Simulation of stromatolite growth using diffusion-limited aggregation and a discrete model of biogeochemical exchanges in microbial mat

Resume : Mieux comprendre les stromatolithes et les tapis microbiens est un sujet important en biogéosciences puisque cela aide à l'étude des premières formes de vie sur Terre, a mieux cerner l'écologie des communautés microbiennes et la contribution des microorganismes a la biominéralisation, et même à poser certains fondements dans les recherches en exobiologie. D'autre part, la modélisation est un outil puissant utilisé dans les sciences naturelles pour appréhender différents phénomènes de façon théorique. Les modèles sont généralement construits sur un système d'équations différentielles et les résultats sont obtenus en résolvant ce système. Les logiciels disponibles pour implémenter les modèles incluent les logiciels mathématiques et les logiciels généraux de simulation. L'objectif principal de cette thèse est de développer des modèles et des logiciels pour aider a comprendre, via la simulation, le fonctionnement des stromatolithes et des tapis microbiens. Ces logiciels ont été développés en C++ en ne partant d'aucun pré-requis de façon a privilégier performance et flexibilité maximales. Cette démarche permet de construire des modèles bien plus spécifiques et plus appropriés aux phénomènes a modéliser. Premièrement, nous avons étudié la croissance et la morphologie des stromatolithes. Nous avons construit un modèle tridimensionnel fondé sur l'agrégation par diffusion limitée. Le modèle a été implémenté en deux applications C++: un moteur de simulation capable d'exécuter un batch de simulations et de produire des fichiers de résultats, et un outil de visualisation qui permet d'analyser les résultats en trois dimensions. Après avoir vérifié que ce modèle peut en effet reproduire la croissance et la morphologie de plusieurs types de stromatolithes, nous avons introduit un processus de sédimentation comme facteur externe. Ceci nous a mené a des résultats intéressants, et permis de soutenir l'hypothèse que la morphologie des stromatolithes pourrait être le résultat de facteurs externes autant que de facteurs internes. Ceci est important car la classification des stromatolithes est généralement fondée sur leur morphologie, imposant que la forme d'un stromatolithe est dépendante de facteurs internes uniquement (c'est-à-dire les tapis microbiens). Les résultats avancés dans ce mémoire contredisent donc ces assertions communément admises. Ensuite, nous avons décidé de mener des recherches plus en profondeur sur les aspects fonctionnels des tapis microbiens. Nous avons construit un modèle bidimensionnel de réaction-diffusion fondé sur la simulation discrète. Ce modèle a été implémenté dans une application C++ qui permet de paramétrer et exécuter des simulations. Nous avons ensuite pu comparer les résultats de simulation avec des données du monde réel et vérifier que le modèle peut en effet imiter le comportement de certains tapis microbiens. Ainsi, nous avons pu émettre et vérifier des hypothèses sur le fonctionnement de certains tapis microbiens pour nous aider à mieux en comprendre certains aspects, comme la dynamique des éléments, en particulier le soufre et l'oxygène. En conclusion, ce travail a abouti à l'écriture de logiciels dédiés à la simulation de tapis microbiens d'un point de vue tant morphologique que fonctionnel, suivant deux approches différentes, l'une holistique, l'autre plus analytique. Ces logiciels sont gratuits et diffusés sous licence GPL (General Public License). Abstract : Better understanding of stromatolites and microbial mats is an important topic in biogeosciences as it helps studying the early forms of life on Earth, provides clues re- garding the ecology of microbial ecosystems and their contribution to biomineralization, and gives basis to a new science, exobiology. On the other hand, modelling is a powerful tool used in natural sciences for the theoretical approach of various phenomena. Models are usually built on a system of differential equations and results are obtained by solving that system. Available software to implement models includes mathematical solvers and general simulation software. The main objective of this thesis is to develop models and software able to help to understand the functioning of stromatolites and microbial mats. Software was developed in C++ from scratch for maximum performance and flexibility. This allows to build models much more specific to a phenomenon rather than general software. First, we studied stromatolite growth and morphology. We built a three-dimensional model based on diffusion-limited aggregation. The model was implemented in two C++ applications: a simulator engine, which can run a batch of simulations and produce result files, and a Visualization tool, which allows results to be analysed in three dimensions. After verifying that our model can indeed reproduce the growth and morphology of several types of stromatolites, we introduced a sedimentation process as an external factor. This lead to interesting results, and allowed to emit the hypothesis that stromatolite morphology may be the result of external factors as much as internal factors. This is important as stromatolite classification is usually based on their morphology, imposing that a stromatolite shape is dependant on internal factors only (i.e. the microbial mat). This statement is contradicted by our findings, Second, we decided to investigate deeper the functioning of microbial mats, We built a two-dimensional reaction-diffusion model based on discrete simulation, The model was implemented in a C++ application that allows setting and running simulations. We could then compare simulation results with real world data and verify that our model can indeed mimic the behaviour of some microbial mats. Thus, we have proposed and verified hypotheses regarding microbial mats functioning in order to help to better understand them, e.g. the cycle of some elements such as oxygen or sulfur. ln conclusion, this PhD provides a simulation software, dealing with two different approaches. This software is free and available under a GPL licence.

