Analysis of LC-MS Data Using Probabilistic-Based Mixture Regression Models

Transcription

1 at 9/2009 Analyss of LC-MS Data Usng Probablstc-Based Mxture Regresson Models Analyse von LC-MS-Daten mt wahrschenlchetsbaserter Mschung von Regressonsmodellen Habtom W. Ressom, Getachew K. Befeadu, Mahlet G. Tadesse, Georgetown Unversty, Washngton DC Dedcated to Prof. Dr.-Ing. habl. Lothar Ltz on hs 60 th brthday Summary A novel framewor of a probablstc-based mxture regresson model (PMRM) s presented for algnment of multple lqud chromatography-mass spectrometry (LC-MS) data wth respect to retenton tme (RT) and mass-to-charge rato (m/z). The expectaton maxmzaton algorthm s used to estmate the jont parameters of splne-based mxture regresson models and pror transformaton densty models. The applcablty of PMRM for algnment of LC-MS data s llustrated through three datasets. The performance of our method s compared wth other approaches ncludng dynamc tme warpng, correlaton optmzed warpng, and contnuous profle model n terms of coeffcent varaton of replcate LC-MS data and accuracy n detectng dfferentally abundant peptdes/protens. Zusammenfassung Baserend auf Regressonsmodellen (probablstc-based mxture regresson models, PMRM) wrd en neues Verfahren zur Analyse von Daten aus der Flüssgchromatographe-Massenspetrometre (LC-MS) vorgestellt. Das Verfahren ann für de Anordnung der LC-MS-Daten m Hnblc auf de Retentonszet (RT) und de Masse zu Ladung (m/z) verwendet werden. Herzu wrd en Erwartungswert-Maxmerungsalgorthmus zur Schätzung der gemensamen Parameter der Splne-baserten Mschung von Regressonsmodellen und Dchten engesetzt. De Anwendbaret der Methode wrd mt dre LC-MS-Datensätzen demonstrert. De Lestung unserer Methode wrd mt anderen Ansätzen (d. h. dynamc tme warpng, correlaton optmzed warpng, und contnuous profle model) m Hnblc auf de Koeffzentenveränderung der mehrfachen LC-MS-Daten und auf de Identfatonsgenauget der dfferentell rechlchen Peptde/Protene verglchen. Keywords Expectaton-maxmzaton, lqud chromatography, mass spectrometer, mxed-regresson model Schlagwörter Erwartungswert-Maxmerungsalgorthmus, Flüssgchromatographe, Massenspetrometre, Regressonsmodelle 1 Introducton In proteomc studes, lqud chromatography coupled wth mass spectrometry (LC-MS) s a common platform to dentfy and determne the abundance of varous peptdes that characterze partcular protens n bologcal samples [1; 2]. A mass spectrometry (MS) taes a sample as nput and produces a measure of abundance of molecules that have partcular mass-to-charge (m/z) values. In mass spectrometry-based proteomcs, the molecules n queston are peptdes (short amno acds sequences), or sometmes whole protens. For proten mxtures that are very complex, such as serum, a separaton step s often used to physcally separate parts of the sample before njecton nto the mass spectrometry. Ths separaton, often conducted by a lqud chromatography (LC), means that a less complex mxture s fed nto the mass spectrometry at any one tme. The LC step spreads out parts of the sample soluton over tme (also called retenton tme) on the bass of some propertes of the molecules. For example, n reverse phase hgh perat Automatserungstechn 57 (2009) 9 / DOI /auto Oldenbourg Wssenschaftsverlag 453

2 454 formance lqud chromatography (RP-HPLC), a tubular column s paced wth some materal made up of hydrophobc molecules. The sample s then loaded onto the paced column, and sample molecules bnd to the paced materal wth dfferent affntes dependng on each molecule s hydrophobc propertes. The soluton s then pumped through the column. Intally, the soluton s mostly water, and s ncreasngly made to contan more and more organc solvent. As ths organc solvent gradent s appled, molecules that are more wealy bound to the paced materal come off frst, and progressvely those that are more and more strongly bound come off. As lqud comes off the column, t accumulates nto a droplet, whch s then vaporzed and onzed as t s ntroduced nto the mass spectrometer. The advantage of usng LC- MS s that at each RT pont a less complex mxture (less complex than f we had njected the whole sample at once nto the MS) s used to obtan mass spectral measurements. Peptdes wth the same m/z values are less lely to analyzed by MS at the same tme, thus reducng potental ambguty s needed. Also, the fewer the number of ons beng smultaneously analyzed by MS at a tme, the less on suppresson comes n to play. In a typcal LC-MS based experment, frst mxtures of protens are solated from bologcal samples and then dgested nto peptdes usng enzymes. Followng ths, the peptdes are separated by one or more steps of hghpressure LC, and are eluted nto an electro-spray on source, where they are nebulzed n small and hghly charged droplets. The output of an LC-MS experment conssts of three dmensons: (1) the eluton tme, also called retenton (RT) pont, (2) the m/z value, and (3) the ntensty (on abundance). Fgure 1a presents three-dmensonal data derved from a typcal LC-MS experment for a sngle run. As shown n the fgure, each LC-MS run generates spectra comprsed of thousands of pea ntenstes for peptdes wth specfc RT and m/z values. Fgure 1b shows a mass spectrum (on abundance vs. m/z) at a partcular RT pont (RT n the fgure s 10 mnutes). Fgure 1c depcts the total on count (TIC) obtaned by calculatng the sum of the on abundances across the m/z dmenson for each RT pont. Although RT s a contnuous varable, the LC-MS system produces mass spectra at a dscrete set of RT ponts, usually a few seconds apart. It s typcal to represent RT ponts by scan ndces, snce there s a one-to-one correspondence between RT ponts and total MS scan numbers. In dfferental proten expresson studes, multple LC-MS runs are compared to dentfy dfferentally abundant peptdes between dstnct bologcal