Analysis of Reciprocal-transfer Experiments to Estimate the Length of Phases having Different Responses to Temperature

Abstract

Background and Aims

The responsiveness of plant ontogeny to temperature may change with plant age. These changes may best be identified by experiments in which individual plants are transferred in a time series from low temperature (LT) to high temperature (HT), and vice versa. Any change in the value of the slope for a plot of the duration taken to complete a developmental phase against time of transfer (either LT to HT or HT to LT) will indicate a change in the temperature responsiveness of development, and the time at which this change occurs. The analysis of this type of reciprocal-transfer experiment is usually performed by regression for each of the visually identified linear sub-phases, separately for the data for LT-to-HT and for HT-to-LT transfers. Here, a mathematical approach is presented using a single curve-fitting procedure.

Methods

Both LT-to-HT and HT-to-LT transfers are combined in a single curve-fitting procedure. This new, combined approach is illustrated using a published data set for three rice (Oryza sativa) cultivars, where the pre-flowering duration is divided into three sub-phases, and temperature responsiveness is generally stronger during the second than the first and third sub-phases.

Results and Conclusions

This new model approach provides an objective method, relative to the separate analyses, for assigning data points to a particular sub-phase. Plausible parameter values can be obtained from capturing the whole data of both sets of transfers, which otherwise could not be obtained from the separate-analysis method. Furthermore, the length of sub-phases identified from the LT-to-HT transfers is consistent, in terms of its response to temperature, with that identified from the HT-to-LT transfers. Re-analysis of the published rice data using the new approach reveals that in addition to temperature sensitivity, the optimum temperature of pre-flowering development may vary with plant age. The new approach gives rise to a generalized model for the analysis of reciprocal transfer experiments to quantify age-dependent changes of response of plants (and potentially insects) to any environmental variables that have a significant impact on their development.

INTRODUCTION

Like photoperiod, temperature is a major environmental factor affecting plant development rate. Various models have been used to quantify responses of phenological events of plants (e.g. time to flowering) to temperature, ranging from a linear temperature-sum (or degree-days) approach (e.g. Bonhomme, 2000) to non-linear models (e.g. Yin et al., 1995). Most models (e.g. Nakagawa et al., 2005) use a single temperature-response function with the same set of parameter values to predict the time for plants to complete a phenological phase (phenophase), assuming that responsiveness of plants to temperature does not vary within that phenophase.

A way to test whether or not the assumption of a constant temperature-response is correct is to conduct a temperature-controlled experiment where plants are grown in phytotron chambers at different temperature settings and to record the durations of sub-phases, as indicated by visible plant morphology. Any change in temperature-response function among the sub-phases will indicate that plant response to temperature varies with stage of development. For example, Slafer and Rawson (1995a) divided the full pre-anthesis period in wheat (Triticum aestivum) into three sub-phases: from the beginning of the experiment to terminal spikelet initiation, from terminal spikelet initiation to heading, and from heading to anthesis. They found that both base temperature (the temperature at or below which the rate of development is considered to be nil) and optimum temperature (the temperature at which the rate of development is maximal) increased progressively with phasic development.

Such an approach is valuable in indicating any stage-dependence of plant responsiveness to temperature. However, it assumes that the change in temperature responsiveness occurs at the moment indicated by a plant morphological event, such as spikelet initiation or heading. If the change in temperature responsiveness occurs somewhere in between the two stages recorded in the experiment, the timing of the change cannot be identified accurately by the use of this approach.

