Predicting germination response to temperature. III. Model validation under field-variable temperature conditions

BACKGROUND AND AIMS: Two previous papers in this series evaluated model fit of eight thermal-germination models parameterized from constant-temperature germination data. The previous studies determined that model formulations with the fewest shape assumptions provided the best estimates of both germination rate and germination time. The purpose of this latest study was to evaluate the accuracy and efficiency of these same models in predicting germination time and relative seedlot performance under field-variable temperature scenarios. METHODS: The seeds of four rangeland grass species were germinated under 104 variable-temperature treatments simulating six planting dates at three field sites in south-western Idaho. Measured and estimated germination times for all subpopulations were compared for all models, species and temperature treatments. KEY RESULTS: All models showed similar, and relatively high, predictive accuracy for field-temperature simulations except for the iterative-probit-optimization (IPO) model, which exhibited systematic errors as a function of subpopulation. Highest efficiency was obtained with the statistical-gridding (SG) model, which could be directly parameterized by measured subpopulation rate data. Relative seedlot response predicted by thermal time coefficients was somewhat different from that estimated from mean field-variable temperature response as a function of subpopulation. CONCLUSIONS: All germination response models tested performed relatively well in estimating field-variable temperature response. IPO caused systematic errors in predictions of germination time, and may have degraded the physiological relevance of resultant cardinal-temperature parameters. Comparative indices based on expected field performance may be more ecologically relevant than indices derived from a broader range of potential thermal conditions.     

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.     

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.     

Maximizing quantitative accuracy of lung airway lumen and wall measures obtained from X-ray CT imaging.

To objectively quantify airway geometry from three-dimensional computed tomographic (CT) images, an idealized (circular cross section) airway model is parameterized by airway luminal caliber, wall thickness, and tilt angle. Using a two-dimensional CT slice, an initial guess for the airway center, and the full-width-half-maximum principle, we form an estimate of inner and outer airway wall locations. We then fit ellipses to the inner and outer airway walls via a direct least squares fit and use the major and minor axes of the ellipses to estimate the tilt and in-plane rotation angles. Convolving the airway model, initialized with these estimates, with the three-dimensional scanner point-spread function forms the predicted image. The difference between predicted and actual images is minimized by refining the model parameter estimates via a multidimensional, unconstrained, nonlinear minimization routine. When optimization converges, airway model parameters estimate the airway inner and outer radii and tilt angle. Results using a Plexiglas phantom show that tilt angle is estimated to within +/-4 degrees and both inner and outer radii to within one-half pixel when a "standard" CT reconstruction kernel is used. By opening up the ability to measure airways that are not oriented perpendicular to the scanning plane, this method allows evaluation of a greater sampling of airways in a two-dimensional CT slice than previously possible. In addition, by combining the tilt-angle compensation with the deconvolution method, we provide significant improvement over the previous full-width-half-maximum method for assessing location of the luminal edge but not the outer edge of the airway wall.     

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.     

Bayesian Inference of Initial Models in Cryo-Electron Microscopy Using Pseudo-atoms

Single-particle cryo-electron microscopy is widely used to study the structure of macromolecular assemblies. Tens of thousands of noisy two-dimensional images of the macromolecular assembly viewed from different directions are used to infer its three-dimensional structure. The first step is to estimate a low-resolution initial model and initial image orientations. This is a challenging global optimization problem with many unknowns, including an unknown orientation for each two-dimensional image. Obtaining a good initial model is crucial for the success of the subsequent refinement step. We introduce a probabilistic algorithm for estimating an initial model. The algorithm is fast, has very few algorithmic parameters, and yields information about the precision of estimated model parameters in addition to the parameters themselves. Our algorithm uses a pseudo-atomic model to represent the low-resolution three-dimensional structure, with isotropic Gaussian components as moveable pseudo-atoms. This leads to a significant reduction in the number of parameters needed to represent the three-dimensional structure, and a simplified way of computing two-dimensional projections. It also contributes to the speed of the algorithm. We combine the estimation of the unknown three-dimensional structure and image orientations in a Bayesian framework. This ensures that there are very few parameters to set, and specifies how to combine different types of prior information about the structure with the given data in a systematic way. To estimate the model parameters we use Markov chain Monte Carlo sampling. The advantage is that instead of just obtaining point estimates of model parameters, we obtain an ensemble of models revealing the precision of the estimated parameters. We demonstrate the algorithm on both simulated and real data.     

