Structure-preserving, or mimetic, discretisations of scalar-valued differential forms have been successfully applied to various fields in physics, engineering and computational modelling. However, scant attention has been paid to higher-order discretisations of differential forms with values in a vector bundle, despite the great advantages such discretisations would confer. A major obstacle to this endeavour is the fact that vectors on a general manifold are each defined in the tangent space to the manifold as a combination of local basis vectors. It is not immediately apparent how vectors on a curved manifold can be interpolated with scalar polynomials, as the local basis varies from point to point.

In this thesis, a mimetic discretization of bundle-valued forms in two dimensions is presented. We show that, for a 0-form, the exterior covariant derivative and the projector commute, and that the second exterior covariant derivative recovers the action of the curvature 2-form in the limit of mesh refinement. Covariant versions of the standard incidence and Hodge matrices are also introduced. These operators are used to compute the exterior covariant derivative and to solve the Hodge Laplacian on the unit 2-sphere, with spectral convergence being achieved in both cases.
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Models governed by systems of ordinary differential equations (ODEs) often produce complex and unpredictable behaviors. To address this, we can use dimension reduction techniques, which simplify these models, allowing for the retention of specific behaviors while greatly decreasing the cost of numerical solutions and, in some cases, enabling analytical derivations of sufficient conditions for the existence of nonzero fixed points of the model’s ODEs. This thesis reviews the state-of-the-art reduction theories and extends the established proofs by Wu et al., by providing necessary assumptions, lemmas, and a novel proof of an important proposition used in their work. We additionally verify and confirm Wu et al.’s findings and predictions for a cooperative version of the Cowan-Wilson model, which describes a population of neurons’ firing activity. We derived a one-dimensional reduction, inspired by Laurence et al., for a generalized Cowan-Wilson model, which we introduced in this thesis. Unlike the original, this generalized model can produce oscillatory behavior without external stimulus. A valuable finding is that the method of reduction does not depend on the specific form of the Cowan-Wilson function, allowing it to be applied to a broader class of nonlinear dynamics. Our reduction is able to predict system behavior effectively, given the network yields a unique reduction. However, the reduction parameter was not unique in approximately half of the networks studied, which saw a 66% increase in average error, suggesting these networks are inherently multi-dimensional. This opens the door for future research into the existence of a multi-dimensional reduction framework that could mitigate this discrepancy. ... Read more

Incorporating living cells into a non-living matrix is one of the many possible steps that can be undertaken to stop climate change. Especially, photosynthetic organisms have a promising future in material design as they can capture atmospheric carbon dioxide and ’ breathe’ oxygen. This research dives into unravelling the properties of a hydrogel-based living material containing C. reinhardtii.
To facilitate a systematic exploration, this goal was divided into three smaller pieces. Firstly, the study delves into the mechanical characteristics, aiming to identify the most suitable bio-ink crosslink technique and composition. To validate the mechanical properties the living material was subjected to rheology and a bridging test. Concluded can be that crosslinking and algae growth improve the mechanical stability of the material, whereas, gelatin did not. The sagging behaviour of the material looked promising.
Secondly, the photosynthetic activity of this living material was researched. It was found that the rise of O2 levels can not be measured accurately, the non-living matrix can release a high amount of CO2 of over 20.000 ppm and airtightness poses a complex challenge in this field of research.
Lastly, the project incorporates attempts to find effective techniques for studying the livingness of this unique material. In this part of the research, inverted optical microscopy, 3D laser scanning microscopy and chlorophyll extraction were discarded as suitable methods to study the livingness of the algae material. It was proven that leveraging the autofluorescence of the algal chlorophyll confocal laser scanning microscopy gives high-resolution images and the livingness of this living material could be studied with this technique in the near future.
This research significantly contributes to our understanding of this hydrogel-based living material and its many challenging properties. It underscores the importance of innovative materials like these in addressing contemporary environmental challenges, particularly in carbon capture. Moreover, it highlights the complexity of characterizing such materials, paving the way for further exploration and development in this relatively new field. ... Read more

