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. 2020 Jun 16;183(4):1424–1427. doi: 10.1104/pp.20.00321

Distance-to-Time Conversion Using Gompertz Model Reveals Age-Dependent Aerenchyma Formation in Rice Roots1,[OPEN]

Takaki Yamauchi 1,2, Mikio Nakazono 2, Yoshiaki Inukai 3, Nobuhiro Tsutsumi 4,3
PMCID: PMC7401129  PMID: 32546569

Abstract

Distance-to-time conversion by using the Gompertz model compensates for the differences in root elongation rates among different genotypes and conditions, and reveals age-dependent aerenchyma formation.

Dear Editor,

Root cells are generated from the stem cells at the root apex (Dolan and Okada, 1999; Petricka et al., 2012; Motte et al., 2019). Although the distance from the root tip roughly reflects the time at which root cells emerged, a deeper understanding of root development requires a time (i.e. age)-dependent evaluation. The Gompertz growth model is a sigmoid model commonly used to interpret population/organism/organ growth. However, it is generally restricted to understanding or predicting patterns of progression (Jeger, 2004; Tjørve and Tjørve, 2017). Here, we use the Gompertz model to convert the distance from the root tip to time after the emergence of root cells, thereby illustrating age-dependent aerenchyma formation in adventitious roots of rice (Oryza sativa).

Aerenchyma in roots is created by cell death and lysis of the cortical cells (Jackson and Armstrong, 1999). Gas exchange through aerenchyma helps plants survive waterlogging (Colmer, 2003). Rice roots constitutively form aerenchyma regardless of the soil oxygen status and further induce aerenchyma formation in response to oxygen deficiency (Colmer and Voesenek, 2009; Yamauchi et al., 2018; Pedersen et al., 2020). Our analysis of the iaa13 mutant, in which a single substitution in the degradation domain of an Auxin/indole-3-acetic acid protein (AUX/IAA; IAA) produces a dominant-negative protein (Kitomi et al., 2012), demonstrated that auxin signaling regulates constitutive aerenchyma formation in rice (Yamauchi et al., 2019).

At the same distance from the adventitious root tips, the iaa13 mutant has significantly less aerenchyma (as a percentage of root cross-sectional area) compared with wild type (Yamauchi et al., 2019). However, the higher root elongation rate of iaa13 (1.06 mm h−1 versus 0.84 mm h−1 in wild type; Fig. 1A) could mask differences in the timing of aerenchyma formation. Accounting for the elongation rates, we calculated that 48 h after the emergence of root cortical cells, the distance from the root tips was 50.8 mm in iaa13 and 40.1 mm in wild type (Fig. 1B). These results indicate that the same distances from the root tips correspond to different time points after root cell emergence in wild type and iaa13.

Figure 1.

Figure 1.

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Conversion of distance from the root tips to time after the root cell emergence in the wild-type (WT) and iaa13 roots. Twenty-day–old aerobically grown wild-type and iaa13 seedlings were further grown under aerated conditions, as described by Yamauchi et al. (2019). A, Elongation rates of the adventitious roots. Boxplots show the median (horizontal lines), 25th to 75th percentiles (edges of the boxes), minimum to maximum (edges of the whiskers), and mean values (dots in the boxes; n = 12 from four independent experiments). B, Scheme of the root elongation and difference in the distances from the root tips between the wild type and iaa13 at 48 h after the start of growth under aerated conditions. The values by the arrows indicate the distance (in millimeters) from the root tips. C, Gompertz curve fitting for the percentages of aerenchyma formation in root cross-sectional areas. The Gompertz model was fitted using the SSgompertz function in the R stats package (v3.5.2; https://rdrr.io/r/stats/stats-package.html) with all the percentage values of aerenchyma formation as the response variables, and distances from the root tips as the explanatory variables. Each dot indicates the actual percentage of aerenchyma formation, and the dots with black borders indicate the mean values (n = 12). D, Time-dependent aerenchyma formation in the adventitious roots. Percentages of aerenchyma formation in the wild-type and iaa13 roots at each time point were calculated using the Gompertz models shown in C by using the distance from the root tips at each time point as the explanatory variable, which was calculated by the corresponding elongation rate shown in A. Values are means ± se (n = 12). E, Differences in aerenchyma formation between the wild type and iaa13 at each position of the roots (magenta dots) or each time point (blue dots). The differences were calculated by using the Gompertz models obtained for the percentages of aerenchyma formation at each position of the roots or at each time point. In A, C, and D, asterisks indicate significant differences between the genotypes by two-sample Student’s t test (*P < 0.01).

