Michaelis–Menten kinetics

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Short description: Model of enzyme kinetics
Curve of the Michaelis–Menten equation labelled in accordance with IUBMB recommendations

In biochemistry, Michaelis–Menten kinetics, named after Leonor Michaelis and Maud Menten, is the simplest case of enzyme kinetics, applied to enzyme-catalysed reactions of one substrate and one product. It takes the form of a differential equation describing the reaction rate [math]\displaystyle{ v }[/math] (rate of formation of product P, with concentration [math]\displaystyle{ p }[/math]) to [math]\displaystyle{ a }[/math], the concentration of the substrate  A (using the symbols recommended by the IUBMB).[1][2][3][4] Its formula is given by the Michaelis–Menten equation:

[math]\displaystyle{ v = \frac{\mathrm{d} p}{\mathrm{d} t} = \frac{V a}{K_\mathrm{m} + a} }[/math]

[math]\displaystyle{ V }[/math], which is often written as [math]\displaystyle{ V_\max }[/math],[5] represents the limiting rate approached by the system at saturating substrate concentration for a given enzyme concentration. The Michaelis constant [math]\displaystyle{ K_\mathrm{m} }[/math] is defined as the concentration of substrate at which the reaction rate is half of [math]\displaystyle{ V }[/math].[6] Biochemical reactions involving a single substrate are often assumed to follow Michaelis–Menten kinetics, without regard to the model's underlying assumptions. Only a small proportion of enzyme-catalysed reactions have just one substrate, but the equation still often applies if only one substrate concentration is varied.

"Michaelis–Menten plot"

Semi-logarithmic plot of Michaelis–Menten data

The plot of [math]\displaystyle{ v }[/math] against [math]\displaystyle{ a }[/math] has often been called a "Michaelis–Menten plot", even recently,[7][8][9] but this is misleading, because Michaelis and Menten did not use such a plot. Instead, they plotted [math]\displaystyle{ v }[/math] against [math]\displaystyle{ \log a }[/math], which has some advantages over the usual ways of plotting Michaelis–Menten data. It has [math]\displaystyle{ v }[/math] as dependent variable, and thus does not distort the experimental errors in [math]\displaystyle{ v }[/math]. Michaelis and Menten did not attempt to estimate [math]\displaystyle{ V }[/math] directly from the limit approached at high [math]\displaystyle{ \log a }[/math], something difficult to do accurately with data obtained with modern techniques, and almost impossible with their data. Instead they took advantage of the fact that the curve is almost straight in the middle range and has a maximum slope of [math]\displaystyle{ 0.576V }[/math] i.e. [math]\displaystyle{ 0.25\ln 10 \cdot V }[/math]. With an accurate value of [math]\displaystyle{ V }[/math] it was easy to determine [math]\displaystyle{ \log K_\mathrm{m} }[/math] from the point on the curve corresponding to [math]\displaystyle{ 0.5V }[/math].

This plot is virtually never used today for estimating [math]\displaystyle{ V }[/math] and [math]\displaystyle{ K_\mathrm{m} }[/math], but it remains of major interest because it has another valuable property: it allows the properties of isoenzymes catalysing the same reaction, but active in very different ranges of substrate concentration, to be compared on a single plot. For example, the four mammalian isoenzymes of hexokinase are half-saturated by glucose at concentrations ranging from about 0.02 mM for hexokinase A (brain hexokinase) to about 50 mm for hexokinase D ("glucokinase", liver hexokinase), more than a 2000-fold range. It would be impossible to show a kinetic comparison between the four isoenzymes on one of the usual plots, but it is easily done on a semi-logarithmic plot.[10]

Model

A decade before Michaelis and Menten, Victor Henri found that enzyme reactions could be explained by assuming a binding interaction between the enzyme and the substrate.[11] His work was taken up by Michaelis and Menten, who investigated the kinetics of invertase, an enzyme that catalyzes the hydrolysis of sucrose into glucose and fructose.[12] In 1913 they proposed a mathematical model of the reaction.[13] It involves an enzyme E binding to a substrate A to form a complex EA that releases a product P regenerating the original form of the enzyme.[6] This may be represented schematically as

[math]\ce{ E{} + A <=>[\mathit{k_\mathrm{+1}}][\mathit{k_\mathrm{-1}}] EA ->[k_\ce{cat}] E{} + P }[/math]

where [math]\displaystyle{ k_\mathrm{+1} }[/math] (forward rate constant), [math]\displaystyle{ k_\mathrm{-1} }[/math] (reverse rate constant), and [math]\displaystyle{ k_\mathrm{cat} }[/math] (catalytic rate constant) denote the rate constants,[14] the double arrows between A (substrate) and EA (enzyme-substrate complex) represent the fact that enzyme-substrate binding is a reversible process, and the single forward arrow represents the formation of P (product).

