Saffman–Delbrück model

The Saffman–Delbrück model describes a lipid membrane as a thin layer of viscous fluid, surrounded by a less viscous bulk liquid. This picture was originally proposed to determine the diffusion coefficient of membrane proteins, but has also been used to describe the dynamics of fluid domains within lipid membranes. The Saffman–Delbrück formula is often applied to determine the size of an object embedded in a membrane from its observed diffusion coefficient, and is characterized by the weak logarithmic dependence of diffusion constant on object radius.

Embedded cylindrical object of radius ${\displaystyle a}$ in a membrane with viscosity ${\displaystyle \eta _{m}}$, height ${\displaystyle h}$, surrounded by bulk fluid with viscosity ${\displaystyle \eta _{f}}$

Origin

In a three-dimensional highly viscous liquid, a spherical object of radius a has diffusion coefficient

${\displaystyle D_{3D}={\frac {k_{B}T}{6\pi \eta a}}}$

by the well-known Stokes–Einstein relation. By contrast, the diffusion coefficient of a circular object embedded in a two-dimensional fluid diverges; this is Stokes' paradox. In a real lipid membrane, the diffusion coefficient may be limited by:

1. the size of the membrane
2. the inertia of the membrane (finite Reynolds number)
3. the effect of the liquid surrounding the membrane

Philip Saffman and Max Delbrück calculated the diffusion coefficient for these three cases, and showed that Case 3 was the relevant effect.[1]

Saffman–Delbrück formula

The diffusion coefficient of a cylindrical inclusion of radius ${\displaystyle a}$ in a membrane with thickness ${\displaystyle h}$ and viscosity ${\displaystyle \eta _{m}}$, surrounded by bulk fluid with viscosity ${\displaystyle \eta _{f}}$ is:

${\displaystyle D_{sd}={\frac {k_{B}T}{4\pi \eta _{m}h}}\left[\ln(2L_{sd}/a)-\gamma \right]}$

where the Saffman–Delbrück length ${\displaystyle L_{sd}={\frac {h\eta _{m}}{2\eta _{f}}}}$ and ${\displaystyle \gamma \approx 0.577}$ is the Euler–Mascheroni constant. Typical values of ${\displaystyle L_{sd}}$ are 0.1 to 10 micrometres.[2] This result is an approximation applicable for radii ${\displaystyle a\ll L_{sd}}$, which is appropriate for proteins (${\displaystyle a\approx }$ nm), but not for micrometre-scale lipid domains.

The Saffman–Delbrück formula predicts that diffusion coefficients ${\displaystyle D_{sd}}$ will only depend weakly on the size of the embedded object; for example, if ${\displaystyle L_{sd}=1\mu m}$, changing ${\displaystyle a}$ from 1 nm to 10 nm only reduces the diffusion coefficient ${\displaystyle D_{sd}}$ by 30%.

Beyond the Saffman–Delbrück length

Hughes, Pailthorpe, and White extended the theory of Saffman and Delbrück to inclusions with any radii ${\displaystyle a}$;[3] for ${\displaystyle a\gg L_{sd}}$,

${\displaystyle D\to {\frac {k_{B}T}{8\eta _{m}h}}{\frac {L_{sd}}{a}}={\frac {k_{B}T}{16\eta _{f}a}}}$

A useful formula that produces the correct diffusion coefficients between these two limits is [2]

${\displaystyle D={\frac {k_{B}T}{4\pi \eta _{m}h}}\left[\ln(2/\epsilon )-\gamma +4\epsilon /\pi -(\epsilon ^{2}/2)\ln(2/\epsilon )\right]\left[1-(\epsilon ^{3}/\pi )\ln(2/\epsilon )+c_{1}\epsilon ^{b_{1}}/(1+c_{2}\epsilon ^{b_{2}})\right]^{-1}}$

where ${\displaystyle \epsilon =a/L_{sd}}$, ${\displaystyle b_{1}=2.74819}$, ${\displaystyle b_{2}=0.51465}$, ${\displaystyle c_{1}=0.73761}$, and ${\displaystyle c_{2}=0.52119}$. Please note that the original version of [2] has a typo in ${\displaystyle b_{2}}$; the value in the correction[4] to that article should be used.

Experimental studies

Though the Saffman–Delbruck formula is commonly used to infer the sizes of nanometer-scale objects, recent controversial[5] experiments on proteins have suggested that the diffusion coefficient's dependence on radius ${\displaystyle a}$ should be ${\displaystyle a^{-1}}$ instead of ${\displaystyle \ln(a)}$.[6] However, for larger objects (such as micrometre-scale lipid domains), the Saffman–Delbruck model (with the extensions above) is well-established [2][7][8]

References

1. P. G. Saffman and M. Delbrück, Brownian motion in biological membranes, Proc. Natl. Acad. Sci. USA, vol. 72 p. 3111–3113 1975
2. Petrov, EP; Schwille, P (2008). "Translational diffusion in lipid membranes beyond the Saffman-Delbruck approximation". Biophys J. 94 (5): L41-3. Bibcode:2008BpJ....94L..41P. doi:10.1529/biophysj.107.126565. PMC 2242757. PMID 18192354.
3. 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. Bibcode:1981JFM...110..349H. doi:10.1017/S0022112081000785.
4. Petrov; Schwille (Jul 2012). "Correction: Translational Diffusion in Lipid Membranes beyond the Saffman-Delbrück Approximation". Biophys. J. 103 (2): 375. Bibcode:2012BpJ...103..375P. doi:10.1016/j.bpj.2012.06.032. PMC 3400766.
5. Weiß; et al. (2013). "Quantifying the Diffusion of Membrane Proteins and Peptides in Black Lipid Membranes with 2-Focus Fluorescence Correlation Spectroscopy". Biophys. J. 105 (2): 455–462. Bibcode:2013BpJ...105..455W. doi:10.1016/j.bpj.2013.06.004. PMC 3714877. PMID 23870266.
6. Gambin, Y.; et al. (2006). "Lateral mobility of proteins in liquid membranes revisited". Proc. Natl. Acad. Sci. USA. 103 (7): 2098–2102. Bibcode:2006PNAS..103.2098G. doi:10.1073/pnas.0511026103. PMC 1413751. PMID 16461891.
7. Klingler, J.F.; McConnell, H.M. (1993). "Brownian motion and fluid mechanics of lipid monolayer domains". J. Phys. Chem. 93 (22): 6096–6100. doi:10.1021/j100124a052.
8. Cicuta, P.; Veatch, S.L.; Keller, S.L. (2007). "Diffusion of Liquid Domains in Lipid Bilayer Membranes". J. Phys. Chem. B. 111 (13): 3328–3331. arXiv:cond-mat/0611492. doi:10.1021/jp0702088. PMID 17388499.