title: A new method for choosing the computational cell in stochastic reaction-diffusion systems.
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.
authors.affiliation: School of Mathematics, University of Minnesota, Twin Cities, MN 55455, USA. hkang@math.umn.edu
authors.forename: Hye-Won
authors.initials: HW
authors.lastname: Kang
authors.forename: Likun
authors.initials: L
authors.lastname: Zheng
authors.forename: Hans G
authors.initials: HG
authors.lastname: Othmer
mesh_terms: D000818:Animals; D019521:Body Patterning / Q000502:physiology*; D003198:Computer Simulation; D004331:Drosophila melanogaster / Q000502:physiology*; D008954:Models, Biological; D015233:Models, Statistical; D013269:Stochastic Processes; D014921:Wings, Animal / Q000502:physiology*
references.citation: Curr Top Dev Biol. 1971;6(6):183-224
references.pmid: 4950136
references.citation: Phys Biol. 2004 Dec;1(3-4):137-51
references.pmid: 16204833
references.citation: Proc Natl Acad Sci U S A. 2003 Aug 5;100(16):9302-7
references.pmid: 12869691
references.citation: Biophys J. 1991 Mar;59(3):742-54
references.pmid: 2049528
references.citation: Proc Natl Acad Sci U S A. 2001 Jul 17;98(15):8614-9
references.pmid: 11438714
references.citation: J Theor Biol. 2004 Aug 21;229(4):501-21
references.pmid: 15246787
references.citation: J Theor Biol. 1971 Sep;32(3):507-37
references.pmid: 5571122
references.citation: Biochemistry. 1975 Dec 2;14(24):5355-73
references.pmid: 1191643
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Dec;80(6 Pt 2):066106
references.pmid: 20365230
references.citation: Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1996 Dec;54(6):6139-6148
references.pmid: 9965833
references.citation: Proc Natl Acad Sci U S A. 2006 Aug 1;103(31):11613-8
references.pmid: 16864795
references.citation: Nature. 2008 Sep 4;455(7209):72-7
references.pmid: 18701888
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Apr;71(4 Pt 1):041103
references.pmid: 15903653
references.citation: Phys Biol. 2009 Aug 21;6(4):046001
references.pmid: 19700812
references.citation: Bull Math Biol. 2010 Nov;72(8):1947-70
references.pmid: 20306147
references.citation: Dev Cell. 2008 Jun;14(6):940-53
references.pmid: 18539121
references.citation: Development. 2003 Oct;130(19):4673-82
references.pmid: 12925593
references.citation: Prog Biophys Mol Biol. 2001;75(1-2):1-17
references.pmid: 11311713
references.citation: J Theor Biol. 1969 Oct;25(1):1-47
references.pmid: 4390734
references.citation: Bull Math Biol. 2005 Sep;67(5):901-46
references.pmid: 15998488
references.citation: Nat Genet. 2002 May;31(1):69-73
references.pmid: 11967532
references.citation: Bioinformatics. 2007 Aug 1;23(15):1969-77
references.pmid: 17537752
references.citation: J Math Biol. 2010 Mar;60(3):387-450
references.pmid: 19350241
references.citation: Curr Top Dev Biol. 2008;81:65-111
references.pmid: 18023724
references.citation: Nature. 1970 Jan 31;225(5231):420-2
references.pmid: 5411117
references.citation: Proc Natl Acad Sci U S A. 2010 Nov 16;107(46):19820-5
references.pmid: 21041672
references.citation: Nature. 2003 Feb 20;421(6925):863-7
references.pmid: 12594518
references.citation: Trends Cell Biol. 2003 Jan;13(1):4-6
references.pmid: 12480334
references.citation: Nature. 1976 Aug 5;262(5568):467-71
references.pmid: 958399