Introduction
The objective of this study was to fit a hydrothermal germination model to germination data for a seedlot of radiata pine (Pinus radiata D. Don), an important plantation forest species. Hydrothermal germination models offer a useful way to understand why and how seed germination varies under different seedbed conditions, and between different seed populations. In hydrothermal models, germination rate and percentage germination are explained by seed population characteristics and by the water potential (Ψ) and temperature (T) of the seed's environment. The hydrothermal germination model was first proposed by Gummerson (Reference Gummerson1986) and can be specified mathematically as:
 t (G)](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151024042816780-0105:S0960258509990031_eqn1.gif?pub-status=live){kind=small}
where θHT is a hydrothermal time constant that has a unique value for the seed population (MPa °C d), Ψb(G) is the base water potential for the Gth percentile of the seed population (MPa), and varies according to a frequency distribution, T b is the base temperature for seed germination (°C) and t(G) is the time for the Gth percentile to germinate (d).
The model predicts that under constant conditions of T and Ψ, the Gth seed in the population will germinate when the seed has accumulated hydrothermal time [calculated from the right-hand side of equation (1)] equal to the population hydrothermal time constant θHT.
A standard assumption in hydrothermal time models is that Ψb(G) varies within the population but T b has a unique value. Hence all seeds in the population will accumulate the same thermal time under any temperature regime in the seed's environment. In contrast, because Ψb(G) is different for each seed percentile, each percentile will accumulate different amounts of ‘hydro-time’ within a specific time period under any moisture regime applied to the whole seed population. Therefore it is the variation in Ψb(G) that results in a spread of seed germination times within the population (Bradford, Reference Bradford2002; Finch-Savage, Reference Finch-Savage, Benech-Arnold and Sanchez2004). Hydrothermal germination models conventionally assume that Ψb(G) is normally distributed, so the median base water potential [Ψb(50)] will equal the mean Ψb(G) for the seed population.
Modelling departures from the conventional hydrothermal germination model
Departures of actual seed germination behaviour at suboptimal temperatures from the conventional hydrothermal model are frequently reported in the literature (Dahal and Bradford, Reference Dahal and Bradford1994; Phelps and Finch-Savage, Reference Phelps, Finch-Savage, Ellis, Black, Murdoch and Hong1997; Kebreab and Murdoch, Reference Kebreab and Murdoch1999; Grundy et al., Reference Grundy, Phelps, Reader and Burston2000). These departures may necessitate changing the assumptions of the model, for example by varying the value of Ψb(G) depending on ambient water potential (Dahal and Bradford, Reference Dahal and Bradford1994).
In addition, the hydrothermal model specified by equation (1) implies that seed germination rate (GR) will increase indefinitely with increasing T, i.e. if the term (T − T b) becomes larger and if θHT is a constant, then t(G) must become smaller. In reality, as temperature increases both GR and germination percentage reach a maximum at the optimum temperature for germination (T o) and then decline to zero at a ceiling temperature (T c). Alvarado and Bradford (Reference Alvarado and Bradford2002) and Rowse and Finch-Savage (Reference Rowse and Finch-Savage2003) have proposed hydrothermal models that account for this decline by adjusting the base water potential of seeds upwards towards zero at supra-optimal temperatures. The Alvarado and Bradford (Reference Alvarado and Bradford2002) and Rowse and Finch-Savage (Reference Rowse and Finch-Savage2003) models are referred to in this paper as the AB and RFS models, respectively.
In both the AB and RFS models, the upwards adjustment in base water potential reduces the rate of hydro-time accumulation by the seed, and so it takes longer to accumulate hydrothermal time equal to θHT, which it must do before it can complete germination. The AB model assumes that the maximum rate of thermal time accumulation occurs at T o, and that supra-optimal temperatures do not contribute additional hydrothermal time, i.e. for T>T o, thermal time = (T o − T b)t(G). It also assumes that the upwards adjustment in seed base water potential scales linearly with supra-optimal temperature, as shown by equation (2):
where Ψb(G)sup is the adjusted seed base water potential at supra-optimal temperatures, and Ψb(G)o is the base water potential of seeds which applies at T = T o. The effects of these two assumptions are that: (1) the relationship between GR and temperature follows a ‘broken-stick’ pattern, with a linear upwards trend in GR from T b to T o followed by an abrupt transition to a linear decline in GR down to T c; and (2) the peak of the ‘broken stick’ occurs at the same temperature (T o) for all seed percentiles (Alvarado and Bradford, Reference Alvarado and Bradford2002).