Under certain assumptions – such as the enzyme concentration being much less than the substrate concentration – the rate of product formation is given by

[math]\displaystyle{ v = \frac{\mathrm{d} p}{\mathrm{d} t} = \frac{V_\max a}{K_\mathrm{m} + a} = \frac{k_\mathrm{cat} e_0 a}{K_\mathrm{m} + a} }[/math]

in which [math]\displaystyle{ e_0 }[/math] is the initial enzyme concentration. The reaction order depends on the relative size of the two terms in the denominator. At low substrate concentration [math]\displaystyle{ a \ll K_\mathrm{m} }[/math], so that the rate [math]\displaystyle{ v = \frac{k_\mathrm{cat} e_0 a}{K_\mathrm{m}} }[/math] varies linearly with substrate concentration [math]\displaystyle{ a }[/math] (first-order kinetics in [math]\displaystyle{ a }[/math]).[15] However at higher [math]\displaystyle{ a }[/math], with [math]\displaystyle{ a \gg K_\mathrm{m} }[/math], the reaction approaches independence of [math]\displaystyle{ a }[/math] (zero-order kinetics in [math]\displaystyle{ a }[/math]),[15] asymptotically approaching the limiting rate [math]\displaystyle{ V_\mathrm{max} = k_\mathrm{cat} e_0 }[/math]. This rate, which is never attained, refers to the hypothetical case in which all enzyme molecules are bound to substrate. [math]\displaystyle{ k_\mathrm{cat} }[/math], known as the turnover number or catalytic constant, normally expressed in s –1, is the limiting number of substrate molecules converted to product per enzyme molecule per unit of time. Further addition of substrate would not increase the rate, and the enzyme is said to be saturated.

The Michaelis constant [math]\displaystyle{ K_\mathrm{m} }[/math] is not affected by the concentration or purity of an enzyme.[16] Its value depends both on the identity of the enzyme and that of the substrate, as well as conditions such as temperature and pH.

The model is used in a variety of biochemical situations other than enzyme-substrate interaction, including antigen–antibody binding, DNA–DNA hybridization, and protein–protein interaction.[17][18] It can be used to characterize a generic biochemical reaction, in the same way that the Langmuir equation can be used to model generic adsorption of biomolecular species.[18] When an empirical equation of this form is applied to microbial growth, it is sometimes called a Monod equation.

Michaelis–Menten kinetics have also been applied to a variety of topics outside of biochemical reactions,[14] including alveolar clearance of dusts,[19] the richness of species pools,[20] clearance of blood alcohol,[21] the photosynthesis-irradiance relationship, and bacterial phage infection.[22]

The equation can also be used to describe the relationship between ion channel conductivity and ligand concentration,[23] and also, for example, to limiting nutrients and phytoplankton growth in the global ocean.[24]

Specificity

The specificity constant [math]\displaystyle{ k_\text{cat}/K_\mathrm{m} }[/math] (also known as the catalytic efficiency) is a measure of how efficiently an enzyme converts a substrate into product. Although it is the ratio of [math]\displaystyle{ k_\text{cat} }[/math] and [math]\displaystyle{ K_\mathrm{m} }[/math] it is a parameter in its own right, more fundamental than [math]\displaystyle{ K_\mathrm{m} }[/math]. Diffusion limited enzymes, such as fumarase, work at the theoretical upper limit of 108 – 1010 M−1s−1, limited by diffusion of substrate into the active site.[25]

If we symbolize the specificity constant for a particular substrate A as [math]\displaystyle{ k_\mathrm{A} = k_\text{cat}/K_\mathrm{m} }[/math] the Michaelis–Menten equation can be written in terms of [math]\displaystyle{ k_\mathrm{A} }[/math] and [math]\displaystyle{ K_\mathrm{m} }[/math] as follows:

[math]\displaystyle{ v = \dfrac{k_\mathrm{A} e_0 a}{1 + \dfrac{a}{K_\mathrm{m}}} }[/math]
The reaction changes from approximately first-order in substrate concentration at low concentrations to approximately zeroth order at high concentrations.

