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.     

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.     

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.     

Properties of GABA-induced transmembrane currents in isolated neurons of the rat spinal ganglia

Using techniques of voltage-clamp in the whole-cell configuration and fast local superfusion, we studied the properties of transmembrane ion currents evoked in freshly isolated neurons of the spinal ganglia of rats by application of γ-aminobutyric acid, GABA, in different concentrations. Increases in the GABA concentration and application time resulted in modification of the amplitude and kinetic parameters of the currents. The dependence between the current amplitude and GABA concentration could be adequately described by the Hill equation. The current rise could be fitted by a sum of two exponential curves with different time constants; the time constant of the second exponent changed with an increase in the GABA concentration, while the first exponent was not sensitive to these changes. The current decay also should be fitted by two exponents. The time constant of the first exponent did not change with increases in the GABA concentration or duration of its application; at the same time the second exponent noticeably depended on the time of GABA application. Our experiments demonstrated that the density of GABA-activated ion channels in the membranes of the studied spinal ganglion cells is relatively high; this finding allows us to suppose possible involvement of these channels in regulation of the transmembrane conductivity in these cells.     

Linked-function origins of cooperativity in a symmetrical dimer

The thermodynamic origins of substrate binding cooperativity in a dimeric enzyme that can bind one substrate (A) and one allosteric ligand (X) to each of two identical subunits are discussed. It is assumed that maximal activity is not subject to allosteric modification and that the substrates and allosteric ligands achieve binding equilibrium in the steady state. Each uniquely ligated form is assumed to be capable of exhibiting unique binding properties, and only the principles of thermodynamic linkage are used to constrain the system further. The explicit relationship between the Hill coefficient, the concentration of X, and the magnitudes of the relevant coupling free energies and dissociation constants is derived. In the absence of X only the homotropic coupling between substrate sites contributes to a nonhyperbolic substrate saturation profile. An allosteric ligand, X, can alter the cooperativity in two distinct ways, one mechanism being manifested when X is saturating and the only only when X is present at saturating concentrations. By evaluating the concentration of substrate required to produce half-maximal velocity as a function of [X], as well as the Hill coefficients when X is absent and fully saturating, the dissociation and coupling constants most important for understanding the mechanisms of allosteric action in an enzyme of this type can be determined.     

Nonessential Sites Improve Phosphorylation Switch

Multisite phosphorylation is a common form of posttranslational protein regulation which has been used to increase the switchlike behavior of the protein response to increasing kinase concentrations. In this letter, we show that the switchlike response of multisite phosphoproteins is strongly enhanced by nonessential phosphorylation sites, a mechanism that is robust to parameter changes and easily implemented in nature. We obtained analytic estimates for the Hill exponent (or coefficient) of the switchlike response, and we observed that a tradeoff exists between the switch and the kinase threshold for activation. This also suggests a possible evolutionary mechanism for the relatively large numbers of phosphorylation sites found in various proteins.     

A new measure of cooperativity in protein-ligand binding

An allosteric binding system consisting of a single ligand and a nondissociating macromolecule having multiple binding sites can be represented by a binding polynomial. Various properties of the binding process can be obtained by analyzing the coefficients of the binding polynomial and such functions as the binding curve and the Hill plot. The Hill plot has an asymptote of unit slope at each end and the departure of the slope from unity at any point can be used to measure the effective interaction free energy at that point. Of particular interest in detecting and measuring cooperativity are extrema of the Hill slope and its value at the half-saturation point. If the binding polynomial is symmetric, then there is an extremum of the Hill slope at the half-saturation point. This value, the Hill coefficient, is a convenient measure of cooperativity. The purpose of this paper is to express the Hill coefficient for symmetric binding polynomials in terms of the roots of the polynomial and to give an interpretation of cooperativity in terms of the geometric pattern of the roots in the complex plane. This interpretation is then applied to the binding polynomials for the MWC (Monod-Wyman-Changeux) and KNF (Koshland-Nemethy-Filmer) models.     

The reactions of Pseudomonas cytochrome c-551 oxidase with potassium cyanide.

The binding of cyanide to both oxidized and ascorbate-reduced forms of Pseudomonas cytochrome c-551 oxidase was investigated. Spectral studies on the oxidized enzyme and its apoprotein showed that the ligand can bind to both the c and d, haem components of the molecule, and kinetic observations indicated that both chromophores reacted, under a variety of conditions, with very similar rates. Cyanide combination velocities were dependent on ligand concentration, and increasing the pH also accelerated the reaction; the second-order rate constant was estimated as approx. 0.2M-1 . s-1 at pH 7.0. The binding of cyanide to the protein was observed to have a considerable influence on reduction of the enzyme by ascorbate. Spectral and kinetic observations have revealed that the species haem d13+-cyanide and any unbound haem c may react relatively rapidly with the reductant, but the behaviour of cyanide-bound haem c indicates that it may not be reduced without prior dissociation of the ligand, which occurs relatively slowly. The reaction of reduced Pseudomonas cytochrome oxidase with cyanide is radically different from that of the oxidized protein. In this case the ligand only binds to the haem d1 component and reacts much more rapidly. Stopped-flow kinetic measurements showed the binding to be biphasic in form. Both the rates of these processes were dependent on cyanide concentration, with the fast phase having a second-order rate constant of 9.3 X 10(5) M-1 . s-1 and the slow phase one of 2.3 X 10(5) M-1 . s-1. The relative proportions of the two phases also showed a dependency on cyanide concentration, the slower phase increasing as the cyanide concentration decreased. Computer simulations indicate that a reaction scheme originally proposed for the reaction of the enzyme with CO is capable of providing a reasonable explanation of the experimental results. Static-titration data of the reduced enzyme with with cyanide indicated that the binding was non-stoicheiometric, the ligand-binding curve being sigmoidal in shape. A Hill plot of the results yielded a Hill coefficient of 2.6.     

Kinetic co-operativity of monomeric mnemonical enzymes. The significance of the kinetic Hill coefficient

The expression of the kinetic Hill coefficient for a two-substrate, two-product mnemonical enzyme has been derived. Its relation with the gamma coefficient, that is the slope of the reciprocal plots for 1/[A]----O, has been established. The variation of this Hill coefficient, as a function of the second substrate and product concentrations, has been studied theoretically. Whereas the gamma coefficient does not vary as a function of the substrate and first product concentrations, the kinetic Hill coefficient does. If the enzyme is positively co-operative, the Hill coefficient increases upon increasing the second substrate concentration and decreases if the first product concentration is increased. The converse is expected to occur if the enzyme displays a negative co-operativity. The last product may either reverse a positive co-operativity into a negative one or, alternatively, strengthen an already negative co-operativity. The co-operativity generated by the mnemonical model has been compared to the kinetic behaviour of a random model. These two models have been shown to be discriminated on the basis of the departure they show with respect to the Michaelis-Menten behaviour. These theoretical considerations have been applied to previously published data, obtained with wheat germ hexokinase LI. This monomeric enzyme has a negative co-operativity with respect to the preferred substrate, glucose. The Hill coefficient decreases with MgATP concentration, increases with MgADP concentration and decreases with glucose-6-phosphate concentration. This is exactly what is to be expected on the basis of the above theory of kinetic co-operativity.     

Prosthetic group content and ligand-binding properties of spinach nitrite reductase

Chemical analysis of the ferredoxin-dependent native form (Mr = 85,000) of spinach nitrite reductase has demonstrated a siroheme content that approaches 2 mol of siroheme/mol of enzyme. A widely studied modified (Mr = 61,000) form of nitrite reductase, that has lost much of the native enzyme's ability to use ferredoxin as an electron donor, contains approximately 1 mol of siroheme/mol of enzyme. Quantitation of the high spin ferri-siroheme EPR signals and of nitrite-binding sites of the two preparations confirmed that the native enzyme's siroheme content is approximately twice that of the modified enzyme. Plots of nitrite and cyanide binding to the native enzyme versus ligand concentration are sigmoidal, with Hill coefficients of 1.6-1.8 and 2.3-2.8, respectively. Plots of enzyme activity versus nitrite concentration for the native enzyme are sigmoidal with a Hill coefficient of 2.4. Cyanide inhibition of enzymatic activity was shown to be not competitive. Addition of cyanide to the native enzyme resulted in a diminution of the high spin ferri-siroheme EPR signal and produced EPR signals with g values of 2.71, 2.33, and 1.49 due to low spin ferri-siroheme.     

Ligand Depletion in vivo Modulates the Dynamic Range and Cooperativity of Signal Transduction

Biological signal transduction commonly involves cooperative interactions in the binding of ligands to their receptors. In many cases, ligand concentrations in vivo are close to the value of the dissociation constant of their receptors, resulting in the phenomenon of ligand depletion. Using examples based on rotational bias of bacterial flagellar motors and calcium binding to mammalian calmodulin, we show that ligand depletion diminishes cooperativity and broadens the dynamic range of sensitivity to the signaling ligand. As a result, the same signal transducer responds to different ranges of signal with various degrees of cooperativity according to its effective cellular concentration. Hence, results from in vitro dose-response analyses cannot be applied directly to understand signaling in vivo. Moreover, the receptor concentration is revealed to be a key element in controlling signal transduction and we propose that its modulation constitutes a new way of controlling sensitivity to signals. In addition, through an analysis of the allosteric enzyme aspartate transcarbamylase, we demonstrate that the classical Hill coefficient is not appropriate for characterizing the change in conformational state upon ligand binding to an oligomeric protein (equivalent to a dose-response curve), because it ignores the cooperativity of the conformational change for the corresponding equivalent monomers, which are generally characterized by a Hill coefficient . Therefore, we propose a new index of cooperativity based on the comparison of the properties of oligomers and their equivalent monomers.     

