Cooperative enzyme kinetics

Different kinetic models have different conventions, and in the case of cooperative enzyme kinetic behavior, the term Ko,s is used in a sense analogous to Km for hyperbolic enzymes. In fact, transforming the original data in Fig. 14.2 to a Hill plot. 

Cooperative enzymes show sigmoid or sigmoidal kinetics because the dependence of the initial velocity on the concentration of the substrate is not Michaelis-Menten-like but gives a sigmoid curve (Fig. 8-7). 

Two other general ways of treating micellar kinetic data should be noted. Piszkiewicz (1977) used equations similar to the Hill equation of enzyme kinetics to fit variations of rate constants and surfactant concentration. This treatment differs from that of Menger and Portnoy (1967) in that it emphasizes cooperative effects due to substrate-micelle interactions. These interactions are probably very important at surfactant concentrations close to the cmc because solutes may promote micellization or bind to submicellar aggregates. Thus, eqn (1) and others like it do not fit the data for dilute surfactant, especially when reactants are hydrophobic and can promote micellization. 

The relationship between substrate concentration ([S]) and reaction velocity (v, equivalent to the degree of binding of substrate to the active site) is, in the absence of cooperativity, usually hyperbolic in nature, with binding behavior complying with the law of mass action. However, the equation describing the hyperbolic relationship between v and [S] can be simple or complex, depending on the enzyme, the identity of the substrate, and the reaction conditions. Quantitative analyses of these v versus [S] relationships are referred to as enzyme kinetics. 

A kinetic parameter, introduced by Koshland, to indicate the ratio of substrate concentrations needed to achieve reaction velocities equal to Q.f max and 0.9Fniax-For an enzyme obeying the Michaelis-Menten equation, o.9/ o.i equals 81, indicating that such enzymes exhibit modest sensitivity of reaction rate relative to changes in the substrate concentration. Many positively cooperative enzymes have So.g/So.i values between five and ten, indicating that they can be turned on or off over a relatively narrow substrate concentration range. 

It is impossible to describe and explain enzyme kinetics unless is explained by an entire book therefore, this chapter describes only briefly some aspects. It is strongly recommended to read once more a textbook on enzymology and enzyme kinetics. Especially the reaction kinetics of enzyme oligomeres, multi-enzyme complexes, and phenomena of cooperation are too complex to explain in just a few pages. 

Reaction velocity as a function of substrate concentration for a first-order enzymic reaction (left) and for a cooperative enzyme with fourth-order kinetics (right). Substrate concentration is expressed as [S]/[S0.s] and velocity as a fraction of maximum velocity. At a [S]/ [S0.5] value of l the substrate concentration is equal to [S0.5] and the reaction velocity is half of the maximum velocity. Note the much greater rate of increase over the same interval for the cooperative enzyme. 

The first two steps in the biosynthesis of tryptophan in Salmonella typhimurium involve the enzyme complex anthranilate synthase-phosphoribosyltransferase, which is a tetramer having two subunits of each enzyme. The anthranilate synthase catalyzes reaction (7) and the phos-phoribosyltransferase catalyzes two reactions the N-terminal portion cleaves glutamine to glutamate giving NH3 for the anthranilate synthase, while the C-terminal portion catalyzes reaction (8).3,1,312 All these reactions require M2+ cations. Orotate phosphoribosyltransferase binds four Mn2+ ions in a cooperative fashion kinetic data have been interpreted in a scheme where both metal-free and metal-containing enzyme catalyze the reaction.313... 

Neet, K. E. (1995) Cooperativity of enzyme function equilibrium and kinetic aspects, in Purich, D. L. (eds.), Methods in Enzymology, 249, Enzyme Kinetics and Mechanisms, Part D, Acad. Press, San-Diego, 519-567. 

There is almost no biochemical reaction in a cell that is not catalyzed by an enzyme. (An enzyme is a specialized protein that increases the flux of a biochemical reaction by facilitating a mechanism [or mechanisms] for the reaction to proceed more rapidly than it would without the enzyme.) While the concept of an enzyme-mediated kinetic mechanism for a biochemical reaction was introduced in the previous chapter, this chapter explores the action of enzymes in greater detail than we have seen so far. Specifically, catalytic cycles associated with enzyme mechanisms are examined non-equilibrium steady state and transient kinetics of enzyme-mediated reactions are studied an asymptotic analysis of the fast and slow timescales of the Michaelis-Menten mechanism is presented and the concepts of cooperativity and hysteresis in enzyme kinetics are introduced. 

