Introduction

Quinoa (Chenopodium quinoa Willd.) is considered a pseudo-grain, and due to its protein quality, has been known as a complete food (Ando et al. 2002). Its nutritional value is due to its protein content (15%) and its great amino acid balance (Aluko and Monu 2003). Quinoa contains minerals, vitamins, and other beneficial compounds such as flavonoids, phytosterols, and polyphenols (Abugoch et al. 2008; James 2009). The Andean region has been cultivating this plant since 5000 years ago (Vega‐Gálvez et al. 2010). Quinoa's strong nutritional content has given it new life as an alternative crop to cereals in recent (Pellegrini et al. 2018).

Seed germination is the most critical step in the life cycle of a grain plant (Karimzadeh Soureshjani et al. 2019). Temperature (T), water potential (Ψ), light, nitrate, and other environmental conditions have varying effects on a seed lot germination (Alvarado and Bradford 2002; Watt et al. 2011). Non-dormant germination of the seed potential is heavily influenced by temperature (Alvarado and Bradford 2002; Bakhshandeh et al. 2017). The T range for seed germinate of a particular species are described by three cardinal Ts (\({T}_{\mathrm{b}}\); base or minimum T, \({T}_{\mathrm{o}}\); optimum T and \({T}_{\mathrm{c}}\); ceiling or maximum T) (Bewley and Black 1994). \({T}_{\mathrm{b}}\) is lowest germination, \({T}_{\mathrm{o}}\) is the most rapid germination, and \({T}_{\mathrm{c}}\) is the highest T germination (Vegis 1964).

Germination rate (GR) and germination time (T) have long been the subject of numerous mathematical theories (e.g., Garcia-Huidobro et al. 1982; Covell et al. 1986; Ellis et al. 1986; Ellis and Butcher 1988). The various models that have been used by different researchers had some strengths and weaknesses, and each of them is more suitable in the specific condition (Soltani et al. 2001; Kamkar et al. 2008; Atashi et al. 2015; Karimzadeh Soureshjani et al. 2019).

The thermal time (TT) concept (or heat unit) can be used to describe the germination timing for sub-optimal Ts (from \({T}_{\mathrm{b}}\) to \({T}_{\mathrm{o}}\)) (Bierhuizen and Wagenvoort 1974). Based on the thermal time concept for each germination percentage, time to that germination percentage (tg) multiplied by the T above \({T}_{\mathrm{b}}\) is constant and equal to the thermal time constant (\({\theta }_{\mathrm{T}}\)(g)), (Eqs. 1 and 2):

$${\theta }_{\mathrm{T}}\left(g\right)=\left(T-{T}_{\mathrm{B}}\right){t}_{\mathrm{g}},$$
(1)
$${\mathrm{GR}}_{\mathrm{g}}=1/{t}_{\mathrm{g}}=(T-{T}_{\mathrm{b}})/{\theta }_{T(g)}.$$
(2)

A linear relationship between the rate of germination and T above \({T}_{\mathrm{b}}\) are predicted using this model, where (\({\mathrm{GR}}_{\mathrm{g}}\) or 1/\({t}_{\mathrm{g}}\)) has a slope of \(1/{\theta }_{T(g)}\). The changes of \({T}_{\mathrm{b}}\) among seeds of a specific species is minimal except when the seeds have a dormancy (Garcia-Huidobro et al. 1982; Covell et al. 1986; Grundy et al. 2000; Kebreab and Murdoch 2000; Batlla and Benech-Arnold 2015; Batlla and Agostinelli 2017).

