Development of a hydrothermal time model that accurately characterises how thermoinhibition regulates seed germination


M. S. Watt. Fax: +64 3 364 2812; e-mail:


Thermoinhibition is the decline in germinability within a seed population as soil temperatures increase above the optimum for germination. Hydrothermal time (HTT) models have been developed that describe the thermoinhibition response as a function of increases in the threshold water potential for seed germination [seed base water potential, Ψb(G)]. Although these models assume a normal distribution of Ψb(G) and a linear upward shift in Ψb(G) with increasing temperature, little research has tested these assumptions. Using germination data obtained from four unrelated plant species, we fitted HTT models that use the Weibull and normal distribution to describe Ψb(G) and compared the accuracy and bias of these two HTT models. For all four species, Ψb(G) and germination were more accurately described by the Weibull than the normal distribution HTT model. At supra-optimal temperatures, Ψb(G) of the earliest germinating seeds showed little thermoinhibition effect so that the seeds germinated very rapidly under moist conditions. However, for the rest of the population, Ψb(G) increased progressively in response to supra-optimal temperatures so that the slower germinating seeds were thermoinhibited. The fitted HTT models reveal aspects of seed thermoinhibition that appear to have adaptational value under variable conditions of soil temperature and moisture.


The optimum temperature for seed germination (To) can be defined as the soil temperature at which the highest germination percentage is achieved by a seed population in the shortest possible period of time (Mayer & Poljakoff-Mayber 1975). At supra-optimal temperatures (T > To), physiological changes occur in seeds, which inhibit both the rate of germination and the proportion of the seed population that will complete germination (Hills & van Staden 2003; Argyris et al. 2008). This progressive decline in germinability within a seed population as soil temperatures increase above To is known as thermoinhibition.

Surprisingly, the optimum germination temperatures for many species are quite low (20 °C or less) (Bradford 2002) so that small increases (<10 °C) above these optima will result in declining germination when temperatures are still favourable for subsequent growth of seedlings (Bloomberg et al. 2009).

Seed germination is a complex physiological process largely determined in non-dormant seeds by temperature and water potential of the seedbed. These two factors have been successfully combined in hydrothermal time (HTT) models to describe the time course of germination for a wide range of plant species, at both sub- and supra-optimal temperatures. The HTT model is a threshold model that simultaneously accounts for germination percentages and germination rates of a seed population (Gummerson 1986; Bradford 1995, 2002; Finch-Savage 2004).

In the HTT model, the order in which seeds germinate is specified by seed percentile (G), such that G = 1 is the first percentile of seeds to germinate, G = 2 is the second percentile to germinate and so on up to G = 100, which represents complete germination of a seed population. The time to complete germination (time from imbibition to radicle emergence) for a specific seed percentile t(G) is specified by


where θHT (Mpa °C d) is the HTT constant for the seed population; T (°C) is a constant temperature of the soil or other surrounding medium, which must be greater than Tb (°C), the base temperature, below which the radicle does not emerge to complete germination; Ψ (MPa) is a constant water potential of the soil or other surrounding medium; and Ψb(G) (MPa) is the base water potential, a threshold value analogous to base temperature.

Unlike Tb, which is relatively constant for all seeds in the population, Ψb(G) is variable, with the lowest (most negative) values corresponding to the lowest seed percentiles of seed germination order (G = 1, 2, 3 . . .) and with values increasing progressively so that the highest (closest to 0 MPa) values correspond to the highest seed percentiles. For seeds germinating at a constant T and Ψ, it is this variation in Ψb(G) that results in a distribution of times to germination, t(G), from the fastest seeds (G = 1, 2, 3) to the slowest seeds (G = 100). The original HTT model proposed by Gummerson (1986) assumes that the variation in Ψb(G) can be described by a normal frequency distribution so that


where Ψb(50) is the 50th percentile of the seed base water potential distribution; probit(G) is the probit function that calculates the standard normal deviate for a specified cumulative frequency (=G); and σΨb is the standard deviation of Ψb values in the population.