The RFS model differs importantly from the AB model in that there is no maximum temperature for calculating thermal time, i.e thermal time accumulates increasingly rapidly with increasing temperature above T o. This means that predicted GR continues to increase above the temperature at which seed base water potential begins to increase as a function of supra-optimal temperature, because the extra thermal time accumulated at higher temperatures counteracts the effect of an upwards adjustment in Ψb(G)sup towards zero (Rowse and Finch-Savage, Reference Rowse and Finch-Savage2003). However, the upwards-adjusting effect of temperature on Ψb(G)sup occurs at a faster rate per unit of temperature than thermal time accumulation, and so there is an inflexion point where GR begins to decline. These assumptions mean that for the RFS model, ‘deviation temperature’ or T d (the temperature threshold where Ψb(G)sup begins to increase as a function of temperature) is substituted in equation (2) for the actual optimum temperature T o, where GR is at a peak. The assumptions of the RFS model also mean that: (1) the plot of predicted GR versus T has a broad, rounded peak (rather than a sharp ‘broken-stick’ profile); and (2) T o occurs at a higher temperature for the lower percentile than for higher percentile seeds (Rowse & Finch-Savage, Reference Rowse and Finch-Savage2003). The next sections describe a germination study using seed from radiata pine. The aim of the study was to test the general hydrothermal model [equation (1)] for germination at suboptimal temperatures, and the AB and RFS models [equation (2)] for germination at supra-optimal temperatures.
Methods
A commercial radiata pine seedlot (seedlot no. 075/780 supplied by Proseed NZ Ltd, Amberley, New Zealand) was incubated for 50 d in incubators at constant temperatures from 12.5 to 32.5°C, and at constant water potentials from 0 to − 1.2 MPa. Pilot experiments with seedlot no. 075/780 suggested that optimum germination was obtained at temperatures of 20–22°C and water potential = 0 MPa (cf. ISTA, 2003) and that base temperature and mean base water potential for germination were approximately 10°C and − 1.5 MPa, respectively.
For each combination of water potential and temperature there were four replicates, each comprising a 750 ml plastic snap-top tray with 25 seeds on hydrated Whatman No. 2 filter papers. Before placement in the trays, filter papers were soaked in trays containing an osmotic solution of the desired water potential for 48 h. The filter papers were then placed on a 5 mm thick glass plate in the trays which were filled with freshly mixed osmotic solution until the solution just made contact with the top of the filter paper. The glass plates were of smaller dimensions than the plastic trays, which meant there was a reservoir of osmotic solution surrounding each plate. This was intended to further buffer the filter paper from changes in water potential due to evaporation of water. Osmotic solutions of water potential = 0, − 0.3, − 0.6, − 0.9 and − 1.2 MPa were made up using polyethylene glycol 8000 (PEG 8000) according to equation (6) in Hardegree and Emmerich (Reference Hardegree and Emmerich1990). The osmotic solution with 0 MPa was reverse osmosis water with no PEG 8000 added.
Seeds were screened to remove any obvious empty or broken seeds and then lightly dusted with fungicide (Thiram) before being carefully placed on to the germination medium. Seeds were expected to have better than 95% germinability, based on seed-testing results provided by Proseed NZ Ltd. Radiata pine seed has no stratification-sensitive seed dormancy (Rimbawanto et al., Reference Rimbawanto, Coolbear and Firth1988), therefore seeds received no stratification or other pre-treatment prior to sowing (ISTA, 2003). Once the seeds were placed in the germination trays, the tops of the trays were sealed with a sheet of cling-wrap and the plastic snap-top lid was firmly placed on the top of the tray. Trays were then placed in eight incubators at constant temperatures of 12.5, 15, 17.5, 20, 22.5, 25, 27.5 and 32.5°C respectively. All water potential treatments had four replicates in each incubator, except for − 1.2 MPa which was only replicated for T = 17.5 and 20°C. Within each incubator, trays were randomly placed and positions were rotated systematically every 3 d to mitigate any small spatial variations in air temperature due to inefficient air circulation. Temperatures were monitored continuously within the incubators using either Tiny-Tag temperature loggers (Gemini Data Loggers, Chichester, UK) or temperature probes interfaced with a HOBO data logger (Onset Computer Corporation, Bourne, Massachusetts, USA). No consequential deviations from target incubator temperatures occurred during the experiment. Water potentials of the germination media were monitored by measuring samples of the osmotic solutions in the trays twice weekly, using a Wescor 5520 vapour pressure osmometer (Wescor Inc., Logan, Utah, USA). Where osmotic potentials deviated from a specified treatment level by >0.05 MPa, that osmotic solution was discarded and replaced with fresh solution.
Germination counts were continued for a period of 50 d. Seeds were inspected twice daily for the first 9 d of the incubation period, and thereafter daily until 27 d, after which seeds were inspected every 2 d. Seeds were considered to have germinated when the radicle protruded more than 2 mm from the seed coat. Germinated seeds were counted and discarded. Seed counting required the trays to be opened, which also served to aerate them.
