A reaction-diffusion system, which can be considered as a generalised Schnakenberg-like model, is studied mathematically in 1D and 2D. This system models an initiation process within a root-hair cell which involves biochemical interactions of the G-proteins, known as Rho of Plants, or ROPs. These proteins attach to the cell membrane prompting a localised patch which, in consequence, induces cell wall softening and subsequently hair growth. This model assumes that the auxin provided is the key catalyst. Auxin is a plant hormone which is known to because of many different features in plant morphogenesis. Also, this hormone is experimentally known to enter the cell including a spatially dependent gradient.. Numerical bifurcation analysis is carried on in order to explore solutions which resemble all features that the G-proteins and auxin arc known to cause. The main bifurcation parameters arc taken to be the overall auxin rate, and the cell length. The analysis is backed up by full numerical simulations and asymptotic analysis using semi-strong interaction theory. The asymptotics not only provides existence of solutions and explains numerical properties, but also sheds light on transition mechanisms via the theory of competition instability and transverse instability of homoclinic stripes. The analytical results are found to agree favourably with numerical simulations, and to give further explanation of the agreement between the model and biological data for different scenarios. From a mathematical point of view, pattern formation of non-homogeneous reaction-diffusion systems is a subject that is not yet well understood. However, upon using the theory of semi-strong interactions, light is shed on the dynamics and instabilities that. spatially dependent coefficients bring about. As a consequence, transitions between different spot-like patterns and the dynamics of their location can be explored and theoretically explained.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:619139 |
| Date | January 2013 |
| Creators | Brena-Medina, Victor Francisco |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.0024 seconds