On the use of non-linear regression with the logistic equation for changes with time of percentage root length colonized by arbuscular mycorrhizal fungi

For the regression of sigmoid-shaped responses with time t of colonization C of roots by arbuscular mycorrhizal fungi, C=C _p/1+[e ^−k(t−ti) ] is the most useful form of the logistic equation. At the time of inflection t _i the slope is maximal and directly proportional to the product of the colonization plateau C _p and the abruptness k of the curve. Coefficient k has a high value when the curve rises abruptly following and preceding long shallow phases. The logistic equation has a curve that is symmetrical about t _i such that C=C _p/2 at inflection. Although the logistic equation can generate a good fit to many data sets for changes in colonization with time, there are cases that are not sigmoid and the logistic equation does not apply. For sigmoid curves, the lag in the development of colonization is directly related to both t _i and k but not to the plateau and not to the value of the maximum slope. Higher values of k or t _i reflect longer lag. When considered alone, t _i and k do not fully summarize the lag in colonization, and so a numerical method to combine them is presented here which allows lag to be compared between curves. In this method, the lag is evaluated by calculating the time during early colonization when the slope equals half of the value of the maximum slope. In summary, use of the logistic equation for regression of sigmoid curves of colonization with time allows numerical comparison between curves of the lag, the period of steep ascent, and the plateau. The logistic equation does not model directly the fundamental processes at work in the development of the mycorrhizae. Instead, it can be used as described here to gain insight into the colonization process by comparing the dynamics of that colonization for different species under various conditions. Accepted: 20 October 2000     

Evaluation of the reference genes for expression analysis using quantitative real‐time polymerase chain reaction in the green peach aphid,Myzus persicae

The green peach aphid, Myzus persicae Sulzer (Hemiptera, Aphididae), is an important cosmopolitan pest. Real time qRT‐PCR has been used for target gene expression analysis on M. persicae. Using real time qRT‐PCR, the expression levels are normalized on the basis of the reliable reference genes. However, to date, the stability of available reference genes has been insufficient. In this study, we evaluated nine candidate reference genes from M. persicae under diverse experimental conditions. The tested candidate genes were comprehensively ranked based on five alternative methods (RefFinder, geNorm, Normfinder, BestKeeper and the comparative ΔC_t method). 18s, Actin and ribosomal protein L27 (L27) were recommended as the most stable reference genes for M. persicae, whereas ribosomal protein L27 (L27) was found to be the least stable reference genes for abiotic studies (photoperiod, temperature and insecticide susceptibility). Our finding not only sheds light on establishing an accurate and reliable normalization of real time qRT‐PCR data in M. persicae but also lays a solid foundation for further studies of M. persicae involving RNA interference and functional gene research.     

Comparing real-time quantitative polymerase chain reaction analysis methods for precision,linearity, and accuracy of estimating amplification efficiency

New methods are used to compare seven qPCR analysis methods for their performance in estimating the quantification cycle (C_q) and amplification efficiency (E) for a large test data set (94 samples for each of 4 dilutions) from a recent study. Precision and linearity are assessed using chi-square (χ²), which is the minimized quantity in least-squares (LS) fitting, equivalent to the variance in unweighted LS, and commonly used to define statistical efficiency. All methods yield C_qs that vary strongly in precision with the starting concentration N₀, requiring weighted LS for proper calibration fitting of C_q vs log(N₀). Then χ² for cubic calibration fits compares the inherent precision of the C_qs, while increases in χ² for quadratic and linear fits show the significance of nonlinearity. Nonlinearity is further manifested in unphysical estimates of E from the same C_q data, results which also challenge a tenet of all qPCR analysis methods — that E is constant throughout the baseline region. Constant-threshold (C_t) methods underperform the other methods when the data vary considerably in scale, as these data do.     

Description of the delayed microbial growth by an extended logistic equation

Logistic equations are suitable for describing microbial growth. By means of VERHULST'S logistic equation, the adaptation to sigmoid-shaped curves of growth improves with a falling ratio C_xo/C_{x, max} < 0.2, if there is no lag-phase. The known logistic equations do not take into account any lag-phase behaviour, so that noticeable deviations in the model adaptation result in this range. Therefore, an extended logistic equation of rate is proposed by which any occuring lag-time is expressed by a 1st-order lag-term. The corresponding time law allows a very good adaptation of curves of delayed growth behaviour, and changes into VERHULST'S logistic equation for a lag-time t_L = 0. Application is facilitated by instructions for the numerical determination.     