There is a comparable model for germination rates at higher than optimum Ts (from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{c}}\)). The germination rate reduces linearly with T increment above \({T}_{\mathrm{o}}\) and reaches zero at \({T}_{\mathrm{c}}\) (Labouriau 1970; Garcia-Huidobro et al. 1982). It was also observed that generally, the \({T}_{\mathrm{c}}\) values of a seed population's different fractions are not the same (Ellis and Butcher 1988). Equations 3 and 4 can explain the variation in \({T}_{\mathrm{c}}\) in different fractions of seed germination (Covell et al. 1986; Ellis et al. 1986; Ellis and Butcher 1988):

$${\theta }_{2}=\left({T}_{\mathrm{c}\left(g\right)}-T\right){t}_{\mathrm{g}},$$
(3)
$${\mathrm{GR}}_{\mathrm{g}}=1/{t}_{\mathrm{g}}=({T}_{\mathrm{c}\left(g\right)}-T)/{\theta }_{2}.$$
(4)

In which \({\theta }_{2}\) presidents the thermal time constant, and \({T}_{\mathrm{c}}\)(g) donates that \({T}_{\mathrm{c}}\) values change in in the seed population among different germination fractions (g).

Although the thermal time model can accurately predict the germination timing at both sub- and supra-optimal Ts, it is unable to explain why the germination rate is reduced at supra-optimal Ts (Alvarado and Bradford 2002). To eliminate such defects, the hydro time (HT) model was introduced by Gummerson (1986) and Bradford (1990). Equations 5 and 6 are used in this model to explain the connection between germination rate (GR) and water potential (Ψ):

$${\theta }_{\mathrm{H}}=\left(\psi -{\psi }_{\mathrm{b}\left(g\right)}\right){t}_{\mathrm{g}},$$
(5)
$${\mathrm{GR}}_{\mathrm{g}}=1/{t}_{\mathrm{g}}=(\psi -{\psi }_{\mathrm{b}\left(g\right)})/{\theta }_{\mathrm{H}}.$$
(6)

In which \({\theta }_{\mathrm{H}}\) indicates the constant of hydro time, ψ represents the water potential, and ψb(g) donates the base or threshold Ψ to prevent germination fraction of seed population. For each T a probit regression (Eq. 7) analyses can also estimate HT parameters at different Ψ (Bradford 1990; Dahal and Bradford 1990):

$$\mathrm{Probit}\left(g\right)=[\psi -({\theta }_{\mathrm{H}}/{t}_{\mathrm{g}})-{\psi }_{\mathrm{b}}]/{\sigma \psi }_{\mathrm{b}}.$$
(7)

The hydrothermal time (HTT) model describes the time of seed germination at different Ψ and T at both sub- (Gummerson 1986) and supra-optimal Ts (Alvarado and Bradford 2002). The HTT equation at sub-optimal Ts can be written as Eq. (8).

$${\theta }_{\mathrm{HTT}}=\left(\psi -{\psi }_{\mathrm{b}\left(g\right)}\right)\left(T-{T}_{\mathrm{b}}\right){t}_{\mathrm{g}},$$
(8)

where \({\theta }_{\mathrm{HTT}}\), \({\psi }_{\mathrm{b}\left(g\right)}\), and tg are hydrothermal time constant, base Ψ, and time to germination fraction, respectively. In this model, ψb(g) changes among seeds in the population, while \({\theta }_{\mathrm{HTT}}\) and \({T}_{\mathrm{b}}\) are assumed to be constant (Gummerson 1986). In sub-optimal Ts (assuming \({\theta }_{\mathrm{HTT}}\) and \({T}_{\mathrm{b}}\) are constant), the probit regression (Eq. 9) can also be used to quantifying seed germination response to Ψ and T (Dahal and Bradford 1994; Bradford 2002):

$$\mathrm{Probit}\left(g\right)=[(\psi -({\theta }_{\mathrm{HTT}}/(T-{T}_{\mathrm{b}}){t}_{\mathrm{g}})-{\psi }_{\mathrm{b}}]/ {\sigma \psi }_{\mathrm{b}},$$
(9)

