Modeling shoot elongation retardation due to daminozide in Chrysanthemum.

R.U. Larsen1 and J.H. Lieth

Department of Environmental Horticulture, University of California, Davis 95616-8587, USA

1address: Swedish University of Agricultural Sciences, Department of Horticultural Science, Box 55, S-23053 Alnarp SWEDEN
 

ABSTRACT

Two mathematical models were developed to simulate shoot elongation of chrysanthemum [Dendranthema × grandiflora (Ramat.) Kitamura] in response to single applications of daminozide at various concentrations. Both models use a modified Richards function to predict the daily increment in shoot growth of the top shoot on pinched plants. The growth retardation effect was described by using a linear and an exponential function. When fitted to collected data, no significant difference was recorded in the accuracy with which the two models represented the daminozide effect on shoot growth. The intial elongation rate inhibition was found to vary with growth regulator concentration. The persistence of the daminozide-effect on shoot elongation was found to be approximately 34 ± 1.2 days. In validation trials both models showed a slight tendency of over-estimating shoot elongation of retarded plants with R2 values ranging from 0.9966 to 0.9996 depending on the treatment studied. The results also showed that the models could be used to simulate the daminozide effect at different times of application. The use of the models as prediction tools is discussed.
 
 

INTRODUCTION

It is a common practice in horticultural potted-plant production to use chemical growth regulators for controlling shoot elongation. Dosage and application recommendations are generally based on empirical trials where the effects on final plant height of various amounts of active ingredient at various times are observed. While this experimental approach has the advantage of producing quick and easily interpreted results, the utility of this information is usually limited to the conditions employed during the studies. The experiments must therefore be repeated when alternate cultural and climate conditions are contemplated. Furthermore, since only final plant height is considered, the collected information cannot be used as a base for short term predictions during the process of shoot growth.

The increased use of computers in greenhouse management has resulted in requests from growers for software, based on mathematical models, for use in prediction or simulation of crop performance associated with various horticultural procedures under various climatic conditions. In addition to describing the final results, such models provide an alternate way of describing information so that dynamic patterns can be visualized and short-term crop progress can be monitored. The goal of this project was to develop such a model for the effect of the growth regulator daminozide (B-Nine or ALAR) on shoot elongation.

B-Nine, the formulation of daminozide marketed for ornamental crops (Uniroyal Chemical Co., Middlebury, CT, USA), has been used for growth retardation for more than twenty five years. It is still one of the most used retardants in pot plant production. It is normally applied to the plant as a foliar spray. Its effect has been related to the inhibition of the synthesis of gibberellic acid in the plant (Zeevaart and Osborne, 1965). Daminozide has been observed to have its highest inhibitory effect immediately upon application, becoming less pronounced thereafter, so that continued retardation is accomplished by reapplication at 10 to 14 day. Daminozide has to be applied more than once in order to give a good retardation in most pot plants (Adriansen, 1972) and, in particular, for Chrysanthemum (Mitlehner, 1966).

Dicks and Charles-Edwards (1973) attempted to use a modeling analysis to explain the effect of daminozide on Chrysanthemum. However, errors in their approach resulted in a model predicting exponentially increasing inhibition over time rather than declining inhibition. Thus their model cannot be used to predict effects of growth regulators (such as daminozide) which decline in their capacity for inhibition with time after application. The objective of our work was to develop a model to describe the inhibitory effect of daminozide on shoot elongation.
 
 

MATERIALS AND METHODS

Growth retardation was studied by tracking the length of the side shoot which forms on the highest remaining node on chrysanthemum after the top is removed (pinched). Two assumptions are basic to the analysis presented here: (1) plants treated with daminozide receive a spray solution containing the active ingredient to the top-most, recently-expanded foliage and (2) the plants are induced to flower after a fixed number of leaves have unfolded, resulting in some maximum shoot length.

This work was begun by selecting a suitable model. The data were collected for estimation of model parameters and testing of the goodness-of-the-fit (verification). A simulation model was then developed from the resulting information. Subsequently, data were collected for the testing of the simulation model as a prediction tool (validation).
 

