Diffusion control in reversible enzyme reactions. Applications to carbonic anhydrase.

The limit to the possible rate of reversible enzymatic reactions set by the diffusional motion has been considered. It is found that not only the diffusion of the reactants to the enzyme but also the diffusion away of the products can be rate limiting. To avoid assumptions about the detailed nature of the enzyme only diffusion in the bulk aqueous medium is treated. By such an approach one obtains an upper limit to the possible rate. In the latter half of the paper the derived general equations are applied to the possible suggested reaction schemes for the enzyme carbonic anhydrase. It is found that a scheme involving HCO3- as substrate for the dehydration process and a direct reaction between buffer and enzyme is comsistent with the limits set by the diffusional motion, while several other possibilities can be ruled out.     

Influence of substrate and product diffusion on the heterogeneous kinetics of enzymic reversible reactions

When a reversible reaction is catalyzed by a surface bound enzyme, the diffusion of both substrate and product can considerably modify the kinetic properties of the reaction. According to this theoretical analysis, the enzyme activity is decreased due to the presence of substrate and product concentration gradients in the enzyme microenvironment, and the relative kinetic importance of the two diffusion steps mainly depend on the value of a dimensionless criterion inversely proportional to the equilibrium constant. Moreover diffusional effects increase with increasing bound enzyme activity, but decrease with increasing substrate and product concentration. Analytical expressions are established for the limiting values of substrate and product concentrations in the enzyme microenvironment, as well as for the increase in half-maximal-activity substrate concentration in the presence of substrate and product diffusional limitations.     

Kinetic parameters of enzymatic reactions in states of maximal activity; an evolutionary approach

A theoretical investigation is presented which allows the calculation of states of maximal reaction rates for single enzymes and for unbranched enzymatic chains. As an extension to previous papers (Heinrich & Holzhütter, 1985, Biomed. biochim. Acta 44, 959-969; Heinrich et al., 1987, Bull. math. Biol. 49, 539-595) a detailed enzymatic mechanism was taken into consideration. Conclusions are drawn for the optimal values of the microscopic rate constants as well as of the maximal activities and Michaelis constants. Ten solutions are found which depend on the equilibrium constant as well as on the concentrations of substrates and products. It is shown that for high equilibrium constants one of the solutions applies to a very large range of the concentrations of the outer reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. In contrast to previous assumptions (Albery & Knowles, 1976b, Biochemistry 15, 5631-5640; Burbaum et al., 1989, Biochemistry 28, 9293-9305) states of maximal reaction rate are not always characterized by the highest possible values of the second-order rate constants which are related to the diffusion of the substrate and the product to the active site of the enzyme. Predictions are made concerning the ratios of maximal activities in optimal states as well as for the adaptation of the Michaelis constants to the concentrations of the outer reactants. Using metabolic control analysis it is shown that the solutions obtained for single enzymes may also be applied in multi-enzyme systems.     

Levels of thermodynamic treatment of biochemical reaction systems.

Equilibrium calculations on biochemical reaction systems can be made at three levels. Level 1 is the usual chemical calculation with species at specified temperature and pressure using standard Gibbs energies of formation of species or equilibrium constants K. Level 2 utilizes reactants such as ATP (a sum of species) at specified T, P, pH, and pMg with standard transformed Gibbs energies of formation of reactants or apparent equilibrium constants K'. Calculations at this level can also be made on the enzymatic mechanism for a biochemical reaction. Level 3 utilizes reactants at specified T, P, pH, and pMg, but the equilibrium concentrations of certain reactants are also specified. The fundamental equation of thermodynamics is derived here for Level 3. Equilibrium calculations at this level use standard transformed Gibbs energies of formation of reactants at specified concentrations of certain reactants or apparent equilibrium constants K". Level 3 is useful in calculating equilibrium concentrations of reactants that can be reached in a living cell when some of the reactants are available at steady-state concentrations. Calculations at all three levels are facilitated by the use of conservation matrices and stoichiometric number matrices for systems. Three cases involving glucokinase, glucose-6-phosphatase, and ATPase are discussed.     

