Models perform better in highly synchronised light–temperature cycles
To examine the attributes of the Wilczek et al. model, we first conducted a detailed analysis of model behaviour alongside the meteorological data. Our analysis showed that the model could more accurately predict the flowering time for some Arabidopsis cohorts than for others. Fig. 2(b) plots the observed vs predicted values, showing that the model produced a good fit for plantings in the summer and spring. For Autumn cohorts, there are three distinctive regions of fit: the Valencia Autumn cohort scattered along the diagonal line, indicating a good match between observed and expected values (Fig. 2e); flowering times of Halle and Cologne Autumn cohorts were mostly overestimated (Fig. 2(f–g)); flowering times of late-flowering genotypes (gi-2, Col-FRI-Sf2, vin3-1 and fve-3) from Norwich Autumn were considerably underestimated (Fig. 2h). To determine the basis for goodness of fit (GoF) variation, the model was analysed using data subsets from cohorts in these different groups, then matched with the respective meteorological data (Fig. 2c–h). The GoF indicator used for subset cohorts was root-mean-squared error (RMSE) expressed in d, which estimates the differences between values predicted by a model and the field data. The RMSE was also normalised by the mean of the observations, which is a measure commonly referred to as the coefficient of variation of the RMSE, or CV(RMSE), and is expressed as a percentage. Larger values of the CV(RMSE) indicate more substantial relative deviations between model predictions and field data.
Our analysis showed that RMSE values were relatively low for Spring, Summer and Valencia Autumn plantings, indicating that the Wilczek et al. model could accurately predict flowering time of these cohorts (Fig. 2c–e). An incremental rise in RMSE was, however, observed for the Autumn cohorts at Halle, followed by Cologne and Norwich, where the discrepancies between model predictions and field data were the greatest (Fig. 2f–h). Analysis of the meteorological data revealed marked differences in the daily temperature trends through the seasons. The Valencia Autumn cohorts, as well as the Spring and Summer cohorts, typically experienced cooler night-time and warmer daytime temperatures (Fig. 2c–e). At the other Autumn sites, the temperature rhythm became less predictable, with occasional peaks at night (Fig. 2f–h). Figs 2(c–h) typify seasonal variation in daily temperature time series. Extended through time, this variation leads to distinctly different patterns of day vs night thermal time accumulation (Supporting Information, Fig. S2). It therefore appears that the model could match the flowering time data with greater accuracy when plants had experienced strong phase synchrony between light and temperature cycles, but was less precise when this was not the case.
Interestingly, the Wilczek et al. model produced comparable GoF for genotypes within each cohort, with the exception of Norwich Autumn where plants fell into two discreet groups: rapid cyclers and winter annuals (Fig. 2f–h). The model prediction for Col wild type (wt), Ler wt and co mutant from this cohort was good. These genotypes were planted in early autumn, compared with later plantings at the Cologne and Halle sites, to coincide with the natural germination flush, which starts earlier in Norwich (Fig. 2a) (Wilczek et al., 2009). During early development, these plants experienced long hours of daylight (11–13 h) and relatively warm daytime temperatures. They therefore adopted a rapid life cycle and bolted in the autumn without vernalisation. However, the model underestimated late-flowering genotypes, gi-2, Col-FRI-Sf2, vin3-1 and fve-3, from Norwich Autumn, which did not flower until the following spring.
The synchrony of thermal and photo-cycles was not explicitly included in the Wilczek et al. photothermal model, but during model optimisation the lowest cost function was achieved when the ‘thermal time’ component (see Fig. 1) considered only daytime temperatures (Wilczek et al., 2009). The model was therefore developed to include a filter P (Eqn 3) that captured the effect of temperature during the photoperiod, and night temperatures were disregarded in the accumulation of degree-days (Fig. 3a). This simple function generated higher MPTUs for genotypes exposed to highly synchronised light–temperature cycles, where temperature was low at night and rose during the day. Meteorological data illustrate that the Cologne and Halle Autumn cohorts did not experience robust daily oscillations; instead the temperature profile was more variable with prolonged periods of relative stability and occasional rises in night temperature relative to day (Fig. 2). As night temperatures can affect floral initiation (Thingnaes et al., 2003), we reasoned that the model may be less accurate at predicting flowering time of winter cohorts, as warm temperatures above the threshold for degree-day accumulation occurred more frequently during the night.
Figure 3. Filter functions P that account for the differential effects of day and night temperatures. The black and white bars represent light–dark cycles. (a) In the Wilczek et al. (2009) model, only day temperatures are considered in the thermal time component by multiplying by a factor of 1 at daytime (Pday) and 0 at night-time (Pnight), thus forming a square waveform. (b) In Model 1 (gradual gating), a ‘triangle’ waveform is used with maximum factor (1) at mid-day and minimum factor (0) at mid-night. A constant factor Pdd is locked to dawn and dusk. (c) In Model 2 (step gating), a square waveform with a non-zero night factor (Pnight) is used.
