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.     

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.     

Thermodynamics of DNA containing very stable chemically modified base pairs

DNA chemical modifications caused by the binding of some antitumor drugs give rise to a very strong local stabilization of the double helix. These sites melt at a temperature that is well above the melting temperatures of ordinary AT and GC base pairs. In this work we have examined the melting behavior of DNA containing very stable sites. Analytical expressions were derived and used to evaluate the thermodynamic properties of homopolymer DNA with several different distributions of stable sites. The results were extended to DNA with a heterogeneous sequence of AT and GC base pairs. The results were compared to the melting properties of DNA with ordinary covalent interstrand cross-links. It was found that, as with an ordinary interstrand cross-link, a single strongly stabilized site makes a DNA's melting temperature (T(m)) independent of strand concentration. However in contrast to a DNA with an interstrand cross-link, a strongly stabilized site makes the DNA's T(m) independent of DNA length and equal to T(infinity), the melting temperature of an infinite length DNA with the same GC-content and without a stabilized site. Moreover, at a temperature where more than 80% of base pairs are melted, the number of ordinary (non-modified) helical base pairs (n) is independent of both the DNA length and the location of the stabilized sites. For this condition, n(T) = (2 omega-a)S/(1-S) and S = exp[DeltaS(T(infinity)-T)/(RT)] where omega is the number of strongly stabilized sites in the DNA chain, a is the number of DNA ends that contain a stabilized site, and DeltaS, T, and R are the base pair entropy change, the temperature, and the universal gas constant per mole. The above expression is valid for a temperature interval that corresponds to n<0.2N for omega=1, and n<0.1N for omega>1, where N is the number of ordinary base pairs in the DNA chain.     

Application of the one- and two-dimensional Ising models to studies of cooperativity between ion channels.

The Ising model of statistical physics provides a framework for studying systems of protomers in which nearest neighbors interact with each other. In this article, the Ising model is applied to the study of cooperative phenomena between ligand-gated ion channels. Expressions for the mean open channel probability, rho o, and the variance, sigma 2, are derived from the grand partition function. In the one-dimensional Ising model, interactions between neighboring open channels give rise to a sigmoidal rho o versus concentration curve and a nonquadratic relationship between sigma 2 and rho o. Positive cooperativity increases the slope at the midpoint of the rho o versus concentration curve, shifts the apparent binding affinity to lower concentrations, and increases the variance for a given rho o. Negative cooperativity has the opposite effects. Strong negative cooperativity results in a bimodal sigma 2 versus rho o curve. The slope of the rho o versus concentration curve increases linearly with the number of binding sites on a protomer, but the sigma 2 versus rho o relationship is independent of the number of ligand binding sites. Thus, the sigma 2 versus rho o curve provides unambiguous information about channel interactions. In the two-dimensional Ising model, rho o and sigma 2 are calculated numerically from a series expansion of the grand partition function appropriate for weak interactions. Virtually all of the features exhibited by the one-dimensional model are qualitatively present in the two-dimensional model. These models are also applicable to voltage-gated ion channels.     

Effects of heterogeneity and cooperativity on the forms of binding curves for multivalent ligands

An exploratory investigation is made of the binding behavior that is likely to be encountered with multivalent ligands under circumstances where a single intrinsic binding constant does not suffice to describe all acceptor-ligand interactions. Numerical simulations of theoretical binding behavior have established that current criteria for recognizing heterogeneity and cooperativity of acceptor sites on the basis of the deviation of the binding curve from rectangular hyperbolic form for univalent ligands also apply to the interpretation of the corresponding binding curves for multivalent ligands. However, for systems in which the source of the departure from equivalence and independence of binding sites resides in the ligand, these criteria are reversed. On the basis of these observations a case is then made for attributing results of an experimental binding study of the interaction between pyruvate kinase and muscle myofibrils to positive cooperativity of enzyme sites rather than to heterogeneity or negative cooperativity of the myofibrillar sites.     

Theory of multivalent binding in one and two-dimensional lattices

Ligand binding to a linear lattice composed of N sites, under general conditions of cooperativity and number of sites covered upon binding, m, is approached in terms of the theory of contracted partition functions. The partition function of the system obeys a recursion relation leading to a generating function that provides an exact analytical solution for any case of interest. Site-specific properties of the lattice are derived from simple transformations of the analytical expressions. The McGhee-von Hippel model is obtained as a special case in the limit N --> infinity. The derivation is straightforward and involves no combinatorial arguments. Partition functions and site-specific properties are also derived for the case of non-cooperative binding to a two-dimensional torus of length N, containing s sites in its section for a total of sN sites. The torus provides a relevant model for ligand binding to double-stranded DNA (s = 2) or protein helices (s = 3,4). It is proved that non-cooperative binding to the two-dimensional torus can mimic cooperative binding to a one-dimensional linear lattice when m = s. The dimensional embedding of the lattice and the geometry of interaction of its sites play a crucial role in defining the binding properties of the system accessible to experimental measurements. Hence, caution must be exercised in the interpretation of Scatchard plots in terms of the one-dimensional McGhee-von Hippel model, especially when m < or = 4 and the geometry of the system is clearly two-dimensional.     

