Design and analysis of protein binding experiments

The design and analysis of protein binding experiments for obtaining precise parameter estimates for a one-site and a two-site model treating fu, the fraction unbound as the experimentally determined quantity was investigated. Total drug concentrations were chosen at which the binding isotherm is determined to yield the most information about the parameters under study. The D-optimization information criterion was used to achieve this although other criteria are also discussed. For both the one-site and the two-site models the number of design points was always equal to the number of parameters being estimated. The results arrived at when dealing with constant variance and unconstrained total drug concentration were rather unique in that in most of the cases studied, all the optimal design points were away from the boundary conditions. For constant relative variance and unconstrained total drug concentrations, one of the design points was always placed at the smallest possible value of fu, the fraction unbound. For the one-site model the second point was always given by K(-1) + nP. The optimal designs arrived at lead to lower theoretical coefficients of variation in the parameters than the corresponding conventional ones. Simulated experiments supported these theoretical findings for both the one-site and the two-site models. For the one-site model, results from nonlinear regression were compared with Scatchard analysis and the optimal designs were also optimal in Scatchard space. We also show that using Scatchard analysis with the conventional strategy leads to poorly determined estimates particularly when the number of observations is low.     

Efficient Estimation of Protein Binding Constants

Assays of cyclic AMP typically employ a Scatchard-like data analysis to estimate the total binding protein R_T and the dissociation constant K_d. The linear regression procedure employed is statistically ad hoc because the response variable (bound counts) occurs on both sides of the Scatchard equation. The alternative analysis presented here is derived from the cyclic AMP binding reaction. The resulting data analysis, employing non-linear least squares, results in estimators that on the basis of Monte Carlo studies have less bias and greater precision than the Scatchard method.     

Simultaneous estimation ofV max,Km, and the rate of endogenous substrate production (R) from substrate depletion data

The nonlinear and 3 linearized forms of the integrated Michaelis-Menten equation were evaluated for their ability to provide reliable estimates of uptake kinetic parameters, when the initial substrate concentration (S₀) is not error-free. Of the 3 linearized forms, the one where t/(S₀–S) is regressed against ln(S₀/S)/(S₀–S) gave estimates ofV _max and K_m closest to the true population means of these parameters. Further, this linearization was the least sensitive of the 3 to errors (±1%) in S₀. Our results illustrate the danger of relying on r² values for choosing among the 3 linearized forms of the integrated Michaelis-Menten equation. Nonlinear regression analysis of progress curve data, when S₀ is not free of error, was superior to even the best of the 3 linearized forms. The integrated Michaelis-Menten equation should not be used to estimateV _max and K_m when substrate production occurs concomitant with consumption of added substrate. We propose the use of a new equation for estimation of these parameters along with a parameter describing endogenous substrate production (R) for kinetic studies done with samples from natural habitats, in which the substrate of interest is an intermediate. The application of this new equation was illustrated for both simulated data and previously obtained H₂ depletion data. The only means by whichV _max, K_m, and R may be evaluated from progress curve data using this new equation is via nonlinear regression, since a linearized form of this equation could not be derived. Mathematical components of computer programs written for fitting data to either of the above nonlinear models using nonlinear least squares analysis are presented.     

Weighting functions and parameter resolvability for oxygenation data subject to error in the independent variable.

Parameter resolvability and bias has been investigated for weighted nonlinear regression of data where the independent variable is subject to instrumental uncertainty. The specific example of cooperative oxygenation of hemoglobin was studied, where fractional saturation is determined spectrophotometrically and the oxygen activity is measured with a Clark polarographic electrode. For this system the instrumental uncertainty in the oxygen electrode was measured directly and the influence of the uncertainties on resolution of oxygen binding parameters was determined by Monte Carlo simulations. Four weighting functions were tested for their ability to minimize parameter uncertainty and bias: (1) uniform weighting; (2) "propagated weighting" whereby uncertainties in the independent variable are propagated into and added to uncertainties of the dependent variable; (3) Hill plot transform, or "end weighting"; and (4) maximum likelihood analysis, where deviations between fitting function and data are minimized as weighted horizontal and vertical distance vectors. Results of the Monte Carlo simulations favor the use of either uniform weighting, propagated weighting, or maximum likelihood weighting methods. Use of the Hill transform as a weighting function produced poorer parameter resolvability and inaccurate representation of the data in general. Bias error was negligible for all weighting functions.     

