Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How patchy landscape affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behaviour to local habitat quality and show preferences for some habitat types over others. In this dissertation, we focus on how landscape composition and the movement behaviour at an interface between habitat patches of different quality affects the steady states of a single species and a predator-prey system. First, we consider a model for population dynamics in a habitat consisting of two homogeneous one-dimensional patches in a coupled ecological reaction-diffusion equation. Several recent publications by other authors explored how individual movement behaviour affects population-level dynamics in a framework of reaction-diffusion systems that are coupled through discontinuous boundary conditions. The movement between patches is incorporated into the interface conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We establish the existence, uniqueness and global asymptotic stability of the steady state, and we classify their qualitative shape depending on movement behaviour. We clarify the role of nonrandom movement in this context, and we apply our analysis to a previous result where it was shown that a randomly diffusing population in a continuously varying habitat can exceed the carrying capacity at steady state. In particular, we apply our results to study the question of why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape. Secondly, we model population dynamics with a predator-prey system in a coupled ecological reaction-diffusion equation in a heterogeneous landscape to study Turing patterns that emerge from diffusion-driven instability (DDI). We derive the DDI conditions, which consist of necessary and sufficient conditions for initiation of spatial patterns in a one-dimensional homogeneous landscape. We use a finite difference scheme method to numerically explore the general conditions using the May model, and we present numerical simulations to illustrate our results. Then we extend our studies on Turing-pattern formation by considering a predator-prey system on an infinite patchy periodic landscape. The movement between patches is incorporated into the interface conditions that link the reaction-diffusion equations between patches. We use a homogenization technique to obtain an analytically tractable approximate model and determine Turing-pattern formation conditions. We use numerical simulations to present our results from this approximation method for this model. With this tool, we then explore how differential movement and habitat preference of both species in this model (prey and predator) affect DDI.

To enhance material performance at different length scales, this study strives to develop a reliable analytical and computational tool with the help of internal state variables spanning micro and macro-level behaviors. First, the practical relevance of a nonlocal damage integral added to an internal state variable (BCJ) model is studied to alleviate numerical instabilities associated within the post-bifurcation regime. The characteristic length scale in the nonlocal damage, which is mathematical in nature, can be calibrated using a series of notch tensile tests. Then the same length scale from the notch tests is used in solving the problem of a high-velocity (between 89 and 107 m/s) rigid projectile colliding against a 6061-T6 aluminum-disk. The investigation indicates that incorporating a characteristic length scale to the constitutive model eliminates the pathological mesh-dependency associated with material instabilities. In addition, the numerical calculations agree well with experimental data. Next, an effort is made rather to introduce a physically motivated length scale than to apply a mathematical-one in the deformation analysis. Along this line, a dislocation based plasticity model is developed where an intrinsic length scale is introduced in the forms of spatial gradients of mobile and immobile dislocation densities. The spatial gradients are naturally invoked from balance laws within a consistent kinematic and thermodynamic framework. An analytical solution of the model variables is derived at homogenous steady state using the linear stability and bifurcation analysis. The model qualitatively captures the formation of dislocation cell-structures through material instabilities at the microscopic level. Finally, the model satisfactorily predicts macroscopic mechanical behaviors - e.g., multi-strain rate uniaxial compression, simple shear, and stress relaxation - and validates experimental results.