Equation (4.31) gives us the equilibrium binding in the case of dual cooperativity, yet it does not tell us what cooperative binding has to do with enzyme kinetics. To illustrate the role of cooperativity in enzyme kinetics, consider the following... 

The previous section illustrated how allosteric cooperativity can result in a sigmoidal relationship between binding saturation and substrate concentration. In this section, we demonstrate how a sigmoidal relationship between product concentration and time can arise from enzyme kinetics with time lags. 

The answer is c. (Murray, pp 48-73. Scriver, pp 4571-4636. Sack, pp 3-17. Wilson, pp 287-317.) Allosteric enzymes, unlike simpler enzymes, do not obey Michaelis-Menten kinetics. Often, one active site of an allosteric enzyme molecule can positively affect another active site in the same molecule. This leads to cooperativity and sigmoidal enzyme kinetics in a plot of [S] versus V The terms competitive inhibition and noncompetitive inhibition apply to Michaelis-Menten kinetics and not to allosteric enzymes. 

Figure 5. Saturation kinetics the dependence of enzyme catalysis on the concentration of substrate. Reaction velocity represents the rate at which product is formed. (A) shows a hyperbolic saturation curve for two hypothetical enzymes. One binds its substrate more tightly than the other and reaches saturation at lower substrate concentration. This enzyme has a lower value, the substrate concentration where the reaction is half of maximum. The other binds the substrate more loosely and reaches the same velocity but requires higher substrate concentrations. (B) shows hypothetical velocities for cooperative enzymes. Although more complex, these enzymes also show the phenomenon of saturation.

Although the MM equation is a powerful kinetic form to which the vast majority of enzyme kinetics has been fitted, one should not forget the assumptions and limitations of the model. As a basic example, feedback inhibition, whereby the product of the reaction inhibits the enzyme-substrate cooperativity, multiple-substrate reactions, allosteric modifications, and other deviations from the reaction scheme in equation (1) are treated only adequately by the MM formalism under certain experimental conditions. In other words, enzyme kinetics are often bent to conform to the MM formalism for the sake of obtaining a set of parameters easily recognizable by most biochemists. The expUcit mathematical and experimental treatment of reaction mechanisms more complex than that shown in equation (1) is highly involved, although a mathematical automated kinetic equation derivation framework for an arbitrary mechanism has been described in the past (e.g., ref. 6). 

In deriving the rate law for a particular radical mechanism, the steady-state assumption in respect of all species with unpaired electrons, i.e. d[Ra ]/dt = 0, can usually be made, since the absolute concentrations are so low. The assumption parallels that made in steady-state enzyme kinetics (d[ES] / dt = 0), but whereas, in the absence of cooperativity, the steady-state approximation in enzyme kinetics always leads to the Michaelis-Menten equation (whatever the significance of or in terms of individual rate constants), the steady-state assumption in radical kinetics often leads to complex expressions. Dimerisations and dissociations in key steps can lead to fractional... 

The inhibition of certain enzymes by specific metabolites is an important element in the regulation of intermediary metabolism and most often occurs with cooperative enzymes that are regulated allosterically. Inhibition of enzymes that obey the Michaelis-Menten equation, noncooperative enzymes, is more commonly used by pharmacists to alter a patient s metabolism. Reversible inhibition of noncooperative enzymes is classified into three groups which can be distinguished kinetically and which have different mechanisms and effects when administered. The classes are called competitive, uncompetitive, and noncompetitive inhibition. Mixed inhibition also occurs. In all these types of inhibition, the inhibitor (usually a small molecule) binds reversibly and rapidly with the enzyme. 

There is some confusion in the literature about the use of the term "allosteric" for an enzyme. Many authors restrict this term to multi-subunit enzymes that show substrate cooperativity and sigmoidal kinetics. Other authors, however, are less specific and their definitions include enzymes that follow Michaelis- Menten kinetics and have non- or uncompetitive inhibitors. Fortunately, in metabolism, regulated enzymes are generally multi-subunit, cooperative, enzymes and fall into the more specific use of the term. 

Figure 11.32 Effect of cooperative substrate binding on enzyme kinetics. 

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