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The combined hydrodynamic and thermodynamic effects of immobilized proteins on the diffusion of mobile transmembrane proteins

  • Rohit R. Singh (a1), Ashok S. Sangani (a2), Susan Daniel (a1) and Donald L. Koch (a1)

Abstract

The plasma membranes of cells are thin viscous sheets in which some transmembrane proteins have two-dimensional mobility and some are immobilized. Previous studies have shown that immobile proteins retard the short-time diffusivity of mobile particles through hydrodynamic interactions and that steric effects of immobile proteins reduce the long-time diffusivity in a model that neglects hydrodynamic interactions. We present a rigorous derivation of the long-time diffusivity of a single mobile protein interacting hydrodynamically and thermodynamically with an array of immobile proteins subject to periodic boundary conditions. This method is based on a finite element method (FEM) solution of the probability density of the mobile protein diffusing with a position-dependent mobility determined through a multipole solution of Stokes equations. The simulated long-time diffusivity in square arrays decreases as the spacing in the array approaches the particle size in a manner consistent with a lubrication analysis. In random arrays, steric effects lead to a percolation threshold volume fraction above which long-time diffusion is arrested. The FEM/multipole approach is used to compute the long-time diffusivity far away from this threshold. An approximate analysis of mobile protein diffusion through a network of pores connected by bonds with resistances determined by the FEM/multipole calculations is then used to explore higher immobile area fractions and to evaluate the finite simulation cell size scaling behaviour of diffusion near the percolation threshold. Surprisingly, the ratio of the long-time diffusivity to the spatially averaged short-time diffusivity in these two-dimensional fixed arrays is higher in the presence of hydrodynamic interactions than in their absence. Finally, the implications of this work are discussed, including the possibility of using the methods developed here to investigate more complex diffusive phenomena observed in cell membranes.

