Claudia Rangel! National institute of genomic medicine!

Transcription

1 Inference and Learning in Compuaional Sysems Biology Claudia Rangel! Naional insiue of genomic medicine! David Wild, Warwick Univ. U.K.! John Angus, Claremon graduae Univ., CA! Francesco Falciani, Universiy of Birmingham, U.K! Zoubin Ghahramani, Univ. of cambridge, U.K.! CIMAT May 5, 21 Gene Expression Our body is a machine regulaed by proeins which are in urn regulaed by genes Genes are found in he nucleous of every cell in our body Undersanding how genes are regulaed meaning being urned on and off is key in undersanding diseases, pahologies, even how our brain operaes Differenial gene expression 1

2 Moivaion Quesions people ask when hey do microarray experimens? High percenage of publicaions ha involve microarray experimens are designed o answer he firs 2 quesions) 1) Are genes differenially expressed? Crol vs Treamen 2) Do hey cluser ogeher? Do hey have common funcions? 3) Wha can we undersand abou he underlying genome proein regulaory neworks Possible scenarios 2

3 LDS / SSM 3

4 Daa Daa is generaed wih a high hroughpu echnology called microarrays These are capable of measure housands of genes simulaneously The echnology is expensive - abou 7 dlls per chip. Having he budge for generaing a reasonable sample size is difficul The echnology is noisy hp:// 4

5 Daa srucure: Time series 1 x 44 x 58 {,2,4,6,8,18,24,48,72} g 1 g 2 g : expression levels replicae : 1 expression levels replicae 2 25,52 daa poins replicae 44 Daa Normalizaion 5

6 Daa Normalizaion Moivaion: Common disribuion of inensiies across replicaes. Algorihm: Quanile Normalizaion [Bolsad e al.] (Based on he Q-Q plos) Biological Sysem Daa Acquisiion Pre-processing Daa Normalizaion LDS model, Hidden saes, Parameer Esimaion Use EM-Algorihm (Kalman Filer, Smoher,ec.) Re-esimae Candidae Models wih Consrains (fuure) Idenify Possible Sub models Boosrap (curren) Diagnosics Consrains Biology / Exper Opinion Finish 6

7 T cell Acivaion The cenral even in he generaion of an immune response is he acivaion of T cells. APC pepide TCR Signaling pahway Infeced cell Cyokines T cell T cell recognizes complex of viral pepide and kills infeced ce ll. T cell acivaion is iniiaed by he ineracion beween he T cell recepor (TCR) and he anigen pepide presened on he surface of an anigen -presening cell. This even riggers a cascade of evens ha couple he simulaory signal received form TCR o gene ranscripion evens in he nucleus. Why Linear Dynamical Sysems (LDS)? Linear Dynamical Sysems or Linear Sae-Space models provide a mehodology for reaing problems in ime series analysis. Mulivariae case is easily handled by simple exensions of univariae heory LDS assume he exisence of a hidden sae variable which evolves wih Markovian dynamics. Hidden variables can model The effecs of genes ha have no been included on he microarray Levels of regulaory proeins The effecs of mrna degradaion Coninuous variables Approach is based on he srucural analysis of he problem. 7

8 Hidden Saes Learning probabilisic models using hidden variables means ha we should accoun for unobserved variables ineracing wih he observables A hidden variable can induce nework srucures or subsrucures improving he accuracy of he nework By adding one or more hidden variables in he srucure can resul in a higher score Having oo many hidden variables makes he model more complex affecing he accuracy of he parameers How do we deermine he number of hidden saes? 8

9 Boosrap Cross Validaion 44-way cross validaion experimen o find he opimal number of hidden saes In general in a R-fold cross-validaion experimen, he daa se is randomly divided ino R muually exclusive subses of equal size. Daa is rained R imes, each ime leaving ou one of he subses from raining, bu using only he omied subse o compue he likelihood. Validaion se (likelihood) Training se LDS Definiion of Model Srucure h h +1 Exogenous on saes B Saes Observaions x x A +1 C y y +1 D u u +1 Exogenous on observaions x +1 = Ax +By + w x +1 = Ax + Bh + w y = Cx + Dy -1 + v y = Cx + Du + v Gene expression daa Saes x -1 A B x C y -1 D Observaions y Assumpions: { w } ~ WN(, Q) { v } ~ WN(, R), { w } { v } 9

