RECONSTRUCTION OF GENE REGULATORY NETWORKS

Transcription

1 RECONSTRUCTION OF GENE REGULATORY NETWORKS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY SİBEL BALCI IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN STATISTICS SEPTEMBER 4

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3 Aroval of the thess: RECONSTRUCTION OF GENE REGULATORY NETWORKS submtted by SİBEL BALCI artal fulfllmet of the requremets for the degree of Doctor of Phlosohy Statstcs Deartmet, Mddle East Techcal Uversty by, Prof. Dr. Caa Özge Dea, Graduate School of Natural ad Aled Sceces Prof. Dr. İc Batmaz Head of Deartmet, Statstcs Prof. Dr. Ayşe Akkaya Suervsor, Statstcs Det., METU Assoc. Prof. Dr. Tolga Ca Co-suervsor, Comuter Egeerg Det., METU Examg Commttee Members: Assst. Prof. Dr. Zeye Kalaylıoğlu Statstcs Det., METU Prof. Dr. Ayşe Akkaya Statstcs Det., METU Assst. Prof. Dr. Yeşm Aydı So Health Iformatcs Det., METU Assst. Prof. Dr. Özlem Türker Bayrak Idustral Egeerg Det., Çakaya Uversty Assst. Prof. Dr. Ceyla Yozgatlıgl Statstcs Det., METU Date:

4 I hereby declare that all formato ths documet has bee obtaed ad reseted accordace wth academc rules ad ethcal coduct. I also declare that, as requred by these rules ad coduct, I have fully cted ad refereced all materal ad results that are ot orgal to ths work. Name, Last ame : Sbel Balcı Sgature : v

5 ABSTRACT RECONSTRUCTION OF GENE REGULATORY NETWORKS Balcı, Sbel Ph.D., Deartmet of Statstcs Suervsor: Prof. Dr. Ayşe Akkaya Co-suervsor: Assoc. Prof. Dr. Tolga Ca Setember 4, 36 ages Wth the develomet of mcroarray techology, t s ow ossble to obta the cocetrato levels of thousads of gees at a gve tme or a gve state. By followg the chages the gee exresso levels, the resosble gees for cell dfferetato or certa dseases ca be detfed. Gee exresso chages are regulated by the teractos betwee the gees ad ther roducts. Gee regulatory etworks GRNs detfy these teractos usg the gee exresso chages. There are a umber of statstcal methods to fer GRNs, however, most of them deed o the ormalty assumto of oses the data. Ths thess cosders the multle lear regresso aalyss for the recostructo of GRNs whe the error term comes from a Webull dstrbuto. Sce oormalty comlcates the data aalyss ad results effcet estmators, t s roosed to use the modfed maxmum lkelhood MML estmato rocedure whch roduces effcet ad robust estmators. Also, exlaatory varables reresetg the gee exresso levels come from a Webull dstrbuto. Therefore, they are cosdered as stochastc ad stochastc multle lear regresso aalyss s used for ferrg GRNs by mlemetg MML method to estmate the model v

6 arameters. Robustess ad ower aalyses for both stochastc ad ostochastc multle lear regresso model arameters are also gve. Keywords: Gee Regulatory Networks, Webull Dstrbuto, Multle Stochastc Lear Regresso, Modfed Maxmum Lkelhood Estmato. v

7 ÖZ GEN DÜZENLEYİCİ AĞLARIN YENİDEN OLUŞTURULMASI Balcı, Sbel Doktora, İstatstk Bölümü Tez Yöetcs: Prof. Dr. Ayşe Akkaya Ortak Tez Yöetcs: Doç. Dr. Tolga Ca Eylül 4, 36 sayfa Mkrodz tekolos gelştrlmesyle, blerce ge kosatrasyo düzeyler belrl br zama ya da belrl br durum ç elde edlmes artık mümkü. Ge fade düzeylerdek değşmler takb le hücre çeştllğe ya da belrl br hastalığa ede ola geler belrleeblmektedr. Ge fadelerdek değşmler, geler ve ge ürüler arasıdak etkleşmlerle düzelemektedr. Ge düzeleyc ağlar GDA, ge fadelerdek değşmler kullaarak bu etkleşmler ortaya çıkarmaktadır. GDA ı çıkarımı ç kullaıla çok sayıda statstksel yötem mevcuttur acak brçoğu verdek hataları ormallk varsayımıa dayamaktadır. Bu tez, GDA ı yede oluşturulması ç hata term Webull dağılımıa sah ola çoklu doğrusal regresyo aalz ele almaktadır. Normal dağılmama durumuu ver aalz zorlaştırmasıda ve etk olmaya tahmclere yol açmasıda dolayı, etk ve güçlü tahmcler ürete uyarlamış e çok olablrlk UEÇO tahm yötem kullaılması öerlmektedr. Ayrıca, ge fade düzeyler göstere açıklayıcı değşkeler de Webull dağılımda gelmektedr. Bu edele, bu değşkeler olasılıksal olduğu düşüülmekte ve GDA ı çıkarımıda model arametreler tahm etmek ç v

8 UEÇO tahm yötem uygulaarak olasılıksal çoklu doğrusal regresyo aalz kullaılmaktadır. Ek olarak, hem olasılıksal hem de olasılıksal olmaya çoklu doğrusal regresyo model arametreler ç sağlamlık ve güç aalzler verlmştr. Aahtar Kelmeler: Ge Düzeleyc Ağlar, Webull Dağılım, Çoklu Olasılıksal Doğrusal Regresyo, Uyarlamış E Çok Olablrlk Tahm. v

9 To My Parets x

10 ACKNOWLEDGEMENTS Frst ad foremost, I would lke to exress my scere grattude to my advsor Prof. Dr. Ayşe Akkaya for her excellet gudace, hel, suort ad atece. She has bee much more tha a advsor to me. I have leared may rofessoal ad ersoal sklls from her. I am deely thakful to my co-advsor Assoc. Prof. Dr. Tolga Ca for beg suortve ad helful throughout the rocess of ths study. He wll always be a role model for me. It s my leasure to ackowledge my commttee members Assst. Prof. Dr. Zeye Kalaylıoğlu, Assst. Prof. Dr. Yeşm Aydı So, Assst. Prof. Dr. Özlem Türker Bayrak ad Assst. Prof. Dr. Ceyla Yozgatlıgl. They have rovded, wth kdess, ther sght ad suggestos, whch are very recous to me. I would lke to reset my grattude to The Scetfc ad Techologcal Research Coucl of Turkey TUBITAK for rovdg the scholarsh that eabled me to comlete my study wthout ay dffcultes. I am also debted to my dear freds Mert, Gül ad Duygu for ther geerous suort ad ecouragemet durg my worst momets. Cem deserves my very secal thaks for helg me every asect. He tolerates me whe eve I caot tolerate myself. All I wat s he stads by my sde. I owe my deeest grattude to my arets. Wthout ther ucodtoal love, I would ot be who I am today. Also, I wll always arecate my beloved brother x