A standard curve based method for relative real time PCR data processing

Background  

Currently real time PCR is the most precise method by which to measure gene expression. The method generates a large amount of raw numerical data and processing may notably influence final results. The data processing is based either on standard curves or on PCR efficiency assessment. At the moment, the PCR efficiency approach is preferred in relative PCR whilst the standard curve is often used for absolute PCR. However, there are no barriers to employ standard curves for relative PCR. This article provides an implementation of the standard curve method and discusses its advantages and limitations in relative real time PCR.     

Multiple source pools for Galápagos plant species richness: a critical analysis of the line of sight connectivity index

An index (C_i*E) combining the number of line‐of‐sight islands (C_i) within a radius i and target island elevation (E) has been proposed as an improved predictive model of plant species richness (S_t) in the Galápagos Archipelago. We examined this index critically and found that several major flaws preclude it from being a useful predictive tool for the archipelago. Although the number of collecting trips to an island was reported over 20 years ago to have substantial predictive value for reported plant species richness in the Galápagos Islands, this relationship was ignored in multiple regression analyses of the index. When we included the number of collecting trips in different multiple regression analyses of the index, C_i*E had less predictive power than collecting trips or ceased to be significant at all. Additionally, the strong significant relationship between elevation and area in the Galápagos Archipelago results in area having a major confounding influence on the C_i*E index. When elevation is removed from the C_i*E index, the predictive power of C_i is far less than area alone. Finally, the data used to construct and correlate the C_i*E index with (S_t) were based only on a subset of the islands and species lists that were incomplete or out of date. Species richness on islands can be related to the interaction of different factors, so development and testing of indices like C_i*E is not inappropriate. However, it is important to examine the interrelationships among the components of these indices thoroughly, and not ignore the effect of factors already known to have high predictive power. We propose several ways in which more meaningful indices of source pool(s) capacity can be constructed.     

Elevated CO2 did not affect the hydrological balance of a mature native Eucalyptus woodland

Elevated atmospheric CO₂ concentration (eC_a) might reduce forest water‐use, due to decreased transpiration, following partial stomatal closure, thus enhancing water‐use efficiency and productivity at low water availability. If evapotranspiration (E_t) is reduced, it may subsequently increase soil water storage (ΔS) or surface runoff (R) and drainage (D_g), although these could be offset or even reversed by changes in vegetation structure, mainly increased leaf area index (L). To understand the effect of eC_a in a water‐limited ecosystem, we tested whether 2 years of eC_a (~40% increase) affected the hydrological partitioning in a mature water‐limited Eucalyptus woodland exposed to Free‐Air CO₂ Enrichment (FACE). This timeframe allowed us to evaluate whether physiological effects of eC_a reduced stand water‐use irrespective of L, which was unaffected by eC_a in this timeframe. We hypothesized that eC_a would reduce tree‐canopy transpiration (E_tree), but excess water from reduced E_tree would be lost via increased soil evaporation and understory transpiration (E_floor) with no increase in ΔS, R or D_g. We computed E_t, ΔS, R and D_g from measurements of sapflow velocity, L, soil water content (θ), understory micrometeorology, throughfall and stemflow. We found that eC_a did not affect E_tree, E_floor, ΔS or θ at any depth (to 4.5 m) over the experimental period. We closed the water balance for dry seasons with no differences in the partitioning to R and D_g between C_a levels. Soil temperature and θ were the main drivers of E_floor while vapour pressure deficit‐controlled E_tree, though eC_a did not significantly affect any of these relationships. Our results suggest that in the short‐term, eC_a does not significantly affect ecosystem water‐use at this site. We conclude that water‐savings under eC_a mediated by either direct effects on plant transpiration or by indirect effects via changes in L or soil moisture availability are unlikely in water‐limited mature eucalypt woodlands.     

