On Process Rate Semantics

On Process Rate Semantics - On Process Rate Semantics Luca...

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2007-12-13 17:48:53 1 On Process Rate Semantics Luca Cardelli Microsoft Research Abstract We provide translations between process algebra and systems of chemical reactions. We show that the translations preserve discrete-state (stochastic) and continuous-state (concentration) semantics, and in particular that the continuous-state semantics of processes corresponds to the differential equations of chemistry based on the law of mass action. The novel semantics of processes so ob- tained equates processes that have the same state occupation dynamics, but that may have different interaction interfaces. 1 Introduction We study stochastic interacting processes : a simple compositional model of stochastic systems, with a natural semantics in terms of continuous time Markov chains. These interacting processes can be translated by an intuitive procedure into a set of chemical reactions from which a continuous seman- tics can be extracted in the form of Ordinary Differential Equations (ODEs). Such a translation estab- lishes a precise connection between process algebra models of biochemical systems, and more tradi- tional models based on chemistry and ODEs. Process algebra interactions are at first sight richer than chemical interaction, so it is not imme- diately clear that the ODEs extracted from the chem- ical translation faithfully represent the behavior of the processes according to the processes own se- mantics. The correspondence is fairly obvious when the process interactions are detangled , meaning when each interaction channel has exactly one source of inputs and one source of outputs. Then, each interac- tion channel corresponds exactly to a chemical reac- tion between two chemical species, and in fact the translation from chemistry back to processes pro- duces detangled systems. In general, though, process interactions can be entangled , meaning that there can be many sources of inputs and outputs on each channel. This is a convenient feature that supports compact ways of organizing models: its effectiveness is indicated by the fact that detangled system can be N 2 bigger than corresponding entangled systems. In this paper we show that these more gener- al process models are still faithful: both the Markov and ODE dynamics of the chemical reactions ex- tracted from process models match the intrinsic dynamics of the processes themselves. A simple example can illustrate the potential problem with such a correspondence. In this intro- duction we limit our discussion to automata , which are those processes th at do not ‚ split ‛ dynamically Figure 1 Automata and chemistry : A r A A A ?a A B A B !a ?a A A A !a a: A+B r A +B a: A+A 2r A +A (a@r) @r (a@r)  
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2007-12-13 17:48:53 2 into more processes, and that can be conveniently drawn as transition diagrams. (Automata are not sufficient to model all of chemistry, however, because a molecule can split into two.) In Figure 1 we have three basic situations and their chemical interpretation as changes in molecule numbers [23][24].
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On Process Rate Semantics - On Process Rate Semantics Luca...

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