The timing of any change in plant response to an environmental variable may best be quantified by a reciprocal-transfer experiment. In such an experiement, plants are initially grown under a favourable environmental condition and are then transferred in series at various times to a less-favourable environmental condition, whilst plants initially grown under the less-favourable environmental condition are simultaneously transferred to the favourable condition. The duration taken for plants to complete a phenophase plotted against the duration from the start of the experiment to the time of individual transfers will indicate whether or not there is a change in responsiveness of development to the environmental variable under study, and the moment in time at which the change occurs. This approach has successfully been used for identifying changes in plant photoperiod sensitivity and the length of photoperiod-sensitive and insensitive sub-phases of phenological development in various crops (e.g. Ellis et al., 1992a; Mozley and Thomas, 1995; Slafer and Rawson, 1995b; Yin et al., 1997b, 2005; Adams et al., 1998a, 2001). These data on the duration of a phenophase (typically, days from seed-sowing to flowering) for various crops reveal a common pattern of plant photoperiod response in that the photoperiod-sensitive sub-phase is flanked by a pre-sensitive juvenile sub-phase and a post-sensitive flower-development sub-phase, although the length of individual sub-phases differs considerably among crops and among cultivars within a crop. For this common pattern, Ellis et al. (1992a), who assumed that the relation between rate of development and photoperiod is linear, first developed a mathematical model for using combined data from both sets of transfers (from long-day to short-day, and from short-day to long-day photoperiods) in one single analysis. Later, Yin et al. (1997b) re-derived the same model without assuming the linearity for the development rate–photoperiod relationship, which, in fact, is typically non-linear over a wide range of photoperiods (Roberts and Carpenter, 1962; Vergara and Lilis, 1967; Yin and Kropff, 1996). The models of Ellis et al. (1992a) and Yin et al. (1997b) assume that the direction per se of photoperiod switch does not have an impact on plant responses to photoperiod. Although this assumption is sometimes not true (Baker et al., 1980; Yin et al., 1997b), the model allows the photoperiod-sensitive sub-phase and insensitive sub-phases of the pre-flowering period to be estimated unambiguously by making use of the combined data of both sets of transfers.

Yin et al. (1997a) were among the first to introduce this reciprocal-transfer experiment approach to examine whether or not there is any change in temperature responsiveness during the pre-flowering period (i.e. the interval from sowing to flowering). In their experiment, rice (Oryza sativa) plants were transferred at regular time intervals after sowing from low temperature (LT) to high temperature (HT) phytotron chambers whilst simultaneously the same numbers of plants were transferred from HT to LT chambers. For the three contrasting rice cultivars studied, they have found that (1) the entire period from sowing to flowering could generally be divided into three sub-phases, roughly corresponding to the three sub-phases identified from the experiments of reciprocal-transfers between two photoperiods, and (2) the second sub-phase was more temperature-sensitive than the first and the third sub-phases, which had a similar temperature sensitivity.

There are few literature reports of experiments involving reciprocal transfers between two temperatures (Adams et al., 1998b; Atkinson et al., 2001), probably due at least in part to the lack of a standard method for quantitative analysis. Yin et al. (1997a) analysed their data by simple regression for each visually identified linear segment (each one corresponding to a sub-phase), separately for LT-to-HT and HT-to-LT transfers. However, problems may arise from the separate-analysis approach. First, it is often difficult to judge which data points should be assigned to a particular sub-phase (especially those points around the transition from one sub-phase to another). Second, the length and sensitivity of sub-phases identified from the LT-to-HT transfers may not be consistent, in terms of response to temperature, with those identified from the HT-to-LT transfers. In the study reported here, a mathematical model is presented such that the data of both LT-to-HT and HT-to-LT transfers can be combined in a single curve-fitting procedure. As will be shown, this new, combined-model approach not only overcomes the problems encountered with the separate-analysis approach, but it can also reveal additional insights into phasic development in response to temperature.

