From binary to multivalued to continuous models: the lac operon as a case study.

F om Bina y o Mul i alued o Con inuous Models: The lac Ope on as a Case S udy Raimo F anke1*, Fabian J. Theis2 and S e en Klam 3 1Depa men o Chemical Biology, Helmhol z Cen e o In ec ion Resea ch, 38124 B aunschweig, Ge many 2Ins i u e o Bioin o ma ics and Sys ems Biology, Helmhol z Cen e Munich - Ge man Resea ch Cen e o En i onmen al Heal h, 85764 Neuhe be g, Ge many 3Max Planck Ins i u e o Dynamics o Complex Technical Sys ems, 39106 Magdebu g, Ge many Summa y Using he lac ope on as a pa adigma ic example o a gene egula o y sys em in p oka yo es, we demons a e how quali a i e knowledge can be ini ially cap u ed using simple disc e e (Boolean) models and hen s epwise e ined o mul i alued logical models and inally o con inuous (ODE) models. A all s ages, signal ansduc ion and ansc ip ional egula ion is in eg a ed in he model desc ip ion. We i s show he po en ial bene i o a disc e e bina y app oach and discuss hen p oblems and limi a ions due o inde e minacy a ising in cyclic ne wo ks. These limi a ions can be pa ially ci cum en ed by using mul ile el logic as gene aliza ion o he Boolean amewo k enabling one o o mula e a mo e ealis ic model o he lac ope on. Ul ima ely a dynamic desc ip ion is needed o ully app ecia e he po en ial dynamic beha io ha can be induced by egula o y eedback loops. As a e y p omising me hod we show how he use o mul i a ia e polynomial in e pola ion allows ans o ma ion o he logical ne wo k in o a sys em o o dina y di e en ial equa ions (ODEs), which hen enables he analysis o key ea u es o he dynamic beha io . 1 In oduc ion Biological ne wo ks can be subdi ided in o me abolic, signal ansduc ion and egula o y ne wo ks. He e he e m " egula o y ne wo k" is used o ansc ip ional and ansla ional egula ion by con en ion, al hough egula o y ea u es a e associa ed wi h all cellula p ocesses [1]. Wi h a e excep ions [2,3] mos o he p e ious compu a ional analyses o cellula ne wo ks ha e ocused on one o hese ne wo k laye s sepa a ely, wi hou conside ing he in e play ha exis s be ween hem. This concep ual di ision can be use ul o he ma hema ical ea men , howe e in he cell all o he componen s wo k in an in eg a ed ashion ha p omo es i ness [1]. A simple example o he in e play o he di e en ne wo k laye s would be a ecep o ha is igge ed by an ex acellula s imulus and induces a signaling cascade, which esul s in he ac i a ion o a ansc ip ion ac o . As a consequence, exp ession o a a ge gene is induced, esul ing in he p oduc ion o a p o ein which inhibi s he signaling cascade a a ce ain s age and he eby p e en s ac i a ion o he ansc ip ion ac o in a nega i e eedback loop. This example shows he in e connec edness o signal ansduc ion and ansc ip ional egula ion. I is desi able o de elop modeling amewo ks ha enable an in eg a ed ea men o all laye s ha make up cellula li e. * To whom co espondence should be add essed. Email: aimo. anke@helmhol z-hzi.de Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 1 Mechanism-based modeling app oaches based on di e en ial equa ions enable a dynamic analysis o cellula ne wo ks and ha e a high p edic i e powe . Howe e , hey a e limi ed o a he small sys ems because hey ely on de ailed knowledge o kine ic laws and pa ame e s o he unde lying biochemical eac ions which is o en no a ailable. Fo modeling la ge-scale ne wo ks, quali a i e modeling app oaches and ne wo k analysis echniques a e usually be e sui ed as hey seek o elucida e unc ional ea u es om he o en well-cha ac e ized ne wo k opologies alone (e.g., wi ing diag am o egula o y ne wo ks o eac ion s oichiome ies in me abolic ne wo ks). Fo example, in