Browsing by Author "Scott, Matthew"

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The Effects of Extrinsic and Intrinsic Noise on a Tumour and a Proposed Metastasis Model
(University of Waterloo, 2015-11-30) Tang, Herbert Hoi Chi; Sivaloganathan, Sivabal; Scott, Matthew
Cancer is a ubiquitous disease that afflicts millions of people worldwide and we will undoubtedly encounter it at some point in our lives, whether it be random strangers on the news, or someone much closer. As such, research into cancer (and cures for cancer) has been an intense area of focus. Malignant tumours show three main characteristics: 1) aggressive and uncontrolled growth, 2) invasion into surrounding tissue, and 3) the ability to leave the primary tumour site and invade another organ (metastasis). Mathematical models of these three aspects began in earnest about half a century ago, for example with Laird\cite{gompertz} in 1964 proposing that tumour growth was gompertzian. With the advent of the modern computer, complex partial differential equation (PDE) models of tumour growth which incorporate the other two features of malignant tumours have become accessible to the average researcher. Utilizing a mixture of both analytical and numerical methods, this thesis aims to add to the schema of cancer research by examining the effects of noise on a well established model of tumour growth, and a promising model for mitosis that we hope to eventually adapt to describe metastasis. The tumour model that we will examine is a single species reaction-diffusion model that captures the first two aspects of tumour growth. The reaction part determines how fast the tumour grows and the diffusion part describes how quickly the cancer cells spread within the domain. Adding noise to the model is a method of describing the disorder that is in a typical tumour, and allows us to determine error bounds on the tumour size and survival time, which gives us a sense of how accurate the estimates of those two quantities are. This method of estimating the uncertainty in survival time also allows us to further determine how modifications to the model (such as changing the diffusivity or adding chemotherapy) affects the error in the system. Our model of mitosis is an excitable system that admits a traveling wave as a possible solution. Mitosis is a process that occurs in a very spatially specific manner within the embryo and this model describes how a signal propagates from the centre of the embryo outwards. When we allow noise to perturb the system we will see that the model allows noise induced traveling waves in what would normally be a stable system in a deterministic setting. This means that the parameter range under which signalling can occur is relaxed and traveling waves occur more readily than expected. In terms of metastasis, where the tumour microenvironment is especially noisy and chaotic, that might explain why metastasis is so prevalent in malignant tumours. Adding noise to well established PDE models gives it an extra layer of fidelity that allows us to extract additional information from the model not available in the traditional deterministic setting. Using both a numerical and an analytical approach, this thesis develops and demonstrates general methods for estimating the uncertainty in the model outputs and also a method for calculating when a system might admit a noise induced instability in an otherwise stable system. More specifically, the error bounds in the estimates of tumour size and survival time in the tumour growth model may be of future use to doctors treating patients, and the noise induced effects of the mitosis model helps us to understand further how metastasis arises out of a developing tumour.
Estimating the Stochastic Bifurcation Structure of Cellular Networks
(Public Library of Science (PLOS), 2010) Song, Carl; Phenix, Hilary; Abedi, Vida; Scott, Matthew; Ingalls, Brian P.; Kaern, Mads; Perkins, Theodore J.
High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.
The Interactions of Graphene with Ionic Solutions and Their Effects on the Differential Capacitance for Sensing Applications
(University of Waterloo, 2019-08-28) Daniels, Lindsey; Miskovic, Zoran; Scott, Matthew
Nano-scale devices continue to challenge the theoretical understanding of microscopic systems. Of particular interest is the characterization of the interface electrochemistry of sensors, which operate as field effect transistors with graphene in contact with the solution. While plenty of experimental research has been conducted in regard to the viability and sensitivity of graphene-based devices, the understanding of the microscopic and macroscopic physics of these sensors has lagged, unlike any other areas of applications for graphene. Although some successful models of these sensors have been developed, relatively little theoretical work to account for the vast extent of experimental work. Typically operated in a regime of high ion concentration and high surface charge density, dielectric saturation, dielectric decrement, and ion crowding become non-negligible at the interface, complicating continuum treatments based upon the Poisson-Boltzmann equation. Modifications to the standard Poisson-Boltzmann theory are explored, with modifications due to dielectric saturation and dielectric decrement considered in tandem with a Bikerman-Friese model to account for the steric effects of ions. In the case of dielectric saturation, a model proposed by Booth is used to characterize the diffuse layer capacitance for both metallic and graphene electrodes immersed in an electrolyte. The dependence of the diffuse layer capacitance on the surface charge density of the electrode exhibits two peaks, in contrast to the experimental results. For dielectric decrement, a dielectric permittivity dependent on the concentration of positive and negative ions is used to determine the diffuse layer capacitance for both metallic and graphene electrodes. The