Validity of quasi-steady-state and transfer-function representations for input-output relation in a Michaelis-Menten reaction

In relation to the input-output characteristics of enzymatic reactions in the cellular metabolism and biochemical reactors, the validity of the quasi-steady-state and transfer-function representations of reaction velocity has been examined for a basic Michaelis-Menten reaction employing computer simulation, that is, numerical integration of the rate equation. The well-known S-v relationship (relationship between substrate concentration and reaction velocity)derived on the quasi-steady-state assumption is found to be in general a good approximation to the actual velocity throughout the temporal progress of the reaction. The validity of the approximation depends on a ratio of the Michaelis constant to the total enzyme concentration in the reaction system rather than on the individual rate constants. A transfer-function representation is derived on assuming an exponential change in the reaction velocity for the indicial response to the substrate influx rate. The representation has a wider valid region with a decrease in influx rate than with an increase in the influx rate. The validity is most dependent on a ratio of total enzyme concentration to the steady-state concentration of the substrate. The analysis of the linear sensitivity of the reaction velocity to rate constants reveals that the characteristics of these valid representations in systems analysis change according to the phase of the reaction.     

A transfer-function representation for regulatory responses of a controlled metabolic pathway

A transfer-function representation for the response of a controlled metabolic pathway to the changes in influx and efflux rates of metabolites is formulated to describe analytically and approximately the regulatory behavior of the pathway around a steady state. The pathway model analysed is an open and homogeneous system which consists of two consecutive enzymatic reactions catalyzed by an allosteric enzyme of Monod-Wyman-Changeux (MWC) dimeric model and a Michaelis-Menten-type enzyme, respectively, and undergoes the feedback inhibition by the end product. The rate equation for the system (a system of ordinary differential equations) is linearized about a steady state, so that the responses of the reaction rates to the changes in influx rate of the substrate and efflux rate of the end product are expressed in a form of transfer function. The formulation leads to the transfer function for the response of production rate of the end product to the change in its efflux rate to clarify the regulatory response of feedback mechanism in controlled metabolic pathways. The relationship among the chemical species in the system at steady stete also supports a reasonable assumption that the regulatory mechanisms in metabolic pathways are to control the production of end product against the change in its demand from the cellular environments.     

Sigmoidicity in Allosteric Models

We show results complementary to papers by Bardsley and Waight [2, 3], Gibson and Levin [13], Goldbeter [14], and Karlin [19] on sigmoidicity as an essential feature of allosteric models, possibly leading to a criterion of choice between these. In particular, we give the explicit form for the second derivative of the saturation function in the MWC case, and also the calculation of the binding polynomial in the circular KNF case. First, we give analytical conditions of sigmoidicity for each characteristic function in the MWC and KNF allosteric models. In the MWC model, the regions of sigmoidicity are different for the state and saturation functions, and when the catalytic activities of the two conformational states are different, the area of sigmoidicity is significantly larger for the steady-state rate function than for the saturation function. Furthermore, we rigorously prove the existence of mixed kinetic cooperativity in certain conditions. In the KNF model, the state and saturation functions are the same and their sigmoidicity depends only on the degree of coupling between subunits and on the relative stability of the asymmetrically induced subunit interactions. Finally, we suggest a theoretical criterion for discrimination between allosteric models.     

Kappa乘积法在寡聚体变构酶反应速度方程中的应用

寡聚体变构酶所催化的反应中,酶与底物的结合是逐步的,形成链式反应.因此,酶与底物结合的每一种形式可用一组相应的速度常数乘积来表示,这是根据平衡理论建立微分方程组所导出来的,本文应用Kappa乘积法简易导出MWC及KNF模式的速度方程,并对这种方法应用范围的扩大作了讨论.     

Computer-based learning of cooperativity and allostery

The aim of this article is to facilitate the understanding ofenzyme cooperativity and allostery by undergraduate and postgraduatestudents with the aid of a graphic microcomputer. For this purposethe molecular models of Monod-Wyman-Changeux (MWC) and of Koshland-Nemethy-Filmer(KNF) are tested by showing how the different plots, direct,reciprocal, Scatchard and Hill, vary as do the parameters consideredin these models. The programs used (one for each model) progressfrom easy aspects to complicated ones without the interventionof the user (student). Nevertheless, ultimately, with the MWCmodel, after the introduction of heterotropic effectors, theusers can select the parameters in order to further their knowledge.This can be useful also for testing the kinetic behaviour ofmultisubunit enzymes which present cooperativity and which havebeen extensively described in the literature. Received on March 28, 1985; accepted on June 20, 1985     

