Legendre’s equation arises when one tries to solve Laplace’s equation in spherical coordinates, m... more Legendre’s equation arises when one tries to solve Laplace’s equation in spherical coordinates, much the same way in which Bessel’s equation arises when Laplace’s equation is solved using cylindrical coordinates. In this lecture we will introduce Legendre’s equation and provide solutions physically meaningful in form of converging series. We will delay the full treatment of Laplace’s equation in spherical coordinates to the end of the lecture, once the tools needed to solve it have been thoroughly introduced. 1 Power series solution of Legendre’s equation Legendre’s equation is one of the important equations in mathematical physics. It is usually written in the following form (1 − x)f ′′(x) − 2xf ′(x) + αf(x) = 0 (1) where α is a real constant. Let us set ourselves to solve equation (1) using a power series expansion in the neighborhood of x = 0, which is a regular point for the equation. We start by postulating the following form for f(x), f(x) = +∞ ∑
This is a very cursory presentation of anomalous scattering in crystallography. A very accessible... more This is a very cursory presentation of anomalous scattering in crystallography. A very accessible but more in-depth introduction to the same subject can be found on the web site maintained by Ethan Merritt at the Biomolecular Structure Centre, University of Washington [1]. Atoms scatter anomalously when the energy, and thus the wavelength, used is close to their resonant values. This means that in general crystals do not scatter anomalously or, better, that the magnitude of anomalous diffraction is negligible when compared to ordinary scattering. At specific wavelengths, though, certain atoms in the crystal resonate and the resulting diffracted beam includes significant anomalous contributions. Their particular nature is exploited for phasing, i.e. to find phases for the scattering factors. Ordinary diffraction is mathematically reflected in the usual factor appearing in the following equation,
It has been said over and over again that Thermodynamics is not an easy subject to learn and unde... more It has been said over and over again that Thermodynamics is not an easy subject to learn and understand. Some students think the mathematics level required to study it is too high for them. This is probably just partly true, as much of the subject requires only derivatives (partial derivatives too) and integrals. What makes Thermodynamics not terribly intuitive is its non-visualizability. This means that to many thermodynamic variables and concepts it is not always easy to associate intuitive and pictorial notions. Speed, force and angular momentum in Mechanics, for instance, are easily imagined in terms of bodies moving under some form of push or pull, and rotating or spinnning. Or consider how, in Electromagnetism, a field is made real by the arrangement of iron filings on a piece of paper held on a natural magnet. But what can we imagine when somebody talks about the entropy of a gas; or, what exactly is Gibbs free energy? The famous italian physicist Enrico Fermi, in his book on...
Acta Crystallographica Section D Structural Biology, 2020
In this article, a new approach to experimental phasing for macromolecular crystallography (MX) a... more In this article, a new approach to experimental phasing for macromolecular crystallography (MX) at synchrotrons is introduced and described for the first time. It makes use of automated robotics applied to a multi-crystal framework in which human intervention is reduced to a minimum. Hundreds of samples are automatically soaked in heavy-atom solutions, using a Labcyte Inc. Echo 550 Liquid Handler, in a highly controlled and optimized fashion in order to generate derivatized and isomorphous crystals. Partial data sets obtained on MX beamlines using an in situ setup for data collection are processed with the aim of producing good-quality anomalous signal leading to successful experimental phasing.
The most important quantitative aspects of computational structural crystallography can be introd... more The most important quantitative aspects of computational structural crystallography can be introduced in a satisfactory way using 1D truncated and periodic Gaussian functions to represent the atoms in a crystal lattice. This paper describes in detail and demonstrates 1D structural crystallography starting with the definition of such truncated Gaussians. The availability of the computer programme CRONE makes possible the repetition of the examples provided in the paper as well as the creation of new ones.
Integral membrane proteins are among the most fascinating and important biomolecules as they play... more Integral membrane proteins are among the most fascinating and important biomolecules as they play a vital role in many biological functions. Knowledge of their atomic structures is fundamental to the understanding of their biochemical function and key in many drug discovery programs. However, over the years, structure determination of integral membrane proteins has proven to be far from trivial, hence they are underrepresented in the protein data bank. Low expression levels, insolubility and instability are just a few of the many hurdles one faces when studying these proteins. X-ray crystallography has been the most used method to determine atomic structures of membrane proteins. However, the production of high quality membrane protein crystals is always very challenging, often seen more as art than a rational experiment. Here we review valuable approaches, methods and techniques to successful membrane protein crystallisation.
