Error propagation on the Euclidean motion group arises in a number of areas such as in dead reckoning errors in mobile robot navigation and joint errors that accumulate from the base to the distal end of kinematic chains such as manipulators and biological macromolecules. We address error propagation in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE(3), can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors that are small (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second-order, whereas prior efforts only considered the first-order case. Our formulation is nonparametric in the sense that it will work for probability density functions of any form (not only Gaussians). Numerical tests demonstrate the accuracy of this second-order theory in the context of a manipulator arm and a flexible needle with bevel tip.
Recursive error propagation; Euclidean group; spatial uncertainty
A fast and easy tool to locate unit-cell matches in the PDB is described.
When embarking upon X-ray diffraction data collection from a potentially novel macromolecular crystal form, it can be useful to ascertain whether the measured data reflect a crystal form that is already recorded in the Protein Data Bank and, if so, whether it is part of a large family of related structures. Providing such information to crystallographers conveniently and quickly, as soon as the first images have been recorded and the unit cell characterized at an X-ray beamline, has the potential to save time and effort as well as pointing to possible search models for molecular replacement. Given an input unit cell, and optionally a space group, Nearest-cell rapidly scans the Protein Data Bank and retrieves near-matches.
Nearest-cell; crystal matches; Protein Data Bank
The purification, crystallization and preliminary X-ray crystallographic analyses of the AAV9 viral capsid are reported.
Adeno-associated virus (AAV) serotype 9, which is under development for gene-delivery applications, shows significantly enhanced capsid-associated transduction efficiency in muscle compared with other AAV serotypes. With the aim of characterizing the structural determinants of this property, the purification, crystallization and preliminary X-ray crystallographic analyses of the AAV9 viral capsid are reported. The crystals diffracted X-rays to 2.8 Å resolution using synchrotron radiation and belonged to the trigonal space group P32, with unit-cell parameters a = b = 251.0, c = 640.0 Å. There are three complete viral capsids in the crystal unit cell. The orientation and position of the asymmetric unit capsid have been determined by molecular-replacement methods and structure determination is in progress.
adeno-associate virus serotype 9; viruses
Crystals of an E. coli AcrB contaminant were grown from 95% pure CorA preparations. This very frequently occurring problem in membrane-protein crystallography laboratories is reported, as well as suggestions to avoid it.
In the course of a crystallographic study of the Methanosarcina mazei CorA transporter, the membrane protein was obtained with at least 95% purity and was submitted to crystallization trials. Small crystals (<100 µm) were grown that diffracted to 3.42 Å resolution and belonged to space group R32, with unit-cell parameters a = b = 145.74, c = 514.0 Å. After molecular-replacement attempts using available CorA structures as search models failed to yield a solution, it was discovered that the crystals consisted of an Escherichia coli contaminating protein, acriflavine resistance protein B (AcrB), that was present at less than 5% in the protein preparations. AcrB contamination is a major problem when expressing membrane proteins in E. coli since it binds naturally to immobilized metal-ion affinity chromatography (IMAC) resins. Here, the structure is compared with previously deposited AcrB structures and strategies are proposed to avoid this contamination.
membrane protein crystallization; contaminants; AcrB
Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups are unimodular. And even in noncommutative cases unimodular Lie groups share many of the useful features of Euclidean space. The rotation and Euclidean motion groups, which are perhaps the most relevant Lie groups to problems in geometric mechanics, are unimodular, as are the unitary groups that play important roles in quantum computing. The extension of core information theoretic inequalities defined in the setting of Euclidean space to this broad class of Lie groups is potentially relevant to a number of problems relating to information gathering in mobile robotics, satellite attitude control, tomographic image reconstruction, biomolecular structure determination, and quantum information theory. In this paper, several definitions are extended from the Euclidean setting to that of Lie groups (including entropy and the Fisher information matrix), and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems. In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed. An example from the field of robotics demonstrates how several of these results can be applied to quantify the amount of information gained by pooling different sensory inputs.
