Summary
- MECHA is a novel mathematical model that computes the flow of water through the walls, membranes and plasmodesmata of each individual cell throughout complete root cross-sections, from a minimal set of cell level hydraulic properties and detailed root anatomical descriptions.
- Using the hydraulic anatomical framework of the Zea mays root reveals that hydraulic principles at the cell and root segment scales, derived independently by Katchalsky and Curran [1967] and Fiscus and Kramer [1975], are fully compatible, irrespective of apoplastic barriers leakiness.
- The hydraulic anatomy model accurately predicts empirical root radial permeability (kr) from relatively high cell wall hydraulic conductivity and low plasmodesmatal conductance reported in the literature.
- MECHA brings novel insights into contradictory interpretations of experiments from the literature by quantifying the impact of intercellular spaces, cortical cell permeability and plasmodesmata among others on root kr, and suggests new experiments efficiently addressing questions of root water relations.
- KPD
- single plasmodesma hydraulic conductance
- kr
- root radial hydraulic conductivity
- kw
- cell wall hydraulic conductivity
- Lp
- cell plasma membrane hydraulic conductivity
1 Introduction
Vascular plant roots develop hydrophobic scaffolds (e.g., Casparian strips) diverting the course of water through cell membranes [Von Wangenheim et al., 2017], which filter as much as 60% of land precipitation on its way back to the atmosphere [Oki and Kanae, 2006]. In addition, root hydraulic and anatomical properties affect the plant water status, leaf growth [Caldeira et al., 2014] and crop water use under water deficit [Schoppach et al., 2014]. Yet, our quantitative understanding of these crucial hydraulic properties is hindered by the complex organization and regulation of sub-cellular elements such as aquaporins, which control the cell membrane permeability (LP) [Maurel and Chrispeels, 2001; Chaumont and Tyerman, 2014] and plasmodesmata, which connect adjacent cells protoplasts [Maule, 2008; Sevilem et al., 2013; Brunkard and Zambryski, 2017].
Due to the high hydraulic conductivity of cell walls, water pathways across root cortex are considered to be dominantly “apoplastic” [Steudle and Boyer, 1985]. In the vicinity of hydrophobic scaffolds, water passively crosses cell membranes through aquaporins and reach the symplasm. In cell layers where a hydrophobic suberin lamellae is embedded in the cell wall, such as the endodermis, the only way between neighbour cells is through plasmodesmata [Enstone and Peterson, 2005] which offer an uninterrupted “symplastic” pathway. Even though the resting diameter of plasmodesmata is approximately 50 10−9 m [Bell and Oparka, 2011], they are partly filled by a desmotubular structure, leaving water pathways that are only one order of magnitude wider than aquaporin aperture (2 to 3 10−9 m versus 2 10−10 m) [Terry and Robards, 1987; Tornroth-Horsefield et al., 2006].
A major difference between the principles of water flow within cell walls, through plasmodesmata and across membranes is the semi-permeability of the latter. Membranes exclude specific solutes, thus generating cell turgor from the same principle as the membrane osmometer [Katchalsky and Curran, 1967]. This semi-permeability is quantified as a “reflection coefficient” (σ, dimensionless) varying between 0 (fully permeable) and 1 (fully selective). In the first case, the water pressure difference (Δψp, MPa) alone drives water flow, while in the second case the osmotic potential difference (Δψo, MPa) contributes as additional driver. Such drivers generate water flow (Q, m3s-1) yet rate-limited by the media pore size distribution, summarized as its hydraulic conductance (K, m3MPa-1s-1): Fiscus and Kramer [1975] propose a similar principle that applies to the complex root structure. Their theory assimilates radial water transport to flow across a single semipermeable membrane: where Js is the water flux at root surface (m3 m-2 s-1), kr is the root radial conductivity (m3 m-2 s-1 MPa-1, note that here conductivity stands for conductance per surface area), ψp,s and ψp,x correspond to the water pressures at the root surface and in xylem vessels relative to atmospheric pressure (MPa), σr is the root reflection coefficient (dimensionless), and ψo,s and ψo,x correspond to the osmotic potential at root surface and xylem vessels, respectively (MPa).
Equation (2) is an effective equation as it merges the multiple cell-scale properties between root surface and xylem vessels into simple effective parameters kr and σr. Yet, whether or not this equation holds for any combination of hydraulic properties, anatomy and environment, is still unclear. In particular, systematically different kr values were reported in hydrostatic and osmotic gradient experiments [Steudle and Frensch, 1989]. Such kr differences were assumed to be the consequence of a substantial fully apoplastic pathway whose hydraulic conductivity would increase in hydrostatic pressure gradient experiments due to the water-filling of intercellular spaces [Steudle and Peterson, 1998].