where ψb shows the base Ψ of the 50th percentile and \({\sigma \psi }_{\mathrm{b}}\) presents the standard deviation of ψb among the seed population. In the supra-optimal Ts, Eqs. (8) and (9) cannot predict the reduction of germination rate (Alvarado and Bradford 2002). Many studies have shown the Ψ and T interaction in the supra-optimal Ts (Alvarado and Bradford 2002; Rowse and Finch‐Savage 2003; Watt et al. 2011; Bakhshandeh et al. 2015). In all of these studies, it was reported that in supra-optimal Ts with an increment of T, ψb was increased (become more positive). It was suggested that these changes of ψb might be the reason for seed germination inhibition in supra-optimal Ts (Alvarado and Bradford 2002). Hence Alvarado and Bradford (2002) modified Eqs. (8) and (9) and written them as Eqs. (10) and (11) for supra-optimal Ts:

$${\theta }_{\mathrm{HTT}}=\left[\psi -{\psi }_{\mathrm{b}\left(g\right)}\right)-({K}_{\mathrm{T}}\left(T-{T}_{\mathrm{o}}\right))](T-{T}_{\mathrm{b}}){t}_{\mathrm{g}},$$
(10)
$$\mathrm{Probit}\left(g\right)=[(\psi -\left({K}_{\mathrm{T}}\left(T-{T}_{\mathrm{o}}\right)\right)-{\theta }_{\mathrm{HTT}}/((T-{T}_{\mathrm{b}}){t}_{\mathrm{g}}))-{\psi }_{\mathrm{b}}]/ {\sigma \psi }_{\mathrm{b}}.$$
(11)

Full details on these strategies can be found in Alvarado and Bradford (2002) and Bradford (2002). True potato seeds (Solanum tuberosum L.) and watermelon (Citrullus vulgaris) were among the many crops studied using the HTT model to determine the germination response to and T. Other crops studied included lemon balm (Melisa officinalis L.), sesame (Sesamum indicum L.) and zucchini (Cucurbita pepo L.) as well as carrots (Daucus carota) and onions.

Quinoa cultivation occurs mainly in marginal lands where such limitations such as temperature stress and water stress are inevitable. Also, there is very little research that quantifies seed germination of quinoa (Chenopodium quinoa Willd.) in the literature. On the other hand, considering the efficiency of HT and HTT models for predicting germination time and percentage at different temperatures and water potentials in the field and laboratory conditions, this study was carried out aiming to quantify the germination response of two cultivars of quinoa (Sajama and Titicaca) to Ψ and T using HT and HTT models.

Materials and methods

The seeds of two quinoa varieties (Sajama and Titicaca) were used in the present study. At 25 °C, 50 seeds of each cultivar were tested for viability in three replications. The viability of Sajama and Titicaca was 98.6 and 97.5%, respectively. The experiment was conducted in 2020 using a germinator at Shahrekord University, Shahrekord, Iran. Eight constant Ts (5–40 °C by 5 °C intervals) were used to incubate the seeds of each cultivar at each following water potentials (0, − 0.3, − 0.6, and − 0.9 MPa). However, the seeds did not germinate in 5 and 40 °C, and the data of those constant Ts were not statistically analyzed. Polyethylene glycol (PEG 6000, Merck, Germany) was used for the water potential treatments, as described by Michel and Kaufmann (1973). Additionally, distilled water was used as a control (0 MPa). For each treatment, the potential of the desired solution determined via a vapor pressure osmometer (Model 5100C: Wescor Inc., Logan, UT, USA).

In a 9 cm diameter Petri dish with 10 ml of distilled water or PEG solutions, 50 seeds were germinated between two Whatman filter paper layers Cat. No. 1001-150. There also were three replications for each T at each water potential treatment. The Petri dishes were then randomly placed in the germinator for each T in darkness except at the time of recording germination. The quantity of germinated seeds was counted multiple times per day (depending on the T and). The radicle length of two mm was the germination criteria (Soltani et al. 2006; Karimzadeh Soureshjani et al. 2019). The seed germination record was completed when no new seeds germinated for three days in a row (Bakhshandeh et al. 2015, 2017).

