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--></style><title>Two species plots from sedimentation coef averages
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<div><font face="Courier">It has been suggested that one cannot
compute the number-, Z- or Z+1- sedimentation coefficient averages in
any meaningful way from g(s*) vs s* patterns.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">After some analysis of this problem with
both exact calculations and simulated data, I think one can safely
conclude that one can expect to be able to get sufficiently accurate
values to be useful, and that they can be used in "two species"
plots. They might be very useful for extrapolation to infinite
concentration to get an estimate of the sedimentation coefficient
largest species (eg. the n-mer in a self-associating system.) This
"extrapolation" is shown in the plots below by the
intersection of the extrapolated data (the red straight line:
theoretical two-species line) to the intersection point with the
hyperbola representing the locus of all single species systems, for
which Sn=Sw=Sz=Sz+1. At infinite concentration only the n-mer would be
present and the condition that Sn=Sw=Sz=Sz+1 is satisfied under those
conditions at the point of intersection.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">Let's take a couple of
examples:</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">Monomer tetramer system: M1 = 40 kDa; S1 =
3.6S and S2 = 9.0S</font></div>
<div><font face="Courier">1. the equations for the various
sedimentation coefficient averages for this system are as
follows</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">Sn = [C(1) +
C(2)]/[C(1)/S1 + C(2)/S2]</font></div>
<div><font face="Courier">Sw = [C(1)S1 +
C(2)S2]/[C(1) + C(2)]</font></div>
<div><font face="Courier">Sz = [C(1)(S1^2) +
C(2)(S2^2)]<b>/</b>[C(1)S1 + C(2)S2]</font></div>
<div><font face="Courier">Sz+1 = [C(1)(S1^3) +
C(2)(S2^3)]<b>/</b>[C(1)(S1^2) + C(2)(S2^2)]</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">One can combine these in pairs to obtained
the following linear relationships (the two species
equations)</font></div>
<div><font face="Courier">Sw = (S1 + S2) -
(S1)(S2)(1/Sn)</font></div>
<div><font face="Courier">Sz = (S1 + S2) -
(S1)(S2)(1/Sw)</font></div>
<div><font face="Courier">Sz+1 = (S1 + S2) -
(S1)(S2)(1/Sz)</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">The<b> question</b> is whether or not all of
these averages (or moments) can be estimated by integrating over the
g(s*) vs s* curves.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">Remember that g(s*) is not a true
sedimentation coefficient distribution [cf Stafford, Anal. Biochem.
1992]. However, it represents a fairly accurate snap shot of the
boundary at a given time and is the derivative of the concentration
profile with respect to the x-axis variable, s*. So, g(s*) vs s* =
dc/ds* vs s*.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">You can show that Sw = Int[
s*(dc/ds*)ds*)/Int[(dc/ds*)ds* (N.B.: This integral must
start and stop in regions where the concentration gradient is zero in
order to be valid.)</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">As Peter Schuck points out, this average (to
the extent that g(s*) represents the boundary's true shape) is the
only one of the three that con be obtained accurately by integrating
over a gaussian g(s*) function.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">So the second question is "How bad will
it be if we do that anyway?".</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><b>Example No. 1.</b></font></div>
<div><font face="Courier"><b>Single species</b>: it should be the case
that Sn=Sw=Sz=Sz+1. but since the g(s) curve is symmetrical (i.e.
closely approximates a gaussian) the moments of the curve will not be
equal and we expect Sn < Sw < Sz < Sz+1; and indeed we see
that. But, as wee will see, the error is not significant.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">So for a case 160 kDa, 9.0S simulated
data (@50,000 RPM).</font></div>
<div><font face="Courier">we get</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"> from
g(s)(dcdt) ratio to true
value</font></div>
<div><font face="Courier">Sn
8.91 <span
></span> 0.99</font></div>
<div><font face="Courier">Sw
9.00 <span
></span> 1.00</font></div>
<div><font face="Courier">Sz
9.08 <span
></span> 1.01</font></div>
<div><font face="Courier">Sz+1
9.17 <span
></span> 1.02</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">and for case 40 kDa, 3.6S</font></div>
<div><font face="Courier">we get</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"> from
g(s) ratio to
true value</font></div>
<div><font face="Courier">Sn 3.34
(3.38) 1.078
(1.065)</font></div>
<div><font face="Courier">Sw 3.53
(3.54) 0.980
(0.983) </font></div>
<div><font face="Courier">Sz 3.70
(3.68) 1.028
(1.022)</font></div>
<div><font face="Courier">Sz+1 3.85
(3.80) 1.069
(1.056)</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">(numbers in parentheses from later time
point and are slightly better.)</font></div>
<div><font face="Courier"><b>Conclusion:</b> as long as the speed
and/or molar mass is high enough (i.e. high enough sigma = w^2(s/D)
),</font></div>
<div><font face="Courier">these averages do not deviate seriously from
their true values</font></div>
<div><font face="Courier">when computed by integrating over g(s*) vs
s* from dcdt analysis.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><b>Example No. 2.</b></font></div>
<div><font face="Courier">Take a monomer-tetramer system<b>
non-interacting</b>, (mixture 0.5 wt fractions).</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">40 kDa, 3.5S; 160 kDa, 9.0S</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">equal mass concentrations.</font></div>
<div><font face="Courier"> </font></div>
<div><font face="Courier"> true
value from
g(s) ratio to true value</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">Sn
5.14 <span
></span>
4.37
0.850</font></div>
<div><font face="Courier">Sw
6.25 <span
></span>
6.15
0.984</font></div>
<div><font face="Courier">Sz
7.46 <span
></span>
7.56
1.013</font></div>
<div><font face="Courier">Sz+1
8.44 <span
></span>
8.43
0.999</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><b>Conclusion</b>: Reasonably reliable
values can be computed from a mixture. (The low value obtained for
1/Sn is expected because of the known skewing of the g(s*) patterns
when the data are taken when the boundary is near the meniscus. This
is the case for the 3.6 S component when the 9.0 S component has not
yet hit the bottom - a necessary condition in order to integrate of
all species present. You can see the effect of this skewing in the
next example of an interacting system spanning multiple loading
concentrations.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><b>Example No. 3.</b></font></div>
<div><font face="Courier">Monomer-tetramer<b> interacting</b>:
simulated with noise +/- 0.005 fringes (3.29 mg/fringe)</font></div>
<div><font face="Courier">monomer 40 kDa and 3.6S</font></div>
<div><font face="Courier">tetramer 9.0S</font></div>
<div><font face="Courier">Ka = 1 x 10^15</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"> Co
g/L Sn* Sn
R Sw*
Sw R Sz*
Sz
R Sz+1 Sz+1
R</font></div>
<div><font face="Courier">1.e-4 4.00 7.49
7.62 0.982 8.17 8.35
0.978 8.50 8.72
0.975 8.71 8.88
0.981</font></div>
<div><font face="Courier">3.e-5 1.20 6.04
6.30 0.959 7.14 7.46
0.957 7.74 8.26
0.937 8.11 8.68
0.934</font></div>
<div><font face="Courier">1.e-5 0.40 4.39
4.83 0.909 5.47 5.90
0.927 6.26 7.11
0.880 6.82 8.04
0.848</font></div>
<div><font face="Courier">3.e-6 0.12 3.18
3.78 0.851 3.74 4.02
0.930 4.19 4.54
0.923 4.57 5.47
0.835</font></div>
<div><font face="Courier">1.e-6 0.04 3.25
3.61 0.900 3.48 3.62
0.961 3.76 3.65
1.030 4.04 3.73
1.083</font></div>
<div><font face="Courier">3.e-7 0.012 3.04
3.60 0.840 3.31 3.60
0.919 3.47 3.60
0.964 3.32 3.60
0.922</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">* = values obtained from integration of
g(s*) vs s*.</font></div>
<div><font face="Courier">Values of Sn,etc without the asterisk are
the exact values according to the set of discrete equations above
using the equilibrium concentrations of monomer and tetramer at the
loading concentration according to mass action.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><b>Conclusion:</b> See FIGURE
below.</font></div>
<div><font face="Courier">This plot is useful if care is taken. The
plot of Sw vs 1/Sn is quite good while the others sag somewhat below
the two-species line near the middle of the range.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">This plot can be quite useful for ruling out
stoichiometries and for detecting non-ideality, with the usual caveat
these these effects can partially compensate for each other. For
example, if you know the sedimentation coefficient of the largest
material in your sample, then all the data must plot in the area
between the hyperbola and the two-species line. If the data plots
above the line, then you know you have larger species present.
(Teller, 1973; discusses the ins and outs of two-species
plots)</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">---------------------------</font></div>
<div><font face="Courier">FIGURE</font></div>
<div><font face="Courier"><b>TOP:</b> exact averages computed from
mass action equations: result not unexpected. This what the bottom
plot would</font></div>
<div><font face="Courier">look like if we could get highly accurate
values.</font></div>
<div><font face="Courier"><b>BOT:</b> computed from simulated data,
analyzed with SEDANAL/DCDT to generate the g(s*)
patterns,</font></div>
<div><font face="Courier">as follows:(Error bars are
plotted.)</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"> Sn =
Int[
(g(s*))ds*]/Int[((1/s*)(g(s*))ds*]</font></div>
<div><font face="Courier"> Sw = Int[
s* (g(s*))ds*]/Int[
(g(s*))ds*]</font></div>
<div><font face="Courier"> Sz = Int[ (s*^2)
(g(s*))ds*]/Int[ s* (g(s*))ds*]</font></div>
<div><font face="Courier"> Sz+1 = Int[ (s*^3)
(g(s*))ds*]/Int[((s*^2)(g(s*))ds*]</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">[figure deleted] jpeg attached.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">A lot more could be said about this, but
I'll stop here for now. Use of these plots might be a helpful starting
point for full-boundary fitting techniques as an aid to choosing a
model.</font></div>
<div><font face="Courier"><br></font></div>
<div><font face="Courier">Walter</font></div>
<x-sigsep><pre>--
</pre></x-sigsep>
<div><font face="Times"
color="#0000FF"
>----------------------------------------------------------</font></div
>
<div><font face="Times" color="#0000FF">Walter Stafford</font></div>
<div><font face="Times"
color="#0000FF">mailto:stafford@bbri.org</font></div>
<div><font face="Times" color="#0000FF">direct dial:
617-658-7808</font></div>
<div><font face="Times" color="#0000FF">receptionist:
617-658-7700</font></div>
<div><font face="Times"
color="#0000FF"
>----------------------------------------------------------</font></div
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