Cooperative effects on binding of proteins to DNA

The Use of the Statistical Entropy in Some New Approaches for the Description of Biosystems

Some problems of describing biological systems with the use of entropy as a measure of the complexity of these systems are considered. Entropy is studied both for the organism as a whole and for its parts down to the molecular level. Correlation of actions of various parts of the whole organism, intercellular interactions and control, as well as cooperativity on the microlevel lead to a more complex structure and lower statistical entropy. For a multicellular organism, entropy is much lower than entropy for the same mass of a colony of unicellular organisms. Cooperativity always reduces the entropy of the system; a simple example of ligand binding to a macromolecule carrying two reaction centers shows how entropy is consistent with the ambiguity of the result in the Bernoulli test scheme. Particular attention is paid to the qualitative and quantitative relationship between the entropy of the system and the cooperativity of ligand binding to macromolecules. A kinetic model of metabolism. corresponding to Schrödinger's concept of the maintenance biosystems by "negentropy feeding", is proposed. This model allows calculating the nonequilibrium local entropy and comparing it with the local equilibrium entropy inherent in non-living matter.

Cooperative kinetics of ligand binding to linear polymers

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Cooperative kinetic equation for large ligands binding to long polymers.

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Cooperativity in general affects binding and release rates.

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Appropriate counting of the available binding sites for a ligand to a linear polymer.

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Positive cooperativity increases polymer coverage by the ligand.

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Large ligand size reduces cooperativity effects.

Ligands change the chemical and mechanical properties of polymers. In particular, single strand binding protein (SSB) non-specifically bounds to single-stranded DNA (ssDNA), modifying the ssDNA stiffness and the DNA replication rate, as recently measured with single-molecule techniques. SSB is a large ligand presenting cooperativity in some of its binding modes. We aim to develop an accurate kinetic model for the cooperative binding kinetics of large ligands. Cooperativity accounts for the changes in the affinity of a ligand to the polymer due to the presence of another bound ligand. Large ligands, attaching to several binding sites, require a detailed counting of the available binding possibilities. This counting has been done by McGhee and von Hippel to obtain the equilibrium state of the ligands-polymer complex. The same procedure allows to obtain the kinetic equations for the cooperative binding of ligands to long polymers, for all ligand sizes. Here, we also derive approximate cooperative kinetic equations in the large ligand limit, at the leading and next-to-leading orders. We found cooperativity is negligible at the leading-order, and appears at the next-to-leading order. Positive cooperativity (increased affinity) can be originated by increased binding affinity or by decreased release affinity, implying different kinetics. Nevertheless, the equilibrium state is independent of the origin of cooperativity and only depends on the overall increase in affinity. Next-to-leading approximation is found to be accurate, particularly for small cooperativity. These results allow to understand and characterize relevant ligand binding processes, as the binding kinetics of SSB to ssDNA, which has been reported to affect the DNA replication rate for several SSB-polymerase pairs.

Influence of Environmental Parameters on the Stability of the DNA Molecule

Fluctuations in viscosity within the cell nucleus have wide limits. When a DNA molecule passes from the region of high viscosity values to the region of low values, open states, denaturation bubbles, and unweaving of DNA strands can occur. Stabilization of the molecule is provided by energy dissipation-dissipation due to interaction with the environment. Separate sections of a DNA molecule in a twisted state can experience supercoiling stress, which, among other things, is due to complex entropic effects caused by interaction with a solvent. In this work, based on the numerical solution of a mechanical mathematical model for the interferon alpha 17 gene and a fragment of the Drosophila gene, an analysis of the external environment viscosity influence on the dynamics of the DNA molecule and its stability was carried out. It has been shown that an increase in viscosity leads to a rapid stabilization of the angular vibrations of nitrogenous bases, while a decrease in viscosity changes the dynamics of DNA: the rate of change in the angular deviations of nitrogenous bases increases and the angular deformations of the DNA strands increase at each moment of time. These processes lead to DNA instability, which increases with time. Thus, the paper considers the influence of the external environment viscosity on the dissipation of the DNA nitrogenous bases' vibrational motion energy. Additionally, the study on the basis of the described model of the molecular dynamics of physiological processes at different indicators of the rheological behavior of nucleoplasm will allow a deeper understanding of the processes of nonequilibrium physics of an active substance in a living cell to be obtained.

