Stochastic Simulation of Reaction-Diffusion Systems
Authors: Paola Lecca, Lorenzo Dematte
Abstract:
Reactiondiffusion systems are mathematical models that describe how the concentration of one or more substances distributed in space changes under the influence of local chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose general form is ÔêétX(x, t) = DΔX(x, t), where X(x, t) is the state vector, D is the matrix of the diffusion coefficients and Δ is the Laplace operator. If the solute move in an homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and of solutes and on local temperature of the medium. In this paper a new stochastic reaction-diffusion model in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces of solvent and solute is presented. Such a model provides a more realistic description of the molecular kinetics in non-homogenoeus and highly structured media as the intra- and inter-cellular spaces. The movement of a molecule A from a region i to a region j of the space is described as a first order reaction Ai k- → Aj , where the rate constant k depends on the diffusion coefficient. Representing the diffusional motion as a chemical reaction allows to assimilate a reaction-diffusion system to a pure reaction system and to simulate it with Gillespie-inspired stochastic simulation algorithms. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the specific speed of reaction and diffusion events. Redi is the software tool, developed to implement the model of reaction-diffusion kinetics and dynamics. It is a free software, that can be downloaded from http://www.cosbi.eu. To demonstrate the validity of the new reaction-diffusion model, the simulation results of the chaperone-assisted protein folding in cytoplasm obtained with Redi are reported. This case study is redrawing the attention of the scientific community due to current interests on protein aggregation as a potential cause for neurodegenerative diseases.
Keywords: Reaction-diffusion systems, Fick's law, stochastic simulation algorithm.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1331659
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[1] J. Elf, A. Doncic, and M. Ehrenberg, "Mesoscopic reaction-diffusion in intracellular signaling," Fluctuation and noise in biological, biophysical and biomedical systems. Procs. of SPIE, vol. 5110, 2003.
[2] P. S. Agutter and D. Wheatley, "Random walks and cell size," BioEssays, vol. 22, pp. 1018-1023, 2000.
[3] P. Agutter, P. Malone, and D. Wheatley, "Intracellular transport mechanisms: a critique of diffusion theory," J. Theor. Biol., vol. 176, pp. 261-272, 1995.
[4] D. Fusco, N. Accornero, B. Lavoie, S. Shenoy, J. Blanchard, R. Singer, and E. Bertrand, "Single mrna molecules demonstrate probabilistic movement in living mammallian cells.," Curr. Biol., vol. 13, pp. 161- 167, 2003.
[5] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular biology of the cell. Garland Science, 4th ed. ed., 2003.
[6] E. R. Kandel, "The molecular biology of memory storage: a dialogue between genes and synapses," Science, vol. 294, pp. 1030-1038, 2001.
[7] D. Gillespie, "Exact stochastic simulation of coupled chemical reactions," Journal of Physical Chemistry, vol. 81, pp. 2340-2361, December 1977.
[8] C. J. Roussel and M. R. Roussel, "Reaction-diffusion models of development with state-dependent chemical diffusion coefficients.," Progress in Biophysics & Molecular Biology, 2004.
[9] K. J. Laidler, J. H. meiser, and B. C. Sanctuary, Physical chemistry. Houghton Mifflin Company Boston New York, 2003.
[10] M. P. Tombs and A. R. Peacocke, The Osmotic Pressure of Biological Macromolecules. Monograph on Physical Biochemistry, Oxford University Press, 1975.
[11] A. Solovyova, P. Schuck, L. Costenaro, and C. Ebel, "Non ideality of sedimantation velocity of halophilic malate dehydrogenase in complex solvent," Biophysical Journal, vol. 81, pp. 1868-1880, 2001.
[12] K. Laidler, J. Meiser, and B. Sanctuary, Physical Chemistry. Houghton Mifflin Company, 2003.
[13] S. Harding and P. Johnson, "The concentration dependence of macromolecular parameters," Biochemical Journal, vol. 231, pp. 543-547,1985.
[14] D. Bernstein, "Simulating mesoscopic reaction-diffusion systems using the gillespie algorithm," PHYSICAL REVIEW E, vol. 71, April 2005.
[15] J. Elf and M. Ehrenberg, "Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases," Syst. Biol., vol. 1, December 2004.
[16] A. R. Kinjo and S. Takada, "Competition between protein folding and aggregation wth molecular chaperones in crowed solutions: insight from mesoscopic simulations," Biophysical Journal, vol. 85, pp. 3521 - 3531, 2003.
[17] H. S. Chan and K. A. Dill, "A simple model of chepronin-mediated protein folding," PROTEINS: Structure, Function, and Genetics, vol. 24, pp. 345-351, 1996.
[18] W. A. Houry, "Chaperone-assisted protein folding," Curr. protein Pept. Sci., vol. 2, no. 3, pp. 227-244, 2001.
[19] J. Frydman and F. U. Hartl, "Principles of chaperone-assisted folding: differences between in vitro and in vivo mechanisms," Science, vol. 272, no. 5667, pp. 1497 - 1502, 1996.
[20] T. Langer, J. martin, E. Nimmesgern, and F. U. Hartl, "The pathway of chaperone-assisted protein folding," Fresenius- Journal of Analytical Chemistry, vol. 343, 1992.
[21] J. Martin and F. U. Hartl, "The effect of macromolecular crowding on chaperonin-mediated protein folding," Proc. Natl. Acad. Sci. USA, vol. 94, pp. 1107-1112, 1997.
[22] D. Thirulamai and G. H. Lorimer, "Chaperonin-mediated protein folding," Ann. Rev. Biophys. Biomol. Struct., vol. 30, pp. 245-268, 2001.
[23] D. Thirumalai and G. H. Lorimer, "Chaperonin-mediated protein folding," Annu. Rev. Biophys. Biomol. Struct., vol. 30, p. 245:269, 2001.
[24] S. A. Isaacson and C. S. Peskin, "Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations," SIAM Journal of Scientific computing, pp. 47-74, 2006.
[25] B. L. Neal, D. Asthagiri, and A. M. Lenhoff, "Molecular origins of osmotic second virial coefficients of proteins," Biophysical Journal, vol. 75, 1998.
[26] A. R. Kinjo and S. Takada, "Effects of macromolecular crowding on protein folding and aggregation studied bu density functional theory: statics," Physical Review E, vol. 66, pp. 031911: 1-9, 2002.
[27] A. R. Kinjo and S. Takada, "Effects of macromolecular crowding on protein folding and aggregation studied by density functional theory: Dynamics," Physical review. E, vol. 66, no. 5, pp. 051902.1-051902.10, 2002.
[28] G. Y. G. Ping and J. M. Yuan, "Depletion force from macromolecular crowding enhances mechanicsl stability of protein molecules," Polymer, vol. 27, p. 2564:2570, 2006.
[29] A. P. Minton, "Molecular crowding: analysis of effects of high concetrations of inert cosolutes on biochemical equilibria and rates in terms of volume exclusion," Methods Enzymol., vol. 295, p. 127:149, 1998.
[30] A. P. Minton, "The influence of macromolecular crowding and macromolecular confinement on biochemical reactions in physiological media," J. Biol. Chem., vol. 276, p. 10577:10580, 2001.
[31] P. Lecca, "A time-dependent extension of gillespie algorithm for biochemical stochastic ¤Ç-calculus," Proceedings of the 2006 ACM symposium on Applied computing, 2001.
[32] P. Lecca, "Simulating the cellular passive transport of glucose using a time-dependent extension of gillespie algorithm for stochastic ¤Ç-calculus.," International Journal of Data Mining and Bioinformatics, vol. 1, no. 4, pp. 315-336, 2007.
[33] L. Dematt'e and T. Mazza, "On parallel stochastic simulation of diffusive systems," in Sixth International Conference on Computational Methods in Systems Biology, 2008.