signaling and egula o y ne wo ks one is ypically in e es ed in (i) de ec ion o ne wo k-wide unc ional in e dependencies be ween ne wo k elemen s, (ii) iden i ica ion o eedback loops, (iii) iden i ica ion o in e en ions ha induce a speci ic esponse and (i ) quali a i e p edic ions on he e ec o pe u ba ions. We in oduced a logical modeling amewo k (Boolean ne wo ks ep esen ed as logical in e ac ion hype g aphs, LIHs [4,5]) ha is ideal o he econs uc ion and quali a i e analysis o cellula ne wo ks wi h signal o in o ma ion lows. Examples ha ha e been s udied include di e se signaling ne wo ks such as T cell signaling [6], cMe signal ansduc ion [7], NF- B signal ansduc ion [8], and EGF signaling [9]. Signaling ne wo ks a e s uc u ed in o inpu , in e media e and ou pu laye , which acili a e c oss alk and in eg a ed decision making and a e o en ea ed as acyclic ne wo k as a i s app oxima ion. Gene egula o y ne wo ks on he o he hand a e s ongly de e mined by hei eedback loops (cyclic ne wo ks). Signaling and egula o y ne wo ks a e in e wined in he cell and accoun ing o he coupling o signaling and gene egula ion is highly desi able owa ds a mo e comple e model o he cell. As a case s udy we chose he well-desc ibed lac ope on, he pa adigma ic example o a gene egula o y sys em in p oka yo es [10-12]. The in ol emen o species om me abolic, signaling and gene egula o y ne wo k laye s and also he exis ence o eedback loops in he egula ion o he lac ope on make i a e y a ac i e sys em o e alua e di e en modeling app oaches. In his s udy we ini ially s a wi h a Boolean model o he lac ope on and e alua e i s po en ial o cap u e essen ial ea u es o he mechanisms in ol ed in he egula ion o he lac ope on. The p oblems and limi a ions ha a ise ollowing a bina y ea men will be shown and ways how o e ine his ep esen a ion o mul i alued logic and hen o quali a i e ODE‟s will be p esen ed. 2 Me hods 2.1 Boolean ne wo ks ep esen ed as logical in e ac ion hype g aphs Fo a de ailed in oduc ion in o he o malism o logical in e ac ion hype g aphs (LIHs) and i s implemen a ion in CellNe Analyze , we e e he eade o ou p e ious publica ions [4,5,13]. In b ie , his Boolean modeling amewo k was ailo ed o s udying he quali a i e inpu -ou pu esponse o signaling ne wo ks. As in all Boolean ne wo ks, nodes in he ne wo k ep esen biomolecula species (e.g. kinases, adap o molecules, ansc ip ion ac o s, o genes) each ha ing an associa ed logical s a e (in he bina y case only “on” (1) o “o ” (0)) exp essing whe he he species is ac i e (o p esen ) o no . Signaling e en s a e encoded as Boolean ope a ions on he ne wo k nodes. In con as o o he wo ks ocusing on disc e e dynamics in Boolean ne wo ks [14,15], he app oach o LIHs was mainly used o s udy he inpu /ou pu beha io o signaling ne wo ks by analyzing he quali a i e (logical) s eady s a e ha esul s om a gi en ex e nal s imula ion o pe u ba ion. LIHs make only use o he Boolean ope a o s AND (·), OR (+), and NOT (!), which a e su icien o ep esen any logical ela ionship. The so-called sum-o -p oduc ep esen a ion, Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 2 whe e AND e ms a e connec ed ia OR ope a o s, makes i possible o ep esen a gi en Boolean ne wo k as a (logical in e ac ion) hype g aph (LIH) [5,13]. To exempli y his o a node in he lac ope on ne wo k: allolac ose is only p oduced (ge s “on”, i.e. alue 1) i lac ose is p esen in he cell AND he -galac osidase LacY enzyme is exp essed, i.e. lac ose_in ( o in e nal lac ose) AND LacY mus be “on” o p oduce allolac ose (see igu e 1). Hence, o he example desc ibed abo e we would w i e lac ose_in AND LacY allolac ose o , sho e , lac ose_in · LacY allolac ose In a g aphical