[RASMB] Theory of S moments

[Peter Schuck]

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
http://www.sciencedirect.com/science/journal/03014622 (Sorry I can't see 
from here if that's freely accessible or if it requires a subscription).

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 sz = 
d(c*sw)/dc, which is a relationship that holds for monomer-n-mer 
self-associations.

It is important that here sz is calculated from the 
concentration-dependence of sw.

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).

2) in the application paper, sz is calculated, again, from the experimental 
dependence of sw on concentration, using above formula.

 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).

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.

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.

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.

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.

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.

Peter


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