Information Theory; Lie Group; Convolution; Inequalities
The catalytic domain of an alkaline mannanase from the alkaliphilic Bacillus sp. N16-5 was expressed in E. coli and purified. Crystallization and preliminarily X-ray crystallographic analysis were performed for the recombinant enzyme.
The catalytic domain of an alkaline β-mannanase from the alkaliphilic Bacillus sp. N16-5 has been expressed and purified. The recombinant enzyme was crystallized using the hanging-drop vapour-diffusion method at 298 K. X-ray diffraction data were collected to 1.6 Å resolution. The crystal belonged to the orthorhombic space group P212121, with unit-cell parameters a = 59.03, b = 63.31, c = 83.34 Å. Initial phasing was carried out by molecular replacement using the three-dimensional structure of a mannanase from the alkaliphilic Bacillus sp. JAMB602 as a search model.
β-mannanases; Bacillus sp. N16-5; alkaliphiles
Motivation: The Backrub is a small but kinematically efficient side-chain-coupled local backbone motion frequently observed in atomic-resolution crystal structures of proteins. A backrub shifts the Cα–Cβ orientation of a given side-chain by rigid-body dipeptide rotation plus smaller individual rotations of the two peptides, with virtually no change in the rest of the protein. Backrubs can therefore provide a biophysically realistic model of local backbone flexibility for structure-based protein design. Previously, however, backrub motions were applied via manual interactive model-building, so their incorporation into a protein design algorithm (a simultaneous search over mutation and backbone/side-chain conformation space) was infeasible.
Results: We present a combinatorial search algorithm for protein design that incorporates an automated procedure for local backbone flexibility via backrub motions. We further derive a dead-end elimination (DEE)-based criterion for pruning candidate rotamers that, in contrast to previous DEE algorithms, is provably accurate with backrub motions. Our backrub-based algorithm successfully predicts alternate side-chain conformations from ≤0.9 Å resolution structures, confirming the suitability of the automated backrub procedure. Finally, the application of our algorithm to redesign two different proteins is shown to identify a large number of lower-energy conformations and mutation sequences that would have been ignored by a rigid-backbone model.
Availability: Contact authors for source code.
A case of imperfect pseudo-merohedral twinning in monoclinic crystals of fungal fatty acid synthase is discussed. A space-group transition during crystal dehydration resulted in a Moiré pattern-like interference of the twinned diffraction patterns.
The recent high-resolution structures of fungal fatty acid synthase (FAS) have provided new insights into the principles of fatty acid biosynthesis by large multifunctional enzymes. The crystallographic phase problem for the 2.6 MDa fungal FAS was initially solved to 5 Å resolution using two crystal forms from Thermomyces lanuginosus. Monoclinic crystals in space group P21 were obtained from orthorhombic crystals in space group P212121 by dehydration. Here, it is shown how this space-group transition induced imperfect pseudo-merohedral twinning in the monoclinic crystal, giving rise to a Moiré pattern-like interference of the two twin-related reciprocal lattices. The strategy for processing the twinned diffraction images and obtaining a quantitative analysis is presented. The twinning is also related to the packing of the molecules in the two crystal forms, which was derived from self-rotation function analysis and molecular-replacement solutions using a low-resolution electron microscopy map as a search model.
imperfect pseudo-merohedral twinning; fungal fatty acid synthase
Site-directed mutagenesis has been applied to improve the overexpression and purification of the icosahedral enzyme lumazine synthase from B. subtilis as well as to produce a new crystal form. The mutant protein crystallizes in space group R3 and diffracts X-rays to 1.6 Å resolution.
