Stochastic modeling to represent intrinsic and extrinsic noise is an important challenge in molecular systems biology. There are numerous ways to model intrinsic noise. One framework for intrinsic noise in gene regulatory networks was recently proposed within the discrete setting. In contrast, extrinsic perturbations were rarely modeled due to the complex mechanisms that contribute to its emergence. Here a discrete framework to model extrinsic noise is proposed. The interacting species of the model are represented by discrete variables and are perturbed to represent extrinsic noise. In particular, they are subject to a discretized lognormal distribution. Additionally, a delay is imposed on the update with a certain probability. These two perturbations represent global extrinsic noise and pathway-specic extrinsic noise. It leads to large variations in the concentration of proteins, which is consistent with an existing continuous way of modeling extrinsic fluctuations. The framework is applied to three different published discrete models: the cell fate of lambda phage infection of bacteria, the lactose utilization system in E. coli, and a signaling network in melanoma cells. The framework captures factors that signicantly contribute to the random decision between lysis and lysogeny as well as explains the bistable switch in the model of the lac operon. Finally, a feed-forward loop analysis is conducted by measuring and comparing the noise level in the target protein of feed-forward loops. This analysis reveals the ability of certain feed-forward loops to attenuate or amplify fluctuations, dependent upon various levels of noise. In conclusion, this thesis aims to resolve the question of how the extrinsic noise can be modeled and how biological systems are able to maintain functionality in the wake of such large variations. / Master of Science

The hybrid activated sludge-biofilm system called Integrated Fixed Film Activated Sludge (IFAS) has recently become popular for enhanced nitrification and denitrification in aerobic zones because it is an alternative to increasing the volume of treatment plant units to accomplish year round nitrification and nitrogen removal. Biomass is retained on the fixed-film media and remains in the aerobic reactor, thus increasing the effective mean cell resident time (MCRT) of the biomass and providing the temperature sensitive, slow growing nitrifiers a means of staying in the system when they otherwise would washout. While the utilization of media in aerobic zones to enhance nitrification and denitrification has been the subject of several studies and full-scale experiments, the effects and performances of fixed film media integrated into the anoxic zones of biological nutrient removal (BNR) systems have not adequately been evaluated as well as the impacts of integrated media upon enhanced biological phosphorus removal (EBPR). Also, user-friendly software designed specifically to simulate the complex mixture of biological processes that occur in IFAS systems are not available. The purpose of this research was to more fully investigate the effects of integrated fixed film media on EBPR, to evaluate the impacts of media integrated into the anoxic zone on system performance, and to develop a software program that could be used to simulate the effects of integrating the various types of media into suspended growth biological nutrient removal (BNR) systems. The UCT type configuration was chosen for the BNR system, and Accuweb rope-like media was selected for integration into the anoxic zones of two IFAS systems. The media also was integrated into the aerobic reactors of one of the systems for comparison and for further investigation of the performance of the Accuweb media on enhanced nitrification and denitrification in the aerobic zones. The experiments were conducted at 10 day total MCRT during the initial phase, and then at 6 days MCRT for the experimental temperature of 10 oC. A13 hour hydraulic retention time (HRT) was used throughout the study. A high and a low COD/TP ratio were used during the investigation to further study the effects of integrated media on EBPR. The PC Windows based IFAS program began with the concepts of IAWQ model No. 2 and a zero-dimensional biofilm model was developed and added to predict the IFAS processes. Experimental data from the initial study and existing data from similar studies performed at high temperatures (>10oC) indicated that there were no significant differences in BNR performances between IFAS systems with media integrated into the anoxic and aerobic or only aerobic zones and a suspended growth control system maintained at the same relative high MCRT and temperature values. Even though greater biological nitrogen removal could not be achieved for the experimental conditions used, the experimental results indicated that the IFAS systems with fixed film media installed in the anoxic zone have a greater potential for denitrification than conventional BNR systems. As much as 30 percent of the total denitrification was observed to occur in the aerobic zones of the system installed the media only anoxic zones and 37% in the system with integrated media in both anoxic and aerobic zones where as no denitrification was observed in the aerobic zones of the control system when the systems were operated at 6 days MCRT and COD/TP of 52. It is statistically confirmed EBPR can be maintained in IFAS systems as well as Control systems, but the IFAS processes tend to have more phosphorus release in the anoxic zones with integrated fixed film installed. Further, the combination of split flow to the anoxic zone and fixed film media in the anoxic zone resulted in the decreased EBPR performances in the IFAS system relative to the control system. / Ph. D.

In a multi-cellular organism, cells constantly receive signals on their internal condition and surrounding environment. In response to various signals, cells proliferate, move around or even undergo suicide. The signal-response is controlled by complex molecular machinery, understanding of which is an important goal of basic molecular biological research. Such understanding is also valuable for clinical application, since lethal diseases like cancer show maladaptive responses to growth-regulating signals. Because the multiple feedbacks in the molecular regulatory machinery obscure cause-effect relations, it is hard to understand these control systems by intuition alone. Here we translate the molecular interactions into differential equations and recapture the cellular physiological properties with the help of numerical simulations and non-linear dynamical tools. The models address the physiological features of programmed cell death, the cell fate decision by p53 and the dynamics of the NF-?B control system. These models identify key molecular interactions responsible for the observed physiological properties, and they generate experimentally testable predictions to validate the assumptions made in the models. / Ph. D.

Protein regulatory networks are the hallmark of many important biological functionalities. Two of these functionalities are mammalian cell cycle progression and near-perfect adaptive responses. Modeling and simulating these functionalities are crucial stages to understanding and predicting them as systems-level properties of cells. In the context of the mammalian cell cycle, the timing of DNA synthesis, mitosis and cell division is regulated by a complex network of biochemical reactions that control the activities of a family of cyclin-dependent kinases. The temporal dynamics of this reaction network is typically modeled by nonlinear differential equations describing the rates of the component reactions. This approach provides exquisite details about molecular regulatory processes but is hampered by the need to estimate realistic values for the many kinetic constants that determine the reaction rates. To avoid this problem, modelers often resort to "qualitative" modeling strategies, such as Boolean switching networks, but these models describe only the coarsest features of cell cycle regulation. In this work, we describe a hybrid approach that combines features of continuous and discrete networks. The model is evaluated in terms of flow cytometry measurements of cyclin proteins in asynchronous populations of human cell lines. Using our hybrid approach, modelers can quickly create quantitatively accurate, computational models of protein regulatory networks found in various contexts within cells. Large-scale protein regulatory networks, such as the one that controls the progression of the mammalian cell cycle, also contain small-scale motifs or modules that carry out specific dynamical functions. Systematic characterization of smaller, interacting, network motifs whose individual behavior is well known under certain conditions is therefore of great interest to systems biologists. We model and simulate various 3-node network motifs to find near-perfect adaptation behavior. This behavior entails that a system responds to a change in its environmental cues, or signals, by coming back nearly to its pre-signal state even in the continued presence of the signal. We let various topologies evolve in their parameter space such that they eventually stumble upon a region where they score well under a pre-defined scoring metric. We find many such parameter sample sets across various classes of topologies. / Ph. D.

