## 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*)):

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:

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:

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*:

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.