Transições entre estados espaciais não homogêneos em sistemas químicos e biológicos - Silva-Dias, Leonardo (Universidade Federal de São Carlos, 2019)
In the article The Chemical Basis of Morphogenesis - 1952, Alan M. Turing proposed that the morphogenesis could be described through a specific reaction mechanism which enables the emergence of inhomogeneous and stationary spatial structures, known as "Turing patterns". Many theoretical and experimental studies revealed that physical and chemical parameters, e.g. radiation, temperature and boundary condition, can interfere on the emergence of those patterns. Among them, the boundary condition is the source of perturbation least explored, however it has a great importance on the study of the dynamics of small systems, e.g. cells. From these perspectives, this work presents the construction of a code, written in FORTRAN 90, capable of solving systems of partial differential equations, reaction-diffusion type, and the results of the theoretical investigation of the effects of the boundary conditions, performed through the perimeter of the system's domain, gradient of temperature and gradient of concentration. This investigation was carried out considering the isothermal and nonisothermal versions of the Brusselator model in a small domain reactor. The outcomes showed that: 1) In isothermal situations the variation of the perimeter enables the emergence of different spatial structures by a possible phenomenon of superposition of Turing patterns. In nonisothermal conditions, the temperature strongly regulates the geometrical formats of the patterns, for any perimeter, preventing the emergence of spatial structures with different configurations. 2) The temperature can be used as parameter of control in the syntonization of different Turing patterns, in a way that a thermal gradient can induce the emergence of different Turing patterns simultaneously. 3) Sources of chemicals defined at the bounders can induce the spatial symmetry breaking of Turing patterns if the following condition is satisfied: The concentration of chemicals, located at the bounders, presents the equivalent value of the equilibrium point of the homogeneous dynamical system. Based on Alan M. Turing ideas and considering the results obtained in this work, a minimal chemical model is proposed to describe the initial steps of the cellular differentiation, embriogenesis, and biogenesis.
Transitions among inhomogeneous spatial states in chemical and biological systems
Keywords: Dinâmica não linear; EDP; Condições de fronteira; Quebra de simetria; Nonlinear dynamics; PDE; Boundary conditions; Symmetry breaking; CIENCIAS EXATAS E DA TERRA::QUIMICA::FISICO-QUIMICA