Steady-state kinetics Michaelis-Menten equation

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

This reaction cycle has more steps than the simple Michaelis-Menten scheme. Nonetheless, the steady-state rate equations describing these reaction cycles have indistinguishable functions, and one cannot determine the number of intermediary steps by steady-state kinetics alone. 

Reversible Inhibition One common type of reversible inhibition is called competitive (Fig. 6-15a). A competitive inhibitor competes with the substrate for the active site of an enzyme. While the inhibitor (I) occupies the active site it prevents binding of the substrate to the enzyme. Many competitive inhibitors are compounds that resemble the substrate and combine with the enzyme to form an El complex, but without leading to catalysis. Even fleeting combinations of this type will reduce the efficiency of the enzyme. By taking into account the molecular geometry of inhibitors that resemble the substrate, we can reach conclusions about which parts of the normal substrate bind to the enzyme. Competitive inhibition can be analyzed quantitatively by steady-state kinetics. In the presence of a competitive inhibitor, the Michaelis-Menten equation (Eqn 6-9) becomes... 

The Michaelis-Menten equation, especially if derived with the steady-state concept as above, is a rigorous rate law which not only fits almost all one-substrate enzyme kinetics, except in the case of inhibition (see Section 5.3), but also allows identification of the kinetic constants with the elementary steps in Eq. (5.1). 

Before we can discuss the measurement of active-site concentration, we need to consider the kinetics of the substrate reaction. The majority of kinetic studies of enzymes are carried out on systems described by Scheme 11.16 where all terms have their usual meanings and where the intermediates have come to a steady-state concentration otherwise, studies of the kinetics of the pre-steady-state conditions usually require the use of specialist, fast reaction, equipment. The Michaelis-Menten equation, Equation 11.12, where all terms again have their usual meanings, can be derived from Scheme 11.16 when the system has reached a steady state at this point the values of [ES] and [P] are still very much less than that of [S] ... 

The steady-state treatment of enzyme kinetics assumes that concentrations of the enzyme-containing intermediates remain constant during the period over which an initial velocity of the reaction is measured. Thus, the rates of changes in the concentrations of the enzyme-containing species equal zero. Under the same experimental conditions (i.e., [S]0 [E]0 and the velocity is measured during the very early stage of the reaction), the rate equation for one substrate reaction (uni uni reaction), if expressed in kinetic parameters (V and Ks), has the form identical to the Michaelis-Menten equation. However, it is important to note the differences in the Michaelis constant that is, Ks = k2/k1 for the quasi-equilibrium treatment whereas Ks = (k2 + k3)/k i for the steady-state treatment. 

It was found that 46 behaves as an exceptional substrate of trypsin, showing a the reaction mode which had not been observed before. Fig. 3 shows the time course of the tryptic catalysis of 46 monitored by the amidinophenol liberation under the condition that the substrate is in much higher concentration than the enzyme. After rapid mixing of enzyme and substrate, a rapid acylation step is observed and a slow deacylation then follows. The kinetics follow a Michaelis-Menten equation strong binding affinity, efficient acylation, and rate-determining slow deacylation steps, which are exactly the same as those of normal-type substrates. As a result, the accumulation of the acyl enzyme intermediate (EA) is realized in the course of the steady-state hydrolysis [cf. Eq. (6)]. 

Fig. 9 2 Lineweaver-Burk graphical procedure for determining the two steady-state kinetic parameters in the Michaelis-Menten equation.

Derive the steady state rate expression of Equations (4.38) and (4.39) from the kinetic mechanism of Equation (4.37). What is the apparent Michaelis-Menten constant for this mechanism ... 

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. 

Under typical experimental conditions, the enzyme system is saturated with O2 and H+. Thus this enzyme system includes four substrates and four products. However, the initial steady state kinetics of this enzyme system obeys a simple Michaelis-Menten equation (a rectangular hyperbolic relation) for each kinetic phase of the two phases at low and high ferrocytochrome c concentrations as described above. This result indicates that the four ferrocytochromes c react with the enzyme in a ping-pong fashion in each substrate concentration range. That is, each ferroferrocytochrome c reacts with the enzyme after the previous cytochrome c in the oxidized state is released from the enzyme. Cytochrome c... 

Several drugs, including salicylate (in overdose), alcohol, and possibly some hydrazines and other drugs which are metabolised by acetylation, have saturable elimination kinetics, but the only significant clinical example is phenytoin. With this drug, capacity-limited elimination is complicated further by its low therapeutic index. A 50% increase in the dose of phenytoin can result in a 600% increase in the steady-state blood concentration, and thus expose the patient to potential toxicity. Capacity-limited pathways of elimination lead to plasma concentrations of drugs which can be described by a form of the Michaelis-Menten equation. In such cases, the plasma concentration at steady state is given by... 

Under certain conditions the steady state has the form of the Michaelis-Menten equation (Section 9.1.2). Nevertheless, the equation for APase contains more factors than that of the simple reaction given in Section 9.1.2. For detailed kinetic considerations, Fernley (1971) and Reid and Wilson (1971) should be consulted. The simplified representation of Fig. 10.3 is also often complicated by other parameters e.g. only one active site per dimer seems to be active for the bacterial enzyme at low substrate coneentrations (<10" M), whereas at higher concentrations both sites are active. Substrate activation, at high substrate concentrations (>10" M), was noted by Heppel et al. (1962) but not at high ionic strength (Simpson and Vallee, 1970). 