At small values of the substrate concentration this approximates to a first-order dependence of the rate on the substrate concentration:

[math]\displaystyle{ v \approx k_\mathrm{A} e_0 a\text{ when } a \rightarrow 0 }[/math]

Conversely it approaches a zero-order dependence on [math]\displaystyle{ a }[/math] when the substrate concentration is high:

[math]\displaystyle{ v \rightarrow k_\mathrm{cat} e_0 \text{ when } a \rightarrow \infty }[/math]

The capacity of an enzyme to distinguish between two competing substrates that both follow Michaelis–Menten kinetics depends only on the specificity constant, and not on either [math]\displaystyle{ k_\text{cat} }[/math] or [math]\displaystyle{ K_\mathrm{m} }[/math] alone. Putting [math]\displaystyle{ k_\mathrm{A} }[/math] for substrate [math]\displaystyle{ \mathrm{A} }[/math] and [math]\displaystyle{ k_\mathrm{A'} }[/math] for a competing substrate [math]\displaystyle{ \mathrm{A'} }[/math], then the two rates when both are present simultaneously are as follows:

[math]\displaystyle{ v_\mathrm{A} = \frac{k_\mathrm{A}e_0 a}{1 + \dfrac{a}{K_\mathrm{m}^\mathrm{A}} + \dfrac{a'}{K_\mathrm{m}^\mathrm{A'}}},\;\;\; v_\mathrm{A'} = \frac{k_\mathrm{A'} e_0 a'}{1 + \dfrac{a}{K_\mathrm{m}^\mathrm{A}} + \dfrac{a'}{K_\mathrm{m}^\mathrm{A'}}} }[/math]

Although both denominators contain the Michaelis constants they are the same, and thus cancel when one equation is divided by the other:

[math]\displaystyle{ \frac{v_\mathrm{A}}{v_\mathrm{A'}} = \frac{k_\mathrm{A}\cdot a}{k_\mathrm{A'}\cdot a'} }[/math]

and so the ratio of rates depends only on the concentrations of the two substrates and their specificity constants.

Nomenclature

As the equation originated with Henri, not with Michaelis and Menten, it is more accurate to call it the Henri–Michaelis–Menten equation,[26] though it was Michaelis and Menten who realized that analysing reactions in terms of initial rates would be simpler, and as a result more productive, than analysing the time course of reaction, as Henri had attempted. Although Henri derived the equation he made no attempt to apply it. In addition, Michaelis and Menten understood the need for buffers to control the pH, but Henri did not.

Applications

Parameter values vary widely between enzymes. Some examples are as follows:[27]

Enzyme [math]\displaystyle{ K_\mathrm{m} }[/math] (M) [math]\displaystyle{ k_\text{cat} }[/math] (s−1) [math]\displaystyle{ k_\text{cat}/K_\mathrm{m} }[/math] (M−1s−1)
Chymotrypsin 1.5 × 10−2 0.14 9.3
Pepsin 3.0 × 10−4 0.50 1.7 × 103
tRNA synthetase 9.0 × 10−4 7.6 8.4 × 103
Ribonuclease 7.9 × 10−3 7.9 × 102 1.0 × 105
Carbonic anhydrase 2.6 × 10−2 4.0 × 105 1.5 × 107
Fumarase 5.0 × 10−6 8.0 × 102 1.6 × 108

Derivation

Equilibrium approximation

In their analysis, Michaelis and Menten (and also Henri) assumed that the substrate is in instantaneous chemical equilibrium with the complex, which implies[13][28]

[math]\displaystyle{ k_{+1} e a = k_{-1} x }[/math]

in which e is the concentration of free enzyme (not the total concentration) and x is the concentration of enzyme-substrate complex EA.

Conservation of enzyme requires that[28]

[math]\displaystyle{ e = e_0 - x }[/math]

where [math]\displaystyle{ e_0 }[/math] is now the total enzyme concentration. After combining the two expressions some straightforward algebra leads to the following expression for the concentration of the enzyme-substrate complex:

[math]\displaystyle{ x= \frac{e_0 a}{K_\mathrm{diss} + a} }[/math]

where [math]\displaystyle{ K_\mathrm{diss} = k_{-1} / k_{+1} }[/math] is the dissociation constant of the enzyme-substrate complex. Hence the rate equation is the Michaelis–Menten equation,[28]

[math]\displaystyle{ v = \frac{k_{+2}e_0 a}{K_\mathrm{diss} + a} }[/math]

where [math]\displaystyle{ k_{+2} }[/math] corresponds to the catalytic constant [math]\displaystyle{ k_\mathrm{cat} }[/math] and the limiting rate is [math]\displaystyle{ V_\mathrm{max} = k_{+2}e_0 = k_\mathrm{cat}e_0 }[/math]. Likewise with the assumption of equilibrium the Michaelis constant [math]\displaystyle{ K_\mathrm{m} = K_\mathrm{diss} }[/math].

