Journal of Nonlinear Science

, Volume 25, Issue 2, pp 389–449 | Cite as

A Multiscale Analysis of Diffusions on Rapidly Varying Surfaces

  • A. B. DuncanEmail author
  • C. M. Elliott
  • G. A. Pavliotis
  • A. M. Stuart


Lateral diffusion of molecules on surfaces plays a very important role in various biological processes, including lipid transport across the cell membrane, synaptic transmission, and other phenomena such as exo- and endocytosis, signal transduction, chemotaxis, and cell growth. In many cases, the surfaces can possess spatial inhomogeneities and/or be rapidly changing shape. Using a generalization of the model for a thermally excited Helfrich elastic membrane, we consider the problem of lateral diffusion on quasi-planar surfaces, possessing both spatial and temporal fluctuations. Using results from homogenization theory, we show that, under the assumption of scale separation between the characteristic length and timescales of the membrane fluctuations and the characteristic scale of the diffusing particle, the lateral diffusion process can be well approximated by a Brownian motion on the plane with constant diffusion tensor \(D\) that depends on a highly nonlinear way on the detailed properties of the surface. The effective diffusion tensor will depend on the relative scales of the spatial and temporal fluctuations, and for different scaling regimes, we prove the existence of a macroscopic limit in each case.


Homogenization Laplace–Beltrami Lateral diffusion Multiscale analysis Helfrich elastic membrane  Effective diffusion tensor 

Mathematics Subject Classification

35Q92 60H30 35B27 



A.D. is grateful to EPSRC for financial support and thanks to the Centre for Scientific Computing @ Warwick for computational resources. G.P. acknowledges financial support from EPSRC Grant Nos. EP/J009636/1 and EP/H034587/1. A.M.S. is grateful to EPSRC and ERC for financial support.


