Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
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
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
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. Dear Dale, I'm not sure that this is true. Let's assume that the Fourier transform of the continuous function is band-limited, and the real-space sampling rate is over twice the Shannon frequency. There are at least *two* different mathematical functions that pass precisely through your sampled values: 1. the original continuous function, and 2. the sampled values themselves. One could perfectly reconstruct the original continuous function using a low pass top-hat filter of width +/-1/2q about the origin in reciprocal space (where q is the real-space sampling interval), thus cutting out the higher resolution ghosts. In real space, this corresponds to convolution of your samples with a sinc function (sinc(x/(q/2)) up to a multiplicative constant). But you could also filter your samples using wider top hats to include higher resolution ghosts (between +/-(2n+1)/2q, where n is an integer), corresopnding to narrower sinc functions in the real-space interplation and therefore resulting in different continuous functions. All these functions will pass though the initial set of sampled values*, but will differ inbetween. For example, in the limit of making your reciprocal space top-hat filter very wide indeed, your sinc function in the real-space interpolation will be delta function-like and will give you a reconstructed continuous function that will look almost like your sequence of sampled values. So I think that even if your function is band-limited and is sampled at a rate greater than twice the Nyquist frequency, there are still an infinite number of functions that can be derived from the samples and that will pass through them. Am I wrong? Joe *The transforms of these continuous functions will have local translational symmetry in reciprocal space that is derived from the periodicity of the transform of the original unfiltered samples. If you now sample these functions at the same positions as with the original function, their transform will be identical to the transform of the original samples (because the periodicity imposed by the sampling will be in register with the translational symmetry mentioned above). So the values obtained from sampling functions derived from the different interpolation schemes must be identical to the original set of samples. 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
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
Hi Dirk I think you're confusing the sampling of the molecular transform with the sampling of the electron density. You say 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. In fact the sampling of the molecular transform has nothing to do with h, it's sampled at points separated by a* = 1/a in the 1-D case. Cheers -- Ian On Fri, Apr 15, 2011 at 12:20 PM, 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 ***
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
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 ***
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
Dear Ian, oh, yes, thank you - you are absolutely right! I really confused the sampling of the molecular transform with the sampling of the electron density in the unit cell! Sometimes I don't see the wood for the trees! Let me then shift my question from the sampling of the molecular transform to the sampling of the electron density within the unit cell. For the 1-dimensional case, this is discretely sampled at a/h for resolution d, which is still 1x sampling and not 2x sampling, as required according to Nyquist-Shannon. Where is my error in reasoning, here? Best regards, Dirk. Am 15.04.11 14:25, schrieb Ian Tickle: Hi Dirk I think you're confusing the sampling of the molecular transform with the sampling of the electron density. You say 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. In fact the sampling of the molecular transform has nothing to do with h, it's sampled at points separated by a* = 1/a in the 1-D case. Cheers -- Ian On Fri, Apr 15, 2011 at 12:20 PM, 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 *** -- *** 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 ***
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
Dear Dirk, The factor of 2 comes from the fact that the diameter of a sphere is twice its radius. The radius of the limiting sphere for data to a certain resolution in reciprocal space is d_star_max. If you sample the electron density at points distant by delta from each other, you periodise the transform of the continuous density at that resolution by a reciprocal lattice of size 1/delta. If you want to avoid aliasing, i.e. corruption of one copy of your data in its sphere of radius d_star_max by the data in a translate of that sphere by 1/delta, you must ensure that 1/delta is larger than 2*d_star_max (the diameter of the limiting sphere. In other words, delta must be less than (1/2)*(1/d_star_max), which is your Shannon/Nyquist criterion, since 1/d_star_max is your d_min or resolution. With best wishes, Gerard. -- On Fri, Apr 15, 2011 at 03:11:41PM +0200, Dirk Kostrewa wrote: Dear Ian, oh, yes, thank you - you are absolutely right! I really confused the sampling of the molecular transform with the sampling of the electron density in the unit cell! Sometimes I don't see the wood for the trees! Let me then shift my question from the sampling of the molecular transform to the sampling of the electron density within the unit cell. For the 1-dimensional case, this is discretely sampled at a/h for resolution d, which is still 1x sampling and not 2x sampling, as required according to Nyquist-Shannon. Where is my error in reasoning, here? Best regards, Dirk. Am 15.04.11 14:25, schrieb Ian Tickle: Hi Dirk I think you're confusing the sampling of the molecular transform with the sampling of the electron density. You say 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. In fact the sampling of the molecular transform has nothing to do with h, it's sampled at points separated by a* = 1/a in the 1-D case. Cheers -- Ian On Fri, Apr 15, 2011 at 12:20 PM, 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