groups. Ths s a challengng tas because of the followng reasons: (1) substantal varaton n RT across multple runs due to the LC nstrument condtons and the varable complexty of peptde mxtures, (2) varaton n m/z values due to occasonal drft n the calbraton of the mass spectrometry nstrument, and (3) varaton n pea ntenstes due to spray condtons (n most cases ths s proportonal to concentraton of peptdes n the sample). Thus, effcent and robust algnment algorthms are needed for quanttatve comparson of multple LC-MS runs. For example, Fg. 2a presents a heatmap of TIC profles representng multple samples. As llustrated n the fgure, there s substantal drft n the RT dmenson that needs to be algned pror to selectng dfferentally abundant ons. Fgure 2b depcts a heatmap of the TIC profles after algnment. Varous methods for algnment of LC-MS data have been descrbed n lterature ncludng dynamc tme warpng (DTW) [3], correlaton optmzed warpng (COW) [3], vectorzed peas [4], (sem-) supervsed algnment usng nonlnear regresson methods [5], statstcal algnment [6], and clusterng [7]. Most of these algorthms are ether lmted to a consensus par-wse combnaton of LC-MS runs or use a reference (template) for algnment. These lmtatons may lead to sub-optmal results compared to global algnment technques. Methods that rely on optmzaton of global fttng functons provde an alternatve soluton to algnment of multple LC-MS runs representng dstnct bologcal groups. For example, a recently ntroduced method called contnuous profle model (CPM) has been appled for algnment of contnuous tme-seres data and for detecton of dfferences n multple LC-MS data [8; 9]. Although CPM s descrbed as a naïve and computatonally ntensve method, the method has notable lmtatons. For example, t creates superfluous sgnal gaps, leadng to non-unform trace ponts across multple LC-MS runs. Also, t s susceptble to fall nto local mnmum solutons due to the sub-optmal problem formulaton. Another lmtaton of CPM algorthm s ts poor performance wth tme complexty scales, requrng substantal computaton tme n modelng hgh resoluton data. Thus, CPM s more sutable for low resoluton of LC-MS data generated from less complex fractonatons. Recently, Morrs et al. developed a Bayesan method for analyss of matrx-asssted laser desorpton onzaton-tme of flght (MALDI-TOF) proteomc data [10]. Ther motvaton extends from earler wor on Bayesan mplementaton of the wavelet-based functonal mxed effects model ntroducedby Morrs and Carroll [11]. The approach s smlar to the splne-based functonal mxed effects model ntroduced by Guo [12], whch nvolves generalzed mxed model equatons to handle potentally rregular data. The method specfcally deals wth the dentfcaton of dfferentally expressed spectral regons across dfferent expermental condtons assumng the algnment ssue has already been taen care of. In ths paper, we propose a probablstc-based mxture regresson model (PMRM) for algnment of LC-MS data. Important features of ths framewor nclude the followng: (1) t s not confned to landmar/feature algnment, (2) t allows contnuous tme algnment, (3) t utlzes a probablstc-based functonal modelng to deal wth potentally rregular data, and (4) t s amenable to model both low and hgh resoluton mass spectra, snce t does

3 Analyss of LC-MS Data Usng Probablstc-Based... not ntroduce any superfluous sgnal gap across multple LC-MS runs. Moreover, the framewor lends tself to an expectaton-maxmzaton (EM) algorthm wth the followng features: () the explct use of transformaton prors for modelng the on abundance (pea ntensty) Fgure 1 LC-MS data obtaned from a typcal LC-MS experment. varablty n both RT and m/z dmensons of the data, () the use of a probablstc metrc that allows estmaton of the dstance among multple spectra nstead of computng par-wse dstances, and () the ablty to extend the method for algnment of LC-MS data nvolvng 455

4 456 multple groups. In addton to the need for algnment of LC-MS data, one may also need to correct for systematc dfferences n the ampltude of the sgnal (on abundaces) at each RT pont and m/z value whch s called the problem of normalzaton for multple LC-MS data. Correctng such systematc varatons n the ampltude of LC-MS data at each RT pont and m/z value s not the focus of ths paper, although the two can be ntertwned,. e., one may perform algnment after normalzaton or the other way. Such normalzaton strategy could be addressed usng ether statstcal models [13; 14] whchwere orgnally developed for gene expresson data analyss or by usng multple nternal standard compounds based on some emprcal rules [15]. The remander of the paper s organzed as follows. In Sect. 2, we outlne the probablstc-based mxture regresson model for algnment of LC-MS data. Ths secton also explans the estmaton algorthm for fndng the maxmum lelhood parameters of the splne-based mxture regresson models and the pror transformaton Fgure 2 Heat maps of TIC profles before (a) and after algnment (b). densty models used for modelng the varablty n RT, m/z, and measurement space (pea ntenstes of peptdes) of the LC-MS data. Secton 3 brefly descrbes the datasets used n ths paper. Results and dscusson llustratng the applcablty of our approach for analyss of multple LC-MS data are gven n Sect. 4. We fnally summarze our fndngs n Sect Methods 2.1 Probablstc-Based Mxture Regresson Models We propose a novel probablstc-based mxture regresson model for analyss of LC-MS data. A partcular advantage of ths approach s ts ablty to model non- Gaussan mult-modal densty functons usng smpler component densty functons that can be defned on non-vector data such as LC-MS data. We assume that the observed dataset D representng multple groups s generated wth the followng features: An ndvdual s randomly drawn from a populaton of M objects (. e., the dataset D).

