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.
In a three-dimensional highly viscous liquid, a spherical object of radius a has diffusion coefficient 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: Philip Saffman and Max Delbrück calculated the diffusion coefficient for these three cases, and showed that Case 3 was the relevant effect.
[1] The diffusion coefficient of a cylindrical inclusion of radius
, surrounded by bulk fluid with viscosity
Typical values of
[2] This result is an approximation applicable for radii
nm), but not for micrometre-scale lipid domains.
The Saffman–Delbrück formula predicts that diffusion coefficients
will only depend weakly on the size of the embedded object; for example, if
from 1 nm to 10 nm only reduces the diffusion coefficient
Hughes, Pailthorpe, and White extended the theory of Saffman and Delbrück to inclusions with any radii
, A useful formula that produces the correct diffusion coefficients between these two limits is [2] where
Please note that the original version of [2] has a typo in
; the value in the correction[4] to that article should be used.
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
[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] The Saffman–Delbrück approach has also been extended in recent works for modeling hydrodynamic interactions between proteins embedded within curved lipid bilayer membranes, such as in vesicles and other structures.
[9][10][11][12] These works use related formulations to study the roles of the membrane hydrodynamic coupling and curvature in the collective drift-diffusion dynamics of proteins within bilayer membranes.
Various models of the protein inclusions within curved membranes have been developed, including models based on series truncations,[9] immersed boundary methods,[11] and fluctuating hydrodynamics.
