Cellular modelling using P systems and process algebra

PROGRESS IN NATURAL SCIENCEVol. 17 ,No. 4 ,April 2007INVITED LECTURESCellular modelling using P systems and process algebraFrancisco J. Romero-C amperol* , Marian Gheorghe- , Gabriel Ciobanu'John M. Auld- and Mario J. Perez-Jimenex'( 1. Research Group on Natural Computing , Department of Computer Science and Artificial Itelligence ，University of Seville，Avda.Reina Mercedes , 41012 Sevila , Spain ; 2. Department of Computer Science, The University of Sheffield Regent Court , PortobelloStreet , Sheffield S1 4DP , UK ;3. Romanian Academy , Institute of Computer Science , Blvd. Carol I nr. 11 , 700505 , Iasi，Romania )In this paper various molecular chemical interactions are modelled under different computational paradigms. P systemsand π-calculus are used to describe intra- cllular reactions like protein-protein interactions and gene regulation control.Keywords : cellular modelling , P systems , process algebra.1 Introductionwhich molecular systems play a key role.' There areprimarily three uses of models in science : understand-The more we learn about cell systems , the less ,ing , integration of partial knowledge and discovery ofparadoxically，we seem to understand. Most systemsnew features ; prediction , capability to estimate thin the living cell involve the interaction of multipledynamics of a system ; and control , to constrain orcomponents and subsystems. This complexity makesmanipulate a system to produce a desirable output orit increasingly difficult to visualize any system as abehaviour. These objectives can be more easily obwhole and requires models that can be used as tools totained if there is a close structural connection betweendisentangle these apparent messy interactions.biological data and the model itself. By structural wemean that the model itself reflects the structure of theModels , abstractions of the reality onto a mathe-system being studied 1].matical/ computational formalism , should not be seenor presented as representations of the truth , but in-The classical approach to modelling is based onstead as a statement of our current knowledge of thethe mathematical theory of differential equations.phenomenon under research. A model is often moreDifferential equations have been used successfully touseful when proved to be wrong , since it shows thatmodel kinetics of conventional macroscopic chemicalour current understanding of the phenomenon studiedreactions. Nevertheless there is an implicit assump-does not match the reality. Thus ,it helps experimen-tion of continuously varying chemical concentrationtalists as a way to decide which experiments are nec-and deterministic dynamics. Two critical characteris-essary to advance understandingf 11.tics of this approach are that the number of moleculesA good model should at least have four proper-of each type in the reaction mix is large and that foreach type of reaction in the system , the number of re-ties : relevance , computability , understandability andactions is large within each observation interval，thatextensibility. A model must be relevant capturingis reactions are fast.the essential properties of the phenomenon investigat-ed ; and computable so it can allow the simulation ofWhen the number of particles of the reactingits dynamic behaviour , and the qualitative and quanti-spec中国煤化工re slow , which is fre-tative reasoning about its properties. An understand-quenMHssion control in bacteriaable model will correspond well to the informal con-andCN M. H Gus presumptions are in-cepts and ideas of molecular biology. Finally ,a goodvalid and the deterministic continuous approach tomodel should be extensible to higher levels of organi-chemical kinetics is questionable. Instead one has tosations, like tissues ，organs ，organisms ，etc. inrecognize that the individual chemical reaction steps:dence should be addressed. E -mail : fran@ us. eswww. tandf. co. uk/ journals Progress in Natural Science Vol. 17 No.42007occur discretely and are separated by time intervals ofcepted model for interacting systems with dynamicallyrandom length.evolving communication topology. The π -calculus al-lows channels to be passed as data along other chan-Previous attempts to model cellular systems fromnels , and this fact provides a channel mobility. Thisa computational point of view include Petri nets , a-mobility is an important feature and increases the ex-gent-based approach ，L-systems，state charts ，pro-pressive power. The π-calculus has a simple seman-cess algebra ,etc. While each of these approaches cap-tics and a tractable algebraic theory. Starting withtures some of the information regarding pathways andatomic actions and simpler processes ，complex pro-their molecular components ，none