 Research
 Open Access
Functional approach to highthroughput plant growth analysis
 Oliver L Tessmer†^{1},
 Yuhua Jiao†^{3},
 Jeffrey A Cruz^{3},
 David M Kramer^{2, 3}Email author and
 Jin Chen^{1, 3}Email author
https://doi.org/10.1186/175205097S6S17
© Tessmer et al.; licensee BioMed Central Ltd. 2013
 Published: 13 December 2013
Abstract
Method
Taking advantage of the current rapid development in imaging systems and computer vision algorithms, we present HPGA, a h ighthroughput p henotyping platform for plant g rowth modeling and functional a nalysis, which produces better understanding of energy distribution in regards of the balance between growth and defense. HPGA has two components, PAE (Plant Area Estimation) and GMA (Growth Modeling and Analysis). In PAE, by taking the complex leaf overlap problem into consideration, the area of every plant is measured from topview images in four steps. Given the abundant measurements obtained with PAE, in the second module GMA, a nonlinear growth model is applied to generate growth curves, followed by functional data analysis.
Results
Experimental results on model plant Arabidopsis thaliana show that, compared to an existing approach, HPGA reduces the error rate of measuring plant area by half. The application of HPGA on the cfq mutant plants under fluctuating light reveals the correlation between low photosynthetic rates and small plant area (compared to wild type), which raises a hypothesis that knocking out cfq changes the sensitivity of the energy distribution under fluctuating light conditions to repress leaf growth.
Availability
HPGA is available at http://www.msu.edu/~jinchen/HPGA.
Keywords
 Wild Type Plant
 Relative Growth Rate
 Leaf Length
 Plant Area
 Functional Data Analysis
Introduction
In order to precisely model plant growth rate, we present a new computational model called HPGA (Highthroughput Plant Growth Analysis) which identifies leaf tips and then uses the short curvature areas around them to estimate the area of each leaf individually, regardless whether they overlap or not. Then a nonlinear model is trained to learn growth characteristics in different development stages.
Plant growth models quantify two kinds of measurements: absolute growth rate (AGR) and relative growth rate (RGR), both of which require measuring biomass or plant area at successive time points. Traditionally, plant growth is often fitted with linear or exponential equations such as a logistic model (one or twoparameter version) [5, 6], comprising of initial exponential growth and a term that reduces RGR as the area increases, resulting in an asymptotic maximum area (Figure 1b) [7]. However, although simple logistic models require relatively few observations, they do not often fit with observations well [4]. In fact, there is an increasing amount of contemporary research suggesting that traditional approaches to modeling growth using linear and exponential models are inadequate due to their oversimplified assumptions [4]. Consequently, researchers have started to apply nonlinear models, including three and fourparameter logistic [8, 9], powerlaw, Gompertz [10, 11] and monomolecular model [11, 12] to provide enough flexibility to obtain the best fit between models and observations [4]. To achieve satisfactory results, all these models need abundant observations [4], which require either a laborintensive protocol to frequently measure plant areas manually, or an automated phenotyping approach using computational measurements of plant area. Furthermore, in largescale screen experiments where hundreds of plants are monitored simultaneously, manual inspection may not be an option. It is necessary to automate the plant area measurement and therefore recognize emergent growth phenotypes.
Taking advantage of the current rapid development in imaging systems and computer vision algorithms, highthroughput computational phenotyping techniques to noninvasively monitor plant growth have been developed [3, 13–15]. In these approaches, topview images are captured periodically and a growth curve is generated using the observed pixels of the plant area over time [3, 16, 17]. However, the observed pixels is remarkably affected by complex leaf overlap during growth (in addition to leaf twisting and curling, and circadian movement), resulting in inaccurate growth patterns. For old plants with many overlapping large leaves, the bias becomes more severe (Supplementary Fig S1). Mokhtarpour et al [18] have setup a threecamera system with two side views and one top view to correct for leaf overlapping areas, but the setup of sideview cameras is not suitable in many cases, e.g. largescale screen of many plants simultaneously.
