Threshold transitions of 15 N fluxes reveal state-dependent root conductance and flux partitioning within steady-state nitrate isotherms

1 Introduction

Nitrogen fertilization has been central to the success of modern agriculture, yet current strategies largely rely on empirical rules that treat roots as black boxes converting soil nitrate into plant nitrogen. The pioneering work of Epstein and his team using nutrient ion tracers revealed ionic isotherms with an important result whose significance was only partially understood (Epstein, 1966; 1976). The isotherm data were interpreted as evidence of phenomenological thermodynamic bifurcation, or dual kinetics, in ion flux through the root. Indeed, above an external ion concentration of 0.5–1 mM, nutrient ion uptake often increases two-to threefold (Siddiqui et al., 1990; Crawford and Glass, 1998; Faure-Rabasse et al., 2002). This “breakpoint” forms the basis of the canonical HATS–LATS (High and Low Affinity Transport Systems) interpretation, which is based on a two-compartment (soil-root) model with transporters inserted in a single membrane under an enzyme-substrate analogy. Despite decades of tracer studies and kinetic analyses, the traditional HATS-LATS model of nitrate uptake has remained remarkably robust. While useful, this framework is based on several assumptions: whole-root influx behaves like an enzyme-substrate reaction, Vm and Km are intrinsic properties of the uptake machinery, and biphasic isotherms primarily reveal two underlying and distinct transport systems.

Recent molecular, biophysical, and modeling studies have challenged these assumptions on several fronts. Moving across spatial and temporal scales - from whole roots to epidermal cells (top-down approach) or vice versa (bottom-up approach) - to elucidate the catalytic machinery behind the isotherm and the origin of its apparent phenomenological thermodynamic bifurcation is far from straightforward. Both approaches require numerous structural and functional approximations and implicit assumptions that are often overlooked. Furthermore, interpreting Michaelis-Menten-like (MM-like) curves using a hyperbolic fit assumes that plant nutrient responses are smooth, continuous, and saturating and that a small set of parameters can capture the system’s causal organization. However, root systems exhibit threshold-like transitions, discontinuities, and history dependence (hysteresis) across scales that hyperbolic forms cannot fully capture. In such systems, a good hyperbolic fit can coexist with an incorrect mechanistic interpretation. Therefore, we propose reinterpreting the “apparent” kinetic parameters (Vm and Km) extracted from curve fitting as emergent readouts of state-dependent conductance and flux partitioning rather than as intrinsic constants. We focus on nitrate as a case study because isotope flux analyses, genetics and physiology repeatedly point to submillimolar concentration ranges where the balance between storage-dominated and export-dominated regimes can shift, even though recent ¹⁵N datasets under homogeneous versus heterogeneous supply indicate that the exact position and even the sharpness of such breakpoints are strongly context-dependent.

The following sections reveal discrepancies between macroscopic observations and microscopic mechanisms (Section 2) and call into question the reliability of apparent kinetic parameters Vm _*app* and Km _*app* as proxies for intrinsic transporter properties (Section 3). Section 4 demonstrates how introducing spatiotemporal plasticity into nitrogen uptake models can provide alternatives based on multi-compartmental root nitrogen flow resistance. The final three sections examine the dynamic mechanisms and critical transition thresholds governing the uptake of ions and water by roots. Section 5 first examines the search for stationary points and threshold transitions in the behavior of ¹⁵N fluxes from steady-state isotherms provided by three experimental treatments: one cotyledon ablation and homogeneous or heterogeneous nitrate supply. The examination uses a three-compartment model to analyze the effects of the treatments. It also explores how these treatments impact the robustness of the transition threshold. Section 6 reviews the literature on which biological components, such as transporters and/or channels, may be involved in threshold transitions. This section also introduces the “varistor metaphor.” Lastly, Section 7 confirms the observed hysteresis behavior of nitrate and water uptake in the literature after inducing and de-inducing plant roots with nitrate. This section aims to highlight how root architecture, radial versus longitudinal resistances, and flux partitioning can jointly shape the nonlinear behavior of nitrate isotherms. The viewpoint points to nonlinear dynamical systems theory as a natural framework for future, more formal analyses of the underlying dynamics. Additionally, it outlines testable hypotheses for future ¹⁵N tracer experiments and modeling work.

2 Ions influx rate: a missed date between plant physiologists and biophysicists

2.1 Use of ion tracers and establishment of ion isotherms

Ion uptake by root epidermal cell transporters is typically assessed using radioactive or stable isotopes in hydroponic solutions under controlled conditions (Figure 1). Influx rates are measured over short periods (5–10 min) in response to varying external ion concentrations (spanning four orders of magnitude) and expressed as μmol per hour per unit of root mass or length. Studies often use entire root systems of young plants under uniform ion supply, isothermal conditions (20–25 °C), and controlled light.

Early models, developed in the 1960s by E. Epstein’s team, used radioactive tracers like ⁸⁶Rb and ⁴²K to study K⁺ uptake in barley (Epstein and Hagen, 1952; Epstein, 1953). The resulting rectangular hyperbolic curves supported an enzyme-like carrier model, inspired by enzyme kinetics theory. Two parameters Vm (maximum velocity) and Km (affinity)- which depend on the external concentration of ions (C), describe the uptake curve using a MM-like according to the following formula:

V C = C × V m / K m + C (1)

This model (Equation 1) assumes the root behaves like a single purified enzyme, with soil and root separated by a transporter-rich membrane. However, there is no evidence that whole root systems follow pure enzymatic kinetics. Despite this, the analogy has prevailed for decades, often blurring the distinction between the macroscopic behavior of roots and the microscopic enzymatic properties of transporters in epidermal cell membranes.

2.2 Biphasic behavior of nitrate isotherms: high- and low-affinity transport systems

The early concept of root catalytic mechanisms (Mechanisms I and II) evolved into two transport systems: HATS (High Affinity Transport Systems) and LATS (Low Affinity Transport Systems). This terminology change recognized the complexity of root uptake, which involves multiple transporters rather than a single carrier (Jackson et al., 1973; Borstlap, 1983; Lee and Drew, 1986; Siddiqi et al., 1989; 1990). Ion isotherms for NO₃ ⁻, K⁺, NH₄ ⁺, Cl⁻, SO₄ ^2-, and PO₄ ^3- often display two phases -dual arcs or an arc followed by a linear segment (Thellier, 1973; Siddiqi et al., 1990; Peuke and Kaiser, 1996). This supported an additive, enzyme-like model where HATS and LATS function together.