5 Analyss of LC-MS Data Usng Probablstc-Based... The ndvdual s assgned to the -th group wth probablty α, where K =1 α = 1. These are the pror weghts correspondng to all K groups, where K M. For an ndvdual where { -th group} there s a densty functon p (y θ ) that generates the observed functonal data y. From the above, t follows that the observed densty on the y s s a mxture model,. e., a convex combnaton of component models p(y θ) = K =1 α p (y θ ). Thus, we estmate the most lely values for the parameters θ and α usng the assumed functonal denstes p (. ) on the observed data y s. Ths s accomplshed by usng the EM algorthm whch s a general procedure for fndng the maxmum-lelhood estmaton of the parameters from the mxture models [16 18]. Thus, ths probablstc based framewor allows us to fnd the best group algnment n tme and measurement spaces from the observed dataset D. In the followng, we consder M LC-MS runs Y = {y } M =1 (. e., observed dataset D), where the -th run y has a length of n, correspondng to the observaton ponts (or tme ponts) x. To defne the mxture regresson model, each observaton y s expressed as a functon of some nown x,. e., each component model s represented by a condtonal densty functon of the form p (y x, θ ). We assume a standard functonal regresson model between y and x as: y = ψ (x, β )+e (1) where e s a zero-mean Gaussan wth varance σ 2 I,. e., e N(0, σ 2I)andψ (.,. ) s are mappng functons of x. Here, θ ncludes both the parameters of the mappng model ψ (x, β ),. e., the regresson coeffcents β and the nose varances σ 2. Thus, the probablty of observng y,gvenx and component model, sassumedtobe a condtonal regresson model. We defne the condtonal densty of the observed data p (y x, θ ) as a mxture densty condtoned on the regresson model parameters as well as on x values: p(y x, θ) = K α p (y =1 x, θ ) (2) where p (y x, θ ) s are the mxture components, α s are the mxng weghts, θ s the parameter set for the component, andθ = {θ } K =1 = {β, σ 2}K =1 s the complete (global) parameter set for the mxture model n Eq. (2). Condtonal ndependence between the functonal data representng multple groups allows the full jont densty to be wrtten as: p(y X, θ) = M K α p (y =1 =1 x, θ ) (3) Thus, the log-lelhood of θ for the gven dataset D can be obtaned drectly from Eq. (3) as follows: l(θ D) =logp(y X, θ) = M log K α p (y =1 =1 x, θ ) (4) 2.2 Expectaton-Maxmzaton Estmaton for the Mxture Regresson Models The goal s to pull out the mxture of components from the full jont densty, usng the observed dataset D as a gude, so that the underlyng group behavor can be nferred. A standard approach to deal wth the hdden data s to utlze the EM algorthm for consstent estmaton. Introducng hdden data, whch correspond to the unnown group membershp of the M functonal data, s more approprate n the log-lelhood formulaton of Eq. (4). Wth ths, let Z be a matrx of reportng vectors z = (z 1, z 2,, z K ) such that z = 1forsome and z t = 0, t =. Ifz = 1, then the -th spectrum s generated from the -th mxture component. Thus, the jont densty of Y and Z gven X can be rewrtten as follows: p(y, Z X, θ) = M p(y =1 z, x, θ) p(z ) (5) Ths follows from the condtonal ndependence assumpton on Y and z s. Thus, the complete-data log-lelhood follows from Eq. (5) as: l(θ/d, Z) =logp(y, Z X, θ) K = M z log α (6) =1 =1 + M K z log p(y =1 =1 x, θ ) To ft the dataset D nto ths framewor for the splnebased mxture regresson model,. e., the regresson model between y and x defned n Eq. (1), s rewrtten as follows: y = B β + e, e N(0, σ 2 I) (7) where B denotes splne-bass matrx evaluated at x. Then, the group specfc condtonal probablty densty functon for y based on the error model can be rewrtten as follows: p (y x, θ ) = N(y B β, σ 2 I) (8) Thus, Eq. (8) defnes the probablstc model for splnebased mxture regresson. Here we remar that the estmaton algorthm s mplemented by tang advantage of the connecton between smoothng B-splnes (at the desgn ponts) and mxture regresson models. Splnes are recommended for data fttng whenever there s no partcular reason for usng a sngle polynomal or other elementary functons. Splne functons have the followng useful propertes: smooth and flexble, easy to evaluate, along wth ther dervatves and ntegrals, and easy to generalze to hgher dmensons. 2.3 Modelng Varablty Along the RT Dmenson In the prevous sectons, we ntroduce a probablstcbased framewor that allows us to ncorporate arbtrary scalng and translaton varablty n tme and measurement spaces of the LC-MS data representng sngle or 457

6 458 multple groups. Ths framewor can be used to algn multple chromatograms nvolvng two dmensons (TIC values vs. RT ponts). In ths secton, we descrbe the steps that defne the prncple upon whch the algnment models are based for the LC-MS data. These steps provde the data generatve model, defne the pror transformaton models, compute the jont probabltes and the log-lelhood functon, and descrbe the assocated EM algorthm. Wth a mnor abuse of notaton, Eq. (9) presents the general model formulaton and the assumed pror transformaton models for scalng and translaton parameters: y = d 1 + c ψ ([a x b 1], β )+e (9) where {a, b } and {c, d } are scalng and translaton varables for tme and measurement spaces, respectvely; and 1 s a column vector of ones wth an approprate dmenson. The posteror probabltes of these parameters are estmated together wth the other regresson model parameters as well as the error varances from the observed dataset D. The defnton gven n Eq. (9) and the correspondng mxture regresson model allow arbtrary scalng and translaton n tme and measurement spaces. Here we remar that the condtonal probablstc densty model p (y x, θ, a, b, c, d ) s also condtoned on the global parameter sets θ = {β, σ 2 } for = 1, 2,..., K. The transformaton prors {a, b } and {c, d } are assocated to each LC-MS run usng the model of Eq. (9). Thus, together wth pror transformaton models, the problem of dervng the resultng jont probabltes can be drectly handled. By consderng {a, b } and {c, d } as random varables wth ther assocated pror probablty densty functons, one can then proceed wth the mxture regresson models for algnment of data representng multple groups. Wth the assumpton of ndependence, the jont pror probablstc model for the tme scalng and translaton p(a, b ) = p(a ) p( b ) should encode the dea that the most lely translaton s the zero-tme translaton and should also dscount large translatons. A zero-mean Gaussan pror s a good ft for ths,. e., b N(0, s 2 )wheres2 s the varance. Moreover, for the tme scalng pror a, a value of one s the most lely value (. e., no scalng at all). Thus, the correspondng Gaussan pror model s gven as a N(1, r 2) I{a > 0}. Thevarancesr 2 and s2 wll be learned from the data wthn the ensung EM algorthm. Smlarly, the jont pror probablstc model for the affne transformaton n measurement space wth ndependence assumpton s gven as p(c, d ) = p(c ) p( d ). Here, we follow a smlar argument as that of the affne tme transformaton above. In other words, we assume that c N(1, u 2 ) I{c > 0} and d N(0, v 2)arethe pror probablstc models and specfy the most lely transformatons. We also remar that affxng to each parameter,. e., θ = {β, σ 2, r2, s2, u2, v2 }, s used to arrve at group-dependent regresson model. After obtanng both the group-specfc condtonal probablty densty functon for y,. e., p (y x, θ, a, b, c, d ), and the complete data lelhood functon l(θ/d, Z) (where θ = {β, σ 2, r2, s2, u2, v2 }K =1 s the complete parameter set for the model n Eq. (9)), the EM algorthm sequentally executes the followng two steps: Expectaton-Step (E-Step): tae the expectaton of l(θ D, Z) wth respect to p(z Y, X, θ t 1 ), where θ t 1 s the current set of parameters, and Maxmzaton-Step (M-Step): maxmze the expected value of the computed log-lelhood from E-Step over the parameters θ to yeld new parameters θ t. For the partcular form of Z chosen here, the expectaton of l(θ D, Z) s K E {l(θ D, Z)} = M h log α =1 =1 + M (10) K h log p (y =1 =1 x, θ ) where h = E{z } p (y z = 1, x, θ t 1 ) p(z = 1) and t corresponds to the posteror probablty of the -th spectra beng generated from the component model. Therefore, the E- and M-steps of the EM estmaton algorthm proceed teratvely to estmate the parameters θ. The followng are the steps nvolved n estmatng the parameters for the splne-based mxture regresson model usng the EM algorthm: Step 1: Intalze randomly the membershp probabltes h = E{z } and estmate the parameter set θ 0 usng these partcular membershp probabltes. Step 2: Compute the new membershp probabltes h = E{z } usng the current parameter set θ t 1. Step 3: Estmate the new parameter set θ t usng the new membershp probabltes from Step 2. Step 4: Repeat Steps 2 and 3 untl a stoppng crteron s met (e. g., maxmum number of teraton or convergence rate). 2.4 Algnment of Three Dmensonal LC-MS Spectra Thus far, we have mplctly assumed that y conssts of two-dmensonal data (. e., the model nvolves RT ponts and TIC values). To tae advantage of the entre LC-MS run for dfferental proten expresson studes, we extend our method for three-dmensonal algnment (. e., algnng wth respect to RT ponts and m/z bns; and performng mplctly normalzaton of pea ntenstes for peptdes). We denote a three-dmensonal LC-MS run as ỹ whch conssts of Q columns such that the contans two-dmensonal data for the -th observaton. In other words, each ỹ (q) corresponds to a standard two-dmensonal profle staced nto a vector. The three-dmensonal nature of each LC-MS run can be ncorporated nto the algnment model by defnng a separate regresson model for each dmenson. Thus, the general regresson model based on the assumed prors for q-th column ỹ (q)

7 Analyss of LC-MS Data Usng Probablstc-Based... scalng and translaton parameters for each of the three dmensons (RT, m/z and pea ntenstes of peptdes) can be presented as follows: ( ) ỹ (q) = d (q) 1 + c (q) ψ (q) [a x b 1], β (q) a N ( 1, r ) 2, b N ( ) 0, s 2 (q), c N ( ) ( ) d (q) N 0, ṽ 2(q), e (q) N 0, σ 2(q) I + e (q) ( 1, ũ 2(q) ), = 1, 2,..., M, q = 1, 2,..., Q, and = 1, 2,..., K (11) where β (q) conssts of the regresson coeffcents for the q-th dmenson (. e., the regresson coeffcents for the q-th column of ỹ (q) ); {a, b } and {c (q), d (q) } are group specfc scalng and translaton varables for RT ponts, m/z bns and measurement space (pea ntenstes of peptdes), respectvely. Note that the scalng and translaton parameters for measurement space vary from dmenson to dmenson. The mxture of drected acyclc graph (DAG) model that