Hydrothermal threshold models can describe the germination response of carrot (Daucus carota) and onion (Allium cepa) seed populations across both sub- and supra-optimal temperatures


  • HR Rowse,

    1. Horticulture Research International, Wellesbourne, Warwick, CV35 9EF, UK
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  • WE Finch-Savage

    Corresponding author
    1. Horticulture Research International, Wellesbourne, Warwick, CV35 9EF, UK
      Author for correspondence: W. E. Finch-Savage Tel: +44 1789470382 Fax: +44 1789470552 Email: bill.finch-savage@hri.ac.uk
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Author for correspondence: W. E. Finch-Savage Tel: +44 1789470382 Fax: +44 1789470552 Email: bill.finch-savage@hri.ac.uk


  • • The effect of temperature on the minimum (base) water potential for seed germination (Ψb) was investigated in Daucus carota and Allium cepa and then described in two hydrothermal threshold models.
  • • Germination was recorded over a wide range of temperatures and water potentials.
  • • At temperatures of 15°C and below the base water potential for germination of the 50th percentile (Ψb(50)) was constant, but in both species, above a temperature (Td) around 16–19 °C, Ψb(50) increased linearly with temperature. Hydrothermal time (HTT) and virtual osmotic potential (VOP) models were altered so that the effective base water potential (Ψb(G,T)) for any percentile of the seed population (G), above Td, was given by Ψb(G)d + m(T – Td), where Ψb(G)d is the uncorrected base water potential for that percentile. The coefficient m is the slope of the linear relationship between Ψb(50) and temperature above Td.
  • • Germination response to all temperatures and water potentials can be adequately described in both the HTT and VOP models by incorporating changes in Ψb(G,T) with temperature.


Seed germination is greatly influenced by both temperature (Roberts, 1988; Probert, 2000) and water potential (Bradford, 1990, 1995) and these factors largely determine the timing of onion and carrot seed germination in the field (Finch-Savage & Phelps, 1993; Finch-Savage et al., 1998). The timing and spread of time to germination within the seed population have a major impact on the efficiency of production in these two crops (Finch-Savage, 1995). Accurate models are therefore required to describe seed responses to these two variables for effective field predictions. Two different population-based threshold modelling approaches are considered in the present work to describe seed responses to the hydrothermal environment. In both approaches, it is assumed that seeds germinate in a set order unaffected by germination conditions, so that each seed can be assigned a value of G, which is the percentage of the population at which it germinates. Gummerson (1986) developed a theory in which germination time t(G) is a function of the extent to which the constant water potential (Ψ) and constant suboptimal temperature (T) of each seed (G) exceed thresholds (bases; Ψb, Tb) below which germination will not occur.

θHT = (Ψ − Ψb(G))(T − Tb)t(G)(Eqn 1)

Hydrothermal time (θHT) and the base temperature (Tb) are assumed constant. Only the base water potential (Ψb) varies with (G) and so the distribution of the germination times of individual seeds within the population are determined by the distribution of this parameter. This approach has been shown to adequately describe germination curves produced in a wide range of suboptimal constant temperatures and water potentials (Gummerson, 1986; Dahal & Bradford, 1994; Finch-Savage et al., 1998; Roman et al., 1999; Shrestha et al., 1999; Allen et al., 2000). Accepting these assumptions it is possible to describe seed response of the whole population in a single equation by incorporating a suitable distribution (usually a normal distribution) of base water potentials within the population (Gummerson, 1986; Dahal et al., 1993; Dahal & Bradford, 1994; Bradford, 1995). In the case of a normal distribution:

Probit (G) = [(Ψ − θHT/(T − Tb)t(G)) − Ψb(50)]/σΨb(Eqn 2)

where Ψb(50) is the base water potential of the 50th percentile and σΨb is the standard deviation of Ψb within the population. The best fit to the model can be obtained by repeated probit regressions varying the values of θHT (e.g. Bradford, 1995). In a range of tomato seed lots this model accounted for 73–93% of the variation in radicle emergence timing across a range of temperatures and water potentials (Cheng & Bradford, 1999).

However, to be useful for field predictions the hydrothermal time model must also be able to describe the reduction in germination rate and nongermination that occurs at supra-optimal temperatures. This factor can be accommodated in the model by employing a shift in the distribution of Ψb with temperature. For example, during thermoinhibition of lettuce (Lactuca sativa) water potential thresholds were found to increase as the temperature approached the upper limit for germination (Bradford & Somasco, 1994). Germination was then prevented as thresholds shifted above ambient water potential. Subsequently, other studies have shown that as temperature increases above the optimum (To) and approaches the ceiling temperature (Tc), the Ψb distribution shifts progressively towards and above 0 MPa (Kebreab & Murdoch, 1999; Meyer et al., 2000; Bradford, 2002). As the distribution shifts above 0 MPa, germination is progressively prevented in the population accounting for the distribution of ceiling temperatures shown by Ellis et al. (1986).

