Is the simplest answer that we indeed do not get all of the information, and are accordingly missing phases? My understanding is that if we were able to sample with higher frequency, we could get phases too. For example, a lone protein in a huge unit cell would enable phase determination. Taken further, I believe the single-particle-FEL-people were envisioning phasing by using direct methods on the continuous transform seen on the detector (or rather the 3D reconstruction of such by combination of many images)
JPK On Fri, Apr 15, 2011 at 6:20 AM, Dirk Kostrewa <kostr...@genzentrum.lmu.de> wrote: > 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. > > -- > > ******************************************************* > Dirk Kostrewa > Gene Center Munich, A5.07 > Department of Biochemistry > Ludwig-Maximilians-Universität München > Feodor-Lynen-Str. 25 > D-81377 Munich > Germany > Phone: +49-89-2180-76845 > Fax: +49-89-2180-76999 > E-mail: kostr...@genzentrum.lmu.de > WWW: www.genzentrum.lmu.de > ******************************************************* > -- ******************************************* Jacob Pearson Keller Northwestern University Medical Scientist Training Program cel: 773.608.9185 email: j-kell...@northwestern.edu *******************************************
