- Open Access
LEAF-E: a tool to analyze grass leaf growth using function fitting
© Voorend et al.; licensee BioMed Central Ltd. 2014
Received: 27 June 2014
Accepted: 26 September 2014
Published: 6 November 2014
In grasses, leaf growth is often monitored to gain insights in growth processes, biomass accumulation, regrowth after cutting, etc. To study the growth dynamics of the grass leaf, its length is measured at regular time intervals to derive the leaf elongation rate (LER) profile over time. From the LER profile, parameters such as maximal LER and leaf elongation duration (LED), which are essential for detecting inter-genotype growth differences and/or quantifying plant growth responses to changing environmental conditions, can be determined. As growth is influenced by the circadian clock and, especially in grasses, changes in environmental conditions such as temperature and evaporative demand, the LER profiles show considerable experimental variation and thus often do not follow a smooth curve. Hence it is difficult to quantify the duration and timing of growth. For these reasons, the measured data points should be fitted using a suitable mathematical function, such as the beta sigmoid function for leaf elongation.
In the context of high-throughput phenotyping, we implemented the fitting of leaf growth measurements into a user-friendly Microsoft Excel-based macro, a tool called LEAF-E. LEAF-E allows to perform non-linear regression modeling of leaf length measurements suitable for robust and automated extraction of leaf growth parameters such as LER and LED from large datasets. LEAF-E is particularly useful to quantify the timing of leaf growth, which forms an important added value for detecting differences in leaf growth development. We illustrate the broad application range of LEAF-E using published and unpublished data sets of maize, Miscanthus spp. and Brachypodium distachyon, generated in independent experiments and for different purposes. In addition, we show that LEAF-E could also be used to fit datasets of other growth-related processes that follow the sigmoidal profile, such as cell length measurements along the leaf axis.
Given its user-friendliness, ability to quantify duration and timing of leaf growth and broad application range, LEAF-E is a tool that could be routinely used to study growth processes following the sigmoidal profile.
Leaf growth has been monitored in a wide variety of grass species such as maize, rice, wheat, barley, Lolium, Miscanthus, Sorghum and Brachypodium making use of leaf length measurements taken at regular time intervals during development[1–8]. Parameters derived from these measurements such as leaf elongation rate (LER) and leaf elongation duration (LED) have been shown to be major determinants of individual and whole plant leaf area[9–14] and can be used to explain differences in final leaf length in response to environmental conditions and/or between genotypes[3, 4, 15].
In plant growth modeling, there is a growing consensus that approaches applying linear and exponential models are inadequate. A linear fit assumes a constant LER over a longer period during leaf development[1, 3, 9, 10] and an exponential or a log-linear relation assumes a constant relative elongation rate (RER). These assumptions limit the utility of the models, as both LER and RER may vary with environmental conditions and developmental stage. The polynomial model does cope with variations in LER and RER during leaf development. However, polynomial functions tend to make spurious upward or downward predictions, especially at the extremes of the data[16, 17]. Nonlinear regression is a more suitable strategy to describe leaf growth and to accommodate temporal variation in growth rates.
The beta sigmoid function, first used to describe whole plant growth, has been successfully applied to model the growth pattern of a single grass leaf[7, 19]. Yin and coworkers compared the performance of the beta sigmoid function with that of some other widely used sigmoid functions, such as Gompertz, Weibull and Richards to analyze datasets from maize, pea and wheat and concluded that the beta sigmoid function is unique in dealing with determinate growth. This is due to the prediction of a zero growth rate at both the start and end of the determinate growth period which is characterized by three sub-phases: an early exponential growth phase, an approximately linear growth phase, followed by a steadily decelerating growth phase. Furthermore, in contrast to other functions, the beta sigmoid function incorporates biologically relevant parameters and is highly flexible for describing various asymmetrical sigmoidal patterns.
In the context of high-throughput leaf phenotyping, there is a need for user-friendly tools that provide rapid and robust analysis of growth parameters from large datasets. Non-linear regression using function fitting is currently imbedded in statistical work packages such as SAS and R rendering the calculation, extraction and visualization of specific leaf growth parameters, such as LED, from large datasets difficult and time-consuming.
Here, we describe LEAF-E, a nonlinear regression-based tool for analyzing grass leaf growth data. The tool can be used to derive biologically relevant parameters such as final leaf length, maximal LER, LED but also parameters for the quantification of the timing of leaf growth, an important asset of this tool. To allow for the analysis of large datasets, the fitting procedure was automated in a user-friendly Microsoft Excel macro, which is innovative. We show how the application of this tool can assist data analysis and interpretation of experiments in which different genotypes or the response of single genotypes to different growth conditions are compared. For this purpose, we quantified and compared leaf growth parameters in published and unpublished datasets of three grass species: Zea mays (maize), Brachypodium distachyon and Miscanthus spp.
Results and discussion
Fitting of kinematic individual leaf length measurements using the beta sigmoid function
Goodness of fit of the beta sigmoid function for maize, Brachypodium and Miscanthus leaf growth
Based on these results, we can conclude that the beta sigmoid function is able to accommodate leaf growth measurements of three grass species with very distinct phenotypic characteristics. Maize and Miscanthus spp. both possess a ‘C4’ metabolism, however, maize is an annual crop characterized by one stem, whereas Miscanthus spp. are rhizomatous perennials that form numerous tillers. Brachypodium is a small, annual ‘C3’ plant used as a model for several temperate grain crops such as wheat and barley. Based upon these findings and the results obtained previously in L. perenne, we can conclude that the beta sigmoid function is probably of broad application for describing leaf growth in both C3 and C4 grass species. For that reason, the beta sigmoid function was further used to develop an automated fitting procedure in an excel macro that we called LEAF-E and to derive biologically relevant parameters from the different datasets.
Biologically relevant function parameters
The Excel macro that we designed, automatically generates the fitted function parameters (the final leaf length Lm, the moment of maximal leaf elongation rate tm, and the moment at which leaf growth ceases te) and all additional parameters in the form of a table. For visual inspection, it also generates a graph showing the original data points, the fitted growth curve and the function parameters for each biological replicate (see Additional files4 and5). Although the tool can be used to fit the measurements of several replicates jointly, fitting data of individual leaves allows statistical analysis, such as estimation of averages and standard deviation values, for all growth parameters, and comparison of genotypes or treatments, a strategy that is both straightforward and statistically correct. The advantage of using the beta sigmoid function in the form of Auzanneau and coworkers is that the function parameters (Lm, tm and te) themselves are biologically relevant when assessing grass leaf growth. This represents a clear advantage over functions that are based on parameters with no biologically relevant meaning or parameters that are difficult to interpret visually such as the Weibull equation. In addition, parameters that allow for the quantification of the timing of leaf growth, such as tm and te, can be estimated, which is not trivial using other methodologies such as a log-linear, polynomial or linear fit.
Effect of GA20ox1 overexpression on maize leaf elongation
AtGA20ox1 OE (mean ± SE)
Control (mean ± SE)
Difference in mean(+)
743 ± 13
535 ± 6
6.2 ± 0.2
4.6 ± 0.1
Thermal time points
107 ± 2
108 ± 4
121 ± 3
111 ± 4
165 ± 3
153 ± 5
180 ± 4
167 ± 5
217 ± 4
203 ± 5
246 ± 5
231 ± 5
Leaf elongation durations
139 ± 4
123 ± 2
96 ± 2
92 ± 2
125 ± 30
120 ± 2
Flexibility to extract additional biologically relevant information from the dataset