fully integratescesses can be constructed in many ways. The processquantitative dynamics，interactions among molecularexpressions are defined by guarded processes , parallelentities and structural organisation of cells.composition P| Q , nondeterministic choice P+ QMembrane computing is an emergent branch ofreplication ! P , and a restriction operator ( ux )Pnatural computing introduced by Gh. Paurt31. Thiscreating a local fresh channel x for a process P. Anew model of computation starts from the assumptionstructural congruence relation providing a static sethat the processes taking place in the compartmentalmantics is defined over the set of processes. I he evo-structure of a living cell can be interpreted as compu-lution of a process is described in π-calculus by a re-tations. The devices of this model are called P sys-duction relation over processes. This relation containstems. Roughly speaking , a P system consists of athose transitions which can be inferred from a set ofcell-like membrane structure , in the compartments ofrules. Different variants have been used to modelwhich one places multisets of objects which evolve ac-molecular interactions 1 , gene networks , and to in-tegrate molecular and gene networkscording to given rules.In the last years there have been attempts to re-Although most research in membrane computingconcentrates on the computational power and efficien-late π-calculus to membrane systems. In[ 13 ], thecy of the devices involved , lately they have been usedtransfer mechanisms involving Na-K pump are deto model biological phenomena within the frameworkscribed step by step and formal verification tools areof computational systems biology being complemen-used to check the validity of the modelling approach.tary and an alternative to more classical approachesIn[ 14 ], the functioning of the same pump is delike ordinary differential equations ( ODEs ) and toscribed and analyzed in the formal framework of Psome recent computational approaches like Petri netssystems. New features such as variable membrane laand π -calculus. In this case P systems are not used asbelling , activation conditions for rules , membrane bilayer and specific communication rules are defined anda computing paradigm , but rather as a formalism fordescribing the behaviour of the system to be mod-utilized to model specific transfer approaches.elled. In this respect several P system models haveIn this paper we will present models developedbeen proposed to describe oscillatory systems 4J , sig-within the framework of membrane computing andnal transductiortsJ , gene regulation control) , quo-process algebra and addressing molecular interactions.rum sensing 6 J and metapopulations 71. These modelsThe aim of the approach is to show differences be-differ in the syntax ( type of the rewriting rules ,tween the above mentioned modelling strategies andmembrane structure ) and the semantic ( strategy ap-their suitability to express local molecular interac-plied to run the rules in the compartments defined bytions. The P system approach relies on Gillespie' s Al-membranes). Some of these models using metabolicgorithnt 15] and uses the tool developed in this realgorithnt 81 , dynamical probabilistic P systems5 7] andspect 16].( multicompartmental ) Gillespie Algorithnt 5] havebeen applied in several case studies'中国煤化工:his paper presents alsointera suitable graphical rep-The π-calculus was introduced by Milner , Par-reserMYHC N M H Grvolved that is supportrow and Walker as a formal language to describe mo-ed by a standard notatiort 16].bile concurrent processesIt is now a widely ac-1 )Pere习穷数据J and Romero-Campero FJ. Moelling gene expression control using P systems : the lac operon ,a case study. To be published.Progress in Natural Science Vol. 17 No. 42007 www. tandf. co. uk/ jourmals3772 Modelling single compartment systemsr2[al→→[ 1Most models of cellular processes so far consist ofprot(r;)= k.lal,i=1 2(2)systems of molecules which interact through a set ofA process is used to specify each molecule of typechemical reactions taking place inside a single fixedvolume or compartment which is assumed to be aa and b in π-calculus. First order reactions are speci-fied using stochastic delays whereas higher order reac-well-stirred solution.tions are specified using communication channelsThere are primarily two different kinds of inter-through which the processes representing the reactantactions inside a single compartment : those known asmolecules interact. In transformation and degrada-protein-protein interactions which comprise basicallytion , a stochastic delay Tk is associated with the pro-transformation , degradation , complex formation andcess a. After this delay the process a behaves eitherdissociation ( dimerisation ), and gene regulation in-as the process b ,in the case of transformation ;or asteractions comprising fundamentally constitutive ex-the empty process in the case of degradation.pression , positive and negative regulation.:=rp.bTp.0prop( τp)= k.