Noncooperative thermodynamics and kinetic models of ligand binding to polymers: Connecting McGhee-von Hippel model with the Tonks gas model

Ligand binding to polymers modifies the physical and chemical properties of the polymers, leading to physical, chemical, and biological implications. McGhee and von Hippel obtained the equilibrium coverage as a function of the ligand affinity, through the computation of the possible binding sites for the ligand. Here, we complete this theory deriving the kinetic model for the ligand-binding dynamics and the associated equilibrium chemical potential, which turns out to be of the Tonks gas model type. At low coverage, the Tonks chemical potential becomes the Fermi chemical potential and even the ideal gas chemical potential. We also discuss kinetic models associated with these chemical potentials. These results clarify the kinetic models of ligand binding, their relations with the chemical potentials, and their range of validity. Our results highlight the inaccuracy of ideal and simplified kinetic approaches for medium and high coverages.

Two-dimensional Ising model for microarray hybridization: cooperative interactions between bound target molecules

The Langmuir adsorption model is widely used for description and quantification of microarray oligo-target hybridization. According to the model, the binding centers for adsorption of target molecules from solution are represented by oligo-probes. However, the Langmuir model does not consider the interactions between the targets adsorbed at the neighboring binding centers, which are possible due to high-density of array-bound probes. We have shown that the two-dimensional Ising model, which takes into account the nearest neighboring target molecules interactions, better describes the experimental data of oligo-target hybridization in comparison with the Langmuir model. Thus, we found an evidence for existence of positive cooperative interactions between adsorbed target molecules: so, binding of the first target molecules facilitates the binding of subsequent ones to the neighboring probes. Communicated by Ramaswamy H. Sarma.

Resolving the contributions of two cooperative mechanisms to the DNA binding of AGT

The O(6)-alkylguanine DNA alkyltransferase (AGT) is a DNA repair enzyme that binds DNA with moderate cooperativity. This cooperativity is important for its search for alkylated bases. A structural model of the cooperative complex of AGT with DNA predicts short-range interactions between nearest protein neighbors and long-range interactions between proteins separated in the array. DNA substrates ranging from 11bp to 30bp allowed us to use differences in binding stoichiometry to resolve short- and long-range protein contributions to the stability of AGT complexes. We found that the short-range component of ΔG°(coop) was nearly independent of DNA length and protein packing density. In contrast the long-range component oscillated with DNA length, with a period equal to the occluded binding site size (4bp). The amplitude of the long-range component decayed from ∼-4 kcal/mole of interaction to ∼-1.2 kcal/mol of interaction as the size of cooperative unit increased from 4 to 7 proteins, suggesting a mechanism to limit the size of cooperative clusters. These features allow us to make testable predictions about AGT distributions and interactions with chromatin structures in vivo.

Calculating transcription factor binding maps for chromatin

Current high-throughput experiments already generate enough data for retrieving the DNA sequence-dependent binding affinities of transcription factors (TF) and other chromosomal proteins throughout the complete genome. However, the reverse task of calculating binding maps in a chromatin context for a given set of concentrations and TF affinities appears to be even more challenging and computationally demanding. The problem can be addressed by considering the DNA sequence as a one-dimensional lattice with units of one or more base pairs. To calculate protein occupancies in chromatin, one needs to consider the competition of TF and histone octamers for binding sites as well as the partial unwrapping of nucleosomal DNA. Here, we consider five different classes of algorithms to compute binding maps that include the binary variable, combinatorial, sequence generating function, transfer matrix and dynamic programming approaches. The calculation time of the binary variable algorithm scales exponentially with DNA length, which limits its use to the analysis of very small genomic regions. For regulatory regions with many overlapping binding sites, potentially applicable algorithms reduce either to the transfer matrix or dynamic programming approach. In addition to the recently proposed transfer matrix formalism for TF access to the nucleosomal organized DNA, we develop here a dynamic programming algorithm that accounts for this feature. In the absence of nucleosomes, dynamic programming outperforms the transfer matrix approach, but the latter is faster when nucleosome unwrapping has to be considered. Strategies are discussed that could further facilitate calculations to allow computing genome-wide TF binding maps.

