Recombinant inbred line population
The same recombinant inbred line (RIL) population used in a previous study (Yin et al., 2000
) was used in the present study. The population consists of 94 RILs, produced by eight generations of single seed descent from a cross of the two-row spring barley cultivars ‘Apex’ and ‘Prisma’. The two parents differ in yielding ability, morphological characteristics, and phenological traits. Plants of ‘Apex’ usually flower earlier than those of ‘Prisma’.
Greenhouse and field experiments
A greenhouse experiment was conducted at Jiangxi Agricultural University, China. Seeds of each RIL were sown on 8 November 2002 in pots arranged on mobile trolleys. The trolleys stayed in the greenhouse between 07.00 h and 17.00 h, after which they were moved into darkrooms attached to the greenhouse. The darkrooms were programmed to create the required photoperiods. There were two photoperiod conditions: long-day (LD) and short-day (SD), which were set to 15 h and 10 h, respectively. The LD photoperiod was created by providing the darkroom with 10 µmol m–2 s–1 supplementary light of 2 h in the morning (05.00–07.00 h) and 3 h in the evening (17.00–20.00 h) of each day. Temperature both in the greenhouse and in the darkrooms was monitored over the experimental period. Initially, half of the pots for each RIL were placed in SD, half in LD. Starting from 17 d after sowing, plants were simultaneously transferred, one pot at a time for each RIL, from one photoperiod to the other, at an interval of 10 d. Once a plant was transferred, it was grown in the new photoperiod until flowering. In total, 14 transfers were implemented. The control plants grown continuously in SD or LD are equivalent to those transferred at 0 d after sowing. The two series of transfer, with 15 transfer times for each series (14 real transfers plus 1 control plant), gave a total of 30 observations for each RIL.
Field experiments were conducted over two growing seasons on the experimental farm of Jiangxi Agricultural University (latitude 28.7° N). In the first season, seeds were sown five times (month/date: 10/25, 11/5, 11/16, 11/25, and 12/4) in 2001. In the second season, there were three sowing dates (10/25, 11/10, and 11/25 in 2002). This created a total of eight field environments. Flowering time was recorded for each RIL of each environment. Daily maximum and minimum temperature over the seasons was obtained from a nearby weather station.
Because some RILs did not flower due to the incomplete appearance of ears from the flag-leaf sheath, the date of awn appearance was considered as the flowering time for all RILs in both greenhouse and field experiments. The greenhouse experiment was used to estimate model parameters. These parameters were then applied to predict flowering time of individual RILs in the two-season field experiments.
Specifying model parameters
The greenhouse experiment does not allow temperature response-related model parameters to be specified for each RIL. Nevertheless, it is a general rule that within a crop species, the genetic difference in temperature sensitivity of phenology is small, compared with the genotypic difference in photoperiod sensitivity (Ellis et al., 1990). Therefore, the same parameter value involving temperature response for all the RILs and parents was assumed. They are determined as: Tb=0, To=21 and Tc=35 °C, based on data of Ellis et al. (1988), Cao and Moss (1989), and Tamaki et al. (2002) for barley. Sensitivity analysis showed that a change in the value of these three cardinal temperatures had only small impact on final model predictions. For photoperiod response, genotypic difference in Po is also small, compared with that in 
, 
1, and 2 (Yin et al., 1997b). It was assumed that all RILs have the same Po of 17 h (Ellis et al., 1988). Thus, the above model has four parameters (fo, 1, 2, and ) to be estimated.
Analysis of data from the greenhouse experiment
Because temperature fluctuated both diurnally and seasonally under these experimental conditions, any impact of these fluctuations should be accounted for in order to obtain an accurate estimate of the four parameters. Equation (2) was used to convert observed days to flowering in the greenhouse experiment to thermal days. A thermal day is equivalent to an actual day only if temperature at any moment of that day is at the optimum value (i.e. 21 °C). g(T) was estimated on an hourly basis and hourly g(T) values were averaged to obtain a daily value. When plants remained in the darkrooms, the hourly temperature was obtained from monitored room temperatures. The hourly temperature outside the darkroom was estimated from the observed daily maximum and minimum temperatures by a sine function assuming the daily maximum occurs at 14.00 h each day (Matthews and Hunt, 1994). Accumulated daily g(T) from sowing to the day when an RIL flowered is the total thermal days for that RIL. Using the calculated thermal days, the model parameter values were estimated.