Negatively cooperative binding of high-density lipoprotein to the HDL receptor SR-BI

Scavenger receptor class B, type I (SR-BI), is a high-density lipoprotein (HDL) receptor, which also binds low-density lipoprotein (LDL), and mediates the cellular selective uptake of cholesteryl esters from lipoproteins. SR-BI also is a coreceptor for hepatitis C virus and a signaling receptor that regulates cell metabolism. Many investigators have reported that lipoproteins bind to SR-BI via a single class of independent (not interacting), high-affinity binding sites (one site model). We have reinvestigated the ligand concentration dependence of (125)I-HDL binding to SR-BI and SR-BI-mediated specific uptake of [(3)H]CE from [(3)H]CE-HDL using an expanded range of ligand concentrations (<1 μg of protein/mL, lower than previously reported). Scatchard and nonlinear least-squares model fitting analyses of the binding and uptake data were both inconsistent with a single class of independent binding sites binding univalent lipoprotein ligands. The data are best fit by models in which SR-BI has either two independent classes of binding sites or one class of sites exhibiting negative cooperativity due to either classic allostery or ensemble effects ("lattice model"). Similar results were observed for LDL. Application of the "infinite dilution" dissociation rate method established that the binding of (125)I-HDL to SR-BI at 4 °C exhibits negative cooperativity. The unexpected complexity of the interactions of lipoproteins with SR-BI should be taken into account when interpreting the results of experiments that explore the mechanism(s) by which SR-BI mediates ligand binding, lipid transport, and cell signaling.     

The relationship between effector binding and inhibition of activity in D-3-phosphoglycerate dehydrogenase

The binding of L-serine to phosphoglycerate dehydrogenase from Escherichia coli displays elements of both positive and negative cooperativity. At pH 7.5, approximately 2 mol of serine are bound per mole of tetrameric enzyme. A substantial degree of positive cooperativity is seen for the binding of the second ligand, but the binding of the third and fourth ligand display substantial negative cooperativity. The data indicate a state of approximately 50% inhibition when only one serine is bound and approximately 80-90% inhibition when two serines are bound. This is consistent with the tethered domain hypothesis that has been presented previously. Comparison of the data derived directly from binding stoichiometry to the binding constants determined from the best fit to the Adair equation, produce a close agreement, and reinforce the general validity of the derived binding constants. The data also support the conclusion that the positive cooperativity between the binding to the first and second site involves binding sites at opposite interfaces over 110 A apart. Thus, an order of binding can be envisioned where the binding of the first ligand initiates a conformational transition that allows the second ligand to bind with much higher affinity at the opposite interface. This is followed by the third ligand, which binds with lesser affinity to one of the two already occupied interfaces, and in so doing, completes a global conformational transition that produces maximum inhibition of activity and an even lower affinity for the fourth ligand, excluding it completely. Thus, maximal inhibition is accomplished with less than maximal occupancy of effector sites through a mechanism that displays strong elements of both positive and negative cooperativity.     

Determination of binding parameters of cyclic AMP and its analogs to cyclic AMP-dependent protein kinase by the fluorescent probe method.

The method for determination of dissociation constants for cyclic AMP and its analogs bound to cyclic AMP-dependent protein kinase from pig brain is described. The technique for measuring the binding parameters of the ligands is based on the changes in the fluorescent spectrum of etheno cyclic AMP once it is bound to protein kinase. The dissociation constants for a number of nonfluorescent cyclic AMP analogs were determined in the competitive displacement of etheno cyclic AMP by these analogs. The number of cyclic AMP-binding sites in the pig brain protein kinase was found to be 2.2; no cooperativity was observed upon binding. The holoenzyme complex (Mr = 180,000) of the protein kinase under study was established to have the stoichiometry of R2C2 type under native conditions.     

