### Abstract

How to choose the computational compartment or cell size for the stochastic simulation of a reaction-diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reaction-diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.

Original language | English (US) |
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Pages (from-to) | 1017-1099 |

Number of pages | 83 |

Journal | Journal of Mathematical Biology |

Volume | 65 |

Issue number | 6-7 |

DOIs | |

State | Published - 2012 |

### Keywords

- Biological networks
- Discretization
- Noise
- Stochastic reaction-diffusion systems

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## Cite this

*Journal of Mathematical Biology*,

*65*(6-7), 1017-1099. https://doi.org/10.1007/s00285-011-0469-6