Growth-induced hormone dilution can explain the dynamics of plant root cell elongation

Abstract

In the elongation zone of the Arabidopsis thaliana plant root, cells undergo rapid elongation, increasing their length by ∼10-fold over 5 h while maintaining a constant radius. Although progress is being made in understanding how this growth is regulated, little consideration has been given as to how cell elongation affects the distribution of the key regulating hormones. Using a multiscale mathematical model and measurements of growth dynamics, we investigate the distribution of the hormone gibberellin in the root elongation zone. The model quantifies how rapid cell expansion causes gibberellin to dilute, creating a significant gradient in gibberellin levels. By incorporating the gibberellin signaling network, we simulate how gibberellin dilution affects the downstream components, including the growth-repressing DELLA proteins. We predict a gradient in DELLA that provides an explanation of the reduction in growth exhibited as cells move toward the end of the elongation zone. These results are validated at the molecular level by comparing predicted mRNA levels with transcriptomic data. To explore the dynamics further, we simulate perturbed systems in which gibberellin levels are reduced, considering both genetically modified and chemically treated roots. By modeling these cases, we predict how these perturbations affect gibberellin and DELLA levels and thereby provide insight into their altered growth dynamics.

Hormone distributions within plant tissues affect plant growth and development (1). Although many studies have investigated the influence of nonuniform distributions of the hormone auxin, gradients in other hormones also govern plant growth (2, 3). In some regions of the plant, cells undergo rapid expansion that dilutes their contents, including hormones. In these regions, organ-scale hormone gradients can arise due to the interplay between dilution, diffusion, production, decay, and receptor binding. Such complex dynamics govern in particular the distribution of the plant hormone gibberellin, which is involved in a diverse range of developmental processes including germination, organ development, and growth (4).

A well-studied context for gibberellin growth regulation is provided by the primary root of the model species Arabidopsis thaliana (3, 5, 6). At the organ level, the Arabidopsis primary root classically presents three distinct morphological zones (ref. 7; Fig. 1A): Cells divide in the meristem, which is located close to the root tip; after a number of divisions, cells then move through the elongation zone, where they rapidly increase in length with negligible change in radius; finally, cells stop growing on entering the mature zone. Gibberellin has been described as a key hormone in regulating both cell division in the root meristem (6) and cell elongation in the elongation zone (5).

Fig. 1.

Model summary. (A) Schematic representation of the Arabidopsis plant root showing the meristem, elongation zone, and mature zone (MZ). The red arrowhead indicates the position of the xylem formation/first root hair bulge, taken to be the end of the elongation zone. (B) Each cell consists of a cytoplasm (white), containing the nucleus (red) and the vacuole (blue), which is enclosed within the tonoplast. The cytoplasm is surrounded by a plasma membrane, which is embedded within the fibrous cell wall (black). (C) The gibberellin signaling network. In the network diagram, circular symbols are used for the hormone gibberellin (GA), rectangular symbols for proteins, and hexagonal symbols for mRNAs.

Gibberellin regulates cell elongation and division by mediating the destabilization of DELLA proteins (8, 9). Gibberellin first binds with its receptor GID1, forming gibberellin–GID1 complexes that can then interact with DELLA proteins, causing their targeted degradation via the SCF/proteosome machinery (refs. 10 and 11; Fig. 1C). It is well known that DELLA proteins function as growth repressors during root development (5, 6), although the molecular mechanisms behind this regulation are not currently understood.

Understanding how gibberellin and DELLA are distributed within a plant organ would be key to characterizing growth regulation. The mathematical model described below seeks to explore how different processes interact to produce the gibberellin distribution and how this distribution is able, through the gibberellin signaling pathway, to determine the DELLA distribution. A multiscale modeling approach is essential to understanding the complexity of such a system (12–14); for example, a genetic mutation may alter both the gibberellin pathway and the growth dynamics, and modeling can integrate these perturbations to predict the DELLA distribution. In this work, we focus on the root elongation zone. Using a mathematical model together with measurements of the growth dynamics, we predict that cell elongation causes significant dilution of gibberellin, resulting in a declining gibberellin concentration along the elongation zone. By modeling the gibberellin signaling network, we predict how this reduction in gibberellin affects the levels of downstream components, including the growth-repressing DELLA proteins. Moreover, by predicting the DELLA levels, we gain unique insight into gibberellin's growth regulation. Although we do not seek to model the growth regulation here, we propose that dilution-induced spatial variations in DELLA concentration can explain the growth dynamics of Arabidopsis root cells.

