The theoretical analysis of kinetic behaviour of “hysteretic” allosteric enzymes V. Relaxation kinetics of dissociating enzyme systems

The expressions for relaxation time as a function of enzyme and specific ligand concentration are deduced for dissociating enzyme system 2p ? P (P is enzyme oligomer which is able to dissociate reversibly forming two identical halves p). It is assumed that ligand binding sites are equivalent and independent in each oligomeric enzyme form and the equilibrium between oligomeric forms develops rather slowly in comparison with the rate of the binding of the ligand. The kinetics of relaxation of the dissociating enzyme system 2p ? P with progressive change of the rate constants for association of oligomeric form p has been analysed in graphic form. The situations when one of the oligomeric enzyme forms is not able to bind the ligand are also considered. The principles of the analysis of relaxation kinetics of dissociating enzyme systems 2p ? P are discussed.     

Kinetic behavior of slowly equilibrating association-dissociation enzyme systems]

The shape of the plots of product accumulation versus time (t) has been analysed for slowly equilibrating association-dissociation enzyme systems of the types 2p in equilibrium P (P is enzyme oligomer which is able to dissociate reversibly forming two identical halves p) and M in equilibrium M2 in equilibrium M2 in equilibrium... (M is monomer which has two association sites overlapping with active sites). It is assumed that the rate of equilibration between oligomeric forms is comparable with the rate of over-all enzymatic reaction and that substrate-oligomer complexes are in rapid equilibrium with free components. It has been shown that characteristic feature of kinetic behavior of slowly equilibrating association-dissociation enzyme systems is that the value of tau depends on enzyme concentration (tau is the intercept on t-axis for linear asymptota of the curve of product concentration versus time at t leads to infinity).     

Specific ligand-induced association of an enzyme. A new model of dissociating allosteric enzyme

The kinetic behavior of dissociative enzyme system of the type inactive monomer in equilibrium active dimer where dimeric form is stabilized by specific ligand (in particular by substrate) which is bound in the region of the contact of monomers has been analysed. It is assumed that the dissociation of dimer results in formation of monomers which retain the subsites for specific ligand binding. The shape of the dependences of enzyme reaction rate (v) on substrate concentration (S) has been characterized using the order of enzyme reaction rate with respect to substrate concentration: ns = d ln v/d ln [S]. When the substrate concentrations are low the dependences of v on [S] have S-shaped form (the maximum value of ns exceeds the unity) at the definite values of the parameters of the enzyme system. The value of ns approaches--2 at sufficiently high substrate concentrations (in the region where the substrate reveals the inhibitory effect due to blocking the association of inactive monomers into active dimer). The methods of calculation of the parameters of the dissociative enzyme system under discussion have been elaborated on the basis of the analysis of the experimental dependences of specific enzyme activity on enzyme concentration obtained at various fixed substrate concentrations.     

Deviations from hyperbolic dependency of transport processes

The rate expression is not simple for the net uptake of a substrate from the cell's environment. Typically, a substrate is passed via a permease, and/or a carrier molecule which traverses the membrane by a process which is possibly coupled to an energy-consuming mechanism, and then finally it is enzymically utilized for cell growth or returned to the environment by some leak process. Mathematically, neither this full case, nor one-way entry, nor one-way efflux, nor the steady-state entry, discharged by independent efflux mechanisms yield simple hyperbolic dependencies on substrate concentration. On the other hand, the calculations presented here show the deviations from hyperbolic kinetics are not large unless possibly extreme choices of certain of the kinetic constants are made.     

New approach to analysis of deviations from hyperbolic law in enzyme kinetics

An empirical equation that describes deviations from Michaelian kinetics is proposed. The equation allows the limiting values of the Michaelis constant at v/Vmax --> 0 and v/Vmax --> 1 to be estimated (v is the rate of the enzymatic reaction and Vmax is the limiting value of v at saturating concentrations of substrate). The applicability of the equation is demonstrated for kinetic data obtained for glutamate dehydrogenases from various sources (negative kinetic cooperativity for coenzyme) and for biosynthetic threonine deaminase from pea seedlings (sharper approaching the limiting value of the enzymatic reaction rate with increasing substrate concentration in comparison with the hyperbolic law). The negative cooperativity for the function of saturation of protein by ligand is also analyzed (data on binding of spin-labeled NAD, NADH, and NADPH by beef liver glutamate dehydrogenase and binding of cupric ions by BSA are used as examples).     

The computation of hyperbolic dependences in enzyme kinetics.

A computer program aimed at analysing results following Michaelis-Menten kinetics can be used unmodified in the treatment of other kinetic results provided that the kinetic equations in these cases can be written in the form of the Michaelis-Menten equation. A list is presented of the parameters to be set instead of substrate concentration and reaction rate, and of constants replacing Km and V, if such a program is applied in analysing enzyme inhibitions, activations and pH-dependences.     

