zbMATH — the first resource for mathematics

Bio-PEPAd: a non-Markovian extension of Bio-PEPA. (English) Zbl 1232.68090
Summary: Delays in biological systems may be used to model events for which the underlying dynamics cannot be precisely observed, or to provide abstraction of some behavior of the system resulting in more compact models. In this paper, we enrich the stochastic process algebra Bio-PEPA, with the possibility of assigning delays to actions, yielding a new non-Markovian stochastic process algebra: Bio-PEPAd. This is a conservative extension meaning that the original syntax of Bio-PEPA is retained and the delay specification which can now be associated with actions may be added to existing Bio-PEPA models. The semantics of the firing of the actions with delays is the delay-as-duration approach, earlier presented in papers on the stochastic simulation of biological systems with delays. This semantics of the algebra is given in the starting-terminating style, meaning that the state and the completion of an action are observed as two separate events, as required by delays. We formally define the encoding of Bio-PEPAd systems in generalized semi-Markov processes (GSMPs), as input for a delay stochastic simulation algorithm (DSSA) and as sets of delay differential equations (DDEs), the deterministic framework for modeling of biological systems with delays. Finally, we prove theorems stating the relation between Bio-PEPA and Bio-PEPAd models. We end the paper with an example model of biological systems with delays to illustrate the approach.

68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
68Q87 Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
92C42 Systems biology, networks
Full Text: DOI
[1] Barbuti, R.; Caravagna, G.; Maggiolo-Schettini, A.; Milazzo, P.; Back, R.; Petre, I.; de Vink, E.P., On the interpretation of delays in delay stochastic simulation of biological systems, Proceedings of the 2nd int. workshop on computational models for cell processes, compmod’09, Eptcs, vol. 6, 17-29, (2009)
[2] Barbuti, R.; Caravagna, G.; Maggiolo-Schettini, A.; Milazzo, P., Delay stochastic simulation of biological systems: a purely delayed approach, Trans. comp. sys. bio. XIII, 6575, 61-84, (2011) · Zbl 1326.92026
[3] Barbuti, R.; Caravagna, G.; Maggiolo-Schettini, A.; Milazzo, P.; Pardini, G.; Bernardo, M.; Degano, P.; Zavattaro, G., The calculus of looping sequences, Formal methods for computational systems biology, SFM 2008, Lncs, vol. 5016, 387-423, (2008)
[4] Barbuti, R.; Maggiolo-Schettini, A.; Milazzo, P.; Pardini, G., Spatial calculus of looping sequences, Int. workshop from biology to concurrency and back, FBTC’08, Entcs, 229, 1, 21-39, (2008) · Zbl 1283.92024
[5] Barrio, M.; Burrage, K.; Leier, A.; Tian, T., Oscillatory regulation of hes1: discrete stochastic delay modelling and simulation, Plos comp. bio., 2, 9, (2006)
[6] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y., Permanence of an SIR epidemic model with distributed time delays, Tohoku math. J., 54, 2, 581-591, (2002) · Zbl 1014.92033
[7] Bio-PEPA web site http://biopepa.org/.
[8] Bratsun, D.; Volfson, D.; Tsimring, L.S.; Hasty, J., Delay-induced stochastic oscillations in gene regulation, Pnas, 102, 41, 14593-14598, (2005)
[9] M. Bravetti, Specification and analysis of stochastic real-time systems, Ph.D. Thesis, Università di Bologna, 2002. · Zbl 1065.68581
[10] Bravetti, M.; Gorrieri, R., The theory of interactive generalized semi-Markov processes, Theoret. comput. sci., 282, 1, 5-32, (2002) · Zbl 0997.68083
[11] Bravetti, M.; Bernardo, M.; Gorrieri, R., Towards performance evaluation with general distributions in process algebras, Proc. of CONCUR’98, Lncs, vol. 1466, 405-422, (1998)
[12] G. Caravagna, Formal modeling and simulation of biological systems with delays, Ph.D. Thesis, Università di Pisa, 2011.
[13] Caravagna, G.; Hillston, J., Modeling biological systems with delays in bio-PEPA, Proc. of the 4-th workshop on membrane computing and biologically inspired process calculi, mecbic 2010, Eptcs, vol. 40, 85-101, (2010)
[14] Ciocchetta, F.; Hillston, J., Bio-PEPA: a framework for the modelling and analysis of biochemical networks, Theoret. comput. sci., 410, 33-34, 3065-3084, (2009) · Zbl 1173.68041
[15] Ciocchetta, F.; Hillston, J.; Bernardo, M.; Degano, P.; Zavattaro, G., Calculi for biological systems, Formal methods for computational systems biology, SFM 2008, Lncs, vol. 5016, 265-312, (2008)
[16] Cox, D.R., The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge phil. soc., 51, 433-440, (1955) · Zbl 0067.10902
[17] Culshaw, R.V.; Ruan, S., A delay-differential equation model of HIV infection of CD4+ T-cells, Math. biosci., 165, 27-39, (2000) · Zbl 0981.92009
[18] D’Argenio, P.R.; Katoen, J.-P., A theory of stochastic systems, part II: process algebra, Inf. and comp., 203, 1, 39-74, (2005) · Zbl 1106.68073
[19] P.R. D’Argenio, J.-P. Katoen, E. Brinksma, A stochastic automata model and its algebraic approach, in: Proc. 5th Int. Workshop on Process Algebra and Performance Modeling, PAPM’97, CTIT technical reports series 97-14, University of Twente, 1997, pp. 1-16.