Both systems are further classified as constitutive or inducible (Figure 2A). The enzyme-substrate model divides nitrate uptake into three components: constitutive HATS (cHATS), inducible HATS (iHATS), and constitutive LATS (cLATS). This model assumes that macroscopic root uptake reflects the microscopic enzymatic properties of individual transporters. It has informed the use of tracers like ¹⁵N and ¹³N to study nitrate uptake under diverse conditions, such as nitrogen starvation, circadian cycles, sugar activation, and inhibition by NH₄ ⁺ or amino acids. Figure 2B shows how ¹⁵NO₃ ⁻ uptake rates at low and high concentrations are typically analyzed from isotherms to interpret data.

2.3 Correlation between nitrate transporter gene expression and nitrate uptake activity

The connection between ion uptake and gene expression became clear in the 1980s–1990s with the discovery of nitrate transporter genes in Arabidopsis. NRT1.1 (formerly CHL1/NPF6.3) was cloned from chlorate-resistant mutants (Tsay, 1993), while NRT2.1 was first identified in Aspergillus and Chlamydomonas before its cloning in Arabidopsis (Unkles et al., 1991; Quesada et al., 1994; Filleur and Daniel-Vedele, 1999). Expression of these genes aligns with nitrate uptake rates at both low and high external concentrations, confirmed by tracer influx studies using ¹⁵N and ¹³N (Lejay et al., 1999; Okamoto et al., 2003). Mutant analyses further established NRT2’s role in HATS and LATS (Filleur et al., 2001; Li et al., 2007) and NRT1’s role in LATS (Tsay, 1993), directly linking gene expression to isotherm components.

2.4 Expression pattern of N-related genes: a misleading analogy

The primary nitrate response triggers rapid transcription of N-related genes, peaking 30–45 min after nitrate exposure (Redinbaugh and Campbell, 1991; Hu et al., 2009). AtNRT2.1 and AtNRT1.1 show biphasic transcription patterns similar to nitrate uptake. However, linking transcription patterns directly to uptake activity is misleading, as gene expression reflects regulatory processes, not transport rates or enzyme kinetics. Extracting Vm and Km from transcriptional data (Hu et al., 2009) is biologically invalid, especially for Km, conflating uptake measurements with gene expression.kinetics represent total nitrate uptake ignores the diverse, non-redundant roles of other transporters (e.g., NRT2.2, NRT2.4, NRT2.5, NRT1.2). Transcriptomic studies at tissue and single-cell levels refute this oversimplification (Okamoto et al., 2003; Lhamo and Luan, 2021). Additionally, why does NRT1.1, a nitrate sensor involved in primary responses, exhibit similar expression patterns to NRT2.1? The cpk8 mutant reduces both genes’ transcript levels at high nitrate (Hu et al., 2009), supporting the theory that Ca²⁺-driven CIPK signaling regulates their transcription across concentrations. This challenges the idea of NRT1.1 as the sole direct nitrate sensor in primary responses (Hu et al., 2009; Ho et al., 2009; Wen and Kaiser, 2018).

3 Are macroscopic kinetic parameters (Vm _*app* and Km _*app*) reliable proxies for microscopic ion transporter parameters (Vm and Km)?

To date, five different experimental approaches have been carried out which do not support this assertion.

3.1 Vm _*app* and Km _*app* values are highly context-dependent

The Vm _*app* and Km _*app* values from ion isotherms are unstable, varying with external factors like light, temperature, plant age, and time of day, as well as internal nitrogen (N) status. For instance, nitrate-starved or pre-induced plants show significant changes in maximum uptake rates due to shifts in gene expression and post-translational regulation (Siddiqui et al., 1989; Okamoto et al., 2003). According to the equation:

l o n i n f l u x r a t e = t r a n s p o r t e r n u m b e r × a c t i v i t y / r o o t s u r f a c e (2)

The ion influx rate is proportional to the number and activity of transporters and inversely related to root surface area (Equation 2). Vm _*app* increases with transporter abundance (via transcription), while Km _*app* may change due to post-translational modifications like phosphorylation. Unlike controlled enzyme assays, root influx rates in hydroponic cultures depend on dynamic N status and spatiotemporal regulation across root segments. Consequently, isotherm-derived parameters (Equation 1) are unreliable for mechanistic models in field conditions, where nitrate fluctuations over time and soil volume are unpredictable (Malagoli et al., 2004; York et al., 2016).

3.2 Mutant-based deconvolution of nitrate isotherms refutes the one gene - one transport system

Isotherm decomposition using Arabidopsis mutants showed that uptake phases are not controlled by single genes. Vm and Km values from isotherms represent aggregate transporter activities (e.g., Vmapp and Kmapp). In nrt2.1 and nrt2.1-nrt2.2 mutants, low-concentration nitrate influx decreased but persisted, indicating other HATS contributors (Filleur et al., 2001; Li et al., 2007). Similarly, high-concentration influx remained in nrt1.1 mutants, suggesting additional LATS contributors (Liu and Tsay, 2003; Li et al., 2016). Recent studies confirm that at least four NRT2 transporters (NRT2.1, NRT2.2, NRT2.4, NRT2.5) shape low-concentration isotherms (Lezhneva et al., 2014; Ye et al., 2019). NRT1.1/NPF6.3 contributes minimally at low nitrate, disproving its previously proposed high-affinity role (Glass and Kotur, 2013).

Thus, Vm _*app* and Km _*app* are operational parameters reflecting aggregated root activities, not intrinsic and microscopic transporter properties. This has obscured the functional significance of isotherm phenomenological bifurcation at the whole-root level.

3.3 The activities HATS or LATS transporters in unicellular systems: oocytes and yeasts

HATS and LATS activities can be studied in heterologous systems like oocytes and yeasts. In oocytes, NRT1.1/NPF6.3 switches between HATS and LATS depending on its phosphorylation status at the T101 site (Liu and Tsay, 2003; Ho et al., 2009). The maize homologue NPF6.6 shows non-biphasic HATS activity in oocytes (Wen et al., 2017), while other homologues (ZmNPF6.6 and ZmNPF6.4) function as native LATS or convert to saturable HATS via pore mutations (Wen et al., 2017). This complexity highlights the challenge of scaling Km and Vm parameters from whole-root kinetics to cellular processes, as shown by recent in silico membrane-level simulations.

3.4 In silico models undermine the HATS/LATS dichotomy

Recent membrane-level computational models challenge the enzymatic HATS/LATS paradigm. Simulations based on thermodynamic principles suggest that apparent high vs. low affinity may reflect transient states -such as transporter density or H⁺-ATPase activity-rather than distinct transporter types (Dreyer et al., 2017; 2024; Dreyer and Michard, 2020; Britto et al., 2021). Biphasic uptake curves may arise from system-level constraints, questioning the enzyme-substrate analogy. However, these models operate on different scales (ms/s and membrane-level) compared to in situ root measurements (minute/h and whole-root), limiting direct comparability.

