Once taken up by roots, mineral elements are transported to upper part with transpiration stream, followed by distributing to different organs and tissues. Understanding of the mechanisms that control the dynamics of both water and mineral elements is an important issue in plant science. Over the past decade, many transporters for uptake of mineral elements have been identified, including those in rice for N, P, K, Mg, B, Mn, Zn, Fe, and Si (Sasaki et al., 2016). These transporters are located in the plasma membrane. Moreover, some these transporters show polar localization. For example, Si influx transporter Lsi1 is localized to the distal side of the root exodermis and endodermis (Ma et al., 2006), while Si efflux transporter Lsi2 is localized to the proximal side of the same cells (Ma et al., 2007).

An important challenge is to develop a mathematical model that can simulate the dynamics of both water and mineral elements in the whole plant to quantitatively understand the complex mineral element transporting systems. Transporter expression depends on mineral concentrations in tissues (Sasaki et al., 2016). To quantify the dynamics of mineral element transport, it is necessary to consider multiple factors simultaneously including transporter activities in roots and shoots, the dynamics of xylem sap and phloem sap, and the expression levels of the transporter genes.

For water flow in plants, the models using an analogy with an electric circuit (Landsberg and Fowkes, 1978) is one of the most historical models. In these models, water potential (potential energy of water per unit volume relative to pure water) is treated like voltage in a circuit. Water flows according to the difference in water potentials. Because the flow of water in the phloem is strongly related to sucrose concentration, some models treat water and sucrose dynamics simultaneously (Daudet et al., 2002; Lacointe and Minchin, 2008; Lobet et al., 2014; Seki et al., 2015). For the transportation of substances in plant, several different concepts are used to model transport of mineral elements and organic pollutants from roots. In the compartment model, the plants are divided into several compartments (such as root and leaf), and substances are assumed to be transported among the compartments according to transition rates that are determined as parameter values (Fryer and Collins, 2003; Fantke et al., 2013; Trapp, 2014). However, these macro-scale models do not consider not only water flow dynamics but also such micro-scale characteristics as the location and activity of the transporters. On the other hand, in models that simulate the dynamics of substances at the micro-scale (Grieneisen et al., 2007, 2012; Sakurai et al., 2015; Yamaji et al., 2015; Foster and Miklavcic, 2016), substance flow is simulated by diffusion and convection in which the location and polarity of the transporters are considered. However, these models simulate dynamics at the cell or tissue level but not in the whole plant level.

The purpose of this study is to develop a new mathematical model that simulate silicon (Si) transport in whole plant in rice. Si is abundant in soils and is beneficial for plant growth (Ma and Takahashi, 2002). Si deposited in plant tissues enhances tolerance to abiotic and biotic stresses via alleviation of water stress, improvement of light interception characteristics by keeping the leaf blade erect, and an increase in resistance to diseases, pests, and lodging (Epstein, 1994; Savant et al., 1997; Ma and Takahashi, 2002). In rice roots, the passive transporter Lsi1 (OsLsi1) and the active transporter Lsi2 (OsLsi2) are involved in Si uptake (Ma et al., 2006, 2007). Both are located in the plasma membranes of the exodermal and endodermal cells, where Casparian strips are located. Lsi1 transports Si along the Si gradient between the plasma membrane, whereas Lsi2 transports Si from the symplast to the apoplast, and this transport is coupled with proton antiport (Ma et al., 2011). In rice nodes, Lsi2, Lsi3 (OsLsi3, active), and Lsi6 (OsLsi6, passive) transport Si from enlarged vascular bundles (EVB) to diffuse vascular bundles (DVB) for preferential distribution of Si to the grains (Yamaji and Ma, 2014; Yamaji et al., 2015). Dehydration stress decreases the expression of Lsi1 and Lsi2 via abscisic acid (ABA) in root (Yamaji and Ma, 2007). The expression of these transporter genes show a diurnal variation (Yamaji and Ma, 2007). More recently, it was reported that the expression of Lsi1 and Lsi2 genes is controlled by Si accumulation in the shoots, not in the roots (Mitani-Ueno et al., 2016). However, the mathematical models that simulate Si dynamics in whole plant have not been constructed yet.

In recent mathematical models of Si transport in both the root and node of rice (Sakurai et al., 2015; Yamaji et al., 2015), Si is assumed to be transported via diffusion and convection. In a previous study (Sakurai et al., 2015), we modeled the dynamics of Si transport from external solution to the cortex and stele of the root using a diffusion equation in which Si diffuses along the gradient of Si concentrations on a two-dimensional grid (4 × 4 μm). Using this micro-scale diffusion model, we could explain the dynamics of the micro-scale Si transport and could account for the characteristics of the spatial location of the transporters. The parameters of the models estimated using statistical computation methods, and the simulation output match the empirical data well. However, it is difficult to extend these models to whole-plant simulation.