Assessment of dormancy
Most seeds germinated within 50 d for the near-optimum treatments. However, for treatments with non-optimal temperatures and/or low water potentials, many seeds remained ungerminated after 50 d. This may have been due to the induction of secondary dormancy. The extent of this possible dormancy was tested after 50 d by placing all remaining ungerminated seeds in trays with water (Ψ = 0 MPa) and incubating them at optimum temperature (20°C) for a further 75 d. Counts of germination were taken at 15 d intervals and, as with the main experiment, germinated seeds were counted and discarded, as were any seeds that were clearly dead (squash test showed loss of structure). Seeds that germinated under ideal conditions within 75 d were considered non-dormant. Seeds that had not germinated by 75 d were considered completely dormant, although they may simply have been non-viable.
Data analysis
Germination rate
The germination rate (GR) of a seed is defined as the inverse of the time taken by the seed to germinate, i.e. 1/GR = t(G). By plotting GR versus Ψ and T, data can be checked for their conformity to the hydrothermal model, i.e. for all percentiles: (1) GR versus T should be linear for T < T o and converge to a single base temperature T b; and (2) GR should increase linearly as water potential increases from Ψ = Ψb (where GR = 0) to Ψ = 0 (where GR reaches its maximum value). In addition, for GR versus Ψ, the trend lines for the different percentiles should be parallel, with each percentile having a common slope but a differing intercept [ = Ψb(G)] with the x-axis (Bradford, Reference Bradford2002).
Estimating hydrothermal germination model parameters
Germination data for the four 25-seed replicates for each temperature/water potential treatment were amalgamated into one set of data (i.e. 100 seeds). Within each treatment, seeds were allocated a percentile number (G), beginning with 1 for the fastest seed to germinate, 2 for the second-fastest and so on. It was assumed that all seeds were potentially germinable, and so the maximum percentile number for any treatment was 100. In order to estimate parameter values for equation (1), it was transformed as follows (Gummerson, Reference Gummerson1986):
![Probit\,(G) = [ \Psi - \Psi _{ b }(50) - \lcub \theta _{ HT }/( T - T _{ b }) t (G)\rcub ]/ \sigma _{ \Psi b }](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151024042816780-0105:S0960258509990031_eqn3.gif?pub-status=live){kind=small}
where Probit (G) is the probit function which calculates the standard normal deviate (z) for a specified cumulative probability ( = G) in a normally distributed population, Ψb(50) is the mean Ψb(G) and σΨb is the standard deviation of Ψb(G) for the seed population.
For the germination data at suboptimal temperatures (12.5–20°C), the Probit (G) statistics for all seeds were calculated, then equation (3) was fitted to the germination data using the ‘nls’ function in programme R (R Development Core Team, 2007). This function simultaneously estimated θHT, Ψb(50), T b and σΨb using an iterative least-squares procedure.
Checking the hydrothermal germination model
Departures from the conventional hydrothermal model may arise when seed base water potential [Ψb(G)] varies from the assumptions that its values are: (1) independent of germination conditions; and (2) normally distributed around a mean Ψb(50) (Dahal and Bradford, Reference Dahal and Bradford1994; Alvarado and Bradford, Reference Alvarado and Bradford2002; Rowse and Finch-Savage, Reference Rowse and Finch-Savage2003). For any mathematical model, model goodness of fit to the data can be checked using the raw residuals ( = observed values for the response variable − predicted values) (Crawley, Reference Crawley2002). Systematic biases or trends in the raw residuals for seed base water potential would indicate that assumptions 1 and 2 above were not valid. The predicted value for seed base water potential is easy to calculate, because if Ψb(G) is normally distributed around a mean [Ψb(50)] then it is axiomatic that the predicted value Ψb(G)pred will be:
This provides a fixed value for Ψb(G) across all environments. However, determining an observed value is less obvious because values of Ψb(G) are not directly observed. To overcome this problem, equation (1) can be rearranged to give:
By substituting model parameters estimated using equation (3) and observed values for t(G), T and Ψ into the right-hand side of equation (5), the calculated value of the left-hand side of equation (5) is a virtual ‘observed value’ for seed base water potential. Unlike Ψb(G)pred, which has a unique constant value for each seed percentile, Ψb(G)′ can vary between different water potential and temperature treatments.
The model raw residuals [denoted by ΔΨb, where ΔΨb = ΔΨb(G)′ − ΔΨb(G)pred] can therefore reveal whether seed base water potential varies systematically with water potential and temperature treatments, as follows:
(1) If seed base water potential values of imbibed seeds change upwards or downwards in response to germination conditions, as suggested by Dahal and Bradford (Reference Dahal and Bradford1994), then raw residuals (ΔΨb) should show a systematic trend upwards or downwards with respect to those germination conditions.
