A METHOD FOR ANALYSING GENE EXPRESSION DATA TEMPORAL SEQUENCE USING PROBABALISTIC BOOLEAN NETWORKS

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1 4th European Sgnal Processng Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyrght by EURASIP A METHOD FOR AALYSIG GEE EXPRESSIO DATA TEMPORAL SEQUECE USIG PROBABALISTIC BOOLEA ETWORKS Stephen Marshall, Le Yu Department of Electronc and Electrcal Engneerng, Unversty of Strathclyde Royal College Buldng, 204 George Street, G XW, Glasgow, Unted Kngdom phone: + (44)448299, fax: + (44)422487, emal: l.yu@eee.strath.ac.uk s.marshall@eee.strath.ac.uk web: ABSTRACT Ths paper descrbes a new method for analysng gene expresson temporal data sequences usng Probablstc Boolean etworks. Swtch-lke phenomena wthn bologcal systems result n dffculty n the modellng of gene regulatory networks. To tackle ths problem, we propose an approach based on so called purty functons to partton the data sequence nto sectons each correspondng to a sngle model wth fxed parameters, and ntroduce a method based on reverse engneerng for the dentfcaton of predctor genes and functons. Furthermore, based on the analyss of Macrophage gene regulaton n the nterferon pathway, we develop a new model extendng the PB concept for the nference of gene regulatory networks from gene expresson tme-course data under dfferent bologcal condtons. In conuncton wth ths, a new approach based on constraned predcton and Coeffcent of Determnaton to dentfy the model from real expresson data s presented n the paper.. ITRODUCTIO In recent years bologcal mcroarray technology has emerged as a hgh-throughput data acquston tool n bonformatcs. It enables the measurement of the expresson level of thousands of genes smultaneously n a cell at a seres of tme ponts n a specfc bologcal process []. It has led to a dramatc revoluton n the feld of systems bology so that computatonal methods for modellng and smulatng gene regulatory networks have been developed to study gene regulaton. Ths has consttuted a key aspect of genomc sgnal processng [2]. Recently, a modellng approach defned as Probablstc Boolean etworks (PBs) has been proposed for modellng gene regulatory networks [3]. Ths technology, an extenson of Boolean etworks [4], s able to capture the tme-varyng determnstc dependences as well as uncertantes relatve to the governng model by a seres of logc based predctor functons. Cells mantan ther phenotype stablty untl the phenotype s swtched n response to an external stmulus. Under the stmulus, there are swtch-lke transtons observed wthn bologcal systems. Therefore, Context-Senstve PB modellng was proposed for nferrng genetc regulatory networks n some gene expresson data wth dfferent contexts for the cell []. The Context-Senstve PB model s a collecton of Boolean etworks (Bs) wth fxed Boolean functons set n a tme-course. Our modellng approach s an extenson of ths concept. For the constructon of a model of gene regulatory behavour for data under dfferent bologcal condtons, t s essental to be able to partton the data nto sectons correspondng to dfferent contexts of the underlyng model. Therefore, we frstly proposed a method for parttonng the sample gene expresson data wth mult-context nto dfferent segments over a tme horzon. Followng partton, the ndvdual consttuent models may be ftted to each parttoned secton respectvely. A method s proposed by whch the parameters of a PB may be nferred drectly from temporal gene expresson data. In ths paper we demonstrate the procedure by reverse engneerng the process and recoverng all the complextes of the generatng model. After that, we apply modellng technques to macrophage gene expresson data under three dfferent bologcal condtons. The results reveal that there exsts a phase-changng phenomenon on the regulator actvtes caused by underlyng mechansms or external nputs. To capture such phenomenon, we have extended the Context-Senstve PB concept to allow the network selecton probablty to vary wth bologcal condtons. Fnally, we have developed an approach usng the Coeffcent of Determnaton (COD) [6] for the nference of the model from the small amounts of expermental data. Our experments are carred out on two knds of data. Frstly, we verfy our technque by conductng experments on a synthetc temporal gene expresson data sequence generated by Genomc Sgnal Processng Laboratory based at Texas A&M Unversty. Accordng to the results of the analyss on these smulated data, we then developed a further experment usng real gene expresson data suppled by the Scottsh Centre for Genomc Technology and Informatcs (GTI), based at Unversty of Ednburgh. The paper s organzed as follows. In Secton 2, a descrpton of an experment on smulated data s provded. In secton 3, we present the experment on real data. Secton 4 contans some concludng remarks. 2. VERIFICATIO WITH SIMULATED DATA The smulated data s modelled by a PB [3] consstng of M Boolean etworks (Bs) B B M. There are n genes

