Mathematical modelling of the uptake and transport of salt in plant roots

A Whole Leaf Comparative Study of Stomatal Conductance Models

We employed a detailed whole leaf hydraulic model to study the local operation of three stomatal conductance models distributed on the scale of a whole leaf. We quantified the behavior of these models by examining the leaf-area distributions of photosynthesis, transpiration, stomatal conductance, and guard cell turgor pressure. We gauged the models' local responses to changes in environmental conditions of carbon dioxide concentration, relative humidity, and light irradiance. We found that a stomatal conductance model that includes mechanical processes dependent on local variables predicts a spatial variation of physiological activity across the leaf: the leaf functions of photosynthesis and transpiration are not uniformly operative even when external conditions are uniform. The gradient pattern of hydraulic pressure which is needed to produce transpiration from the whole leaf is derived from the gradient patterns of turgor pressures of guard cells and epidermal cells and consequently leads to nonuniform spatial distribution patterns of transpiration and photosynthesis via the mechanical stomatal model. Our simulation experiments, comparing the predictions of two versions of a mechanical stomatal conductance model, suggest that leaves exhibit a more complex spatial distribution pattern of both photosynthesis and transpiration rate and more complex dependencies on environmental conditions when a non-linear relationship between the stomatal aperture and guard cell and epidermal cell turgor pressures is implemented. Our model studies offer a deeper understanding of the mechanism of stomatal conductance and point to possible future experimental measurements seeking to quantify the spatial distributions of several physiological activities taking place over a whole leaf.

On the Efficacy of Water Transport in Leaves. A Coupled Xylem-Phloem Model of Water and Solute Transport

In this paper, we present and use a coupled xylem/phloem mathematical model of passive water and solute transport through a reticulated vascular system of an angiosperm leaf. We evaluate the effect of leaf width-to-length proportion and orientation of second-order veins on the indexes of water transport into the leaves and sucrose transport from the leaves. We found that the most important factor affecting the steady-state pattern of hydraulic pressure distribution in the xylem and solute concentration in the phloem was leaf shape: narrower/longer leaves are less efficient in convecting xylem water and phloem solutes than wider/shorter leaves under all conditions studied. The degree of efficiency of transport is greatly influenced by the orientation of second-order veins relative to the main vein for all leaf proportions considered; the dependence is non-monotonic with efficiency maximized when the angle is approximately 45° to the main vein, although the angle of peak efficiency depends on other conditions. The sensitivity of transport efficiency to vein orientation increases with increasing vein conductivity. The vein angle at which efficiency is maximum tended to be smaller (relative to the main vein direction) in narrower leaves. The results may help to explain, or at least contribute to our understanding of, the evolution of parallel vein systems in monocot leaves.

A Comprehensive Biophysical Model of Ion and Water Transport in Plant Roots. III. Quantifying the Energy Costs of Ion Transport in Salt-Stressed Roots of Arabidopsis

Salt stress defense mechanisms in plant roots, such as active Na+ efflux and storage, require energy in the form of ATP. Understanding the energy required for these transport mechanisms is an important step toward achieving an understanding of salt tolerance. However, accurate measurements of the fluxes required to estimate these energy costs are difficult to achieve by experimental means. As a result, the magnitude of the energy costs of ion transport in salt-stressed roots relative to the available energy is unclear, as are the relative contributions of different defense mechanisms to the total cost. We used mathematical modeling to address three key questions about the energy costs of ion transport in salt-stressed Arabidopsis roots: are the energy requirements calculated on the basis of flux data feasible; which transport steps are the main contributors to the total energy costs; and which transport processes could be altered to minimize the total energy costs? Using our biophysical model of ion and water transport we calculated the energy expended in the trans-plasma membrane and trans-tonoplast transport of Na+, K+, Cl-, and H+ in different regions of a salt-stressed model Arabidopsis root. Our calculated energy costs exceeded experimental estimates of the energy supplied by root respiration for high external NaCl concentrations. We found that Na+ exclusion from, and Cl- uptake into, the outer root were the major contributors to the total energy expended. Reducing the leakage of Na+ and the active uptake of Cl- across outer root plasma membranes would lower energy costs while enhancing exclusion of these ions. The high energy cost of ion transport in roots demonstrates that the energetic consequences of altering ion transport processes should be considered when attempting to improve salt tolerance.