Model theory

The length (mm) of the top-most shoot on plants treated with daminozide was denoted with the variable H. The model developed here related this variable to the length of shoots of untreated plants, Hu, over time t (days) measured from the day of pinching. To do this it was assumed that the growth rate of a side shoot of a daminozide-treated plant, dH/dt, was related to the growth rate of untreated shoots, dHu/dt, by:

where g(t) mathematically represents the reduction over untreated growth. dHu/dt represents potential shoot growth and can generally be modeled with a relationship based on Hu and t:
Due to its flexibility, the Richards function (Richards, 1959, 1969) was chosen. Its form of equation [2] may be written as (Causton et al., 1978)
where Huf is the final length of a side shoot on an untreated plant and k and n are parameters related to the shape of the growth curve and the location of the inflection point. When integrated over time, this model describes accumulated growth (shoot length) at any point in time as

where the parameter Hu0 is the height of the untreated shoot at t=0. The function g(t) represents the progression of the degree of inhibition and the return to normal of the growth rate over time (Lieth and Reynolds, 1986). It is also related to the concentration or amount of daminozide applied to the plant, since this is known to affect the degree of inhibition. Thus g(t) varies from zero to unity. For values of time t where g(t) is zero, dH/dt would be zero, representing a situation of total inhibition of growth. When g(t) evaluates to one, the growth rate of a treated shoot will be equal to that of the untreated shoot. Various choices of g(t) can be used to represent different hypotheses regarding the mode of action and impact on the elongation rate.

One possible formulation for g(t) was to model the inhibited-elongation phase with a linear relationship (Fig. 1A):

which can also be written as:
Here tev is the time of daminozide application and trec is the time at which full resumption of the growth rate is achieved. AL is the initial growth rate reduction at the time of application, and cL is the slope while P is the persistence of the effect. It should be noted that trec is not an independent parameter; rather it is related to the other parameters with trec = tev+P.

Another possible expression of g(t) is to use a saturation-type response for the period of recovery:

where AE is the initial elongation rate response at the time of application and cE is a parameter which is related to the rate of recovery. The gE(t) has a high rate of recovery immediately after the initial response (Fig. 1B). However, as time progresses, the rate of recovery slows down as gE(t) asymptotically approaches unity.

The initial growth retardation (AL in gL, or AE in gE) is dependant of the concentration or amount of active ingredient, of daminozide to which the plant has been exposed. This was incorporated into the model by replacing AL or AE in equations 5 and 7 with a decreasing function with daminozide concentration.

Thus the model consisted of equation 4 to describe the shoot length of untreated plants. Equation 3 was then used to compute dHu/dt. After computing g(t), either with equation 5 or 7, dH/dt was computed as in equation 1. This approach implies that whatever portion of the potential growth at a specified time is not fully utilized, is lost and cannot be regained. A similar approach was used by Larsen and Gertsson (1991) when modeling the effect of day and night temperature regimes on shoot elongation in Chrysanthemum.
 

Experimental design

Two experiments were conducted sequentially for model development and parameter estimation. The aim of the first experiment was to observe the growth of plants treated at one daminozide concentration, to verify that the model was feasible, and to determine which of the g(t) models would represent the observed effect best. The purpose of the second experiment was to estimate parameter values of the model, particularly of g(t), and to examine their dependance on daminozide concentration.

In both experiments rooted cuttings of Dendranthema × grandiflora (Ramat.) Kitamura 'Bright Golden Anne' (Yoder Brothers, Barberton, Ohio) were planted in 12 cm-diameter pots in U.C. Mix (1:1:1 peat moss, redwood sawdust, sand, by volume) at a density of 1 cutting per pot. The plants were grown under long-day conditions at 18±2 C in a greenhouse at Davis, CA. Long photoperiods were achieved by exposing the plants to a 4 hour night break, from 10 pm to 2 am, with light from incandescent lamps suspended over the plants. Two weeks after planting the plants were pinched above the fifth leaf to promote the breaking of lateral branches. Starting that same day (t=0) the plants were given 16-hour night periods by covering the plants with black cloth from 4 pm until 8 am to promote flowering. From then on, the length of the highest positioned side shoot was measured once every weekday with a flexible plastic ruler until all flowers were fully open.

During the first experiment sixteen plants were grown in two blocks of 8 plants each. 21 days after pinching, one block was treated with 0.25 % a.i. of daminozide applied as a foliar spray. The remaining eight plants were left untreated. During the second experiment 27 plants were designated for different levels of concentrations of daminozide spray application. Sixteen days after pinching, groups of 3 plants were sprayed with one concentration of daminozide; the following concentrations were used: 0, 0.063, 0.125, 0.250, 0.500, 0.750, 1.00, 1.50, and 2.00 % a.i.. A hood, consisting of a 12-L jar with holes cut into the sides, was placed over the plant. Daminozide was applied by spraying the plant briefly from three directions.
 

Estimation of model parameters (Model Calibration)

Parameters were estimated using the derivative-free nonlinear regression procedure PROC NLIN of PC/SAS (SAS Institute, 1988). The parameters of the Richards function describing the growth of the untreated shoot, Hu0, Huf, k, and n, were estimated by fitting equation 4 to the observed shoot growth data of the untreated plants.

Computation of H using equation 1 required a numerical approximation since the closed-form solution of this differential equation is not available. This was done by computing a set of predictions for H over the experimental time frame and inserting the daily values into an array. Up to t=tev, H was the same as Hu which was computed for each day t with equation 4. Starting with t=tev, dHu was computed by multiplying both sides of equation 3 by the selected time-step dt. Multiplied with g(t) this determined the value of dH (equation 1). The next value of H was computed by adding dH to the previous value of H. By using dt=0.5, every second iteration resulted in daily H values.

While it was possible to carry out this entire simulation every time the regression routine needs to determine H at some time t, it was more efficient to recompute the simulation in an array only when a parameter value changes. This was done during parameter estimation since SAS has the capability to accommodate this programming trick in its nonlinear regression procedure.

The data from the second experiment were analyzed by again fitting equation 4 to the data for the untreated plants. Then, with the resulting parameter values fixed, the parameter pairs AL, P, and AE, cE were estimated with separate fittings for each plant individually. This resulted in a data set of these parameter values for all tested daminozide concentrations which was analyzed for trends in the parameter values with daminozide concentration. Mathematical functions were then tested by fitting them to these data with PC/SAS. The selected equations were then built into gL(t) and gE(t) resulting in the two final models (one with gL and one with gE).

All data from the second experiment were then combined into one large data set including the variable B, the daminozide concentration (% a.i.). The two final models were then fit to these data so that all parameters, including those of equation 4, were estimated at the same time. The parameter values found previously were used as starting values in the iterative nonlinear regression procedure.
 

Test of model prediction (validation)

Two simulation programs, one with the gL(t) and one with gE(t), were written in Turbo Pascal (Borland International, Scotts Valley, CA) following the numerical approximation methodology described above. The parameter values of the final parameter estimations were used.

A separate (third) data set was then collected for validation of the final model. Chrysanthemum plants were grown following the same cultural practice described above. The experiment involved 32 plants that were divided into 4 blocks of 8 plants each. Three of these blocks were sprayed with 0.25 % a.i. of daminozide, and the fourth block was left untreated. This daminozide concentration was chosen since it is commonly used in commercial production of Chrysanthemum. These three blocks were sprayed 16, 26, or 36 days after pinching, respectively. Branch lengths were measured as before.

The parameters of the Richards function, Hu0, Huf, k, and n, were estimated by fitting equation 4 to observed data for the untreated shoots. Replacing these parameter values in the final model, the simulation model was used to predict the response of the treated plants. The resulting predictions were compared against the observed data in a linear regression analysis.
 
 

RESULTS

As expected, the shoot elongation pattern during the first experiment was sigmoid (Fig. 2) starting with an average shoot length (±SE) of 10.4 ± 1.4 mm on day 7. The average final measured length was 380 ± 23 mm on day 55 for the untreated plants. Plants sprayed with 0.25% daminozide had a significantly shorter average final shoot length of 343 ± 14.5 mm. The regressions of the Richards function onto the shoot length data from the untreated plants gave a good fit resulting in an R2 (Regression SSQ / Total SSQ, Table 1) of 0.993. Fitting the model (equations 1, 3, and 5 or 7) resulted in the same R2 value of 0.997 for both gL(t) and gE(t). There was no significant difference in the accuracy with which these two models represented the daminozide effect on shoot growth (Table 1), indicating that either one may be suitable from a goodness-of-fit perspective.

During the second experiment the measured shoots of the untreated plants grew from 8.7 ± 1.5 to 335 ± 7.1 mm from day 7 to 68 after pinching. The daminozide spraying reduced plant heights such that higher concentrations of daminozide resulted in a shorter shoots. Final average shoot lengths ranged from 199 ± 8.1 mm for the 1.5% application to 280 ± 28.3 mm for the 0.063% application. The declining differences confirmed that increasing daminozide concentration causes increased retardation of shoot elongation.

The parameters AL and AE were found to decline with B, the daminozide concentration (% a.i.). The asymptotic correlation matrix of the nonlinear regression result (not shown) indicated that the g(t)-function parameters were strongly correlated for both formulations of g(t). There appeared to be no significant trend of cE with B so that cE was fixed to the average value of the estimated cE values (0.055). The same procedure was employed with the gL(t) parameters. Fittings were then rerun, resulting in A-values which declined with B (Fig 3). The following expression was used to describe the relationship between both AL and AE and B:

where a1 and a2 are parameters. The fitting of the function to the data resulted in parameter estimates for a1 and a2 of 0.367 and 5.916, respectively, (R2= 0.955) for AL and of -0.012 and 2.15, respectively, (R2= 0.856) for AE.

This then resulted in two models, one with gL the other with gE, for predicting shoot length (H) over time (t) in relation to daminozide concentration (B) and time of application (tev). These two final models are called the "L-model" and "E-model", respectively, below. In both models the daminozide effect on the initial growth rate reduction (the A-parameters) is modeled with equation 8.

Fitting the L-model to the entire data set from experiment 2 resulted in parameter values for Huf, Hu0, k, n, P, a1 and a2 (Table 2) with an R2-value of 0.993. This model tracks the observed data for each daminozide concentration quite well (Fig. 4). Fitting the E-model (Table 3) resulted in parameter estimates which differed slightly from those for the L-model, even for the parameter representing the Richards function (the base-line which should theoretically be the same for both models). The 95% asymptotic confidence limits, however, suggest that these differences are not significant. The R2 value was 0.993 for the E-model and resulted in a family of curves for the various tested daminozide concentrations which mimiced the data very well (Fig 5). The total regression sums of squares and mean square errors of the L and E- models were virtually identical indicating that both models represent the data equally well.
 

Test of model prediction (validation)

The fitting of the Richards function to the shoot length data of the untreated plants (Table 4) resulted in an R2-value of 0.998. In the resulting simulations there was a slight tendency of both L or E-models to overestimate the shoot length data (Fig. 6). However, both models gave surprisingly good predictions of the effect of all three daminozide applications in the experiment, thus indicating that both models predict the daminozide effect equally well. The R2-values computed for the goodness-of-fit of these simulation results with respect to the observed validation data were 0.998 and 0.997 for the L and E-models, respectively.
 
 

DISCUSSION

In each of the experiments of this study, the control (untreated) plants showed sigmoid shoot elongation. However, variables such as light and temperature resulted in different final shoot lengths for untreated plants among the experiments. Another factor which affected shoot length was the length of time between breaking of apical dominance (through pinching) and flower initiation since this duration acounts for the number of nodes on the stem. The model developed here was not designed to predict these variations. Rather, assuming these variation, the model predicts the response to daminozide. In other words, the model predicts how one application of daminozide at any concentration and any application date inhibits shoot elongation based on the "normal" (no daminozide) growth of plants.

In the equation describing the effect of daminozide concentration on the intial elongation depression, the parameter a1 is the lower asymptote for the initial response to the daminozide treatment, AL or AE. This asymptotic behavior indicates that as B increased, the relative retardation effect did not continue to increase proportionately. The L-model suggests that no matter how high the daminozide concentration, there will always continue to be some growth since a1 is significantly greater than zero. The negative value of a1 in the E-model (Table 3) is not significantly different from zero.

Both models provide unique insights into the shoot inhibition process through the horticulturally meaningful parameters. The L-model, for example, allows direct estimation of the persistance of the effect, P, and the initial effect of the growth retardant. The persistence of the daminozide effect was estimated to be approximately 34 days. This is surprising since guidelines for the use of daminozide generally suggest that the plants grow out of the effect within 10 to 14 days. This information probably resulted from horticultural observations rather than quantitative analysis, so that the small effects towards the end of the period of impact were not readily detected. This underlines the strength of the L-model in the present approach.

The gE function represents the hypothesis that recovery from the growth regulator application occurs very rapidly shortly after the application. The recovery then tapers off, becoming more gradual. The E-model does not provide a method for estimating persistence directly since the formula technically never reaches 1.0 after tev.

It should be noted that it is possible that other formulations of g(t) could provide equally good results. The exact trajectory of the retardation effect (e.g. Fig 1) is not known. The fact that both gE(t) and gL(t) represent the effect equally well suggests that other similar trajectories might work equally well. Despite the extensive data collection effort, this information could not be resolved with the current appoach. Alternate formulations of g(t) may also be necessary when analyzing the effect of other retardants, particularly if their mode of action is different from that of daminozide.

In the validation experiment the untreated plants had side shoots that were slightly longer than the shoots of the treated plants before the time of daminozide application. Since the predictions made by the g(t) models are based on the growth of the untreated plant, this initial difference affected the relation between actual and predicted shoot growth of the treated plants later on. This may to some extent explain the slight tendency of over-estimation in the shoot length predictions for the treated plants. However, the results show that daminozide has the same effect when it is given at the different application times.

Thus the model can be used to determine the date on which one application at some concentration will have the maximum effect (Fig 7). For B=0.25 %, the concentration recommended by the manufacturer, this results in a curve which declines for tev from day 0 to day 15 and then increases in a saturation pattern for later application dates. The same pattern is also seen for the other daminozide concentrations. For any application date, the relative final height reduction declines with increasing daminozide concentration. Thus for any selected daminozide concentration the maximum effect of a single application can be obtained by making the application within the time frame when the largest height reduction will occur (15<tev<20 days).

While the approach presented here focussed on the top-most shoot of the plant, the results can readily be extended to whole-plant growth. Plant height, measured from the top of the potting medium to the top of the plant is related to this modeling approach by adding the initial plant height to H0 and by replacing the final height Huf with the final height of the untreated plants. In this way, actual plant size can be simulated.

The presented g(t) models can be linked to other models which predict shoot elongation of an untreated plant based on a differential equation for the growth rate. This then results in a prediction tool with potential commercial use. Such a model of shoot elongation was developed for chrysanthemum by Larsen and Gertsson (1991). However, since daminozide is normally applied to chrysanthemum by multiple spraying, strategies for the use the g(t) for multiple daminozide applications have to be tested.
 

Literature Cited

Adriansen, E., 1972. Kemisk vækstregulering af potteplanter. Tidsskr. PlAvl. 76:725-841.

Causton, D.R., Elias, E.C. and Hadley, P., 1978. Biometrical studies of plant growth. I. The Richards function and its application in analyzing the effects of temperature on leaf growth. Plant Cell Environ., 1:163-184.

Dicks, J.W., and Charles-Edwards, D.A., 1973. A Quantitative description of inhibition of stem growth in vegetative lateral shoots of Chrysanthemum morifolium by N-Dimethylamino-succinamic Acid (Daminozide). Planta, 112:71-82.

Larsen, R., and Gertsson, U., 1991. Model analysis of shoot elongation in Chrysanthemum. Scientia Hortic., in press.

Lieth, J.H., and Reynolds, J.F., 1986. Plant growth analysis of discontinuous growth data: A modified Richards function. Scientia Hortic., 28:301-314.

Mitlehner, A.W., 1966. Effect of B-Nine and schedules on Princess Anne chrysanthemums. Proc. 17th Int. Hort. Congr. I, p. 219.

Richards, F.J., 1959. A flexible growth function for empirical use. J. Exp. Bot., 10:290-300.

Richards, F.J., 1969. The quantitative analysis of growth. In: F.C. Steward (Ed), Plant Physiology: A treatise. Academic Press, London, pp.3-76.

SAS Institute, 1988. SAS/STAT Users guide Version 6.03 edition. p. 675-712.

Zeevaart, J.A.D., and Osborne, H.D. 1965. Comparative effects of some Amo-1618 analogs on gibberellin production in Fusarium moniforme and on growth of higher plants. Planta, 66:320-330.
 
 

Figure Legends

Fig. 1. A linear (gL(t)) and an exponential (gE(t)) model describing the relative effect of a growth regulator on shoot elongation in relation to time. The tev indicates the time of application of the growth regulator and trec is the time of recovery from the retardation. The AE and AL indicate the initial effect of the growth regulator at the time of application.

Fig. 2. Comparison between model simulation (lines) and collected data used for estimation of model parameters (points). (A) Indicates simulations made by the linear model gL(t), while (B) indicates simulations made by the exponential model gE(t). Vertical bars indicate standard errors (n=8).

Fig. 3. The observed initial retardation of shoot elongation (AL and AE) daminozide spray concentration (symbols) with the least squares fit of equation 7.

Fig. 4. Comparison between fitted L-model (line) and collected data (symbols) from plants sprayed with different concentrations of daminozide. Treatment was given 15 days after pinching (t=0). Dashed lines represent growth of untreated plants. Vertical bars represent standard errors.

Fig. 5. Comparison between fitted E-model (line) and collected data (symbols) from plants sprayed with different concentrations of daminozide. Treatment was given 15 days after pinching (t=0). Dashed lines represent growth of untreated plants. Vertical bars represent standard errors.

Fig. 6. Validation of the g(t)-models capacity to predict the effect of daminozide (0.250 % a.i.) on shoot elongation when given as an early, a medium, or a late application. In the simulations, the control was used to estimate the elongation of an untreated plant. Model simulations are indicated by lines while dashed lines show the growth of the untreated plant. Points indicate mean shoot length of each treatment (n=8). Vertical range bars indicate standard errors. The time of treatment is indicated by arrows.

Fig. 7. The percent height of daminozide-treated plants with respect to the final height of untreated plants for various combinations of tev and B as predicted by the model.

Table 1. Analysis of variance and parameter estimates from fitting equation 4 to shoot length observations for the untreated plants and from fitting the simulation model (equation 1) with either gL(t) or gE(t) to the shoot length data from the treated plants.
 
Analysis of Variance
Source df Sum of Squares Mean Square
Regression (eqn 4) 4 20143338 5035834
Residual (eqn 4) 292 146694 502
Uncorrected Total 296 20290032
(Corrected Total) 295 5411044
Regression of gL(t) 3 15115033 5038344
Residual after gL(t) 229 39406 172
Regression of gE(t) 3 15114806 5038268
Residual after gE(t) 229 39633 173
Uncorrected Total 232 15154439
(Corrected Total) 231 1746562
 
Table 1 (continued)
Parameter estimation
Sub model Parameter Units Estimate ± A.SE1
Equation 4 Huf mm 394 ± 5.6
k days-1 0.117 ± 0.010
n - 0.366 ± 0.162
Hu0 mm 11.1 ± 2.7
gL(t) AL - 0.441 ± 0.075
cL days-1 0.049 ± 0.014
Hu0 mm 10.2 ± 0.339
gE(t) AE - 0.271 ± 0.146
cE days-1 0.222 ± 0.054
Hu0 mm 10.2 ± 0.372
 1A.SE = asymptotic standard error
 

Table 2. Parameters and estimated values of the final daminozide prediction model based on the linear gL(t) function.
Analysis of Variance
Source df Sum of Squares Mean Square
Regression  7 29322014 4188859
Residual  1112 215593 194
Uncorrected Total 1119 29537608
(Corrected Total) 1118 9118175

 Parameter Estimation
 
table2 continnued: 95% Confidence interval 
Submodel Parameter  Units Estimate ± A.SE1 Lower Upper
dHu/dt Huf mm 316 ± 2.65 311 321
Hu0 mm 12.3 ± 0.774 10.8 13.8
k days-1 0.110 ± 0.005 0.101 0.119
n - 0.847 ± 0.088 0.674 1.02
gL(t) P days 34.2 ± 1.440 31.4 37.1
a1 - 0.124 ± 0.033 0.058 0.189
a2 %-1 1.84 ± 0.087 1.67 2.01

 1A.SE = asymptotic standard error
 

Table 3. Parameters and estimated values of the final daminozide prediction model based on the exponential gE(t) function.
Analysis of Variance
Source df Sum of Squares Mean Square
Regression  7 29319439 4188491
Residual  1112 218169 196
Uncorrected Total 1119 29537608
(Corrected Total) 1118 9118175
 Parameter Estimation
Table 3 continued 95% Confidence interval
Submodel Parameter  Units Estimate ± A.SE1 Lower Upper
dHu/dt Huf mm 324 ± 2.90 318 330
Hu0 mm 12.0 ± 0.822 10.4 13.6
k days-1 0.098 ± 0.005 0.089 0.107
n - 0.683 ± 0.089 -0.507 0.859
gE(t) ce days-1 0.059 ± 0.006 0.047 0.070
a1 - -0.040 ± 0.069 -0.175 0.095
a2 %-1 1.84 ± 0.087 1.67 2.01
1A.SE = asymptotic standard error
 

Table 4. Estimated values of the model of untreated shoot growth (dHu/dt) used during the validation of the g(t) models.
95% Confidence interval
Submodel Parameter  Units Estimate ± A.SE1 Lower Upper
dHu/dt Huf mm 327 ± 2.74 321 332
Hu0 mm 10.9 ± 1.04 8.8 12.9
k days-1 0.105 ± 0.005 0.095 0.116
n - 0.702 ± 0.102 0.501 0.902
1A.SE = asymptotic standard error