The “root composite transport model” proposed by Steudle and Frensch [1989] considers that water follows pathways in parallel (apoplastic and cell-to-cell, the latter including symplastic and transcellular paths) that have different kr and σr values. Root segment properties were linked to cell scale properties by integrating local conductivities and reflection coefficients using simplistic geometrical and hydraulic assumptions for each pathway [Steudle and Boyer, 1985]. Yet, the composite transport model does not constitute an explicit framework to investigate water pathways across individual cells, and for instance cannot explain why cortical cell Lp is not necessarily correlated to kr [Hachez et al., 2012]. Furthermore, a validated root cross-section model may be used to target specific priority cell-level traits in a breeding context.
In this study, we develop a model of explicit root cross-section hydraulic anatomy (MECHA) built on the scientific community’s understanding of hydraulics at the cell scale, interconnected up to the root segment scale. One would expect that hydraulic theories at both scales should be compatible, though it was never explicitly verified. MECHA provides an explicit framework for testing such hypothesis, as it links quantitative hydraulic properties from the cell to the root segment scale. It also offers new insights to revisit conflicting interpretations of root water transport experimental results.
Firstly, we briefly introduce the model whose equations and parameterization are detailed in Notes S1 and section 5.1. The mathematical method to quantify water composite pathways across cell layers is detailed in Notes S3. Secondly, we check the compatibility of hydraulic theories at the cell and root segment scales. Thirdly, we assess the model predictions using literature data of maize (Zea mays) hydraulic properties at the cell and root segment scales. Eventually, we use MECHA to analyse and reinterpret previous statements concerning (1) the strong dependence of kr to the water filling of cortical intercellular spaces [Steudle and Peterson, 1998], (2) the possible independence between kr and cortical cell permeability [Hachez et al., 2012], and (3) the relative contribution of plasmodesmata to radial flow [Hukin et al., 2002].
2 Model
Root anatomical and cell hydraulic information were combined to build a finite difference model explicitly solving water pathways across individual cells of maize (Zea mays) roots.
The cell hydraulic information includes literature data of cell wall hydraulic conductivity (kw, which is an intrinsic conductivity in units of m2s-1MPa-1), cell plasma membrane permeability (Lp, m s-1MPa-1), the hydraulic conductance of individual plasmodesmata (KPD, m3s-1MPa-1) and their frequency in each root tissue (see label “b” in Fig. 1). The cell geometry and connectivity information was collected from a cross-section image of the primary root of maize (Fig. 1a), vectorized with the program CellSet [Pound et al., 2012] (Fig. 1c). Together, they constituted the explicit root cross-section hydraulic anatomy (Fig. 1e). Further details of the parametrization are located in sections 5.1 and 5.2.
Computing water flow in the cross-section hydraulic anatomy yields flow rates across each individual hydraulic conductance (apoplastic, transmembrane and plasmodesmatal), as well as the distribution of water pressure in the apoplast and the symplast. Details on the water flow equations and solving method are given in Notes S1.
Examples of distributed water flow rates across cell walls, plasma membranes and plasmodesmata for relatively high kw [Steudle and Boyer, 1985] and low KPD [Bret-Harte and Silk, 1994] at various stages of apoplastic barrier deposition are described in section 5.1.
Furthermore, in Notes S3 we propose a mathematical method to quantify water radial pathways from local flow rates, which concentrates the essential flow information into a simpler graph (see panel “g” in Fig. 1).
3 Results
3.1 Hydraulic theories at the cell and root segment scales are compatible
Cell-scale hydraulic principles (Eq. (1)) and associated hydraulic properties (Tab. 2) were applied on a maize root cross-section anatomical layout in order to simulate water fluxes under various xylem and root surface boundary conditions (Tab. 3). Fig. 2a shows that the predicted radial fluxes Js at the root surface is proportional to the water potential difference between the root surface and xylem boundaries (ψp,s − ψp,x +σr(ψo,s − ψo,x)) as in the root segment scale theory of Fiscus and Kramer [1975] (Eq. (2)), with an R2 of 1.0000. The proportionality factor could thus be interpreted as equivalent to kr. Theories across scales therefore appear to be compatible, and this is valid for all combinations of cell scale hydraulic properties and apoplastic barriers leakiness.
Because it displays sensitivity to cell-scale hydraulic properties but not to boundary conditions, the upscaled kr seems to be an intrinsic property of the hydraulic anatomy alone. This result suggests that the empirical observation of an impact of boundary conditions on kr [Steudle and Frensch, 1989; Steudle and Peterson, 1998] are due to changes of cell-scale hydraulic properties (e.g., water-filling of intercellular spaces, or aquaporin gating quickly responding to the root osmotic environment [Hachez et al., 2012; Parent et al., 2009; Aroca et al., 2011]).
