[RASMB] Two species plots from sedimentation coef averages -text only

Walter Stafford stafford at bbri.org
Wed Feb 1 10:08:36 PST 2006


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.

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.


Let's take a couple of examples:

Monomer tetramer system: M1 = 40 kDa; S1 = 3.6S and S2 = 9.0S
1. the equations for the various sedimentation coefficient averages 
for this system are as follows

Sn    = [C(1) + C(2)]/[C(1)/S1 + C(2)/S2]
Sw    = [C(1)S1 + C(2)S2]/[C(1) + C(2)]
Sz    = [C(1)(S1^2) + C(2)(S2^2)]/[C(1)S1 + C(2)S2]
Sz+1  = [C(1)(S1^3) + C(2)(S2^3)]/[C(1)(S1^2) + C(2)(S2^2)]

One can combine these in pairs to obtained the following linear 
relationships (the two species equations)
Sw    = (S1 + S2) - (S1)(S2)(1/Sn)
Sz    = (S1 + S2) - (S1)(S2)(1/Sw)
Sz+1  = (S1 + S2) - (S1)(S2)(1/Sz)

The question is whether or not all of these averages (or moments) can 
be estimated by integrating over the g(s*) vs s* curves.

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*.

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.)

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.

So the second question is "How bad will it be if we do that anyway?".


Example No. 1.
Single species: 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.

So for a case 160 kDa, 9.0S  simulated data (@50,000 RPM).
we get

     from g(s)(dcdt)       ratio to true value
Sn      8.91               0.99
Sw      9.00               1.00
Sz      9.08               1.01
Sz+1    9.17               1.02

and for case 40 kDa, 3.6S
we get

       from g(s)          ratio to true value
Sn      3.34  (3.38)        1.078 (1.065)
Sw      3.53  (3.54)        0.980 (0.983) 
Sz      3.70  (3.68)        1.028 (1.022)
Sz+1    3.85  (3.80)        1.069 (1.056)

(numbers in parentheses from later time point and are slightly better.)
Conclusion: as long as the speed and/or molar mass is high enough 
(i.e. high enough sigma = w^2(s/D) ),
these averages do not deviate seriously from their true values
when computed by integrating over g(s*) vs s* from dcdt analysis.


Example No. 2.
Take a monomer-tetramer system non-interacting, (mixture 0.5 wt fractions).

40 kDa, 3.5S; 160 kDa, 9.0S

equal mass concentrations.

     true value           from g(s)      ratio to true value

Sn    5.14                4.37          0.850
Sw    6.25                6.15          0.984
Sz    7.46                7.56          1.013
Sz+1  8.44                8.43          0.999


Conclusion: 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.

Example No. 3.
Monomer-tetramer interacting: simulated with noise +/- 0.005 fringes 
(3.29 mg/fringe)
monomer 40 kDa and 3.6S
tetramer   9.0S
Ka = 1 x 10^15

  Co     g/L    Sn*   Sn     R         Sw*   Sw     R       Sz*  Sz 
R       Sz+1   Sz+1   R
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
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
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
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
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
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

* = values obtained from integration of g(s*) vs s*.
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.

Conclusion: See FIGURE below.
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.

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)

---------------------------
FIGURE
TOP: exact averages computed from mass action equations: result not 
unexpected. This what the bottom plot would
look like if we could get highly accurate values.
BOT: computed from simulated data, analyzed with SEDANAL/DCDT to 
generate the g(s*) patterns,
as follows:(Error bars are plotted.)

  Sn   = Int[        (g(s*))ds*]/Int[((1/s*)(g(s*))ds*]
  Sw   = Int[  s*    (g(s*))ds*]/Int[       (g(s*))ds*]
  Sz   = Int[ (s*^2) (g(s*))ds*]/Int[  s*   (g(s*))ds*]
  Sz+1 = Int[ (s*^3) (g(s*))ds*]/Int[((s*^2)(g(s*))ds*]

[figure deleted] jpeg attached.


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.

Walter
-- 
----------------------------------------------------------
Walter Stafford
mailto:stafford at bbri.org
direct dial:    617-658-7808
receptionist:  617-658-7700
----------------------------------------------------------
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