| a | (3)In what follows we will present how these differ-ent types of reactions are modelled in P systems andFig. 1 shows the time courses obtained using theπ -calculus. Results will be compared with those ob-differential equation in( 1 ),on the left , and the Ptained using ODE. A general reference for biologicalsystems rules in( 2 ),on the right. We may observeconcepts used in this paper is[ 17 ].the characteristic exponential decay and increase ofthe reactant a and the product b.2.1 Protein- protein interactions2.1.1 Transformation and degradation1.00.9AA molecule a can react to produce another0.8-bmoleculeb or it can be degraded by the cell machin-0.7-ery. The dynamics of these reactions can be modelledusing the exponential decay law. This law states that0.sfthe rate of the reaction or its propensity is proportion-0.4|al to the number of molecules of the reactant a .0.3|a.2FClassically , these reactions have been modelled0.1一/using ODEs where the decay of concentration of the100 200 300 一400 500 600substance a , represented by a variable A , is de-Time (s)scribed by the differential equation in(1 ).dA=- kA(1 )dt500-In P systems ，transformation and degradationare represented using rewriting rules where the objecta is replaced with the object b or is simply removed旨300[in the case of degradation. The compartment wherethe molecules are located is also specified using squarebrackets with a label l which identifies the corre-100-/\asponding compartment. A constant k is associated中国煤化工with the rule so that its propensity can be comput-400500 600MYHCN MH Gged!) :rη[al→[b]Fig. 1. Transformation dynamics for k= 10 's-1.1)|a殖数据。number of objects a in the case of P systems or the number of processes a in the case of process algebra.378www. tandf. co. uk/ journals Progress in Natural Science Vol. 17 No.420072.1.2 Complex formation and dissociation0.9Two molecules ,a and b , can collide and stick0.8together through noncovalent interactions to producea complex c. This complex in turn can dissociate0.6-back into a and b. In biochemistry , these reactions邑0.5are referred to as complex formation , more specifical-ly heterodimer formation when a≠b and homodimer3 0.3formation when a =b ; and complex dissociation.0.2Many important cellular processes depend on0.1complex formation and dissociation , since the binding0~51520253035of a molecule to another one can alter ( regulate ) theTime (s)activity of the complex which can be completely dif-Fig. 2. Time course of dimerisation.ferent from the activity of the single molecules.molecules , are replaced with the object c，represent-The dynamics of these reactions follow the massing the complex. In the same way，in the complexaction law , which states that the rate or propensity ofdissociation rule , the object c is replaced with the ob-the reaction is directly proportional to the product ofjectsa and b. The compartment in which the reac-the number of the reactant molecules. Thus，twotions take place is specified using square brackets andconstants k, and k2 are associated with the complexa label l.formation and dissociation reactions respectively sotheir rates or propensities can be computed.The mesoscopic constants k, = 0.048 molec 1s 1 and k2=0.5s-' ,computed from the macroscop-These two chemical reactions lead to the threeic ones ,k1 and k2 ,according to[ 15 ] , are associateddifferential equations in( 4 ),one for the concentra-with the corresponding rules in order to compute theirtion of each chemical species a ,b and c. Followingpropensities. .the mass action law the concentrations of the reac-tants a and b , represented by the variables A andrq:[a + b]→[c]B , decrease according to the term k 1AB ( complex(ki Ia 11b 1ifa≠bformation ); whereas they increase according to theterm k2C ( complex dissociation ), where C repre-prop( r1)=，'。La 1(la 1-1)ifa= b2sents the concentration of c. On the contrary , theconcentration of the complex c is increased by kyABr2 :[c]→[a + bland decreased by k2C.prop( r2)= k2°I cI(5)dA=- k,AB + k2CdThe processesa ,b and c specify the behaviour of thedB(4)corresponding molecules in the π - calculus formalismin( 6 ). The communication channel bindp, representsdC华= k,AB- k2Cthe complex formation reaction ; whereas the stochas-tic delay Tp。 