Predicting germination response to temperature. II. Three-dimensional regression, statistical gridding and iterative-probit optimization using measured and interpolated-subpopulation data

BACKGROUND AND AIMS: Most current thermal-germination models are parameterized with subpopulation-specific rate data, interpolated from cumulative-germination-response curves. The purpose of this study was to evaluate the relative accuracy of three-dimensional models for predicting cumulative germination response to temperature. Three-dimensional models are relatively more efficient to implement than two-dimensional models and can be parameterized directly with measured data. METHODS: Seeds of four rangeland grass species were germinated over the constant-temperature range of 3 to 38 degrees C and monitored for subpopulation variability in germination-rate response. Models for estimating subpopulation germination rate were generated as a function of temperature using three-dimensional regression, statistical gridding and iterative-probit optimization using both measured and interpolated-subpopulation data as model inputs. KEY RESULTS: Statistical gridding is more accurate than three-dimensional regression and iterative-probit optimization for modelling germination rate and germination time as a function of temperature and subpopulation. Optimization of the iterative-probit model lowers base-temperature estimates, relative to two-dimensional cardinal-temperature models, and results in an inability to resolve optimal-temperature coefficients as a function of subpopulation. Residual model error for the three-dimensional model was extremely high when parameterized with measured-subpopulation data. Use of measured data for model evaluation provided a more realistic estimate of predictive error than did evaluation of the larger set of interpolated-subpopulation data. CONCLUSIONS: Statistical-gridding techniques may provide a relatively efficient method for estimating germination response in situations where the primary objective is to estimate germination time. This methodology allows for direct use of germination data for model parameterization and automates the significant computational requirements of a two-dimensional piece-wise-linear model, previously shown to produce the most accurate estimates of germination time.     

Predicting germination response to temperature. I. Cardinal-temperature models and subpopulation-specific regression

BACKGROUND AND AIMS: The purpose of this study was to compare the relative accuracy of different thermal-germination models in predicting germination-time under constant-temperature conditions. Of specific interest was the assessment of shape assumptions associated with the cardinal-temperature germination model and probit distribution often used to distribute thermal coefficients among seed subpopulations. METHODS: The seeds of four rangeland grass species were germinated over the constant-temperature range of 3-38 degrees C and monitored for subpopulation variability in germination-rate response. Subpopulation-specific germination rate was estimated as a function of temperature and residual model error for three variations of the cardinal-temperature model, non-linear regression and piece-wise linear regression. The data were used to test relative model fit under alternative assumptions regarding model shape. KEY RESULTS: In general, optimal model fit was obtained by limiting model-shape assumptions. All models were relatively accurate in the sub-optimal temperature range except in the 3 degrees C treatment where predicted germination times were in error by as much as 70 d for the cardinal-temperature models. CONCLUSIONS: Germination model selection should be driven by research objectives. Cardinal-temperature models yield coefficients that can be directly compared for purposes of screening germplasm. Other model formulations, however, may be more accurate in predicting germination-time, especially at low temperatures where small errors in predicted rate can result in relatively large errors in germination time.     

Dynamic variability in thermal-germination response of squirreltail (Elymus elymoides and Elymus multisetus)

Bottlebrush squirreltail (Elymus elymoides) and big squirreltail (Elymus multisetus) have been identified as high-priority species for restoration and rehabilitation of millions of acres of rangeland in the western United States that have been degraded by wildfire and introduced annual weeds. In this study, squirreltail accessions from Idaho, Colorado, Utah, Arizona and New Mexico were grown in a nursery environment to produce seeds in two different years for germination evaluation at 11 constant temperatures. A statistical-gridding model was used to predict cumulative germination rate of each seedlot for eight simulated planting dates between 1 January and 28 May over a 38-year seedbed-microclimatic simulation. Predicted germination response under simulated conditions of field-variable temperatures yielded a broader ecological basis for the relative ranking of thermal response than was obtained from single-value germination indices derived from either constant-temperature experiments, or from analysis of thermal-time coefficients.     

