The application of robust non-linear regression methods for fitting hyperbolic Scatchard plots

Mathematical approaches in estimating aptamer-target binding affinity

The performance of aptamers as versatile tools in numerous analytical applications is critically dependent on their high target binding specificity and selectivity. However, only the technical or methodological aspects of measuring aptamer-target binding affinities are focused, ignoring the equally important mathematical components that play pivotal roles in affinity measurements. In this study, we aim to provide a comprehensive review regarding the utilization of different mathematical models and equations, along with a detailed description of the computational steps involved in mathematically deriving the binding affinity of aptamers against their specific target molecules. Mathematical models ranging from one-site binding to multiple aptameric binding site-based models are explained in detail. Models applied in several different approaches of affinity measurements such as thermodynamics and kinetic analysis, including cooperativity and competitive-assay based mathematical models have been elaborately discussed. Mathematical models incorporating factors that could potentially affect affinity measurements are also further scrutinized.

Association and transfer functions for analysis of drug binding to proteins

A global association function is proposed that characterizes the apparent overall equilibrium of drug protein binding. It is defined as a function of the total drug concentration and may be directly calculated from each experimental result. The binding transfer function represents the relative rates of change in the bound and free concentrations from the total concentration. Two applications are presented: one is an investigation of the effect of temperature on binding of salicylic acid to proteins in plasma; and the other is a comparison of the phenomenon in two different biological fluids (protein binding of sodium salicylate in plasma and in synovial fluid).

Binding of [3H]flunitrazepam and [3H]spiperone to melanoma cell membrane preparations

Binding of [3H]flunitrazepam and [3H]spiperone to membrane preparations isolated by high speed centrifugation of hamster, rabbit and human melanoma cell homogenates was analyzed. All melanoma cell types expressed a high density of specific binding sites for [3H]flunitrazepam (3-4 pmol/mg protein) with a high affinity (Kd about 30 nM). This binding was independent of melanin content of cells and could be classified, based on competition experiments, as a Ro 5-4864-like binding type. Specific [3H]spiperone binding to these cell lines clearly revealed at least two types of binding sites: a low affinity, high capacity type of binding site (Kd greater than 100 nM, Bmax about 50 pmol/mg protein) and a high affinity, low capacity binding site (Kd less than 1 nm, Bmax 30 fmol/mg protein). Binding of spiperone to the low affinity, high capacity site appeared displaceable by NM 113 and dependent on melanin content of the cells and probably represents binding to melanin. Analysis of drug binding to melanoma membrane cell preparations and correlation with drug effects should include the possible involvement of binding to melanin.

Low density lipoprotein metabolism by endothelial cells from human umbilical cord arteries and veins

Binding and metabolism of low density lipoprotein (LDL) and acetylated LDL were examined in endothelial cells from human umbilical cord arteries and veins. Both high and low affinity LDL interactions were observed. High affinity LDL binding and catabolism were increased five- to sevenfold after preincubation for 18 hours in LPDS containing medium. Subconfluent cells degraded, endocytosed, and bound 1.5 to 2.7 times more LDL by high affinity interaction than confluent cells, when endothelial cell growth supplement (ECGS) was present in the culture system. In the absence of ECGS, these ratios were somewhat less. Low affinity LDL metabolism was less affected by the state of confluency. Binding of LDL and acetylated LDL by venous endothelial cells was more than two- and threefold, respectively, than that by comparable arterial cells. However, the difference in LDL binding was not reflected in an altered LDL catabolism. There apparently is a population of low affinity binding sites not involved in LDL catabolism. LDL metabolism was identical in cells, which were cultured in medium supplemented with 20% to 100% serum or hirudin- or heparin-treated platelet-poor plasma. Without preincubation in LPDS, high affinity adsorptive endocytosis mediated the main part of LDL uptake only at low LDL concentrations (5 to 20 micrograms protein/ml). However, at physiological LDL concentrations (550 micrograms/ml), we estimated that this process mediated only 17% of the LDL uptake. We calculated that fluid endocytosis and low affinity adsorptive endocytosis of LDL accounted for the remaining 12% and 70%, respectively, of the LDL uptake at physiological LDL concentrations.

Difficulties in applying the Scatchard model of ligand binding to proteins--proposal of new mathematical tools--application to salicylates

Ill-considered use of the Scatchard model often leads to unjustified deductions. Since the main difficulty of this model is its number of parameters, new models are proposed that have only two parameters. After checking the models on simulated data, they were applied to real data on the binding of salicylates to albumin.

Fitting biological equations to data using non-parametric methods

Simulated experimental data were generated for these equations: a straight line, the integrated Michaelis--Menten equation, plus a linear term, the Hill equation, a two-exponential function and a double Michaelis--Menten equation. The equations were fitted to the data using (i) least-squares and (ii) non-parametric methods. The precision and accuracy of the parameter estimates obtained by each method were compared and the methods assessed. For several of the equations, non-parametric methods provided robust techniques for parameter estimation. For the remainder, the results were poor. The reasons for this are discussed.

Numerical analysis of binding studies: a direct procedure avoiding the pitfalls of a Scatchard analysis of equilibrium data for unknown binding models

It is shown on theoretical grounds that the straightforward analysis of binding data according to Scatchard may lead to erroneous results, especially when more complicated binding schemes are involved. We have demonstrated this point by presenting Scatchard plots with slight variation of experimental parameters. These inherent difficulties of Scatchard analyses can be avoided by applying a direct procedure. We have developed a program, which compares the measured quantity and the theoretical value directly and which considers the following binding models: (i) independent equivalent binding of n ligands; (ii) independent unequivalent binding of 2 ligands; (iii) positive or negative cooperative binding of 2 ligands. Other binding schemes can easily be implemented. We have used this procedure for the evaluation of equilibrium data on the complex formation of tRNA-Tyr and tyrolyl tRNA synthetase from E. coli in terms of different binding models.