Curves of ligand binding. The use of hyperbolic functions for expressing titration curves

1. The dependences of the concentrations of the non-ligated, uni-ligated and bi-ligated forms of a molecule that binds two molecules of ligand are expressed as functions of the logarithm of free ligand concentration by means of hyperbolic functions. Expressions are also given for the saturation both of an individual site and of the molecule as a whole. This form of expression allows derivation of the following points. 2. The sharpness of bell-shaped curves of concentration of the uni-ligated form is analysed in terms of the heights of their points of inflexion; these can rise to 1/ radical2 of the curve. 3. A single group can exhibit a doubly sigmoid saturation curve if this group and another have comparable affinities for a ligand, and if ligand binding at one of them diminishes the affinity at the other. If the molecular pK values pK(1) and pK(2) for the first and second molecules of ligand are called pK*+/-logm, so that K*(2)=K(1)K(2) and m(2)=K(1)/K(2), then the doubly sigmoid curve can be represented by the sum of two independent one-site saturation curves, in general of unequal height, of pK values pK*+/-log(1/2)[m+ radical(m(2)-4)]. The error in such representation is small either if the mutual interaction between the groups (i.e. m) is large, or if the groups have very similar affinities for the ligand. 4. The sum of two one-site saturation curves, again of pK values of pK*+/-log(1/2)[m+ radical(m(2)-4)] but of equal heights, gives a precise value for the total saturation, provided that the binding of one molecule does not promote the binding of a second, i.e. providing that m>/=2. Hence determinations of saturation cannot distinguish interacting and possibly identical sites from independent and different ones.     

A Bayesian hierarchical non-linear regression model in receiver operating characteristic analysis of clustered continuous diagnostic data

Receiver operating characteristic (ROC) analysis is a useful evaluative method of diagnostic accuracy. A Bayesian hierarchical nonlinear regression model for ROC analysis was developed. A validation analysis of diagnostic accuracy was conducted using prospective multi-center clinical trial prostate cancer biopsy data collected from three participating centers. The gold standard was based on radical prostatectomy to determine local and advanced disease. To evaluate the diagnostic performance of PSA level at fixed levels of Gleason score, a normality transformation was applied to the outcome data. A hierarchical regression analysis incorporating the effects of cluster (clinical center) and cancer risk (low, intermediate, and high) was performed, and the area under the ROC curve (AUC) was estimated.     

Estimation of the initial velocity of enzyme-catalysed reactions by non-linear regression analysis of progress curves.

Most methods for studying the kinetic properties of an enzyme involve the determination of initial velocities. When the reaction progress curve shows significant curvature due to depletion of the substrate, accumulation of inhibitory products or instability of the enzyme, estimation of the initial velocity is a subjective and inexact process. Two methods have been suggested [Cornish-Bowden (1975) Biochem. J. 144, 305-312; Boeker (1982) Biochem J. 203, 117-123] that attempt to eliminate this subjective element. The present study offers a third alternative, which is based on fitting a reparameterized form of the integrated Michaelis-Menten equation to the progress curves by non-linear regression. This method yields estimates and standard errors of the initial velocity and of the time to reach 50% reaction. No prior knowledge of the apparent product concentration at zero time or infinite time is required, since both of these quantities are also estimated from the data. It is shown that this method yields reliable estimates of the initial velocity under a wide range of circumstances, including those where the two previously published methods perform poorly.     

A need for caution in the use of frontal analysis continuous capillary electrophoresis for the determination of ligand binding data

Attention is drawn to a need for caution in the determination of binding data for protein-polyelectrolyte interactions by frontal analysis continuous capillary electrophoresis (FACCE). Because the method is valid only for systems involving comigration of complex(es) and slower-migrating reactant, establishing conformity with that condition is clearly a prerequisite for its application. However, that requirement has not been tested in any published studies thus far. On the basis of calculated FACCE patterns, presented to illustrate features by which such comigration of complex(es) and slower-migrating reactant can be identified, the form of the published pattern for a beta-lactoglobulin-poly(styrenesulfonate) mixture does not seem to signify the migration behavior required to justify its consideration in such terms. Additional experimental studies are therefore needed to ascertain the validity of FACCE as a means of determining binding data for the characterization of protein-polyelectrolyte interactions.     