A study of statistical error in isothermal titration calorimetry

In isothermal titration calorimetry (ITC), the two main sources of random (statistical) error are associated with the extraction of the heat q from the measured temperature changes and with the delivery of metered volumes of titrant. The former leads to uncertainty that is approximately constant and the latter to uncertainty that is proportional to q. The role of these errors in the analysis of ITC data by nonlinear least squares is examined for the case of 1:1 binding, M+X right arrow over left arrow MX. The standard errors in the key parameters-the equilibrium constant Ko and the enthalpy DeltaHo-are assessed from the variance-covariance matrix computed for exactly fitting data. Monte Carlo calculations confirm that these "exact" estimates will normally suffice and show further that neglect of weights in the nonlinear fitting can result in significant loss of efficiency. The effects of the titrant volume error are strongly dependent on assumptions about the nature of this error: If it is random in the integral volume instead of the differential volume, correlated least-squares is required for proper analysis, and the parameter standard errors decrease with increasing number of titration steps rather than increase.     

Modeling Taylor dispersion injections: determination of kinetic/affinity interaction constants and diffusion coefficients in label-free biosensing

A new method based on Taylor dispersion has been developed that enables an analyte gradient to be titrated over a ligand-coated surface for kinetic/affinity analysis of interactions from a minimal number of injections. Taylor dispersion injections generate concentration ranges in excess of four orders of magnitude and enable the analyte diffusion coefficient to be reliably estimated as a fitted parameter when fitting binding interaction models. A numerical model based on finite element analysis, Monte Carlo simulations, and statistical profiling were used to compare the Taylor dispersion method with standard fixed concentration injections in terms of parameter correlation, linearity of parameter error space, and global versus local model fitting. A dramatic decrease in parameter correlations was observed for TDi curves relative to curves from standard fixed concentration injections when surface saturation was achieved. In FCI the binding progress is recorded with respect to injection time, whereas in TDi the second time dependency encoded in the analyte gradient increases resolving power. This greatly lowers the dependence of all parameters on each other and on experimental interferences. When model parameters were fitted locally, the performance of TDis remained comparable to global model fitting, whereas fixed concentration binding response curves yielded unreliable parameter estimates.     

Determination of stoichiometric association constants by a non-iterative computational method

Computational methods for the estimation of stoichiometric associationconstants for multiple-ligand binding systems are currentlybased on non-linear least-squares regression analysis. Thesecomputational methods require sophisticated, iterative algorithmsto assure convergence to a solution, as well as initial parameterand error estimates. A simple procedure, called lambda-invariancetesting (LIT), was developed that provides a single-pass (non-iterative)estimation of stoichiometric association constants. The LITmethod was applied to simulated binding data containing Gaussianerror and to real data drawn from the literature. This methodprovided parameter estimates essentially equivalent to thoseobtained by least-squares regression analysis, with no initialparameter or error estimates required. Received on May 26, 1987; accepted on July 27, 1987     

Comparison of filtration and equilibrium dialysis methods for [3H]imipramine binding to human platelets

Receptor binding of imipramine in human platelets was assessed by filtration through glassfiber filters and by equilibrium dialysis. Both methods yield drug-receptor dissociation constants of similar magnitude (10^?9m) to literature values. However, the density of binding sites (B_max) was fivefold lower by filtration (473 ± 92 fmol/mg protein) compared to equilibrium dialysis (2652 ± 765 fmol/mg protein). Dialysis allows direct assessment of free imipramine and avoids drug loss during the separation step of the filtration assay. Additional advantages were found for computer nonlinear regression analysis of untransformed data to eliminate errors owing to linear transformation in the Scatchard analysis and for simultaneous quantitation of nonspecific and total drug binding.     

Comparative least-squares analysis of hemoglobin oxygen equilibrium curves

The oxygen-binding properties of hemoglobin have been studied at 600 microM protein concentration with organic phosphate, and analyzed by a series of different nonlinear least-squares analysis methods to determine whether reports of negligibly small values of the third overall Adair parameter, A3, are consequences of the data or a product of the data analysis. Data from other laboratories were analyzed as well. The single most important factor in creating a measurement that yields a small A3 is the use of equally weighted fitting in the Adair equation, while end-weighted fitting generally yields a larger A3. Endpoint extrapolation is ruled out as a major cause of abnormal A3 values. Monte Carlo simulations of the 600 microM results suggest that, if a small A3 were present, end weighting is at least as sensitive to a small A3 as equal weighting. We conclude that equally weighted fitting of the tetrameric Adair equation is unable to resolve the upper asymptote of the oxygen-binding data, resulting in an unusually small value for A3.     