The role of water in the thermodynamics of dilute aqueous solutions

Water plays a role in the thermodynamics of dilute aqueous solutions that is unusual in two ways. First, knowledge of hydration equilibrium constants of species is not required in calculations of thermodynamic properties of biochemical reactants and reactions at specified pH. Second, since solvent provides an essentially infinite source of oxygen atoms in a reaction system where water is a reactant, oxygen atoms are not conserved in the reaction system in dilute aqueous solutions. This is related to the fact that H₂O is omitted in equilibrium expressions for dilute aqueous solutions. Calculations of the standard transformed Gibbs energies of formation of total carbon dioxide and total ammonia at specified pH are discussed, and the average bindings of hydrogen ions by these reactants are calculated by differentiation. Since both of these reactants are involved in the urease reaction, the apparent equilibrium constants and changes in the numbers of hydrogen ions bound are calculated for this reaction as functions of pH.     

New relations for steady-state enzyme kinetics and their application to rapid equilibrium assumption

With the use of graph theory, new relations for steady-state enzyme kinetics are derived and strictly proved for an arbitrary mechanism of enzyme-catalyzed reaction containing a reversible segment. Using these relations, a general principle for rapid equilibrium assumption is formulated and proved: the reversible bound segment can be considered as an equilibrium segment only when the values of the base trees that are not proper to this segment can be neglected (within prescribed accuracy) in relation to the values of the base trees that belong to this segment. In contrast with the foreign base trees, the base trees that are proper to the segment have the following properties: the tree that is directed to the base within this segment does not contain edges leaving this segment; and the tree that is directed to the base outside the segment contains only one edge leaving this segment. Equilibrium variations are assessed for steady-state concentrations of intermediates in the equilibrium segment, numerical expressions are obtained for the accuracy of determination of intermediate concentrations as well as for the accuracy of determination of the rate of enzyme-catalyzed reaction under rapid equilibrium assumption.     

Analysis of sedimentation equilibrium distributions reflecting nonideal macromolecular associations

A rigorous statistical-mechanical approach is adopted to derive general quantitative expressions that allow for the effects of thermodynamic nonideality in equilibrium measurements reflecting interaction between dissimilar macromolecular reactants. An analytical procedure based on these expressions is then formulated for obtaining global estimates of equilibrium constants and the corresponding reference thermodynamic activities of the free reactants in each of several sedimentation equilibrium experiments. The method is demonstrated by application to results from an ultracentrifugal study of an electrostatic interaction between ovalbumin and cytochrome c (Winzor, D. J., M. P. Jacobsen, and P. R. Wills. 1998. Biochemistry. 37:2226-2233). It is demonstrated that reliable estimates of relevant thermodynamic parameters are extracted from the data through statistical analysis by means of a simple nonlinear fitting procedure.     

Thermodynamics of the purine nucleotide cycle

Since the standard Gibbs energies of formation are known for all the species in the purine nucleotide cycle at 298.15 K, the functions of pH and ionic strength that yield the standard transformed Gibbs energies of formation of the ten reactants can be calculated. This makes it possible to calculate the standard transformed Gibbs energies of reaction, apparent equilibrium constants, and changes in the binding of hydrogen ions for the three reactions at desired pHs and ionic strengths. These calculations are also made for the net reaction and a reaction that is related to it. The equilibrium concentrations for the cycle are calculated when all the reactants are initially present or only some are present initially. Since the concentrations of GTP, GDP, and P(i) may be in steady states, the equilibrium concentrations are also calculated for the system at specified steady-state concentrations.     

Equilibrium concentrations for pyruvate dehydrogenase and the citric acid cycle at specified concentrations of certain coenzymes

It is of interest to calculate equilibrium compositions of systems of biochemical reactions at specified concentrations of coenzymes because these reactants tend to be in steady states. Thermodynamic calculations under these conditions require the definition of a further transformed Gibbs energy G" by use of a Legendre transform. These calculations are applied to the pyruvate dehydrogenase reaction plus the citric acid cycle, but steady-state concentrations of CoA, acetyl-CoA and succinyl-CoA cannot be specified because they are involved in the conservation of carbon atoms. These calculations require the use of linear algebra to obtain further transformed Gibbs energies of formation of reactants and computer programs to calculate equilibrium compositions. At specified temperature, pH, ionic strength and specified concentrations of several coenzymes, the equilibrium composition depends on the specified concentrations of the coenzymes and the initial amounts of reactants.     

Hydrophobic chromatography of β-galactosidase

The hydrophobic interaction of β-galactosidase with Sepharose 4B substituted with 3,3′-diaminodipropylamine was studied in both batch and column experiments. The equilibrium and the binding rate constants were determined for different phosphate buffer concentrations. The equilibrium constants exhibit a hysteresis effect, i.e., desorption constants are less than adsorption constants, and the higher the ionic strength to start the desorption, the larger the effect. The rate data are not satisfactorily described by a simple reversible first-order model. The column chromatographic data are semiquantitatively described by a local equilibrium theory without axial dispersion or intraparticle diffusion.     

A mathematical theory of enamel solubility and the onset of dental caries: III. Development and computer simulation of a model of caries formation

Part III attempts to develop a diffusion controlled model of caries in the intact enamel employing the kinetic results of the previous two parts. A model of the enamel as a granular bed with a diffusible organic matrix filling the interstices is considered. The basic equations of diffusion and simultaneous reaction are developed under the assumption that all the reactions are so rapid as compared with the diffusion rate, that they are in a quasi-equilibrium state. The resultant system of seven coupled, non-linear parabolic partial differential equations is of such complexity that only numerical solutions could be attempted. Stability restrictions inherent in the problem dictated the use of the DuFort-Frankel numerical solution for parabolic boundary problems. Numerical solutions giving the concentration of all reactants, the rate of mineral loss, and the enamel porosity were obtained for a variety of boundary conditions. It is found that departure from the equilibrium condition expressed in part II is necessary for the occurrence of an attack on the enamel. The rate and pattern of penetration is then determined primarily by the concentrations of undissociated buffer, and salts, together with the rate of diffusion in the surrounding medium. The possibility of a relatively intact surface layer persisting over a demineralized subsurface region due solely to the composition of the demineralizing medium is noted. Remineralization behavior in portions of the carious lesion occurs in the model under certain boundary conditions.     

Sensitivity to values of the rate constants in a neurochemical metabolic model

Equations were derived for the instantaneous relative sensitivities of reaction rates (controllability indices) and metabolite concentrations (response indices) to perturbations in the values of rate constants and were used to analyze the behavior of a model of in vivo glutamate metabolism in rat brain. Controllabilities of reversible reactions were found to increase as the values of the corresponding rate constants (i.e., the rate of approach to equilibrium) increased. Response indices generally declined with the metabolic distance between the metabolite and the rate constant, but they were unexpectedly high for reversible reactions with high controllabilities. The transient response of a given metabolite is most sensitive to reactions involving metabolites which are changing most rapidly relative to their respective pool sizes. Rapidly reversible reactions are most important for communication between metabolite pools.     

Isoelectric focusing of interacting systems. IV. interaction of macromolecules with each other and with ligands

The theoretical isoelectric focusing behavior for rapidly reversible, bimolecular complexing between two macromolecules depends upon the relative value of the isoelectric point of the complex. When it is intermediate in value, the transient patterns exhibit three peaks. As equilibrium is approached the central peak of complex disappears leaving two reactant peaks. When the isoelectric point is acidic or alkaline to both reactants, the equilibrium pattern also shows two peaks; but in this case only one is pure reactant, the other being a reaction zone. The two cases can be distinguished by varying the relative amounts of reactants. Transient patterns for ligand-binding exhibit a peak of unliganded protein and a reaction zone. As the charged ligand is driven out of the focusing column the reaction zone disappears, so that the equilibrium pattern shows only a peak of unliganded protein. In general, the isoelectric point of the complex cannot be determined from the transient patterns.     

Thermodynamics of systems of biochemical reactions

When a reaction system described in terms of species is in a certain state, the Gibbs energy G provides the means for determining whether each reaction will go to the right or the left, and the equilibrium composition of the whole system can be calculated using G. When the pH is specified, a system of biochemical reactions is described in terms of reactants, like ATP (a sum of species), and the transformed Gibbs energy G' provides the means for determining whether each reaction will go to the right or the left. The equilibrium composition of the whole system can be calculated using G'. Since metabolism is complicated, the thermodynamics of systems of reactions like glycolysis and the citric acid cycle can also be considered at specified concentrations of coenzymes like ATP, ADP, NAD(ox), and NAD(red). This is of interest because coenzymes tend to be in steady states because they are involved in many reactions. When the concentrations of coenzymes are constant, the further transformed Gibbs energy G" provides the means for calculating whether each reaction will go to the right or the left, and the equilibrium composition of the whole system can be calculated using G". Under these conditions, a metabolic reaction system can be reconceptualized in terms of sums of reactants; for example, glycolysis can be represented by C(6)=2C(3), where C(6) is the sum of the reactants with six carbon atoms and C(3) is the sum of the reactants with three carbon atoms. These calculations can also be described by use of semigrand partition functions. Semigrand partition functions have the advantage of containing all the thermodynamic information on a series of reactions at specified pH or at specified pH and specified concentrations of coenzymes.     

