General rate equations and rejection criteria for the rapid equilibrium carrier model of cotransport

It is demonstrated that under fixed activator conditions the general flux equation for the rapid equilibrium carrier model of cotransport can be written entirely in terms of five independent kinetic constants. Thus the kinetic parameters from any experiment carried out under the same activator conditions must necessarily be expressible in terms of these five constants. These predicted relationships between experimental kinetic parameters provide rejection criteria for the model, a number of which are derived here. Generalization of the treatment to the case where a competitive substrate is present on both sides of the membrane is also given.     

Dielectric Relaxation of Molecules with Fluctuating Dipole Moment

When a dissolved macromolecule is in chemical equilibrium with a free ionic species, the charge configuration, and hence the dipole vector, of the macromolecule is fluctuating. Expressions for the static dielectric constant and the relaxation spectrum of such a mixture are here derived in terms of the components of the mean moment and the root mean square fluctuation moment, the molecular relaxation time constants, and the chemical rate constants of the ionic binding reaction. Contrary to a previous treatment of this problem by Kirkwood and Shumaker (1), it is shown that fluctuations introduce no independent components into the relaxation spectrum.     

Kinetics of glycogen phosphorylase a with a series of semisynthetic, branched saccharides. A model for binding of polysaccharide substrates.

The requirement of muscle phosphorylase for branched polysaccharide substrates was investigated by kinetic studies on semisynthetic branched saccharides. One series of saccharides was prepared from maltoheptose by oxidizing the reducing group to a carboxyl group and coupling this with an amino group of ethylenediamine. The resulting aminooligosaccharide was coupled with p-nitrophenyl esters of mono-, di-, tetra-, and polycarboxylic aicds to produce saccharides containing one, two, four, and approximately 52 maltodextrin chains per molecule. A similar series of saccharides was prepared from a heterogeneous maltodextrin of average chain length 11.7. Kinetic constants were determined for the reaction with phoshorylase a in the direction of chain elongation. Michaelis constants are equilibrium constants for dissociation of saccharide from the enzyme-AMP-glucose-1P-saccharide complex. The Michaelis constants, expressed in terms of the concentration of nonreducing end groups, are independent of maltodextrin chain length but decrease considerably as the number of chains per molecule increases. Maximum velocities do not differ greatly from that for glycogen. Among the synthetic saccharides, only the polymer behaves similarly to glycogen in exhiiting a decreasing reaction rate as the chains are elongated. The kinetic constants are quantitatively consistent with a model in which two chain termini from the same saccharide molecule bind to the phosphorylase molecule simultaniously, Differences in binding between saccharides having different numbers of equally accessible chains are caused solely by statistical factors in the equilibrium. Highly branched substrates bind better because of their greater multiplicity of two end-group pairs.     

The Oxygen Equilibrium of Mammalian Hemoglobin : Some old and new physicochemical studies

The three chief physicochemical theories of the oxygen-hemoglobin equilibrium in vogue 40 years ago still influence current thought on the problem. Although the Hill theory lost its fundamental basis some 40 years ago, the famous empiric equation to which it gave rise is still much used, as a useful phenomenological expression, only involving two disposable constants. The Haldane theory, of which a difference in aggregation of oxygenated and deoxygenated hemoglobin was a fundamental feature, lay for many years dormant but has recently had an astonishing reawakening through the work on lamprey hemoglobin, which clearly reveals such differences in aggregation. Lamprey hemoglobin might thus be called a "Haldane type" hemoglobin. Adair's four-stage intermediate compound theory still seems applicable in the case of hemoglobins such as those of sheep, whose tetramer molecules do not tend to dissociate into dimers, and which might therefore be called "Adair type" hemoglobins. Horse and human hemoglobins appear to reveal both "Haldane" and "Adair" behaviour. The effects of pH, temperature, and protein concentration on the oxygen-equilibrium of sheep hemoglobin are summarised, and it is shown that, although the equilibrium curves are often isomorphous over their middle range, intensive work at the top and bottom of the curves reveals considerable differences in the relative effects of these factors on the several equilibrium constants of Adair's four intermediate equations. In the last section an account is given of preliminary experimental attempts to interpret the oxygen- and carbon monoxide—equilibrium curves of whole human blood, under physiological conditions in terms of the Adair intermediate compound hypothesis.     

