Simulation of modulated protein crystal structure and diffraction data in a supercell and in superspace

Aperiodic crystals in biology

Biological systems display a broad palette of hierarchically ordered designs spanning over many orders of magnitude in size. Remarkably enough, periodic order, which profusely shows up in non-living ordered compounds, plays a quite subsidiary role in most biological structures, which can be appropriately described in terms of the more general aperiodic crystal notion instead. In this topical review I shall illustrate this issue by considering several representative examples, including botanical phyllotaxis, the geometry of cell patterns in tissues, the morphology of sea urchins, or the symmetry principles underlying virus architectures. In doing so, we will realize that albeit the currently adopted quasicrystal notion is not general enough to properly account for the rich structural features one usually finds in biological arrangements of matter, several mathematical tools and fundamental notions belonging to the aperiodic crystals science toolkit can provide a useful modeling framework to this end.

A new modulated crystal structure of the ANS complex of the St John's wort Hyp-1 protein with 36 protein molecules in the asymmetric unit of the supercell

Superstructure modulation, with violation of the strict short-range periodic order of consecutive crystal unit cells, is well known in small-molecule crystallography but is rarely reported for macromolecular crystals. To date, one modulated macromolecular crystal structure has been successfully determined and refined for a pathogenesis-related class 10 protein from Hypericum perforatum (Hyp-1) crystallized as a complex with 8-anilinonaphthalene-1-sulfonate (ANS) [Sliwiak et al. (2015), Acta Cryst. D71, 829-843]. The commensurate modulation in that case was interpreted in a supercell with sevenfold expansion along c. When crystallized in the additional presence of melatonin, the Hyp-1-ANS complex formed crystals with a different pattern of structure modulation, in which the supercell shows a ninefold expansion of c, manifested in the diffraction pattern by a wave of reflection-intensity modulation with crests at l = 9n and l = 9n ± 4. Despite complicated tetartohedral twinning, the structure has been successfully determined and refined to 2.3 Å resolution using a description in a ninefold-expanded supercell, with 36 independent Hyp-1 chains and 156 ANS ligands populating the three internal (95 ligands) and five interstitial (61 ligands) binding sites. The commensurate superstructures and ligand-binding sites of the two crystal structures are compared, with a discussion of the effect of melatonin on the co-crystallization process.

Supercell refinement: a cautionary tale

Theoretically, crystals with supercells exist at a unique crossroads where they can be considered as either a large unit cell with closely spaced reflections in reciprocal space or a higher dimensional superspace with a modulation that is commensurate with the supercell. In the latter case, the structure would be defined as an average structure with functions representing a modulation to determine the atomic location in 3D space. Here, a model protein structure and simulated diffraction data were used to investigate the possibility of solving a real incommensurately modulated protein crystal using a supercell approximation. In this way, the answer was known and the refinement method could be tested. Firstly, an average structure was solved by using the `main' reflections, which represent the subset of the reflections that belong to the subcell and in general are more intense than the `satellite' reflections. The average structure was then expanded to create a supercell and refined using all of the reflections. Surprisingly, the refined solution did not match the expected solution, even though the statistics were excellent. Interestingly, the corresponding superspace group had multiple 3D daughter supercell space groups as possibilities, and it was one of the alternate daughter space groups that the refinement locked in on. The lessons learned here will be applied to a real incommensurately modulated profilin-actin crystal that has the same superspace group.

Pathological macromolecular crystallographic data affected by twinning, partial-disorder and exhibiting multiple lattices for testing of data processing and refinement tools

Twinning is a crystal growth anomaly, which has posed a challenge in macromolecular crystallography (MX) since the earliest days. Many approaches have been used to treat twinned data in order to extract structural information. However, in most cases it is usually simpler to rescreen for new crystallization conditions that yield an untwinned crystal form or, if possible, collect data from non-twinned parts of the crystal. Here, we report 11 structures of engineered variants of the E. coli enzyme N-acetyl-neuraminic lyase which, despite twinning and incommensurate modulation, have been successfully indexed, solved and deposited. These structures span a resolution range of 1.45-2.30 Å, which is unusually high for datasets presenting such lattice disorders in MX and therefore these data provide an excellent test set for improving and challenging MX data processing programs.

How to assign a (3 + 1)-dimensional superspace group to an incommensurately modulated biological macromolecular crystal

Periodic crystal diffraction is described using a three-dimensional (3D) unit cell and 3D space-group symmetry. Incommensurately modulated crystals are a subset of aperiodic crystals that need four to six dimensions to describe the observed diffraction pattern, and they have characteristic satellite reflections that are offset from the main reflections. These satellites have a non-integral relationship to the primary lattice and require q vectors for processing. Incommensurately modulated biological macromolecular crystals have been frequently observed but so far have not been solved. The authors of this article have been spearheading an initiative to determine this type of crystal structure. The first step toward structure solution is to collect the diffraction data making sure that the satellite reflections are well separated from the main reflections. Once collected they can be integrated and then scaled with appropriate software. Then the assignment of the superspace group is needed. The most common form of modulation is in only one extra direction and can be described with a (3 + 1)D superspace group. The (3 + 1)D superspace groups for chemical crystallographers are fully described in Volume C of International Tables for Crystallography. This text includes all types of crystallographic symmetry elements found in small-molecule crystals and can be difficult for structural biologists to understand and apply to their crystals. This article provides an explanation for structural biologists that includes only the subset of biological symmetry elements and demonstrates the application to a real-life example of an incommensurately modulated protein crystal.

Likelihood-based molecular-replacement solution for a highly pathological crystal with tetartohedral twinning and sevenfold translational noncrystallographic symmetry

Translational noncrystallographic symmetry (tNCS) is a pathology of protein crystals in which multiple copies of a molecule or assembly are found in similar orientations. Structure solution is problematic because this breaks the assumptions used in current likelihood-based methods. To cope with such cases, new likelihood approaches have been developed and implemented in Phaser to account for the statistical effects of tNCS in molecular replacement. Using these new approaches, it was possible to solve the crystal structure of a protein exhibiting an extreme form of this pathology with seven tetrameric assemblies arrayed along the c axis. To resolve space-group ambiguities caused by tetartohedral twinning, the structure was initially solved by placing 56 copies of the monomer in space group P1 and using the symmetry of the solution to define the true space group, C2. The resulting structure of Hyp-1, a pathogenesis-related class 10 (PR-10) protein from the medicinal herb St John's wort, reveals the binding modes of the fluorescent probe 8-anilino-1-naphthalene sulfonate (ANS), providing insight into the function of the protein in binding or storing hydrophobic ligands.