The plant hormones brassinosteroid (BR) and gibberellin (GA) have important roles in a wide range of processes involved in plant growth and development. In this paper we derive and analyse new mathematical models for the BR signalling pathway and for the crosstalk between the BR and GA signalling pathways. To analyse the effects of spatial heterogeneity of the signalling processes, along with spatially-homogeneous ODE models we derive coupled PDE-ODE systems modelling the temporal and spatial dynamics of molecules involved in the signalling pathways. The values of the parameters in the model for the BR signalling pathway are determined using experimental data on the gene expression of BR biosynthetic enzymes. The stability of steady state solutions of our mathematical model, shown for a wide range of parameters, can be related to the BR homeostasis which is essential for proper function of plant cells. Solutions of the mathematical model for the BR signalling pathway can exhibit oscillatory behaviour only for relatively large values of parameters associated with transcription factor brassinazole-resistant1's (BZR) phosphorylation state, suggesting that this process may be important in governing the stability of signalling processes. Comparison between ODE and PDE-ODE models demonstrates distinct spatial distribution in the level of BR in the cell cytoplasm, however the spatial heterogeneity has significant effect on the dynamics of the averaged solutions only in the case when we have oscillations in solutions for at least one of the models, i.e. for possibly biologically not relevant parameter values. Our results for the crosstalk model suggest that the interaction between transcription factors BZR and DELLA exerts more influence on the dynamics of the signalling pathways than BZR-mediated biosynthesis of GA, suggesting that the interaction between transcription factors may constitute the principal mechanism of the crosstalk between the BR and GA signalling pathways. In general, perturbations in the GA signalling pathway have larger effects on the dynamics of components of the BR signalling pathway than perturbations in the BR signalling pathway on the dynamics of the GA pathway. The perturbation in the crosstalk mechanism also has a larger effect on the dynamics of the BR pathway than of the GA pathway. Large changes in the dynamics of the GA signalling processes can be observed only in the case where there are disturbances in both the BR signalling pathway and the crosstalk mechanism. Those results highlight the robustness of the GA signalling pathway.

Auxin has critical roles in plant growth, dependent upon its heterogeneous distribution in tissues. Exactly how auxin signalling and transport, and developmental processes such as growth coordinate to achieve the precise patterns of auxin observed experimentally is not well understood. Here we use mathematical modelling to examine the interplay between these effects and their contribution to formation of patterns in auxin distribution in plant tissues. Mathematical models describing auxin-related signalling pathway, PIN and AUX1 dynamics, auxin transport, and cell growth in plant tissues were derived. Models were analysed and solved numerically to examine the long-time behaviour and auxin distribution. Changes in auxin-related signalling processes were shown to be able to trigger transition between passage and spot type patterns in auxin distribution. The model was also shown to be able to generate isolated cells with oscillatory auxin levels which have been observed experimentally. Cell growth was shown to have strong influence on PIN polarisation and determination of auxin distribution patterns. Numerical simulation results indicate that auxin-related signalling processes can explain the different auxin distributions observed in plant tissues, whereas interplay between auxin transport and growth can explain the 'reverse-fountain' pattern in auxin distribution observed at root tips.

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