Accelerated stochastic and hybrid methods for spatial simulations of reaction–diffusion systems

Abstract Spatial distributions characterize the evolution of reaction–diffusion models of several physical, chemical, and biological systems. We present two novel algorithms for the efficient simulation of these models: Spatial τ -Leaping ( S τ -Leaping), employing a unified acceleration of the stochastic simulation of reaction and diffusion, and Hybrid τ -Leaping ( H τ -Leaping), combining a deterministic diffusion approximation with a τ -Leaping acceleration of the stochastic reactions. The algorithms are validated by solving Fisher’s equation and used to explore the role of the number of particles in pattern formation. The results indicate that the present algorithms have a nearly constant time complexity with respect to the number of events (reaction and diffusion), unlike the exact stochastic simulation algorithm which scales linearly.

[1]  David Bernstein,et al.  Simulating mesoscopic reaction-diffusion systems using the Gillespie algorithm. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  C. Lumsden,et al.  Stochastic Simulation of Coupled Reaction-Diffusion Processes , 1996 .

[3]  B. Ingalls,et al.  Deterministic characterization of stochastic genetic circuits , 2007, Proceedings of the National Academy of Sciences.

[4]  Jaap A. Kaandorp,et al.  Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli , 2006, Bioinform..

[5]  Mauro Ferrari,et al.  Morphologic Instability and Cancer Invasion , 2005, Clinical Cancer Research.

[6]  D. Gillespie Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[7]  Anne Auger,et al.  R-leaping: accelerating the stochastic simulation algorithm by reaction leaps. , 2006, The Journal of chemical physics.

[8]  Linda R Petzold,et al.  Efficient step size selection for the tau-leaping simulation method. , 2006, The Journal of chemical physics.

[9]  J. E. Pearson Complex Patterns in a Simple System , 1993, Science.

[10]  Ronald W. Shonkwiler Mathematical Biology: An Introduction with Maple and Matlab , 2009 .

[11]  D. Gillespie A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[12]  Luis Serrano,et al.  Space as the final frontier in stochastic simulations of biological systems , 2005, FEBS letters.

[13]  B. M. Fulk MATH , 1992 .

[14]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[15]  Johan Hattne,et al.  Stochastic reaction-diffusion simulation with MesoRD , 2005, Bioinform..

[16]  P. Degond,et al.  The weighted particle method for convection-diffusion equations , 1989 .

[17]  D. Gillespie Approximate accelerated stochastic simulation of chemically reacting systems , 2001 .

[18]  J. Elf,et al.  Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. , 2004, Systems biology.

[19]  R. Fisher THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[20]  Nicolas Le Novère,et al.  Particle-Based Stochastic Simulation in Systems Biology , 2006 .

[21]  Eshel Ben-Jacob,et al.  The artistry of nature , 2001, Nature.

[22]  Michael A. Gibson,et al.  Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels , 2000 .

[23]  Ericka Stricklin-Parker,et al.  Ann , 2005 .