11 ad sster for beg there for me whe I eed them. Ad, I am esecally thakful to Öykü, Our ad Defe for gvg me reewed hoe. x

12 TABLE OF CONTENTS ABSTRACT. v ÖZ......v ACKNOWLEDGEMENTS...x TABLE OF CONTENTS...x LIST OF TABLES.....xv LIST OF FIGURES.xv LIST OF ABBREVIATIONS.xv CHAPTERS. INTRODUCTION. Motvato of the Study..6. Am ad Cotrbuto of the Study 7.3 Orgazato of the Study...8. BIOLOGICAL AND HISTORICAL BACKGROUND..... Bologcal Backgroud.. Gee Regulatory Networks Mcroarray Techology Data Aalyss Prearato 7.5 Hstorcal Backgroud...5. Boolea Networks...5. Gaussa Grahcal Models...3 x

13 .5.3 Bayesa Networks Ordary Dfferetal Equatos Network Idetfcato by Multle Lear Regresso 9 3. METHODOLOGY FOR GENE REGULATORY NETWORKS BY MULTIPLE LINEAR REGRESSION ANALYSIS UNDER WEIBULL DISTRIBUTION Least Squares Estmato uder Webull Dstrbuto Modfed Maxmum Lkelhood Estmato uder Webull Dstrbuto.4 4. METHODOGOLY FOR GENE REGULATORY NETWORKS BY STOCHASTIC MULTIPLE LINEAR REGRESSION ANALYSIS UNDER WEIBULL DISTRIBUTION Least Squares Estmato for Stochastc Multle Lear Regresso Modfed Maxmum Lkelhood Estmato for Stochastc Multle Lear Regresso Hyothess Testg for Stochastc Multle Lear Regresso Asymtotc Covarace Matrx for Stochastc Multle Lear Regresso SIMULATION STUDY AND APPLICATION Bas ad Effcecy Comarsos Robustess Comarsos of Estmators Power Comarsos of Test Statstcs Alcato SUMMARY AND CONCLUSIONS... x

14 REFERENCES.5 APPENDICES A. SIMULATION RESULTS FOR LARGE SAMPLE SIZES..7 B. MATLAB CODE FOR ESTIMATION AND HYPOTHESIS TESTING FOR MULTIPLE LINEAR REGRESSION ANALYSIS WITH NONSTOCHASTIC COVARIATES..5 C. MATLAB CODE FOR ESTIMATION AND HYPOTHESIS TESTING FOR STOCHASTIC MULTIPLE LINEAR REGRESSION ANALYSIS CURRICULUM VITAE..35 xv

15 LIST OF TABLES TABLES Table 3. Exresso data..35 Table 3. Gee erturbed trag erturbatos Table 3.3 Skewess ad kurtoss values of Webull dstrbuto...38 Table 5. Mote Carlo averages, varaces, MSEs ad REs for multle lear regresso wth ostochastc covarates;, q 3,, ad,,..,q.7 Table 5. Mote Carlo averages, varaces, MSEs ad REs for re-arameterzed multle lear regresso wth ostochastc covarates;, q 3,, ad,,..,q..7 Table 5.3 Mote Carlo averages, varaces, MSEs ad REs for stochastc multle lear regresso;, q 3 73 Table 5.4 Robustess comarsos for multle lear regresso model wth ostochastc covarates, 3, q 3,, ad,,..,q...75 Table 5.5 Robustess comarsos for stochastc multle lear regresso model; 3, q 3.78 Table 5.6 Power of F ad F tests for multle lear regresso model wth ostochastc covarates; true model We 8,,. 5,, q,, ad 8 Table 5.7 The exact 5% ots of the dstrbutos of F ad F for multle lear regresso model wth ostochastc covarates..8 Table 5.8 Power of F ad F obtaed by usg smulated crtcal values for multle lear regresso model wth ostochastc covarates; true model We 8,,, q,, ad 83 xv

16 Table 5.9 Power of F ad F tests for multle lear regresso model wth stochastc covarates; true model We 8,, q,, 4,,, ad..87 Table 5. The exact 5% ots of the dstrbutos of F ad F for stochastc multle lear regresso model 88 Table 5. Power of F ad F tests obtaed by usg smulated crtcal values for multle lear regresso model wth stochastc covarates; true model We 8,, q,, 4,,, ad 89 Table 5. Costructed multle lear regresso model wth ostochastc covarates for every gee the SOS subetwork.94 Table 5.3 Idvdual t-tests of model costructed for the gee ssb.96 Table 5.4 Costructed multle lear regresso model wth stochastc covarates for gee lexa the SOS subetwork...99 Table A. Mote Carlo averages, varaces, MSEs ad REs for multle lear regresso wth ostochastc covarates; q 3,, ad,,..,q...7 Table A. Mote Carlo averages, varaces, MSEs ad REs for stochastc multle lear regresso; q 3.9 Table A.3 Robustess comarsos for multle lear regresso wth ostochastc covarates, 5, q 3,, ad,,..,q.. Table A.4 Robustess comarsos for stochastc multle lear regresso; 5, q xv

17 LIST OF FIGURES FIGURES Fgure. Cetral Dogma of Molecular Bology Fgure. A examle of gee regulatory etwork: ellses are TFs; boxes are gees; hexagos are the clustered gees 4 Fgure.3 A reresetato of samle Boolea etwork... Fgure.4 A examle for Bayesa etworks.. 6 Fgure.5 A examle for a lear addtve model...9 Fgure 5. Power grahs of the tests for multle lear regresso model wth ostochastc covarates;,, ad 84 Fgure 5. Power grahs of the tests for stochastc multle lear regresso model;,, ad.9 Fgure 5.3 Q-Q Plot of resduals for Webull dstrbuto wth Fgure 5.4 Q-Q Plot of gee exresso for Webull dstrbuto wth xv

18 LIST OF ABBREVIATIONS A AMML ANOVA ARACNE BIC C cdna CGH cs-eqtl DBN DNA DPN FTSS G GA GGM GRAM GRN LOWESS Adee Adatve Modfed Maxmum Lkelhood Aalyss of Varace Algorthm for the Recostructo of Accurate Cellular Networks Bayesa Iformato Crtero Cytose Comlemetary Deoxyrbouclec Acd Comaratve geomc hybrdzato cs-exresso quattatve trat loc Dyamc Bayesa Network Deoxyrbouclec Acd Dyamc Probablstc Networks Fourer Trasform for Stable Systems Guae Geetc Algorthm Gaussa grahcal model Geetc Regulatory Modules Gee Regulatory Network Locally Weghted Regresso ad Smoothg Scatterlots xv

19 LS LSE ML MML mrna MSE MVB MWSLE NCA NIR ODE PBN PCR RE REVEAL RNA RNA-Seq RSS SAGE SDE SEM SML SSE Least Squares Least Squares Estmator Maxmum Lkelhood Modfed Maxmum Lkelhood Messeger RNA Mea Squared Error Mmum Varace Boud Mmum Weght Solutos to Lear Equatos Network Comoet Aalyss Network Idetfcato by Multle Lear Regresso Ordary Dfferetal Equato Probablstc Boolea Networks Polymerase Cha Reacto Relatve Effcecy Reverse Egeerg Algorthm Rbouclec Acd RNA Sequecg Sum of Squares of Resduals Seral Aalyss of Gee Exresso Stochastc Dfferetal Equato Structural Equato Model Sarsty-Aware Maxmum Lkelhood Sum of Squared Errors xx

20 T TBN TdGRN TF TSNI U Thyme Temoral Boolea Network Tme-delayed Gee Regulatory Network Trascrto Factor Tme seres Network Idetfcato Urasl xx

21 CHAPTER INTRODUCTION I all lvg orgasms, there s a herarchcal orgazato of small buldg blocks. Cell s the smallest ut of ths herarchy. Combato of cells havg a secal structure ad fucto forms tssues. Dfferet kds of tssues comose a orga. Several dfferet orgas work together to erform a certa task as a orga system. Ad fally, dfferet orga systems come together ad form the orgasms. All geetc formato whch determes the fucto of a cell s ecoded the Deoxyrbouclec Acd DNA sequece. A gee s a sub segmet of DNA ad all gees the geome comoses the set of structos that orgasm eed to survve. The DNA code of gees s coverted to Rbouclec Acd RNA, whch codes for rote roducts. Amout of the roducts roduced by a artcular gee s the exresso level of that gee. Whle all cells a orgasm have the same geomc DNA, so the same gee sequeces, the exresso levels of may gees dffer dfferet kds of cells ad uder dfferet codtos. Cell dfferetato ad cell fucto are regulated by dfferetal gee exresso. If researchers kow the codtos uder whch gees are exressed at hgh level, they ca get hts about the fucto of those gees. Gee exresso levels are also assocated wth the dsease recurrece. Whe there s a malfucto at ay of the buldg blocks, orgasms caot erform ormally ad dseases occur. Most dseases are result from the abormal actvty of gees the cells. Whle a orgasm s leadg to dseased state from the

22 health state, exresso levels of the gees related to the dsease chage. For examle, oco gees the cacer cells are exressed at hgh levels, but tumor suressor gees are exressed at lower levels. By comarg the exresso levels of the gees dseased ad ormal cells, gees resosble for varous dseases ca be detfed ad ossble theraeutc targets of the drugs ca be determed. Quatfcato of the gee exresso level rofles uder dfferet codtos s a mortat art of the bologcal ad medcal research. Wth the develomet of hgh throughut techologes such as DNA mcroarray ad RNA sequecg RNA-Seq molecular bology, researchers ow ca get the formato about the cocetrato levels of thousads of gees at a gve tme or a gve state of a orgasm. However, detfcato of the resosble gees for cell dfferetato or certa dseases by measurg the gee exresso levels s ot suffcet aloe. It s also mortat to determe how gee roducts are govered. Cells eed every gee roduct ether at the same tme or the same amout. I a cell, gees work together by teractg wth oe aother ad terdeedeces betwee them determe whch, whe ad how much roduct s roduced by a artcular gee, that s, gee exresso levels are regulated by these teractos betwee the gees. Hece, ferrg gee regulatory etworks GRNs becomes a ecessty to uderstad the molecular mechasm of the lfe. GRNs detfy the teractos betwee the gees ad ther roducts usg gee exresso data. They descrbe how the gees are exressed by a cell, whch gees are trascrbed to RNA ad whch of them tur are used for the rote sythess. By GRNs, the relatoshs betwee the gees ad ther regulators ca be vsualzed mag the teractos betwee them oto a grahc. A large umber of studes have bee carred out for ferrg or reverseegeerg GRNs. Kauffma 969 has troduced Boolea etworks to obta

23 GRNs by modelg the gee as a bary devce that ca realze ay oe, but oly oe, of the ossble Boolea fuctos of ts K uts. A geeralzato of the Boolea etworks whch s called the Temoral Boolea Network TBN has bee roosed by Slvescu ad Hoavar 997 to exame the deedeces amog the actvty of gees that sa for more tha oe ut of tme. Fredma et al. 998 have hadled the roblem of learg dyamc robablstc etworks DPN from comlete data by the extedg Bayesa Iformato Crtero BIC scores ad from comlete data by extedg structural equato model SEM algorthm. Lag et al. 998 have vestgated the ossblty of ferrg a comlex regulatory etwork archtecture from ut/outut atter of ts varables ad mlemeted Reverse Egeerg Algorthm REVEAL usg mutual formato measures. Muhy ad Ma 999 have showed that the most of the roosed dscrete tme models reverse-egeerg geetc etworks from tme seres data are all secal cases of a geeral class of models called Dyamc Bayesa Networks DBNs ad revewed the used techques to lear DBNs. Che et al. 999 have roosed a dfferetal equato model for gee exresso ad develoed two methods, Mmum Weght Solutos to Lear Equatos MWSLE ad Fourer Trasform for Stable Systems FTSS, to costruct model from exermetal data. Shmulevch et al. have troduced the model of Probablstc Boolea Networks PBN whch have the rule-based roertes of Boolea etworks, but are robust the face of ucertaty. Also, Bar-oseh et al. 3 have develoed a algorthm called Geetc Regulatory Modules GRAM whch combes formato from geome-wde locato ad exresso data sets to exlore regulatory etworks of gee modules. Kkuch et al. 3 have mroved the method roosed by Tomaga ad Okamoto 998 for the dyamc modelg of comlex bosystems combg a Geetc Algorthm GA ad the S-system ad comared these basc ad modfed methods. Garder et al. 3 have costructed a frst-order model of regulatory teractos a e-gee subetwork of the SOS athway Eschercha col ad dcated the model to detfy correctly the maor regulatory 3

