Modelling photo-modulated internode elongation in growing glasshouse cucumber canopies


Author for correspondence:
Katrin Kahlen
Tel: +49 511 762 2637


  • Growing glasshouse plant canopies are exposed to natural fluctuations in light quantity, and the dynamically changing canopy architecture induces local variations in light quality. This modelling study aimed to analyse the importance of both light signals for an accurate prediction of individual internode lengths.
  • We conceptualized two model approaches for estimating final internode lengths (FILs). The first one is only photosynthetically active radiation (PAR)-sensitive and ignores canopy architecture, whereas the second approach uses a functional–structural growth model for considering variations in both PAR and red : far-red (R : FR) ratio (L-Cucumber). Internode lengths measured in three experiments were used for model parameterization and evaluation.
  • The overall trends for the simulated FILs using the exclusively PAR-sensitive model approach were already in line with the measured FILs, but they underestimated FILs at higher ranks. L-Cucumber provided considerably better FIL predictions under various light conditions and canopy architectures.
  • Both light signals are needed for an accurate estimation of the FILs, and only L-Cucumber is able to consider R : FR signals from the growing canopy. Yet this study highlights the significance of the PAR signal for predicting FILs as neighbour effects increase, which indicates a potential role of photosynthate signalling in internode elongation.


Light quantity and quality are major determinants of internode growth. Reductions in both photosynthetically active radiation (PAR), and red : far-red ratio (R : FR) result in similar shade avoidance responses, such as increased internode elongation (Franklin & Whitelam, 2005; Vandenbussche et al., 2005; Kurepin et al., 2007; Franklin, 2008). In plant communities, the R : FR ratio seems to act as an early competition signal, while at higher leaf area indices (LAIs), the light quantity signal becomes more crucial (Ballaré, 2009). However, most knowledge about the patterns that control plant response to light signals is derived from hypocotyls or, in the case of density effects, even-aged seedling stands (Ballaréet al., 1987, 1989; Aphalo et al., 1999; Shinkle et al., 2004, 2005). In cucumber seedlings, decreasing the R : FR ratio of light perceived at the hypocotyl promotes the growth rate of the hypocotyl. A reduction in the amount of blue light reaching the hypocotyl triggers a similar response (Ballaréet al., 1991a). Different organs are sites of light signal perception. In small Sinapis alba L. plants, for example, both the growing internode and the primary leaf above the internode may perceive R : FR signals, inducing increased stem elongation rates. Internode signal perception results in a rapid response within minutes, whereas the response to the leaf’s signal is delayed by a few hours (Morgan et al., 1980). It is unclear to what extent the various signals from the different organs contribute to internode elongation, especially if we focus our analysis on internode lengths in a glasshouse cucumber canopy grown under naturally varying PAR conditions where the plants are vertically trained and reach a height of 2–5 m (Boonekamp, 2005). One problem to address is the timing and duration of the PAR-sensitive time window for the growth of the individual internode and the contribution of the R : FR signal to final internode length (FIL). Functional–structural plant modelling has already been shown to be useful for understanding the feedback processes among canopy architecture, light distribution and canopy growth (Prusinkiewicz & Rolland-Lagan, 2006). With respect to irradiance and photosignalling in canopies, in particular, Chelle et al. (2007) concluded that three-dimensional plant modelling is superior to the currently used experimental approaches. Several recent studies demonstrated the usefulness of Lindenmayer-systems (L-systems) in constructing functional–structural models (for overviews, see Vos et al., 2007, 2010; Fourcaud et al., 2008; Hanan & Prusinkiewicz, 2008). The formalism of L-systems was introduced by Aristid Lindenmayer as a theoretical model of biological development (Lindenmayer, 1968, 1975). One of the most appealing features of L-systems is the ability to describe the apparent complexity of plants according to a small number of rules (Prusinkiewicz & Lindenmayer, 1990; Room et al., 1996; Prusinkiewicz, 1998; Allen et al., 2005; Lopez et al., 2008).

The objective of this work was to evaluate the significance of both light quantity and quality signals for an accurate prediction of internode lengths in a growing cucumber canopy. We tested the following hypotheses: it is mainly light quantity that controls FIL in a young sparse canopy; and the dynamic transition from a young canopy to a canopy with high LAI is accompanied by varying light quality conditions, which additionally control FILs.

