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] OT: Covalent modification of Cys by reducing agents?
We see BME adducts in all of our estrogen receptor structures, though we don't always put them in the models. Sometimes we only see one or two atoms of the adduct, and in others it is completely ordered. We only see it on the solvent accessible cysteines. We do it on purpose. We used to treat the protein with iodoacetic acid to generate uniform modification of the cysteines, but then we realized we could get then same homogeneity with 20-50mM BME. Kendall Nettles On Apr 15, 2011, at 4:09 PM, "Michael Thompson" wrote: > Hi All, > > I was wondering if anyone knew whether or not it is possible for reducing > agents with thiol groups, such as DTT or beta-mercaptoethanol (BME), to form > covalent S-S bonds with Cys residues, particularly solvent-exposed Cys? I > have some puzzling biochemical results, and in the absence of a structure > (thus far), I was wondering if this might be something to try to control for. > I have never heard of this happening (or seen a structure where there was > density for this type of adduct), but I can't really think of a good reason > for why this wouldn't happen. Especially for something like BME, where the > molecule is very much like the Cys sidechain and seems to me like it should > have similar reactivity. The only thing I can think of is if there is a > kinetic effect taking place. Perhaps the rate of diffusion of these small > molecules is much faster that the formation of the S-S bond? > > Does anyone know whether or not this is possible, and why it does or does not > happen? > > Thanks, > > Mike > > > > > -- > Michael C. Thompson > > Graduate Student > > Biochemistry & Molecular Biology Division > > Department of Chemistry & Biochemistry > > University of California, Los Angeles > > mi...@chem.ucla.edu
Re: [ccp4bb] OT: Covalent modification of Cys by reducing agents?
I've seen BME adducts many times both in structures and via MS. DTT adducts are somewhat less common (due to intramolecular disproportionation into oxidized DTT and free SH) but still observable: http://www.ncbi.nlm.nih.gov/pubmed/11684092 So - not only is this possible, but practically commonplace, especially with BME. In general if something is chemically possible -- it's almost certainly going to be found in biological world. That's not very remarkable since biological systemsplay by the same exact rules as chemical ones. What's perhaps much more remarkable is that biological systems routinely perform chemistries that most synthetic chemists would find to be nearly impossible, or entirely impossible in aqueous environment. This just goes to show that billions of years of molecular (and other) evolution are hard to beat. Artem On Fri, Apr 15, 2011 at 3:09 PM, Michael Thompson wrote: > Hi All, > > I was wondering if anyone knew whether or not it is possible for reducing > agents with thiol groups, such as DTT or beta-mercaptoethanol (BME), to form > covalent S-S bonds with Cys residues, particularly solvent-exposed Cys? I > have some puzzling biochemical results, and in the absence of a structure > (thus far), I was wondering if this might be something to try to control > for. I have never heard of this happening (or seen a structure where there > was density for this type of adduct), but I can't really think of a good > reason for why this wouldn't happen. Especially for something like BME, > where the molecule is very much like the Cys sidechain and seems to me like > it should have similar reactivity. The only thing I can think of is if there > is a kinetic effect taking place. Perhaps the rate of diffusion of these > small molecules is much faster that the formation of the S-S bond? > > Does anyone know whether or not this is possible, and why it does or does > not happen? > > Thanks, > > Mike > > > > > -- > Michael C. Thompson > > Graduate Student > > Biochemistry & Molecular Biology Division > > Department of Chemistry & Biochemistry > > University of California, Los Angeles > > mi...@chem.ucla.edu >
[ccp4bb] William Nunn Lipscomb Jr "The Colonel"
