The reaction-diffusion master equation as an asymptotic approximation of diffusion to a small target

SIAM Journal on Applied Mathematics

Volumn 70, Issue 1, 2009, Pages 77-111

  Isaacson, Samuel A ^a  

^a Boston University   (United States)

Author keywords

Diffusion (limited) vacation; Master equation; Reaction diffusion; Stochastic chemical kinetics

Indexed keywords

ASYMPTOTIC APPROXIMATION; BIMOLECULAR REACTION; BIOLOGICAL PARAMETER; CHEMICAL REACTION PROCESS; CONTINUUM LIMITS; DIFFUSION LIMITED REACTIONS; LATTICE SPACING; MASTER EQUATION; MASTER EQUATIONS; MESH SPACING; PHYSICAL MODEL; REACTION-DIFFUSION; RELATIVE ERRORS; SMALL TARGETS; STOCHASTIC CHEMICAL KINETICS;

ASYMPTOTIC ANALYSIS; BIOLOGICAL SYSTEMS; DIFFUSION IN LIQUIDS; MOLECULES; RANDOM PROCESSES; SYSTEMS ANALYSIS;

REACTION KINETICS;

EID: 67649365828     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/070705039     Document Type: Article

References (48)

1. M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1965.
2. J. L. ADELMAN AND S. ANDREWS, Intracellular pattern formation: A spatial stochastic model of bacterial division site selection proteins MinCDE, in Proceedings of the Santa Fe Institute Summer School, 2004; see http://www.smoldyn.org/andrews/publications.html.
3. S. ALBEVERIO, Z. BRZEŹNIAK, AND L. DA̧BROWSKI, Fundamental solution of the heat and Schrödinger equations with point interaction, J. Funct. Anal., 130 (1995), pp. 220-254.
4. S. ALBEVERIO, F. GESZTESY, R. HØEGH-KROHN, AND H. HOLDEN, Solvable Models in Quantum Mechanics, 2nd ed., AMS Chelsea, New York, 1988.
5. S. ALBEVERIO AND P. KURASOV, Singular Perturbations of Differential Operators, London Math. Soc. Lecture Note Ser. 271, Cambridge University Press, New York, 2000.
6. S. S. ANDREWS AND D. BRAY, Stochastic simulation of chemical reactions with spatial resolution and single molecule detail, Phys. Biol., 1 (2004), pp. 137-151.
7. A. ARKIN AND H. H. MCADAMS, Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci. USA, 94 (1997), pp. 814-819.
8. A. ARKIN, J. ROSS, AND H. H. MCADAMS, Stochastic kinetic analysis of developmental pathway bifurcation in phage X-infected Escherichia coli cells, Genetics, 149 (1998), pp. 1633-1648.
9. F. BARAS AND M. M. MANSOUR, Reaction-diffusion master equation: A comparison with microscopic simulations, Phys. Rev. E, 54 (1996), pp. 6139-6147.
10. F. BEREZIN AND L. FADDEEV, A remark on Schrödinger equation with a singular potential, Soviet Math. Dokl., 2 (1961), pp. 372-375.
11. U. S. BHALLA, Signaling in small subcellular volumes. I. Stochastic and diffusion effects on individual pathways, Biophys. J., 87 (2004), pp. 733-744.
12. W. J. BLAKE, M. KAERN, C. R. CANTOR, AND J. J. COLLINS, Noise in eukaryotic gene expression, Nature, 422 (2003), pp. 633-637.
13. L. M. DELVES AND J. L. MOHAMED, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985.
14. M. DOI, Second quantization representation for classical many-particle system, J. Phys. A: Math. Gen., 9 (1976), pp. 1465-1477.
15. J. ELF AND M. EHRENBERG, Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases, IEE Sys. Biol., 1 (2004), pp. 230-236.
16. R. ERBAN, S. J. CHAPMAN, AND P. K. MAINI, A Practical Guide to Stochastic Simulations of Reaction-Diffusion Processes, lecture notes, available online at http://www2.maths.ox.ac. uk/~ erban/papers/download.html.
17. D. FANGE AND J. ELF, Noise induced min phenotypes in E. coli, PLoS Comput. Biol., 2 (2006), pp. 0637-0638.
18. W. FELLER, An Introduction to Probability Theory and Its Applications, Vol. II, John Wiley & Sons, New York, 1971.
19. E. FERMI, Motion of neutrons in hydrogenous substances, Ricerca Sci., 7 (1936), pp. 13-52.
20. C. W. GARDINER, K. J. MCNEIL, D. F. WALLS, AND I. S. MATHESON, Correlations in stochastic theories of chemical reactions, J. Statist. Phys., 14 (1976), pp. 307-331.
21. D. T. GILLESPIE, Exact stochastic simulation of coupled chemical-reactions, J. Phys. Chem., 81 (1977), pp. 2340-2361.
22. E. HAIRER, CH. LUBICH, AND M. SCHLICHTE, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 532-541.