Gummerson's HTT model does not account for thermoinhibition because it assumes that all parameters are independent of soil temperature and water potential. To account for thermoinhibition in the HTT model, an increase in seed base water potential at supra-optimal temperatures has been proposed (Alvarado & Bradford 2002; Rowse & Finch-Savage 2003). This increased seed base water potential reduces the rate of HTT accumulation at a specific soil water potential, and thus inhibits the rate of germination. This is consistent with physiological changes in the seed at supra-optimal temperatures that raise the threshold soil water potential at which seeds will germinate (Yoshioka et al. 2003; Finch-Savage & Leubner-Metzger 2006). The increase in seed base water potential at supra-optimal temperatures is modelled as a simple rightward shift in the location of the normal frequency distribution for Ψb(G) (Fig. 1). The magnitude of the shift scales with supra-optimal temperature and is specified by k(T − To), where k is a positive constant (Alvarado & Bradford 2002).

Figure 1.

Stylized representation of the assumed shifts to the normal distribution probability density function (pdf) in response to increases in supra-optimal temperature. The lines shown represent the pdf under low (thick solid line) intermediate (thin solid line) and high (long dashed line) supra-optimal temperatures.

The assumption of normality underpinning the Ψb(G) distribution in the original HTT model (Eqns 1 and 2) may not be universally valid. Within the suboptimal temperature range, Ψb(G) for Pinus radiata D. Don and Buddleja davidii Franch. have been shown to have a skewed pattern best described by the Weibull distribution (Weibull 1951) that converges to a closed minimum value of Ψb(G), rather than an open left limit, as described by the normal distribution (Watt, Xu & Bloomberg 2010). Although research has not yet evaluated the utility of the Weibull distribution at describing Ψb(G) and germination in the supra-optimal temperature range, previous observations of germination do suggest that this approach could have merit. At supra-optimal temperatures, germination may be very rapid for the first few seed percentiles, with no evidence of thermoinhibition (Hills & van Staden 2003), but germination times for later percentiles usually scale positively with increasing temperatures. This behaviour implies that Ψb(G) for the initial seed percentiles is very similar for all supra-optimal temperatures, with divergence between Ψb(G) at contrasting supra-optimal temperatures occurring for higher percentiles of the population. This response can be accommodated by the Weibull distribution of Ψb(G), but not the normal distribution, which does not allow for divergence in Ψb(G) between contrasting supra-optimal temperatures across the seed percentile range.

Using data obtained from four unrelated plant species, we (1) fitted HTT models that use the Weibull and normal distribution to describe Ψb(G), and (2) compared the accuracy and bias of these two HTT models at predicting both Ψb(G) and seed germination. We discuss the results from an ecological and an adaptational perspective.


Germination data

Data describing seed germination were obtained for the following four unrelated species: B. davidii Franch. (buddleja), P. radiata D. Don (radiata pine), Allium cepa L. cv. Hyton (onion) and Daucus carota L. cv. Narman (carrot). These data were sourced from three independent studies carried out between 1995 and 2008, and fully described in Rowse & Finch-Savage (2003), Bloomberg et al. (2009) and Watt et al. (2010). The seed populations used in all three studies were not dormant and achieved nearly complete (93–99%) germination under optimum conditions.

Seed germination was undertaken in constant controlled temperatures across a wide range in sub- and supra-optimal temperatures within incubators. Seeds were germinated on media whose water potential was controlled by solutions of polyethylene glycol (PEG). These solutions were mixed to achieve predetermined constant values for water potentials. Experiments were undertaken for each species using a replicated factorial combination of constant temperatures and water potentials. Seeds were considered to have germinated when the radicle protruded more than 2 mm from the seed coat.

The optimum and values of supra-optimal temperatures tested for each species were as follows: B. davidii (To = 25.1 °C; T > To tested: 26.3, 27.0, 31.6 °C), P. radiata (To = 20.0 °C; T > To tested: 22.5, 25.0, 27.5, 32.5 °C), A. cepa (To = 17.0 °C; T > To tested: 20.0, 25.0, 30.0 °C) and D. carota (To = 18.6 °C; T > To tested: 20.0, 25.0, 30.0 °C). Details of suboptimal temperatures used in these experiments can be found in Rowse & Finch-Savage (2003), Bloomberg et al. (2009) and Watt et al. (2010).

Model fitting

All model fitting was undertaken by SAS (SAS Institute Inc. 2000) using the non-linear fitting procedure. Statistical comparisons of model fit to the germination data were made using the coefficient of determination, R2, and root mean square error (RMSE).