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
Dear Dirk, You are getting confused about where the sampling occurs, and this is perhaps because we usually learn about the Shannon criterion from a certain way around (sampling in real/time space - periodicity of the signal transform in frequency/reciprocal space). To see the Shannon criterion in crystallography, you have to look at it the other way around (sampling of the molecular transform in reciprocal space - periodicity of the electron density in space). Twice the signal bandwidth becomes the physical width of the unique portion of your 1D electron density, which is equal to the unit cell repeat by definition. Hence, you are sampling the fourier transform at double the Shannon frequency. Sampling of the electron density makes the sampled molecular transform periodic in reciprocal space, with interval 1/q, where q is your real-space sampling interval. If d is the minimum Bragg spacing, then your molecular transform lies between +/- 1/d in reciprocal space, i.e. has a full-width of 2/d. Thus, in order for the ghost copies of the molecular transform to not overlap, you must have q such that 1/q = 2/d. i.e. q = d/2. Hope that helps, Joe Dear Ian, oh, yes, thank you - you are absolutely right! I really confused the sampling of the molecular transform with the sampling of the electron density in the unit cell! Sometimes I don't see the wood for the trees! Let me then shift my question from the sampling of the molecular transform to the sampling of the electron density within the unit cell. For the 1-dimensional case, this is discretely sampled at a/h for resolution d, which is still 1x sampling and not 2x sampling, as required according to Nyquist-Shannon. Where is my error in reasoning, here? Best regards, Dirk. Am 15.04.11 14:25, schrieb Ian Tickle: Hi Dirk I think you're confusing the sampling of the molecular transform with the sampling of the electron density. You say 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. In fact the sampling of the molecular transform has nothing to do with h, it's sampled at points separated by a* = 1/a in the 1-D case. Cheers -- Ian On Fri, Apr 15, 2011 at 12:20 PM, 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
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
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. -- *** 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 ***
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
Hi Dirk, My interpretation of your question is what is the impact of resolution given by the individual diffraction spots from the electron density sampling and the Nyquist theorem. My explanation would be that the Nyquist theorem gives an upper limit to the frequency information that can be obtained, in the case of crystallography, the highest resolution spot that is possible. Everything with lower resolution, or smaller index, is at a lower frequency than the nyquist limit. The nyquist limit would come from the sampling done in the fourier transform of the frequency domain, which in this case is the transform of reciprocal space to real space. The sampling that is done in real space is limited by the interaction of the X-rays with the electron density of the individual molecules in the lattice. That interaction is nearly continuous across a molecule, leading to a very high/fast sampling rate. The limit of this interaction would be due to the wavelength (~lambda/2) which would result in the diffraction limit in reciprocal space (limiting the largest index that is observable). This my understanding, but I too would like to have a more intuitive understanding of this fundamental limitation. Brett 2011/4/15 Dirk Kostrewa kostr...@genzentrum.lmu.de Dear Ian, oh, yes, thank you - you are absolutely right! I really confused the sampling of the molecular transform with the sampling of the electron density in the unit cell! Sometimes I don't see the wood for the trees! Let me then shift my question from the sampling of the molecular transform to the sampling of the electron density within the unit cell. For the 1-dimensional case, this is discretely sampled at a/h for resolution d, which is still 1x sampling and not 2x sampling, as required according to Nyquist-Shannon. Where is my error in reasoning, here? Best regards, Dirk. Am 15.04.11 14:25, schrieb Ian Tickle: Hi Dirk I think you're confusing the sampling of the molecular transform with the sampling of the electron density. You say 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. In fact the sampling of the molecular transform has nothing to do with h, it's sampled at points separated by a* = 1/a in the 1-D case. Cheers -- Ian On Fri, Apr 15, 2011 at 12:20 PM, 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
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
Dirk Another way of looking at it See slide 7 in http://www.aps.anl.gov/Science/Future/Workshops/Frontier_Science_Using_Soft_Xrays/Presentations/WeierstalTalk.pdf sampling interval 1/W (Bragg sampling) is Shannon sampling if complex Fraunhofer wavefield of object with width W is recorded. If only Fraunhofer intensity of object with width W is recorded, then the FT of the intensity is the autocorrelation with width 2W and the correct (Shannon) sampling interval is 1/2W. Additional issues are present for 2D and 3D but the above gives the basic idea. the sampling of the continuous molecular transform imposed by the crystal lattice is sufficient to get the desired information at a given resolution? Yes, if you have phased amplitudes Regards Colin -Original Message- From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of Dirk Kostrewa Sent: 15 April 2011 12:20 To: CCP4BB@JISCMAIL.AC.UK Subject: [ccp4bb] Lattice sampling and resolution - a seeming paradox? 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 ***