ep esen a ion o he ne wo k such an AND connec ion is displayed as a hype a c (see igu e 1a) indica ing ha all s a nodes o he hype a c (lac ose_in , LacY) mus be in he ”on” s a e in o de o ac i a e he end node o he hype a c (no e ha hype a cs may ha e se e al s a o end nodes). NOT ope a o s o a iables en e ing a hype a c a e allowed and a e g aphically indica ed, e.g., by a ed colo o /and ba s. Fo example, in e ac ion 5 (see able 2 and i s (hype -)g aphical ep esen a ion in igu e 1) eads !PTS-EIIA · lac ose_ex · LacY lac ose_in indica ing ha he unphospho yla ed o m o EIIAglc mus be o (i.e. i is has o be in he phospho yla ed o m EIIAglc~P) AND he pe mease LacY AND he subs a e lac ose mus be a ailable in o de o ge in e nal lac ose. Finally, OR connec ions can be accoun ed o in he hype g aphical ep esen a ion by allowing a node o be independen ly ac i a ed by se e al incoming hype a cs (i.e. by se e al independen AND connec ions). By his hype g aphical ep esen a ion, we can s udy a numbe o use ul p ope ies o he logical ne wo k o i s unde lying in e ac ion g aph [5-9]. One pa icula applica ion o which we will make use he ein is he p edic ion o he inpu -ou pu beha io ha ollows om a gi en inpu s imulus (possibly combined wi h in e nal pe u ba ions such as knockou o ce ain nodes) by compu ing he esul ing logical s eady s a e (LSS). A de ailed desc ip ion o he algo i hm o compu ing he LSS was gi en in [13]; he e we will apply i o he lac ope on model by compu ing he bina y esponse o he in ol ed species o gi en subs a e mix u es (glucose o /and lac ose). The logical models s udied he ein we e implemen ed and analyzed wi h he so wa e ool CellNe Analyze [4]. 2.2 Mul i alued logic As al eady p oposed and applied by o he s (see e.g. [15]), he disc e iza ion o a node‟s ac i a ion le el in mo e han wo (bina y) le els is possible. This mimics he ac , ha in eali y mul iple ele an h eshold alues o a species may exis . I is s aigh o wa d o ex end bina y (Boolean) logic o mul i alued logic. Embedded in he LIH o malism, CellNe Analyze also suppo s mul i alued logic: in addi ion o he bina y on/o -case, o he le els o he species can be de ined. Fo example we can o mula e a logical unc ion like his: !A 2 B 3C. This equa ion means ha "C eaches le el 3 i A is inac i e (le el 0) AND B is a leas a le el 2". The wo d “a leas ” indica es ha CellNe Analyze assumes mono one ela ionships (in p inciple, non-mono one logical unc ions can also be de ined bu a e no conside ed he ein). Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 3 2.3 Mul i a ia e polynomial in e pola ion: ODE y A mo e adi ional model o ansc ip ional egula ion and signaling based on chemical eac ion kine ics in ol es he con inuous desc ip ion o concen a ion changes o he a ious species. The mos common app oach he eby is he use o a sys em o coupled o dina y di e en ial equa ions (ODEs), which essen ially igno e spa ial dimensions as well as ime delays and s ochas ici y o simplici y. In con as o he disc e e sys ems based on mul i alued logic, he use o ODEs allows he eady inclusion o g adual concen a ion changes as well as ully ime- esol ed dynamics. This mo e ine-g ained app oach o cou se comes wi h he cos o many pa ame e s such as eac ion and deg ada ion a es. While his allows a mo e de ailed desc ip ion o he obse a ions and p edic ions, he a es need o be app oxima ed by li e a u e alues o lea ned by i ing he model o da a. He e, we will ake he la e , unbiased app oach. We ha e p e iously desc ibed how o ex end a Boolean logic model o an ODE model [16], which we deno e as ODE y in he ollowing. Gi en a n- a ia e Boolean unc ion B, which is de ined on he e ices o an n-dimensional hype cube, we