Binding stoichiometry of a fluorescent cGMP analogue to membranes of retinal rod outer segments

The high-affinity binding of the cGMP analogue 8-(5-thioacetamidofluorescein)-cGMP to rod outer segment membranes depleted of peripherally bound proteins has been defined by equilibrium dialysis (mean +/- SD): membranes contain about one cGMP binding site per 130 rhodopsin molecules; the concentration of free ligand for half saturation is 2.0 +/- 0.6 microM; the apparent Hill coefficient of the bound versus free ligand relationship is 1.7 +/- 0.5; half saturation of the binding sites is sufficient for 85% activation of calcium permeability. A gating mechanism is proposed.     

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.     

An approach based on diffusion to study ligand-macromolecule interaction

A new approach has been developed to study binding of a ligand to a macromolecule based on the diffusion process. In terms of the Fick's first law, the concentration of free ligand in the presence of a protein can be determined by the measurement of those ligands which are diffused out. This method is applied to the study of binding of methyl-orange to lysozyme in phosphate buffer of pH 6.2, at 30 degrees C. The binding isotherm was determined initially, followed by application of the Hill equation to the data obtained, then binding constant and binding capacity were estimated.     

Application of the rat thymus receptor in a specific, sensitive and simplified assay of 1,25-dihydroxyvitamin D in blood serum

A simple method has been employed to prepare crude nuclear extract from rat thymus, using hypertonic buffer after previous treatment with hypotonic buffer. The preparation is free from serum vitamin D-binding protein and contains a 3.7 S receptor molecule, which specifically binds 1,25-dihydroxyvitamin D-3 (1,25-(OH)2D3). The receptor is of high affinity (KD = 0.85 X 10(-11) M at O degrees C) and low capacity (260-460 fmol/g tissue). The Scatchard analysis of ligand binding results in a concave downward curve. The Hill analysis of the same data gives good linear fitting (r = +0.971) with the Hill coefficient nH = 1.63. These facts indicate positive cooperativity between two ligand binding sites of the rat thymus 1,25-(OH)2D3 receptor. The preparation was used in a competitive protein binding assay of 1,25-(OH)2D in serum extracts, purified on Sep-Pak C18 followed by silica Sep-Pak cartridges. The method was sensitive to 0.5 pg/tube (2.0 ng/l) when 1 ml of serum was extracted. Intra- and interassay coefficients of variation were 9% and 14%, respectively. The serum 1,25-(OH)2D concentration estimated in 33 children (mean age 6.5 +/- 3 years) was 46.6 +/- 18.4 ng/l (mean +/- S.D.).     

Application of Statistical Thermodynamics to the Olfaction Mechanism

The application of the grand canonical ensemble in statisticalthermodynamics to the stimulus adsorption on the olfactory receptorsites, assuming some simplifying hypotheses, leads us to anexpression of the olfactory response R, which is a functionof various physico-chemical parameters involved in the olfactionmechanism, e.g. the stimulus concentration, the saturated vaporpressure, the power law exponent and the partition coefficient.This expression of R is in agreement with the olfactory responseof the Hill model, but is more explicit. Stevens' law and theolfactory threshold expression are easily deduced from R. Theexpression of the threshold we established from R enabled usto explain some empirical relations in the literature betweenthe parameters quoted above. The use of the grand canonicalensemble with the chemical potential notion gave us an interpretationof Stevens' law and a better understanding of the role of someparameters involved in the olfaction mechanism, such as saturatedvapor pressure and power law exponent. Chem. Senses 22: 67–75,1997.     

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.     

Analysis of Hill slopes predicted by the four-ligand exponential model for a regulatory enzyme.

The four-ligand exponential model for a regulatory enzyme is described as it is applied to kinetic studies of yeast pyruvate kinase in which the concentrations of four ligands are systematically varied. The Hill slopes predicted by this model are calculated for the two situations in which the fourth ligand is either a substrate or an effector. It is shown that the individual terms that make up the expression for the Hill slope assist the interpretation of the observed behaviour in terms of the constants employed by the model.