indices (i.e he
aining indices) al eady assigned o he gene combina ions. The end esul is a so ed
o de o he gene combina ions based on he anking sco e lea ned by he SV MRank
algo i hm. Finally, his en i e p ocedu e is compu ed o sensi i i y indices gene a ed
o each and e e y old change a ime poin and de ia ions in old change a di e en
du a ions. Obse ing he changing ank o a pa icula combina ion a di e en imes
and di e en ime pe iods will e eal how a combina ion is beha ing.
No e ha he ollowing is he o de in which he iles should be execu ed in R, in
o de , o ob aining he desi ed esul s (No e ha he code will no be explained he e) - •
use sou ce(”mainSc ip -1-1.R”) wi h a gumen s o Dynamic da a •sou ce(”SVMRank-
Resul s-D.R”), o ank he in e ac ions (again his needs o be done sepa a ely o
di e en kinds o SA me hods), •use sou ce(”Combine-Time- iles.R”), i compu -
ing indices sepa a ely ia p e ious ile, •sou ce(”So -n-Plo -D.R”) o so he in e -
ac ions. No e ha he so ing is chages he in e ac ion anking in ime. Thus •use
sou ce(”In e ac ion-P io i y-In ime.R”) o ind he p io i ized anking o each and e -
e y in e ac ion o e he di e en ime poin s and inally •use sou ce(”P in -Ranking-
AND-In e ac ion-Rank.R”) o p in indi idual anking o he equi ed inpu ac o wi h
o he in e ac ion ac o s.
6. Resul s & Discussion
6.1. Time se ies da a by Guj al and MacBea h [1]
NOTE - Ranking was assigned on sco es ha we e so ed in DECREASING alues.
So, 1 was assigned o highes sco e and ice e sa.
Resul s o he 3 d o de in e ac ions a e p esen ed he e. The esul s i s discuss
7
he beha iou o in e ac ions ac oss he snapsho s o ime using he compu ed sensi-
i i ies on old change measu emen s pe ime snapsho . The analysis was done us-
ing 4 di e en sensi i i y indices. Ou o he 71C3combina ions, I conside /p esen
only hose combina ions ha show a anking wi hin i s 10,000 ou o 57,155. This
choice is libe al and biologis s/oncologis s can ha e a mo e s ic e choice as pe need.
Two obse a ions a e made, • he anking o a pa icula combina ion is conse ed (i.e
wi hin he 10,000 ange) in a pa icula ime poin o in he ea ly phase o la e phase
o WNT3A s imula ion, ac oss he majo i y o he ou sensi i i y me hods, which is a
s ic c i e ia o assessmen o • he anking o a pa icula combina ion is conse ed
ac oss ime poin s/phase (i.e hey a e wi hin he 10,000 ange) and he majo i y o he
ou sensi i i y me hods, which is elaxed c i e ia o assessmen . Applying his il e
helps e eal impo an combina ions o in e es ha migh be wo king syne gis ically
a a highe o de le el in he cell.
Rega ding echnical poin s o implemen a ion, he ankings we e gene a ed wi h-
ou scaling/no malizing he ime se ies da a p o ided by Guj al and MacBea h [1].
Fo es ima ing he sensi i i y indices, a small gaussian dis ibu ion using he unc ion
no m ha gene a es a ec o o no mally dis ibu ed andom a iables gi en a ec o
leng h n (he e 9, he 10 h one is he mean/ eco ded gene egula ion i sel ), a popula ion
mean µand popula ion s anda d de ia ion σ. The syn ax o using no m is as ollows:
no m(n, mean, sd). Fu he , I use he ji e un ion o add a li le bi o noise o he
da a. This helps o see i he gene a ed ankings a e obus o no .