Fluctuating dimension in a discrete model for quantum gravity based on the spectral principle

The spectral principle of Connes and Chamseddine is used as a starting point to define a discrete model for Euclidean quantum gravity. Instead of summing over ordinary geometries, we consider the sum over generalized geometries where topology, metric, and dimension can fluctuate. The model describes the geometry of spaces with a countable number n of points, and is related to the Gaussian unitary ensemble of Hermitian matrices. We show that this simple model has two phases. The expectation value , the average number of points in the Universe, is finite in one phase and diverges in the other. We compute the critical point as well as the critical exponent of . Moreover, the space-time dimension delta is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, < 2.

Self-similarities of periodic structures for a discrete model of a two-gene system

We report self-similar properties of periodic structures remarkably organized in the two-parameter space for a two-gene system, described by two-dimensional symmetric map. The map consists of difference equations derived from the chemical reactions for gene expression and regulation. We characterize the system by using Lyapunov exponents and isoperiodic diagrams identifying periodic windows, denominated Arnold tongues and shrimp-shaped structures. Period-adding sequences are observed for both periodic windows. We also identify Fibonacci-type series and Golden ratio for Arnold tongues, and period multiple-of-three windows for shrimps. (C) 2012 Elsevier B.V. All rights reserved.

A new 3D modelling paradigm for discrete model

Until few years ago, 3D modelling was a topic confined into a professional environment. Nowadays technological innovations, the 3D printer among all, have attracted novice users to this application field. This sudden breakthrough was not supported by adequate software solutions. The 3D editing tools currently available do not assist the non-expert user during the various stages of generation, interaction and manipulation of 3D virtual models. This is mainly due to the current paradigm that is largely supported by two-dimensional input/output devices and strongly affected by obvious geometrical constraints. We have identified three main phases that characterize the creation and management of 3D virtual models. We investigated these directions evaluating and simplifying the classic editing techniques in order to propose more natural and intuitive tools in a pure 3D modelling environment. In particular, we focused on freehand sketch-based modelling to create 3D virtual models, interaction and navigation in a 3D modelling environment and advanced editing tools for free-form deformation and objects composition. To pursuing these goals we wondered how new gesture-based interaction technologies can be successfully employed in a 3D modelling environments, how we could improve the depth perception and the interaction in 3D environments and which operations could be developed to simplify the classical virtual models editing paradigm. Our main aims were to propose a set of solutions with which a common user can realize an idea in a 3D virtual model, drawing in the air just as he would on paper. Moreover, we tried to use gestures and mid-air movements to explore and interact in 3D virtual environment, and we studied simple and effective 3D form transformations. The work was carried out adopting the discrete representation of the models, thanks to its intuitiveness, but especially because it is full of open challenges.

