Multiscale structural optimisation with concurrent coupling between scales

A robust three-dimensional multiscale topology optimisation framework with concurrent coupling between scales is presented. Concurrent coupling ensures that only the microscale data required to evaluate the macroscale model during each iteration of optimisation is collected and results in considerable computational savings. This represents the principal novelty of the framework and permits a previously intractable number of design variables to be used in the parametrisation of the microscale geometry, which in turn enables accessibility to a greater range of mechanical point properties during optimisation. Additionally, the microscale data collected during optimisation is stored in a re-usable database, further reducing the computational expense of subsequent iterations or entirely new optimisation problems. Application of this methodology enables structures with precise functionally-graded mechanical properties over two-scales to be derived, which satisfy one or multiple functional objectives. For all applications of the framework presented within this thesis, only a small fraction of the microstructure database is required to derive the optimised multiscale solutions, which demonstrates a significant reduction in the computational expense of optimisation in comparison to contemporary sequential frameworks. The derivation and integration of novel additive manufacturing constraints for open-walled microstructures within the concurrently coupled multiscale topology optimisation framework is also presented. Problematic fabrication features are discouraged through the application of an augmented projection filter and two relaxed binary integral constraints, which prohibit the formation of unsupported members, isolated assemblies of overhanging members and slender members during optimisation. Through the application of these constraints, it is possible to derive self-supporting, hierarchical structures with varying topology, suitable for fabrication through additive manufacturing processes.Open Acces

Energy Landscapes for Protein Folding

Proteins are involved in numerous functions in the human body, including chemical transport, molecular recognition, and catalysis. To perform their function most proteins must adopt a specific structure (often referred to as the folded structure). A microscopic description of folding is an important prerequisite for elucidating the underlying basis of protein misfolding and rational drug design. However, protein folding occurs on heterogeneous length and time scales, presenting a grand challenge to both experiments and simulations. In computer simulations, challenges are generally mitigated by adopting coarse-grained descriptions of the physical environment, employing enhanced sampling strategies, and improving computing code and hardware. While significant advances have been made in these areas, for numerous systems a large spatiotemporal gap between experiment and simulations still exists, due to the limited time and length scales achieved by simulation, and the inability of many experimental techniques to probe fast motions and short distances. In this thesis, kinetic transition networks (KTNs) are constructed for various protein folding systems, via approaches based on the potential energy landscape (PEL) framework. By applying geometry optimisation techniques, the PEL is discretised into stationary points (i.e.~low-energy minima and the transition states that connect them). Essentially, minima characterise the low-lying regions of the PEL (thermodynamics) and transition states encode the motion between these regions (dynamics). Principles from statistical mechanics and unimolecular rate theory may then be employed to derive free energy surfaces and folding rates, respectively, from the KTN. Furthermore, the PEL framework can take advantage of parallel and distributed computing, since stationary points from separate simulations can be easily integrated into one KTN. Moreover, the use of geometry optimisation facilitates greater conformational sampling than conventional techniques based on molecular dynamics. Accordingly, this framework presents an appealing means of probing complex processes, such as protein folding. In this dissertation, we demonstrate the application of state-of-the-art theory, combining PEL analysis and KTNs to three diverse protein systems. First, to improve the efficiency of protein folding simulations, the intrinsic rigidity of proteins is exploited by implementing a local rigid body (LRB) approach. The LRB approach effectively integrates out irrelevant degrees of freedom from the geometry optimisation procedure and further accelerates conformational sampling. The effects of this approach on the underlying PEL are analysed in a systematic fashion for a model protein (tryptophan zipper\,1). We demonstrate that conservative local rigidification can reproduce the thermodynamic and dynamic properties for the model protein. Next, the PEL framework is employed to model large-scale conformational changes in proteins, which have conventionally been difficult to probe \textit{in silico}. Methods based on geometry optimisation have proved useful in overcoming the broken ergodicity issue, which is associated with proteins that switch morphology. The latest PEL-based approaches are utilised to investigate the most extreme case of fold-switching found in the literature:~the $\alpha$-helical hairpin to $\beta$-barrel transition of the C-terminal domain of RfaH, a bacterial transcription factor. PEL techniques are employed to construct the free energy landscape (FEL) for the refolding process and to discover mechanistic details of the transition at an atomistic level. The final part of the thesis focuses on modelling intrinsically disordered proteins (IDPs). Due to their inherent structural plasticity, IDPs are generally difficult to characterise, both experimentally and via simulations. An approach for studying IDPs within the PEL framework is implemented and tested with various contemporary potential energy functions. The cytoplasmic tail of the human cluster of differentiation 4 (CD4), implicated in HIV-1 infection, is characterised. Metastable states identified on the FEL help to unify, and are consistent with, several earlier predictions.Gates Cambridge Trus