Materials and Methods


Three similar experiments were carried out in experimental glasshouses at the Institute of Biological Production Systems, Leibniz Universität Hannover, Germany (lat. 52°23′N, long. 9°37′E). The setup of Expt 1 was described in detail by Kahlen & Stützel (2007). Cucumber plants (Cucumis sativus L. cv Aramon) were transplanted into the experimental glasshouses at the four-leaf stage in either row or isometric distribution at a density of one and two plants m−2, respectively. In the sparse isometric canopy, the distance from an individual plant to each of its direct neighbours was 108 cm. Each of the three replications covered an area of c. 30 m2. Transplanting took place in the first week of May for the first experiment, and in August for the other two experiments. Plants were trained vertically up to a horizontal wire of c. 2 m height. At transplantation, internodes at rank four had a length of c. 3 cm. In all experiments, the lengths of all internodes (cm) of three plants located in the centre of each replication were measured weekly using a digitizing method described by Kahlen & Stützel (2007). Temperature set points for the glasshouses were 20 : 16°C (day : night) and ventilation was started at 24°C. Daily PAR (μmol m−2 s−1) data were recorded by a weather station located next to the glasshouses.

Model approaches

For model conceptualization and parameterization, we stepwise analysed FILs measured in treatments with isometric distribution at a density of one plant m−2. At first, we tried to quantify the effect of PAR on FILs, resulting in a PAR-sensitive model approach for prediction of FIL, called MA1 (Fig. 1a). In the second step, we tried to evaluate the effect of the R : FR ratio prevailing at the stem level on internode growth and derived a corresponding response function. Model approach 2, called MA2, was achieved by adding a term for this contribution of the R : FR ratio to FIL to MA1 (Fig. 1b).

Figure 1.

Overview of conceptualization, parameterization and evaluation of both model approaches. Step 1 covers the exclusively photosynthetically active radiation (PAR)-sensitive model for final internode length (FIL) prediction (MA1) (a), whereas Step 2 results in a red : far-red (R : FR)-sensitive extension of MA1 (MA2) (b).

Step 1 – PAR-sensitive model approach for predicting FILs

A reduction in PAR above seedlings increases the length of the first internode in sunflower (Kurepin et al., 2007). Similarly, the hypocotyls of cucumber plants respond to reductions in blue light reaching the whole shoot by increased extension growth (Ballaréet al., 1991a). Even though a reduction in R : FR also increases hypocotyl growth (Ballaréet al., 1991a), we assumed in a young crop at a density of one plant m−2 and low LAI (< 0.3 m2 m−2) that the light quality signal from the surrounding neighbours was still negligible. Thus, the underlying hypothesis was that it is mainly light quantity, PAR, that controls FILs of young cucumber plants in a very sparse canopy. In all our experiments, plants were vertically trained, which fixed the apex and growing part of the plant at the top of the canopy. Thus, we ignored the possibly relevant sites of light quantity signal perception and simply considered PAR above the canopy. We analysed the relationship between daily PAR or mean PAR above the canopy for up to 5 d and FILs for internodes 5–9 grown in a sparse isometric cucumber canopy (LAI < 0.3 m2 m−2 as measured 1 wk after transplanting). The exact phase of PAR sensitivity for the FILs is unknown. Studies on light signalling in plants, their transduction pathways and crosstalk mainly focus on the instantaneous changes in growth rates of seedling shoots (Parks et al., 2001). Moreover, the analysis of the light-induced effects on stem elongation in the study of Ballaréet al. (1991a) started within the phase of the maximum growth rate of the cucumber hypocotyl (derived from their Fig. 1). On the other hand, cell division in cucumber leaves has a phase of high PAR sensitivity covering a 6 d period beginning not later than 6 d before leaf unfolding, and final cell number is related to final leaf area (Wilson, 1966; derived from their Fig. 1b,d). If we assume similar periods for internode elongation most sensitive to the PAR, the analysis of the correlation between FILs and PAR data has to be extended to the phase before the maximum internode extension (cm d−1) is reached. Therefore, we also considered periods beginning 10 d before the individual internode has reached its maximum growth rate. The latter was observed to occur at an internode length of c. 3 cm. The end of the analysis period was defined when the internode had reached at least 90% of its final length (Kahlen, 2006). Model approach 1 becomes:


where PAR in terms of the mean PAR (μmol m−2 s−1) of the most sensitive time period, slope and intercept is derived from linear regression analyses. The data of Expt 1 were used for model parameterization, whereas Expts 2 and 3 provided evaluation data (Fig. 1a). For model evaluation, root mean squared deviation (RMSD, cm), bias (cm2) and accuracy (%) were determined:


where xi and yi are measured and simulated values (cm), respectively. The definition of accuracy is a modified approach from Rakocevic et al. (2000).

