Letter: Kinetic negative co-operativity in the allosteric model of Monod, Wyman and Changeux

The co-operativity of homotropic interactions between substrate molecules in oligomeric enzymes is analyzed in the frame of the concerted transition theory of Monod et al. (1965). A discussion of the Hill coefficient n_H allows determination of the conditions for negative co-operativity (n_H < 1). This phenonomenon, usually taken as indicative of a sequential mechanism (Koshland et al., 1966), can be accounted for by the concerted model when the enzyme represents a K-V or V system, i.e. when the two protomer conformational states postulated in the theory differ in their catalytic activity. However, only negative co-operativity for catalysis can be explained by the concerted model, not negative co-operativity of binding.     

Co-operative response of chemically excitable membrane: I. Formulation: Unified theory of co-operativity

The lattice-model of Changeux, Thiery, Tung & Kittel (1966) was extended in order to examine the co-operative response of chemically excitable membrane and the exact mathematical correspondence to the Ising model was shown. In this model, two conformational states S and R with different affinities for the ligand are assumed to be accessible to each protomer, which is interacting with the nearest-neighbor protomers. The model is applicable to any kind of symmetrically interacting system consisting of oligomers and lattices and is an extension of previously proposed models of allosteric protein. It includes the model of Monod, Wyman, & Changeux (1965) and that of Koshland, Némethy & Filmer (1966) as the extreme cases of the oligomer. By assuming that a state-transition from S to R in a protomer is accompanied by a unit increase in conductance, the characteristics of dose-response curves of chemically excitable membrane are examined. The Hill's coefficient n_H of dose-response curve, the measure of the co-operativity, is shown to be proportional to the square of the mean fluctuation of the state function, the fraction of protomers in R state.     

Scaling relationship between tree respiration rates and biomass

The WBE theory proposed by West, Brown and Enquist predicts that larger plant respiration rate, R, scales to the three-quarters power of body size, M. However, studies on the R versus M relationship for larger plants (i.e. trees larger than saplings) have not been reported. Published respiration rates of field-grown trees (saplings and larger trees) were examined to test this relationship. Our results showed that for larger trees, aboveground respiration rates R_A scaled as the 0.82-power of aboveground biomass M_A, and that total respiration rates R_T scaled as the 0.85-power of total biomass M_T, both of which significantly deviated from the three-quarters scaling law predicted by the WBE theory, and which agreed with 0.81–0.84-power scaling of biomass to respiration across the full range of measured tree sizes for an independent dataset reported by Reich et al. (Reich et al. 2006 Nature 439, 457–461). By contrast, R scaled nearly isometrically with M in saplings. We contend that the scaling exponent of plant metabolism is close to unity for saplings and decreases (but is significantly larger than three-quarters) as trees grow, implying that there is no universal metabolic scaling in plants.     

Revisiting Verhulst and Monod models: analysis of batch and fed-batch cultures

The paper re-evaluates Verhulst and Monod models. It has been claimed that standard logistic equation cannot describe the decline phase of mammalian cells in batch and fed-batch cultures and in some cases it fails to fit somatic growth data. In the present work Verhulst, population-based mechanistic growth model was revisited to describe successfully viable cell density (VCD) in exponential and decline phases of batch and fed-batch cultures of three different CHO cell lines. Verhulst model constants, K, carrying capacity (VCD/ml or μg/ml) and r, intrinsic growth factor (h⁻¹) have physical meaning and they are of biological significance. These two parameters together define the course of growth and productivity and therefore, they are valuable in optimisation of culture media, developing feeding strategies and selection of cell lines for productivity. The Verhulst growth model approach was extended to develop productivity models for batch and fed-batch cultures. All Verhulst models were validated against blind data (R² > 0.95). Critical examination of theoretical approaches concluded that Monod parameters have no physical meaning. Monod-hybrid (pseudo-mechanistic) batch models were validated against specific growth rates of respective bolus and continuous fed-batch cultures (R² ≈ 0.90). The reduced form of Monod-hybrid model C_L/(K_L + C_L) describes specific growth rate during metabolic shift (R² ≈ 0.95). Verhulst substrate-based growth models compared favourably with Monod-hybrid models. Thus, experimental evidence implies that the constants in the Monod-hybrid model may not have physical meaning but they behave similarly to the biological constants in Michaelis–Menten enzyme kinetics, the basis of the Monod growth model.     

