Stability of traveling waves in partly parabolic systems

Abstract. We review recent results on stability of traveling waves in partly parabolic reactiondiffusion systems with stable or marginally stable equilibria. We explain how attention to what are apparently mathematical technicalities has led to theorems that allow one to convert spectral calculations, which are used in the sciences and engineering to study stability of a wave, into detailed, theoretically-based information about the behavior of perturbations of the wave

Bistable dynamics underlying excitability of ion homeostasis in neuron models

When neurons fire action potentials, dissipation of free energy is usually not directly considered, because the change in free energy is often negligible compared to the immense reservoir stored in neural transmembrane ion gradients and the long-term energy requirements are met through chemical energy, i.e., metabolism. However, these gradients can temporarily nearly vanish in neurological diseases, such as migraine and stroke, and in traumatic brain injury from concussions to severe injuries. We study biophysical neuron models based on the Hodgkin-Huxley (HH) formalism extended to include time-dependent ion concentrations inside and outside the cell and metabolic energy-driven pumps. We reveal the basic mechanism of a state of free energy-starvation (FES) with bifurcation analyses showing that ion dynamics is for a large range of pump rates bistable without contact to an ion bath. This is interpreted as a threshold reduction of a new fundamental mechanism of 'ionic excitability' that causes a long-lasting but transient FES as observed in pathological states. We can in particular conclude that a coupling of extracellular ion concentrations to a large glial-vascular bath can take a role as an inhibitory mechanism crucial in ion homeostasis, while the Na$^+$/K$^+$ pumps alone are insufficient to recover from FES. Our results provide the missing link between the HH formalism and activator-inhibitor models that have been successfully used for modeling migraine phenotypes, and therefore will allow us to validate the hypothesis that migraine symptoms are explained by disturbed function in ion channel subunits, Na$^+$/K$^+$ pumps, and other proteins that regulate ion homeostasis.Comment: 14 pages, 8 figures, 4 table

Stability estimates for semigroups and partly parabolic reaction diffusion equations

The purpose of my dissertation is the application of the methods of abstract theory of strongly continuous operator semigroups (and of evolution semigroups in particular) to study of the spectral properties of a class of differential operators on the line that appears when one linearizes partial differential equations about such special solutions as steady states or traveling waves. First we discuss the stability of the traveling wave solutions of a reaction-diffusion system with a degenerate diffusion matrix. We demonstrate that under some reasonable assumptions on the system, its spectral stability directly implies the linear stability. In particular, we study asymptotic spectral properties of certain first order matrix differential operators, thus generalizing some results known for the evolution semigroups. We then turn to abstract strongly continuous operator semigroup on Banach spaces, revisit a quantitative version of Datko's Stability Theorem and obtain the estimates for the constant M satisfying the inequalityT(t)≤ M eω t, for all t ≥ 0, in terms of the norm of the convolution, Lp-Fourier multipliers, and other operators involved in Datko's Stability Theorem. This generalizes recent results for the Hilbert spaces on estimating M in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane

Mechanobiological Control of Circular Dorsal Ruffle Dynamics

Dynamic structures of polymerized actin play a crucial role in different cellular processes. These include different kinds of actin waves in a multitude of cell types, like Dictyostelium, neutrophiles, macrophages and fibroblasts. These actin waves are connected to a remodeling of the cytoskeleton, cell protrusion and migration as well as the uptake of extracellular fluids, but their specific functions are still debated. One type of them are circular dorsal ruffles (CDRs), actin-based ring-like membrane undulations on the dorsal cell side of fibroblasts, which emerge after growth factor stimulation. A large number of macromolecules were shown to be localized in CDRs and to be crucial for CDR formation. However, to date, the detailed signaling pathway and the underlying mechanism of CDR formation including their molecular main players remain unknown. Different studies on CDRs described them as actin waves in an excitable system or as wavefronts in a bistable regime between two stable states of actin. However, other studies focused on the interaction between actin polymerization and the cell membrane via the interplay of curved membrane protein complexes. This thesis further investigates the mechanism underlying CDR formation. For this study, the morphology of cells is an essential effector for the dynamics of actin waves. Their complexity and dynamical remodeling pose a challenge to the comparability of data. Therefore, in this work, fibroblasts are shaped into well-defined morphologies by seeding them on disk-like adhesion patterns made of fibronectin. This enables to identify long-range interactions between different CDRs combined with the influence of stochastic perturbations and thus uncovers the important role of the membrane tension in CDR dynamics. In combination with microfluidics, the response of the actin wave machinery to biochemical interference with drugs that target different parts of the actin machinery is investigated. The system allows systematical measurements of CDR velocities, periodicities and lifetimes that are performed to carry out a before/after comparison of the treated cells for examining the influence of actin, PIP3 and N-WASP. It is observed a dependence of CDR velocities, periodicities and lifetimes on the total amount of actin leading to the conclusion of a direct regulating role of actin in CDR formation and propagation. Furthermore, it is found that the actin nucleator N-WASP plays a fundamental role in CDR formation but not in CDR propagation. Numerical solutions of wavefronts in a bistable regime of a model system on an annulus domain resemble experimentally gained data and further uncover a dependence of the stimulation threshold for propagating wavefronts on the total actin concentration. The results underline the hypothesis that CDRs can be considered as wavefronts in a bistable regime between two stable states of actin