Since the observed value from a topview will often cause problems in modeling plant growth, and since there is an emerging research demand for plant highthroughput phenotyping, more advanced approaches for plant growth analysis need to be developed. In this paper, a new computational model HPGA is presented to estimate leaf overlap percentage to measure plant area more precisely. Our approach has the following advantages.

Unlike the existing approaches that simply counts the number of valid pixels in an image [3, 13–15], HPGA estimates plant areas by explicitly taking leaf overlaps into consideration. Specifically, with a leaf development model [19], we address the leaf overlap problem with a fourstep approach: plant center identification, leaf tip identification, leaf area estimation and plant area measurement.

Our approach avoids the leaf segmentation problem to recognize all the leaves of a plant from a topview image, which has been considered to be a challenging problem in the computer vision community due to high planttoplant variations (biodiversity) [20].

With our highthroughput phenotyping technique, researchers are able to generate hundreds or even thousands of observations for every plant automatically. Feeding enough observations to a nonlinear model ensures the robustness and precision of plant growth modeling.

In HPGA, functional data analysis is applied on growth curves for better interpretation of the plant growth scenarios. In our experiment, the coupling of photosynthetic and growth rate phenotypes raises an important hypothesis about gene function.
In summary, our study highlights a costeffective, highthroughput phenotyping approach that, coupled with other phenotyping and genotyping techniques, facilitates the dissection of the dynamics of plant growth and development under varying environmental conditions.
Methods
HPGA is composed of two independent modules, PAE (Plant Area Estimation) and GMA (Growth Modeling and Analysis). In PAE, the plant area is measured in four steps which will be described in the following text. Given the abundant measurements obtained with PAE, in the second module (GMA), a nonlinear growth model is applied to generate a growth curve for each plant, followed by functional data analysis. Modularity in design combines the advantages of standardization with those of customization [21]. In our case, it allows researchers, for example, to explore different nonlinear models without interfering the other parts of the algorithm.
Plant area estimation
Plant center identification
The center of a plant is the geometric location where a plant starts to grow. In general, a plant center is the center of mass because of the symmetry of plants [22]. But in many other cases, due to the loss of leaves or light direction changes, a plant center is different from the geometric center. Therefore, instead of using the whole plant geometric properties, we develop a new method to identify plant center from a topview image. The idea is to subdivide a topview image into constituent regions, recognize a few leaves (≥ 2), and then locate the plant center according to leaf orientations, in that all the leaves arise from the center of a plant.
We first apply Gaussian smoothing [23], a common approach for blurring images and removing details and noises, to reduce the noise level in the topview images (Figure 3b). Then, Laplacian detector [23] is used to identify simple leaves. As a secondorder derivative operator, Laplacian detector is suitable for edge location by looking for zero crossings. With it, the edges of simple leaves can be wellisolated, while the edges of leaf complexes are still crossed. Therefore, the regions containing only simple leaves are correctly detected (Figure 3c). Since the secondorder derivative is extremely sensitive to noise, a fine edge detection approach called Canny edge detector [23, 24] (the most commonly applied edge detector), is employed to accurately identify intensity discontinuities that define the leaf edges of all the simple leaves (Figure 3d).
where Dist(l_{ i }, c) is the perpendicular distance from c to line l_{ i }. In summary, we developed a plant center identification method based on a few segmented simple leaves. It is theoretically more reasonable than to use the center of mass (the square against the triangle in Figure 3f).
Leaf tip identification
Next, starting from a plant center c, a horizontal line is drawn to the right and rotated 360 degrees. For each degree ϕ, we compute the distance from the plant center to the outermost edge of the foreground mask (Figure 4b). This yields a vector of pairs of radius r and degree ϕ which describes, in polar coordinates, the shape of the outermost edge of the foreground mask(Figure 4c), with each peak representing a leaf tip.