To resolve this problem, we first conducted nonlinear regression analyses using the percentages of aerenchyma formation at 10, 20, 30, 40, and 50 mm from the adventitious root tips (Yamauchi et al., 2019), and at 10 mm from the root-shoot junctions of the wild type and iaa13, as the response variables (the average root lengths of the wild type and iaa13 were 82.42 and 93.00 mm, respectively). The quadratic model fitted the percentages of aerenchyma better than the exponential and logarithmic models, as shown by lower Akaike and Bayesian information criteria values (Burnham and Anderson, 2004). However, the graph suggested that better models may exist (Supplemental Fig. S1, A–C). The Gompertz model expresses the number of individuals alive (mortality) at each age (Gompertz, 1825), and we hypothesized that this model could better explain aerenchyma formation, given that the aerenchyma is generated by cortical cell death. Indeed, the Gompertz models for the wild type and iaa13 fitted the wild type and iaa13 better than the other models (Fig. 1C; Supplemental Fig. S1D).

To convert the distance from the root tips to time after the emergence of root cortical cells, we first calculated the distances from the root tips every 6 h, considering the elongation rates of the wild-type and iaa13 roots, and then used the calculated distances as the explanatory variables (x values) in the Gompertz models (Fig. 1D). Moreover, the differences in aerenchyma formation between the wild-type and iaa13 roots were calculated every 6 h (Fig. 1E). The difference in distance-dependent aerenchyma formation peaked at ∼40 mm from the root tips and then gradually decreased toward the basal parts of the roots (Fig. 1E). By contrast, the difference in time-dependent aerenchyma formation peaked at 36 h (at 30.06 and 38.06 mm in wild type and iaa13, respectively) and remained at similar levels until 96 h (Fig. 1E). As the iaa13 mutant protein has a dominant-negative effect on auxin signaling and IAA13 expression is restricted to the apical part of the adventitious roots (Inahashi et al., 2018; Yamauchi et al., 2019), the iaa13 protein may prevent auxin signaling only in the young root cells. If this is the case, the difference in the aerenchyma formation between the wild-type and iaa13 roots must occur mostly in the apical parts of the root. Our modeling analyses support that aerenchyma formation in the adventitious roots of iaa13 is reduced due to delayed initiation of its formation rather than suppression of aerenchyma formation throughout the roots (Fig. 1E).

The initiation of lateral root formation is determined at a more apical part of the adventitious roots than that of aerenchyma formation (Yamauchi et al., 2019). To compare the patterns of time-dependent aerenchyma formation with those of time-dependent lateral root formation, we conducted nonlinear regression analyses for the lateral root numbers at 10, 20, 30, 40, and 50 mm (±5 mm) from the root tips (Yamauchi et al., 2019). The Gompertz models fitted the lateral root numbers better than the other models, even though the logarithmic model was also useful for fitting iaa13 lateral root numbers (Supplemental Fig. S2). The elongation rates of the wild-type and iaa13 roots were 0.86 and 1.02 mm h−1, respectively (Supplemental Fig. S3A), and the distance from the root tips was 48.8 mm in iaa13 and 41.2 mm in wild type 48 h after the emergence of root pericycle cells (Supplemental Fig. S3B).

Then, we converted the distances to time by using the root elongation rates (Supplemental Fig. S3, C and D). Unlike in the aerenchyma formation, the differences in lateral root formation between the wild type and iaa13 mutant calculated by both models showed similar patterns (Supplemental Fig. S3E). This suggests that initiation of lateral root formation is determined at the apical part of the adventitious roots and that the dominant-negative iaa13 protein, predominantly expressed at the apical part of the iaa13 roots (Inahashi et al., 2018; Yamauchi et al., 2019), suppresses lateral root formation throughout the adventitious roots. These results support the idea that local auxin gradients in the cortical cells and pericycle cells are differentially regulated between the beginning of aerenchyma and lateral root formation in adventitious roots of rice (Yamauchi et al., 2019). However, the difference in lateral root numbers increased for at least 48 h after root cell emergence, suggesting that lateral root formation gradually progresses toward the basal part of the adventitious roots even though the initiation of its formation is determined at the most apical part by auxin signaling.