(2) For seeds incubated at supra-optimal temperatures, raw residuals (ΔΨb) can be calculated from equation (2), by assuming that: (i) seed base water potential at T o, [Ψb(G)o] is equal to Ψb(G)pred [equation (4)]; and (ii) Ψb(G)′ [equation (5)] is equal to Ψb(G)sup. Then substituting Ψb(G)′ and Ψb(G)pred into equation (2) gives:
(6)ΔΨb should therefore be linearly related to supra-optimal temperature, because if ΔΨb = Ψb(G)′ − Ψb(G)pred, then ΔΨb = k{T − T o). Note that for the RFS model, Ψb(G)d and T d (where d = deviation temperature) are substituted for Ψb(G)o and T o.
The assumption that seed base water potential is normally distributed can be checked in similar fashion. If the frequency distribution of Ψb(G) is normal, then at suboptimal temperatures Ψb(G)′ is expected to equal Ψb(G)pred and ΔΨb should be distributed around a mean of zero with no bias in the distribution of its values with respect to seed percentile. Conversely, if the frequency distribution of Ψb(G) is non-normal, then at suboptimal temperatures ΔΨb should exhibit a systematic bias in the distribution of its values around its mean, with respect to seed percentile.
ΔΨb for seeds incubated at suboptimal temperature was plotted against seed percentile to check the assumption that seed base water potential was normally distributed. ΔΨb for seeds incubated at supra-optimal temperatures was plotted against temperature, to verify whether seed base water potential increased towards zero with increasing supra-optimal temperatures, as proposed by the AB and RFS models.
Results
Germination rate
Plots of seed GR versus temperature (for optimum Ψ = 0 MPa) and water potential (for T = 20°C) are shown in Fig. 1a and b. Rates are plotted for the 10th, 30th, 50th, 70th and 90th percentiles, to show trends in data across the entire seed population. In general, GRs of the radiata pine seeds at temperatures less than or equal to 20°C conform to the assumptions of the hydrothermal model, i.e. for all percentiles: (1) GR versus T appears to be linear converging to a single base temperature T b; and (2) GR versus Ψ appears to be linear with parallel trend lines for each percentile and with each percentile having a different intercept [ = Ψb(G)] with the x-axis.
Figure 1 Germination rates (GR) for 10th to 90th percentiles versus: (a) temperature when Ψ = 0 MPa; (b) water potential when T = 20°C. Percentile symbols: ● = 10th, ○ = 30th, ▾ = 50th, ▿ = 70th, ■ = 90th. The solid lines are drawn freehand to illustrate trends in GR versus temperature and water potential. Lines in (a) are constrained to pass through a single T b of 9.0°C. Lines in (b) are constrained to a common slope. There is no line drawn for the 90th percentile because 90% germination was only achieved for 20°C and 0 MPa.
For temperatures greater than 20°C, the rate of increase in GR with temperature appears to decline non-linearly until T o (the optimum temperature where GR is at a maximum). Beyond T o, GR declines (quite rapidly for the higher percentiles) to zero at the ceiling temperature T c. However, the values of T o and T c are different for each percentile of the seed population. T o and T c are typically highest for the fastest germinating seeds (i.e. the lowest percentiles in the population). Figure 1a shows that GR for the 10th percentile only just began to decline at 32.5°C and it is difficult to estimate T c for this percentile by extrapolation. These trends strongly suggest that germination of radiata pine at supra-optimal temperatures conforms more closely to the RFS model than to the AB model, because there is a broad range of optimum temperatures which vary between different seed percentiles, rather than a unique value for T o which applies to all percentiles. They also suggest that the deviation temperature (T d) where Ψb begins to decline is ≈ 20°C.
Hydrothermal models at suboptimal temperatures
Estimated hydrothermal model parameters for equation (3) are shown in Table 1 (Model 1). These were used to calculate ΔΨb[ = Ψb(G)′ − Ψb(G)pred], equations (4) and (5), which was then plotted against seed percentile, to verify whether the data conformed to the assumption that the seed base water potential is normally distributed [equation (1)]. Figure 2a (ΔΨb versus seed percentile) shows that although the distribution of ΔΨb is biased, biases in distribution are not consistent with respect to percentile for seeds incubated at different temperatures or water potentials. This suggests that biases were not caused by an underlying non-normal distribution of Ψb(G) in the seed population. However, biases in ΔΨb did appear to be related to the germination time for the seed, suggesting that seed base water potentials changed during the germination time course (Fig. 2b). For seeds that took less than 25 d to germinate, ΔΨb was more or less uniformly distributed around 0, i.e. seed base water potential did not change during this time. For seeds that took longer than 25 d to germinate, ΔΨb increased linearly with respect to germination time, which meant they germinated even slower. Figure 2b also shows that the relationship of ΔΨb versus time was consistent for all temperatures but varied with Ψ. For the driest treatment ( − 1.2 MPa) the relationship became weak. To account for the relationship between ΔΨb and time, two alternative hydrothermal models were fitted with an additional term for the increase of Ψb(G)′ with time to germination, as follows:
Table 1 Model parameters and Akaike Information Criterion (AIC) values for suboptimal hydrothermal models
Figure 2 ΔΨb plotted against (a) seed percentile (G); (b) time to germination; and (c) hydro-time index (HTI). Data are plotted for Ψ treatments (L to R): 0, − 0.3, − 0.6, − 0.9 and − 1.2 MPa. Data for each Ψ treatment include combinations with all suboptimal temperature treatments.
