A hybrid continuous-discrete method for stochastic reaction–diffusion processes

Stochastic fluctuations in reaction–diffusion processes often have substantial effect on spatial and temporal dynamics of signal transductions in complex biological systems. One popular approach for simulating these processes is to divide the system into small spatial compartments assuming that molecules react only within the same compartment and jump between adjacent compartments driven by the diffusion. While the approach is convenient in terms of its implementation, its computational cost may become prohibitive when diffusive jumps occur significantly more frequently than reactions, as in the case of rapid diffusion. Here, we present a hybrid continuous-discrete method in which diffusion is simulated using continuous approximation while reactions are based on the Gillespie algorithm. Specifically, the diffusive jumps are approximated as continuous Gaussian random vectors with time-dependent means and covariances, allowing use of a large time step, even for rapid diffusion. By considering the correlation among diffusive jumps, the approximation is accurate for the second moment of the diffusion process. In addition, a criterion is obtained for identifying the region in which such diffusion approximation is required to enable adaptive calculations for better accuracy. Applications to a linear diffusion system and two nonlinear systems of morphogens demonstrate the effectiveness and benefits of the new hybrid method.

[1]  Sotiria Lampoudi,et al.  The multinomial simulation algorithm for discrete stochastic simulation of reaction-diffusion systems. , 2009, The Journal of chemical physics.

[2]  Q. Nie,et al.  ROBUSTNESS OF MORPHOGEN GRADIENTS WITH "BUCKET BRIGADE" TRANSPORT THROUGH MEMBRANE-ASSOCIATED NON-RECEPTORS. , 2012, Discrete and continuous dynamical systems. Series B.

[3]  M. Elowitz,et al.  Functional roles for noise in genetic circuits , 2010, Nature.

[4]  Samuel A. Isaacson,et al.  Incorporating Diffusion in Complex Geometries into Stochastic Chemical Kinetics Simulations , 2006, SIAM J. Sci. Comput..

[5]  Naama Barkai,et al.  Scaling of morphogen gradients by an expansion-repression integral feedback control , 2010, Proceedings of the National Academy of Sciences.

[6]  R. Erban,et al.  Stochastic modelling of reaction–diffusion processes: algorithms for bimolecular reactions , 2009, Physical biology.

[7]  O. Diekmann,et al.  Comment on "Linking population-level models with growing networks: a class of epidemic models". , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  S. Jonathan Chapman,et al.  Multiscale Reaction-Diffusion Algorithms: PDE-Assisted Brownian Dynamics , 2012, SIAM J. Appl. Math..

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

[10]  Brian Drawert,et al.  The diffusive finite state projection algorithm for efficient simulation of the stochastic reaction-diffusion master equation. , 2010, The Journal of chemical physics.

[11]  H. Othmer,et al.  A new method for choosing the computational cell in stochastic reaction–diffusion systems , 2012, Journal of mathematical biology.

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

[13]  Jifeng Hu,et al.  Stochastic Analysis of Reaction–Diffusion Processes , 2013, Bulletin of Mathematical Biology.

[14]  Samuel A. Isaacson,et al.  The Reaction-Diffusion Master Equation as an Asymptotic Approximation of Diffusion to a Small Target , 2009, SIAM J. Appl. Math..

[15]  Linda R Petzold,et al.  The slow-scale stochastic simulation algorithm. , 2005, The Journal of chemical physics.

[16]  A. Kicheva,et al.  Precision of the Dpp gradient , 2008, Development.

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

[18]  Yuan Lou,et al.  Effects of Sog on Dpp-Receptor Binding , 2005, SIAM J. Appl. Math..

[19]  D. Gillespie,et al.  Avoiding negative populations in explicit Poisson tau-leaping. , 2005, The Journal of chemical physics.

[20]  H. Meinhardt,et al.  Biological pattern formation: fmm basic mechanisms ta complex structures , 1994 .

[21]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[22]  Andreas Hellander,et al.  An adaptive algorithm for simulation of stochastic reaction-diffusion processes , 2010, J. Comput. Phys..

[23]  J. Rawlings,et al.  Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics , 2002 .

[24]  David Fange,et al.  Noise-Induced Min Phenotypes in E. coli , 2006, PLoS Comput. Biol..

[25]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[26]  Chee Meng Tan,et al.  Hybrid simulations of stochastic reaction-diffusion processes for modeling intracellular signaling pathways. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  P. Hanusse,et al.  A Monte Carlo method for large reaction–diffusion systems , 1981 .

[28]  Diego Rossinelli,et al.  Accelerated stochastic and hybrid methods for spatial simulations of reaction–diffusion systems , 2008 .

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

[30]  Q. Nie,et al.  Robust and precise morphogen-mediated patterning: trade-offs, constraints and mechanisms , 2015, Journal of The Royal Society Interface.

[31]  L E Scriven,et al.  Instability and dynamic pattern in cellular networks. , 1971, Journal of theoretical biology.

[32]  Zhen Liu,et al.  Hybrid modeling and simulation of stochastic effects on progression through the eukaryotic cell cycle. , 2012, The Journal of chemical physics.

[33]  A. Oudenaarden,et al.  Cellular Decision Making and Biological Noise: From Microbes to Mammals , 2011, Cell.

[34]  Linda R Petzold,et al.  Adaptive explicit-implicit tau-leaping method with automatic tau selection. , 2007, The Journal of chemical physics.

[35]  Andreas Hellander,et al.  URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries , 2012, BMC Systems Biology.

[36]  Qing Nie,et al.  The measure of success: constraints, objectives, and tradeoffs in morphogen-mediated patterning. , 2009, Cold Spring Harbor perspectives in biology.

[37]  Christian A. Yates,et al.  An adaptive multi-level simulation algorithm for stochastic biological systems. , 2014, The Journal of chemical physics.

[38]  K. McNeil,et al.  Correlations in stochastic theories of chemical reactions , 1976 .

[39]  Georgios Kalantzis,et al.  Hybrid stochastic simulations of intracellular reaction-diffusion systems , 2009, Comput. Biol. Chem..

[40]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .