Dissociation of the lactose repressor-operator DNA complex: effects of size and sequence context of operator-containing DNA

The dissociation kinetics for repressor-32P-labeled operator DNA have been examined by adding unlabeled operator DNA to trap released repressor or by adding a small volume of concentrated salt solution to shift the Kd of repressor-operator interaction. The dissociation rate constant for pLA 322-8, an operator-containing derivative of pBR 322, was 2.4 X 10(-3) s-1 in 0.15 M KCl. The dissociation rate constant at 0.15 M KCl for both lambda plac and pIQ, each of which contain two pseudooperator sequences, was approximately 6 X 10(-4) s-1. Elimination of flanking nonspecific DNA sequences by use of a 40 base pair operator-containing DNA fragment yielded a dissociation rate constant of 9.3 X 10(-3) s-1. The size and salt dependences of the rate constants suggest that dissociation occurs as a multistep process. The data for all the DNAs examined are consistent with a sliding mechanism of facilitated diffusion to/from the operator site. The ability to form a ternary complex of two operators per repressor, determined by stoichiometry measurements, and the diminished dissociation rates in the presence of intramolecular nonspecific and pseudooperator DNA sites suggest the formation of an intramolecular ternary complex. The salt dependence of the dissociation rate constant for pLA 322-8 at high salt concentrations converges with that for a 40 base pair operator. The similarity in dissociation rate constants for pLA 322-8 and a 40 base pair operator fragment under these conditions indicates a common dissociation mechanism from a primary operator site on the repressor.     

On the thermodynamics and kinetics of the cooperative binding of bacteriophage T4-coded gene 32 (helix destabilizing) protein to nucleic acid lattices.

In this paper we summarize a series of thermodynamic, and preliminary kinetic, studies on the molecular details and specificity of interaction of phage T4-coded gene 32-protein (GP32) with nucleic acid lattices. It is shown that the binding of GP32 to short (l = 2--8 residues) oligonucleotides is essentially independent of base composition and sugar-type, as well as of salt concentration. In contrast, cooperative (continuous) or isolated binding of GP32 to single-stranded polynucleotides is base and sugar composition-dependent (binding is tighter to DNA than to RNA) and highly dependent on salt concentrations. Binding constants (K), cooperativity parameters (w), and binding site sizes (n) are determined for binding to various nucleic acid lattices under a variety of environmental conditions. These results are used to show that GP32 can bind to nucleic acid lattices in two different conformations, and to characterize the molecular details of these binding species. Further insight into the molecular origins of binding cooperativity is obtained by determining these thermodynamic parameters also for the specifically proteolytically degraded GP32 fragments GP32 I (C-terminal peptide removed) and GP32 III (C- and N-terminal peptides removed). It is also shown that these GP32-nucleic acid binding measurements can be used to provide a quantitative molecular interpretation of the sequential (competitive) binding equilibria involved in the autogenous translational regulation of GP32 synthesis (Lemaire et al., 1978, J. Mol. Biol. 126:73, 1978), and to illustrate some general principles of the development of interactional specificity in cooperatively binding protein-nucleic acid complexes. Preliminary experiments have also been carried out on the kinetics of GP32 association to, and dissociation from, single-stranded nucleic acid lattices. In particular, fluorescence stopped-flow measurements of the dissociation of GP32 from such lattices as a function of lattice saturation (and protein cluster size) can be interpreted to suggest that the protein may translocate ("slide") on the lattice before dissociation, These studies permit an approach to possible rates and mechanisms of such translocation events.     

Model for the irreversible dissociation kinetics of cooperatively bound protein–nucleic acid complexes

We present a quantitative model for the irreversible dissociation kinetics of cooperatively bound nonspecific protein–nucleic acid complexes. The model assumes that the major pathway of dissociation is via singly contiguously bound protein that “peels” off the ends of clusters of bound protein. It should therefore be most applicable for proteins that bind nucleic acids with high cooperativity (w > 10³). Furthermore, the model assumes that no redistribution of bound protein occurs during the time course of the dissociation. Solutions to the rate equations are presented for the entire time course of the dissociation. Under initial conditions such that the nucleic acid is less than fully saturated with protein, a single-exponential decay is predicted (if w is large). However, when the nucleic acid lattice is initially fully saturated, zero-order kinetics, corresponding to a constant rate of protein dissociation, is predicted. The experimental observation of zero-order dissociation kinetics in a cooperative protein–nucleic acid system is a good qualitative indicator for the dissociation mechanism discussed here. A discussion of the analysis of experimental data that enables one to extract molecular rate constants is presented. Furthermore, comparisons are made between the nonredistributing model presented here and Epstein's model [Epstein, I. R. (1979) Biopolymers 18 , 2037–2050] in which protein can translocate infinitely quickly while bound to the nucleic acid, and hence protein clusters redistribute during dissociation and maintain an equilibrium distribution on the nucleic acid at all times.     

