Skip to main content
Log in

Association rates of diffusion-controlled reactions in two dimensions

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Singer S. J. and Nicolson G. L.: ‘The Fluid Mosaic Model of the Structure of Cell Membranes’, Science 175 (1972), 720.

    Google Scholar 

  2. DeLisi C.: ‘The Biophysics of Ligand-Receptor Interactions’, Quart. Rev. Biophys. 13 (1980) 201–232.

    Google Scholar 

  3. 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.

    Google Scholar 

  4. 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.

  5. Ishizaka K. and Ishizaka T.: ‘Induction of Erithema — Wheal Reaction by Soluble Antigen-Antibody Complexes in Humans’, J. Immunol. 69 (1968), 101.

    Google Scholar 

  6. 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.

    Google Scholar 

  7. 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.

    Google Scholar 

  8. Berg H. C. and Purcell E. M.: ‘Physics of Chemoreception’, Biophys. J. 20 (1977), 193–219.

    Google Scholar 

  9. 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.

    Google Scholar 

  10. 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.

    Google Scholar 

  11. Vanderkooi J. M. and Callis J. B.: ‘Pyrene. A Probe of Lateral Diffusion in the Hydrophobic Region of Membranes’, Biochemistry 13 (1974), 4000–4006.

    Google Scholar 

  12. Szabo A., Schulten K., and Schulten Z.: ‘First Passage Time Approach to Diffusion-Controlled Reactions’, J. Chem. Phys. 72 (1980), 4350–4357.

    Google Scholar 

  13. 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.

    Google Scholar 

  14. 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.

  15. Noyes R. M.: ‘Effects of Diffusion Rates on Chemical Kinetics’, in Progress in Reaction Kinetics, Vol. 1, G.Porter (ed.), Pergamon Press, London, 1961.

    Google Scholar 

  16. 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.

    Google Scholar 

  17. Carslaw H. S. and Jaeger J. C.: Conduction of Heat in Solids, Clarendon Press, Oxford, 1959.

    Google Scholar 

  18. Owen C. S.: ‘Two Dimensional Diffusion Theory: Cylindrical Diffusion Model Applied to Fluorescence Quenching’, J. Chem. Phys. 62 (1975), 3204–3207.

    Google Scholar 

  19. McKean H. P.Jr.: Stochastic Integrals, Academic Press, New York, London, 1969.

    Google Scholar 

  20. 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.

    Google Scholar 

  21. 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.

    Google Scholar 

  22. Del Grosso, G. and Marchetti, F.: ‘Principal Eigenvalues and Exit Times for Diffusions on Manifolds’. Submitted to Syst. An-Mod-Simul.

  23. Mandl P.: Analytical Treatment of One-Dimensional Markov Processes, Springer-Verlag, Berlin, 1968.

    Google Scholar 

  24. 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.

    Google Scholar 

  25. Marchetti F.: ‘Asymptotic Exponentiality of Exit Times’, Stat. Prob. Lett. 1 (1983), 167–170.

    Google Scholar 

  26. DelGrosso G., Gerardi A., and Marchetti F.: ‘A Diffusion Model for Patch Formation on Cellular Surfaces’, Appl. Math. Optim. 7 (1981), 125–135.

    Google Scholar 

  27. Pinsky M. A.: ‘The First Eigenvalue of a Spherical Cap’, Appl. Math. Optim. 7 (1981), 137–139.

    Google Scholar 

  28. Mitra, A. K.: ‘On a Model of Partial Diffusions on Spherical Cell Membranes’, Report of Texas Tech. University, Lubbock, Texas (1980).

  29. Collins F. C. and Kimball G. E.: ‘Diffusion-Controlled Reaction Rates’, J. Colloid. Sci. 4 (1949), 425–437.

    Google Scholar 

  30. Feller W.: ‘The Parabolic Differential Equations and the Associated Semigroups of Transformations’, Ann. Math. 55 (1952), 468–519.

    Google Scholar 

  31. Ikeda N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.

    Google Scholar 

  32. Gerardi, A., Marchetti, F., and Rosa, A. M.: ‘Simulations of Diffusions with Boundary Conditions’, Syst. Contr. Let. (to appear).

  33. Khruslov E. I. and Marchenko V. A.: Boundary Value Problems in Regions with Fine-Grained Boundaries, Naukova Dumka, Kiev, 1974.

    Google Scholar 

  34. Szabo A., Shoup D., Northrup S. H. and McCammon J. A.: ‘Stochastically Gated Diffusion-Influenced Reactions’, J. Chem. Phys. 77 (1982), 4484–4493.

    Google Scholar 

  35. 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.

    Google Scholar 

  36. 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.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints 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

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00052459

AMS (MOS) subject classifications (1980)

Key words

Navigation