Mathematical models have long been used to describe complex biological interactions with the aim of predicting mechanistic interactions hard to distinguish from data. This dissertation uses modeling, mathematical analyses, and data fitting techniques to provide hypotheses on the mechanisms of immune response formation and function. The immune system, comprised of the innate and adaptive immune responses, is responsible for protecting the body against invading pathogens, with disease or vaccine induced immune memory leading to fast responses to subsequent infections. While there is some agreement about the underlying mechanisms of adaptive immune memory, innate immune memory is poorly understood. Stimulation with lipopolysaccharide induces differential phenotypes in innate immune cells depending on the strength of the stimulus, such that a secondary lipopolysaccharide encounter of a constant dose results in either strong or weak inflammatory cytokine expression. We model the biochemical kinetics of three molecules involved in macrophages responses to lipopolysaccharide and find that once a macrophage is programed to show a weak inflammatory response this cannot be reverted. Contrarily, a secondary lipopolysaccharide stimulus of a very high dose or applied prior to waning of the effects of the primary stimulus can induce a phenotype switch in macrophages initially programed to show strong inflammatory responses. Some pathogens, such as the hepatitis B virus, have developed strategies that hinder an efficient innate immune response. Hepatitis B virus infection is a worldwide pandemic with approximately 257 million chronically infected people. One beneficial event in disease progression is the seroclearance of hepatitis B e antigen often in combination with hepatitis B antibody formation. We propose mathematical models of within-host interactions and use them to predict that hepatitis B e antibody formation causes hepatitis B e antigen seroclearance and the subsequent reactivation of cytotoxic T cell immune responses. We use the model to quantify the time between antibody formation and antigen clearance and the average monthly hepatocyte turnover during that time. We further expand the study of hepatitis B infection, by investigating the kinetics of the virus under an experimental drug administered during a clinical trial. Available drugs usually fail to induce hepatitis B s antigen clearance, defined as the functional cure point of chronic hepatitis B infections. Drug therapy clinical trials that combined RNA interference drug ARC-520 with entecavir have shown promising results in reducing hepatitis B s antigen titers. We develop pharmacokinetic-pharmacodynamic models describing the mechanistic interactions of the drugs, hepatitis B virus DNA, and virus proteins. We fit the model to clinical trial data and predict that ARC-520 alone is responsible for the reduction of hepatitis B s and e antigens, while entecavir is the driving force behind viral reduction. This work was supported by Simons Foundation, Grant No. 427115, and National Science Foundation, Grant No. 1813011. / Doctor of Philosophy / Mathematical models have long been used to describe complex biological interactions with the aim of predicting interactions that explain observed data and informing new experiments. This dissertation uses modeling, mathematical analyses, and data fitting techniques to provide hypotheses on the mechanisms of immune response formation and function. The immune system, comprised of the innate and adaptive immune responses, is responsible for protecting the body against invading pathogens, such as viruses, bacteria, or fungi. If an immune response to a secondary pathogen encounter differs from the response when the body first encounters the specific pathogen, this is called immune memory. The mechanisms underlying the memory of immune responses are well understood in the context of adaptive immune responses, but less so for innate immune responses. Stimulation with lipopolysaccharide, a cell wall component of many bacteria, programs innate immune cells, such as macrophages, to be in one of two states, called phenotypes, depending on the strength of the stimulus. Based on their phenotype the macrophages show either a weak or strong inflammatory response upon a secondary lipopolysaccharide encounter of a constant dose. We model the biochemical kinetics of three molecules involved in macrophages responses to lipopolysaccharide. We find that once a macrophage is programed to show a weak inflammatory response this cannot be reverted. Contrarily, a secondary lipopolysaccharide stimulus that is either of a very high dose or applied before the effects of the primary stimulus have waned, can induce a phenotype switch in macrophages initially programed to show strong inflammatory responses. Some pathogens, such as the hepatitis B virus, have developed strategies that hinder an efficient innate immune response. Hepatitis B virus infection is a worldwide pandemic with approximately 257 million chronically infected people. Hepatitis B e antigen is a protein that infected liver cells release into blood and that impairs adaptive immune responses. It is considered a beneficial event in disease progression, and called hepatitis B e antigen clearance, when hepatitis B e antigen becomes indetectable in a patient's blood. We propose mathematical models of interactions between liver cells, the virus, hepatitis B e antigens and hepatitis B e antibodies, which neutralize the antigens. We predict that antibody formation causes antigen clearance and a reactivation of immune responses. We furthermore use the model to quantify the time between antibody formation and antigen clearance and the average number of liver cells killed during that time. We further expand the study of hepatitis B infection, by investigating the kinetics of the virus under an experimental drug administered during a clinical trial. Available drugs rarely induce hepatitis B s antigen clearance, but clinical trials that combined a novel drug, called ARC-520, with the commonly used drug entecavir have shown promising results in reducing hepatitis B s antigen titers in the blood of infected patients. Following the clearance of hepatitis B s antigen, a protein that is released by infected cells and impairs adaptive immunity, the body usually has the capability to control the infection without medication. We develop mathematical models describing the interactions of the drugs, hepatitis B virus, and virus proteins. We fit the model to clinical trial data and predict that ARC-520 alone is responsible for the reduction of hepatitis B s and e antigens, while entecavir is the driving force behind viral reduction.