Complex networks are composed of a large number of interacting nodes. Examples of complex networks include the topology of the Internet, connections between websites or web pages in the World Wide Web (WWW), and connections between participants in social networks.Due to their ubiquity, modeling complex networks is importantfor answering many research questions that cannot be answered without a mathematical model. For example, mathematical models of complex networks can be used to find the most vulnerable nodes to protect during a virus attack in theInternet, to predict connections between websites in the WWW, or to find members of different communities insocial networks. Researchers have analyzed complex networksand concluded that they are distinguished from other networks by four specific statistical properties. These four statistical properties are commonly known in this field as: (i) thesmall world effect,(ii) high average clustering coefficient, (iii) scale-free power law degree distribution, and (iv) emergence of community structure. These four statistical properties are further described later in this dissertation. Mostmodels used to generate complex networks attempt to produce networks with these statistical properties. Additionally, most of these network models generate homogeneous complex networks where all the networknodes are considered to have the same properties. Homogenous complex networks neglect the heterogeneous nature ofthe nodes in many complexnetworks. Moreover, somemodels proposed for generating heterogeneous complexnetworks are not general as they make specific assumptions about the properties of the network.Including heterogeneity in the connection algorithm of a modelwould makeitmore suitable for generating the subset of complex networks that exhibit selective linking.Additionally, all modelsproposed, to date, for generating heterogeneous complex networks do not preserve all four of the statistical properties of complexnetworks stated above. Thus, formulation of a model for the generation of general heterogeneous complex networkswith characteristics that resemble as much as possible the statistical properties common to the real-world networks that have received attention from the research community is still an open research question. In this work, we propose two new types of models to generate heterogeneous complex networks. First, we introduce the Integrated Attribute Similarity Model (IASM). IASM uses preferential attachment(PA) to connect nodes based on a similarity measure for node attributes combined with a node's structural popularity measure. IASM integrates the attribute similarity measure and a structural popularity measure in the computation of the connection function used to determine connectionsbetween each arriving (newly created) node and the existing(previously created or old) network nodes. IASM is also the first model known to assign an attribute vector having more than one element to each node, thus allowing different attributes per node in the generated complex network. Networks generated using IASM have a power law degree distribution and preserve the small world phenomenon. IASM models are enhanced to increase their clustering coefficient using a triad formation step (TFS). In a TFS, a node connects to the neighbor of the node to which it was previously connected through preferential attachment, thus forming a triad. The TFS increases the number of triads that are formed in the generated network which increases the network's average clustering coefficient. We also introduce a second novel model,the Settling Node Adaptive Model (SNAM). SNAM reflects the heterogeneous nature of connectionstandard requirements for nodes. The connectionstandard requirements for a noderefers to the values of attribute similarity and/or structural popularityof old node ythat node new xwould find acceptable in order to connect to node y.SNAM is novel in that such a node connection criterion is not included in any previous model for the generation of complex networks. SNAM is shown to be successful in preserving the power law degree distribution, the small world phenomenon, and the high clustering coefficient of complex networks. Next,we implement a modification to the IASM and SNAM models that results in the emergence of community structure.Nodes are classified into classes according to their attribute values. The connection algorithm is modified to include the class similarity values between network nodes. This community structure model preservesthe PL degree distribution, small world property, and does not affect average clustering coefficient values expected from both IASM and SNAM. Additionally, the model exhibits the presence of community structure having most of the connections made between nodes belonging to the same class with only a small percent of the connections made between nodes of different classes. We perform a mathematical analysis of IASM and SNAM to study the degree distribution for networks generated by both models. This mathematical analysis shows that networks generated by both models have a power law degree distribution. Finally, we completed a case study to illustrate the potential value of our research on the modeling of heterogeneous complex networks. This case study was performed on a Facebook dataset. The case study shows that SNAM, with some modifications to the connection algorithm, is capable of generating a network with almost the same characteristics as found for the original dataset. The case study providesinsight on how the flexibility of SNAM's connection algorithm can be an advantagethat makes SNAM capable of generating networks with different statistical properties. Ideas for future research areas includestudyingthe effect of using eigenvector centrality, instead of degree centrality, on the emergence of community structure in IASM; usingthe nodeindex as an indication for its order of arrival to the network and distributing added connections fairly among networknodes along the life of the generated network; experimenting with the nature of attributesto generatea more comprehensive model; and usingtime sensitive attributes in the models, where the attribute can change its value with time, / Ph. D.

In eukaryotes, distinct nutrient signals are integrated in order to produce robust cellular responses to fluctuations in the environment. This process of signal integration is attributed to the crosstalk between nutrient specific signaling pathways, as well as the large degree of overlap between their regulatory targets. In the budding yeast Saccharomyces cerevisiae, these distinct pathways have been well characterized. However, the significant overlap between these pathways confounds the interpretation of the overall regulatory logic in terms of nutrient-dependent cell state determination. Here, we propose a literature-curated molecular mechanism of the integrated nutrient signaling pathway in budding yeast, focussing on carbon and nitrogen signaling. We build a computational model of this pathway to reconcile the available experimental data with our proposed molecular mechanism. We evaluate the robustness of the model fit to data with respect to the variations in the values of kinetic parameters used to calibrate the model. Finally, we use the model to make novel, experimentally testable predictions of transcription factor activities in mutant strains undergoing complex nutrient shifts. We also propose a novel framework, called BoolODE for utilizing published Boolean models to generate synthetic datasets used to benchmark the performance of algorithms performing gene regulatory network inference from single cell RNA sequencing data. / Doctor of Philosophy / An important problem in biology is how organisms sense and adapt to ever changing environments. A good example of an environmental cue that affects animal behavior is the availability of food; scarcity of food forces animals to search for food-rich habitats, or go into hibernation. At the level of single cells, a range of behaviors are observed depending on the amount of food, or nutrients present in the environment. Moreover, different types of nutrients are important for different biological functions in single cells, and each different nutrient type will have to be available in the right quantities to support cellular growth. At the subcellular level, intricate molecular machineries exist which sense the amounts of each nutrient type, and interpret this information in order to make a decision on how best to respond. This interpretation and integration of nutrient information is a complex, poorly understood process even in a simple unicellular organism like the budding yeast. In order to understand this process, termed nutrient signaling, we propose a mathematical model of how yeasts respond to nutrient availability in the environment. Our model advances the state of knowledge by presenting the first comprehensive mathematical model of the nutrient signaling machinery, accounting for a variety of experimental observations from the last three decades of yeast nutrient signaling. We use our model to make predictions on how yeasts might behave when supplied with different combinations of nutrients, which can be verified by experiments. Finally, the cellular machinery that helps yeasts respond to nutrient availability in the environment is very similar to the machinery in cancer cells that causes them to grow rapidly. Our proposed model can serve as a stepping stone towards the construction of a model of cancer's responses to its nutritional environment.

