Dynamical Systems and Stochastic Programming: To Ordinary Differential Equations and Back Luca Bortolussi 1 and Alberto Policriti 2,3 1 Dept. of Mathematics and Computer Science, University of Trieste, Italy. luca@dmi.units.it 2 Dept. of Mathematics and Computer Science, University of Udine, Italy policriti@dimi.uniud.it 3 Applied Genomics Institute (IGA), Udine, Italy policriti@appliedgenomics.org Abstract. In this paper we focus on the relation between models of biological systems consisting of ordinary differential equations (ODE) and models written in a stochastic and concurrent paradigm (sCCP stochastic Concurrent Constraint Programming). In particular, we de- fine a method to associate a set of ODE’s to an sCCP program and a method converting ODE’s into sCCP programs. Then we study the prop- erties of these two translations. Specifically, we show that the mapping from sCCP to ODE’s preserves rate semantics for the class of biochem- ical models (i.e. chemical kinetics is maintained) and we investigate the invertibility properties of the two mappings. Finally, we concentrate on the question of behavioral preservation, i.e if the models obtained ap- plying the mappings have the same dynamics. We give a convergence theorem in the direction from ODE’s to sCCP and we provide several well-known examples in which this property fails in the inverse direction, discussing them in detail. 1 Introduction The systemic approach to biology is nowadays a fertile and growing research area, considered by many as a promising track to the understanding of life [41, 1]. A key ingredient of systems biology resides in coupling wet lab experiments with mathematical modeling and analysis of bio-systems [33]. Many mathemati- cal instruments have been used for this purpose, some concerned with qualitative analysis, others encapsulating also quantitative data [32]. Quantitative modeling is essentially dominated by two main mathematical tools: (ordinary) differential equations on one side and stochastic processes on the other [32]. Both these meth- ods are concerned with the study of dynamical evolution of systems; however, they differ in the description of the quantities of interest: differential equations represent them as continuous variables, stochastic processes operate, instead, on discrete quantities. Modeling formalisms mixing discrete and continuous ingredi- ents, like hybrid automata [29], have also been used in modeling bio-systems [2].