A lattice model for transcription factor access to nucleosomal DNA

Nucleosomes, the basic repeating unit of chromatin, consist of 147 basepairs of DNA that are wrapped in almost two turns around a histone protein octamer core. Because ∼3/4 of the human genomic DNA is found within nucleosomes, their position and DNA interaction is an essential determinant for the DNA access of gene-specific transcription factors and other proteins. Here, a DNA lattice model was developed for describing ligand binding in the presence of a nucleosome. The model takes into account intermediate states, in which DNA is partially unwrapped from the histone octamer. This facilitates access of transcription factors to up to 60 DNA basepairs located in the outer turn of nucleosomal DNA, while the inner DNA turn was found to be more resistant to competitive ligand binding. As deduced from quantitative comparisons with recently published experimental data, our model provides a better description than the previously used all-or-none lattice-binding model. Importantly, nucleosome-occupancy maps predicted by the nucleosome-unwrapping model also differed significantly when partial unwrapping of nucleosomal DNA was considered. In addition, large effects on the cooperative binding of transcription factors to multiple binding sites occluded by the nucleosome were apparent. These findings indicate that partial unwrapping of DNA from the histone octamer needs to be taken into account in quantitative models of gene regulation in chromatin.

Statistical-mechanical lattice models for protein-DNA binding in chromatin

Statistical-mechanical lattice models for protein-DNA binding are well established as a method to describe complex ligand binding equilibria measured in vitro with purified DNA and protein components. Recently, a new field of applications has opened up for this approach since it has become possible to experimentally quantify genome-wide protein occupancies in relation to the DNA sequence. In particular, the organization of the eukaryotic genome by histone proteins into a nucleoprotein complex termed chromatin has been recognized as a key parameter that controls the access of transcription factors to the DNA sequence. New approaches have to be developed to derive statistical-mechanical lattice descriptions of chromatin-associated protein-DNA interactions. Here, we present the theoretical framework for lattice models of histone-DNA interactions in chromatin and investigate the (competitive) DNA binding of other chromosomal proteins and transcription factors. The results have a number of applications for quantitative models for the regulation of gene expression.

Condensed DNA: condensing the concepts

DNA is stored in vivo in a highly compact, so-called condensed phase, where gene regulatory processes are governed by the intricate interplay between different states of DNA compaction. These systems often have surprising properties, which one would not predict from classical concepts of dilute solutions. The mechanistic details of DNA packing are essential for its functioning, as revealed by the recent developments coming from biochemistry, electrostatics, statistical mechanics, and molecular and cell biology. Different aspects of condensed DNA behavior are linked to each other, but the links are often hidden in the bulk of experimental and theoretical details. Here we try to condense some of these concepts and provide interconnections between the different fields. After a brief description of main experimental features of DNA condensation inside viruses, bacteria, eukaryotes and the test tube, main theoretical approaches for the description of these systems are presented. We end up with an extended discussion of the role of DNA condensation in the context of gene regulation and mention potential applications of DNA condensation in gene therapy and biotechnology.

