Recent advances in biology have revolutionized our understanding of living systems. For the first time, it is possible to study the behavior of individual cells. This has led to the discovery of many amazing phenomena. For example, cells have developed intelligent mechanisms for foraging, communicating, and responding to environmental changes. These diverse functions in cells are controlled through biochemical networks consisting of many different proteins and signaling molecules. These molecules interact and affect gene expression, which in turn affects protein production. This results in a complex mesh of feedback and feedforward interactions. These complex networks are generally highly nonlinear and stochastic, making them difficult to study quantitatively. Recent studies have shown that biochemical networks are also highly modular, meaning that different parts of the network do not interact strongly with each other. These modules tend to be conserved across species and serve specific biological functions. However, detect- ing modules and identifying their function tends to be a very difficult task. To overcome some of these complexities, I present an alternative modeling approach that builds quantitative models using coarse-grained biological processes. These coarse-grained models are often stochastic (probabilistic) and highly non-linear. In this thesis, I focus on modeling biochemical networks in two distinct biological systems: Dictyostelium discoideum and microRNAs. Chapters 2 and 3 focus on cellular communication in the social amoebae Dictyostelium discoideum. Using universality, I propose a stochastic nonlinear model that describes the behavior of individual cells and cellular populations. In chapter 4 I study the interaction between messenger RNAs and noncoding RNAs, using Langevin equations.

This thesis considers the importance of adding stochasticity to models of optimal foraging behaviour. The problem is divided into two elements, risk and information, which are treated separately. Part One: Risk. The theoretical and empirical results concerning animal preferences in risky situations are reviewed. Animals are known to show both risk-averse and risk-prone preferences over food reward. It is shown, theoretically, that a simple optimality model mimimizing the probability of death due to starvation accounts for at least some of the observed patterns of preference. The model is generalized to consider preference from arbitrary combinations of mean and variance. Three limitations of the model are treated in detail, that is the importance of starvation by "ruin", mind-changing about risk preferences, and energetic carry-over are discussed. The implications and limitations of these models are outlined. Part Two: Information. The theoretical literature is reviewed, and the problem of information is divided into three elements. A simple model of environmental tracking is studied. The model suggests that there is a trade-off between sensitivity to change and the costs of sampling. The model is tested using great tits (Parus major) foraging in an aviary. The trend in sampling was as predicted, but the birds were less sensitive to change than predicted. The problem of patch sampling is critically discussed. The value of sampling is defined, and this definition is used to compare the assumptions of previous models. Three such problems are treated: the importance of variance in the mixing distribution of patch sub-types; the importance of alternative and unambiguous patch types; and the importance of patch depression. It is concluded that previous models have often over-valued sampling. A simple and natural model of partial patch recognition is considered, and is shown to have empirical support. Implications and limitations of these models of information are discussed.

Many cell types stochastically switch phenotypes under some conditions, so that genetically identical sister cells may behave quite differently in a common environment. This non-genetic variability likely arises from noise in gene expression, which can be co-opted to allow random fate determination. This thesis examines both phenomena from experimental and theoretical perspectives, starting with a phenotypic switch. Cells of Bacillus subtilis grow either as individual, motile cells, or as connected groups of sessile cells called chains. We constructed an array of microfluidic channels in which we could capture and observe single cells in a constant environment over hundreds of generations of growth. These conditions allow unperturbed observation of decision-making driven only by factors internal to the cell. We observe that switching is asymmetric: transitions from motility to chaining occur with constant probability (memorylessly), but the reverse transition is tightly timed (exhibits memory). These properties are explained by dissecting the genetic circuit underlying switching, which can be quantitatively separated into components responsible for initiation and maintenance of the state. We propose that memory enables transgenerational cooperation between a cell founding a biofilm and its progeny, and that a stochastic sequestration mechanism is the source of random switching. Next, we introduce an exact framework for analyzing noise in gene networks that phrases results in terms of compounded parameters with simple interpretations. We uncover a basic identity that relates fluctuations in the production and degradation rates of one component to those of any other component within the cell. Since the result is exact, it applies to whole classes of gene networks. We identify basic constraints on the ability of negative feedback to suppress noise, and show that suppressing noise in one species generally requires introducing it elsewhere. When applied to the most common model of gene expression, the identity reveals a simple connection between the statistics of proteins and their cognate mRNAs. We reanalyze a recent experimental study of stochastic gene expression and show that the data are inconsistent with this prediction. Thus in contrast to early studies of single genes, there is currently discord between models and measurements of stochastic gene expression.

