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Publication# Incommensurate crystallography without additional dimensions

Abstract

It is shown that the Euclidean group of translations, when treated as a Lie group, generates translations not only in Euclidean space but on any space, curved or not. Translations are then not necessarily vectors (straight lines); they can be any curve compatible with the parameterization of the considered space. In particular, attention is drawn to the fact that one and only one finite and free module of the Lie algebra of the group of translations can generate both modulated and non-modulated lattices, the modulated character being given only by the parameterization of the space in which the lattice is generated. Moreover, it is shown that the diffraction pattern of a structure is directly linked to the action of that free and finite module. In the Fourier transform of a whole structure, the Fourier transform of the electron density of one unit cell (i.e. the structure factor) appears concretely, whether the structure is modulated or not. Thus, there exists a neat separation: the geometrical aspect on the one hand and the action of the group on the other, without requiring additional dimensions.

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The mathematical facet of modern crystallography is essentially based on analytical geometry, linear algebra as well as group theory. This study endeavours to approach the geometry and symmetry of crystals using the tools furnished by differential geometry and the theory of Lie groups. These two branches of mathematics being little known to crystallographers, the pertinent definitions such as differentiable manifold, tangent space or metric tensor or even isometries on a manifold together with some important results are given first. The example of euclidean space, taken as riemannian manifold, is treated, in order to show that the affine aspect of this space is not at all an axiom but the consequence of the euclidean nature of the manifold. Attention is then directed to a particular subgroup of the group of euclidean isometries, namely that of translations. This has the property of a Lie group and it turns out that the action of its elements, as well as those of its Lie algebra, plays an important role in generating a lattice on a manifold and in its tangent space, too. In particular, it is pointed out that one and only one finite and free module of the Lie algebra of the group of translations can generate both, modulated and non-modulated lattices. This last classification therefore appears continuous rather than black and white and is entirely determined by the parametrisation considered. Since a lattice in a tangent space has the properties of a vector space, it always possesses the structure of a finite, free module, which shows that the assignment of aperiodicity to modulated structures is quite subjective, even unmotivated. Thanks to the concept of representation of a lattice or a crystal in a tangent space, novel definitions of the notions of symmetry operation of a space group and point symmetry operation, as well as symmetry element and intrinsic translation arise; they altogether naturally blend into the framework of differential geometry. In order to conveniently pass from one representation of a crystal in one tangent space to another or to the structure on a manifold, an equivalence relation on the tangent bundle of the manifold is introduced. This relation furthermore allows to extend the concept of symmetry operation to the tangent bundle; this extension furnishes, particularly in the euclidean case, a very practical way of representing symmetry operations of space groups completely devoid of any dependence on an origin, or, in other words, in which each and every point may be considered the origin. The investigation of the group of translations having being completed, the study of the linear parts of the isometries comes naturally. Based on the fact that the set of linear parts possesses the structure of a Lie group, several results are proven in a rigorous manner, such as the fact that a rotation angle of π/3 is incompatible with a three-dimensional cubic lattice. Procedures for determining different crystal systems in function of the type of rotation are laid out by way of the study of orthogonal matrices and their relation to the matrix associated with the type of system. Finally, the description of a crystal by its diffraction patterns is taken on. It is shown that the general aspect of such a pattern is directly linked to the action of that free and finite module of the Lie algebra of translations which generates a lattice on a manifold. In the case of modulated crystals, it is demonstrated that the appearance of supplementary spots is caused by the geometry, i.e. by the parametrisation of the manifold in which the crystal exists and not by the action of the module in the Lie algebra. Thus, there exists a neat separation: the geometrical aspect on the one hand, and the action of the group on the other. As the last topic, other ways of interpreting the diffraction pattern of a modulated structure are laid out in order to argue that mere experimental data do not warrant the uniqueness of a model. The goal of this study is by no means an attempt at overthrowing existing structural models such as the superspace-formalism or at revolutionising the methods for determining structures, but is rather aimed at sustaining that the definition of certain notions becomes thoroughly natural within the appropriate mathematical framework, and, that the term aperiodicity assigned to modulated structures no longer has a true meaning.

