Modeling cell movement in anisotropic and heterogeneous network tissues

Cell motion and interaction with the extracellular matrix is studied deriving a kinetic model and considering its diffusive limit. The model takes into account of chemotactic and haptotactic effects, and obtains friction as a result of the interactions between cells and between cells and the fibrous environment. The evolution depends on the fibre distribution, as cells preferentially move along the fibre direction and tend to cleave and remodel the extracellular matrix when their direction of motion is not aligned with the fibre direction. Simulations are performed to describe the behavior of ensemble of cells under the action of a chemotactic field and in presence of heterogeneous and anisotropic fibre networks

From microscopic to macroscopic descriptions of cell\ud migration on growing domains

Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs

Taxis Equations for Amoeboid Cells

The classical macroscopic chemotaxis equations have previously been derived from an individual-based description of the tactic response of cells that use a "run-and-tumble" strategy in response to environmental cues. Here we derive macroscopic equations for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly. We present several models of increasing complexity for which the derivation of population-level equations is possible, and we show how experimentally-measured statistics can be obtained from the transport equation formalism. We also show that amoeboid cells that do not adapt to constant signals can still aggregate in steady gradients, but not in response to periodic waves. This is in contrast to the case of cells that use a "run-and-tumble" strategy, where adaptation is essential.Comment: 35 pages, submitted to the Journal of Mathematical Biolog

Modelling cell motility and chemotaxis with evolving surface finite elements

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction-diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/maskae/CV_Warwick/Chemotaxis.html

Implementing vertex dynamics models of cell populations in biology within a consistent computational framework

The dynamic behaviour of epithelial cell sheets plays a central role during development, growth, disease and wound healing. These processes occur as a result of cell adhesion, migration, division, differentiation and death, and involve multiple processes acting at the cellular and molecular level. Computational models offer a useful means by which to investigate and test hypotheses about these processes, and have played a key role in the study of cell–cell interactions. However, the necessarily complex nature of such models means that it is difficult to make accurate comparison between different models, since it is often impossible to distinguish between differences in behaviour that are due to the underlying model assumptions, and those due to differences in the in silico implementation of the model. In this work, an approach is described for the implementation of vertex dynamics models, a discrete approach that represents each cell by a polygon (or polyhedron) whose vertices may move in response to forces. The implementation is undertaken in a consistent manner within a single open source computational framework, Chaste, which comprises fully tested, industrial-grade software that has been developed using an agile approach. This framework allows one to easily change assumptions regarding force generation and cell rearrangement processes within these models. The versatility and generality of this framework is illustrated using a number of biological examples. In each case we provide full details of all technical aspects of our model implementations, and in some cases provide extensions to make the models more generally applicable

Moments of von Mises and Fisher distributions and applications

The von Mises and Fisher distributions are spherical analogues to the Normal distribution on the unit circle and unit sphere, respectively. The computation of their moments, and in particular the second moment, usually involves solving tedious trigonometric integrals. Here we present a new method to compute the moments of spherical distributions, based on the divergence theorem. This method allows a clear derivation of the second moments and can be easily generalized to higher dimensions. In particular we note that, to our knowledge, the variance-covariance matrix of the three dimensional Fisher distribution has not previously been explicitly computed. While the emphasis of this paper lies in calculating the moments of spherical distributions, their usefulness is motivated by their relationship to population statistics in animal/cell movement models and demonstrated in applications to the modelling of sea turtle navigation, wolf movement and brain tumour growth

Modeling cell movement in anisotropic and heterogeneous network tissues

Cell motion and interaction with the extracellular matrix is studied deriving a kinetic model and considering its diffusive limit. The model takes into account of chemotactic and haptotactic effects, and obtains friction as a result of the interactions between cells and between cells and the fibrous environment. The evolution depends on the fibre distribution, as cells preferentially move along the fibre direction and tend to cleave and remodel the extracellular matrix when their direction of motion is not aligned with the fibre direction. Simulations are performed to describe the behavior of ensemble of cells under the action of a chemotactic field and in presence of heterogeneous and anisotropic fibre networks

Device Modelling of Perovskite Solar Cells

This thesis is primarily concerned with the electrical characterization and modelling of perovskite solar cells. Perovskite cells are a new player in the photovoltaic arena with several intriguing properties. One of these is the presence of intrinsic mobile ions which make these semiconductors simultaneously ionic conductors at room temperature. The presence of mobile ions is significant in that it leads to a number of transient behaviours in optoelectronic measurements, including nominally simple current-voltage measurements where the phenomena are broadly labelled as aspects of ``I-V hysteresis''. The first two-thirds of this thesis describes our work on extended drift-diffusion models which incorporate the presence of mobile ions into the conventional equations of semiconductor physics. These allow us to uncover mechanistic explanations for a variety of transient behaviours which are broadly caused by coupling between electronic and ion dynamics. The first third (Chapter 2) deals with hysteresis in the form of rate-dependent I-V sweeps: a selection of unusual measurements of this type is presented including a temporary enhancement in open-circuit voltage following prolonged periods of negative bias, dramatically S-shaped current-voltage sweeps, decreased current extraction following positive biasing or ``inverted hysteresis'', and non-monotonic transient behaviours in the dark and the light. This initial study is supplemented with a more in-depth investigation of inverted hysteresis and its correlation with band-alignment. The second third (Chapter 3) delves deeper into electrical characterization with a first-principles study of electrical impedance spectroscopy. We focus on accounting for features in the measured capacitance spectrum (sufficient for a full account of the total impedance due to the Kramers-Kronig relations) of standard-structure (non-inverted) perovskite cells. Here our models make clear the necessity of distinguishing fundamental contributions to the measured capacitance due to charging, from those due to currents delayed by slow processes such as ion migration. With this distinction clearly established we provide a detailed account of all the major features observed in impedance measurements of these cells, including the exotic and previously puzzling appearance of giant photo-induced capacitance, loop features and negative capacitance. The final part of this thesis in Chapter 4 concerns the integration of perovskite cells into tandem arrangements with a partner such as the crystalline silicon cell. Of relevance to any thin-film solar cell, and to 4-terminal tandem cells in particular, is the specifications of its transparent conductor layers. We analyze transparent conductor requirements under different regimes of metallization (the addition of metallic bus-bars or fingers). Here a key parameter is the minimal achievable wire width, which dictates the necessary tradeoff between transparency and conductivity in the underlying transparent conductor. We identify \SI{30}{\micro \metre} as a critical width below which many emerging transparent conducting layers such as carbon nanotubes and graphene become competitive with state-of-the-art transparent conductive oxides such as ITO for a stand-alone perovskite cell. We also discuss a novel strategy for integrating perovskite and Si cells into a single monolithic structure without the need for a tunnel junction or recombination layer. This is identified as being possible due to the presence of interfacial sub-gap states which can facilitate high-conductivity ohmic contact between TiO$_2$ and p-type Si, and has significant advantages in terms of reducing optical losses and processing steps