Re: [ccp4bb] A basic question about Fourier Transform
Hi Chen, Here is what I think: Assuming a crystal is perfect and is being shot at 0 K, then the maximum resolution one experiment can achieve is limited by the wavelength of the X-ray. It can’t be better than the half-wavelength under the normal experimental setting (minimum d=lambda/2/sin(90)=lambda/2). With shorter wavelength, we get more reflections (an Ewald sphere with larger radius encloses more lattice points), and that’s more sampling points and more information. With infinitely short wavelength, we get infinitely detailed information. On the other hand, the seemingly continuous, detailed profile we get from a single molecule diffraction is also limited (smeared) by the same X-ray wavelength. So it is only a difference between measuring many discrete points and measuring (smeared) continuity. In other words, the continuous curve we get from the single molecule diffraction experiment does not contain information with frequency higher than that of the X-ray. It only contains information with frequency up to that of the X-ray. Recalling that the minimum d from a crystal diffraction experiment is lambda/2, then for a 1-D crystal’s unit cell (with edge length a), we are sampling it with a frequency of 2a/lambda. I think the sampling theorem says that this sampling frequency is as good as the continuous curve we get from single molecule diffraction with X-ray of wavelength lambda. With real life crystals, which are neither perfect, nor at 0 Kelvin, what makes difference is, by using a single molecule instead of a crystal, we can get away from the conformational differences of molecules found in a crystal, the defects in crystal, the heterogeneity of the crystal (e.g., the mosaicity), and probably even the background generated by solvent atoms (as the single molecule might be floating in vacuum). The packing defects and heterogeneity in a crystal is probably what limits resolution of our protein crystals in most cases. So when we are freed from the situation of having to use a crystal, in theory with short enough X-ray wavelength, by shooting at a single molecule that is not moving too much during the exposure, we can get a very high resolution that would not be achievable using a crystal. Now what weakens our high angle signal, limits the high resolution, and smears our map is the real thermo motion of the atoms, not the heterogeneity of the crystal. Then for the next step, with that highly sensitive detector and our ability of sending small pack of photons to the molecule, we might be able to get very quick snap shots of the molecule, essentially reducing the motion blur. Then in that case, what ultimately limits our resolution (or confidence of the measurement?) is probably the number of photons we can send to an atom before the absorbed energy significantly affect its location – may I say, a situation similar to that faced by the cryoEM people? Zhijie From: Chen Zhao Sent: Tuesday, January 20, 2015 10:18 PM To: CCP4BB@JISCMAIL.AC.UK Subject: [ccp4bb] A basic question about Fourier Transform Dear all, I am sorry about this slightly off-topic question. I am now a graduate TA for crystallography course and one student asked me a question that I didn't ask myself before. I don't have enough knowledge to precisely answer this question, so I am seeking for help here. The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? Thank you so much for your attention and your help in advance! Best, Chen
Re: [ccp4bb] A basic question about Fourier Transform
Phases can be deduced mathematically from a continuous transform, a la David Sayre’s and others’ work. Compared to a crystallographic pattern, a continuous pattern has huge amounts of information—every pixel (roxel?) would be equivalent to a reflection, so instead of having ~10^4-5 data points you would have, say, 10^8-12, all to define ~10^3-4 atoms. And no b-factors to fit at 100K, since the molecule would not be moving at that temp. Of course this would be totally impossible to actually measure, at least for now (!). JPK From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of Chen Zhao Sent: Tuesday, January 20, 2015 11:47 PM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] A basic question about Fourier Transform Dear Steven, Thank you for your reply! I understand that it is nearly impossible to measure the diffraction of a single molecule, and I am just bringing this up as a thought experiment to help understand the basics in crystallography. But I never thought that some molecules actually allow such measurement because you can burn it over and over again without severe damage. Thanks a lot for this piece of information. But for the phase problem, the difference is that, you can have magnetic lens for the electrons in EM, but you cannot have any lenses for X-ray beam. This is why I am still confused about this point. Thanks a lot again, Chen On Tue, Jan 20, 2015 at 11:21 PM, Steven Chou stevezc...