2 4th European Sgnal Processng Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyrght by EURASIP g,, g n, the regulatory actvtes of whch are controlled by a set of Boolean functons of k genes, where k<n. The data s modelled through a sequental process whereby the state of the model at tme t s determned by ts prevous state at tme t-. It s a dscrete-tme stochastc process wth the Markov property. The model parameters, that s the specfc k genes whch control each gene and the set of functons used to determne ther next states are fxed for substantal perods of tme. Under external stmul or hdden mechansms, perodcally a swtch pont occurs where both the functon and the k predctor genes governng each gene change so that gene actvty swtches to another B and therefore a dfferent regulatory rule s adopted. The frst step n the process s to partton the tme course data nto so called pure segments by fndng these swtch ponts. A pure segment corresponds to data generated from a sngle B. 2. Parttonng Data The mportant pont to note here s that the characterstcs of a B may be determned by the parwse transtons of the network. For a B wth fxed parameters the state of the network S t at tme t, must always be followed by the same state, S t+, at tme t+ except when the transton s perturbed by mutaton or nose. Therefore a state transton matrx compled over a pure temporal data sequence and showng whch state S t+, at tme t+ tme follows S t at tme t, wll be sparse contanng only one sgnfcant entry per lne plus some entres due to nose. Typcally the (=2 n ) transton matrx Y(t,t2)would have the overall structure shown below, Current State =S t ext State =S t+ a, a2,... a,... a a2, a22,... a2,... a2... a, a2,... a,... a... a, a 2... a... a where a ndcates the number of state transtons from state to state durng a tme horzon.e. between tme step t and tme step t2. State, are n a fnte state space {, 2, 3,,2 n }. The transtons of the state n the system perodcally cycle n between several attractors. Therefore, the maorty of transtons wll be zero and most non zero values wll correspond to transtons a drven by the model, the remander wll be due to perturbaton or nose. Identfcaton of one sgnfcant transton per state results n a one to one mappng lnkng pars of successve gene states. S = (S ) Assumng that the system conssts of n genes, S=(g, g 2..g n ) then ts state at tme t +, may be wrtten as a functon of a subset of k genes, where k < n. S t+ =(g, g 2..g n ) t + () The state of each gene g at tme t + s a functon of the genes at tme t. More specfcally t s a functon of a subset of k genes S t+ =(F ((g,g 2 g n ) t ),F 2 ((g, g 2 g n ) t ), F n ((g, g 2..g n ) t )) t + Gven suffcent data, the functons and the subset k may be determned by Boolean reducton. The obectve s to dentfy ponts wthn the sequence where the model parameters change and between whch the data s pure,.e. derved from a sngle B wth fxed parameters. We now seek to defne a purty measure whch determnes the lkelhood that the data lyng between two data ponts s pure. The measure proposed s S t t = = (, 2)) (2) [ P Q ] λ ( (3) a where P s the value of the largest (maor) value on each row n column, Q s the value of the second peak and a s the value of the transton matrx. Consder a data sequence wth T ponts generated from two Bs. The frst secton of the sequence, from to t (t [,T]) has been generated from the frst B and the remander of the sequence from t + to T has been generated from the second B. Suppose there s one swtchng pont, the unknown swtchover pont s determned by splttng the sequence nto two parts at pont z. The value of z s a varable whch may take any value from to T and t s an estmate of the value of t. z s vared and the data set before and after t are mapped nto two dfferent transton matrxes W(z)and V(z). W(z)=Y(,z), V(t)=Y(z+,T). The defnton of Matrx Y has been shown n the left. The deal purty factor s one whch s maxmsed for both W(z) and V(z) when z= t. Ths s shown n Fgure. Where a data set conssts of more than two pure sequences then t must be parttoned teratvely n order to determne the set of pure sequences. Once the sequence has been successfully parttoned nto pure segments then the process of dentfyng the functons and nputs can be carred out by Boolean reducton. Tme step W(z) z V(z) Fg. Data sequence dvded by a sldng pont t and Transton matrces produced for data on each sde of the partton T