de ined a con inuous ex ension C o B on he ull hype cube by mul i-linea in e pola ion. In o de o accommoda e di e en le els o ac i i y o each inpu in B, we hen conca ena e C wi h a componen -wise sigmoidal nonlinea i y using he Hill unc ion k,n(x)xn/(xnkn) . He e n is he deg ee o nonlinea i y (which in he ollowing we ix o n=3 o simplici y) and k he swi ching h eshold. Fo n his app oaches a disc e e swi ch a le el x=k, which co esponds o he disc e e unc ion. This in e pola ion echnique can be de i ed om a he modynamical model o gene egula ion [17]. Howe e , o he non-linea unc ions (e.g. logis ic unc ions) could, in p inciple, be used. Using he example o „lac ose_in AND LacY allolac ose‟ om abo e, his is linea ly in e pola ed by he unc ion C(lac ose_in , LacY) = lac ose_in * LacY, because C(x,y)=1 only i bo h x=1 and y=1. The esul ing dynamics is gi en by )1)()(( 1 ))(( 2,21,1 LacY _in lac ose eallolac os d d nknk eallolac os whe e τallolac ose deno es he li e- ime o he species allolac ose, and k1, k2, n1, n2 he pa ame e s o he Hill nonlinea i y. We ha e shown ha he Boolean a ac o s a e conse ed unde his ans o ma ion gi en a su icien ly high deg ee o nonlinea i y [16]. Applica ions o ne wo k in e ence om spa ial pa e ns in neu ode elopmen illus a e ha his echnique is capable o gi ing insigh in o biological dynamics based on ini ial quali a i e in o ma ion [18]. We p o ide he oolbox ODE y as plugin o CellNe Analyze [19], which can be easily used o gene a e he ODE model in a ious o ma s. He e, we op o expo o SBToolbox2 o Ma lab, whe e we can compile he esul ing ODE in bina y o ma o as and e icien e alua ion in he pa ame e i ing pa desc ibed below. O he me hods o ans o ming Boolean in o con inuous ODE models ha e been p oposed and a compa ison o ODE y wi h al e na i e me hods can be ound in [16]. In sho , we can g oup hese me hods in o piecewise-linea in e pola ions, which a e s ill disc e e-ou pu gene aliza ions o s ep- unc ions, uzzy-logic con inua ions, which depend on he choice o deg ee-o -membe ships, and ad-hoc in e pola ions, which do no possess speci ic heo e ical p ope ies, bu e lec a ious bioma hema ical aspec s. We iew he ODE y app oach o Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 4 mul i-linea in e pola ion i s pionee ed by [20] in possible combina ion wi h Hill unc ions as he mos simple nonlinea in e pola ion echnique (in e ms o complexi y measu ed by deg ee o he used polynomial) ha s ill allows a mechanis ic de i a ion as p oposed in [22]. 3 Resul s 3.1 Concep ual model o he lac ope on In he ollowing sec ion a de ailed desc ip ion o a concep ual model o he lac ope on is gi en, which comp ises all he componen s ha a e ansla ed in o he Boolean model. The lac ope on in E. coli consis s o h ee di e en s uc u al genes ha a e ansc ibed as a single mRNA [11] in esponse o a ce ain glucose/lac ose a io. This polycis onic lacZYA mRNA is ansla ed in o h ee p o eins, which a e equi ed o impo and diges he disaccha ide lac ose. The lac ope on is he p o o ype o a single p omo e being unde he con ol by wo di e en ansc ip ion egula o s, he lac ep esso LacI and he ac i a o p o ein CAP (ca aboli e ac i a o p o ein). The basic idea o Jacob and Monod was ha he s uc u al genes o he lac ope on a e egula ed by a ep esso , which ep esses ansc ip ion, un il i in e ac s wi h a chemical "induce " [11]. The lac ope on can be desc ibed as an inducible egula o y sys em, al hough he e m "induce " can be misleading, because he en i onmen al s imulus (lac ose) hal s he ep ession (" ep ession o he ep esso "), which has in essence he e ec o inducing ansc ip ion. Soon a e Jacob and Monod published hei model [12] he lac ose ep esso p o ein