@gmail.commailto:stevezc...@gmail.com wrote: I would say you cannot measure the diffraction pattern of a single biological molecule accurately thus far, because biological molecules are not strong scatters and can be damaged easily. For other molecules, actually you can! In high-resolution electron microscopy, the diffraction pattern in the back focal plane is actually the diffraction pattern of a projection of your sample, which is usually composed of one to several hundred biological molecules. For biological molecules, this pattern usually is dampened to almost zero at a resolution between 30A-4A (actual resolution, not theoretical); for some metal compounds, the resolution can reach up to 1 A, or even better. The diffraction pattern in the back focal plane is the Fourier transform (achieved by a convex lens) of the a 2D projection of your sample. If you apply another Fourier transform (using another convex lens) to the diffraction pattern, you can get the 2D image of your sample (which contains both amplitude and phase). That is, in single particle EM (imaging mode), people don't have the phase problem. In diffraction mode (2D electron crystallography), only the diffraction pattern (intensity) is recorded, so they also have the phase problem. HTH, Steven On Tue, Jan 20, 2015 at 10:18 PM, Chen Zhao c.z...@yale.edumailto:c.z...@yale.edu wrote: Dear all, I am sorry about this slightly off-topic question. I am now a graduate TA for crystallography course and one student asked me a question that I didn't ask myself before. I don't have enough knowledge to precisely answer this question, so I am seeking for help here. The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? Thank you so much for your attention and your help in advance! Best, Chen -- Steven Chou
Re: [ccp4bb] A basic question about Fourier Transform
Dear Ethan, Zhijie and Keller, Thank you so much for your detailed reply! Now I think I have a much deeper view of this problem and understand the relationship between X-ray, XFEL and EM much better. I did learned a lot from your replies! Best, Chen On Wed, Jan 21, 2015 at 7:51 AM, Keller, Jacob kell...@janelia.hhmi.org wrote: Phases can be deduced mathematically from a continuous transform, a la David Sayre’s and others’ work. Compared to a crystallographic pattern, a continuous pattern has huge amounts of information—every pixel (roxel?) would be equivalent to a reflection, so instead of having ~10^4-5 data points you would have, say, 10^8-12, all to define ~10^3-4 atoms. And no b-factors to fit at 100K, since the molecule would not be moving at that temp. Of course this would be totally impossible to actually measure, at least for now (!). JPK *From:* CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] *On Behalf Of *Chen Zhao *Sent:* Tuesday, January 20, 2015 11:47 PM *To:* CCP4BB@JISCMAIL.AC.UK *Subject:* Re: [ccp4bb] A basic question about Fourier Transform Dear Steven, Thank you for your reply! I understand that it is nearly impossible to measure the diffraction of a single molecule, and I am just bringing this up as a thought experiment to help understand the basics in crystallography. But I never thought that some molecules actually allow such measurement because you can burn it over and over again without severe damage. Thanks a lot for this piece of information. But for the phase problem, the difference is that, you can have magnetic lens for the electrons in EM, but you cannot have any lenses for X-ray beam. This is why I am still confused about this point. Thanks a lot again, Chen On Tue, Jan 20, 2015 at 11:21 PM, Steven Chou stevezc...@gmail.com wrote: I would say you cannot measure the diffraction pattern of a single biological molecule accurately thus far, because biological molecules are not strong scatters and can be damaged easily. For other molecules, actually you can! In high-resolution electron microscopy, the diffraction pattern in the back focal plane is actually the diffraction pattern of a projection of your sample, which is usually composed of one to several hundred biological molecules. For biological molecules, this pattern usually is dampened to almost zero at a resolution between 30A-4A (actual resolution, not theoretical); for some metal compounds, the resolution can reach up to 1 A, or even better. The diffraction pattern in the back focal plane is the Fourier transform (achieved by a convex lens) of the a 2D projection of your sample. If you apply another Fourier transform (using another convex lens) to the diffraction pattern, you can get the 2D image of your sample (which contains both amplitude and phase). That is, in single particle EM (imaging mode), people don't have the phase problem. In diffraction mode (2D electron crystallography), only the diffraction pattern (intensity) is recorded, so they also have the phase problem. HTH, Steven On Tue, Jan 20, 2015 at 10:18 PM, Chen Zhao c.z...@yale.edu wrote: Dear all, I am sorry about this slightly off-topic question. I am now a graduate TA for crystallography course and one student asked me a question that I didn't ask myself before. I don't have enough knowledge to precisely answer this question, so I am seeking for help here. The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? Thank you so much for your attention and your help in advance! Best, Chen -- Steven Chou
