Doctor of Philosophy

dissertationThe interplay of dynamics and structure is a common theme in both mathematics and biology. In this thesis, the author develops and analyzes mathematical models that give insight into the dynamics and structure of a variety of biological applications. The author presents a variety of contributions in applications of mathematics to explore biological systems across several scales. First, she analyzes pattern formation in a partial differential equation model based on two interacting proteins that are undergoing passive and active transport, respectively. This work is inspired by a longstanding problem in identifying a biophysical mechanism for the control of synaptic density in C. elegans and leads to a novel mathematical formulation of Turing-type patterns in intracellular transport. The author also demonstrates the persistence of these patterns on growing domains, and discusses extensions for a two-dimensional model. She then presents two models that explore how stochastic processes affect intracellular dynamics. First, the author and her collaborators derive effective stochastic differential equations that describe intermittent virus trafficking. Next, she shows how ion channel fluctuations lead to subthreshold oscillations in neuron models. In the final chapter, she discusses two projects for ongoing and future work: one on modeling parasite infection on dynamic social networks, and another on the bifurcation structure of localized patterns on lattices. All of these projects, presented together, chronicle the journey of the author through her mathematical development and attempts to identify, discover, create, and communicate mathematics that inspires and excites

1991 Summer Study Program in Geophysical Fluid Dynamics : patterns in fluid flow

The GFD program in 1991 focused on pattern forming processes in physics and geophysics. The pricipallecturer, Stephan Fauve, discussed a variety of systems, including our old favorite, Rayleigh-Bénard convection, but passing on to exotic examples such as vertically vibrated granular layers. Fauve's lectures emphasize a unified theoretical viewpoint based on symmetry arguments. Patterns produced by instabilties can be described by amplitude equations, whose form can be deduced by symmetry arguments, rather than the asymptotic expansions that have been the staple of past Summer GFD Programs. The amplitude equations are far simpler than the complete equations of motion, and symetry arguments are easier than asymptotic expansions. Symmetry arguments also explain why diverse systems are often described by the same amplitude equation. Even for granular layers, where there is not a universaly accepted continuum description, the appropnate amplitude equation can often be found using symmetry arguments and then compared with experiment. Our second speaker, Daniel Rothan, surveyed the state of the art in lattice gas computations. His lectures illustrate the great utility of these methods in simulating the flow of complex multiphase fluids, particularly at low Reynolds numbers. The lattice gas simulations reveal a complicated phenomenology much of which awaits analytic exploration. The fellowship lectures cover broad ground and reflect the interests of the staff members associated with the program. They range from the formation of sand dunes, though the theory of lattice gases, and on to two dimensional-turbulence and convection on planetary scales. Readers desiring to quote from these report should seek the permission of the authors (a partial list of electronic mail addresses is included on page v). As in previous years, these reports are extensively reworked for publication or appear as chapters in doctoral theses. The task of assembling the volume in 1991 was at first faciltated by our newly acquired computers, only to be complicated by hurricane Bob which severed electric power to Walsh Cottage in the final hectic days of the Summer.Funding was provided by the National Science Foundation through Grant No. OCE 8901012