Kinetics and mechanism of dissociation of cooperatively bound T4 gene 32 protein-single-stranded nucleic acid complexes. 1. Irreversible dissociation induced by sodium chloride concentration jumps

The dissociation kinetics of cooperatively bound bacteriophage T4 gene 32 protein from a variety of single-stranded homopolynucleotides has been investigated by stopped-flow techniques. Irreversible dissociation of the complexes was induced by rapidly increasing the salt concentration and monitoring the increase in tryptophan fluorescence upon dissociation of the gene 32 protein. The dependence of the apparent dissociation rate constant on initial fractional saturation of the nucleic acid lattice as well as the observation of zero-order kinetics when the lattice is initially fully saturated with protein indicates that dissociation occurs only from the ends of protein clusters and not from doubly contiguous molecules. The data for the entire time course are quantitatively fit by a kinetics model specifying irreversible dissociation of only singly contiguously bound protein [Lohman, T.M. (1983) Biopolymers 22, 1697-1713]. This model is used to extract molecular rate constants for the dissociation of isolated, singly contiguously and doubly contiguously bound protein. It is also shown that the polynucleotide specificity observed for the cooperative binding constant, K omega, and the cooperativity itself are intrinsic properties of the dissociation rate of the various complexes.     

Thermodynamic database for protein-nucleic acid interactions (ProNIT).

MOTIVATION: Protein-nucleic acid interactions are fundamental to the regulation of gene expression. In order to elucidate the molecular mechanism of protein-nucleic acid recognition and analyze the gene regulation network, not only structural data but also quantitative binding data are necessary. Although there are structural databases for proteins and nucleic acids, there exists no database for their experimental binding data. Thus, we have developed a Thermodynamic Database for Protein-Nucleic Acid Interactions (ProNIT). RESULTS: We have collected experimentally observed binding data from the literature. ProNIT contains several important thermodynamic data for protein-nucleic acid binding, such as dissociation constant (K(d)), association constant (K(a)), Gibbs free energy change (DeltaG), enthalpy change (DeltaH), heat capacity change (DeltaC(p)), experimental conditions, structural information of proteins, nucleic acids and the complex, and literature information. These data are integrated into a relational database system together with structural and functional information to provide flexible searching facilities by using combinations of various terms and parameters. A www interface allows users to search for data based on various conditions, with different display and sorting options, and to visualize molecular structures and their interactions. AVAILABILITY: ProNIT is freely accessible at the URL http://www.rtc.riken.go.jp/jouhou/pronit/pronit.html.     

Linkage of EcoRI dissociation from its specific DNA recognition site to water activity, salt concentration, and pH: separating their roles in specific and non-specific binding.

We have measured the dependencies of both the dissociation rate of specifically bound EcoRI endonuclease and the ratio of non-specific and specific association constants on water activity, salt concentration, and pH in order to distinguish the contributions of these solution components to specific and non-specific binding. For proteins such as EcoRI that locate their specific recognition site efficiently by diffusing along non-specific DNA, the specific site dissociation rate can be separated into two steps: an equilibrium between non-specific and specific binding of the enzyme to DNA, and the dissociation of non-specifically bound protein. We demonstrated previously that the osmotic dependence of the dissociation rate is dominated by the equilibrium between specific and non-specific binding that is independent of the osmolyte nature. The remaining osmotic sensitivity linked to the dissociation of non-specifically bound protein depends significantly on the particular osmolyte used, indicating a change in solute-accessible surface area. In contrast, the dissociation of non-specifically bound enzyme accounts for almost all the pH and salt-dependencies. We observed virtually no pH-dependence of the equilibrium between specific and non-specific binding measured by the competition assay. The observed weak salt-sensitivity of the ratio of specific and non-specific association constants is consistent with an osmotic, rather than electrostatic, action. The seeming lack of a dependence on viscosity suggests the rate-limiting step in dissociation of non-specifically bound protein is a discrete conformational change rather than a general diffusion of the protein away from the DNA.     

