Current address: Department of Biology, Texas A & M University, College Station, TX 77843, USA.
A hydrothermal time model explains the cardinal temperatures for seed germination
Article first published online: 18 JUL 2002
DOI: 10.1046/j.1365-3040.2002.00894.x
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How to Cite
Alvarado, V. and Bradford, K. J. (2002), A hydrothermal time model explains the cardinal temperatures for seed germination. Plant, Cell & Environment, 25: 1061–1069. doi: 10.1046/j.1365-3040.2002.00894.x
Publication History
- Issue published online: 18 JUL 2002
- Article first published online: 18 JUL 2002
- Abstract
- Article
- References
- Cited By
Keywords:
- Solanum tuberosum;
- mathematical model;
- potato;
- water potential
Abstract
Temperature (T) and water potential (y) are two primary environmental regulators of seed germination. Seeds exhibit a base or minimum T for germination (T_{b}), an optimum T at which germination is most rapid (T_{o}), and a maximum or ceiling T at which germination is prevented (T_{c}). Germination at suboptimal T can be characterized on the basis of thermal time, or the T in excess of T_{b} multiplied by the time to a given germination percentage (t_{g}). Similarly, germination at reduced y can be characterized on a hydrotime basis, or t_{g} multiplied by the y in excess of a base or threshold y that just prevents germination (y_{b}). Within a seed population, the variation in thermal times to germination among different seed fractions (g) is based on a normal distribution of y_{b} values among seeds (y_{b}(g)). Germination responses across a range of suboptimal T and y can be described by a general hydrothermal time model that combines the T and y components, but this model does not account for the decrease in germination rates and percentages when T exceeds T_{o}. We report here that supra-optimal temperatures shift the ψ_{b}(g) distribution of a potato (Solanum tuberosum L.) seed population to more positive values, explaining why both germination rates and percentages are reduced as T increases above T_{o}. A modified hydrothermal time model incorporating changes in ψ_{b}(g) at T > T_{o} describes germination timing and percentage across all T and ψ at which germination can occur and provides physiologically relevant indices of seed behaviour.
Introduction
Seed germination is a complex physiological process that is responsive to many environmental signals, including temperature (T), water potential (ψ), light, nitrate, smoke, and other factors (Bewley & Black 1994; Baskin & Baskin 1998). Temperature has a primary influence on seed dormancy and germination, affecting both the capacity for germination by regulating dormancy and the rate or speed of germination in non-dormant seeds. It has been recognized since at least 1860 that three cardinal temperatures (minimum, optimum and maximum) describe the range of T over which seeds of a particular species can germinate (Bewley & Black 1994). The minimum or base temperature (T_{b}) is the lowest T at which germination can occur, the optimum temperature (T_{o}) is the T at which germination is most rapid, and the maximum or ceiling temperature (T_{c}) is the highest T at which seeds can germinate. The temperature range between T_{b} and T_{c} is sensitive to the dormancy status of the seeds, often being narrow in dormant seeds and widening as dormancy is lost (Vegis 1964). In particular, low T_{c} values are often associated with seed dormancy, as in relative dormancy or thermo-inhibition exhibited by seeds whose germination is prevented at warm temperatures (Bradford & Somasco 1994). The cardinal temperatures for germination are generally related to the environmental range of adaptation of a given species and serve to match germination timing to favourable conditions for subsequent seedling growth and development.
Mathematical models that describe germination patterns in response to T have been developed (e.g. Garcia-Huidobro, Monteith & Squire 1982; Covell et al. 1986; Ellis & Butcher 1988). For suboptimal temperatures (from T_{b} to T_{o}), germination timing can be described on the basis of thermal time or heat units (Bierhuizen & Wagenvoort 1974). That is, the T in excess of T_{b} multiplied by the time to a given germination percentage (t_{g}), is a constant for that percentage (the thermal time constant, θ_{T}(g)):
- (1)
- (2)
This model predicts that the germination rate for a given seed fraction or percentage g (GR_{g}, or 1/t_{g}) is a linear function of T above T_{b}, with a slope of 1/θ_{T}(g) and an intercept on the T axis of T_{b}. In many cases, T_{b} varies relatively little among seeds in a population within a given species, as predicted by Eqn 1 (Garcia-Huidobro et al. 1982; Covell et al. 1986; Dahal, Bradford & Jones 1990; Kebreab & Murdoch 1999), although there are exceptions to this, particularly when dormancy is present (Labouriau & Osborn 1984; Fyfield & Gregory 1989; Grundy et al. 2000; Kebreab & Murdoch 2000). Nonetheless, the thermal time model (Eqns 1 & 2) has been extensively and successfully applied to describe seed germination timing at suboptimal T.