Model 2
Model 3
![\Psi _{ b }(G)\prime = \Psi _{ b }(G)_{pred} + k \prime \lcub [( \Psi + 1.5) t (G)] - 20\rcub \,for\,[( \Psi + 1.5) t (G)]> 20\hairsp MPa\,d](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151024042816780-0105:S0960258509990031_eqn8.gif?pub-status=live)
where k and k′ are constants. Equation (7) adjusts Ψb(G) as a function of time to germination, t(G). Equation (8) accounts for the effect of water potential on ΔΨb by adjusting Ψb(G) as a function of a ‘hydro-time index’ to germination, [(Ψ+1.5)t(G)]. The base value for the ‘hydro’ component of the index is 1.5 MPa. This is a fixed value chosen so that (Ψ+1.5) is positive for all water potential treatments used in this study. For dry conditions (e.g. − 1.2 MPa), the hydro-time index (HTI) and therefore ΔΨb will be small; for moist conditions (e.g. 0 MPa), the HTI and therefore ΔΨb will be large. Inspection of graphs of ΔΨb versus HTI revealed that the linear increase in ΔΨb occurs from HTI ≈ 20 MPa d onwards (Fig. 2c).
The alternative hydrothermal time models [Model 1, without adjustment to Ψb(G), or Models 2 and 3, with adjustments to Ψb(G), as specified in equations (7) and (8)] were compared for their likelihood using the Akaike Information Criterion (AIC). The AIC is a penalized log-likelihood criterion, i.e. it calculates the likelihoods of alternative models fitted to the same data, but reduces the likelihood of each model in proportion to the number of parameters that it uses (Akaike, Reference Akaike1974; Burnham and Anderson, Reference Burnham and Anderson2001). The lower (more negative) the AIC value, the greater the likelihood of the model.
The AIC values (Table 1) suggest that Model 3 [equation (8)], which adjusts Ψb(G) as a function of HTI, gives the best fit to the measurement data. Model 2 [equation (7)] is also superior to Model 1. Models 2 and 3 also have the advantage compared with Model 1 of offering a mechanistic explanation of departures from normality in the frequency distribution for Ψb(G), i.e. an initial normal distribution is skewed by increases in the Ψb(G) values for the slower-germinating seeds, which occur as the seed proceeds to germination.
Figures 3a, c and e show predicted germination time courses at Ψ = 0 MPa for Models 1, 2 and 3, compared with actual germination data. These clearly show that the model with the HTI-adjusted seed base water potential (Model 3) more correctly predicts germination at all temperatures, whereas the unadjusted model (Model 1) incorrectly predicts higher germination percentages than actually occurred in most cases. Model 2 makes generally correct predictions but incorrectly predicts germination at 12.5°C/0 MPa. The comparisons between models shown in Fig. 3 were consistent with results at other water potentials, as shown by the plots of model residuals (observed values minus predicted values) for germination percentiles of all suboptimal data (Fig. 3b, d and f). In particular, Fig. 3b shows a pronounced curvature in the distribution of residuals for Model 1 when plotted against predicted germination percentile.
Figure 3 Comparison of Models 1, 2 and 3 for suboptimal germination data. (a), (c) and (e) Predicted and actual germination time course for Models 1, 2 and 3 at Ψ = 0 MPa. Data are plotted for each temperature as follows: ● = 12.5°C, ○ = 15°C, ▾ = 17.5°C, ▿ = 20°C. Model predictions are shown by the solid lines. (b), (d) and (f) Residuals [observed minus predicted seed percentile (G)] versus predicted seed percentiles for all suboptimal germination data.