23. M. HAZEWINKEL, ed., Encyclopaedia of Mathematics, Springer-Verlag, New York, Berlin, 2002.
24. K. HUANG AND C. N. YANG, Quantum-mechanical many-body problem with hard-sphere interaction, Phys. Rev., 105 (1957), pp. 767-775.
25. S. A. iSAACSON, A Stochastic Reaction-Diffusion Method for Studying the Control of Gene Expression in Eukaryotic Cells, Ph.D. thesis, Department of Mathematics, New York University, New York, 2005.
26. S. A. ISAACSON, Relationship between the reaction-diffusion master equation and particle tracking models, J. Phys. A, 41 (2008), paper 065003.
27. S. A. ISAACSON AND D. ISAACSON, The Reaction-Diffusion Master Equation, Diffusion Limited Reactions, and Singular Potentials, manuscript.
28. S. A. ISAACSON AND C. S. pESKIN, Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations, SIAM J. Sci. Comput., 28 (2006), pp. 47-74.
29. J. KEIZER, Nonequilibrium statistical thermodynamics and the effect of diffusion on chemical reaction rates, J. Phys. Chem., 86 (1982), pp. 5052-5067.
30. F. KÜHNER, l. T. COSTA, P. M. BISCH, S. THALHAMMER, W. M. HECKL, AND H. E. GAUB, LexA-DNA bond strength by single molecule force spectroscopy, Biophys. J., 87 (2004), pp. 2683-2690.
31. G. LAHAV, N. ROSENFELD, A. SIGAL, N. GEVA-ZATORSKY, A. J. LEVINE, M. B. ELOWITZ, AND U. ALON, Dynamics of the p53-Mdm2 feedback loop in individual cells, Nature Genetics, 36 (2004), pp. 147-150.
32. H. A. lUTHER, An explicit sixth-order Runge-Kutta formula, Math. Comp., 22 (1968), pp. 434-436.
33. H. MAAMAR, A. RAJ, AND D. DUBNAU, Noise in gene expression determines cell fate in Bacillus subtilis, Science, 317 (2007), pp. 526-529.
34. D. A. MCQUARRIE, Stochastic approach to chemical kinetics, J. Appl. Probab., 4 (1967), pp. 413-478.
35. M. MORI AND M. SUGIHARA, The double-exponential transformation in numerical analysis,J. Comput. Appl. Math., 127 (2001), pp. 287-296.
36. M. MUHAMMAD, A. NURMUHAMMAD, M. MORI, AND M. SUGIHARA, Numerical solution of integral equations by means ofthe sinc collocation method based on the double exponential transformation, J. Comput. Appl. Math., 177 (2005), pp. 269-286.
37. E. M. OZBUDAK, M. THATTAI, I. KURTSER, A. D. GROSSMAN, AND A. VAN OUDENAARDEN, Regulation of noise in the expression of a single gene, Nature Genetics, 31 (2002), pp. 69-73.
38. N. PAVIN, H. V. PALJETAK, AND V. kRSTIĆ , Min-protein oscillations in Escherichia coli with spontaneous formation of two-stranded filaments in a three-dimensional stochastic reaction-diffusion model, Phys. Rev. E, 73 (2006), paper 021904.
39. L. PELITI, Path integral approach to birth-death processes on a lattice, J. Physique, 46 (1985), pp. 1469-1483.
40. J. M. RASER AND E. K. O'SHEA, Control of stochasticity in eukaryotic gene expression, Science, 304 (2004), pp. 1811-1814.
41. L. ROSA, fFTCONV.M, MATLAB program, available online at http://www.mathworks.com/ matlabcentral/fileexchange.
42. A. E. SMITH, B. M. SLEPCHENKO, J. C. SCHAFF, L. M. LOEW, AND I. G. MACARA, Systems analysis of Ran transport, Science, 295 (2002), pp. 488-491.
43. M. V. SMOLUCHOWSKI, Mathematical theory of the kinetics of the coagulation of colloidal solutions, Z. Phys. Chem., 92 (1917), pp. 129-168.
44. G. M. SÜEL, J. G. GARCIA-OJALVO, L. M. LIBERMAN, AND M. B. ELOWITZ, An excitable gene regulatory circuit induces transient cellular differentiation, Nature, 440 (2005), pp. 545-550.
45. T. TANG, X. XU, AND J. CHENG, On spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math., 26 (2008), pp. 825-837.
46. U. C. TÄUBER, M. HOWARD, AND B. P. VOLLMAYR-lEE, Applications of field-theoretic renor- malization group methods to reaction-diffusion problems, J. Phys. A, 38 (2005), pp. R79-R131.
47. J. S. vAN ZON, M. J. MORELLI, S. TǍNASE-NICOLA, AND P. R. TEN WOLDE, Diffusion of transcription factors can drastically enhance the noise in gene expression, Biophys. J., 91 (2006), pp. 4350-4367.
48. J. S. VAN ZON AND P. R. TEN WOLDE, Simulating biochemical networks at the particle level and in time and space: Green's function reaction dynamics, Phys. Rev. Lett., 94 (2005), paper 128103.