The use of polynomial regression analysis with indicator variables for interpretation of mercury in fish data

Mercury levels in fish in reservoirs and natural lakes have been monitored on a regular basis since 1978 at the La Grande hydroelectric complex located in the James Bay region of Québec, Canada. The main analytical tools historically used were analysis of covariance (ANCOVA), linear regression of the mercury-to-length relationship and Student-Newman-Keuls (SNK) multiple comparisons of mean mercury levels. Inadequacy of linear regression (mercury-to-length relationships are often curvilinear) and difficulties in comparing mean mercury levels when regressions differ lead us to use polynomial regression with indicator variables.For comparisons between years, polynomial regression models relate mercury levels to length (L), length squared (L²), binary (dummy) indicator variables (B_n), each representing a sampled year, and the products of each of these explanatory variables (L × B₁, L² × B₁, L × B₂, etc.). Optimal transformations of the mercury levels (for normality and homogeneity) were found by the Box-Cox procedure. The models so obtained formed a partially nested series corresponding to four situations: (a) all years are well represented by a single polynomial model; (b) the year-models are of the same shape, but the means may differ; (c) the means are the same, but the year-models differ in shape; (d) both the means and shapes may differ among years. Since year-specific models came from the general one, rigorous statistical comparisons are possible between models.Polynomial regression with indicator variables allows rigorous statistical comparisons of mercury-to-length relationships among years, even when the shape of the relationships differ. It is simple to obtain accurate estimates of mercury levels at standardized length, and multiple comparisons of these estimations are simple to perform. The method can also be applied to spatial analysis (comparison of sampling stations), or to the comparison of different biological forms of the same species (dwarf and normal lake whitefish).     

Bayesian analysis for nonlinear regression model under skewed errors,with application in growth curves

We have considered a Bayesian approach for the nonlinear regression model by replacing the normal distribution on the error term by some skewed distributions, which account for both skewness and heavy tails or skewness alone. The type of data considered in this paper concerns repeated measurements taken in time on a set of individuals. Such multiple observations on the same individual generally produce serially correlated outcomes. Thus, additionally, our model does allow for a correlation between observations made from the same individual. We have illustrated the procedure using a data set to study the growth curves of a clinic measurement of a group of pregnant women from an obstetrics clinic in Santiago, Chile. Parameter estimation and prediction were carried out using appropriate posterior simulation schemes based in Markov Chain Monte Carlo methods. Besides the deviance information criterion (DIC) and the conditional predictive ordinate (CPO), we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. For our data set, all these criteria chose the skew‐t model as the best model for the errors. These DIC and CPO criteria are also validated, for the model proposed here, through a simulation study. As a conclusion of this study, the DIC criterion is not trustful for this kind of complex model.     

Evaluation of the Fermi equation as a model of dose-response curves

 When plotted in linear coordinates, the dose-response curves of microorganisms exposed to a lethal agent, such as radiation or a toxic substance, often have a characteristic sigmoid shape. Irrespective of whether they are very narrow or broad they can be described by the Fermi function, which is a mirror image of the logistic function, i.e. S(X)=1/{1+ exp [(X−X _c)/a]} where S(X) is the fraction of the surviving organisms, X the dose of the lethal agent, X _c a characteristic dose marking the inflection point of S(X), which corresponds to 50% mortality, and a a measure of the steepness of the survival curve around X _c. It is demonstrated that, if the susceptibilities of the individual organisms, expressed in terms of a characteristic lethal dose, have a symmetric unimodal distribution, the dose-response curve of the population has a Fermian sigmoid shape. It is also shown that the mode and variance of the distribution can be estimated from the shape parameters of the Fermian survival curve, X _c and a. Received: 7 November 1995 / Received last revision: 11 April 1996 / Accepted: 29 April 1996     

A semiparametric model for regression analysis of interval-censored failure time data

Left-, right-, and interval-censored response time data arise in a variety of settings, including the analyses of data from laboratory animal carcinogenicity experiments, clinical trials, and longitudinal studies. For such incomplete data, the usual regression techniques such as the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model are inapplicable. In this paper, we present a method for regression analysis which accommodates interval-censored data. We present applications of this methodology to data sets from a study of breast cancer patients who were followed for cosmetic response to therapy, a small animal tumorigenicity study, and a clinical trial.     