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To overcome these deficiencies, two new models which included night temperatures in the accrued degree-days were developed. These models incorporated variations of the filter function P in the Wilczek et al. model that considered night temperatures (Fig. 3). In the Wilczek et al. model, day temperatures were effectively taken into account by multiplying by a factor of 1 (Pday), while night temperatures were disregarded using a factor of 0 (Pnight), thus forming a square waveform (Eqn 3; Fig. 3a). In the first model variant (Model 1), a ‘triangle’ waveform function was used, where 1 and 0 were fixed to the middle of day and night, respectively (Fig. 3b). Such a function allows daytime as well as night-time temperatures to be considered in the accumulation of thermal time units that promote flowering, with the highest effect at midday and the lowest effect in the middle of the night. This function could serve as a proxy for circadian gating of the temperature response that is known to occur in Arabidopsis and other species (Rikin et al., 1993; Fowler et al., 2005; Espinoza et al., 2010). Thus, Model 1 accounts for a gradual gating of temperature effects. Factors at sunrise and sunset were set to a constant value (Pdd) so that the effects at lights-on and lights-off were always the same to allow tracking of dawn and dusk (Edwards et al., 2010). We also created Model 2, where a step gating function was introduced by setting (a priori) Pday as 1 as in the Wilczek et al. model; however, a non-zero night temperature factor (Pnight) applied universally across all plantings was estimated through model-fitting of field data (Fig. 3c). Both model variants contained an additional parameter each, that is, Pdd for Model 1 and Pnight for Model 2.
When compared with the Wilczek et al. model, the new models achieved comparable GoF of Spring, Summer and Valencia Autumn cohort data (with changes in RMSE < 0.5 d). In addition, both new models improved the fit for Autumn data. For the Cologne Autumn cohort, the RMSE was reduced from 23.4 d (20.1%) in the Wilczek et al. model to 16.1 d (13.8%) in Model 1 and 16.4 d (14.0%) in Model 2 (Figs 2, S1). There was also improvement for the Halle Autumn cohort, where RMSE decreased from 14.3 d (10.7%) in the Wilczek et al. model to 8.8 d (6.6%) in Model 1 and 10.2 d (7.7%) in Model 2. Concurring with the observed daily temperature cycles (Fig. 2), thermal time accumulated at a faster rate during the daytime than during the night, at the Spring, Summer and Valencia Autumn sites (Fig. S2). By contrast, thermal time accrued at a more comparable rate during the daytime and night-time at the Halle and Cologne sites. This seasonal difference in thermal time accumulation rate was a result of both reduced day–night amplitudes as well as longer nocturnal durations in the autumn. Collectively, our data indicate that the inclusion of night temperature effects in the thermal time component could be important for determining flowering time of Autumn cohorts.
Interestingly, the new model variants still could not describe the gi-2, Col-FRI-Sf2, vin3-1 and fve-3 late-flowering genotypes in Norwich Autumn. This put forward the possibility that the relatively poor performance of the Wilczek et al. model for Autumn cohorts was only a bias as the optimiser tried to split the difference between divergent genotypes, that is, the four ‘outlier’ genotypes vs the others. If that were true, removing the outliers should improve the fits of both the Wilczek et al. model and the new model variants. We therefore re-parameterised the Wilczek et al. model and our new models without the gi-2, Col-FRI-Sf2, vin3-1 and fve-3 data from Norwich Autumn. As can be seen in Table S1, there was not much improvement in the Wilczek et al. model even without the ‘outlier’ data, but both our new models which incorporated night temperature improved considerably. These results support the inclusion of night temperature in the models to accurately describe Autumn cohort data. As the proportion of temperature data considered by the Wilczek et al. model reduced with the falling photoperiod, our model improvements may simply arise from extending the period during which temperature was considered in the autumn and winter. However, we explored this possibility previously (Wilczek et al., 2009), and the overall fit deteriorated with the incremental inclusion of post-dusk temperature hours. Alternatively, the improved fitting in our new models may reflect seasonal differences in the effectiveness of day and night time temperatures in controlling flowering time.
Both our new model variants displayed improved fit, which was statistically expected with an increase in the number of parameters. We therefore used the second-order Akaike Information Criterion (AICc), which compares model accuracy but penalises for model complexity, to consider whether the additional parameter in each of our new models could justify the improved fit (Table S1a). Lower AICc values indicate the more strongly supported models. In general, both our model variants displayed lower AICc values compared with the original model in all cases except one, indicating that they have strong statistical backing. Model 1 displayed the best improved fit and lowest AICc values compared with Model 2. Nevertheless, owing to the additional flexibility offered by the P function in Model 2 (see phyB mutant section below), this step-gating model variant was selected for subsequent study. The parameter values for Model 2 are listed in Tables S2, S3.