The dynamism of proteins is central to their function, and several proteins have been described a... more The dynamism of proteins is central to their function, and several proteins have been described as flexible, as consisting of multiple domains joined by flexible linkers, and even as intrinsically disordered. Several techniques exist to study protein structures, but small angle X-ray scattering (SAXS) has proven to be particularly powerful for the quantitative analysis of such flexible systems. In the present report, we have used SAXS in combination with X-ray crystallography to highlight their usefulness at characterizing flexible proteins, using as examples two proteins involved in different steps of ribosome biogenesis. The yeast BRCA2 and CDKN1A-interactig protein, Bcp1, is a chaperone for Rpl23 of unknown structure. We showed that it consists of a rigid, slightly elongated protein, with a secondary structure comprising a mixture of alpha helices and beta sheets. As an example of a flexible molecule, we studied the SBDS (Shwachman-Bodian-Diamond Syndrome) protein that is involved in the cytoplasmic maturation of the 60S subunit and constitutes the mutated target in the Shwachman-Diamond Syndrome. In solution, this protein coexists in an ensemble of three main conformations, with the N-and C-terminal ends adopting different orientations with respect to the central domain. The structure observed in the protein crystal corresponds to an average of those predicted by the SAXS flexibility analysis.
The present article describes how to use the computer program BLEND to help assemble complete dat... more The present article describes how to use the computer program BLEND to help assemble complete datasets for the solution of macromolecular structures, starting from partial or complete datasets, derived from data collection from multiple crystals. The program is demonstrated on more than two hundred X-ray diffraction datasets obtained from 50 crystals of a complex formed between the SRF transcription factor, its cognate DNA, and a peptide from the SRF cofactor MRTF-A. This structure is currently in the process of being fully solved. While full details of the structure are not yet available, the repeated application of BLEND on data from this structure, as they have become available, has made it possible to produce electron density maps clear enough to visualise the potential location of MRTF sequences.
KAMO and Blend provide particularly effective tools to automatically manage the merging of large ... more KAMO and Blend provide particularly effective tools to automatically manage the merging of large numbers of data sets from serial crystallography. The requirement for manual intervention in the process can be reduced by extending Blend to support additional clustering options to increase the sensitivity to differences in unit cell parameters and to allow for clustering of nearly complete datasets on the basis of intensity or amplitude differences. If datasets are already sufficiently complete to permit it, apply KAMO once, just for reflections. If starting from incomplete datasets, one applies KAMO twice, first using cell parameters. In this step either the simple cell vector distance of the original Blend is used, or the more sensitive NCDist, to find clusters to merge to achieve sufficient completeness to allow intensities or amplitudes to be compared. One then uses KAMO again using the correlation between the reflections at the common HKLs to merge clusters in a way sensitive to ...
Advances in Experimental Medicine and Biology, 2016
X-ray diffraction from crystals of membrane proteins very often yields incomplete datasets due to... more X-ray diffraction from crystals of membrane proteins very often yields incomplete datasets due to, among other things, severe radiation damage. Multiple crystals are thus required to form complete datasets, provided the crystals themselves are isomorphous. Selection and combination of data from multiple crystals is a difficult and tedious task that can be facilitated by purpose-built software. BLEND, in the CCP4 suite of programs for macromolecular crystallography (MX), has been created exactly for this reason. In this chapter the program is described and its workings illustrated by means of data from two membrane proteins.
Acta Crystallographica Section A Foundations of Crystallography, 2011
Microsymposia C162 are primarily command-line driven, an emphasis has been placed on ease-of-use ... more Microsymposia C162 are primarily command-line driven, an emphasis has been placed on ease-of-use and automation. We have developed a graphical interface for the major components of PHENIX, which currently includes phenix. refine, phenix.xtriage, comprehensive validation tools based on the Molprobity web server, Phaser, and the AutoSol, AutoBuild, AutoMR, and LigandFit automation "wizards". The Python-based framework allows new GUIs to be generated semi-automatically while preserving all of the flexibility of the command-line programs, and supports both Macintosh and Linux. Python extensions for Coot and PyMOL facilitate real-time visualization of refinement and automated model-building, and convenient viewing of results. Transitions between separate modules within PHENIX are simplified or eliminated in the GUI, reducing the amount of manual input required and avoiding the use of command-line tools. Further automation is possible with definition of standard parameter sets and input files for individual programs. Future improvements will include greater use of multiprocessing and clusters, tools for handling multiple structures in parallel, and new automation pipelines.
Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 -... more Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 - 2007
Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 -... more Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 - 2007
Legendre’s equation arises when one tries to solve Laplace’s equation in spherical coordinates, m... more Legendre’s equation arises when one tries to solve Laplace’s equation in spherical coordinates, much the same way in which Bessel’s equation arises when Laplace’s equation is solved using cylindrical coordinates. In this lecture we will introduce Legendre’s equation and provide solutions physically meaningful in form of converging series. We will delay the full treatment of Laplace’s equation in spherical coordinates to the end of the lecture, once the tools needed to solve it have been thoroughly introduced. 1 Power series solution of Legendre’s equation Legendre’s equation is one of the important equations in mathematical physics. It is usually written in the following form (1 − x)f ′′(x) − 2xf ′(x) + αf(x) = 0 (1) where α is a real constant. Let us set ourselves to solve equation (1) using a power series expansion in the neighborhood of x = 0, which is a regular point for the equation. We start by postulating the following form for f(x), f(x) = +∞ ∑
This is a very cursory presentation of anomalous scattering in crystallography. A very accessible... more This is a very cursory presentation of anomalous scattering in crystallography. A very accessible but more in-depth introduction to the same subject can be found on the web site maintained by Ethan Merritt at the Biomolecular Structure Centre, University of Washington [1]. Atoms scatter anomalously when the energy, and thus the wavelength, used is close to their resonant values. This means that in general crystals do not scatter anomalously or, better, that the magnitude of anomalous diffraction is negligible when compared to ordinary scattering. At specific wavelengths, though, certain atoms in the crystal resonate and the resulting diffracted beam includes significant anomalous contributions. Their particular nature is exploited for phasing, i.e. to find phases for the scattering factors. Ordinary diffraction is mathematically reflected in the usual factor appearing in the following equation,
It has been said over and over again that Thermodynamics is not an easy subject to learn and unde... more It has been said over and over again that Thermodynamics is not an easy subject to learn and understand. Some students think the mathematics level required to study it is too high for them. This is probably just partly true, as much of the subject requires only derivatives (partial derivatives too) and integrals. What makes Thermodynamics not terribly intuitive is its non-visualizability. This means that to many thermodynamic variables and concepts it is not always easy to associate intuitive and pictorial notions. Speed, force and angular momentum in Mechanics, for instance, are easily imagined in terms of bodies moving under some form of push or pull, and rotating or spinnning. Or consider how, in Electromagnetism, a field is made real by the arrangement of iron filings on a piece of paper held on a natural magnet. But what can we imagine when somebody talks about the entropy of a gas; or, what exactly is Gibbs free energy? The famous italian physicist Enrico Fermi, in his book on...
Acta Crystallographica Section D Structural Biology, 2020
In this article, a new approach to experimental phasing for macromolecular crystallography (MX) a... more In this article, a new approach to experimental phasing for macromolecular crystallography (MX) at synchrotrons is introduced and described for the first time. It makes use of automated robotics applied to a multi-crystal framework in which human intervention is reduced to a minimum. Hundreds of samples are automatically soaked in heavy-atom solutions, using a Labcyte Inc. Echo 550 Liquid Handler, in a highly controlled and optimized fashion in order to generate derivatized and isomorphous crystals. Partial data sets obtained on MX beamlines using an in situ setup for data collection are processed with the aim of producing good-quality anomalous signal leading to successful experimental phasing.
The most important quantitative aspects of computational structural crystallography can be introd... more The most important quantitative aspects of computational structural crystallography can be introduced in a satisfactory way using 1D truncated and periodic Gaussian functions to represent the atoms in a crystal lattice. This paper describes in detail and demonstrates 1D structural crystallography starting with the definition of such truncated Gaussians. The availability of the computer programme CRONE makes possible the repetition of the examples provided in the paper as well as the creation of new ones.
