<!doctype html public "-//W3C//DTD W3 HTML//EN">
<html><head><style type="text/css"><!--
blockquote, dl, ul, ol, li { padding-top: 0 ; padding-bottom: 0 }
--></style><title>Two species plots from sedimentation coef averages
-resend</title></head><body>
<div>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.</div>
<div><br></div>
<div>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.</div>
<div><br></div>
<div><br></div>
<div>Let's take a couple of examples:</div>
<div><br></div>
<div>Monomer tetramer system: M1 = 40 kDa; S1 = 3.6S and S2 =
9.0S</div>
<div>1. the equations for the various sedimentation coefficient
averages for this system are as follows</div>
<div><br></div>
<div>Sn = [C(1) + C(2)]/[C(1)/S1 + C(2)/S2]</div>
<div>Sw = [C(1)S1 + C(2)S2]/[C(1) + C(2)]</div>
<div>Sz = [C(1)(S1^2) + C(2)(S2^2)]<b>/</b>[C(1)S1 +
C(2)S2]</div>
<div>Sz+1 = [C(1)(S1^3) + C(2)(S2^3)]<b>/</b>[C(1)(S1^2) +
C(2)(S2^2)]</div>
<div><br></div>
<div>One can combine these in pairs to obtained the following linear
relationships (the two species equations)</div>
<div>Sw = (S1 + S2) - (S1)(S2)(1/Sn)</div>
<div>Sz = (S1 + S2) - (S1)(S2)(1/Sw)</div>
<div>Sz+1 = (S1 + S2) - (S1)(S2)(1/Sz)</div>
<div><br></div>
<div>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.</div>
<div><br></div>
<div>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*.</div>
<div><br></div>
<div>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.)</div>
<div><br></div>
<div>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.</div>
<div><br></div>
<div>So the second question is "How bad will it be if we do that
anyway?".</div>
<div><br></div>
<div><br></div>
<div><b>Example No. 1.</b></div>
<div><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.</div>
<div><br></div>
<div>So for a case 160 kDa, 9.0S simulated data (@50,000
RPM).</div>
<div>we get</div>
<div><br></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>and for case 40 kDa, 3.6S</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><br></div>
<div><b>Example No. 2.</b></div>
<div>Take a monomer-tetramer system<b> non-interacting</b>, (mixture
0.5 wt fractions).</div>
<div><br></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><br></div>
<div><b>Example No. 3.</b></div>
<div>Monomer-tetramer<b> interacting</b>: simulated with noise +/-
0.005 fringes (3.29 mg/fringe)</div>
<div>monomer 40 kDa and 3.6S</div>
<div>tetramer 9.0S</div>
<div>Ka = 1 x 10^15</div>
<div><br></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><br></div>
<div>---------------------------</div>
<div>FIGURE</div>
<div><b>TOP:</b> exact averages computed from mass action equations:
result not unexpected. This what the bottom plot would</div>
<div>look like if we could get highly accurate values.</div>
<div><b>BOT:</b> computed from simulated data, analyzed with
SEDANAL/DCDT to generate the g(s*) patterns,</div>
<div>as follows:(Error bars are plotted.)</div>
<div><br></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><br></div>
<div><br></div>
<div><br></div>
<div><img src="cid:p06200719c00698a2a504@[141.154.92.21].1.0"></div>
<div>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.</div>
<div><br></div>
<div>Walter</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
>
</body>
</html>