For example, the leaves of the nine B104 maize plants of dataset 1a (Table 1) reached 100 mm (t100) at 108 ± 4°Cd. In this experiment this was just before the 4th leaf emerged from the pseudo-stem or whorl (the spiral arrangement of leaves forming a cylindrical structure from where newly formed leaves emerge), approximately 8 days after sowing. Knowing that the average final length (Lm) is 535 ± 6 mm, we can state that a considerable share of at least 19% of the final maize leaf length is hidden in the pseudo-stem of this genotype. For t20%, the moment at which the 4th leaf reaches 20% of its final length, we obtained an estimate of 111 ± 4°Cd or 7.9 days after sowing. Exactly 50% of the final leaf length was attained at on average 153 ± 5°Cd (t50%), which is not significantly different from the moment of the maximal LER, tm at 167 ± 5°Cd. The leaf reached 90% of its final leaf length (t90%) at 203 ± 5°Cd or 14.5 days after sowing. The use of parameters that are independent of final leaf length, such as t20%, t50% and t90%, can however be meaningful when comparing genotypes that differ inherently in final leaf lengths. Likewise, we calculated a time window between reaching 20% and 90% of the final leaf length. The parameter was named [LED(20%-90%)] and corresponded to 92 ± 2°Cd or 6.7 days in this experiment (dataset 1a). Depending on the objective of a particular experiment, several other useful parameters can be extracted from the growth curve. Parameters can be deduced from the model, even at time points before empirical evidence can be taken (for maize e.g. t100 and in some cases also t20%) using non-destructive measurements of leaf length over time, as the leaf is then still hidden in the pseudo-stem. However, these are approximations and will need validation using destructive measurements of leaf elongation when one wants to investigate early leaf development, i.e. when the leaf is still hidden in the pseudo-stem[24, 25].
Leaf elongation rate and steady-state growth
Similarly, for any given thermal time t (°Cd), the LER (mm/°Cd) can be estimated using the first derivative of the beta sigmoid function (Equation 2). The maximum of this bell-shaped curve, is denoted as the maximal LER or LERmax (Figure 2). For the nine B104 maize plants (dataset 1a), we estimated a LERmax of 4.6 ± 0.1 mm/°Cd, equivalent to an impressive 2.7 mm/h or 64 mm/day.
Often, the LER profile is derived from calculations of leaf length increases between consecutive measurements divided by the respective thermal time interval. These calculations were performed on a plant-by-plant basis and can be viewed in Additional file1. The LERmax is then defined as the maximal value in that LER profile. A linear regression analysis of the estimated LER and LERmax to the calculated LER and LERmax, respectively showed high R² values (Additional file2), indicating a linear relationship between values obtained by the two approaches. The R² values of the linear regression were highest for the maize dataset (dataset 1a), 0.9067 and 0.9228 for LER and LERmax respectively (see Additional file2 for R² values of all linear regression analyses).
Several studies have assumed that during the growth of a grass leaf a period of constant LER can be defined[5, 15, 25, 26]. This assumption of steady-state growth in grass species such as maize has often been used as an acceptable simplification of the actual growth process[5, 15, 25, 26], although it has been argued that this steady-state growth period is relatively short compared to the total leaf growth period[8, 25, 27]. LEAF-E allows to quantify a steady state growth period. This can be defined as a period in leaf development for which the LER is relatively stable, for example the thermal time window between the points at which LER has a value of 90% or 95% of LERmax. For the maize B104 plants of dataset 1a, the thermal time window for which the LER was higher than 90% and 95% of LERmax was 34.5 ± 0.6°Cd or 2.5 days (from 10.7 until 13.1 days after sowing) and 49.1 ± 0.9°Cd or 3.5 days (from 10.1 until 13.6 days after sowing), respectively. We therefore conclude that, for the investigated dataset, a relatively stable LER is found for a time-span of 2.5 to 3.5 days during leaf growth.
Effect of GA20ox1 overexpression on maize leaf elongation
Comparison of transgenic maize plants overexpressing the GA biosynthesis gene GA20ox1 with non-transgenic plants in a previous study, demonstrated that altering GA levels specifically affects the size of the division zone, resulting in proportional changes in leaf and whole plant growth rates. This published dataset was used to validate the LEAF-E tool.
Using LEAF-E to reanalyze dataset 1a, previous findings were confirmed but this analysis allowed a more detailed study of the timing of leaf growth, thereby facilitating the detection of dissimilarities such as a shift in attaining LERmax in GA20ox1 overexpressing plants, that could not be quantified previously.