Similar models have been proposed to describe germination rates at supra-optimal temperatures (from T_{o} to T_{c}). In many cases, GR_{g} declines linearly with an increase in T between T_{o} and T_{c} (Labouriau 1970; Garcia-Huidobro et al. 1982; Covell et al. 1986). However, it is generally observed that different fractions of the seed population have different T_{c} values. To account for this variation in T_{c} values, Ellis and coworkers (Covell et al. 1986; Ellis et al. 1986; Ellis & Butcher 1988) proposed the following model:
- (3)
- (4)
where θ_{2} is a thermal time constant at supra-optimal T and T_{c}(g) indicates that T_{c} values vary among fractions (g) in the seed population. In this model, differences in GR_{g} for the different seed fractions were a consequence of variation among seeds in their ceiling temperatures (T_{c}(g)), and the total thermal time remained constant in the supra-optimal range of T.
Although this model or subsequent modifications of it have been relatively successful in describing germination timing at supra-optimal T, they do not offer a physiological explanation for this response (i.e. for the decrease in GR_{g} and variation in T_{c}). We propose that seed germination behaviour at supra-optimal T is a consequence of the sensitivity of germination to ψ. The hydrotime model describes the relationship between ψ and seed germination rates in analogy to the thermal time model. Gummerson (1986) defined the hydrotime constant (θ_{H}) as:
- (5)
where ψ_{b}(g) is the base or threshold ψ that will just prevent germination of fraction g of the seed population. In this model, ψ_{b}(g) represents the variation in threshold (ψ_{b}) values among seeds in the population, which often can be described by a normal distribution. Thus, since θ_{H} is a constant, variation in ψ_{b} values is reflected in a proportional variation in t_{g} values among seeds. A normal distribution of ψ_{b}(g) values results in a right-skewed sigmoid cumulative time course of germination events, as is generally observed for seed populations (Bradford 1997). This model can accurately describe germination timing at reduced ψ, simultaneously accounting for reductions in both germination rates and percentages as ψ decreases (Gummerson 1986; Bradford 1990, 1995; Dahal & Bradford 1994).
The hydrotime and thermal time models have been combined into a hydrothermal time model that can describe seed germination patterns across suboptimal T and reduced ψ:
- (6)
where θ_{HT} is the hydrothermal time constant (Gummerson 1986; Bradford 1995). Using this model, seed germination times across the range of suboptimal T and ψ can be described with good accuracy (e.g. Dahal & Bradford 1994). However, the hydrothermal time model (Eqn 6) does not predict a decrease in germination rates as T increases above T_{o}. Interactions have been observed between T and ψ in the supra-optimal range of T, such as for lettuce (Lactuca sativa L.) seeds (Bradford & Somasco 1994), where ψ_{b}(g) values increased (became more positive) with increasing T. Similarly, Kebreab & Murdoch (1999, 2000) found that low ψ restricts the T range for germination in Orobanche seeds. These data suggest that changes in ψ_{b}(g) could be responsible for the delay and inhibition of seed germination in the supra-optimal range of T, as hypothesized previously (Bradford 1996).
Here we report experimental tests of this hypothesis demonstrating that the decrease in germination rates and percentages at supra-optimal T is due to an increase in the ψ_{b}(g) thresholds for germination in a seed population. When modified to account for this effect of supra-optimal T on ψ_{b}(g), the hydrothermal time model can describe seed germination timing and percentages at temperatures from T_{b} to T_{c} and at all ψ at which germination can occur. This model provides both a mathematical description and a physiological rationale for the cardinal temperatures for seed germination.