Target patterns and pacemakers in reaction-diffusion systems

Pattern formation in systems far from thermal equilibrium is a fascinating phenomenon. Reaction-diffusion systems are an important type of system where pattern formation is observed. The target pattern and the associated wave source called pacemaker are typical patterns in such systems. This thesis studies pacemakers and target patterns systematically by analytical and numerical means. The underlying dynamics of the system may be oscillatory or excitable and the pacemakers may either consist of spatial heterogeneities of the medium or be self-organized, i.e. result of intrinsic processes. The investigation of heterogeneous pacemakers in oscillatory systems in the framework of the complex Ginzburg-Landau equation focuses on two aspects. First, the conditions of the creation of pacemakers and extended target patterns versus the creation of wave sinks and localized target patterns are derived systematically. In particular, inward traveling target patterns and large heterogeneities are discussed. Then, pacemakers which emit target waves with high frequencies are considered. In this case, the waves become Eckhaus unstable, causing ring-shaped amplitude defects or other complex patterns. For even larger frequencies, the amplitude defects already take place at the boundary of the heterogeneity, giving rise to a localized desynchronization phenomenon. Moreover, wave sinks can have a significant impact on the spatio-temporal dynamics of the system by breaking the waves arriving from other wave sources. It is well known that oscillatory media close to a Hopf bifurcation are not able to give rise to stable self-organized pacemakers. Therefore, to model such pacemakers, a system close to a pitchfork-Hopf bifurcation is proposed. The normal form and amplitude equations of the pitchfork-Hopf bifurcation are derived. Such a system displays birhythmicity, i.e. bistability of limit cycles, and it is demonstrated analytically that stable self-organized pacemakers are possible. Simulations confirm the existence of stable self-organized pacemakers. In the presence of a parameter gradient, such patterns drift, as shown analytically and numerically. The interaction between pacemakers is studied numerically, giving rise either to coexisting pacemakers or to a new phenomenon called global inhibition: Established pacemakers suppress new cores or merge with them. When the frequencies of the limit cycles differ strongly, the waves may become Eckhaus unstable and the pacemaker may destabilize. Furthermore, kinetic instabilities of the pacemakers are possible, creating breathing and swinging pacemakers. Self-organized pacemakers in excitable media are usually unstable. In this thesis, a three-component activator-inhibitor system on the basis of the FitzHugh-Nagumo model is proposed that gives rise to stable self-organized pacemakers in the excitable regime. The formation of such patterns is demonstrated if several conditions are fulfilled: The system is close to relaxational oscillations, the additional component is strongly diffusive, and the additional component inhibits the inhibitor. Moreover, bistability of pulse solutions is observed in such a system. Different pulses can interact and may create pacemakers. Alternatively, other complex spatio-temporal dynamics is observed. If the diffusion of the activator vanishes, the waves emitted by the wave source are unstable and spatio-temporal chaos appears. Thus, this thesis presents new results on the dynamics of pacemakers with large frequencies and demonstrates for the first time the possibility of stable self-organized pacemakers in birhythmic and excitable systems

