The objective of this investigation is to engender greater confidence in the validity of binding equations derived for multivalent ligands on the basis of reacted-site probability theory. To that end a demonstration of the theoretical interconnection between expressions derived by the classical stepwise equilibria and reacted-site probability approaches for univalent ligands is followed by use of the traditional stepwise procedure to derive binding equations for bivalent and trivalent ligands. As well as demonstrating the unwieldy nature of the classical binding equation for multivalent ligand systems, that exercise has allowed numerical simulation to be used to illustrate the equivalence of binding curves generated by the two approaches. The advantages of employing a redefined binding function for multivalent ligands is also confirmed by subjecting the simulated results to a published analytical procedure that has long been overlooked.

The analysis of the binding of a univalent ligand to equivalent and independent sites on a multivalent acceptor was developed originally (

The aim of the present communication is to bolster confidence in the equations derived from reacted-site considerations by using the traditional approach to derive an expression which predicts the same binding curves as those based on reacted-site probability theory for the simplest multivalentmultivalent system – that in which acceptor and ligand are both bivalent. Thereby demonstrated is the impracticality of adopting the traditional stepwise approach as a general procedure for treating multivalence because of its generation of binding equations involving the ratio of two indefinite multinomial series in free ligand concentration. That undesirable situation can be avoided completely by resorting to reacted-site probability theory, which provides binding equations with closed solutions for all combinations of acceptor and ligand valences.

The derivation of binding equations by either the traditional (_{p}_{i}B_{j}

The central parameters in reacted-site probability theory are _{A}_{B}_{B}_{B}_{B}_{B}_{B}C̄_{B}_{A}_{A}_{A}_{2}, ..., _{p}_{i}_{i}_{i}C̄_{A}

In situations where all interactions involve identical and independent sites on the acceptor, the intrinsic equilibrium constant _{B}C̄_{B}_{A}C̄_{A}_{A}_{A}_{B}_{B}^{–1}. The term in (1 – _{A}_{A}C̄_{A}_{B}C̄_{B}_{B}_{B}/C̄_{B}_{B}C̄_{B}/C̄_{A}

The stepwise equilibria define ^{th} step (_{i}

For interactions involving a univalent ligand there is no particular advantage in switching to the reacted-site probability approach. However, it provides a much simpler means of deriving a binding equation for systems in which the ligand also exhibits multivalence.

Interactions between a _{i}B_{j}_{A}_{B}_{B}

The first term on the right-hand side of _{B}_{B}_{A}_{B}

Although over thirty years have elapsed since the publication of

The interaction between a p-valent acceptor and a bivalent ligand results in the equilibrium coexistence of unbound reactants and an array of _{i}B_{j}_{i}B_{j}_{P}_{2}B,_{2}B_{2},_{2}B_{3},_{2}B_{2p-1}_{3}B_{3},_{3}B_{4},_{3}B_{3p-2}_{4}B_{3},_{4}B_{4},_{4}B_{5},_{4}B_{4p-3}. . . . . . . . . . . .

The simplest approach to quantifying concentrations for these complexes is to obtain expressions for those in the first column of the array, _{i}B_{i–1}, and then to consider the completion of each line of the array. Here we envisage formation of those _{i}B_{i–1} complexes via the addition of

For the interaction of bivalent ligand with _{i}

An expression for the concentration of _{2}_{2}_{3}_{2} may be considered to result from the interaction of univalent _{2}_{i}B_{i–1} species can likewise be determined because all _{i}B_{i–1} species (including

Because each _{i}B_{i–1} complex also possesses (_{i}B_{j}

Consider initially a system in which the acceptor is also bivalent (_{i}B_{i–1}_{B}_{i}B_{i–1}, _{i}B_{i}_{i}B_{i+1}, for each value of _{2,} where _{AB}_{A}α_{AB2} = _{A}α^{2} [from _{A}^{2}. In similar vein,

