Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Stochastic simulations of the Schnakenberg model with spatial inhomogeneities using reactive multiparticle collision dynamics

Department of Mathematics, Ryerson University, 350 Victoria St., Toronto, ON, M5B 2K3, Canada

Special Issues: Applied and Industrial Mathematics in Canada and Worldwide

A numerically efficient globally averaged number density approach is used to simulate a reaction-diffusion system using a particle-based stochastic simulation algorithm called reactive multiparticle collision (RMPC) dynamics. Constant diffusivity of the particles is achieved through a time-varying rotation angle (also called collision angle). Variation in the diffusion coefficient between two different chemical species is achieved in one of two ways: (i) using a different kBT/m value for one species compared to the other, or (ii) using the same kBT/m value for both species, but using a different probability to free-stream for one species compared to another. For smaller diffusivities and larger spatial inhomogeneities, bath particles were necessary for the model to agree with the PDE solution. The latter approach was further used without a bath, and shown to be capable of producing Turing patterns after long simulation times. The significance of our work is that RMPC can serve as a feasible simulation tool for both short and long-term simulations, can handle spatial inhomogeneities, can model a fairly large range of diffusivities in a reaction-diffusion scenario, and is capable of producing Turing patterns. An advantage of this method includes more detailed system information in feasible simulation times.
  Article Metrics

Keywords stochastic simulations; reaction-diffusion; reactive multiparticle collision dynamics; Schnakenberg model

Citation: Alireza Sayyidmousavi, Katrin Rohlf. Stochastic simulations of the Schnakenberg model with spatial inhomogeneities using reactive multiparticle collision dynamics. AIMS Mathematics, 2019, 4(6): 1805-1823. doi: 10.3934/math.2019.6.1805