24 gees ad trascrtoal targets of mtomyc C actvty the subetwork. Perr et al. 3 have dealt wth the detfcato of GRNs from exermetal data usg a statstcal mache learg aroach ad roosed a stochastc model of gee teractos caable of hadlg mssg varables. They have estmated the model arameters by a ealzed lkelhood maxmzato method. Kao et al. 4 have hadled the comlex trascrtoal etworks ad showed the utlty of etwork comoet aalyss NCA determg the multle trascrto factor actvtes. Ott et al. 4 have develoed a algorthm to obta the otmal Bayesa etworks of cosderable sze overcomg the ucertates of heurstc aroaches that makes t dffcult to draw coclusos from etworks estmated by heurstcs. Nachma et al. 4 have reseted fe-graed dyamcal models of gee trascrto ad roosed a algorthm based o DBNs to recostruct these models of GRNs. Laubebacher ad Stgler 4 have roosed a aroach costructg a regulatory etwork as a tmedscrete mult-state dyamcal system after they have descrbed some of the exstg reverse-egeerg methods. Xg ad Laa 5 have descrbed a comrehesve statstcal aroach to obta trascrtoal regulatory etworks usg gee exresso data, trascrto factor bdg stes ad romoter sequeces. Che et al. 5 have troduced a stochastc dfferetal equato SDE model for the trascrtoal regulatory etwork of the tme-course gee exresso datasets. They have aled ths model to the cell-cycle data of buddg yeast Saccharomyces cerevsae ad tred to ft a geeralzed lear model estmatg the trascrto atter of a secfc target gee. Boscolo et al. 5 have addressed the NCA o the bass of some asects. They have used two-stage least square teratve rocedure NCA ad troduced a framework to recostruct multle regulatory subetworks smultaeously. Margol et al. 6 have troduced the Algorthm for the Recostructo of Accurate Cellular Networks ARACNE that s a ovel-theoretc algorthm for ferrg the trascrtoal etworks from mcroarray data. Usg delayed correlatos betwee gees, L et al. 6 have develoed a toolbox called Tme-delayed 4

25 Gee Regulatory Network TdGRN to recostruct regulatory etworks from temoral gee exresso data. Cho et al. 6 have gve S-tree based geetc rogrammg for both the structural ad dyamcal modelg of a bologcal etwork ad estmatg the etwork arameters. Sabatt ad James 6 have troduced a framework whch uses DNA sequece formato ad exresso arrays data cocert to aalyze the effects of a collecto of regulatory rotes o geomc exresso levels. Basal et al. 6 have reseted Tme Seres Network Idetfcato TSNI algorthm whch obta the local etwork of geegee teractos surroudg a gee of terest by erturbg oly oe of the gees the etwork. Basal et al. 7 have comared dfferet algorthms used to fer gee etworks ad showed that these algorthms ca correctly reverse-egeer the gee teractos. Cho et al. 7 have also reseted varous techques of reverse-egeerg GRNs from gee exresso rofles ad bologcal formato ad arraged systematcally these techques based o the requred formato. Kaderal ad Radde 8 have hadled several aroaches gve for dscoverg GRNs ad dscussed ther stregths ad weakesses, also rovded formato o whch models are arorate uder what crcumstaces ad future develomets. Fath et al. 8 have develoed cotext lkelhood of relatedess to obta the trascrtoal regulatory relatos usg trascrtoal rofles of a orgasm across a dverse set of codtos. Scrutzg several kds of comutatoal methods used redctg GRNs mammala cells, Lee ad Tzou 9 have showed how the ower of dfferet kowledge databases of dfferet tyes ca be used to detfy modules ad subetworks. Emmert-Streb et al. have revewed the methods avalable for estmatg the GRNs ad comared two maor aroaches wth cotemorary oes. Sarse structural equato models have bee used Ca et al., 3 to tegrate both gee exresso data ad cs-exresso quattatve trat loc cseqtl, for modelg gee regulatory etworks accordace wth bologcal evdece about gees regulatg or beg regulated by a small umber of gees. A 5

26 systematc ferece method amed sarsty-aware maxmum lkelhood SML has bee also develoed for SEM estmato.. Motvato of the Study The motvato of ths dssertato comes from the work of Garder et al. 3. I ther study, they develo a algorthm called Network Idetfcato by Multle Lear Regresso NIR whch a model of the coectos ad fuctoal relatos betwee elemets a etwork s ferred from measuremets of system dyamcs by alyg multle lear regresso aalyss. They use the method of least squares LS to estmate the arameters of the multle lear regresso model. Whle costructg the model, they assume that the ose term ad exlaatory varables reresetg the exresso levels of gees the model are ormally dstrbuted. However, whe the real data used ther study s examed, t s see that the resduals obtaed by usg LS estmators LSEs ft Webull dstrbuto better. Smlarly, codtoal dstrbutos of exlaatory varables are obtaed as Webull dstrbuto, too. Sce the exlaatory varables are stochastc, LS estmators of the model arameters are ot same wth the maxmum lkelhood ML estmators aymore. I addto, the relatos betwee the exlaatory varables are ot take to cosderato ther study. I our study, t s roosed to use a stochastc multle lear regresso model whe error term ad exlaatory varables come from a Webull dstrbuto cosderg the deedecy betwee the exlaatory varables to costruct GRNs. 6

27 . Am ad Cotrbuto of the Study The ma focus of ths dssertato s to obta a statstcal comutatoal method that ca be used for ferrg gee regulatory etworks from gee exresso data by researchers. It s amed to mrove the NIR algorthm by dealg wth the statstcal assumtos eeded to aly NIR algorthm. Ths study makes the followg cotrbutos: It s kow that o-ormalty comlcates the data aalyss ad results effcet estmators. Therefore, t s very mortat to mrove statstcal rocedures whch are effcet ad robust to devatos from a assumed dstrbuto. Ths study rovdes a robust estmato techque for the multle lear regresso aalyss whe the ose has a Webull dstrbuto by estmatg the model arameters usg the method of modfed maxmum lkelhood MML estmato. As metoed at Secto., exlaatory varables rereset the gee exresso levels ad they are subect to the measuremet errors. Therefore, they are stochastc ad they have a dstrbuto. The arameter estmators obtaed by usg NIR algorthm are ot the ML estmators aymore whe the exlaatory varables are stochastc, whch meas that the estmators of the model arameters obtaed by usg NIR algorthm lost ther good roertes. Ths study hadle ths roblem usg stochastc multle lear regresso aalyss. Lastly, Garder et al. 3 gore the relatoshs betwee the exlaatory varables ther study. However, some exlaatory varables are collear sce the gee exresso levels are regulated by the teractos betwee gees. Therefore, ths study takes to accout the relatoshs betwee the exlaatory varables ad estmate the artal correlato coeffcets betwee the exlaatory varables by 7

28 mlemetg method of MML the stochastc multle lear regresso model..3 Orgazato of the Study Ths thess cossts of sx chaters. I Chater, a bref troducto to gee regulatory etworks s gve by emhaszg the mortace of them molecular bology. Also, the ublcatos related to the GRNs are reseted comrehesvely ad the motvato of the thess s descrbed. Furthermore, the am ad the cotrbutos of the study are stated. I Chater, a bologcal backgroud of gee regulato eeded to uderstad the rest of the thess s rovded. It exlas the gee exresso ad metos some hgh throughut techques used to measure the gee exresso levels. It also revews the exstg methods used for ferrg gee regulatory etworks. Esecally, NIR algorthm s examed detal sce t s the motvato of ths study. Chater 3 gves the theoretcal exlaato of the multle lear regresso models. Sce the exresso data used the regresso model ft a Webull dstrbuto, Webull dstrbuto ad ts roertes are also descrbed ths chater. The, MML ad LS estmators of the arameters the multle lear regresso model wth ostochastc covarates are derved ad the test statstcs based o LS ad MML estmators are obtaed to test sgfcace of model arameters. I Chater 4, stochastc lear regresso model s used to fer GRNs ad model arameters are estmated by usg MML ad LS estmato methods cosderg the relatoshs betwee the exlaatory varables. Test statstcs based o LS ad MML estmators are also obtaed for stochastc lear regresso model. 8