Integral membrane proteins are among the most fascinating and important biomolecules as they play... more Integral membrane proteins are among the most fascinating and important biomolecules as they play a vital role in many biological functions. Knowledge of their atomic structures is fundamental to the understanding of their biochemical function and key in many drug discovery programs. However, over the years, structure determination of integral membrane proteins has proven to be far from trivial, hence they are underrepresented in the protein data bank. Low expression levels, insolubility and instability are just a few of the many hurdles one faces when studying these proteins. X-ray crystallography has been the most used method to determine atomic structures of membrane proteins. However, the production of high quality membrane protein crystals is always very challenging, often seen more as art than a rational experiment. Here we review valuable approaches, methods and techniques to successful membrane protein crystallisation.
The dynamism of proteins is central to their function, and several proteins have been described a... more The dynamism of proteins is central to their function, and several proteins have been described as flexible, as consisting of multiple domains joined by flexible linkers, and even as intrinsically disordered. Several techniques exist to study protein structures, but small angle X-ray scattering (SAXS) has proven to be particularly powerful for the quantitative analysis of such flexible systems. In the present report, we have used SAXS in combination with X-ray crystallography to highlight their usefulness at characterizing flexible proteins, using as examples two proteins involved in different steps of ribosome biogenesis. The yeast BRCA2 and CDKN1A-interactig protein, Bcp1, is a chaperone for Rpl23 of unknown structure. We showed that it consists of a rigid, slightly elongated protein, with a secondary structure comprising a mixture of alpha helices and beta sheets. As an example of a flexible molecule, we studied the SBDS (Shwachman-Bodian-Diamond Syndrome) protein that is involved in the cytoplasmic maturation of the 60S subunit and constitutes the mutated target in the Shwachman-Diamond Syndrome. In solution, this protein coexists in an ensemble of three main conformations, with the N-and C-terminal ends adopting different orientations with respect to the central domain. The structure observed in the protein crystal corresponds to an average of those predicted by the SAXS flexibility analysis.
The present article describes how to use the computer program BLEND to help assemble complete dat... more The present article describes how to use the computer program BLEND to help assemble complete datasets for the solution of macromolecular structures, starting from partial or complete datasets, derived from data collection from multiple crystals. The program is demonstrated on more than two hundred X-ray diffraction datasets obtained from 50 crystals of a complex formed between the SRF transcription factor, its cognate DNA, and a peptide from the SRF cofactor MRTF-A. This structure is currently in the process of being fully solved. While full details of the structure are not yet available, the repeated application of BLEND on data from this structure, as they have become available, has made it possible to produce electron density maps clear enough to visualise the potential location of MRTF sequences.
KAMO and Blend provide particularly effective tools to automatically manage the merging of large ... more KAMO and Blend provide particularly effective tools to automatically manage the merging of large numbers of data sets from serial crystallography. The requirement for manual intervention in the process can be reduced by extending Blend to support additional clustering options to increase the sensitivity to differences in unit cell parameters and to allow for clustering of nearly complete datasets on the basis of intensity or amplitude differences. If datasets are already sufficiently complete to permit it, apply KAMO once, just for reflections. If starting from incomplete datasets, one applies KAMO twice, first using cell parameters. In this step either the simple cell vector distance of the original Blend is used, or the more sensitive NCDist, to find clusters to merge to achieve sufficient completeness to allow intensities or amplitudes to be compared. One then uses KAMO again using the correlation between the reflections at the common HKLs to merge clusters in a way sensitive to ...
Advances in Experimental Medicine and Biology, 2016
X-ray diffraction from crystals of membrane proteins very often yields incomplete datasets due to... more X-ray diffraction from crystals of membrane proteins very often yields incomplete datasets due to, among other things, severe radiation damage. Multiple crystals are thus required to form complete datasets, provided the crystals themselves are isomorphous. Selection and combination of data from multiple crystals is a difficult and tedious task that can be facilitated by purpose-built software. BLEND, in the CCP4 suite of programs for macromolecular crystallography (MX), has been created exactly for this reason. In this chapter the program is described and its workings illustrated by means of data from two membrane proteins.
