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Reaction rates for mesoscopic reaction-diffusion kinetics

Stefan Hellander

Andreas Hellander

Linda Petzold

  • Department of Computer Science, University of California, Santa Barbara, California 93106-5070, Santa Barbara, USA

  • Department of Information Technology, Uppsala University, Box 337, SE-75105, Uppsala, Sweden

  • Department of Computer Science, University of California, Santa Barbara, California 93106-5070, Santa Barbara, USA

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Phys. Rev. E 91, 023312 – Published 23 February, 2015

DOI: https://doi.org/10.1103/PhysRevE.91.023312

Abstract

The mesoscopic reaction-diffusion master equation (RDME) is a popular modeling framework frequently applied to stochastic reaction-diffusion kinetics in systems biology. The RDME is derived from assumptions about the underlying physical properties of the system, and it may produce unphysical results for models where those assumptions fail. In that case, other more comprehensive models are better suited, such as hard-sphere Brownian dynamics (BD). Although the RDME is a model in its own right, and not inferred from any specific microscale model, it proves useful to attempt to approximate a microscale model by a specific choice of mesoscopic reaction rates. In this paper we derive mesoscopic scale-dependent reaction rates by matching certain statistics of the RDME solution to statistics of the solution of a widely used microscopic BD model: the Smoluchowski model with a Robin boundary condition at the reaction radius of two molecules. We also establish fundamental limits on the range of mesh resolutions for which this approach yields accurate results and show both theoretically and in numerical examples that as we approach the lower fundamental limit, the mesoscopic dynamics approach the microscopic dynamics. We show that for mesh sizes below the fundamental lower limit, results are less accurate. Thus, the lower limit determines the mesh size for which we obtain the most accurate results.

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References (42)

