Cooperativity,absolute interaction,and algebraic optimization

We consider a measure of cooperativity based on the minimal interaction required to generate an observed titration behavior. We describe the corresponding algebraic optimization problem and show how it can be solved using the nonlinear algebra tool SCIP. Moreover, we compute the minimal interactions and minimal molecules for several binding polynomials that describe the oxygen binding of various hemoglobins under different conditions. We compare their minimal interaction with the maximal slope of the Hill plot, and discuss similarities and discrepancies with a view towards the shapes of the binding curves.

     

The relationship between zeros and factors of binding polynomials and cooperativity in protein-ligand binding

Cooperativity in the protein-ligand binding process is discussed in terms of the zeros of the binding polynomial and the corresponding possible factorizations of the binding polynomial into polynomials having non-negative coefficients. Particular attention is paid to the case in which the real parts of all zeros are negative (Hurwitz polynomial) and the case in which the binding polynomial admits no positive factorization (positive irreducible polynomial). Such factorizations are then interpreted as site linkage patterns and related to cooperativity. The possible combinations of zeros of the binding polynomials for the MWC and KNF tetrahedral, square and linear models are determined and the corresponding factorization and linkage patterns analyzed. An application and interpretation are then made for data obtained from Trout I hemoglobin.     

Protein-ligand binding with a missing species

Considerable experimental evidence has been produced recently that shows that in the binding of oxygen or carbon monoxide to certain tetrameric hemoglobins, the triply-ligated species is virtually non-existent. The binding polynomial representing this phenomenon for the general case is P(x) = 1 + beta 1x + ... + beta n-1xn-1 + beta nxn, where beta n-1 is nearly zero. The zeros, factorization and associated Hill plots of such binding polynomials with beta n-1 = 0 are investigated for the general case, and are analyzed in detail for n = 3 and n = 4. These results are then compared with the results obtained from experimental data on a number of tetrameric hemoglobins for which beta 3 is small. One concludes that, apart from the slope of the high-saturation asymptote of the Hill plot, a small perturbation of beta 3 from zero produces small changes in other properties associated with the binding process, such as fractional saturation, maximum Hill slope, and zeros and factorization of the binding polynomial.     

Evaluation of Hill slopes and Hill coefficients when the saturation binding or velocity is not known.

1. The Hill coefficient (nH), an often-used measure of deviations from hyperbolic behaviour (nonhyperbolicity) in kinetic and binding systems, is usually estimated from the maximum or minimum slope of the Hill plot. The method depends strongly on the assumed magnitude of the asymptotic velocity (V) or binding (P) whose evaluation may be difficult in nonlinear/co-operative systems. Therefore, alternative procedures were devised for the estimation nH which do not require the prior knowledge of V or P. 2. When pairs of velocity/binding readings (v and w) are obtained at concentrations of c and alpha c, respectively (where alpha is a fixed constant), then the relation between w and v is described by a hyperbola, provided that Hill's equation is valid. In this case, linearizing plots, v/w versus v, w versus, w/v, and 1/w versus 1/v, can be used for estimation of the degree of the equation. However, if the Hill expression is applicable, these methods are not efficient and traditional procedures, particularly nonlinear regression, should be used. 3. The 'linearizing' plots of the Hill equation can be applied advantageously for the evaluation of the Hill slope and of nH also in the general case, when the Hill expression is actually not valid, provided that deviations from hyperbolic behaviour are positive. Appropriately extrapolated intercepts of the first two plots estimate alphanH. Furthermore, the slope of the third plot yields, similarly to the method of Kurganov et al., a continuous measure of the Hill slope (including its maximum) at all concentrations. The agreement is, at positive nonhyperbolicities, excellent theoretical values of Hill slopes and coefficients and those estimated by the proposed methods. 4. A coefficient of nonhyperbolicity (theta) is defined for 2nd-degree rate equations which provides a quantitative measure of positive or negative deviation from first-degree, hyperbolic characteristics. It is closely related to the Hill coefficient.     

