Global Hopf bifurcation in the ZIP regulatory system.

Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by a buffer reaction and analyze the stability of the steady state in dependence of the properties of this reaction. We find that a large enough equilibrium constant of the buffering reaction stabilizes the steady state and prevents the development of oscillations. Hence, our results suggest that buffering has a key role in the dynamics of zinc homeostasis in plant cells.

Implications of a zinc uptake and transport model.

While physicists regularly use mathematical equations to describe natural phenomena, mathematical modeling of biological systems is still not well established and is hampered by communication barriers between experimental and theoretical biologists. In a recent study we developed a mathematical model of zinc uptake and radial transport in Arabidopsis thaliana roots. By refraining from writing many equations in the main text and confining the derivation of formulas to a supplemental file, we attempted to reach both experimentalists and theoreticians likewise. Here, we give a short summary of our results on the accumulation pattern of zinc and the importance of transporter regulation, water flow and geometry. For a better understanding of the dynamics of adaptation to changes in external conditions, we plead for more detailed and frequent measurements. As a new aspect, we analyzed the effect of buffering. Simulations indicate that it dampens oscillations and may therefore play a key role in zinc homeostasis.

Zinc uptake and radial transport in roots of Arabidopsis thaliana: a modelling approach to understand accumulation.

BACKGROUND AND AIMS: Zinc uptake in roots is believed to be mediated by ZIP (ZRT-, IRT-like proteins) transporters. Once inside the symplast, zinc is transported to the pericycle, where it exits by means of HMA (heavy metal ATPase) transporters. The combination of symplastic transport and spatial separation of influx and efflux produces a pattern in which zinc accumulates in the pericycle. Here, mathematical modelling was employed to study the importance of ZIP regulation, HMA abundance and symplastic transport in creation of the radial pattern of zinc in primary roots of Arabidopsis thaliana. METHODS: A comprehensive one-dimensional dynamic model of radial zinc transport in roots was developed and used to conduct simulations. The model accounts for the structure of the root consisting of symplast and apoplast and includes effects of water flow, diffusion and cross-membrane transport via transporters. It also incorporates the radial geometry and varying porosity of root tissues, as well as regulation of ZIP transporters. KEY RESULTS: Steady-state patterns were calculated for various zinc concentrations in the medium, water influx and HMA abundance. The experimentally observed zinc gradient was reproduced very well. An increase of HMA or decrease in water influx led to loss of the gradient. The dynamic behaviour for a change in medium concentration and water influx was also simulated showing short adaptation times in the range of seconds to minutes. Slowing down regulation led to oscillations in expression levels, suggesting the need for rapid regulation and existence of buffering agents. CONCLUSIONS: The model captures the experimental findings very well and confirms the hypothesis that low abundance of HMA4 produces a radial gradient in zinc concentration. Surprisingly, transpiration was found also to be a key parameter. The model suggests that ZIP regulation takes place on a comparable timescale as symplastic transport.

Modeling regulation of zinc uptake via ZIP transporters in yeast and plant roots.

In yeast (Saccharomyces cerevisiae) and plant roots (Arabidopsis thaliana) zinc enters the cells via influx transporters of the ZIP family. Since zinc is both essential for cell function and toxic at high concentrations, tight regulation is essential for cell viability. We provide new insight into the underlying mechanisms, starting from a general model based on ordinary differential equations and adapting it to the specific cases of yeast and plant root cells. In yeast, zinc is transported by the transporters ZRT1 and ZRT2, which are both regulated by the zinc-responsive transcription factor ZAP1. Using biological data, parameters were estimated and analyzed, confirming the different affinities of ZRT1 and ZRT2 reported in the literature. Furthermore, our model suggests that the positive feedback in ZAP1 production has a stabilizing function at high influx rates. In plant roots, various ZIP transporters play a role in zinc uptake. Their regulation is largely unknown, but bZIP transcription factors are thought to be involved. We set up three putative models based on: an activator only, an activator with dimerization and an activator-inhibitor pair. These were fitted to measurements and analyzed. Simulations show that the activator-inhibitor model outperforms the other two in providing robust and stable homeostasis at reasonable parameter ranges.

A model of plasma membrane flow and cytosis regulation in growing pollen tubes.

