Unusual shapes of absorption isotherms in three-component systems

It has been shown that the shape of Scatchard isotherms upon competitive binding of two ligands to the same binding site in the three-component ligand 1-ligand 2-DNA system depends crucially on the binding constant values. The binding isotherm of ligand 2 in the presence of the competitive ligand 1 turns back (has a bow-like form) when the binding constant of the first ligand is larger than the binding constant of the second one.     

Ligand binding isotherm for DNA in the presence of supercoil-induced non-B form: a theoretical analysis

A binding isotherm in the form of a modified McGhee-Von Hippel equation is proposed, on the basis of thermodynamical considerations, to include the non-cooperative binding of extended ligands to supercoiled DNA, where a stretch of non-B form may be present under superhelical stress. It is then studied, on the basis of a non-linear Scatchard plot, how the presence of an intercalating ligand can relax the supercoiled molecule and thus destabilise the non-B stretch, which may be recognised by the existence of a significant kink in the Scatchard plot.     

New plot showing the existence of cooperativity, anticooperativity, and an arbitrary value of the neighbor-exclusion parameter in systems of ligand binding to linear biopolymers

A new way of plotting the isotherm for ligand binding to linear biopolymers is presented. In this plot the isotherm for noncooperative binding of ligands of length n (n-mers) becomes a straight line and the existence of cooperativity and anticooperativity between bound ligands is detected by appearance of opposite convexity of the curved isotherm. It is also usable for cases of fractional n values. Usefulness of the new plot in determining precise mechanisms of binding is shown using experimental data.     

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.     

Calculations of DNA Condensation Caused by Ligand Adsorption

A method for calculating the curves of DNA transition from linear to condensed state upon binding of condensing ligands has been developed. The character of the transition and ligand concentration necessary for condensation have been shown to be governed by the length of DNA molecule, energy and stoichiometry parameters of the DNA–ligand complex (equilibrium constant between linear and condensed form in the absence of ligands, constants for ligand binding to linear and condensed forms, the number of base pairs covered by one ligand, etc.). The results of the calculations indicate that a slight difference in the free energies of these DNA states (less than 6 cal/mol(bp) for a DNA of 500 bp) is sufficient for the existence of a stable linear state in the absence of ligands (in free DNA) and the formation of stable condensed state upon complexation.     

Calculation of DNA condensation, caused by adsorption of ligands

A method for calculating the curves of DNA transition from linear to condensed state upon binding of condensing ligands has been developed. The character of the transition and ligand concentration necessary for condensation have been shown to be governed by the length of DNA molecule, energy and stoichiometry parameters of DNA-ligand complex (equilibrium constant between linear and condensed form in the absence of ligands, constants for ligand binding to linear and condensed forms, the number of base pairs covered by one ligand, etc.). The results of the calculations indicate that only slight difference in the free energies of these states in free DNA (less than 6 cal/mole(bp) for DNA of 500 bp long) is sufficient for the existence of stable linear state in the absence of ligands (in free DNA) and the formation of stable condensed state upon complexation.     

The effect of ligand heterogeneity on the Scatchard plot. Particular relevance to lipoprotein binding analysis

Computer simulations of equilibrium binding studies of a mixture of two labeled ligands binding competitively to a single class of identical and independent sites (receptors) were performed to investigate how ligand heterogeneity affects the observed data in such studies. The simulated data are presented in Scatchard plots. Ligand heterogeneity was generally found to be indistinguishable from the case of a homogeneous ligand when usual experimental conditions applied (that is, Scatchard plots of the data were straight lines). Some factors that increased the probability of recognizing heterogeneity in the system were identified, however. These are 1) a large difference between the dissociation constants of the two ligands, 2) a high concentration of receptors relative to the dissociation constant of the higher-affinity ligand, 3) a high concentration of the lower-affinity ligand relative to that of the higher-affinity ligand, 4) a high specific activity of the lower-affinity ligand relative to that of the higher-affinity ligand, and 5) lack of experimental error. When ligand heterogeneity (under certain conditions) did cause curvilinearity in the Scatchard plot, the curve formed was always concave-downwards. Thus, ligand heterogeneity may occasionally mimic positive cooperativity, but never mimics negative cooperativity or multiple classes of binding sites. Implications of these findings for equilibrium binding studies involving lipoproteins (which are generally isolated as heterogeneous mixtures of particles) are discussed in detail. These findings are also relevant to equilibrium binding studies using ligands which are mixtures of stereoisomers or which contain chemical or radiochemical impurities.     

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.     

Kinetic parameters of calcium binding to tissue structures in human acute hypercalcemia

The kinetics of calcium binding by tissue structures was analyzed in 19 healthy volunteers (13 men and 6 women) from the age group of 33 ± 6.5 years under the conditions of acute hypercalcemia followed by drip intravenous infusion of calcium gluconate for 2.5 h. At the end of each 30-min period, the calcium amount retained by the tissue structures was recorded and the kinetic parameters of calcium binding were determined according to Langmuir and Scatchard. In all volunteers, there was a region of binding isotherm with positive cooperativity (linear regression in Scatchard’s coordinates) with the same buffer capacity (β_tis) for calcium in Langmuir’s coordinates (0.58 ± 0.24 L kg). Half of the volunteers exhibited cooperativity at [Ca²⁺] = 1.3–1.5 mmol/L, while others, at [Ca²⁺] = 1.0–1.3 mmol/L, which corresponded to the differences in the association constant (K_a) and the number of interactive sites (n) with [Ca²⁺] = 1 mmol/L. Additionally, two segments of the binding isotherm were detected with the successive binding of calcium to one set of non-interactive centers with similar kinetic parameters of calcium binding (β_tis, K_a, n). Four different types of calcium binding in healthy volunteers were found. The results of this study may serve as the basis for a functional diagnostic test of disorders of the tissue calcium-binding properties in different pathological conditions.     

An automated continuous-flow dynamic dialysis technique for investigating protein-ligand binding

An automated, continuous-flow dynamic dialysis technique has been developed to investigate protein-ligand binding. The method depends on a comparison of the diffusion of the low molecular mass ligand, in the presence and absence of protein, through a semipermeable membrane. The ligand passes from the sample compartment of a dialysis cell into the sink compartment through which a constant flow of eluting buffer is maintained. Digitized spectrophotometric determinations of the ligand concentration in the eluting buffer at successive, equally spaced time intervals, punched onto paper tape, provide the primary data (normally about 1000 data points). A mathematical treatment of the data based on a model of the diffusion system, whereby the protein-ligand binding isotherm may be evaluated, is discussed. The validity of the method is demonstrated from studies of the binding of phenol red to bovine serum albumin (BSA) at 15, 20, and 25°C. The method yields a large number of points on the binding isotherm (usually several hundred) which, in terms of a Scatchard model, provide values for the number of binding sites on the BSA molecule and binding constants for the phenol red-BSA interaction. The results obtained are consistent with values reported in the chemical literature but which are based on much scantier data.     

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.     

Ligand binding on ladder lattices

The ligand binding problems on two-dimensional ladders, which model many important binding phenomena in molecular biology, are studied in details. The model is represented by four parameters, the interactions between ligands when bound to adjacent sites on opposite legs of the ladder (tau), the interactions between bound ligands in the longitudinal direction of the ladder (sigma), the number of binding sites that are covered by a bound ligand (m), and the intrinsic binding constant (K). The partition functions of ring ladders are approached with the transfer matrix method. A general relation is derived which connects the partition function of a linear ladder with that of a ring ladder. The results obtained apply to the general situation of multivalent binding, in which m>1. Special attention is paid to the case where the ligand covers one site (m=1). In this case explicit formulas are given for the partition functions of ring and linear ladders. Closed-form expressions are obtained for various properties of the system, including the degree of binding (theta), the midpoint in the binding isotherm (1/square root(tau sigma)), the initial and end slopes of the Scatchard plots (2sigma + tau - 4 and -sigma2 tau, respectively). From these closed-form formulas, sigma and tau may be extracted from experimental data. The model reveals certain features which do not exist in one-dimensional models. Using the general method discussed in [1], the recurrence relation is found for the partition functions. The analytical solution found for this model provides test cases to verify the numerical results for more complex two-dimensional models.     