3.5 Scale transition: from cells to whole roots

Measurements of ion influx using ¹³NO₃ ⁻ or ¹⁵NO₃ ⁻ on whole roots or tissues obscure spatial heterogeneity across root zones and axes (Figure 1). For Vm _*app* and Km _*app* to reflect microscopic values, all root regions would need to behave uniformly -which they do not. Studies reveal variable nitrate uptake across root segments in both herbaceous and woody plants (Rao et al., 1997; Taylor and Bloom, 1998; Sorgonà et al., 2011; York et al., 2016). Electrophysiological data in maize show peak uptake in the elongation zone, not the tip or base, indicating functional compartmentalization.

Under patchy field conditions, scaling transporter activity to whole-root behavior is unreliable due to compensatory mechanisms between root zones. Thus, Vm _*app* and Km _*app* represent emergent properties influenced by environment and genetics, not stable transporter-level kinetics. While physiologically relevant, their use in modeling field N uptake should move beyond strict enzymatic analogies.

4 Introducing spatiotemporal plasticity in N uptake modeling

Since most N uptake models rely on static Vm _*app* and Km _*app* values (Barber, 1995; Tinker and Nye, 2000; Ma et al., 2008), recent efforts have aimed to integrate spatiotemporal plasticity into modeling to better reflect dynamic environmental and endogenous regulation of transporter.

4.1 Specific uptake kinetic parameters attributed to different root class

York et al. (2013), (2016) created a model to optimize nitrate uptake by assigning distinct Vm (Imax) and Km values to different maize root segments: seminal, lateral, crown, and brace. The model demonstrates how kinetic variation additively and interactively affects shoot growth. Results highlight that adjusting Imax in seminal and lateral roots at various developmental and fertilization stages yields significant gains.

This approach emphasizes that root uptake plasticity arises from segment-level variation, challenging fixed-parameter models. However, it still assumes enzyme-like kinetics at the transporter level. A key unresolved question is whether breeding should target microscopic kinetic parameters (Vm of specific transporters) or broader emergent parameters like Vm _*app*, which encompass transcriptional, metabolic, and physiological factors.

4.2 Dynamic models using flow–force formalism

A thermodynamic model describes substrate flow (J _*j*) across membranes using root system conductance (L _*j*) and energy-coupling potential (π _*j*) (Thellier et al., 2009; Le Deunff and Malagoli, 2014a; Le Deunff and Malagoli, 2014b; Malagoli and Le Deunff, 2014). These parameters quantify whole-root conductance efficiency for ions, without specifying molecular mechanisms (Supplementary Figure S1).

Applied to oilseed rape (Brassica napus), the model integrates external factors (nitrate levels, soil temperature, PAR) and internal factors (developmental stage, circadian rhythms), dynamically adjusting L _*j* and π _*j* throughout growth (Le Deunff et al., 2019). Both models avoid rigid parameter values: one adjusts Imax at the segment level, while the other uses thermodynamic principles to model root conductance in response to plant-environment interactions. However, a key question remains: why do nitrate isotherms consistently show biphasic shapes?

4.3 Why do ion isotherms have a structurally conserved shape?

NO₃ ⁻-ion isotherms consistently exhibit a phenomenological bifurcation threshold, regardless of species or conditions. This pattern likely reflects thermodynamic optimization of ion transport within the root’s radial architecture. As root meristems are histogenic, the root’s radial structure is a highly invariant, optimized unit for water and ion conductance between branching order, suggesting an evolved and optimized design. Therefore, the robustness of isotherm shapes may arise from biophysical constraints, not just enzymatic properties. The bifurcation in ¹⁵N flux likely corresponds to a structural shift in the transport system, where the limiting resistance for ion flow transitions between the xylem (longitudinal) and the root (radial). Addressing this question requires biophysical and macroscopic approaches -such as constructal theory or nonlinear dynamics-rather than traditional enzyme-substrate models (Bejan, 2005; Laurent et al., 2005; Bejan and Lorente, 2010).

5 Reframing ion and water uptake: beyond enzymatic analogies

According to constructal theory, the tree-like geometry of flow systems, such as plant roots, evolves to optimize access to water while minimizing resistance through its architectural design (Bejan, 2005; 2016; Biondini, 2008; Bejan and Lorente, 2010). In other words, the geometry of the flow network is designed to optimize the distribution of imperfections (e.g., resistances), which determine the geometry of the system. In plants, the root structure is a permeable, porous structure in which the radial flow of soil solution (water and ions) toward the central cylinder is impeded by multiple transverse cell layers and apoplastic barriers, while the longitudinal flow along the xylem experiences less resistance. This model is widely used to study and model water transport in plants (Biondini, 2008). This anisotropy implies that nitrate transport throughout the root cannot be described by a single global flux. Rather, the balance between a radial/storage flux component (v _*app*) and a longitudinal/export flux component (v _*out*) varies according to external conditions and internal regulation. In this context, we propose using a three-compartment model rather than a two-compartment model to analyze the dynamics of ¹⁵N fluxes. This justifies considering the incoming root flux (v _*in*) as the sum of two distinct contributions (v _*app* and v _*out*) rather than a single flux resulting from two successive enzymatic processes at root epidermal membrane level.

5.1 From two-to three-compartment root models: classical instantaneous isotherms vs. steady-state, compartmental isotherms

To explain the biphasic shape of ion isotherms, we shift from a two-compartment model (soil and root) to a three-compartment model (soil, root and shoot; see Figures 3A–D). The enzyme-substrate interpretation implicitly assumes a two-compartment system, with constant nutrient ion export to the aerial parts during measurements and no root or xylem storage (Crawford and Glass, 1998). However, ion concentrations in the stele and vacuole can vary dramatically (Herdel et al., 2001; Peuke et al., 2001; Orieux et al., 2018). This model also oversimplifies the locations of nutrient transporters and channels along the root radius (Figures 3A,C).

Flux measurements are often conducted in a “pseudo-equilibrium” or perturbed state (Table 1) to fit biophysical constraints imposed by enzyme-substrate interpretations (Britto and Kronzucker, 2001b; Ricard, 2003; 2006; Cornish-Bawden et al., 2005; Le Deunff et al., 2016; 2019). To avoid oversimplifying transport as occurring at a single membrane and to integrate pleiotropic regulatory effects (e.g., day-night, feedback, and feed-forward regulations), we instead analyze long-term net apparent flux under steady-state conditions (over hours/days). This approach reflects a stabilized physiological state rather than the maximum uptake capacities (instantaneous flux) measured over short periods (5–10 min).

TABLE 1. Comparison of experimental conditions and implicit assumptions between two- and three-compartment models for nitrate uptake isotherms.

