Browsing by Author "Reed, Michael C"

This dissertation addresses two aspects of dynamical systems arising from biological networks: homeostasis-bifurcation and identifiability.

Homeostasis occurs when a biological quantity does not change very much as a parameter is varied over a wide interval. Local bifurcation occurs when the multiplicity or stability of equilibria changes at a point. Both phenomena can occur simultaneously and as the result of a single mechanism. We show that this is the case in the feedback inhibition network motif. In addition we prove that longer feedback inhibition networks are less stable. Towards understanding interactions between homeostasis and bifurcations, we define a new type of singularity, the homeostasis-bifurcation point. Using singularity theory, the behavior of dynamical systems with homeostasis-bifurcation points is characterized. In particular, we show that multiple homeostatic plateaus separated by hysteretic switches and homeostatic limit cycle periods and amplitudes are common when these singularities occur.

Identifiability asks whether it is possible to infer model parameters from measurements. We characterize the structural identifiability properties for feedforward networks with linear reaction rate kinetics. Interestingly, the set of reaction rates corresponding to the edges of the graph are identifiable, but the assignment of rates to edges is not; Permutations of the reaction rates leads to the same measurements. We show how the identifiability results for linear kinetics can be extended to Michaelis-Menten kinetics using asymptotics.

In perifusion cell cultures, the culture medium flows continuously through a chamber containing immobilized cells and the effluent is collected at the end. In our main applications, gonadotropin releasing hormone (GnRH) or oxytocin is introduced into the chamber as the input. They stimulate the cells to secrete luteinizing hormone (LH), which is collected in the effluent. To relate the effluent LH concentration to the cellular processes producing it, we develop and analyze a mathematical model consisting of coupled partial differential equations describing the intracellular signaling and the movement of substances in the cell chamber. We analyze three different data sets and give cellular mechanisms that explain the data. Our model indicates that two negative feedback loops, one fast and one slow, are needed to explain the data and we give their biological bases. We demonstrate that different LH outcomes in oxytocin and GnRH stimulations might originate from different receptor dynamics. We analyze the model to understand the influence of parameters, like the rate of the medium flow or the fraction collection time, on the experimental outcomes. We investigate how the rate of binding and dissociation of the input hormone to and from its receptor influence its movement down the chamber. Finally, we formulate and analyze simpler models that allow us to predict the distortion of a square pulse due to hormone-receptor interactions and to estimate parameters using perifusion data. We show that in the limit of high binding and dissociation the square pulse moves as a diffusing Gaussian and in this limit the biological parameters can be estimated.

Immunotherapy offers new and promising treatments for many different types of cancer, but the problem of how to practically regulate the immune response in a way that allows it to fight cancer without overwhelming patients with potentially fatal inflammatory toxicity has yet to be solved. This partly results from lingering unanswered questions in immunology and partly results from the inevitable variability between one patient and the next. Here we look specifically at Chimeric Antigen Receptor (CAR)-T cell therapy, explore the mechanisms by which this living drug grows and changes inside the body, and explore how and why this process differs so dramatically between patients.

We develop two ordinary differential equation models -- one which unifies several currently conflicting theories on T cell differentiation pathways, and one which serves as a framework for a Simulated Randomized Clinical Trial (SRCT) to evaluate CAR-T cell therapy as a treatment for patients with leukemia. Our results provide a plausible mechanistic explanation for widely variable patient responses to CAR-T cell therapy seen clinically, and suggest possible ways to improve patient outcomes, even and especially when the patient population is highly heterogeneous. In particular, our results suggest that improvements in patient outcomes can be obtained by reintroducing key characteristics of the endogenous T-cell response that are lost with the dosing protocols currently being used in clinical trials. Our primary conclusion is that single or (homogeneously) split dose protocols should be replaced with protocols for achieving, during the initial stages of therapy, low serum concentrations of effector CAR-T cells, followed by higher serum concentrations of effector CAR-T cells as the tumor burden decreases, and finally substantial serum concentrations of memory CAR-T cells at the end of the treatment period.

This dissertation includes three projects in mathematical biology: (1) Mathematical modeling of the circadian clock and dopamine, (2) Mathematical analysis of a circadian clock model, and (3) Mathematical modeling of sex differences in one-carbon metabolism. Circadian rhythms, dopamine, and metabolism are all important aspects of human physiology and we use methods from dynamical systems and scientific computing to investigate these systems at the molecular level.