Variation in leaf growth behavior in two Miscanthus species
We investigated the leaf growth characteristics of two genotypes of Miscanthus belonging to different species with high potential as bio-energy crops, but with contrasting phenotypic characteristics. M. sinensis ‘Goliath’ is characterized by high shoot densities, whereas M. x giganteus produces less, but thicker and taller shoots[29, 30].
Comparison of leaf elongation in two Miscanthus genotypes from different species
M. Sinensis‘Goliath’ (mean ± SE)
M. x giganteus(mean ± SE)
Difference in mean(+)
1140 ± 46
923 ± 26
4.2 ± 0.2
3.8 ± 0.2
Thermal time points
95 ± 4
167 ± 4
141 ± 7
207 ± 5
230 ± 8
294 ± 8
240 ± 10
320 ± 8
350 ± 10
400 ± 13.8
425 ± 10
461 ± 18
Leaf elongation duration
329 ± 8
295 ± 18
209 ± 5
194 ± 11
283 ± 7
255 ± 16
The analysis with LEAF-E shows that the leaf growth pattern in these two Miscanthus spp. is significantly different and that the differences in total leaf length can be attributed to contrasting early leaf growth.
Variation in leaf growth behavior in different Brachypodium distachyon inbred lines
Fifty Brachypodium distachyon inbred lines are currently being used for a study of natural diversity, which is led by ‘The International Brachypodium Initiative’. We analyzed the leaf growth behavior of four diploid Brachypodium inbred lines that are part of that study: Bd21, Bd21-3, Bd2-3 and Bd3-1 (Figure 3C).
Comparison of leaf elongation in four Brachypodium inbred lines
Bd3-1 (mean ± SE)(+)
Bd21 (mean ± SE)(+)
Bd21-3 (mean ± SE)(+)
Bd2-3 (mean ± SE)(+)
94a ± 3
91a ± 1
113b ± 3
113b ± 2
1.44ab ± 0.05
1.29a ± 0.03
1.52b ± 0.05
1.51b ± 0.03
Thermal time points
162 a ± 3
163a ± 2
148b ± 3
166a ± 2
164a ± 3
166a ± 2
145b ± 3
163a ± 2
189a ± 2
193a ± 2
178b ± 3
197a ± 2
202a ± 2
206a ± 2
192b ± 3
211a ± 2
217ab ± 1
222bc ± 2
210a ± 4
229c ± 2
230.8ab ± 0.8
238bc ± 2
226a ± 4
245c ± 2
Leaf elongation durations
67a ± 3
72a ± 1
81b ± 2
82b ± 2
69a ± 2
75ab ± 1
78b ± 1
79b ± 1
55a ± 2
59ab ± 1
62b ± 1
63b ± 1
Fitting of cell length measurements along the leaf axis of maize overexpressing GA20ox1 using the beta sigmoid function
Effect of GA20ox1 overexpression on the maize cell length profile
AtGA20ox1 OE (mean ± SE)
Control (mean ± SE)
Difference in mean(+)
13.2 ± 1.7
10.9 ± 0.3
118.1 ± 0.6
118.5 ± 3.8
Position along the leaf axis
29.5 ± 2.1
21.3 ± 0.8
56.6 ± 2.5
43.4 ± 0.3
Zones along the leaf axis
division zone (cell < =40 μm)
19.5 ± 0.5
14.6 ± 0.3
elongation zone (cell > 40 μm - Pe)
37.1 ± 1.0
28.7 ± 0.9
Here, we provide a tool that we named LEAF-E for fast and straightforward analysis of grass leaf growth data using nonlinear regression in a simple Microsoft Excel format. We automated the fitting procedure using Excel 2010 and the Solver function (32 bit). The results of fitting the beta sigmoid function to the leaf length measurements are stored in tabular form and can easily be analyzed in standard statistical software programs in search of differences due to the applied treatments or to explore inter-genotypic or inter-population differences.