Materials and methods
True (or botanical) potato seeds (Solanum tuberosum L.), which germinate over a range of T after dormancy is overcome and exhibit clear sub- and supra-optimal ranges of T (Pallais 1995), were used in these studies. Hybrid true potato seeds were produced in Chacas, Perú in 1996 by hand pollination of parental lines Yungay and 104.12LB. After harvest, the seeds were transported to the International Potato Center in Lima, Perú, stored at 15 °C until the seed moisture content was reduced to ∼ 4·5% (fresh weight basis) and subsequently stored at 0 °C in sealed containers. After transfer to the University of California, Davis, the seeds were stored at −20 °C in sealed containers. To control ψ of the germination medium, solutions of polyethylene glycol 8000 were prepared according to Michel (1983). The ψ-values of the solutions were measured using a vapour pressure osmometer (Model 5100C; Wescor Inc., Logan, UT, USA) and corrected for the effect of temperature (Michel 1983). Five replicates of 25 seeds each were placed in 5-cm-diameter Petri dishes on two germination blotters saturated with water (ψ = 0 MPa) or solutions of polyethylene glycol 8000 that maintained specific water potentials (ψ = −0·2 and −0·4 MPa) at 14, 16, 18, 20, 22, 24, 27 and 28 °C. The replicates were randomized within isothermal lanes on a temperature gradient table. Germination was recorded at radicle protrusion to 2 mm, and germinated seeds were removed.
Germination time course data were analysed and the parameters were determined for the thermal time, hydrotime and hydrothermal time models using repeated probit regression analysis as described previously (Bradford 1990; Dahal & Bradford 1990, 1994). Germination rates were calculated as the inverses of the times to radicle emergence, and germination times for specific percentiles of the seed population were calculated by interpolation using curves fit to the time course data. Results for the 16th, 50th and 84th percentiles are reported to represent the median of the population and one standard deviation (σ) above and below it.
Results
Germination responses to temperature and water potential
Germination of true potato seeds in water (0 MPa) progressed more rapidly as T increased in the suboptimal range (Fig. 1a–c). In contrast, germination in water was progressively delayed and final percentages were reduced as T increased above 20 °C (Fig. 2a–e). Plotting the germination rates (1/t_{g}) for the 16th, 50th and 84th percentiles versus T revealed the characteristic linear increases and decreases in GR_{g} below and above T_{o} (Fig. 3a). The lines drawn through these points in Fig. 3a for different germination fractions (based on the model to be described subsequently) converged to a common T_{b} at suboptimal temperatures, but extrapolated to different T_{c} values in the supra-optimal range of T.
When seeds were placed to germinate at reduced ψ in any constant T, a delay in germination was observed relative to the time course in water (Figs 1a–c & 2a–e). At each T, the hydrotime model closely matched actual true potato seed germination time courses at different ψ, as can be seen from comparison of the predicted germination times (curves) and the actual data (symbols). The predicted curves were based upon the ψ_{b}(g) threshold distributions (Figs 1d–f & 2f–j) and the parameter values in Table 1. In the suboptimal range of T, ψ_{b}(50) and σ_{ψb} values were relatively constant, with differences in germination rates being primarily reflected in decreasing θ_{H} values (Table 1A). That is, the time required for germination decreased as T increased, in accordance with the thermal time model, and the seeds’ sensitivity to ψ remained relatively unchanged. However, in the supra-optimal range of T, ψ_{b}(50) values increased (became more positive) with T, rising from −1·39 MPa at 20 °C to only −0·49 MPa at 28 °C (Fig. 2f–j; Table 1C). This was evident in the greater effect of reduced ψ on germination as T increased (Fig. 2a–e). The variation in ψ_{b} among seeds (σ_{ψb}) was relatively constant with T (Table 1C), indicating that the threshold values of all seeds in the population increased by approximately equal amounts as T increased. When the ψ_{b}(g) values estimated by the hydrotime model were plotted versus T for the 16th, 50th and 84th percentiles (Fig. 3b), points in the supra-optimal range showed linear increases that intercepted ψ = 0 MPa at the T_{c} values extrapolated from the GR_{g} data (Fig. 3a). This would be expected if the increase in ψ_{b}(g) is responsible for the decrease in GR_{g}; when the ψ_{b} value of a given seed increases to 0 MPa, the seed would be unable to germinate in water at that T, which is also the definition of T_{c}.