Comparing a discrete and continuum model of the intestinal crypt

The integration of processes at different scales is a key problem in the modelling of cell populations. Owing to increased computational resources and the accumulation of data at the cellular and subcellular scales, the use of discrete, cell-level models, which are typically solved using numerical simulations, has become prominent. One of the merits of this approach is that important biological factors, such as cell heterogeneity and noise, can be easily incorporated. However, it can be difficult to efficiently draw generalizations from the simulation results, as, often, many simulation runs are required to investigate model behaviour in typically large parameter spaces. In some cases, discrete cell-level models can be coarse-grained, yielding continuum models whose analysis can lead to the development of insight into the underlying simulations. In this paper we apply such an approach to the case of a discrete model of cell dynamics in the intestinal crypt. An analysis of the resulting continuum model demonstrates that there is a limited region of parameter space within which steady-state (and hence biologically realistic) solutions exist. Continuum model predictions show good agreement with corresponding results from the underlying simulations and experimental data taken from murine intestinal crypts.

Discrete and continuous models for dry masonry columns

The dynamic response of dry masonry columns can be approximated with finite-difference equations. Continuum models follow by replacing the difference quotients of the discrete model by corresponding differential expressions. The mathematically simplest of these models is a one-dimensional Cosserat theory. Within the presented homogenization context, the Cosserat theory is obtained by making ad hoc assumptions regarding the relative importance of certain terms in the differential expansions. The quality of approximation of the various theories is tested by comparison of the dispersion relations for bending waves with the dispersion relation of the discrete theory. All theories coincide with differences of less than 1% for wave-length-block-height (L/h) ratios bigger than 2 pi. The theory based on systematic differential approximation remains accurate up to L/h = 3 and then diverges rapidly. The Cosserat model becomes increasingly inaccurate for L/h < 2 pi. However, in contrast to the systematic approximation, the wave speed remains finite. In conclusion, considering its relative simplicity, the Cosserat model appears to be the natural starting point for the development of continuum models for blocky structures.

Continuum, Discrete, and Explicit Solvation Models for Describing the Low-Lying Absorption Spectrum of the Pterin Acid in Aqueous Environment

The absorption spectrum of the acid form of pterin in water was investigated theoretically. Different procedures using continuum, discrete, and explicit models were used to include the solvation effect on the absorption spectrum, characterized by two bands. The discrete and explicit models used Monte Carlo simulation to generate the liquid structure and time-dependent density functional theory (B3LYP/6-31G+(d)) to obtain the excitation energies. The discrete model failed to give the correct qualitative effect on the second absorption band. The continuum model, in turn, has given a correct qualitative picture and a semiquantitative description. The explicit use of 29 solvent molecules, forming a hydration shell of 6 angstrom, embedded in the electrostatic field of the remaining solvent molecules, gives absorption transitions at 3.67 and 4.59 eV in excellent agreement with the S(0)-S(1) and S(0)-S(2) absorption bands at of 3.66 and 4.59 eV, respectively, that characterize the experimental spectrum of pterin in water environment. (C) 2010 Wiley Periodicals, Inc. Int J Quantum Chem 110: 2371-2377, 2010

A continuous model for quasinilpotent operators.

A classical result due to Foias and Pearcy establishes a discrete model for every quasinilpotent operator acting on a separable, infinite-dimensional complex Hilbert space HH . More precisely, given a quasinilpotent operator T on HH , there exists a compact quasinilpotent operator K in HH such that T is similar to a part of K⊕K⊕⋯⊕K⊕⋯K⊕K⊕⋯⊕K⊕⋯ acting on the direct sum of countably many copies of HH . We show that a continuous model for any quasinilpotent operator can be provided. The consequences of such a model will be discussed in the context of C0C0 -semigroups of quasinilpotent operators.

On the 'classification' of neurodegenerative disorders:discrete entities, overlap or continuum?

