Asymptotic analysis of multiscale approximations to reaction networks.

*(English)*Zbl 1118.92031Summary: A reaction network is a chemical system involving multiple reactions and chemical species. Stochastic models of such networks treat the system as a continuous time Markov chain on the number of molecules of each species with reactions as possible transitions of the chain. In many cases of biological interest some of the chemical species in the network are present in much greater abundance than others and reaction rate constants can vary over several orders of magnitude.

We consider approaches to approximation of such models that take the multiscale nature of the system into account. Our primary example is a model of a cell’s viral infection for which we apply a combination of averaging and a law of large number arguments to show that the “slow” component of the model can be approximated by a deterministic equation and to characterize the asymptotic distribution of the “fast” components. The main goal is to illustrate techniques that can be used to reduce the dimensionality of much more complex models.

We consider approaches to approximation of such models that take the multiscale nature of the system into account. Our primary example is a model of a cell’s viral infection for which we apply a combination of averaging and a law of large number arguments to show that the “slow” component of the model can be approximated by a deterministic equation and to characterize the asymptotic distribution of the “fast” components. The main goal is to illustrate techniques that can be used to reduce the dimensionality of much more complex models.

##### MSC:

92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |

60J27 | Continuous-time Markov processes on discrete state spaces |

92C37 | Cell biology |

80A30 | Chemical kinetics in thermodynamics and heat transfer |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60F17 | Functional limit theorems; invariance principles |

##### Keywords:

reaction networks; chemical reactions; cellular processes; Markov chains; averaging; scaling limits##### References:

[1] | Athreya, K. B. and Ney, P. E. (2004). Branching Processes . Dover, Mineola, NY. · Zbl 1070.60001 |

[2] | Ball, F. and Donelly, P. (1992). Branching process approximation of epidemic models. Theory Probab. Appl. 37 119–121. · Zbl 0794.92019 · doi:10.1137/1137024 |

[3] | Cao, Y., Gillespie, D. T. and Petzold, L. R. (2005). The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122 014116. · Zbl 1088.80004 |

[4] | Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes . Wiley, New York. · Zbl 0592.60049 |

[5] | Gardiner, C. W. (2004). Handbook of Stochastic Methods for Physics , Chemistry and the Natural Sciences , 3rd ed. Springer, Berlin. · Zbl 1143.60001 |

[6] | Haseltine, E. L. and Rawlings, J. B. (2002). Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117 6959–6969. |

[7] | Kurtz, T. G. (1977/78). Strong approximation theorems for density dependent Markov chains. Stochastic Process. Appl. 6 223–240. · Zbl 0373.60085 · doi:10.1016/0304-4149(78)90020-0 |

[8] | Kurtz, T. G. (1992). Averaging for martingale problems and stochastic approximation. Applied Stochastic Analysis . Lecture Notes in Control and Inform. Sci. 77 186–209. Springer, Berlin. |

[9] | Rao, C. V. and Arkin, A. P. (2003). Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. J. Chem. Phys. 118 4999–5010. |

[10] | Srivastava, R., You, L., Summers, J. and Yin, J. (2002). Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theoret. Biol. 218 309–321. · doi:10.1006/jtbi.2002.3078 |

[11] | Stiefenhofer, M. (1998). Quasi-steady-state approximation for chemical reaction networks. J. Math. Biol. 36 593–609. · Zbl 0945.92030 · doi:10.1007/s002850050116 |

[12] | Van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry . North-Holland, Amsterdam. · Zbl 0511.60038 |

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.