Conformational spread in a ring of proteins: a stochastic approach to allostery

We recently suggested that the sensitivity and range of a cluster of membrane receptors in bacteria would be enhanced by cooperative interactions between neighbouring proteins. Here, we examine the consequences of this "conformational spread" mechanism for an idealised one-dimensional system comprising a closed ring of identical allosteric protomers (protein molecules, or a group of protein domains operating as a unit). We show analytically and by means of Monte Carlo simulations that a ring of allosteric protomers can exhibit a switch-like response to changes in ligand concentration. We derive expressions for the sensitivity and cooperativity of switching and show that the maximum sensitivity is proportional to the number of protomers in the ring. A ring of this kind can reproduce the sensitivity and kinetics of the switch complex of a bacterial flagellar motor, for example, which is based on a ring of 34 FliM proteins. We also compare smaller rings of conformationally coupled protomers to classical allosteric proteins such as haemoglobin and show that the canonical MWC and KNF models arise naturally as limiting cases. Conformational spread appears to be a natural extension of the familiar mechanism of allostery: a physically realistic mechanism that should apply widely to many structures built from protein molecules.     

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.     

A Monod-Wyman-Changeux mechanism can explain G protein-coupled receptor (GPCR) allosteric modulation

The Monod-Wyman-Changeux (MWC) model was initially proposed to describe the allosteric properties of regulatory enzymes and subsequently extended to receptors. Yet despite GPCRs representing the largest family of receptors and drug targets, no study has systematically evaluated the MWC mechanism as it applies to GPCR allosteric ligands. We reveal how the recently described allosteric modulator, benzyl quinolone carboxylic acid (BQCA), behaves according to a strict, two-state MWC mechanism at the M1 muscarinic acetylcholine receptor (mAChR). Despite having a low affinity for the M1 mAChR, BQCA demonstrated state dependence, exhibiting high positive cooperativity with orthosteric agonists in a manner that correlated with efficacy but negative cooperativity with inverse agonists. The activity of BQCA was significantly increased at a constitutively active M1 mAChR but abolished at an inactive mutant. Interestingly, BQCA possessed intrinsic signaling efficacy, ranging from near-quiescence to full agonism depending on the coupling efficiency of the chosen intracellular pathway. This latter cellular property also determined the difference in magnitude of positive cooperativity between BQCA and the orthosteric agonist, carbachol, across pathways. The lack of additional, pathway-biased, allosteric modulation by BQCA was confirmed in genetically engineered yeast strains expressing different chimeras between the endogenous yeast G(pa1) protein and human Gα subunits. These findings define a chemical biological framework that can be applied to the study and classification of allosteric modulators across different GPCR families.     

Comments on Extensions of the Allosteric Model for Haemoglobin

RECENTLY Edelstein¹ has concluded on the basis of a numerical analysis that the sequential model as formulated by Koshland, Nemethy and Filmer (KNF)² describes the oxygen binding curves of a number of species of human haemoglobin less well than does the two-state allosteric model of Monod, Wyman and Changeux (MWC)³. This communication demonstrates that Edelstein's analysis is incomplete and that extension of his analysis reveals that no such conclusion can be drawn from the data considered.     

Control of kinetics by cooperative interactions

Cooperative effects in ligand binding and dissociation kinetics are much less investigated than steady state kinetics or equilibrium binding. Nevertheless, cooperativity in ligand binding leads necessarily to characteristic properties with respect to kinetic properties of the system. In case of positive cooperativity as found in oxygen binding proteins, a typical property is an autocatalytic ligand dissociation behavior leading to a time dependent, apparent ligand dissociation rate. To follow systematically the influence of the various potentially involved parameters on this characteristic property, simulations based on the simple MWC model were performed which should be relevant for all types of models based on the concept of an allosteric unit. In cases where the initial conformational distribution is very much dominated by the R-state, the intrinsic kinetic properties of the T-state are of minor influence for the observed ligand dissociation rate. Even for fast conformational transition rates, the R-state properties together with the size of the allosteric unit and the allosteric equilibrium constant define the shape of the curve. In such a case, a simplified model of the MWC-scheme (the irreversible n-chain model) is a good approximation of the full scheme. However, if in the starting conformational distribution some liganded T-molecules are present (a few percent is enough), the average off-rates can be significantly altered. Thus, the assignment of the initial rates to R-state properties has to be done with great care. However, if the R-state strongly dominates initially it is even possible to get an estimation of the lower limit for the number of interacting subunits from kinetic data: similar to the Hill-coefficient for equilibrium conditions, a measure for "kinetic cooperativity" can be derived by comparing initial and final ligand dissociation rates.     