To further confirm the fit of the Gompertz model to the pattern of aerenchyma formation, we used the percentages of aerenchyma formation at 10, 20, 30, 40, and 50 mm from the adventitious root tips (Yamauchi et al., 2019) and at 10 mm from the root-shoot junctions of the wild type treated with or without the auxin transport inhibitor N-1-naphthylphthalamic acid (NPA; 0.05 µm) for 48 h, when the average root lengths of the wild type with and without NPA were 73.83 and 79.17 mm, respectively. The elongation rates of the wild type with and without NPA were 0.68 and 0.86 mm h−1, respectively (Fig. 2A), and the distance from the root tips was 32.8 mm in the wild type with NPA and 41.3 mm without NPA 48 h after the emergence of root cortical cells (Fig. 2B). Nonlinear regression analyses showed that the Gompertz models fitted both conditions better than the other models (Fig. 2C; Supplemental Fig. S4).

Figure 2.

Figure 2.

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Conversion of distance from the root tips to time after the root cell emergence in the wild type (WT) with or without NPA treatment. Twenty-day–old aerobically grown wild-type seedlings were further grown under aerated conditions with or without 0.05-μm NPA, as described by Yamauchi et al. (2019). A, Elongation rates of the adventitious roots. Boxplots show the median (horizontal lines), 25th to 75th percentiles (edges of the boxes), minimum to maximum (edges of the whiskers), and mean values (dots in the boxes; n = 12 from four independent experiments). B, Scheme of the root elongation and difference in distances from the root tips between the wild type with NPA and that without NPA at 48 h after the start of treatment. The values by the arrows indicate the distance (in millimeters) from the root tips. C, Gompertz curve fitting for the percentages of aerenchyma formation in root cross-sectional areas. The Gompertz model was fitted using the SSgompertz function in the R stats package (v3.5.2) as described in Figure 1C. Each dot indicates the actual percentage of aerenchyma formation, and the dots with black borders indicate the mean values (n = 12). D, Time-dependent aerenchyma formation in the adventitious roots. Percentages of aerenchyma formation in the wild-type roots with and without NPA at each time point were calculated as described in Figure 1D. Values are means ± se (n = 12). E, Differences in aerenchyma formation between the wild type with and without NPA at each position of the roots (magenta dots) or each time point (blue dots). The differences were calculated as described in Figure 1E. In A, C, and D, asterisks indicate significant differences between the conditions by two-sample Student’s t test (*P < 0.01).

Then, we converted the distances to time by using the root elongation rates (Fig. 2, C and D). Although the differences in distance-dependent and time-dependent aerenchyma formation between the wild type with and without NPA showed similar patterns, the peaks differed (Fig. 2E). The difference in time-dependent aerenchyma formation peaked at 48 h (at 32.75 and 41.33 mm with and without NPA, respectively) and gradually decreased thereafter (Fig. 2E). These results suggest that aerenchyma formation is suppressed in the newly emerged cortical cells within 48 h of NPA treatment, but then is stimulated by the elevated auxin level due to the inhibition of auxin transport to the root tips. Taken together, the distance-to-time conversion by using the Gompertz model is a reliable method to evaluate differences in the age-dependent aerenchyma formation among the different growth conditions.

Regulation of root development is essential for the uptake of water and nutrients by plants (Chapman et al., 2012; Lin and Sauter, 2018; Lynch, 2018; Sun et al., 2018). To understand the mechanisms underlying root development and improve the tolerance of crops to environmental stresses, it is essential to precisely evaluate the patterns of root development among diverse genotypes (mutant lines or crop cultivars) with different root elongation rates. Although we found that the Gompertz model provided the best fit for the patterns of aerenchyma formation (and lateral root formation), different models may be required to interpret other phenomena. Nevertheless, we propose that distance-to-time conversion through modeling analyses is a useful approach that enables a deeper understanding of the mechanisms underlying the initiation and/or maintenance of root developmental processes.

SUPPLEMENTAL DATA

The following supplemental materials are available.

  • Supplemental Figure S1. Nonlinear regression analyses of aerenchyma formation in the wild-type and iaa13 roots.

  • Supplemental Figure S2. Nonlinear regression analyses of lateral root formation in the wild type and iaa13 mutant.

  • Supplemental Figure S3. Application of the Gompertz model for the lateral root numbers in the wild type and iaa13 mutant.

  • Supplemental Figure S4. Nonlinear regression analyses of aerenchyma formation in roots of the wild type with or without NPA treatment.

Acknowledgments

We thank Akihiro Tanaka and Hiroki Inahashi for their help in phenotyping the iaa13 mutant.

Footnotes

1

This work was supported by the Japan Science and Technology Agency (Precursory Research for Embryonic Science and Technology grant no. JPMJPR17Q8 to T.Y.).

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