3 4th European Sgnal Processng Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyrght by EURASIP 2.2 Identfcaton of Predctor Genes and Functons Havng parttoned the gene state data nto pure subsequences, t s now necessary to dentfy whch subset, k of the n genes may best predct any gven target gene. Ths was carred out by dentfyng the k genes whch best correlate wth changes n the predcted gene. It s based on a functon whch s mnmzed when the changes n the k genes specfed most accurately concde wth changes n the predcted gene aggregated over each pure data subsequence. In the pure subsequence, the next state of gene s a functon of k genes, = f ( (), (2),., ) (4) where are selecton functons determnng whch k from n genes are used as nputs to the predctor functon. Accordng to the state transton matrx for each segmented pure subsequence of data, t s possble to create a currentnext state table for each pure subsequence. If the states are wrtten n terms of ther ndvdual genes then a smple one to one mappng s produced from n genes to n genes. Ths was set out n Eqn () and (2). In the current-next state table (Table ), each lne presents states of genes correspondng to gene actvty profles n the real measurements. It s clear that the next state of any gene may be wrtten as a functon of all n predctor genes. The problem s to determne whch subset, k out of the n genes may best predct any gven gene. A cost functon R s defned whch s mnmzed when all the output gene values are the same for the same combnaton of k predctor genes. = R ( k) r g ( k) g k The cost functon R conssts of rg k summed over every combnaton of the k predctor genes, where rg k s defned below. f n / 2 r ( k) = (6) g k n f > n / 2 k The value of s ether 0 or as specfed n the truth table. Hence the quantty rg k s mnmzed f the outputs for a partcular combnaton of nputs g k are ether all 0 or all. There are n!/(n-k)!k! ways of selectng k from n nputs. For small numbers of genes the k nputs may be chosen by full search and the cost functon evaluated for every combnaton. For larger sets they are chosen through genetc algorthms to mnmze the cost functon. In some cases the current-next state table s not fully defned. Ths means that the output s tself a don t care term. Tests have shown that the predctor genes may stll be dentfed correctly even for 87.3% of mssng data [7]. Once the subset of k predctor genes has been dentfed, the task of dentfcaton of predctor functons s a straghtforward exercse n logc mnmzaton [8]; made easer by the fact that k s small. 2.3 Experment Results Experments were conducted on synthetc tme-course data, generated at Texas A&M Unversty. The PB conssted of 4 Bs wth n=8 and k 4. The data was processed usng the method descrbed. An example of the purty functon value derved over a secton of the data and clearly contanng swtch ponts n s gven n Fgure 2. In ths way the data was parttoned nto pure segments. The segments were analyzed to determne the k predctor genes and correspondng functons. Table 2 shows the resultng predctor genes and functons derved from the data for B3 as compared to the actual functons. These were derved very accurately despte the fact that only 23% of the transtons for B3 were observed n the data. Swtch Ponts Fg.2. Purty functon values for a 200,000 secton of the data. Gene Actual Predctor genes,2,6,8,2,2,7,8,3,,6 3,4 3,4,8 B3 (sze 4766) Observed transton state: 8 (23%) Predcted Actual Functons Predcted Functons Predctor Genes,2,6,8,2,2,7,8,3,,6 3,4 3,4, ** *000** *000* Table Current-ext state table Table 2. The predctor genes and predctor functons n B3

