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Edited by: Jan Kofod Schjoerring, University of Copenhagen, Denmark

Reviewed by: Miroslav T. Nikolic, University of Belgrade, Serbia; Motohide Seki, Kyushu University, Japan

*Correspondence: Gen Sakurai

This article was submitted to Plant Nutrition, a section of the journal Frontiers in Plant Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Silicon is the second most abundant element in soils and is beneficial for plant growth. Although, the localizations and polarities of rice Si transporters have been elucidated, the mechanisms that control the expression of Si transporter genes and the functional reasons for controlling expression are not well-understood. We developed a new model that simulates the dynamics of Si in the whole plant in rice by considering Si transport in the roots, distribution at the nodes, and signaling substances controlling transporter gene expression. To investigate the functional reason for the diurnal variation of the expression level, we compared investment efficiencies (the amount of Si accumulated in the upper leaf divided by the total expression level of Si transporter genes) at different model settings. The model reproduced the gradual decrease and diurnal variation of the expression level of the transporter genes observed by previous experimental studies. The results of simulation experiments showed that a considerable reduction in the expression of Si transporter genes during the night increases investment efficiency. Our study suggests that rice has a system that maximizes the investment efficiency of Si uptake.

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.,

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.,

For water flow in plants, the models using an analogy with an electric circuit (Landsberg and Fowkes,

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,

In recent mathematical models of Si transport in both the root and node of rice (Sakurai et al.,

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.

To consider the dynamics of water, sucrose, starch, Si, and the signals that control the expression level of Si transporter genes at the whole-plant level, we divided the whole plant into multiple points and connected them like in an electric circuit (Figure

Schematic diagram of the model structure. The model is composed of a phloem network and a xylem network. The two networks are combined at each hydraulic node.

The flow of water and sucrose in the xylem and phloem was calculated following the model proposed by Daudet et al. (

where _{W(i,j)X} is axial water flow in the xylem from the hydraulic node _{X(i)} and Ψ_{X(j)} is xylem water potential at the hydraulic node _{X(i,j)} is xylem flow resistance (Daudet et al., _{W(i)Lat}) is described as:

where Ψ_{X(i)} is water potential in the xylem, Ψ_{P(i)} is water potential in the phloem, and _{Lat(i)} is the sum of the apoplastic pathway resistance between the xylem and phloem (Daudet et al., _{W(i,j)P}] is described as:

where _{P(i)} and _{P(j)} is hydraulic (mainly turgor) pressure in the phloem at the hydraulic node _{P(i,j)} is phloem flow resistance. Because gravity can be ignored when calculating water flow on a small scale, the following equation holds (Daudet et al.,

where Ψ_{P(i)} is water potential in the phloem and Π_{i} is osmotic potential. The latter can be described as:

where _{i} is absolute temperature, and _{S(i)} is sucrose concentration. The axial phloem solute flow [_{S(i,j)}] is described as:

where _{W(i,j)P} is the axial water flow in the phloem. Because the purpose of the model developed in this study was to estimate the dynamics of mineral transport rather than sucrose flow, we ignored lateral sucrose flow for simplicity.

The flow of water should be conserved at any hydraulic node in the xylem and phloem. Therefore, the following equation should hold:

where _{W} is the water flow from the target hydraulic node to the connected nodes. As this equation should hold at any node, we can estimate water potential at each hydraulic node at any time point by solving simultaneous equations under an appropriate boundary condition.