Circadian rhythms in mammals are self-sustained repeating activities driven by the circadian gene expression in cells, which is regulated at both transcriptional and posttranscriptional stages. In this work, we first used mathematical modeling to investigate the transcriptional regulation of circadian gene expression, with a focus on the mechanisms of robust genetic oscillations in the mammalian circadian core clock. Secondly, we built a coarse-grained model to study the post-transcriptional regulation of the rhythmicities of poly(A) tail length observed in hundreds of mRNAs in mouse liver. Lastly, we examined the application of Sobol indices, which is a global sensitivity analysis method, to mathematical models of biological oscillation systems, and proposed two methods tailored for the calculation of circular Sobol indices. In the first project, we modified the core negative feedback loop in a mathematical model of the mammalian genetic oscillator so that the unrealistic tight binding between the repressor PER and the activator BMAL1 is relaxed for robust oscillations. By studying the modified extended models, we found that the auxiliary positive feedback loop, rather than the auxiliary negative feedback loop, makes the oscillations more robust, yet they are similar when accounting for circadian rhythms (~24h period). In the second project, we investigated the regulation of rhythmicities in poly(A) tail length by four coupled rhythmic processes, which are transcription, deadenylation, polyadenylation, and degradation. We found that rhythmic deadenylation is the strongest contributor to the rhythmicity in poly(A) tail length and the rhythmicity in the abundance of the mRNA subpopulation with long poly(A) tails. In line with this finding, the model further showed that the experimentally observed distinct peak phases in the expression of deadenylases, regardless of other rhythmic controls, can robustly cluster the rhythmic mRNAs by their peak phases in poly(A) tail length and abundance of the long-tailed subpopulation. In the last project, we reviewed the theoretical basis of Sobol indices and identified potential problems when it is applied to mathematical models of biological oscillation systems. Based on circular statistics, we proposed two methods for the calculation of circular Sobol indices and compared their performance with the original Sobol indices in several models. We found that though the relative rankings of the contribution from parameters are the same across three methods, circular Sobol indices can better quantitatively distinguish the contribution of individual parameters. Through this work, we showed that mathematical modeling combined with sensitivity analysis can help us understand the mechanisms underlying the circadian gene expression in mammalian cells. Also, testable predictions are made for future experiments and new ideas are provided that can enable potential chronopharmacology research. / Doctor of Philosophy / Circadian rhythms are repeating biological activities with ~24h period observed in most living organisms. Disruption of circadian rhythms in humans has been found to be promote cancer, metabolic diseases, cognitive degeneration etc. In this work, we first used mathematical modeling to study the mechanisms of robust oscillations in the mammalian circadian core clock, which is a molecular regulatory network that drives circadian gene expression at transcriptional stage. Secondly, we built a coarse-grained model to investigate the post-transcriptional regulation of the rhythmicities in poly(A) tail length, which are observed in hundreds of mRNAs in mouse liver. Lastly, we examined the application of Sobol indices, which is a global sensitivity analysis method, to mathematical models of biological oscillation systems, and proposed two methods tailored for the calculation of circular Sobol indices. In the first project, we modified a previous mathematical model of the mammalian genetic oscillator so that it sustains robust oscillation with more realistic parameter values. Our analysis of the model further showed that the auxiliary positive feedback loop, rather than the auxiliary negative feedback loop, makes the oscillations more robust. In the second project, we found that rhythmic deadenylation, among the coupled transcription, polyadenylation, and degradation processes, mostly controls the rhythmicity of poly(A) tail length and mRNA subpopulation with long poly(A) tails. Lastly, we reviewed the theoretical basis of Sobol indices and found potential problems when it is applied to mathematical models of biological oscillation systems. Based on circular statistics, we proposed two circular Sobol indices, which can better distinguish the contribution of individual parameters to model outputs than the original Sobol indices. Altogether, we used mathematical modeling and sensitivity analysis to investigate the regulation of circadian gene expression in mammalian cells, providing testable predictions and new ideas for future experiments and chronopharmacology research.