Measures of cooperativity in the binding of ligands to proteins and their relation to non-additivity in protein-protein interactions

The analogy between cooperativity in the binding of ligands to proteins and non-additivity in protein-protein interactions is demonstrated and discussed in terms of the Wong and the Hill coefficients. A measure of non-additivity, the interaction constant, is rigorously derived for four thermodynamic cycles, involving the binding of small molecules to proteins and protein association. It is the reciprocal of the 'defect factor' of Laskowski et al. in Proteinase inhibitors: medical and biological aspects (ed. N. Katunuma et al.), pp. 55-68 (1983), and its logarithm is the Wong measure of cooperativity. These three measures are thus here given a common theoretical basis. The Hill coefficient for an asymmetric dimer that binds two different ligands which do not compete for the same site, at 50% saturation of each site, is derived. It is shown to be a function of the interaction constant and of the fraction of protein to which ligand is bound at both sites. These relations for protein-ligand interactions are then discussed in the context of non-additivity in protein-protein interactions.     

A danger of low copy numbers for inferring incorrect cooperativity degree

Background  

A dose-response curve depicts the fraction of bound proteins as a function of unbound ligands. Dose-response curves are used to measure the cooperativity degree of a ligand binding process. Frequently, the Hill function is used to fit the experimental data. The Hill function is parameterized by the value of the dissociation constant and the Hill coefficient, which describes the cooperativity degree. The use of Hill's model and the Hill function has been heavily criticised in this context, predominantly the assumption that all ligands bind at once, which resulted in further refinements of the model. In this work, the validity of the Hill function has been studied from an entirely different point of view. In the limit of low copy numbers the dynamics of the system becomes noisy. The goal was to asses the validity of the Hill function in this limit, and to see in what ways the effects of the fluctuations change the form of the dose-response curves.     

An analytic solution to the Monod-Wyman-Changeux model and all parameters in this model.

Starting from the Monod-Wyman-Changeux (MWC) model (Monod, J., J. Wyman, and J. P. Changeux. 1965. J. Mol. Biol. 12:88-118), we obtain an analytical expression for the slope of the Hill plot at any ligand concentration. Furthermore, we derive an equation satisfied by the ligand concentration at the position of maximum slope. From these results, we derive a set of formulas which allow determination of the parameters of the MWC model (kR, C, and L) from the value of the Hill coefficient, nH, the ligand concentration at the position of maximum slope [( A]0), and the value of nu/(n-nu) at this point. We then outline procedures for utilizing these equations to provide a "best fit" of the MWC model to the experimental data, and to obtain a refined set of the parameters. Finally, we demonstrate the applicability of the technique by analysis of oxygen binding data for Octopus hemocyanin.     

On the relationship between the Hill coefficients for steady-state and transient kinetic data: A criterion for concerted transitions in allosteric proteins

A frequently used measure for the extent of cooperativity in ligand binding by allosteric proteins is the Hill coefficient. Hill coefficients can be measured for steady-state kinetic data and also for transient kinetic data. Here, the relationship between the two types of Hill coefficients is analysed. It is shown that a value of 1 for the ratio of the two Hill coefficients is a test for a concerted ligand-induced transition between two conformations of the protein, in accordance with the Monod-Wyman-Changeux model. A value of 1 for this ratio has recently been observed for a series of chaperonin GroEL mutants suggesting that ATP-induced allosteric transitions in this protein are concerted.     

An empirical extremum principle for the hill coefficient in ligand-protein interactions showing negative cooperativity

The Hill coefficient (nH) is a central parameter in the study of ligand-protein interactions, which measures the degree of cooperativity between subunits that bind the ligand in multisubunit proteins. The most common usage of nH is as an estimate of the minimal number of interacting binding sites in positively cooperating systems. In the present study, a statistical interpretation of nH for a generalized system of multiple identical binding sites is developed. This interpretation is then applied to the derivation of an empirical extremum principle for nH in negatively cooperating systems of identical binding sites, which can be used for the estimation of the minimal number of interacting sites in such systems.     

Cooperative binding of calcium to glycerinated skeletal muscle fibers

The binding of ⁴⁵Ca²⁺ to glycerinated rabbit psoas fibers was measured by means of a double isotope technique. With 5 mM Mg²⁺ (no ATP) binding was half-maximal at 1.4 · 10^?6M Ca²⁺ and the maximal amount bound was 1.6 μmol/g protein. At < 50% saturation, the Scatchard plot had a positive slope and the Hill coefficient was 2.2. At greater than 50% saturation, the Scatchard plot was linear with a negative slope (K′ = 0.8 · 10⁶ M^?1) and the Hill coefficient was 1.0. In the absence of Mg²⁺, binding was half-maximal at 3 · 10^?7 M Ca²⁺ and the maximal amount bound was 2.9 μmol/g protein. The Scatchard plot indicated two classes of sites with K′ values of about 2 · 10⁷ and 2 · 10⁶ M^?1. The Hill coefficient in the mid-saturation range was approx. 0.6. The data indicate that in the presence of Mg²⁺ binding to about half of the total Ca²⁺ binding sites is suppressed and there is a strong positive cooperativity involving half of the remaining sites.     