Analysis of teleost hemoglobin by Adair and Monod-Wyman-Changeux models

Summary The allosteric effects of the erythrocytic nucleoside triphosphates (NTP) and of proton concentrations were investigated by precise measurement of Hb–O₂ equilibria of tench hemoglobin (including extreme, high and low saturation ranges) and analysed in terms of the MWC two state model and the Adair four step oxygenation theory.At low concentrations (NTP/Hb ratio=1.0, and pH>7.3) ATP, GTP and protons decrease Hb–O₂ affinity by increasing the allosteric constantL and reducingK _T, the association constant¹ of the deoxy, tense state of the Hb, without significantly affecting that (K _R) of the oxy state, increasing the free energy of cooperativity (G). High concentrations of these effectors, however, also reduceK _R. The greater sensitivity of the half-saturation O₂ tension (P ₅₀) of the Hb to GTP than to ATP at the same concentration, correlates with greater effects of GTP on bothK _T andK _R. The pH and NTP dependence of the four Adair association constants and the calculated fractional populations of Hb molecules in different stages of oxygenation show that the autochthonous NTP effectors and protons stabilize the T structure and postpone the TR transition basic to cooperativity in fish Hb.The possible implications of the findings for aquatic respiration are discussed.Abbreviations ATP adenosine triphosphate - DPG 2,3-diphosphoglycerate (glycerate-2,3-bisphosphate) - GTP guanosine triphosphate - IHP inositol hexaphosphate - NTP nucleoside triphosphates In this paperK _T andK _R are defined as theassociation equilibrium constants instead of dissociation constants (as originally defined by Monod et al. 1965) to facilitate comparison with the Adair constants     

The theoretical analysis of kinetic behaviour of “hysteretic” allosteric enzymes. I. The kinetic manifestations of slow conformational change of an oligomeric enyzme in the Monod,Wyman and Changeux model

The curves of the time (t) dependent product (Pr) accumulation for Monod, Wyman & Changeux model (1965), where the rate of installation of equilibrium between two conformational states of oligomeric enzyme (R ? T) is comparable to that of enzymatic process, are theoretically analysed. It is assumed that the complexes of R and T forms are in rapid equilibrium with the free components. The character of the effective rate constant of conformational transition R ? T and the value of τ (where τ is the intercept of the linear part of the kinetic curve of [Pr] versus t with the time axis) versus the substrate concentration is analysed. It is also shown that slow conformational transition R ? T can be manifested by an unusual shape for Monod et al. model plots of initial velocity of the enzyme reaction v. the substrate concentration (these curves can clearly display expressed inflection points and Hill's cooperativity coefficient less than unity).     

Plasticity of the thermal requirements of exotherms and adaptation to environmental conditions