  1. Abdulle, A., Schwab, C.: Heterogeneous multiscale FEM for diffusion problems on rough surfaces. Multiscale Model. Simul. 3(1), 195–220 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  2. Aizenbud, B.M., Gershon, N.D.: Diffusion of molecules on biological membranes of nonplanar form. II. Diffusion anisotropy. Biophys. J. 48(4), 543–546 (1985)CrossRefGoogle Scholar
  3. Almeida, P.F.F., Vaz, W.L.C.: Lateral diffusion in membranes. Handb. Biol. Phys. 1, 305–357 (1995)CrossRefGoogle Scholar
  4. Ashby, M.C., Maier, S.R., Nishimune, A., Henley, J.M.: Lateral diffusion drives constitutive exchange of AMPA receptors at dendritic spines and is regulated by spine morphology. J. Neurosci. 26(26), 7046–7055 (2006)CrossRefGoogle Scholar
  5. Axelrod, D., Koppel, D.E., Schlessinger, J., Elson, E., Webb, W.W.: Mobility measurement by analysis of fluorescence photobleaching recovery kinetics. Biophys. J. 16(9), 1055–1069 (1976)CrossRefGoogle Scholar
  6. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures, vol. 5. North Holland, NY (1978)zbMATHGoogle Scholar
  7. Berg, H.C.: Random Walks in Biology. Princeton University Press, Princeton (1993)Google Scholar
  8. Borgdorff, A.J., Choquet, D., et al.: Regulation of AMPA receptor lateral movements. Nature 417(6889), 649–653 (2002)CrossRefGoogle Scholar
  9. Bourgeat, A., Jurak, M., Piatnitski, A.L.: Averaging a transport equation with small diffusion and oscillating velocity. Math. Methods Appl. Sci. 26(2), 95–117 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  10. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, Berlin (2008)zbMATHGoogle Scholar
  11. Bressloff, P.C., Newby, J.M.: Stochastic models of intracellular transport. Rev. Mod. Phys. 85(1), 135 (2013)CrossRefGoogle Scholar
  12. Canham, P.B.: The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26(1), 61–81 (1970)CrossRefGoogle Scholar
  13. Coulibaly-Pasquier, K.A.: Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow. In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 47, pp. 515–538. Institut Henri Poincaré (2011)Google Scholar
  14. Crank, J.: The Mathematics of Diffusion. Oxford University Press, Oxford (1979)zbMATHGoogle Scholar
  15. Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  16. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics, vol. 73. Oxford University Press, Oxford (1988)Google Scholar
  17. Duncan, A.B.: Diffusion on Rapidly Varying Surfaces. Ph.D. thesis, University of Warwick (September 2013)Google Scholar
  18. Dykhne, A.M.: Conductivity of a two-dimensional two-phase system. Sov. J. Exp. Theor. Phys. 32, 63 (1971)Google Scholar
  19. Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. Festa, R., d’Agliano, E.G.: Diffusion coefficient for a Brownian particle in a periodic field of force: I. Large friction limit. Phys. A Stat. Mech. Appl. 90(2), 229–244 (1978)CrossRefGoogle Scholar
  21. Friedman, A.: Stochastic Differential Equations and Applications, dover ed. edition. Dover Books on Mathematics. Dover Publications, New York (2006)Google Scholar
  22. Garnier, J.: Homogenization in a periodic and time-dependent potential. SIAM J. Appl. Math. 57(1), 95–111 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  23. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, vol. 224. Springer, Berlin (2001)zbMATHGoogle Scholar
  24. Gonzalez, O., Stuart, A.M.: A First Course in Continuum Mechanics. Cambridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  25. Gov, N.: Membrane undulations driven by force fluctuations of active proteins. Phys. Rev. Lett. 93(26), 268104 (2004)CrossRefGoogle Scholar
  26. Gov, N.S.: Diffusion in curved fluid membranes. Phys. Rev. E 73(4), 041918 (2006)CrossRefGoogle Scholar
  27. Granek, R.: From semi-flexible polymers to membranes: anomalous diffusion and reptation. J. Phys. II 7(12), 1761–1788 (1997)Google Scholar
  28. Gustafsson, S., Halle, B.: Diffusion on a flexible surface. J. Chem. Phys. 106, 1880 (1997)CrossRefGoogle Scholar
  29. Halle, B., Gustafsson, S.: Diffusion in a fluctuating random geometry. Phys. Rev. E 55(1), 680 (1997)CrossRefGoogle Scholar
  30. Helfrich, W., et al.: Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. c 28(11), 693–703 (1973)Google Scholar
  31. Hsu, E.P.: Stochastic Analysis on Manifolds, vol. 38. American Mathematical Society, Providence, RI (2002)zbMATHGoogle Scholar
  32. Jackson, J.L., Coriell, S.R.: Effective diffusion constant in a polyelectrolyte solution. J. Chem. Phys. 38, 959 (1963)CrossRefGoogle Scholar
  33. Karatzas, I.A., Shreve, S.E.: Brownian Motion and Stochastic Calculus, vol. 113. Springer, Berlin (1991)zbMATHGoogle Scholar
  34. Keller, J.B.: Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J. Appl. Phys. 34(4), 991–993 (1963)CrossRefzbMATHGoogle Scholar
  35. Khasminskii, R.Z., Yin, G.: Asymptotic series for singularly perturbed Kolmogorov–Fokker–Planck equations. SIAM J. Appl. Math. 56(6), 1766–1793 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  36. Kim, S., Karrila, S.J.: Microhydrodynamics: Principles and Selected Applications. Dover Publications, NY (1991)Google Scholar