shows the graphcal factorzaton of the jont probablty dstrbuton for ỹ,sshownnfg.3.thsgraphcal model conssts of two components: the structure and the set of local dstrbuton famles. The structure represents condtonal probabltes that assert through a factorzaton of the jont dstrbuton for ỹ,whlethelocal dstrbuton famles assocated wth mxture model are those n Eq. (11). In the followng, we llustrate the dervaton usng only the translaton n measurement space, {d (q) } Q q=1, for three-dmensonal LC-MS data ỹ usng the model specfed n Eq. (11): p ( ỹ, d (1),..., d (Q) ) = Q ( ) N ỹ (q) q=1 B (q) β (q) + d (q) 1, σ 2(q) I ( ) N d (q) 0, ṽ (q) (12) The jont densty can be factored due to the followng necessary condtons: () condtonal ndependence s assumed between the dmensons of ỹ, and () each dmenson s assumed to have ts own set of translaton parameters. Thus, the margnal densty p(ỹ ) s Fgure 3 DAG for probablstc-based mxture regresson models. gven as Q q=1 p(ỹ(q) ) and the resultng log-lelhood of Y = {ỹ } M =1 taes the followng form log p(y) = M log p(ỹ =1 ) = M Q =1 q=1 d (q) log p ( ỹ (q) d (q) ) p ( d (q) ) dd (q) (13) Equaton (13) ncludes all the M Q spectra n the loglelhood functon and analytcally ntegratng Eq. (13) gves the log-lelhood as: log p(y) = M Q ( log N ỹ (q) =1 q=1 B (q) β (q), ṽ(q) 1 ) + σ 2(q) I (14) Thus, for a complete algnment problem n the measurement space ncludng both scalng and translaton parameters, the ndvdual dmensons requre 2Q separate scalng and translaton parameters to be ncorporated n ths framewor. On the other hand, the condton for RT pont algnment needs a dfferent perspectve. Here the assumpton s that the dynamc behavor for each dmenson occurred over the same tme scale. Thus, the RT transformaton parameters need to be shared over the Q dmensons for each group data. For the RT translaton case, each two-dmensonal profle n ỹ shares a sngle translaton parameter b from ts correspondng group behavor and the condtonal densty of ỹ becomes p ( ) Q ỹ /b = p ( ỹ (q) ) q=1 b = Q q=1 N ( ỹ (q) B (q) β (q), σ 2(q) ) (15) I n whch there s only one b for all q n each group. The correspondng margnal densty p(ỹ ) cannot be factored as that of measurement space translaton due to the dependence among dmensons through the translaton parameters b. Therefore the log-lelhood of Y s gven as follows: log p(y) = M =1 log p(ỹ ) = M =1 log b [ p(b ) Q p ( ỹ (q) ) ] q=1 b db (16) The product over the dmenson s now nsde the ntegraton operaton and an approxmate log-lelhood can be computed usng the Monte-Carlo numercal ntegraton technque [19]. Thus, the approxmaton can be computed as follows: log p(y) = [ M R log Q p ( ỹ (q) =1 r=1 q=1 b (r) ) ] M log R (17) 459

8 460 where b (r) N(0, s 2 ) for r = 1, 2,..., R, and R s the sample sze n the Monte-Carlo numercal ntegraton. Moreover, we used the followng approxmaton for the uncondtonal margnal densty of ỹ (q) n arrvng at Eq. (17): p(y (q) ) = α ( ) p y (q) 1 R α ( p y (q),r b (r) ) The dervaton for the scalng c (q) n measurement space and scalng a n RT pont wll follow drectly from the prevous dervatons except for handlng the varous ndvdual q subscrpts on the dmenson dependent parameters. Moreover, the EM algorthm outlned n Sect. 2.3 s then modfed at each teraton to update the followng parameter set θ t = { β (q) 2(q), σ, r 2, s2, ũ2(q), ṽ 2(q) } t for q = 1, 2,..., Q and = 1, 2,..., K. 3 LC-MS Data 3.1 Sngle-class Lstgarten et al. LC-MS Dataset The sngle-class Lstgarten et al. LC-MS dataset was obtaned from They consst of 11 replcate LC-MS runs generated from protens of lysed E.col cells usng a capllary-scale LC coupled on-lne to an on trap mass spectrometer. Each run s represented by a data matrx correspondng to 400 RT ponts ( 55 mn.) and 2400 m/z bns spannng between 400 and 1600 Da. Detaled expermental nformaton can be found n ref. [20]. 3.2 Two-class Lstgarten et al. LC-MS Dataset The two-class Lstgarten et al. dataset (downloaded from was obtaned from human serum samples that represent two classes. The dataset conssts of 14 LC-MS runs generated from two groups of human serum samples. Seven runs were derved from a sample, n whch the human serum was mxed wth three nown peptdes: Angotensn II, Angotensn I, and Substance P wth molecular weghts of , , and Da, respectvely. The remanng seven runs were from a sample consstng of human serum only. The LC-MS data were generated usng hgh performance lqud chromatography (HPLC) pump nterfaced wth a Thermo Fnngan LCQ quadrupole on trap tandem mass spectrometer. Each LC-MS run s represented by a data matrx correspondng to 500 RT ponts ( 55 mn.) and 2400 m/z bns between 400 and 1600 Da. 3.3 Mueller et al. LC-MS Dataset The Mueller et al. LC-MS dataset was obtaned from _Data.php. They consst of 18 LC-MS runs generated from tryptc dgests of sx standard non-human protens (myoglobn, carbonc anhydrase, cytochrome c, lysozyme, alcohol dehydrogenase, and aldolase A) sped wth dfferent concentratons nto a complex sample bacground of human peptdes and solated from serum usng sold-phase Nglycocapture. The LC-MS data were obtaned usng the Fourer transformed-lnear trap quadrupole (FT-LTQ) mass spectrometer. Detaled expermental nformaton can be found n ref. [6]. The 18 LC-MS runs were grouped nto sx classes based on the concentraton of the spe-n protens. Specfcally, Latn square dluton wth dfferent concentratons of the sx standard protens was used. Each run s represented by a data matrx correspondng to 2000 RT ponts ( 55 mn.) and 750 m/z bns between 300 and 1600 Da. 4 Results and Dscusson We appled PMRM to algn one of the above three datasets based on two-dmensonal profles nvolvng TIC and RT ponts. The remanng two datasets were analyzed on the bass of three-dmensonal data nvolvng RT ponts, m/z bns, and pea ntenstes of peptdes. 