Experimental evidence for negative turgor pressure in small leaf cells of Robinia pseudoacacia L versus large cells of Metasequoia glyptostroboides Hu et W.C. Cheng. 2. Höfler diagrams below the volume of zero turgor and the theoretical implication for pressure‐volume curves of living cells

The physiological advantages of negative turgor pressure, P_t, in leaf cells are water saving and homeostasis of reactants. This paper advances methods for detecting the occurrence of negative P_t in leaves. Biomechanical models of pressure‐volume (PV) curves predict that negative P_t does not change the linearity of PV curve plots of inverse balance pressure, P_B, versus relative water loss, but it does predict changes in either the y‐intercept or the x‐intercept of the plots depending on where cell collapse occurs in the P_B domain because of negative P_t. PV curve analysis of Robinia leaves revealed a shift in the x‐intercept (x‐axis is relative water loss) of PV curves, caused by negative P_t of palisade cells. The low x‐intercept of the PV curve was explained by the non‐collapse of palisade cells in Robinia in the P_B domain. Non‐collapse means that P_t smoothly falls from positive to negative values with decreasing cell volume without a dramatic change in slope. The magnitude of negative turgor in non‐collapsing living cells was as low as ?1.3 MPa and the relative volume of the non‐collapsing cell equaled 58% of the total leaf cell volume. This study adds to the growing evidence for negative P_t.     

Sensitivity of canopy transpiration to altered precipitation in an upland oak forest: evidence from a long-term field manipulation study

Climate‐induced changes in regional precipitation could have important implications for the carbon, water, and nutrient cycles of forest ecosystems. However, few studies have examined the response of deciduous forests to increases or decreases in precipitation. Therefore, the throughfall displacement experiment (TDE) was established in 1993 near Oak Ridge, Tennessee to examine the sensitivity of an upland oak (Quercus spp.) forest to ambient, wet (+33%), and dry (?33%) precipitation regimes. Sap flux measurements on co‐occurring tree species were scaled using species‐specific estimates of stand sapwood area to derive daily and seasonal rates of canopy transpiration (E_C) from 2000 to 2003. With the exception of 2003, which was an extremely wet year, daily E_C in the dry plot, and occasionally during extended droughts in the ambient and wet plots, declined as water potential in the upper 0.35 m soil profile approached ‐3.0 MPa. Seasonal patterns of soil water potential and treatment‐specific differences in E_C were dependent on precipitation frequency and intensity. Supplemental precipitation added to the wet plot increased seasonal E_C on average by 9% (range ?1% to 19%), whereas extended periods of drought on the dry plot in 2000, 2001, and 2002 were sufficient to reduce seasonal E_C by 26–30% compared with the ambient plot. There was a strong correlation between seasonal E_C and the water stress integral, a cumulative index of drought severity and duration. A polynomial fitted to these data indicated that reductions in seasonal E_C on the order of 40% were possible given TDE‐imposed reductions in soil water potential. Application of this equation to all years of the TDE (1994–2003) revealed considerable interannual and treatment‐specific variation in canopy transpiration. In general, a 33% removal of throughfall on the dry plot during 1995, 1998, and 2002 resulted in a 23–32% reduction in seasonal E_C compared with the ambient plot. While droughts in deciduous forests are often limited in duration and tend to occur late in the growing season, soil water deficits of the magnitude observed in this study have the potential to impact local and regional forest water budgets.     

A fixed-point algorithm for estimating amplification efficiency from a polymerase chain reaction dilution series

Background

The polymerase chain reaction amplifies and quantifies small amounts of DNA. It is a cyclic process, during each cycle of which each strand of template DNA is copied with probability approaching one: the amount of DNA approximately doubles and this amount can be estimated fluorimetrically each cycle, producing a set of fluorescence values hereafter referred to as the amplification curve. Commonly the biological question of relevance is one of the ratio of DNA concentrations in two samples: a ratio that is deduced by comparing the two amplification curves, usually by way of a plot of fluorescence against cycle number. Central to this analysis is measuring the extent to which one amplification curve is shifted relative to the other, a measurement often accomplished by defining a threshold or quantification cycle, C_q, for each curve: the fractional cycle number at which fluorescence reaches some threshold or at which some other criterion (maximum slope, maximum rate of change of slope) is satisfied.We propose an alternative where position is measured relative to a reference curve; position equates to the cycle shift which maximizes the correlation between the reference and the observed fluorescence sequence. A key parameter of the reference curve is obtained by fixed-point convergence.

Results

We consider the analysis of dilution series constructed for the estimation of qPCR amplification efficiency. The estimate of amplification efficiency is based on the slope of the regression line when the C_q is plotted against the logarithm of dilution. We compare the approach to three commonly used methods for determining C_q; each is applied to publicly accessible calibration data sets, and to ten from our own laboratory. As in the established literature we judge their relative merits both from the standard deviation of the slope of the calibration curve, and from the variance in C_q for replicate fluorescence curves.