Bradford (2002) modified eqn 1 above to describe the response of seed germination to supra-optimal temperatures when the shift in distribution of Ψb(G) with temperature is linear thus:

θHT = [Ψ − Ψb(G)o + kT(T − To)](To − Tb)t(G)(Eqn 3)

where kT is a constant (the slope of the Ψb(G) vs T line when T > (To) and Ψb(G)o is the threshold distribution at To. This equation adjusts Ψb(G)o to higher values as T increases above To and stops the accumulation of thermal time at the value equivalent to that accumulated at To. Thus, temperatures above To do not contribute additional thermal time in the supra-optimal range. Instead, effects on germination are accounted for by the change in Ψb(G). Thus a germination rate response to temperature is described that has a clearly defined optimum at the convergence of two straight lines, as shown in a number of studies (e.g. Garcia-Huidobro et al., 1982; Covell et al., 1986). Germination response can then be described for the full range of temperatures and water potentials using a combination of equations 1 and 3. However, eqn 3 cannot predict the plateau or curved relationship between germination rate and temperature that has been observed around To in a number of species (e.g. Labouriau & Osborn, 1984; Orozco-Segovia et al., 1996). To achieve this a different relationship between Ψb(G) and temperature is needed.

A second population-based threshold modelling approach known as the Virtual Osmotic Potential (VOP) model is being developed to assist simulation of germination response under variable seedbed conditions (Rowse et al., 1999). To initially develop the model, germination of an imbibed seed was considered to result from further water uptake and growth of the radicle. It was assumed that the growth was driven by an increase in turgor pressure exceeding the yield threshold pressure (Y) due to an accumulation of solutes and a consequent decrease in osmotic potential (Ψπ). The model used changes in Ψπ to integrate the history of water potential experienced by the seed. For simplicity, Y was considered as a constant for every seed in the population. However, growth could also result from a decrease in Y. This necessarily means that values of Ψπ used in the model are empirical, and to avoid confusion with real measurable osmotic potentials the term VOP with the symbol Ψπv was used.

The VOP model utilises the concepts of Ψb and Tb, but also incorporates progress towards germination at water potentials below Ψb and above the minimum for metabolic advancement Ψmin (Tarquis & Bradford, 1992) that is not described by the basic hydrothermal time model. It is at these water potentials that priming occurs. As originally derived (Rowse et al., 1999) the VOP model described the effects of water potential on germination time at a single temperature (15°C). In this paper the scope of the VOP model has been extended by assuming a linear relationship between germination rate and suboptimal temperature and therefore replacing the term k0(T)(1 – Ψ/Ψmin) of eqn 9 in Rowse et al. (1999) by R(1 – Ψ/Ψmin)(T/Tb – 1). This gives an equation that describes germination time of a seed fraction as:

image(Eqn 4)

where Ψ, Ψb, Ψmin, T and Tb are as described above and k and Y are constants determined by fitting. A linear relationship between germination rate and suboptimal temperature has been reported both for carrot (Finch-Savage et al., 1998) and onions (Ellis & Butcher, 1988) and is therefore justified in the present work. A problem can arise with this formulation because when Tb = 0, T/Tb= infinity and t(G) = 0, but this situation can be avoided by the use of absolute temperature. However, for most practical purposes this problem can be ignored.

To simulate the effects of a changing environment, advancement towards germination can be determined by integrating changes in Ψπv that are proportional to the history of temperature and water potential experienced by the seed relative to Tb, Ψmin and Ψb. The increase in Ψπv during each finite step of the simulation at a fixed temperature of 15°C is given by Rowse et al. (1999). Modification of this equation to include the effects of suboptimal temperatures yields:

dΨπv(G)/dt = k(1 – Ψ/Ψmin)(T/Tb – 1)(Ψb(G) – Y – Ψπv(G))(Eqn 5)

In the present work the VOP model is further extended to include response to temperature across both sub- and supra-optimal ranges and the same approach is also adopted to describe the later response within the hydrothermal time model.