The simulations indicate that the additivity of pressure and osmotic potentials (after correction by the reflection coefficient σr) is conserved from the cell to the root segment level. This additivity principle might also be preserved from root segment to whole plant scales, as suggested by experimental and modelling studies [Hamza and Aylmore, 1992; Schröder et al., 2014]. Like in Eq. (2), Js is not affected by alterations of cytosolic osmotic potential (ψo,c, see Tab. 3), even though ψo,c was explicitly included in water flow equations across cell membranes (Eq. (S1)). This means that, when uniform, the effect of ψo,c somehow cancels out in the complete hydraulic network. Although radial gradients of root ψo,c were observed in transpiring maize plants [Rygol et al., 1993], they were not considered here. Their impact on water circulation inside roots will be investigated in a future study.
Models of water flow were not always found fully compatible across scales, as was the case in root water uptake models [Feddes et al., 1978; Couvreur et al., 2014; Javaux et al., 2013], or in hydrological models that rely on effective principles [Pokhrel and Gupta, 2010; Brynjarsdóttir and O’Hagan, 2014; Couvreur et al., 2016]. Lacking key processes or processes too complex to be represented with simple physical equations at either scale might explain an incompatibility. In this case, it seems that principles controlling water flow within roots were well captured, so that root segment hydraulics could be explained from underlying principles at the cell scale. Such compatibility opens avenues for the transfer of hydraulic information between cells and root segments.
3.2 Cell scale hydraulic properties injected in the maize hydraulic anatomy match empirical root segment kr
The radial hydraulic conductivity (kr) was frequently measured on the apical part of maize primary roots (up to 14 cm long) using a root pressure probe. Data from Steudle et al. [1987], Zhu and Steudle [1991], Frensch and Steudle [1989], and Bramley et al. [2007] cover one order of magnitude but mostly meet around 2 10−7 m s-1MPa-1 (see coloured areas in Fig. 2b). Root kr data in mature root zones are rare, likely due to the presence of lateral roots that prevent independent measurement of the principal root segment kr and complicate the attachment of the root to the probe. Doussan et al. [1998b] estimated a whole profile of root kr along the maturation gradient by combining inverse modelling and cumulative flow experimental estimations from Varney and Canny [1993] (see dashed line in Fig. 2b, with development of secondary and tertiary endodermal walls between 15 and 19 cm from the tip, and exodermis development between 45 and 60 cm from the tip). MECHA parameterized with empirical cell-scale hydraulic properties yielded radial conductivities (see slopes in Fig. 2a, or symbols in Fig. 2b) that fell in the empirical range of maize root kr. Note that all symbols were paired, the higher and lower kr respectively corresponding to leaky and impermeable apoplastic barriers.
The similarity of hydraulic properties measured and predicted across scales strengthens the validity of the hydraulic anatomy approach, particularly when considering that the reported kr ranges for maize (overall 2 10−8 to 3 10−7 m s-1MPa-1, see boxes in Fig. 2b) appear as relatively narrow in view of the kr values reported in the literature for other plants. For instance, McElrone et al. [2007] measured kr values from 10−5 to 10−4 m s-1MPa-1 with an ultra low flowmeter in deep fine roots of live oak and gum bumelia, while Zarebanadkouki et al. [2016] obtained kr values of 10−6 m s-1MPa-1 in undisturbed lupine primary roots, by combining deuterium tracing, inverse modelling and analytical root hydraulic functions developed by Meunier et al. [2017].
The simulated kr profiles along the root maturity gradient (from black to light grey symbols) display a clear dependence on the cell wall conductivity (kw). The kr profile simulated with the low kw value proposed by Tyree [1968] (stars and circles in Fig. 2b) is indeed quite uniform compared to that obtained with higher kw values measured by Steudle and Boyer [1985]. A higher kw seems therefore more compatible with the important changes of root kr frequently observed along the root of various plants and attributed to apoplastic barrier deposition [Sanderson, 1983; Doussan et al., 1998b; Barrowclough et al., 2000]. Such changes might have resulted from longitudinal alterations of cell hydraulic properties, but this is unlikely as cortical cell membrane conductivities were reported to be axially uniform beyond the elongation zone in maize and onion [Zimmermann et al., 2000; Barrowclough et al., 2000].
The combination of high cell kw and low KPD (Tab. 2) yields the best match with the empirical range of young maize root kr and it reproduces qualitatively the axial trend estimated by Doussan et al. [1998b]. These properties generate a dominantly apoplastic water radial pathway in root cortex as compared to other combinations of cell properties (Fig. 3), which favour the symplastic (continuous blue stripes) and/or transcellular pathways (radial alternation of brown and blue stripes). If the cell-to-cell fraction of cortical radial water transport were to be as low as implied by high kw properties, the plasma membrane aquaporins abundantly expressed in root cortex during the day [Hachez et al., 2006] would hardly participate to radial water flow across the cortex. Still, they may remain necessary for cell turgor regulation [Beauzamy et al., 2014; Chaumont and Tyerman, 2014]. The dynamic regulation of root kr might then be specifically endorsed by those aquaporins highly expressed in the vicinity of apoplastic barriers, such as ZmPIP2;5 [Hachez et al., 2006].