LacI ha con ols he s uc u al genes esponsible o lac ose me abolism was expe imen ally iden i ied [22,23]. The egula o y gene lacI encodes he Lac ep esso LacI, which is capable o inhibi ing ansc ip ion o he s uc u al genes o he lac ope on by binding wi h high a ini y o he lac ope on a a speci ic ope a o DNA sequence (lacO1) nea he lac p omo o [11,23,24]. In addi ion o he p ima y ope a o si e O1, wo auxilia y pseudo-ope a o s we e iden i ied (O2 and O3) [11]. The binding o he LacI ep esso nex o he lac p omo o has he e ec ha RNA polyme ase binding is comp omised [25,26]. As a consequence he lac mRNA le el is s ongly educed bu no o ze o. The lac mRNA is hus no comple ely elimi- na ed by LacI binding, bu educed o a basal le el ( his aspec will la e become impo an o building he mul i alued model). LacI is a e ame ic p o ein, which can in p inciple bind wo ope a o si es simul aneously [11], bu his aspec is no conside ed in ou model. The egula ed s uc u al genes a e lacZ, lacY and lacA, which encode enzymes, ha a e all in ol ed in lac ose me abolism [11]. The lacZ gene encodes he -galac osidase LacZ, a hyd olase enzyme ha ca alyzes he hyd olysis o he disaccha ide lac ose in o he mono- saccha ides glucose and galac ose, which is he i s s ep in lac ose me abolism [27]. The lacY gene encodes β-galac oside pe mease (LacY), a memb ane-bound anspo p o ein ha enables he en y o lac ose in o he cell [28]. The hi d gene lacA, which is also unde he con ol o he lac p omo o , encodes a hiogalac oside ansace ylase, which ans e s an ace yl g oup om coenzymeA (CoA) o he hyd oxyl g oup o galac osides [11]. The ans- ace ylase is no essen ial o lac ose me abolism [29] and he e o e no included in he model. The lac ep esso LacI inhibi s ansc ip ion o he lac s uc u al genes only when i is ac i a ed. LacI is inac i a ed, when i is bound by allolac ose, a by-p oduc o lac ose me abolism (a small ac ion o he clea age p oduc s o lac ose –glucose and galac ose – can econdense o o m allolac ose). In his con ex he basal exp ession le els o lac pe mease and -galac osidase play an essen ial ole. Enzyma ic ac i i y o he pe mease enables anspo o lac ose om he medium in o he cell, whe e -galac osidase con e s a small Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 5 ac ion o lac ose in o 1,6-allolac ose [11,30]. By binding o LacI wi h high a ini y, 1,6- allolac ose lowe s LacI´s a ini y o he ope a o and he eby induces ansc ip ion o he lac s uc u al genes. In he absence o lac ose he Lac ep esso swi ches o he ope on – a mechanism o he cell no o was e ene gy o he p oduc ion o enzymes o he lac ose me abolism, when hey a e no needed. In he p esence o glucose in he medium, E. coli uses i as he sole ca bon sou ce o p oduce ene gy ia espi a ion, e en when bo h lac ose and glucose a e a ailable [31,32]. The p esence o lac ose in he medium alone is hus no su icien o ull induc ion o he lac ope on. Al hough he ep esso LacI does no occupy he ope a o si e, he ope on is ansc ibed in equen ly and emains la gely inac i e, as long as glucose is a ailable [11]. The up ake o glucose in o he cell by he phosphoenol py u a e-dependen phospho ans e ase sys ems (PTS) dec eases he le el o phospho yla ion o one o i s componen s, he enzyme EIIAGlc [33]. The dephospho yla ed EIIAGlc p e en s he up ake o lac ose by binding o he lac pe mease LacY [34,35]. As a consequence when bo h glucose and lac ose a e p esen in he medium, E. coli cells p e e en ially u ilize glucose and he use o lac ose is p e en ed un il he glucose is deple ed [33]. The esul o educing he anspo ac i i y o LacY by unphospho yla ed EIIAGlc is called induce exclusion, because lac ose canno en e he cell, and as a consequence