Cancer is a complex, multi-scale disease marked by unchecked cellular growth and proliferation. As a tumor grows, it is known to lose its capacity to maintain a compact structure. This stage of development, known as invasion, is marked by the disaggregation and dispersion of peripheral cells, and the formation of finger-like margins. This thesis provides an overview of three multi-scale models of tumor growth and invasion. The hybrid discrete-continuum (HDC) model couples a cellular automaton approach, which is used to direct the behavior and interactions of individual cells, with a system of reaction-diffusion-chemotaxis equations that describe the micro-environment. The evolutionary hybrid cellular automaton (EHCA) model maintains the core of the HDC approach, but employs an artificial response network to describe cellular dynamics. In contrast to these two, the immersed boundary (IBCell) model describes cells as fully deformable, viscoelastic entities that interact with each other using membrane bound receptors. As part of this thesis, the HDC model has been modified to examine the role of the ECM as a barrier to cellular expansion. The results of these simulations will be presented and discussed in the context of tumor progression.

Cancer is a complex, multi-scale disease marked by unchecked cellular growth and proliferation. As a tumor grows, it is known to lose its capacity to maintain a compact structure. This stage of development, known as invasion, is marked by the disaggregation and dispersion of peripheral cells, and the formation of finger-like margins. This thesis provides an overview of three multi-scale models of tumor growth and invasion. The hybrid discrete-continuum (HDC) model couples a cellular automaton approach, which is used to direct the behavior and interactions of individual cells, with a system of reaction-diffusion-chemotaxis equations that describe the micro-environment. The evolutionary hybrid cellular automaton (EHCA) model maintains the core of the HDC approach, but employs an artificial response network to describe cellular dynamics. In contrast to these two, the immersed boundary (IBCell) model describes cells as fully deformable, viscoelastic entities that interact with each other using membrane bound receptors. As part of this thesis, the HDC model has been modified to examine the role of the ECM as a barrier to cellular expansion. The results of these simulations will be presented and discussed in the context of tumor progression.

Multi-scale modelling and analysis is becoming increasingly important and relevant. Analysis of the emergent properties from the interactions between scales of multi-scale systems is important to aid in solutions. There is no universally adopted theoretical/computational framework or language for the construction of multi-scale models. Most modelling approaches are specific to the problem that they are addressing and use a hybrid combination of modelling languages to model specific scales. This thesis addresses if process algebra can offer a unique opportunity in the definition and analysis of multi-scale models. In this thesis the generic Process Algebra with Layers (PAL) is defined: a language for multi-scale integration modelling. This work highlights the potential of process algebra to model multi-scale systems. PAL was designed based on features and challenges found from modelling a multi-scale system in an existing process algebra. The unique features of PAL are the layers: Population and Organism. The novel language modularises the spatial scales of the system into layers, therefore, modularising the detail of each scale. An Organism can represent a molecule, organelle, cell, tissue, organ or any organism. An Organism is described by internal species. An internal species, dependent on the scale of the Organism, can also represent a molecule, organelle, cell, tissue, organ or any organism. Populations hold specific types of Organism, for example, life stages, cell phases, infectious states and many more. The Population and Organism layers are integrated through mirrored actions. This novel language allows the clear definition of scales and interactions within and between these scales in one model. PAL can be applied to define a variety of multi-scale systems. PAL has been applied to two unrelated multi-scale system case studies to highlight the advantages of the generic novel language. Firstly the effects of ocean acidification on the life stages of the Pacific oyster. Secondly the effects of DNA damage from cancer treatment on the length of a cell cycle and cell population growth.

Evolutionary game theory has recently emerged as a key paradigm in various behavioral science disciplines. In particular it provides powerful tools and a conceptual framework for the analysis of the time evolution of strategic interdependence among players and its consequences, especially when the players are spatially distributed and linked in a complex social network. We develop various evolutionary game models, analyze these models using appropriate techniques, and study their applications to complex phenomena. In the second chapter, we derive integro-differential equations as deterministic approximations of the microscopic updating stochastic processes. These generalize the known mean-field ordinary differential equations and provide powerful tools to investigate the spatial effects on the time evolutions of the agents' strategy choices. The deterministic equations allow us to identify many interesting features of the evolution of strategy profiles in a population, such as standing and traveling waves, and pattern formation, especially in replicator-type evolutions. We introduce several methods of decomposition of two player normal form games in the third chapter. Viewing the set of all games as a vector space, we exhibit explicit orthonormal bases for the subspaces of potential games, zero-sum games, and their orthogonal complements which we call anti-potential games and anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential game comes either from Rock-paper-scissors type games (in the case of symmetric games) or from Matching Pennies type games (in the case of asymmetric games). Using these decompositions, we prove old (and some new) cycle criteria for potential and zero-sum games (as orthogonality relations between subspaces). We illustrate the usefulness of our decompositions by (a) analyzing the generalized Rock-Paper-Scissors game, (b) completely characterizing the set of all null-stable games, (c) providing a large class of strict stable games, (d) relating the game decomposition to the Hodge decomposition of vector fields for the replicator equations, (e) constructing Lyapunov functions for some replicator dynamics, (f) constructing Zeeman games -games with an interior asymptotically stable Nash equilibrium and a pure strategy ESS. The hierarchical modeling of evolutionary games provides flexibility in addressing the complex nature of social interactions as well as systematic frameworks in which one can keep track of the interplay of within-group dynamics and between-group competitions. For example, it can model husbands and wives' interactions, playing an asymmetric game with each other, while engaging coordination problems with the likes in other families. In the fourth chapter, we provide hierarchical stochastic models of evolutionary games and approximations of these processes, and study their applications