4.1 Sngle-class Lstgarten et al. LC-MS Dataset The sngle-class Lstgarten et al. dataset was converted nto 11 two-dmensonal profles by calculatng the TIC at each RT pont (. e., at each RT pont the sum of all ons across m/z values was calculated). Fgure 4 depcts TIC profles of the 11 replcate LC-MS runs. From the fgure we observe sgnfcant shfts along the expermental RT ponts as well as dstortons n the measurement space (TIC values). The data algned by the PMRM approach are shown n Fg. 5. Fgure 6 depcts the mean TIC profles for the orgnal (before algnment) and those algned by DTW, COW, CPM, and PMRM. From ths fgure, we see that DTW and CPM approaches produce superfluous sgnal gaps, thus stretchng the expermental RT ponts by almost 1.5 and 2 folds, respectvely. Although the stretched RT ponts can be rescaled to the orgnal RT ponts, the dffculty arses n sortng out the nserted gaps durng peptde sequence dentfcaton to accurately match the sequence nformaton wth the expermental RT ponts. On the other hand, the algned tracng ponts obtaned by PMRM and COW are consstent wth the expermental RT ponts. Table 1 Coeffcent of varatons for sngle-class and two-class Lstgarten et al. datasets before algnment (orgnal) and after algnment by DTW, COW, CPM, and PMRM. The result of the method wth the best performance n each category s shown n boldface. Sngle-class Two-class Lstgarten et al. Lstgarten et al. Class 1 Class 2 Orgnal 82% 57% 58% DTW 70% 56% 57% COW 80% 56% 57% CPM 57% 44% 44% PMRM 65% 47% 47%

9 Analyss of LC-MS Data Usng Probablstc-Based... Table 1 presents the coeffcent of varatons (CVs) for the orgnal data and those algned by DTW, COW, CPM, and PMRM. As shown n the table, both CPM and PMRM have yelded better results than DTW and COW. 4.2 Two-class Lstgarten et al. LC-MS Data Algnment. Each LC-MS run n the two-class Lstgarten et al. dataset was reduced from a to a Fgure 4 TIC profles of the orgnal sngleclass Lstgarten et al. dataset. Fgure 5 TIC profles of the sngle-class Lstgarten et al. dataset after algnment by PMRM. Fgure 6 Mean TIC profles for each set of sngle-class Lstgarten et al. dataset (orgnal and the dataset algned by DTW, COW, CPM and PMRM). data matrx by parttonng the m/z dmenson nto eght equal m/z bns. Ths reducton was necessary to handle the computatonal complexty. We appled the PMRM to the reduced matrx and obtaned a set of algned data matrces,. e., for each LC-MS run ( = 1, 2,..., 14), adatamatrxm was obtaned, where M τ, q s the on abundance after algnment at the τ-th algned RT pont (τ = 1, 2,..., 500) and the q-th m/z bn (q = 1, 2,..., 8). 461

10 462 For vsualzaton purpose, we calculated the TIC values from the reduced orgnal and algned matrces. Fgures 7 and 8 depct the TIC profles of the orgnal and algned LC-MS runs, respectvely. Table 1 presents the CVs for the orgnal and those algned by DTW, COW, CPM, and PMRM. As llustrated n the table, both CPM and PMRM have yelded better results than DTW and COW. Although, CV s a good measure for assessng the varablty among replcate datasets, nowledge of such nformaton does not necessarly confrm or drectly ln wth actual nterpretaton of dfferences between the LC-MS datasets. In the followng, we use statstcal measures for detected dfferences between the two classes to assess the performance of varous algnment methods. Dfference detecton. We used the algned data matrces for dfference detecton. Specfcally, we used a multvarate permutaton method mplemented n the BRB-Array-Tools (Natonal Cancer Insttute, Bethesda, MD) to calculate non-parametrc p-values for each RT pont and m/z bn n the algned data matrces represent- ng the two-class dataset. We utlzed the expermentally extracted ground truth nformaton to evaluate the senstvty, specfcty, and accuracy n detectng the truly dfferentally abundant peptdes. Ths s accomplshed by usng a lst of expermentally extracted ground truth peptde m/z values provded n ref. For evaluaton, we consdered three sgnfcance levels (p<0.05, p<0.01, and p<0.001). For each sgnfcance level, we performed a hypothess test to determne whether the on abundances dffer between the two-class datasets. Table 2 presents the computed senstvty, specfcty, accuracy, and precson n detectng dfferentally abundant peptdes usng the LC-MS data algned by DTW, COW, CPM and PMRM. The table also provdes an F-measure, whch s a weghted harmonc mean measure combnng the precson and recall, calculated as follows wth β = 1[21]: F measure = (β2 +1) precson recall β 2 precson + recall As shown n Table 2, the LC-MS data algned by PMRM have yelded overall better performance than those Fgure 7 TIC profles for the two-class Lstgarten et al. dataset before algnment. Fgure 8 TIC profles for the two-class Lstgarten et al. dataset after algnment by PMRM.