The higher estimate of apoplastic barriers leakiness allows a significant fraction of fully apoplastic radial flow from root surface to xylem vessels (10.9 ± 5.6 %, for the 12 hydraulic scenarios reported at the end of section 5.1, not shown in Fig. 3 that has impermeable apoplastic barriers). Leakiness is also responsible for the reduction of σr below 1 (see Fig. 2c) even though all membranes reflection coefficients are equal to 1 (representative of the full selectivity of membranes to mannitol), as hypothesized in several studies [Steudle et al., 1987; Steudle and Frensch, 1989; Steudle and Peterson, 1998]. The experimental range of root σr for mannitol is wide (Fig. 2c) and even controversial as a few studies argue that values significantly lower than 1 stem from erroneous experimental interpretations [Knipfer and Fricke, 2010; Fritz and Ehwald, 2011]. The comparison between measured and upscaled σr is therefore tricky. In the upscaled model, σr appears to be an intrinsic property of the hydraulic anatomy and independent of boundary conditions, as kr.
3.3 Cortical intercellular spaces water-filling may not explain differences between osmotic and hydrostatic kr
Including water-filled intercellular spaces (see sections 5.2 and 5.5) increases the upscaled kr by 2.9 ± 3.3 % on average with impermeable apoplastic barriers (maximum increase: 10%) and by 8.2 ± 5.1 % with leaky apoplastic barriers (maximum increase: 15%) in the 12 hydraulic scenarios. These values are by far lower than the typically observed ten-fold increase [Steudle and Frensch, 1989; Steudle et al., 1987] that Steudle and Peterson [1998] hypothesize to be the consequence of intercellular space water-filling. Despite their high hydraulic conductivity when water-filled, the impact of intercellular spaces on our upscaled kr remains quite limited for the simple reason that they are in series with other hydraulic conductivities that limit the overall radial conductance.
The hydraulic anatomy approach therefore suggests that properties other than intercellular spaces water-filling likely generated the systematic differences between kr measured in osmotic and hydrostatic gradient experiments. In line with this point, Knipfer and Fricke [2010] demonstrated that unstirred layers in the bathing medium are involved in the lowering of kr in osmotic experiments in wheat, as no significant difference between kr estimations in both types of experiments remain when circulating the root medium. Unstirred layers may provoke the underestimation or reduction of kr in two ways: through the erroneous estimation of the osmotic potential difference between root surface and xylem |ψo,s − ψo,x| in Eq. (2), or through alterations of cell hydraulic properties, such as Lp.
A ten-fold change of kr is quite substantial. From the empirical standpoint, closing maize aquaporins with H2O2 typically reduces kr by less than four-fold (60% to 70% decrease [Ye and Steudle, 2006; Boursiac et al., 2008; Parent et al., 2009]). In MECHA, reproducing such alterations of cell hydraulic properties affects the upscaled kr similarly in the youngest region of the root with high kw and low KPD (the most accurate scenario in Fig. 2b). Reducing kAQP by 95% provoked a 56% to 72% decrease of the upscaled kr, for leaky and impermeable apoplastic barriers respectively. In another experiment, maize cortical cell Lp was reported to increase by 380% soon (< 2 hours) after the root medium water potential went from −0.075 to −0.34 MPa [Hachez et al., 2012]. Such alteration of LP in all cells results in a 200 to 260% increase of the upscaled kr. Hence, our quantitative approach suggests that even though LP is the most sensitive property controlling kr before the endodermis suberization (see Fig. 4g), the reported empirical range of LP changes alone is far from sufficient to generate ten-fold kr alterations.
Our hypothesis is that a major part of reported kr differences between hydraulic and osmotic experiments was an artefact stemming from the erroneous estimation of the osmotic driving pressure, as argued by Knipfer and Fricke [2010]. Plasmodesmatal aperture adjustment might also have contributed as it reportedly reacts to the root osmotic environment [Roberts and Oparka, 2003], but quantitative data is lacking so far.
3.4 Why maize root kr correlates with the permeability of some cells and not of others?
Plasma membrane hydraulic conductivity (LP) appears to be the main property controlling kr before the endodermal suberization in the scenario providing the best kr prediction (high kw and low KPD). However, with these hydraulic properties the endodermis is the only tissue whose LP considerably affects the upscaled kr (i.e. 69% of the overall sensitivity, versus 0.7% for cortical LP, see Fig. 4h). This result would explain the experimental observation that LP and ZmPIP aquaporin expression in maize root cortex varied independently of kr during osmotic stress [Hachez et al., 2012]. In MECHA, this feature occurs when the water stream bypasses cortical membranes (see Fig. 3g). The observed independence between cortical LP and kr cannot be reproduced with low cell wall hydraulic conductivities, which generate a high sensitivity of kr to cortical LP (up to 19% see Fig. 4i), and substantial cortical transmembrane flow (see Fig. 3h,i). Our model-assisted interpretations of experimental observations thus support the hypothesis of a higher range of cellulosic walls hydraulic conductivities from both angles of kr sensitivity to apoplastic barrier deposition (section 3.2) and to cell LP (section 3.4).