All living things are driven by chemical reactions. Reactions provide energy and transform matter. Thus, maintaining the system out of equilibrium. However, these chemical reactions have to be organized in space. One way for this spatial organization is via the process of phase separation. Motivated by the recent discovery of liquid-like droplets in cells, this thesis studies the organization of chemical reactions in phase-separated systems, with and without broken detailed balance. After introducing the underlying thermodynamic principles, we generalize mass-action kinetics to systems with homogeneous compartments formed by phase separation. Here, we discuss the constraints resulting from phase equilibrium on chemical reactions. We study the relaxation kinetics towards thermodynamic equilibrium and investigate non-equilibrium states that arise when detailed balance is broken in the rates of reactions such that phase and chemical equilibria contradict each other. We then turn to spatially continuous systems with spatial gradients within formed compartments. We derive thermodynamic consistent dynamical equations for reactions and diffusion processes in such systems. Again, we study the relaxation kinetics towards equilibrium and discuss non-equilibrium states. We investigate the dynamics of droplets in the presence of reactions with broken detailed balance. Furthermore, we introduce active droplet systems maintained away from equilibrium via coupling to reservoirs at their boundaries and organizing reactions solely within droplets. Here, detailed balance is only broken at the boundaries. Nevertheless, stationary chemically active droplets exist in open systems, and droplets can divide. To quantitatively study chemically active droplet systems in multi-component mixtures, we introduce an effective description. Therefore, we couple linearized reaction-diffusion equations via a moving interface within a sharp interface limit. At the interface, the boundary conditions are set by a local phase equilibrium and the continuity of fluxes. Equipped with these tools, we introduce and study protocell models of chemically active droplets. We explicitly model these protocells’ nutrient and waste dynamics, leading to simple models of their metabolism. Next, we study the energetics of these droplets and identify processes responsible for growth or shrinkage and maintaining the system out of equilibrium. Furthermore, we discuss the energy balance leading to the heating and cooling of droplets. Finally, we show why chemically active droplets do not spontaneously divide in two-dimensional systems with bulk-driven reactions. Here, droplets can elongate but do not pinch off. To have a minimal two-dimensional model with droplet division, we introduce additional reactions. When these reactions are localized at the interface and dependent on its mean curvature, droplets robustly divide in 2D. In summary, this thesis contributes to the theoretical understanding of how the existence of droplets changes the kinetics of reactions and, vice versa, how chemical reactions can alter droplet dynamics.:1 Introduction 1.1 Thermodynamics of phase separation 1.1.1 Phase equilibrium in the thermodynamic limit 1.1.2 Relaxation dynamics towards equilibrium 1.1.3 Local stability of homogeneous phases 1.2 Thermodynamics of chemical reactions in homogenous mixtures 1.2.1 Conserved densities and reaction extents 1.2.2 Equilibrium of chemical reactions 1.2.3 Mass-action kinetics towards equilibrium 1.3 Simultaneous equilibrium of chemical reactions and phase separation 1.4 Chemical reactions maintained away from equilibrium 1.5 Structure of this thesis 2 Chemical reactions in compartmentalized systems 2.1 Mass-action kinetics for compartments built by phase separation 2.1.1 Dynamical equations for densities and phase volumes 2.1.2 Relaxation kinetics in a simple example 2.2 Driven chemical reactions in compartmentalized systems 2.2.1 Non-equilibrium steady states at phase equilibrium 2.2.2 The tie line selecting manifold 2.3 Discussion 3 Dynamics of concentration fields in phase-separating systems with chemical reactions 3.1 Reaction-diffusion equations for phase-separating systems 3.2 Relaxation towards thermodynamic equilibrium in spatial systems 3.2.1 Relaxation kinetics and fast diffusion 3.2.2 Relaxation kinetics with spatial gradients 3.3 Driven chemical reactions in phase-separating systems 3.3.1 Driven chemical reaction and fast diffusion 3.3.2 Non-equilibrium steady states and spatial gradients 3.3.3 Droplets growth and ripening with driven chemical reactions 3.4 Boundary-driven chemically active droplets 3.4.1 Droplets in open systems 3.4.2 Non-equilibrium steady droplets and shape instabilities 3.5 Discussion 4 Chemically active droplets in the sharp interface limit 4.1 Droplet dynamics via reaction-diffusion equations coupled by a moving interface 4.2 Stationary interface positions in spherical symmetry 4.2.1 Interface conditions in closed systems 4.2.2 Interface conditions in open systems 4.3 Shape instabilities of spherical droplets 4.4 Discussion 5 Models of protocells and their metabolism as chemically active droplets 5.1 Breaking detailed balance in protocell models 5.1.1 Boundary-driven protocell models 5.1.2 Bulk-driven protocell models 5.2 Protocell dynamics 5.2.1 Steady states droplets 5.2.2 Shape stability of spherical symmetric droplets 5.3 Energetics of protocells 5.3.1 Mass conservation and droplet growth or shrinkage 5.3.2 Energy conservation and droplet heating or cooling 5.4 Discussion 6 The role of dimensionality on droplet division 6.1 Stability of chemically active droplets in 2D vs. 3D 6.1.1 Stationary droplets in 1D, 2D and 3D 6.1.2 Elongation instability 6.1.3 Pinch-off instability 6.2 Pinch-off in 2D via curvature-dependent chemical reactions 6.2.1 Determining the mean curvature of the droplet interface 6.2.2 Chemical reactions at the interface 6.3 Discussion 7 Conclusion and Outlook A Free energy considerations B Surface tension in multi-component mixtures C Figure details Bibliography

