During the last two decades, molecular genetic studies and the completion of the sequencing of the Arabidopsis thaliana genome have increased knowledge of hormonal regulation in plants. These signal transduction pathways act in concert through gene regulatory and signalling networks whose main components have begun to be elucidated. Our understanding of the resulting cellular processes is hindered by the complex, and sometimes counter-intuitive, dynamics of the networks, which may be interconnected through feedback controls and cross-regulation. Mathematical modelling provides a valuable tool to investigate such dynamics and to perform in silico experiments that may not be easily carried out in a laboratory. In this article, we firstly review general methods for modelling gene and signalling networks and their application in plants. We then describe specific models of hormonal perception and cross-talk in plants. This mathematical analysis of sub-cellular molecular mechanisms paves the way for more comprehensive modelling studies of hormonal transport and signalling in a multi-scale setting.

Pollen tubes are tip growing plant cells that display oscillatory growth behavior. It has been demonstrated experimentally that the reduction of the average pollen tube growth rate through elevated extracellular calcium or borate concentrations coincides with a greater amplitude of the growth rate oscillation and a lower oscillation frequency. We present a simple numerical model of pollen tube growth that reproduces these results, as well as analytical calculations that suggest an underlying mechanism. These data show that the pollen tube oscillator is non-isochronous, and is different from harmonic oscillation.

Plants, algae, and fungi are essential for nearly all life on earth. Through photosynthesis, plants and algae convert solar energy to chemical energy in the form of organic compounds that sustains essentially all life on earth. In addition, plants and algae convert the carbon dioxide produced by respiring organisms to oxygen that is needed for respiration. Fungi decompose complex organic compounds produced by respiring organisms so that molecules can be recycled in photosynthesis and respiration. Plants, algae, and fungi have one important feature in common, their cells have walls. Expansive growth and its regulation are central to the life and development of plant, algal, and fungal cells, i.e. cells with walls. In recent decades there has been an explosion of information relevant to expansive growth of cells with walls. Mathematical models have been constructed in an attempt to organize and evaluate this information, to gain insight, to evaluate hypotheses, and to assist in the selection and development of new experimental studies. In this article some of the mathematical models constructed to study expansive growth of cells with walls are reviewed. It is nearly impossible to review all relevant research conducted in this area. Instead, the review focuses on the development of mathematical equations that have been used to model expansive growth, morphogenesis, and growth rate regulation of cells with walls. Also, relevant experimental findings are reviewed, conceptual models are presented, and suggestions for future research are proposed. The authors have attempted to provide an overview that is accessible to researchers that are not working in this field.

Deformable cell model is developed to study pattern formation and to simulate plant tissue growth. Each cell represents a polygon with a number of vertices connected by springs. Some cells in the tissue can grow and divide, other cells are differentiated and do not grow or divide but remain deformable. The model is used to investigate formation of self-similar structures which reproduce the same cell organization during their growth. In numerical experiments we observed that self-similar solutions can exist for a rather precise choice of plant structure and mechanical properties of cell walls. We test the model for simulation of apical meristems functioning which represent self-similar cell structures in plants. At the next stage of modelling, auxin distribution is introduced by means of diffusion and polar transport mechanisms. The existence of steady auxin distribution in a growing root is investigated. Single as well as multiple auxin maxima have been observed in model solutions.

Water flow in plant tissues takes place in two different physical domains separated by semipermeable membranes: cell insides and cell walls. The assembly of all cell insides and cell walls are termed symplast and apoplast, respectively. Water transport is pressure driven in both, where osmosis plays an essential role in membrane crossing. In this paper, a microscopic model of water flow and transport of an osmotically active solute in a plant tissue is considered. The model is posed on the scale of a single cell and the tissue is assumed to be composed of periodically distributed cells. The flow in the symplast can be regarded as a viscous Stokes flow, while Darcy’s law applies in the porous apoplast. Transmission conditions at the interface (semipermeable membrane) are obtained by balancing the mass fluxes through the interface and by describing the protein mediated transport as a surface reaction. Applying homogenization techniques, macroscopic equations for water and solute transport in a plant tissue are derived. The macroscopic problem is given by a Darcy law with a force term proportional to the difference in concentrations of the osmotically active solute in the symplast and apoplast; i.e. the flow is also driven by the local concentration difference and its direction can be different than the one prescribed by the pressure gradient.

Mathematical models of plant growth are generally characterized by a large number of interacting processes, a large number of model parameters and costly experimental data acquisition. Such complexities make model parameterization a difficult process. Moreover, there is a large variety of models that coexist in the literature with generally an absence of benchmarking between the different approaches and insufficient model evaluation. In this context, this paper aims at enhancing good modelling practices in the plant growth modelling community and at increasing model design efficiency. It gives an overview of the different steps in modelling and specify them in the case of plant growth models specifically regarding their above mentioned characteristics.

Different methods allowing to perform these steps are implemented in a dedicated platform PYGMALION (Plant Growth Model Analysis, Identification and Optimization). Some of these methods are original. The C++ platform proposes a framework in which stochastic or deterministic discrete dynamic models can be implemented, and several efficient methods for sensitivity analysis, uncertainty analysis, parameter estimation, model selection or data assimilation can be used for model design, evaluation or application.

Finally, a new model, the LNAS model for sugar beet growth, is presented and serves to illustrate how the different methods in PYGMALION can be used for its parameterization, its evaluation and its application to yield prediction. The model is evaluated from real data and is shown to have interesting predictive capacities when coupled with data assimilation techniques.