[RASMB] Two species plots from sedimentation coef averages

[Peter Schuck]

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.

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.

For the example No. 2:
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.
[]

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.

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.

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.

For the example No. 3:
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).

Below I took Walter's sw values as a function of concentration, plotted, 
and fitted it to a mass action law model:
[]

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.

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.

[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. ]

Peter
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