In tissue development, wound healing and aberrant cancer progression cell–cell interactions drive mixing and segregation of cellular composites. However, the exact nature of these interactions is unsettled. Here we study the dynamics of packed, heterogeneous cellular systems using wound closure experiments. In contrast to previous cell sorting experiments, we find non-universal sorting behavior. For example, monolayer tissue composites with two distinct cell types that show low and high neighbor exchange rates (i.e., MCF-10A & MDA-MB-231) produce segregated domains of each cell type, contrary to conventional expectation that the construct should stay jammed in its initial configuration. On the other hand, tissue compounds where both cell types exhibit high neighbor exchange rates (i.e., MDA-MB-231 & MDA-MB-436) produce highly mixed arrangements despite their differences in intercellular adhesion strength. The anomalies allude to a complex multi-parameter space underlying these sorting dynamics, which remains elusive in simpler systems and theories merely focusing on bulk properties. Using cell tracking data, velocity profiles, neighborhood volatility, and computational modeling, we classify asymmetric interfacial dynamics. We indicate certain understudied facets, such as the effects of cell death & division, mechanical hindrance, active nematic behavior, and laminar & turbulent flow as their potential drivers. Our findings suggest that further analysis and an update of theoretical models, to capture the diverse range of active boundary dynamics which potentially influence self-organization, is warranted.

During animal development numerous organs with functions ranging from fluid transport to signal propagation develop into highly branched shapes and forms. To ensure organ function, the formation of their geometrical and topological as well as size-dependent properties is crucial. For example, organ geometry serves to maximize exchange area with its surroundings and organ topology controls the response to fluctuations and damage. Most importantly, organ size and proportion need to scale throughout animal growth to meet the demands of increasing body size. However, how organ geometry and topology are established and scaled in a self-organized manner, remains poorly understood. In this thesis, we present a novel theoretical framework to study the self-organized growth and scaling of branched organs. In this framework, we represent the organ outline by an infinitely thin interface and consider morphogen-controlled interface evolution in growing domains. We demonstrate that an instability in interface motion can lead to the self-organized formation of complex branched morphologies and show how the interplay between interface motion, morphogen dynamics, and domain growth controls the geometrical, topological, and size-dependent properties of the resulting structures. To understand the formation of branched structures from instabilities in morphogen-controlled interface growth, we first consider a range of different interface growth scenarios in non-growing domains. In a first approach, we present a stochastic lattice model with interface growth driven by a morphogen concentration gradient. We find a range of branched morphologies extending from self-similar fractal structures to almost circular structures with only a few branches depending on the morphogen gradient length scale. We present the Euler characteristic as an example of a topological invariant and employ it to introduce topological constraints into interface growth, leading to the formation of tree-like structures. In a second approach, we study a continuum model for morphogen-controlled interface growth. In this model, the interface has a constant tendency to grow and is inhibited by morphogen concentration. Additionally, we take into account a curvature dependency of interface growth, which leads to an effective stabilization of interface motion at small length scales. We identify branch distance and thickness as key morphological properties and discuss their regulation. We relate branch distance regulation to the interplay of destabilization from morphogen inhibition and stabilization from the curvature dependency of interface growth and explain branch thickness regulation in terms of mutual branch inhibition. By considering interface instability in different scenarios, we overall demonstrate the robustness of our approach. Finally, we apply our theoretical framework to study the branching morphogenesis of the planarian gut. The planarian gut is a highly branched organ that spans the entire organism and is responsible for the delivery of nutrients to the planarian body. Planarians undergo massive body size changes of more than one order of magnitude in organism length and thus constitute an ideal model organism to study the growth and scaling of branched organs. We reconsider our continuum model and include novel features needed to account for the organization of the planarian gut. We take into account external guiding cues that alter the orientation of branches and, most importantly, consider branching morphogenesis in a growing domain. We demonstrate that our model can account for the geometrical and topological properties of the gut and show that gut scaling can arise from to the interplay of branch growth and organism growth. Overall, we present a novel theoretical framework to study the growth and scaling of branched organs. In this framework, we demonstrate the self-organized formation of branched morphologies from instabilities in morphogen-controlled interface growth and show how the interplay of interface motion, morphogen dynamics, and system size determine geometry, topology, and size-dependent properties of the resulting structures.

