Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
I know this question has been answered and Dirk has waved off further discussion but... I have an answer from a different than usual perspective that I've been dieing to try out on someone. Assume you have a one dimensional crystal with a 10 Angstrom repeat. Someone has told you the value of the electron density at 10 equally spaced points in this little unit cell, but you know nothing about the value of the function between those points. I could spend all night with a crayon drawing different functions that exactly hit all 10 points - They are infinite in number and each one has a different set of Fourier coefficients. How can I control this chaos and come up with a simple description, particularly of the reciprocal space view of these 10 points? The Nyquist-Shannon sampling theorem simply means that if we assume that all Fourier coefficient of wave length shorter than 2 Angstrom/cycle (twice our sampling rate) are defined equal to zero we get only one function that will hit all ten points exactly. If we say that the 2 A/cycle reflection has to be zero as well, there are no functions that hit all ten points (except for special cases) but if we allow the next reflection (the h=6 or 1.67 A/cycle wave) to be non-zero we are back to an infinite number of solutions. That's all it is - If you assume that all the Fourier coefficients of higher resolution than twice your sampling rate are zero you are guaranteed one, and only one, set of Fourier coefficients that hit the points and the Discrete Fourier Transform (probably via a FFT) will calculate that set for you. As usual, if your assumption is wrong you will not get the right answer. If you have a function that really has a non-zero 1.67 A/cycle Fourier coefficient but you sample your function at 10 points and calculate a FFT you will get a set of coefficients that hit the 10 points exactly (when back transformed) but they will not be equal to true values. The overlapping spheres that Gerard Bricogne described are simply the way of calculating the manor in which the coefficients are distorted by this bad assumption. Ten Eyck, L. F. (1977). Acta Cryst. A33, 486-492 has an excellent description. If you are certain that your function has no Fourier components higher than your sampling rate can support then the FFT is your friend. If your function has high resolution components and you don't sample it finely enough then the FFT will give you an answer, but it won't be the correct answer. The answer will exactly fit the points you sampled but it will not correctly predict the function's behavior between the points. The principal situations where this is a problem are: Calculating structure factors (Fcalc) from a model electron density map. Calculating gradients using the Agarwal method. Phase extension via ncs map averaging (including cross-crystal averaging). Phase extension via solvent flattening (depending on how you do it). Thank you for your time, Dale Tronrud On 4/15/2011 6:34 AM, Dirk Kostrewa wrote: Dear colleagues of the CCP4BB, many thanks for all your replies - I really got lost in the trees (or wood?) and you helped me out with all your kind responses! I should really leave for the weekend ... Have a nice weekend, too! Best regards, Dirk. Am 15.04.11 13:20, schrieb Dirk Kostrewa: Dear colleagues, I just stumbled across a simple question and a seeming paradox for me in crystallography, that puzzles me. Maybe, it is also interesting for you. The simple question is: is the discrete sampling of the continuous molecular Fourier transform imposed by the crystal lattice sufficient to get the desired information at a given resolution? From my old lectures in Biophysics at the University, I know it has been theoretically proven, but I don't recall the argument, anymore. I looked into a couple of crystallography books and I couldn't find the answer in any of those. Maybe, you can help me out. Let's do a simple gedankenexperiment/thought experiment in the 1-dimensional crystal case with unit cell length a, and desired information at resolution d. According to Braggs law, the resolution for a first order reflection (n=1) is: 1/d = 2*sin(theta)/lambda with 2*sin(theta)/lambda being the length of the scattering vector |S|, which gives: 1/d = |S| In the 1-dimensional crystal, we sample the continuous molecular transform at discrete reciprocal lattice points according to the von Laue condition, S*a = h, which gives |S| = h/a here. In other words, the unit cell with length a is subdivided into h evenly spaced crystallographic planes with distance d = a/h. Now, the discrete sampling by the crystallographic planes a/h is only 1x the resolution d. According to the Nyquist-Shannon sampling theorem in Fourier transformation, in order to get a desired information at a given frequency, we would need a discrete sampling frequency of *twice* that frequency (the Nyquist frequency). In crystallography, this
Re: [ccp4bb] Lattice sampling and resolution - a seeming paradox?
