4.2.2 1 Topological size
4.2.4 Screening for a suitable biological growth function
Due to the non-linearity of biological growth, functions are often used in the analysis of experimental data in both animal and plant studies that record growth by repeated measurements (Hunt, 1 982; Causton, 1 983, Zeide, 1 993b). The biological growth expressed in units of size-mass W (e.g. dry weight, length etc.) is mathematically described as a function of time t:
The function f (t) of biological growth under normal conditions is represented by a logistic, sigmoid, s-shape curve (Thomley and Johnson, 1 990; Causton, 1 99 1 ; Zeide,
1 993b). Decreasing !,Jfowth rate, and thus the shape of the function, is caused by limited substrate availability as the biological unit grows and develops ( Hunt, 1 982). Theoretically, if substrate availability is unlimited, biological growth may continue i n an exponential manner. To some extent, this can be achieved in vitro under a regular sub-culturing regime for plant multiplication purposes. The overall decrease in the function relative growth rate is often affected by the so-called intrinsic or inherent growth rate coefficient (see Section 1 .2.6). The growth parameter of the ' intrinsic' rate coefficient reflects the overall shape and slope ofthe growth rate function. In general, therefore, this single parameter exhibits similar features to growth parameters examining both the function shape and relative growth rate. Predominantly, however, it reports on the slope of the linear part of the growth function.
Several mathematical functions have a sigmoid shape and would potentially be suitable for plant !,rrowth modell ing (Richards, 1 959; Thornley and Johnson, 1 990; Ratkowsky, 1 990; Causton 1 99 1 ; 1 994; Zeide, 1 993b). Preliminarily, the
experimental data were fitted with four different growth functions of the fol lowing general forms:
1 ) Gompertz
Where the intercept(Wo ) and upper asymptote (W =) are given by : Wo
== ce-a
W= = c 2) Richards (Richards, 1 95 9) I W== c(l -deil/}J
(4.4) (4.5a) 1 27Where the intercept (Wo) and upper asymptote (Wo) are given by:
1
Wo = c(l - de
},
W= = cTwo modified forms were also tested for fit to non-normalised data
I W = Ao +
(Wo - Ao
B- { I + eBJ�
I + e al 3) Chapman-Richards W = c(l
-e-alWhere the intercept (Wo ) and asymptote (W = ) are given by :
1Yrl
= 0W= = c
For data fitting a modified form was used:
I
( l - e-al J'
W =
�5
I - ei s-a4) Schnute's (Kort) equation basic form
Where intercept (Wo ) and asymptote (W= ) are given by :
1Yrl
= 0W= = ac
For data fitting a difference form was used:
i W =
[Ch
+ _ cl>]h
- e (4.5b) (4.5c) (4.6a) (4.6b) (4.7a) (4.7b)In all equations, the growth rate coefficient or its equivalent is marked a, and it is always negative as it determines the rate of growth decay (Richards, 1 959). Growth parameter b is found in the power part of the Richards and related functions.
There were advantages and disadvantages to each of these functions, and these were examined in order to select a function suitable to represent the experimental data. The Gompertz function appeared to be flexible enough and convergence with data could be achieved over most treatments, although it did not cope well with data sets that were not approaching the upper asymptote. This was the case for growth of adult plants at the temperature regimes ofl 6/8, 32/24 and 32/8 QC, plantlets at 241 1 6 and 32/42 QC, and juveniles at 241 1 6 and 32/8 Qc.
The original Richards function had good flexibility. However, it would have required more experimental data approaching the upper asymptote to achieve good convergence at this part of function.
The adjusted Richards equation had good flexibility and would have been suitable overall, since it did not require upper asymptote estimation due to the mathematical re-arrangement of the original Richards equation. However, because of the
elimination of the upper asymptote parameter, the function required the estimation of four parameters. Consequently, the high number of evaluated parameters could have compromised the reliability of parameter estimation in data sets where the number of points was of a l imited size, as was the case in these experimental data. Therefore, this function was not used.
The Chapman-Richards function had three parameters to estimate and did not require upper asymptote information from the experimental data. It became clear from initial evaluations, that this function would fit data from most individual plants well, and its three parameters could be estimated from the experimental data range with good confidence intervals. The third parameter of this function was the value of the function at a certain time (set to l 05 days, i .e. the final measurement in the experimental data set). The first derivative was expressed analytically, relative growth rates calculated and further analysed at different time-points of the function.
Similarly, the coefficient of intrinsic growth rate (a) was estimated and further analysed with respect to the experimental factors and the objectives of this study. Estimated growth parameters of the Chapman-Richards function were used to model the growth for each experimental factor in both absolute and relative forms, and the shape of relative growth further analysed using five points along the modelled curve. The Schnute's function did not require upper asymptote inforn1ation either, appeared to be sufficiently flexible, and required an estimate of only two parameters.
However, analytical derivatives of this function were not available. The first
derivatives were necessary for comparisons of growth rates. While this shortcoming could have been overcome by numerical calculation of the growth rates, it would have led to unnecessarily complicated statistical analyses and this function not further examined.