When the endodermis is partly suberized, the sensitivity to endodermal Lp remains higher than that to cortical Lp due to the transcellular water pathway across passage cells (Fig. 4e). It becomes negligible in more mature regions (see Fig. 4b) due to the isolation of cell plasma membranes from the apoplast following the endodermis full suberization.
Our results suggest that, before the suberization of the endodermis, the lack of correlation between cortical Lp and root kr may simply stem from independent fluctuations of cortical Lp and endodermal Lp, as only the latter significantly contributes to root kr. In order to verify it, we recommend to measure root kr, endodermal and cortical Lp before and during osmotic stress. Such experiment might also feed the question of whether separately regulated plasma membrane aquaporins fulfil different functions [Hachez et al., 2006], such as osmoregulation and radial permeability control.
3.5 Plasmodesmata may play a major role on root radial water transport despite their low conductance
Our results show that plasmodesmatal conductance might drastically alter water pathways from dominantly transcellular (Fig. 3c,f,i) to dominantly symplastic (Fig. 3b,e,h) when increasing KPD by a factor 6, which would be achieved by expanding the plasmodesmatal inner spaces by a factor 2.4 only. Despite the relatively low plasmodesmatal conductance in the most accurate scenario (high kw and low KPD), kr sensitivity to KPD is as high as 23% before endodermal suberization (Fig. 4g), and reaches 73% at maturity (Fig. 4a). When passage cells (occupying 4% of the endodermis surface in this cross-section) provide a bypass of plasmodesmata, the apoplastic flow converges toward the exposed plasma membranes of passage cells (see open arrowhead in Fig. S2c), contributing to 10% of water radial pathway (percentage visible in Fig. 3d). The remaining 90% still flows across the endodermis through plasmodesmata (see arrows in Fig. S2c) despite their low conductance.
The hypothesis of a major contribution of plasmodesmata to the water radial pathway has been formulated long ago [Clarkson et al., 1971], but was rejected by Hukin et al. [2002]. In the latter study, carboxyfluorescein was not radially translocated through plasmodesmata from the stele to the cortex in mature root segments (beyond the elongation zone). The authors thus concluded that plasmodesmata must be obstructed after the elongation zone and may not substantially participate to water transport either. However, calculations from Bret-Harte and Silk [1994] demonstrate that sugar transport across plasmodesmata massively relies on water flow, whose direction determines whether or not carboxyfluorescein can be translocated from the stele to the cortex. Considering that water flow across plasmodesmata may have been inwards after the elongation zone in the experiment of Hukin et al. [2002], our results indicating a major contribution of plasmodesmata to water transport remain compatible with the absence of carboxyfluorescein transport toward root cortex.
Advancing our understanding of water flow through plasmodesmata could involve the coupling of water tracing experiments (e.g., with deuterium [Zarebanadkouki et al., 2014]) to quantitative modelling of deuterium diffusion-convection in the root hydraulic anatomy. Each pathway would entail different deuterium distributions inside the root, and reproducing these distributions accurately in MECHA would take us one step further toward the verification of plasmodesmatal contribution to water radial flow across roots.
4 Discussion
MECHA is a novel hydraulic model which computes the flow of water through the walls, membranes and plasmodesmata of each individual cell throughout a complete root cross-section. From this fine scale, the model predicts root reflection coefficient (σr) and radial permeability (kr). Hence it connects hydraulic theories across scales, based on detailed anatomical descriptions and experimental data on the permeability of cell walls (kw), membranes (Lp), and plasmodesmata (KPD). Unlike Zhu and Steudle [1991] who define a single partition of water pathways (apoplastic versus cell-to-cell) for the overall trajectory between root surface and xylem, MECHA quantifies water composite pathways varying radially across successive cell layers, emphasizing the respective hydraulic function of each layer.
Using the hydraulic anatomical framework of a maize root as an example, MECHA reveals that hydraulic principles at the cell and root segment scales (Eqs. (1) and (2), respectively), derived independently by Katchalsky and Curran [1967] and Fiscus and Kramer [1975] are fully compatible, irrespective of apoplastic barriers leakiness. The upscaled maize root kr matched the empirical range of measurements from the literature. In particular, the best reproduction of the kr trend along the root maturation gradient was obtained using relatively high cell kw [Steudle and Boyer, 1985] and low KPD [Bret-Harte and Silk, 1994].
