An understanding of the crystal structure can aid in the rationalisation of physicochemical properties exhibited by a crystalline material. Advances in the area of direct space crystal structure solution means that it is becoming easier to determine crystal structures from powder diffraction data. However, due to the number of structural models generated during structure solution calculations, direct space methods are computationally demanding. Work presented in this thesis reports the optimisation of a differential evolution (DE) algorithm and a cultural differential evolution (CDE) algorithm to reduce the computational demands of direct space methods. Characteristics particular to certain crystal structures are identified as having a significant effect on the efficiency and robustness of structure solution calculations by DE and CDE. The development of a new algorithm that closely mimics the natural evolution of a species is discussed. Results presented in this thesis demonstrate that this new algorithm is significantly more efficient than the DE algorithm. Despite the complexity of powder diffraction patterns recorded for biphasic crystalline materials, in this thesis, the successful development and application of a direct space method to the simultaneous structure solution of two crystals from a biphasic powder pattern is reported.

Research into cocrystals is just a small field within crystallography, though the significant rise of investigation into cocrystals has made it an exciting area of research. This thesis initially introduces the phenomenon of cocrystals, including the design, formation, structure solution and the properties of cocrystals. This thesis presents a number of new multi-component crystalline adducts that have been synthesised using dicarboxylic acids as coformers with isonicotinamide, nicotinamide or adenine. The first results chapter discusses a number of molecular adducts that contain distinct hydrogen bonding networks and includes the discovery of a new polymorph and the potential of adenine to tautomerise. The following two results chapters focus on molecular structures of nicotinamide and isonicotinamide with the dicarboxylic acids in which trends in crystal packing, cocrystal property studies and twisting of the acid backbone are discussed in detail. In conjunction with the crystal structure and property studies, several powder X-ray diffraction datasets were collected from a synchrotron source. The final results chapter shows that a cocrystal structure, such as nicotinamide : succinic acid, can be solved without the need of single crystal X-ray diffraction data.

The successful prediction of the crystal structure and symmetry of a material can give valuable insight into many of its properties, as well as the feasibility of thermodynamically stable poly-morphs to exist. It is not uncommon, however, for numerous theoretical structures to be found within a narrow energy range, making absolute characterisation of the crystal structure impossi-ble. The aim of this project was to investigate a number of structures from this scenario, high-lighting the key differences between three potential methods for the automated comparison of predicted and experimental crystal structures. This work was carried out by comparing the simulated powder diffraction patterns of theoretical predicted crystal structures of small organic materials with their experimental powder diffraction patterns, so that the experimentally identified structure could be automatically singled out from the many calculated. The use of traditional agreement factors (eg. Rwp) was compared with more sophisticated approaches namely PolySNAP, which uses principal-component analysis, and Compare.x, an algorithm based on weighted cross-correlation. Five structures were analysed, two of which had not been previously characterised. As the structure prediction calculations are carried out at 0K, and experimental data were collected over a range of temperatures (10K-293K), the effect of the resulting variations in lattice parameters on the automated processes is discussed. In all cases, Rwp has proven to be a poor and unreliable discriminator in the comparison of pre-dicted and experimental structures. The more contemporary methods based on PolySNAP and Compare.x both gave encouraging results when used to study the three known structures imida-zole, chlorothalonil and 5-azauracil, and they have consequently been used in the successful so-lution of the two previously unknown structures adenine and guanine. A difference in sensitivity in the matching of data collected at different temperatures between the latter approaches was noted. It was found that although there is considerable overlap between the two methods, they are not absolutely interchangeable, and this distinction may be exploited in future work where more case-specific comparisons are carried out. Automated comparison techniques cannot yet replace visual comparison completely, but they reduce it drastically. Ultimately, comparisons made computationally serve as a complement to human judgement, but they may not yet elimi-nate it.