29 I Chater 5, MML ad LS estmato techques are comared by examg the effcecy, robustess ad ower roertes of them through a comrehesve smulato study. I addto, a real lfe alcato s gve ths chater. Fally, the last chater of the thess cocludes the work that has bee doe, suggests some deas about the gee regulatory etworks ad gves the related future work. 9

30

31 CHAPTER BIOLOGICAL AND HISTORICAL BACKGROUND Ths chater resets a bologcal backgroud to elucdate the gee exresso ad the gee regulato comrehesvely by gvg the deftos of some geetc materals such as cell, gees ad DNA etc. Also, t descrbes the hgh throughut techques used to measure gee exresso levels ad exlas the mcroarray techology detal. Furthermore, some commoly used methods for ferrg gee regulatory etworks are revewed ths chater.. Bologcal Backgroud Cells are the mmal uts of all lvg orgasms that cota a multtude of secfc chemcal trasformatos rovdg the eergy eeded by the cells ad coordatg all of the evets Lee, 4. The regulato of gee exresso levels s maybe the most mortat task of cells to meet ther eeds ad to adot the evrometal chages. Macromolecules such as DNA, RNA ad rotes defe the structure of cells ad gover most of the actvtes of lfe Lee, 4 ad esecally lay the ma roles the rocess of the exresso of the geetc formato. DNA s a double-straded ad helcal molecule comosed of four ucleotdes: adee A, guae G, thyme T ad cytose C. The sequece of these four ucleotdes ecodes the geetc formato stored DNA ad hece, gves the geetc structos for the develomet ad the roer fuctog of the

32 orgasms. Sce each strad of the DNA molecule s the comlemetary of the other, the double helx structure of the DNA molecules adds othg to the formato cotaed a sgle strad. I DNA, A ars wth T, ad C wth G. Gees are segmets of DNA ad cota secfc structos whch allow a cell to roduce a secfc roduct. Although every cell of a dvdual orgasm cotas the same DNA, carryg the same formato, dfferet kds of cells are avalable. As metoed Chater, ths dfferetato s resulted from that all the gees are ot exressed the same way all cells Draghc, 3. Cells eed the roducts of some kd of gees called housekeeg gees at all tme. It s assumed that these gees are exressed at costat levels dfferet cell tyes. However, exresso levels of artcular gees are affected from ther crcumstaces ad chages ther exresso levels determe the dstct bologcal characterstcs ad hece cause orgasmal comlexty ad dversty. Dfferetato betwee cells s gve by dfferet atters of gee actvatos whch tur cotrol the roducto of rotes. A gee s actve, or exressed, f the cell roduces the rote ecoded by the gee. If a lot of rote s roduced, the gee s sad to be hghly exressed. If o rote s roduced, the gee s ot exressed or uexressed. The obectve of researchers s to detect ad quatfy gee exresso levels uder artcular crcumstaces Draghc, 3. Gee exresso s the most fudametal level at whch geotye gves rse to the heotye. It s the etre rocess that takes the formato cotaed gees o DNA ad turs that formato to rotes. Gee exresso occurs three maor stages: Relcato, Trascrto ad Prote Sythess or Traslato. I the relcato rocess, a double-straded DNA molecule s dulcated to gve detcal coes. RNA, a sgle-straded molecule whch uses a ucleotde called uracl U stead of thyme reset DNA, s trascrbed from DNA by

33 ezymes called RNA olymerases ad s geerally further rocessed by other ezymes. Ths rocess s called trascrto. I the rocess of rote sythess, RNA sequece s traslated to a sequece of amo acds. A combato of dfferet amo acds forms the rotes. Protes are the comlex orgac comouds cosstg of the mmedate exresso of the geetc formato stored DNA ad attedg varous tasks essetal for survval of the cell. These three stages are all together called the cetral dogma of molecular bology Watso ad Crck, 958; Crck, 97 ad reseted Fgure.. Fgure.: Cetral dogma of molecular bology. Fgure s adated from htt://users.uget.be/~averstr/rcles/cetraldogma.html. 3

34 . Gee Regulatory Networks Regulato of gee exresso cotrols the amout ad tmg of a fuctoal gee roduct. I the rocess of gee exresso, trascrto factors TFs, whch are the secalzed rotes, bd to romoter rego of DNA ad tervee the rate of rote sythess. A crease the rate of rote sythess s called as the actvato or u-regulato of the gee ad a decrease the rate of rote sythess s called as the hbto or dow-regulato of the gee Pase ad Kshrsagar, 3. Gees regulate the exresso levels by teractg each other through gee roducts. A gee regulatory etwork s a collecto of gees ad gee roducts RNAs ad rotes ad descrbes the regulatory relatoshs betwee gees, rotes ad other cellular comoets. Gee regulato ca be vsualzed by grahs whch odes show gees or gee roducts ad drected edges coectg odes show the deedecy betwee them. I Fgure., the grahcal reresetato of a smle gee regulatory etwork s llustrated. Fgure.: A examle of gee regulatory etwork: ellses are TFs; boxes are gees; hexagos are the clustered gees. Fgure s adated from htt://rula.cshl.edu/tred/grn/hif.htm. 4

35 Recostructo of gee regulatory etworks holds great mortace esecally the feld of system bology. Accurate redcto of GRNs rovdes a oortuty to study the dyamcs of secfc gee uder artcular dseased or exermet codtos. It also hels to study dseases that are caused by dysregulated gees. Hece, GRNs eables to develo ew treatmet methods for llesses ad to aalyze the effects of drugs o gees Karlebach ad Shamr, 8; Pase ad Kshrsagar, 3..3 Mcroarray Techology Wth the develomet of hgh throughut techologes, exresso levels of thousads of gees ca be measured smultaeously. Gee exresso data allow researchers to fer gee regulatory etworks by observg chages gee exresso rofles uder varous exermet codtos ad uder dfferet cell cycle stages. Hece, behavors of gees ca be aalyzed. There are several kds of molecular bology techques, as lsted below, to quatfy the gee exresso ad mcroarrays ad ext geerato RNA sequecg are the most curret hgh-throughut techques: Comaratve Geomc Hybrdzato CGH Seral Aalyss of Gee Exresso SAGE RNA Sequecg Real Tme- Polymerase Cha Reacto PCR Mcroarrays I ths subsecto, oly mcroarray aalyss of gee exresso s hadled sce t s the most wdely used techque. 5