For some reason this news affects me deeply. I did not know Bill Lipscomb well, but I interacted closely with James when he was in the UNC Computer Science Department long ago, and members of the Colonel's scientific family have impacted me positively any number of times. So, I share the sorrow of others in this news. The apocryphal stories abound; moving a rotating anode by open-sided sling from one window to another, only to have the weight shift (tragically) in medias res, dropping the Elliot half way into the ground. At one time I equated such stories with Harvard. Today, I can acknowledge that the Colonel's flair played something of a role, too. At the 1971 Cold Spring Harbor Symposium (the only one ever devoted to structural biology - not, as Watson pointedly noted in his opening remarks, because it was important, but because otherwise so many of his friends might die), Lipscomb's group was represented by George Reeke and Don Wiley. Wiley sported a lab T-shirt announcing that there was no law saying... and on the back side a labyrinth with no obvious path into the goal, which was clearly a structure for ATCase ... that there must be a solution. The ATCase structure was eventually solved, and by several others - among them Eric Gouaux and my colleague Hengming Ke. Don's group populated the world with many gifted crystallographers, including grandchildren Ian Wilson and Ed Collins among those I know well and many others I cannot summon. One of his early disciples, Martha Ludwig, passed away recently, leaving many progeny, including I believe, Mark Saper. The Nobel to Tom Steitz renders mention of him superfluous, except that Tom, in turn, has turned out almost countless very gifted protégées while revealing the central dogma, one step at a time. Reviewing this list, as I have been wont to do on many previous occasions, constitutes an open and shut case that the Colonel spawned, if not the first family of US crystallographers, (that might be A.L. Patterson) certainly the most prominent and prolific. As is also true of J.D. Watson, and with apologies for the numerous omissions outside my immediate sphere, at such a moment we all can celebrate what training with the Colonel brought to our community. Charlie On Apr 15, 2011, at 3:06 PM, Peter Moody wrote: > Nobel Laureate William Lipscomb Dies at 91 > By THE ASSOCIATED PRESS > Published: April 15, 2011 at 2:01 PM ET > > I have had this forwarded to me, besides getting a Nobel prize for his > discovery of the bent bonds in boron hydrides, the Colonel was a pioneer in > PX, with work on the role of Zn in carboxypeptidase and the allosteric > mechanism of ATCase perhaps being the best known. Peter > > > > BOSTON (AP) -- A Harvard University professor who won the Nobel chemistry > prize in 1976 for work on chemical bonding has died. William Nunn Lipscomb > Jr. was 91. > > His son, James Lipscomb, said Friday that Lipscomb died Thursday night at a > Cambridge, Mass., hospital of pneumonia and complications from a fall. > > Several of his students also have won Nobels. Yale University professor > Thomas Steitz, who shared the 2009 chemistry prize, says Lipscomb was an > inspiring teacher who encouraged creative thinking. > > The Ohio native grew up in Lexington, Ky., and students affectionately > referred to him as "Colonel" in reference to his upbringing. He graduated > from the University of Kentucky and got a doctorate at the California > Institute of Technology under Nobel laureate Linus Pauling. > > Lipscomb is survived by his wife and three children.
Re: [ccp4bb] OT: Covalent modification of Cys by reducing agents?
We had a case with a BME adduct clearly seen in the structure. PDB ID 1CY5. Leemor On 4/15/11 4:09 PM, "Michael Thompson" wrote: > Hi All, > > I was wondering if anyone knew whether or not it is possible for reducing > agents with thiol groups, such as DTT or beta-mercaptoethanol (BME), to form > covalent S-S bonds with Cys residues, particularly solvent-exposed Cys? I have > some puzzling biochemical results, and in the absence of a structure (thus > far), I was wondering if this might be something to try to control for. I have > never heard of this happening (or seen a structure where there was density for > this type of adduct), but I can't really think of a good reason for why this > wouldn't happen. Especially for something like BME, where the molecule is very > much like the Cys sidechain and seems to me like it should have similar > reactivity. The only thing I can think of is if there is a kinetic effect > taking place. Perhaps the rate of diffusion of these small molecules is much > faster that the formation of the S-S bond? > > Does anyone know whether or not this is possible, and why it does or does not > happen? > > Thanks, > > Mike > > >
[ccp4bb] OT: Covalent modification of Cys by reducing agents?