In this thesis, we establish a locality property for solutions to the Thomas-Fermi-von Weizsäcker (TFW) equations. This is a system of coupled PDEs that models the ground state electron density corresponding to a given nuclear arrangement. The locality property is a pointwise stability estimate for the TFW equations, that demonstrates the exponential response of the electron density to a perturbation of the nuclei. We show that this result holds for the TFW when using either the Coulomb or the Yukawa potential to treat the interaction of charged particles. We then use the locality result to prove several consequences for the TFW ground state, which includes generalising results from [14] regarding the neutrality of infinite systems and also showing the uniform convergence of ground states when passing to the limit from the Yukawa to the Coulomb model. Our main application is the construction of site energies from the TFW energy. The locality result implies that the response of each site energy decays exponentially with respect to a perturbation of the nuclear arrangement. Using these site energies, we then formulate the lattice relaxation problem, that was initially formulated in [23], to consider the response of a perfect crystal lattice to the introduction of a point defect. The site energies allow us to formulate a variational problem over a space of deformations of the lattice, show this problem is well-posed and finally establish the decay properties of minimisers.

Atomic resolution models of proteins are crucial for understanding their biological mechanisms and provide insights into the relationship between protein sequence and structure. Many protein structures incorporate metal ions, exploiting their unique chemistry as reaction centres or for structural stability. This thesis describes the progress made towards solving the structures of three such metal-binding proteins by means of X-ray crystallography. Complement component C1 is a large protein complex that initiates the first line of immune defence and requires calcium for structural stability. Fragments of C1 have already been solved at high resolution, but there are no accurate models of the assembled complex. In this work, a new method for purifying intact C1 from human serum was developed and the purified complex was characterised by various methods. Finally, attempts were made to crystallise native human C1 with a view to obtaining high-resolution structures of the entire complex. Serum amyloid P is another serum protein, also thought to be involved in the immune response. It is often found associated with amyloid deposits, although SAP binds a variety of ligands in a calcium-dependent manner. While the structure of SAP has been determined, its physiological function is still not fully understood. SAP was purified using established methods and its ligand-binding properties were investigated under various conditions using dynamic light scattering, in an attempt to gather more information about the possible function of this molecule. Finally, ZiCo is a small peptide that was designed to switch between a multimeric coiled coil and a monomeric zinc finger fold on binding zinc. The system has been characterised extensively in solution, but high-resolution structures are required to validate the design. ZiCo was crystallised and diffraction data were collected. The structure of the peptide was partially solved, indicating that the multimeric form of the ZiCo peptide is indeed a trimeric coiled coil.

