Kinetics of large-ligand binding to one-dimensional lattices: theory of irreversible binding

Exact solutions are obtained for the time dependence of the extent of irreversible binding of ligands that cover more than one lattice site to a homogeneous one-dimensional lattice. The binding may be cooperative or noncooperative and the lattice either finite or infinite. Although the form of the solution is most convenient when the ligand concentration is buffered, exact numerical or approximate analytical solutions, including upper and lower bounds, can be derived for the case of variable ligand concentration as well. The physical reason behind the relative simplicity of the kinetics of irreversible as opposed to reversible binding in such systems is discussed.     

Cooperative ligand-lattice binding. Approximate Gaussian binding distribution

Nearest-neighbor cooperative binding of a ligand covering n sites and binding with equilibrium constant K and cooperativity factor omega to a large molecule with m binding sites (m much greater than n omega, n/omega) can be approximately described by a Gaussian distribution P(q-qmax), where q is the number of ligands bound and qmax the most probable value of q. The variance of the Gaussian is equal to the derivative dqmax/d ln(L), where L is the free ligand concentration. This variance, sigma 2, is a complicated function of qmax. However, in the limits of very large cooperativity, omega much greater than 1, very large anticooperativity, omega much less than 1, or noncooperativity, omega = 1, simpler expressions for sigma 2 can be given. For qmax = m/(n + 1), where the most probable number of bound ligands equals the number of free binding sites, sigma 2 has a particularly simple form: sigma 2 = 2m omega 1/2/(n + 1)3. The Gaussian and the infinite lattice approximations for the average number of ligands bound are good approximations only if sigma is much smaller than the number of binding sites. The variance may therefore provide an easy check on the validity of the infinite lattice approximation, which is commonly used to analyze experimental binding data.     

On the analysis of linear binding effects associated with curved Scatchard plots.

First the question is examined as to which binding data, especially if given as Scatchard plots, can be described in terms of a basic model mechanism. This referes to a linear lattice of equivalent binding sites (as for example located on a linear biopolymer) which can exert cooperative interaction between nearest neighbors. It is shown that the effect of overlapping of potential site (so-called "multiple-contact"binding), as may occur with larger ligands, will largely be compensated by a higher degree of cooperativity. Therefore, in practice such properties can scarcely be separated by means of oridnary binding experiments. A pronounced inflection point in the Scatchard plot turns out to be clearly indicating a more complex mechanism involving at least two rather antagonistic cooperative interactions which may, however, occur even between equivalent binding sites. Finally some consequences of different classes of bindings sites are considered. In particular a simple approach is introduced by which the binding to mutually exclusive classes of sites may be described. Such a model is of interest for multiple-mode binding of ionic ligands to oppositely charged polymers.     

Consequences of the non-specific binding of a protein to a linear polymer: reconciliation of stoichiometric and equilibrium titration data for the thrombin-heparin interaction

Theoretical aspects of the thermodynamic characterization of cooperative protein interactions with non-specific segments of a linear polymer lattice have been re-examined. This reconsideration has not only provided an alternative derivation of recursive expressions for the stoichiometry of random ligand binding prior to elimination of the parking problem but also extended that treatment to include binding with overlap of additional lattice units. The major obstacle to thermodynamic characterization of non-specific protein-polymer interactions is determination of the lattice capacity for ligand, which in turn defines the length of the polymer segment to which the protein binds. Although these parameters are most readily obtained from studies under conditions that ensure essentially stoichiometric interaction, the endpoint of such a titration is likely to reflect the irreversible rather than the equilibrium binding capacity of the lattice for ligand. Consideration of published results for spectrofluorometric titrations of the thrombin-heparin system under stoichiometric conditions in such terms has permitted their reconciliation with results of a later publication on the interaction under equilibrium conditions.     

Use of binding site neighbor-effect parameters to evaluate the interactions between adjacent ligands on a linear lattice. Effects on ligand-lattice association.