MATERIALS AND METHODS

Experimental data

Yin et al. (1997a) reported in detail on three reciprocal-transfer experiments conducted in six indoor growth chambers (two for each experiment). Only the information relevant to the present model development is summarized here. Three rice cultivars (‘CO36’, ‘Shanyou 63’ and ‘Nipponbare’) were selected on the basis of their contrasting ecotypes and origins, and on results from an earlier study on their response to day and night temperatures (Yin et al., 1996). ‘CO36’ is an indica inbred cultivar from India, ‘Shanyou 63’ is an indica hybrid from China, and ‘Nipponbare’ is a janponica inbred cultivar from Japan. In the first experiment, plants of the three cultivars were mutually transferred between two diurnally constant temperatures (LT = 21 °C and HT = 26 °C) to examine the general response to temperature. Since the effect of temperature on development towards flowering in rice was found to differ between day and night periods (Yin et al., 1996), the second and the third experiments were designed to examine possible specific responses to night and day temperature, respectively. In the second experiment, plants were transferred between chambers with different night temperatures and the same day temperature (LT = 26/16 °C, HT = 26/26 °C for day/night). In the third experiment, plants were transferred between chambers with different day temperatures and the same night temperature (LT = 19/19 °C, HT = 28/19 °C). In chambers where diurnally varying temperatures were used, the duration of day and night temperatures was 12 h each throughout the experiments. Photoperiod was controlled as 12 h d^–1, matching the period of day temperature in each diurnal cycle, so as to avoid confounding effects of an asynchrony between thermoperiod and photoperiod (as reported by Morgan et al., 1987). Transfers were made with an interval of 6 d in the first and second experiments and of 5 d in the third experiment, with two replicate plants at each transfer. However, only one plant was transferred at a time in the third experiment for ‘Shanyou 63’. Once plants had been transferred, they were grown in the new temperature environment thereafter either until flowering or until the experiments were terminated at 175 d after sowing. The mean replicate value of days to flowering at each transfer is the subject of the analysis using the new approach described below.

Model approach

Data from the experiments of Yin et al. (1997a) with reciprocal transfers between LT and HT showed three sub-phases of the pre-flowering period. The model approach that is developed here for analysing the data of LT–HT transfers is similar to the model developed by Ellis et al. (1992a) and Yin et al. (1997b) for experiments involving transfers between two photoperiods. In analogy with their model, the following two assumptions are made here: (1) the transition from one sub-phase to another is abrupt; (2) the effect of temperature on development is accumulative, which means that the direction per se of temperature switch has no impact on the effective thermal-time requirement for plants to complete a phenophase. However, unlike the models of Ellis et al. (1992a) and Yin et al. (1997b), which are based on the fact that photoperiod has its effect only during the middle part of the pre-flowering period, the model presented below has to consider the fact that temperature has its effect throughout the full period. If the response pattern shown by Yin et al. (1997a) that the second sub-phase is more temperature-sensitive than the first and the third sub-phases is typical, then the relationship between the duration from sowing to flowering (f) and the duration from sowing to transfer (t) can be schematically represented by various linear sub-phases (Fig. 1). The objective of the model development is to estimate the length of the first sub-phase under both LT and HT conditions (denoted as I_1L and I_1H, respectively), the length of the second sub-phase (I_2L and I_2H for LT and HT, respectively), and the length of the third sub-phase (I_3L and I_3H for LT and HT, respectively).

Fig. 1.

Schematic representation of the response of duration from sowing to flowering (f) for plants transferred from low-temperature (LT) to high-temperature (HT) conditions (solid lines) or from HT to LT conditions (dashed lines) at various times after sowing (t), if (1) the period from sowing to flowering comprises three sub-phases, and (2) development is hastened by HT more during the second sub-phase than during the other two sub-phases. The three sub-phases under the LT conditions are indicated by the linear segments ‘AB’, ‘BC’, and ‘CD’, respectively, and those under the HT conditions are indicated by linear segments ‘EF’, ‘FG’, and ‘GH’, respectively. In the figure, I_1L and I_1H denote the length of the first sub-phase under LT and HT conditions, respectively; I_2L and I_2H denote the length of the second sub-phase under LT and HT conditions, respectively; and I_3L and I_3H denote the length of the third sub-phase under LT and HT conditions, respectively. Therefore, the duration from sowing to flowering at LT (f_L) can be expressed as f_L = I_1L + I_2L + I_3L, and the duration from sowing to flowering at HT (f_H) can be expressed as f_H = I_1H + I_2H + I_3H. The dashed arrows downwards indicate the position on the t-axis of the points of transition from one sub-phase to another identified from either LT-to-HT or HT-to-LT transfers. The horizontal extension beyond point ‘D’ for the LT-to-HT transfer and the extension beyond point ‘H’ for the HT-to-LT transfer take account of the possibility that some plants flower before the set transfer date.

