Relating continuous and discrete PEPA models of signalling pathways.

*(English)*Zbl 1151.68038Summary: PEPA and its semantics have recently been extended to model biological systems. In order to cope with massive quantities of processes (as is usually the case when considering biological reactions) the model is interpreted in terms of a small set of coupled Ordinary Differential Equations (ODEs) instead of a large state space continuous time Markov chain. So far the relationship between these two semantics of PEPA had not been established. This is the goal of the present paper. After introducing a new extension of PEPA, denoted \(\text{PEPA} +\Pi \), that allows models to capture both mass action law and bounded capacity law cooperations, the relationship between these two semantics is demonstrated. The result relies on Kurtz’s theorem that expresses that a set of ODEs can be, in some sense, considered as the limit of pure jump Markov processes.

##### MSC:

68Q85 | Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) |

92C37 | Cell biology |

92C40 | Biochemistry, molecular biology |

##### Software:

PEPA
PDF
BibTeX
XML
Cite

\textit{N. Geisweiller} et al., Theor. Comput. Sci. 404, No. 1--2, 97--111 (2008; Zbl 1151.68038)

Full Text:
DOI

##### References:

[1] | L. Bortolussi, A. Policriti, Stochastic concurrent constraint programming and differential equations, in: Proceedings of Fifth Workshop on Quantitative Aspects of Programming Languages, Braga, Portugal, 2007 · Zbl 1279.92031 |

[2] | M. Calder, A. Duguid, S. Gilmore, J. Hillston, Stronger computational modelling of signalling pathways using both continuous and discrete-state methods, in: Proceedings of 4th International Workshop on Computational Methods in Systems Biology, Trento, Italy, 18-19th October 2006, pp. 63-77 |

[3] | M. Calder, S. Gilmore, J. Hillston, Modelling the influence of RKIP on the ERK signaling pathway using the stochastic process algebra PEPA, BioConcur, 2004 |

[4] | Calder, M.; Gilmore, S.; Hillston, J., Automatically deriving ODEs from process algebra models of signalling pathways, (), 204-215 |

[5] | M. Calder, J. Hillston, What do scaffold proteins really do? in: Proceedings of the 5th International Workshop on Process Algebras and Stochastically Timed Activities, PASTA’06, Imperial College, London, UK, June 2006, pp. 96-101 |

[6] | Calder, M.; Vyshemirsky, V.; Gilbert, D.; Orton, R., Analysis of signalling pathways using continuous time Markov chains, Transactions on computational systems biology VI, 44-67, (2006) |

[7] | L. Cardelli, From processes to ODEs by chemistry, Unpublished manuscript, 2006 |

[8] | Chiarugi, D.; Curti, M.; Degano, P.; Marangoni, R., VICE: A virtual cell, () · Zbl 1088.68818 |

[9] | Geisweiller, N., An attempt to give a clear semantics of the extension of PEPA for massively parallel processes and biological modelling, (), 36-43 |

[10] | Gillespie, D., Exact stochastic simulation of coupled chemical reactions, Journal of physical chemistry, 81, 25, 2340-2361, (1977) |

[11] | D. Gillespie, L. Petzold, Numerical simulation for biochemical kinetics, 2006 |

[12] | Goss, P.; Peccoud, J., Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets, Proceedings of national Academy of science, USA, 95, 12, (1998), 7650-6755 |

[13] | J. Heath, M. Kwiatkowska, G. Norman, D. Parker, O. Tymchynshyn, Computer assisted biological reasoning: The simulation and analysis of FGF signalling pathway dynamics, in: Proceedings of the Winter Simulation Conference, 2006 |

[14] | J. Hillston, The nature of synchronisation, in: U. Herzog, M. Rettelbach (Eds.), Proceedings of the Second International Workshop on Process Algebras and Performance Modelling, Erlangen, Nov. 1994, pp. 51-70 |

[15] | Hillston, J., A compositional approach to performance modelling, (1996), Cambridge University Press |

[16] | Hillston, J., Fluid flow approximation of PEPA models, (), 33-42 |

[17] | Kurtz, T., Solutions of ordinary differential equations as limits of pure jump Markov processes, J. appl. prob., 7, 49-58, (1970) · Zbl 0191.47301 |

[18] | Kurtz, T., The relationship between stochastic and deterministic models for chemical reactions, Journal of chemical physics, 57, 7, 2976-2978, (1972) |

[19] | Kuttler, C.; Niehren, J., Gene regulation in the pi calculus: simulating cooperativity at the lambda switch, (), 24-55 |

[20] | P. Lecca, C. Priami, C. Laudanna, G. Constantin, A Biospi model of lymphocyte-endothelial interactions in inflamed brain venules, in: Pacific Symposium of Biocomputing, 2004, pp. 521-532 |

[21] | Priami, C.; Regev, A.; Silverman, W.; Shapiro, E., Application of a stochastic name passing process calculus to representation and simulation of molecular processes, Information processing letters, 80, 25-31, (2001) · Zbl 0997.92018 |

[22] | A. Regev, Computational systems biology: A calculus for biomolecular knowledge, 2002 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.