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In this thesis, we introduce several closely related stochastic models which generalize the deterministic Hodgkin-Huxley formalism to an SDE framework. We provide analytical results on the existence and uniqueness of solutions as well as the exact formulas for the moments of a simplified model, with simplifications motivated by the experiments performed by Hodgkin and Huxley in their seminal paper.For more complicated models, we provide an approach for the approximation and simulation of solutions to the corresponding SDEs, and show several realizations of the sample paths and moments of these simulations to verify qualitative behavior in this case. All code for the project is written in the Julia language and can be obtained upon request by the reader.

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Mathematical models of viral infections have been informing virology research for years. Estimating parameter values for these models can lead to understanding of biological values. This has been successful in HIV modeling for the estimation of values such as the lifetime of infected CD8 T-Cells. However, estimating these values is notoriously difficult, especially for highly complex models. We use Bayesian inference and Monte Carlo Markov Chain methods to estimate the underlying densities of the parameters (assumed to be continuous random variables) for three models of influenza. We discuss the advantages and limitations of parameter estimation using these methods. The data and influenza models used for this project are from the lab of Dr. Amber Smith in Memphis, Tennessee. / Master of Science

This dissertation develops algebraic theory for discrete models in systems biology. Many discrete model types can be translated into the framework of polynomial dynamical systems (PDS), that is, time- and state-discrete dynamical systems over a finite field where the transition function for each variable is given as a polynomial. This allows for using a range of theoretical and computational tools from computer algebra, which results in a powerful computational engine for model construction, parameter estimation, and analysis methods. Formal definitions and theorems for PDS and the concept of PDS as models of biological systems are introduced in section 1.3. Constructing a model for given time-course data is a challenging problem. Several methods for reverse-engineering, the process of inferring a model solely based on experimental data, are described briefly in section 1.3. If the underlying dependencies of the model components are known in addition to experimental data, inferring a "good" model amounts to parameter estimation. Chapter 2 describes a parameter estimation algorithm that infers a special class of polynomials, so called nested canalyzing functions. Models consisting of nested canalyzing functions have been shown to exhibit desirable biological properties, namely robustness and stability. The algorithm is based on the parametrization of nested canalyzing functions. To demonstrate the feasibility of the method, it is applied to the cell-cycle network of budding yeast. Several discrete model types, such as Boolean networks, logical models, and bounded Petri nets, can be translated into the framework of PDS. Section 3 describes how to translate agent-based models into polynomial dynamical systems. Chapter 4, 5, and 6 are concerned with analysis of complex models. Section 4 proposes a new method to identify steady states and limit cycles. The method relies on the fact that attractors correspond to the solutions of a system of polynomials over a finite field, a long-studied problem in algebraic geometry which can be efficiently solved by computing Gröbner bases. Section 5 introduces a bit-wise implementation of a Gröbner basis algorithm for Boolean polynomials. This implementation has been incorporated into the core engine of Macaulay 2. Chapter 6 discusses bistability for Boolean models formulated as polynomial dynamical systems. / Ph. D.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: An integro-differential equation describing the dynamical behaviour of phytoplankton cells is considered in which the effects of cell division and aggregation are incorporated by coupling the coagulation-fragmentation equation with growth, and the McKendrick-von Foerster renewal model of an age-structured population. Under appropriate conditions on the model parameters, the associated initial-boundary value problem is shown to be well posed in a physically relevant Banach space using the theory of strongly continuous semigroups of operators, the theory of perturbation of positive semigroups and the semilinear abstract Cauchy problems theory. In particular, we provide sufficient conditions for honesty of the model. Finally, the results on the effects of the growth-fragmentation-coagulation on the overall evolution of the phytoplankton population are summarised. / AFRIKAANSE OPSOMMING: ’n Integro-differensiaalvergelyking wat die dinamiese ontwikkeling van fitoplanktonselle beskryf, word beskou. Die uitwerking van seldeling en -aggregasie is geïnkorporeer deur die vergelyking van koagulasie en fragmentasie met groeiaan die McKendrick-von Foerster hernuwingsmodel van ’n ouderdomsgestruktureerde populasie te koppel. Die teorie van sterk kontinue semigroepe van operatore, steuringsteorie van positiewe semigroepe en die teorie van semilineêre abstrakte Cauchy probleme word aangewend om, onder gepaste voorwaardes met betrekking tot die model se parameters, te bewys dat die geassosieerde beginwaarde-probleem met randvoorwaardes ‘goed gestel’ is in ’n fisies relevante Banach-ruimte. In die besonder word voldoende voorwaardes vir eerlikheid van die model verskaf. Ten slotte word ’n opsomming van die resultate met betrekking tot die gekombineerde uitwerking van groei-fragmentasie- koagulasie op die gesamentlike ontwikkeling van die fitoplanktonpopulasie verskaf.

The ability of cells to move through their environment and spread on surfaces is fundamental to a host of biological processes; including wound healing, growth and immune surveillance. Controlling cell motion has wide-ranging potential for medical applications; including prevention of cancer metastasis and improved colonisation of clinical implants. The relevance of the topic coupled with the naturally arising interplay of biomechanical and biochemical mechanisms that control cell motility make it an exciting problem for mathematical modellers. Two-phase flow models have been widely used in the literature to model cell motility; however, little is known about the mathematical properties of this framework. The majority of this thesis is dedicated to improving our understanding of the two-phase flow framework. We first present the simplest biologically plausible two-phase model for a cell crawling on a flat surface. Stability analyses and a numerical study reveal a number of features relevant to modelling cell motility. That these features are present in such a stripped-down two-phase flow model is notable. We then proceed to investigate how these features are altered in a series of generalisations to the minimal model. We consider the effect of membrane-regulated polymerization of the cell's actin network, the effect of describing the network as viscoelastic, and the effect of explicitly modelling myosin, which drives contraction of the actin network. Validation of hydrodynamical models for cell crawling and spreading requires data on cell shape. The latter part of the thesis develops an image processing routine for extracting the three-dimensional shape of cells settling on a flat surface from confocal microscopy data. Models for cell and droplet settling available in the literature are reviewed and we demonstrate how these could be compared to our cell data. Finally, we summarise the key results and highlight directions for future work.