Systems biology is an interdisciplinary field that focuses on elucidating complex biological processes (systems) by investigating the interactions among its components through an iterative cycle composed of data generation, data analysis and mathematical modeling. Our contributions to systems biology revolve around the following two axes: - Data analysis: Two data analysis projects, which were initiated when I was a co-op at GlaxoSmithKline, are discussed in this thesis. First, next generation sequencing data generated for a phase I clinical trial is analyzed to determine the altered microbial community in human gut before and after antibiotic usage (Chapter 2). To our knowledge, there have not been similar comparative studies in humans on the impacts on the gut microbiome of an antibiotic when administered by different modes. Second, publicly available gene expression data is analyzed to investigate human immune response to tuberculosis (TB) infection (Chapter 3). The novel feature of this study is systematic drug repositioning for the prevention, control and treatment of TB using the Connectivity map. - Mathematical modeling: Polynomial dynamical systems, a state- and time- discrete logical modeling framework, is used to model two biological processes. First, a denitrification pathway in Pseudomonas aeruginosa is modeled to shed light on the reason of greenhouse gas nitrous oxide accumulation (Chapter 4). It is the first mathematical model of denitrification that can predict the effect of phosphate on the denitrification performance of this bacterium. Second, an iron homeostasis pathway linked to iron utilization, oxidative stress response and oncogenic pathways is constructed to investigate how normal breast cells become cancerous (Chapter 5). To date, our intracellular model is the only expanded core iron model that can capture a breast cancer phenotype by overexpression and knockout simulations. / Ph. D.

Antiviral therapy for patients infected with hepatitis B virus is only partially efficient. The field is in high demand for understanding the connections between the virus, immune responses, short-term and long-term drug efficacy and the overall health of the liver. A mathematical model was introduced in 2009 to help elucidate the host-virus dynamics after the start of therapy. The model allows the study of complicated viral patterns observed in HBV patients. In our research, we will analyze this model to determine the biological markers (e.g. liver proliferation, immune responses, and drug efficacy) that determine the different decay patterns. We will also investigate how such markers affect the length of therapy and the amount of liver damage. / Master of Science

Yes / The coupling of thermal (Multi Stage Flash, MSF) and membrane processes (Reverse Osmosis, RO) in desalination systems has been widely presented in the literature to achieve an improvement of performance compared to an individual process. However, very little study has been made to the combined Multi Effect Distillation (MED) and Reverse Osmosis (RO) processes. Therefore, this research investigates several design options of MED with thermal vapor compression (MED_TVC) coupled with RO system. To achieve this aim, detailed mathematical models for the two processes are developed, which are independently validated against the literature. Then, the integrated model is used to investigate the performance of several configurations of the MED_TVC and RO processes in the hybrid system. The performance indicators include the fresh water productivity, energy consumption, fresh water purity, and recovery ratio. Basically, the sensitivity analysis for each configuration is conducted with respect to seawater conditions and steam supply variation. Most importantly, placing the RO membrane process upstream in the hybrid system generates the overall best configuration in terms of the quantity and quality of fresh water produced. This is attributed to acquiring the best recovery ratio and lower energy consumption over a wide range of seawater salinity.