6.2. Enume a ion and anking o 2415 FRAT1-X-X combina ions
om Guj al and MacBea h [1]
In he supplemen a y sec ion, I p esen ou iles, each con aining he ankings o 3 d
o de combina ions, ha wa y in ime (shown o 5 ime poin s). Each ile ep esen s
he ankings compu ed using a pa icula sensi i i y me hod. The changing ankings
in ime o a pa icula combina ion ep esen s he impo ance o con ibu ion/ ole ha
combina ion plays in he cell s imula ed wi h WNT3A. The sensi i i y me hods used
a e Hilbe Schmid Independence C i e ion indices (HSIC) indices (wi h b and linea
ke nel in Da Veiga [15]) and Sobol indicies (wi h 2002 implemen a ion in Sal elli [16]
and ma inez implemen a ion in Ma inez [17] and Baudin e al. [18]).
6.3. Conse ed machine lea ning ankings o es ed FRAT1-X-X
combina ions
A o al o 2415, 3 d o de combina ions in ol ing FRAT1 we e ob ained om a ull se
o 71C3= 57155 combina ions. Fu he , om his selec ed se , using he abo e c i e ia
o conse ed ankings, I epo / abula e he meaning ul combina ions ha migh be
wo king syne gis ically. Tables 2, 3 and 4 show he ankings o he same combina-
ions as in able 1, bu using b ke nel o HSIC, 2002 implemen a ion o SOBOL
and ma inez implemen a ion o SOBOL, espec i ely. As one allies he ankings o
ac oss hese ables o a pa icula combina ion, one inds ha he ole o he combina-
ion o in e es is conse ed. This conse a ion poin s o he exis ence o he biological
8
RANKING @ iUSING HSIC - LINEAR
3 d o de comb. 1 3 6 12 24 3 d o de comb. 1 3 6 12 24
FOSL1-FRAT1-SENP2 24 25444 38488 23401 24549 CXXC4-FRAT1-FRZB 64 19925 35266 282 20625
AES-AXIN1-FRAT1 153 39308 39740 48864 34865 CSNK1D-FGF4-FRAT1 154 4058 50213 49903 6296
CXXC4-FRAT1-TLE2 212 18126 46011 57 25507 CXXC4-FRAT1-FZD1 296 10717 21805 2575 15715
CXXC4-FRAT1-FBXW4 304 26037 23203 450 10918 FRAT1-NLK-SENP2 308 14795 5137 12639 16597
FOSL1-FRAT1-WNT4 372 34237 49971 11736 48629 FRAT1-FZD1-SFRP4 507 39272 1172 47699 40072
CXXC4-FRAT1-TCF7L1 519 13366 30568 2083 18784 FRAT1-NLK-WNT4 534 1131 4271 13789 39755
APC-BCL9-FRAT1 599 22924 30930 45217 55567 DVL1-FOXN1-FRAT1 613 42124 2382 9631 35079
DKK1-FRAT1-SENP2 669 29229 40235 2545 51616 FOSL1-FRAT1-TLE2 675 26389 36474 6816 48298
FRAT1-JUN-WNT2 676 35197 23478 4515 23224 CXXC4-FOSL1-FRAT1 731 20058 29538 14504 27891
CTNNBIP1-FRAT1-WNT5A 734 14176 29898 41568 9680 FBXW2-FGF4-FRAT1 741 34437 49723 51069 4028
FRAT1-JUN-PITX2 748 50033 38113 50412 44660 CTNNBIP1-FRAT1-KREMEN1 758 28255 42529 10768 13750