The timing of differentiation underlies the development of any organ system. In neural development, the expression of the transcription factor Hes1 has been shown to be oscillatory in neural progenitors, but at a low steady state in differentiated neurons. This change in the dynamics of expression marks the timing of differentiation. We previously constructed a mathematical model to test the experimental hypothesis that the topology of the miR-9/Hes1 network and specifically the accumulation of the micro-RNA, miR-9, could terminate Hes1 oscillations and account for the timing of neuronal differentiation, using deterministic delay differential equations. However, biochemical reactions are the result of random encounters between discrete numbers of molecules, and some of these molecules may be present at low numbers. The finite number of molecules interacting within the system leads to inherent randomness, and this is known as intrinsic stochasticity. The stochastic model predicts that low molecular number causes the time to differentiation to be distributed, which is in agreement with recent experimental evidence and considered important to generate cell type diversity. For the exact same model, fewer reacting molecules causes a decrease in the average time to differentiation, showing that the number of molecules can systematically change the timing of differentiation. Oscillations are important for a wide range of biological processes, but current methods for discovering oscillatory genes have primarily been designed for measurements performed on a population of cells. We introduce a new approach for analysing biological time series data designed for cases where the underlying dynamics of gene expression is inherently noisy at a single cell level. Our analysis method combines mechanistic stochastic modelling with the powerful methods of Bayesian nonparametric regression, and can distinguish oscillatory expression in single cell data from random fluctuations of nonoscillatory gene expression, despite peak-to-peak variability in period and amplitude of single cell oscillations. Models of gene expression commonly involve delayed biological processes, but the combination of stochasticity, delay and nonlinearity lead to emergent dynamics that are not understood at a theoretical level. We develop a theory to explain these effects, and apply it to a simple model of gene regulation. The new theory can account for long time-scale dynamics and nonlinear character of the system that emerge when the number of interacting molecules becomes low. Both the absolute length and the uncertainty in the delay time are shown to be crucial in controlling the magnitude of nonlinear effects.

abstract: Synthetic biology is an emerging field which melds genetics, molecular biology, network theory, and mathematical systems to understand, build, and predict gene network behavior. As an engineering discipline, developing a mathematical understanding of the genetic circuits being studied is of fundamental importance. In this dissertation, mathematical concepts for understanding, predicting, and controlling gene transcriptional networks are presented and applied to two synthetic gene network contexts. First, this engineering approach is used to improve the function of the guide ribonucleic acid (gRNA)-targeted, dCas9-regulated transcriptional cascades through analysis and targeted modification of the RNA transcript. In so doing, a fluorescent guide RNA (fgRNA) is developed to more clearly observe gRNA dynamics and aid design. It is shown that through careful optimization, RNA Polymerase II (Pol II) driven gRNA transcripts can be strong enough to exhibit measurable cascading behavior, previously only shown in RNA Polymerase III (Pol III) circuits. Second, inherent gene expression noise is used to achieve precise fractional differentiation of a population. Mathematical methods are employed to predict and understand the observed behavior, and metrics for analyzing and quantifying similar differentiation kinetics are presented. Through careful mathematical analysis and simulation, coupled with experimental data, two methods for achieving ratio control are presented, with the optimal schema for any application being dependent on the noisiness of the system under study. Together, these studies push the boundaries of gene network control, with potential applications in stem cell differentiation, therapeutics, and bio-production. / Dissertation/Thesis / Doctoral Dissertation Biomedical Engineering 2018

abstract: Cancer is a disease involving abnormal growth of cells. Its growth dynamics is perplexing. Mathematical modeling is a way to shed light on this progress and its medical treatments. This dissertation is to study cancer invasion in time and space using a mathematical approach. Chapter 1 presents a detailed review of literature on cancer modeling. Chapter 2 focuses sorely on time where the escape of a generic cancer out of immune control is described by stochastic delayed differential equations (SDDEs). Without time delay and noise, this system demonstrates bistability. The effects of response time of the immune system and stochasticity in the tumor proliferation rate are studied by including delay and noise in the model. Stability, persistence and extinction of the tumor are analyzed. The result shows that both time delay and noise can induce the transition from low tumor burden equilibrium to high tumor equilibrium. The aforementioned work has been published (Han et al., 2019b). In Chapter 3, Glioblastoma multiforme (GBM) is studied using a partial differential equation (PDE) model. GBM is an aggressive brain cancer with a grim prognosis. A mathematical model of GBM growth with explicit motility, birth, and death processes is proposed. A novel method is developed to approximate key characteristics of the wave profile, which can be compared with MRI data. Several test cases of MRI data of GBM patients are used to yield personalized parameterizations of the model. The aforementioned work has been published (Han et al., 2019a). Chapter 4 presents an innovative way of forecasting spatial cancer invasion. Most mathematical models, including the ones described in previous chapters, are formulated based on strong assumptions, which are hard, if not impossible, to verify due to complexity of biological processes and lack of quality data. Instead, a nonparametric forecasting method using Gaussian processes is proposed. By exploiting the local nature of the spatio-temporal process, sparse (in terms of time) data is sufficient for forecasting. Desirable properties of Gaussian processes facilitate selection of the size of the local neighborhood and computationally efficient propagation of uncertainty. The method is tested on synthetic data and demonstrates promising results. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