In exothermal organisms, temperature is an important determinant of the rate of ecophysiological processes, which monotonically increase between the minimum (t_{d min}) and maximum (t_{d max}) temperatures typical for each species. In insects, t_{d min} and t_{d max} are correlated and there is a approximately 20°C interval (thermal window W_T = t_{d max} − t_{d min}) between them over which insects can develop. We assumed that other exotherms have similar thermal windows because the thermal kinetics of their physiological processes are similar. In this study, we determined the thermal requirements for germination in plants. Seeds of 125 species of Central European wild herbaceous and crop plants were germinated at nine constant temperatures between 5 and 37°C, and the time to germination of 50% of the seeds D and rate of germination R (=1/D) were determined for each temperature and the Lactin model used to determine t_{d min}, t_{d max}, and W_T. The average width of the thermal windows for seeds was significantly wider (mean 24°C, 95% CI 22.7–24.2°C), varied more (between 14.5 and 37.5°C) and development occurred at lower temperatures than recorded for insects. The limiting temperatures for germination, t_{d min} and t_{d max}, were not coupled, so the width of the thermal window increased with both a decrease in t_{d min} and/or increase in t_{d max}. Variation in W_T was not associated with taxonomic affiliation, adult longevity, or domestication of the different species, but tends to vary with seed size. Plants are poor at regulating their temperature and cannot move to a more suitable location and as a consequence have to cope with wider ranges in temperatures than insects and possibly do this by having wider thermal windows. Synthesis: The study indicated specificity of W_T in different exotherm taxa and/or their development stages.     

A kinetic study of the pH effect on the allosteric properties of pyruvate kinase from Phycomyces blakesleeanus

This paper reports the pH-dependence of the allosteric kinetics of Phycomyces blakeseeanus pyruvate kinase with phosphoenol pyruvate and Mg2+ ions in the presence and in the absence of fructose 1,6-bisphosphate (allosteric activator) and L-alanine (allosteric inhibitor). Hydrogen ions increase the affinity of the inhibitory binding sites for phosphoenol pyruvate and Mg2+ ions. Assuming partial conformational states of high and low affinity for inhibitory binding sites, the data presented are in good agreement with the predictions postulated by the two-state concerted-symmetry model of Monod, Wyman, and Changeux. Fructose-1,6-bisphosphate and L-alanine show opposite effects on the interactions of phosphoenol pyruvate and Mg2+ ions with their respective catalytic and inhibitory binding sites. At pH 6.0, the regulation of the Phycomyces pyruvate kinase activity by the concentrations of phosphoenol pyruvate and Mg2+ ions is controlled mainly by L-alanine.     

New Insights on the Mechanism of the K+-Independent Activity of Crenarchaeota Pyruvate Kinases

Eukarya pyruvate kinases have glutamate at position 117 (numbered according to the rabbit muscle enzyme), whereas in Bacteria have either glutamate or lysine and in Archaea have other residues. Glutamate at this position makes pyruvate kinases K⁺-dependent, whereas lysine confers K⁺-independence because the positively charged residue substitutes for the monovalent cation charge. Interestingly, pyruvate kinases from two characterized Crenarchaeota exhibit K⁺-independent activity, despite having serine at the equivalent position. To better understand pyruvate kinase catalytic activity in the absence of K⁺ or an internal positive charge, the Thermofilum pendens pyruvate kinase (valine at the equivalent position) was characterized. The enzyme activity was K⁺-independent. The kinetic mechanism was random order with a rapid equilibrium, which is equal to the mechanism of the rabbit muscle enzyme in the presence of K⁺ or the mutant E117K in the absence of K⁺. Thus, the substrate binding order of the T. pendens enzyme was independent despite lacking an internal positive charge. Thermal stability studies of this enzyme showed two calorimetric transitions, one attributable to the A and C domains (T_m of 99.2°C), and the other (T_m of 105.2°C) associated with the B domain. In contrast, the rabbit muscle enzyme exhibits a single calorimetric transition (T_m of 65.2°C). The calorimetric and kinetic data indicate that the B domain of this hyperthermophilic enzyme is more stable than the rest of the protein with a conformation that induces the catalytic readiness of the enzyme. B domain interactions of pyruvate kinases that have been determined in Pyrobaculum aerophilum and modeled in T. pendens were compared with those of the rabbit muscle enzyme. The results show that intra- and interdomain interactions of the Crenarchaeota enzymes may account for their higher B domain stability. Thus the structural arrangement of the T. pendens pyruvate kinase could allow charge-independent catalysis.     