In the reciprocal-transfer experiment, control plants continuously grown at HT are equivalent to plants transferred from LT to HT at sowing, and those continuously grown at LT are equivalent to plants transferred from HT to LT at sowing. These control plants at HT and LT are indicated by points ‘A’ and ‘E’, respectively, in Fig. 1. Let f_H and f_L be the duration from sowing to flowering for the HT and LT control plants, respectively. Expressions for f_H and f_L can be written as:

1A

1B

For plants transferred from LT to HT during the first sub-phase (i.e. t ≤ I_1L, which ends at point ‘B’ in Fig. 1), the duration from sowing to flowering can be expressed as:

2A

where β₁ is the slope factor for the linear segment ‘AB’ in Fig. 1. For plants transferred from LT to HT during the second sub-phase (i.e. I_1L < t ≤ I_1L + I_2L, which begins at point ‘B’ and ends at point ‘C’ in Fig. 1), the sowing–flowering duration is expressed as:

2B

where β₂ is the slope factor for the linear segment ‘BC’ in Fig. 1. For plants transferred from LT to HT during the third sub-phase (i.e. I_1L + I_2L < t ≤ f_L, which begins at point ‘C’ and ends at point ‘D’ in Fig. 1), the sowing–flowering duration can be expressed as:

2C

where β₃ is the slope factor for the linear segment ‘CD’ in Fig. 1. The experiment may also have plants that flower before they are transferred from LT to HT. The theoretical value for their duration from sowing to flowering is f_L, or I_1L + I_2L + I_3L. These expressions are summarized for plants transferred from LT to HT as:

2

Similar logic for the plants transferred from HT to LT gives:

3

where γ₁, γ₂ and γ₃ are the slope factors for the linear segments ‘EF’, ‘FG’ and ‘GH’, respectively, in Fig. 1.

Next, the slope factors have to be solved. For LT-to-HT transfers, the t-axis value for point ‘A’ and point ‘B’ in Fig. 1 can be easily determined as 0 and I_1L, respectively. The f-axis value of point ‘A’ is f_H, but that of point ‘B’ is less obvious. However, the delay in flowering time for point ‘B’, relative to point ‘A’, is simply due to the difference in the length of the first sub-phase between HT and LT; therefore, the f-axis value of point ‘B’ is f_H + I_1L − I_1H. From these t-axis and f-axis values of points ‘A’ and ‘B’, the slope factor β₁ in eqn (2) can be solved as:

4A

The t-axis value for point ‘C’ in Fig. 1 can be easily assigned as I_1L + I_2L. The delay in flowering date for point ‘C’, relative to point ‘B’, is simply due to the difference in the length of the second sub-phase between HT and LT; therefore, the f-axis value of point ‘C’ is f_H + I_1L − I_1H + I_2L − I_2H. From these t-axis and f-axis values of points ‘B’ and ‘C’, the slope factor β₂ in eqn (2) can be solved as:

4B

The t-axis and f-axis values of point ‘D’ in Fig. 1 are both f_L. From these t-axis and f-axis values of points ‘C’ and ‘D’, the slope factor β₃ in eqn (2) can be solved as:

4C

Similar logic for the HT-to-LT transfers results in the t-axis and f-axis values of points ‘E’, ‘F’, ‘G’ and ‘H’ in Fig. 1:

The slope factors γ₁, γ₂, and γ₃ in eqn (3) can then be solved as:

5A

5B

5C

Substituting eqn (1A,B) and eqn (4A–C) into eqn (2) gives the expression of the duration from sowing to flowering for plants transferred from LT to HT:

6

Substituting eqn (1A,B) and eqn (5A–C) into eqn (3) gives the expression of the duration from sowing to flowering for plants transferred from HT to LT:

7

By introducing dummy variables: Z₀ = 1 and Z₁ = 0 for LT-to-HT transfers, and Z₀ = 0 and Z₁ = 1 for HT-to-LT transfers, eqn (6) and eqn (7) can be represented collectively as:

8

Results of the reciprocal-transfer experiment can thus be quantified by the six parameters: I_1L, I_1H, I_2L, I_2H, I_3L and I_3H. These six parameters can be estimated using an iterative procedure of least-squares non-linear regression by standard statistical software. In this analysis, they are estimated by the GAUSS method in the PROC NLIN of the Statistical Analysis Systems Institute version 9·1 (SAS Institute Inc.). The SAS codes for this analysis can be obtained upon request.

