Cancer is an ongoing problem all over the world. To find a cure to this disease, both clinicians and scientists are looking for a reasonable treatment method. According to Hanahan and Weinberg, one of the hallmarks of cancer is evasion of programmed cell death, referred to as apoptosis. Apoptosis is an important cellular process, and is regulated by many different pathways. Proteins in these pathways contribute to either cell death or cell survival depending on the cell stresses. Much research in systems biology has been devoted to understanding these pathways at the molecular level. In this study a mathematical model is built to describe apoptosis, and the pathways involving the related proteins p53 and Akt. The primary purpose of the construction of the kinetic model is to verify that this network can exhibit bistability between cell survival and cell death. Sensitivity and bifurcation analysis are conducted to determine which parameters have the greatest effect on the system behavior.

The MAPK cascade is responsible for transmitting information in the cytoplasm of the cell and regulating important fate decisions like cell division and apoptosis. Due to scarce experimental data and limited knowledge about many complex biochemical processes, existing MAPK pathway models, which exhibit bistability, have a significant structural uncertainty. Often, small perturbations of network interactions or components can reduce the bistable region significantly or make it even disappear and small fluctuations of the input can make the system switch back, which reflects its low robustness. However, real biological systems have developed significant robustness through evolution and this robustness should be reflected by the models. The main goal of the present thesis is the development of a methodology for increasing the robustness of biochemical models, which exhibit bistability. Based on modifying existing network interactions or introducing new interactions to the system, several methods for both internal and external robustification are proposed. Internal robustness is addressed through a sensitivity analysis, which deals with a linearization of the model and can be used sequentially to introduce multiple modifications to the model. The methods for external robustness improvement are based on eigenvalue placement and slope modification (drawing on the linear model) and on the identification of feedback structures (nonlinear model). Further, a way to integrate static interaction changes to the nonlinear model, so that these perturbations have only a local impact on its behavior, is proposed. The application of the methods to existing MAPK models shows that, by introducing small modifications, the internal and external robustness of models can be increased significantly and thus provides knowledge about complex dynamics and interactions that play a key role for the inherent robustness of real biological systems. Furthermore, by employing a robustness analysis, stable steady-state branches can be recovered and bistability can be induced.

Recent advances in systems biology have uncovered detailed mechanisms of  biological processes such as the cell cycle, circadian rhythms, and signaling pathways.  These mechanisms are modeled by chemical reaction networks (CRNs) which are typically simulated by converting to ordinary differential equations (ODEs), so that the goal is to closely reproduce the observed quantitative and qualitative behaviors of the modeled process. This thesis proposes two algorithmic problems related to the construction and comprehension of CRN models. The first problem focuses on reconstructing CRNs from given time series. Given multivariate time course data obtained by perturbing a given CRN, how can we systematically deduce the interconnections between the species of the network? We demonstrate how this problem can be modeled as, first, one of uncovering conditional independence relationships using buffering experiments and, second, of determining the properties of the individual chemical reactions. Experimental results demonstrate the effectiveness of our approach on both synthetic and real CRNs. The second problem this work focuses on is to aid in network comprehension, i.e., to understand the motifs underlying complex dynamical behaviors of CRNs. Specifically, we  focus on bistability---an important dynamical property of a CRN---and propose algorithms to identify the core structures responsible for conferring bistability. The approach we take is to systematically infer the instability causing structures (ICSs) of a CRN and use machine learning techniques to relate properties of the CRN to the presence of such ICSs. This work has the potential to aid in not just network comprehension but also model simplification, by helping  reduce the complexity of known bistable systems. / Ph. D.

Malaria is one of the most devastating infectious diseases, with nearly half of the worlds population currently at risk of infection. Although mathematical models have made significant contributions towards the control and elimination of malaria, it continues to evade control. This thesis focuses on two aspects of malaria that complicate dynamics, helping it persist. The basic reproductive number is one of the most important epidemiological quantities as it provides a foundation for control and elimination. Recently, it has been suggested that R0 should be modified to account for the effects of finite host population on a single disease-generation. In chapter 2, we analytically calculate these finite-population reproductive numbers for both vector-borne and directly transmitted diseases with homogeneous transmission. We find simple, generalizable formula and show that when the population is small, control and elimination may be easier than predicted by R0. In chapter 3, we extend the results of chapter 2 and find expressions for the finite- population reproductive numbers for directly transmitted diseases with different types of heterogeneity in transmission. We also outline a framework for discussing the different types of heterogeneity in transmission. We show that although the effects of heterogeneity in a small population are complex, the implications for control are simple: when R0 is large relative to the size of the population, control or elimination is made easier by heterogeneity. Another basic question in malaria modeling is the effects of immunity on the population- level dynamics of malaria. In chapter 4, we explore the possibility that clinical immunity can cause bistable malaria dynamics. This has important implications for control: in areas with bistable malaria, if malaria could be eliminated until clinical immunity wanes, it would not be able to invade. We built a simple, analytically tractable model of malaria transmission and solved it to find a criterion for when we expect bistability to occur. Additionally, we review what is known about about the parameters underlying the model and highlighted key clinical immunity parameters for which little is known. Building on these results, in chapter 5, we fit the model developed in chapter 4 to incidence data from Kericho, Kenya and estimate key clinical immunity parameters to better understand the role clinical immunity plays in malaria transmission. Finally, in chapter 6, we summarize the key results and discuss the broader implications of these findings on future malaria control. / Thesis / Doctor of Philosophy (PhD)