To obtain the corresponding expression for the concentration of bound ligand, (_{B}_{B}_{AB}_{AB2} = 2_{A}α_{A}α^{2}, and

The binding equation derived by conventional means is clearly not as convenient to use as that deduced from reacted-site probability considerations because the second term on the right-hand side (Ψ) is in the form of a ratio of polynomial series rather than a closed solution. However, it is noted that

Although the same approach can obviously be used for systems with a larger acceptor valence, the derivation becomes increasingly tedious because of the greater number of complexes to which an expression for the concentration has to be assigned. For example, an increase in acceptor valence from _{4}_{j}_{AB}_{A}α_{AB}_{2} = 3_{A}α^{2}, and _{AB}_{3} = _{A}α^{3} [both from _{i}_{A}^{3}. For the corresponding _{2}_{i}_{A}_{B}

Although these traditionally derived expressions for the classical binding function (

The requirement for a revised definition of the binding function for a multivalent ligand surfaced during the development of quantitative affinity chromatography as a means of characterizing the interaction of tetrameric and hence tetravalent glycolytic enzymes with the affinity matrix (

For a system involving the interaction of a _{A}_{q}

In the normal course of events a linear transform is proposed to simplify the characterization of interactions by graphical analysis. However, in this instance the derivation of

An alternative approach to derivation of the general counterpart of the Scatchard equation [

Advantage is taken of the fact that description of the concentration of bound ligand only requires knowledge of the total and free ligand concentrations, whereupon the concentrations of the array of complexes _{i}B_{j}_{B}_{B}_{i}

The term in free acceptor-site concentration (_{A}

New developments arising from the above theoretical considerations have been (i) the generation by the classical stepwise approach of an expression describing the solution composition for an equilibrium mixture of multivalent acceptor and bivalent ligand, and (ii) the consequent derivation of a binding equation for such systems without recourse to reacted-site probability theory. An obvious point to be established is demonstrated agreement between predictions based on the current expressions [

The simulation of normalized binding curves (_{A}_{A}

The results of simulations for the interaction between a bivalent ligand and a bivalent acceptor are summarized in _{A}_{A}_{A}_{A}_{A}_{A}α^{12}. Fortunately, this undesirable and tedious aspect of the classical analysis is countered by the demonstrated equivalence between its predictions and those based on

Despite obvious advantages over its classically derived counterpart, the binding equation emanating from reacted-site probability theory [_{B}_{B}_{A}

Manipulation of the data from _{B}_{B}_{A}_{A}_{B}_{B}_{B}_{B}_{B}_{B}^{1/2} and (_{B}^{1/2} = (^{1/2} required for a plot of the results according to the expression _{A}_{A}

Redefinition of the binding function according to

Despite antibody bivalence, the results from ELISA studies of immunochemical interactions involving multivalent antigens (_{B}_{B}_{B}_{A}_{B}_{B}_{B}_{B}

The purpose of this investigation has been to engender greater confidence in the validity of binding equations derived for multivalent ligands on the basis of reacted-site probability theory rather than the classical stepwise equilibrium method. In addition to demonstration of the theoretical interconnection between the two approaches for a univalent ligand, the classical approach has been employed to derive binding equations for bivalent and trivalent ligands. This action has served not only to demonstrate the unwieldy nature of the classical binding equation for such systems but also to establish by numerical simulation the equivalence of binding curves generated by the two approaches. The advantages of switching to a redefined binding function for multivalent ligands have also been illustrated. It is hoped that these endeavors may lead to experimental adoption of the binding equations derived many years ago for multivalent ligands, and hence to the validity of reported quantitative analyses for antigen–antibody interactions.

Grants from the National Institutes of Health (U54 HL112309, R01 HL082609 and R01 HK040921) and Hemophilia of Georgia, Inc. to P.L. are gratefully acknowledged.

Comparison of simulated binding curves calculated by means of expressions [_{A}_{A}

Amalgamation of the two simulated sets of binding data from