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Probabilistic Model Checking for Systems Biology - PRISM

DRAFT22 Marta Kwiatkowska, Gethin Norman, and David Parkerphosphorylated FGFR in main at time t108642p=1p=0.75p=0.5p=0.25p=000 1 2 3 4 5 6t (hours)(a) 10 simulation runsphosphorylated FGFR in main at time t108642p=1p=0.75p=0.5p=0.25p=000 1 2 3 4 5 6t (hours)(b) 100 simulation runsphosphorylated FGFR in main at time t108642p=1p=0.75p=0.5p=0.25p=000 1 2 3 4 5 6t (hours)(c) model checkingFig. 14. Expected amount of phosphorylated FGFR in main compartment at time tobtained through simulation as opposed to model checking, in Figures 14(a) and14(b) we present the results obtained with **PRISM**’s simulator when averagingover 10 and 100 runs respectively and Figure 14(c) presents the same resultswhen using model checking. The graphs show, that increasing the chance ofbeing relocated to the recycling compartment as opposed to the degradationcompartment will increase the amount of phosphorylated FGFR receptors inthe main compartment. This is due to the fact that as FGFR receptors from themain compartment relocated to the degradation compartment will not return tothe main compartment while those relocated to the recycling compartment willeventually return to the main compartment.Consider the difference between the plots, we see a large fluctuation in theamount of phosphorylated FGFR receptors in the main compartment when onlya small number of runs are considered (Figure 14(a)) and as the number ofruns increases these fluctuations diminish (Figure 14(b)), while employing themodel checking approach we obtain ‘smooth’ curves (Figure 14(c)). This canbe attributed to the fact that, when we consider an individual run, a reaction(e.g. a binding, phosphorylation or relocation) occurs (with probability 1) at aspecific time point and there**for**e its influence can be seen at that specific timepoint, while in model checking, an average over all possible runs is considered,and hence the probability of the reaction occurring at a certain time point isalso taken into account.5.2 Exercises1. Based on the model in Figure 12 and the reward structures in Figure 13,write CSL specifications **for** the following properties:(a) ‘the probability that eventually all FGFR receptors get degraded’;(b) ‘if there are l 1 free FGF ligands and l 2 free FGFR receptors in the maincompartment, the probability that the first degradation of an FGFRreceptor occurs after time t is less than 0.1’;(c) ‘if in the main compartment there are more than l FGF:FGFR compoundsphosphorylated, then the expect number of relocations occurringby time t is at least 5.6’.

DRAFT**Probabilistic** **Model** **Checking** **for** **Systems** **Biology** 232. Construct an appropriate reward structure **for** calculating the expected numberof dephosphorylations that occur in the recycling and degradation compartmentsof the model in Figure 12 and write a CSL specification **for** theexpected number of dephosphorylations in the recycling and degradationcompartments by time t.Hint: You will need to add extra action labels. Make sure sure that these areall distinct to avoid unwanted synchronisations between modules.3. Extend the model in Figure 12 with variables to count the number of receptorsthat get relocated to the recycling compartment and the number ofreceptors that return from the recycling compartment to the main compartment.Write CSL specifications **for** the following properties:(a) ‘if l 1 receptors have entered the recycling compartment and nothing hasreturned to the main compartment then, with probability at least 0.55,l 2 receptors will return within the next t seconds’;(b) ‘the probability that l 1 receptors return from the main compartmentbe**for**e the total number of relocations is l 2 ’.Hint: Since nothing leaves the degradation compartment you can use thevariables in this compartment to determine the number of relocations tothis compartment.4. Rewrite the **PRISM** description of Figure 12 using a single **PRISM** module.Check that the new model has the same number of states and transitionsas the original. In addition, using the simulation engine, generate graphssimilar to those presented in Figures 14(a) and 14(b).6 Related WorkIn [8], **PRISM** has been used to study a more detailed model of the FGF (FibroblastGrowth Factor) signalling pathway. The model corresponds to a singleinstance of the pathway, i.e. there can be at most one of each molecule or species.This has the advantage that the resulting state space is relatively small, howeverthe model is still highly complex due to the large number of different interactionsthat can occur in the pathway and is sufficiently rich to explain the rolesof the components in the pathway and how they interact. In [4], **PRISM** is usedto model the RKIP-inhibited ERK pathway using an approximate ‘population’approach to modelling in which concentrations are modelled by discrete abstractquantities. Also modelling the RKIP-inhibited ERK pathway, [3] demonstrateshow the stochastic process algebra PEPA [9] can be used to model biologicalpathways. The stochastic π-calculus [18] has also been proposed as a modellanguage **for** biological systems [20, 19]; this approach has so far been used inconjunction with stochastic simulation, **for** example through the tools BioSpi[19] and SPiM [16]. A translation from the stochastic π-calculus to **PRISM** hasalso been developed [14].An alternative is to use the language SBML [11], a computer-readable languagebased on XML **for** representing models of biochemical reaction networks,and the translator from SBML to the **PRISM** modelling language [23]. SBML is

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