A | T (°C) | θ_{H} (MPa h) | ψ_{b}(50) (MPa) | σ_{ψb} (MPa) | r^{2} | ||
---|---|---|---|---|---|---|---|
14 | 203 | −1·64 | 0·24 | 0·92 | |||
16 | 164 | −1·56 | 0·19 | 0·92 | |||
18 | 137 | −1·46 | 0·23 | 0·89 |
B | T (°C) | θ_{HT} (MPa°h) | ψ_{b}(50) (MPa) | σ_{ψb} (MPa) | T_{b} (°C) | r^{2} | |
---|---|---|---|---|---|---|---|
14–18 | 2090 | −1·54 | 0·23 | 3·2 | 0·87 |
C | T (°C) | θ_{H} (MPa h) | ψ_{b}(50) (MPa) | σ_{ψb} (MPa) | r^{2} | ||
---|---|---|---|---|---|---|---|
20 | 130 | −1·39 | 0·24 | 0·94 | |||
22 | 130 | −1·24 | 0·31 | 0·90 | |||
24 | 130 | −1·06 | 0·26 | 0·89 | |||
27 | 130 | −0·63 | 0·26 | 0·96 | |||
28 | 130 | −0·49 | 0·19 | 0·92 |
D | T (°C) | θ_{H} (MPa h) | ψ_{b}(50) (MPa) | σ_{ψb} (MPa) | T_{o} (°C) | k_{T} (MPa °C^{−1}) | r^{2} |
---|---|---|---|---|---|---|---|
| |||||||
20–28 | 130 | −1·54 | 0·26 | 19·3 | 0·12 | 0·88 |
A hydrothermal time model of germination across all temperatures
Bradford (1990) derived a factor allowing the normalization of germination time courses across a range of ψ. This factor, [1 − (ψ/ψ_{b}(g))] t_{g}, normalizes the germination time of a seed fraction at any ψ to the corresponding germination time that would occur in water, given the ψ_{b} value for that seed fraction. In essence, this factor removes the effect of reduced ψ on the germination time course. If application of this factor normalizes time courses at different ψ to a common predicted time course in water, it indicates that the model parameters have accurately described the sensitivity to ψ of the seed population. Using the hydrothermal time model (Eqn 6) and this normalization factor, all of the germination time courses at suboptimal T (Fig. 1a–c) were plotted on a common normalized thermal time scale (Fig. 4a). That is, the time courses at different ψ at each T were normalized to the equivalent time course in water at that T, and then these were plotted on a thermal time scale. These data at suboptimal T and reduced ψ are described well (r^{2} = 0·87) by a common set of hydrothermal time parameters (Table 1B).
This approach clearly would not work at supra-optimal T, as the ψ_{b}(g) distributions shifted positively as T increased (Fig. 2f–j), precluding a common set of model parameters. However, the value of ψ_{b}(g) can be easily modified as follows to account for its linear increase as a function of T above T_{o}:
- (7)
where ψ_{b}(g)_{T>To} is the ψ_{b}(g) threshold distribution at T above T_{o}, ψ_{b}(g)_{To} is the ψ_{b}(g) distribution at T_{o}, and k_{T} is the slope of the relationship between ψ_{b}(g) and T in the supra-optimal range of T (Fig. 3b). The value of ψ_{b}(g) is simply increased linearly as T increases above T_{o}. This modified value of ψ_{b}(g)_{T>To} can then be used in the hydrotime model (Eqn 5) to predict germination timing. It should therefore be possible to combine together all of the data shown in Fig. 2(a–e) and determine the parameter values that describe germination in this range of T.
