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Drag coefficient of a liquid domain with distinct viscosity in a fluid membrane

Published online by Cambridge University Press:  13 December 2017

Hisasi Tani Open the ORCID record for Hisasi Tani [Opens in a new window]  and
Youhei Fujitani
Hisasi Tani*
Affiliation:
Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University, Kawasaki 214-8571, Kanagawa, Japan School of Fundamental Science and Technology, Keio University, Yokohama 223-8522, Kanagawa, Japan
Youhei Fujitani
Affiliation:
School of Fundamental Science and Technology, Keio University, Yokohama 223-8522, Kanagawa, Japan
*
†Email address for correspondence: [email protected]
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Abstract

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

JFM classification

Drops and Bubbles: Drops Biological Fluid Dynamics: Membranes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 910 - 931
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Copyright
© 2017 Cambridge University Press 

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References

Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K. & Walter, P. 2007 Molecular Biology of the Cell, 5th edn. Garland Science.CrossRefGoogle Scholar
Aliaskarisohi, S., Tierno, P., Dhar, P., Khattari, Z., Blaszczynski, M. & Fischer, T. M. 2010 On the diffusion of circular domains on a spherical vesicle. J. Fluid Mech. 654, 417–451.CrossRefGoogle Scholar
Aris, R. 1989 Vectors Tensors and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
Bertseva, E., Grebenkov, D., Schmidhauser, P., Gribkova, S., Jeney, S. & Forró, L. 2012 Optical trapping microrheology in cultured human cells. Eur. Phys. J. E 35, 63.Google ScholarPubMed
Bian, X., Kim, C. & Karniadakis, G. E. 2016 111 years of Brownian motion. Soft Matt. 12, 6331–6346.CrossRefGoogle ScholarPubMed
Camley, B. A. & Brown, F. L. H. 2014 Fluctuating hydrodynamics of multicomponent membranes with embedded proteins. J. Chem. Phys. 141, 075103.CrossRefGoogle ScholarPubMed
Cicuta, P., Keller, S. L. & Veatch, S. L. 2007 Diffusion of liquid domains in lipid bilayer membranes. J. Phys. Chem. 111, 3328–3331.CrossRefGoogle ScholarPubMed
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.Google Scholar
Dietrich, C., Bagatolli, L. A., Volovyk, Z. N., Thompson, N. L., Levi, M., Jacobson, K. & Gratton, E. 2001 Lipid rafts reconstituted in model membranes. Biophys. J. 80, 1417–1428.CrossRefGoogle ScholarPubMed
Einstein, A. 1905 Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen. Ann. Phys. (Leipzig) 322, 549–560.Google Scholar
Franosch, T., Grimm, M., Belushkin, M., Mor, F. M., Foffi, G., Forró, L. & Jeney, S. 2011 Resonances arising from hydrodynamic memory in Brownian motion. Nature 478, 85–88.CrossRefGoogle ScholarPubMed
Fujitani, Y. 1994 Dynamics of the lipid-bilayer membrane taking a vesicle shape. Physica A 203, 214–242; (Erratum 237, 346 (1997)).Google Scholar
Fujitani, Y. 2005 Small deformation of a nearly circular lipid-raft in the stagnation flow. J. Phys. Soc. Japan 74, 642–647.Google Scholar
Fujitani, Y. 2011 Drag coefficient of a liquid domain in a fluid membrane. J. Phys. Soc. Japan 80, 074609.Google Scholar
Fujitani, Y. 2012 Drag coefficient of a liquid domain in a fluid membrane almost as viscous as the domain. J. Phys. Soc. Japan 81, 084601.CrossRefGoogle Scholar
Fujitani, Y. 2013a Drag coefficient of a liquid domain in a fluid membrane surrounded by confined three-dimensional fluids. J. Phys. Soc. Japan 82, 084403.Google Scholar
Fujitani, Y. 2013b Drag coefficient of a raftlike domain embedded in a fluid membrane being a near-critical binary mixture. J. Phys. Soc. Japan 82, 124601; (Erratum 83, 088001 (2014)).Google Scholar
Fujitani, Y. 2013c Hydrodynamic effect on concentration fluctuation in a two-component fluid membrane with a spherical shape. J. Phys. Soc. Japan 82, 014601.CrossRefGoogle Scholar
Fujitani, Y. 2016 Relaxation rate of the shape fluctuation of a fluid membrane immersed in a near-critical binary fluid mixture. Eur. Phys. J. E 39, 31.Google Scholar
Gambin, Y., Lopez-Esparza, R., Reffay, M., Sierecki, E., Gov, N. S., Genest, M., Hodges, R. S. & Urbach, W. 2006 Lateral mobility of proteins in liquid membranes revisited. Proc. Natl. Acad. Sci. USA 103 (7), 2098–2102.Google Scholar
Grebenkov, D. S., Vahabi, M., Bertseva, E., Forró, L. & Jeney, S. 2013 Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium. Phys. Rev.  E 88, 040701.Google Scholar