4 4th European Sgnal Processng Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyrght by EURASIP 3. EXPERIMETS O REAL DATA 3. Data Analyss and Model Defnton In the experments on real data, the gene expresson data are taken from a hybrdzatons mcroarray study, n whch bone marrow derved macrophages were exposed to three dfferent bologcal condtons over 2 hours, wth measurements made n 30 mnute ntervals. The three condtons were nterferon treatment only (IFg), vral nfecton wth nterferon treatment (C3X_IFg) and vral nfecton only (C3X). All target genes and predctor genes come from a functonal group contanng target genes n the data (Il2b, Cybb, Gp2, Itgam, Fcer2a). Fgure 3 shows how the regulatory actvtes of the genes vary under the 3 dfferent bologcal condtons. We found that most gene actvtes n nterferon gamma treated samples were consstent wth the consensus pathway (for example, Fcer2a=Irf4 stat6.e. target gene Fcer2a s affected by the addng of Irf4 and stat6). However, ths was less so for gene actvtes n vral nfecton wth nterferon treatment and nterferon treatment wthout nfecton. Table 3 shows the proportons of whch gene actvtes ft the known bologcal pathway from samples n 3 bologcal condtons respectvely. It s clear that ther proportons vary correspondng to the dfferent bologcal condtons. The results on the analyss of macrophage gene expresson data under three dfferent bologcal condtons show that nterferon treatment establshes the cognate pathway connectons whle nfecton leads to a lmted engagement of the regulatory network. It revealed that there s a phase-changng phenomenon on the regulator actvtes caused by underlyng mechansms or external nputs. To capture such phenomenon, we defne a dynamc model whch extends the PB concept to allow network selecton probablty to vary wth bologcal condtons. Fg.3. ormalzed log-ratos (n log2(test/control)) of gene (Il2b) under 3 bologcal condtons. Target genes and logc predctor functons % of gene actvtes fttng known bologcal pathway IFg C3X_IFg C3X Fcer2a (Fcer2a=Irf4 stat6) 84% 80% 2% Itgam (Itgam=Irf8) 88% 2% 36% Il2b (Il2b=(Irf8 Irf) Irf2) 84% 64% 2% Gp2 (Gp2=(Irf8 Sfp- 76% 68% 6% Irf4 Irf2) Irf) Cybb (Cybb=Irf8 Sfp Irf Crebbp) 92% 84% 64% Table 3. The proportons of gene actvtes fttng the known bologcal pathway from samples n 3 bologcal condtons The proposed model conssts of a number of PBs and the system behavour swtches between these on a stochastc bass. The PBs nferred from the data n 3 dfferent bologcal condtons have the same structure but dfferent parameters (the functon selectng probablty). 3.2 Model Identfcaton ext we present an approach based on reverse engneerng for the nference of ths model from the data. To smulate the real condton, a small number of synthetc data from the supposed model under 3 bologcal condtons wll be used for the model dentfcaton. The number of target genes s set as 7. In the experment, because of the lmted amount of data, the predctor functons are constraned to fall nto the class of functons known as Canalzng functons [4]. The maxmum network connecton s defned as k=3. By analyzng the data and employng logc reducton technques from dgtal electroncs the predctor functons and the predctor genes for each target gene may be recovered. After that, we select as the predctors, the k genes whch best correlate wth the data for each of the 3 bologcal condtons. In some cases, several genes ft equally well. Table 4 shows the domnant predctor genes nferred from the data n the 3 bologcal condtons respectvely. The numbers of the entry n the table represent the ndces of the genes. After that, the selecton probabltes of predctor functons a m () ( s the ndex of target gene, s the ndex of the predctor functon, m s the ndex of the bologcal condton) n the 3 bologcal condtons wll be determned. Because of the lmted amount of data avalable, a method based on the Coeffcent of Determnaton (COD) [6] s used. The coeffcent measures the degree to whch the transcrptonal levels of an observed gene set can be used to mprove the predcton of the transcrptonal state of a target gene relatve to the best possble predcton n the absence of observatons. The ndex of genes Predctor genes of the consttuent predctor functons Bologcal Condton Bologcal Condton2 Bologcal Condton3 g,6,7,3, 3,,7 g 2,3,6 2,3,,2, g 3,2,7 2,3,7 2,3,7,2,,2,3 g 4,4, 4,6,7,4, 4,,7 3,4,7 2,4,7,4,7 g 4,,6 3,4,,,6 g 6 4,6,7,6,7 2,,6,3,7,3,6,2,7 g 7 4,6,7,4,6 3,4,7 3,6,7 Table 4. The predctor genes nferred from samples n 3 bologcal condtons.