A model of cytosis regulation in growing pollen tubes is developed and simulations presented. The authors address the question on the minimal assumptions needed to describe the pattern of exocytosis and endocytosis reported recently by experimental biologists. Biological implications of the model are also treated. Concepts of flow and conservation of membrane material are used to pose an equation system, which describes the movement of plasma membrane in the tip of growing pollen tubes. After obtaining the central equations, relations describing the rates of endocytosis and exocytosis are proposed. Two cytosis receptors (for exocytosis and endocytosis), which have different recycling rates and activation times, suffice to describe a stable growing tube. Simulations show a very good spatial separation between endocytosis and exocytosis, in which separation is shown to depend strongly on exocytic vesicle delivery. In accordance to measurements, most vesicles in the clear zone are predicted to be endocytic. Membrane flow is essential to maintain cell polarity, and bi-directional flow seems to be a natural consequence of the proposed mechanism. For the first time, a model addressing plasma membrane flow and cytosis regulation were posed. Therefore, it represents a missing piece in an integrative model of pollen tube growth, in which cell wall mechanics, hydrodynamic fluxes and regulation mechanisms are combined.

Photosynthesis in lightfleck areas of homobaric and heterobaric leaves.

Leaves within a canopy are exposed to a spatially and temporally fluctuating light environment which may cause lateral gradients in leaf internal CO(2) concentration and diffusion between shaded and illuminated areas. In previous studies it was hypothesized that lateral CO(2) diffusion may support leaf photosynthesis, but the magnitude of this effect is still not well understood. In the present study homobaric leaves of Vicia faba or heterobaric leaves of Glycine max were illuminated with lightflecks of different sizes, mimicking sunflecks. Photosynthetic properties of the lightfleck areas were assessed with combined gas exchange measurements and chlorophyll fluorescence imaging. Lateral diffusion in homobaric leaves with an interconnected intercellular air space stimulated photosynthesis and the effect was largest in small lightfleck areas, in particular when plants were under drought stress. Such effects were not observed in the heterobaric leaves with strongly compartmented intercellular gas spaces. It is concluded that lateral diffusion may significantly contribute to photosynthesis of lightfleck areas of homobaric leaves depending on lightfleck size, lateral diffusivity, and stomatal conductance. Since homobaric leaf structures have been reported for many plant species, it is hypothesized that leaf homobary may have an impact on overall plant performance under conditions with a highly heterogeneous light environment.

Modelling in vitro growth of dense root networks.

Hairy roots are plants genetically transformed by Agrobacterium rhizogenes, which do not produce shoots and are composed mainly by roots. Hairy roots of Ophiorrhiza mungos Linn. are currently gaining interest of pharmacologists, since a secondary product of their metabolism, camptothecin, is used in chemotherapy. To optimize the production of valuable secondary metabolites it is necessary to understand the metabolism and growth of these roots systems. In this work, a mathematical model for description of apical growth of a dense root network (e.g. hairy roots) is derived. A continuous approach is used to define densities of root tips and root volume. Equations are posed to describe the evolution of these and are coupled to the distribution of nutrient concentration in the medium and inside the network. Following the principles of irreversible thermodynamics, growth velocity is defined as the sum over three different driving forces: nutrient concentration gradients, space gradients and root tip diffusion. A finite volume scheme was used for the simulation and parameters were chosen to fit experimental data from O. mungos Linn. hairy roots. Internal nutrient concentration determines short-term growth. Long-term behavior is limited by the total nutrient amount in the medium. Therefore, mass yield could be increased by guaranteeing a constant supply of nutrients. Increasing the initial mass of inoculation did not result in higher mass yields, since nutrient consumption due to metabolism also rose. Four different growth strategies are compared and their properties discussed. This allowed to understand which strategy might be the best to increase mass production optimally. The model is able to describe very well the temporal evolution of mass increase and nutrient uptake. Our results provide further understanding of growth and density distribution of hairy root network and therefore it is a sound base for future applications to describe, e.g., secondary metabolite production.

Photosynthesis can be enhanced by lateral CO2 diffusion inside leaves over distances of several millimeters.

This study examines the extent to which lateral gas diffusion can influence intercellular CO(2) concentrations (c(i)) and thus photosynthesis in leaf areas with closed stomata. Leaves were partly greased to close stomata artificially, and effects of laterally diffusing CO(2) into the greased areas were studied by gas-exchange measurement and chlorophyll fluorescence imaging. Effective quantum yields (Delta F/F(m)') across the greased areas were analysed with an image-processing tool and transposed into c(i) profiles, and lateral CO(2) diffusion coefficients (D(C'lat)), directly proportional to lateral conductivities (), were estimated using a one-dimensional (1D) diffusion model. Effective CO(2) diffusion distances in Vicia faba (homobaric), Commelina vulgaris (homobaric) and Phaseolus vulgaris (heterobaric) leaves clearly differed, and were dependent on D(C'lat), light intensity, [CO(2)], and [O(2)]: largest distances were approx. 7.0 mm for homobaric leaves (with high D(C'lat)) and approx. 1.9 mm for heterobaric leaves (low D(C'lat)). Modeled lateral CO(2) fluxes indicate large support of photosynthesis over submillimeter distances for leaves with low D(C'lat), whereas in leaves with large D(C'lat), photosynthesis can be stimulated over distances of several millimeters. For the plant species investigated, the surplus CO(2) assimilation rates of the greased leaf areas (A(gr)) differed clearly, depending on lateral conductivities of the respective leaves.