Step 2 – PAR- and R : FR-sensitive model for predicting FILs

The second step is based on the assumption that light quality is the key factor controlling the variability in FILs with respect to the growing canopy. Thus, the extension of MA1, model approach 2 (MA2), becomes:


with f (RFR) (cm) representing a function of the R : FR contribution to FIL. In cucumber, neither the exact timing of R : FR sensitivity nor the exact elongation response of the individual internode to the R : FR signal is known. Moreover, the relevance of specific sites of signal perception is not unambiguously verified (Morgan et al., 1980; Thompson, 1995), but there is clear evidence for the role of the vertically oriented internodes as sites for light quality signal perception (Ballaréet al., 1991a). Therefore, we considered the R : FR ratios perceived by the individual internode at its maximum growth rate and the whole stem’s R : FR ratio in the same period as the light quantity signal. The mean value of these two ratios is taken as R : FR signal. In the early growth phase of the canopy, where we assumed that it is mainly PAR that controls the FILs as indicated by Step 1, the prevailing R : FR conditions at the internode or the whole stem do not additionally affect FIL. Thus, above an upper threshold of the R : FR ratio, R : FRu, the function f(R : FRu) should become 0. In uniform stands at high LAI, again the light quantity signal plays an important role in shade avoidance responses of the stem (Ballaréet al., 1991b; Vandenbussche et al., 2005; Ballaré, 2009). This could mean that the R : FR signal exerted by neighbours has reached its maximum influence on FILs. This hypothesis is supported by findings of Aphalo et al. (1999), which showed that above a threshold density, an increase in canopy density in silver birch seedlings does not increase plant height. In these cases, the amount of structural dry matter available for the internode growth or biomechanical constraints might become the limiting factors (Schmitt et al., 2003; Pearcy et al., 2005; Mazzella et al., 2008; Hamant & Traas, 2010). Therefore, the influence of the R : FR signal had to be limited for values lower than a threshold value, R : FRt. In between these thresholds, we assumed a linear relationship between the R : FR ratio and its contribution to the FILs.

Estimation of light quality conditions for the parameterization of the R : FR response curve f

We initially analysed R : FR ratios prevailing at the growing internode and at the whole stem in just one treatment of Expt 2. These light quality data were provided by a functional–structural plant model of cucumber, L-Cucumber (Kahlen et al., 2008), where measured FILs of the sparse isometric canopy with corresponding PAR data of Expt 2 were used as input data. The threshold R : FR ratios, R : FRu and R : FRt, were estimated from these simulated R : FR data. R : FRu was set to the maximum simulated R : FR signal. The maximum difference between the length of an internode and the corresponding simulated FIL based on MA1 in Expt 1, dmax, was used as the upper limit for the R : FR contribution to FIL below a threshold, R : FRt. Subsequently, R : FRt was estimated using a nonlinear algorithm for optimization (Generalized Reduced Gradient, underlying algorithm of the solver function in Excel, 2007) to minimize the squared differences between measured and simulated FILs. Slope and intercept of f(RFR) were calculated from its two solutions f(R : FRt) = dmax and f(R : FRu) = 0.

L-Cucumber is based on a parametric L-system (Prusinkiewicz & Lindenmayer, 1990; Prusinkiewicz, 1999). It is a typical functional–structural plant model, because the production of dry matter and growth of the leaves are dependent on the local light conditions of each individual leaf within the canopy, which are provided by a coupled light model (Kahlen et al., 2008). In previous work, we showed that this model simulates canopy architecture realistically (Kahlen, 2006, 2007; Kahlen et al., 2008). Originally, internodes simply elongate from a minimum value at internode appearance to a constant FIL. The increase in internode length per d is equal to 50% of the difference between actual internode length and FIL. For this study, the code was translated into the L + C language allowing for fast simulation runs performed by the lpfg plant modelling program (Karwowski & Lane, 2008) in L-studio (4.2.13), while using the light simulation program QuasiMC of Cieslak et al. (2008). To consider the daily changes in the light, each time step now comprises 1 d.