Predicting the bolting times of phyA and phyB mutants
One advantage of a genetically informed model is its ability to describe different genotypes by making simple changes to relevant model components. The photothermal model was parameterised using field data of seven genotypes impaired in the photoperiod, vernalisation or autonomous pathways. Here we explore the ability of the model to describe genotypes not previously used in model optimisation, that is, photoreceptor mutants. We first compared the published leaf number data (Giakountis et al., 2010) of phyA-201 and phyB-1 mutants with that of the Ler wt (Fig. S3a). These mutant alleles were also included in the field study (Wilczek et al., 2009). We used leaf number data for phenotypic comparison, as this indicator has been shown to be tightly coupled to bolting time within a wide photoperiod window (Koornneef et al., 1991; Pouteau et al., 2006). Leaf number data are also more widely available in the literature. We modified the parameters in the photoperiod component (CSDL, CLDL, DSD and DLD in Eqn 1) in Model 2 based on the proportional differences between the mutants and the wild type (Fig. S3a). While a proportional rate informed by leaf number data may not be quantitatively accurate, the data displayed a qualitative photoperiod response that supports the role of these mutants in the photoperiod pathway (see Fig. S3a and below). This qualitative response also concurs with early flowering time phenotype that has been reported for phyB mutants under both long- and short-day conditions (Mockler et al., 1999; Cerdan & Chory, 2003).
According to Fig. S3(a), the rate to bolting for phyA-201 during long days (with photoperiod above 10 h) was lower, correlating with a loss of phyA activity in stabilising CO protein (Valverde et al., 2004). However, the maximum rate was not altered, as there are other layers of CO regulation by FKF1 and GI (Salazar et al., 2009). For phyB-1, the rate was higher in general, following the role of activated phyB in promoting the degradation of CO protein in the morning (Valverde et al., 2004). In our model adjustment, we assumed that the phyB-1 rate to bolting would achieve its maximum at photoperiods of 16 h or above. Adjusted parameters are listed in Table S2. Fig. S3(b) shows that the modified model could predict the bolting times of phyA-201 mutant grown in the same field plantings in Wilczek et al. (2009) (Table S4), with a RMSE of 7.4 d (14.2%). The RMSE for phyB-1 was 7.6 d (19.1%) but deviations were uneven, with all Spring/Summer cohorts underestimated while Autumn cohorts were overestimated.
Published data have shown that phyB has a temperature-dependent role in flowering (Halliday et al., 2003; Halliday & Whitelam, 2003). As our earlier results suggested that the phase relationship between photoperiod and temperature cycles was significant, we sought to establish if our thermal-gating model might reveal any information regarding the dual role of phyB in light and temperature signalling. Using the new photoperiod parameters for phyB-1 as described earlier (Fig. S3a), we re-estimated both the day and night factors (Pday and Pnight) in step-gating Model 2 (Fig. 3c) to fit the phyB-1 field data while holding all other parameters to the values estimated earlier for the wild type. Intriguingly, we achieved optimal fitting when Pday and Pnight were at values of 0.5959 and 0.6856, respectively. This unexpected result suggested that temperature gating was almost abolished in phyB-deficient plants. To test this, we re-parameterised the model for phyB-1 by constraining Pday and Pnight to be equal and constant (Fig. S3c), in other words abolishing the gating effect. The modified model, with a constant ‘gating’ factor of 0.6279, showed a marked improvement with a RMSE of 4.9 d (12.5%) (Fig. 6) compared with 7.6 d (19.1%) in Fig. S3(b). Nevertheless, this improvement could be a mathematical artefact to compensate for the changes made in the photoperiod component. However, using Model 2 alone without any modification, that is, the model for Ler wt, resulted in a RMSE of 19.8 d (50.0%), suggesting that modification(s) was indeed required to describe phyB-1 field data. To further investigate this, we repeated the estimation of Pday and Pnight but without altering the photoperiod parameters. In this case, phyB-1 mutants experienced fully accelerated rate on days of intermediate (14 h) and not just very long (16 h) photoperiods. The optimised values for Pday and Pnight were 0.8291 and 0.6992, respectively, which again displayed a reduced gating effect. Comparison of AICc values for different modification schemes (Table S1b) supported the notion of constant gating for phyB-1. In addition, the model with double modifications that embodied a constant gating showed the lowest set of RMSE and AICc values. These results suggested that in order to describe the phyB-1 mutant field data, both the photoperiod and thermal-gating modifications were required.
Figure 6. Predicted vs observed bolting times of Arabidopsis thaliana phyA-201 (open circles) and phyB-1 (closed circles) mutants using Model 2 (step gating). The photoperiod component was modified according to Supporting Information Fig. S2(a) for both mutants. For phyB-1, a constant gating of 0.6279 was adopted. The observed values are field data from the same plantings in Wilczek et al. (2009). The diagonal line represents the perfect fit. Error bars represent one standard error. CV(RMSE), coefficient of variation of the root-mean-squared error.
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