Spatio-temporal quantification of differential growth processes in root growth zones based on a novel combination of image sequence processing and refined concepts describing curvature production.

Differential growth processes in root and shoot growth zones are governed by the transport kinetics of auxin and other plant hormones. While gene expression and protein localization of hormone transport facilitators are currently being unraveled using state-of-the-art techniques of live cell imaging, the quantitative analysis of growth reactions is lagging behind because of a lack of suitable methods. A noninvasive technique, based on digital image sequence processing, for visualizing and quantifying highly resolved spatio-temporal root growth processes was applied in the model plant Arabidopsis thaliana and was adapted to provide precise information on differential curvature production activity within the root growth zone. Comparison of root gravitropic curvature kinetics in wild-type and mutant plants altered in a facilitator for auxin translocation allowed the determination of differences in the location and in the temporal response of curvature along the growth zone between the investigated plant lines. The findings of the quantitative growth analysis performed here confirm the proposed action of the investigated transport facilitator. The procedure developed here for the investigation of differential growth processes is a valuable tool for characterizing the phenomenology of a wide range of shoot and root growth movements and hence facilitates elucidation of their molecular characterization.

Quantification of curvature production in cylindrical organs, such as roots and hypocotyls.

Differential growth curvature rate (DGCR), defined as the spatial derivative of the tropic speed, was derived as a measure of curvature production in cylindrical organs. Its relation to usual concepts, such as curvature (kappa), rate of curvature (dkappa/dt) and differential growth profiles, was determined. A root gravitropism model, testing the hypothesis of one and two motors, exemplified its capabilities.DGCR was derived using cylindrical geometry and its meaning was obtained through a curvature conservation equation. The root gravitropism model was solved using a discrete difference method on a computer.DGCR described curvature production independently of growth, and was superior to dkappa/dt, which underestimated production. Moreover, DGCR profiles were able to differ between one and two motors, while profiles of kappa and dkappa/dt were not. The choice of the measure of curvature production has a large impact on experimental results, in particular when spatial and temporal patterns of differential growth need to be determined. DGCR was shown to fulfill the accuracy needed in the quantification of curvature production and should thus serve as a helpful tool for measurements.

Nocturnal changes in leaf growth of Populus deltoides are controlled by cytoplasmic growth.

Growing leaves do not expand at a constant rate but exhibit pronounced diel growth rhythms. However, the mechanisms giving rise to distinct diel growth dynamics in different species are still largely unknown. As a first step towards identifying genes controlling rate and timing of leaf growth, we analysed the transcriptomes of rapidly expanding and fully expanded leaves of Populus deltoides Bartr. ex. Marsh at points of high and low expansion at night. Tissues with well defined temporal growth rates were harvested using an online growth-monitoring system based on a digital image sequence processing method developed for quantitative mapping of dicot leaf growth. Unlike plants studied previously, leaf growth in P. deltoides was characterised by lack of a base-tip gradient across the lamina, and by maximal and minimal growth at dusk and dawn, respectively. Microarray analysis revealed that the nocturnal decline in growth coincided with a concerted down-regulation of ribosomal protein genes, indicating deceleration of cytoplasmic growth. In a subsequent time-course experiment, Northern blotting and real-time RT-PCR confirmed that the ribosomal protein gene RPL12 and a cell-cycle gene H2B were down-regulated after midnight following a decrease in cellular carbohydrate concentrations. Thus, we propose that the spatio-temporal growth pattern in leaves of P. deltoides primarily arises from cytoplasmic growth whose activity increases from afternoon to midnight and thereafter decreases in this species.

Primary root growth: a biophysical model of auxin-related control.

Plant hormones control many aspects of plant development and play an important role in root growth. Many plant reactions, such as gravitropism and hydrotropism, rely on growth as a driving motor and hormones as signals. Thus, modelling the effects of hormones on expanding root tips is an essential step in understanding plant roots. Here we achieve a connection between root growth and hormone distribution by extending a model of root tip growth, which describes the tip as a string of dividing and expanding cells. In contrast to a former model, a biophysical growth equation relates the cell wall extensibility, the osmotic potential and the yield threshold to the relative growth rate. This equation is used in combination with a refined hormone model including active auxin transport. The model assumes that the wall extensibility is determined by the concentration of a wall enzyme, whose production and degradation are assumed to be controlled by auxin and cytokinin. Investigation of the effects of auxin on the relative growth rate distribution thus becomes possible. Solving the equations numerically allows us to test the reaction of the model to changes in auxin production. Results are validated with measurements found in literature.