Thesis (MSc)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Having caused the deaths of more than 10 million individuals since 2000 with most of them occurring in Africa, malaria remains a serious disease that requires undivided attention. To this end a detailed kinetic model of Plasmodium falciparum glycolysis was constructed, validated and used to determine potential drug targets for the development of novel, effective antimalarial therapies. The kinetic model described the behaviour of the glycolytic enzymes with a set of ordinary differential equations that was solved to obtain the steady state fluxes and concentrations of internal metabolites. The model included a glycerol branch represented in a single fitted equation. This present study set out to detect, characterise, and incorporate into the model the enzymes that constitute the glycerol branch of P. falciparum glycolysis. The kinetic parameters of glycerol 3-phosphate dehydrogenase (G3PDH), the first enzyme in the branch and catalyst of the dihydroxyacetone phosphosate (DHAP) reducing reaction, was determined and added to the detailed kinetic model. The model was subsequently validated by comparing its prediction of steady state fluxes with experimentally measured fluxes. Once it was evident that the predictions of the unfitted model agreed with experimentally measured fluxes, metabolic control analysis was performed on this branched system to ascertain the distribution of control over the steady state flux through the glycerol branch. The control G3PDH exercised over its own flux was less than expected due to the enzyme’s sensitivity to changes in NADH and thus the redox balance of the cell. Attempts were made to detect the enzymes responsible for the conversion of glycerol 3-phosphate (G3P) to glycerol. Very low levels of glycerol kinase activity was observed. Although G3P-dependent release of inorganic phosphate was detected results were inconclusive as to whether a non-specific phosphatase also mediated the conversion. Overall, the expansion of the model to include G3PDH did not affect the steady state metabolite concentrations and flux adversely. / AFRIKAANSE OPSOMMING: Vanaf die jaar 2000 het malaria die dood van meer as 10 miljoen mense veroorsaak. Die meeste sterftes het in Afrika voorgekom —’n aanduiding van hoe ernstige siekte dit is en een wat onverdeelde aandag moet geniet. Om hierdie rede is ’n gedetaileerde kinetiese model van glikoliese in Plasmodium falciparum gebou, gevalideer en gebruik om potensiële dwelm teikens te identifiseer vir die ontwikkeling van nuwe, meer effektiewe anti-malaria terapieë. Die kinetiese model beskryf die gedrag van die glikolitiese ensieme in terme van gewone differensiële vergelykings wat opgelos is om die bestendige toestand fluksies en interne metaboliet konsentrasies te bepaal. Die model sluit ’n gliserol-tak in wat deur ’n enkele aangepaste vergelyking verteenwoordig word. Hierdie studie het voorgeneem om die ensieme van die gliserol-tak van P. falciparum glikoliese te identifiseer, karakteriseer en in die model te inkorporeer. Ons het die kinetiese parameters van die eerste ensiem in die gliserol-tak, gliserol 3-fosfaat dehidrogenase (G3PDH), die katalis van die dihidroksiasetoon fosfaat(DHAP) reduserende reaksie, bepaal. Die kinetiese parameters is by die gedetaileerde model gevoeg. Validering het plaasgevind deur die model se voorspellings met eksperimenteel bepaalde waardes te vergelyk. Toe dit duidelik geword het dat die voorspellings van die model met die eksperimenteel bepaalde fluks ooreenstem, is metaboliese kontrole analiese op die vertakte sisteem uitgevoer. Dit is gedoen om vas te stel hoe die bestendige toestand fluks deur die gliserol-tak beheer word. G3PDH het nie volle beheer oor sy eie fluks nie, in teenstelling met ons vergewagtinge. Daar is gepoog om vas te stel watter ensieme verantwoordelik is vir die produksie van gliserol vanuit gliserol 3-fosfaat (G3P). ’n Lae gliserolkinase aktiwiteit is waargeneem. Alhoewel G3P afhanklike vrystelling van anorganise fosfaat waargeneem is, is dit nie duidelik vanuit die resultate of die proses deur ’n nie-spesifieke fosfatase uitgevoer word nie. Die uitbreiding van die model om ’n G3PDH vergelyking in te sluit het nie die bestendige toestand metaboliet konsentrasies en fluks negatief geaffekteer nie.