A method using binding site "neighbor-effect" parameters (NEPs) is introduced to evaluate the effects of interaction between adjacent ligands on their binding to an infinite linear lattice. Binding site overlap is also taken into account. This enables the conditional probability approach of McGhee & von Hippel to be extended to more complex situations. The general equation for the isotherm is v/LF = SFKF, where v is the ratio of bound ligands to lattice residues, LF is the free ligand concentration, SF is the fraction of binding sites that are free, and KF is the average association constant of a free site. Solutions are derived for three cases: symmetric ligands, and asymmetric ligands on isotropic or anisotropic lattices. For symmetric ligands there is one NEP, E, which is the ratio of the average binding affinity of a free site if the status of the lattice residue neighboring one end of the site is unspecified (left to chance) to the affinity when this residue is free (holding the other neighbor constant). Thus KF is KE2, where K is the affinity of an isolated site. If a site is n residues long, SF is f ffn-1, where f = 1 - nv is the fraction of residues that are free and ff is the conditional probability that a free residue is bordered on a given side by another free residue. The expression for ff is 1/(1 + x/E), where x is v/f, E is (1 - x + [(1 - x)2 + 4x omega]1/2)/2, and omega is the co-operativity parameter. The binding of asymmetric ligands to an isotropic lattice is described by two NEPs; the last case involves four NEPs and a bound ligand orientation parameter. For each case, the expected length distribution of clusters of bound ligands can be calculated as a function of v. When Scatchard plots with the same intercepts and initial slope are compared, it is found that ligand asymmetry lowers the isotherm (relative to the corresponding symmetric ligand isotherm), whereas lattice anisotrophy raises it.     

Cooperative effects during the binding of large ligands with DNA. Non-contact interaction between adsorbed ligands

Equations are derived to describe the cooperative binding of large ligands to DNA. A mathematical approach is developed which enables one to give a simple probabilistic interpretation of binding equations and to solve them in the general case when long-range interactions are allowed between bound ligands. These interactions can be mediated by conformation changes induced in the DNA in the course of binding process and transformed over some distances beyond the DNA region immediately covered by a bound ligand molecule (allosteric effect of DNA). Interactions between ligand molecules can be formally described in terms of model potential characterizing pairwise interactions between bound ligands. A procedure is developed which allows one to determined the form of such potential from experimentally measured binding isotherms. It is based on a comparison of experimental binding isotherms with the appropriate curves calculated for the case of non-interacting ligands.     

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.     

Interactions of Escherichia coli primary replicative helicase DnaB protein with nucleotide cofactors.

Interactions between the Escherichia coli primary replicative helicase DnaB protein and nucleotide cofactors have been studied using several fluorescent nucleotide analogs and unmodified nucleotides. The thermodynamically rigorous fluorescent titration technique has been used to obtain true binding isotherms, independently of the assumptions of any relationships between the observed quenching of protein fluorescence and the degree of nucleotide binding. Fluorescence titrations using several MANT derivatives of nucleoside diphosphates (MANT-ADP, 3',2'-O-(N-methylantraniloyl)adenosine-5'-diphosphate; MANT-GDP, 3',2'-O(N-methylantraniloyl)guanosine-5'-diphosphate; MANT-CDP, 3',2'-O-(N-methylantraniloyl)cytidine-5'-diphosphate; MANT-UDP, 3',2'-O-(N-methylantraniloyl)uridine-5'-diphosphate) have shown that the DnaB helicase has a preference for purine nucleotides. Binding of all modified nucleotides is characterized by similar negative cooperativity, indicating that negative cooperative interactions are base-independent. Thermodynamic parameters for the interactions of the unmodified nucleotides (ADP, GDP, CDP, and UDP) and inorganic phosphate (P(i)) have been obtained by using the competition titration approach. To analyze multiple ligand binding to a finite circular lattice, for a general case in which each lattice binding site can exist in different multiple states, we developed a matrix method approach to derive analytical expressions for the partition function and the average degree of binding for such cases. Application of the theory to competition titrations has allowed us to extract the intrinsic binding constants and cooperativity parameters for all unmodified ligands. This is the first quantitative estimate of affinities and the mechanisms of binding of different unmodified nucleotides and inorganic phosphate for a hexameric helicase. The intrinsic affinities of all of the studied ATP analogs are lower than the intrinsic affinities of the corresponding ADP analogs. The implications of these results for the mechanism of helicase action are 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.     

Thermodynamic model of cooperativity in a dimeric protein: unique and independent parameters formulation.

