Kor. J. Hort. Sci. Technol. 27(2):239-243, 2009
Modeling Approaches for Estimating Cardinal Temperatures by Bilinear, Parabolic, and Beta Distribution Functions 1
2
3*
Young Yeol Cho , Myung-Min Oh , and Jung Eek Son 1
Department of Horticulture, Faculty of Bioscience and Industry, Jeju National University, Jeju 690-756, Korea 2 Department of Horticulture, Forestry and Recreation Resources, College of Agriculture, Kansas State University, Manhattan, Kansas 66506, USA 3 Department of Plant Science, Research Institute for Agriculture and Life Sciences, Seoul National University, Seoul 151-921, Korea
Abstract. Estimation of the cardinal temperatures – the base, optimum and maximum temperatures – is indispensable because plant growth and development are affected by temperature. Although several models including linear and nonlinear functions are available to estimate the temperatures, a model suitable to the specific crop should be selected. The objectives of this study were to analyzed the estimated the cardinal temperatures for germination of spinach (Spinachia oleracea cv. Gwibin) by bilinear, parabolic and beta distribution models and to find a model reflecting the plant response to temperature adequately. Seeds of spinach were germinated in a growth chamber at constant temperatures of 2, 4, 8, 12, 16, 20, 24, 28, 32, and 36℃. Radicle emergence of 1 mm was scored as germination. The time course of germination was fitted using a logistic function. The base, optimum, and maximum temperatures were estimated by regression of the inverse time to 50% germination rate against temperature gradient. We obtained the base, optimum and maximum temperatures of -4.0, 17.1 and 33.2℃ in bilinear function; -0.9, 15.8 and 32.5℃ in parabolic function; and -2.6, 16.6 and 32.6℃ in beta distribution function, respectively. Among the three functions, a beta distribution 2 function had a good agreement with the plant response to temperature showing the highest R (coefficient of determination) and the lowest RMSE. Additional key words: base temperature, germination test, modeling, spinach, temperature
Introduction Temperature is an important environmental factor directly affecting plant growth and development. Temperature influences all biological processes, and the responses can be summarized in terms of cardinal temperatures: the base temperature, optimum temperature, and maximum temperature (Jami Al-Ahmadi and Kafi, 2007; Yan and Hunt, 1999). Estimation of the cardinal temperatures is necessary for seed germination because germination rate increases between the base temperature and the optimum, decreases between the optimum and the maximum, and ceases below the base and above the maximum. The germination time was selected for estimating the cardinal temperatures because facilities were available for experiments over a range of constant temperatures (Roché et al., 1997). The cardinal temperatures are often determined by the extrapolation of inverses of the germination rate (Jami Al-Ahmadi and Kafi, 2007; Montieth, 1981; Seefeldt et al., 2002). *Corresponding author:
[email protected],
[email protected] ※ Received 24 December 2008; Accepted 24 February 2009.
Knowledge of cardinal temperatures for plant growth and development is important for successful prediction of its cultivation, maturity, and yield in a particular environment. Several models have been applied to seeding material to explain the effect of temperature on germination rate (Aflakpui et al., 1998; Cho et al., 2008; Jami Al-Ahmadi and Kafi, 2007; Roché et al., 1997; Seefeldt et al., 2002). Generally, temperature response functions are found to be a linear, a hyperbola or a nonlinear function. Within a limited range of temperature, the rate of development or growth showed a linear function of the temperature (Aflakpui et al., 1998). However, temperatures exceed the optimum temperature in natural conditions. To satisfy this situation, many researchers have applied to a bilinear equation with two different linear equations. This model has been successfully applied to several crops. Curvilinear relationship was reported to be the typical shape for expressing plant development time with temperature (Cho et al., 2008; Roché et al., 1997; Seefeldt et al., 2002; Yan and Hunt, 1999; Yin et al., 1995). Parabolic equation has applied to several crops as temperature response function (Cho et al., 2008; Yin et al., 1995). Recently the beta Kor. J. Hort. Sci. Technol. 27(2), June 2009
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Materials and Methods Spinach (Spinachia oleracea cv. Gwibin) cultivar (Nong Woo Bio. Co., Korea) was used in the study. The types of ‘Gwibin’ spinach seeds are round. One thousand seed weight of Gwibin cultivar was 7.2 ± 0.12 g. The seeds of spinach were germinated in a growth chamber at constant temperatures of 2, 4, 8, 12, 16, 20, 24, 28, 32, and 36℃. At each temperature, four replicates of 100 seeds were placed on two layers of Whatman No. 2 filter paper on the fabric. Germination test was evaluated in the germination bed designed by Liebenberg (Cho et al., 1996). Radicle emergence of 1 mm was scored as germination and the germinated seeds were removed daily. Germination was fitted using the following a logistic function: -1
Y = M×[1+exp(-k×(t-L))]
(1)
where, Y is cumulative percentage germination at time t, M is maximum potential germination, L is time scale (lag related) constant, and k is the rate of increase (Roché et al., 1997). Parameters were estimated using the Gauss-Newton algorithm, a nonlinear least squares technique. The base, optimum, and maximum temperatures were estimated by regressing the inverse of the time to 50% germination rate (GR50) against the temperature gradient. Eqs. 2 and 3 are employed in the bilinear model to predict the cardinal temperatues.
analysis. The base and maximum temperatures were obtained by extrapolation to the intercept with the abscissa. Equation 3 was fitted to the parabolic equation: 2
GR50 = a+b×T+c×T
(4)
where, a, b, and c are the intercept, first and second order regression coefficients, respectively. Parameters were estimated using the regression analysis. The base and maximum temperatures were obtained by extrapolation to the intercept with the abscissa. Eqs. 5 and 6 were obtained by the beta distribution equation. GR50 = exp(k)×(T-Tbase)α×(Tmax-T)β
(5)
Topt = (α×Tmax+β×Tbase)/(α+β)
(6)
where, k, α and β are the model parameters; and Tbase, Topt and Tmax are the base, optimum and the maximum (ceiling) temperatures, respectively. Parameters were estimated using the Gauss-Newton algorithm, a nonlinear least squares technique. Statistical computations were carried out using the SigmaPlot (SPSS Inc., USA) and the SAS (SAS Institute Inc., USA) software. A completely randomized block design was used as the experimental design with four replications. The experimental results were subjected to an analysis of variance (ANOVA). When significant differences occurred, the means were separated using Duncan multiple range test at 5% level.
Results The germination curves of spinach under constant temperature 100
Cumulative germination (%)
distribution equation, nonlinear function based on the response of enzymatic reactions to temperature, was introduced to describe the temperature response of crop development (Yan and Hunt, 1999). For most temperate species, the base and optimum temperatures lie between 0 to 5℃, and 20 to 25℃, respectively (Monteith, 1981). Minimum germination temperature in spinach was reported to be -3.0℃ (Røeggen, 1984) and average temperature was known as 16-18℃. However, cardinal temperatures vary with growth stage or cultivars (Jami Al-Ahmadi and Kafi, 2007; Monteith, 1981; Seefeldt et al., 2002) and their estimated values are different with models. The objectives of this study were to estimate the cardinal temperatures for germination of spinach by bilinear, parabolic and beta distribution models and to find a model reflecting the plant response to temperature most adequately among them.
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GR50 = a1+b1×T for T