Assume you have a one dimensional crystal with a 10 Angstrom repeat. Someone has told you the value of the electron density at 10 equally spaced points in this little unit cell, but you know nothing about the value of the function between those points. I could spend all night with a crayon drawing different functions that exactly hit all 10 points - They are infinite in number and each one has a different set of Fourier coefficients. How can I control this chaos and come up with a simple description, particularly of the reciprocal space view of these 10 points? The Nyquist-Shannon sampling theorem simply means that if we assume that all Fourier coefficient of wave length shorter than 2 Angstrom/cycle (twice our sampling rate) are defined equal to zero we get only one function that will hit all ten points exactly. If we say that the 2 A/cycle reflection has to be zero as well, there are no functions that hit all ten points (except for special cases) but if we allow the next reflection (the h=6 or 1.67 A/cycle wave) to be non-zero we are back to an infinite number of solutions. Dear Dale, I'm not sure that this is true. Let's assume that the Fourier transform of the continuous function is band-limited, and the real-space sampling rate is over twice the Shannon frequency. There are at least *two* different mathematical functions that pass precisely through your sampled values: 1. the original continuous function, and 2. the sampled values themselves. One could perfectly reconstruct the original continuous function using a low pass top-hat filter of width +/-1/2q about the origin in reciprocal space (where q is the real-space sampling interval), thus cutting out the higher resolution ghosts. In real space, this corresponds to convolution of your samples with a sinc function (sinc(x/(q/2)) up to a multiplicative constant). But you could also filter your samples using wider top hats to include higher resolution ghosts (between +/-(2n+1)/2q, where n is an integer), corresopnding to narrower sinc functions in the real-space interplation and therefore resulting in different continuous functions. All these functions will pass though the initial set of sampled values*, but will differ inbetween. For example, in the limit of making your reciprocal space top-hat filter very wide indeed, your sinc function in the real-space interpolation will be delta function-like and will give you a reconstructed continuous function that will look almost like your sequence of sampled values. So I think that even if your function is band-limited and is sampled at a rate greater than twice the Nyquist frequency, there are still an infinite number of functions that can be derived from the samples and that will pass through them. Am I wrong? Joe *The transforms of these continuous functions will have local translational symmetry in reciprocal space that is derived from the periodicity of the transform of the original unfiltered samples. If you now sample these functions at the same positions as with the original function, their transform will be identical to the transform of the original samples (because the periodicity imposed by the sampling will be in register with the translational symmetry mentioned above). So the values obtained from sampling functions derived from the different interpolation schemes must be identical to the original set of samples. That's all it is - If you assume that all the Fourier coefficients of higher resolution than twice your sampling rate are zero you are guaranteed one, and only one, set of Fourier coefficients that hit the points and the Discrete Fourier Transform (probably via a FFT) will calculate that set for you. As usual, if your assumption is wrong you will not get the right answer. If you have a function that really has a non-zero 1.67 A/cycle Fourier coefficient but you sample your function at 10 points and calculate a FFT you will get a set of coefficients that hit the 10 points exactly (when back transformed) but they will not be equal to true values. The overlapping spheres that Gerard Bricogne described are simply the way of calculating the manor in which the coefficients are distorted by this bad assumption. Ten Eyck, L. F. (1977). Acta Cryst. A33, 486-492 has an excellent description. If you are certain that your function has no Fourier components higher than your sampling rate can support then the FFT is your friend. If your function has high resolution components and you don't sample it finely enough then the FFT will give you an answer, but it won't be the correct answer. The answer will exactly fit the points you sampled but it will not correctly predict the function's behavior between the points. The principal situations where this is a problem are: Calculating structure factors (Fcalc) from a model electron density map. Calculating gradients using the Agarwal method. Phase extension via ncs map
Re: [ccp4bb] OT: Covalent modification of Cys by reducing agents?
Dear Mike BME readily autooxidizes (need for metal traces and dissolved O2). Is yours a metalloprotein? Is your buffer contaminated with metals? Those situations would make the case a bit different. If not, unless your BME stock is already oxidized, blocking of the accesible thiols with BME should take some time. If you treat your protein for 40 min with fresh BME you should not observe thiol blocking. If you let the preparation to stay for several days, even at 4-6 °C you may observe the blocking that you may be observing. If you want to prevent Cys blocking you can also change to DTT (it is a dithiol, does not readily form mixed disulfides) and use it with caution (for thiol reduction it is advisable to use stoichiometric DTT (with respect to the number of Cys you need to reduce) and 10 fold excess of BME, look for their redox potentials). Take care of not over-reducing your protein if internal disulfide bonds are expected. Once reduced I suggest you to remove any reducing agent and store the protein at -80 °C. External Cys can be easily oxidized, they are highly expossed to metals and oxidants (H2O2, BME disulfides, etc). Diffusion is for sure much faster than SS bond formation, although some cys react at almost diffusion-controlled rates with oxidants (is yours a thiol'dependen t peroxidase?) You can take a look at the following reference (advertising): 2011. Factors Affecting Protein Thiol Reactivity and Specificity in Peroxide Reduction. Chem Res Toxicol. Metals can contaminate bad quality materials (water, salts, buffers, etc), take care of that too. If you need to control the redox state of your protein you should use DTNB (Ellman´s reagent), or DTDPy, to measure accesible reduced thiol groups. Good luck! Horacio Quoting Kendall Nettles knett...@scripps.edu: 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 mi...@chem.ucla.edu 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?
Dear Horacio, How does TECEP compare to BME or DTT? People claim it is better, but I want some crystallographers' opinion? Nian On Sat, Apr 16, 2011 at 4:24 PM, Horacio Botti hbo...@pasteur.edu.uywrote: Dear Mike BME readily autooxidizes (need for metal traces and dissolved O2). Is yours a metalloprotein? Is your buffer contaminated with metals? Those situations would make the case a bit different. If not, unless your BME stock is already oxidized, blocking of the accesible thiols with BME should take some time. If you treat your protein for 40 min with fresh BME you should not observe thiol blocking. If you let the preparation to stay for several days, even at 4-6 °C you may observe the blocking that you may be observing. If you want to prevent Cys blocking you can also change to DTT (it is a dithiol, does not readily form mixed disulfides) and use it with caution (for thiol reduction it is advisable to use stoichiometric DTT (with respect to the number of Cys you need to reduce) and 10 fold excess of BME, look for their redox potentials). Take care of not over-reducing your protein if internal disulfide bonds are expected. Once reduced I suggest you to remove any reducing agent and store the protein at -80 °C. External Cys can be easily oxidized, they are highly expossed to metals and oxidants (H2O2, BME disulfides, etc). Diffusion is for sure much faster than SS bond formation, although some cys react at almost diffusion-controlled rates with oxidants (is yours a thiol'dependen t peroxidase?) You can take a look at the following reference (advertising): 2011. Factors Affecting Protein Thiol Reactivity and Specificity in Peroxide Reduction. Chem Res Toxicol. Metals can contaminate bad quality materials (water, salts, buffers, etc), take care of that too. If you need to control the redox state of your protein you should use DTNB (Ellman´s reagent), or DTDPy, to measure accesible reduced thiol groups. Good luck! Horacio Quoting Kendall Nettles knett...@scripps.edu: 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 mi...@chem.ucla.edu 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
