Mathematical aspects of molecular replacement. III. Properties of space groups preferred by proteins in the Protein Data Bank

2015 ◽  
Vol 71 (2) ◽  
pp. 186-194 ◽  
Author(s):  
G. Chirikjian ◽  
S. Sajjadi ◽  
D. Toptygin ◽  
Y. Yan
Keyword(s):  
Rigid Body ◽  
Protein Data Bank ◽  
Data Bank ◽  
Continuous Group ◽  
Molecular Replacement ◽  
Macromolecular Crystallography ◽  
Coset Space ◽  
Space Groups ◽  
The Third ◽  
Rigid Body Motions

The main goal of molecular replacement in macromolecular crystallography is to find the appropriate rigid-body transformations that situate identical copies of model proteins in the crystallographic unit cell. The search for such transformations can be thought of as taking place in the coset space Γ\Gwhere Γ is the Sohncke group of the macromolecular crystal andGis the continuous group of rigid-body motions in Euclidean space. This paper, the third in a series, is concerned with viewing nonsymmorphic Γ in a new way. These space groups, rather than symmorphic ones, are the most common ones for protein crystals. Moreover, their properties impact the structure of the space Γ\G. In particular, nonsymmorphic space groups contain both Bieberbach subgroups and symmorphic subgroups. A number of new theorems focusing on these subgroups are proven, and it is shown that these concepts are related to the preferences that proteins have for crystallizing in different space groups, as observed in the Protein Data Bank.

scholarly journals Quasi-Groups and the Configuration Space of Symmetry-Constrained Motions

2014 ◽  
Vol 70 (a1) ◽  
pp. C1421-C1421
Author(s):  
Gregory Chirikjian
Keyword(s):  
Rigid Body ◽  
Mathematical Formulation ◽  
Compact Subgroup ◽  
Continuous Group ◽  
Full Group ◽  
Coset Space ◽  
Space Group ◽  
Generalized Symmetry ◽  
Rigid Body Motions ◽  
Biomolecular Crystallography

In this work, the set of all possible positions and orientations of a large molecule within the crystallographic asymmetric unit is equated to the coset space of the continuous group of proper rigid-body motions modulo the chiral space group of the macromolecular crystal. Since every chiral space group is a co-compact subgroup in the full group of rigid-body motions (which is a six-dimensional Lie group), the resulting coset space is a compact 6D manifold. However, since none of the crystallographic groups are normal in the full group of rigid-body motions, this coset space is not a group. However, it can be endowed with an operation that satisfies all of the group axioms except for associativity, thereby giving it the structure of a quasi-group. The quasi-group properties of such spaces are explored in this work as an example of generalized symmetry. The mathematical formulation presented here, which builds on the author's prior work cited below, is relevant to both the Molecular Replacement (MR) method in biomolecular crystallography, and in the design of new engineered crystals.

scholarly journals Quantizing Euclidean Motions via Double-Coset Decomposition

Research ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Christian Wülker ◽  
Sipu Ruan ◽  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Search Algorithm ◽  
Double Coset ◽  
Continuous Group ◽  
Robot Motion ◽  
Space Groups ◽  
Sampling Schemes ◽  
The World ◽  
Rigid Body Motions ◽  
A Finite Group

Concepts from mathematical crystallography and group theory are used here to quantize the group of rigid-body motions, resulting in a “motion alphabet” with which robot motion primitives are expressed. From these primitives it is possible to develop a dictionary of physical actions. Equipped with an alphabet of the sort developed here, intelligent actions of robots in the world can be approximated with finite sequences of characters, thereby forming the foundation of a language in which robot motion is articulated. In particular, we use the discrete handedness-preserving symmetries of macromolecular crystals (known in mathematical crystallography as Sohncke space groups) to form a coarse discretization of the space SE(3) of rigid-body motions. This discretization is made finer by subdividing using the concept of double-coset decomposition. More specifically, a very efficient, equivolumetric quantization of spatial motion can be defined using the group-theoretic concept of a double-coset decomposition of the form Γ\SE(3)/Δ, where Γ is a Sohncke space group and Δ is a finite group of rotational symmetries such as those of the icosahedron. The resulting discrete alphabet is based on a very uniform sampling of SE(3) and is a tool for describing the continuous trajectories of robots and humans. An efficient coarse-to-fine search algorithm is presented to round off any motion sampled from the continuous group of motions to the nearest element of our alphabet. It is shown that our alphabet and this efficient rounding algorithm can be used as a geometric data structure to accelerate the performance of other sampling schemes designed for desirable dispersion or discrepancy properties. Moreover, the general “signals to symbols” problem in artificial intelligence is cast in this framework for robots moving continuously in the world.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces

2017 ◽  
Vol 73 (5) ◽  
pp. 387-402 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Sajdeh Sajjadi ◽  
Bernard Shiffman ◽  
Steven M. Zucker
Keyword(s):  
General Theory ◽  
Rigid Body ◽  
Three Dimensional ◽  
Molecular Replacement ◽  
Translation Group ◽  
Common Occurrence ◽  
Space Group ◽  
Space Groups ◽  
Relevant Case ◽  
The Subject

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

scholarly journals Crystallographic models of SARS-CoV-2 3CLpro: in-depth assessment of structure quality and validation

IUCrJ ◽  
2021 ◽  
Vol 8 (2) ◽  
pp. 238-256
Author(s):  
Mariusz Jaskolski ◽  
Zbigniew Dauter ◽  
Ivan G. Shabalin ◽  
Miroslaw Gilski ◽  
Dariusz Brzezinski ◽  
...  
Keyword(s):  
Rapid Determination ◽  
Data Bank ◽  
Molecular Replacement ◽  
Structure Quality ◽  
Space Groups ◽  
Important Target ◽  
Density Maps ◽  
Main Protease ◽  
Experimental Approaches

The appearance at the end of 2019 of the new SARS-CoV-2 coronavirus led to an unprecedented response by the structural biology community, resulting in the rapid determination of many hundreds of structures of proteins encoded by the virus. As part of an effort to analyze and, if necessary, remediate these structures as deposited in the Protein Data Bank (PDB), this work presents a detailed analysis of 81 crystal structures of the main protease 3CLpro, an important target for the design of drugs against COVID-19. The structures of the unliganded enzyme and its complexes with a number of inhibitors were determined by multiple research groups using different experimental approaches and conditions; the resulting structures span 13 different polymorphs representing seven space groups. The structures of the enzyme itself, all determined by molecular replacement, are highly similar, with the exception of one polymorph with a different inter-domain orientation. However, a number of complexes with bound inhibitors were found to pose significant problems. Some of these could be traced to faulty definitions of geometrical restraints for ligands and to the general problem of a lack of such information in the PDB depositions. Several problems with ligand definition in the PDB itself were also noted. In several cases extensive corrections to the models were necessary to adhere to the evidence of the electron-density maps. Taken together, this analysis of a large number of structures of a single, medically important protein, all determined within less than a year using modern experimental tools, should be useful in future studies of other systems of high interest to the biomedical community.

Kinematics Meets Crystallography: The Concept of a Motion Space

2014 ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Molecular Structures ◽  
New Drugs ◽  
Dimensional Structure ◽  
Biological Macromolecules ◽  
Molecular Replacement ◽  
X Ray Diffraction ◽  
X Ray ◽  
Rigid Body Motions ◽  
Diffraction Patterns

In this paper, it is shown how rigid-body kinematics can be used to assist in determining the atomic structure of proteins and nucleic acids when using x-ray crystallography, which is a powerful method for structure determination. The importance of determining molecular structures for understanding biological processes and for the design of new drugs is well known. Phasing is a necessary step in determining the three-dimensional structure of molecules from x-ray diffraction patterns. A computational approach called molecular replacement (MR) is a well-established method for phasing of x-ray diffraction patterns for crystals composed of biological macromolecules. In MR, a search is performed over positions and orientations of a known biomolecular structure within a model of the crystallographic asymmetric unit, or, equivalently, multiple symmetry-related molecules in the crystallographic unit cell. Unlike the discrete space groups known to crystallographers and the continuous rigid-body motions known to kinematicians, the set of motions over which molecular replacement searches are performed does not form a group. Rather, it is a coset space of the group of continuous rigid-body motions, SE(3), with respect to the crystallographic space group of the crystal, which is a discrete sub-group of SE(3). Properties of these ‘motion spaces’ (which are compact manifolds) are investigated here.

Instantaneous Properties of Trajectories Generated by Planar, Spherical, and Spatial Rigid Body Motions

1982 ◽  
Vol 104 (1) ◽  
pp. 39-50 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Roth
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Spatial Motion ◽  
Third Order ◽  
Local Shape ◽  
Motion Trajectories ◽  
The Third ◽  
Moving Body ◽  
Rigid Body Motions ◽  
Point Trajectories

This paper develops relationships between the instantaneous invariants of a motion and the local shape of the trajectories generated during the motion. We consider the point trajectories generated by planar and spherical motions and the line trajectories generated by spatial motion. Those points or lines which generate special trajectories are located on (and define) so-called boundary loci in the moving body. These boundary loci define regions, within the body, for which all the points or lines generate similarly shaped trajectories. The shapes of these boundaries depend directly upon the invariants of the motion. It is shown how to qualitatively determine the fundamental trajectory shapes, analyze the effect of the invariants on the boundary loci, and how to combine these results to visualize the motion trajectories to the third order.