Use of DNA length variation to detect periodicities in positively cooperative, nonspecific binding

The experiments described here demonstrate ways in which DNA length can be used as an experimental variable for the characterization of positively cooperative, sequence nonspecific DNA binding. Examples are drawn from recent studies of the interactions of O(6)-alkylguanine DNA alkyltransferase (AGT) with duplex DNAs (Melikishvili et al. (2008). Interactions of human O(6)-alkylguanine-DNA alkyltransferase (AGT) with short double-stranded DNAs. Biochemistry 47, 13754-13763). Oscillations in binding density and apparent binding site size (S(app)) are predicted by models in which a single cooperative assembly forms on each DNA molecule and in which enzyme molecules bind full-length binding sites, but not partial ones. These oscillations provide an accurate, DNA-length independent measure of the occluded binding site size (the length of DNA that one protein molecule occupies to the exclusion of others). In addition, length-dependent oscillations in association constant (K) and cooperativity (ω) reveal the degree to which substrate length can influence these parameters.

General transfer matrix formalism to calculate DNA-protein-drug binding in gene regulation: application to OR operator of phage lambda

The transfer matrix methodology is proposed as a systematic tool for the statistical-mechanical description of DNA-protein-drug binding involved in gene regulation. We show that a genetic system of several cis-regulatory modules is calculable using this method, considering explicitly the site-overlapping, competitive, cooperative binding of regulatory proteins, their multilayer assembly and DNA looping. In the methodological section, the matrix models are solved for the basic types of short- and long-range interactions between DNA-bound proteins, drugs and nucleosomes. We apply the matrix method to gene regulation at the O(R) operator of phage lambda. The transfer matrix formalism allowed the description of the lambda-switch at a single-nucleotide resolution, taking into account the effects of a range of inter-protein distances. Our calculations confirm previously established roles of the contact CI-Cro-RNAP interactions. Concerning long-range interactions, we show that while the DNA loop between the O(R) and O(L) operators is important at the lysogenic CI concentrations, the interference between the adjacent promoters P(R) and P(RM) becomes more important at small CI concentrations. A large change in the expression pattern may arise in this regime due to anticooperative interactions between DNA-bound RNA polymerases. The applicability of the matrix method to more complex systems is discussed.

Ligand Binding With Continuous Modification of Binding Sites

Simultaneous formulation binding and structure modification of the binding site leads to binding process that can be analyzed within the framework of the non-linear theory of dynamic systems. Such an approach allows us to obtain several properties of the binding center: plurality of stationary (stable and unstable) states at binding, recognition of bistable and hysteretic binding modes. It is also shown that adsorption centre deformation leads to a S-shaped adsorption curve.

Adsorption of ligands on macromolecules in the fluctuating medium

In the present work fluctuations of number of ligands adsorbed on macromolecule are investigated. We have taken into account the adsorption and desorption of ligands under the circumstance of some adsorption centers fluctuations affected by medium fluctuation. The correlation function and spectral density of number of ligands adsorbed on macromolecule are calculated. The properties of these fluctuations which allow identifying a noisemaker are determined. It has been shown, thatfas andsluggis adsorption can be distinguished by properties of dispersion and spectral density. It has been also shown, that comparison of experimental and theoretical correlation functions (or spectral densities) allows to calculate constants of ligand - adsorption center binding and unbinding.

Sequence distribution and intercooperativity detection for two ligands simultaneously binding to DNA

A method for detecting and quantifying the cooperativity in the simultaneous binding of two ligands, A and B, to DNA (intercooperativity; omega(AB)) is proposed. This involves the determination of an apparent affinity constant K(app) for one of the ligands (A) in the limit of its null saturation (nu(Alpha) --> 0), in the presence of the second one (B). A definition for this constant is given and an expression is derived corresponding to a simple model of competitive binding to an unbranched three-state homogeneous polar lattice with nearest-neighbor interactions (Markov chain). The ratio between the apparent and intrinsic affinity constants of one ligand in the maximum saturation limit of the other one becomes omega(2)(AB), and thus can be graphically obtained from K(app)(A) vs nu(B) plots. All the frequencies that define the sequence distribution of ligands can be easily calculated by introducing some generalized statistical weights for the free lattice monomer in a standard sequence generating function procedure. A model of fluorescence quenching emission is obtained from such frequencies under the hypothesis of a short-range electron transfer mechanism of the deactivation; it confirms, as suggested by the binding model, an outstanding influence of the intercooperativity on the distribution.