High kw also yield root kr that do not correlate with cortical cell LP as observed experimentally [Hachez et al., 2012]. As the patterns of radial transmembrane flow and of kr sensitivity to tissue LP were comparable to the patterns of expression of specific aquaporins (ZmPIP2;5 [Hachez et al., 2006]), our results support the view that the function of specific aquaporins might be closely related to their localization. Regarding plasmodesmata, the model suggests that, despite their low conductance, they would play a major role in water radial transport across the suberized endodermis even in the presence of passage cells offering a direct transmembrane bypass, as argued by Clarkson et al. [1971].
The explicit hydraulic anatomy framework of MECHA offers the possibility to challenge the assumption of Steudle and Peterson [1998] that differences between osmotic and hydrostatic kr stem from intercellular spaces water-filling. Our results do not support this hypothesis, as upscaled kr were barely sensitive to such water-filling and independent of boundary conditions. Altered cell LP coupled to inaccurate estimations of the osmotic driving pressure might explain the observation.
This enrichment of the composite transport model sheds new light on the field of plant water relations. Cutting-edge research on hydropatterning [Bao et al., 2014], hydrotropism [Dietrich et al., 2017], and apoplastic barriers [Barberon et al., 2016; Doblas et al., 2017] recently highlighted the need for a quantitative hydraulic framework to address questions related to pressure distribution and water flow direction at the cell scale. We expect MECHA to become a tool that will bridge the gap between protein regulatory pathways operating at the cell level and hydraulic behaviour at higher levels, pushing forward the potential of combined modelling and experimental approaches. Its compatibility with a root anatomical software [Pound et al., 2012] and functional-structural-plant-models [Javaux et al., 2008] has already opened avenues to investigate the relation between root architecture, anatomy and water availability [Passot et al., 2017].
The model comes with an online visualisation interface, available at http://plantmodelling.shinyapps.io/mecha.
5 Methods
5.1 Cell level hydraulic parametrization
The hydraulic conductance of individual plasmodesma (KPD) was evaluated from their geometry by Bret-Harte and Silk [1994] in maize, accounting for its partial occlusion by the desmotubule and increased viscosity of water in channels at the nanometre scale (3.4 mPa s). The obtained KPD values range from 3.05 10−19 to 1.22 10−18 m3s-1MPa-1 (geometrical average: 6.1 10−19 m3s-1MPa-1, here referred to as “low KPD”). Their estimation is up to five orders of magnitude lower than previously calculated for barley roots by Clarkson et al. [1971] (10−16 to 10−14 m3s-1MPa-1) who neglected the presence of desmotubules, and an order of magnitude less than estimated experimentally by Ginsburg and Ginzburg [1970] in maize (3.54 10−18 m3s-1MPa-1, here referred to as “high KPD”). Note that in the latter case, a plasmodesmatal frequency of 0.48 μm-2 between the endodermis and the pericycle (as in Ma and Peterson [2001]) was assumed to turn the conductivity into a conductance per plasmodesma.
Single plasmodesmata conductances were assumed to be uniform (no data available on distributed plasmodesmata aperture) and scaled to the cell level through multiplications by plasmodesmata frequency (μm-2) and shared wall surface (μm2) of neighbouring cells. Plasmodesmatal frequency data in maize roots after cell elongation, within and between tissue types, were measured by Warmbrodt [1985] and Clarkson et al. [1987] (as reported by Ma and Peterson [2001], see Tab. 1). Shared wall surface estimations were based on the discretized root cross-section anatomy (see section 5.2: Root geometry).
Cell membrane hydraulic conductivity (i.e. conductance per membrane surface unit, associated to the minuscule “k” case) was measured with a pressure probe in maize root cortical cells by Ehlert et al. [2009]. As acid loading provokes the closure of aquaporins due to their protonation [Tournaire-Roux et al., 2003], the difference between hydraulic conductivities of control and acid load treatments (kAQP, 5.0 10−7 m s-1MPa-1) was attributed to aquaporins (the fraction of aquaporins that remained opened after the acid load was assumed negligible), while the remaining conductivity (2.7 10−7 m s-1MPa-1) includes the parallel contributions of the phospholipid bilayer (km) and plasmodesmata (kPD). Note that the impact of plasmodesmatal conductivity cannot be distinguished from that of cell membrane conductivity with a cell pressure probe [Zhu and Steudle, 1991]. Hence, we used the geometrical-averaged KPD with default plasmodesmatal frequency to obtain a default kPD (2.44 10−7 m s-1MPa-1) and estimate the remaining km (2.6 10−8 m s-1MPa-1). For convenience, the sum of km and kAQP was referred to as cell Lp in this paper.