[20] Danos, V.; Feret, J.; Fontana, W.; Harmer, R.; Krivine, J., Rule-based modelling of cellular signalling, Proc. of CONCUR’07, Lncs, vol. 4703, 17-41, (2007) · Zbl 1151.68723
[21] Dematté, L.; Priami, C.; Romanel, A., Modelling and simulation of biological processes in blenx, SIGMETRICS perform. eval. rev., 35, 4, 32-39, (2008) · Zbl 1160.68678
[22] Galpin, V.; Hillston, J., A semantic equivalence for bio-PEPA based on discretisation of continuous values, Theoret. comput. sci., 412, 21, 2142-2161, (2011) · Zbl 1215.68140
[23] German, R., Performance analysis of communication systems with non – markovian stochastic Petri nets, (2000), John Wiley & Sons, Inc. · Zbl 0953.68015
[24] Gillespie, D., Exact stochastic simulation of coupled chemical reactions, J. phys. chem., 81, 2340, (1977)
[25] van Glabbeek, R.J.; Vaandrager, F.W., Petri net models for algebraic theories of concurrency, Lncs, 259, 224-242, (1987)
[26] P.W. Glynn, On the role of generalized semi-markov processes in simulation output analysis, in: Proc. of the 15th Conference on Winter Simulation, Volume 1, 1983, pp. 39-44.
[27] Hillston, J., A compositional approach to performance modelling, (1996), Cambridge University Press
[28] Heath, J.; Kwiatkowska, M.; Norman, G.; Parker, D.; Tymchyshyn, O., Probabilistic model checking of complex biological pathways, Theoret. comput. sci., 391, 239-257, (2008) · Zbl 1133.68043
[29] Lopez, N.; Nunez, M., NMSPA: a non-Markovian model for stochastic processes, (), 33-40
[30] J. Markovski, Real and stochastic time in process algebras for performance evaluation, Ph.D. Thesis, University of Eindhoven, 2008.
[31] P. Milazzo, Qualitative and quantitative formal modeling of biological systems, Ph.D. Thesis, Università di Pisa, 2007.
[32] Milner, R.; Toft, M.; Harper, R., The definition of standard ML, (1990), MIT Press
[33] Mura, I.; Prandi, D.; Priami, C.; Romanel, A., Exploiting non-Markovian bio-processes, Entcs, 253, 3, 83-98, (2009)
[34] L. Paulevé, S. Youssef, M.R. Lakin, A. Phillips, A generic abstract machine for stochastic process calculi, in: Proceedings of the 8th International Conference on Computational Methods in Systems Biology CMSB vol. 10, 2010, pp. 43-54.
[35] Pedersen, M.; Plotkin, G., A language for biochemical systems: design and formal specification, Trans. comp. sys. bio. XII, 5945, 3, 77-145, (2010) · Zbl 1275.92020
[36] Plotkin, G., The origins of structural operational semantics, J. log. alg. progr., 60-61, 3-15, (2004) · Zbl 1072.68063
[37] Priami, C., Stochastic \(\pi\)-calculus, The comp. J., 38, 7, 578-589, (1995)
[38] Priami, C.; Regev, A.; Shapiro, E.; Silverman, W., Application of a stochastic name-passing calculus to representation and simulation of molecular processes, Inform. process. lett., 80, 25-31, (2001) · Zbl 0997.92018
[39] Regev, A.; Panina, E.M.; Silverman, W.; Cardelli, L.; Shapiro, E.Y., Bioambients: an abstraction for biological compartments, Theoret. comput. sci., 325, 1, 141-167, (2004) · Zbl 1069.68569
[40] Regev, A.; Silverman, W.; Shapiro, E., Representation and simulation of biochemical processes using the pi-calculus process algebra, (), 459-470
[41] Prism web site http://www.prismmodelchecker.org/.
[42] Villasana, M.; Radunskaya, A., A delay differential equation model for tumor growth, J. math. biol., 47, 270-294, (2003) · Zbl 1023.92014
[43] Zhanga, F.; Lia, Z.; Zhangc, F., Global stability of an SIR epidemic model with constant infectious period, Appl. math. comp., 199, 1, 285-291, (2008)
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.