To facilitate fitting this model by the repeated probit regression method (Bradford 1990), we used the fact that the model responds only to the difference between ψ and ψ_{b}(g), so a positive shift in ψ_{b}(g) has the same effect as an equivalent negative shift in ψ. Thus for calculation purposes, ψ can be shifted negatively instead of shifting ψ_{b}(g) positively in order to fit the model and determine the parameter values. The model for fitting was therefore:
- (8)
where θ_{H} is the hydrotime value at T_{o}. We fit this model by changing systematically T_{o}, k_{T} and θ_{H} until the ψ_{b}(50) for this model was similar to the ψ_{b}(50) of the hydrothermal time model at suboptimal T (–1·54 MPa). The values that best fit the model are shown in Table 1D. Normalization of all the data of Fig. 2(a–e) using these parameters illustrated that the model worked well to predict germination timing and extent under these assumptions (Fig. 4b). Assuming further that thermal time accumulation is maximal at T_{o} and no additional thermal time accrues at T > T_{o}, the normalized data at supra-optimal T were multiplied by T_{o} − T_{b} to put them on a thermal time basis (Fig. 4c). These normalized thermal time courses then coincided with the normalized data from the hydrothermal time model at suboptimal T (Fig. 4c). By accounting separately for germination behavior at sub- and supra-optimal ranges of T, the model could normalize all data from T_{b} to T_{c} and across ψ-values to a single common thermal time course.
The hydrotime parameters at each supra-optimal T mentioned previously in Fig. 2(f–j) and Table 1C were actually derived by fitting the data at each T using the common hydrotime constant (θ_{H}) predicted by the supra-optimal model across 20–28 °C (130 MPa h; Table 1D). The ψ_{b}(g) distributions calculated at each T using this common θ_{H} value are shown in Fig. 2(f–j) and were used to generate the predicted germination time courses at each T and ψ combination shown in Fig. 2(a–e). Similarly, the lines drawn in Fig. 3 are derived from the sub- and supra-optimal model parameters for the 16th, 50th and 84th germination percentiles (Tables 1B & D).
A modified hydrothermal time model can describe and predict both germination timing and percentage across all constant T and ψ at which germination can occur according to:
- (9)
where [k_{T}(T − T_{o})] applies only when T > T_{o}, and in this range of T the value of ψ_{b}(g) is equal to ψ_{b}(g)_{To} and T − T_{b} is equal to T_{o} − T_{b}.
Discussion
It has long been recognized that seed germination is characterized by minimum (T_{b}), optimum (T_{o}) and maximum (T_{c}) temperatures (Bewley & Black 1994). It has also been known for many years that the timing of germination at suboptimal T conforms to a heat units or thermal time model (Labouriau 1970; Bierhuizen & Wagenvoort 1974). The thermal time approach is well understood and has wide applicability for modelling developmental rates of plants, insects and other poikilothermic organisms (Ritchie & NeSmith 1991). However, this model (Eqn 1) does not predict the decrease in germination rates and percentages that occurs at T > T_{o}. Empirical models have been proposed that can match the observed changes in germination in this temperature range (e.g. Garcia-Huidobro et al. 1982; Ellis et al. 1986; Orozco-Segovia et al. 1996; Kebreab & Murdoch 2000). For purposes of ecological modelling of seed germination and seedling emergence, an empirical approach may be satisfactory (Forcella et al. 2000). However, if the model is to be used to guide further biochemical investigation into mechanisms controlling germination responses to environmental factors, a physiologically based model is preferable to a purely empirical one (Vleeshouwers & Kropff 2000; Benech-Arnold et al. 2000).