To estimate light quality conditions needed for the parameterization of the R : FR response curve f, we ran a specific simulation scenario representing the growing isometric cucumber stand at a density of one plant m−2 of Expt 2. The virtual canopy consisted of 19 plants, arranged in an isometric distribution, with 108 cm distance between direct neighbours. This canopy composition was established in the axiom of the L-system. In agreement with the cultivation practice in the real experiments, virtual plants were located on a white floor, which reflects 80% of the incoming light. Fig. 2(a) shows a top view of such a canopy with each plant having 20 leaves.

Figure 2.

The graphical output of the light simulation program QuasiMC of Cieslak et al. (2008) shows a virtual canopy of 19 plants with isometric distribution at a density of one plant m−2 in a top view (a) and side view (b). The distribution of the light sources is controlled by the settings in QuasiMC. It represents the mean light distribution in Hannover, Germany, on a clear day in mid-July. Brighter light sources emit more light than darker ones. The data of the plant in the centre of the virtual canopy are used for the analyses.

New elements were added to L-Cucumber, which now provide local light data of the internodes. Four virtual light sensors were located around each internode, so that each sensor was oriented perpendicular to one cardinal point (east, south, west or north compass point). An internode sensor was represented by a rhombus with the length of the associated internode and constant diagonal of 1 cm (Fig. 3). The rhombus was selected, because it is the simplest geometric object with two axes of symmetry that can be dealt with in the light model. At each time step, the geometrical data of all leaves and internode sensors were sent to the light simulation model QuasiMC, which estimated the local light conditions and returned these data to the plant simulator program lpfg in L-Studio (Cieslak et al., 2008; Cieslak, 2009). A hemispherical approximation of the sky based on the CIE standard clear sky model (CIE 110, 1994) was used as light source, which accounts for a bright region around the sun and a slight brightening around the horizon (Fig. 2). To mimic the prevailing light conditions in the real experiment, the sky model considered the location of the experiments, Hannover, Germany, by its longitude and latitude, and was set up for the path of the sun on a specific day, Julian day 200, which represents the middle of the cultivation period in Expt 2. The daily variation in PAR is accounted for by adjusting the radiation intensity of each source. The leaf canopy interacts with the light model, because each virtual leaf reflects 6% and transmits 7% of the incident PAR and red light (R, 660 nm), whereas leaf reflectance and transmittance of far-red light (FR, 730 nm) are 38 and 45%, respectively (Kahlen et al., 2008). For the stochastic sampling in QuasiMC, the randomized quasi-Monte Carlo method with Korobov point set generator is selected. The path tracing algorithm is executed 10 times, each with 1 048 576 rays tracing through the virtual scene. It uses one ray per spectral range, and a local light model with Lambertian distribution function is applied to determine the direction and energy of a ray intersecting with the canopy (for details, see Cieslak et al., 2008; Cieslak, 2009). The internode sensor data from QuasiMC, here incident R and FR light on the outer surface of the rhombus (Fig. 3), are used to calculate the individual R : FR for each sensor and then averaged over the four sensors per internode to determine the internode-specific R : FR ratio. The stem R : FR ratio is defined as the average R : FR ratio over all internodes weighted by their lengths. To avoid model artefacts, simulations were repeated five times, each run with a slightly different initial orientation (within [−30°, +30°]) for each plant in the virtual canopy. Moreover, a random factor was added, resulting in a uniformly distributed deviation (within [−5°, +5°]) of the originally fixed phyllotaxis angle. Therefore, average light quality data from the five simulations were used for analyses.

Figure 3.

Schematic representation of the virtual plant (light grey) and the internode sensors. There are four rhombuses per internode with the same length, diameter and height above ground. To indicate that each of them is oriented towards a compass direction, south and west are represented by arrows. Red : far-red (R : FR) ratio of incident light on the outer surface is used to calculate internode R : FR values.

Evaluation of the PAR- and R : FR-sensitive model for predicting FILs

For model evaluation, we extended the internode growth model used in L-Cucumber by including the light-sensitive FIL-prediction model, MA2, and compared both simulated FILs and length of growing internodes with measured data. Simulation runs with L-Cucumber were done for plant densities and plant distributions corresponding to the remaining treatments of Expts 2 and 3 (Fig. 1b). In all these simulation runs, only daily PAR data were used as model input, whereas the local R : FR ratios were affected by the interplay of canopy architecture and optical light properties of the leaves.


