[RASMB] 3mm centerpieces - the SVENSSON EQUATION

Tue Sep 19 05:28:08 PDT 2006

Greetings everyone -

I come into this discussion a little late - having been away for a week. But
reading through some very informative contributions, it does appear that one
significant point has been missed, when we are talking about the effect on
focussing of the optics in the context of 3 mm path length c'pieces.

The need to focus on the 2/3rds plane to avoid ('Wiener') skewing of the
fringe pattern is obvious when you look at the Svensson (1954) equation,
which gives the optical difference delta-S (between solution and solvent
sectors) as

delta-S = a*dn + a^2(beta*dn/dr)(1-2*r)/2*n0  +  a^3((dn/dr)^2*(2-3r))/6n0

where a is the optical path length, beta is the angular aperture, n0 is the
refractive index of the solvent and 0<r<1 is measured from the front face of
the cell window. dn/dr is the refractive index gradient.

The second term - which to eliminate one would have to focus on the
mid-point of the cell - contributes a degree of 'blurring' to the fringe
pattern when it is non-zero, but this one can live with. The 3rd term is the
killer one. Obviously it becomes zero when the optics are focussed exactly
on the 2/3rds position (r = 0.666667). This may not be easy to achieve with
12 mm p'length cells, and - one might suppose - becomes 4x harder if 3 mm
p'length cells are used.

However - let us look at that equation a bit more closely. The 3rd term
involves the 3rd power (cube) of the optical p'length (= a). So - if we
decrease 'a' 4-fold, then the 3rd term becomes 4^3 = 64 times smaller! OK -
it is not actually as good as that, since it is the ratio of the 3rd to the
1st term which matters, so in those terms the skewing arising from the 3rd
term becomes 4^2 =16 times less important. In other words, a given amount of
error in focussing (epsilon-r) will produce 16 times less problem if you are
using 3 mm p'length c'pieces as compared to 12 mm ones.

But how big are these effects in real life? Tom correctly recalls that the
bigger the gradient of refraction, the worse the effect (note: the 3rd term
has the square of the gradient in it): and gives a figure of around 50
fringes/mm for what one can tolerate. Let us have a look at some figures to
see what the actual size of the effect can be under plausible conditions.

Peter Lloyd long ago computed that in a 10 mm optical path length cell,
optics focussed on the mid-point position (i.e.error of 0.166667 in r), one
had to keep the gradient down to <76 fringes/mm** in order to avoid errors
in fringe shift of greater than 0.02 fringes. In modern practice the latter
level of error is quite unacceptable. In SV work it would result in TD
(time-dependent) noise, whilst in SE work one would be for ever identifying
small amounts of ill-defined - and in fact non-existent - 'oligomers' as
being present.

As a ball-park, to avoid systematic problems arising from this effect, one
might go for limiting the problem to stay within the gradient levels which
one can actually measure by keeping the error in r to within around ± 0.01,
for 12 mm cells: thus confining fringe errors to 0.002 fringes. This means
focussing to  ± 0.12 mm in real space. Or ± 0.48 mm for 3 mm cells (4x not
16x, because r is scaled relative to a).

The question is: are we happy that these levels of precision are
attainable/attained in the XL-I?

Arthur

**Peter's book has a typo in it - the limit of 76 fringes/cm is given. I
long ago re-computed this, and checked that the actual figure is 76
fringes/mm for the conditions quoted.

Peter H Lloyd (1974) "Optical Methods in Ultracentrifugation Electrophoresis
and Diffusion" Clarendon Press, Oxford, pp. 87=88.