  • 1. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361.    
  • 2. D. T. Gillespie, Approximate accelerated stochastic simulation of reacting systems, J. Phys. Chem., 115 (2001), 1716-1733.    
  • 3. A. Stundzia, C. Lumsden, Stochastic simulation of coupled reaction-diffusion processes, J. Comput. Phys., 127 (1996), 196-207.    
  • 4. S. Isaacson, C. Peskin, Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations, SIAM J. Sci. Comput., 28 (2006), 47-74.    
  • 5. D. Fange, J. Elf, Noise-induced min phenotypes in E. coli, PLoS Comput. Biol., 2 (2006), e80.
  • 6. L. Ferm, A. Hellander, P. Lotstedt, An adaptive algorithm for simulation of stochastic reaction-diffusion processes, J. Comput. Phys., 229 (2010), 343-360.    
  • 7. J. M. A. Padgett, S. Ilie, An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems, AIP Adv., 6 (2016), 035217.
  • 8. R. Strehl, S. Ilie, An adaptive tau-leaping method for stochastic systems with slow and fast dynamics, J. Chem. Phys., 143 (2015), 234108.
  • 9. C. A. Smith, C. A. Yates, Spatially extended hybrid methods: A review, J. R. Soc. Interface, 15 (2018), 20170931.
  • 10. M. B. Flegg, S. J. Chapman, R. Erban, The two regime method for optimizing stochastic reaction-diffusion simulations, J. R. Soc. Interface, 9 (2012), 859-868.    
  • 11. A. Hellander, S. Hellander, P. Lotstedt, Coupled mesoscopic and microscopic simulation of stochastic reaction-diffusion processes in mixed dimensions, Multiscale Model. Sim., 10 (2012), 585-611.    
  • 12. M. B. Flegg, S. Hellander, R. Erban, Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations, J. Chem. Phys., 289 (2015), 1-17.
  • 13. J. Hattne, D. Fange, J. Elf, Stochastic reaction-diffusion simulation with MesoRD, Bioinformatics, 21 (2005), 2923-2924.    
  • 14. S. S. Andrews, D. Bray, Stochastic simulation of chemical reactions with spatial resolution and single molecule detail, Phys. Biol., 1 (2004), 137-151.    
  • 15. J. S. van Zon, P. R. ten Wolde, Greens-function reaction dynamics: A particle-based approach for simulating biochemical networks in time and space, J. Chem. Phys., 123 (2005), 234910.
  • 16. J. S. van Zon, 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), 128103.
  • 17. S. J. Chapman, R. Erban, S. A. Isaacson, Reactive boundary conditions as limits of interaction potentials for Brownian and Langevin dynamics, SIAM J. Appl. Math., 76 (2016), 368-390.    
  • 18. H. G. Othmer, S. R. Dunbar, W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.    
  • 19. Y. Cao, R. Erban, Stochastic Turing patterns: Analysis of compartment-based approaches, B. Math. Biol., 76 (2014), 3051-3069.    
  • 20. M. B. Flegg, Smoluchowski reaction kinetics for reactions of any order, SIAM J. Appl. Math., 76 (2016), 1403-1432.    
  • 21. A. Malevanets, R. Kapral, Mesoscopic model for solvent dynamics, J. Chem. Phys., 110 (1999), 8605-8613.    
  • 22. A. Malevanets, R. Kapral, Solute molecular dynamics in a mesoscale solvent, J. Chem. Phys., 112 (2000), 7260-7269.    
  • 23. K. Tucci, R. Kapral, Mesoscopic model for diffusion influenced reaction dynamics, J. Chem. Phys., 120 (2004), 8262-8270.    
  • 24. K. Tucci, R. Kapral, Mesoscopic multiparticle collision dynamics of reaction-diffusion fronts, J. Chem. Phys. B, 109 (2005), 21300-21304.    
  • 25. K. Rohlf, S. Fraser, R. Kapral, Reactive multiparticle collision dynamics, Comput. Phys. Commun., 179 (2008), 132-139.    
  • 26. K. Rohlf, Stochastic phase-space description for reactions that change particle numbers, J. Math. Chem., 45 (2009), 141-160.    
  • 27. J. M. Yeomans, Mesoscale simulations: Lattice Boltzmann and particle algorithms, Physica A, 369 (2006), 159.
  • 28. P. J. Hoogerbrugge, J. M. V. A. Koelman, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, EPL, 19 (1992), 155.
  • 29. S. Bedkihal, J. C. Kumaradas, K. Rohlf, Steady flow through a constricted cylinder by multiparticle collision dynamics, Biomech. Model. Mechan., 12 (2013), 929-939.    
  • 30. T. Akhter, K. Rohlf, Quantifying compressibility and slip in multiparticle collision (MPC) flow through a local constriction, Entropy, 16 (2014), 418-442.    
  • 31. K. Rohlf, Compressible slip flow through constricted cylinders with density-dependent viscosity, Dynam. Cont. Dis. Ser. B, 25 (2018), 233-257.
  • 32. T. Ihle, D. M. Kroll, Stochastic rotation dynamics: A Galilean-invariant mesoscopic model for fluid flow, Phys. Rev. E, 63 (2001), 020201.
  • 33. T. Ihle, D. M. Kroll, Stochastic rotation dynamics: I. Formalism, Galilean invariance, and Green-Kubo relations, Phys. Rev. E, 67 (2003), 066705.
  • 34. T. Ihle, D. M. Kroll, Stochastic rotation dynamics: II. Transport coefficients, numerics, and long-time tails, Phys. Rev. E, 67 (2003), 066706.
  • 35. H. Noguchi, G. Gompper, Transport coefficients of off-lattice mesoscale-hydrodynamics simulation techniques, Phys. Rev. E, 78 (2008), 016706.
  • 36. E. Tüzel, T. Ihle, D. M. Kroll, Dynamic correlations in stochastic rotation dynamics, Phys. Rev. E, 74 (2006), 056702.
  • 37. R. Strehl, K. Rohlf, Multiparticle collision dynamics for diffusion-influenced signaling pathways, Phys. Biol., 13 (2016), 046004.
  • 38. A. Sayyidmousavi, K. Rohlf, Reactive multi-particle collision dynamics with reactive boundary conditions, Phys. Biol., 15 (2018), 046007.
  • 39. V. Mortazavi, M. Nosonovsky, Friction-induced pattern formation and Turing systems, Langmuir, 27 (2011), 4772-4779.    
  • 40. D. A. Garzon-Alvarado, C. H. Galeano, J. M. Mantilla, Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: A first example, Appl. Math. Model., 36 (2012), 5029-5045.    
  • 41. J. Wei, M. Winter, Flow-distributed spikes for Schnakenberg kinetics, J. Math. Biol., 64 (2012), 211-254.    


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved