Modeling and Simulating Reaction-Diffusion Systems with State-Dependent Diffusion Coefficients
The present models and simulation algorithms of intracellular stochastic kinetics are usually based on the premise that diffusion is so fast that the concentrations of all the involved species are homogeneous in space. However, recents experimental measurements of intracellular diffusion constants indicate that the assumption of a homogeneous well-stirred cytosol is not necessarily valid even for small prokaryotic cells. In this work a mathematical treatment of diffusion that can be incorporated in a stochastic algorithm simulating the dynamics of a reaction-diffusion system is presented. The movement of a molecule A from a region i to a region j of the space is represented as a first order reaction Ai k- ! Aj , where the rate constant k depends on the diffusion coefficient. The diffusion coefficients are modeled as function of the local concentration of the solutes, their intrinsic viscosities, their frictional coefficients and the temperature of the system. 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 intrinsic reaction kinetics and diffusion dynamics. To demonstrate the method the simulation results of the reaction-diffusion system of chaperoneassisted protein folding in cytoplasm are shown.
[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] R. J. E. B. van den Berg and C. M. Dobson, "Effects of macromolecular
crowding on protein folding and aggregation," The EMBO Journal,
vol. 18, p. 6927:6933, 1999.
[15] J. J. Z. Hu and R. Rajagopalan, "Effects of macromolecular crowding
on biochemical reaction equilibria: A molecular thermodynamic perspective,"
Biophysical Journal, vol. 93, pp. 1464-1473, 2007.
[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] 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.
[18] 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.
[19] D. Bernstein, "Simulating mesoscopic reaction-diffusion systems using
the gillespie algorithm," PHYSICAL REVIEW E, vol. 71, April 2005.
[20] J. Elf and M. Ehrenberg, "Spontaneous separation of bi-stable biochemical
systems into spatial domains of opposite phases," Syst. Biol., vol. 1,
December 2004.
[21] 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.
[22] W. A. Houry, "Chaperone-assisted protein folding," Curr. protein Pept.
Sci., vol. 2, no. 3, pp. 227-244, 2001.
[23] 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.
[24] 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.
[25] 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.
[26] D. Thirulamai and G. H. Lorimer, "Chaperonin-mediated protein folding,"
Ann. Rev. Biophys. Biomol. Struct., vol. 30, pp. 245-268, 2001.
[27] D. Thirumalai and G. H. Lorimer, "Chaperonin-mediated protein folding,"
Annu. Rev. Biophys. Biomol. Struct., vol. 30, p. 245:269, 2001.
[28] 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.
[29] B. L. Neal, D. Asthagiri, and A. M. Lenhoff, "Molecular origins of
osmotic second virial coefficients of proteins," Biophysical Journal,
vol. 75, 1998.
[30] 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.
[31] 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.
[32] 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.
[33] P. Lecca, "A time-dependent extension of gillespie algorithm for biochemical
stochastic -calculus," Proceedings of the 2006 ACM symposium
on Applied computing, 2001.
[34] P. Lecca, "Simulating the cellular passive transport of glucose using
a time-dependent extension of gillespie algorithm for stochastic -
calculus.," International Journal of Data Mining and Bioinformatics,
vol. 1, no. 4, pp. 315-336, 2007.
[35] L. Dematt'e and T. Mazza, "On parallel stochastic simulation of diffusive
systems," in Sixth International Conference on Computational Methods
in Systems Biology, 2008.
[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] R. J. E. B. van den Berg and C. M. Dobson, "Effects of macromolecular
crowding on protein folding and aggregation," The EMBO Journal,
vol. 18, p. 6927:6933, 1999.
[15] J. J. Z. Hu and R. Rajagopalan, "Effects of macromolecular crowding
on biochemical reaction equilibria: A molecular thermodynamic perspective,"
Biophysical Journal, vol. 93, pp. 1464-1473, 2007.
[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] 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.
[18] 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.
[19] D. Bernstein, "Simulating mesoscopic reaction-diffusion systems using
the gillespie algorithm," PHYSICAL REVIEW E, vol. 71, April 2005.
[20] J. Elf and M. Ehrenberg, "Spontaneous separation of bi-stable biochemical
systems into spatial domains of opposite phases," Syst. Biol., vol. 1,
December 2004.
[21] 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.
[22] W. A. Houry, "Chaperone-assisted protein folding," Curr. protein Pept.
Sci., vol. 2, no. 3, pp. 227-244, 2001.
[23] 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.
[24] 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.
[25] 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.
[26] D. Thirulamai and G. H. Lorimer, "Chaperonin-mediated protein folding,"
Ann. Rev. Biophys. Biomol. Struct., vol. 30, pp. 245-268, 2001.
[27] D. Thirumalai and G. H. Lorimer, "Chaperonin-mediated protein folding,"
Annu. Rev. Biophys. Biomol. Struct., vol. 30, p. 245:269, 2001.
[28] 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.
[29] B. L. Neal, D. Asthagiri, and A. M. Lenhoff, "Molecular origins of
osmotic second virial coefficients of proteins," Biophysical Journal,
vol. 75, 1998.
[30] 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.
[31] 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.
[32] 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.
[33] P. Lecca, "A time-dependent extension of gillespie algorithm for biochemical
stochastic -calculus," Proceedings of the 2006 ACM symposium
on Applied computing, 2001.
[34] P. Lecca, "Simulating the cellular passive transport of glucose using
a time-dependent extension of gillespie algorithm for stochastic -
calculus.," International Journal of Data Mining and Bioinformatics,
vol. 1, no. 4, pp. 315-336, 2007.
[35] L. Dematt'e and T. Mazza, "On parallel stochastic simulation of diffusive
systems," in Sixth International Conference on Computational Methods
in Systems Biology, 2008.
@article{"International Journal of Chemical, Materials and Biomolecular Sciences:49600", author = "Paola Lecca and Lorenzo Dematte and Corrado Priami", title = "Modeling and Simulating Reaction-Diffusion Systems with State-Dependent Diffusion Coefficients", abstract = "The present models and simulation algorithms of intracellular stochastic kinetics are usually based on the premise that diffusion is so fast that the concentrations of all the involved species are homogeneous in space. However, recents experimental measurements of intracellular diffusion constants indicate that the assumption of a homogeneous well-stirred cytosol is not necessarily valid even for small prokaryotic cells. In this work a mathematical treatment of diffusion that can be incorporated in a stochastic algorithm simulating the dynamics of a reaction-diffusion system is presented. The movement of a molecule A from a region i to a region j of the space is represented as a first order reaction Ai k- ! Aj , where the rate constant k depends on the diffusion coefficient. The diffusion coefficients are modeled as function of the local concentration of the solutes, their intrinsic viscosities, their frictional coefficients and the temperature of the system. 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 intrinsic reaction kinetics and diffusion dynamics. To demonstrate the method the simulation results of the reaction-diffusion system of chaperoneassisted protein folding in cytoplasm are shown.
", keywords = "Reaction-diffusion systems, diffusion coefficient, stochastic simulation algorithm.", volume = "2", number = "8", pages = "140-21", }
