From: Jo Butler <pjgb@mrc-lmb.cam.ac.uk>
  To  : RASMB Noticeboard <rasmb@alpha.bbri.org>
  Date: Fri, 23 Apr 1999 18:18:57 +0100

Units of K from sedimentation equilibrium

Dear RASMBers,

In March I wrote to RASMB asking about the concentration units of K's
fitted to sedimentation equilibrium data by the standard programs,
particularly that supplied by Beckman with the XL-A/I.
I have received a number of replies - thank you to all of you.
Some have shared my concern at the lack of clarity on this point in the
normal documentation and I also received the suggestion that it would be
helpful if I would summarise any helpful comments, for the general benefit.
Hence this e-mail.

The most general comment on this whole area came from Allen Minton, who
helpfully drew to my attention his 1997 article in Progr. Colloid Polym.
Sci. 107, 11-19, which I am afraid I had missed previously.  In this Allen
treats aggregation generally, but simplifying to just homodimerisation (for
convenience of notation in the limited format of e-mail), one gets:
Kc = C2/(C1)**2 = {(M1)**2/M2} Kw = {(M1)/2} Kw ; where K's are association
constants, subscript c or w corresponds to concentration or weight; C and M
are concentration or molecular mass, subscript 1 or 2 corresponding to
monomer or dimer.
More generally, if k is the index of a signal S, such that Sn = En x Wn ;
where E is the conversion factor and W the weight concentration; then:
Kk = S2/(S1)**2 = {E2/(E1)**2} Kw = (2/E1) Kw ; assuming that E2 = 2 x E1.
Hence: Kw = {(E1)/2} Kk
Combining these equations:
Kc = {(E1 x M1)/4} Kk

Much of the fundamental discussion about fitting c against r data directly
(e.g. Johnson & Straume in Modern Analytical Ultracentrifugation (Schuster
& Laue, eds.), pp.37-65) does not raise the importance of the concentration
units, nor have I found it mentioned in the Beckman writeup supplied with
the program for running in Origin.
Jack Correia and Joel Mackay told me that Nonlin handles this properly.
Jack wrote that "Nonlin uses a monomer to n-mer formalism in the units of
detection, absorbance or fringes.  When you plug in the conversion to
molarity (e.g. C(M) = OD/l*extinction where l = 1.2 cm ...)" and pointed
out that, with the assumption that E2 = 2x E1, this converts to the
standard units.  Joel wrote that "Nonlin and Origin do take account of the
difference in extinction coefficient - all these programs make the
assumption that the value for an n-mer is n times the value for the
monomer, and this is included in the equations used".  Thank you for this
information - this was the first time I had been told it about the Beckman
Origin software, even though I had been on the Beckman course at Palo Alto!
However, one still does appear to be in absorbence (or fringe) units with
the Beckman version, as it does not appear to ask for the extinction (or
refractive) coefficient.

I have found the responses to my query to RASMB very helpful and thought
provoking.  They have certainly also left me with the conclusion that it is
absolutely essential to know EXACTLY what the software you are planning to
use will do on this issue.  It is not always transparent and it is vital to
know the units in which K is returned.  There is also the general point
that the biophysists who write programs for analysing ultracentrifuge runs
tend to work in association constants (Ka), while many biochemists or
molecular biologists (like myself) prefer dissociation constants (Kd).

Many thanks to all those who wrote to me,
Yours,

Jo


Dr P.J.G. Butler,
MRC Laboratory of Molecular Biology,
Hills Road,
Cambridge, CB2 2QH,
UK.
Tel. 01223 248011 (or 01223 402296 DDI)
FAX. 01223 213556