Using seedlings grown on agarose gel provides a physically stable root environment closer to natural soil conditions. Diffusion is more controlled, minimizing convection and turbulence around the roots and reducing experimental artifacts (perturbational conditions). This uniform and stable environment improves reproducibility, simplifies handling, and slows ion translocation to the shoots (reduced transpiring conditions), enabling more accurate kinetic measurements necessary for establishing reliable ion isotherms.

By employing steady-state, compartmental isotherms, we interpret isotherms not in terms of isolated transporters but as integrated transport processes that combine metabolic pathways (e.g., reduction/assimilation), membrane transporters, and ion channels (e.g., influx/efflux). The quantities v _*in*, v _*app*, and v _*out* describe an emergent property of these “integrated transport processes” (Figures 3C,D), from which switch-transition in overall conductance (in terms of flow and force) can emerge, modeled by a transition threshold or bottleneck between flows (storage vs. export). This property is inferred from apparent ¹⁵N fluxes at the organ or whole-plant scale, rather than from the activity of any single or several nitrate carriers in root epidermis. In contrast, classical instantaneous isotherms, measured under pseudo-equilibrium conditions, are interpreted in terms of instantaneous flow rates, which determine the kinetic parameters (Km and Vm) of transport systems. However, they obscure downstream processes, including metabolism, vacuolar storage, and xylem translocation.

The overlap of v _*app* and v _*out* captures radial apparent flow from the root surface to the central cylinder and longitudinal apparent flow via the xylem to shoots (Figure 3C). The biphasic v _*in* curve arises from these dynamics (Figures 3C,D). In the original B. napus agarose system, a change in dominance from v _*app* -dominated to v _*out* -dominated behavior was observed in the sub millimolar range and could be modelled as a transition threshold near 0.5 mM external nitrate. We interpret this as a prototypical example of a regime shift from predominantly radial/storage fluxes to predominantly longitudinal/export fluxes in a specific experimental context, rather than as a universal internal constant of the nitrate uptake machinery.

5.2 Dissecting and adjusting the apparent ¹⁵N fluxes: v _*in*, v _*app* and v _*out*

In the three-compartment soil-root-shoot model, the non-linear dynamics of ¹⁵N fluxes between root and shoot were studied in B. napus seedlings under steady-state conditions, exposed to a 400-fold range of external nitrate concentrations. The net ¹⁵NO₃ ⁻ uptake rate (vᵢ _*n*) in roots was decomposed into two flows: apparent ¹⁵N inflow (v _*app*) and translocation flow to shoots (v _{*ou* *t*}) (Figure 3D). vᵢ _*n* represents total ¹⁵N influx, while v _*app* and v _{*ou* *t*} describe partitioning into storage/assimilation and translocation, respectively. This reflects nitrogen compartmentalization and recycling between root and shoot. v _*app* quantifies the apparent retention of ¹⁵N within root tissues, including storage and assimilation processes into organic molecules, forming immobilized (proteins) or mobilizable (amino acids) fractions. v _{*ou* *t*} represents apparent flux of ¹⁵N (nitrate or organic forms) translocated to the shoot via the xylem (Figure 3D). The relationship between v _*app* and v _{*ou* *t*} captures the balance between radial transport within the root and longitudinal transport toward to shoots (Figure 3C). This balance determines the internal allocation dynamics of nitrate and contributes to non-linear behavior of vᵢ _*n* (Figures 3C,D).

At the stationary point (v _*app* = v _{*ou* *t*}), the root ¹⁵N pool is dynamically stable, with no net ¹⁵N accumulation in the root, while vᵢ _*n* remains the sum of both fluxes. Notably, a transition threshold was observed around ∼0.5 mM external nitrate, where v _{*ou* *t*} progressively exceeds v _*app*,. This shift reflects a change in the dominant mode of nitrate handling, from root retention to enhanced translocation, consistent with the patterns observed in the v _*app* /v _*out* ratio (see below). Rather than a purely continuous response, these results suggest a threshold-like reorganization of flux partitioning within the root system.

5.3 Search for stationary point and threshold transition

We analyzed the dynamics of vᵢ _*n*, v _*app* and v _{*ou* *t*} across a wide range of external nitrate concentrations. V _*app* exhibited relatively consistent responses across experimental conditions, while v _*out* showed marked variations depending on the experimental context (Figure 4). Statistical analyses (ANOVA followed by Tukey’s test) confirmed the significant effect of nitrate concentration on v _*app* , v _*out* and v _*in*, although the strength of discrimination between treatments varied among variables and conditions (Supplementary Table S1). Log-log representations revealed a stable region around 0.3–0.5 mM, where v _*app* is approximately equal to v _{*ou* *t*} (v _*app* = v _{*ou* *t*}), as well as a transition threshold near 0.5 mM (Figures 4A–D). Within this region, v _{*ou* *t*} progressively exceeded v _*app* , indicating a shift in the relative contribution of the two fluxes.

5.4 The effects of treatment on the robustness of the bifurcation threshold

This behavior was better reflected in the v _*app* /v _{*ou* *t*} ratio, which decreased sharply within the same concentration range (Figure 5A). Using an ohmic formulation (J = Δµ/R), the v_app/v_out ratio qualitatively indicates the balance between longitudinal and radial resistances (V_app/V_out ∼ R_longitudinal/R_radial). This ratio shows how resistance changes with external nitrate concentration without quantifying absolute values. When V _*app*/V _*out* > 1, longitudinal resistance dominates, and when V _*app*/V _*out* < 1, radial transport limits nitrate movement.To allow for comparison across experimental conditions, the ratio was normalized to its value at 10 mM nitrate (Figure 5B). This normalization was used for visualization purposes only (statistical analyses were performed on the raw data, see Supplementary Table S1). Across all treatments, the ratio decreased with increasing external nitrate concentration, indicating a progressive shift in the relative contributions of v _*app* and v _{*ou* *t*}. Despite some variability between replicates, the transition consistently occurred within a narrow concentration window of ∼0.3–0.5 mM. Under homogeneous conditions, the decrease in the ratio was relatively abrupt. However, under heterogeneous nitrate supply, the decrease in the v _*app* /v _{*ou* *t*} ratio was more gradual and extended over a broader concentration range. In contrast, cotyledon ablation resulted in a more abrupt transition. These patterns suggest that spatial nitrate supply heterogeneity and plant physiological status influence the position and sharpness of the transition in flux partitioning dynamics.

