Abstract
The DIF concept states that equal internode length can be achieved with the same difference between day and night temperature irrespective of the mean 24 h temperature. However, the physiological background of the DIF concept is unclear. An attempt to model internode elongation is presented based on three plausible processes, namely (1) the accumulation of elongation requirements during the day, (2) elongation during the night using elongation requirements and (3) the limitation of internode length due to low turgor pressure unable to counter cell wall elasticity. Each reaction rate constant, one per process, depends on temperature according to Arrhenius’ Law. The resulting process‐based model describes internode elongation in time and was calibrated on a chrysanthemum data set. Chrysanthemum plants were grown in growth chambers with rigorously defined day and night temperatures. In total, 16 temperature treatments were applied, resulting from the combination of four day and four night temperatures (16, 20, 24 and 28 °C). Internode elongation was measured for the tenth internode in ten plants per treatment. The percentage variance accounted for, R2adj, was almost 91 %. Transferability of model parameters was shown to exist by cross validation. Simulation of the internode length in time as function of mean 24 h temperature and DIF showed that the DIF concept is not apparent after a growing period of 10 d, but is visible after 20 d. This model structure for describing internode elongation might also be applicable for other plants that show the DIF concept.
Key words: Chrysanthemum, day temperature, DIF, elongation, internode length, model, night temperature
INTRODUCTION
The DIF concept states that plants grown with the same difference between day (DT) and night (NT) temperature will have equal final internode length, regardless of the mean temperature (Jacobson and Willits, 1998; Carvalho et al., 2002). In several species including tomato, lilium and chrysanthemum, temperature combinations resulting in a negative DIF produced shorter plants with smaller internodes, compared with plants growing under a positive DIF (Erwin et al., 1989; Karlsson et al., 1989; Jacobsen and Amsen, 1992; Bertram and Karlsen, 1994; Cockshull et al., 1995). Total stem length results from the average internode length multiplied by the number of internodes (Pearson et al., 1995), and the DIF effect on stem length is mainly a result of its influence on internode elongation (Langton and Cockshull, 1997; Carvalho et al., 2002).
Despite being the object of a large number of studies, the physiological background of the DIF concept remains an open question (Bertram and Karlsen, 1994; Langton, 1998). Many descriptive models have been developed for stem length in chrysanthemum (e.g. Karlsson et al., 1989; Pearson et al., 1995; Khattak and Pearson, 1997). However, these models lack insight into the internode elongation process and only a few have included the DIF effect on stem elongation (e.g. Jacobson and Willits, 1998; Carvalho et al., 2002). The aim of the present work was to develop a process‐based model to describe internode elongation as a function of temperature. This model was calibrated by re‐analysing recently published data on chrysanthemum (Carvalho et al., 2002).
MATERIALS AND METHODS
Conceptual model
The model was developed using a system of problem decomposition (Sloof, 2001). This system is oriented towards underlying processes that cause the observed phenomena, rather than towards the phenomena themselves.
Chrysanthemum, capsicum, salvia and petunia plants have higher stem elongation rates (SERs) during the night than during the day (Bertram and Karlsen, 1994). The main difference in SER between chrysanthemum plants grown under constant daily temperature and under a negative DIF occurred at night (Bertram and Karlsen, 1994). In chrysanthemum, capsicum, salvia and petunia, SER is affected by the irradiance level, such that high irradiance during the day induces high SERs during the following night (Bertram and Karlsen, 1994). Apparently, the presence of sufficient levels of elongation requirements (ERs), obtained during the preceding day, affects stem elongation during the night.