Equilibrium calculations on systems of biochemical reactions at specified pH and pMg.

The use of G' in discussing the thermodynamics of biochemical reactions at a specified pH and pMg is justified by use of a Legendre transform of the Gibbs energy G. When several enzymatic reactions occur simultaneously in a system, the standard transformed Gibbs energies of reaction delta rG'0 can be used in a computer program to calculate the equilibrium composition that minimizes the transformed Gibbs energy at the specified pH and pMg. The calculation of standard transformed Gibbs energies of formation of reactants at pH 7 and pMg 3 is described. In addition a method for calculating the equilibrium concentrations of reactants is illustrated for a system with steady state concentrations of some reactants like ATP and NAD.     

Analysis of coupled bimolecular reaction kinetics and diffusion by two-color fluorescence correlation spectroscopy: enhanced resolution of kinetics by resonance energy transfer

In two-color fluorescence correlation spectroscopy (TCFCS), the fluorescence intensities of two fluorescently-labeled species are cross-correlated over time and can be used to identify static and dynamic interactions. Generally, fluorophore labels are chosen that do not undergo F?rster resonance energy transfer (FRET). Here, a general TCFCS theory is presented that accounts for the possibility of FRET between reactants in the reversible bimolecular reaction, [reaction: see text] where k(f) and k(b) are forward and reverse rate constants, respectively (dissociation constant K(d) = k(b)/k(f)). Using this theory, we systematically investigated the influence on the correlation function of FRET, reaction rates, reactant concentrations, diffusion, and component visibility. For reactants of comparable size and an energy-transfer efficiency of approximately 90%, experimentally measurable cross-correlation functions should be sensitive to reaction kinetics for K(d) > 10(-8) M and k(f) >or= approximately 10(7) M(-1)s(-1). Measured auto-correlation functions corresponding to donor and acceptor labels are generally less sensitive to reaction kinetics, although for the acceptor, this sensitivity increases as the visibility of the donor increases relative to the acceptor. In the absence of FRET or a significant hydrodynamic difference between reactant species, there is little effect of reaction kinetics on the shape of auto- and cross-correlation functions. Our results suggest that a subset of biologically relevant association-dissociation kinetics can be measured by TCFCS and that FRET can be advantageous in enhancing these effects.     

Control of enzymatic velocity under near-equilibrium conditions

Algebraic derivations demonstrate that if a multireactant enzyme system is poised at equilibrium and the concentration of one of the reactants is then changed by a small fraction, the resulting net reaction velocity is hyperbolically related to the fractional perturbation rather than the initial or final absolute concentration of that reactant. For small fractional perturbations the velocity is almost identical regardless which reactant is perturbed. Similar results are obtained even if the reaction system is already displaced by up to 30% from equilibrium at the time of the perturbation. These conclusions are independent of the relationships between the reactant concentrations and the kinetic constants for the enzyme. Thus under any near-equilibrium condition each of the reactants for a multireactant enzyme system shares almost equally in control of the net reaction velocity.     

Investigation of a Monte Carlo model for chemical reactions

Monte Carlo computer simulations are in use at a number of laboratories for calculating time-dependent yields, which can be compared with experiments in the radiolysis of water. We report here on calculations to investigate the validity and consistency of the procedures used for simulating chemical reactions in our code, RADLYS. Model calculations were performed of the rate constants themselves. The rates thus determined showed an expected rapid decline over the first few hundred ps and a very gradual decline thereafter out to the termination of the calculations at 4.5 ns. Results are reported for different initial concentrations and numbers of reactive species. Generally, the calculated rate constants are smallest when the initial concentrations of the reactants are largest. It is found that inhomogeneities that quickly develop in the initial random spatial distribution of reactants persist in time as a result of subsequent chemical reactions, and thus conditions may poorly approximate those assumed from diffusion theory. We also investigated the reaction of a single species of one type placed among a large number of randomly distributed species of another type with which it could react. The distribution of survival times of the single species was calculated by using three different combinations of the diffusion constants for the two species, as is sometimes discussed in diffusion theory. The three methods gave virtually identical results. Received: 6 May 1998 / Accepted in revised form: 5 July 1998