The traditional method of classifying neurodegenerative diseases is based on the original clinico-pathological concept supported by 'consensus' criteria and data from molecular pathological studies. This review discusses first, current problems in classification resulting from the coexistence of different classificatory schemes, the presence of disease heterogeneity and multiple pathologies, the use of 'signature' brain lesions in diagnosis, and the existence of pathological processes common to different diseases. Second, three models of neurodegenerative disease are proposed: (1) that distinct diseases exist ('discrete' model), (2) that relatively distinct diseases exist but exhibit overlapping features ('overlap' model), and (3) that distinct diseases do not exist and neurodegenerative disease is a 'continuum' in which there is continuous variation in clinical/pathological features from one case to another ('continuum' model). Third, to distinguish between models, the distribution of the most important molecular 'signature' lesions across the different diseases is reviewed. Such lesions often have poor 'fidelity', i.e., they are not unique to individual disorders but are distributed across many diseases consistent with the overlap or continuum models. Fourth, the question of whether the current classificatory system should be rejected is considered and three alternatives are proposed, viz., objective classification, classification for convenience (a 'dissection'), or analysis as a continuum.

Different Approaches for Modeling Grouped Survival Data: A Mango Tree Study

Interval-censored survival data, in which the event of interest is not observed exactly but is only known to occur within some time interval, occur very frequently. In some situations, event times might be censored into different, possibly overlapping intervals of variable widths; however, in other situations, information is available for all units at the same observed visit time. In the latter cases, interval-censored data are termed grouped survival data. Here we present alternative approaches for analyzing interval-censored data. We illustrate these techniques using a survival data set involving mango tree lifetimes. This study is an example of grouped survival data.

Transition probabilities and duration analysis among disability states: Some evidence from Spanish data

In this paper we study the disability transition probabilities (as well as the mortalityprobabilities) due to concurrent factors to age such as income, gender and education. Althoughit is well known that ageing and socioeconomic status influence the probability ofcausing functional disorders, surprisingly little attention has been paid to the combined effectof those factors along the individuals' life and how this affects the transition from one degreeof disability to another. The assumption that tomorrow's disability state is only a functionof the today's state is very strong, since disability is a complex variable that depends onseveral other elements than time. This paper contributes into the field in two ways: (1) byattending the distinction between the initial disability level and the process that leads tohis course (2) by addressing whether and how education, age and income differentially affectthe disability transitions. Using a Markov chain discrete model and a survival analysis, weestimate the probability by year and individual characteristics that changes the state of disabilityand the duration that it takes its progression in each case. We find that people withan initial state of disability have a higher propensity to change and take less time to transitfrom different stages. Men do that more frequently than women. Education and incomehave negative effects on transition. Moreover, we consider the disability benefits associatedto those changes along different stages of disability and therefore we offer some clues onthe potential savings of preventive actions that may delay or avoid those transitions. Onpure cost considerations, preventive programs for improvement show higher benefits thanthose for preventing deterioration, and in general terms, those focussing individuals below65 should go first. Finally the trend of disability in Spain seems not to change among yearsand regional differences are not found.