  1. D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22, 403 (1976).
  2. C. W. Gardiner, K. J. McNeil, D. F. Walls and I. S. Matheson, Correlations in stochastic theories of chemical reactions, J. Stat. Phys. 14, 307 (1976).
  3. F. Baras and M. Malek Mansour, Reaction-diffusion master equation: A comparison with microscopic simulations, Phys. Rev. E. 54, 6139 (1996).
  4. M. J. Lawson, B. Drawert, M. Khammash, and L. Petzold, Spatial stochastic dynamics enable robust cell polarization, PLoS Comput. Biol. 9, e1003139 (2013).
  5. M. Howard and A. D. Rutenberg, Pattern formation inside bacteria: Fluctuations due to the low copy number of proteins, Phys. Rev. Lett. 90, 128102 (2003).
  6. D. Fange and J. Elf, Noise induced Min phenotypes in E. coli, PLoS Comput. Biol. 2, e80 (2006).
  7. M. Sturrock, A. Hellander, A. Marsavinos, and M. Chaplain, Spatial stochastic modeling of the hes1 pathway: Intrinsic noise can explain heterogeneity in embryonic stem cell differentiation, J. Roy. Soc. Interface 10, 20120988 (2013).
  8. M. Sturrock, A. Hellander, S. Aldakheel, L. Petzold, and M. Chaplain, The role of dimerisation and nuclear transport in the hes1 gene regulatory network, Bull. Math. Biol. 76, 766 (2013).
  9. J. Elf and M. Ehrenberg, Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases, Syst. Biol. 1, 230 (2004).
  10. J. S. van Zon and P. R. ten Wolde, Green's-function reaction dynamics: A particle-based approach for simulating biochemical networks in time and space, J. Chem. Phys. 123, 234910 (2005).
  11. 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, 128103 (2005).
  12. S. Hellander and P. Lötstedt, Flexible single molecule simulation of reaction-diffusion processes, J. Comput. Phys. 230, 3948 (2011).
  13. J. Hattne, D. Fange, and J. Elf, Stochastic reaction-diffusion simulation with MesoRD, Bioinformatics 21, 2923 (2005).
  14. B. Drawert, S. Engblom, and A. Hellander, URDME: A modular framework for stochastic simulation of reaction-transport processes in complex geometries, BMC Syst. Biol. 6, 76 (2012).
  15. I. Hepburn, W. Chen, S. Wils, and E. De Schutter, STEPS: Efficient simulation of stochastic reaction-diffusion models in realistic morphologies, BMC Syst. Biol. 6, 425 (2012).
  16. D. Rossinelli, B. Bayati, and P. Koumoutsakos, Accelerated stochastic and hybrid method for spatial simulations of reaction-diffusion systems, Chem. Phys. Lett. 451, 136 (2008).
  17. T. T. Marquez-Lago and K. Burrage, Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics, J. Chem. Phys. 127, 104101 (2007).
  18. S. Lampoudi, D. Gillespie, and L. R. Petzold, The multinomial simulation algorithm for discrete stochastic simulation of reaction-diffusion systems, J. Chem. Phys. 130, 094104 (2009).
  19. L. Ferm, A. Hellander, and P. Lötstedt, An adaptive algorithm for simulation of stochastic reaction-diffusion processes, J. Comput. Phys. 229, 343 (2010).
  20. A. Hellander and P. Lötstedt, Incorporating active transport in mesoscopic reaction-transport models inside living cells, Multiscale Model. Simul. 8, 1691 (2010).
  21. B. Bayati, Fractional diffusion-reaction stochastic simulations, J. Chem. Phys. 138, 104117 (2013).
  22. S. A. Isaacson and C. S. Peskin, Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations, SIAM J. Sci. Comput. 28, 47 (2006).
  23. S. Engblom, L. Ferm, A. Hellander, and P. Lötstedt, Simulation of stochastic reaction-diffusion processes on unstructured meshes, SIAM J. Sci. Comput. 31, 1774 (2009).
  24. M. v. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Z. Phys. Chemie 92, 129 (1917).
  25. M. Tomita, K. Hashimoto, K. Takahashi, T. S. Shimizu, Y. Matsuzaki, F. Miyoshi, K. Saito, S. Tanida, K. Yugi, J. C. Venter, and C. A. Hutchison, E-cell: Software environment for whole-cell simulation, Bioinformatics 15, 72 (1999).
  26. S. S. Andrews, N. J. Addy, R. Brent, and A. P. Arkin, Detailed simulations of cell biology with Smoldyn 2.1, PLoS Comput. Biol. 6, e1000705 (2010).
  27. F. C. Collins and G. E. Kimball, Diffusion-controlled reaction rates, J. Colloid Sci. 4, 425 (1949).
  28. D. T. Gillespie, A diffusional bimolecular propensity function, J. Chem. Phys. 131, 164109 (2009).
  29. S. A. Isaacson, The reaction-diffusion master equation as an asymptotic approximation of diffusion to a small target, SIAM J. Appl. Math. 70, 77 (2009).
  30. R. Erban and J. Chapman, Stochastic modeling of reaction-diffusion processes: Algorithms for bimolecular reactions, Phys. Biol. 6, 046001 (2009).
  31. S. Hellander, A. Hellander, and L. R. Petzold, Reaction-diffusion master equation in the microscopic limit, Phys. Rev. E 85, 042901 (2012).
  32. D. Fange, O. G. Berg, P. Sjöberg, and J. Elf, Stochastic reaction-diffusion kinetics in the microscopic limit, Proc. Natl. Acad. Sci. USA 107, 19820 (2010).
  33. A. Hellander, S. Hellander, and P. Lötstedt, Coupled mesoscopic and microscopic simulation of stochastic reaction-diffusion processes in mixed dimensions, Multiscale Model. Simul. 10, 585 (2012).
  34. I. C. Agbanusi and S. A. Isaacson, A comparison of bimolecular reaction models for stochastic reaction-diffusion systems, Bull. Math. Biol. 76, 922 (2014).
  35. S. S. Andrews and D. Bray, Stochastic simulation of chemical reactions with spatial resolution and single molecule detail, Phys. Biol. 1, 137 (2004).
  36. D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A 188, 404 (1992).
  37. C. W. Gardiner, Handbook of Stochastic Methods, Springer Series in Synergetics, 3rd ed. (Springer-Verlag, Berlin, 2004).
  38. D. T. Gillespie, A. Hellander, and L. R. Petzold, Perspective: Stochastic algorithms for chemical kinetics, J. Chem. Phys. 138, 170901 (2013).
  39. S. A. Isaacson, Relationship between the reaction-diffusion master equation and particle tracking models, J. Phys. A: Math. Theor. 41, 065003 (2008).
  40. D. T. Gillespie, L. R. Petzold, and E. Seitaridou, Validity conditions for stochastic chemical kinetics in diffusion-limited systems, J. Chem. Phys. 140, 054111 (2014).
  41. K. Takahashi, S. Tănase-Nicola, and P. R. ten Wolde, Spatio-temporal correlations can drastically change the response of a MAPK pathway, Proc. Natl. Acad. Sci. USA 107, 2473 (2010).
  42. S. A. Isaacson, A convergent reaction-diffusion master equation, J. Chem. Phys. 139, 054101 (2013).

Outline

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Phys. Rev. E 91, 023312– Published 23 February, 2015
  • Received 1 November 2014
Article part of CHORUS

©2015 American Physical Society

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