he induce o he lac ope on exp ession allolac ose is no p oduced. An addi ional con ol mechnism by he ca aboli e ac i a o p o ein (CAP), egula ed by cyclic AMP (cAMP), con ibu es as well o he selec i e u iliza ion o me aboli es. In bac e ia, cAMP is low when glucose is used as ca bon sou ce. This occu s h ough inhibi ion o he cAMP-p oducing enzyme, adenyl cyclase, as a side-e ec o glucose anspo in o he cell [36]. Glucose is anspo ed in o he E. coli cell by he PTS [37]. A phospha e g oup is ans e ed o m phosphoenol py u a e h ough a se ies o in e media y p o eins o he EIIAGlc complex, which inally ans e s a phospha e g oup o glucose, ha en e s he cell as glucose-6-phopha e [37]. The EIIAGlc subuni o he EII complex is also in ol ed in he ac i a ion o he adenyla e cyclase (AC). Wi h glucose in he medium, he EIIAGlc-phospha e will be used o supply phospha e o he glucose and he amoun o EIIAGlc-phospha e will hus be educed. As only he phospho yla ed o m o EIIAGlc s imula es he adenyla e cylclase ac i i y, he cAMP le el will all [37]. When glucose, he p e e ed ca bon sou ce o E. coli, is no longe a ailable in he medium, he in acellula concen a ion o cAMP ises. The change o he cAMP le el signals o he bac e ium ha glucose is no longe a ailable and ha i has o swi ch o lac ose me abolism. This is achie ed by binding o cAMP o CAP, which om a cAMP-CAP complex. In u n binding o his complex o a DNA sequence in he p omo o egion jus ups eam om he lac p omo e enhances a ini y o he RNA polyme ase o he p omo o and he eby ini ia es ull ansc ip ion o he lac s uc u al genes [11,31,32]. Wi hou he binding o he ac i a o CAP, he lac p omo o is only ma ginally able o bind and posi ion he RNA polyme ase esul ing in a low le el o ansc ip ion (50- old educed) [31,32]. When glucose is p esen and cAMP concen a ion is low in he cell, he lac ope on is swi ched o , because he ac i a o p o ein CAP can only bind o DNA i i is bound o cAMP. Glucose up ake also in e e es wi h LacY ac i i y, leading o induce exclusion as discussed abo e [37]. Induce exclusion and he d op o he cAMP le el in esponse o glucose up ake oge he media e ca aboli e ep ession - he abili y o glucose o inhibi lac exp ession [37,38]. The logical model should be able o explain how ep ession and ac i a ion con ol mechanisms accoun o he selec i e u iliza ion o he a ailable me aboli es. Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 6 3.2 A Boolean model o he lac ope on The Boolean ne wo k model o he o lac ope on was cons uc ed based on he biological knowledge desc ibed in he p e ious sec ion. Figu e 1a shows he esul ing model in hype g aphical no a ion. The igu e was d awn by using he So wa e CellDesigne ollowing SBGN (Sys ems Biology G aphical No a ion) con en ions as a as possible. Figu e 1b shows again he ules o depic Boolean ga es in a hype g aphical ep esen a ion (as LIH). Table 1 and 2 lis s he ne wo k species and in e ac ions o he logical model. Figu e 1: (a) Boolean model o he lac ope on, hype a cs a e numbe ed acco ding o he anno a ions in able 2. (b) Legend o g aphical ep esen a ions o logic ope a o s. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 b. a. Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 7 Symbol Explana ion AC adenyla e cyclase allolac ose allolac ose cAMP cyclic adenosine monophopha e CAP ca aboli e ac i a o p o ein glucose_ex ex e nal glucose in he g ow h medium LacI-bound LacI ep esso bound o he lac p omo o lac ose_ex ex e nal lac ose in he g ow h medium lac ose_in lac ose inside o he cell LacY LacY pe mease lacZYA he s uc u al genes o he lac ope on: lacZ, lacY and lacA LacZ -galac osidase LacZ lacZYA_mRNA polycis onic lacZYA mRNA PTS-EIIA unphospho yla ed EIIAGlc subuni o he phospho ans e ase sys em Table 1: Species o he Boolean lac ope on model. Hype a c Anno a ion 1 glucose_ex PTS-EIIA When glucose is impo ed, phospha e ans e om he PTS subuni EIIAGlc o glucose is induced. Ac i e PTS-EIIA in he model co esponds o he unphospho yla ed o m o EIIA. 2 !PTS-EIIA AC Phospho yla ed EIIAGlc ac i a es AC, hus PTS-EIIA mus be o . 