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Probabilistic Model Checking for Systems Biology - PRISM

DRAFT2 Marta Kwiatkowska, Gethin Norman, and David Parkerthe temporal logic CSL and the probabilistic model checking tool **PRISM**. In Sections3, 4 and 5, we consider three examples to demonstrate both how to modelbiological pathways in the **PRISM** language and how probabilistic model checkingcan be used to analyse such pathways. In Section 3 we start with a simpleset of reactions. Then, in Sections 4 and 5, we extend this simple example, firstby increasing the number of components of each molecular species in the systemand then by extending the possible reactions of the system. Finally, Section 6outlines related topics of interest.2 **Probabilistic** **Model** **Checking** and **PRISM****Probabilistic** model checking is a technique **for** the modelling and analysis of systemswhich exhibit stochastic behaviour. This technique is a variant of modelchecking, a well-established and widely used **for**mal method **for** ascertaining thecorrectness of real-life systems. **Model** checking requires two inputs: a descriptionof the system in some high-level modelling **for**malism (such as a Petri netor process algebra), and specification of one or more desired properties of thatsystem, usually in temporal logic (e.g. CTL or LTL). From the **for**mer, a modelof the system is constructed, typically a labelled state-transition system in whicheach state represents a possible system configuration and the transitions representthe evolution of the system from one configuration to another over time. Itis then possible to automatically verify whether or not each property is satisfied,based on a systematic and exhaustive exploration of the model.In probabilistic model checking, the models are augmented with quantitativein**for**mation regarding the likelihood that transitions occur and the timesat which they do so. In practice, these models are typically Markov chainsor Markov decision processes. In this chapter, we use continuous-time Markovchains (CTMCs), in which transitions between states are assigned (positive,real-valued) rates, which are interpreted as the rates of negative exponentialdistributions. Properties, while still expressed in temporal logic, are now quantitativein nature. For example, rather than verifying that ‘the protein alwayseventually degrades’, we may ask ‘what is the probability that the protein eventuallydegrades?’ or ‘what is the probability that the protein degrades within thours?’. The most common temporal logic **for** this purpose is is CSL (ContinuousStochastic Logic). Furthermore, by adding rewards to a CTMC, we canalso specify properties such as ‘what is the expected energy dissipation throughprotein binding within the first t time units?’ and ‘what is the expected numberof binding reactions be**for**e relocation occurs?’.In the remainder of this section we present an introduction to CTMCs, CSL,and the software tool **PRISM**, which provides support **for** probabilistic modelchecking of CTMCs using CSL. For further details, see e.g. [13].2.1 Continuous-Time Markov ChainsContinuous-time Markov chains (CTMCs), frequently used in per**for**mance analysis,model both continuous real time and probabilistic choice. This is done by

DRAFT**Probabilistic** **Model** **Checking** **for** **Systems** **Biology** 3specifying the rate at which a transition between two states occurs. Formally,we have the following definition.Definition 1. A CTMC is a tuple C = (S, R, L) where: S is a finite set ofstates; R : S × S → R ≥0 is a transition rate matrix; L : S → 2 AP is a labellingfunction.The transition rate matrix R assigns rates to each pair of states, which are usedas the parameter of an exponential distribution. A transition can only occurbetween states s and s ′ if R(s, s ′ )>0 and, in this case, the probability of thistransition being triggered within t time-units equals 1 − e −R(s,s′ )·t . Typically, ina state s, there is more than one state s ′ **for** which R(s, s ′ )>0, this is known as arace condition and the first transition to be triggered determines the next state.The choice of successor state s ′ from state s is thus probabilistic. The probabilityof moving to s ′ is R(s,s′ )E(s) , where E(s) = ∑ s ′ ∈S R(s, s′ ) is the exit rate of state s.The total time spent in s be**for**e any transition occurs is exponentially distributedwith rate E(s). The labelling function L assigns atomic propositions from a setAP to each state of the CTMC. These are used to label states with propertiesof interest.A CTMC can be augmented with reward structures which are used to annotateits states and/or transitions with additional quantitative in**for**mation.Formally, a reward structure **for** a CTMC is a pair (ρ, ι) where:– ρ : S → R ≥0 is the state reward function;– ι : S × S → R ≥0 is the transition reward function.A path of a CTMC C = (S, R, L) is a non-empty sequence s 0 t 0 s 1 t 1 s 2 . . . wheres i ∈ S, t i ∈ R >0 and R(s i , s i+1 )>0 **for** all i≥0. The value t i represents theamount of time spent in the state s i and we denote by ω@t the state occupiedat time t, i.e. s j where j is the smallest index **for** which ∑ ji=0 t i ≥ t. We denoteby Path(s) the set of all (infinite and finite) paths of the CTMC C starting instate s. A probability measure Pr s over Path(s) can then be derived [2].Two traditional properties of CTMCs are transient behaviour, which relatesto the state of the model at a particular time instant; and steady-statebehaviour, which describes the state of the CTMC in the long-run. For a CTMCC = (S, R, L), the transient probability π s,t (s ′ ) is defined as the probability,having started in state s, of being in state s ′ at time instant t. The steady-stateprobability π s (s ′ ) is the probability of, having started in state s, being in state s ′in the long-run. The steady-state probability distribution, i.e. the values π s (s ′ )**for** all s ′ ∈ S, can be used to infer the percentage of time, in the long-run, thatthe CTMC spends in each state.2.2 Continuous Stochastic LogicThe temporal logic CSL originally introduced by Aziz et al. [1] and since extendedby Baier et al. [2] is based on the temporal logics CTL [5] and PCTL[7] and provides a powerful means to specify both path-based and traditional

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