In this thesis, a proof of concept was established for the use of a novel coupled QM-MD approach to modelling metallic (copper) electrode-electrolyte interfaces. SCC-DFTB calculations of the instantaneous electronic structure of a copper electrode were coupled to a classical MD simulation of an electrode-electrolyte interface. The applied QM-MD method was described rigorously, and used to investigate the compound distribution and dynamics at the interface, relative to a fully classical MD simulation. Polarisation effects were observed to bring about a significant increase in the attraction between cations and the cathode. Moreover, local polarisation of the cathode was found to immobilise adsorbed cations, and induce an increased orientational preference of the nearby water dipoles. The secondary goal of this thesis was to explore to what extent neural networks are able to replicate SCC-DFTB calculations of the electronic charge density on a metallic electrode. Using a computer vision approach, qualitative evidence was obtained indicating that neural networks can be used to replicate SCC-DFTB predictions on periodic metallic surfaces. ... Read more

In animal cells, cell shape is primarily regulated by the actin cortex, a thin filament network connected to the plasma membrane. The architecture of the cortex is considered a key regulator of its function. Visualising the cortex is difficult due to the high density of numerous small-sized proteins involved in its formation. Consequently, the structure of the cortex at the membrane remains poorly studied. This study aims to gain insight into the organisation at the membrane of two key cortical components, human septin and septin-recruited actin. To study these filament structures, we reconstitute minimal cortices on supported lipid layers as model-membranes, allowing for imaging with electron and atomic force microscopy. We show that membrane binding of human septin results in ordered organisations of filaments. However, we find septin organises into arrays of paired filaments when incubated on a lipid monolayer and networks of bundles when incubated on a lipid bilayer. In addition, we showed a proof of concept for actin cortex reconstitution on lipid monolayers, which allowed us to see actin recruitment by septin meshworks. ... Read more

In this thesis, the diffusive limit of active particle motion in R^d is studied via a technique based on homogenisation. Thereafter, this study is extended to active particle motion on a Riemannian manifold.

Furthermore, as an application of active particle motion, a connection is made with the Dirac equation. On the basis of this connection, a Monte Carlo method is developed to find the ground state of a Dirac equation with static potential. The core idea of this method is based on the Diffusion Monte Carlo method for the Schrödinger equation. ... Read more

Implementing a correction to vesicle fluctuation analysis that accomodates non-zero exposure times, analysing the effects of the changes. ... Read more

Predicting off-target cleavage sites for CRISPR/Cas nucleases on the human genome based on biophysical modelling and high-throughput biochemical profiling. ... Read more

In this thesis the CRISPR-Cas9 mechanism, a promising mechanism for geneediting, is considered. Closed form expressions are derived for the probability and time to cleave or unbind for the associated Cas9 protein. The mechanism can be modelled mathematically by a birth and death process, therefore the expressions could be derived using Markov chains and semigroups. The expressions are compared to simulations and interpreted using the model of hybridization kinetics. Finally the moment generating function of the stopping time is derived for two special Markov processes, i.e. a random walk and a Brownian motion with drift. This was done using martingales. ... Read more

Cells of the most common organisms like plants and animals are filled with polymeric networks that fulfil important functions of the cell. There is however no analytically solvable model that describes diffusion in such a cell. This thesis presents a model for diffusion in polymeric environments, and some predictions about the behaviour of the model are made and confirmed by simulations.
Furthermore, the Fokker-Planck equation of this problem is studied, in order to solve the problem. Certain approximations are presented and solved, and it is investigated when the approximations are sound. Moreover, a method is described that can derive a solution to an equation with certain boundary conditions, from a solution to the same equation with different boundary conditions. This thesis also shows how this novel method can be applied to this model to find a non-approximated solution to the equation, where this was not
possible without this method. Finally, it is described how a multitude of partial differential equations that are linked to each other via the boundary conditions can be solved, can be solved using the method described. ... Read more