10 Model Parameers Saes x -1 A x x +1 = Ax +By + w B C y = Cx + Dy -1 + v y -1 D Observaions y { w } ~ WN(, Q) { v } ~ WN(, R), { w } { v } A: K x K ransiion marix (K is he number of hidden saes) B: K x 58 inpu o sae marix C: 58 x K influence of hidden saes on gene expression a each ime poin D: 58 x 58 gene o gene expression level influence a a consecuive ime poins Noes: 1. We are ineresed in he CB+D marix bu ha does no involve addiional parameer esimaion. 2. K=9 General Srucural Properies 1

11 Idenifiabiliy P θ {P θ :θ Θ} P θ {P θ :θ Θ} Imporance The idenifiabiliy problem has been sudied exensively for he linear dynamic sysem model of h form x +1 = Ax +Bu + w y = Cx + Du + v Taking he unknown parameer θ o be he composie of A,B,C,D, Q,R, i is known ha wihou any resricions on he parameer, his model is no idenifiable. In fac, i is easily seen ha by a coordinae ransformaion of he sae variable x, 11

12 Anoher way of Represening he Gene Expression Model The gene expression model can be expressed in a simpler saespace form ~ x A ~ x w~ + 1 = + g = Hx ~ where, ~ x x = y ; A A = CA B CB + D ; H = [ ] ~ I ; w = Cw + v +1 w and he whie noise erm in he sae equaion now has variance ~ Q Q = CQ QC' CQC' + R Simpler form allows us o address sabiliy, conrollabiliy, observabiliy and idenifiabiliy in erms of known resuls. Definiion of Model Srucure h h +1 Exogenous on saes B Saes x x A +1 C x +1 = Ax +By + w Observaions y y +1 D u u +1 Exogenous on observaions y Saes x -1 = Cx + Dy -1 + v A x x +1 = Ax + Bh + w y = Cx + Du + v ~ x A ~ x w~ + 1 = + g = Hx ~ Gene expression daa Saes x~ A ~ x + 1 H g g +1 B C y -1 D y Observaions { w } ~ WN(, Q) { v } ~ WN(, R), { w } { v } Idenify Model Properies Observaions 12

13 13

14 Therefore, In he geneic model we have ha Bu for each ransformaion we have i is clear ha D remains idenifiable, in some sense, as i is invarian o he choice of T. By inspecion, oher invarians can be seen o include CB + D, CB, and CA k B, k = 1, 2,... Model Properies - Geneic Model Sabiliy (parameers) he sae variable does no explode exponenially - The Model will be sable iff he marix A A = CA has specral radius less han one, B CB + D Conrollabiliy (inpus) abiliy o move he sae from any given iniial value o a predeermined final value by manipulaion of he noise - The model will be conrollable iff he marix 2 K 1 [ I, A, A,..., A ] K = dim( ~ x ) is full rank, Observabiliy (oupus) abiliy o deermine he iniial sae from a sequence of noiseless observaions The model will be observable iff he marix is full rank. 2 K 1 [ H HA HA HA ] T K = dim( ~ x ) 14

15 Mehodology Expecaion Maximizaion (EM) algorihm The moivaion for using EM algorihm is ha i ieraively compues he MLE for incomplee daa ses. Filering Filering is aimed a updaing our knowledge of he sysem as each observaion y comes in Smoohing Smoohing enables us o base our esimaes of quaniies of ineres on he enire sample y 1,,y T. Boosrapping Boosrap mehods can be used for esimaing confidence bounds for nework oupus EM Algorihm x +1 = Ax +By + w w ~ N(,Q) y = Cx + Dy -1 + v v ~ N(,R) E-sep M-sep Use x, P, A, B, C, D, Q, R Use x ˆ, P Kalman Filer Smooher To Re-esimae x ˆ, P x, P, Aˆ, Bˆ, Cˆ, Dˆ, Qˆ, Rˆ Compue he expeced log likelihood given he daa By maximizing he log likelihood 15

