From: Jose Garcia de la Torre <jgt@um.es>
  To  : minton@helix.nih.gov
  Date: vie, 07 sep 2001 11:35:31 +0200

hydrodynamics of cylindrical rods

Dear Allen and RAMSB colleages,

Here is a little help from hydrodynamics.

First of all, let me stress that representing a cylinder by a
long prolate ellipsoid is not correct. Unlike the cylinder, and
supposedly your particle, the ellipsoid has no uniform cross section.
This can be particularly bad if you are interested in characterizing
effects arising from the thickness, i.e., the diameter of the rod.

The cylinder is a simple and common geometry and its hydrodynamic
has been well studied (as a problem different from that of ellipsoids).
>From basic fluid-mechanic theory, it is known that the translational
friction coefficient of a cylindrical rod is given by

f = 3 pi eta_0 L / [ln p + C]

where pi=3.14... eta_0=solvent viscosity, p=L/d, L=diameter and
d=diameter. For VERY LONG cylinders C is a numerical constant, 
whose numerical was, long ago, the subject of some controversy.
Anhow, this is only valid for extremely large values of the aspect 
(length-to-diameter) ratio, p. This excludes most cases presented
by biological macromolecules, even for long filamentous viruses: 
for instance TMV has p approx. 20 and this is not long enough.

For moderately long or appreciably short cylinders, C is not a
constant but depends on p. The C(p) funtion has been also the subject 
of some studies (you may have heard of some calculations by Broersma
that were proved to be wrong). The definitive result for C(p), which
has become the standard reference during the last 20 years was
derived by Mercedes Tirado and myself. The equation is very simple:

C(p) = 0.312 + 0.565/p + 0.100/p^2

The primary references are two: (a) M.M.Tirado and J. Garcia de la
Torre, "Translational friction coefficients of rigid, symmetric
top macromolecules. Application to circular cylinders" J. Chem. Phys. 
71, 2581-2587 (1979), where the numerical work was done, and
(b) J. Garcia de la Torre and V.A. Bloomfield, "Hydrodynamic properties
of complex, rigid, biological macromolecules: theory and applications"
Quart. Rev. Biophys. 14, 81-139 (1981), where the equation of
C(p) was explicitely presented. (please consult the original references
to discard possible typos in this message).

This results is valid for moderately long and even quite short cylinders;
the accuracy is very good for p>2, i.e., L>2d. Presently I am 
considering the extension of the theory to even short cylinders and disks
with p<1, and I have some preliminary values. By the way, if someone
is interested in cases that could be representable by such flat cylinders
or disk, please contact me.

Thanks to Steve Harding for addressing the problem to me. If any
of you have further questions, you would be most welcome.

Best regards,
Jose.



 particle has a uniform
cross section while the ellipsoid

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Jose Garcia de la Torre

Departamento de Quimica Fisica     Phone: + 34 - 968 -367426
Universidad de Murcia              Fax:   + 34 - 968 -364148
30071 Murcia, Spain

WEB: <http://leonardo.fcu.um.es/macromol>
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