The nature of the ERs is unclear. As irridiance during the preceding day is important for elongation during the night, it might be that ERs are photosynthates. On the other hand, gibberellin (GA) metabolism is involved in the stem elongation process. The average stem length of tomato plants grown for 14 d in contrasting day–night temperature regimes increased considerably following addition of exogenous GA4+7, irrespective of the DIF regime applied and whether GA‐deficient mutants or wild‐type plants were used (Langton, 1998). It has been suggested that temperature affects elongation by influencing the GA concentration and not by changing the sensitivity of the plant (Langton, 1998). Temperature regimes that stimulate extension growth for bellflower showed an increase of physiologically active GA1 and its precursors GA19 and GA44. Reciprocal temperature regimes were accompanied by an increase in the inactive hydroxalated form of GA53, the precursor of GA44 (Jensen et al., 1996). Exactly how GA promotes stem elongation is not completely clear. It was reported for pea that GA changes the orientation of microtubules and cellulose microfibrils, making the cells swell more in length (Duckett and Lloyd, 1994). Another effect of GA is that the activity of xyloglucan endotransglycosylase, the enzyme that hydrolyses and then re‐links hemicellulose, is increased, enhancing wall stretching in pea (Potter and Fry, 1993).
Phytochrome‐mediated changes in GA production triggered stem elongation in cowpea (Fang et al., 1991). Langton (1998) hypothesized that a light‐on or a light‐off signal triggers active GA production or interconversion. This might suggest that parallel to the accumulation of photosynthates during the day, accumulation of inactive GAs takes place, which are converted to the active form by a light‐off signal. The first proposition for the internode length model is that accumulation of ERs is governed only by DT. The second proposition is that only NT governs the conversion of ERs into elongation.
GA enhances the synthesis of enzymes such as invertase, which converts sucrose to glucose and fructose, thereby lowering the water potential which helps to maintain turgor (Miyamoto et al., 1993, 2000). As stem elongation progresses, it will become increasingly hard to sustain turgor in the stem cells as their volume increases. It can be suggested that elongation may be limited by low turgor pressure insufficient to counter cell wall elasticity. The third proposition for the internode length model is that the elongation process is limited by cell wall elasticity, which, as it is mainly a physical process, is governed both by DT and NT. The third proposition implies that, during the day, internode length decay may occur. Indeed, SERs appeared to become negative for small periods during the day when chrysanthemum, salvia and capsicum were grown in a glasshouse, and these periods of apparent negative SER increased considerably when plants were grown under long‐day compared with short‐day treatments (Bertram and Karlsen, 1994). This indicates that stem elongation in glasshouses, started during the night, is sometimes turned into a stem length decay during the day; the shorter the night, the stronger the effect.
Mathematical model
The model is based on kinetic mechanisms, assumed and plausible for that particular process, and which have been developed using well known rules of chemical kinetics (Segel, 1993; Whitaker, 1994). The three propositions for the internode length model are indicative for the processes of accumulation of elongation requirements (ER) [eqn (1)], internode length (L) growth [eqn (2)], and the limitation of L [eqn (3)], respectively.
where ker, kf and kd are the reaction rate constants for the formation of ERs, the formation of L and the decay of L, respectively. These processes describe a system that is expressed by the following set of differential equations [eqns (4) and (5)]:
This set of equations can be solved analytically for constant external conditions [eqn (6)].
where E0 and L0 are the amounts of ERs and L when internode elongation starts. As internode length was not visible when the treatments started, L0 = 0 and therefore also E0 = 0. The internode length development in time can then be expressed according to eqn (7).
Equation (8) describes the development of the internode length in time, starting at a length of zero (t = 0) and finishing at ker/kd (mm, t = + ∞). However, the experiment did not start at t = 0, but earlier when internode elongation was still absent. To take into account that every internode starts to grow at a different point in time, a time shift factor, tshift, is introduced, defined as t = tstart – tshift, where tstart is the experimental point in time when internode elongation starts. Incorporating the time shift factor results in the internode elongation equation used in the data analysis [eqn (8)].
