Re: [ccp4bb] difference density ripples around Hg atoms

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CCP4bb <-- 1999 <-- November 1999 <-- 30 November 1999

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Subject: Re: difference density ripples around Hg atoms
From: "George M {- dot -} Sheldrick" gsheldr {- at -} SHELX {- dot -} UNI-AC {- dot -} GWDG {- dot -} DE
Date: 2007-08-01

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Subject: Re: difference density ripples around Hg atoms
From: Bart Hazes bart {- dot -} hazes {- at -} UALBERTA {- dot -} CA
Date: 2007-08-01

Subject: Re: difference density ripples around Hg atoms
From: Kay Diederichs Kay {- dot -} Diederichs {- at -} UNI-KONSTANZ {- dot -} DE
Date: 2007-08-01

Bart Hazes schrieb:
...
> W.r.t. Kay's reply I think the argument does not hold since it depends
> on how badly the data is truncated. E.g. truncated near the limit of
> diffraction will give few ripples whereas a data set truncated at I/SigI
> of 5 will have much more servious effects.
>
> Bart

Bart,

if you truncate at the limit of diffraction (i.e. where there is no more
signal) you will not get any ripple at all !

Of course, if you truncate at a resolution where there is significant
signal (and I do agree with you in that respect: many people truncate
their datasets at too low resolution) there _will_ be Fourier ripples.
However, a ripple is never as high than the peak itself.

To get a quantitative picture of the worst-case scenario, consider the
following: truncation means multiplication of the data with a Heaviside
function (that is 1 up to the chosen resolution limit, and 0 beyond). In
real space, this translates into a series of ripples, arising by
convolution of the true electron density with the Fourier transform of
the Heaviside function. The Fourier transform of a one-dimensional
Heaviside function is the function sin(x)/x . Convolution with sin(x)/x
has the effect of
a) broadening (or "smearing") the true electron density, resulting in a
low-resolution electron density map instead of the true one
b) adding ripples at certain distances (which can be calculated from the
resolution) around each peak. The first negative ripple has an absolute
value of less than 1/4 of the peak height, and the first positive ripple
about 1/8 of the peak height.

So in the worst case (one-dimensional truncation of data) my estimate of
12% was wrong - I estimated the height of the first positive ripple
whereas Klemens reported the first negative ripple!

On the other hand, if I remember correctly, the Fourier transform of the
3-dimensional Heaviside function (a filled sphere) is a Bessel function
that has ripples which (I think) are lower than those of the
one-dimensional Heaviside function. Surely somebody knows the function,
and its peak heights?

best,

Kay
--
Kay Diederichs http://strucbio.biologie.uni-konstanz.de
email: Kay.Diederichs@uni-konstanz.de Tel +49 7531 88 4049 Fax 3183
Fachbereich Biologie, Universität Konstanz, Box M647, D-78457 Konstanz

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CCP4bb <-- 1999 <-- November 1999 <-- 30 November 1999

Next message:
Subject: Re: difference density ripples around Hg atoms
From: Bart Hazes bart {- dot -} hazes {- at -} UALBERTA {- dot -} CA
Date: 2007-08-01