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Since this is a scientific discussion forum: I think these examples
provide a good basis for discussion. I also believe Walter's
analysis is correct in that for these cases the number-average,
z-average, and z+1 average can be computed, and the limitations in their
precision may not be too much of a problem in those cases for making a
two-species plot. <br><br>
However, for clarity about these hypothetical situations and to reconcile
with what I argued earlier, it is of interest to look at the alternatives
for the systems considered. <br><br>
<b>For the example No. 2</b>:<br>
This was a non-interacting two-species system with 3.5 S and 9.0 S.
Below is the ls-g*(s) in black, and the c(s) in blue. <br>
<img src="cid:.0" width=443 height=330 alt="[]"><br>
This seems to be in line with what I argued earlier: if you get
baseline-separated peaks, then the errors in sn, sz, and sz+1 don't
matter that much. <br><br>
However, I'm not sure what I would learn from the averages in this
case, since simply looking at the peaks - no matter if they come
from dcdt, ls-g*(s), or c(s) - would directly give a precise
representation of the sedimentation coefficients of the underlying
species. <br><br>
This example is a good contrast to the example I proposed before, where
one would get baseline separated peaks only in c(s) but not in g*(s), and
therefore the limited precision in sn, sz, and sz+1 is more of a
problem. Generally, with systems of smaller s-values, I suspect
errors will also be more pronounced.<br><br>
<b>For the example No. 3: <br>
</b>This was the interacting system of monomer-tetramer self-association,
also with very different s-values for the different species (s1 = 3.6 S,
s4 = 9.0 S). <br><br>
Below I took Walter's sw values as a function of concentration, plotted,
and fitted it to a mass action law model: <br>
<img src="cid:.1" width=307 height=227 alt="[]"><br>
This is a traditional, simple and perfectly rigorous thermodynamic
analysis, from which you not only get a good estimate of s1 = 3.4 and s4
= 8.9, but also a perfectly good estimate of log(K) = 14.8. Looking
at the literature, it appears to me most people would stop here, but of
course one could take this values and proceed with direct Lamm equation
modeling, if desired. <br><br>
As in the Example No. 2, it is not obvious to me what additional
information the computation of sn, sz, and sz+1 and the two-species plot
would provide. <br><br>
[They do appear to have an advantage that they don't require modeling
with a known n-mer, which may seem to permit first estimating the
oligomer s-value, fixing it, and then determining which n-mer is
generated by modeling of the isotherm. However, this would be wrong
since it would introduce bias from the moderately precise estimate of
s(n-mer) from the analysis of sn, sz, and sz+1 into the further analysis,
which will be highly sensitive to the detailed s(n-mer) value, and errors
in any s(n-mer) constraint might easily result in wrong n values.
To be clear, nobody suggested this. ] <br><br>
Peter<br>
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