Irreversible first step

When studying urease at about the same time as Michaelis and Menten were studying invertase, Donald Van Slyke and G. E. Cullen[29] made essentially the opposite assumption, treating the first step not as an equilibrium but as an irreversible second-order reaction with rate constant [math]\displaystyle{ k_{+1} }[/math]. As their approach is never used today it is sufficient to give their final rate equation:

[math]\displaystyle{ v = \frac{k_\mathrm{+2}e_0 a}{k_{+2}/k_{+1} + a} }[/math]

and to note that it is functionally indistinguishable from the Henri–Michaelis–Menten equation. One cannot tell from inspection of the kinetic behaviour whether [math]\displaystyle{ K_\mathrm{m} }[/math] is equal to [math]\displaystyle{ k_{+2}/k_{+1} }[/math] or to [math]\displaystyle{ k_{-1}/k_{+1} }[/math] or to something else.

Steady-state approximation

G. E. Briggs and J. B. S. Haldane undertook an analysis that harmonized the approaches of Michaelis and Menten and of Van Slyke and Cullen,[30][31] and is taken as the basic approach to enzyme kinetics today. They assumed that the concentration of the intermediate complex does not change on the time scale over which product formation is measured.[32] This assumption means that [math]\displaystyle{ k_{+1} e a = k_{-1}x + k_\mathrm{cat} x = (k_{-1} + k_\mathrm{cat})x }[/math]. The resulting rate equation is as follows:

[math]\displaystyle{ v = \frac{k_\mathrm{cat}e_0 a}{K_\mathrm{m} + a} }[/math]

where

[math]\displaystyle{ k_\mathrm{cat} = k_{+2} \text { and } K_\mathrm{m} = \frac{k_{-1} + k_\mathrm{cat}}{k_{+1}} }[/math]

This is the generalized definition of the Michaelis constant.[33]

Assumptions and limitations

All of the derivations given treat the initial binding step in terms of the law of mass action, which assumes free diffusion through the solution. However, in the environment of a living cell where there is a high concentration of proteins, the cytoplasm often behaves more like a viscous gel than a free-flowing liquid, limiting molecular movements by diffusion and altering reaction rates.[34] Note, however that although this gel-like structure severely restricts large molecules like proteins its effect on small molecules, like many of the metabolites that participate in central metabolism, is very much smaller.[35] In practice, therefore, treating the movement of substrates in terms of diffusion is not likely to produce major errors. Nonetheless, Schnell and Turner consider that is more appropriate to model the cytoplasm as a fractal, in order to capture its limited-mobility kinetics.[36]

Estimation of Michaelis–Menten parameters

Graphical methods

Determining the parameters of the Michaelis–Menten equation typically involves running a series of enzyme assays at varying substrate concentrations [math]\displaystyle{ a }[/math], and measuring the initial reaction rates [math]\displaystyle{ v }[/math], i.e. the reaction rates are measured after a time period short enough for it to be assumed that the enzyme-substrate complex has formed, but that the substrate concentration remains almost constant, and so the equilibrium or quasi-steady-state approximation remain valid.[37] By plotting reaction rate against concentration, and using nonlinear regression of the Michaelis–Menten equation with correct weighting based on known error distribution properties of the rates, the parameters may be obtained.

Before computing facilities to perform nonlinear regression became available, graphical methods involving linearisation of the equation were used. A number of these were proposed, including the Eadie–Hofstee plot of [math]\displaystyle{ v }[/math] against [math]\displaystyle{ v/a }[/math],[38][39] the Hanes plot of [math]\displaystyle{ a/v }[/math] against [math]\displaystyle{ a }[/math],[40] and the Lineweaver–Burk plot (also known as the double-reciprocal plot) of [math]\displaystyle{ 1/v }[/math] against [math]\displaystyle{ 1/a }[/math].[41] Of these,[42] the Hanes plot is the most accurate when [math]\displaystyle{ v }[/math] is subject to errors with uniform standard deviation.[43] From the point of view of visualizaing the data the Eadie–Hofstee plot has an important property: the entire possible range of [math]\displaystyle{ v }[/math] values from [math]\displaystyle{ 0 }[/math] to [math]\displaystyle{ V }[/math] occupies a finite range of ordinate scale, making it impossible to choose axes that conceal a poor experimental design.

However, while useful for visualization, all three linear plots distort the error structure of the data and provide less precise estimates of [math]\displaystyle{ v }[/math] and [math]\displaystyle{ K_\mathrm{m} }[/math] than correctly weighted non-linear regression. Assuming an error [math]\displaystyle{ \varepsilon (v) }[/math] on [math]\displaystyle{ v }[/math], an inverse representation leads to an error of [math]\displaystyle{ \varepsilon (v)/v^2 }[/math] on [math]\displaystyle{ 1/v }[/math] (Propagation of uncertainty), implying that linear regression of the double-reciprocal plot should include weights of [math]\displaystyle{ v^4 }[/math]. This was well understood by Lineweaver and Burk,[41] who had consulted the eminent statistician W. Edwards Deming before analysing their data.[44] Unlike nearly all workers since, Burk made an experimental study of the error distribution, finding it consistent with a uniform standard error in [math]\displaystyle{ v }[/math], before deciding on the appropriate weights.[45] This aspect of the work of Lineweaver and Burk received virtually no attention at the time, and was subsequently forgotten.

The direct linear plot is a graphical method in which the observations are represented by straight lines in parameter space, with axes [math]\displaystyle{ K_\mathrm{m} }[/math] and [math]\displaystyle{ V }[/math]: each line is drawn with an intercept of [math]\displaystyle{ -a }[/math] on the [math]\displaystyle{ K_\mathrm{m} }[/math] axis and [math]\displaystyle{ v }[/math] on the [math]\displaystyle{ V }[/math] axis. The point of intersection of the lines for different observations yields the values of [math]\displaystyle{ K_\mathrm{m} }[/math] and [math]\displaystyle{ V }[/math].[46]

Weighting

Many authors, for example Greco and Hakala,[47] have claimed that non-linear regression is always superior to regression of the linear forms of the Michaelis–Menten equation. However, that is correct only if the appropriate weighting scheme is used, preferably on the basis of experimental investigation, something that is almost never done. As noted above, Burk[45] carried out the appropriate investigation, and found that the error structure of his data was consistent with a uniform standard deviation in [math]\displaystyle{ v }[/math]. More recent studies found that a uniform coefficient of variation (standard deviation expressed as a percentage) was closer to the truth with the techniques in use in the 1970s.[48][49] However, this truth may be more complicated than any dependence on [math]\displaystyle{ v }[/math] alone can represent.[50]

Uniform standard deviation of [math]\displaystyle{ 1/v }[/math]. If the rates are considered to have a uniform standard deviation the appropriate weight for every [math]\displaystyle{ v }[/math] value for non-linear regression is 1. If the double-reciprocal plot is used each value of [math]\displaystyle{ 1/v }[/math] should have a weight of [math]\displaystyle{ v^4 }[/math], whereas if the Hanes plot is used each value of [math]\displaystyle{ a/v }[/math] should have a weight of [math]\displaystyle{ v^4/a^2 }[/math].

Uniform coefficient variation of [math]\displaystyle{ 1/v }[/math]. If the rates are considered to have a uniform coefficient variation the appropriate weight for every [math]\displaystyle{ v }[/math] value for non-linear regression is [math]\displaystyle{ v^2 }[/math]. If the double-reciprocal plot is used each value of [math]\displaystyle{ 1/v }[/math] should have a weight of [math]\displaystyle{ v^2 }[/math], whereas if the Hanes plot is used each value of [math]\displaystyle{ a/v }[/math] should have a weight of [math]\displaystyle{ v^2/a^2 }[/math].

Ideally the [math]\displaystyle{ v }[/math] in each of these cases should be the true value, but that is always unknown. However, after a preliminary estimation one can use the calculated values [math]\displaystyle{ \hat v }[/math] for refining the estimation. In practice the error structure of enzyme kinetic data is very rarely investigated experimentally, therefore almost never known, but simply assumed. It is, however, possible to form an impression of the error structure from internal evidence in the data.[51] This is tedious to do by hand, but can readily be done in the computer.

