Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.Comment: Final submitted version; the printed version can be found at http://dx.doi.org/10.1038/nature06095 Supplementary movies are available at http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie1.AVI and http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie2.AV

Controlling Chimeras

Coupled phase oscillators model a variety of dynamical phenomena in nature and technological applications. Non-local coupling gives rise to chimera states which are characterized by a distinct part of phase-synchronized oscillators while the remaining ones move incoherently. Here, we apply the idea of control to chimera states: using gradient dynamics to exploit drift of a chimera, it will attain any desired target position. Through control, chimera states become functionally relevant; for example, the controlled position of localized synchrony may encode information and perform computations. Since functional aspects are crucial in (neuro-)biology and technology, the localized synchronization of a chimera state becomes accessible to develop novel applications. Based on gradient dynamics, our control strategy applies to any suitable observable and can be generalized to arbitrary dimensions. Thus, the applicability of chimera control goes beyond chimera states in non-locally coupled 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

Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans

In this paper, we are interested in the spatio-temporal dynamics of the transmembrane potential in paced isotropic and anisotropic cardiac tissues. In particular, we observe a specific precursor of cardiac arrhythmias that is the presence of alternans in the action potential duration. The underlying mathematical model consists of a reaction–diffusion system describing the propagation of the electric potential and the nonlinear interaction with ionic gating variables. Either conforming piecewise continuous finite elements or a finite volume-element scheme are employed for the spatial discretization of all fields, whereas operator splitting strategies of first and second order are used for the time integration. We also describe an efficient mechanism to compute pseudo-ECG signals, and we analyze restitution curves and alternans patterns for physiological and pathological cardiac rhythms

Protein Pattern Formation

Protein pattern formation is essential for the spatial organization of many intracellular processes like cell division, flagellum positioning, and chemotaxis. A prominent example of intracellular patterns are the oscillatory pole-to-pole oscillations of Min proteins in \textit{E. coli} whose biological function is to ensure precise cell division. Cell polarization, a prerequisite for processes such as stem cell differentiation and cell polarity in yeast, is also mediated by a diffusion-reaction process. More generally, these functional modules of cells serve as model systems for self-organization, one of the core principles of life. Under which conditions spatio-temporal patterns emerge, and how these patterns are regulated by biochemical and geometrical factors are major aspects of current research. Here we review recent theoretical and experimental advances in the field of intracellular pattern formation, focusing on general design principles and fundamental physical mechanisms.Comment: 17 pages, 14 figures, review articl

An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum

Motivated by recent experimental studies, we derive and analyze a twodimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a model of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The model equations for the poroelastic medium are obtained from continuum force-balance equations that include the relevant mechanical fields and an incompressibility relation for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stresses. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a chemomechanical feedback. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.Comment: Additional supplemental material is supplie