Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-18T09:52:13.249Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Sampling in Event-Driven Algorithms for Reaction-Diffusion Processes

Published online by Cambridge University Press:  03 June 2015

Mohammad Hossein Bani-Hashemian ,
Stefan Hellander  and
Per Lötstedt
Mohammad Hossein Bani-Hashemian*
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, P. O. Box 337, SE-75105 Uppsala, Sweden
Stefan Hellander*
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, P. O. Box 337, SE-75105 Uppsala, Sweden
Per Lötstedt*
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, P. O. Box 337, SE-75105 Uppsala, Sweden
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

In event-driven algorithms for simulation of diffusing, colliding, and reacting particles, new positions and events are sampled from the cumulative distribution function (CDF) of a probability distribution. The distribution is sampled frequently and it is important for the efficiency of the algorithm that the sampling is fast. The CDF is known analytically or computed numerically. Analytical formulas are sometimes rather complicated making them difficult to evaluate. The CDF may be stored in a table for interpolation or computed directly when it is needed. Different alternatives are compared for chemically reacting molecules moving by Brownian diffusion in two and three dimensions. The best strategy depends on the dimension of the problem, the length of the time interval, the density of the particles, and the number of different reactions.

Keywords

65C0565C3582C8002.50.Ey02.70.Uu83.10.Rs87.10.RtEvent-driven algorithmchemical reactionsdiffusionefficient sampling
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 4 , April 2013 , pp. 958 - 984
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Andrews, S. S., Addy, N. J., Brent, R., and Arkin, A. P.Detailed simulations of cell biology with Smoldyn 2.1. PLoS Comput. Biol., 6(3):e1000705,2010.Google Scholar
[2]Bani-Hashemian, M. H.Accurate and Efficient Solution of the Smoluchowski Equation. Master’s thesis, Department of Information Technology, Uppsala University, Uppsala, Sweden, 2011. Available online at: http: //urn.kb. se/resolve?urn=urn:nbn: se:uu: diva-156440.Google Scholar
[3]Carlsson, T., Ekholm, T., and Elvingson, C.Algorithm for generating a Brownian motion on a sphere. J. Phys. A: Math. Theor., 43:505001,2010.Google Scholar
[4]Carslaw, H. S. and Jaeger, J. C.Conduction of Heat in Solids. Oxford University Press, Oxford, second edition, 1959.Google Scholar
[5]Coggan, J. S., Bartol, T. M., Esquenazi, E., Stiles, J. R., Lamont, S., Martone, M. E., Berg, D. K., Ellisman, M. H., and Sejnowski, T. J.Evidence for ectopic neurotransmission at a neuronal synapse. Science, 309:446–451,2005.Google Scholar
[6]Collins, F. C. and Kimball, G. E.Diffusion-controlled reaction rates. J. Colloid. Sci., 4:425-437, 1949.Google Scholar
[7]Donev, A.Asynchronous event-driven particle algorithms. Simulation, 85(4):229–242,2009.CrossRefGoogle Scholar
[8]Donev, A., Bulatov, V. V., Oppelstrup, T., Gilmer, G. H., Sadigh, B., and Kalos, M. H.A first- passage kinetic Monte Carlo algorithm for complex diffusion-reaction systems. J. Comput. Phys., 229:3214–3236,2010.Google Scholar
[9]Hellander, S. and Lotstedt, P.Flexible single molecule simulation of reaction-diffusion processes. J. Comput. Phys., 230(10):3948–3965,2011.CrossRefGoogle Scholar
[10]Kim, H. and Shin, K. J.Exact solution of the reversible diffusion-influenced reaction for an isolated pair in three dimensions. Phys. Rev. Lett., 82(7):1578–1581,1999.Google Scholar
[11]Kim, H., Yang, M., and Shin, K. J.Dynamic correlation effect in reversible diffusion-influenced reactions: Brownian dynamics simulation in three dimensions. J. Chem. Phys., 111(3):1068–1075,1999.CrossRefGoogle Scholar
[12]Plimpton, S. J. and Slepoy, A.Microbial cell modeling via reacting diffusive particles. J. Physics: Conf. Series, 16(1):305,2005.Google Scholar
[13]Scala, A., Voigtmann, Th., and De Michele, C.Event-driven Brownian dynamics for hard spheres. J. Chem. Phys., 126(13):134109,2007.Google Scholar
[14]Smoluchowski, M.Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem., 92:129–168,1917.Google Scholar
[15]Strang, G.On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5:506–517,1968.Google Scholar
[16]Tăkahashi, K., Tanase-Nicola, S., and ten Wolde, P. R.Spatio-temporal correlations can drastically change the response of a MAPK pathway. Proc. Natl. Acad. Sci. USA, 107(6):2473–2478, 2010.Google Scholar
[17]Torre, J. Dalla, Bocquet, J.-L., Doan, N. V., Adam, E., and Barbu, A.JERK, an event-based Kinetic Monte Carlo model to predict microstructure evolution of materials under irradiation. Phil. Mag., 85(4):549–558,2005.Google Scholar
[18]Tulovsky, V. and Papiez, L.Formula for the fundamental solution of the heat equation on the sphere. Appl. Math. Lett., 14:881–884,2001.Google Scholar
[19]van Zon, J. S. and ten Wolde, P. R.Green’s-Function Reaction Dynamics: A particle-based approach for simulating biochemical networks in time and space. J. Chem. Phys., 123(23):234910,2005.Google Scholar
[20]Winitzki, S.Uniform approximations for transcendental functions. In Kumar, V., Gavrilova, M., Tan, C., and L’Ecuyer, P., editors, Computational Science and Its Applications - ICCSA 2003, volume 2667 of Lecture Notes in Computer Science, pages 780–789. Springer, 2003.Google Scholar
[21]Yosida, K.Brownian motion on the surface of the 3-sphere. Ann. Math. Statistics, 20:292-296, 1949.Google Scholar