A model of the cooperative interaction of ligand binding to a dimeric protein is presented based upon the unique and independent parameters (UIP) thermodynamic formulation (Gutheil and McKenna, Biophys. Chem. 45 (1992) 171-179). The analysis is developed from an initial model which includes coupled conformational and ligand binding equilibria. This completely general model is then restricted to focus on conformationally mediated cooperative interactions between the ligands and the expressions for the apparent ligand binding constant and the apparent ligand-ligand interaction constant are derived. The conditions under which there is no cooperative interaction between the ligands are found as roots to a polynomial equation. Consideration of the distribution of species among the various conformational states in this general model leads to a set of inequalities which can be represented as a two dimensional plot of boundaries. By superimposing a contour plot of the value of the apparent ligand-ligand interaction constant over the plot of boundaries a complete graphical representation of this system is achieved similar to a phase diagram. It is found that the parameter space homologous to Koshland-Nemethy-Filmer type of model is most consistent with both positive and negative cooperativity in this model. The maximal amount of positive and negative cooperativity are found to be simple functions of Kc, the equilibrium constant associated with the change of a subunit and ligand from the unligated to ligated conformation. It is shown that under certain limiting conditions the apparent allosteric interaction between ligands is equal to the conformational interaction between subunits. The methods presented are generally applicable to the theoretical analysis of thermodynamic interactions in complex systems.     

Cooperation effects in binding of large ligands to DNA. II. Contact interactions between adsorbed ligands

Cooperative effects arising upon binding of biologically active ligands to DNA are considered. Equations are derived which enable one to describe the binding of two different ligands to DNA. We also consider the case when ligand can form two type of DNA complexes. The cooperative binding of the ligand in the vicinity of saturation level of binding can be described with a good accuracy by equation derived for the non-cooperative adsorption of the same ligand with some effective binding constant Keff. It is shown that cooperative effects arising upon binding of proteins and other ligands to DNA can be divided into two groups depending on the symmetry of interactions between the bound ligand molecules. In particular, if such interactions favor the formation of dimeric ligand species on the DNA, Keff approximately a1/2, where a is the ligand-ligand interaction constant. If cooperative interactions favor the formation of aggregates of unrestricted size, then Keff approximately aL+Y, where L is the size of the binding site for the ligand on DNA.     

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.     

Binding modes of 6,7 di-substituted 4-anilinoquinoline-3-carbonitriles to EGFR

4-Anilino-3-cyanoquinolines were reported to have irreversible binding to epidermal growth factor receptor kinase (EGFRK) by forming a covalent linkage to C773. Our initial docking studies gave results inconsistent with the in vitro data and showed two different binding modes. To perceive the exact mode of binding of these ligands, two models of the ligand-EGFR complexes were considered: (1) reversible binding mode in which the ligand had hydrogen bond interactions at the binding site and (2) irreversible binding mode wherein the ligand's Michael acceptor side chain has proximity to the sulfhydryl group of C773 of EGFR, thereby enabling a covalent interaction. The irreversible binding mode correlated better than reversible binding mode with respect to in vitro data. However, our results indicate that both modes are being adopted by the ligands and could be utilized to design more potent EGFRK inhibitors.     

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.     

Ligand binding to proteins: the binding landscape model.

Models of ligand binding are often based on four assumptions: (1) steric fit: that binding is determined mainly by shape complementarity; (2) native binding: that ligands mainly bind to native states; (3) locality: that ligands perturb protein structures mainly at the binding site; and (4) continuity: that small changes in ligand or protein structure lead to small changes in binding affinity. Using a generalization of the 2D HP lattice model, we study ligand binding and explore these assumptions. We first validate the model by showing that it reproduces typical binding behaviors. We observe ligand-induced denaturation, ANS and heme-like binding, and "lock-and-key" and "induced-fit" specific binding behaviors characterized by Michaelis-Menten or more cooperative types of binding isotherms. We then explore cases where the model predicts violations of the standard assumptions. For example, very different binding modes can result from two ligands of identical shape. Ligands can sometimes bind highly denatured states more tightly than native states and yet have Michaelis-Menten isotherms. Even low-population binding to denatured states can cause changes in global stability, hydrogen-exchange rates, and thermal B-factors, contrary to expectations, but in agreement with experiments. We conclude that ligand binding, similar to protein folding, may be better described in terms of energy landscapes than in terms of simpler mass-action models.