Step 1 – PAR-sensitive model approach for predicting FILs

Parameterization  The exclusively PAR-sensitive model approach, MA1, became:


where PAR4d is the mean PAR (μmol m−2 s−1) of 4 d starting 6 d before the internode has reached its maximum growth rate and < 0.01 for slope and intercept (Fig. 4).

Figure 4.

Light quantity (mean photosynthetically active radiation (PAR) of 4 d (PAR4d)) response curve for final internode lengths (FILs) of cucumber. Closed symbols represent the mean of measured data of five internodes (ranks 5–9) in the sparse isometric canopy of Expt 1; bars represent standard deviations of the mean data (= 9). The curve is the result of a linear regression approach, FIL = 13.40 − 0.014 · PAR4d (Table 1 and Model approach 1 (MA1)), with a standard error of estimate of 0.36 cm, and standard error of the slope and intercept of 0.0023 and 0.78, respectively.

The RMSDs varied between 0 and 1 cm, an example of which is shown for daily PAR and mean PAR of 4 d (PAR4d) in Table 1. Switching from single day PAR data to PAR4d increased the mean coefficient of determination from 0.37 to 0.62, if only data sets with negative slopes were considered. Using PAR4d, the best fit with similar RMSDs and the same coefficients of determination were obtained 6 d before (N = −6) and 1 d after (N = 1) the internodes reached their maximum growth rates (Table 1). In both cases, shifts of 1 or 2 d backward or forward resulted in severe reductions of the coefficients of determination. To assess whether both time windows are relevant for predicting FILs, both parameter sets were tested for the corresponding PAR4d of the full range of measured FILs from rank 5 to 20. While the early time window resulted in a RMSD of 0.8 cm, an accuracy of 90% and a slight underestimation of FILs, the data of the later time window significantly increased RMSD to 1.8 cm with an accuracy of 78% and an overestimation of 1.3 cm. Therefore, we selected the earlier time window for MA1.

Table 1.   Statistical analysis of the relationship between final internode length (FIL) in cucumber and light quantity above the canopy
NPAR (μmol m−2 s−1)PAR4d (μmol m−2 s−1)
  1. RMSD, root mean squared deviation; PAR, daily photosynthetically active radiation; PAR4d, mean PAR data of 4 d. N, number of days after the internode had reached a length of 3 cm.

  2. The model was: FIL = ax b, where x is PAR or PAR4d. FILs were measured data at ranks 5–9 from Expt 1.


Evaluation  The application of MA1 to the corresponding PAR data of both experiments revealed an overall trend for the simulated FILs, which was in line with the trend in measured FILs of plants grown in any considered canopy from both years; and a good agreement of FILs at lower ranks followed by a systematic underestimation of FILs at higher ranks (shown for the sparse isometric and the dense row canopies in Fig. 5). The experiments showed considerable differences in the time course and variation of PAR (Fig. 6a,b). However, the fluctuating environmental conditions in Expt 3 neither reduced the accuracy of the predictions nor increased the bias of the simulated FILs compared with Expt 2 (Table 2).

Figure 5.

Measured final internode lengths (FILs), in Expt 2 (a, c) and Expt 3 (b, d) (closed circles) are compared with simulated FILs (open symbols). Model approach 1 (MA1) estimates FIL based only on light quantity data above the canopy (open triangles), whereas MA2 considers both light quantity and quality (open circles). The canopy compositions are: isometric distribution at a density of one plant m−2 (a, b) and row distribution at a density of two plants m−2 (c, d). Bars represent standard deviations of measured and simulated values in MA2. All data are evaluation data except for measured FILs of the sparse isometric canopy in the case of MA2. These data are used for parameterization of MA2 (see also Fig. 1).

Figure 6.

Time course of daily photosynthetically active radiation (PAR) (closed stars) and mean PAR of 4 d (PAR4d, open stars) in Expts 2 (a) and 3 (b).

Table 2.   Evaluation of the two model approaches for final internode length (FIL) prediction in cucumber
 Expt 2Expt 3
  1. RMSD, root mean squared deviation. I1, isometric distribution at a density of one plant m−2; I2, isometric distribution at a density of two plants m−2; R1, row distribution at a density of one plant m−2; R2 row distribution at a density of two plants m−2.