Acta Crystallographica Section A Foundations of Crystallography, 2011
Microsymposia C162 are primarily command-line driven, an emphasis has been placed on ease-of-use ... more Microsymposia C162 are primarily command-line driven, an emphasis has been placed on ease-of-use and automation. We have developed a graphical interface for the major components of PHENIX, which currently includes phenix. refine, phenix.xtriage, comprehensive validation tools based on the Molprobity web server, Phaser, and the AutoSol, AutoBuild, AutoMR, and LigandFit automation "wizards". The Python-based framework allows new GUIs to be generated semi-automatically while preserving all of the flexibility of the command-line programs, and supports both Macintosh and Linux. Python extensions for Coot and PyMOL facilitate real-time visualization of refinement and automated model-building, and convenient viewing of results. Transitions between separate modules within PHENIX are simplified or eliminated in the GUI, reducing the amount of manual input required and avoiding the use of command-line tools. Further automation is possible with definition of standard parameter sets and input files for individual programs. Future improvements will include greater use of multiprocessing and clusters, tools for handling multiple structures in parallel, and new automation pipelines.
Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 -... more Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 - 2007
Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 -... more Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 - 2007
Macromolecular crystallography (MX) is the most powerful technique available to structural biolog... more Macromolecular crystallography (MX) is the most powerful technique available to structural biologists to visualize in atomic detail the macromolecular machinery of the cell. Since the emergence of structural genomics initiatives, significant advances have been made in all key steps of the structure determination process. In particular, third-generation synchrotron sources and the application of highly automated approaches to data acquisition and analysis at these facilities have been the major factors in the rate of increase of macromolecular structures determined annually. A plethora of tools are now available to users of synchrotron beamlines to enable rapid and efficient evaluation of samples, collection of the best data, and in favorable cases structure solution in near real time. Here, we provide a short overview of the emerging use of collecting X-ray diffraction data directly from the crystallization experiment. These in situ experiments are now routinely available to users at a number of synchrotron MX beamlines. A practical guide to the use of the method on the MX suite of beamlines at Diamond Light Source is given.
Assembling data from multiple crystals with BLEND
In recent times, in macromolecular crystallogr... more Assembling data from multiple crystals with BLEND
In recent times, in macromolecular crystallography, it has become increasingly common at synchrotrons to obtain complete merged datasets using partial data wedges from multiple crystals. There are several reasons for the emergence of this recent trend, all pointing to the fact that completeness and redundancy from many crystals are, in general, superior to those coming from single crystals. When dealing with multiple crystals, though, users face two problems: crystals isomorphism and the large number of possible datasets combinations. Both problems can be addressed using cluster analysis. The program BLEND [1], developed at the Diamond Light Source synchrotron and available in CCP4 [2], makes use of cluster analysis and of the POINTLESS [3] and AIMLESS [4] programs to help users assembling useful and complete data from multiple crystals. BLEND acts by default in a semi-automated way, but it is most effective when used interactively in repeated sessions. BLEND has been employed successfully to assemble data for a variety of macromolecules, especially membrane proteins and protein complexes.
[1] Foadi J, Aller P, Alguel Y, Cameron A, Axford D (2013) Clustering procedures for the optimal selection of data sets from multiple crystals in macromolecular crystallography. Acta Crystallogr D Biol Crystallogr 69(8):1617-1632.
[2] Winn MD, Ballard CC, Cowtan KD, Dodson EJ, Emsley P et al (2011) Overview of the CCP4 suite and current developments. Acta Crystallogr D Biol Crystallogr 67(4): 235-242.
[3] Evans PR (2006) Scaling and assessment of data quality. Acta Crystallogr D Biol Crystallogr 62: 72-82.
[4] Evans PR.and Murshudov GN (2013) How good are my data and what is the resolution? Acta Crystallogr D Biol Crystallogr 69(7):1204-1214.
• What is x-ray crystallography? • Statistical applications in x-ray crystallography • Use of R i... more • What is x-ray crystallography? • Statistical applications in x-ray crystallography • Use of R in crystallographic research • Description of cRy Statistical applications in x-ray crystallography The probability distribution of x-ray intensities A. J. Wilson (1949) Traitment statistique des erreurs dans la determination des structures cristallines V. Luzzati (1952) Probability distribution connected with structure amplitudes of two related crystals G. N. Ramachandran, R. Srinivasan and R. Sarma (1963) On the application of phase relationships to complex structures. The additional use of statistical information in tangent-formula refinement. S. E. Hull and M. J. Irwin (1978) Maximum entropy and the foundations of direct methods G. Bricogne (1984)
I have prepared this material for a 2-days "intensive" course directed to scientists at the Diamo... more I have prepared this material for a 2-days "intensive" course directed to scientists at the Diamond Light Source Ltd, in Oxfordshire, in 2013. You are welcome to use the material for your own purpose (and, please, acknowledge my contribution in case it's useful to you).