5.5 Temporal robustness of flux partitioning dynamics

To determine if the observed transition in flux partitioning depended on the time scale of the analysis, v _*app*, v _*out* and v _*in* were recalculated over successive 24-h intervals (72–96 h and 96–120 h) based on ¹⁵N accumulation and total root length (Figure 5C). Under steady-state conditions, a transition in the v _*app* /v _{*ou* *t*} ratio around ∼0.5 mM nitrate was consistently observed across these time windows (Figure 5C). These results suggest that the shift in flux partitioning is consistent over time, independent of the integration period’s duration. Similar patterns have been reported in short-term steady-state experiments involving 1.5-h steady-state treatments followed by 5-min measurements of ¹⁵NO₃ ⁻ instantaneous fluxes (Le Deunff et al., 2020), suggesting that these dynamics can be captured across different temporal scales.

6 Candidate components of flux partitioning dynamics

6.1 Could some transporters or channels act as biological “varistors” on the v _*out* flux?

Several transport and signaling modules are plausible contributors to the threshold-like behavior on v _*out* flux inferred from the three-compartment analysis. At the plasma membrane, NRT1.1/NPF6.3 is weakly active at external nitrate concentrations below 0.5 mM (Glass and Kotur, 2013) but can switch to higher-affinity and higher-capacity transport modes depending on its phosphorylation and dimerization state (Liu and Tsay, 2003; Parker and Newstead, 2014; Rashid et al., 2018; 2020). Its histochemical expression at both ends of nitrate transport in epidermal and stele tissues (Guo et al., 2002; 2003) and its role in bidirectional transport during uptake and translocation (Léran et al., 2013) suggest that it may be involved in v _*out* flow. Likewise NRT1.5/NPF7.2 transporter has been shown to mediate nitrate translocation through ¹⁵NO₃ ⁻ tracer studies (Lin et al., 2008; Léran et al., 2013) that it is consistent with its histochemical localization (Guo et al., 2002; 2003; Remans et al., 2006). It also suggests that it may be involved in v_out flow. In parallel, xylem-parenchyma anion channels such as X-QUAC exhibit broad activation windows where modest voltage changes produce large current variation (Köhler and Raschke, 2000; Köhler et al., 2002). X-QUAC functions as a biological varistor, becoming active above a certain voltage threshold and enabling bidirectional nitrate flux depending on the electrochemical gradient (Figures 6A,B). This switch-like behavior is modulated by apoplastic nitrate and positions X-QUAC as a regulatory valve for nitrate partitioning between root retention and xylem translocation (Figure 6B). Furthermore, aquaporins and Ca²⁺-dependent CBL–CIPK signaling can rapidly modulate radial and longitudinal conductance through the NRT1.1 phosphorylation pattern. Together, these elements provide a mechanistic substrate on which system-level, varistor-like shifts in nitrate partitioning could emerge, without implying that any single protein behaves as a perfect electrical varistor. In fact, this candidate list can be used as a starting point for targeted experiments rather than as a closed mechanistic model (Figure 6C).

6.2 Hormonal regulation of radial-longitudinal resistance and v _*out* flux transition threshold

In plants, free indole-3-acetic acid (IAA, auxin) is mainly synthesized in the root and shoot via the tryptophan (Trp)/indolepyruvic acid (IPyA) pathway. Thus, Trp is successively converted to IPyA by TAA1/TAR aminotransferases, and then to IAA by YUCCA monooxygenases. While the concentration of Trp continuously decreases in the shoot, reflecting IAA production linked to nitrate-stimulated cell expansion, a threshold transition occurs in the roots at around 0.5 mM of external nitrate, where the ratio of Trp_root/Trp_shoot ≈ 1 (Figure 7A).). Above this threshold, root cell elongation is inhibited and cotyledon expansion is promoted (Le Ny et al., 2013; Le Deunff et al., 2020). This indicates that, at around 0.5 mM, IAA is involved in the balance between radial and longitudinal resistance, redirecting systemic fluxes towards shoot growth as revealed by the power law (r ² = 0,99) found between the v _*in* /v _*out* ratio of ¹⁵N flux rates and the ratio of tryptophan content in root and shoot tissues (Figure 7B).

At 1 mM K¹⁵NO₃, the ethylene precursor ACC further modulates the partitioning of ¹⁵N between roots and shoots (Figure 7C). Low ACC levels (0.1–1 µM) slightly increased total ¹⁵N uptake and maintained balanced v _*app* /v _*out* ratios, consistent with ethylene-induced enhancement of root hair development and radial conductance (Lemaire et al., 2013). In contrast, 10 µM ACC drastically reduced both ¹⁵N uptake and root length, suggesting that excessive ethylene production reinforces apoplastic barriers and restricts nitrate translocation to the stele (Figure 7C). Therefore, ethylene acts in parallel with auxin to fine-tune the radial/longitudinal resistance ratio, thereby extending hormonal control of systemic nitrate flux stability beyond the transition threshold.

7 Dynamics of nitrate-induced water and nitrate fluxes

Since the macroscopic longitudinal flux of v _*out* ¹⁵N is coupled to water flux, the reversibility (hysteresis) of nitrate-induced effects on both water and nitrate fluxes was further analyzed from literature data.

7.1 Hysteresis in water and nitrate fluxes following abrupt changes in nitrate availability

Since longitudinal nitrate fluxes (v _*out*) are tightly coupled to water fluxes (Le Ny et al., 2013; Leblanc et al., 2013; Tyerman et al., 2017), the reversibility of these fluxes was assessed in order to characterize potential hysteresis effects (Laurent et al., 2005). Root water conductance displays a clear hysteresis in sunflower when external nitrate abruptly shifts (Gloser et al., 2007; Gorska et al., 2008): an increase from 0.2 to 6 mM induces a gradual increase in root water conductance, whereas a reverse shift (6–0.2 mM) triggers a rapid drop (Figure 8A). The lag in root water conductance increase suggests structural reorganization of water pathways, involving the recruitment of new xylem elements (Windt et al., 2006; Schulze-Till et al., 2009) and the aquaporin activation (Li et al., 2016; Chu et al., 2021). In contrast, the rapid decline upon nitrate removal may be due to aquaporin dephosphorylation.

Similar hysteresis occurs in nitrate uptake in Arabidopsis across abrupt 0–1 mM or 1-0 mM transitions (Figure 8B). Notably, nitrate and water fluxes are inversely modulated during induction and de-induction phases (Figures 8A,B), indicating a short-term coordinated mechanism. These dynamic mechanisms could buffer nitrate fluctuations in the xylem, an hypothesis supported by the regulation of root PIPs by AtNRT2.1 (Li et al., 2016), and the varistor-like role of NRT1.1/X-QUAC and NRT1.1-like transporters in nitrate uptake and xylem loading (cf. §V.2).