The comparable order of magnitude between interatomic distances in a crystal and the wavelength of X-rays make X-ray crystallography the ideal analytical tool to gain insight into the structure of crystalline material, including biomolecules. Nevertheless, biomolecular crystallography has until now relied on the successful growth of single crystals of suitable size and quality. These remain the exception rather than the rule, since biomolecules often produce polycrystalline precipitate instead. Yet, an interest in making use of the once-discarded polycrystalline material, through the technique of powder diffraction, has only recently emerged. This can be accounted for by the information deficit which powder diffraction data suffers from in comparison with that of single crystal. The paucity of information in powder data stems from the compression of the three-dimensional reciprocal space onto the one-dimension of a powder pattern. In spite of this, powder diffraction holds the potential for application in biomolecular crystallography as is shown in the two different studies presented herein. Both studies were carried out with methods which do not rely on employing previously determined crystal-structures as molecular models. This therefore allowed the objective assessment of the quality of information that powder diffraction data can contribute to the structural investigation of biomolecules. In the first project, the traditional single-crystal structure-solution process is applied to data extracted from protein powder diffraction patterns measured on a synchrotron source. The use of models is avoided by employing the de novo phasing method of isomorphous replacement. With two protein test-cases, namely hen egg white lysozyme and porcine pancreatic elastase, it is demonstrated that protein powder diffraction data can afford structural information up to medium resolution. Indeed, a single isomorphous replacement analysis generated molecular envelopes accurately describing the crystal packing of both protein systems, while a multiple isomorphous replacement experiment, carried out only on lysozyme, revealed an electron density map in which elements of the secondary structure could be located. In fact, the resolution of the latter was discovered to be sufficient to determine the chirality of the protein molecule it represented. In addition to being encouraging, these results do not reflect the full potential of biomolecular powder diffraction, due, in large part, to the ultimately unsuitable nature of one type of phasing method, single crystal, being applied to another type of data, powder. An alternative approach to extract information from protein powder diffraction data is to employ powder-specific structure-solution techniques, such as global optimization methods. Although these methods make use of a starting "model", it is a molecular description of the system under study based on known chemical quantities rather than a related molecular configuration based on a previously determined crystal structure. Since their conception, global optimization methods have continuously been developed to enable the tackling of increasingly complex crystal structures. However, the immense complexity of biomacromolecules has kept proteins well out of reach of such methods. In an attempt to further reduce the gap separating the two levels of complexity, the second study reported herein puts forth the implementation of Ramachandran plot restraints into the algorithm of a global optimization method, i.e. that of simulated annealing. More specifically, the Ramachandran plot was approximated using a two-dimensional Fourier series which was subsequently expressed as a penalty function and incorporated into the search algorithm.

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Several methods have been proposed and realized to increase the lateral resolution in microscopy and ultimately to beat the actual diffraction limit as stated by Abbe. Structured illumination for example enhances the lateral resolution by virtually extending the spatial frequency support of the system transfer function. Higher spatial frequencies are in fact mapped into the range covered by the detection transfer function. The set of spatial frequencies can be further enhanced by decoupling illumination and detection and by employing concepts similar to synthetic apertures. We present a novel method for diffractionless microscopic imaging – Fourier Domain or k-Microscopy[1] (FDM). Again we decouple illumination from detection to achieve a lateral resolution that is independent of the detection numerical aperture. Similar to structured illumination we produce a set of fringes covering a band of spatial frequencies on the sample. However, instead of detecting with an area sensor we use a simple PIN detector: By tuning the spatial frequency of the illumination we obtain the Fourier coefficient from the measured total backscattered or transmitted intensity for the particular spatial frequency in the sample dimension normal to the fringe orientation. The fringe pattern is obtained from two collimated coherent beams that are superimposed. The spatial frequency of the pattern is controlled via tuning the wavelength. In order to obtain an unambiguous Fourier representation of the one dimensional sample structure it is necessary to record at least two π/2 phase shifted copies of the frequency response signal. The lateral resolution is then determined by the tuning bandwidth of the laser as well as the subtended angle and the central wavelength of the laser. This relation is in fact very similar to the concept of Fourier Domain Optical Coherence tomography, only that it acts on the transverse instead of the longitudinal resolution. We give first experimental proof of the concept in one dimension using a slowly tuning broad- bandwidth Ti:Sapph laser. We resolve the defined structure of a resolution test target. A particular feature of the method is that the reconstructed sample will appear spatially band-pass filtered due to the finite spectral support given by the tuned fringe frequencies. Hence only the edges of the test target lines are visible. We believe that the concept of FDM will greatly benefit from new developments in rapidly- wavelength tuning source technology that recently entered the field of OCT. 1. M. Geissbuehler, T. Lasser, and R. A. Leitgeb, "K- Microscopy - resolution beyond the diffraction limit," in Progress in Biomedical Optics and Imaging - Proceedings of SPIE(2008).

2011