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Die vorliegende Arbeit befasst sich mit einer Reihe von Fragestellungen, die Forschungsfeldern wie rauschinduziertem Verhalten, Strukturbildung in aktiven Medien und Synchronisation nichlinearer Oszillatoren erwachsen. Die verwendeten nichtlinearen Modelle verfügen über erregbare, oszillatorische und bistabile Eigenschaften. Zusätzliche stochastische Fluktuationen tragen wesentlich zur Entstehung komplexer Dynamik bei. Modelliert wird, auf welche Weise sich extrazelluläre Kaliumkonzentration, gespeist von umliegenden Neuronen, auf die Aktivität dieser Neuronen auswirkt. Neben lokaler Dynamik wird die Ausbildung ausgedehnter Strukturen in einem heterogenem Medium analysiert. Die raum-zeitlichen Muster umfassen sowohl Wellenfronten und Spiralen als auch ungewöhnliche Strukturen, wie wandernde Cluster oder invertierte Wellen. Eine wesentliche Rolle bei der Ausprägung solcher Strukturen spielen die Randbedingungen des Systems. Sowohl für diskret gekoppelte bistabile Elemente als auch für kontinuierliche Fronten werden Methoden zur Berechnung von Frontgeschwindigkeiten bei fixierten Rändern vorgestellt. Typische Bifurkationen werden quantifiziert und diskutiert. Der Rückkopplungsmechanismus aus dem Modell neuronaler Einheiten und deren passiver Umgebung kann weiter abstrahiert werden. Ein Zweizustandsmodell wird über zwei Wartezeitverteilungen definiert, welche erregbares Verhalten widerspiegeln. Untersucht wird die instantane und die zeitverzögerte Antwort des Ensembles auf die Rückkopplung. Im Fall von Zeitverzögerung tritt eine Hopf-Bifurkation auf, die zu Oszillationen der mittleren Gesamtaktivität führt. Das letzte Kapitel befasst sich mit Diffusion und Transport von Brownschen Teilchen in einem raum-zeiltich periodischen Potential. Wieder sind es Synchronisationsmechanismen, die nahezu streuungsfreien Transport ermöglichen können. Für eine erhöhte effektiven Diffusion gelangen wir zu einer Abschätzung der maximierenden Parameter. / In this work problems are investigated that arises from resarch fields of noise induced dynamics, pattern formation in active media and synchronisation of self-sustained oscillators. The applied model systems exhibit excitable, oscillatory and bistable behavior as basic modes of nonlinear dynamics. Addition of stochastic fluctuations contribute to the appearance of complex behavior. The extracellular potassium concentration fed by surrounding activated neurons and the feeback to these neurons is modelled. Beside considering the local behavior, nucleation of spatially extended structures is studied. We find typical fronts and spirales as well as unusal patterns such as moving clusters and inverted waves. The boundary conditions of the considered system play an essential role in the formation process of such structures. We present methods to find expressions of the front velocity for discretely coupled bistable units as well as for the countinus front interacting with boundary values. Canonical bifurcation scenarios can be quantified. The feedback mechanism from the model for neuronal units can be generalized further. A two-state model is defined by two waiting time distributions representing excitable dynamics. We analyse the instantaneous and delayed response of the ensemble. In the case of delayed feedback a Hopf-bifurcation occur which lead to oscillations of the mean activity. In the last chapter the transport and diffusion of Brownian particles in a spatio-temporal oscillating potential is discussed. As a cause of nearly dispersionless transport synchronisations mechanisms can be identified. We find an estimation for parameter values which maximizes the effective diiffusion.