The normal distribution HTT model was fitted to the data using


Following Watt et al. (2010), the Weibull distribution HTT model was fitted to the germination data by


where the parameter µ describes the location of the lowest value of Ψb(G) [=Ψb(0)], γ is the shape parameter and α is the scale parameter for the distribution of Ψb(G).

Using methods fully described in Watt et al. (2010), and the suboptimal germination data described in Rowse & Finch-Savage (2003), Bloomberg et al. (2009) and Watt et al. (2010), we fitted the two HTT models to the suboptimal germination data. Briefly, for the normal distribution HTT model, we fitted Eqn 3, without the modifier k(T − To), while for the Weibull distribution HTT model, we fitted Eqn 4 to the data. The HTT model based on the Weibull distribution of Ψb(G) more precisely described suboptimal germination data (i.e. higher R2 and lower RMSE) than the HTT model based on the normal distribution for all four species (see Watt et al. 2010 for comparative statistics).

Using the suboptimal parameter values for the normal distribution (Tb, θHT, Ψb(50), σΨb, Table 1), Eqn 3 was then fitted to the data in the supra-optimal T range to estimate values of k. In fitting the Weibull distribution HTT model to the supra-optimal T range, the parameters Tb, θHT and µ in Eqn 4 were held at the values determined from the suboptimal fit (see Table 1 for parameter values). Eqn 4 was modified to account for thermoinhibition by fitting the scale parameter (α) as a function of temperature above the optimum, using either a polynomial [α = a + b (T − To+ c(T − To)2] for D. carota or an exponential function {α = a + b exp[c (T − To)]} for the remaining three plant species. Values of a, b and c, and the shape parameter, γ, were empirically estimated by fitting the non-linear function to the data. The shape (γ) and location (µ) were not modified as functions of T or Ψ in the model as, in contrast to the scale parameter, they showed no discernible trend with either T or Ψ (data not shown). The full set of parameter values used to describe germination in the supra-optimal T range for the normal and Weibull HTT models are given in Table 1.

Table 1.  Summary of model parameters for the normal and Weibull distribution hydrothermal time models used to describe germination in the supra-optimal temperature range
 B. davidiiP. radiataA. cepaD. carota
  1. Model parameters given for Buddleja davidii, Pinus radiata and Allium cepa were used in the following equation to describe the scale parameter: α = a + b exp[c (T − To)], while parameter values given for Daucus carota were used in the following equation: α = a + b (T − To) + c (T − To)2. Also shown are model statistics that include the root mean square error (RMSE) and coefficient of determination (R2) for models fitted to germination data.

Normal distribution    
 Tb (°C)
 θHT (MPa °C d)102.0165.560.3972.58
 Ψb (50) (MPa)−1.10−1.42−1.13−1.21
 σΨb (MPa)0.320.450.250.28
 k (MPa °C−1)0.1180.07670.05570.0457
Model statistics    
 RMSE (%)7.209.3518.5014.00
Weibull distribution    
 Tb (°C)
 θHT (MPa °C d)101.6162.963.076.7
 µ (MPa)−1.64−2.23−1.73−1.71
Model statistics    
 RMSE (%)7.006.477.2311.58

In both the normal and the Weibull HTT models, base water potential [Ψb(G)] is a key variable influencing model performance. Base water potential determines the frequency distribution of time to germination within the seed population. The thermoinhibition response across the supra-optimal temperature range is also controlled by how Ψb(G) changes in response to changing temperatures. These two critical aspects of model performance were graphically assessed by plotting observed Ψb(G) and model predictions of Ψb(G) against actual germination percentile. Observed Ψb(G) was determined for the supra-optimal T data by rearranging Eqn 1 as


using estimated parameter values for θHT and Tb (Table 1). The plots of observed and predicted Ψb(G) provide a useful means of determining how well the models describe the observed population distribution of Ψb(G) across a range of supra-optimal temperatures. Using the final developed models, probability density functions of Ψb(G) were generated to examine changes in the distribution of Ψb(G) as a function of temperature.