Principles and theory of protein-based pattern formation

Biological systems perform functions by the orchestrated interplay of many small components without a "conductor." Such self-organization pervades life on many scales, from the subcellular level to populations of many organisms and whole ecosystems. On the intracellular level, protein-based pattern formation coordinates and instructs functions like cell division, differentiation and motility. A key feature of protein-based pattern formation is that the total numbers of the involved proteins remain constant on the timescale of pattern formation. The overarching theme of this thesis is the profound impact of this mass-conservation property on pattern formation and how one can harness mass conservation to understand the underlying physical principles. The central insight is that changes in local densities shift local reactive equilibria, and thus induce concentration gradients which, in turn, drive diffusive transport of mass. For two-component systems, this dynamic interplay can be captured by simple geometric objects in the (low-dimensional) phase space of chemical concentrations. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. Moreover, we introduce the notion of regional (in)stabilities, which allows one to characterize the dynamics in the highly nonlinear regime reveals an inherent connection between Turing instability and stimulus-induced pattern formation. The insights gained for conceptual two-component systems can be generalized to systems with more components and several conserved masses. In the minimal setting of two diffusively coupled "reactors," the full dynamics can be embedded in the phase-space of redistributed masses where the phase space flow is organized by surfaces of local reactive equilibria. Building on the phase-space analysis for two component systems, we develop a new approach to the important open problem of wavelength selection in the highly nonlinear regime. We show that two-component reaction–diffusion systems always exhibit uninterrupted coarsening (the continual growth of the characteristic length scale) of patterns if they are strictly mass conserving. Selection of a finite wavelength emerges due to weakly broken mass-conservation, or coupling to additional components, which counteract and stop the competition instability that drives coarsening. For complex dynamical phenomena like wave patterns and the transition to spatiotemporal chaos, an analysis in terms of local equilibria and their stability properties provides a powerful tool to interpret data from numerical simulations and experiments, and to reveal the underlying physical mechanisms. In collaborations with different experimental labs, we studied the Min system of Escherichia coli. A central insight from these investigations is that bulk-surface coupling imparts a strong dependence of pattern formation on the geometry of the spatial confinement, which explains the qualitatively different dynamics observed inside cells compared to in vitro reconstitutions. By theoretically studying the polarization machinery in budding yeast and testing predictions in collaboration with experimentalists, we found that this functional module implements several redundant polarization mechanisms that depend on different subsets of proteins. Taken together, our work reveals unifying principles underlying biological self-organization and elucidates how microscopic interaction rules and physical constraints collectively lead to specific biological functions.Biologische Systeme führen Funktionen durch das orchestrierte Zusammenspiel vieler kleiner Komponenten ohne einen "Dirigenten" aus. Solche Selbstorganisation durchdringt das Leben auf vielen Skalen, von der subzellulären Ebene bis zu Populationen vieler Organismen und ganzen Ökosystemen. Auf der intrazellulären Ebene koordiniert und instruieren proteinbasierte Muster Funktionen wie Zellteilung, Differenzierung und Motilität. Ein wesentliches Merkmal der proteinbasierten Musterbildung ist, dass die Gesamtzahl der beteiligten Proteine auf der Zeitskala der Musterbildung konstant bleibt. Das übergreifende Thema dieser Arbeit ist es, den tiefgreifenden Einfluss dieser Massenerhaltung auf die Musterbildung zu untersuchen und Methoden zu entwickeln, die Massenerhaltung nutzen, um die zugrunde liegenden physikalischen Prinzipien von proteinbasierter Musterbildung zu verstehen. Die zentrale Erkenntnis ist, dass Änderungen der lokalen Dichten lokale reaktive Gleichgewichte verschieben und somit Konzentrationsgradienten induzieren, die wiederum den diffusiven Transport von Masse antreiben. Für Zweikomponentensysteme kann dieses dynamische Wechselspiel durch einfache geometrische Objekte im (niedrigdimensionalen) Phasenraum der chemischen Konzentrationen erfasst werden. Auf dieser Phasenraumebene können physikalische Erkenntnisse durch geometrische Kriterien und grafische Konstruktionen gewonnen werden. Darüber hinaus führen wir den Begriff der regionalen (In-)stabilität ein, der es erlaubt, die Dynamik im hochgradig nichtlinearen Regime zu charakterisieren und einen inhärenten Zusammenhang zwischen Turing-Instabilität und stimulusinduzierter Musterbildung aufzuzeigen. Die für konzeptionelle Zweikomponentensysteme gewonnenen Erkenntnisse können auf Systeme mit mehr Komponenten und mehreren erhaltenen Massen verallgemeinert werden. In der minimalen Fassung von zwei diffusiv gekoppelten "Reaktoren" kann die gesamte Dynamik in den Phasenraum umverteilter Massen eingebettet werden, wobei der Phasenraumfluss durch Flächen lokaler reaktiver Gleichgewichte organisiert wird. Aufbauend auf der Phasenraumanalyse für Zweikomponentensysteme entwickeln wir einen neuen Ansatz für die wichtige offene Fragestellung der Wellenängenselektion im hochgradig nichtlinearen Regime. Wir zeigen, dass "coarsening" (das stetige wachsen der charakteristischen Längenskala) von Mustern in Zweikomponentensystemen nie stoppt, wenn sie exakt massenerhaltend sind. Die Selektion einer endlichen Wellenlänge entsteht durch schwach gebrochene Massenerhaltung oder durch Kopplung an zusätzliche Komponenten. Diese Prozesse wirken der Masseumverteilung, die coarsening treibt, entgegen und stoppen so das coarsening. Bei komplexen dynamischen Phänomenen wie Wellenmustern und dem Übergang zu raumzeitlichen Chaos bietet eine Analyse in Bezug auf lokale Gleichgewichte und deren Stabilitätseigenschaften ein leistungsstarkes Werkzeug, um Daten aus numerischen Simulationen und Experimenten zu interpretieren und die zugrunde liegenden physikalischen Mechanismen aufzudecken. In Zusammenarbeit mit verschiedenen experimentellen Labors haben wir das Min-System von Escherichia coli untersucht. Eine zentrale Erkenntnis aus diesen Untersuchungen ist, dass die Kopplung zwischen Volumen und Oberfläche zu einer starken Abhängigkeit der Musterbildung von der räumlichen Geometrie führt. Das erklärt die qualitativ unterschiedliche Dynamik, die in Zellen im Vergleich zu in vitro Rekonstitutionen beobachtet wird. Durch die theoretische Untersuchung der Polarisationsmaschinerie in Hefezellen, kombiniert mit experimentellen Tests theoretischer Vorhersagen, haben wir herausgefunden, dass dieses Funktionsmodul mehrere redundante Polarisationsmechanismen implementiert, die von verschiedenen Untergruppen von Proteinen abhängen. Zusammengenommen beleuchtet unsere Arbeit die vereinheitlichenden Prinzipien, die der intrazellulären Selbstorganisation zugrunde liegen, und zeigt, wie mikroskopische Interaktionsregeln und physikalische Bedingungen gemeinsam zu spezifischen biologischen Funktionen führen