Study of the binding of thyroxine by thyroxine-binding prealbumin by a dialysis method using a fixed free concentration of ligand (author's transl)]

In order to study ligand-protein binding in solution, a dialysis method was used in which the free concentration of ligand can be controlled. The method has certain advantages and was applied to the binding of thyroxine by thyroxine-binding prealbumin, a system about which the results found in the literature are not in good agreement. From the isotherm drawn according to the Scatchard plot, it was found that thyroxine-binding prealbumin only presents a single binding site for thyroxine per molecule, the association constant being 1.7 . 10(8) M-1.     

Short-range interactions and size of ligands bound to DNA strongly influence adsorptive phase transition caused by long-range interactions

Long-range interaction between all the ligands bound to DNA molecule may give rise to adsorption with the character of phase transition of the first kind (D. Y. Lando, V. B. Teif, J. Biomol. Struct & Dynam. 18, 903-911 (2000)). In this case, the binding curve, c(c(o)), is characterized by a sudden change of the relative concentration of bound ligands ((c)) at a critical concentration of free (unbound) ligands, c(o)=c(ocr), from a low c value to a high one where c(o) is molar concentration of free ligands. Such a transition might be caused by some types of DNA condensation or changes in DNA topology. For the study of the conditions necessary for adsorption with the character of phase transition, a calculation procedure based on the method of the free energy minimum is developed. The ligand size and two types of interactions between ligands adsorbed on DNA molecule are taken into consideration: long-range interaction between all the ligands bound to DNA and contact interactions between neighboring ligands. It was found that a) Stronger long-range interaction is required for longer ligands to induce phase transition that is occurred at greater c(ocr) values; b) Pure contact interaction between neighboring ligands can not itself initiate phase transition. However contact cooperativity strongly decreases the threshold value of energy of long-range interaction necessary to give rise to the transition.     

Distribution functions, describing the binding of extended ligands with DNA molecules. Possible use for cases of DNA condensation

Due to noncooperative binding of ligands to DNA molecules, DNA molecules are in equilibrium with different numbers of adsorbed ligands. This equilibrium for a given concentration of the free ligand in the solution is characterized by the distribution function, which describes the probability of revealing the DNA molecule with a definite number of adsorbed ligands. If polycations act as ligands, DNA molecules with the number of ligands sufficient for neutralizing the charges on phosphates may undergo a phase transition. One example of this transition is the formation of liquid-crystalline dispersions during the binding of DNA to chitosan. We analyzed the binding of chitosan to DNA on the assumption that this binding is due to equilibrium adsorption. At a definite concentration of chitosan in solution, DNA molecules are in equilibrium with different numbers of adsorbed molecules of chitosan. If the number of adsorbed ligands exceeds some critical value, the DNA molecule covered with chitosan becomes capable of interacting with other DNA molecules. As a result of this interaction (attraction), liquid-crystalline dispersions can form. Equations describing the dependence of the concentration of DNA molecules on the concentration of the ligand in solution were derived. It was shown that, at given parameters of the model, it is possible to describe experimental data characterizing the formation of cholesteric liquid-crystalline dispersions. The analysis of the data makes it possible to reconstitute both the size of the binding site occupied by chitosan on the DNA and the energy of interaction of chitosan with DNA.     

Interaction of phenylthiolato-(2,2'',2"-terpyridine)platinum(II) cation with DNA.