The incorporation of cytochrome oxidase into newly-forming yeast mitochondrial membranes

Cytochrome c oxidase extracted from a yeast auxotroph grown on different fatty acid supplements, and assayed at various temperatures (0→37°), gives transition points (T_t) when the data are expressed in an Arrhenius plot: T_t for linoleic, oleic and elaidic acids were 7.7°, 10.2° and 21.8° respectively. Lipid anlaysis of isolated mitochondria established that there is a change in lipid profile when yeast is first grown aerobically on linoleic acid and then transferred to elaidic acid under anaerobic consitions. These two observations bave been used as a basis for the investigation of the assembly of cytochrome c oxidase when cells are oxygen induced for mitochondriogenesisvia a linoleic, N₂ elaidic, O₂ transfer experiment. A three-stage assembly process is proposed consisting of (i) 0.0→0.25 h, auto-assembly of oxidase from preformed, chloramphenicol-sensitive precursors and newly synthesized cycloheximide-sensitive precursors, (ii) 0.25 h→0.5h, a continuation of (i) plusde novo synthesis of new precursors of both kinds and (iii) 0.5 h onwards; normal, synchronisedde novo synthesis. This model predicts one population of oxidase molecules at 0.0 h (linoleic T_t) and two populations of oxidase at 0.5 h (end of stage (ii)); one from the anaerobic precursors (linoleic T_t) and one for thede nove, aerobic oxidase (elaidic T_t): these predictions are confirmed by Arrhenius profiles constructed from samples taken at 0.0, 0.5, 1.0 and 3.0 h after oxygen challenge.     

Interaction of thyroid hormones and cyclic AMP in the stimulation of hepatic gluconeogenesis

The possible interaction of l-3,3′,-5-triiodthyronine (T₃) and cycli AMP on hepatic gluconeogenesis was investigated in perfused livers isolated from hypothyroid rats starved for 24 h. T₃ (1·10^?6) and cyclic AMP (2·10^?4 M) increased hepatic gluconeogenesis from alanine within 30–60 min perfusion time (+85%/ + 90%), both were additive in their action (+191%). Concomitantly, α-amino[¹⁴C]isobutyric acid as well as net alanine uptake and urea production were elevated by T₃ and by cyclic AMP. T₃ increased the oligomycin-sensitive O₂ consumption and the tissue ‘overall’ ATP/ADP ratio, whereas cyclic AMP showed only a minor effect on cellular energy metabolism. As was observed recently for cyclic AMP, the stimulating action of T₃ on hepatic gluconeogenesis was independent of exogenous Ca²⁺ concentration. T₃ by itself affected neither the total nor the protein-bound hepatic cyclic AMP contents, pyruvate kinese (v:0.15 mM) activation nor the tissue levels of gluconeogenic intermediates. In contrast, cyclic AMP itself — although less effective than in euthyroid livers — decreased pyruvate kinase activity in hypothyroid livers with a concomitant increase in hepatic phosphoenolpyruvate concentration. This resulted in a ‘crossover’ between pyruvate and phosphoenolpyruvate. Cyclic AMP action was not affected by the further addition of T₃. Glucagon (1·10^?8 M) was less effective in hypo-than in euthyroid livers in increasing endogenous cyclic AMP content, deactivating pyruvate kinase and stimualting glucose production; this is normalized by the further addition of 1-methyl-3-isobutylxanthine (50 μM). It is concluded that T₃ stimulats hepatic gluconeogenesis by a cyclic-AMP-independent mechanism. In addition, the stimulatory action of cyclic AMP and glucagon with respect to hepatic gluconeogenesis is reduced in hypothyroidism. This may be explained by an increase in hepatic phosphodiesterase activity.     