We will restrict our description to two functions, x(t) and y(t). Things can be generalized quite... more We will restrict our description to two functions, x(t) and y(t). Things can be generalized quite straightforwardly. These type of systems are generally written in the following form:
In this document I have collected a few general results concerning homogeneous linear secondorder... more In this document I have collected a few general results concerning homogeneous linear secondorder differential equations, i.e. equations of form,
The solution of non-homogeneous equations is possible when a particular solution, y p , of the eq... more The solution of non-homogeneous equations is possible when a particular solution, y p , of the equation can be found. If y c is the general solution of the associated homogeneous equation, then we know that y = y c + y p is the general solution of the non-homogeneous equation. The method of variation of parameters is a powerful general method that can be used to find such a particular solution. To be more specific (and focus on second order linear equations) we would like to find a particular solution for,
Among ordinary differential equations of order greater than 1, the linear homogeneous equations w... more Among ordinary differential equations of order greater than 1, the linear homogeneous equations with constant coefficients are the most easily solvable. It is not worth to spend many words on their solution. This is a very straightforward task. First the equation is turned into an algebraic equation. Then the root of the equations are found and they become coefficients of x in the argument of an exponential. If a root has a certain multiplicity, the first independent solution is the exponential, but then a multiplication of the exponential by increasing powers of x is needed for the other independent solutions. A few examples will clarify the procedure.
Legendre's equation arises when one tries to solve Laplace's equation in spherical coordinates, m... more Legendre's equation arises when one tries to solve Laplace's equation in spherical coordinates, much the same way in which Bessel's equation arises when Laplace's equation is solved using cylindrical coordinates. In this lecture we will introduce Legendre's equation and provide solutions physically meaningful in form of converging series. We will delay the full treatment of Laplace's equation in spherical coordinates to the end of the lecture, once the tools needed to solve it have been thoroughly introduced.
Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found m... more Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems. Being able to change all variables and expression involved in a given problem, when a different coordinate system is chosen, is one of those skills a physicist, and even more a theoretical physicist, needs to possess. In this lecture a general method to express any variable and expression in an arbitrary curvilinear coordinate system will be introduced and explained. We will be mainly interested to find out general expressions for the gradient, the divergence and the curl of scalar and vector fields. Specific applications to the widely used cylindrical and spherical systems will conclude this lecture.
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Papers by James Foadi
In recent times, in macromolecular crystallography, it has become increasingly common at synchrotrons to obtain complete merged datasets using partial data wedges from multiple crystals. There are several reasons for the emergence of this recent trend, all pointing to the fact that completeness and redundancy from many crystals are, in general, superior to those coming from single crystals. When dealing with multiple crystals, though, users face two problems: crystals isomorphism and the large number of possible datasets combinations. Both problems can be addressed using cluster analysis. The program BLEND [1], developed at the Diamond Light Source synchrotron and available in CCP4 [2], makes use of cluster analysis and of the POINTLESS [3] and AIMLESS [4] programs to help users assembling useful and complete data from multiple crystals. BLEND acts by default in a semi-automated way, but it is most effective when used interactively in repeated sessions. BLEND has been employed successfully to assemble data for a variety of macromolecules, especially membrane proteins and protein complexes.
[1] Foadi J, Aller P, Alguel Y, Cameron A, Axford D (2013) Clustering procedures for the optimal selection of data sets from multiple crystals in macromolecular crystallography. Acta Crystallogr D Biol Crystallogr 69(8):1617-1632.
[2] Winn MD, Ballard CC, Cowtan KD, Dodson EJ, Emsley P et al (2011) Overview of the CCP4 suite and current developments. Acta Crystallogr D Biol Crystallogr 67(4): 235-242.
[3] Evans PR (2006) Scaling and assessment of data quality. Acta Crystallogr D Biol Crystallogr 62: 72-82.
[4] Evans PR.and Murshudov GN (2013) How good are my data and what is the resolution? Acta Crystallogr D Biol Crystallogr 69(7):1204-1214.