A comparison of the binding of endothelin to various tissues from spontaneously hypertensive and Wistar-Kyoto rats

The binding affinities for endothelin-1 and endothelin-3 to membrane preparations of various tissues of spontaneously hypertensive rats and normotensive Wistar-Kyoto rats were compared by competition binding of the peptides with [¹²⁵I]endothelin-1. Endothelin-1 binding data obtained using membrane preparations from brain, heart, kidney, liver, lung and spleen of both strains were better fit with a one-site model. The brain tissue demonstrated the highest affinity for endothelin-1 in both strains with the same IC₅₀ of 0.11 nM, while the kidney and lung tissues showed the lowest affinities in both strains with IC₅₀ values ranged between 1.4 and 4.1 nM. Only the kidney tissues of these two strains showed a statistically significant difference in binding affinities for endothelin-1; the IC₅₀ values were 1.4 ± 0.1 nM (mean ± SE, N = 3) and 3.2 ± 0.4 nM (n = 4) for the spontaneously hypertensive and normotensive rats, respectively. Endothelin-3 binding data obtained using membrane preparations from brain, kidney and lung of both strains were also better fit with a one-site model. In contrast, a two-site model was more suitable for analyzing endothelin-3 binding results obtained using membrane preparations from heart, liver and spleen of both strains. Again, only the kidney tissues of the two strains showed a statistically significant difference in binding affinities for endothelin-3. The ratio of IC₅₀ value of the major endothelin-3 binding site to that of endothelin-1 in each tissue varied from approx. 1.5 in brain, kidney and liver to greater than 500 in heart and spleen of both strains. Scatchard analysis of saturation binding data showed that [¹²⁵I]endothelin-1 bound to a single class of binding sites in brain, heart, liver and spleen of both rat strains and in kidney of the spontaneously hypertensive rats. Specific binding to the kidney membrane preparation of the normotensive rats was not saturable at radioligand concentrations up to about 2 nM. These results suggest that the tissues of both strains investigated have different affinities as well as different selectivities for endothelin-1 and endothelin-3. Furthermore, kidney is the only tissue examined which showed higher binding affinity in the spontaneously hypertensive rats than that of the normotensive ones.     

Age-related increase of β1-adrenergic receptor gene expression in rat liver: a potential mechanism contributing to increased β-adrenergic receptor density and responsiveness during aging

In this study we examined whether the levels of gene expressions of the three β- adrenergic receptor (βAR) subtypes, β₁, β₂, and β₃, contribute to age-related increase in βAR density. Liver membranes and total RNA were prepared from young (4- to 6-month-old) and old (24-month-old) male Fischer 344 rats. βAR density (B_max) in liver membranes was measured by a radioligand receptor binding assay using the receptor subtype nonselective βAR antagonist ¹²⁵I-pindolol as the radioligand. Steady-state levels of β₂AR mRNA in rat liver were measured by Northern blot analysis; because of the low abundance of β₁AR and β₃AR mRNA in rat liver, the expressions of these genes were measured by a semiquantitative RT-PCR or an RT-PCR. Scatchard analysis of saturation binding curves of the binding assay confirmed an age-related increase in B_max (young: 7.1?±?0.8?fmol/mg protein vs. old: 18.1?±?4.3?fmol/mg protein). No age-related differences were found in the levels of β₂AR mRNA. However, semiquantitative RT-PCR revealed an approximately twofold increase in β₁AR mRNA level between young and old rats (P?<?0.05). β₁AR mRNA levels were also correlated with B_max values for ¹²⁵I-pindolol binding sites in individual rats (r = 0.67; P?=?0.012). β₃AR mRNA, which was demonstrable in rat white adipose tissue by RT-PCR, was generally not detected in livers from young or old rats, with the exception of two old rats with the highest B_max. These results suggest that an age-related increase of β₁AR gene expression contributes to increased βAR density and β adrenergic responsiveness in rat liver during aging.     

Novel differential elimination method for determining kinetic coefficients under substrate self-inhibition

A differential elimination method (DEM) is developed to determine the kinetic coefficients for substrate self-inhibition. Finite differentiation of the equation eliminates either K_I or K_S, which enables the equation to be linearized so that [^(\textq)] {\hat{\text{q}}} , K_S, and K_I can be estimated without using nonlinear least square regression (NLSR). The DEM options that eliminate K_I or K_S computed the parameter values exactly when the data did not contain any errors. If one-point or random errors were not too large, both DEM options worked as well as NLSR when data were acquired with geometric intervals for substrate concentration. The DEM was more accurate for fitting the data for the smallest and largest values of S, but relatively weaker in estimating the observed maximum substrate utilization rate, q_max. The estimates for S_max, the concentration at which the maximum specific substrate utilization rate is observed, were relatively invariant among the methods, even when K_S and K_I differed. When the intervals were arithmetic (i.e., equal intervals of substrate concentration) and the data contained errors, the DEM and NLSR estimated the parameters poorly, indicating that collecting data with an arithmetic interval greatly increases the risk of poor parameter estimation. Parameter estimates by DEM fit very well experimental data from nitrification or photosynthesis, which were taken with geometric intervals of substrate concentration or light intensity, but fit poorly phenol-degradation data, which were obtained with arithmetic substrate intervals. Besides providing a reasonable substitute for NLSR, the DEM also can be used as a tool to diagnose the quality of experimental data by comparing its estimates between the DEM options, or, more rigorously, to those from NLSR.     