A kinetic investigation of the binding of azo dyes to cyclo-maltohexaose

The equilibrium constants and rate constants for the formation of azo dye-cyclomaltohexaose complexes have been determined at 26.6° and an ionic strength of 0.15. The effect of changing the substituents on the dyes is explained in terms of the sizes (steric factors) of the substituents and the charges on the substituents     

Minireview: Multiple receptor populations: Binding isotherms and their numerical analysis

Abstract

This paper reviews present models and methods of parameter estimation in relation-ships describing receptor-ligand interactions in equilibrium, as used in the author's laboratory. The state-of-the-art and the present experience can be summarized as follows: 1) Binding isotherms (relationships of bound and free/total ligand concentrations) are superpositions of several elementary terms describing the ligand binding to individual binding sites (receptors) present in the biological material investigated. The “nonspecific binding” is usually represented by a linear term. 2) The elementary terms are most frequently described by a rectangular hyperbola, Hill (power) function, or a rational function (binding of several ligand molecules to one molecule of receptor). Heterologous displacement requires specific functions which, however, can be transformed into one of the elementary terms. 3) Parameters (binding capacities, dissociation constants, Hill coefficients, etc.) can most reliably be estimated by nonlinear regression methods. However, these methods frequently fail to yield physically relevant values if initial estimates are far from “real” values, or if the data are strongly scattered. Some of the available routines (e.g., LIGAND) offer manifold tools to solve these difficulties. 4) The “affinity spectrum”, a relationship between binding capacities and equilibrium constants, shows the presence of individual binding sites in the binding system in question. The spectrum can be constructed either by Fourier analysis, or by a stepwise procedure (computation of binding capacity for several dissociation constants). The former way of analysis is demanding; software tools are rare. 5) The “STEP” routine based on Hill/Scatchard linearization routines yields profiles similar to affinity spectra, but offers, in addition, values of Hill coefficients of individual binding populations. Values obtained can be used as initial estimates for nonlinear regression. 6) Selection of a suitable model, its testing, numerical procedures, statistical estimates, etc. frequently entail severe difficulties which are approached in the available software packages in different ways, none of them is usually optimal.     

Conformational species of gramicidin A in non-polar solvent. A kinetic and thermodynamic treatment in the absence and presence of phosphatidylcholine as studied by high-performance liquid chromatography

A kinetic and thermodynamic study has been carried out to characterize quantitatively the conformational equilibrium of gramicidin A (GA) in tetrahydrofuran at different peptide concentrations in the absence and presence of egg yolk phosphatidylcholine by using size-exclusion high-performance liquid chromatographic analysis. In the absence of lipid, the experimental data fit a simple dimer-monomer equilibrium, the rate and equilibrium constants for the dissociation process being (1.6 +/- 0.7) X 10(-7) s-1 and (8.5 +/- 0.3) X 10(-6) M, respectively. A higher extent of monomerization and a decrease in the time required for reaching equilibrium are detected in the presence of phospholipid, the kinetic and thermodynamic effects depending on both lipid and GA concentrations. In order to account for these observations a cyclic equilibrium mechanism is proposed which is analysed in terms of four conformational species, namely, free monomer, free dimer, lipid-bound monomer and lipid-bound dimer. The results obtained are discussed in relation to recent literature data on lipid-protein interactions.     

Average quantities accessible to the analysis of sedimentation equilibrium data for self-association systems

This paper shows that analysis of sedimentation equilibrium data, searching for average molecular weights, gives quantities which are dependent on the total protein concentration of the samples, while knowledge of the molecular weight of the monomeric subunits allows a more meaningful search for the concentrations of the individual polymeric components of the system. From these the various average molecular weights can be construed, and the various dissociation equilibrium constants evaluated. Also, in this paper considerations are proposed on the meaning of nonideality terms in associating systems and possible ways for estimating them. As an example the proposed procedures have been applied to measurements of sedimentation equilibrium in carbonmonoxyhemoglobin.     

Distribution and diffusivity of a hydrophobic probe molecule in the interior of a membrane: theory and simulation.