Electrophoretic pattern of cytosolic pyruvate kinase fractions A and B (type L and M2) from normal rat liver and Morris hepatoma 7777

Cytosolic pyruvate kinase fractions A and B obtained by salting out procedure from normal rat liver and Morris hepatoma 7777, purified by affinity chromatography on Blue Sepharose CL-6B, have shown similar electrophoretic patterns in polyacrylamide gel at pH 8.3, to previously studied pyruvate kinase extracts from chromatin of cell nuclei. Three variants (α₁, β₁, γ₁) from normal liver pyruvate kinase fraction A (type L) had the greatest electrophoretic mobility, showed sigmoidal kinetics in relation to 2-phosphoenolpyruvate (PEP), and sensitivity to ATP and fructose 1,6-diphosphate (FDP). The fraction A dominated over normal liver fraction B (type M₂), which in electrophoresis showed a slower y2 variant, similar to the fraction A of hepatoma. All variants from fractions B of normal liver and A of hepatoma had linear kinetics and were sensitive to ATP but not to FDP. The greatest differences showed pyruvate kinase fraction B from Morris hepatoma. Its all variants α₂, β₂, γ₃ were more cathodic and had linear kinetics in relation to PEP. They all were insensitive to normal signal molecules (ATP and FDP). The γ₃ alkaline variant acquired sensitivity to inhibition by l-cysteine. Showing several-fold higher activity, much greater affinity to the main substrate, and a lack of sensitivity to feed-back inhibition by ATP, it was responsible for a high rate of aerobic glycolysis and diminution of the Pasteur effect in metabolic studies. It was probably encoded during oncogene activation and plays a special role in different metabolic strategies of tumour cells.     

Soybeans and radish leaves contain only one of the sulfonium diastereoisomers of S-adenosylmethionine

Using a liquid chromatography method that separates the two sulfonium diastereoisomers of adenosylmethionine, we have found that immature soybeans, soybean callus culture, radish leaves, yeast and rat liver contain only the (S)-sulfonium form of S-adenosylmethionine. Our findings contradict the suggestion by Stolowitz and Minch that 10–20% of naturally-occurring adenosylmethionine may have the (R)-configuration at the sulfonium pole. Absence of the (R)-sulfonium isomer of adenosylmethionine in biological materials indicates that the (R)-sulfonium form of adenosylmethionine present in commercial adenosylmethionine samples is an artifact of the isolation procedure. Our method of measuring the isomers of adenosylmethionine enabled us to readily determine the rate of racemization and hydrolysis of adenosylmethionine. Our rate constants for racemization (K_r) and hydrolysis (K_h) were 2.4 × 10^?6 sec^?1 and 12.3 × 10-^?6 sec^?1, respectively; values which are noticeably different from those of Wu and co-workers which were obtained with a more complicated method (K_r = 8 × 10^?1 sec^?1; K_h = 6 × 10^?6 sec^?1). We believe the absence of the (R)-isomer in vivo is best explained by stabilization of the (S)-isomer as suggested by Wu et al. Although the tissues we have analysed contained the (S)-sulfonium form of adenosylmethionine exclusively, when ethionine-resistant soybean cell lines were given ethionine, they accumulated both sulfonium diastereoisomers of adenosylethionine.     

Purification of porcine liver pyruvate dehydrogenase complex and characterization of its catalytic and regulatory properties.