Our approach ensures the successful identification of all the outer leaf tips, as shown in Figure 4d. Its performance may be reduced if a plant is overgrown which, in fact, rarely happens in leaf based scientific experiments.
Leaf area estimation
In early leaf development, because of cell proliferation, the primordium (formed by apical and marginal meristems) undergoes a slow limited expansion phase, followed by a rapid dramatic expansion phase, principally because of cell expansion [19]. Starting with this synthetic model, recent studies reveal that there is a strong relationship between leaf area and leaf properties such as primary and secondary vein density [18, 19]. In Sack et al., a loglinear relationship between leaf area and leaf dimension (length and width) has been discovered with a significant pvalue [19]. However, in our situation it is difficult to measure the leaf width because of the challenging leaf segmentation problem. Therefore, we develop a new leaf lengthtoarea model to infer the leaf area with leaf length and a small area around leaf tip.
We first define the leaf length as the distance from the base of the petiole of the leaf to the outermost point on the leaf. In our model, we assume that all the leaves have petiole bases at the same point, i.e., the center of the plant. Therefore, a leaf length is estimated as being the distance from the plant center to the leaf tip, which is the radius (r). We include the petiole length because of two reasons. First, it is difficult to identify the leaf bottom because of the leaf overlap problem. Second, the leaf length and petiole length are usually proportional, except for a few genetic or natural variations. Consequently, the proportion can be modeled in the leaf area inference.
Due to the diversity of leaf shapes, leaves with the same length can have very different areas. To this end, we define "curvature ratio" cr = (r  r_{ s })/r, where r is the leaf length and r_{ s } is the smoothed leaf length defined in Eq 2, to describe the shape of the small area around leaf tip (Figure 5a). Curvature ratio is an indicator of leaf shape, because the leaf edge is usually smooth, the small area around leaf tip, which is the easiest to obtain, has the complete information of leaf shape.
where I is the leaf area, cr is the leaf curvature ratio and r is the leaf length. The 95% confidence intervals (CI) for cr < 0.07 are (29.9, 36.0) and (1012.0, 527.6); 95% CI for cr ≥ 0.07 are (406.1, 621.6) and (0.0130, 0.0162), meaning 24.8% and 39.7% maximal leaftoleaf variances respectively.
Plant area measurement
where a_{ l } is the summarized leaf area defined as ${a}_{l}=\phantom{\rule{0.3em}{0ex}}{\sum}_{i=1}^{m}{l}_{i}$(l_{ i } is the area of the i th identified leaf and m is the number of leaf tips of a plant), a_{ t } is the observed value of plant area from a topview, and p_{ overlap } is the leaf overlap percentage which equals the absolute difference between a_{ t } and a_{ l } divided by the maximum value of the two. In Eq 4, if p_{ overlap } = 0, plant area is exactly the summary of all the leaf areas whose tips are identified; otherwise, plant area is estimated with ${a}_{l}$ and ${a}_{t}$: if ${a}_{l}>{a}_{t},a=2\cdot {a}_{l}{a}_{t}$, else $a=2\cdot {a}_{l\phantom{\rule{0.3em}{0ex}}}\phantom{\rule{0.3em}{0ex}}{a}_{l}^{2}/{a}_{t}$.
Growth modeling and analysis
Given the abundant plant area measures obtained with PAE, a nonlinear model is applied to generate precise growth curves which are suitable for the subsequent functional data analysis.