Conclusions

The approach does not require modification of experimental protocols, and can be applied retrospectively to existing data. We recommend that it be added to the methodological toolkit with which laboratories interpret their real-time PCR data.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-014-0372-4) contains supplementary material, which is available to authorized users.     

Quantifying uncertainty in estimates of C emissions from above-ground biomass due to historic land-use change to cropping in Australia

Quantifying continental scale carbon emissions from the oxidation of above‐ground plant biomass following land‐use change (LUC) is made difficult by the lack of information on how much biomass was present prior to vegetation clearing and on the timing and location of historical LUC. The considerable spatial variability of vegetation and the uncertainty of this variability leads to difficulties in predicting biomass C density (t_C ha^?1) prior to LUC. The issue of quantifying uncertainties in the estimation of land based sources and sinks of CO₂, and the feasibility of reducing these uncertainties by further sampling, is critical information required by governments world‐wide for public policy development on climate change issues. A quantitative statistical approach is required to calculate confidence intervals (the level of certainty) of estimated cleared above‐ground biomass. In this study, a set of high‐quality observations of steady state above‐ground biomass from relatively undisturbed ecological sites across the Australian continent was combined with vegetation, topographic, climatic and edaphic data sets within a Geographical Information System. A statistical model was developed from the data set of observations to predict potential biomass and the standard error of potential biomass for all 0.05° (approximately 5 × 5 km) land grid cells of the continent. In addition, the spatial autocorrelation of observations and residuals from the statistical model was examined. Finally, total C emissions due to historic LUC to cultivation and cropping were estimated by combining the statistical model with a data set of fractional cropland area per land grid cell, f_Ac (Ramankutty & Foley 1998). Total C emissions from loss of above‐ground biomass due to cropping since European colonization of Australia was estimated to be 757 Mt_C. These estimates are an upper limit because the predicted steady state biomass may be less than the above‐ground biomass immediately prior to LUC because of disturbance. The estimated standard error of total C emissions was calculated from the standard error of predicted biomass, the standard error of f_Ac and the spatial autocorrelation of biomass. However, quantitative estimates of the standard error of f_Ac were unavailable. Thus, two scenarios were developed to examine the effect of error in f_Ac on the error in total C emissions. In the first scenario, in which f_Ac was regarded as accurate (i.e. a coefficient of variation, CV, of f_Ac = 0.0), the 95% confidence interval of the continental C emissions was 379–1135 Mt_C. In the second scenario, a 50% error in estimated cropland area was assumed (a CV of f_Ac = 0.50) and the estimated confidence interval increased to between 350 and 1294 Mt_C. The CV of C emissions for these two scenarios was 25% and 29%. Thus, while accurate maps of land‐use change contribute to decreasing uncertainty in C emissions from LUC, the major source of this uncertainty arises from the prediction accuracy of biomass C density. It is argued that, even with large sample numbers, the high cost of sampling biomass carbon may limit the uncertainty of above‐ground biomass to about a CV of 25%.     

Chronopharmacokinetics of Imipenem in the Rat

There are no studies indicating a possible modification of imipenem pharmacokinetics related to the hour (i.e., circadian time) of its administration. The aim of this study was to evaluate the influence of different times of intramuscular imipenem administration on its disposition in Wistar AF EOPS rats. Four groups of eight animals were given a single intramuscular injection of 140 mg/kg of imipenem either at 10∶00, 16∶00, 22∶00, or 04∶00 h. Blood samples were collected 0.5, 1, 2, 3, 4, 6, and 8 h after drug injection, and the main pharmacokinetic parameters determined were C_max, T_max, elimination half‐life (t_1/2), area under the concentration‐versus‐time curve (AUC), total serum clearance (CL/F), and volume of distribution (V/F). Circadian variation of C_max (49%), T_max (92%), and AUC (19%) was observed leading to variability of imipenem exposure. Clearance and volume of distribution were modified according to the circadian time of drug injection but did not reach statistical significance. The results suggest that varying the time of administration induces intra‐individual variability.     