Procedures are described for isolating highly purified porcine liver pyruvate and α-ketoglutarate dehydrogenase complexes. Rabbit serum stabilized these enzyme complexes in mitochondrial extracts, apparently by inhibiting lysosomal proteases. The complexes were purified by a three-step procedure involving fractionation with polyethylene glycol, pelleting through 12.5% sucrose, and a second fractionation under altered conditions with polyethylene glycol. Sedimentation equilibrium studies gave a molecular weight of 7.2 × 10⁶ for the liver pyruvate dehydrogenase complex. Kinetic parameters are presented for the reaction catalyzed by the pyruvate dehydrogenase complex and for the regulatory reactions catalyzed by the pyruvate dehydrogenase kinase and pyruvate dehydrogenase phosphatase. For the overall catalytic reaction, the competitive K_i to K_m ratio for NADH versus NAD⁺ and acetyl CoA versus CoA were 4.7 and 5.2, respectively. Near maximal stimulations of pyruvate dehydrogenase kinase by NADH and acetyl CoA were observed at NADH:NAD⁺ and acetyl CoA:CoA ratios of 0.15 and 0.5, respectively. The much lower ratios required for enhanced inactivation of the complex by pyruvate dehydrogenase kinase than for product inhibition indicate that the level of activity of the regulatory enzyme is not directly determined by the relative affinity of substrates and products of catalytic sites in the pyruvate dehydrogenase complex. In the pyruvate dehydrogenase kinase reaction, K⁺ and NH⁺₄ decreased the K_m for ATP and the competitive inhibition constants for ADP and (β,γ-methylene)adenosine triphosphate. Thiamine pyrophosphate strongly inhibited kinase activity. A high concentration of ADP did not alter the degree of inhibition by thiamine pyrophosphate nor did it increase the concentration of thiamine pyrophosphate required for half-maximal inhibition.     

Pyruvate kinase variants of the Alaskan king-crab. Evidence for a temperature-dependent interconversion between two forms having distinct and adaptive kinetic properties

1. Pyruvate kinase of Alaskan king-crab leg muscle exists in two kinetically distinct forms, each of which displays a different temperature-dependence in the K_m for phosphoenolpyruvate. 2. A `cold' variant of the enzyme has hyperbolic kinetics and exhibits a minimal K<sub>m</sub> for substrate at 5°. At physiological concentrations of phosphoenolpyruvate the `cold' enzyme is active only below 10°. A `warm' pyruvate kinase has a minimal K<sub>m</sub> for substrate at about 12°. This enzyme displays sigmoidal kinetics and is likely to be inactive, at physiological substrate concentrations, at temperatures below 9°. 3. The combined activities of these two pyruvate kinases yield highly temperature-independent rates of catalysis, at physiological substrate concentrations, over the range of habitat temperatures encountered by the organism, namely 4–12°. 4. The two variants of pyruvate kinase do not appear to be isoenzymes in the conventional sense. Electrophoretic and electrofocus analyses revealed only single peaks of activity. 5. The results suggest that the `warm' pyruvate kinase and the `cold' pyruvate kinase are formed by a temperature-dependent interconversion of one protein species. This interconversion has major adaptive significance: as the temperature is lowered the `warm' enzyme is converted into the `cold' enzyme; the opposite situation obtains when the temperature is raised. Temperature changes thus mimic the effects noted for fructose 1,6-diphosphate on certain mammalian pyruvate kinases.     

Functional analysis, overexpression, and kinetic characterization of pyruvate kinase from Plasmodium falciparum

The important role of pyruvate kinase during malarial infection has prompted the cloning of a cDNA encoding Plasmodium falciparum pyruvate kinase (pfPyrK), using mRNA from intraerythrocytic-stage malaria parasites. The full-length cDNA encodes a protein with a computed molecular weight of 55.6 kDa and an isoelectric point of 7.5. The purified recombinant pfPyrK is enzymatically active and exists as a homotetramer in its active form. The enzyme exhibits hyperbolic kinetics with respect to phosphoenolpyruvate and ADP, with K_m of 0.19 and 0.12 mM, respectively. pfPyrK is not affected by fructose-1,6-bisphosphate, a general activating factor of pyruvate kinase for most species. Glucose-6-phosphate, an activator of the Toxoplasma gondii enzyme, does not affect pfPyrK activity. Similar to rabbit pyruvate kinase, pfPyrK is susceptible to inactivation by 1 mM pyridoxal-5′-phosphate, but to a lesser extent. A screen for inhibitors to pfPyrK revealed that it is markedly inhibited by ATP and citrate. Detailed kinetic analysis revealed a transition from hyperbolic to sigmoidal kinetics for PEP in the presence of citrate, as well as competitive inhibitory behavior for ATP with respect to PEP. Citrate exhibits non-competitive inhibition with respect to ADP with a K_i of 0.8 mM. In conclusion, P. falciparum expresses an active pyruvate kinase during the intraerythrocytic-stage of its developmental cycle that may play important metabolic roles during infection.     