The time interval from sowing to flowering (f) of each RIL from the SD–LD mutual transfer experiment can be quantified by the following summary relations (Yin et al., 1997a):
 |
(4) |
where
t is the time interval from sowing to each transfer,
I1 is the duration of the juvenile phase when plants are not yet responsive to photoperiod,
I2I and
I2N are the durations of the photoperiod-sensitive phase under the inductive and non-inductive photoperiod conditions, respectively (15 h and 10 h, respectively, in this experiment),
I3 is the duration of post-photoperiod-sensitive phase,
Z0 and
Z1 are dummy variables:
Z0=0 and
Z1=1 refers to transfers from inductive to non-inductive photoperiods, and
Z0=1 and
Z1=0 refers to transfers from non-inductive to inductive photoperiods. Readers are recommended to refer to an earlier work (Yin
et al., 1997
a) for details of
equation (4). With this model,
I1,
I2I,
I2N, and
I3 can then be estimated using the iterative procedure of the PROC NLIN of the Statistical Analysis Systems (SAS Institute Inc.). The advantage of such an approach is that data from both sets of transfers (LD to SD and SD to LD) can be combined in one single analysis (Adams
et al., 2001
).
Conversion of I1, I2I, I2N, and I3 into the four model parameters fo, 1, 2, and
Equations are derived here for the conversion of I1, I2I, I2N, and I3 into fo, 1, 2, and , according to whether or not the inductive photoperiod (PI) in the experiment agrees with the optimum photoperiod Po.
If PI equals Po or is above Po for long-day plants and below Po for short-day plants, then fo, 1, and 2 can be straightforwardly calculated by:
 |
(5a) |
 |
(5b) |
 |
(5c) |
where
I2o is the minimum duration of photoperiod-sensitive phase, equivalent to
I2I in the case. The photoperiod sensitivity parameter
in
equation (3) can be estimated by:
 |
(6) |
where
PN is the non-inductive photoperiod used in the transfer experiment.
Equation (6) is derived by considering the accumulation of daily development rate over the photoperiod-sensitive phase at the optimum temperature condition. First, in terms of definition of the phenology model,
equation (1), this accumulation for the case of non-inductive photoperiod gives:
 |
(7) |
where
I2o/
fo is the result by substituting
equation (5b,
c) for
1 and
2.
Equation (7) means that
 |
(8) |
Secondly, the accumulation for the case of the optimum photoperiod is:
 |
(9) |
Equation (9) means that
 |
(10) |
Because
h(
Po)=1, the ratio of
equation (8) to
equation (10) results in:
 |
(11) |
Substituting
equation (11) into
equation (3) and solving for
results in
equation (6).
If PI is below Po for long-day plants and above Po for short-day plants, equation (5a–c) is still valid; but first, I2o in these equations has to become known. The ratio of I2I to I2N can be formulated as:
 |
(12) |
On the basis of
equation (3) and
equation (11),
equation (12) can be further written as:
 |
(13) |
Solving for
from
equation (13) gives the photoperiod sensitivity parameter
:
 |
(14) |
The value of
I2o can then be estimated by either of the following two:
 |
(15a) |
 |
(15b) |
where
is calculated by
equation (14).
For the present greenhouse experiment, PI is 15 h, less than the pre-set Po (=17 h). Therefore, equation (14) and equation (15) are used first to obtain and I2o, respectively. Equation (5a– c) is then applied to obtain the estimates for fo, 1, and 2.
Predicting flowering dates of field experiments
On the basis of the four parameters fo, 1, 2, and , equations (1– 3) were used to predict the flowering dates obtained from field experiments with eight sowing dates of the two growing seasons. The calculation used hourly temperatures, with the approach mentioned earlier. The photoperiod required for each day under field conditions of the growing seasons was the astronomic daylength calculated from the equation presented by Goudriaan and van Laar (1994). For each genotype grown in each environment, calculated daily development rates were accumulated to obtain the value of development stage ; the day at which is 
1.0 is the predicted date of flowering for that genotype grown in that particular environment.
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