El genoma mitocondrial humano es una molécula circular de doble cadena localizada en la matriz mitocondrial. La transcripción del DNAmt se inicia a partir de tres promotores localizados en el D-loop, uno para la cadena ligera (L) y dos para la cadena pesada (H1 y H2). Las unidades de transcripción que se inician en H2 y en L producen una especie policistrónica gigante que cubre casi toda la cadena. En cambio, la transcripción iniciada en H1 se termina en el extremo 3’ del rRNA 16S. El papel principal de esta atenuación se debe al factor de terminación de la transcripción, mTERF. mTERF, es una proteína codificada en núcleo, cuya forma madura tiene 342 aminoácidos. Se ha descrito que se une a varios lugares estratégicos del DNA mitocondrial, donde controla procesos como la terminación de la transcripción o la pausa de la maquinaria de replicación mitocondrial. De estos lugares, el que presenta mayor afinidad es la secuencia de 28 pares de bases (DNA28pb) en el gen del tRNALeu(UUR) mitocondrial. Ensayos in vitro han demostrado que la unión de mTERF a esta región provoca la terminación de la transcripción del DNA mitocondrial en ambos sentidos. En este trabajo se estudia, desde un punto de vista estructural y mediante técnicas de cristalografía de proteínas, el complejo de mTERF con el DNA que contiene la secuencia de terminación. Durante la producción de mTERF recombinante, la proteólisis espontánea de la forma entera de la proteína (Arg56-Ala399) dio lugar a una variante corta, mTERF-ΔN (Arg99-Ala399) que mantenía la unión al DNA. Se determinó la estructura tridimensional del complejo con DNA de mTERF-ΔN a 2.4 Å de resolución. La disposición del DNA en el eje z del cristal provocó que los cristales de mTERF-ΔN en complejo con un DNA29pb, con un parámetro c de la celda unidad demasiado pequeño para albergarlo, presentaran graves problemas de anisotropía y desorden, dando lugar a mapas de densidad electrónica muy distorsionados. Se realizaron entonces ensayos de cristalización con oligonucleótidos de menor longitud, pudiéndose trazar y refinar el DNA y la proteína completa con los cristales de mTERF-ΔN en complejo con el DNA12pb. La estructura del fragmento ΔN se usó para resolver la estructura del complejo de mTERF entera con el DNA15pb a 3.1 Å de resolución. En estos cristales, la proteína interacciona mediante los subdominio N-terminal y C-terminal con dos moléculas de DNA simétricas que no interaccionan entre ellas y cuya posición relativa viene dada por un ángulo de ~36º. mTERF se une a la largo del surco mayor del DNA, estableciendo contactos con los fosfatos mediante 19 residuos y con bases nitrogenadas mediante 5 residuos. La estructura cristalográfica de mTERF consiste en nueve repeticiones estructurales, de TERF-I a IX, flanqueadas por un segmento N-terminal y una hélice α C-terminal. Cada motivo TERF está formado por unos 35 aminoácidos que se estructuran en tres hélices α (H1, H2 y H3) formando una superhélice trigonal levógira con un núcleo hidrofóbico central. La combinación de la conectividad en superhélice levógira con la propagación en tándem del motivo para formar un solenoide es única de las proteínas de la familia MTERF. En base a la estructura cristalográfica se generaron modelos para estudiar los complejos en solución mediante la técnica de SAXS. La representación gráfica del error asociado al ajuste entre la curva teórica de dispersión para cada modelo y la curva medida experimentalmente, presenta un mínimo, indicando que la unión de mTERF al DNA28pb se da preferentemente sobre una zona del DNA. Los ensayos de unión y especificidad de mTERF y las formas truncadas de ésta con el DNA28pb mediante geles nativos de retardo muestran que mTERF-ΔC y ΔN-mTERF-ΔC no unen DNA y que mTERF y mTERF-ΔN lo unen específicamente. / The human mitochondrial genome is a closed circular molecule located within the mitochondrial matrix. mtDNA transcription is initiated from tree promoters in the D-loop, one for L-strand (L) and two for the H-strand (H1 and H2). Transcription from H2 and L proceeds around the mtDNA circle, creating a polycistronic RNA. On the other hand, transcription initiated in H1 ends at the 3’end of rRNA 16S. The main role of this attenuation is due to the mitochondrial transcription termination factor mTERF. mTERF, is a nucleus-encoded protein whose mature form has 342 amino acids. It has been described that mTERF binds to several strategic mtDNA sites, where it controls processes such as transcription termination or pausing of the replication machinery. Among them, the 28bp sequence from the tRNALeuUUR gene shows the highest affinity of binding to mTERF. In vitro assays have shown that binding of mTERF to this site promotes termination of transcription bidirectionally. We study, from a structural point of view, the mTERF complex with the oligonucleotide containing the termination sequence. During recombinant mTERF production, spontaneous proteolysis of the full-length protein (Arg56–Ala399) yielded a shorter variant (Arg99–Ala399; mTERF-ΔN) yet DNA-binding form. The three-dimensional struc¬ture of mTERF-ΔN was determined from SeMet-derivatized crystals at 2.4Å resolution. In the first mTERF-ΔN-DNA crystals, the arrangement in the z-axis of the large 29bpDNA molecule, compared to the c parameter of the unit cell, caused severe problems of anisotropy and disorder, which resulted in low quality electron density maps. Then crystallization assays were performed with shorter oligonucleotides, being able to trace and refine the complex structure with mTERF-ΔN-12bpDNA crystals. The structure of ΔN fragment was used to solve the full-length mTERF structure at 3.1Å resolution in complex with 15bpDNA. In these crystals, two crystallographically related identical oligonucleotides related by ~36° are bound by two distinct regions of a single protein moiety, the N-terminal and C-terminal subdomains. mTERF wraps round the DNA along the major groove through 19 nonspecific interactions with both dsDNA backbone phosphates and 5 interactions with DNA nitrogen bases. mTERF consists of nine structural repeats, TERF-I–TERF-IX flanked by an N-terminal extended segment and a C-terminal α-helix C. Within each repeat, roughly 35 residues fold into three α-helices (H1, H2 and H3) that are arranged in a left-handed triangular superhelix around a central hydrophobic core. The combination of left-handed helical connec¬tivity with tandem propagation is unique to mTERF. Based on the crystallographic structure, we constructed models to study the complex in solution by small-angle X-ray scattering (SAXS). The graphical representation of agreement of each of the models to the experimental SAXS curve presents a minimum, indicating the specificity of the binding between mTERF and the 28bpDNA. The 28bpDNA binding affinity and specificity of mTERF and truncated forms, assayed by native gels, shows that mTERF-ΔC and ΔN-mTERF-ΔC do not bind DNA while mTERF and mTERF-ΔN bind it specifically.