  2. MA1 estimates FILs only from light quantity data (Step 1), whereas MA2 also considers changing light quality conditions in terms of the red : far-red (R : FR) ratio perceived at the stem (Step 2). FILs at rank 5–20 were considered.

RMSD 0.860.700.790.741.390.871.34
Accuracy 0.910.920.920.920.860.910.87
Bias 0.01−

Simulated FILs for the sparse canopies (one plant m−2) showed similar prediction qualities (Table 2, Fig. 5a,b). The comparison of simulated data with FILs measured in denser canopies (two plants m−2) with either isometric plant distribution or row distribution (Fig. 5c,d) showed significantly increased RSMDs, reduced accuracies and biases of up to 2 cm. At this density, deviations were the largest for row canopies (Table 2). This treatment revealed the only systematic difference in the comparison between measured and simulated data. In the third experiment, there was also a systematic underestimation of measured FILs at the lower ranks (Fig. 5d), while in Expt 2 a similar agreement was obtained as in all other treatments (Fig. 5c).

Step 2 – PAR- and R : FR-sensitive model for predicting FILs

Parameterization  The estimated R : FR ratio at the whole stem decreased monotonically with increasing rank, whereas a cyclic pattern of the R : FR at the internode at its maximum growth rate was apparent (Fig. 7a). In the latter case, there was no overall decreasing trend. Taking both signals together in terms of a mean value resulted in an overall decreasing trend of this combined R : FR signal for ranks above seven (Fig. 7a).

Figure 7.

Simulated red : far-red (R : FR) ratios perceived at the whole stem (closed circles) and the growing internode (triangles) 1 d after appearance in the virtual canopy at an age of 20°C d (a). R : FR data (circles) are output data from five simulation runs with L-Cucumber, each run with a slightly different single plant orientation, for a virtual canopy representing an isometric cucumber stand with a density of one plant m−2. Measured final internode lengths (FILs) of Expt 2 and corresponding radiation data (mean photosynthetically active radiation (PAR) of 4 d (PAR4d)) were used as input data. Bars represent standard deviations of the mean data of five simulation runs. (b) Estimated R : FR-contribution to FILs (solid line). The fitting procedure for the response curve is described in the text. The dotted and dashed lines represent maximum and minimum R : FR contributions to FILs, respectively.

The estimated thresholds for the R : FR signal were R : FRu = 0.91 and R : FRt = 0.24 with f(R : FRt) = 2.6 cm. The slope and intercept of f(RFR) within these thresholds were −3.87 and 3.52 cm, respectively (Fig. 7b). This resulted in a standard deviation of the simulated FILs of 0.7 cm for the parameterization data.

Incorporating the light-quality-sensitive FIL model MA2 into L-Cucumber showed considerably better FIL predictions for the sparse isometric canopy in Expt 2 (Fig. 5a), with an improved accuracy and a decrease of the RMSD by almost 40% compared with MA1, and negligible bias (Table 2). The trend in FILs over the ranks was now almost perfectly mimicked. Only at rank 9, 13 and 16 did small deviations occur. Interestingly, the standard deviations for the simulated data at these ranks were the highest. This indicated that small changes in the plant orientations in the virtual canopies resulted in a large variation of the simulated FILs for these internodes. On the other hand, the trend in the FIL data seemed to show a one-rank shift over the range of ranks 8–14, that is, a reduction in simulated data from rank 8 to 9, while measured FILs showed a reduction from rank 9 to 10, and so on.

Evaluation  L-Cucumber with MA2 significantly improved FIL predictions for all canopy compositions compared with the exclusively PAR-sensitive model approach, MA1, both for light conditions in the parameterization (e.g. Fig. 5c) and for entirely different light conditions (Fig. 5b,d).

In all cases, there was a general agreement between measured and simulated data using MA2 (Table 2). For the sparse isometric canopy in Expt 3 (Fig. 5b), for example, the RMSD was reduced and the accuracy increased (Table 2). Even though both daily PAR and mean PAR of 4 d showed large variations and alternating patterns (Fig. 6b), the FIL estimates based on MA2 showed almost no bias. Deviations occurred at ranks 9, 10, 12, 16 and 20. For the upper five internodes, the increasing trend in FILs of the measured data seemed to be excessively pronounced by the simulated data. For all other treatments, small phases of overestimation or underestimation occurred, for example, in the high-density treatments in Expt 2 at ranks 10–13 (Fig. 5c). Nevertheless, in Expt 2, simulated data were of uniform quality, while under different light conditions the dense treatments showed slightly reduced prediction qualities (Table 2).