Yes / Adequate drug distribution through tumours is essential for treatment to be effective. Palbociclib is a cyclin-dependent kinase (CDK) 4/6 inhibitor approved for use in patients with hormone receptor (HR) positive, HER2 negative metastatic breast cancer (BC). It has unusual physicochemical properties, which may significantly influence its distribution in tumour tissue. We studied the penetration and distribution of palbociclib in vitro, including the use of multicellular three-dimensional models and mathematical modelling. MCF-7 and DLD-1 cell lines were grown as single cell suspensions (SCS) and spheroids; palbociclib uptake and efflux were studied using liquid chromatography-tandem mass spectrometry (LC-MS/MS). Intracellular concentrations of palbociclib for MCF-7 SCS (Cmax 3.22 µM) and spheroids (Cmax 2.91 µM) were 32 and 29 fold higher and in DLD-1, 13 and 7 fold higher, respectively than the media concentration (0.1 µM). Total palbociclib uptake was lower in DLD-1 cells than MCF-7 cells both in SCS and in spheroids. Both uptake and efflux of palbociclib were slower in spheroids than SCS. These data were used to develop a mathematical model of palbociclib transport that quantifies key parameters determining drug penetration and distribution. The model reproduced qualitatively most features of the experimental data and distinguished between SCS and spheroids, providing additional support for hypotheses derived from the experimental data. Mathematical modelling has the potential for translating in vitro data into clinically relevant estimates of tumour drug concentrations. / Grant for Translational Research and a grant from Leeds NHS Charitable Trustees.

This dissertation unravels intricate biological dynamics in three distinct biological systems as the following. These studies combine mathematical models with experimental data to enhance our understanding of these complex processes. 1. Bipolar Spindle Assembly: Mitosis relies on the formation of a bipolar mitotic spindle, which ensures an even distribution of duplicated chromosomes to daughter cells. We address the issue of how the spindle can robustly recover bipolarity from the irregular forms caused by centrosome defects/perturbations. By developing a biophysical model based on experimental data, we uncover the mechanisms that guide the separation and/or clustering of centrosomes. Our model identifies key biophysical factors that play a critical role in achieving robust spindle bipolarization, when centrosomes initially organize a monopolar or multipolar spindle. These factors encompass force fluctuations between centrosomes, balance between repulsive and attractive inter-centrosomal forces, centrosome exclusion from the cell center, proper cell size and geometry, and limitation of the centrosome number. 2. Chromosome Oscillation: During mitotic metaphase, chromosomes align at the spindle equator in preparation for segregation, and form the metaphase plate. However, these chromosomes are not static; they exhibit continuous oscillations around the spindle equator. Notably, either increasing or decreasing centromeric stiffness in PtK1 cells can lead to prolonged metaphase chromosome oscillations. To understand this biphasic relationship, we employ a force-balance model to reveal how oscillation arises in the spindle, and how the amplitude and period of chromosome oscillations depend on the biological properties of spindle components, including centromeric stiffness. 3. Pattern Formation in Bacterial-Phage Systems: The coexistence of bacteriophages (phages) and their host bacteria is essential for maintaining microbial communities. In resource-limited environments, mobile bacteria actively move toward nutrient-rich areas, while phages, lacking mobility, infect these motile bacterial hosts and disperse spatially through them. We utilize a combination of experimental methods and mathematical modeling to explore the coexistence and co-propagation of lytic phages and their mobile host bacteria. Our mathematical model highlights the role of local nutrient depletion in shaping a sector-shaped lysis pattern in the 2D phage-bacteria system. Our model further shows that this pattern, characterized by straight radial boundaries, is a distinctive indicator of extended coexistence and co-propagation of bacteria and phages. Such patterns rely on a delicate balance among the intrinsic biological characteristics of phages and bacteria, which have likely arisen from the coevolution of cognate pairs of phages and bacteria. / Doctor of Philosophy / Mathematical modeling is a powerful tool for studying intricate biological dynamics, as modeling can provide a comprehensive and coherent picture about the system of interest that facilitates our understanding, and can provide ways to probe the system that are otherwise impossible through experiments. This dissertation includes three studies of biological dynamics using mathematical modeling: 1. Bipolar Spindle Assembly: Mitotic spindle is a bipolar subcellular structure that self-assembles during cell division. The spindle ensures an even distribution of duplicated chromosomes into two daughter cells. Certain perturbations can cause the spindle to assemble abnormally with one pole or more than two poles, which would cause the daughter cells to inherit incorrect number of chromosomes and die from the error. However, the cell is surprisingly good at correcting these spindle abnormalities and recovering the bipolar spindle. Here we build a model to explore how the cell achieves such recoveries and preferentially form a bipolar spindle to rescue itself. 2. Chromosome Oscillation: In mitotic metaphase, chromosomes are aligned at the spindle equator before they segregate. Interestingly, unlike the cartoon images in textbooks, the aligned chromosomes often move rhythmically around the spindle equator. We used a mathematical model to unravel how the chromosome oscillation arises and how it depends on the biological properties of the spindle components, such as stiffness of the centromere, the structure that connects the two halves of duplicated chromosomes. 3. Pattern Formation in Bacterial-Phage System: Phages are viruses that hijack their host bacteria for proliferation and spreading. In this study we developed a mathematical model to elucidate a common lysis pattern that forms when expanding host bacterial colony encounters phages. Interestingly, our model revealed that such a lysis pattern is a telltale sign that the bacterium-phage pair have achieved a delicate balance between each other and are capable of spreading together over a long period of time.

