The cooperative binding of large ligands to a one-dimensional lattice: the steric hindrance effect

The cooperative binding of monomeric ligands to a long lattice of a linear polymer with complete or partial steric hindrance is treated using a matrix method. Results and typical calculations of the model are represented. Non-saturated cooperative binding as well as two-step (biphasic) binding isotherms can be interpreted by the steric hindrance model. This is applicable to the analysis of the binding of surfactants to polymer. The usefulness and the limitation also are discussed.     

On the cooperative binding of large ligands to a one-dimensional homogeneous lattice: the generalized three-state lattice model

We present the general secular equation for three-state lattice models for the cooperative binding of large ligands to a one-dimensional lattice. In addition, a closed-form expression for the isotherm is also obtained, that can be used with all values of the cooperativity parameter omega(0 less than omega less than infinity) thus eliminating the need for multiple equations.     

Cooperative and non-cooperative binding of large ligands to a finite one-dimensional lattice: A model for ligand-ougonucleotide interactions

A combinatorial approach is employed to calculate exact expressions for the extent of binding to a finite one dimensional lattice of ligands which cover more than one lattice site. The binding may be either cooperative or non-cooperative. It is found that the assumption of an effectively infinite lattice is generally a good one, except with relatively low concentrations of strongly cooperative ligands. An approach to analyzing experimental data is suggested which makes explicit use of the lattice length dependence of binding to extract more information about the binding parameters than can be obtained using the infinite lattice approximation. It is shown that irreversible binding cannot be viewed as a limiting case of reversible binding- The reasons for this difference are discussed, and expressions for the extent of irreversible binding are derived.     

Methods for the analysis of the kinetics of cooperative binding of ligands to a two-site lattice

Solutions are obtained that describe the time dependence of the reversible binding of a ligand to a two-site lattice. The binding may be cooperative. Three methods are used to obtain these solutions: the separation of on/off processes with a variable transformation, the asymptotic series analysis, and the singular perturbation procedure. Applications to parameter calculation from experimental data are presented. This kinetic system is such that it is difficult to extract all kinetic parameters from data analysis, and the implications for each method are also discussed.     

Theoretical aspects of DNA-protein interactions: co-operative and non-co-operative binding of large ligands to a one-dimensional homogeneous lattice

The interaction of proteins binding non-specifically to DNA, as well as the properties of many other interacting ligand-lattice systems important in molecular biology, requires a fundamentally different type of theoretical analysis than that provided by the classical Scatchard independent-binding-site treatment. Exact and relatively simple equations describing the binding of both non-interacting and interacting (co-operative) ligands to a homogeneous one-dimensional lattice are derived in terms of ligand site size, intrinsic binding constant and ligand-ligand co-operativity (equations (10) and (15) in the text). The mathematical approach is based on simple conditional probabilities, and reveals some largely unrecognized characteristics of such lattice binding systems. The results indicate that the binding of any non-interacting ligand covering more than one lattice residue results in non-linear (convex downward) Scatchard plots. The introduction of positive ligand-ligand co-operativity antagonizes this non-linearity, and eventually leads to plots of the opposite curvature. The maxima, limiting slopes, and intercepts of such plots can be used to estimate the required binding parameters. The method can be extended to systems involving heterogeneous ligands, and some types of heterogeneous lattices. Procedures for applying the method to a variety of interacting systems are presented, and a preliminary analysis is carried out for some selected sets of data from the literature.     

Kinetic mechanism of rat polymerase beta-dsDNA interactions. Fluorescence stopped-flow analysis of the cooperative ligand binding to a two-site one-dimensional lattice

Kinetics of cooperative binding of rat polymerase beta to a double-stranded DNA has been studied using the fluorescence stopped-flow techniques. The data have been analyzed by an approach developed to examine complete kinetics of cooperative large ligand binding to a one-dimensional lattice. The method is based on using the smallest possible system that preserves key ingredients of cooperative binding; i.e., at saturation, the lattice can accept only two ligand molecules. It allows the identification of collective amplitudes as well as amplitudes describing particular normal modes of the reaction. The mechanism of the intrinsic binding of pol beta to the dsDNA is different from the analogous mechanism for the ssDNA. The difference originates from different enzyme orientations in the corresponding complexes. Intrinsic binding to the dsDNA includes only two sequential steps: a very fast bimolecular association followed by an energetically favorable conformational transition of the complex. The transition following the bimolecular step does not facilitate the engagement of the enzyme in cooperative interactions. Its role seems to be reinforcing the affinity of the bimolecular step. Salt and magnesium cations affect both the bimolecular step and the conformational transition. As a result, the bimolecular step is less sensitive to the increased salt concentration, allowing the enzyme to preserve its initial dsDNA affinity. The changing character of cooperative interactions between bound enzyme molecules as a function of NaCl concentration and MgCl(2) concentration does not affect the binding mechanism. The engagement in cooperative interactions is approximately 3-4 orders of magnitude slower than the conformational transition of the DNA-bound polymerase. The importance of the obtained results for the pol beta activities is discussed.     

