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

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

Plants, algae, and fungi are essential for nearly all life on earth. Throughphotosynthesis, plants and algae convert solar energy to chemical energy in the form oforganic compounds that sustains essentially all life on earth. In addition, plants andalgae convert the carbon dioxide produced by respiring organisms to oxygen that is neededfor respiration. Fungi decompose complex organic compounds produced by respiring organismsso that molecules can be recycled in photosynthesis and respiration. Plants, algae, andfungi have one important feature in common, their cells have walls. Expansive growth andits 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 informationrelevant to expansive growth of cells with walls. Mathematical models have beenconstructed in an attempt to organize and evaluate this information, to gain insight, toevaluate hypotheses, and to assist in the selection and development of new experimentalstudies. In this article some of the mathematical models constructed to study expansivegrowth of cells with walls are reviewed. It is nearly impossible to review all relevantresearch conducted in this area. Instead, the review focuses on the development ofmathematical equations that have been used to model expansive growth, morphogenesis, andgrowth rate regulation of cells with walls. Also, relevant experimental findings arereviewed, conceptual models are presented, and suggestions for future research areproposed. The authors have attempted to provide an overview that is accessible toresearchers that are not working in this field.

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

Water flow in plant tissues takes place in two different physical domains separated bysemipermeable membranes: cell insides and cell walls. The assembly of all cell insides andcell walls are termed symplast and apoplast,respectively. Water transport is pressure driven in both, where osmosis plays an essentialrole in membrane crossing. In this paper, a microscopic model of water flow and transportof an osmotically active solute in a plant tissue is considered. The model is posed on thescale of a single cell and the tissue is assumed to be composed of periodicallydistributed 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 interfaceand by describing the protein mediated transport as a surface reaction. Applyinghomogenization techniques, macroscopic equations for water and solute transport in a planttissue are derived. The macroscopic problem is given by a Darcy law with a force termproportional to the difference in concentrations of the osmotically active solute in thesymplast and apoplast; i.e. the flow is also driven by the local concentration differenceand 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 ofinteracting processes, a large number of model parameters and costly experimental dataacquisition. Such complexities make model parameterization a difficult process. Moreover,there is a large variety of models that coexist in the literature with generally anabsence of benchmarking between the different approaches and insufficient modelevaluation. In this context, this paper aims at enhancing good modelling practices in theplant growth modelling community and at increasing model design efficiency. It gives anoverview of the different steps in modelling and specify them in the case of plant growthmodels specifically regarding their above mentioned characteristics.

Different methods allowing to perform these steps are implemented in a dedicated platformPYGMALION (Plant Growth Model Analysis, Identification and Optimization). Some of thesemethods are original. The C++ platform proposes a framework in which stochastic ordeterministic discrete dynamic models can be implemented, and several efficient methodsfor sensitivity analysis, uncertainty analysis, parameter estimation, model selection ordata 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 toillustrate 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 realdata and is shown to have interesting predictive capacities when coupled with dataassimilation techniques.