36 A DNA mcroarray, also kow as DNA ch, has bee troduced by Schea et al. 995 for the frst tme ad become a very oular techque to detfy the gee exresso chages resulted from a varety codtos such as develomet, agg ad dseases or drugs Alzadeh et al., ; Blba et al., ; Taaka et al., ; Youg,. It cossts of a small membrae or glass slde cotag samles of may gees arraged a regular atter. The surface of a mcroarray s sotted wth olgoucleotdes that are the small arts of DNA molecules u to 5 ucleotdes, comlemetary DNA cdna or small fragmets of olymerase cha reacto roducts that rereset secfc gee codg regos. There are thousads of mcroscoc sot kow as robes o a mcroarray ad each robe corresods to a artcular gee Amaratuga ad Cabrera, 4. Mcroarrays ca be classfed as sgle-chael oe-color ad two-chael twocolor arrays ad both tyes of mcroarray are used for the hybrdzato exermets. Oe-color mcroarray measures the testy of oly oe hybrdzed bologcal samle whle two-color mcroarray measures exresso ratos betwee two hybrdzed samles. DNA mcroarray techology deeds o the arallel hybrdzato of labeled target to mmoblzed robes Schea et al., 995. Dfferetal gee exresso s determed by usg a two-color scheme. Frstly, the messeger RNA mrna s solated from the exermetal samles such as healthy or tumor tssue samle ad reverse-trascrbed to more stable comlemetary DNA. The cdna samles are labeled by a fluorescet dye geerally healthy cdna s labeled gree ad tumor cdna s labeled red ad combed samle s hybrdzed to the mcroarray ch. Target samle bdg to a robe geerates a sgal ad ts stregth deeds uo the amout of target samle bdg to that robe. The, fluorescet testy o each robe s measured ad coverted to the raw data by usg a secal scaer. 6

37 Actve gees roduce may mrna molecules, hece, may labeled cdna samles, ad geerate a very brght fluorescet sots. Gees that are less actve roduce fewer mrna molecules, thus, less labeled cdna samles, ad geerate dmmer fluorescet sots. If there s o fluorescece, oe of the messeger molecules have hybrdzed to the DNA whch dcates that the gee s actve. Alcato areas of mcroarrays ca be summarzed as follows: Gee dscovery: Mcroarray techology s used to detfy gees ad to determe ther fucto ad exresso levels at the artcular codto Cho et al., 998; Chu et al., 998; Tao et al., 999; Laub et al., ; We et al., ; Cha et al., 3. Gee regulato studes: Mcroarray techology s used to fer gee regulatory etworks descrbg the regulatory relatoshs betwee gees ad gee roducts de Sazeu et al., ; Gross et al., ; Arf et al., ; Kuh et al., ; Brtto et al.,. Dsease dagoss: Mcroarray techology s used to determe dsease by the detfcato of chages the exresso levels of artcular gees Ggeras et al., 998; va t Veer et al., ; Macoska. Drug dscovery ad toxcology: Mcroarray techology s used to develo treatmets for llesses by studyg the theraeutc resoses to drugs. It s also used to search the macts of toxs o the cells Wlso et al., 999; Bammert ad Fostel, ; Clarke et al.,.4 Data Aalyss Prearato As metoed Secto.3, mcroarray techology measures the labelled fluorescet testes whch rereset the amout of mrna molecules solated from the exermetal samles ad coverts these testes to the gee exresso data. However, mcroarray exermets are geerally subect to some 7

38 sources of varatos ad these varatos mask the bologcal sgals of the actual terest, whch meas that fluorescet testy chages may ot always show the actual exresso chages. Measuremet errors that affect the exresso data ca be classfed to two categores: systematc error ad radom error. Systematc error s a bas resultg from array sottg, scag, labellg, hybrdzato etc., ad t reflects the accuracy of exermet measuremets Clavere, 999; Schuchhardt et al., ; Lou et al., ; Tseg et al., ; Yue et al.,. For examle, a well-kow systematc error s fluorescet dye bas. Whe two detcal samles are labelled wth red ad gree colors ad hybrdzed to same slde, t s exected that gree testes ad red testes are at the same level sce there s o dfferetal exresso. However, red testes geerally ted to be lower tha gree testes Smyth et al., 3. Oce systematc errors are detfed ad removed, t s cosdered that the remag measuremet errors are radom. Radom error s a measure of ucertaty the measuremets ad reflects the recso of exresso data. It costtutes a ose whch revets the chages bologcal sgals to be determed correctly. Chages the exresso levels ca be dstgushed from radom ose by usg some statstcal tests. Systematc errors ca be removed or cotrolled by usg strct exermetal rocedures ad emloyg ormalzato methods. After all backgroud correctos are carred out, ormalzato of mcroarray data have to be erformed to make observatos comarable each other. I mcroarray data aalyss, t s geerally amed to detfy the dfferetally exressed gees by comarg the exresso levels of gees uder dfferet codtos, for ths reaso, gee exressos are rereseted as the rato of two florescet testes Parmga et al., 3. Although ratos rovde a tutve measure of exresso chages, they treat u-regulated ad dow-regulated gees dfferetly. The testy ratos usually have a skewed dstrbuto sce the ratos of dow-regulated gees take 8

39 the values the terval, whereas the ratos of u-regulated gees take the values the terval,. For examle, gees u-regulated by a factor of have a exresso rato of whereas those dow-regulated by the same factor have a exresso level of -.5. To overcome ths dsadvatage of the ratos, exresso data eed to be trasformed before the ormalzato. The most commoly used trasformato s the logarthm base trasformato. It rovdes the gees u-regulated ad dow-regulated by a factor of to have a log rato of ad -, resectvely Quackebush,. A umber of ormalzato aroaches have bee troduced to remove the systematc errors. The most well-kow aroaches ca be lsted as Global Normalzato Total RNA Normalzato Self-Normalzato Housekeeg Gee Normalzato Locally Weghted Regresso ad Smoothg Scatterlots LOWESS Global ormalzato s the frst roosed method for the ormalzato of the mcroarray data. Ths aroach relates the red testy to the gree testy by a multlcatve costat ad shfts the ceter of the dstrbuto of trasformed exresso ratos to zero Yag et al.,. Total RNA ormalzato assumes that amout of total RNA carred by each cell does ot chage over tme Fag et al., 3. The self-ormalzato method removes the systematc error by alyg a subtract oerato to the data sce the error o log scale s assumed to be addtve. The housekeeg gee ormalzato evaluates the labellg ad samle hybrdzato by sottg a set of housekeeg gees o the array. I ths aroach, t s assumed that the housekeeg gees are exressed at a costat level uder dfferet exermetal codtos Yag et al.,. Trasformed exresso ratos have some testy-deedet varatos ad LOWESS 9