Cell wall hydraulic conductivity (kw) was seldom quantified, but generally considered as least limiting the hydraulic media for water radial flow. Steudle and Boyer [1985] measured in soybean a value of 7.7 10−8 m2s-1MPa-1 (here referred to as “high kw”). They concluded that this relatively high value observed in hydrostatic pressure gradient experiments was likely due to the dominance of water flow in intercellular spaces, which was previously not accounted for. For instance, kw was observed to be orders of magnitude lower in maize (2.5 10−10 to 6.1 10−9 m2s-1MPa-1 [Zhu and Steudle, 1991]) and isolated Nitella cell walls (1.4 10−10 m2s-1MPa-1 [Tyree, 1968], here referred to as “low kw”).
Casparian strips and suberin lamellae being hydrophobic, they were attributed null hydraulic conductivities, except in scenarios investigating the hypothesis of leaky apoplastic barriers. Lignified cell wall pores have the order of magnitude of 1 nanometre [Deng et al., 2016], which is small enough to fractionate between mannitol (C6H14O6) and ribitol (C5H12O5) [Fritz and Ehwald, 2011]. From Poiseuille law, we estimated that their intrinsic hydraulic conductivity could be no more than 10−11 m2s-1MPa-1, which was used as higher range for leaky apoplastic barriers. A summary of cell scale hydraulic properties is displayed in Tab. 2.
5.2 Root geometry
The hydraulic network geometry reproduced the anatomy of an aeroponically grown maize principal root, five centimeters from the tip (above the elongation zone, 0.9 mm diameter). A cross-section stained by immunocytochemistry for the cellular distribution of ZmPIP2;1/2;2, two plasma membrane aquaporins, pictured with an epifluorescent Leica DMR microscope (Wetzlar, Germany) as in Hachez et al. [2006], was segmented with the program CellSet [Pound et al., 2012]. Individual cells and structures were labelled manually (xylem, epidermis, exodermis, cortex, endodermis, pericycle and stelar parenchyma) based on their shape and position in the cross-section (Fig. 5a). Phloem elements and their companion cells could not be identified at that resolution. However, as they do not bear apoplastic barriers and represent a relatively small fraction of the cross-section, we assumed that their specific properties did not significantly impact root kr, and labelled them as their neighbour stelar parenchyma cells. Cortical intercellular spaces (204 on total in this crosssection) were recognized based on their relatively small size as compared to surrounding cells. They were attributed specific properties so that water could freely cross them when water-filled, while this path was blocked in scenarios with air-filled intercellular spaces (see blue polygons, i.e. null apoplastic fluxes, in Fig. 5b).
Labelled cells were grouped in layers, based on cell type (epidermis, exodermis, endodermis, and pericycle each constitute a cell layer) and cell connectivity. Cortical cells directly connected to the exodermis (through plasmodesmata) were assumed to belong to the first cortical cell layer. Similarly, cortical cells directly connected to the endodermis belonged to the last cortical cell layer. This process applied sequentially at each end of the not-yetclassified cortical cells. Classifying stelar cells was more straightforward as it started from the pericycle only. This layered classification is used in Fig. 3.
Three stages of apoplastic barrier development (structured deposition of hydrophobic material in cell walls) were selected for this study: (i) a simple endodermal Casparian band, (ii) an endodermis covered with suberin lamellae, except for three passage cells located in front of early metaxylem vessels, and (iii) a fully suberized endodermis (no passage cells) combined with an exodermal Casparian strip. In order to isolate the impact of apoplastic barrier deposition on root kr and water pathways, the network geometry was conserved and cell wall hydraulic properties were adjusted at apoplastic barriers locations.
The cross-section was given a pseudo three-dimensional representation by attributing a height of 200 microns (typical length of maize elongated root cells) to the system. Transversal cell walls were thus included on top and at the bottom of the cross-section. In consequence, the top and bottom of cells were aligned in our model. As the mere presence of transversal walls affects kr by less than 1.1% in all scenarios, we assumed that transversal wall alignment did not significantly change root hydraulic conductivity and water pathways, as compared to a more realistic non-aligned system. Note that only horizontal components of water flow were accounted for.
An example of distributed apoplastic water fluxes (i.e., flow rates per normal surface area) obtained from impermeable apoplastic barriers, high kw and low KPD hydraulic properties (Tab. 2) at the third stage of apoplastic barrier development (full endodermis and exodermal Casparian strip) is shown in Fig. 5b (other stages visible in Fig. S2). High velocities appear in red, mostly in the radial direction, corresponding to the direction of the water potential gradient. Low velocities in blue are mostly located at apoplastic barriers, where the apoplastic pathway is interrupted, and in tangential cell walls. While this detailed representation is appropriate to emphasize particular flow pathways in the vicinity of special features such as passage cells, the need for conciseness and clarity fostered the development of an alternative view of results: the radial representation of water pathways (see Notes S3, and Fig. 3).
