Ajit K. Thakur

Graphical analysis of ligand-binding systems: evaluation by Monte Carlo studies.

Abstract Graphical methods have traditionally been the principal means for estimation of parameters (e.g., affinity constants, cooperativity parameters, and concentrations of receptor sites) in enzymology and ligand-binding problems. The present report provides a review of these methods as well as new results, as applied to three coordinate systems popularly used in ligand-binding studies: B F vs [Bound]. B F vs [Free], and B F vs [Total]. We consider two extremely general models, the statistical mechanical model and the Adair model for equilibrium ligand binding. We also consider a very specialized case of receptor interaction wherein the equilibrium constannt of dissociation is linearly related to receptor occupancy. We collect previously described equations and derive new ones, to enable the user to estimate the parameters of the models in terms of relatively easily measurable graphical characteristics. We have evaluated the performance of these methods in representative cases using Monte Carlo studies. The results indicate the kind of precision and accuracy which can be obtained with typical experimental designs. Depending upon the magnitude of experimental error, the graphical methods can provide dependable values for the binding parameters. However, in general, the results obtained by the graphical methods should be regarded as reasonable initial estimates for further refinement by weighted nonlinear least-squares curve fitting.

Quantitative characterization of hormone receptors

Most workers characterize steroid (and other hormone) receptors by graphical analysis of Scatchard plots or by simple linear regression. Unfortunately, these methods are suboptimal from a statistical point of view. The Scatchard plot, B/F vs. [Bound], does not satisfy the assumptions underlying simple linear regression: both variables are subject to error, and these errors are intimately interdependent. Accordingly, neither B/F nor [Bound] is an appropriate independent variable. Furthermore, both variables (B/F and [Bound]) show non‐uniformity of variance. Thus, even when the Scatchard plot is linear, one should estimate the binding parameters (affinity, K, and binding capacity, R) by means of weighted nonlinear least‐squares regression, using the Total ligand concentration as the independent variable, and either B/T or [Bound] as the dependent variable.

Characterization of ligand-binding systems by continuous affinity distributions of arbitrary shape

Abstract We have developed computer programs for characterization of ligand-binding systems in terms of continuous affinity distributions of arbitrary shape based on a numerical finite difference method. This method provides an excellent initial estimate of the affinity distribution, which can be further refined by means of nonlinear least-squares curve fitting. The method has been extensively tested for several cases including receptor heterogeneity, cooperativity, and for several examples of experimental design (e.g., ligand concentrations), and various levels of random and systematic experimental errors. The results provide a guide to experimental design, and indicate limits to the resolution obtained by ligand-binding studies, irrespective of the method of analysis.

Graphical aids to interpretation of Scatchard plots and dose-response curves.

Abstract General expressions are derived for the limiting slopes and intercepts of graphical representations of experimental binding data in either the Scatchard or the “dose-response” co-ordinate systems. We apply a previously formulated general model that includes heterogeneity and/or cooperativity of receptor affinity. One must establish or assume a physical chemical mechanism in order to fully interpret these limiting slopes and intercepts. However, they do provide satisfactory initial estimates for the binding parameters, for use in a non-linear least squares curve fitting approach using an exact model.

On the stochastic theory of compartments: III. General time-dependent reversible systems

General formulation of stochastic single- and multi-compartment reversible systems with time-dependent transitions is made. The correspondence between the stochastic mean and the deterministic value is established in case of time-dependence and it is shown how the consequence of this can be utilized to compute the distribution and the moments of each individual compartment of the system. A two-compartment reversible system previously proposed by Cardenas and Matis (1975a) is analyzed on the basis of the theory.

Antigen binding to receptors on immunocompetent cells: II. Thermodynamic and biological implications of the receptor cross-linking requirement for B-cell activation

Abstract If one assumes that receptor cross-linking is a necessary, but not sufficient condition for cellular activation, a number of predictions can be made bearing on the physical chemical properties of the cells selected. In this paper we show that the following response characteristics can be consequences of a cross-linking requirement. (1) Small sparsely haptenated antigens such as DNP10BSA are expected to elicit a response that matures, and such maturation can occur even with antigenic determinant density in excess over the concentration of cellular receptors. (2) There is an optimal concentration for activation of cells with a given affinity, and therefore an optimally immunogenic dose. (3) The optimal dose is relatively insensitive to antigen valence. (4) Increasing valence increases the breadth of the affinity distribution. (5) For supra optimal doses of antigen, unresponsiveness will be preferentially induced in high affinity cells. (6) Small densely haptenated antigens (e.g. DNP40BSA) are not expected to elicit responses that mature as quickly as those that are lightly coupled. (7) Large polymeric antigens are not expected to induce responses that mature. (8) Antigens with low determinant density may induce tolerance in vivo but not in vitro. The predictions are briefly discussed in the context of relevant experimental data.

An analysis of the limits of resolution of binding experiments as assays for affinity heterogeneity.

Abstract An analysis is presented of the limits of resolution of Scatchard and Sips plots as methods for detecting affinity heterogeneity in receptor populations. We estimate using statistical criteria for defining resolution that Scatchard plots can resolve two groups present at equal concentration, provided they differ by more than a factor of about six in affinity, and that resolution drops rapidly as the concentration difference between the groups increases. In addition, because of the non linear relation between affinity and free energy, and because the Sips analysis must necessarily be formulated in terms of free energies, the Scatchard plot is the more sensitive of the two methods for detecting heterogeneity.

Development of Compartmental Concepts

The first compartmental models were used in Physics for the description of radioactive decay. After Becquerel (1896) discovered the radioactivity, Rutherford and Soddy (1902) found experimentally that Thorium X decays in time according to an exponential law, i.e. that the number of radioactive atoms decaying per unit time is proportional to the number of radioactive atoms present. If X(t0) and X(t) are the quantities of radioactive substance present at time to and t respectively, the law of radioactive decay is

The use and abuse of models

{\rm{dX/dt}}\,{\rm{ = }}\,{\rm{ - }}\,{\rm{K}}{\rm{.X,}}