Re: [ccp4bb] A basic question about Fourier Transform
I would say you cannot measure the diffraction pattern of a single biological molecule accurately thus far, because biological molecules are not strong scatters and can be damaged easily. For other molecules, actually you can! In high-resolution electron microscopy, the diffraction pattern in the back focal plane is actually the diffraction pattern of a projection of your sample, which is usually composed of one to several hundred biological molecules. For biological molecules, this pattern usually is dampened to almost zero at a resolution between 30A-4A (actual resolution, not theoretical); for some metal compounds, the resolution can reach up to 1 A, or even better. The diffraction pattern in the back focal plane is the Fourier transform (achieved by a convex lens) of the a 2D projection of your sample. If you apply another Fourier transform (using another convex lens) to the diffraction pattern, you can get the 2D image of your sample (which contains both amplitude and phase). That is, in single particle EM (imaging mode), people don't have the phase problem. In diffraction mode (2D electron crystallography), only the diffraction pattern (intensity) is recorded, so they also have the phase problem. HTH, Steven On Tue, Jan 20, 2015 at 10:18 PM, Chen Zhao c.z...@yale.edu wrote: Dear all, I am sorry about this slightly off-topic question. I am now a graduate TA for crystallography course and one student asked me a question that I didn't ask myself before. I don't have enough knowledge to precisely answer this question, so I am seeking for help here. The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? Thank you so much for your attention and your help in advance! Best, Chen -- Steven Chou
Re: [ccp4bb] A basic question about Fourier Transform
The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? This question depends on how you set your criteria for the goodness of the single-molecule data; in a way by saying the data quality is comparable to crystals, the question is tautological, since the cases would by stipulation be the same. If you mean the resolution would be equal, and all amplitudes would be of SNR similar to crystal data, of course the continuously-sampled version would be far better. There would be no phase problem (sampling theorem), and maps would be outstanding. This type of thing was, I think, the original vision of the XFEL projects—you could check out those early papers for more details. JPK
Re: [ccp4bb] A basic question about Fourier Transform
Dear Steven, Thank you for your reply! I understand that it is nearly impossible to measure the diffraction of a single molecule, and I am just bringing this up as a thought experiment to help understand the basics in crystallography. But I never thought that some molecules actually allow such measurement because you can burn it over and over again without severe damage. Thanks a lot for this piece of information. But for the phase problem, the difference is that, you can have magnetic lens for the electrons in EM, but you cannot have any lenses for X-ray beam. This is why I am still confused about this point. Thanks a lot again, Chen On Tue, Jan 20, 2015 at 11:21 PM, Steven Chou stevezc...@gmail.com wrote: I would say you cannot measure the diffraction pattern of a single biological molecule accurately thus far, because biological molecules are not strong scatters and can be damaged easily. For other molecules, actually you can! In high-resolution electron microscopy, the diffraction pattern in the back focal plane is actually the diffraction pattern of a projection of your sample, which is usually composed of one to several hundred biological molecules. For biological molecules, this pattern usually is dampened to almost zero at a resolution between 30A-4A (actual resolution, not theoretical); for some metal compounds, the resolution can reach up to 1 A, or even better. The diffraction pattern in the back focal plane is the Fourier transform (achieved by a convex lens) of the a 2D projection of your sample. If you apply another Fourier transform (using another convex lens) to the diffraction pattern, you can get the 2D image of your sample (which contains both amplitude and phase). That is, in single particle EM (imaging mode), people don't have the phase problem. In diffraction mode (2D electron crystallography), only the diffraction pattern (intensity) is recorded, so they also have the phase problem. HTH, Steven On Tue, Jan 20, 2015 at 10:18 PM, Chen Zhao c.z...@yale.edu wrote: Dear all, I am sorry about this slightly off-topic question. I am now a graduate TA for crystallography course and one student asked me a question that I didn't ask myself before. I don't have enough knowledge to precisely answer this question, so I am seeking for help here. The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? Thank you so much for your attention and your help in advance! Best, Chen -- Steven Chou