Hi All, I was wondering if anyone knew whether or not it is possible for reducing agents with thiol groups, such as DTT or beta-mercaptoethanol (BME), to form covalent S-S bonds with Cys residues, particularly solvent-exposed Cys? I have some puzzling biochemical results, and in the absence of a structure (thus far), I was wondering if this might be something to try to control for. I have never heard of this happening (or seen a structure where there was density for this type of adduct), but I can't really think of a good reason for why this wouldn't happen. Especially for something like BME, where the molecule is very much like the Cys sidechain and seems to me like it should have similar reactivity. The only thing I can think of is if there is a kinetic effect taking place. Perhaps the rate of diffusion of these small molecules is much faster that the formation of the S-S bond? Does anyone know whether or not this is possible, and why it does or does not happen? Thanks, Mike -- Michael C. Thompson Graduate Student Biochemistry & Molecular Biology Division Department of Chemistry & Biochemistry University of California, Los Angeles mi...@chem.ucla.edu
[ccp4bb] William Nunn Lipscomb Jr "The Colonel"
Nobel Laureate William Lipscomb Dies at 91 By THE ASSOCIATED PRESS Published: April 15, 2011 at 2:01 PM ET I have had this forwarded to me, besides getting a Nobel prize for his discovery of the bent bonds in boron hydrides, the Colonel was a pioneer in PX, with work on the role of Zn in carboxypeptidase and the allosteric mechanism of ATCase perhaps being the best known. Peter BOSTON (AP) -- A Harvard University professor who won the Nobel chemistry prize in 1976 for work on chemical bonding has died. William Nunn Lipscomb Jr. was 91. His son, James Lipscomb, said Friday that Lipscomb died Thursday night at a Cambridge, Mass., hospital of pneumonia and complications from a fall. Several of his students also have won Nobels. Yale University professor Thomas Steitz, who shared the 2009 chemistry prize, says Lipscomb was an inspiring teacher who encouraged creative thinking. The Ohio native grew up in Lexington, Ky., and students affectionately referred to him as "Colonel" in reference to his upbringing. He graduated from the University of Kentucky and got a doctorate at the California Institute of Technology under Nobel laureate Linus Pauling. Lipscomb is survived by his wife and three children.
Re: [ccp4bb] Searching for very radiation sensitive crystals
Hi Robert, If you don't mind the crystals being rather small, then polyhedra crystals might be what you're after (in fact, many microcrystals satisfy your criteria). There is a mini review here: http://dx.doi.org/10.1080/0889311X.2010.527964 These articles should also help. Ji, X.; Sutton, G.; Evans, G.; Axford, D.; Owen, R.; Stuart, D.I. How Baculovirus Polyhedra Fit Square Pegs into Round Holes to Robustly Package Viruses. EMBO J. 2010, 29 (2), 505–514. Coulibaly, F.; Chiu, E.; Ikeda, K.; Gutmann, S.; Haebel, P.W.; Schulze-Briese, C.; Mori, H.; Metcalf, P. The Molecular Organization of Cypovirus Polyhedra. Nature 2007, 446 (7131), 97–101. Cheers -- David On 12 April 2011 03:06, Robert Thorne wrote: > Dear CCP4 Community, > > We are trying to find protein or virus crystals that diffract to reasonably > high resolution (2.5 Angstroms or better) and that are very radiation > sensitive at room temperature. "Very radiation sensitive" in this case > means that the diffraction dies after a few frames, for crystals that don't > contain heavy atoms. (If you're calculating doses, the diffraction should > die after 0.1 MGy or less). > > Typically, these crystals have high solvent contents (80% or more) and weak > packing interactions. > > > So, we would be very interested in collaborating if you have: > > - crystals that you know are very radiation sensitive at room temperature; > and/or > > - crystals with solvent contents >80%. > > Any advice/suggestions would be appreciated! > > Rob > > > -- > Robert E. Thorne > Professor of Physics > 529A Clark Hall > Cornell University > Ithaca, NY 14853 > phone: (607) 255 6487 > fax: (607) 697 0400 > > > > >
Re: [ccp4bb] Question about BALBES R(free) fraction