Materials and Methods

Carrot (Daucus carota L. cv. Narman) and onion (Alium cepa L. cv. Hyton) used in these experiments had 95 and 98% viability, respectively, in germination tests (ISTA, 1999). Seed germination (radicle emerged) was recorded on four replicates of 100 seeds of both species in all combinations of six temperatures (5, 10, 15, 20, 25 and 30 ± 0.5°C) and nine nominal water potentials (0, −0.2, −0.3, −0.4, −0.5, −0.7, −0.9, −1.1 and −1.3 MPa). Water potentials were maintained using the appropriate strength solution of polyethylene glycol (PEG) (molecular weight, 20 000 Da). Solution strength was originally determined using the calibration of PEG concentration and water potential given by Williams & Shaykewich (1969). The potentials of solutions used were subsequently measured using a Wescor vapour pressure osmometer (model 5100C; Wescor Inc., Logan, UT, USA).

PEG solutions were placed in a tubular semipermeable membrane that was 100 mm wide when laid flat (Dialysis tubing – visking, Medicell International Ltd, London, UK, size 15). Seeds were placed under the membrane, but on top of stainless steel mesh in the base of a sealed polyethylene sandwich box (170 × 90 mm and depth 70 mm). When laid flat the membrane containing the PEG was just narrower than the width of the box. This replicate unit, described in more detail elsewhere (Rowse et al., 1999), exposed seeds to accurate water potentials and adequate aeration. The molecular weight cut-off of the membrane was quoted as 12 000–14 000 Da so that it should retain the PEG 20 000, but allow free passage of water to the seeds. The membranes containing the solutions were weighed each week and any loss of water replaced. Seeds were removed daily as they germinated and counting continued for at least 100 d.



The percentage of seeds germinating in water was constant across the temperature range used in carrot and up to 25°C in onion. In both species, percentage germination was progressively reduced by water potentials below −0.67 MPa at temperatures of 15°C and below (data not shown). As temperatures increased above 15°C, percentage germination was reduced at progressively higher water potentials. At 30°C, percentage germination of carrots was reduced by a water potential of −2.8 Mpa, whereas at 30°C in onion, no germination occurred at −0.18 MPa and germination in water was reduced to 27%. Thus onion seeds were more sensitive to high temperature than carrot (data not shown). In both species, rate of germination decreased with water potential and increased with temperature to a maximum (To) before decreasing (Fig. 1). In general, To occurred at lower temperatures in later germinating percentiles (data not shown) and with increasing water stress (Fig. 1). This response was particularly clear in onions at 25°C where germination rate was greatly reduced from that in water by a water potential of −0.18. The relationship between germination rate and temperatures of 15°C and below, at all water potentials, appeared to be linear and could be constrained to a single base temperature (Tb) within the seed population (Fig. 1).

Figure 1.

The effect of temperature and water potential on the germination rate (1/T50) of carrot (a,c) and onion (b,d) seeds. Lines are fitted according to eqns 6 and 7 in which the hydrothermal time (a,b) and VOP (c,d) models, respectively, have been modified to take account of changes in Ψb(G) with temperature. Cardinal temperatures are shown as closed circles, 0; open circles, 0.18; closed squares, 0.28; open squares 0.39; triangles, 0.51 MPa, respectively.

Modelling germination

HTT (eqn 2) and VOP (eqn 4) models were fitted to data collected at 15°C and below and the full range of water potential treatments. Data for carrot at 10°C was not used as a drift in temperature had influenced the results. For fitting, a computer programme was used to optimise the parameters of both the HTT and VOP models, so as to minimise the residual sum of squares of the difference between the measured and the modelled germination percentages (Table 1a). Using these parameters, both models adequately described germination of onions and carrots at temperatures of 15°C and below at the full range of water potentials (Fig. 2, data with solid symbols). However, description of germination by both models was progressively worse as temperature increased to 20°C and above (Fig. 2, open symbols).

Table 1.  (a) Parameters for the HTT time (eqn 2) and VOP (eqn 4) models determined from the data collected in all temperature and water potential treatments used in experiments. (b) Additional parameters used in the extended HTT (eqn 6) and VOP (eqn 7) models
Tb (°C) 1.9 2.0 1.2 1.2
Ψb(50) (MPa)−0.84−0.86−0.83−0.87
σΨb (MPa) 0.13 0.19 0.13 0.21
k0 (day−1) 0.030 0.014
Y (MPa) 1.00 0.71
Ψmin (MPa) (not fitted)−2.01−4.18
θHT (MPa°C day)48.246.7
Td (°C)18.617.517.016.1
m (MPa°C−1) 0.030 0.039 0.039 0.051
Figure 2.