Matrix-based techniques for (flow-)transition studies

In this thesis, numerical techniques for the computation of flow transitions was introduced and studied. The numerical experiments on a variety of two- and three- dimensional multi-physics problems show that continuation approach is a practical and efficient way to solve series of steady states as a function of parameters and to do bifurcation analysis. Starting with a proper initial guess, Newton’s method converges in a few steps. Since solving the linear systems arising from the discretization takes most of the computational work, efficiency is determined by how fast the linear systems can be solved. Our home-made preconditioner Hybrid Multilevel Linear Solver(HYMLS) can compute three-dimensional solutions at higher Reynolds numbers and shows its robustness both in the computation of solutions as well as eigenpairs, due to the iteration in the divergence-free space. To test the efficiency of linear solvers for non-flow problems, we studied a well-known reaction-diffusion system, i.e., the BVAM model of the Turing problem. The application to the Turing system not only proved our program’s ability in doing nonlinear bifurcation analysis efficiently but also provided insightful information on two- and three- dimensional pattern formation

Pattern formation with a conservation law

The evolution of many pattern-forming systems is strongly influenced by the presence of a conserved quantity. Diverse physical phenomena such as magnetoconvection, rotating fluid convection, binary fluid convection, vibrated granular and fluid layers, filament dynamics and sandbank formation, all possess a conservation law which plays a central role in their nonlinear dynamics. In this thesis, this influence of a conserved quantity is examined through analyses of three distinct pattern-formation problems. Firstly, the consequences of conservation of mass are investigated in a phenomenological model of a vibrated granular layer. A new weakly nonlinear analysis is performed that reveals the existence of modulational instabilities for patterns composed of either stripes and squares. The nonlinear evolution of these instabilities is numerically studied and a plethora of patterns and localised arrangements are exhibited. The second component of this work concerns an oscillatory bifurcation in the presence of a conserved quantity. Building upon existing theory for the corresponding stationary bifurcation, universal amplitude equations are constructed through symmetry and asymptotic considerations. Subsequently, the stability properties of travelling and standing waves are found to be significantly altered and new modulational instabilities are uncovered. Numerical simulations reveal that, in the presence of a conserved quantity, travelling and standing waves lose stability to spatially localised patterns, either coherent, time-periodic or chaotic. Finally, wave-like behaviour of large-scale modes is examined through an analysis of a model for Faraday waves, that has been modified to account for flnite fluid depth. Several approaches to the weakly nonlinear analysis are considered and two sets of amplitude equations are derived that account for the unusual wave-like behaviour of large-scale modes. Numerical simulations reveal amplitude-modulated and localised patterns away from the small-amplitude, weak-viscosity limit