5.3 Compatibility of hydraulic theories across scales
In order to verify the compatibility between hydraulic theories at the cell and root segment scales, we checked whether Eq. (2) holds when water flow across a root segment is simulated from cell scale hydraulic principles (Eq. (1)). If Eq. (2) holds in the aggregated system, Js should scale linearly with the potential difference “ψp,s−ψp,x+σr(ψo,s−ψo,x)”, for any given cell-scale hydraulic properties. The tested boundary conditions (Tab. 3) varied around experimental conditions reported in Steudle et al. [1987]: ψo,s the osmotic potential of the “Johnson-solution” (−0.02 MPa), ψp,x the equilibrated xylem pressure (+0.16 MPa), and ψo,x (−0.18 MPa) set to generate no flow when ψp,x equals +0.16 MPa. Cell cytosol osmotic potentials (ψo,c) were set to the average value measured in maize cortex (−0.7 MPa) by Enns et al. [2000]. The verification was carried out for the combinations of cell hydraulic properties reported in Tab. 2, with both leaky and impermeable apoplastic barriers.
Note that in this study, stelar cell walls osmotic potentials were assumed equal to that of xylem vessels (ψo,x), other walls osmotic potentials were assumed equal to that of the root surface (ψo,s), and membranes reflection coefficients were set to unity for all cells. A future study will focus on the higher levels of complexity implied by non-uniform reflection coefficients and osmotic potentials on water flow distribution in the root.
5.4 Compatibility of hydraulic properties across scales
An essential step strengthening the validity of the approach was carried out by verifying whether maize kr empirical values from the literature match kr predictions obtained using a typical root cross-section picture and empirical values of cell scale hydraulic parameters. This verification is novel as the present model is the first to have the ability to predict the root segment permeability from root section anatomy and cell hydraulic property distribution. The predicted cross-section radial hydraulic conductivity (kr, m s-1MPa-1) was calculated from the simulated root surface water fluxes (Js, m s-1) under default boundary conditions (see Tab. 3), using Eq. (2). Note that as argued by Fiscus and Kramer [1975] it would make sense to calculate kr relative to the endodermis surface area. However, kr empirical data reported in the literature is generally relative to the outer surface of the root. Hence, we complied with the latter approach.
5.5 Impact of intercellular spaces on root radial conductivity
In practice, hydraulic theories between cell and root segment scales may be compatible despite a non-linear relation between Js and boundary conditions, at the condition that kr was altered by boundary conditions. For instance, Steudle and Peterson [1998] systematically measured maize kr values that are one order of magnitude higher under hydrostatic than osmotic water potential gradients. Their interpretation is that in hydrostatic gradient experiments intercellular spaces get filled with water, thus increasing kr. Here we test the extent of kr changes when intercellular spaces get filled with water in the model of the hydraulic anatomy, as compared with the air-filled case (i.e., infinite versus null conductivity of intercellular spaces). The analysis of kr sensitivity to intercellular spaces water-filling was carried out for all cell hydraulic properties reported in Tab. 2, for both leaky and impermeable apoplastic barrier scenarios, the latter allowing a fraction of purely apoplastic flow, as hypothesized by Steudle and Peterson [1998].
5.6 Impact of cell plasmodesmata and plasma membranes permeabilities on root radial conductivity
Eventually we tested whether (and in which conditions) root kr could be non-correlated to cortical cell membrane permeability as observed by Hachez et al. [2012], by running a sensitivity analysis of root kr to various cell scale properties. The analysed properties were (i) cell membranes and plasmodesmatal hydraulic conductivities (for all cells simultaneously), and (ii) cell membrane conductivities (in specific tissues). This analysis was carried out for all cell hydraulic properties reported in Tab. 2, using the default boundary conditions from Tab. 3. Details of the sensitivity quantification were developed in Notes S4.
Author contribution
V.C. and M.F. developed the code of MECHA. V.C., G.L., M.J., F.C., and X.D. gathered ideas and participated to the writing. G.L. developed the ShinyApp.
Acknowledgements
All information about the model, including the source code, is available here: https://mecharoot.github.io/
A permanent version of the code used in the model is available here: ZENODO LINK
We also developed a web application to visualize typical outputs of our model. It was developed using the R Shiny framework and uses the following packages: ggplot2, ddply, reshape2. The web application is accessible here: https://plantmodelling.shinyapps.io/mecha/
MECHA was released under the GPL.2 open source licence.
We would like to thank Prof. C. Hachez for providing the root-cross section anatomical image. This work was supported by the Belgian National Fund for Scientific Research (FNRS), the Interuniversity Attraction Poles Programme-Belgian Science Policy (grant IAP7/29), and the “Communauté française de Belgique-Actions de Recherches Concertées” (grants ARC11/16-036 and ARC16/21-075).
V.C. and M.F. were supported by post-doctoral grants on the PAI MARS P7/29 project.