A Linear Model to Describe Branching and Allometry in Root Architecture

Root architecture is a complex structure that comprises multiple traits of the root phenotype. Novel platforms and models have been developed to better understand root architecture. In this methods paper, we introduce a novel allometric model, named rhizochron index (m), which describes lateral root (LR) branching and elongation patterns across the primary root (PR). To test our model, we obtained data from 16 natural accessions of Arabidopsis thaliana at three stages of early root development to measure conventional traits of root architecture (e.g., PR and LR length), and extracted the rhizochron index (m). In addition, we tested previously published datasets to assess the utility of the rhizochron index (m) to distinguish mutants and environmental effects on root architecture. Our results indicate that rhizochron index (m) is useful to distinguish the natural variations of root architecture between A. thaliana accessions, but not across early stages of root development. Correlation analyses in these accessions showed that m is a novel trait that partially captures information from other root architecture traits such as total lateral root length, and the ratio between lateral root and primary root lengths. Moreover, we found that the rhizochron index was useful to distinguish ABA effect on root architecture, as well as the mutant pho1 phenotype. We propose the rhizochron index (m) as a new feature of the root architectural system to be considered, in addition to conventional traits in future investigations.

A Comprehensive Biophysical Model of Ion and Water Transport in Plant Roots. II. Clarifying the Roles of SOS1 in the Salt-Stress Response in Arabidopsis

SOS1 transporters play an essential role in plant salt tolerance. Although SOS1 is known to encode a plasma membrane Na⁺/H⁺ antiporter, the transport mechanisms by which these transporters contribute to salt tolerance at the level of the whole root are unclear. Gene expression and flux measurements have provided conflicting evidence for the location of SOS1 transporter activity, making it difficult to determine their function. Whether SOS1 transporters load or unload Na⁺ from the root xylem transpiration stream is also disputed. To address these areas of contention, we applied a mathematical model to answer the question: what is the function of SOS1 transporters in salt-stressed Arabidopsis roots? We used our biophysical model of ion and water transport in a salt-stressed root to simulate a wide range of SOS1 transporter locations in a model Arabidopsis root, providing a level of detail that cannot currently be achieved by experimentation. We compared our simulations with available experimental data to find reasonable parameters for the model and to determine likely locations of SOS1 transporter activity. We found that SOS1 transporters are likely to be operating in at least one tissue of the outer mature root, in the mature stele, and in the epidermis of the root apex. SOS1 transporter activity in the mature outer root cells is essential to maintain low cytosolic Na⁺ levels in the root and also restricts the uptake of Na⁺ to the shoot. SOS1 transporters in the stele actively load Na⁺ into the xylem transpiration stream, enhancing the transport of Na⁺ and water to the shoot. SOS1 transporters acting in the apex restrict cytosolic Na⁺ concentrations in the apex but are unable to maintain low cytosolic Na⁺ levels in the mature root. Our findings suggest that targeted, tissue-specific overexpression or knockout of SOS1 may lead to greater salt tolerance than has been achieved with constitutive gene changes. Tissue-specific changes to the expression of SOS1 could be used to identify the appropriate balance between limiting Na⁺ uptake to the shoot while maintaining water uptake, potentially leading to enhancements in salt tolerance.

A Comprehensive Biophysical Model of Ion and Water Transport in Plant Roots. I. Clarifying the Roles of Endodermal Barriers in the Salt Stress Response