11 Analyss of LC-MS Data Usng Probablstc-Based... Table 2 Senstvty, specfcty, accuracy, precson and F-measure n detectng dfferentally abundant peptdes after algnng n the Two-class Lstgarten etal.datasetbythedtw,cow,cpm,andpmrm(forp<0.05, p<0.01, and p<0.001). The result of the method wth the best performance n each category s shown n boldface. P-values Senstvty Specfcty Accuracy Precson F-measure DTW 60% 55% 58% 60% 60% COW 64% 57% 60% 62% 63% 0.05 CPM 59% 83% 70% 73% 64% PMRM 65% 79% 73% 73% 69% DTW 63% 63% 61% 64% 63% COW 63% 61% 62% 77% 69% 0.01 CPM 45% 90% 68% 73% 57% PMRM 62% 81% 72% 73% 67% DTW 63% 64% 64% 61% 62% COW 67% 66% 66% 63% 65% 0.001* CPM PMRM 59% 83% 71% 73% 64% * The condton for p<0.001 s too strngent for CPM. algned by DTW, COW and CPM. For p<0.001 sgnfcant level, CPM dd not yeld any dfferentally abundance ons. Possble reasons for ths could be due to over smoothng and attenuaton of peas by CPM. 4.3 Mueller et al. LC-MS Dataset Algnment. The Mueller et al. LC-MS dataset was reduced from a to data matrx by parttonng the m/z dmenson nto eght equal m/z bns. After applyng PMRM to ths reduced matrx, we obtaned a set of algned data matrces,. e., for each LC-MS run ( = 1, 2,..., 18), a data matrx M was obtaned, where M τ, q s the on abundance obtaned after algnment at τ- th algned RT ponts (τ = 1, 2,..., 2000) and the q-th m/z bn (q = 1, 2,..., 8). Fgures 9 and 10 depct TIC profles of the orgnal and algned LC-MS spectra, respectvely, for RT ponts between 800 and Dfference detecton. The algned data matrces were used for dfference detecton. Snce the orgnal LC-MS dataset conssted of the dluted seres of sx protens wth dfferent concentraton n each sample, we determned the peptdes wth statstcally sgnfcant dfference n ther expresson level based on an F-statstc. To accomplsh Table 3 Senstvty, specfcty, accuracy, precson and F-measure n detectng dfferentally abundant peptdes n the Mueller et al. dataset bythedtw,cow,cpm,andpmrmforp<0.01. The result of the method wth the best performance n each category s shown n boldface. Senstvty Specfcty Accuracy Precson F-measure DTW 76% 70% 85% 70% 73% COW 83% 75% 79% 76% 79% CPM 90% 75% 83% 84% 80% PMRM 91% 78% 85% 83% 87% Fgure 9 TICproflesfortheorgnalMueller et al. LC-MS dataset. 463

12 464 ths, we frst generated a lst of peptdes assumng no ms-cleavage from trypsn dgeston of sx non-human protens (myoglobn, carbonc anhydrase, cytochrome c, lysozyme, alcohol dehydrogenase, and aldolase A) usng the MS-Dgest software from the Unversty of Calforna, San Francsco ( Then, we used the permutaton method to examne each RT pont and m/z bn. We compared the statstcally sgnfcantly dfferent RT-m/z pars (p < 0.01) aganst the lst of peptdes that represent the sx protens to calculate senstvty, specfcty, accuracy, precson, and F-measure. Table 3 presents these results for the data algned by DTW, COW, CPM and PMRM. The table llustrates that PMRM yelds overall better results than DTW, COW, and CPM. 4.4 Reproducblty Study We examne the reproducblty behavor of the PMRM n algnng LC-MS data. To accomplsh ths, t s necessary that the dentfablty property holds wth respect to the mxture models gven by Eq. (2) (oreq.(3) and the general model defnton gven by Eq. (9)). However, the general statement on the dentfablty property Table 4 The expected values of the scalng parameters a. Confdence Intervals E{a } 95% 99% 99.9% [1.059, 1.073] [1.056, 1.076] [1.054, 1.078] [1.084, 1.099] [1.082, 1.102] [1.079, 1.105] [1.094, 1.108] [1.091, 1.111] [1.088, 1.113] [1.065, 1.080] [1.063, 1.083] [1.060, 1.085] [1.100, 1.115] [1.098, 1.118] [1.095, 1.121] [1.123, 1.138] [1.120, 1.140] [1.117, 1.143] [1.093, 1.108] [1.090, 1.110] [1.087, 1.113] [1.088, 1.103] [1.085, 1.105] [1.083, 1.108] [1.056, 1.070] [1.053, 1.072] [1.051, 1.075] [1.021, 1.035] [1.018, 1.037] [1.016, 1.039] [1.074, 1.088] [1.072, 1.091] [1.069, 1.093] Fgure 10 TIC profles for the Mueller et al. LC-MS dataset after algnment by PMRM. requres mnmzng dstrbutons usng Kullbac-Lebler dvergence from the true mxng dstrbuton [22; 23]. Snce the dentfablty property of mxng dstrbutons s of nterest n ts own rght, we focus here on examnng the consstency of the PMRM parameters by runnng the algorthm multple tmes for the same set of LC-MS data. Table 4 presents the 95%, 99% and 99.9% confdence ntervals for the mean of algnment scalng feld (. e., the expected posteror for scalng a parameters) obtaned n algnng the sngle-class Lstgarten et al. LC-MS data. From ths table, we can see that estmated values show farly narrow ntervals, ndcatng consstent estmaton of the parameters across multple runs. Ths analyss also provdes smlar mean posteror estmates for the other algnment feld parameters. Moreover, t too approxmately nne mnutes for the algorthm to complete one run on a PC wth an Intel Core 2 Duo 64 bt 2.66 GHz CPU and 8 GB RAM. 5 Concluson Ths paper proposes a probablstc-based mxture regresson model (PMRM) for algnment of multple LC-MS data. We utlze the well nown maxmum lelhoodbased EM algorthm for estmatng the mxture regresson models and the pror transformaton (scalng and translaton parameters) models of the LC-MS data. The latter accounts for the varablty n RT ponts, m/z values, and pea ntenstes. The proposed framewor allows algnment wth respect to RT ponts and m/z bns; and t mplctly performs normalzaton of pea ntenstes n multple LC-MS runs. The performance of the approach s assessed through three LC-MS datasets: replcate LC- MS dataset generated from protens of lysed E.col cells, LC-MS dataset representng two classes of human serum samples (wth and wthout spe-n peptdes), and spectra representng sx classes, where sx protens are sped at dfferent concentratons nto a complex sample bacground of human peptdes. Through these datasets, t s demonstrated that PMRM acheves good coeffcent