An improved model for determining degree-day values from daily temperature data

Although using hourly weather data offers the greatest accuracy for estimating growing degree-day values, daily maximum and minimum temperature data are often used to estimate these values by approximating the diurnal temperature trends. This paper presents a new empirical model for estimating the hourly mean temperature. The model describes the diurnal variation using a sine function from the minimum temperature at sunrise until the maximum temperature is reached, another sine function from the maximum temperature until sunset, and a square-root function from then until sunrise the next morning. The model was developed and calibrated using several years of hourly data obtained from five automated weather stations located in California and representing a wide range of climate conditions. The model was tested against an additional data-set at each location. The temperature model gave good results, the root-mean-square error being less than 2.0 °C for most years and locations. The comparison with published models from the literature showed that the model was superior to the other methods. Hourly temperatures from the model were used to calculate degree-day values. A comparison between degree-day estimates determined from the model and those obtained other selected methods is presented. The results showed that the model had the best accuracy in general regardless of the season. Received: 25 October 2000 / Revised: 2 July 2001 / Accepted: 2 July 2001     

Regression analysis of longitudinal binary data with time-dependent environmental covariates: bias and efficiency

Generalized estimating equations (Liang and Zeger, 1986) is a widely used, moment-based procedure to estimate marginal regression parameters. However, a subtle and often overlooked point is that valid inference requires the mean for the response at time t to be expressed properly as a function of the complete past, present, and future values of any time-varying covariate. For example, with environmental exposures it may be necessary to express the response as a function of multiple lagged values of the covariate series. Despite the fact that multiple lagged covariates may be predictive of outcomes, researchers often focus interest on parameters in a 'cross-sectional' model, where the response is expressed as a function of a single lag in the covariate series. Cross-sectional models yield parameters with simple interpretations and avoid issues of collinearity associated with multiple lagged values of a covariate. Pepe and Anderson (1994), showed that parameter estimates for time-varying covariates may be biased unless the mean, given all past, present, and future covariate values, is equal to the cross-sectional mean or unless independence estimating equations are used. Although working independence avoids potential bias, many authors have shown that a poor choice for the response correlation model can lead to highly inefficient parameter estimates. The purpose of this paper is to study the bias-efficiency trade-off associated with working correlation choices for application with binary response data. We investigate data characteristics or design features (e.g. cluster size, overall response association, functional form of the response association, covariate distribution, and others) that influence the small and large sample characteristics of parameter estimates obtained from several different weighting schemes or equivalently 'working' covariance models. We find that the impact of covariance model choice depends highly on the specific structure of the data features, and that key aspects should be examined before choosing a weighting scheme.     

大气向下长波辐射参数化模型在长白山地区的适用

利用中国科学院长白山森林生态系统定位站的近地面气象观测数据,分析评价了目前被广泛使用的8个晴天与8个云天大气长波辐射参数化模型的模拟性能.结果表明： 晴天时Satterlund模型最适用,其偏差(BIAS)与均方根误差(RMSE)分别是-23.34和28.55 W·m^-2;系数校正后,虽然其参数值变化不大,但其模拟效果有很大提高,BIAS与RMSE分别降低为-6.33和18.08 W·m^-2;云天时Jacobs模型最准确,BIAS和RMSE只有0.38和29.29 W·m^-2.对模型中大气发射率的敏感性分析表明,大气发射率对水汽压的变化最敏感,对温度的变化不敏感.应用优选模型(晴天和云天)得到的模拟值与观测值的日变化趋势基本一致,但在云量发生突变的节点上模拟效果不太理想.
     

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     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

Comparison of Linear,Nonlinear and Semiparametric Mixed‐effects Models for Estimating HIV Dynamic Parameters

The potency of antiretroviral agents in AIDS clinical trials can be assessed on the basis of an early viral response such as viral decay rate or change in viral load (number of copies of HIV RNA) of the plasma. Linear, parametric nonlinear, and semiparametric nonlinear mixed‐effects models have been proposed to estimate viral decay rates in viral dynamic models. However, before applying these models to clinical data, a critical question that remains to be addressed is whether these models produce coherent estimates of viral decay rates, and if not, which model is appropriate and should be used in practice. In this paper, we applied these models to data from an AIDS clinical trial of potent antiviral treatments and found significant incongruity in the estimated rates of reduction in viral load. Simulation studies indicated that reliable estimates of viral decay rate were obtained by using the parametric and semiparametric nonlinear mixed‐effects models. Our analysis also indicated that the decay rates estimated by using linear mixed‐effects models should be interpreted differently from those estimated by using nonlinear mixed‐effects models. The semiparametric nonlinear mixed‐effects model is preferred to other models because arbitrary data truncation is not needed. Based on real data analysis and simulation studies, we provide guidelines for estimating viral decay rates from clinical data. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Predicting and testing functional responses: An example from a tardigrade–nematode system

Numerous studies have empirically measured consumer functional responses or theoretically developed response models, but whether these models can quantitatively predict observed data has hardly been tested. We perform such a test for the terrestrial predator–prey system Macrobiotus richtersi (Tardigrada)–Acrobeloides nanus (Nematoda). For two different size classes of A. nanus, we report a functional response as measured in the laboratory and quantitatively compare it to predictions of three models with different degrees of complexity: (1) the disc equation which does not include satiation effects; (2) the steady-state satiation (SSS) equation which assumes a constant level of predator satiation; and (3) the satiation model which accounts for prey depletion and increasing predator satiation over the course of the experiments. We parameterized these models with data that were measured independently of the functional response experiments. In both prey-size classes, the predictions of the satiation model matched the observations best, and the match came close to that of logistic regressions fitted to the observations. Thus, the parameterized satiation model seems to include the most important determinants of the functional response in our focal system. For understanding functional responses, we need more studies that compare data to independently derived model predictions.     