3 AC cAMP AC p oduces cAMP h ough con e sion o ATP. 4 cAMP CAP cAMP binds o he ecep o molecule CAP o o m he ac i a o CAP complex. 5 !PTS-EIIA · lac ose_ex · LacY lac ose_in En y o lac ose om he medium in o he cell is enabled by he LacY pe mease, unphospho yla ed EIIA blocks he impo o lac ose. 6 lac ose_in · LacZ allolac ose I galac osidase LacZ is exp essed and lac ose is p esen in he cell, allolac ose is p oduced as a byp oduc o lac ose me abolism. 7 !allolac ose LacI-bound Binding o allolac ose o LacI inac i a es he ep esso . 8 !LacI-bound · CAP · lacZYA lacZYA_mRNA Only when LacI is inac i e and no bound o lacO and he lacZYA s uc u al genes a e p esen and CAP binds, lacZYA_mRNA is p oduced. 9 lacZYA mRNA LacY T ansla ion o lacZYA mRNA p oduces he LacY pe mease. 10 lacZYA mRNA LacZ T ansla ion o lacZYA mRNA p oduces he LacZ -galac osidase. 11 lacZYA mRNA LacA (dashed a cs a e no inco po a ed in he model) T ansla ion o lacZYA mRNA p oduces he LacA hiogalac oside ansace ylase (no included in he model; only o illus a ion). 12,13,14 dashed a cs a e no inco po a ed in he model lac ose_in , glucose-phospha e and LacZ en e he lac ose and glucose me abolism ha a e no conside ed in he model. Table 2: Hype a cs o he Boolean lac ope on model. Exclama ion ma k wi hin he equa ions deno e a logical NOT, do s indica e AND ope a ions. As alida ion o he ne wo k model, well-known expe imen al scena ios should be ep oduced by he model. As desc ibed unde 3.1, he p esence o glucose and lac ose in he g ow h Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 8 medium con ol he s a e o he lac ope on. We app oxima e he beha iou s by on and o s a es o he ne wo k nodes. Fou (22) combina ions o glucose/lac ose in he medium a e possible when conside ing he bina y case These ou scena ios we e simula ed wi h he Boolean model by se ing he inpu cues glucose and lac ose app op ia ely. The esul ing logical s eady s a es we e compu ed wi h CellNe Analyze . The s eady s a e alues o he mos impo an nodes a e shown in able 3 oge he wi h he expe imen ally obse ed s a e o he lac ope on (measu ed by lacY exp ession). bina y model +Glc / +Lac -Glc / -Lac +Glc / -Lac -Glc / +Lac cAMP 0 1 0 1 allolac ose 0 0 0 # LacI-bound 1 1 1 # lacZYA mRNA 0 0 0 # LacZ 0 0 0 # LacY 0 0 0 # lac ose_in 0 0 0 # Obse ed lac ope on o o o on Table 3: Logical s eady s a es o he key nodes in he bina y model and he obse ed lac ope on s a e o ou di e en scena ios wi h glucose/lac ose as subs a es. Fo he i s h ee scena ios he model simula ions a e in ag eemen wi h he expec ed ou come om li e a u e. The s a us o he lac ope on can be deduced om he s eady s a e alue o lacZYA mRNA, i.e. whe he mRNA is p oduced o no . In all hese h ee cases, he model p edic s co ec ly ha he lac ope on is swi ched o . Fo he las scena io, whe e only lac ose is used as a s imulus (glucose is o ), a unique logical esponse o his se o inpu s imuli canno be esol ed (see sc eensho o CellNe Analyze in igu e 2). A logical s eady s a e ( he inal esponse o he sys em) canno be calcula ed o all ne wo k nodes, because in he simula ion he signal canno be u he p opaga ed om lac ose_ex o lac ose_in wi hou he LacY pe mease being in on-s a e. The limi a ions o a bina y iew o biology a e made ob ious: when lac ose is sensed in he en i onmen , he sys em should