represents the complex dissociation reac-The graph in Fig. 2 represents the time coursetion.of dimerisation obtained after solving the systems ofODEsin(4)for k=0.3 μuM-'s-1 ,k2=0.5 s 1. The processes a and b have complementaryand initial conditions[A ]=[B]=1 pμM and[ C]=中国煤化工rindp; ? and bindp; !,throwith the rate charac-0 μM.terisMYHc N M H Grter the conmunicationIn P systems , the complex formation and disso-the process a behaves as the process c，which repre-ciation reactions are specified using the rewriting rulessents the complex , and the process b is replaced within( 5 ) which take the same name as the reactionsthe empty process. The complex dissociation reactionthey represent, In the complex formation rule the ob-is specified in the process c using the stochastic delayjects a and 效周representing the correspondingTk。. Once this delay is completed the processc is re-Progress in Natural Science Vol. 17 No. 42007 www. tandf. co. uk/ jourmals37960p (a)placed with the processes a and b representing theunbound molecules.50fa := bindp, ?cprop( bindp)= ki |a1I b |10fb := bindg: !0prof(τp,)= k2 |c 1c := τp..(aI b)30}(6)The case of homodimer formation ( complex for-mation with two molecules of the same chemical5一10152025. 30 35specie) is treated in π-calculus in a different way.Time (s)The processes a , representing molecules of type a.600 (b)have both complementary channels bindk; ? and5000bindk; ! Observe that , in this case the constant is40002halved since the natural encoding of heterodimer for-3000mation in π -calculus produces an artificial doubling ofineracions lal(a|- 1)inteadofla(lgl-1),1000-hence by halving the constant we compensate thisreplication in order to get the expected propensity ，05方10方.202530355kila|(|a|-1 )a := bindk; ?c + bindk; .0Fig.3. Evolution of the number of molecules over time for com-plex formation and dissociation dynamics in a Golgi body( a ) and(7 )yeast cell( b ). Blue line represents molecules A and B , red line rep-c := rp.(a| a)resents the molecule C.Although the level of noise differs considerably inIn Fig. 3 we study the effect on the dynamics ofdifferent volumes , the equilibrium is reached approxi-dimerisation of the volume of the compartment wheremately at the same point in time.the reactions take place. The volumes considered arethat of a golgi body≈10- 16l and yeast cell≈10 141.Finally , let us highlight that these results sup-The constants associated to the complex formationport the use of different Gillespie' s algorithms ( mul-rule for the different volumes are ki = 4.8x 10ticompartmental Gillespie' s algorithnt5J )in the com-molec-'s-1 and k' =4.8X 10-5molec 1s-1 ; thepartments of a cell system , since the difference in vol-ume highly influences the level of noise of the system.constant associated to the complex dissociation doesnot depends on the volume of the compartment and so2.2 Gene regulation controlit does not change according to[ 15 ]We now discuss how gene regulation control inNote that the smaller the volume , the higher theprokraryotes can be specified in different formalisms.level of noise is. Observe the noisy behaviour in theWe deal only with prokraryotes since we are focusinggolgi body( Fig. 3( a)) and how the dynamics getson single compartment systems. In contrast tosmoother in a yeast cell as the volume increases and soprokaryotic cells ，in eukaryotes there are two com-the number of molecules( Fig.3( b)). In the case ofpartments involved in gene regulation , namely , thelarge volume( considered by ODE models ) , and largenuc中国煤化工and more complex pro-number of molecules ，the time series given by thecesseplace , and the cyto-ODEs in( 4 )and plotted in Fig. 2 will describe faith-plasr:AHCNM H.Gost trascripional con-fully the dynamics in dimerization. Nevertheless , fortrol and translation occur.small volumes ,as it is the case in the organella of eu-In this paper for simplicity transcription andkaryotic cells ，the approach based on ODEs is ques-translation are represented as individual reactions.tionable.Nevertheless , in living cells transcription and transla-380www. tandf. co. uk/ journals Progress in Natural Science Vol. 17 No.42007tion involve many interactions between RNA poly-dP= kzRNA- kP(8)merase , DNA , mRNA and ribosomes that take placein a concurrent manner ; for example , before a geneFig.4( left ) shows the solution of the differen-has been completely transcribed , ribosomes can starttial equations in( 8 ). Note that the protein concen-transcribing the growing mRNA. In the frameworkof P systems and π -calculus , transcription and trans-tration reaches equilibrium and the cell keeps it con-lation are explicitly represented as taking place in par-stant to that value. Also observe that the protein con-allelI18].centration is considerably higher than the mRNA con-centration.The central dogma of molecular biology statesThe processes of transcription ，translation andthat genetic information is stored in the DNA , tran-scribed into messenger RNA ( mRNA ), and thendegradation are represented in P systems by rewritingtranslated into proteins. This picture is much morerules with the same names as the processes they rep-complicated since certain proteins , called transcriptionresent , see( 9 ). Note that , since all these reactionsfactors, act as regulators in the transcriptions ofare assumed to be first other reactions , we can use thesame constants as in differential equations accordinggenes , either positively or negatively ; that is ,an in-crease in the amount of transcription factor leads toto[ 15 ].either more or less gene expression. This provides ar1 :[genel-→[ gene + rna ]feedback pathway by which genes can regulate theprop( r)= k,expression of other genes and so of the production ofthe proteins encoded by them.r2:[rmal-[ 1prof( r2)= k21 rna |2.3 C onstitutive expression(9)r3:[ rna l→[rna + p]We start by modelling genes whose level of ex-prop( r3)= kz1 rna |pression does not depend on transcription factors.These genes are called constitutive genes or house-r4:[pl°→[ ]keeping genes. This kind of genes are transcribedprop(r4)= k41 p|continually at a relatively constant level compared tofacultative genes , which are only transcribed whenThe time courses obtained when simulating the Pneeded.system rules in( 9 ) are represented in Fig. 4( b ).In this case from the gene encoded in the DNANote that these results are in accordance with the pre-the mRNA is transcribed. This mRNA is then trans-vious results from differential equations plotted inlated into the protein product associated to the geneFig.4( a ). Nonetheless ODEs do not capture theand the mRNA is also degraded by the cell machin-noise , we can see in Fig.4( b) ,due to the low num-ber of molecules. This type of noise is also referred inery.physics as shot noise ; this noise occurs when theThis system is represented by the two differen-finite number of particles that carry energy is smalltial equations in( 8 ) , where the variable RNA repre-enough to give rise to detectable statistical fluctua-sents the concentration of mRNA and P representstions in a measurement 20].the concentration of the protein product encoded byIn the π -calculus specification in( 10 ), the pro-the gene. Transcription is assumed to take place at aconstant rate k,. Degradation of the mRNA follows ,cess gene represents transcription by producing theas before , the exponential decay law characterized byprocesses gene and rna after a stochastic delaythe term k2 RNA. Translation is assumed to be pro-中国煤化工astic delay Tk2 andzportional to the concentration of mRNA, kjRNA.reprC N M H Gradation respectively.Finally , protein degradation is modelled in the termFinally ,the process p represents the protein productk.pi9].of the gene and specifies degradation using thedRNA= kq- k2RNAstochastic delay Tk。.dt1 ) See fotnote on page 376.Progress in Natural Science Vol. 17 No. 42007 www. tandf. co. uk/ jourmals3810.6r (a)Michaelis-Menten kinetics are often used inODEs to model positive regulatiort 191of a gene by anactivator whose concentration is represented by a vari-0.4able A. The three processes , binding , debinding andtranscription , presented in( 11 ), are simplified such0.3-that the rate of transcription is assumed to depend onthe concentration of the activator A. This is repre-sented on the so- called Michaelis constant Km , whichreflects the affinity between the activator and promot-er ,and on the maximum turn-over rate , Vm , which020406080100120140160180200in this case corresponds to the maximum transcriptionTime (s)450r (b)rate. Thus the rate of transcription is given by the400_Ak,+ k,350term VmKm+A, where Km =k.00-A + gene之A. geneA.gene+rna(11)200The two equations in( 12 ) model gene positive100regulation. For the mRNA we have the correspond-ing Michaelis Menten dynamics, VmK + A ,minus20406080100120140160180200Time (S)the exponential decay , ky RNA , specifying the mR-NA degradation. The rate of translation is assumed tobe proportional to the concentration of mRNA ,of constitutive gene expression by using P systems( b). Blue linerepresents the protein P and black line represents mRNA.k2 RNA and protein degradation follows the exponen-tial decay law ,k3P.gene := Tr,.