A two-scale model consisting of ceramic grain scale and composite scale are developed to systematically evaluate the effects of microstructures (e.g., residual pores, grain size, texture) and geometry on the piezoelectric responses of the polarized triply periodic bi-continuous (TPC) piezocomposites. These TPC piezocomposites were fabricated by a recently developed additive manufacturing (AM) process named suspension-enclosing projection-stereolithography (SEPS) under different process conditions. In the model, the Fourier spectral iterative perturbation method (FSIPM) and the finite element method will be adopted for the calculation at the grain and composite scale, respectively. On the grain scale, a DL approach based on stacked generative adversarial network (StackGAN-v2) is proposed to reconstruct microstructures. The presented modeling approach can reconstruct high-fidelity microstructures of additively manufactured piezoceramics with different resolutions, which are statistically equivalent to original microstructures either experimentally observed or numerically predicted. Design maps for hydrostatic piezoelectric charging coefficients dh show they can achieve optimal performance at wide ranges of micro-porosity and geometry parameter u for the proposed TPC piezocomposites. In addition, geometry parameter u plays a dominant role in determining the intensity of hydrostatic voltage coefficient gh and hydrostatic figure of merit (HFOM) of all the presented TPC piezocomposites in the vicinity of the starting point of three-dimension (3D) interconnectivity. Within this range, these properties would increase first with the increasing of micro-porosity volume fraction (VF) and start to decrease once they reach peak values. The presented TPC piezocomposites exhibit a superb hydrostatic properties, with the same 20% VF of ceramics and 2% VF of micro-porosity with respect to composites and ceramics, respectively, TPC of face center cubic (FCC) demonstrates 327-fold enhancement of HFOM than that of the piezocomposite with three intersecting ceramic cuboids. The piezoelectric properties of FCC are superior to those of body center cubic (BCC) and simple cubic (SC). The calculated piezoelectric charging constants d33 and relative permittivity κ33 were then compared with the data measured from the products fabricated by the SEPS under different process conditions. The calculation results at both grain scale and composite scale were found to agree well with experimental results.

Les Formes Réactives de l'Oxygène (FRO) regroupent des molécules comme les radicaux superoxide ($O_2^{\bullet -}$)et hydroxyle ($HO^\bullet$) ou le peroxyde d'hydrogène ($H_2O_2$) produites au sein des cellules en aérobiose. Malgré des systèmes de défense, des FRO peuvent réagir fortuitement avec des protéines, des lipides ou l'ADN provoquant des dommages cellulaires dont les mécanismes ne sont pas encore entièrement élucidés. Afin d'appréhender ce``stress oxydant'', cette thèse présente des simulations numériques de la dynamique de FRO en utilisant la bactérie E. coli comme organisme modèle. Dans un premier temps, les simulations numériques sont réalisées de façon déterministe sur un ensemble de cellules. L'étude de la mortalité de E. coli exposé à $H_2O_2$ montre que le fer intracellulaire libre et la densité cellulaire, deux facteurs potentiellement impliqués dans la dynamique des FRO, jouent un rôle primordial dans l'interprétation expérimentale comme par exemple le comportement bi-modal de E. coli opposé à $H_2O_2$. Nous avons également évalué les rôles relatifs des principales défenses mises en place contre $H_2O_2$ à savoir la membrane cellulaire et les enzymes. Une étude détaillée indique que leur implication dépend non linéairement de la concentration en $H_2O_2$.Dans une seconde approche nous réduisons l'échelle d'étude pour nous ramener à la cellule unique dans les conditions physiologiques. Il apparaît ainsi que la stochasticité intrinsèque des réactions chimiques associées aux FRO permet à certaines bactéries de se différencier en vue d'un futur stress. / The Reactive Oxygen Species (ROS) are molecules (superoxide $O_2^{\bullet -}$, hydrogen peroxide $H_2O_2$ and hydroxyl radical $HO^\bullet$) generated in living cells as a consequence of aerobic life. They are partially eliminated by scavenging systems. Nevertheless, ROS can unfortunately react with cellular proteins, lipids or DNA leading to cell damage. The mechanisms of such lesions is still being studied: we are talking about``oxidative stress''. Using Escherichia coli as a model organism this thesis is concerned with the numerical simulation of ROS dynamics.In the first part of this work, simulations were performed in a deterministic way to predict the behaviour of a set of cells. By studying killing of E. coli by exposure to $H_2O_2$, we show that intracellularavailable iron and cell density, two factors potentially involved in ROSdynamics, play a major role in the prediction of experimental results in particular in bimodal cell killing.We then evaluate the relative roles of major defences against $H_2O_2$. Although the key actors in celldefence are enzymes and membrane, a detailed analysisshows that their involvement depends on the $H_2O_2$ concentration level. In the second part, we study more closely the fate of the single cell with a stochastic point of view in physiological conditions.We show that elementary chemical stochasticity allows bacteria to segregate specialized cells in prevision of possible stress challenge. Actually, whereas ROS distribution does not activate defence regulation without exogenous stress, we demonstrate that this distribution may activate DNA repair mechanisms.