7.2 Nitrate-induced phase shift in diel water uptake for shoot growth

Although it is known that nitrate enhances root water uptake (Guo et al., 2002; 2007; Gorska et al., 2008; Leblanc et al., 2008; Orieux et al., 2018), the influence of nitrate on stomatal conductance is still debated (Clarkson et al., 2000; Guo et al., 2003). Nevertheless, gravimetric studies (Orieux et al., 2018) demonstrate that nitrate-driven ¹⁵N translocation is associated with osmotic water uptake, supporting shoot growth over diel cycles (Figure 9A). Increasing nitrate from 0.5 to 5 mM shifts the peak water accumulation toward the beginning of the photoperiod (Figure 9B). This shift is accentuated under low transpiration when seedlings act as osmometers under turgor pressure. These findings suggest that nitrate can rapidly alter the partitioning of water and nutrients to modulate shoot growth. However, despite the importance of nitrate regulation in systemic water transport, the underlying signaling pathways remain to be elucidated.

8 Discussion

The combined analysis of v _*app*, v _*out* and their ratio reveals that nitrate responses in B. napus cannot be fully described by a simple continuous and saturating uptake model. While statistical analyses (ANOVA followed by Tukey’s test) confirmed significant treatment effects, they did not fully capture the structure of the response.

In contrast, the v _*app* /v _*out* ratio provided a direct and integrative descriptor of N flux partitioning in the considered compartments. It revealed a sharp decrease within a narrow concentration range (∼0.3–0.5 mM) and indicated a transition from a root retention-dominated regime to a translocation-dominated regime.

This transition was not associated with changes in apparent v _*app* flux, which remained relatively stable (Figure 4), but rather with strong modulation of v _*out* translocation flux. This suggests that regulation of nitrate allocation occurred predominantly at the level of export toward the shoot.

A central question raised by these observations is how local transport processes operating at the cellular level give rise to such a coordinated response at the whole-root scale.

In this context, the biphasic behavior observed in steady-state nitrate isotherms may not be fully explained by the classical interpretation based on two distinct uptake systems inserted in the plasma membrane of root epidermis (HATS/LATS paradigm). Instead, this may reflects a shift in the functional regime of nitrate flux partitioning within an integrated system. This perspective builds on the electrokinetic framework proposed by Thellier (1973), which described ion absorption as a non-linear, force-flow coupled process governed by thermodynamic constraints. However, it extends his framework by incorporating the spatial organization of the root system.

The present study suggests that the threshold transition observed in ¹⁵N steady-state isotherms arises from the interplay between transport kinetics and the structural organization of root tissues, in particular the balance between radial processes (uptake, storage, assimilation) and longitudinal transport (xylem export). Within this framework, the condition v _*app* ≈ v _*out* does not represent a static equilibrium, but rather a dynamically maintained balance between competing fluxes. The transition around ∼0.5 mM nitrate reflects a reorganization of this balance, in which translocation progressively dominates over local retention. This interpretation is consistent with the observed robustness of the transition across temporal scales and experimental conditions, including heterogeneous nitrate supply and cotyledon ablation, which respectively modulate the smoothness and abruptness of the response.

Altogether, these findings support a model in which nitrate fluxes are governed by a context-dependent regulatory balance integrating external nitrate availability, root system structure, and whole-plant nitrogen status.

8.1 Nitrate-induced responses depend on physiological and spatial context

A mechanistic model, combining a saturating absorption term (Michaelis–Menten-type) for v _*app* (Equation 3) and a gating-regulated translocation process for v _*out* (Equation 4) that depends on both v _*app* and external nitrate concentration, successfully described the entire dataset (Supplementary Figure S3). In this coupled model, two parameters describe the v _*app* curve and three parameters describe the v _*out* curve, according to the following equations:

v a p p = V m a x / K a + N O 3 − e x t (3)

v o u t = ρ v a p p N O 3 − e x t h / ( K g h + N O 3 − e x t h (4)

with v i n = v a p p + v o u t (5)

According to the plant treatment conditions, the evolution of the model parameters indicates that the shape of the v _*in* isotherm response (Equation 5) is not intrinsic, but rather, results from the context (Supplementary Figure S3). In other words, the results demonstrate that nitrate responses depend not only on the external nitrate concentration, but also on how the signalization is organized spatially. Three modes of functioning emerge: (i) a threshold mode in the presence of homogeneous nitrate input, (ii) a constrained mode during cotyledon ablation, and (iii) a distributed mode in the presence of heterogeneous nitrate input, where the response is gradual and reflects the integration of local signals.

Furthermore, under all conditions, v _*out* consistently shapes v _*in* dynamics. This indicates that absorption alone does not determine system behavior, and that regulation primarily occurs at the level of N internal redistribution. The Hill-type parameter h and the gating parameter ρ govern the shape of the v _*out* response. Under homogeneous conditions, h ∼ 2 and ρ = 5.92; following cotyledon ablation, h = 1.88 and ρ = 3; and under heterogeneous input, where translocation dominates, h = 2.35 and ρ = 7.61. These results identify translocation as a central control point in nitrate flux regulation.

Additionally, the normalized v _*app* /v _*out* ratio shows that cotyledon ablation not only reduces the flux amplitude, but also fundamentally alters the response structure (Figure 5). This leads to a reduction in discriminatory capacity and the emergence of constrained, threshold-like behavior. These results suggest that systemic regulation depends on the integrity of the entire plant and likely involves source-sink coordination. Consistent with this hypothesis, cotyledon ablation resulted in a loss of nitrogen dilution in root and shoot tissues, as demonstrated by the correlations between nitrogen (N) and carbon (C) contents during the ablation treatment (Supplementary Figure S4). Furthermore, contrasting responses between treatments support the idea that the root system is a distributed, functionally integrated sensing network.

8.2 Toward a non-linear, structure-dependent view of nitrate transport

Previous work has proposed non-linear dynamic frameworks to explain biphasic nitrate isotherms, linking them to cooperative behavior of v _*app* and v _*out* fluxes (Le Deunff et al., 2020). Building on this perspective, the present results suggest that the observed transition in nitrate fluxes may reflect threshold-like responses emerging from the interaction between transport kinetics and root structural organization. In this context, certain transport systems, such as NRT1.1/NPF6.3-like or anion channels (X-QUAC), may exhibit behavior analogous to non-linear conductance elements, where flux increases sharply beyond a critical range of external nitrate concentration.

Rather than being solely determined by intrinsic transporter affinity, nitrate uptake dynamics may therefore emerge from the balance between transport processes and structural constraints within the root system. This perspective provides a conceptual framework linking molecular transport properties, tissue-level organization, and whole-plant nitrogen allocation, and may help reinterpret classical models of nitrate uptake in a more integrative manner.