Comparison of the Effects of Ions and GTP on Substance P Binding to Membrane-Bound and Solubilized Specific Sites

Mg2+ increased but Na+ and GTP decrease [3H]substance P (SP) binding to rat cerebral cortical membranes and to 10 mM 3-[(3-cholamidopropyl)dimethylammonio]-1-propane sulfonate (CHAPS)-solubilized membrane fraction. To determine the binding parameters that are modified by the cations and GTP, inhibition experiments of [3H]SP binding by unlabeled SP were performed in both of the preparations. Nonlinear least-squares regression analysis of data in the membrane fraction indicated that optimal fitting of the inhibition curves in the presence of 10 mM MgCl2 was attained with a two-site model, corresponding to a "high-affinity (H)" and a "low-affinity (L)" state. By omitting MgCl2, or by addition of NaCl and GTP, the [3H]SP specific binding was decreased, the H state disappeared, and the L state and a new "super-low affinity (SL)" state observed. The SP/[3H]SP inhibition curves in the cerebral cortical membranes by in vivo treatment with pertussis toxin (islet-activating protein) were similar to that in the presence of GTP in control membranes. The effects of MgCl2, NaCl, and GTP were greater in the CHAPS-solubilized fraction than in the membrane fraction. In contrast to the membrane fraction, the inhibition curves of [3H]SP binding by unlabeled SP in the presence of MgCl2 in the CHAPS-solubilized fraction were best fitted to a one-site model. The KD value was relatively close to that of the low-affinity state in the membrane fraction. Even with the addition of NaCl or GTP, or by reducing MgCl2 concentration to 1 mM, although the inhibition curves consistently fit the one-site model, the KD values changed only slightly.     

单木生物量模型估计区域尺度生物量的不确定性

采用系统抽样体系江西省固定样地杉木连续观测数据和生物量数据,通过Monte Carlo法反复模拟由单木生物量模型推算区域尺度地上生物量的过程,估计了江西省杉木地上总生物量。基于不同水平建模样本量n及不同决定系数R~2的设计,分别研究了单木生物量模型参数变异性及模型残差变异性对区域尺度生物量估计不确定性的影响。研究结果表明:2009年江西省杉木地上生物量估计值为(19.84±1.27)t/hm~2,不确定性占生物量估计值约6.41%。生物量估计值和不确定性值达到平稳状态所需的运算时间随建模样本量及决定系数R~2的增大而减小;相对于模型参数变异性,残差变异性对不确定性的影响更小。     

Life‐history constraints on maximum population growth for loggerhead turtles in the northwest Atlantic

Conservation planning for protected species often relies on estimates of life‐history parameters. A commonly used parameter is the instantaneous maximum population growth rate (r_max) that can be used to limit removals and design recovery targets. Estimation of r_max can be challenging because of limited availability of species‐ and population‐specific data and life‐history information. We applied a method proposed by Neil and Lebreton, originally developed for birds, to loggerhead turtles. The method uses age‐at‐first‐reproduction and adult survival to estimate r_max. We used a variety of datasets and matrix population models to confirm an allometric assumption required by the method, and to generate estimates of age‐at‐first‐reproduction and adult survival. A meta‐analysis was applied to parameters from reported growth curves, which were then combined with the size distribution of neophyte nesters to derive estimates of age‐at‐first‐reproduction. Adult survival rates were obtained from an existing matrix population model. Monte Carlo simulation was then used to combine the estimates of the allometric coefficients, age‐at‐first‐reproduction, and adult survival to obtain a probability distribution of approximate r_max values. Estimated annual maximum population growth rates averaged 0.024, with a mode of 0.017 and a 95% highest density interval of 0.006–0.047. These estimates were similar to values reported by others using different methods and captured the variability in positive, annual change estimates across nesting beach sites for the northwest Atlantic loggerhead population. The use of life‐history parameters has a long history in wildlife and fisheries management and conservation planning. Our estimates of r_max, while having some biases and uncertainty, encompassed values presently used in recovery planning for loggerhead turtles and offer additional information for the management of endangered and threatened species.     

