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The determination of protein structure and the exploration of protein folding landscapes are two of the key problems in computational biology. In order to address these challenges, both a protein model that accurately captures the physics of interest and an efficient sampling algorithm are required. The first part of this thesis documents the continued development of CRANKITE, a coarse-grained protein model, and its energy landscape exploration using nested sampling, a Bayesian sampling algorithm. We extend CRANKITE and optimize its parameters using a maximum likelihood approach. The efficiency of our procedure, using the contrastive divergence approximation, allows a large training set to be used, producing a model which is transferable to proteins not included in the training set. We develop an empirical Bayes model for the prediction of protein β-contacts, which are required inputs for CRANKITE. Our approach couples the constraints and prior knowledge associated with β-contacts to a maximum entropy-based statistic which predicts evolutionarily-related contacts. Nested sampling (NS) is a Bayesian algorithm shown to be efficient at sampling systems which exhibit a first-order phase transition. In this work we parallelize the algorithm and, for the first time, apply it to a biophysical system: small globular proteins modelled using CRANKITE. We generate energy landscape charts, which give a large-scale visualization of the protein folding landscape, and we compare the efficiency of NS to an alternative sampling technique, parallel tempering, when calculating the heat capacity of a short peptide. In the final part of the thesis we adapt the NS algorithm for use within a molecular dynamics framework and demonstrate the application of the algorithm by calculating the thermodynamics of allatom models of a small peptide, comparing results to the standard replica exchange approach. This adaptation will allow NS to be used with more realistic force fields in the future.

In this thesis, we describe periodic 2D supramolecular networks using a simple polyomino model with nearest-neighbour interactions. In particular, we focus on design rules for ordered molecular tilings, describing how the realised molecular tiling depends on the parameters of the system, such as the interactions, molecular shape, temperature, and defects. A major component of our analysis is in keeping the interaction parameters free and exploring the polyomino system from the perspective of the space of interaction counts. The design principles and methods outlined in this thesis include several different themes, which give a complementary view on the properties of 2D supramolecular networks. Within the thesis we describe our algorithm for enumerating polyomino patterns and identifying their symmetries, making use of group theory methods that are specific for polyomino tilings. This information is used to analyse properties such as chirality, lowest energy states, degeneracy, and heat capacity curves, as depending on system parameters. The domino tilings are considered in terms of enumeration by periodicity, and in terms of the kinetically accessible subsets of configurations. The generalisation to polyominoes gives a wider sense in which many of the techniques can be used, showing similarities and some differences with the domino system. A probabilistic version of algorithm DLX is described and tested that allows us to gather sample statistics of domino configurations for larger unit cells. Finally, fixed defects are considered and their effect on the kinetically accessible domino subsets is elucidated.

In the chemical industry, the control of pH is a well-known problem that presents difficulties due to the large variations in its process dynamics and the static nonlinearity between pH and concentration. pH control requires the application of advanced control techniques such as linear or nonlinear adaptive control methods. Unfortunately, adaptive controllers rely on a mathematical model of the process being controlled, the parameters being determined or modified in real time. Because of its characteristics, the pH control process is extremely difficult to model accurately. Fuzzy logic, which is derived from Zadeh's theory of fuzzy sets and algorithms, provides an effective means of capturing the approximate, inexact nature of the physical world. It can be used to convert a linguistic control strategy based on expert knowledge, into an automatic control strategy to control a system in the absence of an exact mathematical model. The work described in this thesis sets out to investigate the suitability of fuzzy techniques for the control of pH within a continuous flow titration process. Initially, a simple fuzzy development system was designed and used to produce an experimental fuzzy control program. A detailed study was then performed on the relationship between fuzzy decision table scaling factors and the control constants of a digital PI controller. Equation derived from this study were then confirmed experimentally using an analogue simulation of a first order plant. As a result of this work a novel method of tuning a fuzzy controller by adjusting its scaling factors, was derived. This technique was then used for the remainder of the work described in this thesis. The findings of the simulation studies were confirmed by an extensive series of experiments using a pH process pilot plant. The performance of the tunable fuzzy controller was compared with that of a conventional PI controller in response to step change in the set-point, at a number of pH levels. The results showed not only that the fuzzy controller could be easily adjusted to provided a wide range of operating characteristics, but also that the fuzzy controller was much better at controlling the highly non-linear pH process, than a conventional digital PI controller. The fuzzy controller achieved a shorter settling time, produced less over-shoot, and was less affected by contamination than the digital PI controller. One of the most important characteristics of the tunable fuzzy controller is its ability to implement a wide variety of control mechanisms simply by modifying one or two control variables. Thus the controller can be made to behave in a manner similar to that of a conventional PI controller, or with different parameter values, can imitate other forms of controller. One such mode of operation uses sliding mode control, with the fuzzy decision table main diagonal being used as the variable structure system (VSS) switching line. A theoretical explanation of this behavior, and its boundary conditions, are given within the text. While the work described within this thesis has concentrated on the use of fuzzy techniques in the control of continuous flow pH plants, the flexibility of the fuzzy control strategy described here, make it of interest in other areas. It is likely to be particularly useful in situations where high degrees of non-linearity make more conventional control methods ineffective.