Temperature dependence
Each of the reaction rate constants mentioned (ker, kf and kd) depends on temperature, presumably according to Arrhenius’ Law [eqn (9)].
with Rgas being the gas constant (8·314 J mol–1K–1). The parameter ki,ref stands for the reaction rate constant at the arbitrarily chosen reference temperature Tref (K). The energy of activation Ei expresses the dependence of the reaction rate constant ki on temperature, with i = er, f or d. T (K) is the DT or NT of the growth chambers applicable for the accompanying process. According to the propositions, ker is governed by DT only, kf by NT only, and kd by both DT and NT. The application of kd being dependent on night and day temperature is solved by using two sub‐parameters, kd DT and kd NT, which are only dependent on day and night temperature, respectively, but have a common value of kd,ref.
Experimental set‐up and plant measurements
Cuttings of Chrysanthemum ‘Reagan improved’ were obtained from a commercial propagator and planted in 14 cm pots containing a peat‐based commercial potting compost on 16 May 2001 (replication 1) and 13 Jun. 2001 (replication 2). Plants were selected for uniformity (8 ± 1 leaves per plant; 12 ± 2 cm stem length). After 2 d in a common glasshouse environment (18 °C DT, 16 °C NT and 18 h light), temperature treatments were imposed. A total of 16 temperature treatments was applied, resulting from the combination of four DTs and four NTs (16 ± 0·1, 20 ± 0·1, 24 ± 0·1 and 28 ± 0·1 °C). Plants were grown in growth chambers (2·90 m × 2·20 m × 3·15 m), as individual plants, under ambient CO2 and at a constant vapour pressure deficit of 0·57 kPa. Fluorescent tubes [Philips TL‐58W, colour 84, 99 µmol m–2 s–1 photosynthetically active radiation (PAR) at average plant level] were used for 12 h per day. This light level was similar to that received by plants growing in commercial glasshouses during winter in the Netherlands. Every 5 min, temperature was recorded using a commercial computer system (Hoogendoorn, Vlaardingen, The Netherlands). Since only four growth chambers were available, plants were shifted at the start and end of each day according to their DT and NT treatment. No effect on stem length of moving the plants was observed (Carvalho et al., 2002). A more detailed description is given in Carvalho et al. (2002).
Length of internode 10 of ten plants per temperature treatment was measured non‐destructively using a digital calliper. This internode was chosen because it was not visible when the treatments started. Measurements were repeated for replication 1 on days 0, 5, 10, 17, 21 and 26 and for replication 2 on days 0, 5, 10 ,17, 24 and 28 after the start of the treatments. The experiments ended before day 30 as internode 10 had apparently reached its final length for all temperature treatments as deduced by the lack of any clear increase in internode length over time. Each replicate consisted of ten plants per treatment divided between two trays. Plants were randomly assigned to treatments and trays.
Statistical analysis
Experimental data on internode elongation were analysed statistically using the non‐linear regression routine of Genstat 5 (release 3·2; Lawes Agricultural Trust, Rothamsted Experimental Station, UK). The equations and mathematical description of the model were developed using Maple V (release 4; Waterloo Maple Software, Waterloo, Canada). The data set of the first replication was analysed using the model formulation of eqn (8) together with the temperature dependence according to the Arrhenius equation [eqn (9)]. These data were analysed using temperature and time simultaneously as explaining variables estimating the time shift factor (tshift) per plant and the kinetic parameters (ker, kf, kd, Eer, Ef and Ed) in common for all plants in one optimization. The internode length data set of the second replication was analysed using the model formulation of eqn (8) and the kinetic parameters obtained from the analysis of the first replication data set, estimating tshift per plant. The reference temperature for the Arrhenius equation (Tref) in both analyses was 289 K (16 °C).