Hydrothermal models at supra-optimal temperatures
Raw residuals [ΔΨb, calculated from equations (4) and (5)] were calculated for the germination data at supra-optimal temperatures. Residuals were used to test the RFS model assumption that thermal time accumulation continues to increase linearly above T d. Inspection of Fig. 1a suggests that T d = 20°C. This was the value used in subsequent analysis. Figure 4a shows ΔΨb at supra-optimal temperatures, plotted against temperature. In Fig. 4a, ΔΨb increased in magnitude with increasing temperatures above 20°C as predicted by the RFS model. Equation (6) was therefore fitted to the data using least squares regression. The estimated RFS model was ΔΨb = 0.0729(T − T d) where T d = 20°C, where T d = 20°C. However, Fig. 4a also shows that actual ΔΨb values varied widely around the estimated model at all temperature levels. It appears that temperature did not fully account for variation in ΔΨb at supra-optimal temperature. For suboptimal temperatures, ΔΨb increased as seeds proceeded towards germination. A similar increase may occur at supra-optimal temperatures. Figure 4b shows ΔΨb versus time to germination, for each supra-optimal temperature. This shows that ΔΨb did increase non-linearly with time to germination but the form of the relationship varied with temperature. To account for this, ΔΨb was plotted against thermal time accumulated above T d, i.e. [T − 20]t(G), for each level of water potential in the experiment (Fig. 4c). Doing this revealed that ΔΨb was strongly correlated with thermal time, although the form of the relationship varied between water potential treatments. This variation between water potential treatments followed a consistent trend, with ΔΨb increasing at a slower rate per unit of thermal time as water potential became drier. Therefore ΔΨb was also plotted against a ‘supra-optimal hydrothermal time index’ [SOHTI = (Ψ+1.5)(T − T d)t(G)]. Note that this supra-optimal hydrothermal time index (SOHTI) differs from the population hydrothermal time constant θHT, because it is a variable, not a model parameter. It is calculated from a base temperature of T d = 20°C rather than T b, and from the time to germination t(G), which varies for each seed. The base water potential (1.5 MPa) used to calculate the SOHTI is a fixed value, arbitrarily chosen so that (Ψ+1.5) is positive for all water potential treatments used in this study. This arbitrary value was used because attempts to analytically derive a base water potential for SOHTI did not produce a convergent solution. The relationship between ΔΨb and SOHTI was quite consistent for all data, and appeared to have an asymptotic exponential form with an asymptote of 1.5 MPa (Fig. 4d). This suggests that the temperature-driven shift in Ψb(G) proposed by Alvarado and Bradford (Reference Alvarado and Bradford2002) and by Rowse and Finch-Savage (Reference Rowse and Finch-Savage2003) was (1) not instantaneous but occurred over time; and (2) was also influenced by the water potential of the seed's environment.
Figure 4 ΔΨb plotted against (a) temperature. The solid line is ΔΨb = 0.0729(T − 20), where T = temperature; (b) time to germination for each temperature level (● = 22.5°C, ○ = 25°C, ▾ = 27.5°C, ▿ = 32.5°C); (c) thermal time to germination (T − 20)t(G) for each water potential (● = 0 MPa, ○ = − 0.3 MPa, ▾ = − 0.6 MPa, ▿ = − 0.9 MPa, ■ = − 1.2 MPa); (d) supra-optimal hydrothermal time index (SOHTI). The solid line is ΔΨb = 1.448 − {1.562 exp [ − 0.00 596(SOHTI)]}.
A regression was fitted to the relationship between ΔΨb and SOHTI using an asymptotic exponential equation (Crawley, Reference Crawley2002). The model was fitted using the ‘nls’ function in program R (R Development Core Team, 2007). This function simultaneously estimated all model parameters using an iterative least-squares procedure. The estimated model was ΔΨb = 1.448 − {1.562 exp [ − 0.00 596(SOHTI)]}, therefore the following equation was used to adjust seed Ψb(G) [equation (4)] as a function of SOHTI:
![\Psi _{ b }(G)\prime = \Psi _{ b }(G)_{pred} + 1.448 - \lcub 1.562\hairsp exp\,[ - 0.00\,596(SOHTI)]\rcub](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151024042816780-0105:S0960258509990031_eqn9.gif?pub-status=live){kind=small}
The adjusted Ψb(G)′ was then used as an input into a hydrothermal time model, with other model parameters derived from Model 3 for suboptimal temperatures, i.e. θHT = 176 MPa °C d, T b = 9.0°C. Figures 5a and c show predicted germination time courses at Ψ = 0 MPa for the RFS and SOHTI models compared with actual germination data. The RFS model correctly predicted germination up to 10 d, but markedly overpredicted germination from 10 d onwards (Fig. 5a). In contrast, the SOHTI model more accurately predicted reduced germination at all supra-optimal temperatures (Fig. 5c). The comparisons between models at Ψ = 0 (Fig. 5a and c) were consistent with results at other water potentials, as shown by: (1) AIC values (Akaike, Reference Akaike1974) of 81.8 and − 744 for the RFS and SOHTI models, respectively; and (2) plots of model residuals for germination percentiles of all supra-optimal data (Fig. 5b and d). Figure 5b shows how the residuals for the RFS model were strongly biased when plotted against predicted germination percentile. Figure 5d shows how this bias was corrected using the SOHTI model.
