During development, organs with different shape and functionality form from a single fertilized egg cell. Mechanisms that control shape, size and morphology of tissues pose challenges for developmental biology. These mechanisms are tightly controlled by an underlying signaling system by which cells communicate to each other. However, these signaling networks can affect tissue size and morphology through limited processes such as cell proliferation, cell death and cell shape changes,which are controlled by cell mechanics and cell adhesion. One example of such a signaling system is the network of interacting proteins that control planar polarization of cells. These proteins distribute asymmetrically within cells and their distribution in each cell determines of the polarity of the neighboring cells. These proteins control the pattern of hairs in the adult Drosophila wing as well as hexagonal repacking of wing cells during development. Planar polarity proteins also control developmental processes such as convergent-extension. We present a theoretical study of cell packing geometry in developing epithelia. We use a vertex model to describe the packing geometry of tissues, for which forces are balanced throughout the tissue. We introduce a cell division algorithm and show that repeated cell division results in the formation of a distinct pattern of cells, which is controlled by cell mechanics and cell-cell interactions. We compare the vertex model with experimental measurements in the wing disc of Drosophila and quantify for the first time cell adhesion and perimeter contractility of cells. We also present a simple model for the dynamics of polarity order in tissues. We identify a basic mechanism by which long-range polarity order throughout the tissue can be established. In particular we study the role of shear deformations on polarity pattern and show that the polarity of the tissue reorients during shear flow. Our simple mechanisms for ordering can account for the processes observed during development of the Drosophila wing.

During development, organs with different shape and functionality form from a single fertilized egg cell. Mechanisms that control shape, size and morphology of tissues pose challenges for developmental biology. These mechanisms are tightly controlled by an underlying signaling system by which cells communicate to each other. However, these signaling networks can affect tissue size and morphology through limited processes such as cell proliferation, cell death and cell shape changes,which are controlled by cell mechanics and cell adhesion. One example of such a signaling system is the network of interacting proteins that control planar polarization of cells. These proteins distribute asymmetrically within cells and their distribution in each cell determines of the polarity of the neighboring cells. These proteins control the pattern of hairs in the adult Drosophila wing as well as hexagonal repacking of wing cells during development. Planar polarity proteins also control developmental processes such as convergent-extension. We present a theoretical study of cell packing geometry in developing epithelia. We use a vertex model to describe the packing geometry of tissues, for which forces are balanced throughout the tissue. We introduce a cell division algorithm and show that repeated cell division results in the formation of a distinct pattern of cells, which is controlled by cell mechanics and cell-cell interactions. We compare the vertex model with experimental measurements in the wing disc of Drosophila and quantify for the first time cell adhesion and perimeter contractility of cells. We also present a simple model for the dynamics of polarity order in tissues. We identify a basic mechanism by which long-range polarity order throughout the tissue can be established. In particular we study the role of shear deformations on polarity pattern and show that the polarity of the tissue reorients during shear flow. Our simple mechanisms for ordering can account for the processes observed during development of the Drosophila wing.

No description available.

Nós obtemos o hamiltoniano associado ao modelo de oito vértices simétrico, tomando o limite de tempo contínuo em um modelo equivalente (modelo de Ashkin-Teller). O resultado é um hamiltoniano de Heisenberg com coeficientes Jx , Jy e Jz idênticos àqueles encontrados por Sutherland, na região crítica. A mudança nos operadores é acompanhada explicitamente e a relação entre o operador \"crossover\" do modelo de Ashkin-Teller e o operador energia do modelo de Baxter é obtida de forma transparente. / We obtain the associated Harniltonian for the symmetric eight-vertex model by taking the time-continuous limit in an equivalent Ashkin-Teller model. The result is a Heisenberg Hamiltonian with coefficients Jx , Jy e Jz identical to those found by Sutherland for choices of the parameters a, b, c and d that bring the model close to the transition. The change in the operators is accomplished explicitly, the relation between the crossover operator for the Ashkin-Teller model and the energy operator for the eight-vertex model being obtained in a transparent form.

