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Probabilistic Model Checking for Systems Biology - PRISM

DRAFT18 Marta Kwiatkowska, Gethin Norman, and David Parkerprobability l relocations by time t10.80.60.40.2l=0l=2l=4l=6l=8l=1000 5 10 15 20 25 30t (minutes)(a)expected phosphorylated FGFR at time t108642N=10N=8N=6N=4N=200 5 10 15 20 25 30t (seconds)(b)expected phosphorylated FGFR at time t108642N=10N=8N=6N=4N=200 5 10 15 20 25 30t (minutes)(c)Fig. 11. Transient properties **for** the population example: (a) probability of l relocationsby time t; (b/c) expected phosphorylated FGFR at time t (seconds/minutes).– (fgfp + bndp=l) ⇒ R{“phos”} ≥6.8 [ I =t ] - ‘if l FGFR receptors are phosphorylated,the expected number of FGFR receptors phosphorylated t secondslater is at least 6.8’;– R{“bind”} =? [ F reloc=l ] - ‘the expected number of times that an FGF ligandbinds with an FGFR receptor be**for**e l FGFR receptors have relocated’.As in the previous section, we also give an illustration of the kind of quantitativeresults than can be obtained. Figure 11(a) shows the probability of l relocationsby time t **for** a range of different values of l when there are initially both 10 FGFligands and 10 FGFR receptors. Figure 11(b)–(c) shows the expected amount ofphosphorylated FGFR at time t **for** a variety of initial configurations and twodifferent time scales (seconds and minutes). The latter graphs demonstrate thatafter a rapid increase in phosphorylated FGFR there is a gradual decrease as theamount of relocated FGFR increases and starting from a larger concentration ofFGFR and FGF leads to an increase in the amount of phosphorylated FGFR.4.4 Exercises1. Based on the model in Figure 9 and the reward structures in Figure 10, writeCSL specifications **for** the following properties:(a) ‘the probability that the number of free FGF ligands does not drop belowl 1 until after l 2 FGFR receptors have been relocated’;(b) ‘if the number of phosphorylated FGFR receptors is at least l, then theprobability that all FGFR receptors have been relocated by time t isgreater than 0.6’;(c) ‘the probability that no FGF ligands are bound to FGFR receptors untiltime t 1 and, be**for**e another t 2 seconds pass, a binding occurs’;(d) ‘if at most l FGFR receptors are bound, then the expected number ofbindings that occur within t seconds is at least 12.4’.2. Construct appropriate reward structures **for** properties of the model in Figure9 relating to the expected time that at least l FGFR receptors are phosphorylatedand the expected number of phosphorylations.Hint: You may first need to change the **PRISM** model description **for** this.Using these reward structures, write specifications **for** calculating:

DRAFT**Probabilistic** **Model** **Checking** **for** **Systems** **Biology** 19(a) ‘the expected time that at least l FGFR receptors are phosphorylatedup until time t’;(b) ‘the expected time that at least l FGFR receptors are phosphorylatedbe**for**e M−l receptors have been relocated’;(c) ‘the expected number of phosphorylations be**for**e a relocation occurs’;(d) ‘the expected number of phosphorylations by time t’.3. Extend the model in Figure 9 with variables to count the number of timesthat an FGF ligand binds and the number of times that an FGF ligandunbinds with an FGFR receptor and write CSL specifications **for** the properties:(a) ‘the probability that at most l reactions between FGF ligands and FGFRreceptors (i.e. bindings or unbindings) occur within the first t seconds’;(b) ‘if no FGF ligands have bound to FGFR receptors, then the probabilitythat l bindings occur be**for**e any of unbindings occur is at least 0.5’.5 A more complex model: CompartmentsIn this section we extend the population model of the previous section to includebehaviour relating to the relocated FGFR. The model is motivated by the factthat FGF ligands can only interact with FGFR receptors at the cell surface;relocation causes FGFR to move inside the cell where it loses sight of the ligand.Inside the cell we suppose that either the FGFR is recycled back to the surfaceor it is destroyed by degradation. To incorporate this behaviour into our modelwe divide the system into the following compartments:main compartment: in this compartment FGF ligands interact with FGFRreceptors (i.e. at the cell surface);recycling compartment: this compartment receives FGFR receptors from themain compartment and at a rate of k 6 (=1/30 min −1 ) re-introduces theFGFR receptors back into the main compartment;degradation compartment: this compartment also receives FGFR receptorsfrom the main compartment, but in this case the proteins are degraded at arate of k 7 (=1/15 min −1 ).In this model, we keep the assumption that the ligand FGF disappears if it isbound to FGFR when relocation occurs. In addition, we suppose that the choicebetween whether the proteins reach either the recycling or degradation compartmentfrom the main compartment is probabilistic: the recycling compartmentis reached with probability p and the degradation compartment is reached withprobability 1−p. Considering the rates in the CTMC representation of this modelit follows that if r is the rate in the CTMC that FGFR receptors leave the maincompartment, then the rate of FGFR being relocated to the recycling compartmentis p·r and the rate of being relocated to the degradation compartment is(1−p)·r.

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