To estimate cardinal Ts of both cultivars, a segmented model (Eq. 12) was fitted to germination data at Ψ = 0 MPa (Karimzadeh Soureshjani et al. 2019). Based on the results of this fit, the \({T}_{\mathrm{b}}\), \({T}_{\mathrm{o}}\) and \({T}_{\mathrm{c}}\) in Sajama were estimated at 6.9, 21.9, and 34.9 °C, respectively, and in Titicaca were estimated 8.0, 21.8, and 33.6 °C, respectively. Hence to fit the HT and HTT models, Ts below 22 °C were considered as sub-optimal Ts, and Ts above 22 °C were considered supra-optimal Ts:

$$ \begin{aligned} f\left( T \right) & = \frac{{T - T_{{\text{b}}} }}{{T_{{\text{o}}} - T_{{\text{b}}} }}\quad {\text{If }}T_{{\text{b}}} { } < { }T{ } < T_{{\text{o}}} \\ f\left( T \right) & = 1 - \left( {\frac{{T - T_{{\text{o}}} }}{{T_{{\text{c}}} - T_{{\text{o}}} }}} \right)\quad {\text{If }}T_{{\text{o}}} { } \le {\text{ T }} < { }T_{{\text{c}}} \\ f\left( T \right) & = 0 \quad {\text{If }}T{ } \le { }T_{{\text{b}}} { }\;{\text{or}}\;{ }T{ } \ge T_{{\text{c}}} , \\ \end{aligned} $$
(12)

f(T) donates the temperature function as a reduction factor, which ranges from 0 at the base and ceiling T to one at optimum T. T; temperature, \({T}_{\mathrm{b}}\); base T, \({T}_{\mathrm{o}}\); optimum T, \({T}_{\mathrm{c}}\); ceiling T.

The NLIN procedure in SAS environment software was used to fit the HT and HTT models to germination data (SAS 9.4). The Coefficient of determination (R2) and root mean square error (RMSE) were used to compare the parameters estimated by the two models and evaluate the model's goodness of fit using R2 and RMSE, respectively. Analysis of variance was also performed in SAS 9.4 environment, using GLM procedure:

$${R}^{2}=1-\frac{\mathrm{SSR}}{\mathrm{SST}}.$$
(13)

In which, SSR is the sum of squares of residuals and SST shows the total sum of squares:

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}{({O}_{i}-{S}_{i})}^{2}}.$$
(14)

In which, n shows number of samples, Oi donates the observed value, and Si represents the predicted values.

Results

The results showed that the T, Ψ, cultivar, and interaction of T* Ψ significantly affected the percentage of maximum germination (MGP) (Fig. 1). Mean MGP averaged over all germination conditions was significantly higher in Titicaca, so that, mean MGP of Titicaca and Sajama was 56 and 54.5%, respectively. The quinoa cultivar response to T depended on Ψ. In each T level, the highest MGP was observed in Ψ = 0 MPa and, decreasing of Ψ caused a significant reduction of MGP (Fig. 1). The results also showed that the highest MGP was recorded in 20 and 25 °C for all levels of Ψ (Fig. 1). In other words, with the T exceeding 25 °C and the T decreasing from 20 °C, the MGP reduced in both cultivars. It was also observed that in all T levels, the MGP of both cultivars did not reach 50% in Ψ = − 0.9 MPa, and it did not exceed 60% in Ψ = − 0.6 MPa.

Fig. 1
figure 1

Analysis of variance and maximum germination percentage (MGP) of two quinoa cultivars affected by different temperatures (T) and water potentials (Ψ) (ns, ** and ***; non-significance difference, Significant difference at P ≤ 0.01 and Significant difference at P ≤ 0.001, respectively)

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The HT and HTT models were fitted to germination data of both quinoa cultivars to quantify quinoa germination, and the parameters were estimated (Tables 1, 2). Results showed that the HT model had high accuracy in predicting germination results so that except for Sajama in 10 °C (R2 = 0.79), in the other cases, it had an R2 greater than 0.90 (Table 1). The lowest \({\theta }_{\mathrm{H}}\) was observed in 20 and 25 °C Ts and T greater than 25 °C or lower than 20 °C increased \({\theta }_{\mathrm{H}}\). In sub- and supra-optimal Ts \({\sigma \psi }_{\mathrm{b}}\) was relatively constant, but ψb increased by changing the T from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{b}}\) and \({T}_{\mathrm{c}}\) (Table 1).