O(N) methods in electronic structure calculations

Linear scaling methods, or O(N) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high performance computers. The linear scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas is then discussed. The applications of linear scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear scaling methods are discussed.Comment: 85 pages, 15 figures, 488 references. Resubmitted to Rep. Prog. Phys (small changes

A simplicial homology algorithm for Lipschitz optimisation

The simplicial homology global optimisation (SHGO) algorithm is a general purpose global optimisation algorithm based on applications of simplicial integral homology and combinatorial topology. SHGO approximates the homology groups of a complex built on a hypersurface homeomorphic to a complex on the objective function. This provides both approximations of locally convex subdomains in the search space through Sperner's lemma (Sperner, 1928) and a useful visual tool for characterising and e ciently solving higher dimensional black and grey box optimisation problems. This complex is built up using sampling points within the feasible search space as vertices. The algorithm is specialised in nding all the local minima of an objective function with expensive function evaluations e ciently which is especially suitable to applications such as energy landscape exploration. SHGO was initially developed as an improvement on the topographical global optimisation (TGO) method rst proposed by T orn (1986; 1990; 1992). It is proven that the SHGO algorithm will always outperform TGO on function evaluations if the objective function is Lipschitz smooth. In this dissertation SHGO is applied to non-convex problems with linear and box constraints with bounds placed on the variables. Numerical experiments on linearly constrained test problems show that SHGO gives competitive results compared to TGO and the recently developed Lc-DISIMPL algorithm (Paulavi cius and Zilinskas, 2016) as well as the PSwarm and DIRECT-L1 algorithms. Furthermore SHGO is compared with the TGO, basinhopping (BH) and di erential evolution (DE) global optimisation algorithms over a large selection of black-box problems with bounds placed on the variables from the SciPy (Jones, Oliphant, Peterson, et al., 2001{) benchmarking test suite. A Python implementation of the SHGO and TGO algorithms published under a MIT license can be found from https://bitbucket.org/upiamcompthermo/shgo/.Dissertation (MEng)--University of Pretoria, 2017.Chemical EngineeringMEngUnrestricte

Colloidal Systems Confined to Curved Surfaces

Surface curvature plays a vital role in many biological processes. Examples include the organising of proteins in cell membranes, the tiling of cells in epithelial layers and the growth of virus capsids. A major technological benefit of micro- and nanoscale curvature is that it can guide colloidal self-assembly, a property which is of major importance in fields such as drug-delivery, biosensor fabrication, and the development of meta-materials. Recent advances in fabrication techniques, such as 3D-printing, have made a rich library of geometries available to experimentalists and engineers seeking to realise these complex and fascinating systems. Here, we use bespoke simulations and theoretical models to study the effects of surface curvature on two-dimensional systems of isotropically attractive colloids. We identify four important properties: the finite but boundary-free area of closed surfaces, the minimum perimeter of a patch with a given area, the difference between the Euclidean and geodesic separation of points on the surface, and the frustration of the hexagonal lattice in regions of non-zero Gaussian curvature. We also show that competition between these effects produces a range of novel behaviours. Starting from the simplest example of a sphere, we show that surface curvature has a strong effect on the gas-liquid nucleation profile and the size of the critical nucleus, as well as destroying the equivalence of the canonical and grand canonical ensembles. Then, focussing on surfaces with non-uniform curvature, we use tori to demonstrate that the different thermodynamic phases are localised to specific regions of the surface, and the transitions between them involve the translation of the colloidal assemble. Finally, we investigate the cone, where there is no Gaussian curvature but the mean curvature varies. We find that surface curvature can stabilise chiral and achiral crystals, and the ground states depend on both the range of the potential and whether it acts through Euclidean space or along geodesics

