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

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.

Are the St John's wort Hyp-1 superstructures different?

Two commensurately modulated structures (PDB entries 4n3e and 6sjj) were solved using translational noncrystallographic symmetry (tNCS). The data required the use of large supercells, sevenfold and ninefold, respectively, to properly index the reflections. Commensurately modulated structures can be challenging to solve. Molecular-replacement software such as Phaser can detect tNCS and either handle it automatically or, for more challenging situations, allow the user to enter a tNCS vector, which the software then uses to place the components. Although this approach has been successful in solving these types of challenging structures, it does not make it easy to understand the underlying modulation in the structure or how these two structures are related. An alternate view of this problem is that the atoms and associated parameters are following periodic atomic modulation functions (AMFs) in higher dimensional space, and what is being observed in these supercells are the points where these higher dimensional AMFs intersect physical 3D space. In this case, the two 3D structures, with a sevenfold and a ninefold superstructure, seem to be quite different. However, describing those structures within the higher dimensional superspace approach makes a strong case that they are closely related, as they show very similar AMFs and can be described with one unique (3+1)D structure, i.e. they are two different 3D intersections of the same (3+1)D structure.

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.