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I'd like to add some information to these references. For those
interested, the journal has a complete set of issues online as pdf
at<br>
<a href="http://www.sciencedirect.com/science/journal/03014622" eudora="autourl">
http://www.sciencedirect.com/science/journal/03014622</a> (Sorry I can't
see from here if that's freely accessible or if it requires a
subscription).<br><br>
1) the first paper shows theoretical relationships between the different
s-values, but does not address the question how one would determine those
values experimentally, except to the extent of determining sw,
eliminating hydrodynamic concentration dependent sedimentation, and using
<b>sz = d(c*sw)/dc</b>, which is a relationship that holds for
monomer-n-mer self-associations. <br><br>
It is important that here sz is calculated from the
concentration-dependence of sw.<br><br>
A similar but a little bit more elaborate formula is given for the
number-average s-value, which can be related to an integral over the
concentration-dependence sw(c) (weighted with the weight-average molar
mass). <br><br>
2) in the application paper, sz is calculated, again, <b>from the
experimental dependence of sw on concentration</b>, using above
formula.<br><br>
From a modern point of view, one could argue that compared to the direct
modeling of the concentration-dependence of sw(c) directly, the
differentiation of sw(c) will not add any new information. However,
at the time, probably it was a good question to ask if the derivative of
the weight-average is a good quantity to analyze, maybe having in mind
another experimental method to determine it more directly (such as
difference sedimentation).<br><br>
More to the point of our discussion, I think it is a misconception to
identify the z-average or number-average value derived in theory from the
concentration-dependence of the weight-average with those 'formal'
z-averages and number-averages that one would get from integrating the
sedimentation coefficient distributions. This is not a criticism on
the original papers, since at that time the g(s*) distribution have not
been widely used (and neither dcdt nor ls-g*(s) was around at that time),
and the authors do not comment on sedimentation coefficient
distributions. This is simply a problem of making unvalidated
assumptions about the properties of 'modern' sedimentation coefficient
distributions.<br><br>
In particular with g*(s) (be it in the incarnation of g(s*) via dcdt or
equally with ls-g*(s) or dcdr), we've seen from the previous discussion
that there are significant problems with the accuracy in the presence of
the diffusionally-broadened distributions. c(s) may perform better,
but the number- and z-averages would still not be even close to rigorous
except for trivial cases (in contrast to sw-averages). Some
may be willing to accept these values, and in some cases (like large
species with low diffusion) it may work OK, but to my knowledge it still
has not been clarified in the literature if there is error amplification
when equating the averages with the 'ideal' theoretical averages, how
that would affect the final results of an analysis, and when it works and
when it doesn't. <br><br>
Regarding the question: Has this method been recently applied to
sedimentation experiments? (Not to be confused with the analysis of
sw(c)!) Interestingly, the ISI Web of Science has a record for the
second paper. It lists a total of 3 citations in other works:
1) in a 1988 NMR-paper on lactoglobulin, without reference to any AUC
work; 2) Jack Correia's review (p.97) , where he discusses the z-average
from g(s*) as a potentially useful quantity to analyze
interactions. In this review, in reference to the practical paper
Jack stated "while this technique has been available for 25 years it
has only been applied to simulated data and a single test case,
b-lactoglobulin" (this was written in the year 2000), and the use of
moments to aid the reliability of the analysis (of a ligand-induced
selfassociation) is referenced by Jack as "Correia et al, manuscript
in preparation". 3) in our own work (Dam et al., the recent BJ
paper on c(s) and Gilbert-Jenkins theory Biophys J. 89(2005)651-666), in
which we clarified (p664) in reference to Jack's suggestion that the
Weirich and Beckerdite papers actually use sz = d(c*sz)/dc, not the
'formal' integrals over a sedimentation coefficient
distribution. <br><br>
The reason for making this further comment is that I feel, at this point
in time, in the absence of further publications on this topic, the new
AUC users would be misguided to look at these papers as the way to
analyze self-associating systems in conjunction with the modern
sedimentation coefficient distributions. We're fortunate to have
Lamm equation solutions at hand (which Jack also stated in his previous
comment), and we can get good and rigorous analyses based on the
concentration-dependence of sw directly. <br><br>
In terms of using this approach for initial model-building and deriving
starting estimates for the rigorous analyses, which is what I believe
Jack suggests, this is of course a reasonable idea, since getting hints
for possible parameter ranges is always a good thing. However, in
the final analysis one would have to float these values anyway, in order
not to introduce bias. The situation is more tricky if that
preliminary analysis is used to guide model selection. I think
without a theoretical framework for what the quality of these initial
estimates would be, one may be better off using the more rigorous methods
and examining (and perhaps accepting) the flexibility of the models, for
example for sw(c), in order not to end up biasing the analysis and
excluding realistic scenarios. <br><br>
Peter<br><br>
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