Nonlinear growth model
Among the basic functional forms for plant growth modelling, logistic model is the most commonly utilized asymptotic form [6, 7, 11]. It has one, two, three, four and fiveparameter versions, where each version uses a logistic function to relate examinee ability and the parameter(s) to the growth responding to time [27]. The simple logistic models (one and twoparameter versions, called 1PLM and 2PLM) do not often fit with observations well [4], although they require relatively few observations for the training.
where t is time, A(t) is plant area at time t (modelled value, different from the plant area observation (a)), A_{0} is the initial plant area, A_{ a } indicates the upper horizontal asymptotes, and γ is an acceleration or deceleration parameter related to time. If t = 0, A = A_{0}A_{ a } /(A_{0} + (A_{ a }  A_{0})) = A_{0}; if t → ∞, A = A_{0}A_{ a } /(A_{0} + (A_{ a }  A_{0})·0) = A_{ a }; if A = (A_{ a }  A_{0})/ 2, A(t)" = 0.
The fourparameter logistic model (4PLM) looses one or the other of the constraints in 3PLM [8, 28]. For some data, the additional flexibility of the fourparameter version greatly increases the variance explained by the model, although 3PLM provides a more parsimonious and equally adequate fit in other situations. The most general form of this is the fiveparameter logistic model (5PLM) [9], which provides maximum flexibility and alleviates both restrictions.
Note that inappropriate functional forms will often fail to converge; or in other cases, the wrong form can result in convergence with unreasonable parameter estimates [4]. Therefore, we choose 3PLM for plant growth modeling to avoid overparameterization, and use nonlinear least squares to fit the plant area observations to 3PLM.
Functional data analysis
Plants are selfassembled systems for solar harvesting. In the early stages of plant growth, harvested energy is used primarily for the creation of new light capture facilities (leaves), resulting in an exponential growth rate. As a plant matures, an increasing percentage of the energy captured can be redirected to storage for later harvest. Studying plant growth over time is essential towards the understanding of how plants manage resources at different ages or under different environmental stresses.
The rate of change of the plant area sometimes is more interesting than its actual value. To this end, we need to study what alters velocity (the first order derivative of the growth curve, Eq 8) and acceleration (the second order derivative of the growth curve, Eq 9) which is instantaneous curvature in a growth curve [29]. The smoothed growth curve generated with 3PLM is capable of giving a qualified impression of the velocity and acceleration of the plant growth.
Experiments
cfq is an AAAtype ATPase family protein which is involved in ATP synthase regulation [31, 32]. It harbors a mutation on the gamma subunit of the ATP synthase, which accelerates ATP synthetic activity at the cost of accumulating pmf (and consequently the ΔpH required for photoprotective, q_{ E }). Equilibrium redox titration revealed that this mutation makes the regulatory sulfhydryl group energetically much more difficult to reduce relative to the wild type [33]. The growth of the mutant, however, is not significantly impaired under standard laboratory growth conditions (constant light at 150 μmol m^{2} s^{1}) [33]. Here we investigate how the mutant affects growth under nonstatic light conditions, in that overlapping regulatory mechanisms can compensate for loss of some processes under artificially static lab conditions, but each underlying process may have different dynamic responses and may be activated under different sets of environmental conditions. In the following text, we first evaluate HPGA in terms of measuring plant area, followed by the dissection of the different growth patterns of the cfq and wild type plants.
Results of plant area estimation
Results on leaf tip identification (true positive rate).
Days  cfq  WT  

Sample 1  Sample 2  Sample 3  Avg ( cfq )  Sample 1  Sample 2  Sample 3  Avg(WT)  
Day 15  100%  97%  99%  99%  98%  97%  94%  97% 
Day 610  97%  96%  92%  95%  97%  89%  94%  93% 
Day 1115  94%  88%  92%  91%  89%  89%  82%  86% 
Results on growth modeling and analysis
Parameters of 3PLM and their upper and lower bounds with 95% confidence.