To calculate osmotic potential, we need sucrose concentration. In this model, we simply input the photosynthetic and transpiration rates as the boundary condition. Following photosynthesis, starch is synthesized in the leaf. The starch is dehydrated to sucrose and gradually loaded into the phloem, which is conveyed by the flow in the phloem, which follows the hydraulic pressure. The models of the dynamics of starch and sucrose in leaves and other tissues, which are similar to that of Daudet et al. (

To understand the dynamics of Si in the whole shoot, we developed a simple two-compartment model that emulates the transport of Si in root. If we assume the compartmentation of the root cortex between the external solution (soil) and root stele, then the flow of Si from external solution (soil) to the cortex can be described as:

where _{M(o:c)} is the flow of Si from external solution (soil) to the cortex, α regulates the expression level of the transporter (from 0 to 1), _{exo} is the transportation capability at the maximum expression level in exodermal cells, _{M:out} and _{M:cor} are Si concentrations in external solution and the cortex, respectively, and _{cm} is the permeability of the cell membrane (Sakurai et al.,

where _{M(c:s)} is the flow of Si from the cortex to the stele, _{end} is the transportation capability at the maximum expression level in the endodermis, _{M(nr)} is the Si concentration in the stele (i.e., in the hydraulic node of the root), and

Note that both _{exo} (Equation 8) and _{end} (Equation 9) include the activity of both Lsi1 and Lsi2. The change of Si concentration can be described as:

where _{cor} is the tissue volume of the cortex (assumed to be 1 ml for simplicity), _{M(nr,nr−1)} is the flow of Si from the root to the hydraulic node above the root, and _{st(i)} is the tissue volume of the sieve tube.

We assumed that Si absorbed in the root is transported with the flow of water only in the xylem. Therefore, the following equation holds for any node:

where _{M(i,j)} is the axial Si flow in the xylem and _{M(i)} is Si concentration. The model assumes only transpiration as the force driving the water in the xylem; it does not consider the case when _{W(i,j)X} is negative.

Using a diffusion equation for Si transport between EVB and DVB, we previously revealed the importance of the apoplastic barrier at the bundle sheath cells and suggested that transporters generate large differences in Si concentration between DVB and EVB to enable rice to transfer sufficient Si upward (Yamaji et al.,

where _{M(i,j)} and _{M(i,k)} are Si flow, _{W(i,j)} and _{W(i,k)} are water flow, _{M}(_{DVB} and _{M(i)EVB} are Si concentrations in DVB and EVB, respectively, and ρ_{i} determines how Si concentration increases in DVB. We assume that hydraulic node _{i}.

We assumed that Si is unloaded in each tissue according to the following equation:

where _{M(i)unload} is Si flow from the xylem to tissue cells, _{M:unload} is a parameter, and _{con(i)} is the tissue volume of the conduit. The Si concentration in tissue cells, _{M(i)cyt}, is calculated as:

where _{cyt(i)} is the volume of the cytoplasm in the tissue.

Lsi2 expression is decreased by high Si accumulation in the shoot through an unknown signal from shoots to roots (Yamaji and Ma,

where _{c}, _{J}, and _{r} are parameters that determine the generation rate of signal in response to the target factor, _{M(i)cyt} is Si concentration in leaf cells, and

where _{R(i)} is signal concentration. We assumed that the generated signal is transferred via the phloem water flow. When the signal reaches the root, the expression level of the Si transporter is regulated according to the signal concentration as follows:

where α is a factor that regulates transporter expression level, _{R(nr,t)} is signal concentration at time

We set the values of transpiration rate, photosynthesis rate, and temperature as input data. We set the standard transpiration rate to 0.4 ml cm^{−2} day^{−1} (Kuwagata et al., ^{−2} s^{−1}. We assumed (i) 10-h night, (ii) 2-h peaks of photosynthesis and transpiration (Figure

Time series of transpiration rate and photosynthesis rate that was used as a boundary condition in simulation.

To estimate the values of _{exo}, _{end}, and _{cm}, we used time-series Si concentrations in xylem sap of 1-month-old seedlings exposed to 1.0 mM Si solution and measured every 5 min (Sakurai et al.,

where _{t} is the standard deviation of the error distribution at time _{M(nr),t}(θ) is the Si concentration in xylem sap at time _{M(nr),t} is the observed Si concentration. We assumed that σ_{t} is equivalent to the standard deviation of the observed data. The initial setting for _{M:cor} (Equation 8) was 0.0 mM. The observed time-series data for xylem sap and estimated data are shown in Supplementary Figure