The interaction between a novel aromatic thiolato derivative from the family of DNA-intercalating platinum complexes, phenylthiolato-(2,2',2"-terpyridine)platinum(II)-[PhS(ter py)Pt+], and nucleic acids was studied by using viscosity, equilibrium-dialysis and kinetic measurements. Viscosity measurements with sonicated DNA provide direct evidence for intercalation, and show that at binding ratios below 0.2 molecules per base-pair PhS(terpy)Pt+ causes an increase in contour length of 0.2 nm per bound molecule. However, helix extension diminishes at greater extents of binding, indicating the existence of additional, non-intercalated, externally bound forms of the ligand. The ability of PhS(terpy)Pt+ to aggregate in neutral aqueous buffers at a range of ionic strengths and temperatures was assessed by using optical-absorption methods. Scatchard plots for binding to calf thymus DNA at ionic strength 0.01 (corrected for dimerization) are curvilinear, concave upward, providing further evidence for two modes of binding. The association constant decreases at higher ionic strengths, in accord with the expectations of polyelectrolyte theory, although the number of cations released per bound unipositive ligand molecule is substantially greater than 1. Stopped-flow kinetic measurements confirm the complexity of the binding reaction by revealing multiple bound forms of the ligand whose kinetic processes are both fast and closely coupled. Thermal denaturation of DNA radically alters the shapes of binding isotherms and either has little effect on, or enhances, the affinity of potential binding sites, depending on experimental conditions. Scatchard plots for binding to natural DNA species with differing nucleotide composition show that the ligand has a requirement for a single G X C base-pair at the highest-affinity intercalation sites.     

Isotherm of ligand adsorption on DNA at multiplicative noise

Fluctuations of the number of ligands adsorbed on macromolecules are considered in the case when the number of ligands in the solution fluctuates under the action of fluctuations of the external medium (external noise). For the case of small filling, the multiplicative type of stochastic differential equation is obtained, describing the time variation of the number of ligands adsorbed on macromolecules. The isotherm of adsorption of ligands on DNA is obtained. It is shown that at small ligand concentrations, for some relations between adsorption parameters and the intensity of the external noise, no macromolecule adsorption of ligands takes place.     

Quantifying force-dependent and zero-force DNA intercalation by single-molecule stretching

We used single DNA molecule stretching to investigate DNA intercalation by ethidium and three ruthenium complexes. By measuring ligand-induced DNA elongation at different ligand concentrations, we determined the binding constant and site size as a function of force. Both quantities depend strongly on force and, in the limit of zero force, converge to the known bulk solution values, when available. This approach allowed us to distinguish the intercalative mode of ligand binding from other binding modes and allowed characterization of intercalation with binding constants ranging over almost six orders of magnitude, including ligands that do not intercalate under experimentally accessible solution conditions. As ligand concentration increased, the DNA stretching curves saturated at the maximum amount of ligand intercalation. The results showed that the applied force partially relieves normal intercalation constraints. We also characterized the flexibility of intercalator-saturated dsDNA for the first time.     

Affinity characterization-mass spectrometry methodology for quantitative analyses of small molecule protein binding in solution

Affinity characterization by mass spectrometry (AC–MS) is a novel LC–MS methodology for quantitative determination of small molecule ligand binding to macromolecules. Its most distinguishing feature is the direct determination of all three concentration terms of the equilibrium binding equation, i.e., (M), (L), and (ML), which denote the macromolecule, ligand, and the corresponding complex, respectively. Although it is possible to obtain the dissociation constant from a single mixing experiment, saturation analyses are still valuable for assessing the overall binding phenomenon based on an established formalism. In addition to providing the prerequisite dissociation constant and binding stoichiometry, the technique also provides valuable information about the actual solubility of both macromolecule and ligand upon dilution and mixing in binding buffers. The dissociation constants and binding mode for interactions of DNA primase and thymidylate synthetase (TS) with high and low affinity small molecule ligands were obtained using the AC–MS method. The data were consistent with the expected affinity of TS for these ligands based on dissociation constants determined by alternative thermal-denaturation techniques: TdF or TdCD, and also consistent enzyme inhibition constants reported in the literature. The validity of AC–MS was likewise extended to a larger set of soluble protein–ligand systems. It was established as a valuable resource for counter screen and structure–activity relationship studies in drug discovery, especially when other classical techniques could only provide ambiguous results.