Results

Model Description.

To study the effect of gibberellin dilution in the plant-root elongation zone, we developed a multiscale mathematical model (Fig. 1). The model provides understanding of the interplay between processes that occur on the network, cellular, and organ scales, and in particular it demonstrates the importance of dilution in creating spatial gradients in hormone, protein, and mRNA levels.

In constructing the model, we exploited the simple architecture of the Arabidopsis root and, following ref. 15, represented the elongation zone as a single cell file (Fig. 1A). We focused on mature plant roots in which the sizes of the meristem and elongation zone were stable. For Arabidopsis plants, this stabilization occurs ∼6 d after germination in our growth conditions (6). Thus, at approximately regular time intervals, the elongation zone gains a new cell from the meristem and loses one to the mature zone, maintaining a constant number of cells. The anisotropy of the cell growth is captured by prescribing each cell to have time-dependent length but constant radius.

We modeled gibberellin movement on the subcellular scale, treating each cell as comprising four compartments: the vacuole, nucleus, cytoplasm, and adjacent cell wall (Fig. 1B and SI Appendix). Within each compartment, gibberellin is present in both protonated and anionic forms, with the proportion of each depending on the compartment's pH and on gibberellin's dissociation constant, pK (16, 17). Both protonated and anionic gibberellin can be transported between the cytoplasm and nucleus, but only protonated gibberellin is able to cross the plasma membrane and tonoplast to move between the cytoplasm and the cell wall and vacuole. This movement of protonated gibberellin results in a slow diffusive flux through the tissue. Considering the different compartments within each cell allows the accurate modeling of gibberellin dilution: During cell elongation, it is predominantly the vacuole that expands; the resulting reduction in the vacuolar gibberellin concentration causes protonated gibberellin to cross the tonoplast, from the cytoplasm to the vacuole, which in turn drives gibberellin from the nucleus to the cytoplasm. Thus, expansion of the vacuole causes a reduction in gibberellin concentration throughout the cell.

Gibberellin affects the cell's concentration of DELLA proteins via an interaction network that forms part of the gibberellin biosynthesis and signaling network (18). Transcriptomic (19) and reporter analysis (refs. 20 and 21; SI Appendix, Fig. S1) suggest that gibberellin biosynthesis is negligible in the root elongation zone, occurring predominantly in the meristem. Thus, our network model encompasses only gibberellin signaling (with cells entering the elongation zone containing a given amount of gibberellin), and we simulate the following reactions (Fig. 1C). First gibberellin binds to its receptor, GID1; because these species are both soluble, this binding occurs both in the cytoplasm and the nucleus. Once bound to gibberellin, the GID1 proteins undergo a reversible conformational change whereby their lid closes (22). This conformational change enables gibberellin–GID1 complexes to bind the DELLA proteins. This binding can take two forms: a relatively unstable interaction via the DELLA/TVHYNP binding site and a more stable version that also involves the GRAS domain of the DELLA proteins (23). The latter can dissociate into a gibberellin–GID1 complex and a ubiquitin-tagged DELLA protein that is primed for degradation via the SCF–proteosome machinery (23). DELLA proteins localize to the nucleus; thus, in the model, these binding reactions are assumed to occur only in this compartment. The DELLA proteins activate GID1 and repress DELLA transcription (24–26). In Arabidopsis, three genes encoding GID1 proteins (27) and five encoding DELLA proteins (4) have been identified; however, for simplicity, we consider a reduced network model and treat each gene family as a single representative species.

Gibberellin can also be deactivated by members of GA2-oxidase family; however, transcriptomics data for dissected regions of the root (28) revealed that the GA2-oxidase is not expressed in the elongation zone [with AtGA2ox6, the main GA2-oxidase expressed in Arabidopsis roots (29), being expressed only once the cells leave this zone]. We therefore considered gibberellin degradation to be negligible in the model. We further confirmed this model assumption by analyzing the morphology of ga2ox quintuple mutants (29), in which five of the GA2-oxidases are no longer expressed (including AtGA2ox6). Measurements of the root growth rate, elongation zone length, and cell lengths were similar to those in wild-type plants (SI Appendix, Fig. S2), and we concluded that gibberellin degradation has a negligible effect on the elongation-zone growth dynamics.