In this study, we propose a new mathematical model that simulates Si transport in the whole plant using empirical data and knowledge from previous modeling studies. Using the model, we examined the factors and mechanisms affecting the expression of Si transporter genes. We assumed three possible signaling mechanisms that control the expression of Si transporter genes in this model: accumulation control, shortage control, and water stress control. Under accumulation control, expression is reduced by excess Si concentration in leaf cells. Under shortage control, expression is increased by low Si concentration in leaf cells. Under water stress control, expression responds to water stress (indicated by transpiration rate in this model). We compared the expression levels simulated by the models with those observed. Finally, using the model, we investigated the reason for the diurnal change of transporter gene expression levels in rice from the point of view of investment efficiency.

In this study, we proposed a new mathematical model that simulate the Si dynamics in whole plant in rice and investigated the possible mechanisms underlying diurnal variation of the expression level of the transporter genes. To simulate the dynamics of mineral nutrients in rice, we have to simulate not only water flows in the xylem and phloem but also the transport and distribution of mineral nutrients via transporters. Models have been developed that simulate water flows in the xylem and phloem (Daudet et al., 2002; Lacointe and Minchin, 2008; Lobet et al., 2014; Seki et al., 2015) and transport of mineral nutrients from roots or mineral distribution at nodes (Sakurai et al., 2015; Yamaji et al., 2015). However, no study has been conducted on modeling the dynamics of mineral nutrients by considering both water flow and transporter expression level.

The conceptual characteristics of the model proposed here are as follows: (1) it can simulate the dynamics of a mineral nutrient in a whole rice plant while considering plant morphology (multiple leaves, nodes, and stems); (2) the model can simulate mineral transport from roots and its distribution at nodes; and (3) the model can simulate the control of the expression level of the transporter genes in roots. This concept can also be applied to other mineral nutrients and crops if the experimental data on the absorption and distribution of the target mineral nutrients can be obtained.

In the present study, we assumed that three mechanisms control transporter expression levels. The first mechanism is accumulation control, in which a signaling substance is generated in response to Si concentration in leaf cells and is then transported to roots through phloem sap flow. The model based on this mechanism reproduces the experimental data to some extent: the Lsi1 expression level gradually decreases after root exposure to Si solution of sufficient concentration (Figure 5). The best parameter sets that agree with the empirical data were dec = 0.05, slp_c = 0.01 and dec = 0.1, slp_c = 0.02 (Figure 5). For all parameter sets, the estimated Si concentration in xylem sap was nearly zero during the night because a low transpiration rate at night decreases the differences between water potentials of hydraulic nodes. Interestingly, the concentration of the signaling substance in the leaf xylem showed diurnal variation (Figure 3C) despite the steady increase in the generation rate of the signaling substance in response to the Si concentration in leaf cells (Figure 3B). This phenomenon may be related to the dynamics of phloem sap. At dawn, the difference in the potential between hydraulic nodes in phloem became small (Supplementary Figure 7) because of the depletion of starch in leaves by this time. Therefore, the flow of phloem sap from top to bottom decreased and the signaling substance remained in the leaves. In the root, the concentration of the signaling substance decreased at dawn (Figure 4). The diurnal variation in the transport rate of the signaling substance generated the diurnal variation in the expression level of the transporter genes (Figure 5).

Under the assumption of shortage control, the expression level of the transporter genes gradually decreased, as under the assumption of accumulation control, but local minima were reached at dawn (Figure 6). The downward convex curve of the expression level under shortage control may be attributable to the mechanisms that determine how fast the expression is suppressed. Under accumulation control, the decrease in the suppression rate depends mainly on the rate of generation of the signaling substance in leaves, whereas under shortage control, it depends mainly on the rate of decay of the signaling substance. The convex curve of the expression level under shortage control appears to fit the data reported by Yamaji and Ma (2011). Therefore, shortage control may be a more likely mechanism for the control of Si dynamics than accumulation control from the aspect of the shape of the curve (but see below).

Simulation under the assumption of water stress control shows diurnal variation of the expression level of the transporter genes (Figure 7). Local minima were reached around midnight. All parameter sets produced similar expression cycles, but the amplitudes differed depending on slp parameter values. The parameter set of dec = 0.2, slp = 0.2 fit well the experimental data of Yamaji and Ma (2007), which show a decrease in the Lsi1 expression level at midnight to one-third of that at daytime.