CLIP: a method for identifying protein-RNA interaction sites in living cells

Nucleic-acid binding proteins constitute nearly one-fourth of all functionally annotated human genes. Genome-wide analysis of protein-nucleic acid contacts has not yet been performed for most of these proteins, restricting attempts to establish a comprehensive understanding of protein function. UV cross-linking is a method typically used to determine the position of direct interactions between proteins and nucleic acids. We have developed the cross-linking and immunoprecipitation assay, which exploits the covalent protein-nucleic acid cross-linking to stringently purify a specific protein-RNA complex using immunoprecipitation followed by SDS-PAGE separation. In this way, the vast majority of non-specific contaminating RNA, which can bind to co-immunoprecipitated proteins or beads, can be removed. Here, we present an improved protocol that performs RNA linker ligation before the SDS-PAGE step, and describe its application to the specific purification and amplification of RNA ligands of Nova in neurons.     

Theory of electrostatically regulated binding of T4 gene 32 protein to single- and double-stranded DNA

Bacteriophage T4 gene 32 protein (gp32) is a single-stranded DNA binding protein, which is essential for DNA replication, recombination, and repair. In a recent article, we described a new method using single DNA molecule stretching measurements to determine the noncooperative association constants K(ds) to double-stranded DNA for gp32 and *I, a truncated form of gp32. In addition, we developed a single molecule method for measuring K(ss), the association constant of these proteins to single-stranded DNA. We found that in low salt both K(ds) and K(ss) have a very weak salt dependence for gp32, whereas for *I the salt dependence remains strong. In this article we propose a model that explains the salt dependence of gp32 and *I binding to single-stranded nucleic acids. The main feature of this model is the strongly salt-dependent removal of the C-terminal domain of gp32 from its nucleic acid binding site that is in pre-equilibrium to protein binding to both double-stranded and single-stranded nucleic acid. We hypothesize that unbinding of the C-terminal domain is associated with counterion condensation of sodium ions onto this part of gp32, which compensates for sodium ion release from the nucleic acid upon its binding to the protein. This results in the salt-independence of gp32 binding to DNA in low salt. The predictions of our model quantitatively describe the large body of thermodynamic and kinetic data from bulk and single molecule experiments on gp32 and *I binding to single-stranded nucleic acids.     

Dissecting the metal ion dependence of DNA binding by PvuII endonuclease.

Divalent cations can provide an effective means of modulating the behavior of nucleic acid binding proteins. As a result, there is strong interest in understanding the role of metal ions in the function of both nucleic acid binding proteins and their enzymes. We have applied complementary fluorescence spectroscopic and nitrocellulose filter binding assays to quantitate the role of metal ions in mediating DNA binding and sequence specificity by the representative PvuII endonuclease. At pH 7.5 in the presence of the catalytically nonsupportive Ca(II), this enzyme binds the PvuII target sequence with a K(d) of 50 pM. Under strict metal-free conditions, the enzyme exhibits a K(d) of only 300 nM for the cognate sequence, an affinity which is weak relative to those measured for other systems in the absence of metal ions. This represents a 6000-fold increase in PvuII affinity for cognate DNA upon the addition of Ca(II). The pH dependences of both metal ion-dependent and metal ion-independent DNA binding are remarkably shallow throughout the physiological range; other characterized restriction enzymes exhibit more pronounced pH dependences of DNA binding even in the absence of metal ions. Similar measurements with noncognate sequences indicate that divalent metal ions are not important to nonspecific DNA binding; K(d) values are approximately equal to 200 nM throughout the physiological pH range, a behavior shared with other endonucleases. While some of these results extend somewhat the range of expected behavior for restriction enzymes, these results indicate that PvuII endonuclease shares with other characterized systems a mechanism by which cognate affinity and sequence discrimination are most effectively achieved in the presence of divalent metal ions.     

Kinetics of nonspecific binding reactions of proteins with DNA flexible coils: site-based and molecule-based association reactions

Domain effects on the pseudo-first-order kinetics of the reversible and irreversible association of proteins or other ligands with nucleic acids containing multiple binding sites are treated using the classical reaction-diffusion equation applied to a spherical cell model of the nucleic acid solution and a diffuse-sphere model for the nucleic acid chain molecule. Both uniform and Gaussian distributions of chain segments are analyzed. In general, the details of the segment distribution do not have a major effect on the kinetics of association. Domain effects are best examined experimentally by determining the effect of the molecular weight of the nucleic acid on the kinetics of the association reaction. A theoretical framework is presented that permits such data to be analyzed simply. Kinetic studies over a wide range of nucleic acid molecular weights are required in order to separate the contributions of diffusion and reaction to the observed kinetics, and to determine the contributions of site-based and molecule-based elements to the rate constants.     

