From: Allen MInton <minoton@protein.osaka-u.ac.jp>
  To  : rasmb@bbri.harvard.edu
  Date: Mon, 07 Jul 1997 09:37:14 +0900

Frictional and virial coefficients

Hello out there in RASMBland --  

At Walter's request I am amplifying my response to Julius 
Clauwaerts' question regarding the calculation of frictional and 
second virial coefficients for cylindrical particles.

The frictional coefficient of an ellipsoid of rotation is given 
by the Perrin (not Simha -- I misspoke) equation, which may be 
found in standard textbooks of physical biochemistry (eg Tanford, 
Cantor & Schimmel).  A sufficiently long cylinder may be modeled 
by a prolate ellipsoid of rotation having major and minor axes 
the same as the length and diameter of the cylinder, or, 
alternatively, the same diameter or length and volume.  
However, there is a complication that should be considered.  The 
Perrin equation represents an average over all particle 
orientations with respect to the axis of translational motion.  
It is entirely appropriate for calculation of the frictional 
coefficient for diffusion.  However, in sedimentation there is a 
net directional flow of particles relative to solvent.  If long, thin 
particles are sedimenting suffiently rapidly under high g-force, 
they will preferentially orient with their long axes tending to 
line up along the direction of motion to minimize friction.  In 
this case the average frictional coefficient will be smaller, 
and the particles will sediment more rapidly, than calculated 
using the Perrin equation.  I believe that this effect has been 
treated in the literature but I can't recall exactly where.  It's 
a very complex hydrodynamic problem. 

The second virial coefficient for a suspension of cylinders (or 
any hard convex particle) may be calculated using the equation of 
Kihara, which is presented most completely in Kihara, Reviews of 
Modern Physics 25, 831 (1953).  I think that equations are also 
presented for third virial coefficients for certain types of 
particles, but I don't have the paper in front of me to verify this.

As long as I'm here, I'd like to take advantage of this "bully 
pulpit" to say a few words about the treatment of nonideality in 
sedimentation equilibrium.  One must be very very cautious about 
using a single second virial coefficient to allow for 
nonideality.  The commonly (and often uncritically) used equation
               M,a = M/(1 + B2*c)     (1)
is a truncated form of the full expression
               M,a = M/(1 + c*(d ln gam/dc))    (2)
where gam is the activity coefficient of solute.

Point 1 -  Equation (2) - and hence eqn (1) - is only 
theoretically valid for a solution of a single homogeneous 
solute.  It does not apply to solutions of multiple species.  It 
MAY be an acceptable approximation under certain conditions, but 
it is incumbent upon the user to verify that this is so.  I refer 
interested parties to Chatelier & Minton, Biopolymers 26: 507 and 
1097 (1987) for further analysis.  Also, see the very interesting 
analysis of sedimentation equilibrium in a nonideal solution of a 
self-associating solute presented by Wills and Winzor in 
Biophysical Chemistry, December 1995.

Point 2 - The truncated form eqn (1) is only valid for slightly 
nonideal solutions, and its validity should be demonstrated by 
showing that the identical result is obtained when an additional 
term is included, i.e.,
               M,a = M/(1 + B2*c + 2*B3*c^2)    (3)
using physically appropriate values of B2 and B3, i.e., not just 
arbitrarily adjustable fitting fitting parameters.  In other 
words, it should be demonstrated that the last term in the 
denominator of eqn (3) really is negligible, but that the second 
term really is not negligible.                                     

Allen Minton