Kinetics of Bacterial Growth on Chlorinated Aliphatic Compounds

With the pure bacterial cultures Ancylobacter aquaticus AD20 and AD25, Xanthobacter autotrophicus GJ10, and Pseudomonas sp. strain AD1, Monod kinetics was observed during growth in chemostat cultures on 1,2-dichloroethane (AD20, AD25, and GJ10), 2-chloroethanol (AD20 and GJ10), and 1,3-dichloro-2-propanol (AD1). Both the Michaelis-Menten constants (K_m) of the first catabolic (dehalogenating) enzyme and the Monod half-saturation constants (K_s) followed the order 2-chloroethanol, 1,3-dichloro-2-propanol, epichlorohydrin, and 1,2-dichloroethane. The K_s values of strains GJ10, AD20, and AD25 for 1,2-dichloroethane were 260, 222, and 24 μM, respectively. The low K_s value of strain AD25 was correlated with a higher haloalkane dehalogenase content of this bacterium. The growth rates of strains AD20 and GJ10 in continuous cultures on 1,2-dichloroethane were higher than the rates predicted from the kinetics of the haloalkane dehalogenase and the concentration of the enzyme in the cells. The results indicate that the efficiency of chlorinated compound removal is indeed influenced by the kinetic properties and cellular content of the first catabolic enzyme. The cell envelope did not seem to act as a barrier for permeation of 1,2-dichloroethane.     

A three-state MWC analysis of oxygenation in tench (Tinca tinca) hemoglobin

Summary Oxygen equilibria in tench hemoglobin were analysed according to a three-state MWC model. In addition to theT andR states of the traditionally used two-state model, the three-state model introduces an additional state, theS state, when organic phosphates bind to theT-structure hemoglobin. Under conditions covering natural red cell pH values and nucleoside triphosphate-hemoglobin ratios, it was possible to closely fit experimental data to the three-state equation with constant values of the association constantsK _R ,K _T , andK _S , and with only the allosteric constantsL andM varying with effector conditions. Thus, in contrast to a twostate analysis of oxygen equilibria, the three-state analysis was consistent with the basic assumption of the MWC model, that heterotropic ligands only affect allosteric constants and not association constants. The temperature-dependence of the three-state parameter values showed that in the presence of nucleoside triphosphate the dominance of theS state over theT state was most pronounced at low temperatures. Furthermore, the numerical values of the enthalpy and entropy change of oxygenation were lower in theS state than in theT andR states, and the enthalpy and entropy change for the allostericSR transition were much larger than for theTR transition.Abbreviations Hb hemoglobin - Y fractional O₂ saturation - ATP adenosine triphosphate     

A method to calculate binding equilibrium concentrations in the allosteric ternary complex model that supports ligand depletion

The allosteric ternary complex model is frequently used in pharmacology to represent the interaction of a receptor R with two ligands A and B. Certain well-known formulas are routinely used to calculate the fractions of the receptor bound at equilibrium with A only, B only, and both A and B. However, it is often omitted that these classical formulas presume that there is no ligand depletion, i.e. that the equilibrium concentrations [A] and [B] of the ligands are well approximated by their total concentrations [A]_T and [B]_T. We present a calculation method which is applicable without this or any restrictions. The equilibrium concentration [R] of the receptor is implicitly characterized by an equation which is solved with a very simple convergent numerical algorithm. The concentrations [A] and [B] are given by explicit formulas in terms of [R]. The required parameters are the equilibrium dissociation constants K_A and K_B, the cooperativity factor α, and the total concentrations [R]_T, [A]_T and [B]_T.