Enzyme kinetics single-substrate reactions

Most enzymes catalyse reactions and follow Michaelis-Menten kinetics. The rate can be described on the basis of the concentration of the substrate and the enzymes. For a single enzyme and single substrate, the rate equation is ... 

It expresses the velocity (v) of a single-substrate reaction (Equation C1. 1.1) in terms of substrate concentration at time zero ([S]) and the kinetic constants KM and V. is defined as the limiting maximal velocity for the reaction, which is observed when all of the enzyme is present as ES. KM, known as the Michaelis constant, is a pseudoequilibrium constant, which equals the concentration of substrate at which the reaction velocity equals one-half Vtrax (Figure Cl. 1.1). 

As we have seen, the catalytic cycle flux provides a useful metric for analyzing enzyme kinetics. In this section, we analyze the turnover time for catalytic cycles and show that the quasi-steady rate law arises from the mean cycle time [151]. In addition, we show that for arbitrary mechanisms for a single-substrate reaction, the steady state rate law can always be expressed using the Michaelis-Menten form... 

The kinetic parameters and Umax are estimated from the Michaelis-Menten equation and provide quantitative information regarding enzyme function. or the Michaelis constant is operationally defined as the concentration of substrate at which half-maximal velocity of the reaction is achieved (Fig. 4.1). With respect to the single substrate reaction scheme (Scheme 4.1), it should be realized that is equal to k + k2)lkx and thus is the amalgamation of several rate constants. With respect to affinity, unfortunately, is frequently (and incorrectly) used interchangeably with which is the substrate dissociation constant. Though may sometimes approximate the two do not have to be equal and numerous examples exist where these parameter values vary dramatically. 

Assigning instantaneous substrate concentration to S, instantaneous reaction time to t, steady-state kinetics of Michaelis-Menten enzyme on single substrate follows Equ.(l). 

Equation 11-15 is known as the Michaelis-Menten equation. It represents the kinetics of many simple enzyme-catalyzed reactions, which involve a single substrate. The interpretation of as an equilibrium constant is not universally valid, since the assumption that the reversible reaction as a fast equilibrium process often does not apply. 

While many enzymes have a single substrate, many others have two—and sometimes more than two—substrates and products. The fundamental principles discussed above, while illustrated for single-substrate enzymes, apply also to multisubstrate enzymes. The mathematical expressions used to evaluate multisubstrate reactions are, however, complex. While detailed kinetic analysis of multisubstrate reactions exceeds the scope of this chapter, two-substrate, two-product reactions (termed Bi-Bi reactions) are considered below. 

In the presence of sucrose alone as the single substrate, initial reaction rates follow Michaelis-Menten kinetics up to 200 mM sucrose concentration, but the enzyme is inhibited by higher concentrations of substrate.30 The inhibitor constant for sucrose is 730 mM. This inhibition can be overcome by the addition of acceptors.31,32 The enzyme activity is significantly enhanced, and stabilized, by the presence of dextran, and by calcium ions. 

A kinetic description of large reaction networks entirely in terms of elementary reactionsteps is often not suitable in practice. Rather, enzyme-catalyzed reactions are described by simplified overall reactions, invoking several reasonable approximations. Consider an enzyme-catalyzed reaction with a single substrate The substrate S binds reversibly to the enzyme E, thereby forming an enzyme substrate complex [/iS ]. Subsequently, the product P is irreversibly dissociated from the enzyme. The resulting scheme, named after L. Michaelis and M. L. Menten [152], can be depicted as... 

Reversible inhibition occurs rapidly in a system which is near its equilibrium point and its extent is dependent on the concentration of enzyme, inhibitor and substrate. It remains constant over the period when the initial reaction velocity studies are performed. In contrast, irreversible inhibition may increase with time. In simple single-substrate enzyme-catalysed reactions there are three main types of inhibition patterns involving reactions following the Michaelis-Menten equation competitive, uncompetitive and non-competitive inhibition. Competitive inhibition occurs when the inhibitor directly competes with the substrate in forming the enzyme complex. Uncompetitive inhibition involves the interaction of the inhibitor with only the enzyme-substrate complex, while non-competitive inhibition occurs when the inhibitor binds to either the enzyme or the enzyme-substrate complex without affecting the binding of the substrate. The kinetic modifications of the Michaelis-Menten equation associated with the various types of inhibition are shown below. The derivation of these equations is shown in Appendix S.S. 

The steady-state kinetics of a simple single-substrate, single-binding site, single-intermediate-enzyme catalysed reaction in the presence of competitive inhibitor are shown in Scheme A5.5.1. 

The simplest enzymatic system is the conversion of a single substrate to a single product. Even this straightforward case involves a minimum of three steps binding of the substrate by the enzyme, conversion of the substrate to the product, and release of the product by the enzyme (Scheme 4.6). Each step has its own forward and reverse rate constant. Based on the induced fit hypothesis, the binding step alone can involve multiple distinct steps. The substrate-to-product reaction is also typically a multistep reaction. Kinetically, the most important step is the rate-determining step, which limits the rate of conversion. 

