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.

Intra- and interdimeric caspase-8 self-cleavage controls strength and timing of CD95-induced apoptosis.

Apoptosis in response to the ligand CD95L (also known as Fas ligand) is initiated by caspase-8, which is activated by dimerization and self-cleavage at death-inducing signaling complexes (DISCs). Previous work indicated that the degree of substrate cleavage by caspase-8 determines whether a cell dies or survives in response to a death stimulus. To determine how a death ligand stimulus is effectively translated into caspase-8 activity, we assessed this activity over time in single cells with compartmentalized probes that are cleaved by caspase-8 and used multiscale modeling to simultaneously describe single-cell and population data with an ensemble of single-cell models. We derived and experimentally validated a minimal model in which cleavage of caspase-8 in the enzymatic domain occurs in an interdimeric manner through interaction between DISCs, whereas prodomain cleavage sites are cleaved in an intradimeric manner within DISCs. Modeling indicated that sustained membrane-bound caspase-8 activity is followed by transient cytosolic activity, which can be interpreted as a molecular timer mechanism reflected by a limited lifetime of active caspase-8. The activation of caspase-8 by combined intra- and interdimeric cleavage ensures weak signaling at low concentrations of CD95L and strongly accelerated activation at higher ligand concentrations, thereby contributing to precise control of apoptosis.

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.