RESULTS
Kinetic parameters of the internode length model
The data set of the first replication was used to calibrate the model parameters per tray. Internode length measurements were used, without transformation, for L(t) in eqn (8). Separate analysis per tray showed that a high percentage of variance (R2adj) was accounted for, on average more than 92 %. Using the average parameter values, R2adj was almost 91 % for both analyses (Table 1). Experimental and simulated data, i.e. applying the estimated parameters from Table 1 and estimated tshift values for all plants in the first tray of replication 1, are shown in Fig. 1. At the moment the internode started growing, a sigmoidal behaviour in time was encountered depending on DT and NT in combination with tshift. For some treatments, for instance 16 °C DT and 16 °C NT, it is obvious that the final internode length is not reached by day 30.
Table 1.
Parameter estimates and their standard error (s.e.) for the analysis of the data set of replication 1
| Parameter | Value | s.e. |
| ker,ref (mm d–1) | 2·256 | 0·151 |
| kf,ref (d–1) | 0·1053 | 0·0204 |
| kd,ref (d–1) | 0·0422 | 0·0086 |
| Eer (J mol–1) | 63 810 | 7487 |
| Ef (J mol–1) | 57 011 | 8323 |
| Ed (J mol–1) | 94 696 | 2935 |
| Tref (K) | 289 (16 °C) | |
| R2adj (%) | 90·8 | |
| n* | 780 |
* Number of observations.
Fig. 1. Experimental (symbols) and fitted (solid lines) internode elongation over time for five plants for each combination of day and night temperature.
To validate the model formulation, the model parameters estimated on the data set of the first replication (Table 1) were applied to the data set of the second replication in a separate analysis. In this analysis only the tshift values were estimated. The scatterplot, showing observed against expected internode length data for all temperature treatments, had an R2adj value of 92 % (Fig. 2). A comparison between the distribution of the estimated start day of elongation, tshift, per tray and per replication (Fig. 3) showed that the distributions per replication were more closely related than those per tray. The difference between the tshift distributions of the two replications was about 1 d, indicating that internodes from the first replication started growing, on average, 1 d earlier. Indeed, for the first replication, the length of internode 9 was generally greater at the time when measurements of internode 10 began (data not shown).
Fig. 2. Observed and expected internode length for chrysanthemum of replication 2. Symbols represent the different combinations of day and night temperatures (DT–NT).
Fig. 3. Distribution of tshift, the estimated point in time when elongation starts, encountered during the analysis of the data sets per tray and per replication. White and light grey bars indicate the distribution of tshift for tray 1 and tray 2 of replication 1, and the black and dark grey bars those for tray 1 and tray 2 of replication 2.
Simulation of the internode length
After calibration of the model parameters on the internode length data, it was possible to simulate final internode length as function of DT and NT (Fig. 4). The maximum internode length was achieved by combining a DT of 22 °C and the lowest NT (16 °C). Extrapolating the simulation to temperature ranges just below the experimental temperature range resulted in the maximum internode length after 30 d being achieved by a combination of 22 °C DT and 13 °C NT (data not shown).
Fig. 4. Simulation of the internode length at 30 d after the start of the temperature treatments, as a function of day temperature (DT) and night temperature (NT).
Internode length development as function of DIF (°C) and the mean 24 h temperature, MT24h (°C), is shown at 10 (Fig. 5A and B), 20 (Fig. 5C and D) and 30 d (Fig. 5E and F). Over the range of MT24h, the maximum difference between the minimum and maximum internode length within one DIF was 9 mm at 10 d (Fig. 5B). At 20 d, this maximum difference within one DIF decreased to about 4 mm (Fig. 5D), and increased again at 30 d to about 6 mm (Fig. 5F). The differences between minimum and maximum internode lengths at 10, 20 and 30 d are rather small as the range of internode lengths observed over DIF and MT24h was about 18 mm at 30 d (Fig. 5E and F). However, at 10 d, the range in internode lengths over DIF and MT24h was about 9 mm and this difference was also encountered within one DIF (Fig. 5B). So, at 10 d, no clear DIF response was observed, as was also found by Langton and Cockshull (1997). However, at 20 and 30 d, the range in internode lengths over DIF and MT24h increased rapidly, whereas the variation in internode length within one DIF was much smaller (4 mm at 20 d, Fig. 5D; 6 mm at 30 d, Fig. 5F) than the variation in the internode length at 10 d. Thus, when internode elongation is observed for more than 20 d, the variation in internode length is primarily explained by DIF.