FRAT1-PORCN-SFRP4 816 23261 18955 23358 33807 CXXC4-FRAT1-FZD8 827 10881 39520 3401 24688
FOSL1-FRAT1-FBXW4 832 28714 21515 30364 38569 FRAT1-JUN-SENP2 876 27146 34459 14780 13983
FRAT1-FZD1-FZD7 888 26968 7671 38732 41048 FRAT1-PORCN-SENP2 915 19198 9986 14436 18176
AXIN1-FOXN1-FRAT1 990 1738 4384 15268 22220 FRAT1-NLK-WNT3A 1005 19771 7884 9479 10249
FRAT1-FZD1-LRP5 1051 34518 43038 31699 24508 FRAT1-PORCN-RHOU 1077 31453 10861 21530 17749
CCND1-FGF4-FRAT1 1105 52382 24371 40896 23694 FRAT1-JUN-SFRP4 1177 38200 26202 16198 34929
FOSL1-FRAT1-TCF7L1 1246 18255 19691 45038 20105 DKK1-FRAT1-FRZB 1283 35064 37595 2698 45083
APC-FOXN1-FRAT1 1418 596 2065 13524 33264 CXXC4-FRAT1-WNT3A 1421 24998 29856 17406 1342
DKK1-FGF4-FRAT1 1462 21926 23959 17544 31567 DIXDC1-FGF4-FRAT1 1533 9050 41138 51810 43824
FBXW11-FOXN1-FRAT1 1538 7360 6561 15328 24410 FRAT1-NLK-FBXW4 1570 6400 25113 13146 24103
EP300-FGF4-FRAT1 1588 28152 39282 54116 41791 FOSL1-FRAT1-LRP5 1606 10313 45532 19498 50080
FRAT1-JUN-TCF7L1 1626 39276 40450 30425 12391 FRAT1-PORCN-WNT2B 1796 26246 45830 36444 11708
CXXC4-FRAT1-LRP5 1810 3252 48463 383 12590 EP300-FOXN1-FRAT1 1814 1637 5045 2015 815
FGF4-FOSL1-FRAT1 1842 39537 26432 55090 52175 BCL9-FGF4-FRAT1 1875 5789 48461 56251 35603
FRAT1-JUN-KREMEN1 1937 45988 42307 30915 9629 DKK1-FRAT1-FZD1 1941 7211 29023 2680 47286
DAAM1-FOXN1-FRAT1 1981 44098 29909 11384 47997 FRAT1-PORCN-WNT4 1985 2835 18017 16053 23730
FZD5-FGF4-FRAT1 1988 17464 37325 36960 11187 APC-BTRC-FRAT1 1993 4234 3760 19668 20988
CTNNB1-FRAT1-TLE1 2116 31975 20159 13284 55089 FRAT1-GSK3A-RHOU 2147 48585 3413 51070 8121
DAAM1-FGF4-FRAT1 2191 52861 50322 53931 30508 FRAT1-WNT1-WNT2B 2208 3244 1753 8833 56838
FRAT1-GSK3A-SFRP4 2238 47737 14694 38145 13213 DKK1-FOSL1-FRAT1 2248 20750 40597 26220 53540
FRAT1-FZD7-SFRP4 2265 6984 42886 29035 50691 CSNK2A1-CTNNB1-FRAT1 2318 44804 22708 39543 10094
FRAT1-JUN-PPP2R1A 2465 21726 41887 23096 41465 CSNK1G1-CXXC4-FRAT1 2496 24384 49297 7946 49881
FRAT1-NLK-TLE2 2530 6281 12625 8358 38149 FRAT1-JUN-PPP2CA 2547 32152 28002 1650 1276
CTNNB1-FRAT1-WIF1 2625 18691 37238 24416 4230 FRAT1-GSK3A-PPP2CA 2638 38412 1286 35232 13767
DVL1-FGF4-FRAT1 2664 45629 31255 34356 19416 FRAT1-JUN-SLC9A3R1 2692 5241 18995 27536 35730
CSNK1D-CXXC4-FRAT1 2744 18327 34701 8594 2661 CTBP1-FRAT1-WNT4 2821 37523 56525 1417 51685
CXXC4-FGF4-FRAT1 2842 9594 30978 23790 51147 FRAT1-JUN-LRP5 2861 38548 27221 17003 13998
FRAT1-JUN-WNT3A 2877 41865 29807 34170 11250 CXXC4-FRAT1-WNT2 2894 23698 18152 32 32646