Dear Gregory, You need a sufficient number "n" of free reflections to get a statistically valid Rfree value. 500 should be ok. I normally select 5% free reflections. However, I always hate it when programs decide that they know better than the user. Nobody prevents you from using your original mtz, which was used as an input for BALBES, for further refinement and ignore the mtz produced by BALBES. The only result from molecular replacement you really need is the positioned and oriented pdb file. You may have to set the space group in your mz to the space group found by BALBES. Good luck! Herman From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of Gregory T Costakes Sent: Friday, April 15, 2011 4:20 PM To: CCP4BB@JISCMAIL.AC.UK Subject: [ccp4bb] Question about BALBES R(free) fraction My name is Greg Costakes and I am a graduate student for Cynthia Stauffacher at Purdue University. I am attempting to solve the structure of an 18kDa protein domain using molecular replacement to obtain proper phasing (44% identical / 78% homologous to an existing structure). Crystals diffracted to roughly 2 angstroms and are in the space group P212121. There are 10,200 unique reflections with 5.7 fold redundancy. I chose to make an R(free) set containing 10% of the reflections so that I will have just over 1000 in the set. After failing to obtain a valid solution from Phaser and Molrep, I turned to BALBES which was able to successfully perform the molecular replacement and give me final R/R(free) values of 0.3/0.34. However, I noticed the MTZ file that BALBES gave me contained an R(free) set of 5% containing only 500 reflections. I was wondering why BALBES changed the percentage of reflections pegged for the R(free) set. Is there a way to prevent BALBES from changing your R(free) fraction size? Also, is 500 reflections enough for an R(free) set? I had previously been told that R(free) should contain between 1000-1500 reflections. Please let me know. Thank you. --- Greg Costakes PhD Candidate Department of Structural Biology Purdue University Hockmeyer Hall, Room 320 240 S. Martin Jischke Drive, West Lafayette, IN 47907
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 > ***
[ccp4bb] Question about BALBES R(free) fraction
My name is Greg Costakes and I am a graduate student for Cynthia Stauffacher at Purdue University. I am attempting to solve the structure of an 18kDa protein domain using molecular replacement to obtain proper phasing (44% identical / 78% homologous to an existing structure). Crystals diffracted to roughly 2 angstroms and are in the space group P212121. There are 10,200 unique reflections with 5.7 fold redundancy. I chose to make an R(free) set containing 10% of the reflections so that I will have just over 1000 in the set. After failing to obtain a valid solution from Phaser and Molrep, I turned to BALBES which was able to successfully perform the molecular replacement and give me final R/R(free) values of 0.3/0.34. However, I noticed the MTZ file that BALBES gave me contained an R(free) set of 5% containing only 500 reflections. I was wondering why BALBES changed the percentage of reflections pegged for the R(free) set. Is there a way to prevent BALBES from changing your R(free) fraction size? Also, is 500 reflections enough for an R(free) set? I had previously been told that R(free) should contain between 1000-1500 reflections. Please let me know. Thank you. --- Greg Costakes PhD Candidate Department of Structural Biology Purdue University Hockmeyer Hall, Room 320 240 S. Martin Jischke Drive, West Lafayette, IN 47907
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 > 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 >> 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-Shann
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?
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 >> 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 >>>
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 >> 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 yo
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 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 ***
[ccp4bb] Lattice sampling and resolution - a seeming paradox?
Hi Dirk, If you have a N points of a 1D real discrete function, there will be Fourier coefficients indexed h=0,1,2,...,N-1. Taking N as odd, there will be int(N/2)+1 independent Fourier coefficients but your h(max) will in fact be 'N-1'. In crystallography we write h(N-1) as h(-1) etc and tend to ignore these Freidel mates. If cell length is then 'a' the sampling distance is a/N which is related to h(max) but you need to be careful how to defined h(max) Adam
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 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?