40 ormalzato removes these varatos by alyg a smoothg adustmet Clevelad, 979; Quackebush,. Ulke systematc errors, radom errors caot be removed etrely but they ca be estmated from the observed data. By the relcato of the exermet, radom errors ca be mmzed sce t s exected that relcates gve same results uder the same codto excet for radom error Nado ad Shoemaker,..5 Hstorcal Backgroud Sce the modellg of gee regulatory etworks has become a very useful tool for the aalyss of the gee teractos, umerous methods have bee roosed to costruct gee regulatory etworks the lterature. These methods ca be classfed as hyscal aroach ad fluece aroach. I the hyscal aroach, the rotes regulatg the trascrto ad DNA motfs to whch they bd are detfed, ad hece, true molecular teractos are determed. However, the fluece aroach does ot seek true hyscal teractos, stead, t descrbes the regulatory flueces betwee RNA trascrts by observg the chages the trascrto levels. For the modellg of gee regulatory etworks, the fluece aroach s more referable to the hyscal aroach sce the hyscal aroach eeds more ror kowledge ad secfc data. Models used for ferrg gee regulatory etworks are also dvded to two grous as dyamc ad statc. Dyamc models whch cota a tme-comoet are used whe the dyamc behavor of the etwork s requred Hecker, 7. I ths study, the most wdely used methods are revewed. These are lsted as follows:

41 Boolea Networks Gaussa Grahcal Models Bayesa Networks Ordary Dfferetal Equatos Network Idetfcato by Multle Lear Regresso.5. Boolea Networks Boolea etworks have bee used frstly by Kauffma 969 to model behavor of the large ets of radomly tercoected bary gees. I the modellg of gee regulato by Boolea etworks, each ode reresets a gee ad drected edge reresets bologcal teracto betwee two gees. A Boolea etwork ca be exlaed by the defto gve below: Defto Boolea Network: A Boolea etwork s a tule G X,B where X x s a vector of Boolea varables, ad B s a set,x,...,x {, } of Boolea fuctos B { f, f,..., f }, f : {, } {, } Kaderal ad Radde, 8. I the gee regulatory etworks, x reresets the state of gee ad reresets the teractos betwee them. It s assumed that each gee ca be ether state o or off. State o meas that gee s exressed above some threshold whle state off meas that gee s exressed below that threshold. If the gee s the state o, state off, x takes the value of. x takes the value of ad f the gee s the f By usg Boolea fuctos, the states of all gees are udated at a dscrete tme ste:

42 x t f x t,x t,...,x t. Here t s assumed that gees udate ther states smultaeously. At ay gve tme t, the exresso or state of the etwork are rereseted by the states of all odes as follows: xt x t,x t,...,x t. A examle of the Boolea etworks s show Fgure.3. Here, oted array shows a actvato. For examle, f gee B s actve, the t wll actvate gee A. If gee A s actve, the t wll actvate gee B. Also f ether gee A or gee B s actve, the gee C wll be actvated. A Iut Outut A B C A B C C B Fgure.3: A reresetato of samle Boolea etwork. Because of the dyamc roertes of Boolea etworks, ths method s qute oular for the recostructo of gee regulatory etworks. However, formg a accurate etwork s ot a easy ssue sce t s mossble to determe the values

43 of states a Boolea etwork wth odes. Istead, Kauffma 969 have troduced NK Boolea etworks whch studes a class of Boolea etworks of odes. I ths aroach, each ode has a radomly selected k uts from odes ad has! / k! ossble combato of k uts. I addto, there are ossble fuctos for each ode. Hece, the umber of ossble etworks s obtaed as follows: k k!.3 k! As the umber of odes creases, the umber of the ossble etworks grows exoetally. To solve ths roblem, the umber of edges drected to a ode s bouded by a costat..5. Gaussa Grahcal Models Gaussa grahcal model GGM s a very oular aroach to the recostructo of gee regulatory etworks Dobra et al., 4. I ths aroach, t s assumed that the avalable data come from a multvarate Gaussa dstrbuto Whttaker, 99. Hece, the am s to determe the codtoal deedeces amog gees by dervg the artal correlatos the ot robablty dstrbuto of exresso data. Gaussa grahcal models gve the drect assocato betwee gees but drect assocatos ca also be obtaed easly Wag et al., 3. GGM s descrbed by a grah G V,E where V {,,..., } corresods to the ode sets reresetg the varables ad E e corresods to the edge set reresetg codtoal deedeces betwee odes. If there s o edge betwee two odes 3

44 e, the these two odes are codtoally deedet gve all other odes. I the GGM, X X,X,...,X reresets the real valued states of odes ad follow a multvarate Gaussa dstrbuto wth mea ad covarace matrx. Hece, artal correlatos ca be obtaed by the verse covarace matrx { w }. They are obtaed as follows: w.4 w w where s the artal correlato betwee gee ad gee gve all other gees. If e s, the w becomes ad valued elemets the verse covarace matrx gve the codtoally deedet gees the etwork..5.3 Bayesa Networks Bayesa etwork model has bee roosed by Fredma et al. ad Hartemk et al. for ferrg gee regulatory etworks. It determes the robablstc relatoshs betwee the odes of the etwork by establshg a drected acyclc grah. Drected acyclc grah s deoted by G X,A where the odes X X,X,...,X corresod to the radom varables reresetg the exressos of gees ad the drected edges A rereset the robablstc deedeces betwee the radom varables. A edge from deedecy of X o X. I ths case, X s called a aret of X to X shows the X. Therefore, has a codtoal robablty dstrbuto deoted by x arets x where X arets x s the set of arets for X. If X does ot have a aret, the t s 4

45 ucodtoal robablty dstrbuto x. I a Bayesa etwork, t s assumed that each radom varable s deedet of ts o-descedats. Hece, the ot robablty dstrbuto fucto of follows: X,X,...,X ca be wrtte as x, x,..., x x arets x.5 Fgure.4 gves a smle Bayesa etwork cosstg of four odes A, B, C ad D wth dscrete states o ad off. It s see that A s the aret of both B ad C, ad C s the aret of D. D s assumed to be codtoally deedet from A gve C. By the gve robabltes, ot robabltes ca be comuted from the grah. For examle; P A, B, C, D P A P B A P C A.6 P D C

46 A A P A.6.4 B C A P B A P B A A P C A P C A D C P D C P D C Fgure.4: A examle for Bayesa etworks. Iferrg Bayesa etworks cossts of two arts. I the frst art, the best grah G s foud gve the observed data D. I the secod art, the best codtoal robabltes are obtaed gve the grah G ad observed data D. Model structure G s samled from the osteror robablty of a etwork toology gve by D G G G D.7 D where P G s the ror dstrbuto over etwork structures. Here codtoal dstrbuto D G ca be comuted as follows: D G D q,g q G dq.8 6