CTNNBIP1-FRAT1-RHOU 2910 38029 54971 19935 51723 CTNNBIP1-FRAT1-FZD1 2935 24095 30151 22445 24470
FRAT1-FBXW4-WNT3A 2950 1626 150 55124 17849 FRAT1-PORCN-TCF7 3001 18766 42465 28851 20276
FRAT1-JUN-TCF7 3032 30172 40041 39760 42770 CTNNB1-FRAT1-WNT4 3049 31688 29436 13852 32929
CSNK1G1-FGF4-FRAT1 3087 3833 51164 49600 29524 FRAT1-JUN-PYGO1 3094 24225 44587 43882 21321
CCND1-FRAT1-FZD8 3095 44826 16342 57124 29952 DAAM1-FRAT1-FZD1 3098 48916 26712 40838 53076
FOSL1-FRAT1-FZD7 3142 34552 21454 20799 55831 FRAT1-WIF1-WNT4 3157 3802 25184 32202 1523
FZD5-CTNNBIP1-FRAT1 3164 40626 31787 13769 20789 FRAT1-GSK3A-KREMEN1 3217 49686 3676 49905 14302
FZD5-FOXN1-FRAT1 3239 1614 3233 20068 5777 DIXDC1-DVL2-FRAT1 3310 15749 16154 12064 34806
FRAT1-WIF1-WNT3A 3328 15377 26382 55074 7682 FGF4-FRAT1-LEF1 3378 28956 25427 26696 11917
FRAT1-JUN-TLE2 3416 30940 881 14082 28776 FGF4-FRAT1-T 3423 44186 35087 36000 23427
CXXC4-FRAT1-KREMEN1 3542 41044 29584 5685 51990 CSNK1D-FRAT1-FZD1 3569 4219 52992 30046 47083
FRAT1-PORCN-PPP2R1A 3641 17091 37502 41949 11356 BTRC-FOXN1-FRAT1 3675 26618 6170 28752 42168
FRAT1-FZD1-TCF7 3848 22218 19562 49291 40637 CSNK1G1-CTNNBIP1-FRAT1 3861 2169 25070 15462 45122
FRAT1-FZD1-WNT4 3864 17916 29682 40787 38438 FRAT1-GSK3A-WNT2B 3884 47252 4540 55696 35586
CCND1-FRAT1-NLK 3929 35214 17635 44956 38660 CCND3-FGF4-FRAT1 3979 45710 31230 52987 44235
FOSL1-FRAT1-WNT5A 3989 11766 16292 43205 16137 CXXC4-FRAT1-WNT2B 4026 15233 43434 56478 23693
DKK1-FOXN1-FRAT1 4029 12980 50095 12148 39677 FOSL1-FRAT1-WNT3A 4035 2654 13405 47524 18665
FBXW2-FOXN1-FRAT1 4037 3650 18095 22100 31935 APC-CCND1-FRAT1 4041 16166 3819 56694 45787
FRAT1-NLK-RHOU 4074 18571 5441 22722 18400 FRAT1-NLK-PPP2R1A 4096 7150 14882 4380 42041
FRAT1-JUN-SFRP1 4122 39180 11075 7077 30063 FRAT1-PYGO1-WNT3A 4132 14097 12408 18181 22475
FRAT1-NKD1-WNT2B 4145 29853 29809 31718 30950 CSNK1G1-DVL2-FRAT1 4160 46593 24765 28311 27782
FRAT1-PORCN-PPP2CA 4204 33092 39327 31761 14446 CCND3-FOXN1-FRAT1 4217 35968 4531 12204 31006
CTNNBIP1-FRAT1-PPP2R1A 4272 14548 33170 16600 48598 FRAT1-PORCN-FBXW4 4346 15930 19660 14012 11065
FRAT1-GSK3A-PITX2 4363 56251 2247 18588 44926 FRAT1-GSK3A-SLC9A3R1 4367 13258 9450 48551 44914
Table 1: Rankings o RHOU-X-X. A lis o app oxima ely i s 125 combina ions wi h ankings below
10,000 ou o 57,155. SA - HSIC; Ke nel - linea
syne gy, whe he he combina ion has been es ed o unexplo ed/un es ed.
9
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16