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 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 > *** >
[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 ***
Re: [ccp4bb] XXII IUCr Congress and General Assembly - Madrid (Spain)
Hi Gerard Sorry - what Royal Wedding? On 15 Apr 2011, at 11:48, Gerard Bricogne wrote: > Dear Harry, > > Thank you for the reminder. Upon connecting to the URL given in > Martin's e-mail, however, one immediately gets a pop-up saying: > > "New deadline fo abstract submission: April 28" > > (no doubt to enable everyone to work until the last minute prior to the > royal wedding ...). Can you confirm? > > > With best wishes, > > Gerard. > > -- > On Fri, Apr 15, 2011 at 11:39:14AM +0100, Harry Powell wrote: >> Hi folks >> >> today is the last day to get your abstract in! >> >> On 7 Apr 2011, at 14:24, Martin M. Ripoll wrote: >> >>> Dear colleagues, >>> >>> This is just to remind you that the “XXII Congress and General Assembly” of >>> the IUCr (International Union of Crystallography) will be held in Madrid >>> (Spain) from 22-30 August 2011 and that your Spanish colleagues kindly >>> invite you to participate not only in the most important crystallographic >>> event until 2014, but also to enjoy a fruitful meeting in a sunny and full >>> of life city! >>> >>> All information is to be found in: >>> http://www.iucr2011madrid.es/ >>> although two important dates to remember are: >>> April 15, 2011: Deadline for abstracts submission >>> May 15, 2011: Deadline for "Early bird" registration >>> >>> All the best and see you in Madrid! >>> >>> On behalf of the Organizing Committee, >>> >>> Martin (Vice-Chairman) >>> >>> Dr. Martin Martinez-Ripoll >>> Research Professor >>> xmar...@iqfr.csic.es >>> Department of Crystallography & Structural Biology >>> www.xtal.iqfr.csic.es >>> Telf.: +34 917459550 >>> Consejo Superior de Investigaciones Científicas >>> Spanish National Research Council >>> www.csic.es >>> >>> >> >> Harry >> -- >> Dr Harry Powell >> Chairman ECA SIG9 (Crystallographic Computing) >> Acting Chairman IUCr Commission on Crystallographic Computing >> >> http://www.iucr.org/resources/commissions/crystallographic-computing/schools/mieres2011 >> > > -- > > === > * * > * Gerard Bricogne g...@globalphasing.com * > * * > * Global Phasing Ltd. * > * Sheraton House, Castle Park Tel: +44-(0)1223-353033 * > * Cambridge CB3 0AX, UK Fax: +44-(0)1223-366889 * > * * > === Harry -- Dr Harry Powell, MRC Laboratory of Molecular Biology, MRC Centre, Hills Road, Cambridge, CB2 0QH http://www.iucr.org/resources/commissions/crystallographic-computing/schools/mieres2011
Re: [ccp4bb] XXII IUCr Congress and General Assembly - Madrid (Spain)
Dear Gerard, you are right, the abstract submission has been extended. Best regards, Maria Quoting Gerard Bricogne: > Dear Harry, > > Thank you for the reminder. Upon connecting to the URL given in > Martin's e-mail, however, one immediately gets a pop-up saying: > >"New deadline fo abstract submission: April 28" > > (no doubt to enable everyone to work until the last minute prior to the > royal wedding ...). Can you confirm? > > > With best wishes, > > Gerard. > > -- > On Fri, Apr 15, 2011 at 11:39:14AM +0100, Harry Powell wrote: >> Hi folks >> >> today is the last day to get your abstract in! >> >> On 7 Apr 2011, at 14:24, Martin M. Ripoll wrote: >> >> > Dear colleagues, >> > >> > This is just to remind you that the “XXII Congress and General >> Assembly” of the IUCr (International Union of Crystallography) will >> be held in Madrid (Spain) from 22-30 August 2011 and that your >> Spanish colleagues kindly invite you to participate not only in the >> most important crystallographic event until 2014, but also to enjoy >> a fruitful meeting in a sunny and full of life city! >> > >> > All information is to be found in: >> > http://www.iucr2011madrid.es/ >> > although two important dates to remember are: >> > April 15, 2011: Deadline for abstracts submission >> > May 15, 2011: Deadline for "Early bird" registration >> > >> > All the best and see you in Madrid! >> > >> > On behalf of the Organizing Committee, >> > >> > Martin (Vice-Chairman) >> > >> > Dr. Martin Martinez-Ripoll >> > Research Professor >> > xmar...