[RASMB] ellipsoids of revolution

[Tom Laue]

Hi-
Only one hydrodynamic parameter may be extracted rigorously from the 
sedimentation coefficient extrapolated to zero solute concentration, and 
that is the frictional coefficient, f, from which the Stokes radius, Rs, 
may be calculated:
f = 6 pi eta Rs, where pi = 3.14, eta is the solvent viscosity and Rs is 
the Stokes radius.

There are several things to note about this-

   1. f = 6 pi eta Rs for a solute that exhibits "stick" conditions with
      the solvent (i.e. there are favorable solvent/solute surface
      interactions). For "slip" conditions, f = 4 pi eta Rs. Where it
      has been tested, biological molecules exhibit stick behavior.
      Given what is known about the solvent interactions of
      biomolecules, I am comfortable that there is no need to worry
      about using 6. FYI, for rotary diffusion, the 4 pi term is the one
      to use.
   2. The solvent viscosity is used because the calculation of Rs from f
      becomes extremely complicated at higher concentrations. By using
      very dilute solutions, or extrapolating s to c=0, there is no
      ambiguity about what viscosity should be used. At high
      concentrations, the bulk viscosity is not an appropriate term for
      interpreting the frictional coefficient. To learn more, I
      recommend Fujita's monograph.
   3. The Stokes radius comes from the solution of the Navier equation,
      using Stokes approximation (an incompressible solvent). Solving
      the Navier-Stokes equation involves setting a boundary that
      encloses one volume of solution from another, then applying the
      conservation principles for mass, energy, linear momentum and
      rotary momentum. There is little that is instructive in going
      through the math involved in the solution. One important point is
      that conservation of energy includes conservation of charge, and
      that conservation of momentum includes ion-ion momentum transfer
      through space.
   4. The Stokes radius has been interpreted further, but only by
      imposing a model. Perrin determined solutions to Stokes equations
      from which the axial ratios for _solid, uncharged_ molecules could
      be calculated from the Stokes radius.
   5. Calculation of axial ratios from the Stokes radius inevitably
      results in ratios that are larger than would be expected (or that
      are known from other data). A rational explanation for the
      discrepancy is that there is a layer of solvent carried with the
      molecule, and that this solvent layer adds to the molecular size.
      Hence, the inclusion of the "solvation" layer, usually described
      in terms of the number of grams of water bound per gram of
      protein. But this approximation needs to be viewed with care since
      it is not a rigorous interpretation of Rs as it is not developed
      from the Navier-Stokes equation. For starters, there is only one
      parameter accessible from s, and now we are trying to get two- the
      axial ratio and the extent of solvation. One of these two terms
      must be set.
   6. There is some confusion about the difference between solvation and
      hydration. Sednterp estimates the _hydration_ of a protein using
      data from Kuntz and Kauffman obtained by proton NMR. Steve Perkins
      has shown that the hydration of pretty much every protein is about
      0.3 g/g. However, it is the _solvation_ of the protein that
      matters in Rs (and even that is a bit of a kludge), and the
      solvation includes the through-space momentum transfer due to
      charge coupling. Hence, if you use the hydration values calculated
      by Sednterp, or if you use Steve's value for hydration, you will
      almost inevitably end up with axial ratio estimates that are large
      compared to estimates from other methods.

The long and short of it is that Rs is a valid and valuable quantity 
describing both the asymmetry of a protein and its interactions with the 
solvent. Comparisons of Rs values for a protein acquired in different  
solvents, or between molecules is valid. One should be careful, though, 
about interpreting Rs solely in terms of the shape of a molecule.
Best wishes,
Tom
mitrana at mail.utexas.edu wrote:
> Hi all
>
> I have a very basic question. How does one get from determining the a/b ratio to
> the gross molecular dimensions of the protein molecule. The molecular weight
> should place a limit but I am a bit fuzzy as to how exactly this is carried
> out. 
>
> Mitra
>   

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