RESULTS AND DISCUSSION

Model fit to the combined data

Using simple linear regression for each sub-phase separately, Yin et al. (1997a) showed that the slope for the first sub-phase and the third sub-phase, identified either from LT-to-HT or from HT-to-LT transfers, did not differ significantly (Figs 2–4). A notable exception is the data for the HT-to-LT transfers for ‘Nipponbare’ in Experiment 2 (Fig. 3C), where the first sub-phase originally had a positive slope whereas the third sub-phase had a negative slope. The positive slope may suggest that HT resulted in a slower development than LT for the first sub-phase, but this reasoning is not supported by the data of the equivalent LT-to-HT transfers. Therefore, I considered the point of the third HT-to-LT transfer to be an outlier, due to unknown reasons. Excluding this data point for analysis, the slopes for the first sub-phase and the third sub-phase were similar for all cultivar–experiment combinations. This means that β₁ = β₃ and γ₁ = γ₃, both yielding [cf. eqns (4A,C) and (5A,C)]:

9

Fig. 2.

Days from sowing to flowering for plants of three rice cultivars transferred from diurnally constant low temperature (LT, 21 °C) to high temperature (HT, 26 °C) (squares) or from HT to LT (circles) (Experiment 1). The full and dotted lines represent the model fit, eqn (8; cf. Table 1), to the observations of LT-to-HT and HT-to-LT transfers, respectively. The closed symbols and thick lines correspond to observations and model fit, respectively, for those plants that flowered before transfer.

Fig. 3.

Days from sowing to flowering for plants of three rice cultivars transferred from low night temperature (LT, 26/16 °C day/night) to high night temperature (HT, 26/26 °C) (squares) or from HT to LT (circles) (Experiment 2). Those plants of ‘CO36’ of early HT-to-LT transfers (with arrow upwards) had not flowered when the experiment terminated at 175 d after sowing and those of ‘Nipponbare’ at the third HT-to-LT transfer (the circled point) were apparently an outlier, and were therefore not included for curve-fitting (see text). Further details as for Fig. 2.

If eqn (9) is combined with eqn (8), only five parameters need to be estimated. An F-test showed that, relative to the five-parameter model, the six-parameter model did not lead to a significant improvement in fit to the data for all cases. Therefore, the results of curve-fitting using the five-parameter model are presented here (Table 1, Figs 2–4).

Table 1.

Duration of the first sub-phase (I_1L and I_1H for LT and HT conditions, respectively), of the second sub-phase (I_2L and I_2H), and of the third sub-phase (I_3L and I_3H) of development from sowing to flowering in three rice cultivars. The duration was estimated by non-linear curve-fitting based on eqns (8) and (9) to data obtained either from Experiment 1, where plants of the three cultivars were transferred between two growth chambers of different diurnally constant temperatures (LT = 21/21 °C and HT = 26/26 °C for day/night, cf. Fig. 2); from Experiment 2, where plants were transferred between two growth chambers of different night temperatures with the same day temperature (LT = 26/16 °C and HT = 26/26 °C, cf. Fig. 3); or from Experiment 3, where plants were transferred between two growth chambers of different day temperatures with the same night temperature (LT = 19/19 °C and HT = 28/19 °C, cf. Fig. 4).