The hydrotime and hydrothermal time models provide insight into how physiological and environmental factors interact to regulate the germination behavior of seed populations (Bradford 1995, 2002). These models have revealed that at a given T, the timing and percentage of germination in a seed population are controlled by the difference between a physiologically determined ψ threshold (which can vary among individual seeds in the population) and the ψ of the seed. Studies have found that seed dormancy is a reflection of high (more positive) values of the ψ_{b}(g) threshold, and that conditions that break dormancy (after-ripening, hormones, etc.) shift the ψ_{b}(g) distribution to lower (more negative) values (Ni & Bradford 1992, 1993; Bradford 1996; Christensen, Meyer & Allen 1996; Allen & Meyer 1998; Meyer, Debaene-Gill & Allen 2000; V. Alvarado and K.J. Bradford, unpublished results). ψ_{b}(g) values have also been reported to be at a minimum around the optimum T and to become more positive at supra-optimal T (Bradford & Somasco 1994; Dutta & Bradford 1994; Christensen et al. 1996; Kebreab & Murdoch 1999; Meyer et al. 2000).
We have demonstrated here that when T exceeds T_{o}, the ψ_{b}(g) distribution of a true potato seed population shifts to higher values (Figs 2 & 3). This has the consequence of delaying germination for all seeds in the population, and of preventing germination in those seeds whose ψ_{b} thresholds now exceed the ψ of the environment. As T increases further above T_{o}, different fractions of the seed population will have different T_{c} values, or temperatures at which ψ_{b}(g) for the particular fraction is equal to 0 MPa. This explains the common observations that GR_{g} values decrease as T exceeds T_{o} and that T_{c} values vary among seeds in a population, often in a normal distribution (e.g. Ellis et al. 1986; Ellis & Butcher 1988) that reflects the normal distribution of ψ_{b}(g) values. Treatments such as seed priming or ethylene, which can expand the high T range for germination, do so by increasing the T at which ψ_{b}(g) begins to be affected (Bradford & Somasco 1994; Dutta & Bradford 1994). Thus, T acts on seed germination in the supra-optimal range primarily by causing a positive shift in the ψ_{b}(g) values of the seed population (Figs 2 & 3).
Kebreab & Murdoch (1999) proposed an alternative interpretation and model for interactions between T and ψ where reduced ψ modifies the upper and lower temperature limits and germination rate relationships within an essentially thermal time model. However, a limitation to this model is that it predicts that the seed population will eventually achieve 100% germination under all conditions within the temperature limits, which is not the case. A subsequent version of the model was developed to predict the final germination percentages at different T and ψ combinations, but these two models have to be combined sequentially in order to describe both germination rate and percentage (Kebreab & Murdoch 2000). The hydrotime model, however, predicts both germination rates and percentages and closely matches the actual time courses, which asymptote at different final percentages depending upon the values of ψ and ψ_{b}(g) (Fig. 2; Bradford 1995). Shifting ψ_{b}(g) distributions in response to T automatically adjusts both germination rates and percentages. The parameters of the hydrothermal time model also have clear physiological meaning, which is not always the case with other empirical models. Thus, we believe that the approach proposed here wherein ψ_{b}(g) distributions shift in response to changes in T is a more accurate and parsimonious description of actual seed behavior than the alternative of modifying thermal coefficients in response to changes in ψ.
Application of the population-based hydrothermal time model also allows normalization of germination time courses across all T and ψ at which germination can occur. In the suboptimal range of T, a common T_{b} among seeds and essentially constant values of ψ_{b}(50) and σ_{ψb} made the application of this model straightforward. The factor developed by Bradford (1990) was used with the hydrothermal time model to normalize the effects of ψ on germination across suboptimal T (Fig. 4a; Dahal and Bradford, 1994). When the effect of supra-optimal T on ψ_{b}(g) was taken into account, germination time courses at these T could be normalized onto a common scale (Fig. 4b). As shown here by our application of Eqn 8 to fit the model, and in the final formulation of the model (Eqn 9), the effect on germination of increasing T above T_{o} was equivalent to the effect of a reduction in ψ. The k_{T} value of 0·12 MPa °C^{−1} indicates that for every degree that T increased above T_{o}, the effect on germination was as if the seed ψ was reduced by 0·12 MPa. As ψ_{b}(g) increased further at higher T, germination became more and more sensitive to ψ and eventually was prevented even in water when T = T_{c}(g), or the T at which ψ_{b}(g) was equal to 0 MPa (Figs 2a–e & 3b). Furthermore, when the normalized t_{g} values in Fig. 4b were multiplied by T_{o} − T_{b}, they coincided with the normalized time courses at suboptimal temperatures (Fig. 4c). This demonstrates that thermal time accumulation per unit actual time increases as T exceeds T_{b}, but further accumulation stops when T > T_{o}. In the supra-optimal range of T, the purely thermal effect of T on germination rates is maximal, and germination behaviour is governed by the values of the ψ_{b}(g) threshold distribution relative to the ψ of the environment.