We propose a simple model for the distribution of position and orientation and the diffusion of a hydrophobic probe molecule embedded in a membrane. The molecule experiences both a Maier-Saupe orienting potential as well as an enclosing potential of repulsion from the membrane walls. A statistical thermodynamics treatment of the model provides predictions of the location and orientation of the molecule within the membrane. In particular, we evaluate the order parameter of the molecule in terms of the model constants. The diffusivity of the probe is studied by Brownian dynamics simulation. For rotational diffusion, we check an available analytical approximate treatment that allows for the prediction of the dynamics in terms of equilibrium quantities. We also pay attention to quantities related to the initial and mean reorientational rate of the probe. For translational diffusion, we use the simulation results to analyze some general aspects of lateral and transversal diffusion.     

Determination of protein-ligand binding constants at equilibrium in biological samples.

Protein-ligand complexes can be separated functionally into two classes. "Specific" binding is characterized, in relative terms, by a high affinity for the ligand and a low binding capacity. "Non-specific" binding is characterized by a low affinity and a very large capacity. The calculation of equilibrium binding constants for any specific protein-ligand interaction requires the exact determination of the unbound ligand concentration and the specifically bound ligand concentration. These determinations usually require corrections for the contribution of non-specific binding. The use of two correction terms, kn and f, is proposed: kn is the product of the affinity constant k times the number of binding sites n of the non-specific components, while f is the fraction of the non-specific binding included in the experimental estimates of bound ligand. Several theoretical solutions using these terms are proposed for the calculation of specific binding constants. The practical choice of the correction factor may be different when the simultaneous measurement of the affinity constant and maximum number of binding sites, or when only the latter, is desired. In the case of complex binding systesm containing more than one specific component, the individual constants can be determined by non-graphical methods, using computer-aided iterative statistical calculations. A complete solution is given for a system containing two specific plus non-specific interactions and actual experiments are reported for steroid hormone-receptro complexes.     

Computerized optimization of radioimmunoassays for hCG and estradiol: an experimental evaluation

The mathematical and statistical theory of radioimmunoassays (RIAs) has been used to develop a series of computer programs to optimize sensitivity or precision at any desired dose level for either equilibrium or nonequilibrium assays. These computer programs provide for the calculation of the equilibrium constants of association and binding capacities for antisera (parameters of Scatchard plots), the association and dissociation rate constants, and prediction of optimum concentrations of labeled ligand and antibody and optimum incubation times for the assay. This paper presents an experimental evaluation of the use of these computer programs applied to RIAs for human chorionic gonadotropin (hCG) and estradiol. The experimental results are in reasonable semiquantitative agreement with the predictions of the computer simulations (usually within a factor of two) and thus partially validate the use of computer techniques to optimize RIAs that are reasonably “well behaved”, as in the case of the hCG and estradiol RIAs. Further, these programs can provide insights into the nature of the RIA system, e.g., the general nature of the “sensitivity” and “precision” surfaces. This facilitates empirical optimization of conditions.     

Kinetics of sickle hemoglobin polymerization. II. A double nucleation mechanism

A double nucleation mechanism for the polymerization of sickle hemoglobin is described. The mechanism accounts for all of the major kinetic observations: the appearance of a delay, the high concentration dependence of the delay time, and the stochastic behavior of slowly polymerizing samples in small volumes. The mechanism postulates that there are two pathways for polymer formation: polymerization is initiated by homogeneous nucleation in the solution phase, followed by nucleation of additional polymers on the surface of existing ones. This second pathway is called heterogeneous nucleation. Since the surface of polymers is continuously increasing with time, heterogeneous nucleation provides a mechanism for the extreme autocatalysis that is manifested as an apparent delay in the kinetic progress curves. In this mechanism, each spherulitic domain of polymers is considered to be initiated by a single homogeneous nucleation event. The mechanism explains the irreproducibility of the delay time for single domain formation as arising from stochastic fluctuations in the time at which the homogeneous nucleus for the first polymer is formed. Integration of the linearized rate equations that describe this model results in a simple kinetic form: A[cosh(Bt)-1] (Bishop & Ferrone, 1984). In the accompanying paper (Ferrone et al., 1985) it was shown that the initial 10 to 15% of progress curves, with delay times varying from a few milliseconds to over 10(5) seconds, is well fit by this equation. In this paper, we present an approximate statistical thermodynamic treatment of the equilibrium nucleation processes that shows how the nucleus sizes and nucleation equilibrium constants depend on monomer concentration. The equilibrium model results in expressions for B and B2A as a function of monomer concentration in terms of five adjustable parameters: the bimolecular addition rate of a monomer to the growing aggregate, the fraction of polymerized monomers that serve as heterogeneous nucleation sites, the free energy of intermolecular bonding within the polymer, and two parameters that describe the free energy change as a function of size for the bonding of the heterogeneous nucleus to a polymer surface. This model provides an excellent fit to the data for B and B2A as a function of concentration using physically reasonable parameters. The model also correctly predicts the time regime in which stochastic behavior is observed for polymerization in small volumes.     