Cultivar	I1L (d)	I1H (d)	I2L (d)	I2H (d)	I3L (d)	R2	na	Ib3H	fLc	fHd
Experiment 1
‘CO36’	48·06 (6·36)	44·65 (4·86)	58·35 (8·78)	37·01 (7·40)	8·92 (4·35)	0·925	42	8·29	115·33	89·95
‘Shanyou 63’	34·37 (3·74)	40·16 (2·82)	64·81 (4·71)	35·72 (4·34)	5·63 (2·12)	0·929	38	6·58	104·81	82·46
‘Nipponbare’	25·12 (4·00)	20·00 (2·98)	17·70 (6·00)	7·40 (4·29)	23·37 (2·71)	0·977	38	18·61	66·19	46·01
Experiment 2
‘CO36’	54·01 (4·85)	41·88 (2·26)	143·10 (16·67)	20·63 (2·61)	39·23 (3·07)	0·969	33	30·42	236·34	92·93
‘Shanyou 63’	47·47 (3·39)	42·57 (1·19)	81·00 (4·08)	19·43 (1·89)	22·50 (1·79)	0·983	38	20·18	150·97	82·18
‘Nipponbare’	28·23 (2·54)	19·99 (0·62)	38·99 (4·09)	6·82 (1·07)	26·96 (2·31)	0·988	29	19·09	94·18	45·90
Experiment 3
‘CO36’	31·58 (4·04)	19·37 (2·44)	31·65 (6·79)	38·50 (4·83)	98·18 (4·19)	0·903	46	60·22	161·41	118·09
‘Shanyou 63’	74·24 (6·32)	74·51 (4·33)	52·01 (8·33)	23·08 (7·29)	15·49 (4·87)	0·808	44	15·55	141·74	113·14
‘Nipponbare’	18·96 (5·14)	15·04 (2·88)	80·95 (4·88)	43·43 (3·02)	0·00e	0·986	44	0·00	99·91	58·47

s.e. given in parentheses.

^a Number of data points used for curve-fitting (cf. Figs 2–4).

^b Calculated according to eqn (9), i.e. I_3H = I_3LI_1H/I_1L.

^c Calculated according to eqn (1B).

^d Calculated according to eqn (1A).

^e Fixed at zero in the curve-fitting (see text).

The five-parameter model described the data satisfactorily (Figs 2–4), with R² ranging from 0·81 to 0·99 (Table 1). Three sub-phases were identified from curve-fitting in all cases except for ‘Nipponbare’ in Experiment 3 (Fig. 4C). For this particular case, the value of I_3L was initially estimated as –1·85 d (with s.e. = 4·60), not significantly different from zero. Therefore, the parameter value for ‘Nipponbare’ in Experiment 3 (Table 1) was estimated from fixing I_3L at zero in the second-round fitting; thus only two sub-phases were identified.

Fig. 4.

Days from sowing to flowering for plants of three rice cultivars transferred from low day temperature (LT, 19/19 °C day/night) to high day temperature (HT, 28/19 °C) (squares) or from HT to LT (circles) (Experiment 3). Further details as for Fig. 2.

From eqns (4A–C) and (5A–C), it can easily be obtained that:

10

where i = 1, 2 and 3. Equation (10) allows a statistical test of the validity of the assumption upon which the new model method was based, i.e. that the direction per se of temperature switch has no impact on the effective thermal-time requirement for plants to complete a phenophase. Equation (10) holds if this assumption is true. Of the 27 experiment × cultivar × sub-phase combinations, the data for 20 occasions are sufficient to allow the statistical test. Of these 20 occasions, only in five did β_i differ significantly (P < 0·05) from 1 – 1/(1 − γ_i), largely due to the scatter of data. This test indicates that, overall, the assumption underlying the new approach is theoretically justified.

Advantages of the combined-model analysis

Relative to the separate-analysis approach, the advantages of the combined approach are obvious. First, only five or six parameters (I_1L, I_1H, I_2L, I_2H, I_3L and/or I_3H) are needed to represent the whole combined data, whereas the separate-analysis approach needs a maximum of 12 parameters (= 2 transfer series × 3 sub-phases × 2 parameters per sub-phase per transfer series). From these six parameters, the slope factor of each sub-phase can be calculated from eqns (4A–C) and (5A–C). Secondly, in the combined analysis using eqn (8), all the data points are used for curve-fitting, whereas the separate-analysis approach does not utilize data points for those plants that flower before transfer (i.e. the solid symbols in Figs 2–4). Thirdly, the combined approach results in an inherent consistent solution, which ensures that the length of sub-phases identified from the LT-to-HT transfers is compatible, in terms of plant response to temperature, with that identified from the HT-to-LT transfers. This is achieved because individual linear sub-phases were fitted simultaneously to result in the parameter values that give the best overall fit to the combined data. As a result, the model may not describe exactly the individual linear segments of the sub-phases of either LT-to-HT or HT-to-LT transfers.