Figure 5 Comparison of the RFS with the SOHTI model. (a) and (c) Predicted and actual germination time course for Ψ = 0 MPa. Data are plotted for each temperature as follows: ● = 22.5°C, ○ = 25°C, ▾ = 27.5°C, ▿ = 32.5°C. Model predictions are plotted as 
= 22.5°C, — — — = 25°C, – – – – – = 27.5°C, 
= 32.5°C. (b) and (d) Residuals [observed minus predicted seed percentile (G)] versus predicted seed percentiles for all supra-optimal germination data.
Ungerminated seeds
For treatments at low temperatures (12.5 or 15°C) and for all supra-optimal temperatures, the higher seed percentiles did not germinate. The unadjusted hydrothermal model [equation (1)] predicted that most of these seeds should have germinated, because they should have accumulated sufficient hydrothermal time. In contrast, the hydrothermal models using adjusted seed base water potentials [equations (7), (8), (9)] correctly predicted no germination by the higher seed percentiles, because they adjusted seed base water potential upwards, as described in the previous section. This shift in base water potential towards zero resulted in the high percentile seeds failing to germinate because they accumulated insufficient hydrothermal time to germinate, even under moist conditions. An alternative explanation for this lack of germination might be the induction of secondary dormancy at both suboptimal and supra-optimal temperatures. The extent of dormancy in all remaining seeds was tested after 50 d by placing the seeds in germination trays with moist germination substrate (Ψb = 0 MPa) and incubating at optimum temperature (20°C) for a further 75 d. For ungerminated seed from the suboptimal T and Ψ treatments, >90% germinated within 20 d of being placed in optimal conditions. Seeds from the driest treatments ( − 1.2 and − 0.9 MPa) were slower to respond, typically requiring more than 30 d to reach >90% germination. This suggests that the lack of germination during the experimental period of 50 d was due to insufficient hydrothermal time being accumulated by the seeds. For temperature treatments of 27.5°C and 32.5°C, and for all supra-optimal temperature treatments where Ψ was ≤ − 0.6 MPa, seed germination during the 50 d incubation period was markedly reduced (Fig. 5). After a further 75 d under optimum conditions, seeds from treatments where Ψ ≥ − 0.6 MPa achieved ~80% germination, except for the 32.5°C treatments. Towards the end of the 75 d, ungerminated seeds began to decompose, especially seeds from 27.5°C and 32.5°C treatments.
Discussion
Implications for models of germination at suboptimal temperatures
At suboptimal temperatures, a conventional hydrothermal model [equation (1)] gave a reasonable fit to the germination data, but overpredicted germination rates for slower-germinating seeds and consequently overpredicted germination percentages. Modified models which adjusted Ψb(G) upward as a linear function of time or hydro-time index (HTI) to germination were markedly more accurate for all combinations of Ψ and suboptimal T used in this study. While departures of actual seed behaviour at suboptimal temperatures from the conventional hydrothermal model are frequently reported in the literature, the hypothesis that these departures are caused by Ψb(G) varying as a function of germination time course has not been widely tested. Allen et al. (Reference Allen, Meyer, Khan, Black, Bradford and Vasquez-Ramos2000) reported that for a range of dry climate plant species, Ψb(50) of seeds appears to increase towards zero when incubated at temperatures close to T b. This behaviour was successfully modelled by allowing Ψb(50) to increase under these conditions, although the authors do not describe the methods used to calculate that increase.
A more detailed paper by Gianinetti and Cohn (Reference Gianinetti and Cohn2007) describes the induction of secondary dormancy in red rice (Oryza sativa L.) in terms of an upwards shift in Ψb(G). This upwards shift was modelled as a function of hydro-time accumulated during germination, an approach equivalent to Model 3 in this study. More commonly, departures from the conventional hydrothermal model are achieved by specifying separate models for different levels of ambient T and Ψ. Ni and Bradford (Reference Ni and Bradford1992) and Dahal and Bradford (Reference Dahal and Bradford1994) reported a downwards adjustment in Ψb(G) for tomato (Lycopersicon aesculentum Mill.) seeds incubated under low water potentials, so that seeds became more germinable under these conditions. Dahal and Bradford (Reference Dahal and Bradford1994) accounted for this adjustment by fitting two separate hydrothermal models to the data, one for germination data where Ψ was drier than − 0.5 MPa [with a lower mean Ψb(G)] and one for data when Ψ was moister than − 0.5 MPa [with a higher mean Ψb(G)].