In summary, the model incorporates gibberellin movement between different compartments, gibberellin dilution, and the signaling network. The dynamics within each cell are described by using a system of 10 ordinary differential equations for 10 variables (SI Appendix, Model Description and Table S2). The protein–protein interactions are captured by the law of mass action; Hill functions are used for gene transcription; and Fickian flux terms model gibberellin movement between subcellular compartments (30). The dynamics depend on 27 parameter values; the majority of these values are available from the literature, and we make particular use of the estimates of the binding and transcription rates recently obtained by Middleton et al. (18) (SI Appendix, Table S1). We were unable to source estimates for four of the parameters; however, as discussed in SI Appendix, the choices of these values have little effect on our model results.

Prescribing the Cells’ Growth Dynamics.

Simulating the governing equations described above requires us to prescribe the cells’ elongation rate and their passage through the elongation zone; these growth dynamics were calculated from experimental measurements of root growth rates and cell lengths (Figs. 2A, 3B, and 4B and SI Appendix, Fig. S3). Assuming growth is constant on the organ scale, we calculated the time interval between successive cells entering (and leaving) the elongation zone, denoted by c, by dividing the mature cell length by the root growth rate (31). We then determined the cells’ growth rates having entered the elongation zone by relating organ-scale cell-length measurements to the elongation rates of individual cells (in effect moving from a continuum to discrete formulation of the growth dynamics).

Fig. 2.

The elongation zone (EZ) dynamics in Arabidopsis plant roots. (A) Experimental data of cortical cell lengths through the elongation zone, l(x), for wild-type roots (Columbia ecotype). Data from each file of cortical cells are plotted with a different symbol. The line shows the average of the data. (B) Evolution of each cell's length as it passes through the elongation zone, L_i(t), obtained via Eq. 1. (C) Predicted fold changes in gibberellin (GA) and DELLA concentrations and GID1 and DELLA mRNA levels. (D) Measured mRNA expression of the GID1 family members, GID1a and GID1b, and of the two DELLAs important in the root, RGA and GAI, for the elongation zone (EZ; from the top of the lateral root cap to the first visible root hair bulge, ∼850 μm from the top of the lateral root cap) and the decelerating elongation zone (DZ; from the first root hair bulge to the first fully elongated root hair). Within each zone, gene expression is normalized relative to ACT2 (a standard housekeeping gene). We present fold changes of these values relative to the elongation zone values.

Fig. 3.

The elongation zone dynamics in paclobutrazol-treated Arabidopsis plant roots. (A) Six-day-old Arabidopsis seedlings (Columbia ecotype) treated with 0, 1, and 5 μM paclobutrazol (PAC). (B) Averaged cell length measurements through the elongation zone, l(x), with error bars showing the SE. (C) Time interval between successive cells entering the elongation zone, with error bars showing the SE. (D) Evolution of each cell's length, L_i(t). (E) Evolution of each cell's RER, 1/L_idL_i/dt. (F) Predicted gibberellin concentrations. (G) Predicted DELLA concentrations. (H) Predicted fold change in DELLA. (I) Relationship between fold change in DELLA and RER in control, paclobutrazol-treated, and mutant plant roots.

Fig. 4.

The elongation zone dynamics in gibberellin-deficient Arabidopsis plant roots. (A) Six-day-old Arabidopsis seedlings, Landsberg ecotype (Control); gibberellin-biosynthesis mutants, ga1-3; and mutants in both gibberellin biosynthesis and loss of DELLA function, ga1-3/gai-t6/rga24. (Bar: 1 cm.) (B) Averaged cell length measurements through the elongation zone, l(x), with error bars showing the SE. (C) Time interval between successive cells entering the elongation zone, with error bars showing the SE. (D) Evolution of each cell's length, L_i(t). (E) Evolution of each cell's RER, 1/L_idL_i/dt. (F) Predicted gibberellin concentrations. (G) Predicted DELLA concentrations. (H) Predicted fold change in DELLA.