In this paper, we present a detailed and comprehensive mathematical model of active and passive ion and water transport in plant roots. Two key features are the explicit consideration of the separate, but interconnected, apoplastic, and symplastic transport pathways for ions and water, and the inclusion of both active and passive ion transport mechanisms. The model is used to investigate the respective roles of the endodermal Casparian strip and suberin lamellae in the salt stress response of plant roots. While it is thought that these barriers influence different transport pathways, it has proven difficult to distinguish their separate functions experimentally. In particular, the specific role of the suberin lamellae has been unclear. A key finding based on our simulations was that the Casparian strip is essential in preventing excessive uptake of Na⁺ into the plant via apoplastic bypass, with a barrier efficiency that is reflected by a sharp gradient in the steady-state radial distribution of apoplastic Na⁺ across the barrier. Even more significantly, this function cannot be replaced by the action of membrane transporters. The simulations also demonstrated that the positive effect of the Casparian strip of controlling Na⁺ uptake, was somewhat offset by its contribution to the osmotic stress component: a more effective barrier increased the detrimental osmotic stress effect. In contrast, the suberin lamellae were found to play a relatively minor, even non-essential, role in the overall response to salt stress, with the presence of the suberin lamellae resulting in only a slight reduction in Na⁺ uptake. However, perhaps more significantly, the simulations identified a possible role of suberin lamellae in reducing plant energy requirements by acting as a physical barrier to preventing the passive leakage of Na⁺ into endodermal cells. The model results suggest that more and particular experimental attention should be paid to the properties of the Casparian strip when assessing the salt tolerance of different plant varieties and species. Indeed, the Casparian strip appears to be a more promising target for plant breeding and plant genetic engineering efforts than the suberin lamellae for the goal of improving salt tolerance.

Modeling Root Zone Effects on Preferred Pathways for the Passive Transport of Ions and Water in Plant Roots

We extend a model of ion and water transport through a root to describe transport along and through a root exhibiting a complexity of differentiation zones. Attention is focused on convective and diffusive transport, both radially and longitudinally, through different root tissue types (radial differentiation) and root developmental zones (longitudinal differentiation). Model transport parameters are selected to mimic the relative abilities of the different tissues and developmental zones to transport water and ions. For each transport scenario in this extensive simulations study, we quantify the optimal 3D flow path taken by water and ions, in response to internal barriers such as the Casparian strip and suberin lamellae. We present and discuss both transient and steady state results of ion concentrations as well as ion and water fluxes. We find that the peak in passive uptake of ions and water occurs at the start of the differentiation zone. In addition, our results show that the level of transpiration has a significant impact on the distribution of ions within the root as well as the rate of ion and water uptake in the differentiation zone, while not impacting on transport in the elongation zone. From our model results we infer information about the active transport of ions in the different developmental zones. In particular, our results suggest that any uptake measured in the elongation zone under steady state conditions is likely to be due to active transport.

RootAnalyzer: A Cross-Section Image Analysis Tool for Automated Characterization of Root Cells and Tissues

The morphology of plant root anatomical features is a key factor in effective water and nutrient uptake. Existing techniques for phenotyping root anatomical traits are often based on manual or semi-automatic segmentation and annotation of microscopic images of root cross sections. In this article, we propose a fully automated tool, hereinafter referred to as RootAnalyzer, for efficiently extracting and analyzing anatomical traits from root-cross section images. Using a range of image processing techniques such as local thresholding and nearest neighbor identification, RootAnalyzer segments the plant root from the image's background, classifies and characterizes the cortex, stele, endodermis and epidermis, and subsequently produces statistics about the morphological properties of the root cells and tissues. We use RootAnalyzer to analyze 15 images of wheat plants and one maize plant image and evaluate its performance against manually-obtained ground truth data. The comparison shows that RootAnalyzer can fully characterize most root tissue regions with over 90% accuracy.

On the competitive uptake and transport of ions through differentiated root tissues

We simulate the competitive uptake and transport of a mixed salt system in the differentiated tissues of plant roots. The results are based on a physical model that includes both forced diffusion and convection by the transpiration stream. The influence of the Casparian strip on regulating apoplastic flow, the focus of the paper, is modelled by varying ion diffusive permeabilities, hydraulic reflection coefficients and water permeability for transport across the endodermis-pericycle interface. We find that reducing diffusive permeabilities leads to significantly altered ion concentration profiles in the pericycle and vascular cylinder regions, while increased convective reflectivities affect predominantly ion concentrations in the cortex and endodermis tissues. The self-consistent electric field arising from ion separation is a major influence on predicted ion fluxes and accumulation rates.