Homogenization-Based Approach to Modeling Plant Tissue Dynamics

  • Magni, Francesco (Scuola Italiana Superiore di Studi Avanzati)
  • Mondini, Nicolò (Politecnico di Milano)
  • Pederzoli, Valenitina (Politecnico di Milano)
  • Ferro, Nicola (Università Ca' Foscari, Venezia Mestre)
  • Avitabile, Daniele (Vrije Universiteit Amsterdam)
  • Perotto, Simona (Politecnico di Milano)

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Multiple internal mechanisms, such as hormonal activity, play an important role in regulating plant growth. Many of such mechanisms rely on spatio-temporal signals to convey information across portions of tissue. Thus, in the case of multi-cellular tissue, the evolution dynamics of such signals is assumed to follow a multi-scale behavior. From a modeling viewpoint, tissue is split into multiple cellular domains where intra-cellular dynamics are accounted for by state variables that evolve according to ODE models [1]. This kind of approach fails to retain spatial information at a sub-cellular level. In the case of numerous cells individually affecting different aspects of spatially-related phenomena (e.g., reactive and diffusive contributions), such feat could prove highly challenging from a numerical viewpoint. To account for the spatial information while maintaining a manageable 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 treat a multi-scale problem through homogenization theory. Thus, a coarse- grained averaged value of the individual cell contribution can serve as a good approximation of the real spatially heterogeneous coefficients. The proposed 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 available in the literature. This intermediate step is used to validate the new model against a theoretical solution. Building on this, we extend the model to a system of coupled equations. Finally, we apply the proposed approach to existing biochemical models, validating against state-of-the art literature [2].