16 Kalman Filering & Smoohing The likelihood can be calculaed by a rouine applicaion of he Kalman filer, considered he opimal linear esimaor. The Kalman filer esimaes he curren value of our variables incorporaing all informaion available. Knowledge of he sysem The saisical descripion of any uncerainy of he dynamics of he model Noises and measuremen errors Iniial condiions The Smooher solves he problem of esimaing he sae a ime given he parameers and he observaions. Boosrapping 16

17 Usamos resulados del Boosrapping Usamos resulados del Boosrapping 17

18 Usamos resulados del Boosrapping Resuls Simulaed Daa : 4 samples, 1 ime poins, 5 genes

19 Resuls Simulaed Daa: 5 and 11 nodes 11 genes (nodes) 19

20 39 Nodes Arificial ime series are no saionary for a few ime poins sample -> bias Diagnosics on Fied Model Common Mehods Examinaion of sandardized innovaions for lack of correlaion / paern Check ha esimaes of A, B, C, D are in he observable, conrollable, sable region of he parameer space: 2

21 Il-2 Main cellular funcions modulaed during T cell acivaion Acivaion (1) Proliferaion (2) Gene Regulaory Nework Aciv. IL-2Rϒ, IL-4Rα, IL-3Rα Proliferaion gene: Cyclin A2 Il-2 Cell deah (3) Acivaion (1) Proliferaion (2) Main cellular funcions modulaed during T cell acivaion 21

22 Gene Regulaory Nework IL-2Rϒ, IL-4Rα, IL-3Rα Proliferaion gene: Cyclin A2 Apoposis response Il-2 gene: Clusering Cell deah (3) Acivaion (1) Proliferaion (2) Main cellular funcions modulaed during T cell acivaion Gene Regulaory Nework IL-2Rϒ, IL-4Rα, IL-3Rα Proliferaion gene: Cyclin A2 Apoposis response gene: Clusering Early T-cell acivaion marker: CD69 (acivación ardía) Il-2 Cell deah (3) Acivaion (1) Proliferaion (2) Main cellular funcions modulaed during T cell acivaion 22

23 Gene Regulaory Nework IL-2Rϒ, IL-4Rα, IL-3Rα Proliferaion gene: Cyclin A2 Apoposis response gene: Clusering Early T-cell acivaion marker: CD69 TF involved in T-cell anigen reg: GATA Il-2 Cell deah (3) Acivaion (1) Proliferaion (2) Main cellular funcions modulaed during T cell acivaion Gene Regulaory Nework No presen in he microarray and i is considered hidden variable in ACTIVATION Il-2 Cell deah (3) Acivaion (1) Proliferaion (2) Main cellular funcions modulaed during T cell acivaion 23

24 Gene Regulaory Nework TCR phosphoriles FYN Targe gene of NFKB is IL-2 TCR NFKB Il-2 Main cellular funcions modulaed during T cell acivaion Cell deah (3) Acivaion (1) Proliferaion (2) Follow-up Research VBSSM Variaional Bayesian Sae-Space Model Synheic Daa Genome Research Dirk Husmeier Consrains Learning and Inference in Compuaional Biology MIT press

25 Incorporaing Biological Knowledge, Knocking Ou: Implemening Consrains By hinking of each elemen in D as he connecion srengh wih which gene i influences gene j over ime, allows he marix D o be consrained o have zero values where here is no connecion beween wo genes. Two ypes of consrains on D of he form DF=G (*) Fvec(D) = G (**) Consrained model address he esimaion of fewer number of parameers Implemened by Lagrange mulipliers by doing consrained maximizaion in he M sep. * Shumway and Soffer 1982 ** New resul 25