Abstract
A detailed analysis of the main results concerning mathematical models of diffusion-controlled reactions in two dimensions is presented. Specific emphasis is placed on methods for evaluating association rates. After a review of planar models, the effects due to the curvature of the ambient space are investigated. Finally, different possible choices of boundary conditions are considered, and suggestions are given on their aptness to model different physicochemical situations.
Similar content being viewed by others
References
Singer S. J. and Nicolson G. L.: ‘The Fluid Mosaic Model of the Structure of Cell Membranes’, Science 175 (1972), 720.
DeLisi C.: ‘The Biophysics of Ligand-Receptor Interactions’, Quart. Rev. Biophys. 13 (1980) 201–232.
Yahara I. and Edelman G. M.: ‘Modulation of Limphocyte Receptor Mobility by Locally Bound Con A’, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 1579–1583.
Krakauer, H. and Peacock, J. S., Archer, B. G., and Krakauer, T.: ‘The Interaction of Surface Immunoglobulins of Lymphocytes with Highly Defined Synthetic Antigens’, in Physical Chemical Aspects of Cell Surfaces Events in Cellular Regulations, Elsevier/North-Holland, 1979, pp. 345–362.
Ishizaka K. and Ishizaka T.: ‘Induction of Erithema — Wheal Reaction by Soluble Antigen-Antibody Complexes in Humans’, J. Immunol. 69 (1968), 101.
Siriganian R. P., Hook W. A., and Levine R. B.: ‘Specific in Vitro Histamine Release from Basophils by Bivalent Haptens: Evidence for Activation by Simple Bridging of Membrane Bound Antibody’, Immunochem. 12 (1975), 149–157.
Adam G. and Delbrück M.: ‘Reduction of Dimensionality in Biological Diffusion Processes’, in Structural Chemistry and Molecular Biology, A.Rich and N.Davidson (eds.), W. H. Freeman, San Francisco, 1968, pp. 198–215.
Berg H. C. and Purcell E. M.: ‘Physics of Chemoreception’, Biophys. J. 20 (1977), 193–219.
Hochman J. H., Schindler M., Lee J. G., Ferguson-Miller S.: ‘Lateral Mobility of Cytochrome c on Intact Mitochondrial Membranes as Determined by Fluorescence Redistribution after Photobleaching’, Proc. Nat. Acad. Sci. U.S.A. 79 (1982), 6866–6870.
Yguerabide J., Dillon M. A., and Burton M.: ‘Kinetics of Diffusion-Controlled Processes in Liquids. Theoretical Consideration of Luminescent Systems: Quenching and Excitation Transfer in Collision’, J. Chem. Phys. 40 (1964) 3040.
Vanderkooi J. M. and Callis J. B.: ‘Pyrene. A Probe of Lateral Diffusion in the Hydrophobic Region of Membranes’, Biochemistry 13 (1974), 4000–4006.
Szabo A., Schulten K., and Schulten Z.: ‘First Passage Time Approach to Diffusion-Controlled Reactions’, J. Chem. Phys. 72 (1980), 4350–4357.
DeLisi C. and Wieger F. W.: ‘Effect of Nonspecific Forces and Finite Receptor Number on Rate Constants of Ligand-All Bound-Receptor Interactions’, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 5569–5572.
Wank, S. A., De Lisi, C., and Metzger, H.: ‘Analysis of the Rate-Limiting Step in Ligand-Cell Receptor Interaction: The Immunoglobulin E System’, Biochemistry (Feb. 15, 1983), 954–959.
Noyes R. M.: ‘Effects of Diffusion Rates on Chemical Kinetics’, in Progress in Reaction Kinetics, Vol. 1, G.Porter (ed.), Pergamon Press, London, 1961.
Razi Naqvi K.: ‘Diffusion-Controlled Reactions in Two-Dimensional Fluids: Discussion of Measurements of Lateral Diffusion of Lipids in Biological Membranes’, Chem. Phys. Lett. 28 (1974), 280–284.
Carslaw H. S. and Jaeger J. C.: Conduction of Heat in Solids, Clarendon Press, Oxford, 1959.
Owen C. S.: ‘Two Dimensional Diffusion Theory: Cylindrical Diffusion Model Applied to Fluorescence Quenching’, J. Chem. Phys. 62 (1975), 3204–3207.
McKean H. P.Jr.: Stochastic Integrals, Academic Press, New York, London, 1969.
Papanicolaou G. and Varadhan S. R. S.: ‘Diffusions in Regions with Many Small Holes’, in Stochastic Differential Systems, Proc. Vilnius Conference in Probability, B.Grigelionis (ed.), Lecture Notes in Control and Information Sciences No. 25, Springer Verlag, Berlin, 1980, pp. 190–206.
Wiegel F. W. and DeLisi C.: ‘Evaluation of Reaction Rate Enhancement by Reduction in Dimensionality”, Am. J. Physiol. 243 (Regulatory Integrative Comp. Physiol. 12) (1982), R475-R479.
Del Grosso, G. and Marchetti, F.: ‘Principal Eigenvalues and Exit Times for Diffusions on Manifolds’. Submitted to Syst. An-Mod-Simul.
Mandl P.: Analytical Treatment of One-Dimensional Markov Processes, Springer-Verlag, Berlin, 1968.
DelGrosso G. and Marchetti F.: ‘Asymptotic Estimates on the Principal Eigenvalue of the Laplacian in a Geodesic Ball’, Appl. Math. Optim. 10 (1983), 37–50.
Marchetti F.: ‘Asymptotic Exponentiality of Exit Times’, Stat. Prob. Lett. 1 (1983), 167–170.
DelGrosso G., Gerardi A., and Marchetti F.: ‘A Diffusion Model for Patch Formation on Cellular Surfaces’, Appl. Math. Optim. 7 (1981), 125–135.
Pinsky M. A.: ‘The First Eigenvalue of a Spherical Cap’, Appl. Math. Optim. 7 (1981), 137–139.
Mitra, A. K.: ‘On a Model of Partial Diffusions on Spherical Cell Membranes’, Report of Texas Tech. University, Lubbock, Texas (1980).
Collins F. C. and Kimball G. E.: ‘Diffusion-Controlled Reaction Rates’, J. Colloid. Sci. 4 (1949), 425–437.
Feller W.: ‘The Parabolic Differential Equations and the Associated Semigroups of Transformations’, Ann. Math. 55 (1952), 468–519.
Ikeda N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
Gerardi, A., Marchetti, F., and Rosa, A. M.: ‘Simulations of Diffusions with Boundary Conditions’, Syst. Contr. Let. (to appear).
Khruslov E. I. and Marchenko V. A.: Boundary Value Problems in Regions with Fine-Grained Boundaries, Naukova Dumka, Kiev, 1974.
Szabo A., Shoup D., Northrup S. H. and McCammon J. A.: ‘Stochastically Gated Diffusion-Influenced Reactions’, J. Chem. Phys. 77 (1982), 4484–4493.
Nicolson G. L. and Poste G.: ‘The Cancer Cell: Dynamic Aspects and Modifications in Cell Surface Organization I, II’, New England J. Med. 295 (1976), 197–203, 253–258.
Lang R. and Nguyen Xuan Xanh: ‘Smoluchovski's Theory of Coagulation in Colloids Holds Rigorously in the Boltzmann-Grad-Limit’, Z. Warsch. verw. Geb. 54 (1980), 227–280.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gandolfi, A., Gerardi, A. & Marchetti, F. Association rates of diffusion-controlled reactions in two dimensions. Acta Appl Math 4, 139–155 (1985). https://doi.org/10.1007/BF00052459
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00052459