5 4th European Sgnal Processng Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyrght by EURASIP Defne g as the target gene, the predctor genes come from a set of genes g, g 2, g n. For g, the COD of a selected predctor gene set g () s gven as θ = ( ε ε ) ε () where ε s denoted as the error of best estmaton for g. opt ε s the optmal mnmum error. From the tranng data n all 3 bologcal condtons, d (), whch s the selecton probablty of a certan predctor functon f () to target gene g can be nferred from COD values wth the range of 0 to by the () ( ) followng expresson [3] d = θ θ (2) opt k k = where () s the number of possble predctor functons for target gene g. Smlarly, a () m, the selecton probablty of the predctor functon under a certan bologcal condton can be nferred from the data. We defne c () m as the context selecton probablty for the functon sets n the 3 bologcal condtons. d () can be represented as () M d = c () () m a m m= M and c () m = m= where M s the number of the bologcal condtons (M=3 n our experment). Therefore, c m () can be obtaned from equaton (3). We defne mean square error (MSE) w m () as the cost functon of w m () = E [ (d () - d () ) 2 ] (4) where d () s the value of equaton (3) gven a seres of c m (). Once w m () dsplays the mnmum value, the optmal realzaton of c m () wll have been found. 3.3 Experment Results State at tme t State at tme t+ Fg.4. An example of the selected predctor functons under 3 bologcal condtons for g 2 f f 2 73% 4% 3% Bologcal Condton f g f 2 g f 2 g 24% 68% Fg.. An example of a m (), the selecton probablty of predctor functons under 3 bologcal condtons for g. 8% Bologcal Condton2 f % 22% 27% Bologcal Condton3 (3) Fgure 4 shows the consttuent predctor functons whch are n the class of canalzng functons for the target gene g 2. Fgure shows the selecton probabltes o predctor functons for the target gene g. The probabltes dsplay qute dfferent values for the dfferent bologcal condtons. In ths model, the structure of the consttuent Boolean networks remans stable, whereas the network selecton probabltes change n response to the dfferent bologcal condtons. It mples a fxed fnte dscrete state space but dfferent statonary dstrbutons. Ths model keeps the rule-based structure whle allowng for uncertanty. It s sutable for the nference of the gene regulatory network from gene expresson data n dfferent bologcal condtons. 4. COCLUSIO In ths paper, a method of fttng multple Boolean etworks to tme course gene expresson profles has been successfully appled. The parameters of the models used to generate the data have been accurately recovered despte the fact that only a small percentage of all possble transtons have been observed from smulated data. We also proposed a novel approach to model the gene regulator actvtes under dfferent bologcal condtons. A method dentfyng the model parameters from tme course gene expresson profles has been successfully appled. All these results are appled n the context of pathway bology to the analyss of an nterferon gene nteracton network. ACKOWLEDGEMET We would lke to acknowledge, Yufe Xao and Edward R. Dougherty at Genomc Sgnal Processng Laboratory, Texas A&M Unversty, Thorsten Forster and P. Ghazal at Scottsh Centre for Genomc Technology and Informatcs, Unversty of Ednburgh for ther support and help n ths research. REFERECES [] DJ Duggan, M Btter, Y Chen, P Meltzer and JM. Trent (999) Expresson Proflng usng cda Mcroarrays. atural Genetcs. 2(Sup), 0-4 [2] ER Dougherty, A Datta, C Sma, (200) Research Issues n Genomc Sgnal Processng. IEEE Sgnal Processng Magazne. ov, [3] I Shmulevch, ER Dougherty, S Km, W Zhang, (2002a) Probablstc Boolean etworks: A Rule-Based Uncertanty Model for Gene Regulatory etworks. Bonformatcs. 8(2) [4] SA Kauffman, (993) The Orgns of Order: Self-organzaton and Selecton n Evoluton. ew York: Oxford Unversty Press. [] ER Dougherty, A Datta, (200) Genomc Sgnal Processng: Dagnoss and Therapy. IEEE Sgnal Processng Magazne. Jan, [6] ER Dougherty, S Km, Y Chen, (2000) Coeffcent of Determnaton n onlnear Sgnal Processng. Sgnal Proc 80, [7] S Marshall, L Yu, Y Xao, ER Dougherty (2006), Inference of a Probablstc Boolean etwork from a Sngle Observed Temporal Sequence, submtted EURASIP Bo Journal n July [8] CH Roth, (99) Fundamentals of Logc Desgn, 4 th edton, Brooks Cole, 99.

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