Re: [ccp4bb] A basic question about Fourier Transform
Hi Jacob, Thanks a lot for your reply! Yes, by comparable data quality I did mean the comparable resolution and SNR. I now understand the original question and kinda confirm what I thought. But I am also learning myself and I don't quite get why the continuous sampling would get rid of the phase problem. My first thought would be that mathematically speaking, the difference between solving the structure from a crystal diffraction pattern and solving the structure from a single molecule diffraction pattern only means the transition from a discrete Fourier summation to a continuous Fourier integral, but the phase term is always there. I am sorry for my lack of knowledge and I only knew the very basics in the sampling theorem many years ago. Another related question is, do different crystal lattices sample the reciprocal space equally well (or lose the same amount of information)? My intuition is yes because of the symmetry in the real space stays in the reciprocal space, but maybe I am wrong. Sorry for keeping bothering, Chen On Tue, Jan 20, 2015 at 10:41 PM, Keller, Jacob kell...@janelia.hhmi.org wrote: The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? This question depends on how you set your criteria for the goodness of the single-molecule data; in a way by saying the data quality is comparable to crystals, the question is tautological, since the cases would by stipulation be the same. If you mean the resolution would be equal, and all amplitudes would be of SNR similar to crystal data, of course the continuously-sampled version would be far better. There would be no phase problem (sampling theorem), and maps would be outstanding. This type of thing was, I think, the original vision of the XFEL projects—you could check out those early papers for more details. JPK
Re: [ccp4bb] A basic question about Fourier Transform
On Tuesday, 20 January 2015 10:18:35 PM Chen Zhao wrote: Dear all, I am sorry about this slightly off-topic question. I am now a graduate TA for crystallography course and one student asked me a question that I didn't ask myself before. I don't have enough knowledge to precisely answer this question, so I am seeking for help here. The question is, as I rephrased it, assuming we are able to measure the diffraction pattern of a single molecule with acceptable accuracy and precision (comparable to what we have now for the common crystals), is it better than we measure the diffraction spots from a crystal, given that the spots are just a sampling of the continuous pattern from a single molecule and there is loss of information in the space between the spots that are not sampled by the lattice? While it is true that there is a loss of information because of the space between the Bragg reflections, this is not as bad as you might think. The Nyquist theorem tells us that we can reconstruct a Fourier term exactly if we can sample at one half the period of that term. So for any given resolution of Bragg spots, the continuous transform to half that resolution can be reconstructed. Here can be reconstructed implicitly includes ... if we know the phase. So it comes back to the phase problem. If we could measure the phase, it would only matter to a factor of 2 in resolution that we are not measuring the continuous transform. By the way, as Jacob Keller alluded to earlier, XFEL diffraction from nanocrystals introduces a situation half way between the two cases. Because there are only a small number of unit cells in each direction, the observed diffraction pattern indeed contains information in between the Bragg peaks. One approach to interpreting this data is to treat the measured diffraction pattern as a continuous transform of a single particle, where that single particle just happens to be a nanocrystal containing a small number of identical unit cells. Ethan Of course this is more of a thought experiment, so we don't need to consider that all measurement is discrete in nature owing to the limitation of the pixel size. I kinda agree with him and I have a feeling that this is related to the sampling theorem. I do appreciate your valuable comments. If this is not true, why? If this is true, what is its effect on electron density? Thank you so much for your attention and your help in advance! Best, Chen -- mail: Biomolecular Structure Center, K-428 Health Sciences Bldg MS 357742, University of Washington, Seattle 98195-7742
