Speaker
Description
Hormonal activity, along with genetic regulation and vascular transport, plays a crucial role in regulating plant growth. In such mechanism, spatio-temporal signals are used to spread information across different portions of the plant. When multi-cellular tissues are involved in the transmission process, the evolution dynamics of such signals shows a multi-scale behavior. To model the information exchange, common mathematical approaches subdivide the considered tissue into multiple cellular domains where intra-cellular dynamics are accounted for by state variables that evolve according to ordinary differential equation models. However, this approach fails to retain spatial information at a sub-cellular level. Moreover, the numerical discretization of these models becomes demanding, especially in the case of high numbers of cells individually affecting different aspects of spatially-related phenomena (e.g., reaction and diffusion of chemicals).
To account for the spatial information while maintaining a manageable computational effort, we propose a new method that surrogates the effect of individual cell contributions on the macroscopic domain. To this aim, under the assumption that cells are arranged periodically within plant tissue, we treat a multi-scale problem through homogenization theory so that a coarse-grained averaged value of the individual cell contribution can be used as a good approximation of the real spatially heterogeneous coefficients.
The numerical test phase for the proposed approach is organized in multiple steps of increasing complexity. We start by considering a linear reaction-diffusion equation with different boundary conditions. Then, considering a variation, keeping the same coefficients as in the previous case, of the nonlinear Liouville-Bratu-Gelfand equation, for which analytical solutions are available in the literature, we validate the new model against a theoretical solution. Building on this, we extend the model to a system of coupled equations, and, finally, we apply the proposed approach to existing state-of-the art biochemical models.