To visualize the overall quality of the resulting internode growth model, simulation results for internode lengths at two different time steps for the youngest six internodes are shown for the sparse isometric canopy in Expt 3 (Fig. 8). Measured and simulated data were in good agreement even for growing internodes such as those at ranks 13, 14, 15 in measurement 1, M1, and 19, 20, 21 in measurement 2, M2.

Figure 8.

Measured internode lengths (closed symbols, = 3) of six internodes at the top of the plants with a shift of 7 d between the two measurements, M1 and M2, in the third experiment and corresponding simulated data based on simulation runs with L-Cucumber using MA2. Here, calculated final internode lengths (FILs), depend on both light quality and quantity (open symbols for simulated lengths, = 5). Bars represent standard deviations of measured and simulated values, respectively. In the virtual and real experiments, plants were grown in canopies with isometric distribution at a density of one plant m−2.


The presented model approaches mainly depend on the response curves of the FILs to the different light signals, as well as on the underlying assumption about the time window for signal sensitivity.

The postulated linearity of the light quantity signal response to FILs (Fig. 4) is not directly supported by data in the literature, except for being the simplest model approach. It is in line with the empirical model approach of Gautier et al. (2000) for FILs in white clover. Even though their model approach is parameterized for a general polynomial function with respect to PAR and R : FR, the resulting shape of their response curve is nearly linear for constant R : FR within the domain of the response curve in Fig. 4. The measured PAR4d data during the initiation of leaves five to nine showed a slightly decreasing trend in Expt 2 and exceeded the domain of the parameterization data set, while in Expt 3, light data covered c. 50% of this domain with changing pattern. Results of previous studies on the effect of light on tall cucumber plants (Hao & Papadopoulos, 1999; Ménard et al., 2006; Trouwborst et al., 2010) do not allow the timing of the PAR-sensitive window to be determined, because of no or too little variation in daily PAR over time. Thus, we compared different hypotheses concerning the timing of the PAR signal and then selected the time window resulting in both biologically meaningful parameter sets and the closest correspondence between measured and simulated data. A similar method for using a model to test scientific hypotheses has been recommended by Wallach (2006). Our results highlight that the early growth phase of an internode, that is, four successive days starting 1 wk before the internode has reached its maximum growth rate, seems to be particularly sensitive to changes in the PAR signal. This coincides with the PAR sensitivity of the cucumber leaf in its early growth phase. High light and, in consequence, enhanced photosynthate cause greater cell division rates and larger final leaf areas (Wilson, 1966). But in the case of internodes, higher PAR4d values, which correspond to higher integrals of PAR, and presumably larger amounts of accumulated photosynthate, cause a decrease in elongation. Almost all studies on the signalling effects of PAR or blue light on internode elongation demonstrate the sensitivity of the maximum growth rate (Ballaréet al., 1991a). Moreover, the response to blue light is known to be immediate (Parks et al., 2001). Thus, another signalling system might be involved in internode elongation. The fact that accumulated PAR has explanatory power (Fig. 5) might indicate that photosynthate plays a role in internode elongation.

The response of the FILs to the light quality signal was derived from a functional–structural plant model. This noninvasive and nondestructive method is in line with recently proposed applications of the three-dimensional plant modelling approach (Chelle et al., 2007). In a simulation study using a three-dimensional virtual model, Evers et al. (2007) tested different types of response curves describing local R : FR effects on tillering of wheat. Based on simulation results on the number of tillers, they finally suggested a specific type of response function. Our internode length data allowed for a parameterization of the R : FR response curve, because, in contrast to bud fate, internode length is a quantitative trait. Although photobiology predicts a hyperbolic relationship between R : FR signals and plant response, we have chosen a triphasic linear response of FILs for reasons of simplicity (Smith, 1982; Ballaréet al., 1989). Future work should try to improve the consistency of the model with photobiological evidence, as well as reduce uncertainties about the sites of signal perception and the time window of sensitivity.