An important protective feature of the cerebral circulation is its ability to maintain sufficient cerebral blood flow and oxygen supply in accordance with the energy demands of the brain despite variations in a number of external factors such as arterial blood pressure, heart rate and respiration rate. If cerebral autoregulation is impaired, abnormally low or high CBF can lead to cerebral ischemia, intracranial hypertension or even capillary damage, thus contributing to the onset of cerebrovascular events. The control and regulation of cerebral blood flow is a dynamic, multivariate phenomenon. Sensitive techniques are required to monitor and process experimental data concerning cerebral blood flow and metabolic rate in a clinical setting. This thesis presents a model simulation study and 4 related signal processing studies concerned with CBF regulation. The first study models the regulation of the cerebral vasculature to systemic changes in blood pressure, dissolved blood gas concentration and neural activation in a integrated haemodynamic system. The model simulations show that the three pathways which are generally thought to be independent (pressure, CO₂ and activation) greatly influence each other, it is vital to consider parallel changes of unmeasured variability when performing a single pathway study. The second study shows how simultaneously measured blood gas concentration fluctuations can improve the accuracy of an existing frequency domain technique for recovering cerebral autoregulation dynamics from spontaneous fluctuations in blood pressure and cerebral blood flow velocity. The third study shows how the continuous wavelet transform can recover both time and frequency information about dynamic autoregulation, including the contribution of blood gas concentration. The fourth study shows how the discrete wavelet transform can be used to investigate frequency-dependent coupling between cerebral and systemic cardiovascular dynamics. The final study then uses these techniques to investigate the systemic effects on resting BOLD variability. The general approach taken in this thesis is a combined analysis of both modelling and data analysis. Physiologically-based models encapsulate hypotheses about features of CBF regulation, particularly those features that may be difficult to recover using existing analysis methods, and thus provide the motivation for developing both new analysis methods and criteria to evaluate these methods. On the other hand, the statistical features extracted directly from experimental data can be used to validate and improve the model.

Cancer is one of the leading causes of premature deaths across the world today, and there is an urgent need for imaging techniques that can help in early diagnosis and treatment planning for cancer patients. In the last four decades, magnetic resonance imaging (MRI) has emerged as one of the leading modalities for non-invasive imaging of tumours. By using dynamic contrast-enhanced magnetic resonance imaging (DCEMRI), this modality can be used to acquire information about perfusion and vascularity of tumours, which can help in predicting response to treatment. There are many factors that complicate the analysis of DCE-MRI data, and make clinical predictions based on it unreliable. During data acquisition, there are many sources of uncertainties and errors, especially patient motion, which result in the same image position being representative of many different anatomical locations across time. Apart from motion, there are also other inherent uncertainties and noise associated with the measurement of DCE-MRI parameters, which contribute to the model-fitting error observed when trying to apply pharmacokinetic (PK) models to the data. In this thesis, a probabilistic, model-based registration and parameter estimation (MoRPE) framework for motion correction and PK-parameter estimation in DCE-MRI sequences is presented. The MoRPE framework is first compared with conventional motion correction methods on simulated data, and then applied to data from a clinical trial involving twenty colorectal cancer patients. On clinical data, the ability of MoRPE to discriminate between responders and non-responders to combined chemoand radiotherapy is tested, and found to be superior to other methods. The effect of incorporating different arterial input functions within MoRPE is also assessed. Following this, a quantitative analysis of the uncertainties associated with the different PK parameters is performed using a variational Bayes mathematical framework. This analysis provides a quantitative estimate of the extent to which motion correction affects the uncertainties associated with different parameters. Finally, the importance of estimating spatial heterogeneity of PK parameters within tumours is assessed. The efficacy of different measures of spatial heterogeneity, in predicting response to therapy based on the pre-therapy scan alone are compared, and the prognostic value of a new derived PK parameter the 'acceleration constant' is investigated. The integration of uncertainty estimates of different DCE-MRI parameters into the calculation of their heterogeneity measures is also shown to improve the prediction of response to therapy.

As mathematical modeling gains popularity in K-12 classrooms, it is important to define what this entails for both students and teachers. The following study reviews various definitions of mathematical modeling and how these definitions are relevant for middle grades (5-9) teachers. Following a professional development workshop on mathematical modeling, four middle school teachers expressed their views about teaching mathematics through modeling tasks. This study documents the teachers' perceptions of what it means to model with mathematics, which tasks are most appropriate for their students, and why this is important in each of their classrooms. Although the teachers varied in their views depending on the context and circumstances surrounding each modeling task, they agreed that mathematical modeling helps students build critical thinking skills and provides an opportunity to align mathematics concepts with engaging, realistic phenomena.