Kebreab and Murdoch (Reference Kebreab and Murdoch1999) described results for germinating broomrape (Orobanche aegyptiaca Persoon), where: (1) T b apparently varied with ambient water potential, in contrast to the conventional hydrothermal model assumption of a single T b value for the whole seed population; and (2) the mean base water potential Ψb(50) was found to vary with temperature, being − 2 MPa between 14 and 23°C but increasing towards zero at both higher and lower temperatures. These variations in T b and Ψb for O. aegyptiaca were modelled by the authors as functions of ambient water potential. However, neither Dahal and Bradford (Reference Dahal and Bradford1994) nor Kebreab and Murdoch (Reference Kebreab and Murdoch1999) consider the effect of the longer germination times that occur when ambient T and Ψ are not optimal. One may speculate, for example, that apparent shifts in Ψb(50) with low temperature described by Kebreab and Murdoch (Reference Kebreab and Murdoch1999) arise because of slower germination at these temperatures. Similarly, the shifts in Ψb(G) described by Dahal and Bradford (Reference Dahal and Bradford1994) for tomato at low water potentials also occurred at low temperatures, suggesting that slower germination times may have been a factor in causing these shifts.
Implications for models of germination at supra-optimal temperatures
At suboptimal temperatures, imbibed radiata pine seeds whose Ψb(G) distribution shifts upwards because conditions do not allow rapid germination, may remain ungerminated until they experience sustained optimal conditions. This seems to be a useful ‘bet-hedging’ adaptation for radiata pine seeds which are otherwise fully germinable when mature (Kao and Rowan, Reference Kao and Rowan1970; Rimbawanto et al., Reference Rimbawanto, Coolbear and Firth1988).
However, at supra-optimal temperatures, the adaptive value of reduced germination under moist conditions is less clear, because germination could proceed rapidly under these conditions. In this study, plots of GR versus T indicated that radiata pine seeds can accumulate thermal time linearly up to 27.5°C (Fig. 1a), suggesting that seeds would be able to germinate very rapidly at this temperature, in the absence of an upwards adjustment in seed base water potential caused by temperatures where T>T d. Furthermore, after germination young radiata pine seedlings can grow successfully under moist conditions with daytime temperatures ≈ 30°C (Hellmers and Rook, Reference Hellmers and Rook1975). More generally, Bradford (Reference Bradford2002) comments that the reasons for the decrease in seed GR and inhibition of germination at supra-optimal temperatures are not obvious, and that the most commonly offered explanation, i.e. non-linearity in thermal time accumulation due to denaturation of proteins, is unlikely to occur at temperatures as low as 20°C.
A possible explanation is offered by the model for germination at supra-optimal temperatures proposed in this paper [equation (9)]. In contrast to the RFS and AB models, which explain reduced germination solely in terms of supra-optimal temperatures, equation (9) shows that reduced germination occurs as an asymptotic function of time and is also dependent on ambient Ψ, with the rate of reduction occurring fastest under moist conditions. It seems counter-intuitive that seeds would most rapidly inhibit their germination when ambient moisture is ideal for germination and seedling growth. However, it makes more sense if the upwards adjustment in seed base water potential is an adaptation to cope with a ‘false break’, a feature of mediterranean and other summer-dry climates where unseasonal late summer rains may trigger seed germination but germinated seedlings subsequently die because the rains do not persist (Norman et al., Reference Norman, Cocks, Smith and Nutt1998). Radiata pine originates from the ‘fog belt’ along the coast of central California, where the climate is classified as cool mediterranean (Dallman, Reference Dallman1998) and summer droughts are the norm. One may speculate that radiata pine seeds that imbibe and proceed towards germination during a ‘false break’ in late summer (1) will not germinate due to a rapid upwards shift in seed base water potential caused by supra-optimal temperatures combined with moist soils; but (2) will germinate once autumn rains bring cooler temperatures, when seedling survival is more likely. This rapid shift in seed base water potential is not needed if the ‘false break’ does not occur, even though soil temperatures will be supra-optimal much of the time, because ambient soil moisture potentials will be drier than the unadjusted seed base water potentials and the seeds will not germinate anyway.
However, this speculation must be qualified by saying that the time-, temperature- and moisture-dependent adjustments to seed base water potential reported here would need to be observed under field conditions similar to the natural habitat of radiata pine on the Central Californian coast – if only to verify that they are a real feature of the seed ecology of radiata pine, and not an artefact of the constant incubation conditions used in this study.
Acknowledgements
Lissy Kajer, Keith Pollock, Stephen Stilwell and Rosemary Weaver provided technical assistance with the germinations. Dr Bill Finch-Savage and Kath Phelps (WHRI, University of Warwick) provided comments and advice at various times during this study. The manuscript also benefited from the comments of two anonymous reviewers.
