Mathematical modelling of transport processes in plant tissues with consideration of the internal microstructure
Final Report Abstract
The main goal of the project was to increase the knowledge on transport processes in plants and plant tissues by using an interdisciplinary approach based on quantitative mathematical models. The project focused on three processes: transport from the soil (uptake), plant internal transport (distribution) and gas exchange with the atmosphere (photosynthesis). A general approach to model transport processes in plant tissues was established. The basic idea is similar to cell-based models used in organ development and computational morphodynamics. A tissue is decomposed into a set of interacting/interfacing cells in which regulatory mechanisms are modeled by ordinary differential equations. The fluxes into and out of a cell are assumed to depend on the expression level of influx and efflux transporters on the plasma membrane (interface between symplast and apoplast). These regulatory models are then combined with physical models of e.g. mass conservation, which describe the fluxes of water and solutes inside the tissue with partial differential equations. This general modelling framework was used to model the regulation of uptake and transport of the heavy metal zinc in roots of Arabidopsis thaliana. A dynamical model based on ordinary differential equations was developed to study the regulation of the uptake transporters (ZIPs). For this, a general model of zinc homeostasis was posed, which was also used to describe uptake in yeast cells quantitatively. Then, this model was coupled to a radial transport model, which accounts for the structure of the root by considering the different cell layers, and includes effects of water flow, diffusion, and cross-membrane transport via transporter proteins. The experimentally known pattern of zinc accumulation close to the center of the root and the dependence of the pattern on the efflux transporters (HMAs), were confirmed using simulations of the model. While biologists thought the levels of efflux transporters (HMAs) to be the only agent responsible for this pattern, our models suggest that the water flow velocity is also a key parameter. Following the same general approach of decomposing a tissue into single interacting cells, a mesoscopic model of water flow was derived using homogenization techniques. One of the largest challenges was to choose properly the transmission conditions between the apoplastic and symplastic flows (Darcy and Stokes flows separated by a semipermeable membrane). The mesoscopic model obtained rigorously by homogenization is based on a Darcy law - which describes the flow through a porous medium - with a force term that depends on the local difference in symplastic and apoplastic solute concentration. Thus, the model accounts very well for effects of osmolarity, such as that the solute is able to drive the flow either by changing the local turgor pressure or directly by gradients of its concentration. Finally, the general approach was used to pose a cell-based model of a leaf cross section. The tissue surrounding a stoma was described by a set of cells interacting physically through fluxes and forces, where each cell, cell wall piece and surrounding interstitial space has an own set of differential equations. Theoretical considerations concerning guard cell and epidermis cell mechanics were conducted to account for the effect of water loss on mechanical status of the tissue. Fluxes in the interstitial gas space were modeled separately as multicomponent Maxwell-Stefan diffusion combined with a flow equation to account for advection. The gradients of pressure on the surface of leaves were estimated to be used as boundary conditions for the flow. An all-embracing physically sound model of leaf gas exchange could be obtained in the future by combining these sub-models.
Publications
- (2012). Modeling regulation of zinc uptake via ZIP transporters in yeast and plant roots. PLoS One 7: e37193
Claus, Chavarría-Krauser
(See online at https://doi.org/10.1371/journal.pone.0037193) - 2012). Zinc Uptake and Radial Transport in Roots of Arabidopsis thaliana: A modelling approach to understand accumulation. Ann Bot-London 112: 369–380
Claus, Bohmann, Chavarría-Krauser
(See online at https://doi.org/10.1093/aob/mcs263) - (2013). A model of mechanics and gas exchange in a neighborhood of a single stoma. Proceedings of 7th Int Conf Funct-Struct Plant Model, p. 134
Bohmann, Chavarría-Krauser
- (2013). Homogenization approach to water transport in plant tissues with periodic microstructures. Math Model Nat Phenom 8: 80–111
Chavarría-Krauser, Ptashnyk
(See online at https://doi.org/10.1051/mmnp/20138406) - (2013). Implications of a zinc uptake and transport model. Plant Signal Behav 8: e24167
Claus, Chavarría-Krauser
- (2013). Modelling transport processes in tissues and organs at a mesoscopic scale. Proceedings of 7th Int Conf Funct-Struct Plant Model, pp. 115–117
Bohmann, Claus, Chavarría-Krauser
- (2014). Modelling of zinc uptake and transport in plant roots. PhD-Thesis, Universität Heidelberg. Electronic copy at heiDOK
Claus
(See online at https://dx.doi.org/10.11588/heidok.00017881) - (2015). Global Hopf bifurcation in the ZIP regulatory system. J Math Biol 71: 795–816
Claus, Ptashnyk, Bohmann, Chavarría-Krauser
(See online at https://doi.org/10.1007/s00285-014-0836-1)
