We present stochastic process algebra models of a Circadian clock mechanism used in many biological organisms to regulate time-based behaviour. We compare modelling techniques from different modelling paradigms, PEPA and stochastic $pi$-calculus

Bradley, Jeremy T.

Thorne, Thomas

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Stochastic Process Algebra models ofa Circadian ClockJeremy T. Bradley Thomas ThorneDepartment of Computing, Imperial College London180 Queen’s Gate, London SW7 2BZ, United KingdomEmail: jb@doc.ic.ac.ukAbstractWe present stochastic process algebra models of a Circadian clockmechanism used in many biological organisms to regulate time-basedbehaviour. We compare modelling techniques from different modellingparadigms, PEPA and stochastic pi-calculus.1 IntroductionMany biological systems make use of a Circadian clock to keep track of the pas-sage of time. The Circadian clock has evolved to create periodic concentrationsof chemicals, in such a way that cells can regulate its behaviour according tothe time of day or season of the year [1, 2].With recent innovations in the use of stochastic process algebras to generate sys-tems of ODEs or simulation models for huge state spaces, we will use the modelof the Circadian clock to compare the biological modelling process in PEPAand stochastic pi-calculus. This will be with a view to comparing simulationand numerical solution results of the same model in due course.2 Circadian ClockFigure 1 (taken from [1]) shows a biological graphical description of a Circadianclock with two DNA molecules for proteins A and R interacting through theirrespective mRNA molecules.In the diagram, there are two DNA molecules, DA and DR, which describeproteins A and R. These DNA molecules generate mRNA molecules which inturn generate their respective proteins. High concentrations of the R proteinabsorb the A molecules and therefore R acts as a repressor for A. In the absenceof A, R will degrade naturally. However, as A also acts as an activator for thegeneration of mRNA for both A and R molecules, we have several opportunities1Dagstuhl Seminar Proceedings 06161Simulation and Verification of Dynamic Systemshttp://drops.dagstuhl.de/opus/volltexte/2006/705Fig. 1. The biological network for the Circadian clock.for constructive and destructive feedback within the system. The result is thatthe concentration of A should oscillate in anti-phase to the concentration of R.The spikes in the concentration of A act as the ticks of a clock for an organism.We use Figure 1 to create both stochastic pi-calculus and PEPA models of thegene network.3 Model3.1 Modelling experience3.1.1 Molecule creationThe major difference between modelling in stochastic pi-calculus and PEPAis the way in which new molecules are generated. In stochastic pi-calculus,it is succinct to have new molecules spontaneously appear in parallel out ofindividual molecule descriptions, as in:DAdef= ταA .(DA |MA) (1)Here, after an exponential delay at rate αA, a DA molecule becomes a DA andan MA molecule. In effect, this means that the DA molecule remains and anMA molecule is spontaneously created.In contrast, PEPA has a notion of a bounded component structure which encour-ages the creation of independent molecule lifecycles which capture an individual2ADADRADA MAADR MR RCbindCbindAbindRγRγAθAθRαAαRαR′ βR δAαA′ βAγCδMR δRδMAδAFig. 2. An equivalent unbounded Petri net model of the gene/protein networkmolecule’s state, even if one of those states is just the potential to create themolecule. The PEPA equivalent of Equation (1) is given by:DAdef= (transA, αA).DAM ′Adef= (transA,>).MA (2)Here the state M ′A represents the concept of there being sufficient resources inthe system that, when driven by a transA action from the DNA molecule DA,an MA molecule is instantiated.3.1.2 Molecule creation: RamificationsThis single modelling difference between the formalisms has large implications.1. To start with the stochastic pi-calculus model can grow unboundedly, gen-erating an indefinite number of MA molecules. Whereas in the PEPAmodel, we would have to pre-specify the number of MA molecules thatthe system was capable of creating using the following system equation:DA ¤¢{transA}(M ′A || · · · ||M ′A)︸ ︷︷ ︸nSuch a system would have the capacity to generate n molecules of MAand no more.3As to which approach is appropriate, that will depend on the modellingsituation and the facets of the system that the modeller is trying to cap-ture.2. The unbounded nature of the stochastic pi-calculus model generates an in-finite stochastic state space, which would make probabilistic model check-ing, in all but the most fortunate of cases, virtually impossible. So if toolssuch as PRISM, PEPA Workbench or ETMCC are