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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 [[Brownian motion|diffusion coefficient]], and is characterized by the weak logarithmic dependence of diffusion constant on object radius.
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[[Image:sd-illus-wiki.png|frame|Embedded cylindrical object of radius <math>a</math> in a membrane with viscosity <math>\eta_m</math>, height <math>h</math>, surrounded by bulk fluid with viscosity <math>\eta_f</math>|right]]
 
==Origin==
In a three-dimensional highly viscous liquid, a spherical object of radius ''a'' has diffusion coefficient
 
: <math>
D_{3D} = \frac{k_B T}{6 \pi \eta a}
</math>
 
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:
 
# the size of the membrane
# the inertia of the membrane (finite [[Reynolds number]])
# 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.<ref name="sd">[http://www.pnas.org/content/72/8/3111.abstract P. G. Saffman and M. Delbrück, ''Brownian motion in biological membranes'', Proc. Nat. Acad. Sci., USA, vol. 72 p. 3111–3113 1975]</ref>
 
==Saffman–Delbrück formula==
The diffusion coefficient of a cylindrical inclusion of radius <math>a</math> in a membrane with thickness <math>h</math> and [[viscosity]] <math>\eta_m</math>, surrounded by bulk fluid with viscosity <math>\eta_f</math> is:
 
: <math>
D_{sd} = \frac{k_B T}{4 \pi \eta_m h} \left[\ln(2 L_{sd} / a) - \gamma\right]
</math>
 
where the Saffman–Delbrück length <math>L_{sd} = \frac{h \eta_m}{2 \eta_f}</math> and <math>\gamma\approx 0.577</math> is the [[Euler–Mascheroni constant]]. Typical values of <math>L_{sd}</math> are 0.1 to 10 micrometres.<ref name="petrovschwille">[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2242757/ E.P. Petrov and P. Schwille, ''Translational Diffusion in Lipid Membranes beyond the Saffman–Delbrück Approximation'', Biophys. J. vol. 94, pL41–L43 2008]</ref> This result is an approximation applicable for radii <math>a \ll L_{sd}</math>, which is appropriate for proteins (<math>a\approx</math> nm), but not for micrometre-scale lipid domains.
 
The Saffman–Delbrück formula predicts that diffusion coefficients <math>D_{sd}</math> will only depend weakly on the size of the embedded object; for example, if <math>L_{sd} = 1 \mu m</math>, changing <math>a</math> from 1&nbsp;nm to 10&nbsp;nm only reduces the diffusion coefficient <math>D_{sd}</math> by&nbsp;30%.
 
==Beyond the Saffman–Delbrück length==
Hughes, Pailthorpe, and White extended the theory of Saffman and Delbrück to inclusions with any radii <math>a</math>;<ref name="hpw">[http://dx.doi.org/10.1017/S0022112081000785 B.D. Hughes, B.A. Pailthorpe, and L.R. White, ''The translational and rotational drag on a cylinder moving in a membrane'', J. Fluid Mech. vol. 110, p. 349–372 1981]</ref> for <math>a \gg L_{sd}</math>,
 
: <math>
D \to \frac{k_B T}{8 \eta_m h a} \frac{L_{sd}}{a}
</math>
 
A useful formula that produces the correct diffusion coefficients between these two limits is <ref name="petrovschwille"/>
 
: <math>
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}
</math>
 
where <math>\epsilon = a / L_{sd}</math>, <math>b_1 = 2.74819</math>, <math>b_2 = 0.51465</math>, <math>c_1 = 0.73761</math>, and <math>c_2 = 0.52119</math>.
 
==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 <math>a</math> should be <math>a^{-1}</math> instead of <math>\ln(a)</math>.<ref name="gambin">[http://dx.doi.org/10.1073/pnas.0511026103  Y. Gambin et al., ''Lateral mobility of proteins in liquid membranes revisited'', Proc. Nat. Acad. Sci., USA, vol. 103, pp. 2098–2102, 2006]</ref> However, for larger objects (such as micrometre-scale [[lipid rafts|lipid domains]]), the Saffman–Delbruck model (with the extensions above) is well-established <ref name = "petrovschwille"/><ref name = "klingler">[http://dx.doi.org/10.1021/j100124a052 J.F. Klingler and H.M. McConnell 1993, ''Brownian motion and fluid mechanics of lipid monolayer domains'' J. Phys. Chem. vol. 93, p. 6096–6100, 1993]</ref><ref name = "cicuta">[http://dx.doi.org/10.1021/jp0702088/ P. Cicuta, S.L. Veatch, and S.L. Keller, ''Diffusion of Liquid Domains in Lipid Bilayer Membranes'' J. Phys. Chem. B, vol. 111, p. 3328–3331, 2007]</ref>
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Saffman-Delbruck Model}}
[[Category:Biophysics]]
[[Category:Proteins]]
[[Category:Membrane biology]]

Revision as of 01:28, 1 March 2014

Machetes are a kind of survival, or tactical knife They sometimes have one edge, with a blade that widens on the tip. They're used extensively in many tropical areas for clearing brush, and chopping by way of jungle development. Many cultures rely on them for farming purposes. Having mentioned that, there are those that do use them for weapons. Width of the cardboard spine will be so long as sides of your box Facet A + Facet B + Aspect A + Side B  + a couple of centimeters further to allow for easy folding + closing flap of the field



The Work Sharp Knife and Software Sharpener is a singular belt sander that makes use of a half inch huge sanding belt of varied grits to sharpen both software blades and knives to maintain your shop and kitchen tools at their peak performing edge. Work Sharp manufactures software and knife sharpeners that supply knowledgeable edge to your chopping tools utilizing abrasive grit sandpaper either in a round disc or belt sander. The Work Sharp Knife and Tool Sharpener is a handheld sharpener that makes use of a half inch vast six inch sanding belt to sharpen instruments and knives.

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The M1903A3 Springfield rifle, otherwise known as the United States Rifle, Caliber30-06, was a service rifle issued to U.S. troops from 1905 to 1937. The 1903A3 is a bolt-action, clip-fed rifle with an effective vary of 650 yards. With such a rich American historical past, the rifle is usually collected by navy surplus fanatics. As with all weapons, periodic upkeep is essential Thebestpocketknifereviews.Com in order for the rifle to stay in optimum condition. This will likely embrace eradicating the barrel from the receiver. A barrel wrench - if not accessible - will be constituted of scratch with applicable machine tools.

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