Table 1 Hydrotime model parameters, characterizing germination of two quinoa cultivar seeds at four water potentials (0, − 0.3, − 0.6 and − 0.9 MPa) at sub- (10, 15 and 20 °C) and supra-optimal (25, 30 and 35 °C) temperatures
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Table 2 Hydrothermal time model parameters, characterizing germination of two quinoa cultivar seeds at four water potentials (0, − 0.3, − 0.6 and − 0.9 MPa) at sub- (10, 15 and 20 °C) and supra-optimal (25, 30 and 35 °C) temperatures
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The use of the HTT model makes it possible to estimate the \({T}_{\mathrm{b}}\) and \({T}_{\mathrm{o}}\) in different Ψ in addition to the parameters of the HT model. Using this model, the \({T}_{\mathrm{b}}\) was estimated by 8.28 and 8.39 °C for Sajama and Titicaca, respectively. The \({T}_{\mathrm{o}}\) also was estimated by 26.96 and 27.21 °C for Sajama and Titicaca, respectively. Estimated parameters were not different for the two cultivars, and the \({\theta }_{\mathrm{HTT}}\) was also lower in sub-optimal Ts (Table 2). The HTT model had acceptable accuracy for both cultivars at sub-optimal Ts (R2 greater than 0.90), while at high Ts, its accuracy was reduced (R2 lower than 0.80).

To compare these two models, the predicted germination rate (for the 50% of germination, GR50, using parameters presented in Tables 1 and 2) of both cultivars were plotted versus observed germination rate (Fig. 2). It was observed that at the highest germination rate (\({T}_{\mathrm{o}}\)), the predicted amounts of GR50 were slightly higher than observed GR50. However, the HT model had a higher R2 and a lower RMSE than the HTT model; therefore, it is more accurate in predicting the germination rate.

Fig. 2
figure 2

Predicted GR50 (predicted by HT and HTT models) vs observed GR50 of two quinoa cultivars affected by different temperatures (T) and water potentials (Ψ)

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The predicted (using the HT model) and observed cumulative germination percentage of Titicaca and Sajama are shown in Figs. 3 and 4, respectively, for sub-optimal Ts using parameters presented in Table 1. The germination of both cultivars’ seeds at sub-optimal Ts was increased with T increasing (Figs. 3, 4a–c). Therefore, at each T, MGP and germination rates were reduced by reduction of Ψ. In other words, MGP and germination rate was increased caused by Ψ increment at each sub-optimal T. The normal distribution of ψb was plotted at each T using Table 1 parameters. The increment of T from \({T}_{\mathrm{b}}\) to \({T}_{\mathrm{o}}\) resulted in ψb increases at sub-optimal Ts (Figs. 3, 4d–f). In other words, the seeds can germinate at more negative ψb at \({T}_{\mathrm{o}}\), and the time to germination reduced at this condition for all Ψ.

Fig. 3
figure 3

Germination time courses of Sajama seeds at sub-optimal temperatures. a–c Germination time courses at 10, 15 and 20 °C and 0, − 0·3, − 0.6 and − 0·9 MPa. The symbols are the actual data, and the lines are the time courses predicted by the hydrotime model using the values shown in Table 1. d–f Normal distributions showing the relative frequencies of Ψb values at each temperature. The median or Ψb values are shown (see Table 1)

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Fig. 4
figure 4

Germination time courses of Titicaca seeds at sub-optimal temperatures. a–c Germination time courses at 10, 15 and 20 °C and 0, − 0·3, − 0.6 and − 0·9 MPa. The symbols are the actual data, and the lines are the time courses predicted by the hydrotime model using the values shown in Table 1. d–f Normal distributions showing the relative frequencies of Ψb values at each temperature. The median or Ψb values are shown (see Table 1)

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At supra-optimal Ts also the MGP and germination rates were decreased in both cultivars due to T increment from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{c}}\) (Figs. 5, 6a–c). The reduction of Ψ at each supra-optimal T also reduced MGP and germination rates of both cultivars. The results also showed that the increase of T from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{c}}\) caused the seeds to need a more positive ψb for germination (Figs. 5, 6d–f). It seems as the T rises above the \({T}_{\mathrm{o}}\), the seeds cannot absorb water at more negative Ψ. In this condition, the time to germination was increased for all Ψ.

