Hybrid semantics for Bio-PEPA

Vashti Galpin
2014 Information and Computation  
This paper investigates the stochastic, continuous and instantaneous (hybrid) modelling of systems defined in Bio-PEPA, a quantitative process algebra for biological modelling. This is achieved by mapping a Bio-PEPA model to a model in stochastic HYPE, a process algebra that models these three behaviour types in a compositional and structured manner. The novel mapping between process algebras provides another method of analysis for Bio-PEPA models and presents the modeller with a
more » ... stochastic HYPE model which can itself be easily modified and is only a small constant larger in size than the Bio-PEPA model. The structure of the stochastic HYPE model generated has desirable properties and also gives a general framework for modelling biochemical systems where the advantages of both stochastic and deterministic simulation are required. Thresholds are introduced for each reaction, and when all values are above these thresholds, the reaction is treated deterministically. However, if a relevant value is below a threshold, the reaction is treated stochastically (as are the changes in species quantities as a result of that reaction). It is proved that in the purely deterministic case and in the purely stochastic case, the stochastic HYPE model has the same behaviour as the Bio-PEPA model when considered purely deterministically and purely stochastically, respectively. Furthermore, addition of instantaneous events in the style of Bio-PEPA with events is illustrated, and a proposal for mapping Bio-PEPA with delays (Bio-PEPAd) to stochastic HYPE is presented. the behaviour of the system. To obtain an idea of average behaviour, multiple traces must be obtained. This approach is closer to our understanding of biochemistry than the ODE approach but can be computationally expensive. The low cost of ODE approaches has led to this approach being dominant in past decades. However, as computer power has increased and become more parallel, stochastic simulation is now frequently used. It is an example of an embarrassingly parallel problem as each simulation can be executed independently of another, hence allowing multiple simulation runs to be spread over many processors. Hybrid approaches have also been proposed as a way to mitigate the cost of stochastic simulation where part of the simulation is run stochastically and part is run by other faster methods including ODEs. Some of these approaches are static in nature in that the identification of the part of the system that will be run stochastically is done in advance of simulation. Others are dynamic and the simulation algorithm switches between stochastic and other approaches as appropriate, defined by thresholds on the values of molecule numbers, reaction speeds or propensities. Examples of these are discussed in two recent survey papers [36, 58] . A recent development in systems biology is the use of formal languages to describe reaction systems. These languages or methods were originally developed in computer science for the specification, verification and performance modelling of human-created systems such as networks and computer systems. The languages used for biological modelling are translated into the mathematical structures of interest: ODEs for input into an appropriate solver or input for a general stochastic simulation program. One of the advantages of these languages is the ability to provide an unambiguous description which is separate from programs that perform analysis and simulation. Examples are κ-calculus [33], π-calculus [63, 61, 6], Beta-binders [60], Bio-Ambients [62], sCCP [11], continuous π-calculus [54] and LBS [59]. This paper focusses on Bio-PEPA [30] which was developed from the stochastic process algebra PEPA [48] . Bio-PEPA models can be analysed in a number of ways including stochastic simulation and deterministic simulation, but it has not been possible to analyse it using a combination of these methods. This analysis will be provided by mapping a Bio-PEPA model to a stochastic HYPE model. Stochastic HYPE is a process algebra encompassing instantaneous, stochastic and continuous behaviour 1 , and whose semantics are defined by transition-driven stochastic hybrid automata (TDSHA) [13] . The decision to map to stochastic HYPE rather than its semantics is due to the fact that the mapping provides a well-structured language-based model which is easy to modify and also provides a general framework for describing reaction systems as stochastic hybrid models. When a Bio-PEPA model is expressed in stochastic HYPE, it is possible to treat each reaction (and changes to quantities of species involved in the reaction) stochastically or deterministically. The treatment of a reaction is determined by thresholds on reactant species and the reaction rate. If any value is below its threshold, the reaction will be treated stochastically and if all values are above their thresholds, the reaction and the species will be treated deterministically. Note that a species can be treated stochastically in one reaction and deterministically in another, thus allowing for stochastic treatment only when necessary. The structure of this paper is as follows. In the next section, the advantages and the disadvantages of the continuous deterministic and the stochastic discrete approaches are considered, motivating the combination of the two approaches. After that Bio-PEPA is introduced with an example which will be used through this paper to illustrate various notions. Stochastic HYPE and the mapping of Bio-PEPA to stochastic HYPE is given followed by the properties of the HYPE model. The next section is a case study after which a discussion of the general approach appears. Next, the addition of events in the style of Bio-PEPA with events and a proposal to extend HYPE to allow the mapping of Bio-PEPA with delays are described and finally related work, further work and conclusions are presented. Deterministic, stochastic and inbetween Recent articles have surveyed the different approaches that are available for deterministic, stochastic and hybrid modelling [58, 36] and give guidelines for choosing between the two main approaches of deterministic and stochastic [64, 70] . Criteria include the specific objective of the model, limitations in terms of computational power and time, availability of experimental data, and the level of detail or accuracy required. As usual, Box's statement applies, "all models are wrong but some are useful" [15] .
doi:10.1016/j.ic.2014.01.016 fatcat:4wbcyfznwzbqdfma2z3lgauzsq