During development, cells consume energy, divide, rearrange, and die. Bulk properties such as viscosity and elasticity emerge from cell-scale mechanics and dynamics. Order appears, for example in patterns of hair outgrowth, or in the predominately hexagonal pattern of cell boundaries in the wing of a fruit fly. In the past fifty years, much progress has been made in understanding tissues as living materials. However, the physical mechanisms underlying tissue-scale behaviour are not completely understood. Here we apply theories from statistical physics and fluid dynamics to understand mechanics and order in two-dimensional tissues. We restrict our attention to the mechanics and dynamics of cell boundaries and vertices, and to planar polarity, a type of long-ranged order visible in anisotropic patterns of proteins and hair outgrowth. Our principle tool for understanding mechanics and dynamics is a vertex model where cell shapes are represented using polygons. We analytically derive the ground-state diagram of this vertex model, finding it to be dominated by the geometric requirement that cells be polygons, and the topological requirement that those polygons tile the plane. We present a simplified algorithm for cell division and growth, and furthermore derive a dynamic equation for the vertex model, which we use to demonstrate the emergence of quasistatic behaviour in the limit of slow growth. All our results relating to the vertex model are consistent with and build off past calculations and experiments. To investigate the emergence of planar polarity, we develop quantification methods for cell flow and planar polarity based on confocal microscope images of developing fly wings. We analyze cell flow using a velocity gradient tensor, which is uniquely decomposed into terms corresponding to local compression, shear, and rotations. We argue that a pattern in an inhomogeneously flowing tissue will necessarily be reorganized, motivating a hydrodynamic theory of polarity reorientation. Using such a coarse-grained theory of polarity reorientation, we show that the quantified patterns of shear and rotation in the wing are consistent with the observed polarity reorganization, and conclude that cell flow reorients planar polarity in the wing of the fruit fly. Finally, we present a cell-scale model of planar polarity based on the vertex model, unifying the themes of this thesis.

The development and maintenance of tissues and organs depend on the careful regulation and coordinated motion of large numbers of cells. There is substantial evidence that many complex tissue functions, such as cell division, collective cell migration and gene expression, are directly regulated by mechanical forces. However, relatively little is known about how mechanical stress is distributed within a tissue and how this may guide biochemical signalling. Working in the framework of a popular vertex-based model, we derive expressions for stress tensors at the cell and tissue level to build analytic relationships between cell shape and mechanical stress. The discrete vertex model is upscaled, providing exact expressions for the bulk and shear moduli of disordered cellular networks, which bridges the gap to traditional continuum-level descriptions of tissues. Combining this theoretical work with new experimental techniques for whole-tissue stretching of Xenopus laevis tissue, we separate the roles of mechanical stress and cell shape in orienting and cueing epithelial mitosis. We find that the orientation of division is best predicted by the shape of tricellular junctions, while there appears to be a more direct role for mechanical stress as a mitotic cue.

Nós obtemos o hamiltoniano associado ao modelo de oito vértices simétrico, tomando o limite de tempo contínuo em um modelo equivalente (modelo de Ashkin-Teller). O resultado é um hamiltoniano de Heisenberg com coeficientes Jx , Jy e Jz idênticos àqueles encontrados por Sutherland, na região crítica. A mudança nos operadores é acompanhada explicitamente e a relação entre o operador \"crossover\" do modelo de Ashkin-Teller e o operador energia do modelo de Baxter é obtida de forma transparente. / We obtain the associated Harniltonian for the symmetric eight-vertex model by taking the time-continuous limit in an equivalent Ashkin-Teller model. The result is a Heisenberg Hamiltonian with coefficients Jx , Jy e Jz identical to those found by Sutherland for choices of the parameters a, b, c and d that bring the model close to the transition. The change in the operators is accomplished explicitly, the relation between the crossover operator for the Ashkin-Teller model and the energy operator for the eight-vertex model being obtained in a transparent form.

During development, cells consume energy, divide, rearrange, and die. Bulk properties such as viscosity and elasticity emerge from cell-scale mechanics and dynamics. Order appears, for example in patterns of hair outgrowth, or in the predominately hexagonal pattern of cell boundaries in the wing of a fruit fly. In the past fifty years, much progress has been made in understanding tissues as living materials. However, the physical mechanisms underlying tissue-scale behaviour are not completely understood. Here we apply theories from statistical physics and fluid dynamics to understand mechanics and order in two-dimensional tissues. We restrict our attention to the mechanics and dynamics of cell boundaries and vertices, and to planar polarity, a type of long-ranged order visible in anisotropic patterns of proteins and hair outgrowth. Our principle tool for understanding mechanics and dynamics is a vertex model where cell shapes are represented using polygons. We analytically derive the ground-state diagram of this vertex model, finding it to be dominated by the geometric requirement that cells be polygons, and the topological requirement that those polygons tile the plane. We present a simplified algorithm for cell division and growth, and furthermore derive a dynamic equation for the vertex model, which we use to demonstrate the emergence of quasistatic behaviour in the limit of slow growth. All our results relating to the vertex model are consistent with and build off past calculations and experiments. To investigate the emergence of planar polarity, we develop quantification methods for cell flow and planar polarity based on confocal microscope images of developing fly wings. We analyze cell flow using a velocity gradient tensor, which is uniquely decomposed into terms corresponding to local compression, shear, and rotations. We argue that a pattern in an inhomogeneously flowing tissue will necessarily be reorganized, motivating a hydrodynamic theory of polarity reorientation. Using such a coarse-grained theory of polarity reorientation, we show that the quantified patterns of shear and rotation in the wing are consistent with the observed polarity reorganization, and conclude that cell flow reorients planar polarity in the wing of the fruit fly. Finally, we present a cell-scale model of planar polarity based on the vertex model, unifying the themes of this thesis.