Structures algébriques, systèmes superintégrables et polynômes orthogonaux

Cette thèse est divisée en cinq parties portant sur les thèmes suivants: l’interprétation physique et algébrique de familles de fonctions orthogonales multivariées et leurs applications, les systèmes quantiques superintégrables en deux et trois dimensions faisant intervenir des opérateurs de réflexion, la caractérisation de familles de polynômes orthogonaux appartenant au tableau de Bannai-Ito et l’examen des structures algébriques qui leurs sont associées, l’étude de la relation entre le recouplage de représentations irréductibles d’algèbres et de superalgèbres et les systèmes superintégrables, ainsi que l’interprétation algébrique de familles de polynômes multi-orthogonaux matriciels. Dans la première partie, on développe l’interprétation physico-algébrique des familles de polynômes orthogonaux multivariés de Krawtchouk, de Meixner et de Charlier en tant qu’éléments de matrice des représentations unitaires des groupes SO(d+1), SO(d,1) et E(d) sur les états d’oscillateurs. On détermine les amplitudes de transition entre les états de l’oscillateur singulier associés aux bases cartésienne et polysphérique en termes des polynômes multivariés de Hahn. On examine les coefficients 9j de su(1,1) par le biais du système superintégrable générique sur la 3-sphère. On caractérise les polynômes de q-Krawtchouk comme éléments de matrices des «q-rotations» de U_q(sl_2). On conçoit un réseau de spin bidimensionnel qui permet le transfert parfait d’états quantiques à l’aide des polynômes de Krawtchouk à deux variables et on construit un modèle discret de l’oscillateur quantique dans le plan à l’aide des polynômes de Meixner bivariés. Dans la seconde partie, on étudie les systèmes superintégrables de type Dunkl, qui font intervenir des opérateurs de réflexion. On examine l’oscillateur de Dunkl en deux et trois dimensions, l’oscillateur singulier de Dunkl dans le plan et le système générique sur la 2-sphère avec réflexions. On démontre la superintégrabilité de chacun de ces systèmes. On obtient leurs constantes du mouvement, on détermine leurs algèbres de symétrie et leurs représentations, on donne leurs solutions exactes et on détaille leurs liens avec les polynômes orthogonaux du tableau de Bannai-Ito. Dans la troisième partie, on caractérise deux familles de polynômes du tableau de Bannai-Ito: les polynômes de Bannai-Ito complémentaires et les polynômes de Chihara. On montre également que les polynômes de Bannai-Ito sont les coefficients de Racah de la superalgèbre osp(1,2). On détermine l’algèbre de symétrie des polynômes duaux -1 de Hahn dans le cadre du problème de Clebsch-Gordan de osp(1,2). On propose une q - généralisation des polynômes de Bannai-Ito en examinant le problème de Racah pour la superalgèbre quantique osp_q(1,2). Finalement, on montre que la q -algèbre de Bannai-Ito sert d’algèbre de covariance à osp_q(1,2). Dans la quatrième partie, on détermine le lien entre le recouplage de représentations des algèbres su(1,1) et osp(1,2) et les systèmes superintégrables du deuxième ordre avec ou sans réflexions. On étudie également les représentations des algèbres de Racah-Wilson et de Bannai-Ito. On montre aussi que l’algèbre de Racah-Wilson sert d’algèbre de covariance quadratique à l’algèbre de Lie sl(2). Dans la cinquième partie, on construit deux familles explicites de polynômes d-orthogonaux basées sur su(2). On étudie les états cohérents et comprimés de l’oscillateur fini et on caractérise une famille de polynômes multi-orthogonaux matriciels.

Modelagem linear e identificação do modelo dinâmico de um robô móvel com acionamento diferencial

This work presents a modelling and identification method for a wheeled mobile robot, including the actuator dynamics. Instead of the classic modelling approach, where the robot position coordinates (x,y) are utilized as state variables (resulting in a non linear model), the proposed discrete model is based on the travelled distance increment Delta_l. Thus, the resulting model is linear and time invariant and it can be identified through classical methods such as Recursive Least Mean Squares. This approach has a problem: Delta_l can not be directly measured. In this paper, this problem is solved using an estimate of Delta_l based on a second order polynomial approximation. Experimental data were colected and the proposed method was used to identify the model of a real robot

Controlling dispersal dynamics of Aedes aegypti

The dengue virus is transmitted in regions previously infested with the mosquito Aedes aegypti. To assess the spreading and establishment of the dengue disease vector, a mathematical model is developed that takes into account the diffusion and advection phenomena. A discrete model based on the cellular automata approach, which is a good framework to deal with small populations, is also developed to be compared with the continuous modeling.

Neighborhood Interactions and Larval Dispersal Behavior in Blowflies

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)