Kinetics and mechanism of dissociation of cooperatively bound T4 gene 32 protein-single-stranded nucleic acid complexes. 2. Changes in mechanism as a function of sodium chloride concentration and other solution variables

The dissociation kinetics of bacteriophage T4 coded gene 32 protein-single-stranded nucleic acid complexes have been examined as a function of monovalent salt concentration, temperature, and pH in order to investigate the details of the dissociation of cooperatively bound protein. Fluorescence stopped-flow techniques were used, and irreversible dissociation was induced by a combination of [NaCl] jumps and mixing with excess nucleic acid competitor. This made it possible to directly investigate the irreversible dissociation process over a wide range of NaCl concentrations [e.g., from 50 mM to 0.60 M for the gene 32 protein-poly(A) complex], in the absence of reassociation. Over the entire salt range, the only dissociable species observed is the singly contiguously bound gene 32 protein which dissociates from the ends of protein clusters. However, the [NaCl] dependence of the dissociation rate constant suggests that two competing pathways exist for dissociation of cooperatively bound gene 32 protein from the ends of protein clusters. At high monovalent salt concentrations, dissociation is dominated by a single-step process, with log ke/log [NaCl] = 6.5 +/- 0.5; i.e., the dissociation rate constant increases with increasing NaCl concentration due to the uptake of approximately six monovalent ions upon dissociation. This indicates that singly contiguous protein dissociates directly into solution. However, at much lower [NaCl] the data suggest that gene 32 protein, when bound at the end of a protein cluster, dissociates by first sliding off the end to form a noncooperatively bound intermediate which subsequently dissociates. A quantitative model which incorporates the sliding pathway [Berg, O. G., Winter, R. B., & von Hippel, P. H. (1981) Biochemistry 20, 6929-6948] in the dissociation mechanism fits the data reasonably well and suggests that noncooperatively bound monomers of gene 32 protein may be capable of one-dimensional translocation along single-stranded nucleic acids as suggested by independent kinetic data on the association reaction [Lohman, T. M., & Kowalczykowski, S. C. (1981) J. Mol. Biol. 152, 67-109]. It is also observed that both the absolute dissociation rate constant for T4 gene 32 protein and its salt dependence are sensitive to the average molecular weight and polydispersity of the nucleic acid sample used. This is a general phenomenon exhibited by proteins that bind to nucleic acids in a highly cooperative manner.     

Real time kinetic studies of the interaction between folded antisense and target RNAs using surface plasmon resonance

Antisense RNAs interact with their complementary target RNAs as folded structures. The formation of early binding intermediates is the most important step in determining the overall rates of stable complex formation in vitro and the efficiency of control in vivo. In the case of CopA and CopT (antisense/target RNA pair of plasmid R1), recent studies have identified a four-way junction structure as the major binding intermediate. Previously, the kinetics of antisense/target RNA interaction was studied by indirect methods. Here we have used surface plasmon resonance to follow the binding of CopI (a truncated variant of CopA) to CopT in real time. A protocol was developed that permitted the determination of association and dissociation rate constants for wild-type and mutant CopI-CopT pairs. The K(D)-values calculated from these rate constants were in good agreement with the results obtained by indirect methods. In comparison to earlier model studies of interactions between simple complementary nucleic acids, we observe a different temperature dependence for dissociation rate constants. This may be indicative of the complexity of the steps required for interacting folded RNAs; intramolecular structure competes with intermolecular helix progression during complex formation. The association rate constants were not significantly dependent on temperature. The analysis presented shows that the stability of a kissing complex is not the primary determinant of the rate of stable CopA/CopT complex formation.     

Sequence specific recognition of ssDNA by a lupus autoantibody: kinetics and mechanism of binding.

