Abstract
Many models and simulation algorithms of intracellular 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, 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 chapter, we first present a mathematical description of the diffusion induced by concentration gradient. The diffusion coefficients and mechanical quantities as frictional forces are dependent on the local values of solutes concentration. We then present an algorithm implementing the model and simulating a reaction–diffusion system. The algorithm is an efficient modification of the well-known Gillespie algorithm, adapted for systems that are both reactive and diffusive. Second, we present mathematical models of drug release and drug diffusion.
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References
P.S. Agutter, P.C. Malone, D.N. Wheatley, Intracellular transport mechanisms: a critique of diffusion theory. I. Theor. Biol. 176, 261–272 (1995)
P.S. Agutter, D.N. Wheatley, Random walks and cell size. BioEssays 22, 1018–1023 (2000)
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, 4th edn (Garland Science, 2003)
D. Bernstein, Exact stochastic simulation of coupled chemical reactions. Phys. Rev. E 71 (2005)
J. Crank, The Mathematics of Diffusion, 2nd edn. (Oxford Universuty Press, Oxford, 1975)
J. Elf, M. Ehrenberg, Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. Syst. Biol. 1(2), 230–236 (2004)
Y. Fu, W.J. Kao, Drug release kinetics and transport mechanisms of non-degradable and degradable polymeric delivery systems. Exp. Opin. Drug Delivery 7(4), 429–444 (2010)
D. Fusco, N. Accornero, B. Lavoie, S. Shenoy, J. Blanchard, R. Singer, E. Bertrand, Single mRNA molecules demonstrate probabilistic movement in living mammallian cells. Curr. Biol. 13, 161–167 (2003)
D.T. Gillespie, Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)
S.E. Harding, P. Johnson, The concentration dependence of macromolecular parameters. Biochem. J. 231, 543–547 (1985)
T. Higuchi, Mechanism of sustained-action medication. theoretical analysis of rate of release of solid drugs dispersed in solid matrices. J. Pharm. Sci. 52(12), 1145–1149 (1963)
E.R. Kandel, The molecular biology of memory storage: a dialogue between genes and synapses. Science 294, 1030–1038 (2001)
K.J. Laidler, J.H. Meiser, B.C. Sanctuary, Physical Chemistry (Houghton Mifflin Company, 2003)
P. Lecca, L. Demattè, Stochastic simulation of reaction diffusion systems. J. Med. Biol. Eng. 1(4), 211–231 (2008)
P. Lecca, L. Demattè, C. Priami, Modeling and simulating reaction-diffusion systems with state-dependent diffusion coefficients, in International Conference on Bioinformatics and Biomedicine 2008, vol. 34, p. 361 (World Academy of Science, Engineering and Technology, 2008)
P. Lecca, A.E.C. Ihekwaba, L. Dematté, C. Priami, Stochastic simulation of the spatio-temporal dynamics of reaction-diffusion systems: the case for the bicoid gradient. J. Integr. Bioinform. 7(1), 140–171 (2010)
P. Macheras, A. Iliadis, Modelling in Biopharmaceutics, Phamacokinetics and Pharmacodynamics. Homogenoeus and Heterogeneous Approaches, 1st edn. (Springer, Berlin, 2006)
D.R. Paul, Elaborations on the higuchi model for drug delivery. Int. J. Pharm. 418(1), 13–17 (2011)
J. Siepmann, N.A. Peppas, Higuchi equation: Derivation, applications, use and misuse. Int. J. Pharm. 418(1), 6–12 (2011)
J. Siepmanna, N.A. Peppas, Higuchi equation: Derivation, applications, use and misuse. Int. J. Pharm. 418(1), 6–12 (2011)
A. Solovyova, P. Schuck, L. Costenaro, C. Ebel, Non ideality of sedimantation velocity of halophilic malate dehydrogenase in complex solvent. Biophys. J. 81, 1868–1880 (2001)
M.P. Tombs, A.R. Peacocke, The Osmotic Pressure of Biological Macromolecules. Monograph on Physical Biochemistry (Oxford University Press, Oxford, 1975)
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Lecca, P., Carpentieri, B. (2023). Reaction–Diffusion Systems. In: Introduction to Mathematics for Computational Biology. Techniques in Life Science and Biomedicine for the Non-Expert. Springer, Cham. https://doi.org/10.1007/978-3-031-36566-9_6
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DOI: https://doi.org/10.1007/978-3-031-36566-9_6
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