-- 
*************************
Arthur Rowe
Lab at Sutton Bonington
tel: +44 115 951 6156
fax: +44 115 951 6157
*************************

Kristian,
we've actually made these spacers and systematically compared data quality
and boundary shapes observed with the 3 mm centerpieces in different
positions.  In our hands, our conclusion was that for most cases there was
no difference in data quality or information.  We will make this data
available soon.
Peter

At 09:34 AM 9/18/2006, you wrote:
>Hi-
>Yes you can calculate (or measure) the correct 2/3rds position for the
>centerpiece, make unequal spacers and drill the cell housing holes at that
>level. Be aware that the 2/3 position is not the mechanical location, but
>the optical. These positions differ by a mm or more, and are different for
>sapphire and quartz window. Those cells would be useful for the
>interference optics, so it would be best to determine the 2/3 position for
>sapphire windows. The absorbance optics are focused at the midpoint, so
>the new housings would not be helpful for them.
>Best wishes,
>Tom
>
>Kristian Schilling wrote:
>>Tom and Borries,
>>
>>I am wondering whether we might modify cell housings to overcome that
>>problem. There is no other need for equal spacers but to align the
>>centerpiece with the filling holes, so we might just drill the filling
>>holes at a modified position. Of course that would mean that these
>>housings can be used only with 3-mm-cps, but the first question is: can
>>we match the ideal position for the centerpiece and enhance data quality?
>>
>>Kristian
>>
>>At 01:07 15.09.2006 -0400, you wrote:
>>>Hi-
>>>The spacers for the 3 mm centerpieces place the centerpiece at the
>>>mid-point of the cell, which is out of focus for the interference
>>>system, which is focused at the 2/3 plane of a 12-mm centerpiece, with
>>>sapphire windows and a solvent refractive index of water. Mis-focusing
>>>will lead to mis-registration of the true radial position with the
>>>apparent radial position in regions of the image with steep gradients in
>>>the concentration (1). For the 3 mm centerpieces, the focal plane is
>>>below the sample 2/3 plane, causing steeper fringes than might be
>>>expected... i.e. analysis will reveal what appears to be a weak
>>>association or a heterogeneous system. This problem would also affect
>>>the absorbance system, but the absorbance optics cannot trace a steep
>>>enough gradient.
>>>So long as you keep gradients low enough (as I recall, about 50 fringes
>>>displacement per mm), your data will be OK.
>>>Best wishes,
>>>Tom
>>>
>>>(1)   D. A. Yphantis. Equilibrium Ultracentrifugation of Dilute
>>>Solutions. Biochemistry 3:297-317, 1964.
>>>
>>>Borries Demeler wrote:
>>>>Question about 3 mm centerpieces - the spacers that come with them seem
>>>>to take it out of the focal plane for the interference system, at least
>>>>we cannot get 3 mm CPs to work in interference mode. Are there spacers
>>>>that will correct this issue, or is there something else we should try?
>>>>
>>>>thanks, -Borries
>>>>_______________________________________________
>>>>RASMB mailing list
>>>>RASMB at rasmb.bbri.org
>>>>http://rasmb.bbri.org/mailman/listinfo/rasmb
>>>>
>>>
>>>--
>>>Department of Biochemistry and Molecular Biology
>>>University of New Hampshire
>>>Rudman Hall 379
>>>46 College Road
>>>Durham, NH 03824-3544
>>>Phone: 603-862-2459
>>>FAX:  603-862-0031
>>>www.camis.unh.edu
>>>www.bitc.unh.edu
>>>
>>>
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>>>http://rasmb.bbri.org/mailman/listinfo/rasmb
>>
>>
>>
>>=================================
>>Nanolytics
>>Gesellschaft fuer Kolloidanalytik mbH
>>Dr. Kristian Schilling
>>
>>Hauptstr. 20
>>D-14624 Dallgow
>>
>>Tel.:       +49 3322 24200-5
>>Fax:       +49 3322 24200-6
>>e-mail:   schilling at nanolytics.de
>>Internet:   www.nanolytics.de
>>
>>
>>_______________________________________________
>>RASMB mailing list
>>RASMB at rasmb.bbri.org
>>http://rasmb.bbri.org/mailman/listinfo/rasmb
>
>--
>Department of Biochemistry and Molecular Biology
>University of New Hampshire
>Durham, NH 03824-3544
>Phone: 603-862-2459
>FAX:  603-862-0031
>E-mail: Tom.Laue at unh.edu
>www.bitc.unh.edu
>www.camis.unh.edu
>
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>http://rasmb.bbri.org/mailman/listinfo/rasmb
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