Additional insights from the combined-model analysis

Several additional insights were revealed from the curve-fitting to the combined data of LT-to HT and HT-to-LT transfers. Firstly, on two occasions, the estimated length of a sub-phase was somewhat higher under HT than LT conditions, i.e. I_1H > I_1L for ‘Shanyou 63’ in Experiment 1 and I_2H > I_2L for ‘CO36’ in Experiment 3 (Table 1), although an F-test showed that the differences in both cases were not significant (P > 0·05). As a result, the pattern for the first sub-phase in Fig. 2B and the second sub-phase in Fig. 4A contrast with the general pattern as illustrated in Fig. 1. Since the relationship between development rate and temperature is typically bell-shaped over a wide range of conditions (e.g. Ellis et al., 1992b; Yin et al., 1995), the longer estimated duration of a sub-phase under HT than LT conditions suggests that the HT used in the experiment might have been supra-optimal for the sub-phase, such that the HT yielded slightly slower development than the LT did. This reasoning does not necessarily infringe the results for these cultivars in other experiments, because the relative difference between HT and LT varied in those experiments. A change in the optimum temperature with genotype and with sub-phase of development is in line with the experimental reports of Slafer and Rawson (1995a).

Secondly, although the timing for the end of the last sub-phase at LT was often not observed from LT-to-HT transfers (Figs. 3–4), the length of the last sub-phase at LT (i.e. I_3L) can still be estimated confidently by the combined analysis (Table 1), mainly by deploying the information from those plants of early HT-to-LT transfers. It is not possible to estimate I_3L from the separate analysis (Yin et al. 1997a). For ‘CO36’ in Experiment 2, the estimated f_L was 236·3 d (Table 1), and the estimated days from sowing to flowering for the first nine HT-to-LT transfers were 235·8, 234·0, 232·3, 230·5, 228·8, 227·1, 225·3, 211·6 and 176·0 d, respectively; so, correctly predicting that plants of these transfers did not flower at 175 d after sowing when the experiment terminated (Fig. 3A).

Thirdly, data points of ‘CO36’ and ‘Shanyou 63’ in Experiment 3 were quite irregular, which Yin et al. (1997a) explained to be probably due to more chilling damage by the constant LT (i.e. 19 °C) throughout the experiment in the two indica rice cultivars than in the japonica ‘Nipponbare’. The separate-analysis approach, as used by Yin et al. (1997a), relies on the pre-identification of sub-phases by visual inspection of data points. Therefore, for these two cultivars in Experiment 3, Yin et al. (1997a) were unable to perform an analysis. However, using the combined approach, the model resulted in a decent fit to the data (Fig. 4A,B), especially for ‘CO36’, although uncertainties existed for ‘Shanyou 63’ for which the final estimated parameter values depended on the initial values given for the iteration procedure of non-linear regression. For this cultivar, the best final fit of several runs has been shown in Fig. 4B and Table 1.

Obviously, curve-fitting to the combined data based eqn (8) is a more objective and powerful approach than the separate linear-regression analysis used by Yin et al. (1997a). The latter approach cannot estimate the duration for many sub-phases for the rice data set. For the sub-phases whose duration was estimated by both approaches, the difference in the estimates by the two approaches was often substantial (Fig. 5).

Fig. 5.

Durations of sub-phases at low temperature (LT, open circles) or at high temperature (HT, closed circles) estimated from using eqn (8) fit to the combined data compared with those estimated by Yin et al. (1997a) using separate simple linear regression on visually identified individual sub-phases. The diagonal line represents the 1:1 relationship.