Grimm, M., Franosch, T. & Jeney, S. 2012 High-resolution detection of Brownian motion for quantitative optical Tweezers experiments. Phys. Rev. E 86, 021912.Google Scholar
Guigas, G. & Weiss, M. 2006 Size-dependent diffusion of membrane inclusions. Biophys. J. 91, 2393–2398.Google Scholar
Hadamard, J. S. 1911 Motion of liquid drops. C. R. Acad. Sci. Paris 152, 1735–1738.Google Scholar
Honerkamp-Smith, A. R., Cicuta, P., Collins, M. D., Veatch, S. L., den Nijs, M., Schick, M. & Keller, S. L. 2008 Line tensions, correlation lengths, and critical exponents in lipid membranes near critical points. Biophys. J. 95, 236–246.Google Scholar
Honerkamp-Smith, A. R., Machta, B. B. & Keller, S. L. 2012 Experimental observations of dynamic critical phenomena in a lipid membrane. Phys. Rev. Lett. 108, 265702.Google Scholar
Honerkamp-Smith, A. R., Veatch, S. L. & Keller, S. L. 2009 An introduction to critical points for biophysicists; observations of compositional heterogeneity in lipid membranes. Biochim. Biophys. Acta 1788, 53–63.Google Scholar
Hughes, B. D., Pailthorpe, B. A. & White, L. R. 1981 The translational and rotational drag on a cylinder moving in a membrane. J. Fluid Mech. 110, 349–372.Google Scholar
Inaura, K. & Fujitani, Y. 2008 Concentration fluctuation in a two-component fluid membrane surrounded with three-dimensional fluids. J. Phys. Soc. Japan 77, 114603.CrossRefGoogle Scholar
Klingler, J. F. & McConnell, H. M. 1993 Brownian motion and fluid mechanics of lipid monolayer domains. J. Phys. Chem. 97, 6096–6100.Google Scholar
De Koker, R.1996 The program in biophysics. PhD thesis, Stanford University.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lee, C. C. & Petersen, N. O. 2003 The lateral diffusion of selectively aggregated peptides in giant unilamellar vesicles. Biophys. J. 84, 1756–1764.Google Scholar
Loth, E. 2008 Quasi-steady shape and drag of deformable bubbles and drops. Intl J. Multiphase Flow 34, 523–546.Google Scholar
Merkel, R., Sackmann, E. & Evans, E. 1989 Molecular friction and in supported bilayers epitactic coupling monolayers. J. Phys. (Paris) 50, 1535–1555.Google Scholar
Okamoto, R., Fujitani, Y. & Komura, S. 2013 Drag coefficient of a rigid spherical particle in a near-critical binary fluid mixture. J. Phys. Soc. Japan 82, 084003.Google Scholar
Peters, R. & Cherry, R. J. 1982 Lateral and rotational diffusion of bacteriorhodopsin in lipid bilayers: experimental test of the Saffman–Delbrück equations. Proc. Natl Acad. Sci. USA 79, 4317–4321.Google Scholar
Powers, T. R. 2010 Dynamics of filaments and membranes in a viscous fluid. Rev. Mod. Phys. 82, 1607–1631.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
Quemeneur, F., Sigurdssone, J. K., Renner, M., Atzberger, P. J., Bassereau, P. & Lacoste, D. 2014 Shape matters in protein mobility within membranes. Proc. Natl. Acad. Sci. USA 111, 5083–5087.Google Scholar
Rao, V. L. & Das, S. L. 2015 Drag force on a liquid domain moving inside a membrane sheet surrounded by aqueous medium. J. Fluid Mech. 779, 468–482.Google Scholar
Rybczynski, W. 1911 On the translatory motion of a fluid sphere in a viscous medium. Bull. Acad. Sci. Cracovie Ser. A 40.Google Scholar
Saffman, P. G. 1976 Brownian motion in thin sheets of viscous fluid. J. Fluid Mech. 73, 593–602.Google Scholar
Saffman, P. G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl. Acad. Sci. USA 72, 3111–3113.Google Scholar
Shiba, H., Noguchi, H. & Fournier, J. B. 2016 Monte Carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area. Soft Matt. 12, 2373–2380.Google Scholar
Simons, K. & Ikonen, E. 1997 Functional rafts in cell membranes. Nature 387, 569–572.Google Scholar
Smeulders, J. B. A. F., Blom, C. & Mellema, J. 1990 Linear viscoelastic study of lipid vesicle dispersions: hard-sphere behavior and bilayer surface dynamics. Phys. Rev. A 42, 3483–3498.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–14.Google Scholar
Sutherland, W. 1905 A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. Phil. Mag. 9, 781–785.Google Scholar
Veatch, S. L. & Keller, S. L. 2002 Organization in lipid membranes containing cholesterol. Phys. Rev. Lett. 89, 268101.CrossRefGoogle ScholarPubMed
Veatch, S. L. & Keller, S. L. 2005 Miscibility phase diagrams of giant vesicles containing sphingomyelin. Phys. Rev. Lett. 94, 148101.Google Scholar
Waigh, T. A. 2005 Microrheology of complex fluids. Rep. Prog. Phys. 68, 685–742.Google Scholar
Yanagisawa, M., Imai, M., Masui, T., Komura, S. & Ohta, T. 2007 Growth dynamics of domains in ternary fluid vesicles. Biophys. J. 92, 115–125.Google Scholar