A new method for choosing the computational cell in stochastic reaction–diffusion systems

• Published in: Journal of mathematical biology Vol. 65; no. 6-7; pp. 1017 - 1099
• Main Authors: Kang, Hye-Won, Zheng, Likun, Othmer, Hans G.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.12.2012; Springer Nature B.V
• Subjects: Animals; Applications of Mathematics; Body Patterning - physiology; Computer Simulation; Drosophila melanogaster - physiology; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Models, Biological; Models, Statistical; Stochastic Processes; Wings, Animal - physiology; 92C45; Stochastic reaction–diffusion systems; Discretization; Noise; 60J28; Biological networks; 80A30; 35Q92
• ISSN: 0303-6812; 1432-1416; 1432-1416
• DOI: 10.1007/s00285-011-0469-6

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.