Mixture Markov regression model with application to mosquito surveillance data analysis

A mixture Markov regression model is proposed to analyze heterogeneous time series data. Mixture quasi‐likelihood is formulated to model time series with mixture components and exogenous variables. The parameters are estimated by quasi‐likelihood estimating equations. A modified EM algorithm is developed for the mixture time series model. The model and proposed algorithm are tested on simulated data and applied to mosquito surveillance data in Peel Region, Canada.     

A smooth test of goodness-of-fit for growth curves and monotonic nonlinear regression models

We propose a goodness-of-fit test for growth curves based on an adaptation of the data-driven smooth test paradigm. It is simple to apply and can assess the fit of a model to a set of growth experiences. A simulation study shows that for small samples, the test holds its level. Moreover, its power is found to be generally greater than existing tests. The article concludes by revisiting the long-standing problem of validating a model for the growth of human stature.     

Inference methods for the conditional logistic regression model with longitudinal data

This paper considers inference methods for case-control logistic regression in longitudinal setups. The motivation is provided by an analysis of plains bison spatial location as a function of habitat heterogeneity. The sampling is done according to a longitudinal matched case-control design in which, at certain time points, exactly one case, the actual location of an animal, is matched to a number of controls, the alternative locations that could have been reached. We develop inference methods for the conditional logistic regression model in this setup, which can be formulated within a generalized estimating equation (GEE) framework. This permits the use of statistical techniques developed for GEE-based inference, such as robust variance estimators and model selection criteria adapted for non-independent data. The performance of the methods is investigated in a simulation study and illustrated with the bison data analysis.     

The use of a mixture model in the analysis of count data

A mixture model is presented for the analysis of data on premature ventricular contractions. The analysis is shown to be straightforward and the conclusions relatively simple.     

DESIGN: computerized optimization of experimental design for estimating Kd and Bmax in ligand binding experiments. II. Simultaneous analysis of homologous and heterologous competition curves and analysis blocking and of "multiligand" dose-response surfaces

We have developed a computer program, DESIGN, for optimization of ligand binding experiments to minimize the "average" uncertainty in all unknown parameters. An earlier report [G. E. Rovati, D. Rodbard, and P. J. Munson (1988) Anal. Biochem. 174, 636-649] described the application of this program to experiments involving a single homologous or heterologous dose-response curve. We now present several advanced features of the program DESIGN, including simultaneous optimization of two or more binding competition curves optimization of a "multiligand" experiment. Multiligand designs are those which use combinations of two (or more) ligands in each reaction tube. Such designs are an important and natural extension of the popular method of "blocking experiments" where an additional ligand is used to suppress one or more classes of sites. Extending the idea of a dose-response curve, the most general multiligand design would result in a "dose-response surface". One can now optimize the design not only for a single binding curve, but also for families of curves and for binding surfaces. The examples presented in this report further demonstrate the power and utility of the program DESIGN and the nature of D-optimal designs in the context of more complex binding experiments. We illustrate D-optimal designs involving one radioligand and two unlabeled ligands; we consider one example of homogeneous and several examples of heterogeneous binding sites. Further, to demonstrate the virtues of the dose-response surface experiment, we have compared the optimal surface design to the equivalent design restricted to traditional dose-response curves. The use of DESIGN in conjunction with multiligand experiments can improve the efficiency of estimation of the binding parameters, potentially resulting in reduction of the number of observations needed to obtain a desired degree of precision in representative cases.     

Comparison of surface plasmon resonance binding curves for characterization of protein interactions and analysis of screening data

Label-free technologies, such as surface plasmon resonance, are typically used for characterization of protein interactions and in screening for selection of antibodies or small molecules with preferred binding properties. In characterization, complete binding curves are normally fitted to defined interaction models to provide affinity and rate constants, whereas report points indicative of binding and stability of binding are often used for analysis of screening data. As an alternative to these procedures, here we describe how the analysis, in certain cases, can be simplified by comparison with upper and lower limit binding curves that represent expected or wanted binding profiles. The use of such profiles is applied to the analysis of kinetically complex IgG–Fc receptor interactions and for selection of antibody candidates. The comparison procedure described may be particularly useful in batch-to-batch comparisons and in comparability and biosimilar studies of biotherapeutic medicines. In screening, more informed selections may become possible as entire binding profiles and not a few report points are used in the analysis and as each new sample is directly compared with a predefined outcome.