Most measurements of glycosidase kinetics are carried out under steady-state conditions. Substrate is in large excess over enzyme and the reaction is monitored on a time-scale that is long compared with the reciprocals of the rate constants for individual molecular events, so that changes in the concentrations of various liganded and unliganded forms of the enzyme can be set to zero. If only one substrate is involved and the active sites are independent, eqn. (5.1), the Michaelis-Menten equation, holds ... 

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... 

Simple steady state kinetics and Michaelis-Menten equation... 

The Michaelis-Menten equation (8.8) and the irreversible Uni Uni kinetic scheme (Scheme 8.1) are only really applicable to an irreversible biocatalytic process involving a single substrate interacting with a biocatalyst that comprises a single catalytic site. Hence with reference to the biocatalyst examples given in Section 8.1, Equation (8.8), the Uni Uni kinetic scheme is only really directly applicable to the steady state kinetic analysis of TIM biocatalysis (Figure 8.1, Table 8.1). Furthermore, even this statement is only valid with the proviso that all biocatalytic initial rate values are determined in the absence of product. Similarly, the Uni Uni kinetic schemes for competitive, uncompetitive and non-competitive inhibition are only really applicable directly for the steady state kinetic analysis for the inhibition of TIM (Table 8.1). Therefore, why are Equation (8.8) and the irreversible Uni Uni kinetic scheme apparently used so widely for the steady state analysis of many different biocatalytic processes A main reason for this is that Equation (8.8) is simple to use and measured k t and Km parameters can be easily interpreted. There is only a necessity to adapt catalysis conditions such that... 

In this section, we shall begin to see how the Briggs-Haldane steady state approach can be enlarged to derive steady state kinetics equations appropriate to more complex kinetic schemes. In doing this, there will be some pleasant surprises in that the form of these new steady state kinetic equations will follow the form of Michaelis-Menten equation (8.8) with a few adaptations not unlike those seen in the Uni Uni steady state kinetic scheme adapted to fit the presence of inhibitors (see Section 8.2.4). 

Unfortunately, the form of Equation (8.53) is a little way off the form of the Michaelis-Menten equation. For this reason, the King-Altman approach is usually supplemented by an approach developed by Cleland. The Cleland approach seeks to group kinetic rate constants together into numbers (num), coefficients (Coef) and constants (const) that themselves can be collectively defined as experimental steady-state kinetic parameters equivalent to fccat> Umax and fCm of the original Michaelis-Menten equation. After such substitutions, the result is that equations may be algebraically manipulated to reproduce the form of the Michaelis-Menten equation (8.8). Use of the Cleland approach is illustrated as follows. 

In this case, v is the velocity of the reaction, [S] is the substrate concentration, Vmax (also known as V or Vj ) is the maximum velocity of the reaction, and is the Michaelis constant. From this equation quantitative descriptions of enzyme-catalyzed reactions, in terms of rate and concentration, can be made. As can be surmised by the form of the equation, data that is described by the Michaelis-Menten equation takes the shape of a hyperbola when plotted in two-dimensional fashion with velocity as the y-axis and substrate concentration as the x-axis (Fig. 4.1). Use of the Michaelis-Menten equation is based on the assumption that the enzyme reaction is operating under both steady state and rapid equilibrium conditions (i.e., that the concentration of all of the enzyme-substrate intermediates (see Scheme 4.1) become constant soon after initiation of the reaction). The assumption is also made that the active site of the enzyme contains only one binding site at which catalysis occurs and that only one substrate molecule at a time is interacting with the binding site. As will be discussed below, this latter assumption is not always valid when considering the kinetics of drug metabolizing enzymes. 

The steady-state kinetic treatment of random reactions is complex and gives rise to rate equations of higher order in substrate and product terms. For kinetic treatment of random reactions that display the Michaelis-Menten (i.e. hyperbolic velocity-substrate relationship) or linear (linearly transformed kinetic plots) kinetic behavior, the quasi-equilibrium assumption is commonly made to analyze enzyme kinetic data. 

This textbook for advanced courses in enzyme chemistry and enzyme kinetics covers the field of steady-state enz5mie kinetics from the basic principles inherent in the Michaelis-Menten equation to the expressions that describe the multi-substrate enzyme reactions. The purpose of this book is to provide a simple but comprehensive framework for the study of enzymes with the aid of kinetic studies of enzyme-catalyzed reactions. The aim of enzyme kinetics is twofold to study the kinetic mechanism of enz5mie reactions, and to study the chemical mechanism of action of enzymes. 

Michaelis-Menten equations for the monosubstrate reactions (Eqs. (3.9) and (3.27)) in the forward direction (A -> P), have four fundamental kinetic constants or steady-state kinetic constants ... 

Rat equation in Enzyme kinetics (see), an equation expressing the rate of a reaction in terms of rate constants and the concentrations of enzyme spedes, substrate and product. When it is assumed that steady state conditions obtain, the Michaelis-Menten equation (see) is a suitable approximation. R.e. are represented graphically (see Enzyme graph) they may be derived by the King-Altman method (see). 

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