Over the supra-optimal data range, germination for all four species was more accurately described by the Weibull than the normal distribution HTT model. Gains in the coefficient of determination averaged 0.11, ranging from 0.01 (0.89 versus 0.88) for B. davidii to 0.28 (0.95 versus 0.67) for A. cepa (Table 1). Increases in model accuracy were also demonstrated by reductions in the RMSE. Compared with the normal distribution HTT model, RMSE for the Weibull distribution HTT model was, on average, 4.2% lower, ranging from 0.2 (7.2 versus 7.0%) for B. davidii to 11.3% (18.5 versus 7.2%) for A. cepa (Table 1).

The normal distribution model for supra-optimal temperatures (Eqn 3) assumes that the increase in Ψb(G) as a function of increasing temperature is constant for all germination percentiles. However, when observed, Ψb(G) (Eqn 5) was plotted against germination percentile for all four species there appeared to be very little increase in Ψb(G) for the earliest germinating percentiles, which had similar values at all supra-optimal temperatures (Fig. 2). Secondly, these plots show a divergence between observed Ψb(G) at mid to high germination percentiles with increasing supra-optimal temperature. As neither of these properties was captured by the normal distribution model, it provided a relatively poor and biased fit to the observed Ψb(G) for all four species (Fig. 2a–d).

Figure 2.

Relationship between germination percentile and observed (symbols) and predicted (lines) base water potential in the supra-optimal temperature range. Figures are shown for (a, e) Buddleja davidii, (b, f) Pinus radiata, (c, g) Allium cepa and (d, h) Daucus carota. (a–d) show predicted base water potential (multiple lines) using the normal distribution hydrothermal time (HTT) model, while (e–h) show predicted base water potential (multiple lines) determined from the Weibull distribution HTT model. Symbols and lines represent the lowest (blue), intermediate (green), upper intermediate (pink, for P. radiata only) and highest (red) supra-optimal temperatures. Values representing lowest, lower intermediate, upper intermediate and highest supra-optimal temperatures for each species are described in the Materials and Methods section.

In contrast, increases in observed Ψb(G) with both temperature and germination percentiles were accurately modelled by the modified Weibull model (Eqn 4). This function fitted the observed Ψb(G) for all four species well and, importantly, described the divergence in the observed Ψb(G) between contrasting supra-optimal temperatures that occur with increasing germination percentile (Fig. 2e–h). Predicted increases in the scale parameter (α) with increasing temperature above the optimum showed a similar non-linear form for the four species studied here (Fig. 3).

Figure 3.

Relationship between the scale parameter (α) in the Weibull distribution hydrothermal time model and the temperature above the optimum. Lines are shown for Buddleja davidii (thin solid line), Pinus radiata (short dashed line), Allium cepa (long dashed line) and Daucus carota (thick solid line). Values are shown over the supra-optimal data range used for each species in the described germination experiments. The functional form of each equation and parameter values are given in Table 1.

For the fitted Weibull distribution, the probability density function of Ψb(G) over the supra-optimal temperature range shows a compression and a right-hand shift with increases in temperatures above To (Fig. 4). For B. davidii, P. radiata and A. cepa, the result of these changes is that with increasing temperature a greater proportion of the seed population has a Ψb(G) shifted above 0 MPa, so that seeds cannot germinate even under moist conditions (Fig. 4). In contrast, Ψb(G) did not increase above 0 MPa for any part of the D. carota population, even at the highest supra-optimal temperature.

Figure 4.

Probability distribution functions (pdf) of base water potential. Figures are shown for (a) Buddleja davidii, (b) Pinus radiata, (c) Allium cepa and (d) Daucus carota. Lines denote the pdf for the optimal temperature (thick solid line) and the highest supra-optimal temperature used to germinate the seed (short dashed line). Also shown are pdf values for equally spaced lower (thin solid line) and upper (long dashed line) intermediate temperatures. Shown on all figures is the base water potential of 0, above which germination is not possible under moist conditions (= 0 MPa).


The HTT model assumes the timing, rate and percentage of seed germination for a constant T to be controlled by the difference between Ψ of the seedbed and the Ψb for a given percentile (Kebreab & Murdoch 1999; Alvarado & Bradford 2002; Rowse & Finch-Savage 2003). Most previous research has assumed that Ψb(G) is normally distributed. One of the advantages of this assumption is that the parameters σψb and mean Ψb(G), along with θHT and Tb, provide a complete description of the likely germination behaviour of a seed population and allow the comparison of germination behaviour of different species (Allen, Meyer & Khan 2000; Köchy & Tielbörger 2007).