Thermodynamic properties of enzyme-catalyzed reactions involving cytosine, uracil, thymine, and their nucleosides and nucleotides

The standard Gibbs energies of formation of species in the cytidine triphosphate series, uridine triphosphate series, and thymidine triphosphate series have been calculated on the basis of the convention that Delta(f)G=0 for the neutral form of cytidine in aqueous solution at 298.15 K at zero ionic strength. This makes it possible to calculate apparent equilibrium constants for a number of reactions for which apparent equilibrium constants have not been measured or cannot be measured because they are too large. This paper adds fifteen reactants to the database BasicBiochemData3 at MathSource that includes 199 reactants. The standard transformed Gibbs energies of formation of these fifteen reactants are used to calculate apparent equilibrium constants at 298.15 K, ionic strength 0.25 M, and pHs 5, 6, 7, 8, and 9 for thirty two reactions. The pKs, standard Gibbs energies of hydrolysis, and standard Gibbs energies of deamination are given for these fifteen reactants.     

Conformational equilibria of alpha-L-iduronate residues in disaccharides derived from heparin.

The disaccharides IdoA(2SO3)-anManOH(6SO3) and IdoA-anManOH (where IdoA represents alpha-L-iduronate, anManOH represents 2,5-anhydro-D-mannitol and SO3 represents sulphate ester) were prepared from bovine lung heparin using HNO2 depolymerization, borohydride reduction and desulphation, and were examined by 400 MHz 1H-n.m.r. spectroscopy. Three-bond proton-proton coupling constants around the IdoA ring were determined under a range of experimental conditions. For unsulphated IdoA all four proton-proton coupling constants varied markedly as a function of temperature, pH and solvent, providing clear evidence for a rapid conformational equilibrium. These data were analysed in terms of the three most energetically stable IdoA conformers: 1C4, 4C1, and 2S0. Predicted coupling constants for these conformers were determined using a modified Karplus-type relationship. For unsulphated IdoA in dimethyl sulphoxide the equilibrium was provoked strongly in favour of a slightly distorted 4C1 'chair' IdoA conformer for which coupling constants have not previously been reported. For sulphated IdoA in aqueous conditions and at low pH the equilibrium is strongly in favour of the alternative 1C4 chair conformer. Under many conditions, however, significant contributions from all three conformers occur for the non-reducing terminal IdoA in these disaccharides.     

Nonlinear Regression Analysis of the time Course of Ligand Binding Experiments

Abstract

The binding and release of hormones and growth factors are often relatively slow processes under biological conditions. Consequently, a knowledge of the underlying rate constants may be of greater physiological relevance than the equilibrium constant. Here we show how, by following a single time course of binding, the rate constants for both binding and release can be determined. The ratio of these rate constants allows the binding constant to be calculated. A nonlinear regression computer program is described which facilitates these calculations and which provides estimates and standard errors of the constants determined. The method is illustrated by the binding of human growth hormone to the human growth hormone binding protein, and the binding of ovine prolactin to the rabbit prolactin receptor.     

A growth mixture theory for cartilage with application to growth-related experiments on cartilage explants

In this paper, we present a growth mixture model for cartilage. The main features of this model are illustrated in a simple equilibrium boundary-value problem that is chosen to illustrate how a mechanical theory of cartilage growth may be applied to growth-related experiments on cartilage explants. The cartilage growth mixture model describes the independent growth of the proteoglycan and collagen constituents due to volumetric mass deposition, which leads to the remodeling of the composition and the mechanical properties of the solid matrix. The model developed here also describes how the material constants of the collagen constituent depend on a scalar parameter that may change over time (e.g., crosslink density); this leads to a remodeling of the structural and mechanical properties of the collagen constituent. The equilibrium boundary-value problem that describes the changes observed in cartilage explants harvested at different stages of a growth or a degenerative process is formulated. This boundary-value problem is solved using existing experimental data for developing bovine cartilage explants harvested at three developmental stages. The solution of the boundary-value problem in conjunction with existing experimental data suggest the types of experimental studies that need to be conducted in the future to determine model parameters and to further refine the model.     