Icosahedral macromolecules have a wide spectrum of potential nanotechnological applications, the success of which relies on the level of accuracy at which the molecular structure is known. Lumazine synthase from Bacillus subtilis forms a 150 Å icosahedral capsid consisting of 60 subunits and crystallizes in space group P6322 or C2. However, the quality of these crystals is poor and structural information is only available at 2.4 Å resolution. As classical strategies for growing better diffracting crystals have so far failed, protein engineering has been employed in order to improve the overexpression and purification of the molecule as well as to obtain new crystal forms. Two cysteines were replaced to bypass misfolding problems and a charged surface residue was replaced to force different molecular packings. The mutant protein crystallizes in space group R3, with unit-cell parameters a = b = 313.02, c = 365.77 Å, α = β = 90.0, γ = 120°, and diffracts to 1.6 Å resolution.
lumazine synthase; icosahedral capsid; site-directed mutagenesis; crystal quality
Fluctuations in the bending angles at internal irregularities of DNA and RNA (such as symmetric loops, bulges, and nicks/gaps) have been observed from various experiments. However, little effort has been made to computationally predict and explain the statistical behavior of semi-flexible chains with internal defects. In this paper, we describe the general structure of these macromolecular chains as inextensible elastic chains with one or more internal joints which have limited ranges of rotation, and propose a method to compute the probability density functions of the end-to-end pose of these macromolecular chains. Our method takes advantage of the operational properties of the non-commutative Fourier transform for the group of rigid-body motions in three-dimensional space, SE(3). Two representative types of joints, the hinge for planar rotation and the ball joint for spatial rotation, are discussed in detail. The proposed method applies to various stiffness models of semi-flexible chain-like macromolecules. Examples are calculated using the Kratky-Porod model with specified stiffness, angular fluctuation, and joint locations. Entropic effects associated with internal angular fluctuations of semi-flexible macromolecular chains with internal joints can be computed using this formulation. Our method also provides a potential tool to detect the existence of internal irregularities.
The production, purification, crystallization and crystallographic analysis of H-1 Parvovirus, a gene-therapy vector, are reported.
Crystals of H-1 Parvovirus (H-1PV), an antitumor gene-delivery vector, were obtained for DNA-containing capsids and diffracted X-rays to 2.7 Å resolution using synchrotron radiation. The crystals belonged to the monoclinic space group P21, with unit-cell parameters a = 255.4, b = 350.4, c = 271.6 Å, β = 90.34°. The unit cell contained two capsids, with one capsid per crystallographic asymmetric unit. The H-1PV structure has been determined by molecular replacement and is currently being refined.
H-1 Parvovirus; viruses; antitumor gene delivery
A galactose specific lectin was purified from the seeds of a tropical plant, Spatholobus parviflorus. Its X-ray crystallographic structure was solved by the molecular replacement method.
A galactose-specific seed lectin was purified from the legume Spatholobus parviflorus and crystallized using the hanging-drop vapour-diffusion technique. The crystals belonged to space group P1, with unit-cell parameters a = 60.998, b = 60.792, c = 78.179 Å, α = 101.32, β = 91.38, γ = 104.32°. X-ray diffraction data were collected under cryoconditions (100 K) to a resolution of 2.04 Å using a MAR image-plate detector system mounted on a rotating-anode X-ray (Cu Kα) generator. Molecular replacement using legume-lectin coordinates as a search model gave a tetrameric structure.
galactose-specific lectins; Spatholobus parviflorus; seed lectins
X-ray diffraction data from the targeting (FAT) domain of focal adhesion kinase (FAK) were collected from a single crystal that diffracted to 1.99 Å resolution.
X-ray diffraction data from the targeting (FAT) domain of focal adhesion kinase (FAK) were collected from a single crystal that diffracted to 1.99 Å resolution and reduced to the primitive orthorhombic lattice. A single molecule was predicted to be present in the asymmetric unit based on the Matthews coefficient. The data were phased using molecular-replacement methods using an existing model of the FAK FAT domain. All structures of human focal adhesion kinase FAT domains solved to date have been solved in a C-centered orthorhombic space group.
focal adhesion kinase; targeting domain
A β-glucosidase A (BglA) from the thermophile Halothermothrix orenii has been cloned, purified and crystallized in an orthorhombic space group. X-ray diffraction data have been collected to 3.5 Å resolution, and the structure was solved by molecular replacement, revealing the presence of two molecules in the asymmetric unit.