Binding of ligands to a one-dimensional heterogeneous lattice. I. General model for the calculation of binding isotherms by a Monte Carlo approach

A Monte Carlo method is presented to calculate equilibria for the binding of ligands to one-dimensional heteropolymers. Equivalency with other methods suitable for particular cases was verified (i.e., matrix and combinatorial methods). The principal interest of this Monte Carlo method is in its facility for adaptation to any physically conceivable binding model and that it gives access to the parameters accounting for partial binding to each different type of site. General properties of binding isotherms with excluded-site effects and relations between partial binding ratios and partial free site ratios are discussed. An effective calculation is presented for illustration of the method.     

Binding of ligands to a one-dimensional heterogeneous lattice. II. Intercalation of tilorone with DNA and polynucleotides

The interaction of tilorone with DNA and five synthetic polydeoxyribonucleotides [(I): poly[d(A-T)]·poly[d(A-T)]; (II): poly[d(A-C)]·poly[d(G-T)]; (III): poly[d(G-C)]·poly[d(G-C)]; (IV): poly(dG)·poly(dC); and (V): poly(dA)·poly(dT)] has been investigated. Binding isotherms for the homopolymers were obtained by microdialysis equilibria using ¹⁴C-labeled tilorone and interpreted with different models: exclusion effect, associated or not associated with cooperativity, or variable exclusion. Affinity appears to be related more to local structure than to base composition and decreases in the following order: (I) > (II) > (III) > (IV) > (V). Intercalation in circular DNA was demonstrated by electrophoresis migration and electron microscopy, which yielded an average unwinding angle of 7° per bound dye. The behavior observed in CD and UV spectroscopy shows a sequence similar to the affinities. Tilorone seems to be less intercalated in (IV) and not at all in (V). The experimental binding isotherm of tilorone to DNA was well fitted on the basis of a model where DNA acts as a heterogeneous lattice built with the six different possible couples of adjacent base pairs, each potential site behaving as if it were in the corresponding homopolymer. The results are discussed in terms of specificity of alternating Pyr-Pur sequences and related to theoretical calculations on intercalation energies of DNA.     

Linear cooperative binding of large ligands involving mutual exclusion of different binding modes

A rigorous treatment is given for mutually exclusive multiple mode cooperative binding on a linear structure of equivalent binding "contacts". This will be of special interest with regard to larger ligands implicating the possibility that there are different kinds of binding interactions with more than one monomeric sub unit of a linear biopolymer. Quantitative evaluation of binding properties is shown to be essentially based on calculating the largest root of an algebraic equation. The whole procedure can be practically executed by means of a fairly simple computer program. Various typical examples comprising only two modes are discussed in more detail. For some nucleotide-polylysine systems definite binding parameters have been determined from pertinent experimental data.     

A general formulation of the free energy of interaction in cooperative ligand binding: application to hemoglobin

A general method for calculating the change in free energy of cooperative interaction upon partial or complete liganding of a protein containing an arbitrary number of binding sites is presented. This method is used to estimate the change in free energy of interaction upon oxygenation of hemoglobin under physiological conditions, which is compared to the internal energy change upon binding oxygen to the non-interacting binding site.     

Energetics of protein–DNA interactions: An exact calculation for binding of ligands to a lattice of overlapping sites

Exact equal ions are developed for analyzing the binding of ligands to a linear lattice of overlapping sites in which occupied–unoccupied as well as occupied–occupied interactions are included for the analysis of the binding isotherms. We demonstrate that positive cooperativity on the binding of ligands to multiple sites may derive from either occupied–unoccupied or occupied–occupied interactions. When the binding of proteins to linear polynucleotides and DNA has exhibited positive cooperativity protein–protein (occupied–occupied), interactions have heretofore been invoked as the sole energetic source in determining the cooperative effect. Models and equations developed previously for the analysis of these binding isotherms have included only the protein–protein interactions (usually characterized with the symbol ω). The exact equations of this paper are capable of analyzing binding data in a manner to evaluate the relative importance of both occupied–unoccupied and occupied–occupied interactions Relations derived here are employed to analyze some existing data, and the resulting parameter values are compared to those developed with equations employing only the protein–protein (occupied–occupied) interactions. The resulting parameter values are qualitatively different. Values of the binding constants differ by about three orders of magnitude. When only protein–protein interactions are taken into account, the resulting free energy of interaction is negative, indicating attractive forces between bound protein molecules; when both occupied–unoccupied and occupied–occupied interactions are applied, the resulting free energies of interaction are positive, indicating destabilizing forces acting primarily on the polynucleotide lattice. © 1995 John Wiley & Sons, Inc.     