47 whch q s the arameter vector for the codtoal dstrbutos, D q,g s the lkelhood fucto ad q G s the ror dstrbuto of the arameters. If the structure of grah G ad observed data D are assumed to be gve, the the detals of the codtoal dstrbutos ca be ehaced by obtag the values of arameters of the codtoal dstrbutos. The osteror dstrbuto of the arameters q s gve by D q,g q G q D,G..9 D G Bayesa etwork modellg s a fascatg method to costruct gee regulatory etworks sce they are stochastc ad thus they ca deal wth the osy measuremets..5.4 Ordary Dfferetal Equatos Ulke Bayesa etworks, ordary dfferetal equatos ODEs rovde a determstc asect the recostructo of gee regulatory etworks. Usg ordary dfferetal equatos, cocetratos of RNAs, rotes ad other cellular molecules ca be modelled by a dscrete or cotuous tme-deedet varable. Chages the exresso level of a gee at a artcular tme s exlaed by a rate equato whch has the mathematcal form dx f x,x,...,x,,u. dt 7

48 whch s the exresso level of gee at tme t, s the umber x of gees, s the arameter set of the etwork ad u s the exteral erturbato to the etwork. The fucto f ca be lear, ecewse lear or olear. ODEs have bee used frstly by Che et al. 999 to model gee regulato. They form the gee etwork by a smle lear fucto f xt Axt. where A s gee ad gee. matrx of elemets a defg the regulatory relato betwee The most wdely used class of the ODEs s S-systems. They have bee used to recostruct gee regulatory etworks by Kkuch et al. 3. S-systems are descrbed as follows: dxt g h x t x t. dt Here g ad h are the ketc orders ad ad are the rate costats. The frst ad secod terms at the rght had sde descrbe the effects of actvators ad hbtors, resectvely. I Fgure.5, a lear addtve model s gve as a examle of the lear ordary equatos. Nodes rereset the exresso levels of gees, dashed les rereset the hbtg relatos, full les rereset the actvatg relatos ad reresets the stregth of effect of gee o gee. For examle, the exresso level of gee A at tme t+ ca be obtaed by the equato q 8

49 x A t x t q x t q x t..3 A CA C DA D x A t x B t A q CA x C t C q BC B q DA q CD q BD D x D t Fgure.5: A examle for a lear addtve model..5.5 Network Idetfcato by Multle Lear Regresso Garder et al. 3 have develoed a algorthm called Network Idetfcato by Multle Lear Regresso NIR to fer fuctoal relatoshs betwee the gees. Ths algorthm models the behavor of a gee regulatory etwork by frstorder lear equatos descrbg the rate of accumulato of each etwork seces resultg from a trascrtoal erturbato: d x / dt Ax u.4 where x reresets the mrna cocetratos of gees, dx / dt reresets the rate of accumulato of mrna cocetratos, A reresets the etwork model descrbg regulatory relatos betwee mrna cocetratos ad u reresets the set of exteral erturbatos. 9

50 For each gee the etwork, Equato.4 ca be wrtte the followg form: dx dt l N a x l u l,,..., N, l,..., M,.5 a T x u l l where x l s the mrna cocetrato of gee followg the erturbato exermet l; a reresets the fluece of gee o gee ; ad u l s a exteral erturbato to the exresso of gee exermet l. By usg matrx otato, the rate of accumulato for all N gees the etwork s gve by d x dt l A.x u, l,..., M,.6 l l where x s a N vector of mrna cocetratos of the N gees l exermet l, A s a l N N coectvty matrx, comosed of elemets a, ad u s a N vector of the erturbatos aled to each of the N gees exermet l. Near a steady-state ot whch meas that gee exresso does ot chage d x substatally over tme l, the followg equato s obtaed: dt A.X U,.7 where X s a N M matrx comosed of colums x l ; U s a N M wth each colum, u l. 3

51 Equato.7 ca be solved oly f M N. To satsfy ths codto, umber of exermets ca be creased, but the, A wll be extremely sestve to ose the erturbatos ad be urelable. To overcome ths roblem, t s assumed that the maxmum umber of regulators of each gee, k, s less tha M,.e., the etwork s ot fully coected. Sce the data both X ad U are osy, two ose terms are added to Equato.7 ad the followg multle lear regresso model s obtaed for each ossble combato of k out of N weghts: y T T T b. Z e.8 where y s a M vector of measuremets of yl ul, b s a k vector reresetg oe of N choose k ossble combatos of the elemets of s a a, Z K M matrx, where each colum s the vector z l for oe of the M exermets z, reresets ormally dstrbuted measuremet ose l l l x l o the mrna cocetratos exermet l; ad e s a M vector of ose, l reresets ormally dstrbuted measuremet ose o T el l b l erturbatos of gee exermet l. By alyg method of LS, NIR algorthm obtas the estmator of multle lear regresso model gve by Equato.8 as follows: b the b ~ Z.Z T. Z..9 y To obta the best estmate for b, combatos of weghts for gee ad the estmate b ~ s calculated for each of the N choose k b ~ wth the smallest sum of 3

52 squared errors s selected as the best aroxmato of Sum of squared errors SSE lost fucto s defed by a Equato.5. SSE k M l M ~ T y ~ y y b. z.. l l l l l From Equato.9, the redctor for y gve the data matrx Z s ~ T ~ T y b.z.. Garder et al. 3 assumes that the ose term the model gve by Equato.8 s ormally dstrbuted ad the least square estmators are the most effcet for ormal data. They also assume that the regressors are ucorrelated ad roose to use rdge regresso whe some of the regressors are collear. Furthermore, they states that b ~ s ot the maxmum lkelhood estmator of the model arameter b. 3

53 CHAPTER 3 METHODOLOGY FOR GENE REGULATORY NETWORKS BY MULTIPLE LINEAR REGRESSION ANALYSIS UNDER WEIBULL DISTRIBUTION I ths chater, the algorthm of etwork detfcato by multle lear regresso develoed by Garder et al. 3 s cosdered ad t s amed to mrove ths algorthm by hadlg the ormalty assumto eeded to aly the algorthm. For the recostructo of gee regulatory etworks, NIR algorthm forms a multle lear regresso model for each gee the etwork as follows: y x x... q xq e,,.., 3. whch y u reresets the exteral erturbato to the exresso level of a artcular gee exermet,,,..,;,,...,q reresets the x exresso level wth ose of gee followg the erturbato exermet,,,...,q reresets the effect of gee o a artcular gee ad e reresets the error term. I the study of Garder et al. 3, NIR algorthm s aled to a e trascrt subetwork of the SOS athway regulatg the cell survval ad rearg after DNA damage Eschercha col. Ths subetwork s chose to clude the gees called lexa, reca, ssb, recf, d, umudc, rod, roh ad ros. It s kow that 33