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

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, and is characterized by the weak logarithmic dependence of diffusion constant on object radius.

Origin

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:

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

Philip Saffman

Philip Geoffrey Saffman was an applied mathematician, the Theodore von Karman Professor of Applied Mathematics and Aeronautics at the California Institute of Technology.-Life, career and honors:...

 and Max Delbrück

Max Delbrück

Max Ludwig Henning Delbrück was a German-American biophysicist and Nobel laureate.-Biography:Delbrück was born in Berlin, German Empire...

 calculated the diffusion coefficient for these three cases, and showed that Case 3 was the relevant effect .

Saffman–Delbrück formula

The diffusion coefficient of a cylindrical inclusion of radius in a membrane with thickness and viscosity

Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

 , surrounded by bulk fluid with viscosity is:

where the Saffman–Delbrück length and is the Euler–Mascheroni constant

Euler–Mascheroni constant

The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

. Typical values of are 0.1 to 10 micrometres . This result is an approximation applicable for radii , which is appropriate for proteins ( 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 , changing from 1 nm to 10 nm only reduces the diffusion coefficient by 30%.

Beyond the Saffman–Delbrück length

Hughes, Pailthorpe, and White extended the theory of Saffman and Delbrück to inclusions with any radii ; for ,

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

where , , , , and .

Experimental studies

Though the Saffman–Delbruck formula is commonly used to infer the sizes of nanometer-scale objects, recent experiments on proteins have suggested that the diffusion coefficient's dependence on radius should be instead of . However, for larger objects (such as micrometre-scale lipid domains), the Saffman–Delbruck model (with the extensions above) is well-established

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