swi ch on he lac ope on, which is achie ed by binding o allolac ose o he ep esso . The only p oblem is ha he genes o he pe mease and he -galac osidase, which a e needed (i) o enable lac ose o en e he cell and (ii) o p oduce allolac ose a e bo h s uc u al genes wi hin he lac ope on whose ac i i y, as men ioned abo e, depends in u n on he p esence o allolac ose (whose ini ial s a e is no gi en). To esol e his cyclic causali y (induced by a posi i e eedback loop which is also he sou ce o bis able beha io discussed in a la e sec ion) we need a basal exp ession le el o he lac ope on o p oduce some mRNA which will make he sys em wo k o he Glc /Lac+ scena io. Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 9 in es iga e he dynamic e ec s o his addi ional egula o y loop. Again we i ed he model o he da a poin s as in igu e 4a. Figu e 8a shows a ypical ime cou se o he s a e a iables when swi ching om glucose o lac ose u iliza ion ( o a pa ame e se i ing he model well o he LacY da a om igu e 4a). We also s udied he capabili y o his al e ed wi ing diag am o induce bis able beha io (Figu e 8b). As o he o iginal model, he e exis pa ame e se s ha do no induce bis abili y a all whe eas o he s ha e he po en ial. The e is also no clea answe whe he he looped model inc eases he capabili y o he sys em o bis abili y: al hough sligh ly mo e samples om he looped-model gi e be e i o he da a while keeping s ong bis abili y alues (=dis ances o he wo s eady s a es), his is no ye su icien o a gue ha based on he da a we should choose he looped model o e he non-looped one. The samples ha e been ob ained by unning mul iple (connec ed) local op imiza ions, so hey do no ep esen independen samples o he cos unc ion‟s landscape. In o de o mo e obus ly in e model pa ame e s and o pe o m model selec ion, we a e wo king on implemen ing Bayesian easoning and in e ence wi hin he p oposed model. Howe e , his is ou o he scope o his con ibu ion. Figu e 8: Simula ions wi h he “looped model”. The looped model was de i ed om a mul i alued logical model ha accoun s o he nega i e e ec o he lac ose me abolism on he cAMP le el ( he a c AC cAMP in he non-looped (“s anda d”) model in igu e 3 is eplaced by he hype a c AC !lac ose_in cAMP). a) Simula ions o he looped ODE Model. In con as o igu e 4b, he cAMP le el inc eases du ing he ansi ion om glucose o lac ose u iliza ion bu dec eases again when his shi has been accomplished. b): Po en ial o he looped and non-looped model o show bis abili y. Bis able beha io is quan i ied as he dis ance o he wo s eady s a es eached when g owing on low lac ose le els (0.2 in he ODE y model) a e p ecul u ing ei he on lac ose o glucose ( he dis ance is ze o i only one s eady s a e exis s). In bo h he s anda d and he looped model, we can iden i y pa ame e s ha i he da a (low dis ance o LacY) and also exhibi s ong bis able beha io . The cu en sampling me hods a e no su icien o di e en ia e hese wo models. 5 Conclusion In summa y, we used he lac ope on as a pa adigma ic example o demons a e how quali a i e knowledge can be ini ially cap u ed using simple disc e e (Boolean) models and hen s epwise e ined o mul i alued logical models and inally o con inuous (ODE) models. Each modeling o malism and he ( o wa d) ans o ma ions be ween hem a e suppo ed by CellNe Analyze enabling one o swi ch be ween di e en ypes o models wi hou he need o ees ablish he whole model building p ocess. Jou nal o In eg a i e Bioin o ma ics, 7(1):151, 2010 h p://jou nal.imbio.de doi:10.2390/biecoll-jib-2010-151 16 Acknowledgemen s We a e g a e ul o Ka ja Be enb ock o help ul commen s du ing he p epa a ion o he manusc ip . 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