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Corresponding author

Email address for correspondence: dlk15@cornell.edu

References

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Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235 (1200), 6777.
Banchio, A. J. & Brady, J. F. 2003 Accelerated Stokesian dynamics: Brownian motion. J. Chem. Phys. 118 (22), 1032310332.10.1063/1.1571819
Brady, J. F. 1994 The long-time self-diffusivity in concentrated colloidal dispersions. J. Fluid Mech. 272, 109134.10.1017/S0022112094004404
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111157.10.1146/annurev.fl.20.010188.000551
Brannigan, G. & Brown, F. L. H. 2006 A consistent model for thermal fluctuations and protein-induced deformations in lipid bilayers. Biophys. J. 90 (5), 15011520.10.1529/biophysj.105.075838
Brenner, H. 1980 Dispersion resulting from flow through spatially periodic porous media. Phil. Trans. R. Soc. Lond. A 297 (1430), 81133.10.1098/rsta.1980.0205
Brenner, H. & Adler, P. M. 1982 Dispersion resulting from flow through spatially periodic porous media II. Surface and intraparticle transport. Phil. Trans. R. Soc. Lond. A 307 (1498), 149200.10.1098/rsta.1982.0108
Brinkman, H. C. 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow Turbul. Combust. 1 (1), 2734.10.1007/BF02120313
Brown, F. L. H. 2003 Regulation of protein mobility via thermal membrane undulations. Biophys. J. 84 (2), 842853.10.1016/S0006-3495(03)74903-0
Bussell, S. J., Hammer, D. A. & Koch, D. L. 1994 The effect of hydrodynamic interactions on the tracer and gradient diffusion of integral membrane proteins in lipid bilayers. J. Fluid Mech. 258, 167190.10.1017/S0022112094003289
Bussell, S. J., Koch, D. L. & Hammer, D. A. 1992 The resistivity and mobility functions for a model system of two equal-sized proteins in a lipid bilayer. J. Fluid Mech. 243, 679697.10.1017/S002211209200288X
Bussell, S. J., Koch, D. L. & Hammer, D. A. 1995 Effect of hydrodynamic interactions on the diffusion of integral membrane proteins: diffusion in plasma membranes. Biophys. J. 68 (5), 18361849.10.1016/S0006-3495(95)80360-7
Charlaix, E., Guyon, E. & Roux, S. 1987 Permeability of a random array of fractures of widely varying apertures. Trans. Porous Med. 2 (1), 3143.
Dodd, T. L., Hammer, D. A., Sangani, A. S. & Koch, D. L. 1995 Numerical simulations of the effect of hydrodynamic interactions on diffusivities of integral membrane proteins. J. Fluid Mech. 293, 147180.10.1017/S0022112095001674
Ermak, D. L. & McCammon, J. A. 1978 Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69 (4), 13521360.10.1063/1.436761
Fujiwara, T., Ritchie, K., Murakoshi, H., Jacobson, K. & Kusumi, A. 2002 Phospholipids undergo hop diffusion in compartmentalized cell membrane. J. Cell Biol. 157 (6), 10711081.10.1083/jcb.200202050
Geuzaine, C. & Remacle, J.-F. 2009 Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Intl J. Numer. Meth. Engng 79 (11), 13091331.10.1002/nme.2579
Grassia, P. S., Hinch, E. J. & Nitsche, L. C. 1995 Computer simulations of Brownian motion of complex systems. J. Fluid Mech. 282, 373403.10.1017/S0022112095000176
Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (02), 317328.10.1017/S0022112059000222
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 28 (11–12), 693703.10.1515/znc-1973-11-1209
Jin, S., Haggie, P. M. & Verkman, A. S. 2007 Single-particle tracking of membrane protein diffusion in a potential: simulation, detection, and application to confined diffusion of CFTR cl-channels. Biophys. J. 93 (3), 10791088.10.1529/biophysj.106.102244
Johnson, E. M., Berk, D. A., Jain, R. K. & Deen, W. M. 1996 Hindered diffusion in agarose gels: test of effective medium model. Biophys. J. 70 (2), 10171023.10.1016/S0006-3495(96)79645-5
Lin, L. C.-L. & Brown, F. L. H. 2004 Dynamics of pinned membranes with application to protein diffusion on the surface of red blood cells. Biophys. J. 86 (2), 764780.10.1016/S0006-3495(04)74153-3
Lodish, H., Berk, A., Kaiser, C. A., Scott, M. P., Bretscher, A., Ploegh, H. & Matsudaira, P. 2008 Molecular Cell Biology. W. H. Freeman.
Medina-Noyola, M. 1988 Long-time self-diffusion in concentrated colloidal dispersions. Phys. Rev. Lett. 60 (26), 27052708.10.1103/PhysRevLett.60.2705
Nicolau, D. V., Hancock, J. F. & Burrage, K. 2007 Sources of anomalous diffusion on cell membranes: a Monte Carlo study. Biophys. J. 92 (6), 19751987.10.1529/biophysj.105.076869
Niehaus, A. M. S., Vlachos, D. G., Edwards, J. S., Plechac, P. & Tribe, R. 2008 Microscopic simulation of membrane molecule diffusion on corralled membrane surfaces. Biophys. J. 94 (5), 15511564.10.1529/biophysj.107.106484