Effects of locally rare taxa on the precision and sensitivity of RIVPACS bioassessment of freshwaters

1. An overall aim in freshwater bioassessment is to use biological methods, metrics and forms of indices which are precise, in that they give repeatable results between replicate samples, but which are also sensitive to changes in environmental impacts and stresses. Here we studied the effects of excluding taxa with site‐specific River Invertebrate Prediction and Classification System (RIVPACS)‐type model expected probabilities less than (or equal to) a threshold P_t (0.0, 0.1, 0.2,…,0.9) on the value, precision and power to detect biological effects of environmental stress using the observed to expected ratios (O/E) of biotic indices used to assess the ecological status of U.K. river sites. 2. Amongst the 614 high quality GB RIVPACS reference sites, excluding taxa with low expected probabilities of occurrence gave less total variation (i.e. lower SD) in the estimates O/E for number of taxa (O/E_TAXA) and the average score per taxon (O/E_ASPT). 3. A separate analysis of a replicated sampling study of sites from a wide range of physical types and qualities revealed that sampling variances in O/E for reference condition sites decreased as more locally rare taxa were excluded (but only up to P_t = 0.5 for O/E_ASPT). However, for moderately impacted and poor quality sites, estimates of both O/E_TAXA and O/E_ASPT based on all (P_t = 0.0) or most taxa (i.e. P_t ≤ 0.3) had lower sampling variances and were more precise. 4. Within a very large independent set of test sites with a wide range of perceived levels of environmental stress, increasing the threshold P_t led to systematic compression of the realised O/E scale towards unity. Specifically, with increasing threshold, O/E values >1 are on average reduced, while O/E values <1 have a tendency to be higher and closer to unity (with the exception of O/E_ASPT for the most severely stressed sites). 5. Accuracy and statistical power to detect environmental stress (measured by the percentage of stressed sites with O/E below the lower 10‐percentile value for reference sites) was very similar using O/E_TAXA for P_t up to 0.7. Using O/E_ASPT, power to detect overall general stress decreased slowly as P_t was increased; the rate of fall in power was slightly faster when restricted to sites subject to moderate or severe stress from organic inputs. 6. Taxa which are more sensitive to (organic) stresses [i.e. have high Biological Monitoring Working Party (BMWP) scores] tend to be naturally less widespread (i.e. amongst reference sites) and thus were found to have considerably lower average site‐specific expected probabilities; this may explain why the use of higher thresholds P_t can exclude more such sensitive taxa and lead to underestimation of the extent of impacts. 7. The standard U.K. RIVPACS sampling and sample processing procedures aim to identify all taxa within a sample. This may lead to a longer distribution tail of rarer (low probability) taxa than sampling methods based on a fixed count subsample and influence the practical effects of excluding rare taxa with low expected probabilities from bioassessments.     

On the need to incorporate sensitivity to CO2 transfer conductance into the Farquhar–von Caemmerer–Berry leaf photosynthesis model

Virtually all current estimates of the maximum carboxylation rate (V_cmax) of ribulose‐1,5‐bisphosphate carboxylase/oxygenase (Rubisco) and the maximum electron transport rate (J_max) for C₃ species implicitly assume an infinite CO₂ transfer conductance (g_i). And yet, most measurements in perennial plant species or in ageing or stressed leaves show that g_i imposes a significant limitation on photosynthesis. Herein, we demonstrate that many current parameterizations of the photosynthesis model of Farquhar, von Caemmerer & Berry (Planta 149, 78–90, 1980 ) based on the leaf intercellular CO₂ concentration (C_i) are incorrect for leaves where g_i limits photosynthesis. We show how conventional A–C_i curve (net CO₂ assimilation rate of a leaf –A_n– as a function of C_i) fitting methods which rely on a rectangular hyperbola model under the assumption of infinite g_i can significantly underestimate V_cmax for such leaves. Alternative parameterizations of the conventional method based on a single, apparent Michaelis–Menten constant for CO₂ evaluated at C_i[K_m(CO₂)_i] used for all C₃ plants are also not acceptable since the relationship between V_cmax and g_i is not conserved among species. We present an alternative A–C_i curve fitting method that accounts for g_i through a non‐rectangular hyperbola version of the model of Farquhar et al. (1980 ). Simulated and real examples are used to demonstrate how this new approach eliminates the errors of the conventional A–C_i curve fitting method and provides V_cmax estimates that are virtually insensitive to g_i. Finally, we show how the new A–C_i curve fitting method can be used to estimate the value of the kinetic constants of Rubisco in vivo is presented     