scholarly journals New tools for the analysis and validation of cryo-EM maps and atomic models

2018 ◽  
Vol 74 (9) ◽  
pp. 814-840 ◽  
Author(s):  
Pavel V. Afonine ◽  
Bruno P. Klaholz ◽  
Nigel W. Moriarty ◽  
Billy K. Poon ◽  
Oleg V. Sobolev ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Protein Data Bank ◽  
Computational Methods ◽  
Data Bank ◽  
Macromolecular Crystallography ◽  
Electron Cryomicroscopy ◽  
New Methods ◽  
Atomic Models ◽  
Electron Microscopy Data ◽  
Microscopy Data

Recent advances in the field of electron cryomicroscopy (cryo-EM) have resulted in a rapidly increasing number of atomic models of biomacromolecules that have been solved using this technique and deposited in the Protein Data Bank and the Electron Microscopy Data Bank. Similar to macromolecular crystallography, validation tools for these models and maps are required. While some of these validation tools may be borrowed from crystallography, new methods specifically designed for cryo-EM validation are required. Here, new computational methods and tools implemented inPHENIXare discussed, includingd99to estimate resolution,phenix.auto_sharpento improve maps andphenix.mtriageto analyze cryo-EM maps. It is suggested that cryo-EM half-maps and masks should be deposited to facilitate the evaluation and validation of cryo-EM-derived atomic models and maps. The application of these tools to deposited cryo-EM atomic models and maps is also presented.

scholarly journals Molecular replacement with a large number of molecules in the asymmetric unit

2014 ◽  
Vol 70 (9) ◽  
pp. 1296-1302 ◽  
Author(s):  
Chacko Jobichen ◽  
Kunchithapadam Swaminathan
Keyword(s):  
Protein Data Bank ◽  
Structure Determination ◽  
Protein Structures ◽  
Data Bank ◽  
Lessons Learned ◽  
Molecular Replacement ◽  
Asymmetric Unit ◽  
Exponential Increase ◽  
Crystallographic Symmetry ◽  
Input Model

The exponential increase in protein structures deposited in the Protein Data Bank (PDB) has resulted in the elucidation of most, if not all, protein folds, thus making molecular replacement (MR) the most frequently used method for structure determination. A survey of the PDB shows that most of the structures determined by molecular replacement contain less than ten molecules in the asymmetric unit and that it is predominantly virus and ribosome structures that contain more than 20 molecules in the asymmetric unit. While the success of the MR method depends on several factors, such as the homology and the size of an input model, it is also a well known fact that this method can become significantly difficult in cases with a large number of molecules in the asymmetric unit, higher crystallographic symmetry and tight packing. In this paper, five representative structures containing 16–18 homomeric molecules in the asymmetric unit and the strategies that have been used to solve these structures are described. The difficulties faced and the lessons learned from these structure-determination efforts will be useful for selected and similar future situations with a large number of molecules in the asymmetric unit.

scholarly journals Nearest-cell: a fast and easy tool for locating crystal matches in the PDB

2012 ◽  
Vol 68 (12) ◽  
pp. 1697-1700 ◽  
Author(s):  
V. Ramraj ◽  
G. Evans ◽  
J. M. Diprose ◽  
R. M. Esnouf
Keyword(s):  
Protein Data Bank ◽  
Large Family ◽  
Data Bank ◽  
Crystal Form ◽  
Measured Data ◽  
Unit Cell ◽  
Molecular Replacement ◽  
X Ray Diffraction ◽  
Input Unit ◽  
X Ray

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-cellrapidly scans the Protein Data Bank and retrieves near-matches.

Collision-free configuration-spaces in macromolecular crystals

Robotica ◽  
2016 ◽  
Vol 34 (8) ◽  
pp. 1679-1704 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Bernard Shiffman
Keyword(s):  
Rigid Body ◽  
A Priori ◽  
Two Dimensions ◽  
Computational Method ◽  
Configuration Spaces ◽  
Molecular Replacement ◽  
Macromolecular Crystallography ◽  
Single Body ◽  
Protein Molecules ◽  
Explicit Expressions

SUMMARYMolecular replacement (MR) is a well-established computational method for phasing in macromolecular crystallography. In MR searches, spaces of motions are explored for determining the appropriate placement of rigid single-body (or articulated multi-rigid-body) models of macromolecules. By determining a priori which portions of motion space correspond to non-physical packing arrangements with symmetry mates in collision, it becomes feasible to construct more efficient MR techniques which avoid searching in these non-realizable regions of motion space. This paper investigates which portion of the motion space is physically realizable, given that packing of protein molecules in a crystal are subject to the constraint that they cannot interpenetrate, and gives explicit expressions for the volume of the non-realizable regions for crystals in two-dimensions.

Sign in / Sign up

Export Citation Format

Share Document