Predicting germination response of four cool-season range grasses to field-variable temperature regimes

Germination of non-dormant seeds under variable-temperature conditions can be predicted from constant-temperature germination data if it is assumed that instantaneous germination rate is independent of thermal history. Thermal-response models of this type have not been validated under simulated field-variable temperature conditions that vary in diurnal pattern, diurnal range and longer-term trends in mean–daily temperature. The purpose of this experiment was to evaluate germination response of thickspike wheatgrass (Elymus lanceolatus), bluebunch wheatgrass (Pseudoroegneria spicata), Sandberg bluegrass (Poa sandbergii) and bottlebrush squirreltail (Elymus elymoides) under both constant and field-variable temperature regimes in the laboratory. It was hypothesized that the thermal history assumption was valid and that constant-temperature data could be used to accurately estimate field-variable temperature response. Seeds were germinated at seven constant temperatures between 5 and 35°C, and under 18 variable-temperature regimes simulating six planting dates at three field sites. Predictions of germination time under variable-temperature conditions were accurate to within a fraction of 1 day up to a cumulative germination percentage of 70% for thickspike wheatgrass, 60% for bluebunch wheatgrass, 55% for Sandberg bluegrass and 70% for bottlebrush squirreltail. It was concluded that, for the variable-temperature regimes tested in this experiment, the thermal-history assumption was valid for earlier-germinating subpopulations.     

A mathematical model that uses Gaussian distribution to analyze the germination of Manfreda brachystachya (Agavaceae) in a thermogradient

Germination of nondormant seeds of Manfreda brachystachya (Agavaceae) was analyzed at temperatures ranging from 11–35°C. Maximum germination (95%) occurred at 25°C. An exponential sigmoid relationship was found between time and cumulative germination. Germination rate for every subpopulation (10–90% germination) was estimated by means of a normal distribution analysis. The kurtosis indicated die amplitude of the range of temperatures where the highest germination rates were concentrated, and the skew indicated sharply inhibitory temperatures in the range of temperatures used. Based on analysis of the normal distribution models for each subpopulation, we calculated a theoretical function which described germination rate over the temperature range considered: F(T,χ) = A × exp[−B(C−1)²], where A is the function that describes germination rate for each subpopulation (characterized by the percentage [χ] at optimal temperature); B is a shape parameter, 1/(σG²); and C is the ratio between each germination temperature (T) and the optimal germination temperature. The Gaussian curves were used to calculate thermal time, and base and ceiling temperatures. Germination thermal time ranged from 1 333 to 2 373°C h, and base and ceiling temperatures were 10.44 ± 0.7°C and 39.54 ± 0.7°C, respectively. There was a linear relationship between thermal time and cumulative percentage of germination of the subpopulations. Based on fitted curves for each subpopulation, the use of a general model for all the subpopulations has been proven: F8 = A × exp[−5.9437(C−1)²], where changes in the curves for each subpopulation depended on temperature only.     

A mathematical model that uses Gaussian distribution to analyze the germination of Manfreda brachystachya (Agavaceae) in a thermogradient

Germination of nondormant seeds of Manfreda brachystachya (Agavaceae) was analyzed at temperatures ranging from 11–35°C. Maximum germination (95%) occurred at 25°C. An exponential sigmoid relationship was found between time and cumulative germination. Germination rate for every subpopulation (10–90% germination) was estimated by means of a normal distribution analysis. The kurtosis indicated die amplitude of the range of temperatures where the highest germination rates were concentrated, and the skew indicated sharply inhibitory temperatures in the range of temperatures used. Based on analysis of the normal distribution models for each subpopulation, we calculated a theoretical function which described germination rate over the temperature range considered: F(T,x) = A × exp[-B(C−1)²], where A is the function that describes germination rate for each subpopulation (characterized by the percentage [x] at optimal temperature); B is a shape parameter, 1/(σ²); and C is the ratio between each germination temperature (T) and the optimal germination temperature. The Gaussian curves were used to calculate thermal time, and base and ceiling temperatures. Germination thermal time ranged from 1333 to 2373°C h, and base and ceiling temperatures were 10.44 ± 0.7°C and 39.54 ± 0.7°C, respectively. There was a linear relationship between thermal time and cumulative percentage of germination of the subpopulations. Based on fitted curves for each subpopulation, the use of a general model for all the subpopulations has been proven: F8 = A × exp[−5.9437(C−1)²], where changes in the curves for each subpopulation depended on temperature only.     