11F8 is a pathogenic anti-ssDNA monoclonal autoantibody isolated from a lupus-prone mouse. Previous studies have established that 11F8 is sequence specific. To determine the basis for the observed binding specificity, stopped-flow fluorescence spectroscopy was used to measure the kinetic parameters and establish the mechanisms for the association of 11F8 with its target sequence, noncognate, and nonspecific ssDNA ligands. The data revealed that sequence-specific binding follows a two-step mechanism where the initial association step is second order. Values of k(1) are fast and above the modified Smoluchowski limit for a diffusion limited interaction (10(5)-10(6)M(-1)s(-1)). The dependency of k(1) on [salt] and solvent polarity indicates that electrostatic steering is responsible for this rapid association rate. The second association step is rate limiting and is characteristic of an isomerization process during which binding interfaces are optimized. This step apparently is driven by the desolvation of hydrophobic surfaces within the binding interface. The differences in the rate of dissociation for the various DNA ligands suggest that specificity is governed primarily through the dissociation of the final complexes.     

Analytic binding isotherms describing competitive interactions of a protein ligand with specific and nonspecific sites on the same DNA oligomer

Many studies of specific protein-nucleic acid binding use short oligonucleotides or restriction fragments, in part to minimize the potential for nonspecific binding of the protein. However, when the specificity ratio is low, multiple nonspecifically bound proteins may occupy the region of DNA corresponding to one specific site; this situation was encountered in our recent calorimetric study of binding of integration host factor (IHF) protein to its specific 34-bp H' DNA site. Here, beginning from the analytical McGhee and von Hippel infinite-lattice nonspecific binding isotherm, we derive a novel analytic isotherm for nonspecific binding of a ligand to a finite lattice. This isotherm is an excellent approximation to the exact factorial-based Epstein finite lattice isotherm even for short lattices and therefore is of great practical significance for analysis of experimental data and for analytic theory. Using this isotherm, we develop an analytic treatment of the competition between specific and nonspecific binding of a large ligand to the same finite lattice (i.e., DNA oligomer) containing one specific and multiple overlapping nonspecific binding sites. Analysis of calorimetric data for IHF-H' DNA binding using this treatment yields enthalpies and binding constants for both specific and nonspecific binding and the nonspecific site size. This novel analysis demonstrates the potential contribution of nonspecific binding to the observed thermodynamics of specific binding, even with very short DNA oligomers, and the need for reverse (constant protein) titrations or titrations with nonspecific DNA to resolve specific and nonspecific contributions. The competition treatment is useful in analyzing low-specificity systems, including those where specificity is weakened by mutations or the absence of specificity factors.     

Distamycin A affects the stability of NF-kappaB p50-DNA complexes in a sequence-dependent manner

The effect of two different DNA minor groove binding molecules, Hoechst 33258 and distamycin A, on the binding kinetics of NF-kappaB p50 to three different specific DNA sequences was studied at various salt concentrations. Distamycin A was shown to significantly increase the dissociation rate constant of p50 from the sequences PRDII (5'-GGGAAATTCC-3') and Ig-kappa B (5'-GGGACTTTCC-3') but had a negligible effect on the dissociation from the palindromic target-kappaB binding site (5'-GGGAATTCCC-3'). By comparison, the effect of Hoechst 33258 on binding of p50 to each sequence was found to be minimal. The dissociation rates for the protein--DNA complexes increased at higher potassium chloride concentrations for the PRDII and Ig-kappaB binding motifs and this effect was magnified by distamycin A. In contrast, p50 bound to the palindromic target-kappaB site with a much higher intrinsic affinity and exhibited a significantly reduced salt dependence of binding over the ionic strength range studied, retaining a K(D) of less than 10 pM at 150 mM KCl. Our results demonstrate that the DNA binding kinetics of p50 and their salt dependence is strongly sequence-dependent and, in addition, that the binding of p50 to DNA can be influenced by the addition of minor groove-binding drugs in a sequence-dependent manner.     

Kinetic and thermodynamic characterization of the interaction between Q beta-replicase and template RNA molecules

The specific binding of the RNA polymerase Q beta-replicase to some of its RNA template molecules, the single-stranded RNA variant MDV and also Q beta-RNA, was studied under various conditions by using a gel-retardation assay as well as filter retention. The dissociation of the replicase-RNA complex proceeds with first-order kinetics. The dependence of the dissociation rate constant on the concentration of monovalent ions suggests that there are three contacts between the midivariant (MDV) RNA and the replicase. Through analysis of the temperature dependence of the dissociation rate constant, values of 35 and 43 kJ/mol were obtained for the activation energies of complex dissociation between Q beta-replicase and the minus (-) and plus (+) strands of MDV, respectively. The bimolecular association is of second order with high rate constants that increase when the temperature is raised and decrease at higher salt concentrations. The equilibrium constants vary between 4.10(11) M-1 and 5.10(7) M-1, according to the reaction conditions. The temperature dependence of Ka gives delta H = -39 kJ/mol for MDV- and -47 kJ/mol for MDV+. Under nearly all conditions, distinct differences in the association and dissociation rates of plus and minus strands of MDV are observed. The binding of the small variant MDV to Q beta-replicase is three orders of magnitude stronger than the binding of the natural template Q beta-RNA.     