Cumulative carrot (a,c) and onion (b,d) germination data collected at all temperatures and water potentials mapped on to a hydrothermal time (HTT) scale (a,b) which has units of MPa°C d, or to the following VOP function: inline image (c,d) which has units of time (d). The models were fitted with a fixed value for Ψb as in eqns 2 and 4. For clarity all water potentials at the same temperature are given the same symbol. Closed circles, 5; closed squares, 10; closed triangles, 15; open circles, 20; open squares, 25; and open triangles, 30°C, respectively.

When model parameters were determined separately for the range of water potentials at each temperature, it appeared that the value of base water potential of the 50th percentile (Ψb(50)) remained approximately constant below 15°C, but increased linearly with temperature at higher temperatures (Fig. 3). The temperature at which Ψb(50) starts to change (Td), and the slope of the line at higher temperatures (m) were estimated by fitting (Table 1b). This relationship between Ψb(50) and temperature, when determined by either model, is very similar for both species. Estimates of the ceiling temperature made from extrapolation of the linear relationship to the temperature axis are lower from the VOP model than the HTT model (Fig. 3), but estimates from both models of Tb, Td and Ψb(50) are similar (Table 1a).

Figure 3.

The effect of temperature on Ψb(50) for carrot (a,c) and onion (b,d) seed germination calculated according to the hydrothermal time (a,b) and VOP (c,d) models. Vertical lines are the standard deviation of Ψb. Td is defined in Fig. 3(a).

For modelling purposes, it is assumed that the normal distribution of Ψb(G) within the population remains constant. This assumption appears reasonable because there is no consistent difference between the standard deviation of Ψb(G) calculated at each temperature (Fig. 3). In this case, the effective base water potential for any percentile of the seed population at temperatures above Td is given by Ψb(G)d + m(T – Td), where Ψb(G)d is the uncorrected base water potential for that percentile. Therefore, the HTT and VOP models can be extended to describe both sub- and supra-optimal ranges thus:

image(Eqn 6 HTT)
image(Eqn 7 VOP)

Where (for both models) when:

Tb < T ≤ Td; Ψb(G,T) = Ψb(G)d
Td < T < Tc; Ψb(G,T) = Ψb(G)d + m(T – Td)

In these extended models, Ψb(50) increases linearly at temperatures above Td. Therefore the increase in rate due to the increase in T – Tb above Td is offset by an increase in Ψb(G) reducing Ψ – Ψb(G). The resulting germination rate response with temperature above Td from the models is a curve that has To at its crown (Fig. 1), in contrast to the sharply defined optimum described by eqn 3.

Germination response to the full range of temperatures and water potentials, used in the present work, was accommodated in both the temperature corrected HTT (eqn 6) and VOP (eqn 7) models (Figs 1 and 4). With onions, the pattern of germination rate with temperature differed between seed in water and those at negative water potentials and this resulted in a large reduction in To with only a mild stress of −0.18 MPa (Fig. 1). Inspection of the data revealed that 25°C was optimal for lower percentiles of the population, but percentiles greater than 60% germinated more slowly at 25 than at 20°C (data not shown). The poor fit of the models to germination rate of the 50th percentile of the onion seed population at 25°C in Fig. 1(b,d) may have resulted from the presence of these two subpopulations in the seed lot. The poor fit to this percentile at 25°C occurs because the fitting procedure provides the best fit of the model to the whole population of seeds to provide a good description of the full data set (Fig. 4).

Figure 4.

Cumulative carrot (a,c) and onion (b,d) germination data collected at all temperatures and water potentials mapped on to a hydrothermal time scale (a,b) or to a VOP function (c,d) as in Fig. 2. However, in contrast to Fig. 2 models were fitted so that Ψb(G) changes with temperature (eqns 6 and 7) to account for the effect of supra-optimal temperatures as described in the text. Closed circles, 5; closed squares, 10; closed triangles, 15; open circles, 20; open squares, 25; and open triangles, 30°C, respectively.


The motivation for the present work was to develop models with sufficient flexibility to describe germination under variable field conditions. The threshold models used here provide a robust framework in which to describe seed responses to the environment. By accommodating changes in Ψb(G) with temperature, both models were shown to adequately describe carrot and onion seed response over both sub- and supra-optimal temperature ranges at a range of water potentials. By this approach, both models retain the ability to describe these responses for the whole seed population and therefore have the potential to predict both the distribution of germination in time and percentage germination. For simulation purposes, the VOP model developed here has the potential to describe seed response to temperature at the full range of water potentials where seeds are physiologically active (Ψ > Ψmin) in one equation. At present the HTT model presented here requires the additional use of hydrothermal priming time (Cheng & Bradford, 1999) to cover the range Ψb > Ψ > Ψmin. However, there may be mathematical solutions to combining the different components of the HTT approach (Battaglia, 1997).