La première partie de ma thèse est consacrée à l’étude de l’auto-organisation de souches génétiquement modifiées de bactéries Escherichia coli. Ce projet, réalisé en collaboration avec des biologistes synthétiques de l’Université de Hong Kong, a pour objectif l’exploration et le décryptage d’un nouveau mécanisme d’auto-organisation dans des colonies bactériennes multi-espèces. Cela a été inspiré par la question fascinante de comment les écosystèmes bactériens comprenant plusieurs espèces de bactéries peuvent s’auto-organiser dans l’espace. En considérant des systèmes dans lesquels deux souches de bactéries régulent mutuellement leurs motilités, j’ai pu montrer que le contrôle de densité réciproque est une voie générique de formation de motifs: si deux souches tendent à faire augmenter mutuellement leur motilité (la souche A se déplace plus vite quand la souche B est présent, et vice versa), ils subissent un processus de formation de motifs conduisant à la démixtion entre les deux souches. Inversement, l’inhibition mutuelle de la motilité conduit à la formation de motifs avec colocalisation. Ces résultats ont étévalidés expérimentalement par nos collaborateurs biologistes. Par la suite, j’ai étendu mon étude à des systèmes composés de plus de deux espèces en interaction, trouvant des règles simples permettant de prédire l’auto-organisation spatiale d’un nombre arbitraire d’espèces dont la motilité est sous contrôle mutuel. Cette partie de ma thèse ouvre une nouvelle voie pour comprendre l’auto-organisation des colonies bactériennes avec des souches concurrentes, ce qui est une question importante pour comprendre la dynamique des biofilms ou des écosystèmes bactériens dans les sols. Le deuxième problème traité dans ma thèse est inspiré par le comportement collectif des moteurs moléculaires se déplaçant le long des microtubules dans le cytoplasme des cellules eucaryotes. Un modèle pertinent pour le mouvement des moteurs moléculaires est donné par un système paradigmatique de non-équilibre appelé Processus Asymmetrique d’Exclusion Simple, en anglais Asymmetric Simple Exclusion Process (ASEP). Dans ce modèle sur réseau unidimensionnel, les particules se déplacent dans les sites voisins vides à des taux constants, avec un biais gauche-droite qui déséquilibre le système.Lorsqu’il est connecté à ses extrémités à des réservoirs de particules, l’ASEP est un exemple prototypique de transitions de phase unidimensionnelles guidées par les conditions aux limites. Les exemples réalistes, cependant, impliquent rarement une seule voie:les microtubules sont constitués de plusieurs pistes de tubuline auxquelles les moteurs peuvent s’attacher. Dans ma thèse, j’explique comment on peut théoriquement prédire le comportement de phase de systèmes à plusieurs voies complexes, dans lesquels les particules peuvent également sauter entre des voies parallèles. En particulier, je montre que la transition de phase unidimensionnelle vue dans l’ASEP survit cette complexité supplémentaire mais implique de nouvelles caractéristiques telles que des courants transversaux stables non-nulles et une localisation de cisaillement. / The first part of my thesis is devoted to studying the self-organization of engineered strains of run-and-tumble bacteria Escherichia coli. This project, carried out in collaboration with synthetic biologists at Hong Kong University, has as its objective the exploration and decipherment of a novel self-organization mechanism in multi-species bacterial colonies. This was inspired by the fascinating question of how bacterial ecosystems comprising several species of bacteria can self-organize in space. By considering systems in which two strains of bacteria mutually regulate their motilities, I was able to show that reciprocal density control is a generic pattern-formation pathway: if two strains tend tomutually enhance their motility (strain A moves faster when strain B is present, and conversely),they undergo a pattern formation process leading to demixing between the two strains. Conversely, mutual inhibition of motility leads to pattern formation with colocalization. These results were validated experimentally by our biologist collaborators. Subsequently, I extended my study to systems composed of more than two interacting species, finding simple rules that can predict the spatial self-organization of an arbitrary number of species whose motility is under mutual control. This part of my thesis opens up a new route to understand the self-organization of bacterial colonies with competing strains, which is an important question to understand the dynamics of biofilms or bacterial ecosystems in soils.The second problem treated in my thesis is inspired by the collective behaviour ofmolecular motorsmoving along microtubules in the cytoplasm of eukaryotic cells. A relevant model for the molecularmotors’ motion is given by a paradigmatic non-equilibrium system called Asymmetric Simple Exclusion Process (ASEP). In this one-dimensional lattice- based model, particles hop on empty neighboring sites at constant rates, with a leftright bias that drives the systemout of equilibrium. When connected at its ends to particle reservoirs, the ASEP is a prototypical example of one-dimensional boundary driven phase transitions. Realistic examples, however, seldom involve only one lane: microtubules are made of several tubulin tracks to which the motors can attach. In my thesis, I explained how one can theoretically predict the phase behaviour of complex multilane systems, in which particles can also hop between parallel lanes. In particular, I showed that the onedimensional phase transition seen in the ASEP survives this additional complexity but involves new features such as non-zero steady transverse currents and shear localization.