This thesis consists of five studies on the applications of the molecular field theory to model systems of biaxial molecules which form biaxial and uniaxial nematic and smectic A phases. The first study extends the original theory for biaxial nematic phases of D2h symmetry to allow the phase symmetry to be C2h. In the second study, a dipolar interaction is introduced to the original model of biaxial nematic phases formed from V-shaped molecules to explain a disagreement between theory and experiment. This leads to the stabilisation of the novel polar biaxial nematic phase. In the third study, we introduce molecular flexibility at a simplified level into an existing model of V-shaped molecules to investigate its effects on the stability of the biaxial nematic phases. The fourth study aims to explain and predict various effects of magnetic field on the uniaxial nematic to isotropic phase transition for a system of rigid V-shaped molecules. In the fifth study, we develop a model for biaxial smectic A phases. The theory is simplified by using several approximations which facilitates the calculations.

In this thesis we study the interactions between light and matter in photorefractive liquid crystal cells. To model the liquid crystal alignment we develop a fast and accurate approximation of the normally stiff equations which minimise the Landau-deGennes free energy of a nematic liquid crystal. The resulting equations are suitable for all configurations in which defects are not present, making them ideal for device simulation. Specifically, they offer an increase in computational efficiency by a factor of 100 while maintaining an error of order (10−4) when compared to the full stiff equations. As this approximation is based on aQ–tensor formalism, the sign reversal symmetry of the liquid crystal is respected. We consider both the simplified case, where the director is restricted to a plane, and the full three-dimensional case. An approximation of the error is also given. We use the liquid crystal model to understand two different optical effects. The first of these is optical coupling. This effect is observed in liquid crystals in both the Bragg and Raman–Nath regimes. To account for this behaviour we develop an extension to the coupled wave theory which is suitable for all regimes of coupling. The model assumes that the refractive index grating, generated by the liquid crystal, has an arbitrary profile in one direction and is periodic (but not necessarily sinusoidal) in the other. Higher order diffracted terms are considered and appropriate mismatch terms dealt with. It is shown that this model is analytically equivalent to both the Bragg and Raman–Nath regime coupling models under an appropriate set of assumptions. This model is also verified through comparison to finite element simulations of Maxwell’s equations. The second effect we model is the coupling of surface plasmon polaritons at the interface between a metal layer and a photorefractive liquid crystal cell. We implement existing numerical models to gain a thorough understanding of the system. These models are qualitatively compared with experimental observations. Analytic approximations to describe the coupling of surface plasmon polaritons at the surface of the liquid crystal cell are developed. These expressions provide a great deal of insight into the coupling mechanisms and will be of fundamental importance in optimising these systems.

This thesis is concerned with magnetism at the surfaces of frustrated magnets, and in particular with magnetism on face-centred cubic (FCC) lattice systems. Normally, magnets do not react to a surface. Frustrated magnets do, however, and we consider two cases: Magnetic relaxation and the more unusual magnetic reconstruction phenomenon. Magnetic relaxation involves the extinction near the surface of a magnetic order that is present in the bulk and exists as a type of magnetic domain wall. Calculations of the ground state configuration of a semi-infinite system of uranium dioxide, an FCC triple-q magnet, show a solitonic solution corresponding to this relaxation. Fluctuations of this domain wall are considered in order to explain the unusual disordering observed experimentally in the near-surface region at a temperature below the bulk ordering transition temperature. The rarer case of reconstruction involves completely new magnetic order from the bulk appearing at the surface spontaneously and at a temperature below the bulk ordering transition temperature. Analysis of this phenomenon is undertaken via the construction of a phase diagram for a frustrated square lattice Heisenberg model. Regions of the phase diagram are found to exist in which the reconstruction is expected to occur, and furthermore the results can be mapped directly to type-1 FCC lattice systems.

In this thesis, thermodynamic-based mathematical modelling is combined with experimental in vitro extracellular flux analysis in order to assess drug redox cycling and cellular bioenergetics. It is widely accepted that pharmacological activity of certain classes of drugs (e.g. anticancer, antimalarial) is related to their ability to accept one or two electrons. However, pharmacological activity via redox cycling is an understated mechanism of toxicity associated with many classes of drugs. In particular, oxidative stress as a result of redox cycling plays a pivotal role in the cause of cardiac toxicity. For example, doxorubicin is an anti-neoplastic drug used to treat cancer. It has strong links to redox cycling-induced cardiac toxicity associated directly with elevated levels of reactive oxygen species (ROS) and oxidative stress within the mitochondria. The underlying mechanisms of redox cycling is very difficult to elucidate, due to the fleeting existence of the radical species. However, assessment of such cellular bioenergetics can be ameliorated with the aid of computational assistance. In chapter 2 the development of a novel thermodynamic-based in silico model of doxorubicin redox cycling is described, which is parameterized using data from in vitro extracellular flux analysis. The model is used to simulate mitochondrial-specific ROS, with its outputs confirmed against in vitro data. Chapter 3 describes construction of a pH-dependent thermodynamic model of hepatic glycolytic flux, used to determine the role of the monocarboxylate transporter 1 flux during extracellular acidification. Finally, chapter 4 describes a thermodynamic-based in silico model of mitochondrial bioenergetics, capable of simulating oxygen consumption rates of a cohort of in vitro human primary hepatocyte data. The model is finally used to simulate perturbations in key bioenergetic variables and reaction fluxes, illustrating the resulting changes on mitochondrial pH, membrane potential and subsequent oxygen consumption rates.