The effect of gap junctional coupling on the spatiotemporal patterns of Ca2+ signals and the harmonization of Ca2+-related cellular responses

The calcium ion (Ca²⁺), a universal signaling molecule, is widely recognized to play a fundamental role in the regulation of various biological processes. Agonist–evoked Ca²⁺ signals often manifest as rhythmic changes in the cytosolic free Ca²⁺ concentration (ccyt) called Ca²⁺ oscillations. Stimuli intensity was found to be proportional to the oscillation frequency and the evoked down-steam cellular response. Stochastic receptor expression in individual cells in a cell population inevitably leads to individually different oscillation frequencies and individually different Ca²⁺-related cellular responses. However, in many organs, the neighboring cells have to overcome their individually different sensitivity and produce a synchronized response. Gap junctions are integral membrane structures that enable the direct cytoplasmic exchange of Ca²⁺ ions and InsP₃ molecules between neighboring cells. By simulations, we were able to demonstrate how the strength of intercellular gap junctional coupling in relation to stimulus intensity can modify the spatiotemporal patterns of Ca²⁺ signals and harmonize the Ca²⁺-related cellular responses via synchronization of oscillation frequency. We demonstrate that the most sensitive cells are the wave initiator cells and that a highly sensitive region plays an important role in the determination of the Ca²⁺ phase wave direction. This sensitive region will then also progressively determine the global behavior of the entire system

Advancing cell signaling interrogation using theoretical and experimental approaches in eukaryotic model systems

Understanding how cells use intracellular signaling to detect environmental changes and alter behaviors is essential for understanding a wide range of biological processes. The current gap of understanding resides in time-changing signals in individual cells (signaling dynamics) and cell-cell communication in multicellular contexts. Here, we used theoretical and experimental approaches to study cell signaling in two eukaryotic model systems, with a specific focus on processes that involve signaling dynamics and cell-cell communication. First, we focus on the starvation-induced population-level signaling oscillations in the social amoebae, Dictyostelium discoideum. By constructing a unifying theoretical framework, we were able to directly compare existing models and experimental data. From this systematic investigation, we identified that the key features in single-cell signaling networks that coordinate population-level oscillations are adaptive spiking and fold-change detection. We then applied experimental approaches to interrogate how temporal changes in a signaling molecule modulate cell behaviors ("signal decoding") and how environmental cues modulate the dynamics of a signaling molecule ("environment encoding") in mammalian fibroblast cells. First, we explored the impact of transient, direct activation of the cAMP pathway on cell migration using an optogenetic tool. We found that cell migration is inhibited by repetitive transient activation of the cAMP pathway, and the inhibitory effect depends on the extent of activation. By characterizing a series of single-cell behaviors, we found that transient activation of the cAMP pathway induces reversible cell contractile force relaxation and actin cytoskeleton reorganization, both of which can potentially mediate migration inhibition. Further, we confirmed that the induced actin cytoskeleton reorganization is mediated by calcium signaling. Next, we investigated cytosolic calcium dynamics in the presence of a common culture media supplement, serum. We found serum induces trains of calcium spikes and further identified a major serum component mediating this response as lysophosphatidic acid (LPA). Although features of calcium spiking display a great amount of variability among cells, the faction of spiking cells and spiking frequency generally encode the concentration of environmental LPA. Through a series of pharmaceutical inhibitor experiments, we identified major sources of calcium ions as well as other pathways that shape calcium spiking. This body of work demonstrates the different utilities of theoretical modeling and experiments in understanding cell signaling which provides an advanced understanding of biological processes that involve signaling dynamics or cell-cell communication

High-performance computing for computational biology of the heart

This thesis describes the development of Beatbox — a simulation environment for computational biology of the heart. Beatbox aims to provide an adaptable, approachable simulation tool and an extensible framework with which High Performance Computing may be harnessed by researchers. Beatbox is built upon the QUI software package, which is studied in Chapter 2. The chapter discusses QUI’s functionality and common patterns of use, and describes its underlying software architecture, in particular its extensibility through the addition of new software modules called ‘devices’. The chapter summarises good practice for device developers in the Laws of Devices. Chapter 3 discusses the parallel architecture of Beatbox and its implementation for distributed memory clusters. The chapter discusses strategies for domain decomposition, halo swapping and introduces an efficient method for exchange of data with diagonal neighbours called Magic Corners. The development of Beatbox’s parallel Input/Output facilities is detailed, and its impact on scaling performance discussed. The chapter discusses the way in which parallelism can be hidden from the user, even while permitting the runtime execution user-defined functions. The chapter goes on to show how QUI’s extensibility can be continued in a parallel environment by providing implicit parallelism for devices and defining Laws of Parallel Devices to guide third-party developers. Beatbox’s parallel performance is evaluated and discussed. Chapter 4 describes the extension of Beatbox to simulate anatomically realistic tissue geometry. Representation of irregular geometries is described, along with associated user controls. A technique to compute no-flux boundary conditions on irregular boundaries is introduced. The Laws of Devices are further developed to include irregular geometries. Finally, parallel performance of anatomically realistic meshes is evaluated