Application of the model

The present model, eqn (8), was illustrated using the experimental data of Yin et al. (1997a), in which both HT and LT treatments started immediately from sowing. It is likely that in an experiment all plants are grown first under the same conditions (e.g. natural, open-air environment) before the temperature treatment starts. For such a case, eqn (8) is still valid provided that f in the model refers to the duration from the start of temperature treatment to the end of a phenophase under study, and t refers to the duration from the start of temperature treatment to transfer. Similarly, phenological events can be measured not only using number of days but also by other traits such as main-stem leaf number (e.g. Tollenaar and Hunter, 1983; Adams et al., 1998b). So long as the relation can be represented by linear sub-phases, eqn. (8) can also be applied if f in the model refers to the final main-stem leaf number and t refers to the main-stem leaf number at the time of transfer. However, unlike photoperiod, temperature generally has a strong effect on leaf initiation and appearance rates but little, or only a small effect, on the final leaf number (e.g. Miglietta, 1991); thus, use of the final leaf number recorded in a reciprocal-transfer experiment to identify changes in temperature response may not be very reliable.

Equation (8) presumes that a phenophase can be divided into three sub-phases in terms of temperature responsiveness. In cases where only two sub-phases are identified, the model can be applied by pre-fixing I_3H = I_3L = 0, as for ‘Nipponbare’ in Experiment 3 (Fig. 4C). What about more than three sub-phases? In the case of four sub-phases, eqn (8) can be easily extended using the principles presented here by including two additional parameters, I_4L and I_4H – the length of the fourth sub-phase under LT and HT conditions, respectively. The model then contains a total of eight parameters: a simultaneous estimation of such a high number of parameters would not personally be recommended because it can easily yield over-fitting to any data set. Additional information, e.g. eqn (9), has to be found to reduce the number of parameters to be estimated if, indeed, there are more than three sub-phases observed.

The approach of Ellis et al. (1992a) and Yin et al. (1997b) for the combined analysis of data from two-series reciprocal transfers between two photoperiods assumes that plants during the first and the third sub-phases of the sowing–flowering period are photoperiod-insensitive. Their models are clearly the special case of eqn (8) presented here if I_1H = I_1L and I_3H = I_3L, presuming that HT is equivalent to a long-day photoperiod for long-day plants or a short-day photoperiod for short-day plants and LT is equivalent to a short-day photoperiod for long-day plants or a long-day photoperiod for short-day plants. To this end, eqn (8) is a generalized version of the models of Ellis et al. (1992a) and Yin et al. (1997b), thereby providing a model with which the question of whether the first and/or third sub-phases of the pre-flowering period are indeed photoperiod-insensitive can be tested statistically. Furthermore, eqn (8) is flexible in that it can be fitted where LT hastens development, thereby potentially offering a good approach to quantify the timing and sensitivity of a vernalization response. In principle, eqn (8) can be applied to identify changes in developmental response of any species (including, for example, insects) to any environmental variables (e.g. water stress), provided that two contrasting levels of these variables or stresses can be maintained consistently throughout the experiment and can result in a significant difference in the development of the species under study.

The curve-fitting to data for several cases (e.g. ‘CO36’ in Experiment 1, ‘Shanyou 63’ in Experiments 2 and 3) yielded an estimated I_1H that was nearly identical to the estimated I_1L (Table 1; difference not significant, P > 0·05). This indicates that either the development was insensitive to temperature, or that LT and HT in the experiments straddled the optimum temperature (thereby yielding similar development rates) during the first sub-phase. A possible change of the optimum temperature with plant age is also supported by the previously mentioned occasions where a delay of development caused by HT was detected (i.e. Figs 2B and 4A). Hence, re-analysis of the data from Yin et al. (1997a) for rice by use of the new approach provided a suggestion that either the temperature sensitivity or optimum temperature of the developmental response varies with plant age within the pre-flowering period of this crop. However, to enhance our understanding of rice flowering physiology, a further experimental study using good temperature-controlled facilities, combined with an analysis using the technique presented here, would be needed to elucidate whether it was the optimum (day and night) temperature or the thermal sensitivity per se (or both) that caused in the changes in the slope values as shown in Figs 2–4.