Binding of calcium ions to the isolated asialo-glycoprotein receptor. Implications for receptor function in suspended hepatocytes

The binding of calcium ions by the isolated asialoglycoprotein receptor of hepatocytes and the inter-relationship between the calcium ion concentration and receptor function have been studied. The isolated receptor binds calcium ions only in the presence of asialoglycoprotein. The asialo-glycoprotein receptor complex binds 4 calcium ions; the binding exhibits marked positive cooperativity, and the association constant at half-saturation of the binding sites was of the order of 10*5) M-1 as determined from a Hill plot. The isolated receptor was almost saturated at a calcium ion concentration of 0.1 mM. The binding capacity of isolated hepatocytes for asialo-glycoproteins increased, however, even when the calcium concentration was increased above this level. This may be explained by the exposure of increasing numbers of functional receptors on the surface of the cell with increasing membrane potential, and this explanation is supported by analogous observations in the presence of 5 mM La3+.     

Cooperativity: over the Hill

Cooperativity, the ability of ligand binding at one site on a macromolecule to influence ligand binding at a different site on the same macromolecule, is a fascinating biological property that is often poorly explained in textbooks. The Hill coefficient is commonly used in biophysical studies of cooperative systems although it is not a quantitative measure of cooperativity. The free energy of interaction between binding sites (ΔΔG) is a more stringent definition of cooperativity and provides a direct quantitative measure of how the binding of ligand at one site affects the ligand affinity of another site.     

Hill coefficient for estimating the magnitude of cooperativity in gating transitions of voltage-dependent ion channels

A frequently used measure for the extent of cooperativity in ligand binding by an allosteric protein is the Hill coefficient, obtained by fitting data of initial reaction velocity (or fractional binding saturation) as a function of substrate concentration to the Hill equation. Here, it is demonstrated that the simple two-state Boltzmann equation that is widely used to fit voltage-activation data of voltage-dependent ion channels is analogous to the Hill equation. A general empiric definition for a Hill coefficient (n(H)) for channel gating transitions that is analogous to the logarithmic potential sensitivity function of Almers is derived. This definition provides a novel framework for interpreting the meaning of the Hill coefficient. In considering three particular and simple gating schemes for a voltage-activated cation channel, the relation of the Hill coefficient to the magnitude and nature of cooperative interactions along the reaction coordinate of channel gating is demonstrated. A possible functional explanation for the low value of the Hill coefficient for gating transitions of the Shaker voltage-activated K(+) channel is suggested. The analogy between the Hill coefficients for ligand binding and for channel gating transitions further points to a unified conceptual framework in analyzing enzymes and channels behavior.     

The binding of nucleotides and bivalent cations to the calcium-and-magnesium ion-dependent adenosine triphosphatase from rabbit muscle sarcoplasmic reticulum.

The binding of MgATP to purified Ca2+Mg2+-dependent adenosine triphosphatase from rabbit muscle sarcoplasmic reticulum was studied by using a flow-dialysis method. Phosphoryl-enzyme formation and catalytic activity were also measured, and all three processes demonstrated negative co-operativity, with half-saturation of all three parameters at a MgATP concentration of 40-50muM, and a Hill coefficient (h) of 0.8. The variation of the binding constant with with pH was measured and showed tighter binding of MgATP with increasing pH over the range 6.8-8.5. Binding parameters for ATP analogues were also measured. The binding of Ca2+ in the presence and absence of ATP analogues gave half saturation at a Ca2+ concentration of 1.2-1.3muM. Hill plots of Ca2+-binding data gave a slope of 0.8. These results show that the binding of MgATP and Ca2+ can occur in a random manner, with neither substrate influencing the affinity of the enzyme for the other.     

Cooperativity of ligand binding as a function of monomer-dimer equilibrium parameters and acceptor concentration

A general monomer-dimer equilibrium system involving ligand interactions ispresented. Cooperativity features of specific limited models are analyzed by selecting the appropriate family of equilibrium constants from this general scheme. Each system is then characterized in terms of Hill coefficient dependency on alterations in values of equilibrium constants and total acceptor concentration. This method permits comparison of predicted cooperativity trends between systems. Contrasting reports concerning cooperativity dependencies for certain defined equilibrium systems are compared and the discrepancies resolved. Characteristics of cooperativity binding patterns are shown to include symmetry about dimerization association constant values, both positive and negative cooperativity for a single set of parameters, and significant changes in cooperativity features with relatively small changes in equilibrium parameters.     