The β-glucosidase A gene (bglA) has been cloned from the halothermophilic bacterium Halothermothrix orenii and the recombinant enzyme (BglA; EC 220.127.116.11) was bacterially expressed, purified using metal ion-affinity chromatography and subsequently crystallized. Orthorhombic crystals were obtained that diffracted to a resolution limit of 3.5 Å. The crystal structure with two molecules in the asymmetric unit was solved by molecular replacement using a library of known glucosidase structures. Attempts to collect higher resolution diffraction data from crystals grown under different conditions and structure refinement are currently in progress.
β-glucosidases; halothermophiles; Halothermothrix orenii; BglA
Virtually all docking methods include some local continuous minimization of an energy/scoring function in order to remove steric clashes and obtain more reliable energy values. In this paper, we describe an efficient rigid-body optimization algorithm that, compared to the most widely used algorithms, converges approximately an order of magnitude faster to conformations with equal or slightly lower energy. The space of rigid body transformations is a nonlinear manifold, namely, a space which locally resembles a Euclidean space. We use a canonical parametrization of the manifold, called the exponential parametrization, to map the Euclidean tangent space of the manifold onto the manifold itself. Thus, we locally transform the rigid body optimization to an optimization over a Euclidean space where basic optimization algorithms are applicable. Compared to commonly used methods, this formulation substantially reduces the dimension of the search space. As a result, it requires far fewer costly function and gradient evaluations and leads to a more efficient algorithm. We have selected the LBFGS quasi-Newton method for local optimization since it uses only gradient information to obtain second order information about the energy function and avoids the far more costly direct Hessian evaluations. Two applications, one in protein-protein docking, and the other in protein-small molecular interactions, as part of macromolecular docking protocols are presented. The code is available to the community under open source license, and with minimal effort can be incorporated into any molecular modeling package.
Crystals of human mitochondrial tyrosyl-tRNA synthetase lacking the C-terminal S4-like domain diffract to 2.7 Å resolution and are suitable for structure determination.
Human mitochondrial tyrosyl-tRNA synthetase and a truncated version with its C-terminal S4-like domain deleted were purified and crystallized. Only the truncated version, which is active in tyrosine activation and Escherichia coli tRNATyr charging, yielded crystals suitable for structure determination. These tetragonal crystals, belonging to space group P43212, were obtained in the presence of PEG 4000 as a crystallizing agent and diffracted X-rays to 2.7 Å resolution. Complete data sets could be collected and led to structure solution by molecular replacement.
tyrosyl-tRNA synthetase; aminoacyl-tRNA synthetases
A truncated mutant missing the first 62 residues of the N-terminal, cytoplasmic domain of the sodium-bicarbonate NBCe1-A cotransporter crystallizes in space group P31 with pseudo-P3121 symmetry and a hemihedral twin fraction of 33.0%. Twinned fractions and twin-pair statistics over binned resolutions confirm that the calculated twin fraction is associated with hemihedral twinning and not to non-crystallographic symmetry.
NBCe1-A membrane-embedded macromolecules that cotransport sodium and bicarbonate ions across the bilayer serve to maintain acid–base homeostasis throughout the body. Defects result in a number of renal and eye disorders, including type-II renal tubular acidosis and cataracts. Here, crystals of a human truncated mutant of the cytoplasmic N-terminal domain of NBCe1 (Δ1–62NtNBCe1-A) are reported that diffract X-rays to 2.4 Å resolution. The crystal symmetry of Δ1–62NtNBCe1-A is of space group P31 with pseudo-P3121 symmetry and it has a hemihedral twin fraction of 33.0%. The crystals may provide insight into the pathogenic processes observed in a subset of patients with truncating and point mutations in the gene encoding NBCe1.