Conventional desalination plant, municipal water supply and wastewater treatment system are among the most electricity-intensive facilities. Microbial Desalination Cell (MDC) has emerged as a promising technique to capture the chemical energy stored in wastewater directly for desalination, which has the potential to solve the high energy consumption issue in desalination industry as well as wastewater treatment system. The MDC is composed of two critical components, the electrodes (anode and cathode), and the ion-exchange membranes separating the two electrodes which drive anions migrate towards the anode, and cations migrate towards the cathode. The multiple components allow us to manipulate the configuration to achieve most efficient desalination performance. By coupling with Donnan Dialysis or Microbial Fuel Cell, the device can effectively achieve boron removal which has been a critical issue in desalination plants. The uncertainty of water quality of the final desalinated water caused by contaminant back diffusion from the wastewater side can be theoretically explained by two mechanisms, Donnan exchange and molecule transport which are controlled by bioelectricity and concentration gradient. Scaling and fouling is also a factor needs to be taken into consideration when operating the MDC system in real world. With mathematical modeling, we can provide insight to bridge the gap between lab-scale experiments and industrial applications. This study is expected to provide guidance to enhance the efficiency as well as the reliability and controllability of MDC for desalination. / Ph. D. / Water and energy are the world’s most valuable resources. The recent emerging technology, Microbial Desalination Cell (MDC), however, can achieve wastewater treatment, desalination for fresh water production, and energy generation simultaneously. Owing to the anodophilic microorganisms working as organic matter consumer and electron generator, the wastewater can be cleaned and the device can generate electricity through electron flow to drive ion separation for salt removal in the solution. The MDC can be constructed in versatile configurations. Decoupled configuration of anode and cathode allows flexibility of operation and maintenance. Although the MDC has wastewater adjacent to seawater which are separated by a piece of anion exchange membrane, the microorganisms and viruses are effectively blocked by the membrane which has tiny pore size around 1 nm. Back diffusion of contaminants in wastewater into the desalinated water is minimal under bioelectricity generation condition. The MDC has proved to successfully remove various inorganic ions by itself as well as remove non-dissociable boron when coupled to other devices, such as Donnan Dialysis or Microbial Fuel Cell. The water product quality can meet irrigation guideline. Through mathematical modeling tools, we can better understand the MDC process, analyze it, and make informative predictions.

