Implications of a zinc uptake and transport model

Abstract

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.

model noun a system of postulates, data and inferences presented as a mathematical description of an entity or state of affairs.¹

Biologists, applied mathematicians and physicists working on biological systems have the shared objective of creating models to reduce complexity and allow better understanding. However, their approach and the outcome can be quiet different despite of working on the same problem. Biologists might focus rather on the “data and inferences,” while mathematicians concentrate on the “mathematical description” and physicists end up somewhere in between. The different approaches lead to communication problems, exemplified recently by an analysis of the impact of equation usage in biological articles. Fawcett and Higginson found that the density of equations has a significant negative impact on citation rates, explainable by a sharp drop in citations from nontheoretical papers.² As a conclusion, they suggest to change the presentation of mathematical models; essentially to “hide” these in an appendix or supplemental file. Though not analyzed, nontheoretical articles will suffer from a similar problem but to a much lesser extent. Theorists, including ourselves, prefer nicely presented reviews over papers with many technical details on the experimental procedures. We learn, it is a matter of presentation and the communities have to trust each other mutually when it comes to the details. After seeing Fawcett and Higginson's analysis, we decided to follow their suggestions in our last paper on modeling zinc uptake and transport in roots.³ Most equations are presented in a supplemental file to “hide” the mathematical details. Their suggestions worked and the reward came promptly as an invitation to write this addendum. Instead of bringing here more mathematical details, we concentrate on the limitations and implications of our model. We present a critical review of the time scale of regulation and the role of buffering and plead for higher frequencies when measuring the time course of expression levels of zinc regulating genes.

Our model is roughly speaking composed of two main packages: regulation and transport.³ The novelty lies in coupling these two to analyze the role of regulation in zinc accumulation, i.e., to bridge the gap between the cellular and organ levels and to predict phenotypes from gene expression profiles. This way, we hope to contribute in a longer term to the understanding of hyperaccumulation (see ref. ⁴ for review). Figure 1 presents a scheme with details of the processes accounted for by the transport model. The regulation model is based on a regulatory network with a dimerizing activator-inhibitor pair (see refs. ³ and ⁵ for details). Epidermis, cortex and endodermis cells are assumed to express their own level of a ZIP-like influx transporter, while a constant level of HMA4-like efflux transporters was assumed in the epidermis and pericycle. Simulated ZIP expression levels behaved inversely proportional to the accumulation pattern and water uptake and the cylindrical geometry were found to be key parameters for creation of the pattern.³

Figure 1. Scheme of root tissues from epidermis (right) to xylem (left) and processes considered in the transport model. Blue arrows indicate the direction of water flow; gray dots and arrows are used for zinc.

The model is still in the beginning and misses some key points. Though possible, we decided in a first attempt not to consider buffering by chelators or by the vacuole. The reason for this decision was simple: each new aspect included needs more biological details and introduces more unknown parameters. To estimate the effect of buffering, we extended the model and included a simple buffering mechanism (Fig. 1 and ref. ⁶) where C is an arbitrary zinc-buffering chelator and ZnC denotes the complex:

Zn²⁺ + C ↔ ZnC

A pool of immobile chelators is assumed with a concentration much greater than the free zinc concentration. Although no quantitative data are available, it is generally believed that most of the intracellular zinc is chelated or sequestered to avoid toxicity.⁷ Therefore, the ratio of bound to free zinc K has to be large. Note that K is related to the equilibrium constant.

The dynamics of buffering are determined by K and a characteristic time scale, defined as the time needed to reach 95% of the steady-state of the buffer reaction. The effects of varying K and this time scale are shown in Figure 2 for simulations of a zinc resupply from 1 µM to 10 µM in the external medium. Without buffering and under slow regulation, this scenario leads to oscillations (right panel in Fig. 2, also shown in ref. ³, Fig. 9, green line). In the presence of a buffer, however, the oscillations are dampened or even extinguished, optimally under short characteristic times (less than 3 sec) and high K values (at least 100 to 1,000). Since such values are not unlikely in plants, our simulations suggest that buffering is indeed a key mechanism to prevent potentially toxic oscillations.

Figure 2. Variation of buffering parameters with constant characteristic time of regulation (10 min). Curves show the dynamic behavior of the internal zinc concentration without buffering (right) and with buffering (left) under different values of the ratio of bound to free zinc, K, and characteristic time of buffering. Oscillations are increasingly dampened with higher K and lower characteristic times.

As a second effect besides the dampening of oscillations, buffering can slow down the adaptation of the entire root system to external changes. In reference ³ we argued that due to fast diffusion and transport, equilibration takes place quickly, within seconds to minutes. Therefore, the regulation of ZIP transporters might also take place faster than thought until now and their expression levels might have to be measured more frequently than every two hours, which was the interval used by Talke et al.⁸ Although buffering allows slower regulation without oscillations, our plea for more frequent data points is supported by new simulations (Fig. 3). They show that even with strong and rapid buffering (K = 1,000 and characteristic time 0.003 sec), adaptation of ZIP activity is faster than two hours. Varying the time scale of regulation by a factor of 10³ leads to no measurable differences in ZIP activity after one hour. To be able to determine a more realistic time scale of ZIP regulation, measurements need to be taken at shorter intervals of preferably less than 10 min.

Figure 3. Dynamics of ZIP activity under different time scales of regulation. Characteristic time of 6 sec (magenta), 1 min (green), 10 min (blue) and 100 min (red) were simulated. The buffering parameters were kept constant at K = 1,000 and a characteristic time of 0.003 sec.

At the moment, the model is still limited in some aspects. Buffering could be captured more accurately by including, e.g., mobile diffusive buffers and sequestration into the vacuole. Also, the water flow model needs to be extended to account for more details of symplastic and apoplastic water flows.⁹ Nevertheless, our model in its current state helps to interpret and understand biological phenomena and to point out directions for future experiments. In this sense, we hope to contribute hand in hand with experimentalists to a profound understanding of metal uptake and homeostasis in plants.

Glossary

Abbreviations:

ZIP

ZRT-IRT-like proteins

HMA

heavy metal ATPase

Disclosure of Potential Conflicts of Interest

No potential conflicts of interest were disclosed.