Kinetic and equilibrium analysis of a threading intercalation mode: DNA sequence and ion effects

The interaction of a symmetric naphthalene diimide with alkylamino substituents at each imide position was investigated with the alternating sequence polymers, poly[d(A-T)]2 and poly[d(G-C)]2. Spectrophotometric binding studies indicate strong binding of the diimide to both sequences although the GC binding constant is 20-25 times larger than the AT binding constant. Analysis of the effects of salt concentration on the binding equilibria shows that the diimide forms two ion pairs in its complex with both polymers as expected for a simple dication. Stopped-flow kinetics experiments demonstrate that the diimide both associates and dissociates from DNA more slowly than classical intercalators with similar binding constants. Analysis of salt concentration effects on dissociation kinetics rate constants (kd) reveals that slopes in log kd versus log [Na+] plots are only approximately half the value obtained for classical dicationic intercalators that have both charged groups in the same groove. These kinetics results support a threading intercalation model, with one charged diimide substituent in each of the DNA grooves rather than with both side chains in the same groove, for the diimide complex with DNA. In the rate-determining step of the mechanism for dissociation of a threading complex only one ion pair is broken; the free side chain can then slide between base pairs to put both diimide side chains in the same groove, and this is followed by rapid full dissociation of the diimide. This sequential release of ion pairs makes the dissociation slope for dicationic threading intercalators more similar to the slope for classical monocationic intercalating ligands.(ABSTRACT TRUNCATED AT 250 WORDS)     

Interactions of cationic ligands and proteins with small nucleic acids: analytic treatment of the large coulombic end effect on binding free energy as a function of salt concentration

For nonspecific binding of oligopeptides and other cationic ligands, including proteins, to nucleic acid oligomers, we develop a model capable of quantifying and predicting the salt concentration dependence of the binding free energy (deltaG(o)obs) by way of an analytic treatment of the Coulombic end effect (CEE). Ligands, nucleic acids, and their complexes (species j of valence Zj) are modeled as finite lattices with absolute value(Zj) charged residues; the CEE is quantified by its characteristic length Ne (specified in charged residues) and its consequences for the free energy and ion association of the oligomer. Expressions are developed for the individual site binding constants Ki as a function of position (site number i) of a bound ligand on a nucleic acid and for the observed binding constant Kobs as an ensemble average of Ki. Analysis of deltaG(o)obs = -RT ln Kobs and Sa Kobs identical with (partial differential ln Kobs)/(partial differential ln a(+/-)) for binding of the oligopeptide KWK6 (ZL = +8) to single-stranded (ss) dT(pdT)(absolute value(ZD) oligomers (dT-mers) where ZD = {-6, -10, -11, -14, -15} in the range 0.1-0.25 M Na+ yields Ne = 9.0 +/- 0.8 residues at each end, demonstrating that both KWK6 and the above dT-mers are sufficiently short so that the CEE extends over the entire molecule. The dependences of Kobs and of Sa Kobs on absolute value(ZD) for a given ZL are determined by the difference between 2Ne and the net number of charged residues Q in the complex (Q identical with absolute value(ZD) - ZL). For Q < 2Ne, characteristic of complexes of KWK6 with this set of dT-mers, the distribution of binding free energies deltaG(o)obs = -RT ln Ki for sites along the DNA oligomer is parabolic, and Kobs and Sa Kobs are strongly dependent on absolute value(ZD). For Q > or = 2Ne, the distribution of binding free energies deltaG(o)obs is trapezoidal, and the dependence of Kobs and Sa Kobs on absolute value(ZD) is weaker. Application of the model to nonspecific binding of human DNA polymerase beta to ssDNA demonstrates the significance of the CEE in determining Kobs and Sa Kobs of binding of a cationic site on a protein to a DNA oligomer.