Modeling Auxin Distribution in Plant Roots: a Homogenization-based Approach
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Hormonal activity plays an important role in the regulation of plant growth. In particular, auxin is linked to plant structure development and cell wall modification. In light of this, auxin behavior in root hair cells becomes the object of inquiry. Unraveling the mechanisms that impact it might prove crucial to improve the performances in agricultural applications. From a modeling viewpoint, auxin distribution in plant roots can be interpreted as a system of reaction-diffusion partial differential equations. Reconstructing auxin distribution in a portion of plant tissue, in which numerous cells individually affect reactive and diffusive phenomena, is highly challenging. Indeed, the variability and complexity of such phenomenon requires substantial computational resources. To reduce the computational effort, we propose a new method that surrogates the effect of individual cells on the macroscopic domain. To this aim, we assume cells to be arranged periodically within the plant tissue. Consequently, we can treat auxin behavior as a multi-scale problem. By homogenization theory, an average value of the individual cell contribution can serve as a good approximation. Our approach is organized in multiple steps of increasing modeling complexity. We start by testing a linear reaction-diffusion equation with different boundary conditions. Then, we consider a version of the Liouville-Bratu-Gelfand equation, a nonlinear equation for which analytical solutions are already present in literature. Building on this, we extend the model to a system of coupled equations. Finally, we apply the proposed model for auxin distribution in plants, validating against state-of-the art literature.