When predicting germination time, using threshold models (biological time), errors have greatest impact in optimum environments when progress towards germination is most rapid. It is therefore essential that any model used for prediction fits the data most accurately near the optimum. In the generally accepted thermal time theory, germination rate has a positive linear relationship with suboptimal and negative linear relationship with supra-optimal temperatures so that To is clearly defined at their convergence (Garcia-Huidobro et al., 1982; Covell et al., 1986). For many purposes, such as comparison of genotypes or treatments (Covell et al., 1986; Dahal et al., 1993) this approach describes the data presented and has been accommodated into the hydrothermal time model (eqn 3; Bradford, 2002). In the current work, the relationship of germination rate with temperature appeared curved around To in agreement with data on onions elsewhere (Ellis & Butcher, 1988). The width of this plateau can differ between seed lots (Ellis & Butcher, 1988). In many other data sets, a plateau or curved relationship has also been reported near to the optimum (i.e. Labouriau & Osborn, 1984; Orozco-Segovia et al., 1996). For example, the comprehensive data set and analysis conducted by Labouriau & Osborn (1984) on tomato seeds showed linear relationships between germination rate and temperature in both sub- and supra-optimal ranges, but the optimum occurred over a range of temperatures between 25.9 and 29.5. This plateau response has been accommodated in a further development of the thermal time model based on Gaussian curves, which describes the germination response across both sub- and supra-optimal temperature ranges (Orozco-Segovia et al., 1996). However, a wide range of responses to both temperature and water potential can be accommodated within the HTT and VOP models if it is no longer assumed that Ψb(G) is independent of temperature.

According to the models developed here, Td the temperature at which Ψb(G) begins to increase with temperature must be less than the optimum temperature (To), although they can be very close. At temperatures greater than Td, the increased rate of biological time accumulation resulting from higher temperature (i.e. increase in T – Tb) as the optimum is approached would be offset by a concurrent increase in Ψb (reducing Ψ − Ψb). By incorporating this linear increase in Ψb(G) at temperatures below To, both the HTT and the VOP models predicted a curved germination rate response to temperature around To that described carrot and onion data in the present work. Data reported for fully after-ripened seeds of Elymus elymoides were also explained in this way, as Ψb(G) was found to increase linearly with temperature (10–30°C) resulting in little difference in germination rates over this range of temperature (Meyer et al., 2000). Shifts in Ψb(G) have also been used to describe other patterns of seed behaviour as dormancy is lost during after-ripening (Christensen et al., 1996; Bauer et al., 1998). However, the current VOP or HTT models are not yet flexible enough to describe all germination behaviour. For example, Kebreab & Murdoch (1999) have found that in Orobanche aegyptiaca seeds, Ψb(G) varied with T, both above and below the optimum, and Tb varied with Ψ. Grundy et al. (2000) have also found that in Stellaria media there is a normal distribution of Tb as well as Tc. Alternative approaches have been developed to describe this data by modelling thresholds and rates separately (Kebreab & Murdoch, 1999, 2000; Grundy et al., 2000). However, it is yet to be seen if this type of data can be fully accommodated within the type of population-based threshold models used here, by allowing Ψb(G) to vary with temperature. This approach maintains the advantage of predicting germination rate and percentage germination in the same model.

Extrapolation to the temperature axis of the relationship between Ψb(50) and temperatures of 20°C and above provides an estimate of Tc for the 50th percentile. For onions, Tc is estimated as 38.5 and 33°C for the HTT and VOP models, respectively (Figs 1 and 3). In experiments, germination did not reach 50% at 30°C during the recording period, indicating that these estimates may be high. However, they are in general agreement with an estimate of 35.5°C elsewhere for another cultivar (Ellis & Butcher, 1988). The estimate of Tc for carrot was also higher (46°C) from the HTT model than that from the VOP model (39°C). Again these values are in agreement with estimates of between 42 and 47°C reported for three other cultivars (Corbineau et al., 1995).

Both population-based threshold models used in the present work can describe germination data over a wide range of temperatures and water potentials and give similar estimates of the parameters Ψb(50), Tb and Tc. This gives further weight to the view that, although these models are fitted empirically, the thresholds determined appear to have a physiological basis (Welbaum et al., 1998; Meyer et al., 2000). These modelling approaches are afforded much greater flexibility to describe seed responses to the hydrothermal environment when Ψb(G) is no longer considered independent of temperature.


We thank J. M. T. McKee, E. C. Higgs and J. R. A. Steckel for technical support and the Department for Environment, Food and Rural Affairs for funding.