to be employed to per-form probabilistic analysis on biological systems, it would seem that thePEPA style of modelling has more potential.It should be noted though that if explicit state-space representation tech-niques are used by the tool, then even if a bounded and finite model isgenerated, only a very small version will be capable of being analysed asthe state space quickly becomes unmanageable.The only practical way to analyse such large models is through continu-ous state-space representation via numerical ODE solution or stochasticsimulation. As yet there is no model checking framework in which thesetechniques can be used.3.1.3 Predefined synchronisation rateAgain comparing the same snippets of process model from Equation (1) and (2),we note that the rate of delay prior to molecule generation is defined as αA. Asdiscussed earlier this process is a succinct way of representing a synchronisa-tion between the environment (the amino acids that are the building blocks ofproteins and mRNA) and the DNA molecule. It could be said that as therewas no explicit definition of how the individual processes participated in thesynchronisation, that this does not produce a composable model. However, acounter argument would quite reasonably suggest that the action was τ -actionanyway and not observable by other processes and that the above example wasan abstraction of underlying cooperation.3.2 Stochastic pi-Calculus ModelDAdef= bindAγA .DA′ + ταA .(DA |MA)DA′def= τθA .(DA | A) + ταA′ .(DA′ |MA)DRdef= bindRγR .DR′ + ταR .(DR |MR)DR′def= τθR .(DR | A) + ταR′ .(DR′ |MR)MAdef= τδMA .Ø+ τβA .(MA | A)MRdef= τδMR .Ø+ τβR .(MR | R)Adef= bindAγA .Ø+ bindRγR .Ø+ bindCγC .Ø+ τδA .ØRdef= bindCγC .C + τδR .ØCdef= τδA .R43.3 PEPA ModelDAdef= (bindADA , γA).ADA + (mkMA, αA).DAADAdef= (unbindADA , θA).DA + (mkMA, αA′).ADADRdef= (bindADR , γR).ADR + (mkMR, αR).DRADRdef= (unbindADR , θR).DR + (mkMR, αR′).ADRM ′Adef= (mkMA,>).MAMAdef= (decayMA , δMA).M′A + (mkA, βA).MAM ′Rdef= (mkMR,>).MRMRdef= (decayMR , δMR).M′R + (mkR, βR).MRA′ def= (mkA,>).AAdef= (bindADA , γA).ADA + (bindADR , γR).ADR + (bindAR,>).AC+ (decayA, δA).A′ADAdef= (unbindADA ,>).AADRdef= (unbindADR ,>).AACdef= (unbindAR,>).A′R′ def= (mkR,>).RRdef= (bindAR, γC).C + (decayR, δR).R′Cdef= (unbindAR, δA).RThe different process definitions represent the different states of the moleculesin the system. The states M ′A, M′R, A′ and R′ represent potential to create themolecules MA, MR, A and R. The system would start in the state with thepotential to create nX molecules of X for X ∈ {MA,MR, A,R}.Circadian def= (DA || DR)¤¢L ((M ′A[nMA ] ||M ′R[nMR ])¤¢M (A′[nA]¤¢N R′[nR]))L = {bindADA , unbindADA , bindADR , unbindADR ,mkMA,mkMR}M = {mkA,mkR}N = {bindAR, unbindAR}3.4 ParametersThe initial conditions and parameter values for the Circadian clock models aretaken directly from [1]: DA = DR = 1 mol, D′A = D′R = MA = MR =A = R = C = 0, which require that the cell has a single copy of the ac-tivator and repressor genes: DA + D′A = 1 mol and DR + D′R = 1 mol.αA = 50h−1αA′ = 500h−1αR = 0.01h−1αR′ = 50h−1βA = 50h−1βR = 5h−1δMA = 10h−1δMR = 0.5h−1δA = 1h−1δR = 0.2h−1γA = 1mol−1hr−1γR = 1mol−1hr−1γC = 2mol−1hr−1θA = 50h−1θR = 100h−154 Work in progressWe have generated and solved the ODE systems for both stochastic pi-calculusand PEPA models and have reproduced the same results as obtained by Vilar etal. [1], in both cases. We note that restricting the capacity of the PEPA modelto make key proteins upsets the phase of the Circadian rhythm, but does notdestroy it altogether. We plan to explore this issue in a future publication.We also intend to study further how the models differ from each other giventhe differing philosophies behind bounded (PEPA) and unbounded (stochasticpi-calculus) modelling techniques. We would also like to see how the boundedcapacity rate semantics of PEPA (use of the min function in synchronisation)compare to the mass action semantics of most biological and chemical modellingparadigms. We would like to know, in particular, what cooperation semanticsin PEPA correspond to the mass action semantics in the final ODE/simulationmodel.AcknowledgementsJeremy Bradley was supported in part by the Nuffield Foundation under grantNAL/00805/G.References[1] J. M. G. Vilar, H. Y. Kueh, N. BarkaiDagger, and S. Leibler, “Mechanisms ofnoise-resistance in genetic oscillators,” PNAS, vol. 99, pp. 5988–5992, April2002.[2] N. Barkai and S. Leibler, “Biological rhythms: Circadian clocks limited bynoise,” Nature, vol. 403, pp. 267–268, January 2000.6

Stochastic Process Algebra Models of a Circadian Clock

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Stochastic Process Algebra Models of a Circadian Clock

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