Experimental evidence for allosteric modifier saturation as predicted by the bi-substrate Hill equation

The cooperative enzyme reaction rates predicted by the bi-substrate Hill equation and the bi-substrate Monod-Wyman-Changeux (MWC) equation when allosterically inhibited are compared in silico. Theoretically, the Hill equation predicts that when the maximum inhibitory effect at a certain substrate condition has been reached, an increase in allosteric inhibitor concentration will have no effect on reaction rate, that is the Hill equation shows allosteric inhibitor saturation. This saturating inhibitory effect is not present in the MWC equation. Experimental in vitro data for pyruvate kinase, a bi-substrate cooperative enzyme that is allosterically inhibited, are presented. This enzyme also shows inhibitor saturation, and therefore serves as experimental evidence that the bi-substrate Hill equation predicts more realistic allosteric inhibitor behaviour than the bi-substrate MWC equation.     

Dynamic dissociating homo-oligomers and the control of protein function

Homo-oligomeric protein assemblies are known to participate in dynamic association/disassociation equilibria under native conditions, thus creating an equilibrium of assembly states. Such quaternary structure equilibria may be influenced in a physiologically significant manner either by covalent modification or by the non-covalent binding of ligands. This review follows the evolution of ideas about homo-oligomeric equilibria through the 20th and into the 21st centuries and the relationship of these equilibria to allosteric regulation by the non-covalent binding of ligands. A dynamic quaternary structure equilibria is described where the dissociated state can have alternate conformations that cannot reassociate to the original multimer; the alternate conformations dictate assembly to functionally distinct alternate multimers of finite stoichiometry. The functional distinction between different assemblies provides a mechanism for allostery. The requirement for dissociation distinguishes this morpheein model of allosteric regulation from the classical MWC concerted and KNF sequential models. These models are described alongside earlier dissociating allosteric models. The identification of proteins that exist as an equilibrium of diverse native quaternary structure assemblies has the potential to define new targets for allosteric modulation with significant consequences for further understanding and/or controlling protein structure and function. Thus, a rationale for identifying proteins that may use the morpheein model of allostery is presented and a selection of proteins for which published data suggests this mechanism may be operative are listed.     

The Tertiary Origin of the Allosteric Activation of E. coli Glucosamine-6-Phosphate Deaminase Studied by Sol-Gel Nanoencapsulation of Its T Conformer

The role of tertiary conformational changes associated to ligand binding was explored using the allosteric enzyme glucosamine-6-phosphate (GlcN6P) deaminase from Escherichia coli (EcGNPDA) as an experimental model. This is an enzyme of amino sugar catabolism that deaminates GlcN6P, giving fructose 6-phosphate and ammonia, and is allosterically activated by N-acetylglucosamine 6-phosphate (GlcNAc6P). We resorted to the nanoencapsulation of this enzyme in wet silica sol-gels for studying the role of intrasubunit local mobility in its allosteric activation under the suppression of quaternary transition. The gel-trapped enzyme lost its characteristic homotropic cooperativity while keeping its catalytic properties and the allosteric activation by GlcNAc6P. The nanoencapsulation keeps the enzyme in the T quaternary conformation, making possible the study of its allosteric activation under a condition that is not possible to attain in a soluble phase. The involved local transition was slowed down by nanoencapsulation, thus easing the fluorometric analysis of its relaxation kinetics, which revealed an induced-fit mechanism. The absence of cooperativity produced allosterically activated transitory states displaying velocity against substrate concentration curves with apparent negative cooperativity, due to the simultaneous presence of subunits with different substrate affinities. Reaction kinetics experiments performed at different tertiary conformational relaxation times also reveal the sequential nature of the allosteric activation. We assumed as a minimal model the existence of two tertiary states, t and r, of low and high affinity, respectively, for the substrate and the activator. By fitting the velocity-substrate curves as a linear combination of two hyperbolic functions with K _t and K _r as K_M values, we obtained comparable values to those reported for the quaternary conformers in solution fitted to MWC model. These results are discussed in the background of the known crystallographic structures of T and R EcGNPDA conformers. These results are consistent with the postulates of the Tertiary Two-States (TTS) model.     

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.     