We estimated sucrose dynamics under several parameter settings. The estimated parameter values were 0.1, 0.15, and 0.2 for k4 and 2.0e-5, 4.0e-5, 6.0e-5, 8.0e-5, 10.0e-5, 12.0e-5, 14.0e-5, 16.0e-5, 18.0e-5, and 20.0e-5 for k1 (see Supplementary Information for k1 and k4). The parameter set where sucrose has the stable cyclic dynamics (k1 = 8.0e-5 and k4 = 0.1) was used for the simulation experiment (Supplementary Figure

To estimate Si dynamics, we used Equations (19), (20), or (21). In each case, we simulated Si dynamics with several parameter sets for

To evaluate investment efficiency, we defined it as:

where _{M(i,t)unload} is Si flow from the xylem to the cells of top leaf at time _{t} is a factor that regulates transporter expression level at time

Finally, we simulated Si dynamics under the natural environmental condition. In the above simulation setting, the artificial pattern of transpiration rate and photosynthesis rate were used (Figure _{2} Exchange) but also LAI (Leaf Area Index) were measured. We used LE and NEE for the estimation of transpiration and photosynthesis rates, respectively, assuming that the effects of evaporation from the water surface and heterotrophic respiration from the soil on the observed fluxes were negligible in our analysis. We used the data during day 33 and 35 after transplanting.

Using Equation (19), we simulated the model with several parameter sets (0.05, 0.1, and 0.2 for _{c}). During the day, Si concentration in the xylem of the top leaf rapidly increased and then decreased (Figure _{c} values (rapid generation of the signaling substance). Si concentration in leaf cells gradually increased, and the differences among parameter sets became apparent after 24 h (Figure _{c} = 0.02, and was only about half of that at _{c} = 0.005. The pattern was the same in the lowest leaf (Supplementary Figure _{c} = 0.005, and was about one tenth of that at _{c} = 0.02) for the roots. The expression level of the transporter genes in roots gradually decreased with time (Figure _{c} = 0.01; _{c} = 0.02) fit best the observed expression levels (mean mRNA levels) of Lsi1 measured using real-time RT-PCR by Yamaji and Ma (

Si concentration in xylem sap _{c} in Supplementary Table

Signal level in xylem sap of the root simulated with multiple parameter sets. Parameter _{c} in Supplementary Table

Time series of the expression levels of Si transporter genes simulated with multiple parameter sets under the accumulation control assumption. Parameter _{c} in Supplementary Table

Using Equation (20) and the same parameter sets, we simulated the expression level of the transporter genes in roots (Figure

Time series of the expression levels of Si transporter genes simulated with multiple parameter sets under the shortage control assumption. Parameter _{r} in Supplementary Table

Using Equation (21) and the same parameter sets, we simulated the dynamics of Si concentration in the xylem (Supplementary Figure

Time series of the expression level of Si transporter genes simulated with multiple parameter sets under the water stress control assumption. Parameter _{J} in Supplementary Table

We investigated the reason why the expression level of the transporter genes shows diurnal variation in the point of view of investment efficiency. We used _{J} = 0.05 (low sensitivity of Si transporter expression to water stress), 0.1 (intermediate), or 0.2 (high) under the water stress control assumption (see Figure

Ratio of the investment during the night to that during the day for each parameter set _{J} = 0.0), low sensitivity (_{J} = 0.05), intermediate sensitivity (_{J} = 0.1), and high sensitivity (_{J} = 0.2) under the water stress control assumption (see Figure

Patterns of photosynthesis and transpiration rates were similar between the artificial input data and empirical data (compare Figures

Time-series of measured transpiration and photosynthesis rates

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.,

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 _{c} = 0.01 and _{c} = 0.02 (Figure

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

Simulation under the assumption of water stress control shows diurnal variation of the expression level of the transporter genes (Figure

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

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

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.,

To confirm the result, we simulated the model with field data and found a similar diurnal pattern of transporter expression (Figure

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.,

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,

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 (

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

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at: