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

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.

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, *H*_{u},
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, d*H*/d*t*, was related to the growth rate of untreated
shoots, d*H*_{u}/d*t*, by:

where the parameter *H*_{u0} 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, d*H*/d*t*
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):

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

The initial growth retardation (*A*_{L} in *g*_{L},
or *A*_{E} in *g*_{E}) 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 *A*_{L}
or *A*_{E} 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 d*H*_{u}/d*t*.
After computing *g*(*t*), either with equation 5 or 7, d*H*/d*t*
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, *H*_{u0},
*H*_{uf}, *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*=*t*_{ev},
*H* was the same as *H*_{u} which was computed for each
day *t* with equation 4. Starting with *t*=*t*_{ev},
d*H*_{u} was computed by multiplying both sides of equation
3 by the selected time-step d*t*. Multiplied with *g*(*t*)
this determined the value of d*H* (equation 1). The next value of
*H* was computed by adding d*H* to the previous value of *H*.
By using d*t*=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 *A*_{L}, P, and *A*_{E},
*c*_{E} 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 *g*_{L}(*t*) and *g*_{E}(*t*)
resulting in the two final models (one with *g*_{L} and one
with *g*_{E}).

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 *g*_{L}(*t*)
and one with *g*_{E}(*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, *H*_{u0}, *H*_{uf},
*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.

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 *A*_{L} and *A*_{E} 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 *c _{E}* with

This then resulted in two models, one with *g*_{L} the
other with *g*_{E}, for predicting shoot length (*H*)
over time (*t*) in relation to daminozide concentration (*B*)
and time of application (*t*_{ev}). 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 *H*_{uf}, *H*_{u0}, *k*,
*n*, *P*, *a*_{1} and *a*_{2}
(Table 2) with an *R*^{2}-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 *R*^{2} 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 *R*^{2}-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 *R*^{2}-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.

In the equation describing the effect of daminozide concentration on
the intial elongation depression, the parameter *a*_{1} is
the lower asymptote for the initial response to the daminozide treatment,
*A*_{L} or *A*_{E}. 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 *a*_{1} is significantly greater than
zero. The negative value of *a _{1}* 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 *g*_{E} 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 *t*_{ev}.

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 *g*_{E}(*t*)
and *g*_{L}(*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 t_{ev} 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<t_{ev}<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 H_{0} and by replacing the final height H_{uf} 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.

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.

Fig. 1. A linear (*g*_{L}(*t*)) and an exponential
(*g*_{E}(*t*)) model describing the relative effect of
a growth regulator on shoot elongation in relation to time. The *t*_{ev}
indicates the time of application of the growth regulator and *t*_{rec}
is the time of recovery from the retardation. The *A*_{E}
and *A*_{L} 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 *g*_{L}(*t*), while (B) indicates
simulations made by the exponential model *g*_{E}(*t*).
Vertical bars indicate standard errors (n=8).

Fig. 3. The observed initial retardation of shoot elongation (*A*_{L}
and *A*_{E}) 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 *t*_{ev}
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 *g*_{L}(*t*)
or *g*_{E}(*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 g_{L}(t) |
3 | 15115033 | 5038344 |

Residual after g_{L}(t) |
229 | 39406 | 172 |

Regression of g_{E}(t) |
3 | 15114806 | 5038268 |

Residual after g_{E}(t) |
229 | 39633 | 173 |

Uncorrected Total | 232 | 15154439 | |

(Corrected Total) | 231 | 1746562 |

Table 1 (continued)

Parameter estimation | |||

Sub model | Parameter | Units | Estimate ± A.SE^{1} |

Equation 4 | H_{uf} |
mm | 394 ± 5.6 |

k |
days^{-1} |
0.117 ± 0.010 | |

n |
- | 0.366 ± 0.162 | |

H_{u0} |
mm | 11.1 ± 2.7 | |

g_{L}(t) |
A_{L} |
- | 0.441 ± 0.075 |

c_{L} |
days^{-1} |
0.049 ± 0.014 | |

H_{u0} |
mm | 10.2 ± 0.339 | |

g_{E}(t) |
A_{E} |
- | 0.271 ± 0.146 |

c_{E} |
days^{-1} |
0.222 ± 0.054 | |

H_{u0} |
mm | 10.2 ± 0.372 |

Table 2. Parameters and estimated values of the final daminozide prediction
model based on the linear *g*_{L}(*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.SE^{1} |
Lower | Upper |

dH_{u}/dt |
H_{uf} |
mm | 316 ± 2.65 | 311 | 321 |

H_{u0} |
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 | |

g_{L}(t) |
P |
days | 34.2 ± 1.440 | 31.4 | 37.1 |

a_{1} |
- | 0.124 ± 0.033 | 0.058 | 0.189 | |

a_{2} |
%^{-1} |
1.84 ± 0.087 | 1.67 | 2.01 |

^{1}A.SE = asymptotic standard error

Table 3. Parameters and estimated values of the final daminozide prediction
model based on the exponential *g*_{E}(*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 |

Table 3 continued | 95% Confidence interval | ||||

Submodel | Parameter | Units | Estimate ± A.SE^{1} |
Lower | Upper |

dH_{u}/dt |
H_{uf} |
mm | 324 ± 2.90 | 318 | 330 |

H_{u0} |
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 | |

g_{E}(t) |
c_{e} |
days^{-1} |
0.059 ± 0.006 | 0.047 | 0.070 |

a_{1} |
- | -0.040 ± 0.069 | -0.175 | 0.095 | |

a_{2} |
%^{-1} |
1.84 ± 0.087 | 1.67 | 2.01 |

Table 4. Estimated values of the model of untreated shoot growth (d*H*_{u}/d*t*)
used during the validation of the *g*(*t*) models.

95% Confidence interval | |||||

Submodel | Parameter | Units | Estimate ± A.SE^{1} |
Lower | Upper |

dH_{u}/dt |
H_{uf} |
mm | 327 ± 2.74 | 321 | 332 |

H_{u0} |
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 |