David Rodbard

Ligand: a versatile computerized approach for characterization of ligand-binding systems.

We have developed a general strategy and a versatile computer program for analysis of data from ligand-binding experiments (e.g., radioreceptor assay systems for hormones, neurotransmitters, drugs). This method provides optimal (weighted least squares) estimates of “binding parameters” (affinity constants, binding capacities, nonspecific binding) for any number of ligands reacting simultaneously with any number of receptors. This approach provides two major advantages compared with other available methods: (i): It uses an exact mathematical model of the ligand-binding system, thereby avoiding the possible biases introduced by several commonly used approximations. (ii) It uses a statistically valid, appropriately weighted least-squares curve-fitting algorithm with objective measurement of goodness of fit, thereby avoiding the subjective graphical or simplified statistical methods which may introduce bias. Additional important features include the following. (i) The level of nonspecific binding is regarded as an unknown parameter, subject to uncertainty, which must be estimated simultaneously with other parameters of the system by appropriate statistical methods. This approach provides a more accurate and precise estimate of the parameters and their standard errors. (ii) Selected parameters can be forced to share a common value, or be fixed at any desired constant value. This feature facilitates hypothesis testing by appropriate statistical methods e.g., testing whether a particular experimental manipulation results in a change in affinity (K), binding capacity (R), or both parameters. (iii) One can combine results from multiple experiments by introduction of explicit scaling or “correction” factors which compensate for the commonly observed large degree of between-experiment variation of the overall binding capacity (Bmax) while other properties of the system (e.g., K values, relative binding capacities for high- and low-affinity sites) are highly reproducible. (iv) One can characterize complex cross-reacting systems involving any number of ligands reacting simultaneously with any number of binding sites. This enables one to pool results from several curves obtained using several different ligands.

Estimation of molecular radius, free mobility, and valence using polyacrylamide gel electrophoresis☆

Relative mobility values for macromolecules in polyacrylamide gel electrophoresis at various gel concentrations are used to compute the retardation coefficient, KR, molecular radius, R, free mobility, M0, and valence, V. Automated data processing and formal statistical analysis are applied. Results obtained with 29 proteins in 9 electrophoretic systems illustrate the utility of this approach, and also provide estimates of radius and total length of the acrylamide polymer.

Polyacrylamide gel electrophoresis in sodium dodecyl sulfate-containing buffers using multiphasic buffer systems: properties of the stack, valid Rf- measurement, and optimized procedure.

SDS-proteins can be stacked in sharp starting zones in SDS-PAGE in multiphasic buffer systems, Stacking of SDS-proteins has been possible with a lower stacking limit of up to 0.300, above neutral pH in “nonrestrictive” gels. Under the same conditions, SDS is stacked. Stacking of SDS, derived from both the SDS content of the sample and the upper buffer, broadens the stack in proportion to SDS load and duration of electrophoresis. Such broadening produces an inordinately wide starting zone in the stacking gel, leading to impaired resolution, unless one limits the SDS load. In the resolving gel, this zone broadening makes the trailing edge of the stack, and the tracking dye migrating there [e.g., bromphenol blue at low gel concentration (%T)] unsuitable as a valid reference for Rf. However, the leading edge of the stack and the tracking dye migrating there (e.g., pyronin-Y-SDS at low %T) are a valid Rf reference. At high %T, both SDS and pyronin-Y-SDS unstack, whereas bromphenol blue now moves to the leading edge of the stack and becomes, for that %T range, a valid reference of Rf. An optimized system for SDS-PAGE is described which contains no SDS in the gel, 0.03% SDS in the upper buffer, 150 μg or less SDS in the sample, and a single set of gel buffer constituents in stacking and resolving gels. Valid Rf measurement is a requisite for linear Ferguson plots (log Rf vs %T). Since the slope of these plots, KR, is a measure of molecular weight the procedure of SDS-PAGE proposed in this study should provide improved reliability of melecular weight estimates.

Statistical estimation of the minimal detectable concentration ("sensitivity") for radioligand assays.

Abstract A method is presented for calculation of the minimal detectable dose (MDD) or minimal detectable concentration (MDC) for a radioimmunoassay or related radioligand assay (e.g., immunoradiometric assays, “sandwich” assays, etc.). It is necessary to consider the uncertainty in the estimate of the “true” response for zero dose, as well as the error in the mean response for replicates of an “unknown” sample. The MDC decreases as the degree of replication for the unknown increases, approaching a nonzero finite lower limit as the number of replieates for the unknown gets large. “Pooling” of information about the variability of response at zero dose results in improved utilization of data, with increased degrees of freedom and smaller MDC. A one-tailed, rather than two-tailed. Students t statistic should be used. The considerations reviewed here should be applicable irrespective of the nature of the dose-response curve or the particular model used for curve fitting.