Azide binding to the cytochrome c ferric heme octapeptide. A model for anion binding to the active site of high spin ferric heme proteins.

Equilibrium constants for the binding of azide to the ferric heme c octapeptide in 50% ethylene glycol 50% buffer were measured spectrophotometrically. The equilibrium constant for azide binding at 20 degrees C and pH* 7.4 is 29.2, which is approximately 3 to 4 orders of magnitude lower than that observed for azide binding to various ferric hemeproteins. The equilibrium constant was indepent of pH* in the range from 7 to 8. Equilibrium constants at several temperatures exhibited an apparent van't Hoff relationship yielding thermodynamic values of delta H0 = -26,100 J/mol (-6240 cal/mol) and delta S0 = -61.5 J/0K mol (-14.7 e.u.). Comparison of these values to the values for the heme proteins enables one to explain the differences in equiliberium constants in terms of differences in the polarity of the heme environments. The results are consistent with the concept that the oxygen affinity of heme complexes increases with the polarity of the heme environment. The data also suggest that an increase in the polarity of the heme environment should result in a corresponding increase in the susceptibility of ferrous heme dioxygen complexes toward oxidation by the dioxygen.     

Time-dependent 31P saturation transfer in the phosphoglucomutase reaction. Characterization of the spin system for the Cd(II) enzyme and evaluation of rate constants for the transfer process

Time-dependent 31P saturation-transfer studies were conducted with the Cd2+-activated form of muscle phosphoglucomutase to probe the origin of the 100-fold difference between its catalytic efficiency (in terms of kcat) and that of the more efficient Mg2+-activated enzyme. The present paper describes the equilibrium mixture of phosphoglucomutase and its substrate/product pair when the concentration of the Cd2+ enzyme approaches that of the substrate and how the nine-spin 31P NMR system provided by this mixture was treated. It shows that the presence of abortive complexes is not a significant factor in the reduced activity of the Cd2+ enzyme since the complex of the dephosphoenzyme and glucose 1,6-bisphosphate, which accounts for a large majority of the enzyme present at equilibrium, is catalytically competent. It also shows that rate constants for saturation transfer obtained at three different ratios of enzyme to free substrate are mutually compatible. These constants, which were measured at chemical equilibrium, can be used to provide a quantitative kinetic rationale for the reduced steady-state activity elicited by Cd2+ relative to Mg2+ [cf. Ray, W.J., Post, C.B., & Puvathingal, J.M. (1989) Biochemistry (following paper in this issue)]. They also provide minimal estimates of 350 and 150 s-1 for the rate constants describing (PO3-) transfer from the Cd2+ phosphoenzyme to the 6-position of bound glucose 1-phosphate and to the 1-position of bound glucose 6-phosphate, respectively. These minimal estimates are compared with analogous estimates for the Mg2+ and Li+ forms of the enzyme in the accompanying paper.     

Determination of kinetic parameters from chemical relaxation data discrimination between coupled two-step binding reactions differing in stoichiometry

The chemical relaxation times of two different two-step equilibrium reactions, characterized by a 1:1 binding process followed by a subsequent rearrangement step and a stepwise 1:2 binding reaction, are analyzed for the purpose of qualitative model discrimination and quantitative determination of kinetic parameters. The equations describing the dependences of the two reciprocal relaxation times on suitable concentrations are given for both models in the general case as well as for four different limiting situations which are characterized by well separated relaxation times. The conditions corresponding to the limiting cases are expressed in terms of strong, weak and no coupling between the two partial equilibrium steps involved in both models. The coupling strength depends on the rate constants as well as on the total concentrations of the reactants. Criteria to discriminate between these two reaction models under defined limiting conditions are developed. In the general case, the product of both reciprocal relaxation times can be used to distinguish both models. If only one relaxation time can be resolved experimentally, it is possible under conditions described to determine only a reduced set of individual rate constants for most of the limiting cases considered. If both relaxation times are observed, all rate constants are determinable in the general case as well as in most of the limiting cases discussed.