Mixed mode of ligand-DNA binding results in S-shaped binding curves

S-shaped binding curves often characterize interactions of ligands with nucleic acid molecules as analyzed by different physico-chemical and biophysical techniques. S-shaped experimental binding curves are usually interpreted as indicative of the positive cooperative interactions between the bound ligand molecules. This paper demonstrates that S-shaped binding curves may occur as a result of the "mixed mode" of DNA binding by the same ligand molecule. Mixed mode of the ligand-DNA binding can occur, for example, due to 1) isomerization or dimerization of the ligands in solution or on the DNA lattice, 2) their ability to intercalate the DNA and to bind it within the minor groove in different orientations. DNA-ligand complexes are characterized by the length of the ligand binding site on the DNA lattice (so-called "multiple-contact" model). We show here that if two or more complexes with different lengths of the ligand binding sites could be produced by the same ligand, the dependence of the concentration of the complex with the shorter length of binding site on the total concentration of ligand should be S-shaped. Our theoretical model is confirmed by comparison of the calculated and experimental CD binding curves for bis-netropsin binding to poly(dA-dT) poly(dA-dT). Bis-netropsin forms two types of DNA complexes due to its ability to interact with the DNA as monomers and trimers. Experimental S-shaped bis-netropsin-DNA binding curve is shown to be in good correlation with those calculated on the basis of our theoretical model. The present work provides new insight into the analysis of ligand-DNA binding curves.

Determination of constant rates of adsorption of ligand on DNA: analysis of correlation functions

Correlation functions and spectral density of the number of molecules of ligand bound to DNA are calculated theoretically. Kinetics of rates of formation and decomposition of the complex are determined by calculating the dependence of correlation function on concentration of ligand in solution. The analysis of spectral density allows to distinguish "fast" and "slow" adsorption of ligands on macromolecule.

Hairpin polyamides that use parallel and antiparallel side-by-side peptide motifs in binding to DNA

Pt-bis-netropsin is a synthetic sequence-specific DNA-binding ligand comprizing two netropsin-like fragments which are linked in a tail-to-tail manner via a cis-diammineplatinum (II) residue. The CD studies and thermodynamic characterization of the DNA-binding properties exhibited by this compound reveal that it forms two types of complexes with poly[d(AT)].poly[d(AT)] and DNA oligomers containing nucleotide sequences 5'-CC(TA)n CC-3', with n = 4, 5 and 6. The first type corresponds to the binding of Pt-bis-netropsin in the extended conformation and is characterized by the saturating ratio of one bound Pt-bis-netropsin molecule per 9 AT-base pairs. The second type of the complex corresponds to the binding of Pt-bis-netropsin to DNA in the folded hairpin form. The binding approaches saturation level when one Pt-bis-netropsin molecule is bound per four or five AT-base pairs. The hairpin form of Pt-bis-netropsin complex is built on the basis of parallel side-by-side peptide motif which is inserted in the minor DNA groove. The CD spectral profiles reflecting the binding of Pt-bis-netropsin in the hairpin form are different from those observed for binding of another bis-netropsin with the sequence Lys-Gly-Py-Py-Gly-Gly-Gly-Py-Py-Dp, where Py is a N-propylpyrrole amino acid residue and Dp is a dimethylaminopropylamino residue. The hairpin form of this bis-netropsin is formed on the basis of antiparallel side-by-side peptide motif. The CD spectra obtained for complexes of this polyamide in the hairpin form with poly[d(AT)].poly[d(AT)] exhibit positive CD band with a peak at 325 nm, whereas the CD spectral profiles for the second complex of Pt-bis-Nt with poly[d(AT)].poly[d(AT)] and short DNA oligomers have two intense positive CD bands near 290 nm and 328 nm. This reflects the fact that two bis-netropsins use different structural motifs on binding to DNA in the hairpin form.