  37. King, M.R.: Apparent 2-D diffusivity in a ruffled cell membrane. J. Theor. Biol. 227(3), 323–326 (2004)CrossRefGoogle Scholar
  38. Kohler, W., Papanicolaou, G.C.: Bounds for the effective conductivity of random media. In: Macroscopic Properties of Disordered Media, pp. 111–130. Springer, Berlin (1982)Google Scholar
  39. Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation, vol. 345. Springer, Berlin (2012)Google Scholar
  40. Larsson, S., Thomée, V.: Partial Differential Equations with Numerical Methods, vol. 45. Springer, Berlin (2009)zbMATHGoogle Scholar
  41. Leitenberger, S.M., Reister-Gottfried, E., Seifert, U.: Curvature coupling dependence of membrane protein diffusion coefficients. Langmuir 24(4), 1254–1261 (2008)CrossRefGoogle Scholar
  42. Lifson, S., Jackson, J.L.: On the self-diffusion of ions in a polyelectrolyte solution. J. Chem. Phys. 36, 2410 (1962)CrossRefGoogle Scholar
  43. Lin, L.C.L., Gov, N., Brown, F.L.H.: Nonequilibrium membrane fluctuations driven by active proteins. J. Chem. Phys. 124(7), 074903–074903 (2006)CrossRefGoogle Scholar
  44. Lin, L.C.L., Brown, F.L.H.: Dynamics of pinned membranes with application to protein diffusion on the surface of red blood cells. Biophys. J. 86(2), 764–780 (2004)CrossRefMathSciNetGoogle Scholar
  45. Lindblom, G., Orädd, G.: NMR studies of translational diffusion in lyotropic liquid crystals and lipid membranes. Prog. Nucl. Magn. Reson. Spectrosc. 26, 483–515 (1994)CrossRefGoogle Scholar
  46. Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)Google Scholar
  47. Mattingly, J.C., Stuart, A.M.: Geometric ergodicity of some hypo-elliptic diffusions for particle motions. Markov Process. Relat. Fields 8(2), 199–214 (2002)zbMATHMathSciNetGoogle Scholar
  48. Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Appl. 101(2), 185–232 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  49. Mendelson, K.S.: A theorem on the effective conductivity of a two-dimensional heterogeneous medium. J. Appl. Phys. 46(11), 4740–4741 (1975)CrossRefGoogle Scholar
  50. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  51. Naji, A., Brown, F.L.H.: Diffusion on ruffled membrane surfaces. J. Chem. Phys. 126, 235103 (2007)CrossRefGoogle Scholar
  52. Papanicolaou, G.C.: Introduction to the asymptotic analysis of stochastic equations. Mod. Model. Contin. Phenom. (Ninth Summer Sem. Appl. Math., Rensselaer Polytech. Inst., Troy, NY, 1975) 16, 109–147 (1977)MathSciNetGoogle Scholar
  53. Pardoux, É.: Homogenization of linear and semilinear second order parabolic pdes with periodic coefficients: a probabilistic approach. J. Funct. Anal. 167(2), 498–520 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  54. Pavliotis, G.A., Stuart, A.M., Zygalakis, K.C.: Homogenization for inertial particles in a random flow. Commun. Math. Sci. 5(3), 507–531 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  55. Pavliotis, G.A., Stuart, A.M.: Multiscale Methods: Averaging and Homogenization. Springer, Berlin (2008)Google Scholar
  56. Poo, M., Cone, R.A., et al.: Lateral diffusion of rhodopsin in the photoreceptor membrane. Nature 247(441), 438 (1974)CrossRefGoogle Scholar
  57. Reister, E., Seifert, U.: Lateral diffusion of a protein on a fluctuating membrane. EPL (Europhys. Lett.) 71(5), 859 (2007)CrossRefGoogle Scholar
  58. Reister-Gottfried, E., Leitenberger, S.M., Seifert, U.: Hybrid simulations of lateral diffusion in fluctuating membranes. Phys. Rev. E 75(1), 011908 (2007)CrossRefGoogle Scholar
  59. Reister-Gottfried, E., Leitenberger, S.M., Seifert, U.: Diffusing proteins on a fluctuating membrane: analytical theory and simulations. Phys. Rev. E 81(3), 031903 (2010)CrossRefGoogle Scholar
  60. Risken, H.: The Fokker–Planck equation: methods of solution and applications, vol. 18. Springer, Berlin (1996)zbMATHGoogle Scholar
  61. Saxton, M.J., Jacobson, K.: Single-particle tracking: applications to membrane dynamics. Annu. Rev. Biophys. Biomol. Struct. 26(1), 373–399 (1997)CrossRefGoogle Scholar
  62. Sbalzarini, I.F., Hayer, A., Helenius, A., Koumoutsakos, P.: Simulations of (an) isotropic diffusion on curved biological surfaces. Biophys. J. 90(3), 878 (2006)CrossRefGoogle Scholar
  63. Seifert, U.: Configurations of fluid membranes and vesicles. Adv. Phys. 46(1), 13–137 (1997)CrossRefGoogle Scholar
  64. Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numer. 19(1), 451–559 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  65. Van Den Berg, M., Lewis, J.T.: Brownian motion on a hypersurface. Bull. Lond. Math. Soc. 17(2), 144–150 (1985)CrossRefzbMATHGoogle Scholar
  66. Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam (2007)Google Scholar
  67. Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • A. B. Duncan
    • 1
    Email author
  • C. M. Elliott
    • 2
  • G. A. Pavliotis
    • 1
  • A. M. Stuart
    • 2
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Mathematics InstituteWarwick UniversityCoventryUK

Personalised recommendations