Several recent conceptual models resonate with our viewpoint while addressing different scales of nitrate transport. CATE-based analyses (Compartmental Analysis by Tracer Efflux) have emphasized the remarkable stability of cytosolic nitrate turnover times under fluctuating external supply, attributing this “high fidelity” to compensatory adjustments within the cellular acquisition network, including coordinated changes in influx, efflux, vacuolar transport and reduction (Britto and Kronzucker, 2001a; 2001b; 2003). More recently, membrane-level “ion homeostat” models have shown how non-linear combinations of transporters and channels can maintain near-steady state ion gradients at minimal energetic cost (Dreyer et al., 2024). The proposed varistor-like, three-compartment framework complements, rather than replaces, these approaches. It focuses on how local homeostatic mechanisms embedded in an anisotropic root architecture can produce threshold-like shifts in flux partitioning between root storage and shoot export at the organ scale. At this stage, we view “homeostat” and “varistor” as two compatible organizing metaphors for different levels of description. Formal multiscale modeling will be necessary to determine how cellular turnover dynamics and whole-root flux threshold transition constrain each other in vivo.

8.3 Experimentally testable predictions derived from the varistor model

Steady-state ¹⁵N isotherms performed on seedlings provide new readouts that could help identify key nodes of nitrate root conductance that control homeostasis and translocation. If the threshold transition of v _*out* and v _*in* at approximately 0.5 mM nitrate reflects a fundamental root transport property, the following tests should be prioritized: (i) measuring v _*in*, v _*app*, and v _*out* isotherms in Arabidopsis NRT1.1/NPF6.3 phospho-mutants (e.g., T101A/D), NRT1.4/NPF6.2, and NRT1.5/NPF7.2 mutant lines to assess whether the position or existence of the threshold transition could be altered; (ii) manipulating root hydraulic conductivities through the NRT2.1 mutant line to test whether changes in radial versus longitudinal resistance shift the v _*app*/v _*out* transition; (iii) performing split-root experiments under heterogeneous nitrate supply to examine how local changes in transport and signaling impact whole-plant flux bifurcation and hysteresis; and (iv) using Casparian strip mutants (Roppolo et al., 2011) or vascular tissue mutants (Lefebvre et al., 2011) to examine how disruption of root radial structure shifts the threshold transition. Similarly, mutants with alterations in the CBL-CIPK signaling network (e.g., cipk23, cipk8, cbl1 and cbl9) could be used to test their impact on the threshold transition. These kinases regulate the expression and phosphorylation of NRT2.1/NRT1.1, likely influencing root conductance.

Additionally, variation in the transition threshold may enable modulation of nitrogen use efficiency (NUE) in response to soil water and nitrate conditions. Screening natural populations for variations in threshold concentration could help identify alleles associated with improved NUE under low-nitrogen conditions. These experiments are all essential to determine whether the threshold transition and the “varistor metaphor” reflect a robust organizational principle of nitrate transport, or if other descriptions of the nonlinear organization of fluxes in roots are more appropriate.

9 Conclusion

This viewpoint reinterprets nitrate isotherms using a three-compartment model and a varistor-like metaphor to reveal that threshold transition in root nitrogen flow is an emergent systemic phenomenon arising from the interplay between root architecture (radial vs. longitudinal resistance balance), nonlinear transporter kinetics (e.g., NRT1.1/NPF6.3-like and X-QUAC channel activation) and hormonal signaling (auxin and ethylene-mediated regulation of N flux). The abruptness of this threshold-like shift helps to explain why aligning nitrogen uptake with crop requirements has long been unsuccessful. The proposed framework provides a macroscopic and mechanical explanation for plants’ abrupt response to nitrate concentration thresholds, very different from the enzyme-substrate analogy (e.g., HATS and LATS). This mechanism is pivotal for crop adaptation in the face of challenges posed by the transition to agroecology. Future experimental validations that combine genetics, electrophysiology, and dynamic modeling will refine the model and maximize its potential to improve NUE in crop plants.

10 Materials and methods

10.1 Seedlings culture in agarose gel

The B. napus L. winter oilseed rape seeds (cv. Capitol) used in this study were treated for germination according to Leblanc et al. (2008). Forty-eight hours after sowing in sterile conditions at room temperature, four seedlings were selected based on radicle length (5–6 mm) and transferred to 12 × 12 cm square Petri dishes filled with 50 mL of agar culture medium. The basic medium contained 0.4 mM KH₂PO₄, 0.15 mM K₂HPO₄, 1 mM K₂SO₄, 0.5 mM MgSO₄, 3 mM CaCl₂, 0.2 mM Fe-Na EDTA of macronutrients, and 14 µM H₃BO₃, 5 µM MnSO₄, 3 µM ZnSO₄, 0.7 µM CuSO₄, 0.7 µM (NH₄)₆Mo₇O₂₄, and 0.1 µM CoCl₂ of micronutrients, and it was solidified with 0.8% (w/v) agar (Sigma 1–7,002), pH 6.75. The basal medium was supplemented with KNO₃ as the sole nitrogen source at the indicated concentration for each experiment. The Petri dishes were then half-sealed with adhesive tape and incubated vertically in a growth chamber at 22 °C under a 16/8 light/dark regimen with a photon flux density of 250 μmol m^-2 s^-1.

10.2 ¹⁵NO₃ uptake and isotope analysis

The net uptake of KNO₃ was obtained through homogeneous or heterogeneous labeling with K¹⁵NO₃ (¹⁵N atom percentage: 1% or 5%) via serial sampling at 24-, 48-, 72-, 96-, and 120-h time points. The only changing variable was the KNO₃ concentration, which varied 50-fold from 0.2 to 10 mM. The ¹⁵N that accumulated in the different plant organs was analyzed using isotope ratio mass spectrometry (IRMS). Prior to IRMS analysis, the excised organs of three or four seedlings from each of the four agar plate replicates (N = 4) were weighed for fresh weight at each sampling time. Organs from seedlings on the same agar plate were pooled, then dried at 60 °C for 72 h. The dry tissues were then ground into a fine powder with 5 mm stainless steel beads in an oscillating grinder (Retch Mixer Mill MM301) for 2 minutes. The ¹⁵N accumulated in the roots and shoots was then analyzed using an elemental analyzer (EA 3000 Analyzer, Eurovector, Milan, Italy) coupled with an isotope ratio mass spectrometer (IRMS, Isoprime X, GV Instruments).