Fig. 5
figure 5

Germination time courses of Sajama seeds at supra-optimal temperatures. (a–c) Germination time courses at 25, 30 and 35 °C and 0, − 0·3, − 0.6 and − 0·9 MPa. The symbols are the actual data, and the lines are the time courses predicted by the hydrotime model using the values shown in Table 1. d–f Normal distributions showing the relative frequencies of Ψb values at each temperature. The median or Ψb values are shown (see Table 1)

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Fig. 6
figure 6

Germination time courses of Titicaca seeds at supra-optimal temperatures. a–c Germination time courses at 25, 30 and 35 °C and 0, − 0.3, − 0.6 and − 0.9 MPa. The symbols are the actual data, and the lines are the time courses predicted by the hydrotime model using the values shown in Table 1. d–f Normal distributions showing the relative frequencies of Ψb values at each temperature. The median or Ψb values are shown (see Table 1)

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Discussion

The current experiment results showed that T, Ψ, cultivar, and their interaction had a significant effect on MGP. In general, reduction of Ψ and change of T from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{b}}\) and \({T}_{\mathrm{c}}\) reduced MGP of both cultivars. Other plants, such watermelon (Bakhshandeh et al. 2015), sesame (Bakhshandeh et al. 2013), and potato (Alvarado and Bradford 2002) have shown reduced germination percentages and rates due to a decrease in Ψ. In the present study, not only MGP was reduced by T increment from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{c}}\) or T reduction from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{b}}\) but also the water potential at which growth is initiated is increased.

Since the thermal time model cannot predict the germination reduction (germination percentage and rate) at supra-optimal Ts (Alvarado and Bradford 2002; Rowse and Finch‐Savage 2003), the uses of HT and HTT models can compensate for this defect. These two models explain how the difference of Ψ in inside the seed and the surrounding environment controls germination of a seed batch at a certain T (Gummerson 1986; Alvarado and Bradford 2002). Although, shifts in the \({\psi }_{\mathrm{b}}\) thresholds and shifts in response to multiple factors, including dormancy level and temperature that are more likely physiologically controlled by the seeds, have effects on germination reduction at supra-optimal Ts. The results of this study showed that the lowest estimation of \({\theta }_{\mathrm{H}}\) was observed at \({T}_{\mathrm{o}}\), and it increased with T increment over \({T}_{\mathrm{o}}\) or T reduction from \({T}_{\mathrm{o}}\). The increase in \({\theta }_{\mathrm{H}}\) due to T reduction (from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{b}}\)) was reported in many plants such as tomato (Dahal and Bradford 1994), potato (Alvarado and Bradford 2002), winterfat (Wang et al. 2005). As a result, \({\theta }_{\mathrm{H}}\) can be utilized as a measure of seed vigor and physiological quality in seed lot (Dahal and Bradford 1990). The amount of \({\theta }_{\mathrm{H}}\) in large seeds was found to be lower than that in small seeds, especially at lower Ts (Wang et al. 2005).