diffuse layer capacitance shows a strong interplay between ion polarizability and steric effects, while exhibiting a single peak. A self-consistent and parameter-free method for the inclusion of a Stern layer is used in both cases, which eliminates the spurious secondary peak in the case of dielectric saturation and reduces the overall magnitude of the capacitance of the diffuse layer in both dielectric saturation and dielectric decrement. When a graphene electrode is used, the total capacitance in all modifications is dominated by V-shaped quantum capacitance of graphene at low potentials, which is a manifestation of the Dirac cone structure of the graphene $\pi$-electron bands. A broad peak develops in the total capacitance at high potentials, which is sensitive to the ion size at dielectric saturation, but is stable with dielectric decrement. In addition to the interactions of graphene with an electrolyte, considerable interest has recently been shown in studying the electric double layer that arises at the interface of doped graphene and a class of electrolytes known as ionic liquids. Ionic liquids are a class of molten ionic salts at room temperature that have low volatility and high ionic concentration, and are characterized by the overscreening and overcrowding effects in their electric double layer. A mean field model for ionic liquids is presented, which takes into account both the ion correlation and the finite ion size effects, in order to calculate the differential capacitance of the ionic liquid interface with single-layer graphene. Besides choosing ion packing fractions that give rise to the camel-shaped and bell-shaped capacitances of the diffuse layer in ionic liquids, the regimes of ``dilute electrolytes'' and asymmetric ionic liquids are considered. As in the case of electrolytes, the main effect of a graphene electrode arises due to its V-shaped quantum capacitance. As a result, the total capacitance of a graphene--ionic liquid interface exhibits a camel-shaped dependence on the total applied potential, even for large ion packing fractions and finite ion correlation lengths. While the minimum at the neutrality point in the total capacitance is``inherited” from the quantum capacitance of graphene, the two peaks that occur at applied potentials of $\sim \pm1$ V are sensitive to the presence of the ion correlation and a Stern layer, which both tend to reduce the height and flatten the peaks in the camel-shaped total capacitance. It is also determined that the largest fraction of the applied potential goes to charging the graphene electrode. When considering the sensitivity of graphene-based sensors to ion concentration and/or pH of the surrounding environment, a site binding model which allows hydrogen and hydroxyl groups to adsorb onto the surface of the device is proposed. Both a regime in which bare graphene is exposed to the electrolyte and a regime where a functionalized oxide, which contains a density of charged impurities to facilitate ion binding, is situated between graphene and the electrolyte are proposed. With regard to the dependence on ion concentration, comparisons between the model and experimental data show good agreement when the finite size of ions is included in the electrolyte. In the case of pH dependence, comparisons between the model and experimental data show excellent agreement, particularly when steric effects are included in the electrolyte. The favourable comparisons here are the first steps in developing a comprehensive model of graphene based biological and chemical sensors.
Mutation rates of Escherichia coli with different balanced growth rates: a new fluctuation test protocol and phenotypic lag adjustments
(University of Waterloo, 2020-10-23) Barna, Christian Henderson; Scott, Matthew
Bacteria are the oldest, most abundant life form on the planet, and every other organism’s livelihood is dependent on them. The bacteria Escherichia coli (E. coli) is commonly used in microbiology as a model organism to give insight into the functions of bacteria and cells in general. Of particular interest in these studies is the methods with which bacteria grow and evolve. Growth is what propagates a bacteria's species; whereas evolution is what allows them to adapt to the ever-changing world. Evolution is made possible by mutations which change a bacterium's DNA. In 1943, Luria and Delbrück developed a method, called a "fluctuation test", to estimate mutation rates from the number of mutants in a collection of parallel cultures exposed to a selecting agent after growth. The original fluctuation test methodology suffers from two major limitations. First, the bacteria are not in a reproducible, balanced state of growth throughout the test. Second, the new phenotype resulting from a mutation may not be immediately expressed (referred to as "phenotypic lag") resulting in an underestimated mutation rate. To overcome these issues, I developed a refined experimental protocol that ensures cells are in balanced growth and a suite of analysis tools that account for the effects of phenotypic lag. To test the methodology, I compared the mutation rate and phenotypic lag in fast growing E. coli (23 minutes per doubling) and slow growing E. coli (48 minutes per doubling). It is found that when not accounting for phenotypic lag, fast growing E. coli have a markedly lower mutation rate than slow growing E. coli, but when phenotypic lag is accounted for, the faster growing cells have a longer phenotypic lag, resulting in an indistinguishable mutation rate for fast and slow growing populations. The implications of mutation rate being coupled to growth rate, as well as possible explanations for why it and phenotypic lag would be growth rate dependent are discussed. Finally, possible ways to improve the experimental methodology and analysis protocols, in addition to future experiments that can be performed to further explore mutation rate - growth rate coupling are proposed.