A filter assay for binding of labeled ligands to membrane-bound receptors

A filter assay was developed for the specific binding of labeled ligands to the acetylcholine receptor in an unpurified suspension of membrane fragments from Torpedo californica. Binding to the filter membrane and the standard deviation of replicates were studied, and it was possible to reduce relative standard deviation to about 3% by accounting for the variable weight of the particles dispensed, stopping the loss of particles to the filter apparatus, regulating the transmembrane pressure during filtration, controlling the time the filter remains subject to suction after all the liquid has passed through the filter, and correcting for variable counting efficiency. A method for dispensing equal aliquots of suspended particulates is also described.     

Theoretical aspects of specific and non-specific equilibrium binding of proteins to DNA as studied by the nitrocellulose filter binding assay. Co-operative and non-co-operative binding to a one-dimensional lattice

The analysis of equilibrium binding isotherms obtained by methods such as the nitrocellulose filter binding assay, which measure the fraction. θ, of DNA to which at least one protein molecule is bound, as a function of the free protein concentration (L_F) require a different type of theoretical framework from that required for analysis of conventional equilibrium binding data, in which the number of moles of protein bound per mole of DNA, θ_c is measured as a function of L_F. The theoretical framework required to analyse equilibrium binding data generated by measuring θ(L_F) is developed for co-operative and non-co-operative binding of a protein to a large number of non-specific sites and to a specific sites(s) in the presence of a large number of non-specific sites on a DNA molecule. The theory is simple to apply, equations for θ(L_F) being easy to derive and evaluate, and is suitable for least-squares analysis. Two examples of the application of the theory to the analysis of experimental data are provided for the specific and non-specific binding of the EcoRI restriction endonuclease to bacteriophage λ DNA, and for the specific and non-specific binding of the enzyme dihydrofolate reductase from Lactobacillus casei to pBR322 and pWDLcB1 DNA, the latter differing from the former only in a 2.9 × 10³ base-pair insert containing the L. casei dihydrofolate reductase structural gene. The theoretical and experimental advantages and disadvantages of measuring θ(L_F) rather than θ_c(L_F) are discussed.     

Evidence for a quaternary structure change in the cooperative binding of cyanide to ferrihemoglobin A.

At pH 6.5 in a 0.05 $\text{M}$ bis-Tris-0.1 $\text{M}$ Cl^? buffer, tetra aquo ferrihemoglobin A (HbA⁺) binds CN^? with a Hill coefficient of n = 1.4. The Hill coefficient increases slightly and the average CN^? affinity decreases in the presence of excess spin labeled triphosphate (SLTP). This is probably the result of the finding that the SLTP exhibits a twofold higher affinity for HbA⁺ than for tetra cyano HbA⁺. Over the course of heme saturation with CN^?, a certain fraction of the SLTP is specifically released. This shows linkage between organic phosphate binding and heme ligation. These findings bear a marked resemblance to the ligand binding phenomena in hemoglobin A (HbA) and provide good evidence that under these experimental conditions, HbA⁺ is undergoing a quaternary conformation change as the hemes become saturated.     

Map analysis of ligand binding to a linear lattice

A one-dimensional mapping of the binding properties of a linear lattice offers an exact analytical solution for the site-specific properties of the lattice once the length N and the parameter for nearest neighbor interactions are specified. The solution is derived independent of the definition of the partition function or the transfer matrix, nor does it involve combinatorial arguments. This result provides a simple and effective way of analyzing experimental data for protein-ligand interactions and broadens our understanding of site-specific properties in biological macromolecules.     

A cooperative model for ligand binding to biological macromolecules as applied to oxygen carriers

This paper presents a model describing the thermodynamics of cooperative ligand binding to multimeric biological macromolecules and integrating some of the features of the two-state and induced-fit models. The protein is taken to be partitioned into a number of noninteracting functional constellations, each one existing in two possible quaternary conformations. Furthermore, the model postulates that a functional constellation is organized in several subsets of sites (called cooperons), in which subunits interact according to an induced-fit mechanism. In the present version the number of subunits forming a cooperon has been limited to two and the total number of parameters used for fitting experimental data is four, all having a precise physical meaning. Although the present application is limited to oxygen-carrying proteins (hemoglobins, hemocyanins, erythrocruorins), the model appears suitable to describe other biological macromolecules with functional interactions.