Michaelis-Menten equation, derivation

Linear Plots Can Be Derived from the Michaelis-Menten Equation... 

Because of the hyperbolic shape of versus [S] plots, Vmax only be determined from an extrapolation of the asymptotic approach of v to some limiting value as [S] increases indefinitely (Figure 14.7) and is derived from that value of [S] giving v= V(nax/2. However, several rearrangements of the Michaelis-Menten equation transform it into a straight-line equation. The best known of these is the Lineweaver-Burk double-reciprocal plot ... 

The Km and Vj1iax of the Michaelis-Menten equation are actually made up of sums and products of little k s. You only have to look in most biochemistry texts to see a description of the derivation of the Michaelis-Menten equation in terms of little k s. The little k s are like quarks and leptons—you ve heard the names, but you re not quite sure what they are and even less sure about how they work. There s a section later (actually last) in the book if you haven t heard or can t remember about rate constants. 

This is an assumption used to derive the Michaelis-Menten equation in which the velocity of ES formation is assumed to be equal to the velocity of ES breakdown. 

This is one of the two equations you might have to deal with in introductory biochemistry. Like the other equation (the Michaelis-Menten equation), this one took two people to derive (wonder which one was... 

The so-called Michaelis-Menten equation (Eq. 1) [22] follows independently of the approximation chosen to solve the differential equation resulting from Scheme 10.1. Its derivation in detail can, for example, be found in [23]. 

When these assumptions hold, Thellier et al. [264] and others [267,268] have shown that the Michaelis-Menten equation (equation (35)) can be rigorously derived from the conservation equations for the various forms of the carrier. By... 

In the Briggs-Haldane derivation of the Michaelis-Menten equation, the concentration of ES is assumed to be at steady state, i.e., its rate of production [Eq. (3.12)] is exactly counterbalanced by its rate of dissociation [Eq. (3.13)]. Since the rate of formation of ES from E -(- P is vanishingly small, it is neglected. Equating the two equations and rearranging yields Eq. (3.14), where KM replaces (k2 + h)/k and is known as the Michaelis-Menten... 

Together with the net reaction for product formation v = k 2 [US ] — k 2[E][P], the reversible Michaelis Menten equation can be derived ... 

The Michaelis-Menten equation can also be derived by applying the steady state assumption to the following scheme ... 

In the ground-breaking scientific paper that presented their work, Menten and Michaelis also derived an important mathematical formula. This formula describes the rate at which enzymes break down their substrates. It correlates the speed of the enzyme reaction with the concentrations of the enzyme and the substrate. Called the Michaelis-Menten equation, it remains fundamental to our understanding of how enzymes catalyze reactions. 

In the absence of an enzyme, the reaction rate v is proportional to the concentration of substance A (top). The constant k is the rate constant of the uncatalyzed reaction. Like all catalysts, the enzyme E (total concentration [E]t) creates a new reaction pathway, initially, A is bound to E (partial reaction 1, left), if this reaction is in chemical equilibrium, then with the help of the law of mass action—and taking into account the fact that [E]t = [E] + [EA]—one can express the concentration [EA] of the enzyme-substrate complex as a function of [A] (left). The Michaelis constant lknow that kcat > k—in other words, enzyme-bound substrate reacts to B much faster than A alone (partial reaction 2, right), kcat. the enzyme s turnover number, corresponds to the number of substrate molecules converted by one enzyme molecule per second. Like the conversion A B, the formation of B from EA is a first-order reaction—i. e., V = k [EA] applies. When this equation is combined with the expression already derived for EA, the result is the 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-... 

Km and Vmax are mostly determined from linearized plots derived from conversions of the Michaelis-Menten equation. Some of these linearizations are given in Table 9.3. 

In chapter 8 the most generally nsed kinetic eqnations for describing the consnmption of snbstrate as a resnlt of biocatalysis have been given and/or derived. In biocatalysis, in the absence of limitation of the rate of consnmption by diffusion of substrate, the Michaelis-Menten equation usually is a good description ... 

Here we develop the basic logic and the algebraic steps in a modern derivation of the Michaelis-Menten equation, which includes the steady-state assumption introduced by Briggs and Haldane. The derivation starts with the two basic steps of the formation and breakdown of ES (Eqns 6-7 and 6-8). Early in the reaction, the concentration of the product, [P], is negligible, and we make the simplifying assumption that the reverse reaction, P—>S (described by k 2), can be ignored. This assumption is not critical but it simplifies our task. The overall reaction then reduces to... 

Other transformations of the Michaelis-Menten equation have been derived, each with some particular advantage in analyzing enzyme kinetic data. (See Problem 11 at the end of this chapter.)... 

The rate equations for this process can be derived exactly as for enzyme-catalyzed reactions (Chapter 6), yielding an expression analogous to the Michaelis-Menten equation ... 

The initial reaction velocity, vQ, of an enzyme-catalyzed reaction varies with the substrate concentration, [S], as shown in Figure E5.1. The Michaelis-Menten equation has been derived to account for the kinetic properties of enzymes. (Consult a biochemistry textbook for a derivation of this equation and for a discussion of the conditions under which the equation is valid.) The common form of the equation is... 

B 8. Show the mathematical steps required to derive the Lineweaver-Burk equation beginning with the Michaelis-Menten equation. 

The initial rate of the enzyme-catalyzed reaction is directly proportional to [S] (Equation El 1.3). Most clinical assays using enzymes are performed under the conditions of Equation El 1.3. From further study of this equation, you will note that also depends on enzyme concentration, since there is an enzyme concentration term hidden in Vmax. (If you have forgotten this, review the derivation of the Michaelis-Menten equation in your biochemistry textbook.) This can be used to advantage, because if a reaction used for a clinical analysis is very slow (it probably will be, since [S] is low), extra enzyme can be used so that the reaction will proceed to completion in a reasonable period of time. 

For most enzymes, the rate of reaction can be described by the Michaelis-Menten equation which was originally derived in 1913 by Michaelis and MENTEN 21 . Its derivation can be achieved by making one of two assumptions, one of which is a special case of the more general Briggs-Haldane scheme, whilst the alternative is the rapid-equilibrium method given in Appendix 5.3(2 ). 

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. 

As stated above, many two-substrate reactions obey the Michaelis-Menten equation with respect to one substrate while the concentration of the other substrate remains constant. This is true of reactions involving only one site, or those involving several sites provided that there is no interaction between sites. Alberty derived the following general equation for the reaction 23 ... 

Appendix 5.3. Derivation of the Michaelis-Menten Equation using the Rapid Equilibrium Assumption... 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

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