13 Analyss of LC-MS Data Usng Probablstc-Based... of varaton among replcate LC-MS data whle preservng the orgnal expermental retenton tme,. e. wthout ntroducng any superfluous sgnal gap across multple LC-MS spectra. Also, through spe-n peptdes, the paper demonstrates that PMRM leads to more accurate dentfcaton of dfferentally abundant peptdes than DTW, COW, and CPM. Future wor wll focus on extendng PMRM for algnment and normalzaton of multple LC- MS data that wll account varablty among LC-MS runs representng samples from dstnct bologcal groups as well as modelng the heterogenety wthn group. Acnowledgements Ths wor was supported n part by the Natonal Scence Foundaton Grant (IIS ), Natonal Cancer Insttute (NCI) R21 CA Grant, NCI Early Detecton Research Networ Assocate Membershp Grant, and the Prevent Cancer Foundaton Grant awarded to HWR. References [1] Aebersold R, Mann M: Mass spectrometry-based proteomcs. Nature 2003, 422(6928): [2] Lebler D C: Introducton to proteomcs: tools for the new bology. Totowa, NJ: Humana Press, [3] Tomas G, van den Berg F, Andersson C: Correlaton Optmzed Warpng and Dynamc Tme Warpng as Preprocessng Methods for Chromatographc Data. Journal of Chemometrcs and Intellgent Laboratory Systems 2004, 18: [4] Hastngs C A, Norton S M, Roy S: New algorthms for processng and pea detecton n lqud chromatography/mass spectrometry data. Rapd Commun Mass Spectrom 2002, 16(5): [5] Fscher B, Grossmann J, Roth V, Grussem W, Bagnsy S, Buhmann J M: Sem-supervsed LC/MS algnment for dfferental proteomcs. Bonformatcs 2006, 22(14):e [6] Wang P, Tang H, Ftzgbbon M P, McIntosh M, Coram M, Zhang H, YE, Aebersold R: A statstcal method for chromatographc algnment of LC-MS data. Bostatstcs 2007, 8(2): [7] Mueller L N, Rnner O, Schmdt A, Letarte S, Bodenmller B, Brusna M Y, Vte O, Aebersold R, Muller M: SuperHrn a novel tool for hgh resoluton LC-MS-based peptde/proten proflng. Proteomcs 2007, 7(19): [8] Lstgarten J, Neal R M, Rowes S T, Wong P, Eml A: Dfference detecton n LC-MS data for proten bomarer dscovery. Bonformatcs 2007, 23(2):e [9] Lstgarten J, Neal R M, Rowes S T, Emly A: Multple Algnment of Contnuous Tme Seres. Neural Informaton Processng Systems 2005, 17: [10] Morrs J S, Brown P J, Herrc R C, et al.: Bayesan analyss of mass spectrometry proteomc data usng wavelet-based functonal mxed models. Bometrcs 2008, 64(2): [11] Morrs J S, Carroll R J: Wavelet-based functonal mxed models. Journal of the Royal Statstcal Socety Seres B 2006, 68(2): [12] Guo W: Functonal data analyss n longtudnal settngs usng smoothng splnes. Stat Methods Med Res 2004, 13(1): [13] Wagner M S, Graham D J, Ratner B D Castner D G: Maxmzng nformaton obtaned from secondary on mass spectra of organc thn flms usng multvarate analyss. Surface Scence 2004, 570: [14] Hartemn A J, Gfford D K, Jaaola T S, Young R A: Maxmumlelhood estmaton of optmal scalng factors for expresson array normalzaton. Presented at Mcroarrays: Optcal Technologes and Informatcs, San Jose, CA, USA, [15] Sys-Aho M, Katajamaa M, Yetuur L, Oresc M: Normalzaton method for metabolomcs data usng optmal selecton of multple nternal standards. BMC Bonformatcs 2007, 8:93. [16] Dempster A P, Lard N M, Rubn D B: Maxmum Lelhood from Incomplete Data va the EM Algorthm. Journal of the Royal Statstcal Socety Seres B (Methodologcal) 1977, 39(1):1 38. [17] Jordan M I, Jacobs R A: Herarchcal mxtures of experts and the EM algorthm. Neural Comput 1994, 6(2): [18] Redner R A, Waler H F: Mxture Denstes, Maxmum Lelhood and the Em Algorthm. SIAM Revew 1984, 26(2): [19] Lange K: Numercal analyss for statstcans. New Yor: Sprnger, [20] Lstgarten J: Analyss of sblng tme seres data: algnment and dfference detecton. Ph. D. Thess, Toronto: Unversty of Toronto, [21] Van Rjsbergen C J: Informaton retreval, 2nd ed. London; Boston: Butterworths, [22] Henna J: On estmatng of the number of consttuents of a fnte mxture of contnuous dstrbutons. Annals of the Insttute of Statstcal Mathematcs 1985, 37(1): [23] McLachlan G J, Basford K E: Mxture models: nference and applcatons to clusterng. New Yor, N.Y.: M. Deer, Receved: March 13, 2009 Dr.-Ing. Habtom W. Ressom s an Assocate Professor and Drector of the Genomcs and Epgenomcs Shared Resource at the Lombard Comprehensve Cancer Center, Georgetown Unversty. Hs research s focused on the applcaton of statstcal and machne learnng methods for analyss of hgh dmensonal omcs data. Address: Department of Oncology, Lombard Comprehensve Cancer Center, Georgetown Unversty, Sute 173, Buldng D, 4000 Reservor Road NW, Washngton DC 20057, Tel.: , Fax: , e-mal: hwr@georgetown.edu Dr.-Ing. Getachew K. Befeadu s a Postdoctoral Research Fellow wth the Department of Oncology at Georgetown Unversty. Hs research nterests focus on developng computatonal technques ncludng appled mathematcs, nformatcs, statstcs, and machne learnng methods for analyss of proteomcs and metabolomcs data. Address: Department of Oncology, Lombard Comprehensve Cancer Center, Georgetown Unversty, 176 Buldng D, 4000 Reservor Road NW, Washngton DC 20057, Tel.: , Fax: , e-mal: gb8@georgetown.edu Dr. Mahlet G. Tadesse s an Assstant Professor n the Department of Mathematcs at Georgetown Unversty. Her research focuses on the development of statstcal and computatonal tools for the analyss of large-scale genomc data. She s partcularly nterested n stochastc search methods and Bayesan nferental strateges to dentfy structures and relatonshps n hgh-dmensonal data sets. Address: Department of Mathematcs, Georgetown Unversty, 308 St. Mary s Hall, 3700 Reservor Road NW, Washngton DC 20057, Tel.: , Fax: , e-mal: mgt26@georgetown.edu 465