La première partie de cette thèse porte sur les automates cellulaires probabilistes (ACP) sur la ligne et à deux voisins. Pour un ACP donné, nous cherchons l'ensemble de ces lois invariantes. Pour des raisons expliquées en détail dans la thèse, ceci est à l'heure actuelle inenvisageable de toutes les obtenir et nous nous concentrons, dans cette thèse, surles lois invariantes markoviennes. Nous établissons, tout d'abord, un théorème de nature algébrique qui donne des conditions nécessaires et suffisantes pour qu'un ACP admette une ou plusieurs lois invariantes markoviennes dans le cas où l'alphabet E est fini. Par la suite, nous généralisons ce résultat au cas d'un alphabet E polonais après avoir clarifié les difficultés topologiques rencontrées. Enfin, nous calculons la fonction de corrélation du modèleà 8 sommets pour certaines valeurs des paramètres du modèle en utilisant une partie desrésultats précédents. / The first part of this thesis is about probabilistic cellular automata (PCA) on the line and with two neighbors. For a given PCA, we look for the set of its invariant distributions. Due to reasons explained in detail in this thesis, it is nowadays unthinkable to get all of them and we concentrate our reections on the invariant Markovian distributions. We establish, first, an algebraic theorem that gives a necessary and sufficient condition for a PCA to have one or more invariant Markovian distributions when the alphabet E is finite. Then, we generalize this result to the case of a polish alphabet E once we have clarified the encountered topological difficulties. Finally, we calculate the 8-vertex model's correlation function for some parameters values using previous results.The second part of this thesis is about infinite iterations of stochastic processes. We establish the convergence of the finite dimensional distributions of the α-stable processes iterated n times, when n goes to infinite, according to parameter of stability and to drift r. Then, we describe the limit distributions. In the iterated Brownian motion case, we show that the limit distributions are linked with iterated functions system.

Submitted by Alison Vanceto (alison-vanceto@hotmail.com) on 2016-10-05T14:14:54Z No. of bitstreams: 1 TeseRSV.pdf: 1391601 bytes, checksum: fb3e58d9db6c377161785dede432eeee (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-05T19:33:46Z (GMT) No. of bitstreams: 1 TeseRSV.pdf: 1391601 bytes, checksum: fb3e58d9db6c377161785dede432eeee (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-10-05T19:34:21Z (GMT) No. of bitstreams: 1 TeseRSV.pdf: 1391601 bytes, checksum: fb3e58d9db6c377161785dede432eeee (MD5) / Made available in DSpace on 2016-10-07T18:13:48Z (GMT). No. of bitstreams: 1 TeseRSV.pdf: 1391601 bytes, checksum: fb3e58d9db6c377161785dede432eeee (MD5) Previous issue date: 2015-05-15 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / In this dissertation we made an analytic study of the Bethe Ansatz equations for the XXZ six vertex model with periodic boundary conditions. We had show that the Bethe Ansatz equations deduced from the algebraic and coordinate Bethe Ansatze are related by a conformal map. This allowed us to reduce the Bethe Ansatz equations to a system of polynomial equations. For the one, two and three magnon sectors, we succeeded in decouple these equations, so that the solutions could be expressed in terms of the roots of some self-inversive polynomials, Pa (z). Through new theorems deduced here about the distribution of the roots of self-inversive polynomials in the complex plane, we did a thorough analysis of the distribution of the Bethe roots for the two-magnon sector. This analysis allowed us to show that the Bethe Ansatz is indeed complete for this sector, except at some critical values of the anisotropy parameter A, in which the polynomials Pa (z) may have multiple roots. Finally, an unexpected connection between the Bethe Ansatz equations and the Salem polynomials was found and a new algorithm for search small Salem numbers was elaborated. / Nesta tese fizemos um estudo analítico das equações de Bethe para o modelo de seis vértices XXZ com condições de contorno periódicas. Mostramos que as equações de Bethe deduzidas pelo Ansatz algébrico estão relacionadas com as equações de Bethe do Ansatz de coordenadas por uma transformação conforme. Isso nos permitiu reduzir as equações de Bethe a um sistema de equações polinomiais. Para os setores de um, dois e três mágnons, mostramos que essas equações podem ser desacopladas, de modo que as suas soluções podem ser expressas em termos das raízes de certos polinómios auto-inversivos, Pa(z). Deduzimos aqui novos teoremas acerca da distribuição das raízes dos polinómios auto-inversivos no plano complexo, o que nos permitiu fazer uma análise minuciosa da distribuição das raízes de Bethe para o setor de dois mágnons. Esta análise nos permitiu mostrar que o Ansatz de Bethe é de fato completo para este setor, exceto para alguns valores críticos do parâmetro de anisotropia A, no qual os polinómios Pa(z) podem apresentar raízes múltiplas. Por fim, uma inesperada conexão entre as equações de Bethe e os polinómios de Salem foi encontrada e um novo algoritmo para se procurar por números de Salem pequenos foi elaborado.