The structure and physical properties of the lead-free piezoelectric niobate perovskite material sodium potassium niobate, KxNa1-xNbO3 (KNN) have been investigated via a variety of experimental techniques. High- and low-temperature X-ray and neutron powder diffraction has been performed on polycrystalline samples of KNN. Rietveld refinement has been carried out on data for samples of KNN from x = 0:1 to x = 0:85, with the structure determined for each sample. The phase diagram from x=0:1 to x=1 has been determined, with particular attention paid to the oxygen octahedra tilt systems. The proposed compositionally-driven phase transitions were also investigated, and it has been found that changes in the tilt system occur at x = 0:2 and x = 0:4 at room temperature, which is a correction to previously published work. The first-order phase transition at x = 0:5 from a monoclinic structure to an orthorhombic structure has also been confirmed. To complement the polycrystalline structural analysis, single-crystal samples of KNN have been fabricated to confirm the exact nature of the tilt systems. It has been found that all crystals possess the same merohedral twinning that is present for other perovskite systems that follow a similar phase transition path from the prototypic cubic phase. The tilt systems above and below the the x=0:2 boundary have been found to be of the type a0b+c0 and a-b+c- respectively. Low-temperature single-crystal diffraction also confirmed the structure of the low-temperature trigonal (rhombohedral) phase. To complement the diffraction work, the short-range structure of KNN has been investigated with 1D and 2D 23Na NMR experiments. It has been found that chemical shift dispersion is present for all samples of KNN because of the shared A-site occupancy of potassium and sodium atoms. The quadrupolar parameters have been determined, and an increase in the electric field gradient for one sodium site can be seen to coincide with an increase in the distortion in the structure because of the tilting of the oxygen octahedra. Birefringence data have been collected for single-crystal samples of KNN using the Metripol technique. No obvious domain structure was seen. The nature of the phase transition from the prototypic cubic phase and the tetragonal phase was investigated, and the analysis suggests that the transition is at a tricritical point between a first-order and a second-order phase transition. Finally, an optical study was performed on samples of KNbO3 (KN) using the Metripol technique. Large self-organised domains similar to those previously observed in barium titanate were seen only on the first heating run. Very small (< 5μm) domains were observed in the samples, and remained in the orthorhombic phase after several heating runs. Single-crystal X-ray diffraction was implemented to determine the structure of the sample. It was also found that same twin law as in barium titanate is present for KN.

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