DNA bis-intercalation: Theoretical calculation of binding curves

A statistical mechanical calculation of the binding properties of DNA bis-intercalators is presented, based on the sequence-generating function method of Lifson. The effects of binding by intercalation of one or both chromophores of a bifunctional intercalating agent are examined. The secular equation for a general model that includes the effects of neighbor (nearest and non-nearest) exclusion and/or cooperativity in the binding of both singly and doubly intercalated ligands is derived. Numerical results for binding curves are presented for a more restricted model in which each type of bound ligand rigorously excludes its nearest neighbor and the total number of sites covered by a doubly intercalated ligand is variable. At low values of free ligand concentration bis-intercalation dominates the binding process, while at high value of free ligand concentration, intercalation of only one chromophore per ligand becomes significant due to the unavailability of contiguous free sites required for bis-intercalation. Also, depending on the binding parameters, the free energy of the system can be lowered by a loss of doubly intercalated ligands in favor of singly intercalated ones. Corresponding to this transition in binding mode, the average number of sites occupied by a bound ligand decreases from that characteristic of bis-intercalation to that characteristic of mono-intercalation as free ligand concentration increases. An analysis of Scatchard plots describing bis-intercalation is presented.     

Long-range interactions between DNA-bound ligands

We have studied the interaction of the A:T specific minor-groove binding ligand 4′, 6-diamidino-2-phenylindole (DAPI) with synthetic DNA oligomers containing specific binding sites in order to investigate possible long-range interactions between bound ligands. We find that DAPI binds cooperatively to the oligomers. The degree of cooperativity increases with increasing number of binding sites and decreases with the separation between them. This dependence is paralleled by changes in the induced circular dichroism spectrum of DAPI, which decreases in intensity at 335 nm and increases at 365 nm. These results are consistent with an allosteric interaction of DAPI with DNA, where bound ligands cooperatively alter the structure of the DNA molecule. This structural change seems possible to induce under various conditions, including physiological. One consequence of allosteric binding is that ligands bound at a distance from each other sense each other's presence and influence each other's properties. If some regulatory proteins the same conformational change as DAPI, novel mechanisms for controlling gene expression can be anticipated.     

Thermodynamic models of binding ligands to nucleic acids

Models of adsorption were considered, which describe the binding of biologically active ligands on DNA templates. The binding is described most comprehensively and in greatest detail by the distribution function, which determines the probability of detecting the preset number of adsorbed ligands on the template. In the case of noncooperative binding, this function corresponds to the Gaussian distribution and is characterized by two quantities: the mean value of the occupation of the template by ligands and the dispersion of occupation. The accuracy of the occupation of the template by ligands is inversely proportional to dispersion. As the length of the template and the number of reaction sites covered by one ligand upon binding increase, the accuracy of the occupation of the template by ligands increases. An important characteristic of binding is the degree of coverage of the template by ligands. This characteristic represents the portion of template reaction sites covered by all ligands adsorbed on the template. If polycations are bound to nucleic acid molecules, the coverage of the template determines the transition of nucleic acids to a compact state. The degree of template coverage for extended ligands depends only slightly on the binding constant in a wide range of concentrations of a free ligand in solution. Different adsorption models are considered from the unified point of view. The classification of cooperative interactions for a wide class of systems is given, from situations when several ligands are bound on nucleic acid templates to a situation when templates change by the action of ligands and begin to interact with each other.     

The binding of caldesmon to actin and its effect on the ATPase activity of soluble myosin subfragments in the presence and absence of tropomyosin

The binding of 125I- and 14C-caldesmon to actin and actin-tropomyosin was studied using a cosedimentation technique and was analyzed by the method of McGhee and von Hippel [1974) J. Mol. Biol. 86, 469-489) for the binding of large ligands to a homogeneous lattice. The binding was adequately described by a single class of binding sites with a stoichiometry between 1:7 and 1:10. The binding exhibited a small degree of positive cooperativity (omega = 5-6) which was the same in the presence and absence of tropomyosin. The association constant for the binding of caldesmon to an isolated binding site was enhanced, from about 6 X 10(5) to about 1.4 X 10(6) M-1, by the presence of smooth muscle tropomyosin. Caldesmon inhibited the actin-activated ATPase activity of skeletal myosin subfragment 1 in both the absence and presence of tropomyosin. Maximum inhibition of ATPase activity occurred when one caldesmon molecule bound to seven actin monomers. A greater degree of inhibition was observed in the presence of tropomyosin than in the absence. This greater inhibition cannot be explained totally by the increased strength of binding of caldesmon to actin in the presence of tropomyosin. Finally, Ca2+-calmodulin completely reversed the binding of caldesmon to actin.     

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.     