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

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

Cooperative binding to the Ca2+-specific sites of troponin C in regulated actin and actomyosin

The Ca2+-binding component of troponin (TnC) and its proteolytic fragments containing Ca2+-binding sites I-III (TH1) or sites III and IV (TR2C) have been labeled with the fluorescent probes dansylaziridine (DANZ) at methionine 25 or 5-(iodoacetamidoethyl)amino-naphthalene-1-sulfonic acid (AEDANS) at cysteine-98. These probes report binding of Ca2+ to the low and high affinity sites, respectively. Fluorescence changes as a function of [Ca2+] were measured for the free peptides, their complexes with troponin I + troponin T, and these complexes bound to actin-tropomyosin in the presence of Mg2+ and ATP with and without myosin. An apparent Hill coefficient of 1.0-1.1 has been obtained for the Ca2+-induced fluorescence changes in TnC, its fragments, and their ternary complexes regardless of the label used. When a ternary complex containing appropriately labeled TnC or its fragment is bound to the actin-tropomyosin complex, the Hill coefficient for the titration of the low affinity sites increases to 1.5-1.6 and further increases to greater than 2 in the presence of myosin. To interpret the apparent Hill coefficients, we used a model containing two binding sites and a single reporter of the conformational change. Hill coefficients between 1.0 and 1.2 can be obtained for the fluorescence change without true cooperativity in metal binding, depending on the mechanism of the fluorescence change; i.e. the contribution of the singly or doubly occupied species to the fluorescence change. A Hill coefficient between 1.2 and 2, however, always indicates cooperativity in binding independently of the mechanism. Thus, our finding that fluorescence titrations of Ca2+ binding to TnCDANZ bound to actin-tropomyosin exhibit a Hill coefficient of 1.5 in the absence of myosin and 2.4 in its presence indicates the existence of true positive cooperativity in metal binding to sites I and II. No cooperativity was observed for AEDANS-labeled complexes that reflect Ca2+-binding to the high affinity sites. Plots of the Ca2+ dependence of myosin ATPase activity activated by actin-tropomyosin in the presence of any of the troponin complexes used had apparent Hill coefficients of approximately 4. The higher value suggests cooperative interactions in the activation of ATPase beyond those involved in Ca2+-binding to the Ca2+-specific sites.     

The Hill analysis and co-ion–driven transporter kinetics

ConclusionIn mechanistic models for binding of multiple ligands to a biological unit, the saturation behavior depends on the experimental readout. In the binding model, the Hill analysis does not provide information on the number of binding sites N. In contrast, the Hill analysis of the response model does contain this information, in which case the slope of the curve in the lower concentration range corresponds to the number of binding sites. In neither model does the Hill coefficient—defined as the slope of the curve at half-maximal saturation—report this number. In the binding model, the Hill coefficient varies between a value of 1 in the absence of interaction and a value of N in case of extremely strong interaction. In the response model, it varies between a number larger than 1 and N. In both models, the derived Hill coefficient is a measure of the cooperativity and sets a lowest possible number of sites.Ion-coupled transporters are of the response model type, and the saturation behavior of the rate with the co-ion in the lower concentration limit contains the information on the number of cotransported ions. Additionally, in the case of an ordered-binding mechanism, in which the co-ions bind before the transported substrate, the Hill coefficient of co-ion binding is a function of the substrate concentration. The apparent interaction between the co-ion sites increases with the substrate concentration and, consequently, the Hill coefficient extrapolates to the number of co-ions. Measurements of the Hill coefficient over the entire range of substrate concentrations provide information on both the extent of interaction between the sites and the number of sites.

Online supplemental material

Five supplemental texts accompany this review: (1) derivation of the equations for the binding and response models; (2) derivation of the equations for the ordered-binding transporter mechanism; (3) derivation of the equations for the Hill analysis of the saturation level functions; (4) derivation of the equations for substrate-dependent kinetics of mechanistic transporter models; (5) data analysis by curve fitting. The online supplemental material is available at http://www.jgp.org/cgi/content/full/jgp.201411332/DC1.