We applied LEAF-E to three datasets containing leaf length measurements of maize, Miscanthus and Brachypodium. In Miscanthus and Brachypodium, we have shown that LEAF-E is an appropriate tool for data analysis and that the analyzed species and genotypes display distinct leaf growth characteristics. In maize, the changes in both leaf elongation and cell length profile along the leaf axis as a result of enhanced GA levels, previously demonstrated by Nelissen and coworkers, were confirmed. In addition, we demonstrated that using LEAF-E, the timing of leaf growth can be studied, thereby facilitating the detection of dissimilarities in the timing of leaf growth that could not be quantified using other approaches. Furthermore, analysis with LEAF-E allows for a robust calculation of LERmax and LED, which leads to reliable detection of significant changes. Therefore, we propose LEAF-E as an excellent tool for comparing leaf growth behavior in different genotypes or to analyze the response of specific genotypes to a treatment.
Datasets used for validation
The dataset on leaf elongation of the maize B104 inbred line overexpressing the Arabidopsis thaliana GIBBERELLIC ACID 20 OXIDASE1 (GA20ox1) gene, previously described by Nelissen and coworkers was reanalyzed here. A segregating population produced by backcrossing the overexpression line (hemizygous for the transgenic event) to the wild-type B104 inbred line and consisting of 9 non-transgenic and 11 transgenic plants was used.
To determine leaf elongation rates, the length of the 4th leaf of transgenic and non-transgenic plants was measured daily until complete development, as previously described. The plants were grown in a growth chamber at 24°C. Here we used a base temperature of 10°C for thermal time (Growing degree days, GDD) calculations. For further details about growth conditions see.
The same segregating population, used to generate dataset 1a, was previously used by Nelissen and coworkers for the analysis of cell lengths along the leaf axis, based upon methods previously described. In short, the 4th leaf was harvested two days after appearance from the pseudo-stem (stem-like structure composed of concentric rolled or folded blades and sheaths that surround the growing point). At this time point, the ligule is only a few mm away from the base of the plant. The length of cell files adjacent to stomatal rows along the proximal-distal axis was measured using a DIC microscope (AxioImager, Zeiss, USA), and image analysis software (AxioVision, Zeiss, USA). The size of the division zone was determined as the distance between the base and the most distally observed mitotic figure in DAPI-stained leaves along the proximal-distal axis, with a fluorescence microscope (AxioImager, Zeiss, USA). Here we reanalyzed the cell length measurements.
Eight Miscanthus x giganteus and nine M. sinensis ‘Goliath’ plants were grown at 20°C (average temperature over the measuring period was 19.1°C) in a greenhouse, in Melle, Belgium, in September 2012 with no supplementary light. Plants were grown from rhizome cuttings in 2-l pots and were hand-watered and not fertilized during the experiment. The rhizomes were excavated during the winter of 2011 and stored in a cold room at 3°C until the start of the experiment. The length of the 4th leaf was measured five times a week (from leaf tip to soil level). Based on our observations at this developmental stage, the contribution of internode elongation to the height measurements as leaf lengths can be neglected and transition to flowering has not yet occurred. The measurements were spread over a time period of approximately four weeks. The calculation of thermal time was based on the average air temperature in the greenhouse taking into account a base temperature of 8°C, based on Farrell and coworkers.
The Brachypodium distachyon inbred lines Bd21, Bd2-3 and Bd3-1 were provided by David F. Garvin from the USDA-ARS (Minnesota, US), and line Bd21-3 was provided by Richard Sibout from INRA-IJPB (Versailles, France). Plants were grown in rootrainers (Haxnicks®, UK) in biological replicates (n = 10, 10, 7, 9 respectively) in a greenhouse at an average temperature of 21°C in Melle, Belgium, August 2012 with no supplementary light. To calculate the thermal time, a base temperature of 10°C was used. Fertilizer was added with the water supply: conductivity Ec = 1mS/cm; water soluble fertilizer Poly-feed (Haifa, Belgium) (N, P2O5, K20; 20:5:20 + 3 MgO). Measurements were taken from the tip of the 3rd leaf to its basal level on a daily basis, for a period of 10 days. Based on our observations at this developmental stage, the contribution of internode elongation to the height measurements as leaf lengths can be neglected and transition to flowering has not yet occurred.
A mathematical function for fitting leaf length measurements
Equation 2. Leaf elongation rate function, modified from.