It has long been recognized that germination of seeds occurs in the range of cardinal Ts, including minimum (\({T}_{\mathrm{b}}\)), optimum (\({T}_{\mathrm{o}}\)), and maximum (\({T}_{\mathrm{c}}\)) (Bewley and Black 1994; Alvarado and Bradford 2002). The \({T}_{\mathrm{b}}\) of quinoa was estimated at 8.28 °C for Sajama and 8.39 °C for Titicaca in the current work. \({T}_{\mathrm{o}}\) also was 26.96 and 27.21 °C for Sajama and Titicaca, respectively. González et al. (2017) used a binomial model to estimate cardinal Ts of quinoa. They observed that the \({T}_{\mathrm{o}}\) of quinoa was lower than 33 °C. Mamedi et al. (2017) also studied the effect of different Ts on quinoa's seed germination. They reported that \({T}_{\mathrm{o}}\) ranged between 18 to 36 °C. As can be seen, the range of T data estimated in these studies is extensive. However, in the present study, the cardinal Ts (\({T}_{\mathrm{b}}\) and \({T}_{\mathrm{o}}\)) have been accurately estimated using the HTT model. As Alvarado and Bradford (2002) reported, if thermal time accumulation stopped at \({T}_{\mathrm{o}}\), as in the present study (Fig. 7), the modified HTT model can be used in which (\({T}_{\mathrm{o}}\) − \({T}_{\mathrm{b}}\)) is placed instead of (T − \({T}_{\mathrm{b}}\)) in Eq. 10. Using the modified HTT resulted in more accurate parameter estimation in both genotypes so that \({\theta }_{\mathrm{HTT}}\) in Sajama and Titicaca were estimated by 496.8 and 484.7, respectively. \({T}_{\mathrm{o}}\) was also estimated 27.46 in Sajama and 27.73 in Titicaca. \({\sigma \psi }_{\mathrm{b}}\) was 0.53 and 0.52 in Sajama and Titicaca, respectively, while \({K}_{\mathrm{T}}\) was equal (0.07) in both genotypes. Also, R2 in Sajama and Titicaca were estimated by 0.84 and 0.83, respectively.

Fig. 7
figure 7

Cumulative germination (%) vs. normalized thermal time in Sajama and Titicaca at sub- and supra optimal temperatures. A normalization factor [\(1-(\frac{\psi }{{\psi }_{\mathrm{b}\left(g\right)}})\)]\({t}_{\mathrm{g}}\), was used for the normalization of germination time courses across a range of Ψ

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The variations in mean Ψb values that occurred as a result of a reduction in T and an increase in T were observed in the current investigation. The seeds could not absorb water at a higher negative Ψ when they were farther away from \({T}_{\mathrm{o}}\). Seeds just can germinate at more positive Ψ in this condition. At low temperatures, prior studies found that Ψb remained constant. The germination rate and germination percentage were found to be reduced at supra-optimal Ts because of a linear increase in b, which was found to increase with T (Dahal and Bradford 1994; Alvarado and Bradford 2002; Rowse and Finch‐Savage 2003; Atashi et al. 2014). However, our results showed that not only Ψb was less negative with T increment at supra-optimal Ts, but also it was less negative with T reduction at sub-optimal Ts. In other words, increasing the Ψb in addition to supra-optimal Ts, at sub-optimal Ts, is also the reason for the decrease in germination rate and percentage. Similar to our result, the reduction of T from \({T}_{\mathrm{o}}\) to \({T}_{\mathrm{b}}\) increased Ψb of watermelon (Bakhshandeh et al. 2015) and winterfat (Wang et al. 2005). The current study results showed that for each degree Celsius decrease of T at sub-optimal Ts, the Ψb increased by 0.032 and 0.034 MPa in Sajama and Titicaca, respectively (Fig. 8). Each degree Celsius increase of T at supra-optimal Ts also increased Ψb by 0.021 MPa in Sajama and 0.020 MPa in Titicaca (Fig. 8).

Fig. 8
figure 8

Relationship between temperature (T) and base water potential (Ψb) of two quinoa cultivars

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Conclusion

In conclusion, T and Ψ affected MFP and germination rates of both quinoa cultivars. To quantify quinoa's germination response to T and Ψ, we used HT and HTT. The results showed that these models could accurately predict the germination (percentage and rate) changes under varying T and Ψ. Using these models can also help us to estimate cardinal Ts and threshold (Ψb) water potential for seed germination of quinoa. The results of this experiment provide data for future simulating models of quinoa growth and development.

Author contribution statement

HKS: designed and performed experiment, analyzed data and wrote the paper. MB: performed experiment and collected the data. MRT: provided essential materials and equipment. AGD: performed experiment and collected the data. HKS wrote the first draft of the manuscript and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.