Protein structure prediction and modelling

The prediction of protein structures from their amino acid sequence alone is a very challenging problem. Using the variety of methods available, it is often possible to achieve good models or at least to gain some more information, to aid scientists in their research. This thesis uses many of the widely available methods for the prediction and modelling of protein structures and proposes some new ideas for aiding the process. A new method for measuring the buriedness (or exposure) of residues is discussed which may lead to a potential way of assessing proteins' individual amino acid placement and whether they have a standard profile. This may become useful in assessing predicted models. Threading analysis and modelling of structures for the Critical Assessment of Techniques for Protein Structure Prediction (CASP2) highlights inaccuracies in the current state of protein prediction, particularly with the alignment predictions of sequence on structure. An in depth analysis of the placement of gaps within a multiple sequence threading method is discussed, with ideas for the improvement of threading predictions by the construction of an improved gap penalty. A threading based homology model was constructed with an RMSD of 6.2A, showing how combinations of methods can give usable results. Using a distance geometry method, DRAGON, the ab initio prediction of a protein (NK Lysin) for the CASP2 assessment was achieved with an accuracy of 4.6Å. This highlighted several ideas in disulphide prediction and a novel method for predicting which cysteine residues might form disulphide bonds in proteins. Using a combination of all the methods, with some like threading and homology modelling proving inadequate, an ab initio model of the N-terminal domain of a GPCR was built based on secondary structure and predictions of disulphide bonds. Use of multiple sequences in comparing sequences to structures in threading should give enough information to enable the improvements required before threading can be-come a major way of building homology models. Furthermore, with the ability to predict disulphide bonds: restraints can be placed when building models, ab initio or otherwise

Hydrogen transfer in hydrogen bonded solid state materials

The investigation of strongly hydrogen bonded solid state materials and the hydrogen transfer processes therein are the subject of the present work. Strong hydrogen bonds are found whenever the hydrogen bonded species compete for the hydrogen atom, and are thereby on the verge of showing hydrogen transfer. Consequently, the strongly hydrogen bonded solid state materials investigated in this work are synthesised by co-crystallising chemical compounds which have a similar affinity for the proton. The molecular complexes of isonicotinamide with oxalic acid crystallise in two hydrogenous polymorphs and, upon substituting the acidic hydrogen for deuterium, in two deuterated polymorphs, neither being isostructural to the hydrogenous forms. This phenomenon is known as isotopomeric polymorphism and is rarely observed in molecular materials. The four polymorphic forms are found to exhibit different degrees of hydron transfer. The hydrogenous forms show strong hydrogen bonding between the acid and the pyridine base. The nature of these strong hydrogen bonds is characterised by combined X ray charge density and single crystal neutron diffraction studies and found to be covalent in nature. The covalent hydroxyl bonds are considerably elongated, to an extent that in one polymorph the hydrogen atom occupies a near central position in the strong hydrogen bond. The structural work has been complemented by ab-initio computational studies, using the plane wave and localised atomic orbital methods, to evaluate the nature and the dynamics of the strong hydrogen bonds, and to establish an energy scale for polymorphism. It is found that the atomic orbital calculations yield results in good agreement with the experiment, while the plane wave calculations fail to reproduce the experimental hydrogen bond geometries. A strong electronic delocalisation is observed in the difference electron densities of strong acid – pyridine base hydrogen bonds. The major contribution to the delocalisation is found to originate from the nitrogen lone pair density which in this type of strong hydrogen bond is found to be observed to low experimental resolutions in standard X-ray diffraction experiments. As a consequence, such hydrogen bonds are susceptible to misinterpretation, and can be misinterpreted as hydrogen bonds with a disordered hydrogen, altering the descriptive character of materials significantly from being neutral to being ionic. It is shown that a careful examination of the difference electron densities, with the knowledge of the presence of the nitrogen lone pair density, allows a reasonably accurate determination of nuclear hydrogen positions from X-ray diffraction experiments alone. The hydrogen transfer behaviour in a series of strongly hydrogen bonded materials has been studied. For the molecular complexes of pentachlorophenol with the series of dimethylpyridines, a correlation is established between the dissociation constants determined in solution and the degree of hydrogen transfer from phenol to the pyridine bases in the solid state. The influence of additional strong and weak hydrogen bonding interactions in the vicinity of the strong hydrogen bonds on the hydrogen transfer behaviour is rationalised. Similar studies have been carried out on the molecular complexes of oxalic acid and fumaric acid with the dimethylpyridines, and on the molecular complexes of pentachlorophenol with 1,4-diazabicyclo[2.2.2]octane. The design approach leading to these materials and the hydrogen transfer behaviour observed in these materials is critically analysed