<p>Many of the earth’s ecosystems are experiencing large species losses due to human impacts such as habitat destruction and fragmentation, climate change, species invasions, pollution, and overfishing. Due to the complex interactions between species in food webs the extinction of one species could lead to a cascade of further extinctions and hence cause dramatic changes in species composition and ecosystem processes. The complexity of ecological systems makes it difficult to study them empirically. The systems often consist of large species numbers with lots of interactions between species. Investigating ecological communities within a theoretical approach, using mathematical models and computer simulations, is an alternative or a complement to experimental studies. This thesis is a collection of theoretical studies. We use model food webs in order to explore how biodiversity (species number) affects the response of communities to species loss (Paper I-III) and to environmental variability (Paper IV).</p><p>In paper I and II we investigate the risk of secondary extinctions following deletion of one species. It is shown that resistance against additional species extinctions increases with redundancy (number of species per functional group) (Paper I) in the absence of competition between basal species but decreases with redundancy in the presence of competition between basal species (Paper II). It is further shown that food webs with low redundancy run the risk of losing a greater proportion of species following a species deletion in a deterministic environment but when demographic stochasticity is included the benefits of redundancy are largely lost (Paper II). This finding implies that in the construction of nature reserves the advantages of redundancy for conservation of communities may be lost if the reserves are small in size. Additionally, food webs show higher risks of further extinctions after the loss of basal species and herbivores than after the loss of top predators (Paper I and II).</p><p>Secondary extinctions caused by a primary extinction and mediated through direct and indirect effects, are likely to occur with a time delay since the manifestation of indirect effects can take long time to appear. In paper III we show that the loss of a top predator leads to a significantly earlier onset of secondary extinctions in model communities than does the loss of a species from other trophic levels. If local secondary extinctions occur early they are less likely to be balanced by immigration of species from local communities nearby implying that secondary extinctions caused by the loss of top predators are less likely to be balanced by dispersal than secondary extinctions caused by the loss of other species. As top predators are vulnerable to human-induced disturbances on ecosystems in the first place, our results suggest that conservation of top predators should be a priority. Moreover, in most cases time to secondary extinction is shown to increase with species richness indicating the decay of ecological communities to be slower in species-rich than in species-poor communities.</p><p>Apart from the human-induced disturbances that often force species towards extinction the environment is also, to a smaller or larger extent, varying over time in a natural way. Such environmental stochasticity influences the dynamics of populations. In paper IV we compare the responses of food webs of different sizes to environmental stochasticity. Species-rich webs are found to be more sensitive to environmental stochasticity. Particularly, species-rich webs lose a greater proportion of species than species-poor webs and they also begin losing species faster than species-poor webs. However, once one species is lost time to final extinction is longer in species-rich webs than in species-poor webs. We also find that the results differ depending on whether species respond similarly to environmental fluctuations or whether they are totally uncorrelated in their response. For a given species richness, communities with uncorrelated species responses run a considerable higher risk of losing a fixed proportion of species compared with communities with correlated species responses.</p>

Fluid dynamics models provide a powerful deterministic technique to approximate stochasticity in a variety of application areas. In this paper, we study two classes of fluid models, investigate their relationship as well as some of their applications. This analysis allows us to provide analytical models of travel times as they arise in dynamically evolving environments, such as transportation networks as well as supply chains. In particular, using the laws of hydrodynamic theory, we first propose and examine a general second order fluid model. We consider a first-order approximation of this model and show how it is helpful in analyzing the dynamic traffic equilibrium problem. Furthermore, we present an alternate class of fluid models that are traditionally used in the context of dynamic traffic assignment. By interpreting travel times as price/inventory-sojourn-time relationships, we are also able to connect this approach with a tractable fluid model in the context of dynamic pricing and inventory management. Finally, we investigate the relationship between these two classes of fluid models. / Singapore-MIT Alliance (SMA)