The dependence of flux,sensitivity and response of the flux on the concentrations of substrate and modifiers for cooperative enzymes

The sensitivity (change of flux per unit change in the concentration of substrate) and response (change of flux per unit change in the concentration of modifier) are studied for a two-site Adair model in which cooperativity arises from both binding and catalytic interactions. For positive cooperativity, the sensitivity is weakly dependent on the Hill coefficient for the binding case, but can increase without limit for the catalytic case. Negatively cooperative enzymes (binding only) give very large sensitivities compared with positively or non-interacting systems, but the sensitivity rapidly decreases as the saturation increases above 25%. Modifiers greatly enhance the sensitivity; large changes in flux can be obtained for small changes in the concentrations of substrates and modifiers. In general, increasing the degree of kinetic cooperativity decreases the degree of binding cooperativity; selective pressure to maximize the sensitivity and response of allosteric enzymes may act to optimize cooperativity of binding modifiers and kinetic cooperativity of substrate turnover. The initial velocity equations including modifiers can be extended to bi-substrate, cooperative kinetics. The kinetics of methanol dehydrogenase are discussed.     

Allosteric control in Limulus polyphemus hemocyanin: functional relevance of interactions between hexamers

Hemocyanin of the horseshoe crab Limulus polyphemus is composed of 48 oxygen-binding subunits, which are arranged in eight hexameric building blocks. Allosteric interactions in this oligomeric protein have been examined by measurement of high-precision oxygen-equilibrium curves, using an automated Imai cell. Several models were compared in numerical analysis of the data. A number of conclusions can be drawn with confidence. (1) Oxygen binding by Limulus hemocyanin cannot satisfactorily be described by the two-state MWC model [Monod, J., Wyman, J., & Changeux, J.P. (1965) J. Mol. Biol. 12, 88-118] for allosteric transitions with either the hexamer or dodecamer as the allosteric unit. (2) Of the models tested, the data sets can be best described by an extended MWC model that allows for an equilibrium, within the 48-subunit ensemble, between cooperative hexamers and cooperative dodecamers. The model invokes T and R states for both hexamers (T6 and R6) and dodecamers (T12 and R12). Allosteric effectors modulate oxygen affinity and cooperativity by affecting the R to T equilibria within hexamers and dodecamers and by shifting the equilibria between hexamers and dodecamers. (3) The fitted model parameters show that under most conditions the intersubunit contacts within T-state hexamers are more constrained than those within T-state dodecamers. (4) The oxygen affinities of the hexameric and dodecameric R states are the same, but under all conditions examined the conformation of the fully oxygenated molecule is that of the dodecameric R state. (5) Between pH 7.4 and pH 8.5 the dodecameric T state has a higher affinity for oxygen than the hexameric T state, allowing for "T-state cooperativity".(ABSTRACT TRUNCATED AT 250 WORDS)     

Thermodynamic analysis of carbon monoxide binding by hemoglobin trout I.

Calorimetric measurements at 25 degrees of the differential heat of CO binding by hemoglobin trout I have been examined together with the CO binding isotherms for the protein at 4 degrees and 20 degrees. Simultaneous treatment of these data sets by a statistically rigorous technique permits evaluation of all the thermodynamic parameters for both the Adair and the Monod, Wyman, Changeux (MWC) models. The results show the details of the unusual temperature dependent cooperativity which this hemoglobin exhibits. In the Adair formalism the increasingly favorable free energy change for successive steps of ligand binding are nearly linearly paralleled by increasingly negative enthalpy changes for these steps. This causes the enhanced cooperativity observed as the temperature is decreased. For the MWC case, lowering the temperature increases the stability of the unligated T state relative to the unligated R state since the enthalpy of the T leads to R transition is 29.4 kcal mol-1. Simultaneously, the favorability of ligating R forms relative to T is enhanced since R form ligation is 14.1 kcal (mol CO)-1 more exothermic than that of T. The balance between these opposing effects is to increase ligand binding cooperativity at low temperatures. The predicted temperature dependence of the Hill coefficient for the MWC and Adair models is identical at low and intermediate temperatures, but, interestingly, would show a strong divergence at high temperatures where negative cooperativity is suggested for the Adair case and positive cooperativity for the MWC case.     

Stabilization of the T-state of hemoglobin

The effect of inositol hexaphosphate and bezafibrate on binding of O2 and CO to HbAO at high concentrations (1 mM) has been evaluated using thin layer optical techniques. Data analysis shows 1) the occurrence of greatly reduced ligand dependent cooperativity (Hill slope of 2.23 for CO and 1.51 for O2), and 2) the presence of significant triply ligated species. The data fits a nested allosteric two-state MWC model in which the T state consists of two allosteric substrates, Tt and Tr, where Tt binds only to the alpha chains and Tr binds to both alpha and beta chains. The model indicates that the triply ligated species consists of a predominant amount of T form, agreeing with kinetic observations of CO ligated hemoglobin. The maximum amount of triply ligated R molecules (CO or O2) implicated is less than 1%, a result similar to that found previously for binding studies made in the absence of BZF and IHP.