This thesis describes work on a new high-level mathematical modeling language and framework called PDEModelica for modeling with partial differential equations. It is an extension to the current Modelica modeling language for object-oriented, equation-based modeling based on differential and algebraic equations. The language extensions and the framework presented in this thesis are consistent with the concepts of Modelica while adding support for partial differential equations and space-distributed variables called fields. The specification of a partial differential equation problem consists of three parts: 1) the description of the definition domain, i.e., the geometric region where the equations are defined, 2) the initial and boundary conditions, and 3) the actual equations. The known and unknown distributed variables in the equation are represented by field variables in PDEModelica. Domains are defined by a geometric description of their boundaries. Equations may use the Modelica derivative operator extended with support for partial derivatives, or vector differential operators such as divergence and gradient, which can be defined for general curvilinear coordinates based on coordinate system definitions. The PDEModelica system also allows the partial differential equation models to be defined using a coefficient-based approach, where PDE models from a library are instantiated with different parameter values. Such a library contains both continuous and discrete representations of the PDE model. The user can instantiate the continuous parts and define the parameters, and the discrete parts containing the equations are automatically instantiated and used to solve the PDE problem numerically. Compared to most earlier work in the area of mathematical modeling languages supporting PDEs, this work provides a modern object-oriented component-based approach to modeling with PDEs, including general support for hierarchical modeling, and for general, complex geometries. It is possible to separate the geometry definition from the model definition, which allows geometries to be defined separately, collected into libraries, and reused in new models. It is also possible to separate the analytical continuous model description from the chosen discretization and numerical solution methods. This allows the model description to be reused, independent of different numerical solution approaches. The PDEModelica field concept allows general declaration of spatially distributed variables. Compared to most other approaches, the field concept described in this work affords a clearer abstraction and defines a new type of variable. Arrays of such field variables can be defined in the same way as arrays of regular, scalar variables. The PDEModelica language supports a clear, mathematical syntax that can be used both for equations referring to fields and explicit domain specifications, used for example to specify boundary conditions. Hierarchical modeling and decomposition is integrated with a general connection concept, which allows connections between ODE/DAE and PDE based models. The implementation of a Modelica library needed for PDEModelica and a prototype implementation of field variables are also described in the thesis. The PDEModelica library contains internal and external solver implementations, and uses external software for mesh generation, requisite for numerical solution of the PDEs. Finally, some examples modeled with PDEModelica and solved using these implementations are presented.

The brain accounts for around 2 % of human adult bodyweight but consumes 20 % of the resting oxygen available to the whole body. The brain is dependent on a constant supply of oxygen to tissue, transported from the heart via the vasculature and carried in blood. An interruption to flow can lead to ischaemia (a reduced oxygen supply) and prolonged interruption may result in tissue death, and permanent brain damage. The cerebral vasculature consists of many, densely packed, micro-vessels with a very large total surface area. Oxygen dissolved in blood enters tissue by passive diffusion through the micro-vessel walls. Imaging shows bursts of metabolic activity and flow in localised brain areas coordinated with brain activity (such as raising a hand). An appropriate level of oxygenation, according to physiological demand, is maintained via autoregulation; a set of response pathways in the brain which cause upstream or downstream vessels to expand or contract in diameter as necessary to provide sufficient oxygen to every region of the brain. Further, autoregulation is also evident in the response to pressure changes in the vasculature: the perfusing pressure can vary over a wide range from the basal-state with only a small effect on flow due to the constriction or dilation of vessels. Presented here is a new vasculature model where diameter and length are calculated in order to match the data available for flow velocity and blood pressure in different sized vessels. These vessels are arranged in a network of 6 generations each of bifurcating arterioles and venules, and a set of capillary beds. The input pressure and number of generations are the only specifications required to describe the network. The number of vessels, and therefore vessel geometry, is governed by how many generations are chosen and this can be altered in order to create more simple or complex networks. The flow, geometry and oxygen concentrations are calculated based on the vessel resistance due to flow from geometry based on Kirchoff circuit laws. The passive and active length-tension characteristics of the vasculature are established using an approximation of the network at upper and lower autoregulation limits. An activation model is described with an activation factor which governs the contributions of elastic andmuscle tension to the total vessel tension. This tension balances with the circumferential tension due to pressure and diameter and the change in activation sets the vessel diameter. The mass transport equation for oxygen is used to calculate the concentration of oxygen at every point in the network using data for oxygen saturation to establish a relationship between the permeability of the vessel wall to oxygen and the geometry and flow in individual vessels. A tissue compartment is introduced which enables the modelling of metabolic control. There is evidence for a coordinated response by surrounding vessels to local changes. A signal is proposed based on oxygen demand which can be conducted upstream. This signal decays exponentially with vessel length but also accumulates with the signal added from other vessels. The activation factor is therefore set by weighted signals proportional to changes in tissue concentration, circumferential tension, shear stress and conducted oxygen demand. The model is able to reproduce the autoregulation curve whereby a change in pressure has only a small effect on flow. The model is also able to replicate experimental results of diameter and tissue concentration following an increase in oxygen demand.