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.

Im Fokus dieser Untersuchung steht der Einfluss von Fluktuationen auf gekoppelte erregbare Systeme. Wir betrachten numerisch die stationäre Wahrscheinlichkeitsverteilung und den -fluss für ein einzelnes FitzHugh--Nagumo (FHN) System mit Rauschen. Abhängig von der Rauschintensität treten unterschiedliche Kombinationen von Extrema in der Verteilung auf. In einer Kombination finden wir Reminiszenzen an Kohärente Resonanz. Zur Untersuchung von gekoppelten Ensembles erregbarer Systeme nutzen wir eine Methode die auf der Dynamik der zentralen Momente der zugehörigen Verteilungen basiert. Wir leiten einen allgemeinen Ausdruck für ein System mit N Variablen her und diskutieren die Qualität verschiedener Näherungsmethoden. Rauschen kann die Dynamik eines ursprünglich nicht erregbaren Systems derart verändern, dass dieses Erregbarkeit zeigt. Dies demonstrieren wir durch Verallgemeinerung eines bekannten Modells für rauschinduzierte Phasenübergänge. Mit Hilfe der Momentenmethode erhalten wir das Bifurkationsdiagramm. Es zeigt Regionen rauschinduzierter Oszillationen und auch rauschinduzierter Erregbarkeit. Wenn wir additives Rauschen auf ein global gekoppeltes Ensemble mit FHN Kinetik anwenden, beobachten wir ein kompliziertes Übergangsregime hin zu Oszillationen des Mittelwerts. Neben Periodenverdopplung, Chaos und anderen Dynamiken finden wir ein plötzliches starkes Ansteigen der Ausdehnung eines chaotischen Attraktors. Dieses Phänomen ist bei Grenzzyklen als Canard Explosion bekannt. Wir demonstrieren die Möglichkeit von Musterbildung mit Hilfe von dichotomen Fluktuationen an Hand eines lokal gekoppelten Systems mit FHN Kinetik. Abhängig von räumlichen und zeitlichen Korrelationen tritt die Formation von Structure Patterns durch unterschiedliche Mechanismen und in unterschiedlichen Parameterregionen auf. / The focus of this study is on the influence of fluctuations on coupled excitable systems. We examine numerically the stationary probability distribution and the flux for an individual FitzHugh--Nagumo (FHN) system with noise. Depending on noise intensity different combinations of extrema of the distribution occur. In one combination we find reminiscences of coherence resonance. For the investigation of coupled ensembles of excitable systems we use a method based on the central moment dynamics of the corresponding probability distribution. We derive a general expression for a system with N variables and discuss the quality of different approximation techniques. Noise can alter dynamics that are formerly not excitable in a way that it becomes excitable. We demonstrate this using a generalization of a well known model for noise-induced phase transitions. With the help of the moment dynamics we obtain the phase diagram. It shows regions of noise induced oscillations and noise-induced excitability. When applying additive noise to a globally coupled ensemble with FHN kinetics a complicated transition regime towards oscillations of the mean is observed. Besides period-doubling, chaos, and other regimes we find a quick increase of the size of a chaotic attractor. This phenomenon is known from limit cycle oscillations as Canard explosion. We demonstrate the possibility of pattern formation with the help of dichotomous fluctuations using a system with nearest neighbor coupling obeying FHN kinetics. Depending on the spatial and temporal correlations we find structure pattern formation via different mechanisms in different parameter regions.

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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.

No description available.