The prediction quality of the R : FR-sensitive model approach, MA2, also depends on how accurately the virtual crop model predicts canopy architecture and on the precision of the R : FR signals provided by the coupled light model. The evaluation of these aspects was beyond the scope of this work, but clearly needs further investigation. In the extended L-Cucumber model, only final lengths of internodes were predicted using MA2. It is also possible to model the adaptations of the daily or linear growth rate to the environmental conditions (Fournier & Andrieu, 1999; Tardieu et al., 2000; Schouten et al., 2002; Fournier et al., 2005; Dauzat et al., 2008 or Song et al., 2008), but none of these studies considered light as a signal for internode elongation. Moreover, such an approach requires a high resolution in the measurement sequence, and in our experiments, plants were digitized only once a week, whereas the internode growth process finished within 2 wk (see also Fig. 8). Interesting model approaches exist that explicitly include the effect of hormones on internode elongation (Buck-Sorlin et al., 2005, 2008). However, parameterization at the biochemical level is difficult and there are interactions not only with light quantity and quality, but also with plant water status.

For all light conditions, plant densities and distributions considered, simulated FILs based on MA2 were significantly better than those based on the exclusively PAR-sensitive model approach, MA1. The latter resulted in an underestimation of measured FILs at almost all higher ranks, which became more severe in canopies with high plant density (Table 2, Fig. 5). The stepwise model conceptualization allows the underestimation to be attributed to the neglect of canopy growth, that is, the reduced R : FR ratio reflecting from the surrounding plants. This is in line with shade avoidance theory (Vandenbussche et al., 2005). Here, a comparison of the simulation results showed that, for example, in a fully grown sparse isometric canopy c. 20% of individual FILs can be attributed to the R : FR signal. Despite all limitations, the FIL predictions by the exclusively PAR-sensitive model followed the overall trends of the measured FILs over all ranks with increasing and decreasing pattern inverse to the PAR4d trend. This characteristic was clearly identifiable in all treatments and light conditions and therefore seems to indicate the fundamental role of the PAR signal in predicting variation in FILs. It particularly supports the results about the timing and duration of the PAR-sensitive phase, which is the most relevant for an accurate prediction of FILs in growing glasshouse cucumber canopies.

Photosynthetically active radiation also influences leaf area (Wilson, 1966; Cookson & Granier, 2006), which in turn might influence the proximity signal from the neighbours before canopy closure (Ballaréet al., 1990; Franklin & Whitelam, 2005). In contrast to the final leaf area profile along the stem, which shows a clear pattern with three distinct phases of smaller, larger and again smaller leaves from the bottom upwards (for cucumber (Kahlen, 2006) and other species (Granier & Tardieu, 2009)), there is no such characteristic profile of internode lengths along the stem. The leaf profile is essentially attributed to the relationship between leaf position and the development of the reproductive organs (Granier & Tardieu, 2009), whereas our results highlight the importance of the effect of fluctuating light quantity and quality conditions on FILs. Position-specific data are not needed to estimate internode length accurately.

Furthermore, variations in other environmental factors might alter the internode growth process (Ballaréet al., 1991a). In particular, in a glasshouse production system, the interplay of CO2, temperature and light of specific wavelengths might have an affect on canopy architecture (de Koning, 1992; Xiong et al., 2002; Christophe et al., 2006; Folta & Maruhnich, 2007; Cowan & Reekie, 2008; Patil & Moe, 2009; Trouwborst et al., 2010).


The assumed hypotheses – in MA1, that mainly PAR controls FILs of plants in young sparse canopies, and in MA2 that varying light quality conditions additionally control FILs – were not unequivocally validated by this study. But both light signals are needed for an accurate estimation of the FILs and only L-Cucumber is capable of considering R : FR signals from the growing canopy. Interestingly, our results highlight the significance of the PAR signal in predicting FILs as neighbour effects increase, which indicates a potential role of photosynthate signalling in internode elongation. Thus, it would be fruitful to conduct further research on the effect of light on internode elongation in a growing canopy. The focus should be on the various signalling systems; the interplay of timing, sensitivity and site of signal perception; and the up-scaling on whole canopy level.


We would like to thank Mikolaj Cieslak, Brendan Lane, Przemyslaw Prusinkiewicz and Dirk Wiechers for their support and inspiration, and both anonymous referees for the very valuable and constructive comments on the manuscript. This project has been supported by the German Research Foundation (DFG).