Cette thèse porte sur la mise en place et le développement d’une approche expérimentale pour l’étude de la dynamique spatio-temporelle de réseaux de réactions à base d’ADN. Nos résultats démontrent la capacité des réseaux d’ADN à se spatialiser sous la forme d’ondes progressives. Nous avons également pu obtenir des motifs stationnaires à base d’ADN et d’assemblages de billes. Ce travail contribue donc à la conception de motifs spatio-temporels de réactions chimiques et de matériaux par le biais de réseaux réactionnels biochimiques programmables. Nous apportons également de nouvelles données sur l’émergence d’ordre spatio-temporel à partir de processus de réaction-diffusion. De ce fait, cette étude contribue à une meilleure compréhension des principes fondamentaux qui régissent l’apparition d’une auto-organisation moléculaire dans un système chimique hors-équilibre. De plus, la combinaison de réseaux synthétiques d’ADN, du contrôle du coefficient de diffusion de plusieurs espèces d’ADN et de la micro-fluidique peut donner lieu à des motifs spatiaux stables, comme par exemple, les fameuses structures de Turing, ce qui tend à confirmer le rôle de celles-ci dans la morphogénèse. / This PhD work is devoted to developing an experimental framework to investigate chemical spatiotemporal organization through mechanisms that could be at play during pattern formation in development. We introduce new tools to increase the versatility of DNA-based networks as pattern-forming systems. The emergence of organization in living systems is a longstanding fundamental question in biology. The two most influential ideas in developmental biology used to explain chemical pattern formation are Wolpert's positional information and Turing's reaction-diffusion self-organization. In the case of positional information, the pattern emerges from a pre-existing morphogen gradient across space that provides positional values as in a coordinate system. Whereas, the Turing mechanism relies on self-organization by driving a system of an initially homogeneous distribution of chemicals into an inhomogeneous pattern of concentration by a process that involves solely reaction and diffusion. Although numerical simulations and mathematical analysis corroborate the incredible potential of reaction-diffusion mechanisms to generate patterns, their experimental implementation is not trivial. And despite of the exceptional achievements in pattern formation with Belousov–Zhabotinsky systems, these are difficult to engineer, thus limiting their experimental implementation to few available mechanisms. In order to engineer reaction-diffusion systems that display spatiotemporal dynamics the following three key elements must be controlled: (i) the topology of the network (how reactions are linked to each other, i.e. in a positive or negative feedback manner), (ii) the reaction rates and (iii) the diffusion coefficients. Recently, using nucleic acids as a substrate to make programmable dynamic chemical systems together with the lessons from synthetic biology and DNA nanotechnology has appeared as an attractive approach due to the simplicity to control reaction rates and network topology by the sequence. Our experimental framework is based on the PEN-DNA toolbox, which involves DNA hybridization and enzymatic reactions that can be maintained out of equilibrium in a closed system for long periods of time. The programmability and biocompatibility of the PEN-DNA toolbox open new perspectives for the engineering of the reaction-diffusion chemical synthesis, in particular in two directions. Firstly, to study biologically-inspired pattern-forming mechanisms in simplified, yet relevant, experimental conditions. Secondly to build new materials that would self-build by a process inspired from embryo morphogenesis. We worked towards the goal of meeting the two requirements of Turing patterning, transferring chemical spatiotemporal behavior into material patterns, and imposing boundary conditions to spatiotemporal patterns. Therefore, the structure of this document is divided into four specific objectives resulting in four chapters. In chapter 1 we worked on testing a DNA-based reaction network with an inhibitor-activator topology. In chapter 2 we focused on developing a strategy to tune the diffusion coefficient of activator DNA strands. In chapter 3 we explored how chemical patterns determine the shape of a material. Finally, in chapter 4 we addressed the issue of controlling the geometry over a DNA-based reaction-diffusion system. Overall, we have expanded the number of available tools to study chemical and material pattern formation and advance towards Turing patterns with DNA.

Cette thèse porte sur la dynamique de fronts de réaction-diffusion en présence de défauts localisés. Nous étudions des non-linéarités bistable et monostable pour lesquelles il existe des solutions exactes en milieu homogène. L'équation aux dérivées partielles est résolue numériquement et la solution est approchée en utilisant des solutions exactes. Parallèlement, nous développons une analyse en coordonnées collectives, position et largeur du front, basée sur des lois d'équilibre. Pour les deux non-linéarités, l'analyse approchée est en bon accord avec la solution numérique. Il est de plus possible de prédire l'arrêt du front dans le cas bistable. L'étude révèle des différences qualitatives entre les deux types de non linéarités. Elle montre l'importance des dimensions caractéristiques du défaut et du front. Enfin, elle fournit un modèle standardisé qui peut servir en théorie du contrôle ou pour la détermination de paramètres à partir de séries temporelles. / We study reaction-diffusion fronts in presence of a localized defect. We consider bistable and monostable nonlinearities for which exact solutions exist in the homogeneous case. The partial differential equation is solved numerically and the solution is fitted using these exact solutions. We also develop a collective coordinate analysis for the position and width of a front, based on balance laws. For both non linearities, the approximate analysis agrees well with the numerical solution. We cab predict the pinning of the front in the bistable case. The sudy reveals qualitative differences between the two nonlinearities. It shows the importance of the characteristic lenghts of the defect and the front. Finally it provides a reduced model, useful for control theory or for the determination of parameters from time-series.

Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

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.

Recent experiments revealed micrometer (µm)-sized Selenium (Se)-doped Germanium (Ge) particles forming a network of inactive phase (Li-Ge-Se) bring superior performance in cycling stability and capacity over un-doped Ge particles. Therefore, based on two states of Li (one for diffusion and another for alloyed reaction), a phaseield model (PFM) is developed incorporating both chemical reaction and Li diffusion to investigate remaining elusive underpinning mechanism. The reaction-diffusion PFM enables us to directly determine the conditions under which the lithiation process is diffusion- and/or reaction-controlled. Moreover, coupling the elasto-plastic deformation, the model allows us to investigate the role of the inactive phase in morphology and stress variation of Se-doped Ge electrode upon lithiation. The numerical results reveal that the tensile hoop stress at the surface of the particles is significantly suppressed due to softness of the inactive Li-Ge-Se phase, in line with the experimental observation of surface fractureree behavior. Further, we find that the soft Li-Ge-Se phase reduces a compressive mean stress at the reaction front, thus alleviating the stress retardation effect on the lithiation kinetics. And, the high Li diffusivity of the amorphous Li-Ge-Se network provides an effective Li diffusion path for inter-particle diffusion, reducing stress difference between the surfaces of neighboring particles. Besides, the constraint between the adjacent particles induces a higher compressive stress at the reaction front impeding the mobile Li insertion during lithiation. Though small c-Ge nano-particle in the Ge0.9Se0.1 microparticle is lithiated faster than large one, the compressive stress is generated at the center of small one for stress equilibrium which causes more retardation effect. Meanwhile, the size difference between adjacent particles increases the principle and shear stresses in the inactive Li-Ge-Se network near adjacent surfaces, which could potentially lead to mechanical failure and debonding of the amorphous network. We believe that the results of this investigation can shed some light on the optimization design of electrodes.

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This dissertation presents a detailed account of the study undertaken on the subject of heat and mass transfer phenomena in porous media. The current work specifically targets the general reaction-diffusion systems arising in separation processes using porous sorbent particles. These particles are comprised of pore channels spanning length scales over almost three orders of magnitude while involving a variety of physical processes such as mass diffusion, heat transfer and surface adsorption-desorption. A novel methodology is proposed in this work that combines models that account for the multi-scale and multi-physics phenomena involved. Pore-resolving DNS calculations using an immersed boundary method (IBM) framework are used to simulate the macro-scale physics while the phenomena at smaller scales are modeled using a sub-pore modeling technique. The IBM scheme developed as part of this work is applicable to complex geometries on curvilinear grids, while also being very efficient, consuming less than 1% of the total simulation time per time-step. A new method of implementing the conjugate heat transfer (CHT) boundary condition is proposed which is a direct extension of the method used for other boundary conditions and does not involve any complex interpolations like previous CHT implementations using IBM. Detailed code verification and validation studies are carried out to demonstrate the accuracy of the developed method. The developed IBM scheme is used in conjunction with a stochastic reconstruction procedure based on simulated annealing. The developed framework is tested in a two-dimensional channel with two types of porous sections - one created using a random assembly of square blocks and another using the stochastic reconstruction procedure. Numerous simulations are performed to demonstrate the capability of the developed framework. The computed pressure drops across the porous section are compared with predictions from the Darcy-Forchheimer equation for media composed of different structure sizes. The developed methodology is also applied to CO2 diffusion studies in porous spherical particles of varying porosities. For the pore channels that are unresolved by the IBM framework, a sub-pore modeling methodology developed as part of this work which solves a one-dimensional unsteady diffusion equation in a hierarchy of scales represented by a fractal-type geometry. The model includes surface adsorption-desorption, and heat generation and absorption. It is established that the current framework is useful and necessary for reaction-diffusion problems in which the adsorption time scales are very small (diffusion-limited) or comparable to the diffusion time scales. Lastly, parametric studies are conducted for a set of diffusion-limited problems to showcase the powerful capability of the developed methodology. / Ph. D.