野豌豆属4种植物种子萌发的积温模型分析

以青藏高原野豌豆属窄叶野豌豆(Vicia angustifolia)、山野豌豆(V. amoena)、歪头菜(V. unijuga) 3种野生植物与一种当地栽培植物救荒野豌豆(箭筈豌豆) (V. sativa) ‘兰箭3号’种子为材料, 在5、10、15、20、25及30 ℃下进行萌发实验, 应用种子萌发的积温模型对上述4种植物萌发对温度的响应特征进行了比较分析。结果表明: 1)基于萌发速率(1/T_g)对种子萌发温度最低温T_b值的估计受萌发率(g)的影响较小; 与此不同, 除‘兰箭3号’种子外, 对萌发最高温T_c值的估计, 受到g的显著影响。 这表明种群内所有种子个体萌发的T_b值相对恒定, 但T_c值在有些物种中变异较大; 2)基于重复概率单位回归分析估计的种子萌发T_b值与基于萌发速率估计的值较为接近; 而由此方法估计的T_c值则与萌发率为50%时的估计值较为接近; 3)相比多年生豆科植物歪头菜和山野豌豆, 一年生豆科植物箭筈豌豆‘兰箭3号’与窄叶野豌豆具有相对较低的T_b与T_c值; 4)积温模型可准确地预测休眠破除后豆科植物种子在不同温度条件下的萌发进程。     

Estimation of base temperatures for nine weed species

Experiments were conducted to test several methods for estimating low temperature thresholds for seed germination. Temperature responses of nine weeds common in annual agroecosystems were assessed in temperature gradient experiments. Species included summer annuals (Amaranthus albus, A. palmeri, Digitaria sanguinalis, Echinochloa crus-galli, Portulaca oleracea, and Setaria glauca), winter annuals (Hirschfeldia incana and Sonchus oleraceus), and Conyza canadensis, which is classified as a summer or winter annual. The temperature below which development ceases (Tbase) was estimated as the x-intercept of four conventional germination rate indices regressed on temperature, by repeated probit analysis, and by a mathematical approach. An overall Tbase estimate for each species was the average across indices weighted by the reciprocal of the variance associated with the estimate. Germination rates increased linearly with temperature between 15 degrees C and 30 degrees C for all species. Consistent estimates of Tbase were obtained for most species using several indices. The most statistically robust and biologically relevant method was the reciprocal time to median germination, which can also be used to estimate other biologically meaningful parameters. The mean Tbase for summer annuals (13.8 degrees C) was higher than that for winter annuals (8.3 degrees C). The two germination response characteristics, Tbase and slope (rate), influence a species' germination behaviour in the field since the germination inhibiting effects of a high Tbase may be offset by the germination promoting effects of a rapid germination response to temperature. Estimates of Tbase may be incorporated into predictive thermal time models to assist weed control practitioners in making management decisions.     

Modelling the effects of water stress and temperature on germination rate of Orobanche aegyptiaca seeds

Orobanche aegyptiaca seeds were germinated at a range of water potentials and temperatures and the progress of germination within the seed population was modelled. Base water potentials (at which the rate of progress towards germination is zero) varied between individual seeds according to a normal distribution with a mean of -1.96 MPa and standard deviation of 0.33 MPa at 20C. Contrary to the underlying assumption of the hydrothermal time model in the literature, the median base water potential varied significantly with temperature, being c. -2 MPa at 14-23C and increasing at both higher and lower temperatures. Thermal times to germination also varied according to a normal distribution between individual seeds with a mean of 49°Cd and standard deviation of 18°Cd in water. The median thermal time to germination varied with water potential. Again, however, an assumption of the hydrothermal time model was found to be invalid since the base temperature for rate of germination also varied significantly with water potential. The relationship of both base temperature and thermal time to water potential were linear such that germination progress curves in 33 different hydrothermal environments (8-26°C and 0 to -1.2 MPa) could be described according to a new modified thermal time model which accounted for 78% of the variation in the data.Keywords: Water stress, temperature, Orobanche aegyptiaca, germination rate, hydrothermal time models.      

