A Type System for a Stochastic CLS

Abstract

The Stochastic Calculus of Looping Sequences is suitable to describe the evolution of microbiological systems, taking into account the speed of the described activities. We propose a type system for this calculus that models how the presence of positive and negative catalysers can modify these speeds. We claim that types are the right abstraction in order to represent the interaction between elements without specifying exactly the element positions. Our claim is supported through an example modelling the lactose operon. The Calculus of Looping Sequences (CLS for short) [4, 5, 19], is a formalism for describing biological systems and their evolution. CLS is based on term rewriting, given a set of predefined rules modelling the activities one would like to describe. The model has been extended with several features, such as a commutative parallel composition operator, and some semantic means, such as bisimulations [5, 7], which are common in process calculi. This permits to combine the simplicity of notation of rewrite systems with the advantage of a form of compositionality. A Stochastic version of CLS (SCLS for short) is proposed in [6]. Rates are associated with rewrite rules in order to model the speed of the described activities. Therefore, transitions derived in S CLS are driven by a rate that models the parameter of an exponential distribution and characterizes the stoch astic behaviour of the transition. The choice of the next rule to be applied and of the time of its applicatio n is based on the classical Gillespie’s algorithm [15]. Defining a stochastic semantics for CLS requires a correct en umeration of all the possible and distinct ways to apply each rewrite rule within a term. A single pattern may have several, though isomorphic, matches within a CLS term. In this paper, we simplify the counting mechanism used in [6] by imposing some restrictions on the patterns modelling the rewrite rul es. Each rewrite rule states explicitly the types of the elements whose occurrence are able to speed-up or slow-down a reaction. The occurrences of the elements of these types are then processed by a rate function (instead of a rate constant) which is used to compute the actual rate of a transition. We show how we can define patterns in our typed stochastic framework to model some common biological activities, and, in particular, we underline the possibility to combine the modelling of positive and negative catalysers within a single rule by reproducing a general case of osmosis. Finally, as a complete modelling application, we show the expressiveness of our formalism by describing the lactose operon in Escherichia Coli.