In this thesis I study the effects of gap junctions on pattern formation in a neural field model for working memory. I review known results for the base model (the “Amari model”), then see how the results change for the “gap junction model”. I find steady states of both models analytically and numerically, using lateral inhibition with a step firing rate function, and a decaying oscillatory coupling function with a smooth firing rate function. Steady states are homoclinic orbits to the fixed point at the origin. I also use a method of piecewise construction of solutions by deriving an ordinary differential equation from the partial integro-differential formulation of the model. Solutions are found numerically using AUTO and my own continuation code in MATLAB. Given an appropriate level of threshold, as the firing rate function steepens, the solution curve becomes discontinuous and stable homoclinic orbits no longer exist in a region of parameter space. These results have not been described previously in the literature. Taking a phase space approach, the Amari model is written as a four-dimensional, reversible Hamiltonian system. I develop a numerical technique for finding both symmetric and asymmetric homoclinic orbits. I discover a small separate solution curve that causes the main curve to break as the firing rate function steepens and show there is a global bifurcation. The small curve and the global bifurcation have not been reported previously in the literature. Through the use of travelling fronts and construction of an Evans function, I show the existence of stable heteroclinic orbits. I also find asymmetric steady state solutions using other numerical techniques. Various methods of determining the stability of solutions are presented, including a method of eigenvalue analysis that I develop. I then find both stable and transient Turing structures in one and two spatial dimensions, as well as a Type-I intermittency. To my knowledge, this is the first time transient Turing structures have been found in a neural field model. In the Appendix, I outline numerical integration schemes, the pseudo-arclength continuation method, and introduce the software package AUTO used throughout the thesis.

The present work is concerned with the mathematical modelling of a bed of granular material in a gravitational field vertically fluidised by a vibrating surface. The particles are in rapid motion, and lose energy by inelastic collisions. The steady state is maintained by a balance of the rate of dissipation of energy in inelastic particle collisions and the rate of transfer of energy due to particle collisions with the vibrating surface. The limit where the energy dissipation due to inelastic collisions is small compared to the mean kinetic energy of the particles is considered. This non-equilibrium steady state is similar to a dilute gas at equilibrium with a uniform temperature and an exponentially decaying density, obtained from the ideal gas equation of state. From the analysis of this state, four non-dimensional numbers are derived which uniquely specify the state of the system. A perturbative analysis about the uniform temperature state is carried out and analytical solutions to the macroscopic variables of the system are obtained using two types of approximations. The first is a hydrodynamic model using constitutive relations from the general kinetic theory of granular media, and the second is a kinetic theory formulation derived exclusively for the vibro-fluidised bed. The latter permits an anisotropy between the horizontal and vertical directions due to the anisotropic nature of the source of energy at the bottom wall. The kinetic theory is extended to incorporate the corrections due to the high density effects, which is similar to the Enskog correction to dense gases. An event driven (ED), or hard sphere molecular dynamic (MD), simulation of the vibrated bed is carried out. The quantitative predictions of the theories are validated by the simulation. A systematic probing of the parameter space within the ED simulations revealed two new phenomena in a vibro-fluidised bed which are inhomogeneous in the horizontal direction. These are convection rolls similar to the Rayleigh-Benard instability in fluids, and a clustering instability leading to a phase separation. The instabilities are characterised using a phase diagram. The homogeneous states close to these new states are adequately described by the models developed here. An analysis of the stability of this state could have implications in understanding the instabilities in driven granular materials (such as in sheared media and fluidised beds) in general, and pattern formation in vibrated beds in particular.