Because we are considering growth to be steady (i.e., growth of mature plant roots), the difference in length between two adjacent cells equals the amount that the rootward cell has grown during time interval c (31), implying

In the left-hand side of Eq. 1, L_i(t) is the length of cell i at time t since entering the elongation zone (for i = 1,2,3…), and the right-hand side is evaluated in terms of distance x from the start of the elongation zone, with l(x) being a smooth function that interpolates cell lengths over x (see SI Appendix, Characterizing the Dynamics of Cell Growth for clarification). Eq. 1 is evaluated when cell i is at location x. Cells enter the elongation zone with prescribed length, l_init. With Eq. 1, we used our interpolated measurements of cell lengths through the elongation zone, l(x), to determine the evolution of each cell's length as it moves through the elongation zone, L_i(t).

Gibberellin Dilution Produces Significant DELLA Gradients that Explain the Growth Dynamics in the Elongation Zone of Wild-Type Roots.

We measured the cell lengths through the elongation zone and the root growth rate (Fig. 2A and SI Appendix, Fig. S3C) and calculated that cells enter the elongation zone every ∼0.46 h. These data are consistent with reported growth dynamics (7, 32), with the cells’ elongation rate being approximately constant throughout the elongation zone and abruptly reducing to zero on progression to the mature zone (SI Appendix, Analytical Model Solutions). Fitting a smooth function, l(x), to the cell length data and using Eq. 1, we computed the growth dynamics of each cell, L_i(t) (Fig. 2B). By prescribing these growth dynamics in the mathematical model described above, our model predicted a significant reduction in gibberellin concentration as cells pass through the elongation zone (Fig. 2C). The reduction in gibberellin causes an increase in DELLA concentration because fewer DELLAs are degraded at lower gibberellin levels (Fig. 2C). As DELLA proteins induce the expression of GID1 genes and repress DELLA genes, GID1 mRNA levels were predicted to increase through the elongation zone and DELLA mRNA levels to decrease (Fig. 2C).

These predictions were validated by using transcriptomic data for dissected regions of the root (28), which showed that GID1 mRNA levels increased between the elongation zone and the mature zone, whereas the mRNA levels of RGA and GAI (the main DELLA in roots) decreased (Fig. 2D). Extending the spatial domain of the model to the mature zone, we predicted that the general trend in GID1 and DELLA mRNA levels would continue once cells left the elongation zone (SI Appendix, Fig. S4). Thus, the observed gradients in the mRNA levels agreed with the model predictions and were consistent with a reduction of gibberellin levels.

The predicted spatial distribution of DELLA (Fig. 2C) provides understanding of gibberellin's growth regulation: It is well known that DELLAs repress growth (5, 6), and the prediction that DELLA levels are high at the end of the elongation zone is consistent with the reduction in cell elongation at this location.

Gibberellin Dilution Explains the Growth Dynamics in the Elongation Zone of Paclobutrazol-Treated Roots.

The multiscale model suggests that cell elongation affects gibberellin levels significantly, implying the presence of a feedback loop in which changes in subcellular components affect cell elongation, which, in turn, via gibberellin dilution, affects those same subcellular components. To test such implications further, the effects of lower initial gibberellin levels owing to reduced gibberellin biosynthesis in the meristem were explored.

Gibberellin biosynthesis can be inhibited by chemically treating the roots with paclobutrazol (33). Experimental measurements revealed that paclobutrazol caused a reduction in root length, mature cell length, elongation zone length, and root growth rate (Fig. 3 A and B and SI Appendix, Fig. S3 B and C). We calculated the corresponding increase in the time interval of successive cells entering the zone (Fig. 3C) (which is likely to be due to paclobutrazol reducing the cell division rate in the meristem; ref. 6).

Computing the growth dynamics of the individual cells via Eq. 1 showed that, in paclobutrazol-treated roots, cells elongated normally for ∼3 h after entering the elongation zone, but then elongation slowed (Fig. 3 D and E). In modeling the paclobutrazol-treated plants, we imposed the cell growth dynamics (Fig. 3D) and captured the reduction in gibberellin biosynthesis in the meristem by prescribing a reduced initial gibberellin concentration in the cells as they entered the elongation zone (SI Appendix, Table S4). The model predicted that paclobutrazol treatment would reduce the gibberellin concentration throughout the elongation zone (Fig. 3F) and illustrated how the reduced gibberellin levels resulted in a higher DELLA concentration throughout the zone (Fig. 3G).

Studying the relationship between DELLA levels and the cells’ relative elongation rate (RER) (Fig. 3 E and G) provided insight into how DELLA regulates cell growth. Interestingly, the model results suggest that the RER does not correspond to the absolute DELLA levels but appears to reflect the fold change in DELLA as cells traverse the elongation zone (Fig. 3H). Once the fold change in DELLA reaches a threshold, the RER appears to rapidly reduce (Fig. 3I), and, thus, the dilution-induced DELLA gradient may determine the cell elongation dynamics.

