I know this question has been answered and Dirk has waved off further
discussion but... I have an answer from a different than usual perspective
that I've been dieing to try out on someone.
Assume you have a one dimensional crystal with a 10 Angstrom repeat.
Someone has told you the value of the electron density at 10 equally
spaced points in this little unit cell, but you know nothing about the
value of the function between those points. I could spend all night
with a crayon drawing different functions that exactly hit all 10 points -
They are infinite in number and each one has a different set of Fourier
coefficients. How can I control this chaos and come up with a simple
description, particularly of the reciprocal space view of these 10
points?
The Nyquist-Shannon sampling theorem simply means that if we assume
that all Fourier coefficient of wave length shorter than 2 Angstrom/cycle
(twice our sampling rate) are defined equal to zero we get only one
function that will hit all ten points exactly. If we say that the 2 A/cycle
reflection has to be zero as well, there are no functions that hit all ten
points (except for special cases) but if we allow the next reflection (the
h=6 or 1.67 A/cycle wave) to be non-zero we are back to an infinite number
of solutions.
That's all it is - If you assume that all the Fourier coefficients of
higher resolution than twice your sampling rate are zero you are guaranteed
one, and only one, set of Fourier coefficients that hit the points and the
Discrete Fourier Transform (probably via a FFT) will calculate that set for
you.
As usual, if your assumption is wrong you will not get the right answer.
If you have a function that really has a non-zero 1.67 A/cycle Fourier
coefficient but you sample your function at 10 points and calculate a
FFT you will get a set of coefficients that hit the 10 points exactly
(when back transformed) but they will not be equal to "true" values.
The overlapping spheres that Gerard Bricogne described are simply the
way of calculating the manor in which the coefficients are distorted by
this bad assumption. Ten Eyck, L. F. (1977). Acta Cryst. A33, 486-492
has an excellent description.
If you are certain that your function has no Fourier components higher
than your sampling rate can support then the FFT is your friend. If your
function has high resolution components and you don't sample it finely
enough then the FFT will give you an answer, but it won't be the correct
answer. The answer will exactly fit the points you sampled but it will
not correctly predict the function's behavior between the points.
The principal situations where this is a problem are:
Calculating structure factors (Fcalc) from a model electron density map.
Calculating gradients using the Agarwal method.
Phase extension via ncs map averaging (including cross-crystal averaging).
Phase extension via solvent flattening (depending on how you do it).
Thank you for your time,
Dale Tronrud
On 4/15/2011 6:34 AM, Dirk Kostrewa wrote:
Dear colleagues of the CCP4BB,
many thanks for all your replies - I really got lost in the trees (or wood?)
and you helped me out with all your kind responses!
I should really leave for the weekend ...
Have a nice weekend, too!
Best regards,
Dirk.
Am 15.04.11 13:20, schrieb Dirk Kostrewa:
Dear colleagues,
I just stumbled across a simple question and a seeming paradox for me in
crystallography, that puzzles me. Maybe, it is also
interesting for you.
The simple question is: is the discrete sampling of the continuous molecular
Fourier transform imposed by the crystal lattice
sufficient to get the desired information at a given resolution?
From my old lectures in Biophysics at the University, I know it has been
theoretically proven, but I don't recall the argument,
anymore. I looked into a couple of crystallography books and I couldn't find
the answer in any of those. Maybe, you can help me out.
Let's do a simple gedankenexperiment/thought experiment in the 1-dimensional
crystal case with unit cell length a, and desired
information at resolution d.
According to Braggs law, the resolution for a first order reflection (n=1) is:
1/d = 2*sin(theta)/lambda
with 2*sin(theta)/lambda being the length of the scattering vector |S|, which
gives:
1/d = |S|
In the 1-dimensional crystal, we sample the continuous molecular transform at
discrete reciprocal lattice points according to the
von Laue condition, S*a = h, which gives |S| = h/a here. In other words, the
unit cell with length a is subdivided into h evenly
spaced crystallographic planes with distance d = a/h.
Now, the discrete sampling by the crystallographic planes a/h is only 1x the
resolution d. According to the Nyquist-Shannon
sampling theorem in Fourier transformation, in order to get a desired
information at a given frequency, we would need a discrete
sampling frequency of *twice* that frequency (the Nyquist frequency).
In crystallography, this Nyquist frequency is also used, for instance, in the
calculation of electron density maps on a discrete
grid, where the grid spacing for an electron density map at resolution d should be
<= d/2. For calculating that electron density
map by Fourier transformation, all coefficients from -h to +h would be used,
which gives twice the number of Fourier coefficients,
but the underlying sampling of the unit cell along a with maximum index |h| is
still only a/h!
This leads to my seeming paradox: according to Braggs law and the von Laue
conditions, I get the information at resolution d
already with a 1x sampling a/h, but according to the Nyquist-Shannon sampling
theory, I would need a 2x sampling a/(2h).
So what is the argument again, that the sampling of the continuous molecular
transform imposed by the crystal lattice is
sufficient to get the desired information at a given resolution?
I would be very grateful for your help!
Best regards,
Dirk.