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.

Thermodynamics of ligand-nucleic acid interactions

Ligand-and protein-DNA equilibria are extremely sensitive to solution conditions (e.g., salt, temperature, and pH), and, in general, the effects of different solution variables are interdependent (i.e., linked). As a result, an assessment of the basis for the stability and specificity of ligand-or protein-DNA interactions requires quantitative studies of these interactions as a function of a range of solution variables. Many of the most dramatic effects on the stability of these interactions result from changes in the entropy of the system, caused by the preferential interaction of small molecules, principally ions which are released into solution on complex formation. A determination of the contributions of these entropy changes to the stability and specificity of protein-and ligand-DNA interactions requires thermodynamic approaches and cannot be assessed from structural studies alone.

Calculation of site affinity constants and cooperativity coefficients for binding of ligands and/or protons to macromolecules. I. Generation of partition functions and mass balance equations

The thermodynamics of binding of a ligand A and/or proton H to a macromolecule M is treated by the partition function method. In complex systems, the representation of the equilibria by means of cumulative constants beta PQR used as coefficients in partition functions ZM, ZA, and ZH is ill-suited to least-squares refinement procedures because the cumulative constants are interrelated by common cooperativity functions gamma j(i) and common site affinity constants kappa j. There is therefore the need to express ZM, ZA, ZH as functions of site constants kappa j and cooperativity coefficients bj. This is done by developing an algebra of partition functions based on the following concepts: (i) factorability of partition functions; (ii) binary generating function Jj = (1 + kappa j[Y])i tau for each class j of sites, represented by column (Jj) and row (Jj) vectors; (iii) cooperativity between sites of one class described by functions gamma j(i), represented by diagonal matrices gamma j; (iv) probability of finding microspecies represented by elements of tensor product matrix Ll = (J1)[J2]; (v) statistical factors mij obtained from Newton polynomials, Jj; (vi) power operators Oi', O(i-l)', and O(i tau-l)', transforming vectors Jj; and (vii) operators Oi or O(i-l) indicating tensor products of i or (i-l) vectors Jj. Vectors Jj combined in tensors Ll give rise to both an affinity/cooperativity space and a parallel index space. The partition functions ZM, ZA, and ZH and the total amounts TM, TA, and TH can be obtained as an appropriate sum of elements of matrices Ll, each of which is represented in an index space by a combination p1, p2,...q1, q2,...r1, r2,... of indices ij. From these indices the contribution of that element to partition function ZM, ZA, or ZH and to total amount TM, TA, or TH is calculated in the affinity/cooperativity space as product of factors: [i tau !/i !(i tau-i)!]kappa ij(exp[bj (i-1)i])[X]i, i being any index p, q, r and X any component M, A, or H. Future applications of this algorithm to practical problems of macromolecule-ligand-proton equilibria are outlined.

A general secular equation for cooperative binding of n-mer ligands to a one-dimensional lattice

The binding of n-mer ligands to a one-dimensional lattice involving many ligand species and complex multiple-binding mechanisms is studied. We show that, when derived using the sequence-generating function method of Lifson, the secular equation of any binding system with a finite number of "elementary units" can be expressed in a matrix determinant form that is very symmetric and easy to construct. In other words, for any binding system whose elementary units are known, the secular equation of the system can be obtained readily without going through the formal derivation of the equation. We also show that the "determinant" secular equation obtained using the present procedure can be employed directly to the calculation of binding isotherms.

Binding of n-mers to one-dimensional lattices with longer than close-contact interactions

A general formalism is derived for the evaluation of binding isotherms of n-mers (ligands) to one-dimensional polymers in the presence of ligand-ligand interactions which extend over several binding sites with distance-dependent interaction energies (multi-parameter model). This is an extension of the usual n-mer binding theory developed by several investigators in which ligand-ligand interaction occurs only when two ligands are in close contact (one-parameter model). The difference in binding isotherms between a one-parameter model and a multi-parameter model is studied numerically using the present formalism.