Gibberellin Dilution Explains the Growth Dynamics in the Elongation Zone of Roots with Mutations in the Gibberellin Biosynthesis and Signaling Pathways.

Gibberellin levels can also be reduced genetically by using gibberellin biosynthetic mutants, namely the single mutant ga1-3 (34) and the triple mutant ga1-3/gai-t6/rga-24 (35)—the latter also having loss of function in the two DELLA proteins important in the root, GAI and RGA. As expected, the measured growth dynamics in the ga1-3 mutant were similar to those of the paclobutrazol-treated plants: Measurements indicated a reduction in the root length, mature cell length, elongation zone length, and root growth rate (Fig. 4 A and B and SI Appendix, Fig. S3 D and E) and an increase in the time interval of successive cells entering the elongation zone (Fig. 4C). Using these data to calculate the cell length dynamics, via Eq. 1, we showed that cells in the ga1-3 mutant grew normally for the first 4 h after entering the elongation zone but that cell elongation reduced much earlier than in wild type (Fig. 4 D and E). In contrast, the ga1-3/gai-t6/rga-24 mutant exhibited a normal root phenotype (Fig. 4A), and measurements showed that the cell lengths and the time interval of successive cells entering were very similar to those in wild type (Fig. 4 B and C), resulting in similar growth dynamics of the individual cells (Fig. 4 D and E). To simulate the dynamics in the ga1-3 and ga1-3/gai-t6/rga-24 mutants, the measured cell growth dynamics were prescribed (by using data in Fig. 4 C and D), and the gibberellin concentration in the cells entering the elongation zone were reduced. For mutant ga1-3/gai-t6/rga-24, we also encompassed the loss of function of the DELLA proteins: Accounting for the remaining members of the DELLA family (RGL1-3), we prescribed a greatly reduced DELLA translation rate (i.e., considering only the production of functional DELLA; SI Appendix, Table S4). In both mutants, the gibberellin concentrations throughout the elongation zone were predicted to be much lower than in wild type (Fig. 4F). However, although the concentration of DELLA was higher in the ga1-3 mutant (as in the paclobutrazol-treated plants), the gai-t6/rga-24 mutation counteracted this effect, so that on the assumption that the reductions in gibberellin biosynthesis and DELLA translation are suitably balanced, the ga1-3/gai-t6/rga-24 mutant was predicted to have functional DELLA levels that are very close to those in wild type (Fig. 4G). Thus, the effects of reduced gibberellin biosynthesis and loss of DELLA function can cancel each other out: Because DELLA regulates growth (5, 6), this model prediction is consistent with the ga1-3/gai-t6/rga-24 mutant exhibiting normal cell elongation.

Our results with the ga1-3 and ga1-3/gai-t6/rga-24 mutants also support our hypothesis that cell elongation reflects the fold change in DELLA, with the predicted relationship between the RER and DELLA fold change agreeing with that for the paclobutrazol-treated roots (Fig. 3I).

Discussion

In the elongation zone of plant roots, cells undergo rapid anisotropic growth, increasing in length by ∼10-fold in 5 h while maintaining a constant radius. Although many authors have investigated how this growth is regulated, little consideration has been given to how this growth affects the key hormones that provide the regulation. Focusing on the hormone gibberellin and the model plant species Arabidopsis, our mathematical model simulates how cell growth in the root elongation zone affects the distribution of gibberellin and the downstream proteins and mRNA. A key finding is the importance of dilution in creating a significant reduction in gibberellin levels as cells pass through the elongation zone.

Although a few studies have investigated the role of dilution in other systems (36, 37), and dilution has been included in previous plant models (for example, ref. 15), its importance during plant growth has not previously been highlighted. In many regions of plant tissue, cells undergo rapid expansion, making dilution of their contents a passive and unavoidable process. Thus, dilution may be key to understanding organ scale distributions, generating gradients of hormones, proteins, and mRNA without transcriptional or proteomic regulation. Diffusion reduces dilution-generated gradients, so the effect depends critically on the rate of hormone transport. Our modeling suggests that the slow movement of gibberellin between adjacent cells leads to diffusion being unable to smooth out the dilution-induced gibberellin gradient within the root elongation zone. In any region of plant tissue where cells grow rapidly, we would expect dilution to produce similar gradients in both gibberellin and other slow-moving phytohormones, such as abscisic acid (16). In contrast, auxin moves rapidly between cells (due to influx and efflux carriers present on some cell membranes), resulting in large tissue-scale velocities of ∼1 cm/h (38); the effects of dilution are then predicted to be negligible.