Niemelä, P. S., Miettinen, M. S., Monticelli, L., Hammaren, H., Bjelkmar, P., Murtola, T., Lindahl, E. & Vattulainen, I. 2010 Membrane proteins diffuse as dynamic complexes with lipids. J. Am. Chem. Soc. 132 (22), 75747575.10.1021/ja101481b
Niemelä, P. S., Ollila, S., Hyvönen, M. T., Karttunen, M. & Vattulainen, I. 2007 Assessing the nature of lipid raft membranes. PLoS Comput. Biol. 3 (2), e34.10.1371/journal.pcbi.0030034
Oppenheimer, N. & Diamant, H. 2009 Correlated diffusion of membrane proteins and their effect on membrane viscosity. Biophys. J. 96 (8), 30413049.10.1016/j.bpj.2009.01.020
Oppenheimer, N. & Diamant, H. 2010 Correlated dynamics of inclusions in a supported membrane. Phys. Rev. E 82 (4), 041912.
Oppenheimer, N. & Diamant, H. 2011 In-plane dynamics of membranes with immobile inclusions. Phys. Rev. Lett. 107 (25), 258102.10.1103/PhysRevLett.107.258102
Phillips, R. J., Deen, W. M. & Brady, J. F. 1989 Hindered transport of spherical macromolecules in fibrous membranes and gels. AIChE J. 35 (11), 17611769.10.1002/aic.690351102
Phillips, R. J. 2000 A hydrodynamic model for hindered diffusion of proteins and micelles in hydrogels. Biophys. J. 79 (6), 33503354.10.1016/S0006-3495(00)76566-0
Przybylo, M., Skora, J., Humpolíková, J., Benda, A., Zan, A. & Hof, M. 2006 Lipid diffusion in giant unilamellar vesicles is more than 2 times faster than in supported phospholipid bilayers under identical conditions. Langmuir 22 (22), 90969099.10.1021/la061934p
Ratto, T. V. & Longo, M. L. 2003 Anomalous subdiffusion in heterogeneous lipid bilayers. Langmuir 19 (5), 17881793.10.1021/la0261803
Saffman, P. G. 1976 Brownian motion in thin sheets of viscous fluid. J. Fluid Mech. 73 (04), 593602.10.1017/S0022112076001511
Saffman, P. G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. USA 72 (8), 31113113.10.1073/pnas.72.8.3111
Sangani, A. S. & Behl, S. 1989 The planar singular solutions of Stokes and Laplace equations and their application to transport processes near porous surfaces. Phys. Fluids A 1 (1), 2137.10.1063/1.857544
Sangani, A. S. & Yao, C. 1988 Transport processes in random arrays of cylinders. II. Viscous flow. Phys. Fluids 31 (9), 24352444.10.1063/1.866596
Sangani, A. S., Zhang, D. Z. & Prosperetti, A. 1991 The added mass, basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small-amplitude oscillatory motion. Phys. Fluids A 3 (12), 29552970.10.1063/1.857838
Saxton, M. J. 1987 Lateral diffusion in an archipelago. The effect of mobile obstacles. Biophys. J. 52 (6), 989997.10.1016/S0006-3495(87)83291-5
Saxton, M. J. 1990 Lateral diffusion in a mixture of mobile and immobile particles. A Monte Carlo study. Biophys. J. 58 (5), 13031306.10.1016/S0006-3495(90)82470-X
Saxton, M. J. 1994 Anomalous diffusion due to obstacles: a Monte Carlo study. Biophys. J. 66 (2), 394401.10.1016/S0006-3495(94)80789-1
Schütz, G. J., Schindler, H. & Schmidt, T. 1997 Single-molecule microscopy on model membranes reveals anomalous diffusion. Biophys. J. 73 (2), 10731080.10.1016/S0006-3495(97)78139-6
Skaug, M. J., Faller, R. & Longo, M. L. 2011 Correlating anomalous diffusion with lipid bilayer membrane structure using single molecule tracking and atomic force microscopy. J. Chem. Phys. 134 (21), 06B602.
Stauffer, D. & Aharony, A. 1994 Introduction to Percolation Theory. CRC Press.
Sung, B. J. & Yethiraj, A. 2006 Lateral diffusion and percolation in membranes. Phys. Rev. Lett. 96 (22), 228103.10.1103/PhysRevLett.96.228103
Sung, B. J. & Yethiraj, A. 2008 Lateral diffusion of proteins in the plasma membrane: Spatial tessellation and percolation theory. J. Phys. Chem. B 112 (1), 143149.10.1021/jp0772068
Tomishige, M., Sako, Y. & Kusumi, A. 1998 Regulation mechanism of the lateral diffusion of band 3 in erythrocyte membranes by the membrane skeleton. J. Cell Biol. 142 (4), 9891000.10.1083/jcb.142.4.989
Weigel, A. V., Simon, B., Tamkun, M. M. & Krapf, D. 2011 Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking. Proc. Natl Acad. Sci. USA 108 (16), 64386443.10.1073/pnas.1016325108
Zhou, H.-X. 2009 Crowding effects of membrane proteins. J. Phys. Chem. B 113 (23), 79958005.10.1021/jp8107446
Zimmerman, R. W., Chen, D.-W. & Cook, N. G. W. 1992 The effect of contact area on the permeability of fractures. J. Hydrol. 139 (1–4), 7996.10.1016/0022-1694(92)90196-3
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The combined hydrodynamic and thermodynamic effects of immobilized proteins on the diffusion of mobile transmembrane proteins

  • Rohit R. Singh (a1), Ashok S. Sangani (a2), Susan Daniel (a1) and Donald L. Koch (a1)

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