As Equation 1 is a continuous function, it allows calculating the leaf length L (mm) at any given moment in the leaf elongation period t, and vice versa (Figure 2). Therefore, in addition to the parameters Lm, t0, tm and te, we estimated a set of parameters that can be biologically relevant. For the maize dataset 1a we estimated the time point t100, the moment at which the leaf length is 100 mm. The t100 time point, which is early in development, was chosen since it is close to the moment at which the leaf emerges from the pseudo-stem in maize non-transgenic B104 plants. Furthermore, we estimated t20%, t50% and t90% (°Cd), which are the moments at which the leaf reaches 20%, 50% and 90% of its final length, respectively. For Brachypodium (dataset 2), we replaced the t100 parameter by t20 (°Cd), the moment at which the leaf reaches 20 mm in length, to accommodate the smaller size of the Brachypodium leaf. These extra parameters allow the comparison of responses to different treatments or detect inter-genotypic differences in leaf growth development.
Expressing leaf growth as durations of thermal time, the leaf elongation duration or LED, allows describing leaf growth in a fluctuating environment.
On that account, various LEDs were explored. For example, LED(20%-90%) defines the leaf growth duration between reaching 20% and 90% of its final length. Parameters LED(100-e) and LED(20%-e) were defined similarly.
A mathematical function for fitting cell length measurements
Equation 3. Extended version of the beta sigmoid function modified from to fit cell length measurements along the leaf axis.
LEAF-E: function fitting using a Microsoft excel spreadsheet and the SOLVER function
The fitting of leaf growth data using the beta sigmoid function was performed using Excel 2010 and the Solver function (32 bit) according to Brown. The automation of the procedure, as described below, in the form of a macro is innovative. Each row in the datasheet contains the data of one individual leaf (ordered in a time series), the starting values of the parameters of the model, and the formulae to extract the necessary statistical components for the calculation of the least square estimates following Neter and coworkers. First, the macro checks for non-empty rows. When a non-empty row is found, the model is fitted to the data by minimizing the sum of squares of the errors iteratively, and changing the starting values of the parameters at each step. Per row (=leaf), all values described in the previous sections are calculated and stored in a tabular form for further statistical analysis. In addition, a graph showing the original data points, the fitted growth curve and the function parameters is generated automatically. This enables the evaluation of the correlation coefficient and visual interpretation of the goodness of fit. It also provides an easy way to check for miss fits or errors in the data. Miss fitting can occur when the Solver function fails to minimize the sum of squares of the errors using a particular set of starting values. Accordingly, the procedure can be repeated using more appropriate starting values. This way of working guarantees a fast analysis, robust estimation of growth parameters and provides a table in standardized format containing the resulting parameters and derived parameters per leaf. The output data can easily be analyzed in search of differential responses using standard statistical software, depending on the experimental setup. We gave this Excel tool the name LEAF-E. LEAF-E is included as additional file to this article (Additional files4 and5, with a user manual on the first sheet). Additional file4 includes test data of M. sinensis ‘Goliath’ leaf growth and default starting values prior running the macro and Additional file5 includes the results of the fitting procedure and graphs after running the automated fitting procedure.
The statistical analysis of the derived growth parameters comprised a student t-test on datasets containing only two genotypes (datasets 1a, 1b and 2) and an ANOVA followed by post hoc Scheffé tests for datasets containing more than two genotypes (dataset 3). All these analyses were carried out in the software package STATISTICA version 11 (Statsoft Inc., USA). For the analysis of the goodness of fit of cell length data of dataset 1b, the fitting of the beta sigmoid and three other commonly used growth functions (Weibull, Gompertz and Logistic) and linear regression of the predicted to the observed values was carried out in SIGMAPLOT version 12_5 (Systat Software Inc., USA).
We thank the Agency for Innovation by Science and Technology in Flanders (IWT) for funding and Simon Fonteyne (ILVO, Belgium) for providing leaf measurements of Miscanthus spp. We also like to thank David Garvin (USDA-ARS, US) and Richard Sibout (INRA-IJPB, France) for providing Brachypodium seed stocks.
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