Escherichia coli E-39 ADPglucose synthetase has different activation kinetics from the wild-type allosteric enzyme

Kinetic and binding studies have shown that Lys39 of Escherichia coli ADPglucose synthetase is involved in binding of the allosteric activator. In order to study structure-function relationships at the activator binding site, this lysine residue was substituted by glutamic acid (Lys39----Glu) by site-directed mutagenesis. The resultant mutant enzyme (E-39) showed activation kinetics different from those of the wild-type enzyme. The level of activation of the E-39 enzyme by the major activators of E. coli ADPglucose synthetase, 2-phosphoglycerate, pyridoxal phosphate, and fructose-1,6-phosphatase was only approximately 2-fold compared to activation of 15- to 28-fold respectively, for the wild-type enzyme. NADPH, an activator of the wild-type enzyme, was unable to activate the mutant enzyme. In addition, the concentrations of the above activators necessary to obtain 50% of the maximal stimulation of enzyme activity (A0.5) were 5-, 9-, and 23-fold higher, respectively, than those for the wild-type enzyme. The E-39 enzyme also had a lower apparent affinity (S0.5) for the substrates ATP and MgCl2 than the wild-type enzyme and the values obtained in the presence or absence of activator were similar. The concentration of inhibitor giving 50% of enzyme activity (I0.5) was also similar for the E-39 enzyme in the presence or absence of activator. These results indicate that the E-39 mutant enzyme is not effectively activated by the major activators of the E. coli ADPglucose synthetase wild-type enzyme, and that this amino acid substitution also prevents the allosteric effect that the activator has on the wild-type enzyme kinetics, either increasing its apparent affinity for the substrates or modulating the enzyme's sensitivity to inhibition.     

Dihydroorotase from Clostridium oroticum is an allosteric enzyme.

Dihydroorotase from $\text{Clostridium}$$\text{oroticum}$ exhibits allosteric behavior with respect to both of its substrates. L-dihydroorotate dependence reflects a positive homotropic interaction for which the Hill coefficient is 1.3–1.6, depending upon the preparation. Conversely, a negative homotropic response is observed when L-ureidosuccinate serves as substrate, as characterized by a Hill coefficient of 0.65–0.75. Interaction between L-dihydroorotate binding sites is a labile characteristic lost during enzyme purification. Negative cooperativity of ureidosuccinate binding appears to be more stable. The effects of purification and medium are also discussed.     

The kinetics of enzyme systems involving activation of zymogens

A general model of zymogen activation is proposed and explicit kinetic equations for the time courses of the various species and products involved are given. These equations are valid for the whole course of the reaction and therefore for both the transient phase and the steady state. This model is sufficiently general to include mechanisms possessing one or more steps of zymogen activation besides possible steps of inhibition (reversible or irreversible) or inactivation.     

Cooperative coupling in allosteric systems

Allosterism of the Monod type applies only to systems with more than one binding site and two (noncooperative) states with intrinsic binding constants. Allosterism is then defined by an interconversion constant Lo (greater than 1) and a ratio of intrinsic binding constants, c (less than 1). The value of c determines whether weak or strong cooperativity among binding sites prevails. Cooperativity is weak, if 1 greater than c greater than 0.1, and strong, when c less than or equal to 0.1. Cooperativity is the stronger, the smaller c. Cooperativity may exist only between a restricted number of binding sites. The binding of Ca2+ to calmodulin shows this behavior under certain conditions. An (internal) indicator for binding may signal binding to both states or to only one. The results would be quite different with the extent of the difference determined by the extent of cooperativity (c in relation to the particular Li near 1). The size of Lo cannot be ignored in reference to the size of c. Effectors external to the ligand could alter Lo to shift cooperative behavior. Effectors could also make Lo too small or too large for any allosteric behavior to appear.     

Steady state enzyme kinetics for systems with three enzyme-binding species

The steady state enzyme kinetics of those systems are discussed, which involve three species binding to enzymes. Two specific systems are considered. In one system, all three species bind only once to the enzyme. In the other system, two species bind once and one binds twice to the enzyme. The species are labeled S, A and B. The general case is considered, in which all possible complexes involving enzyme E and species S generate product P. Species A and B may become co-substrates, activators or inhibitors. The steady state enzyme kinetic equations for the general case for both systems are presented. These equations are further discussed for a number of special cases, which may be of interest to enzymologists and others using enzymes.     

NMR spin exchange kinetics at equilibrium in membrane transport and enzyme systems

A set of differential equations is formulated to describe the rapid exchange (time scale, approximately 0.01 to approximately 10 s) of a labelled solute across the membranes of cells in suspension. The labelling is achieved with nuclear magnetic resonance by exposure of the system to a high intensity radio-frequency pulse, and the excited nuclei relax to the equilibrium state with a short half life. An analytical expression for the decay of the magnetic resonance signal is presented; the solution involves the determination of eigenvalues, of an array of Laplace-Carson transformed differential equations, by use of the general solution of a quartic polynomial. Simulations of the behaviour of the exchange system using various conditions of cell number, rate constants and nuclear magnetic relaxation times are presented. The marked concentration dependence of the extent of reaction at a given time has not previously been reported for nuclear magnetic resonance exchange systems and is a feature anticipated from the known saturability of several membrane transport systems including glucose transport into human erythrocytes. The theory is readily generalized to other model systems by appropriate reinterpretation of the physical meaning of various parameters; the general form of the solution holds in many biological contexts other than membrane transport and includes equilibrium enzyme kinetics.     

A general solution for the steady-state kinetics of immobilized enzyme systems

A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.