Department of Computing Imperial College London Continuous and spatial extension of stochastic π-calculus Final Year Project Anton Stefanek Marker: Dr. Maria Grazia Vigliotti Second Marker: Dr. Jeremy Bradley June 26, 2009 ii Abstract In this project, we work towards a continuous and spatial extension of stochastic π calculus. The continuous semantics is a useful alternative to the discrete semantics and has been recently provided for other process algebras. Ability to express spatial properties of the models is an important practical extension, specially in Systems Biology. Inspired by previous work [7][20], after showing results on process aggregation in stochastic π calculus (Sπ) in form of multisets, we formulate and informally justify the continuous semantics. We show that this is tractable (in the sense that the set of resulting ordinary differential equations (ODEs) is finite) for the case of a subset of stochastic π calculus called Chemical Ground Form (CGF) defined in [7]. We attempt to tackle the problem of potentially infinite set of ODEs. We define two notions of finiteness, one allowing a direct analysis and another allowing further investigation of convergence results. We also provide an algorithm translating models in stochastic Sπ into CGF in case the finiteness is satisfied. We give a syntactical restriction of Sπ which guarantees finiteness. We intuitively and informally describe another condition on Sπ models guaranteeing finiteness. We explore the relationship between the continuous and discrete semantics. We experimentally look at the effect of scaling populations of processes in various existing models. We define a simple spatial extension of Sπ. We bring the aggregation results to this exten- sion and define an extended continuous semantics. We give an original example demonstrating advantages of this extension. As an essential co-product, we develop an efficient, user friendly and portable tool implementing the above formalisms, with comparable simulation performance with the state of the art Stochastic Pi Machine (SPiM) simulator[32]. We also collect some of the available models in stochastic π calculus from Systems Biology, whose analysis can be enriched by the additional continuous semantics. iii Acknowledgments Firstly, I would like to thank my supervisor Dr. Maria Vigliotti for her support and enthusiasm. I would also like to thank to Dr. Jeremy Bradley for his helpful discussions. I would like to thank to all my friends for looking after me during the recent months. Finally, I am grateful to my parents for their total love and support. iv CONTENTS v Contents 1 Introduction 1 1.1 Project Aim . .1 1.2 Contributions . .2 2 Process algebras in Systems Biology 5 2.1 Mathematical background . .6 2.1.1 Exponential distribution . .7 2.1.2 Sampling from random variables . .9 2.1.3 Markov chains . 10 2.1.4 Simulation of Markov chains . 10 2.1.5 Gillespie Algorithm . 10 2.1.6 Next reaction method . 12 2.1.7 Numerical algorithms for solving systems of ODEs . 13 2.2 Stochastic process algebras . 13 2.2.1 Stochastic π calculus . 14 2.2.2 Continuous π calculus . 15 2.2.3 Bio-PEPA . 15 2.3 Spatial extensions . 16 2.4 Summary . 16 3 Stochastic π calculus 17 3.1 Syntax . 17 3.2 Substitution and alpha congruence . 20 3.3 Semantics . 22 3.4 Structural congruence . 25 3.5 Simulation . 26 3.6 Prime processes . 28 3.7 Summary . 29 4 Continuous semantics of stochastic π calculus 31 4.1 Translation to CGF . 36 4.2 Summary . 39 5 Finiteness conditions and convergence investigations 41 5.1 Conditions for finiteness . 43 5.1.1 Syntactic restriction . 43 5.1.2 Restriction on private names . 44 5.2 Relationship between continuous and discrete semantics . 45 6 Spatial extension of stochastic π calculus 53 6.1 Syntax . 54 6.2 Semantics . 54 6.3 Simulation . 55 6.4 Continuous semantics . 61 6.5 Relationship to Sπ .................................... 62 6.6 Summary . 63 vi CONTENTS 7 Implementation 65 7.1 Architecture overview . 66 7.1.1 Used libraries . 67 7.2 Implementation details . 67 7.2.1 ANTLR grammars . 67 7.2.2 Process representation . 68 7.2.3 Higher level collections . 68 7.2.4 Commands . 71 7.3 Spatial extension . 72 7.4 Testing . 74 7.5 Benchmarking . 74 8 Evaluation and Future work 75 8.1 Multiset representation . 75 8.2 Continuous semantics . 75 8.3 Finiteness conditions . 75 8.4 Relationship between the two semantics . 76 8.5 Spatial extension . 76 8.6 Implementation . 76 8.7 Collection of models . 77 8.8 Future work . 77 8.9 Conclusion . 78 A JSPiM 83 A.1 Language definition . 83 A.1.1 Core . 83 A.1.2 Spatial extension . 84 A.2 Screenshots . 84 B Collection of basic examples 87 B.1 Circadian clock . 88 B.1.1 Model . 88 B.1.2 Results . 88 B.2 Circadian clock in CGF . 89 B.2.1 Model . 89 B.2.2 Results . 89 B.3 Oregonator 1 . 91 B.3.1 Model . 91 B.3.2 Results . 91 B.4 Oregonator 2 . 93 B.4.1 Model . 93 B.4.2 Results . 93 B.5 MAPK 1 . 95 B.5.1 Model . 95 B.5.2 Results . 95 B.6 MAPK 1 in CGF . 97 B.6.1 Model . 97 B.6.2 Results . 97 B.7 Bistable . 99 B.7.1 Model . 99 B.7.2 Results . 99 B.8 Bistable in CGF . 101 B.8.1 Model . ..