Statistical properties of a non-polarized active tissue

Abstract

This thesis aims to develop a framework to analyze the properties of epithelial tissues, taking them as an example of active matter. Understanding their mechanical behavior is of interest for biologists, physicists, and others. Several works have exploited the idea of polarized activity as an internal force, producing, for example, cellular migration. Here, instead, we use a non-polarized system in which the interaction between the elements gives the direction of motion and where novel active scalar terms are considered. By performing analytical calculations using the vertex model, we find instabilities characterized by the coupling of some particular deformation modes that appear when the system is under stress or when cellular activity is considered, both applied homogeneously. We show an excellent agreement with numerical simulation, where the non-convexity of the cells is a geometrical proxy of the instabilities. When we consider activity in a single cell of the system, mimicking apical constrictions observed in several biological processes, we observe different geometrical responses depending on which cellular region is the active one: if it is the medial region, the active cell takes an anisotropic shape; if it is the perimeter, then the active cell tends to get an isotropic shape. We apply this analysis to the study of apical contraction pulses observed experimentally on frustrated cellular divisions in the blastula stage of the annual killifish Austrolebias nigripinnis. From an optimization process of geometrical observables, we can discriminate the type of activity that better describes the evolution of the system and also obtain the best fit of all parameters of the vertex model, globals, and specific for each active event. To compare the micro and macro-scale, we describe a continuum model reported in the literature, inspired by the vertex model, obtaining a continuum version of the stress tensor. We include the cellular activity in the model, both medial and perimeter, in an active Gaussian region and study the states in mechanical equilibrium and the dynamical evolution of the system, both with the continuum model and numerical simulations of the vertex model. We show an excellent agreement between both descriptions when rescaling the time-scales between them, and the active contractility of the perimeter. To analyze the self-organization, we perform numerical simulations of a non-polarized-fluctuating tissue, in which activity is inspired in the continuum reconstruction of the cytoskeleton. We calculate several static structure factors and find a scale separation that differentiates between a micro and a macro-scale in the model. Also, the response shows compression waves in the long-wavelength regime and an inverse energy cascade. In the end, we present the general conclusions of our work and its perspective in the future.