The kinetics of enzyme reactions was first established by Michaelis and Menten, following the earlier work of Henri [23]. The famous Michaelis-Menten equation for the kinetics of an enzyme reaction with a single substrate is often written [23]... 

In 1913, Michaelis and Men ten presented a general theory for enzyme kinetics, extended later by Briggs and Haldane, which accounts for the velocity curve shown in Figure 5.5. This theory for reactions catalyzed by enzymes having a single substrate assumes that the substrate S binds to the active site of the enzyme E to form the enzyme-substrate complex ES, which yields the product P and the free enzyme E ... 

The most common enzymatic reactions are those with two or more substrates and as many products. But many of the simpler single-substrate schemes are valuable for the development of kinetic ideas concerning effects of pH, temperature, etc., on enzyme reaction rates. Although the mechanisms of multisubstrate reactions are complicated, their kinetics can often be described by an equation of the form ... 

While the majority of these concepts are introduced and illustrated based on single-substrate single-product Michaelis-Menten-like reaction mechanisms, the final section details examples of mechanisms for multi-substrate multi-product reactions. Such mechanisms are the backbone for the simulation and analysis of biochemical systems, from small-scale systems of Chapter 5 to the large-scale simulations considered in Chapter 6. Hence we are about to embark on an entire chapter devoted to the theory of enzyme kinetics. Yet before delving into the subject, it is worthwhile to point out that the entire theory of enzymes is based on the simplification that proteins acting as enzymes may be effectively represented as existing in a finite number of discrete states (substrate-bound states and/or distinct conformational states). These states are assumed to inter-convert based on the law of mass action. The set of states for an enzyme and associated biochemical reaction is known as an enzyme mechanism. In this chapter we will explore how the kinetics of a given enzyme mechanism depend on the concentrations of reactants and enzyme states and the values of the mass action rate constants associated with the mechanism. 

Previous sections of this chapter have focused on developing general principles for enzyme-catalyzed reactions based on analysis of single-substrate enzyme systems. Yet the majority of biochemical reactions involve multiple substrates and products. With multiple binding steps, competitive and uncompetitive binding interactions, and allosteric and covalent activations and inhibitions possible, the complete set of possible kinetic mechanisms is vast. For extensive treatments on a great number of mechanisms, we point readers to Segel s book [183], Here we review a handful of two-substrate reaction mechanisms, with detailed analysis of the compulsory-order ternary mechanism and a cursory overview of several other mechanisms. 

The electron transfer from cytochrome c to O2 catalyzed by cytochrome c oxidase was studied with initial steady state kinetics, following the absorbance decrease at 550 nm due to the oxidation of ferrocyto-chrome c in the presence of catalytic amounts of cytochrome c oxidase (Minnart, 1961 Errede ci a/., 1976 Ferguson-Miller ei a/., 1976). Oxidation of cytochrome c oxidase is a first-order reaction with respect to ferrocytochrome c concentration. Thus initial velocity can be determined quite accurately from the first-order rate constant multiplied by the initial concentration of ferrocytochrome c. The initial velocity depends on the substrate (ferrocytochrome c) concentration following the Michaelis-Menten equation (Minnart, 1961). Furthermore, a second catalytic site was found by careful examination of the enzyme reaction at low substrate concentration (Ferguson-Miller et al., 1976). The Km value was about two orders of magnitude smaller than that of the enzyme reaction previously found. The multiphasic enzyme kinetic behavior could be interpreted by a single catalytic site model (Speck et al., 1984). However, this model also requires two cytochrome c sites, catalytic and noncatalytic. 

The various kinds of reversible inhibition that have been identified all depend on non-covalent binding, but inhibitors differ in how they act, with consequent differences in their kinetic effects. Figure 8-6 depicts a general scheme for enzyme inhibition of a simple single substrate-single product reaction. 

It is not always immediately apparent from an enzymatic reaction whether it should be treated with single- or dual-substrate enzyme kinetic expressions. Lactose hydrolase, which catalyzes the hydrolysis of lactose according to... 

Note that the ternary complex EAB can be formed in two different ways. If the formation of EAB can occur with either substrate binding first, the reaction is known as a random single-displacement reaction. Many reactions catalyzed by phosphotransferases are of this type. If a particular substrate must bind first with the enzyme before the second substrate can bind, the reaction is known as an ordered single-displacement reaction. Many reactions catalyzed by dehydrogenases are of this type. The values for Km and Vmax for each substrate can be obtained from experiments in which the concentration of one substance is held constant at saturating levels while the concentration of the second substrate is varied. Kinetic analyses can distinguish between these types of reactions. 

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