Plant  Parameters  Lower bound of 95% Cl  Upper bound of 95% Cl  

γ  A _{0}  A _{ a }  γ  A _{0}  A _{ a }  γ  A _{0}  A _{ a }  
cfq Sample 1  0.03711  129.9  17440  0.0334  93.55  12780  0.04081  166.3  22100 
cfq Sample 2  0.04296  127.2  24850  0.03984  95.37  21300  0.04607  159.1  28390 
cfq Sample 3  0.04128  167.3  22590  0.03916  139.9  20540  0.04340  194.7  24650 
WT Sample 1  0.03823  210.1  53640  0.03502  156.2  35500  0.04144  263.9  71770 
WT Sample 2  0.05020  127.5  38570  0.04835  107.7  36640  0.05205  147.3  40500 
WT Sample 3  0.03907  304.8  66060  0.03656  243.8  52480  0.04159  365.9  79640 
The growth model reveals two interesting results. First, unlike the growth pattern under normal static light conditions, which is similar for wild type and cfq plants [33], the upper horizontal asymptotes of the wild type plants (~65, 000 pixels) is almost three times that of the cfq plants (~24, 000 pixels) (Table 2), suggesting that by knocking out the cfq gene, plants have much less potential to grow under fluctuating light conditions. The slower growth rate of the mutant correlates well with diminished photosynthetic efficiency compared to the wild type (Figure 6), consistent with the higher energy requirement for sustained activation of the ATP synthase and a lower capacity of ATP synthesis in the mutant [33]. The efficiency of cfq is continuously repressed or unable to recover under fluctuating light. This is probably because of the significantly lower lightinduced ATPase and ATP synthase activity in the mutant compared with the wild type [33]. In summary, the low photosynthetic rates correlates well with the small plant area of the cfq plants, raising a hypothesis that knocking out the cfq gene changes the sensitivity of the energy distribution under fluctuating light conditions to repress leaf growth.
Second, in the growth curve, the wild type plants reach their upper horizontal asymptotes at around 25 days from the start of our experiment, which is 35 days from seedling. This matches with the Arabidopsis growth stage description, which states that the rosette growth of wild type plants completes after 29.3 days from seedling with a standard deviation of 3.5 days [30]. The cfq plants reach the upper horizontal asymptotes at around 20 days from the start of our experiment. This shorter growth period (5 days less), combined with the decreased growth rate, results in a smaller rosette leaf surface area in cfq. The values of the cfq sample 1 and wild type sample 2 are statistically different from the rests (pvalue 7.1E14 and 3.6E09 respectively with a twotailed ttest with unequal variance [34]; Supplementary Fig S2), probably because of natural biological variability. Therefore, they were excluded in the downstream growth analysis.
The AGR and RGR of cfq and wild type plants were calculated using Eq 6 and 7 (respectively) with Δt = 0.35 days. The averaged results shown in Figure 10a, b reveal distinctive growth patterns between the two kinds of plants. The wild type plant is almost three times of cfq when reaching the upper horizontal asymptote, and it have significantly higher AGR than cfq (Figure 10c). The peak of AGR of the wild type plants is 2.28 days later than the peak of cfq. The RGR of cfq is slight higher (0.003) than that of the wild type plants during the first a few days. The decreasing rates of RGR of both kinds of plants are very similar, which means the RGR of the cfq plants was shifted for about 2.5 days to the left, limiting the plant to grow at a fast rate for a shorter period of time (Figure 10d).
Discussion
The experiments on Arabidopsis thaliana wild type and cfq mutant plants show that HPGA reduces the error rate of measuring plant area by half in average if compared with the existing approaches. The low photosynthetic rates and small plant area of cfq suggests that knocking out cfq changes the sensitivity of the energy distribution under fluctuating light conditions to repress leaf growth. If it is true, a key question regarding to growth is whether the plant size difference changes linearly all through the developmental stages, or it actually varies and there is a dominant period. To answer this question, the following functional analysis on the growth curves was conducted.