Mapping ligand binding domains in chimeric fibroblast growth factor receptor molecules. Multiple regions determine ligand binding specificity

Fibroblast growth factors (FGFs) mediate essential cellular functions by activating one of four alternatively spliced FGF receptors (FGFRs). To determine the mechanism regulating ligand binding affinity and specificity, soluble FGFR1 and FGFR3 binding domains were compared for activity. FGFR1 bound well to FGF2 but poorly to FGF8 and FGF9. In contrast, FGFR3 bound well to FGF8 and FGF9 but poorly to FGF2. The differential ligand binding specificity of these two receptors was exploited to map specific ligand binding regions in mutant and chimeric receptor molecules. Deletion of immunoglobulin-like (Ig) domain I did not effect ligand binding, thus localizing the binding region(s) to the distal two Ig domains. Mapping studies identified two regions that contribute to FGF binding. Additionally, FGF2 binding showed positive cooperativity, suggesting the presence of two binding sites on a single FGFR or two interacting sites on an FGFR dimer. Analysis of FGF8 and FGF9 binding to chimeric receptors showed that a broad region spanning Ig domain II and sequences further N-terminal determines binding specificity for these ligands. These data demonstrate that multiple regions of the FGFR regulate ligand binding specificity and that these regions are distinct with respect to different members of the FGF family.     

Signalling via rat dopamine D2-receptors expressed in mouse fibroblasts is not influenced by compounds binding to the sigma sites of these cells

The mouse fibroblast cell line LZR-1 is a well-established test system used to characterize the intrinsic activity of dopamine D₂-receptor ligands (Neve et al., Mol. Pharmac., 1989). This cell line is transfected with a eucaryotic expression vector containing the gene of the rat dopamine D_2B-receptor (short isoform of the D₂-receptor) (Bunzow et al., Nature, 1988). In addition to the expression of high levels of the rat dopamine D₂-receptor, these mouse cells also express sigma binding sites. Binding affinities of BMY 14,802, DTG, haloperidol, ( + )-pentazocine, ( + )-3-PPP, and ( + )-SKF 10,047 for the sigma binding sites as well as for the D₂-dopamine receptor in membranes of LZR-1 cells were determined. Using intact LZR-1 cells, it was found that the influence of sigma ligands on signalling via the dopamine D₂-receptor can be explained by their affinity for the latter receptor. Specific sigma ligands did not influence dopaminergic signal transduction in LZR-1 cells. It is therefore concluded that the LZR-1 cell is a suitable test model for determination of the intrinsic activity of dopamine D₂-receptor ligands even if these compounds have affinity for sigma binding sites.     

Equilibrium studies on the association of the nuclear poly(A) binding protein with poly(A) of different lengths

The nuclear poly(A) binding protein (PABPN1) binds the growing poly(A) tail during pre-mRNA 3'-end processing, stimulating its elongation and controlling its final length. Here we report binding studies of PABPN1 to poly(A) in solution. Quantitative fluorescence titration was used to determine the stoichiometry, intrinsic affinity, and cooperativity of binding to a series of size-fractionated poly(A). The intrinsic association constant K(i) was about 2 x 10(6) M(-1) for oligo(A) and all size classes of poly(A). The binding of PABPN1 to poly(A) was enhanced by protein-protein interactions which were, however, weak (cooperativity parameter omega < 50). No significant change of cooperativity could be detected with increasing polynucleotide length in the range of 140-450 nucleotides. An average binding site size n of 11-14 was found for all poly(A) lengths, which is close to the minimal site size m found for binding to oligo(A). The data are discussed with respect to the previous observation of two different forms of the poly(A)-PABPN1 complex.     

Determination of binding constants for cooperative site-specific protein-DNA interactions using the gel mobility-shift assay

We have investigated the question of whether the gel mobility-shift assay can provide data that are useful to the demonstration of cooperativity in the site-specific binding of proteins to DNA. Three common patterns of protein-DNA interaction were considered: (i) the cooperative binding of a protein to two sites (illustrated by the Escherichia coli Gal repressor); (ii) the cooperative binding of a bidentate protein to two sites (illustrated by the E. coli Lac repressor); and (iii) the cooperative binding of a protein to three sites (illustrated by the lambda cI repressor). A simple, rigorous, and easily extendable statistical mechanical approach to the derivation of the binding equations for the different patterns is presented. Both simulated and experimental data for each case are analyzed. The mobility-shift assay provides estimates of the macroscopic binding constants for each step of ligation based on its separation of liganded species by the number of ligands bound. Resolution of the binding constants depends on the precision with which the equilibrium distribution of liganded species is determined over the entire range of titration of each of the sites. However, the evaluation of cooperativity from the macroscopic binding constants is meaningful only for data that are also accurate. Some criteria that are useful in evaluating accuracy are introduced and illustrated. Resolution of cooperative effects is robust only for the simplest case, in which there are two identical protein binding sites. In this case, cooperative effects of up to 1,000-fold are precisely determined. For heterogeneous sites, cooperative effects of greater than 1,000-fold are resolvable, but weak cooperativity is masked by the heterogeneity. For three-site systems, only averaged pair-wise cooperative effects are resolvable.