The model also provides insight into the growth regulation of several genetic mutants. Although several studies have considered the ga1-3/gai-t6/rga-24 mutant, it has not previously been suggested that the ga1-3 and gai-t6/rga-24 mutations have opposing effects on the levels of functional DELLA and could have the consequence that the functional DELLA levels are normal in the triple ga1-3/gai-t6/rga-24 mutant. It is likely that similar behavior occurs in the quadruple mutant rga-28/gid-1a/1b/1c, which also exhibits normal growth: If we reduce GID1 transcription (via the gid-1a/1b/1c knockout), fewer DELLAs are degraded, but the translation of functional DELLA is reduced via the rga-28 knockout.

The study illustrates the power of multiscale modeling in understanding how subcellular and cellular processes interact to produce organ-scale distributions of key regulatory proteins. Modeling is essential for assimilating our biological knowledge of such complex systems and deducing the dominant features of the dynamics. Furthermore, high-resolution spatiotemporal measurements of gibberellin levels are currently impractical in Arabidopsis; by simulating the network, we can test our predictions using amenable experimental data on mRNA distributions, providing confidence in our predictions of the distributions of “unseen” components, such as gibberellin itself. Although DELLA's regulation of growth is likely to be complex, involving numerous interactions downstream of DELLA, modeling can also provide insights into this regulation. Plotting the predicted DELLA levels against the RER enabled us to deduce a phenomenological relationship between these variables, and we propose that the RER may depend on the fold changes in DELLA, with the fold change reaching a threshold to reduce cell elongation on reaching the end of the elongation zone. This prediction raises questions as to the nature of DELLA's growth regulation, suggesting that the structure of the network downstream of DELLA may enable cells to adapt to the basal DELLA levels as they enter the elongation zone and respond to subsequent fold changes.

The model provides a framework for understanding how growth affects the distribution of other plant hormones in other plant organs and demonstrates the benefits of combining experimental data and model predictions to investigate the relationship between key regulators, such as DELLA, and cell elongation rates. The model also provides a building block for more sophisticated root models that consider realistic cell geometries, the role of gibberellin in different cell types (for example, considering the importance of the endodermal cells in gibberellin's growth regulation; ref. 5), and the cross-talk between gibberellin signaling and other hormone pathways. Further understanding could be gained by linking this model with a biomechanical model of cell growth (for example, ref. 39), taking into account phenomenological models of DELLA's regulation of cell-wall remodeling enzymes and how these enzymes in turn affect the biomechanical properties of the cell wall. These extensions to our work would enable us to investigate the interplay between gibberellin's growth regulation, the influence of other hormones, and biomechanical effects such as cellulose fiber reorientation.

In conclusion, our study demonstrates that dilution reduces a cell's gibberellin level, resulting in an increase in DELLA concentration. The DELLA proteins are key growth repressors, and therefore the increase in their concentration may explain the reduction in the cell elongation rate seen experimentally as cells move toward the end of the elongation zone. Thus, the interplay between cell growth causing dilution and the dilution affecting the regulation of cell growth can explain the root growth dynamics. Many authors have observed the characteristic pattern of the RER close to the tip of plant roots of many species (7, 31, 32). Explaining this growth pattern in terms of gibberellin levels being reduced by dilution provides a key insight.

Materials and Methods

Experimental Materials and Methods.

A. thaliana gibberellin mutants were grown and used as indicated in ref. 6, SI Appendix, Experimental Materials and Methods and Results, and the main text. Root growth rate was assessed as described in ref. 6 and in SI Appendix. Analysis of elongation zone and cortical cell lengths was performed as described (5), and the cell length data were smoothed and averaged as described (40).

Model Description.

Representing the root elongation zone as a file of N cells, the model dynamics can be described by 10N coupled ordinary differential equations, given in SI Appendix. The model involves 27 parameters; the parameter values and how these values influence the predicted concentrations are discussed in SI Appendix.