Fig. 5. Simulations of internode length as a function of DIF and the mean 24 h temperature, MT24h, at 10 (A and B), 20 (C and D) and 30 d (E and F) after the start of the temperature treatments.
DISCUSSION
Regression models
Karlsson et al. (1989) used a regression‐based approach to describe final internode length as a quadratic function of mean 24 h temperature and DIF. Pearson et al. (1995) described final internode length by using a regression‐based approach for internode elongation based on the weighted 24 h mean temperature. Both descriptions lack biological significance as plants do not react primarily to DIF or weighted 24 h mean temperature with regard to internode elongation. Carvalho et al. (2002) used the same regression‐based approach and described final internode length as a quadratic combination of DT and NT. This approach was successful for the temperature range examined, but gave poor results when extrapolated to slightly higher or lower temperatures. The advantage of the proposed model is that it is process‐based, indicating that only physiologically viable parameters are included; this means that only actual DT, NT and time are included. As these processes are also likely to dominate at slightly higher and slightly lower temperatures, extrapolation to temperatures just outside the experimental range is more likely to yield plausible results.
The Richards function
The observed sigmoidal behaviour over time encountered for internode elongation (Fig. 1) has been described before. Modelling efforts of this sigmoidal behaviour have centred on the Richards function (Berghage and Heins, 1991; Larsen, 1990; Karlsson and Heins, 1994; Jacobson and Willits, 1998). This empirical equation showed flexibility in describing final internode elongation of different internodes and as a function of photosynthetically active radiation in a glasshouse. Difficulties arise when small differences in the data produce large differences in the predicted parameters, making interpretation of the relationships between parameters and the resulting growth difficult (Jacobson and Willits, 1998). Jacobson and Willits (1998) used an approach where the coefficients of the Richards function were assumed to be independent of environmental factors, and included growth factors to modify the Richards function. Although growth factors were linked to physiological parameters, the application was carried out on an ad hoc basis. This implicates that for a batch, as defined by the products of one harvest, one glasshouse and one cultivar (Schouten et al., 2002), these coefficients and growth factors may be fitted satisfactorily, but that the applicability of the model over batches of chrysanthemum is likely to be limited. The proposed internode elongation model is based on plausible physiological processes occurring in chrysanthemum elongation and is therefore of a more fundamental nature. As these processes occur at the same rate in other batches of the same cultivar and growth conditions, transferability of model parameters is possible. This was shown by applying the model parameters estimated on the data set of replicate 1 to describe the internode length elongation of chrysanthemums of replicate 2 (cross validation, Fig. 2). Application of the same growth conditions on chrysanthemums of other cultivars should be possible, as the same processes are likely to determine internode elongation. The only difference would be the value of the kinetic parameters, but the model structure would be identical. Application of a process‐based model for colour development in cucumbers was shown to be transferable to different cultivars (Schouten et al., 2002). Furthermore, as the processes underlying the internode elongation model also occur in other plant species, the model structure might be applicable for a host of other plant species that show a clear DIF response.
A limitation of the proposed internode length model is that it was formulated and tested for chrysanthemums grown in growth chambers, with rigorously defined light and temperature conditions, and not in a commercial glasshouse. However, this model may be the basis for an expanded version that incorporates dynamically changing temperature and light conditions. For instance, incorporation of dynamically changing temperature conditions is a matter of approaching the statistical analysis differently by applying rate sums for all the reaction rates constants in the model (Tijskens and Verdenius, 2000).
Supplementary Material
Received: 15 March 2002; Accepted: 7 May 2002
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