10.3 Steady-state nitrate isotherms and compartmental analyses

Steady-state nutritional conditions for absorption and growth are defined as a situation in which net fluxes into and out of root compartments remain stable with respect to external nitrate concentrations. Steady-state conditions are reached when inflows (v _*in* and v _*app*) and outflows (v _*out*) of ¹⁵N to the shoot are no longer altered except by threshold-like transitions caused by a change in the flow dynamic regime. The inflow and outflow rates are determined by the accumulation of ¹⁵N in the root and shoot compartments during the serial sampling period and the total root length of the seedlings on the sampling date. These rates are expressed in units of µmol ¹⁵N per hour per root cm. V _*in* is the velocity of substrate S _*j* uptake into the roots (i.e., the net ¹⁵NO₃ ⁻ uptake rate) and represents the sum of v _*app* and v _*out* (v _*in* = v _*app* + v _*out*), while v _*out* is the velocity of net elimination of substrate S _*j* from the roots (e.g., the net translocation rate of ¹⁵NO₃ ⁻ and ¹⁵N reduced forms). V _*app* flow captures the ¹⁵N stored in root tissues or assimilated into immobilized or mobilizable organic molecules, but it also depends on the release of ¹⁵N by the roots (i.e., efflux). These apparent fluxes reflect nitrogen compartmentalization and recycling between roots and shoots and the overlap of v _*app* and v _{*ou* *t*} captures radial apparent flow from root surface to central cylinder and longitudinal apparent flow via xylem to shoots. Therefore, the steady-state compartmental isotherms obtained cannot be interpreted in terms of isolated transporters derived from instantaneous flow rates. Rather, they can be interpreted in terms of integrated transport processes that combine metabolic pathways (e.g., reduction/assimilation), membrane transporters, and ion channels. Thus, v _*in*, v _*app*, and v _*out* describe emergent properties of the set of “transport processes” from which changes in overall conductance (in terms of flow and force) can emerge. This approach allowed us to identify nonlinear dynamics in ¹⁵N fluxes, across the entire range of external nitrate concentrations and the identification of stationary states and bifurcation thresholds (storage versus export) as previously described by Le Deunff et al., 2020. A stationary point is reached under steady-state conditions with K¹⁵NO₃ availability after a treatment duration when:

d 15 N / dt = v app − v out = 0

To evaluate whether the bifurcation threshold observed in v _*out* and v _*in* can coincides with a change in flow resistance from xylem (longitudinal) to root (radial) flux we analyzed the v _*app*/v _*ou*t ratio of ¹⁵N fluxes to compare different plant treatments (homogeneous and heterogeneous nitrate supply). The goal was to trace the bottleneck between flows (storage versus export) as an integrated systemic property of flux dynamics.

10.4 Experiment 1: dose-responses to homogeneous nitrate supply

To characterize nitrate uptake kinetics, the net uptake of KNO₃ was obtained by homogeneous labeling with K¹⁵NO₃ by using serial sampling at time points of 24 h, 48 h, 72 h, 96 h and 120 h after nitrate treatments. The amounts of ¹⁵N obtained after IRMS analyses were measured on roots or shoots organs of four seedlings mixed together and provides by four agar plates carrying four seedlings (i.e., four replicates, N = 4). Two similar experiments where conducted. For the first experiment, seedlings were treated with KNO₃ concentration varied from 0.05 to 20 mM (i.e., 0.05, 0.2, 0.5, 1, 5, 10 and 20 mM). In second experiment seedlings were treated with KNO₃ concentration varied from 0.2 to 10 mM (i.e., 0.2, 0.5, 1, 2.5, 5 and 10 mM).

10.5 Experiment 2: dose-responses after one cotyledon ablation

Experiment 2 was conducted under conditions similar to those of Experiment 1. However, the net ¹⁵N uptake and accumulation rates in the root and shoot compartments were obtained over 120 h of K¹⁵NO₃ treatment. The KNO₃ concentration varied from 0.2 to 20 mM (i.e., 0.2; 0.5; 1; 2.5; 5; 10 and 20 mM). In one set of seedlings, one cotyledon was removed with sterile scissors after 48 h of sowing (ablation treatment). The other set of seedlings was kept as a control group without cotyledon ablation.

10.6 Experiment 3: dose-responses to heterogeneous nitrate supply

To study localized nitrate responses, we used segmented agar plates that were divided into three compartments, each measuring approximately 4 cm, and separated by parafilm. Two-day-old B. napus seedlings, germinated under homogeneous 0.2 mM KNO₃, were transferred to these plates (3 seedlings/plate) so that their primary roots extended across all segments. The upper and lower segments contained 0.2 mM KNO₃, while the middle segment was supplied with K¹⁵NO₃ (¹⁵N atom percentage: 5%) at concentrations ranging from 0.2 to 10 mM. Total root lengths and ¹⁵N accumulation were measured at 72 h after transfer on four agar plates carrying three seedlings (N = 4).

10.7 Experiment 4: effects of ACC treatment on ¹⁵N flux partitioning

Since ethylene controls root elongation and cell wall biosynthesis, promoting root radial expansion, which results in a thicker epidermis and thinner cortex (Zhang et al., 2026), we used ACC treatments to modify root radial conductance for K¹⁵NO_3. The v _*app* /v _*out* ratio was estimated after 120 h of root treatment with different concentrations of ACC, ranging from 0.1 to 10 μM, on seedlings supplied with 1 mM K¹⁵NO₃ (Lemaire et al., 2013), in order to evaluate the hormonal control of systemic nitrate flux behavior through induced modifications of root radial resistance beyond the bifurcation threshold.

10.8 Morphometric analyses

A transparent slide was placed behind each agar plate, and the daily root elongation of each root seedling treated with a homogeneous or heterogeneous supply was tracked manually with a permanent marker. Then, the slides were analyzed using the WinRHIZO scan system (Regent Instruments, Inc.), and the total root length (primary and lateral roots) in the different root segments and at the different sampling time points was estimated.

10.9 Amino acids analysis by UPLC analysis

Amino acids were extracted, derivatized and analyzed using an ACQUITY UPLC (Ultra Performance Liquid Chromatography) system (Waters Corp.) following the method previously described in Renault et al. (2010) and Le Ny et al. (2013).

10.10 Statistical analyses

All statistical analyses and curve fitting were performed using R (RStudio environment). For each treatment, values are presented as mean ± standard error (SE) of three to four biological replicates. Each replicate corresponded to an independent agar plate containing three to four seedlings. For each experiment, the effect of external nitrate concentration on v _*app* , v _*out* , and v _*in* was assessed using one-way analysis of variance (ANOVA), followed by Tukey’s post hoc test for multiple comparisons. Treatments not sharing a common letter were considered significantly different (P < 0.05). Assumptions of normality and homogeneity of variances were evaluated using Shapiro–Wilk and Levene’s tests, respectively (Supplementary Table S1). Although minor deviations from these assumptions were observed in some cases (notably v _*out* under cotyledon ablation), the balanced experimental design and the known robustness of ANOVA to moderate violations supported the validity of the analyses.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frbis.2026.1677942/full#supplementary-material

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Footnotes

Edited by: Olga Vinogradova, University of Connecticut, United States

Reviewed by: Lars Hendrik Wegner, Foshan University, China

Xiaorong Fan, Nanjing Agricultural University, China