NBCe1; bicarbonate transport
This work demonstrates how the digestion of a surface-exposed loop of B. fragilis type III glutamine synthetase results in crystal-packing rearrangements and explains how these changes facilitated the first structure solution.
This work details the intentional modifications that led to the first structure of a type III glutamine synthetase enzyme (GSIII). This approach followed the serendipitous discovery of digestion caused by an extracellular protease from a contaminating bacterium, Pseudomonas fluorescens. The protease only cleaves the GSIII protein at a single site, leaving the oligomer intact but allowing the protein to crystallize in a different space group. This transition from space group P1 to space group C2221 is accompanied by improved growth characteristics, more reproducible diffraction and enhanced mechanical stability. The crystallographic analyses presented here provide the structural basis of the altered molecular packing in the full-length and digested crystal forms and suggest modifications for future structural studies.
glutamine synthetase III; Bacteroides fragilis
A glyoxalase II from L. infantum was cloned, purified and crystallized and its structure was solved by X-ray crystallography.
In trypanosomatids, trypanothione replaces glutathione in all glutathione-dependent processes. Of the two enzymes involved in the glyoxalase pathway, glyoxalase I and glyoxalase II, the latter shows absolute specificity towards trypanothione thioester, making this enzyme an excellent model to understand the molecular basis of trypanothione binding. Cloned glyoxalase II from Leishmania infantum was overexpressed in Escherichia coli, purified and crystallized. Crystals belong to space group C2221 (unit-cell parameters a = 65.6, b = 88.3, c = 85.2 Å) and diffract beyond 2.15 Å using synchrotron radiation. The structure was solved by molecular replacement using the human glyoxalase II structure as a search model. These results, together with future detailed kinetic characterization using lactoyltrypanothione, should shed light on the evolutionary selection of trypanothione instead of glutathione by trypanosomatids.
Leishmania infantum; glyoxalase II; trypanothione
The ubiquitin-like domain of human hHR23A protein was crystallized by the hanging-drop vapour-diffusion method in space group P6522 and diffraction data were collected to 1.97 Å resolution. Structure solution by molecular replacement is described.
Human homologue A of Rad23 (hHR23A) plays dual roles in DNA repair as well as serving as a shuttle vehicle targeting polyubiquitinated proteins for degradation. Its N-terminal ubiquitin-like (UbL) domain interacts with the 19S proteasomal cap and provides the docking mechanism for protein delivery. Pyramidal crystals of the UbL domain of hHR23A were obtained by the hanging-drop vapour-diffusion method with ammonium sulfate as the crystallizing agent. The crystals diffracted to beyond 2 Å resolution and belonged to the hexagonal space group P6522, with unit-cell parameters a = b = 78.48, c = 63.57 Å. The structure was solved by molecular replacement using the UbL domain of yeast Dsk2 as the search model.
hHR23A; Rad23; ubiquitin-like domains; ubiquitin proteasome system; DNA excision repair
VSP1 from Arabidopsis thaliana was expressed in E. coli, purified and crystallized. X-ray diffraction data were collected to 1.9 Å resolution.
VSP1 is a defence protein in Arabidopsis thaliana that may also be involved in control of plant development. The recombinant protein has been overexpressed in Escherichia coli, purified and crystallized using the sitting-drop vapour-diffusion method. The crystal diffracted to 1.9 Å resolution and a complete X-ray data set was collected at 100 K using Cu Kα radiation from a rotating-anode X-ray source. The crystals belonged to space group C2. As there are no related structures that could be used as a search model for molecular replacement, work is in progress on experimental phasing using heavy-atom derivatives and selenomethionine derivatives.
VSP1; Arabidopsis thaliana; defence proteins
Crystals of the C-terminal 10 kDa lid subdomain from the C. elegans chaperone Hsp70 have been obtained that diffract X-rays to ∼3.5 Å and belong to space group I212121. Analysis of X-ray data and initial heavy-atom phasing reveals 24 monomers in the asymmetric unit related by 432 non-crystallographic symmetry.
Hsp70 is an important molecular chaperone involved in the regulation of protein folding. Crystals of the C-terminal 10 kDa helical lid domain (residues 542–640) from a Caenorhabditis elegans Hsp70 homologue have been produced that diffract X-rays to ∼3.4 Å. Crystals belong to space group I212121, with unit-cell parameters a = b = 197, c = 200 Å. The Matthews coefficient, self-rotation function and Patterson map indicate 24 monomers in the asymmetric unit, showing non-crystallographic 432 symmetry. Molecular-replacement studies using the corresponding domain from rat, the only eukaryotic homologue with a known structure, failed and a mercury derivative was obtained. Preliminary MAD phasing using SHELXD and SHARP for location and refinement of the heavy-atom substructure and SOLOMON for density modification produced interpretable maps with a clear protein–solvent boundary. Further density-modification, model-building and refinement are currently under way.
Hsp70; chaperone; C. elegans
The FAF1 UBX domain was crystallized. X-ray diffraction data were collected to 3.00 Å resolution and the crystals belonged to space group F4132.
Fas-associated factor 1 (FAF1) is a multifunctional pro-apoptotic protein that is involved in Fas-mediated apoptosis, NF-κB signalling and the ubiquitin–proteasome pathway. In the ubiquitin–proteasome pathway, FAF1 binds to the N domain of p97/VCP, a molecular chaperone that acts in complex with the proteasome, through its C-terminal UBX domain and inhibits the proteasomal protein-degradation process. In an effort to elucidate the structural basis of the function of FAF1 in modulating p97/VCP activity related to proteasomal protein degradation, crystallographic analysis of the FAF1 UBX domain and the p97/VCP N domain was initiated. Following the recently reported crystallization of the FAF1 UBX domain bound to the p97/VCP N domain, the unbound FAF1 UBX domain was also crystallized for purposes of structural comparison. X-ray data were collected to 3.00 Å resolution and the crystals belonged to space group F4132, with unit-cell parameters a = b = c = 176.40 Å. The Matthews coefficient and solvent content were estimated to be 3.04 Å3 Da−1 and 59.5%, respectively, assuming that the asymmetric unit contained two molecules of the UBX domain, which was subsequently confirmed by molecular-replacement calculations.
Fas-associated factor 1; FAF1; UBX domain
T. maritima mannitol dehydrogenase has been crystallized in space group P212121 with a = 84.43, b = 120.61, c = 145.76 Å. The crystals diffracted to 3.3 Å resolution at the Canadian Light Source.
Diffraction data have been collected from a crystal of Thermotoga maritima mannitol dehydrogenase at the Canadian Light Source. The crystal diffracted to 3.3 Å resolution and belongs to space group P212121, with unit-cell parameters a = 83.43, b = 120.61, c = 145.76 Å. The structure is likely to be solved by molecular replacement.
mannitol dehydrogenase; Thermotoga maritima
Conditions are reported for the crystallization of the flexible nuclear import receptor importin-β. Preliminary X-ray diffraction data indicate that the structure can be solved using molecular replacement.
The transport of macromolecules across the nuclear envelope is an essential eukaryotic process that enables proteins such as transcription factors, polymerases and histones to gain access to the genetic material contained within the nucleus. Importin-β plays a central role in the nucleocytoplasmic transport process, mediating nuclear import through a range of interactions with cytoplasmic, nuclear and nuclear pore proteins such as importin-α, Ran, nucleoporins and various cargo molecules. The unliganded form of the full-length yeast importin-β has been expressed and crystallized. The crystals were obtained by vapour diffusion at pH 6.5 and 290 K. The crystals belonged to space group P21 (unit-cell parameters a = 58.17, b = 127.25, c = 68.52 Å, β = 102.23). One molecule is expected in the asymmetric unit. The crystals diffracted to 2.4 Å resolution using a laboratory X-ray source and were suitable for crystal structure determination.
nucleocytoplasmic transport; importins; karyopherins