Mathematical description of the seed germination dependency on time and temperature

Abstract. A mathematical model which describes the germination percentage dependency on time and temperature of a seed population was derived from the experimental results with a seed population of Amaranthus patulus Bertol. under sub-optimal temperature conditions (Washitani & Takenaka, 1984). The equation of the model which is a modified thermal time model is where G is germination percentage at a certain time after the start of imbibition ( t ) at a certain temperature ( T ), μ _{T 1} and σ _{T 1} are the mean and standard deviation of lower limit temperature among the seeds belonging to the seed population, and Tb, m , and A are the parameters characterizing the linear relationship between the rate and temperature, namely, Tb is the base temperature, m , the median of the required thermal time and A , a parameter determining the pattern of the variation of the required thermal time within seed population, respectively. The equation yields time courses for germination which are very similar to those observed by experiment.     

New mathematical model to estimate tissue blood perfusion, thermal contact resistance and core temperature

Analytical solutions were developed based on the Green's function method to describe heat transfer in tissue including the effects of blood perfusion. These one-dimensional transient solutions were used with a simple parameter estimation technique and experimental measurements of temperature and heat flux at the surface of simulated tissue. It was demonstrated how such surface measurements can be used during step changes in the surface thermal conditions to estimate the value of three important parameters: blood perfusion (w(b)), thermal contact resistance (R"), and core temperature of the tissue (T(core)). The new models were tested against finite-difference solutions of thermal events on the surface to show the validity of the analytical solution. Simulated data was used to demonstrate the response of the model in predicting optimal parameters from noisy temperature and heat flux measurements. Finally, the analytical model and simple parameter estimation routine were used with actual experimental data from perfusion in phantom tissue. The model was shown to provide a very good match with the data curves. This demonstrated the first time that all three of these important parameters (w(b), R", and T(core)) have simultaneously been estimated from a single set of thermal measurements at the surface of tissue.     

A simple method to predict body temperature of small reptiles from environmental temperature

To study behavioral thermoregulation, it is useful to use thermal sensors and physical models to collect environmental temperatures that are used to predict organism body temperature. Many techniques involve expensive or numerous types of sensors (cast copper models, or temperature, humidity, radiation, and wind speed sensors) to collect the microhabitat data necessary to predict body temperatures. Expense and diversity of requisite sensors can limit sampling resolution and accessibility of these methods. We compare body temperature predictions of small lizards from iButtons, DS18B20 sensors, and simple copper models, in both laboratory and natural conditions. Our aim was to develop an inexpensive yet accurate method for body temperature prediction. Either method was applicable given appropriate parameterization of the heat transfer equation used. The simplest and cheapest method was DS18B20 sensors attached to a small recording computer. There was little if any deficit in precision or accuracy compared to other published methods. We show how the heat transfer equation can be parameterized, and it can also be used to predict body temperature from historically collected data, allowing strong comparisons between current and previous environmental temperatures using the most modern techniques. Our simple method uses very cheap sensors and loggers to extensively sample habitat temperature, improving our understanding of microhabitat structure and thermal variability with respect to small ectotherms. While our method was quite precise, we feel any potential loss in accuracy is offset by the increase in sample resolution, important as it is increasingly apparent that, particularly for small ectotherms, habitat thermal heterogeneity is the strongest influence on transient body temperature.     

Development of a Threshold Model to Predict Germination of Populus tomentosa Seeds after Harvest and Storage under Ambient Condition

Effects of temperature, storage time and their combination on germination of aspen (Populus tomentosa) seeds were investigated. Aspen seeds were germinated at 5 to 30°C at 5°C intervals after storage for a period of time under 28°C and 75% relative humidity. The effect of temperature on aspen seed germination could not be effectively described by the thermal time (TT) model, which underestimated the germination rate at 5°C and poorly predicted the time courses of germination at 10, 20, 25 and 30°C. A modified TT model (MTT) which assumed a two-phased linear relationship between germination rate and temperature was more accurate in predicting the germination rate and percentage and had a higher likelihood of being correct than the TT model. The maximum lifetime threshold (MLT) model accurately described the effect of storage time on seed germination across all the germination temperatures. An aging thermal time (ATT) model combining both the TT and MLT models was developed to describe the effect of both temperature and storage time on seed germination. When the ATT model was applied to germination data across all the temperatures and storage times, it produced a relatively poor fit. Adjusting the ATT model to separately fit germination data at low and high temperatures in the suboptimal range increased the models accuracy for predicting seed germination. Both the MLT and ATT models indicate that germination of aspen seeds have distinct physiological responses to temperature within a suboptimal range.     