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Cellular processes critical to sustaining physiology, including growth, division and differentiation, are carefully governed by intricate control systems. Deregulations in these systems often result in complex diseases such as cancer. Hence, it is crucial to understand the interactions between molecular players of these control systems, their emergent network dynamics, and, ultimately, the overall contribution to cellular physiology. In this dissertation, we have developed a mathematical framework to understand two such cellular systems: an early checkpoint (START) in the budding yeast cell cycle (Chapter 1), and the canonical Wnt signaling pathway involved in cell proliferation and differentiation (Chapter 2). START transition is an important decision point where the cell commits to one round DNA replication followed by cell division. Several years of experimental research have gone into uncovering molecular details of this process, but a unified understanding is yet to emerge. In chapter one, we have developed a comprehensive mathematical model of START transition that incorporates several findings including information about the phosphorylation state of key START proteins and their subcellular localization. In the second chapter, we focus on modeling the canonical Wnt signaling pathway, a cellular circuit that plays a key role in cell proliferation and differentiation. The Wnt pathway is often deregulated in colon cancers. Based on some evidence of bistability in the Wnt signaling pathway, we proposed the existence of a positive feedback loop underlying the activation and inactivation of the core protein complex of the pathway. Bistability is a common feature of biological systems that toggle between ON and OFF states because it ensures robust switching back and forth between the two states. To study and explain the behavior of this dynamical system, we developed a mathematical model. Based on experimentally determined interactions, our simple model recapitulates the observed phenomena of bimodality (bistability) and hysteresis under the effects of the physiological signal (Wnt), a Wnt-mimic (LiCl), and a stabilizer of one of the key members of core complex (IWR-1). Overall, we believe that cell biologists and molecular geneticists can benefit from our work by using our model to make novel quantitative predictions for experimental verification. / Ph. D.

Acousto-optic hybrid bistable devices have been studied previously. However, previous studies are limited only to the first-Bragg regime involving two diffracted orders. No actual comparison has been made between experimental results and theoretical predictions. A model including both acousto-optic diffraction and a nonlinear feedback path is studied in this thesis. Theoretical results based on diffraction involving two and four diffracted orders have been obtained and compared. Experimental results confirm the validity of the theoretical model. The principle of operation is discussed along with experimental results. The performance of the bistable system is then studied. In the investigation, the Klein-Cook parameter, Q, has been introduced into the study. Methods to improve the performance of the system with a low Q acousto-optic device by adjusting the effective feedback gain and the operation point are suggested. Finally, a technique to measure the effective feedback gain has been derived. Future topics are suggested along with a modified and improved model. / Master of Science

Statistical mechanics has been applied to a wide range of systems in physics, biology, medicine and even anthropology. This theory has been recently used to model the complex biochemical processes of gene expression and regulation. In particular, genetic networks offer a large number of interesting phenomena, such as multistability and oscillatory behaviour, that can be modelled with statistical mechanics tools. In the first part of this thesis we introduce gene regulation, genetic switches, and the colonization of a spatially structured media. We also introduce statistical mechanics and some of its useful tools, such as the master equation and mean- field theories. We present simple examples that are both pedagogical and also set the basis for the study of more complicated scenarios. In the second part we consider the exclusive genetic switch, a fundamental example of genetic networks. In this system, two proteins compete to regulate each other's dynamics. We characterize the switch by solving the stationary state in different limits of the protein binding and unbinding rates. We perform a study of the bistability of the system by examining its probability distribution, and by applying information theory techniques. We then present several versions of a mean field theory that offers further information about the switch. Finally, we compute the stationary probability distribution with an exact perturbative approach in the unbinding parameter, obtaining a valid result for a wide range of parameters values. The techniques used for this calculation are successfully applied to other switches. The topic studied in the third part of the thesis is the propagation of a trait inside an expanding population. This trait may represent resistance to an antibiotic or being infected with a certain virus. Although our model accounts for different examples in the genetic context, it is also very useful for the general study of a trait propagating in a population. We compute the speed of expansion and the stationary population densities for the invasion of an established and an expanding population, finding non-trivial criteria for speed selection and interesting speed transitions. The obtained formulae for the different wave speeds show excellent agreement with the results provided by simulations. Moreover, we are able to obtain the value of the speeds through a detailed analysis of the populations, and establish the requirements for our equations to present speed transitions. We finally apply our model to the propagation in a position-dependent fitness landscape. In this situation, the growth rate or the maximum concentration depends on the position. The amplitudes and speeds of the waves are again successfully predicted in every case.

The fundamental components for processing all-optically represented data, namely optical switches and logic gates are investigated. Improved techniques for fabricating nonlinear Fabry-Perot etalons containing GaAs have brought a proliferation of GaAs optical bistable devices. These devices show significant improvements in speed, power requirements, operating temperature and thermal stability. Experiments verify predictions that one can operate a single nonlinear etalon as optical logic gates or two such etalons as a flip-flop. Optimization of the logic gates is then discussed from a systems approach.