La complexité des structures géologiques en Moyenne Durance rend difficile la caractérisation et l’évaluation du risque sismique dans cette région. La compréhension 3D des structures nécessite l’utilisation de techniques modernes de modélisation numérique pour réaliser des modèles 3D du substratum géologique qui soient cohérents à différentes échelles, pour effectuer des simulations des mouvements sismiques. Pour réaliser des modèles 3D géologiques, nous avons harmonisé des bases de données géologiques et géophysiques hétérogènes (forages pétroliers, sondages géotechniques, profils sismiques, profils géophysiques H/V, cartographie géologique haute et basse résolution, datations biostratigraphiques, ...) dans le géomodeleur 3D gOcad. La réalisation des modèles 3D permet de caractériser la géométrie 3D des principales structures tectoniques en Moyenne Durance, notamment la Zone de failles de la Moyenne Durance (ZFMD) ainsi que la géométrie haute résolution du substratum géologique de la vallée de Cadarache, notamment de son remplissage sédimentaire tertiaire. Notre étude permet de préciser la géométrie 3D de la ZFMD et son rôle sur les structures géologiques adjacentes. La ZFMD constitue une zone de transfert qui limite l’extension latérale et modifie la géométrie des plis et chevauchements. Les structures tectoniques en Moyenne Durance sont caractérisées par une déformation de type thin-skin. La tectonique salifère joue un rôle important dans cette déformation (niveau de décollement, formation et géométrie 3D des plis). A l’échelle de la vallée de Cadarache, l’étude géologique des affleurements couplée à l’étude géostatistique des sondages permet de réaliser une simulation en 3D de la répartition spatiale des faciès et de proposer un modèle sédimentaire 3D du remplissage tertiaire de la vallée. Il est caractérisé par une sédimentation en deux étapes : une première phase de remplissage par des apports détritiques importants en provenance du Sud-Est, liés au démantèlement du massif des Maures-Estérel, une deuxième phase de remplissage par des dépôts d’environnement fluviatile méandriforme à forte sinuosité et à dominante de sables. Enfin, l’imbrication multi-échelle des modèles 3D a permis d’expliquer la formation de la vallée de Cadarache dans le contexte géodynamique de la région de la Moyenne Durance au Tertiaire, et notamment sous l’influence de la compression pyrénéenne et de la ZFMD sur la région de la Moyenne Durance et sur la vallée de Cadarache. / The complexity of geological structures in Middle Durance region makes difficult the characterization and evaluation of seismic risk in this region. Understanding these structures in 3D requires to use modern techniques of 3D digital modelling in order to achieve the 3D geological models of the bedrock with coherence on different scales, to perform ground motion simulations. Building 3D geological models need to normalize heterogeneous geophysical and geological databases (oil boreholes, geotechnical boreholes, seismic profiles, H/V geophysical profiles, high and low resolution geological mapping, biostratigraphic dating,...) in the 3D software gOcad. The realization of 3D geological models allows to characterize the 3D geometry of main the tectonic structures in Middle Durance region, including the Middle Durance Fault Zone (ZFMD) and the high resolution geometry of geological bedrock of the Cadarache Valley, and in particular its tertiary sedimentary fill. Our study allows to specify the ZFMD 3D geometry and its role on the adjacent geological structures. The ZFMD is a transfer zone that limits lateral extension and changes the geometry of folds and thrusts. The Middle Durance tectonic structures are characterized by a thin-skin deformation. Salt tectonics plays an important role in this deformation (detachment level, training and 3D geometry of folds). At the scale of the Valley of Cadarache, outcrop geological study coupled with boreholes geostatistical study allows to perform a 3D simulation of the spatial distribution of facies and to propose a 3D sedimentary model for the Tertiary filling of the Cadarache Valley. It is characterized by sedimentation in two steps: a first step of filling by important detrital inputs from the Southeast, related to the erosion of the Maures-Esterel massif, a second step of filling by deposits of fluvial meandering sand-dominant environment with high-sinuosity. Finally, building multi-scale 3D models allows explaining the formation of the Valley of Cadarache in the geodynamic context of the Middle Durance region at Tertiary times, and particularly under the influence of the Pyrenean compression and ZFMD on the Middle Durance region and the Valley of Cadarache.