Although we used a single distribution of ψ_{b}(g) to describe germination at suboptimal T, there was a small increase in ψ_{b}(50) values in this range also as T increased (Table 1A; Fig. 3b). This could potentially be accounted for with another slope constant to adjust ψ_{b}(g) values relative to T in this suboptimal range. Since a common estimate of the hydrothermal time model parameters did an adequate job of accounting for the data in this example (e.g. Fig, 4a), we did not pursue this further. However, there are cases where variation in apparent T_{b} among seeds in a population or in germination sensitivity to ψ at low T were observed (Orozco-Segovia et al. 1996; Kebreab & Murdoch 1999, 2000; Grundy et al. 2000), and changes in both the upper and lower temperature limits for germination are often associated with the imposition and release of dormancy (e.g. Kruk & Benech-Arnold 2000). Thus, further modification of the hydrothermal time model to allow variation in ψ_{b}(g) at both sub- and supra-optimal temperatures may be necessary to describe germination behaviour of other species (Bradford 2002).
The conclusion that shifts in ψ_{b}(g) distributions determine the germination behaviour of seeds in the supra-optimal temperature range has a number of significant consequences. It further confirms that rather than being fixed values, ψ_{b}(g) thresholds of seed populations are under physiological control in response to environmental and hormonal conditions (Ni & Bradford 1992, 1993; Bradford & Somasco 1994; Dahal & Bradford 1994; Christensen et al. 1996; Meyer et al. 2000). As noted above, the common observation that the temperature limits for germination widen as dormancy is released and narrow as dormancy is imposed (Vegis 1964) is likely to be due to corresponding shifts in the ψ_{b}(g) distributions of the seed populations in response to dormancy-regulating factors. We can predict that additional factors that regulate germination under natural conditions, such as after-ripening, light or nitrate (Hilhorst & Toorop 1997; Benech-Arnold et al. 2000), will act via this mechanism. Although variable ψ_{b}(g) values complicate some applications of the hydrothermal time model for predicting seedling emergence in the field, this may also explain why models assuming fixed values of ψ_{b}(g) could not adequately describe all aspects of seed behaviour across variable environmental conditions (e.g. Kebreab & Murdoch 1999; Finch-Savage et al. 2000; Grundy et al. 2000; Bradford 2002). Although the model does not identify the biochemical mechanism(s) by which ψ_{b}(g) is itself regulated, it clearly points further biochemical and molecular investigations in this direction. A number of candidate genes and processes have been identified that could be involved in the initiation of germination (Bradford et al. 2000), at least some of which are regulated and expressed in a manner consistent with a role in establishing ψ_{b} thresholds (Chen & Bradford 2000; Nonogaki, Gee & Bradford 2000; Chen, Dahal & Bradford 2001, Chen, Nonogaki & Bradford 2002). Finally, we anticipate that this population-based threshold model can be applied to describe the responses of other biochemical or physiological processes in cell populations to changing T, ψ, hormonal, or developmental conditions (Bradford & Trewavas 1994).
In conclusion, we have extended the hydrothermal time model to describe germination timing and percentage across all T and ψ at which germination can occur. This comprehensive, physiologically based model accounts for all three of the cardinal temperatures for seed germination. The parameters of the model can be used to quantitatively characterize and compare the physiological status of seed populations under different environmental conditions or having different genetic backgrounds. In addition, the model targets the processes by which seed water potential thresholds are determined for further biochemical and molecular investigation.
Acknowledgments
This project was supported in part by Western Regional Research Project W-168. We thank Dr Noel Pallais of the International Potato Center, Lima, Perú, for providing the true potato seeds and encouraging this investigation.
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