Time, Temperature and Germination of Pearl Millet (Pennisetum typhoides S. & H.): II. ALTERNATING TEMPERATURE

Methods of analysing the response of germination to constanttemperature are extended to alternating temperatures. The analysisis illustrated for seeds of pearl millet (Pennisetum typhoidesS. & H.) germinated on a thermal gradient plate in pairsof alternating temperatures ranging from 15?C to 47?C. Alternatingtemperatures had a small but systematic effect on germinationrate such that below 42 ?C, alternations in temperature hadno effect on the maximum fraction of seeds which germinatedin the population, but increases in temperature amplitude fromzero to ?8 ?C caused a small but systematic increase in therate of germination.     

Germination responses of a seed population of Taraxacum officinale Weber to constant temperatures including the supra-optimal range

Abstract The germination responses of a nondormant fraction of a seed population of Taraxacum officinale Weber at constant temperatures in the range 7–34°C were analysed through a time-course study. Maximal percentage germination (approximately 90%) was attained at temperatures 10–18°C, where simple linear relationships were observed between the temperature and the germination rates, i.e. the reciprocals of the time taken to germinate by subpopulations with 20–80% germination. There was a variation in the required ‘thermal times’ (θ) which characterized the linear relationships, the distribution of which could be approximated for the seed population by the following distribution function: where m is the median of the distribution, and A is a shape parameter characterizing the pattern of the distribution. Final percentage germination decreased with increasing temperature from 20 to 32°C, where the final percentage germination vs. temperature plotted on a normal probability scale yielded a straight line, indicating the normality of the distribution of the upper limit temperature in the seed population. The estimated mean and standard deviation were 27.25 ± 3.75°C. The rate of germination for the subpopulation with 20–80% germination also decreased with increases in the temperature from 22 to 30°C. If the relationships between the temperature within this range and the rate for the subpopulations with 20–80% germination were approximated by the regression lines, the negative ‘thermal time’ characterizing the yielded linear relationship would have a distribution which could be approximated by the same function with the required thermal time for the relationship of suboptimal range. The parameters m and A for the negative ‘thermal time’ were determined to be 2870 K h and 1.7 × 10^-10 K^-3 h^-3.     

A Mathematical Model to Utilize the Logistic Function in Germination and Seedling Growth

Several models have been proposed to describe germination rates,but most are limited in statistical analysis and biologicalmeaning of indices. Therefore, a mathematical model is proposedto utilize the logistic function. The function was defined asan overall response including time, temperature, and the interactionbetween time and temperature. Cumulative germination percentagesover time were used to develop the model. Germination tests were conducted on indiangrass (Sorghastrumnutans (L.) Nash) strain ‘IG-2C-F1’, at constanttemperatures of 9, 12, 15, 20, 25, and 30 °C. The functionfitted the observed data over six temperatures at r² = 0.99.Time to reach 10% of final germination (Gt₁₀) increased from2.5 d at 30 °C to 44.0 d at 9 °C, and Gt₅₀ (time toreach 50% of final germination) increased from 3.6 d at 30 °Cto 53.8 d at 9 °C. True germination rate (% d^–1) foreach temperature was maximum at Gt₅₀. A linear model of 1/Gt₅₀versus temperature was used to estimate the base temperatureof 8.3 °C for germination. An Arrhenius plot indicated achange occurred between 20 °C and 25 °C for temperatureresponse of germination. Published data on hypocotyl growthof Cucumis melo L. were recalculated using the model. Absolutegrowth rates showed a temperature response similar to the publishedweighted-mean elongation rates. Base temperature for hypocotylgrowth of C. melo was estimated as 8.8 °C. The proposedmodel proved to be useful in calculating and interpreting germinationand growth kinetics. Key words: Indiangrass, Sorghastrum nutans (L.) Nash, Germination rate, Threshold temperature, Arrhenius plot, Growth rate, Cucumis melo L     

Kinetics of dormancy release and the high temperature germination response in Aesculus hippocastanum seeds

