A semi-quantitative equivalence for abstracting from fast reactions

Semantic equivalences are used in process algebra to capture the notion of similar behaviour, and this paper proposes a semi-quantitative equivalence for a stochastic process algebra developed for biological modelling. We consider abstracting away from fast reactions as suggested by the Quasi-Steady-State Assumption. We define a fast-slow bisimilarity based on this idea. We also show congruence under an appropriate condition for the cooperation operator of Bio-PEPA. The condition requires that there is no synchronisation over fast actions, and this distinguishes fast-slow bisimilarity from weak bisimilarity. We also show congruence for an operator which extends the reactions available for a species. We characterise models for which it is only necessary to consider the matching of slow transitions and we illustrate the equivalence on two models of competitive inhibition.

[1]  D. Herries.  Enzyme Kinetics: Behaviour and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems: By Irwin H. Segel. John Wiley & Sons, 1975. pp xxii + 957. Boards, £15.00 , 1976 .

[2]  Leila Ribeiro,et al.  Modelling, property verification and behavioural equivalence of lactose operon regulation , 2007, Comput. Biol. Medicine.

[3]  Jane Hillston,et al.  A Simple Time Scale Decomposition Technique for Stochastic Process Algebras , 1995, Comput. J..

[4]  Kishor S. Trivedi,et al.  An Aggregation Technique for the Transient Analysis of Stiff Markov Chains , 1986, IEEE Transactions on Computers.

[5]  Monika Heiner,et al.  Petri Nets for Systems and Synthetic Biology , 2008, SFM.

[6]  G. Verghese,et al.  Enhanced identification and exploitation of time scales for model reduction in stochastic chemical kinetics. , 2008, The Journal of chemical physics.

[7]  Maria Luisa Guerriero,et al.  Modelling Biological Compartments in Bio-PEPA , 2008, MeCBIC.

[8]  Ian Stark,et al.  The Continuous pi-Calculus: A Process Algebra for Biochemical Modelling , 2008, CMSB.

[9]  Allan Clark,et al.  On verifying Bio-PEPA models , 2010, CMSB '10.

[10]  Corrado Priami,et al.  Beta Binders for Biological Interactions , 2004, CMSB.

[11]  L. Segel,et al.  Extending the quasi-steady state approximation by changing variables. , 1996, Bulletin of mathematical biology.

[12]  D. Gillespie,et al.  Accelerated stochastic simulation of the stiff enzyme-substrate reaction. , 2005, The Journal of chemical physics.

[13]  Hans G. Kaper,et al.  Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics , 2004, J. Nonlinear Sci..

[14]  E. Shapiro,et al.  Cellular abstractions: Cells as computation , 2002, Nature.

[15]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[16]  Jane Hillston,et al.  Process Algebras in Systems Biology , 2008, SFM.

[18]  Alberto Policriti,et al.  Modeling Biological Systems in Stochastic Concurrent Constraint Programming , 2008, Constraints.

[19]  Paolo Milazzo,et al.  Bisimulations in calculi modelling membranes , 2008, Formal Aspects of Computing.

[20]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[21]  L. A. Segel,et al.  The Quasi-Steady-State Assumption: A Case Study in Perturbation , 1989, SIAM Rev..

[22]  Vashti Galpin,et al.  Equivalences for a biological process algebra , 2011, Theor. Comput. Sci..

[23]  Luca Cardelli,et al.  BioAmbients: an abstraction for biological compartments , 2004, Theor. Comput. Sci..

[24]  Cosimo Laneve,et al.  Formal molecular biology , 2004, Theor. Comput. Sci..

[25]  Satish K. Tripathi,et al.  Automated time scale decomposition and analysis of stochastic Petri nets , 1993, Proceedings of 5th International Workshop on Petri Nets and Performance Models.

[26]  Cosimo Laneve,et al.  A Simple Calculus for Proteins and Cells , 2007, Electron. Notes Theor. Comput. Sci..

[27]  Jane Hillston,et al.  A compositional approach to performance modelling , 1996 .

[28]  Jane Hillston,et al.  Bio-PEPA: A framework for the modelling and analysis of biological systems , 2009, Theor. Comput. Sci..

[29]  Linda R Petzold,et al.  The slow-scale stochastic simulation algorithm. , 2005, The Journal of chemical physics.

[30]  Allan Clark,et al.  Formal methods for checking the consistency of biological models. , 2012, Advances in experimental medicine and biology.

[31]  Jane Hillston,et al.  A semantic equivalence for Bio-PEPA based on discretisation of continuous values , 2011, Theor. Comput. Sci..

[32]  Luca Cardelli,et al.  A Compositional Approach to the Stochastic Dynamics of Gene Networks , 2006, Trans. Comp. Sys. Biology.

[33]  Marta Simeoni,et al.  Taming the complexity of biochemical models through bisimulation and collapsing: theory and practice , 2004, Theor. Comput. Sci..

[34]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[35]  Gabriel Ciobanu,et al.  On the relationship between membranes and ambients , 2008, Biosyst..

[36]  Christel Baier,et al.  Comparative branching-time semantics for Markov chains , 2005, Inf. Comput..

[37]  Kevin R. Sanft,et al.  Legitimacy of the stochastic Michaelis-Menten approximation. , 2011, IET systems biology.

[38]  Luca Cardelli.  A Compositional Approach to the Stochastic Dynamics of Gene Networks , 2005, CONCUR.

[39]  Corrado Priami,et al.  Application of a stochastic name-passing calculus to representation and simulation of molecular processes , 2001, Inf. Process. Lett..