@iqfr.csic.es >> > Department of Crystallography & Structural Biology >> > www.xtal.iqfr.csic.es >> > Telf.: +34 917459550 >> > Consejo Superior de Investigaciones Científicas >> > Spanish National Research Council >> > www.csic.es >> > >> > >> >> Harry >> -- >> Dr Harry Powell >> Chairman ECA SIG9 (Crystallographic Computing) >> Acting Chairman IUCr Commission on Crystallographic Computing >> >> http://www.iucr.org/resources/commissions/crystallographic-computing/schools/mieres2011 >> > > -- > > === > * * > * Gerard Bricogne g...@globalphasing.com * > * * > * Global Phasing Ltd. * > * Sheraton House, Castle Park Tel: +44-(0)1223-353033 * > * Cambridge CB3 0AX, UK Fax: +44-(0)1223-366889 * > * * > === >
Re: [ccp4bb] XXII IUCr Congress and General Assembly - Madrid (Spain)
Dear Harry, Thank you for the reminder. Upon connecting to the URL given in Martin's e-mail, however, one immediately gets a pop-up saying: "New deadline fo abstract submission: April 28" (no doubt to enable everyone to work until the last minute prior to the royal wedding ...). Can you confirm? With best wishes, Gerard. -- On Fri, Apr 15, 2011 at 11:39:14AM +0100, Harry Powell wrote: > Hi folks > > today is the last day to get your abstract in! > > On 7 Apr 2011, at 14:24, Martin M. Ripoll wrote: > > > Dear colleagues, > > > > This is just to remind you that the “XXII Congress and General Assembly” of > > the IUCr (International Union of Crystallography) will be held in Madrid > > (Spain) from 22-30 August 2011 and that your Spanish colleagues kindly > > invite you to participate not only in the most important crystallographic > > event until 2014, but also to enjoy a fruitful meeting in a sunny and full > > of life city! > > > > All information is to be found in: > > http://www.iucr2011madrid.es/ > > although two important dates to remember are: > > April 15, 2011: Deadline for abstracts submission > > May 15, 2011: Deadline for "Early bird" registration > > > > All the best and see you in Madrid! > > > > On behalf of the Organizing Committee, > > > > Martin (Vice-Chairman) > > > > Dr. Martin Martinez-Ripoll > > Research Professor > > xmar...@iqfr.csic.es > > Department of Crystallography & Structural Biology > > www.xtal.iqfr.csic.es > > Telf.: +34 917459550 > > Consejo Superior de Investigaciones Científicas > > Spanish National Research Council > > www.csic.es > > > > > > Harry > -- > Dr Harry Powell > Chairman ECA SIG9 (Crystallographic Computing) > Acting Chairman IUCr Commission on Crystallographic Computing > > http://www.iucr.org/resources/commissions/crystallographic-computing/schools/mieres2011 > -- === * * * Gerard Bricogne g...@globalphasing.com * * * * Global Phasing Ltd. * * Sheraton House, Castle Park Tel: +44-(0)1223-353033 * * Cambridge CB3 0AX, UK Fax: +44-(0)1223-366889 * * * ===
Re: [ccp4bb] XXII IUCr Congress and General Assembly - Madrid (Spain)
Hi folks today is the last day to get your abstract in! On 7 Apr 2011, at 14:24, Martin M. Ripoll wrote: > Dear colleagues, > > This is just to remind you that the “XXII Congress and General Assembly” of > the IUCr (International Union of Crystallography) will be held in Madrid > (Spain) from 22-30 August 2011 and that your Spanish colleagues kindly invite > you to participate not only in the most important crystallographic event > until 2014, but also to enjoy a fruitful meeting in a sunny and full of life > city! > > All information is to be found in: > http://www.iucr2011madrid.es/ > although two important dates to remember are: > April 15, 2011: Deadline for abstracts submission > May 15, 2011: Deadline for "Early bird" registration > > All the best and see you in Madrid! > > On behalf of the Organizing Committee, > > Martin (Vice-Chairman) > > Dr. Martin Martinez-Ripoll > Research Professor > xmar...@iqfr.csic.es > Department of Crystallography & Structural Biology > www.xtal.iqfr.csic.es > Telf.: +34 917459550 > Consejo Superior de Investigaciones Científicas > Spanish National Research Council > www.csic.es > > Harry -- Dr Harry Powell Chairman ECA SIG9 (Crystallographic Computing) Acting Chairman IUCr Commission on Crystallographic Computing http://www.iucr.org/resources/commissions/crystallographic-computing/schools/mieres2011
