From: Prof.Dr.M.D.Lechner <lechner@rz.Uni-Osnabrueck.DE>
  To  : rasmb@bbri.harvard.edu
  Date: Thu, 29 Feb 1996 11:07:08 +0100

Calculation of Mn, Mw, Mz, molar mass distribution

To: T. Aerts, Dept. of  Biochemistry, University of Antwerp
To: All XLA users

in his message from 23. Feb 1996  T. Aerts states that for calculating Mz 
one has to calculate the weight average molar masses and the
concentrations 
at the bottom Mw,b and Cb and at the meniscus Mw,m and Cm of the cell:

Mz = (Mwb Cb - Mwm Cm)/(Cb - Cm)

Mw(r) = (1/L)[d ln C/d(r^2)]

L = [(1 - v2 rho1) omega^2]/(2 R T)  ;   r = distance from center
of rotation

The determination of  Mz requires the extrapolation of the experimental 
obtained concentration C(r) as a function of  r to the bottom and to the 
meniscus; the concentration C(r) may be calculated in the cases of 
absorption optics, interference optics and schlieren optics with the
usual 
equations.

The extrapolation of C(r) = f(r) to the meniscus and to the bottom is
done 
by a special regression spline procedure. The method has been tested with 
many systems. The results for Mz are satisfactory if the measurements are 
within distinct experimental errors. Additionally you can calculate Mn if 
you have a meniscus depletion run (see H. Fujita, Foundations in 
Ultracentrifugal Analysis).

We have developed the programs „XLASE“ (absorption optics) and
„EQUIL“ 
(interference optics, schlieren optics) for calculation of  Mn, Mw, and
Mz. 
The program „XLASE“ processes directly the XLA - files. If you are 
interested, I can send you the programs on a diskette together with a 
complete documentation. Another possibility is that you send me your XLA
- 
files; the evaluation is then done by us and send to you. We are very
much 
interested in measurments on (bio)polymers with relatively broad molar
mass 
distributions.

Additionally to the above described procedure it would be possible to 
determine the molar mass distribution of  (bio)polymers including Mn, Mw, 
and Mz. The method is based on inverse transformation or optimization 
procedures. (see H. Fujita, Foundations in Ultracentrifugal Analysis; 
M.D.Lechner, W.Mächtle, Makromol.Chem.,Rapid.Commun. 13(1992)555). This 
method has also been tested with many systems. Our feeling is that this 
method is slightly better than the extrapolation procedure. The reason
may 
be, that you don’t need the error producing determination of  the
slopes at 
the meniscus and at the bottom of the C(r) curves of  the cell.

We have developed the program „DIRECT“ (absorption optics,
interference 
optics, schlieren optics) for calculation of the molar mass distribution
and 
Mn, Mw, and Mz. This program accepts directly the XLA - files. If you are 
interested, I can send you this program together with a
complete documentation.

In case of any troubles or questions, I would kindly ask you to contact me.
M.D.Lechner