The thermodynamic characteristic of four-heme cytochrome c in Rhodopseudomonas viridis reaction centers,as derived from a quantitative analysis of the differential absorption spectra in α-domain

A method of decomposing of the absorption spectrum of four-heme cytochrome of a Rhodopseudomonas viridis reaction center preparation into spectra of individual components was used to estimate the degree of the reduction of hemes as a function of redox potential in the medium. The method enables an evaluation of the shape of redox-titration curves of each heme. The redox-titration curves derived by this approach are approximated well by a Nernst equation with n = 1 and E_m-values of 360 mV, 312 mV, 20 mV and less than −50 mV. For all of the redox species the values of midpoint potential estimates by the above method are in good agreement with those determined earlier using another procedure [Dracheva et al. (1988) Eur. J. Biochem. 171, 253-264]. The accuracy of deconvolution of data is within the experimental errors of the redox potential measurement.     

Urea interactions with protein groups: A volumetric study

We determined the partial molar volumes and adiabatic compressibilities of N‐acetyl amino acid amides, N‐acetyl amino acid methylamides, N‐acetyl amino acids, and short oligoglycines as a function of urea concentration. We analyze these data within the framework of a statistical thermodynamic formalism to determine the association constants for the reaction in which urea binds to the glycyl unit and each of the naturally occurring amino acid side chains replacing two waters of hydration. Our determined association constants, k, range from 0.04 to 0.39M. We derive a general equation that links k with changes in free energy, ΔG_tr, accompanying the transfer of functional groups from water to urea. In this equation, ΔG_tr is the sum of a change in the free energy of cavity formation, ΔΔG_C, and the differential free energy of solute–solvent interactions, ΔΔG_I, in urea and water. The observed range of affinity coefficients, k, corresponds to the values of ΔΔG_I ranging from highly favorable to slightly unfavorable. Taken together, our data support a direct interaction model in which urea denatures a protein by concerted action via favorable interactions with a wide range of protein groups. Our derived equation linking k to ΔG_tr suggests that ΔΔG_I and, hence, the net transfer free energy, ΔG_tr, are both strongly influenced by the concentration of a solute used in the experiment. We emphasize the need to exercise caution when two solutes differing in solubility are compared to determine the ΔG_tr contribution of a particular functional group. © 2010 Wiley Periodicals, Inc. Biopolymers 93: 866–879, 2010.     

Comparison of thermoluminescence characteristics in γ‐ray and C5+ ion beam‐irradiated LiCaAlF6:Ce phosphor

We compare the thermoluminescence (TL) behavior of Ce³⁺ ion‐activated LiCaAlF₆ exposed to γ‐rays and a carbon ion beam. The reported phosphor is synthesized using an in‐house precipitation method with varying concentrations of activator ion and is characterized by X‐ray diffraction (XRD) and TL. Rietveld refinement is performed to study the structural statistics. The TL glow curve consists of a prominent glow peak at 232°C with three shoulders at 115, 159 and 333°C when exposed to γ‐rays from a ⁶⁰Co source. When exposed to a C⁵⁺ ion beam, the TL glow curve consists of five peaks with peak temperatures near 156, 221, 250, 287 and 330°C, and is found to vary slightly with changing fluence. Glow curve convolution deconvolution (GCCD) functions are applied to the TL curves for complete analysis of the glow curve structure and TL traps. The order of kinetics (b), activation energy (E) and frequency factor are determined using Chen's peak shape method and theoretical curves are drawn using GCCD functions. A track interaction model (TIM) is used to explain the sublinearity/saturation at higher fluences. Ion beam parameters are analyzed using Monte‐Carlo simulation‐based SRIM‐2013 code. Copyright © 2016 John Wiley & Sons, Ltd.     

SENSORY THRESHOLDS: ESTIMATION FROM DOSE-RESPONSE CURVES

Three methods of threshold estimation are compared for the detection of 1,8-cineole added to Concord grape juice presented in a semi-ascending paired difference design (Lundahl et al. 1986). Method A tests the significance of obtaining a nonzero response, method B determines the transition point of a 2-phase linear model fitted to the data, method C fits a logistic model to the data and defines threshold as the point of maximum curvature, C_max, of the curve. Method A yields interval estimates of thresholds, the midpoints of which systematically decrease with increasing sample size. Methods B and C yield point estimates of thresholds which do not systematically decrease with increasing sample size, are a feature of the dose-response curve, and are similar in value to each other.