Why does rice have a control system that generates the diurnal pattern of the expression level of the transporter genes? To answer this question, we compared the investment efficiency at different parameter values. A decrease in the transporter expression level during the night decreased the relative investment into the expression of the transporter genes (Figure 8A). As a result, the investment efficiency was highest in the simulation that had the largest amplitude of the diurnal variation of expression (Figure 8B) because of the difference in the transpiration rates between day and night. During the day, the transpiration rate is high, xylem sap flow is large, and Si absorbed in the root is efficiently transported to the upper tissues. During the night, the transpiration rate is low, xylem sap flow is small, and Si is not efficiently transported. In other words, when a conveyor belt is moving rapidly, many pieces of baggage can be loaded, but loading many pieces of baggage on a slow conveyor belt is not a good strategy. Interestingly, the increasing rate of the Si concentrations in the tissue cells of the roots during the night were higher than during the day (Supplementary Figure 8). It is because “many pieces of baggage” fell from the slow conveyor. As the results, the average Si concentration of all hydraulic nodes constantly increased even during the night to some extent (Supplementary Figure 9). This result is consistent with the previous experimental studies in which the rate of Si uptake did not slow down during the night (Ma and Takahashi, 2002).

We also investigated the investment efficiencies of accumulation control and shortage control. Under accumulation control, the investment efficiencies do not change greatly among parameter sets (Supplementary Figure 10A). Under shortage control, on the other hand, they were greatly decreased at all settings (Supplementary Figure 10B), perhaps because the level of expression of the transporters decreased during daytime rather than nighttime. Therefore, accumulation control may be preferable for rice from the aspect of investment efficiency. Although, it was reported that dehydration stress decreases the expression of Lsi1 and Lsi2 via ABA in root (Yamaji and Ma, 2007), the mechanism that gradually decreases the expression of transporter genes is not well-understood. Evaluating whether accumulation control or shortage control is the actual mechanism in rice is a task for future study.

The transpiration rate during the night used for the present simulation setting (10% of the daytime transpiration rate) may be large from the actual night-time transpiration rate. However, if the actual transpiration rate during the night is lower than 10% (e.g., Nakano et al., 2010), the conclusion discussed above would not change. It is because lower transpiration rate during the night should slow the xylem sap flow.

To confirm the result, we simulated the model with field data and found a similar diurnal pattern of transporter expression (Figure 9). The investment efficiency was highest when the model was simulated with the most sensitive (large-amplitude) parameters (dec = 0.2, slp = 0.2).

A previous study suggested that the localization and polarity of transporters observed in rice roots provide highest investment efficiency among all possible patterns evaluated (Sakurai et al., 2015). The present study suggests that rice maximizes the investment efficiency in terms of not only the spatial pattern but also the temporal pattern. A gradual decrease in the expression level of Si transporter genes in response to Si concentration in leaf cells might be the mechanism that increases investment efficiency. In rice, many positive effects of Si have been reported with no detectable negative effects of excess Si intake (Ma and Takahashi, 2002). However, the control of the transporter expression level in response to Si concentration in shoot should improve the efficiency of resource allocation.

In the current model, the processes of Si transport in roots and distribution in nodes are simplified. Including more detailed processes will be needed if the aim is to focus on the dynamics of Si at finer scales, such as the dynamics inside and outside of the cell membrane or the localization and polarity of transporters. However, as the current model was designed to describe the dynamics of Si at the whole-plant scale, its degree of simplicity is appropriate. Moreover, the photosynthate dynamics modeled in this study would be a general pattern of plants and does not include characteristic partitioning processes of carbohydrates found in grasses. Grasses store carbohydrates in mainly stem tissue when carbohydrates from the source is greater than whole plant demand (Slewinski, 2012). However, the non-structural carbohydrates in the stem is mainly expended during reproductive growth period (Slewinski, 2012) and leaves would be the main source of carbohydrates during daytime and nighttime in rice (Eom et al., 2012). Therefore, it could be assumed that the downward transport of sucrose is predominant during nighttime at least in the early vegetative period. This was the case of our simulation in the present study. If the model is applied for the Si dynamics during the reproductive period, it might have to be modified so as to include the effect of the storage.

In the current study, the model structure and values of resistance may be oversimplified. The purpose of this study was to propose a new model to investigate qualitatively why rice controls the expression of Si transporter genes. For quantitative understanding of mineral transport, more realistic structure and resistance of water flow values should be reflected in the model, which would be require a large amount of additional experimental data.

We developed a new model that simulates the dynamics of Si in a whole rice plant by considering Si transport in the roots, its distribution at the nodes, and the control of the expression level of Si transporter genes by a signaling substances. The model reproduced a gradual decrease and diurnal variation of the expression level of the transporter genes observed by Yamaji and Ma (2007, 2011). Our modeling suggests that a considerable reduction in the expression level of Si transporter genes during the night increases investment efficiency (the amount of Si accumulated in top leaf divided by the total expression level of Si transporter genes). Our study suggests that rice has a system that maximizes the investment efficiency for Si uptake in terms of not only the spatial pattern (Sakurai et al., 2015) but also the temporal pattern.

GS, NY, NM, MY, and JFM designed the study. GS performed the simulations. KO measured field data. All authors contributed to drafting the paper.