Cette thèse concerne l’analyse et le développement de schémas d’intégration numérique de la Dynamique des Particules Dissipatives. Une présentation et une analyse de convergence faible de schémas existants est présentée, suivie d’une présentation et d’une analyse similaire de deux nouveaux schémas d’intégration facilement parallélisables. Une analyse des propriétés de conservation d’énergie de tous ces schémas est effectuée suivie d’une étude comparative de leurs biais sur l’estimation des valeurs moyennes d’observables physiques pour des systèmes à l’équilibre. Les schémas sont ensuite testés sur des systèmes choqués de fluides DPDE, où l’on montre que nos deux nouveaux schémas apportent une amélioration dans la précision de la description du comportement de tels systèmes par rapport aux schémas facilement parallélisables existants.Finalement, nous présentons une tentative d’accélération d’un schéma d’intégration de référence s’appliquant aux simulations séquentielles de la DPDE / This thesis is about the development and analysis of numerical schemes forthe integration of the Dissipative Particle Dynamics with Energy conservation. A presentation and a weak convergence analysis of existing schemes is performed, as well as the introduction and a similar analysis of two new straightforwardly parallelizable schemes. The energy preservation properties of all these schemes are studied followed by a comparative study of their biases on the estimation of the average values of physical observables on equilibrium simulations. The schemes are then tested on shock simulations of DPDE fluids, where we show that our schemes bring an improvement on the accuracy of the description of the behavior of such systems compared to existing straightforwardly parallelizable schemes. Finally, we present an attempt at accelerating a reference DPDE integration scheme on sequential simulations

Atomistic modeling and simulation methods enable a modern molecular approach to bio-medical research. Issues addressed range from structure-function relationships to structure-based drug design. The ability of these methods to address biologically relevant problems is largely determined by their accurate treatment of electrostatic interactions in the target biomolecular structure. In practical molecular simulations, the electrostatic charge density of molecules is approximated by an arrangement of fractional "point charges" throughout the molecule. While chemically intuitive and straightforward in technical implementation, models based exclusively on atom-centered charge placement, a major workhorse of the biomolecular simulations, do not necessarily provide a sufficiently detailed description of the molecular electrostatic potentials for small systems, and can become prohibitively expensive for large systems with thousands to millions of atoms. In this work, we propose a rigorous and generally applicable approach, Optimal Point Charge Approximation (OPCA), for approximating electrostatic charge distributions of biomolecules with a small number of point charges to best represent the underlying electrostatic potential, regardless of the distance to the charge distribution. OPCA places a given number of point charges so that the lowest order multipole moments of the reference charge distribution are optimally reproduced. We provide a general framework for calculating OPCAs to any order, and introduce closed-form analytical expressions for the 1-charge, 2-charge and 3-charge OPCA. We demonstrate the advantage of OPCA by applying it to a wide range of biomolecules of varied sizes. We use the concept of OPCA to develop a different, novel approach of constructing accurate and simple point charge water models. The proposed approach permits a virtually exhaustive search for optimal model parameters in the sub-space most relevant to electrostatic properties of the water molecule in liquid phase. A novel rigid 4-point Optimal Point Charge (OPC) water model constructed based on the new approach is substantially more accurate than commonly used models in terms of bulk water properties, and delivers critical accuracy improvement in practical atomistic simulations, such as RNA simulations, protein folding, protein-ligand binding and small molecule hydration. We also apply our new approach to construct a 3-point version of the Optimal Point Charge water model, referred to as OPC3. OPCA can be employed to represent large charge distributions with only a few point charges. We use this capability of OPCA to develop a multi-scale, yet fully atomistic, generalized Born approach (GB-HCPO) that can deliver up to 2 orders of magnitude speedup compared to the reference MD simulation. As a practical demonstration, we exploit the new multi-scale approach to gain insight into the structure of million-atom 30-nm chromatin fiber. Our results suggest important structural details consistent with experiment: the linker DNA fills the core region and the H3 histone tails interact with the linker DNA. OPC, OPC3 and GB-HCPO are implemented in AMBER molecular dynamics software package. / Ph. D.