[RASMB] averages from SV

[Walter Stafford]

Hi Holger,
	Peter's comments notwithstanding, the computation of these 
averages (in addition to Sw) can be quite useful and was inspired by 
the so-called "two-species plot" introduced by Sophianopoulos, AJ, 
and VanHolde KE., (1964)  J Biol Chem.239:2516-24, and further 
developed by Roark, DE and Yphantis, DA (1969) Ann N Y Acad Sci. 
164:245-78.

It can be shown that for a two species system, we can combine these 
averages in pairs (two equations and two unknowns) to solve for the 
sedimentation coefficients of the two species. If you look at the 
equations in either of these papers, you can see that you can 
substitute s(i) for M(i) where i = n, w, z, ...in each equation. 
These are the averages that are computed by DCDT(SedAnal), and DCDT+. 
The 2 species relationships are linear (see below) and extrapolate to 
the hyperbola Mz=Mw=Mn to give the s values for the two species. If 
you make this plot for a single species system, the points will 
cluster on the hyperbola at the value of s for that species. If you 
have 2 species in the sample, the plot will be linear and if you have 
more than two species present, the plot will have upward curvature.

Because these other "moments" of g(s*) are easy to compute, I decided 
to include them all in the g(s*) output file of DCDT when Sw is 
computed.  You can choose to ignore them if you don't find them 
useful. The plots can be useful for extrapolating data to infinite 
concentration to get s for the n-mer - corresponding to the intercept 
with the hyperbola at the upper end. There is quite a bit of 
(apparently) forgotten literature from the 60's, 70's and 80's in 
which these plots were used. Sn, Sz and Sz+1 are more sensitive to 
the choice of baseline offset than Sw, but with good data, combined 
in a "two-species plot" they can be quite helpful in ruling out 
hypotheses.

The equation using Sw and Sz is     Sz = (S1 + S2) - S1*S2 (1/Sw)
and that using     Sn and Sw is     Sw = (S1 + S2) - S1*S2 (1/Sn)

For a series of runs, with the averages computed at different loading 
concentrations, these will plot as a straight line. (Simply solving 
these equations for a single data set will not work very well. The 
relationship is a quadratic whose solutions are S1 and S2)

The corresponding one using Sz+1 follows the same pattern but is 
often too noisy to be useful. For a two species interacting system 
these three plots will partially overlap at their respective ends to 
provide a plot over a wide range.

There are also two other "non-ideal" two species plots that can be 
used to treat non-ideality. But that story (also in the literature) 
is for another day.

Walter


At 9:47 -0500 1/26/06, Peter Schuck wrote:
>Holger,
>
>to be honest, to me these averages do not make much sense, with the 
>exception of the weight-average sw.
>
>Sw, of course, reflects mass balance and therefore a 
>thermodynamically well-defined value of the sedimenting molecules. 
>I have studied this for differential sedimentation coefficient 
>distributions (Anal Biochem 320 (2003) 104-124), and shown that they 
>can actually be derived from differential sedimentation coefficient 
>distributions, provided that they imply a faithful representation of 
>the boundary (or the area under the boundary).  This condition is 
>*not* automatically fulfilled, for example, with dcdt or ls-g*(s), 
>but requires comparison with the original sedimentation data.  (To 
>facilitate this test, I've build into sedfit a tool that can take 
>dcdt-derived distributions and compare them with direct boundary 
>data.)  As far as I know, nobody has demonstrated a theoretical 
>connection of a thermodynamically well-defined sw with what you 
>would get from taking a formal weight average over the van-Holde 
>Weischet distribution, and I doubt there is such a connection 
>because of the neglect of absolute boundary height information in 
>the original method (perhaps one could empirically compare).
>
>As far as the other averages are concerned, I believe they do not 
>actually represent well the sedimenting molecules, but
>depend very much on the analysis method and the data acquisition. 
>For example, the width and shape of the peaks will be dominated by 
>signal-noise ratio, rotor speed, and  the extent to which diffusion 
>was deconvoluted, regularization level (if used), and for dcdt and 
>ls-g*(s) the timepoints from which the distribution is derived. 
>There might be situations where it can be useful to empirically 
>compare the width of a peak from samples that are run side-by-side 
>in the same experiment and detected and analyzed in exactly the same 
>way.  For that purpose, there's a second central moment calculated 
>in the integration tool in sedfit.
>
>Just to be specific, this applies to sedimentation coefficient 
>distributions and the way they can be determined in AUC.  An 
>exception is the case of very large particles, such as cells or 
>dispersions, where things are easier because diffusion is virtually 
>absent on the time-scale of sedimentation.  If the sedimentation 
>coefficient distribution is not distorting, and data acquisition 
>takes place with good signal-noise ratio, then these averages could 
>be calculated.  But even then, I would rather compare simply the 
>shape of the whole distribution, which in my opinion is more 
>informative than extracting averages.
>
>Generally, I think it is important to realize that taking formal 
>averages does not necessarily mean that the numbers reflect a real 
>property of the sedimenting sample!
>
>Sorry if that comment will generate a flood of email in your 
>inbox...  maybe somebody else knows some recent examples where these 
>averages - of course other than sw - have been useful?
>
>Regards,
>Peter
>
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Walter Stafford
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