Second, in Figure 11b, a phaseplane plot of acceleration against velocity describes a basic harmonic process that bounces between two states: potential and kinetic [29]. In terms of plant growth, potential corresponds to resources that are available to bring about some growth activity such as cell proliferation, and kinetic corresponds to the process when the resources are consuming (e.g. cell expansion). At the point that a plant starts to grow from its lower horizontal asymptote, its potential and kinetic are both zero (Figure 11b point a); it overlaps with the point that a plant reaches its upper horizontal asymptote. Potential is also zero when kinetic is maximized (Figure 11b point c) which happens when plant area increase is half of its maximum value ((A_{ a }  A_{0})/2), i.e., A(t)" = 0. In the same figure, point b means no kinetic but maximal potential, and point d means no kinetic but maximal negative potential. The two points represent two critical time points of growth, probably related to the turnover points of cell proliferation, suggesting that point b relates to the most active cell proliferation and point d relates to the most inhibition to cell proliferation.
In our experiments, the wild type plants reach point b at acceleration 8.6 and velocity 384.7 on the 12th day and reach point d at acceleration 8.5 and velocity 389.1 on the 20th day, with the absolute acceleration values more than twice as high as that of cfq (4.0 on the 10th day and 4.1 on the 17th day), suggesting the regulation of cell proliferation in cfq is much less active than wild type under fluctuating light. The period from point b to d is usually defined as the fast growing period. While the wild type plants spent 8 days in the period, cfq plants only spent 7 days and have much lower acceleration and velocity, resulting in much smaller plant areas. The covered area in the phaseplane plot is proportional to the amount of energy transferred during the process. The area ratio 4.88 (WT 6819 against cfq 1398) indicates that in cfq much less energy has been distributed to the growth.
Conclusion
HPGA is a highthroughput phenotyping platform for plant growth modeling and functional analysis. It has two components, PAE (Plant Area Estimation) and GMA (Growth Modeling and Analysis). In PAE, by taking the complex leaf overlap problem into consideration, the area of every plant is measured from topview images in four steps. In GMA, a nonlinear growth model is applied to generate growth curves, followed by functional data analysis. HPGA addresses the leaf overlap problem by counting leaves and then measuring leaf lengths. It avoids segmenting all the leaves from a topview image, which is extremely challenging [20]. Feeding enough and highquality plant area measures to a nonlinear model (3PLM) ensures the robustness and precision of nonlinear plant growth estimation.
The major contribution in HPGA is a new plant area measurement which takes leaf overlap into consideration. We also noticed that the common approach in fluorescence image segmentation is global thresholding [16–18, 35]. However, in a dynamic/natural condition, fluorescence intensity varies from time to time, so that applying a fixed global threshold for all the images may result in significant artifacts and consequently not practical for our study. In HPGA, we processed the images with a reasonable assemble of current images processing techniques that are clearly better than the global thresholding approach.
In HPGA, the leaf lengthtoarea model is genome specific. Changing from one species to another needs to train the model again with the new leaves. Seeking new ways to relate leaf area to detectable attributes is an essential future work. HPGA is developed for plant science research focusing on 2D plant Arabidopsis thaliana. Without knowing the height of each leaf, it is impossible to apply HPGA on any 3D plants that are seen more often in the field. By adding more cameras, we plan to extend HPGA to model the growth of 3D plants such as camelina, tobacco, tomato and bean. Furthermore, it is frequently desirable to develop a growth model with biologically interpretable parameters, which shall be addressed in our future model as well.
Notes
Declarations
Acknowledgements
We thank Dr Gregg Howe, Dr Thomas Sharkey, Dr Xiaoming Liu, Dr Yiying Tong and Dr Jun Li for providing inspiring ideas to improve HPGA. We thank Dr Linda Savage for managing the experiment. The project is supported by Center for Advanced Algal and Plant Phenotyping, Michigan State University to DMK, and Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (grant no. DEFG0291ER20021) to DMK and JC.
Declarations
The funding to support the publication fees is Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (grant no. DEFG0291ER20021) to DMK and JC.
This article has been published as part of BMC Systems Biology Volume 7 Supplement 6, 2013: Selected articles from the 24th International Conference on Genome Informatics (GIW2013). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/7/S6.
Authors’ Affiliations
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