Recent developments for making gastric evacuation and daily ration determinations

Synopsis Gastric evacuation rate estimates often suffer from an important bias caused by fitting experimentally-derived data distributions that are inherently constricted by the X-axis (Y = 0). Monte Carlo simulations were used to evaluate the bias. Truncating constricted distributions prior to curve fitting was suggested as a means to circumvent the problem. A food consumption model developed by D.S. Robson was presented. It employs the integral of the function fit to percentage gastric evacuation data, and does not require an a priori assumption of exponential gastric evacuation. The methods were illustrated using experimental gastric evacuation data and stomach contents data for fishery-caught yellowfin tuna, Thunnus albacares.     

Analysis of ligand-binding data without knowledge of bound or free ligand molar concentration

A method is proposed to set the parameters in a nonlinear regression procedure to determine the equilibrium dissociation constant (Kd) and the high affinity receptor concentration (Bmax) of systems consisting of one ligand, one high affinity receptor, and n low affinity binding sites. This method is suitable when neither bound or free ligand formal concentrations nor the maximum of the binding signal can be deduced from the experimental data. The method makes use of (i) the abscissa of the first inflection point of the plot of any signal proportional to the binding of ligand to receptors versus the logarithm of the total ligand concentration, and (ii) the initial slope of the saturation curve plotted in direct coordinates. We first demonstrate that when such an inflection point exists, its abscissa lies between Bmax/2 + Kd(1 + d) and Bmax + 2Kd(1 + d), where d is a parameter representative of the binding to the low affinity sites. Second, we demonstrate that the initial slopes of two saturation curves in direct coordinates, where Bmax varies by a known factor, allows an estimation of the Bmax/Kd ratio, within certain limits. From these two sets of data it is subsequently possible to define a precise window for the values of both Bmax and Kd. The performance of the method has been evaluated in representative cases using Monte Carlo studies. The results establish conditions for the existence of an inflection point as well as the influence of low affinity binding, whether or not proportional to Bmax.     

Two Bayesian methods for estimating parameters of the von Bertalanffy growth equation

The von Bertalanffy growth equation (VBGE) is commonly used in ecology and fisheries management to model individual growth of an organism. Generally, a nonlinear regression is used with length-at-age data to recover key life history parameters: L _∞ (asymptotic size), k (the growth coefficient), and t ₀ (a time used to calculate size at age 0). However, age data are often unavailable for many species of interest, which makes the regression impossible. To confront this problem, we have developed a Bayesian model to find L _∞ using only length data. We use length-at-age data for female blue shark, Prionace glauca, to test our hypothesis. Preliminary comparisons of the model output and the results of a nonlinear regression using the VBGE show similar estimates of L _∞ . We also developed a full Bayesian model that fits the VBGE to the same data used in the classical regression and the length-based Bayesian model. Classical regression methods are highly sensitive to missing data points, and our analysis shows that fitting the VBGE in a Bayesian framework is more robust. We investigate the assumptions made with the traditional curve fitting methods, and argue that either the full Bayesian or the length-based Bayesian models are preferable to classical nonlinear regressions. These methods clarify and address assumptions␣made in classical regressions using von Bertalanffy growth and facilitate more detailed stock assessments of species for which data are sparse.     

New Methods for Characterization of Complex Receptor Systems Involving 3 or More Binding Sites: Application to Brain Opiate Receptors

Abstract

The problems of evaluating results based on the analysis of complex binding models are considered and new methods are proposed to provide independent confirmation of the existence of multiple sites. A new plotting format for the results from experiments involving two ligands is introduced, and its utility is demonstrated for a) finding initial estimates for nonlinear least squares curve fitting; b) presenting the results of multiple experiments; and c) giving a new means for evaluating the significance of a third site. The general problem of finding initial estimates for models involving three classes of sites, and strategies for using nonlinear least squares curve fitting algorithms to optimize the fit are considered.     

Dose Response Analysis Using Robust Covariance Estimation

A dose response analysis is robustified by estimating the asymptotic covariance of the fitted model parameters by the approximate information sandwich (a sandwich statistic) under a heterogeneous variance. The robust method is described by using a nonlinear four‐parameter regression model. The usual, robust, bootstrap, and jackknife estimates of the asymptotic variance are examined for the bioassay data. Under the response of a normal distribution with changing variances over the dose levels, the performance of the usual and robust variances is investigated by Monte Carlo study. It confirms the robustness of the sandwich estimate and shows the non‐accuracy of the usual asymptotic variance estimates of fitted model parameters under the different forms of nonconstant variance structures.