( gene| rna )dRNAprop(τk, )= kydK,+ A- ky RNA( 12)rna := Tp.0+ rp,(rma 1 p)k,RNA - kgPd.prop( k, )= k21 rnaNote that this approach does not model explicitlyp :=Tk.0the binding and debinding of the activator to the pro-prop( Tk, )= k; | rnamoter of the gene and so it does not capture theprop(Tk, )= kg1 p|( 10)boolean nature of the activation of genes. These reac-tions are specified explicitly in P systems using thetwo first rules in( 13 ) which follow the same dynam-Constitutive gene expression is modelled in aics as complex formation and dissociation. The othersimilar way using π-calculus in[ 12 ]; although，inrules represent transcription ，RNA degradation ,contrast to( 10 ) , transcription and translation are nottranslation and protein degradation in the same waymodelled explicitly ，instead the protein product isas in the constitutive expression of genes.produced from the gene after a stochastic delay.r1:[a+gene]>[a. gene ]2.4 Positive regulation中国煤化工Unlike constitutive genes , facultative genes areonly expressed when needed according to some signalsMHCN MHGr + gene ]received by the cell from its surroundings. Activatorsprop( r2)= kare transcription factors which bind to the promoterr3:[a.gene1->[a. gene + rna ]of genes and activate their expression by recruitingpolymeras:prop( r3)= k,382www. tandf. co. uk/ journals Progress in Natural Science Vol. 17 No.42007r:[ma]°→[卫tion and dissociation using the processes a and gene .T ranscription，translation and degradation are mod-prop( r4)= k, | rnaelled using similar processes and stochastic delays as inrs:[ mal-→[ra + p]the case of constitutive expression.a := bind; ?(prop( rs)= k, I rna |prot( bindp;)= k;la 1r。:[p]°→[ ]prop( rg)= k31 p|( 13)gene := bindk, !.a. gene .prot(τkg )= k,The time courses of positive gene regulation ob-tained using our simulator , available from[ 16 ], anda.gene := re(a | gene )+ rp(a.gene | rna )the rules in( 13 ) are depicted in Fig. 5(b). As the .prop(rp。)= k,case of constitutive expression , our results are in a-greement with those of ODEs presented in Fig. 5 .rna := Tp.0+ Tp,.( rna I p)( a). Nevertheless , ODEs do not represent the levelprop( r,)= k, | rnaof noise ; which is more noticeable here than in theprevious case. This is due to the boolean nature of thep :=Tp.0activation of a gene by an activator which producesprop( Tk, )= k2 | rnathe so-called burst in transcription. This noise is re-ferred to as telegraph noise in physics , in analogy toprop(rk, )= k31 p|(14)the telegraph which is either silent or in a sendingIn[ 12 ] positive gene expression is modelled us-state as the operator taps.ing π-calculus ; although , unlike( 10 ), binding anda)debinding of the activator to the gene promoter , tran-0.5-scription and translation are not modelled explicitly ,instead a communication channel , through which the.4Fprocesses representing the activator and the gene0.3}communicate are used to produce the protein productdirectly from the gene..26 |2.5 Negative regulation0.19As opposed to positive regulation in some condi-20406080100120140160180200tions cells do not need the protein product encoded byTime (s)a gene ;in this case this gene is turned off or repressed350[by transcription factors called repressors. Repressorsoofbind to the promoter site of genes blocking it so thatMwMWh μNwpolymerase cannot bind to it , thus preventing genes250-from being transcribed.00-In this case Michaelis Menten dynamics is also50-used usually to model gene repression or negative reg-ooulatiort191For the mRNA , a basal transcriptionrate , leakiness ,is assumed to take place at a rate k,0 50100150200250300350400450500simi中国煤化工on. Here the term rep-rese1dynamics has negativesign:YHCc N M H Ggulaion ,an icre ofFig. 5. Time course of positive gene expression( a ) and dynamicsrepressors produces on the transcription of the gene.of positive gene regulation using P systems( b). Blue line representsFinally , mRNA degradation follows the exponentialthe protein P and black line represents mRNA.decay law characterised by the constant k2.The w-calcylyus specification also represents bind-ing and debindingin the same way as complex forma-The dynamics of the protein product of the geneProgress in Natural Science Vol. 17 No. 42007 www. tandf. co. uk/ jourmals383is similar to the case of constitutive expression andthe intra-cellular level. The capabilities of the twopositive regulation.modelling approaches are contrasted with respect todRNAR_classical continuous models based on ordinary differ-dt= k1-kIR+R- k2RNA( 15)ential equations.dP= kzRNA- k.PReferences1 Bower J and Bolouri H. Computationl modelling of genetic and bio-Similar to positive regulation the P system andchemical networks. J. Comput. Neurosci. ,2001 ,13 :217- -235π-calculus specifications represent explicitly the bind-2 Regev A and Shapiro E. 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T ransactions on Computational Systems Biol-3 Conclusionsogy VI , LNBI 4220 ,2006,113-149中国煤化工ig tasriptionl regualatoryIn this paper some fundamental chemical interac--1129tions occurring at the cellular level are described by20HCNM H Ged CA. White noise in MOStransistors and resistors. IEEE Circuits Devices Mag. ，1993 ,using two modelling approaches , P systems and π2329.calculus. The interactions analyzed in the paper cover