First, we create a mathematical model to understand the influences of the mammalian circadian clock on the neurotransmitter dopamine (DA). The circadian clock circuitry in the suprachiasmatic nucleus (SCN) drives 24-hour rhythms in many important physiological processes, including the dopaminergic system. DA imbalances have been linked to various neurological and psychiatric conditions such as Parkinson's disease, attention-deficit/hyperactivity disorder (ADHD), and mood disorders. Previous studies have shown that these conditions are often accompanied by disrupted circadian rhythms, but it has not been well understood why. We use our mathematical model to explain the mechanisms by which the circadian clock influences DA in the brain. We show that the model corresponds well with a host of experimental data and accurately predicts daily variation in extracellular DA.

The model is comprised of a system of nonlinear ordinary differential equations with solutions displaying a remarkably robust 24-hour period consistent with the biology. We further investigate different dynamical behaviors in the model, including periodicity, quasiperiodicity, and decoupling. We show that these behaviors are biologically meaningful and may help to explain clinically observed circadian or dopaminergic disruptions. In addition, experiments have suggested several links between the mammalian clock and one-carbon metabolism (OCM). OCM is essential for the synthesis of DNA and proteins, and we use mathematical modeling to understand how the biochemical reactions in OCM are influenced by sex hormones and micronutrients like folate, vitamin B12, and vitamin B6.

Evolution by natural selection can occur at multiple biological scales. This is particularly the case for host-pathogen systems, where selection occurs both within each infected host as well as through transmission between hosts. Despite there being established mathematical models for understanding evolution at a single biological scale, fewer tractable models exist for multiscale evolutionary dynamics. Here I present mathematical approaches using tools from probability and stochastic processes as well as dynamical systems to handle multiscale evolutionary systems. The first problem I address concerns the antigenic evolution of influenza. Using a combination of ordinary differential equations and inhomogeneous Poisson processes, I study how immune selection pressures at the within-host level impact population-level evolutionary dynamics. The second problem involves the more general question of evolutionary dynamics when selection occurs antagonistically at two biological scales. In addition to host-pathogen systems, such situations arise naturally in the evolution of traits such as the production of a public good and the use of a common resource. I introduce a model for this general phenomenon that is intuitively visualized as a a stochastic ball-and-urn system and can be used to systematically obtain general properties of antagonistic multiscale evolution. Lastly, this ball-and-urn framework is in itself an interesting mathematical object which can studied as either a measure-valued process or an interacting particle system. In this mathematical context, I show that under different scalings, the measure-valued process can have either a propagation of chaos or Fleming-Viot limit.

We consider stochastic hybrid systems that stem from evolution equations with right-hand sides that stochastically switch between a given set of right-hand sides. To begin our study, we consider a linear ordinary differential equation whose right-hand side stochastically switches between a collection of different matrices. Despite its apparent simplicity, we prove that this system can exhibit surprising behavior.

Next, we construct mathematical machinery for analyzing general stochastic hybrid systems. This machinery combines techniques from various fields of mathematics to prove convergence to a steady state distribution and to analyze its structure.

Finally, we apply the tools from our general framework to partial differential equations with randomly switching boundary conditions. There, we see that these tools yield explicit formulae for statistics of the process and make seemingly intractable problems amenable to analysis.

Biochemical reaction networks with a sufficiently large number of molecules may be represented as systems of differential equations. Many networks receive inputs that fluctuate continuously in time. These networks may never settle down to a static equilibrium and are of great interest both mathematically and biologically. Biological systems receive inputs that vary on multiple time scales. Hormonal and neural inputs vary on a scale of seconds or minutes; inputs from meals and circadian rhythms vary on a scale of hours or days; and long term environmental changes (such as diet, disease, and pollution) vary on a scale of years. In this thesis, we consider the limiting behavior of networks in which the input is on a different time scale compared to the reaction kinetics within the network.

We prove analytic results of how the variance of reaction rates within a system compares to the variance of the input when the input is on a different time scale than the reaction kinetics within the network. We consider the behavior of simple chains, single species complex networks, reversible chains, and certain classes of non-linear systems with time-scaled stochastic input, as the input speeds up and slows down. In all cases, as the input fluctuates more and more quickly, the variance of species within the system approaches to zero. As the input fluctuates more and more slowly, the variance of the species approaches the variance of the input, up to a normalization factor.