Modeling the evolution of the human mitochondrial genome.

Mitochondrial DNA data have been used extensively to study evolution and early human origins. These applications require estimates of the rate at which nucleotide substitutions occur in the DNA sequence. We consider the problem of estimating substitution rates in the presence of site-to-site rate variation. A coalescent model is presented that allows for different substitution rates for purines and pyrimidines, as well as more detailed models that allow fast and slow rates within each of the purine and pyrimidine classes. A method for estimating such rates is presented. Even for these simple models of site heterogeneity, there are, typically, insufficient data to obtain reliable estimates of site-specific substitution rates. However, estimates of the average rate across all sites appear to be relatively stable even in the presence of site heterogeneity. Simulations of models with site-to-site variation in mutation rate show that hypervariable sites can produce peaks in the pairwise difference curves that have previously been attributed to population dynamics.     

Reaction kinetic parameters for ion transport from steady-state current-voltage curves.

This study demonstrates possible ways to estimate the rate constants of reaction kinetic models for ion transport from steady-state current-voltage data as measured at various substrate concentrations. This issue is treated theoretically by algebraic reduction and extension of a reaction kinetic four-state model for uniport. Furthermore, an example for application is given; current-voltage data from an open K+ selective channel (Schroeder, J.I., R. Hedrich, and J.M. Fernandez, 1984, Nature (Lond.), 312:361-362) supplemented by some new data have been evaluated. The analysis yields absolute numerical estimates of the 14 rate constants of a six-state model, which is discussed in a wider context.     

Predicting germination in semi-arid wildland seedbeds II. Field validation of wet thermal-time models

Accurate prediction of germination for species used for semi-arid land revegetation would support selection of plant materials for specific climatic conditions and sites. Wet thermal-time models predict germination time by summing progress toward germination subpopulation percentages as a function of temperature across intermittent wet periods or within singular wet periods. Wet periods may be defined by any reasonable seedbed water potential above which seeds are expected to imbibe sufficiently to germinate. These models may be especially applicable to the Artemisia steppe of the western U.S.A. where water availability limits germination in summer and early fall while cool temperatures limit germination in late fall, winter, and spring when soil water is available. To test accuracy of wet thermal-time models we placed seedbags with seeds of five species commonly used in wildland revegetation, as well as two collections of the invasive annual grass, Bromus tectorum L. into Artemisia tridentata Nutt. ssp. wyomingensis Beetle and Young zone seedbeds for 19 field incubation periods over four seasons. Hourly surface (1–3 cm) soil temperatures and soil water potentials were measured near the seedbags. These data were input into thermal-time models which predicted time to germination for each seedbag retrieval date. Binomial data representing agreement (1) or lack of agreement (0) of predicted and actual germination for each retrieval date were analyzed using logistic regression. Thermal summation method, season, water potential threshold, and species most affected accuracy of predictions (P < 0.0002). A model which defined a wet period as ≥−1.5 MPa soil water potential and summed progress toward germination across intermittent wet periods was most accurate in predicting actual germination by a retrieval date. Across all species, this model correctly predicted that germination would occur in seedbags 75–95% of the time over the latewinter to mid-spring seasons, but only 50–71% of the time for the fall-early winter season when time of soil water availability was least. Although the wet thermal-time model overestimated time to germination for some species and seasons, its accuracy should be high enough to evaluate germination potential by mid-spring for different species, sites, and climatic conditions.     

Estimating kinetic constants from single channel data.

The process underlying the opening and closing of ionic channels in biological or artificial lipid membranes can be modeled kinetically as a time-homogeneous Markov chain. The elements of the chain are kinetic states that can be either open or closed. A maximum likelihood procedure is described for estimating the transition rates between these states from single channel data. The method has been implemented for linear kinetic schemes of fewer than six states, and is suitable for nonstationary data in which one or more independent channels are functioning simultaneously. It also provides standard errors for all estimates of rate constants and permits testing of smoothly parameterized subhypotheses of a general model. We have illustrated our approach by analysis of single channel data simulated on a computer and have described a procedure for analysis of experimental data.