Computer Simulation of Plasma Insulin and Glucose Dynamics After Subcutaneous Insulin Injection

We developed a computer program for the simulation of plasma insulin and glucose dynamics after subcutaneous injection of insulin. The program incorporates a pharmacokinetic model to calculate the time courses of plasma insulin for various combinations of popular preparations (regular, NPH, lente, and ultralente). With the use of a pharmacodynamic model describing the dependence of glucose dynamics on plasma insulin and glucose levels, the program can predict the expected time course of plasma glucose in response to a change in carbohydrate intake, insulin dose, timing, or regimen. A set of typical parameters has been obtained by analysis of data from the literature. The results of several computer simulations are presented showing the effect on a 24-h insulin and glucose profile of systematically changing insulin regimen, dose, timing of meals, or timing of preprandial insulin administration. The program can be used to explore on a theoretical basis the impact of various factors associated with glycemic control in insulin-dependent diabetes mellitus. As an educational tool, the program provides a realistic environment for demonstration of the combined or isolated effects of insulin and diet on glycemia

Mathematical theory of cross-reactive radioimmunoassay and ligand-binding systems at equilibrium

Abstract A mathematical basis for radioimmunoassays and radio-ligand assays in multicomponent cross-reactive systems at equilibrium has been defined, formulated, and solved for the case of several antigens (or ligands) and several antibodies (or binding proteins) with different pairwise affinity constants. This formulation makes it possible to describe complex binding systems quantitatively, predicting such assay parameters as response level, sensitivity, and specificity under conditions of cross-reactivity, nonidentity of labeled and unlabeled hormone, or other cases of multiplicity and diversity of reactants. Computer programs have been developed for making these quantitative predictions. Experimental applications are discussed and illustrated.

Pore gradient electrophoresis.

Abstract Gel gradient electrophoresis has been proposed as a method for improving resolution and for measuring molecular size. Margolis and Slater have suggested that each macro-ion will reach a “pore limit” or “dead stop” and that the protein pattern would remain essentially stable thereafter. However, no quantitative treatment of the migration of charged macromolecules in a gel gradient has been reported to date. The present report provides an analysis of the behavior of macro-ions in both linear and nonlinear gel gradients. This is made possible by combining Fergusons relationship for electrophoretic mobility versus gel concentration with the equation describing the gel gradient. Solution of these two equations yields the instantaneous velocity and position for each band as a function of time. This analysis provides insight into situations in which pore gradients may improve resolution and conditions for optimal resolution of multicomponent systems (macromolecular mapping). Position and instantaneous velocity of migration for any macromolecule on gel concentration (“pore”) gradient electrophoresis may be calculated for a linear gradient: χ= log e (ba 2 u o t + exp (bT o )) − bT o ba 2 (iva) ν= u o (ba 2 u o t + exp (bT o )) (va) This makes it possible to calibrate or standardize pore gradient electrophoresis. This analysis indicates that the proteins do not come to a “pore limit” or “dead stop.” Pore gradient electrophoresis is not indicated for analysis or separation of one, two, or three components, nor for charge fractionation. It is useful for simultaneous analysis of components of multicomponent mixtures (macromolecular mapping); and a “transverse” gradient presents a promising approach to obtaining Ferguson plots.

Precision of sodium dodecyl sulfate-polyacrylamide-gel electrophoresis for the molecular weight estimation of a membrane glycoprotein: Studies on bovine rhodopsin

Abstract Criteria for assessing the precision and accuracy of methods for estimation of molecular weight for proteins using sodium dodecyl sulfate-polyacrylamide-gel electrophoresis have been applied to rhodopsin from bovine visual cell outer segment membranes. Various methods of preparing this hydrophobic protein for electrophoresis differ in their ability to solubilize and disaggregate polypeptide constituents of the outer segment membrane, with resultant variations in the pattern of protein bands and the apparent molecular weight of rhodopsin. Even with optimal solubilization and disaggregation, the behavior of rhodopsin relative to a series of standard proteins is such that the apparent molecular weight decreases systematically from 40,400 to 34,500 as the acrylamide concentration increases from 4 to 10%. As demonstrated by Ferguson plots of log R f vs gel concentration and split gel experiments, this discrepancy is explained by the fact that the extrapolated R f for zero gel concentration ( Y 0 ) for rhodopsin is significantly lower than the Y 0 s for the soluble proteins used as molecular weight standards. In such cases, a possibly more reliable molecular weight estimate is obtained by plotting the retardation coefficient ( K R ) vs molecular weight. This method yields a value of 29,500 ± 1000 for bovine rhodopsin if only the errors in measurement of R f are considered and a quadratic relationship between K R and molecular weight is used. Using weighted linear regression for K R vs molecular weight, we obtain a molecular weight estimate of 32,700 ± 5000 when the uncertainty in the calibration curve is considered. Because of uncertainties regarding the detergent-binding properties of rhodopsin and the relationship of its Stokes radius to its molecular weight by comparison with the soluble protein standards, these values must be viewed with caution.

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.