The model that we propose retains the HTT model structure, including the parameters θHT and Tb. However, attempting to fit a normal frequency distribution for Ψb(G) to skewed data will result in biased predictions of time to germination. Furthermore, the mean Ψb(G) will not be equal to the median [Ψb(50)] nor will it necessarily be the mode, and σψb will not be an unbiased estimate of dispersal for Ψb(G).

Our results suggest that the Weibull distribution may be more suitable than the normal distribution for modelling Ψb(G) in the supra-optimal temperature range. Use of a closed left limit to model Ψb(G) gives an exact, relatively high and more realistic Ψb(G) value for the start of germination [Ψb(0)]. This starting point [Ψb(0)] is defined by the location parameter. By fitting a single value of the location parameter across all temperatures, the Weibull model accurately captures the observed relative invariance of Ψb(G) to supra-optimal temperature for the initial germination percentiles. The divergence between Ψb(G) for contrasting supra-optimal temperatures, observed here with increasing germination percentile, above [Ψb(0)], can be captured well by linking the scale parameter of the Weibull function to the temperature above the optimum.

The Weibull HTT model presented here shows how thermoinhibition may work as a bet-hedging adaptation (sensuSlatkin 1974; Seger & Brockmann 1987). In the absence of thermoinhibition, HTT will rapidly accumulate after a summertime ‘false break’– a germination-inducing summer rainfall event (Chapman & Asseng 2001), when the soil is both moist and warm. This, in turn, will lead to rapid germination of the whole seed population. The risk that arises from rapid mass germination after a false break is that the evapotranspiration rates will be high, and the soil, although temporarily moist, will rapidly dry out. This means that the risk of newly germinated seedlings dying of desiccation is high. The seed population therefore ‘hedges its bets’ by allowing only the earliest percentiles to rapidly germinate without significant thermoinhibition, in response to favourable temperature and moisture. In contrast to the normal distribution, the model based on the Weibull distribution presented here accurately simulates this behaviour by anchoring the left-hand limit of the distribution of Ψb(G) at a single value, which does not vary with temperature.

However, the progressively larger upwards adjustment of Ψb(G) towards zero, with increasing G and supra-optimal temperature modelled here, prevents rapid germination of the remainder of the seed population. The rate of the increase in Ψb(G) with respect to G positively scales with supra-optimal temperature – arguably because at warmer temperatures, evapotranspiration rates are higher and also because thermal time accumulation is more rapid. The risk of en masse seedling germination followed by mortality due to desiccation is therefore higher at warmer temperatures, so that a larger and more rapid thermoinhibition response is required.

At the same time, it is important that at least some seeds germinate rapidly. Firstly, because under warm temperature conditions, these newly germinated seedlings may be able to sustain rapid downwards growth of their radicle and therefore maintain contact with the ‘moisture front’ as this moves deeper into the soil profile (Monteith 1986). Rapid germination and seedling development after a ‘false break’ is therefore one way to avoid the risk of seedling desiccation under subsequent high evapotranspiration rates. Secondly, rapid germination of a part of the seed population is important because to wait for safer, cooler germination conditions runs the risk of ‘top-down’ or asymmetric competition for light with established plants of other competing species (Wilson 1988) that are not so conservative in their germination behaviour. It also allows for a greater degree of seedling establishment and growth before the onset of cold winter conditions.

Our results were obtained from seed populations that were not summer-dormant. However, thermoinhibition has also been demonstrated for seed populations with summer dormancy when collected in autumn or winter and germinated at supra-optimal temperatures under moist conditions (Facelli & Chesson 2008). Summer dormancy is progressively reduced and lost because of processes such as dry after-ripening, and thus, as seeds become non-dormant, they are also at risk of seedling desiccation after germination in a late false break. Thermoinhibition may therefore be a general bet-hedging mechanism, useful even in species with summer dormancy.


We thank Kent Bradford, Graeme Bourdôt, Phil Hulme, Darren Kriticos and Mark Kimberley for comments on the manuscript. We also thank Hugh Rowse for access to germination data and assistance with its interpretation. Funding was provided by the New Zealand Foundation for Research, Science and Technology programme Undermining Weeds (Contract No. C10X0811).