The kinetics of primary dormancy loss were investigated in seeds of horse chestnut (Aesculus hippocastanum L.) harvested in four different years. Freshly collected seeds from 1991 held for up to 1 year at temperatures between 2C and 42C exhibited two peaks in germination (radicle growth), representing a low temperature (2-8°C) and a high temperature response (31-36°C). Germination at 36°C generally occurred within 1 month of sowing, but was never fully expressed in the seedlots investigated. At low temperatures (2-8°C), germination started after around 4 months. Generally, very low levels of termination were observed at intermediate temperatures (11-26°C). Stratification at 6°C prior to germination at warmer temperatures increased the proportion of seeds that germinated, and the rate of germination for all seedlots. Within a harvest, germination percentage (on a probit scale) increased linearly with stratification time and this relationship was independent of germination temperature (16-26°C). However, inter-seasonal differences in the increases in germination capacity following chilling were observed, varying from 0.044 to 0.07 probits d^-1 of chilling at 6°C. Increased sensitivity to chilling was associated with warmer temperatures during the period of seed filling. The estimated base temperature for germination, Tb, for newly harvested seeds varied slightly between collection years but was close to 25°C. For all seedlots, Tb decreased by 1°C every 6 d of chilling at 6°C. This systematic reduction in Tb with chilling ultimately facilitated germination at 6°C after dormancy release.     

Patterns of seed after-ripening in Bromus tectorum L

For grass seeds that lose dormancy through after ripening indry storage, the probability of germination following a particularwetting event can be predicted only if the relationship betweenstorage temperature and change in after-ripening status is known.This study examined patterns of seed dormancy loss in Bromustectorum L., quantifying changes in germination percentage,speed, and uniformity through time. Seed collections from threesemi-arid habitats were stored at temperatures from 10–40C. At monthly intervals, subsamples were incubated at 5/15,10/20, 15/25, and 20/30 C. For recently harvested seeds, germinationpercentage, mean germination time, and days between 10% and90% of total germination (D90–D10) ranged from 1–75%,10–24 d, and 10–20 d, respectively. Recently harvestedseeds were generally most dormant, slowest to germinate andleast uniform at high incubation temperatures. In contrast,after ripened seeds for all collections had nearly 100% germination,mean germination times <5 d, and D90–D10 values <5d. Three indices were used to characterize after-ripening ratesfor each seedlot at each incubation temperature. The mean dormancyperiod, the mean rate index, and the mean uniformity index definedthe storage period required for seedlots to become half as dormantas at harvest, to progress half-way to the fastest speed, andto progress half-way to the greatest uniformity, respectively.Seeds required longer storage to germinate uniformly than togerminate completely or quickly, because germination time-coursecurves for incompletely after-ripened seeds were positivelyskewed rather than sigmoidal. Mathematically, the three indiceswere described as negative exponential functions of storagetemperature, which suggests that after-ripening is likely completedin late summer or early autumn regardless of summer conditions. Key words: Seed dormancy, germination timing     

Case study of skin temperature and thermal perception in a hot outdoor environment

Focusing on the understanding and the estimation of the biometeorological conditions during summer in outdoor places, a field study was conducted in July 2010 in Athens, Greece over 6 days at three different sites: Syntagma Square, Ermou Street and Flisvos coast. Thermo-physiological measurements of five subjects were carried out from morning to evening for each site, simultaneously with meteorological measurements and subjective assessments of thermal sensation reported by questionnaires. The thermo-physiological variables measured were skin temperature, heat flux and metabolic heat production, while meteorological measurements included air temperature, relative humidity, wind speed, globe temperature, ground surface temperature and global radiation. The possible relation of skin temperature with the meteorological parameters was examined. Theoretical values of mean skin temperature and mean radiant temperature were estimated applying the MENEX model and were compared with the measured values. Two biometeorological indices, thermal sensation (TS) and heat load (HL)—were calculated in order to compare the predicted thermal sensation with the actual thermal vote. The theoretically estimated values of skin temperature were underestimated in relation to the measured values, while the theoretical model of mean radiant temperature was more sensitive to variations of solar radiation compared to the experimental values. TS index underestimated the thermal sensation of the five subjects when their thermal vote was ‘hot’ or ‘very hot’ and overestimated thermal sensation in the case of ‘neutral’. The HL index predicted with greater accuracy thermal sensation tending to overestimate the thermal sensation of the subjects.