Closed form equation

Santiago Schnell and Claudio Mendoza suggested a closed form solution for the time course kinetics analysis of the Michaelis–Menten kinetics based on the solution of the Lambert W function.[52] Namely,

[math]\displaystyle{ \frac{a}{K_\mathrm{m}} = W(F(t)) }[/math]

where W is the Lambert W function and

[math]\displaystyle{ F(t) = \frac{a}{K_\mathrm{m}} \exp\!\left(\frac{a_0}{K_\mathrm{m}} - \frac{Vt}{K_\mathrm{m}} \right) }[/math]

The above equation, known nowadays as the Schnell-Mendoza equation,[53] has been used to estimate [math]\displaystyle{ V }[/math] and [math]\displaystyle{ K_\mathrm{m} }[/math] from time course data.[54][55]

Reactions with more than one substrate

Only a small minority of enzyme-catalysed reactions have just one substrate, and even the number is increased by treating two-substrate reactions in which one substrate is water as one-substrate reactions the number is still small. One might accordingly suppose that the Michaelis–Menten equation, normally written with just one substrate, is of limited usefulness. This supposition is misleading, however. One of the common equations for a two-substrate reaction can be written as follows to express [math]\displaystyle{ v }[/math] in terms of two substrate concentrations [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math]:

[math]\displaystyle{ v = \frac{Vab}{K_\mathrm{iA}K_\mathrm{mB} + K_\mathrm{mB}a + K_\mathrm{mA}b + ab} }[/math]

the other symbols represent kinetic constants. Suppose now that [math]\displaystyle{ a }[/math] is varied with [math]\displaystyle{ b }[/math] held constant. Then it is convenient to reorganize the equation as follows:

[math]\displaystyle{ v = \frac{Vb \cdot a}{K_\mathrm{iA}K_\mathrm{mB}+ K_\mathrm{mA}b +(K_\mathrm{mB} + b)a} = \dfrac{\dfrac{Vb }{K_\mathrm{mB}+b}\cdot a}{\dfrac{K_\mathrm{iA}K_\mathrm{mB}+ K_\mathrm{mA}b}{K_\mathrm{mB}+b} +a} }[/math]

This has exactly the form of the Michaelis–Menten equation

[math]\displaystyle{ v = \frac{V^\mathrm{app} a}{K^\mathrm{app}_\mathrm{m} + a} }[/math]

with apparent values [math]\displaystyle{ V^\mathrm{app} }[/math] and [math]\displaystyle{ K^\mathrm{app}_\mathrm{m} }[/math] defined as follows:

[math]\displaystyle{ V^\mathrm{app} = \dfrac{Vb}{K_\mathrm{mB}+b} }[/math]
[math]\displaystyle{ K^\mathrm{app}_\mathrm{m} = \dfrac{K_\mathrm{iA}K_\mathrm{mB}+ K_\mathrm{mA}b}{K_\mathrm{mB}+b} }[/math]

Linear inhibition

The linear (simple) types of inhibition can be classified in terms of the general equation for mixed inhibition at an inhibitor concentration [math]\displaystyle{ i }[/math]:

[math]\displaystyle{ v = \dfrac{Va}{K_\mathrm{m}\left(1 + \dfrac {i}{K_\mathrm{ic}} \right) + a\left(1 + \dfrac {i}{K_\mathrm{iu}} \right)} }[/math]

in which [math]\displaystyle{ K_\mathrm{ic} }[/math] is the competitive inhibition constant and [math]\displaystyle{ K_\mathrm{iu} }[/math] is the uncompetitive inhibition constant. This equation includes the other types of inhibition as special cases:

  • If [math]\displaystyle{ K_\mathrm{iu} \rightarrow \infty }[/math] the second parenthesis in the denominator approaches [math]\displaystyle{ 1 }[/math] and the resulting behaviour[56] is competitive inhibition.
  • If [math]\displaystyle{ K_\mathrm{ic} \rightarrow \infty }[/math] the first parenthesis in the denominator approaches [math]\displaystyle{ 1 }[/math] and the resulting behaviour is uncompetitive inhibition.
  • If both [math]\displaystyle{ K_\mathrm{ic} }[/math] and [math]\displaystyle{ K_\mathrm{iu} }[/math] are finite the behaviour is mixed inhibition.
  • If [math]\displaystyle{ K_\mathrm{ic} = K_\mathrm{iu} }[/math] the resulting special case is pure non-competitive inhibition.

Pure non-competitive inhibition is very rare, being mainly confined to effects of protons and some metal ions. Cleland recognized this, and he redefined noncompetitive to mean mixed.[57] Some authors have followed him in this respect, but not all, so when reading any publication one needs to check what definition the authors are using.

In all cases the kinetic equations have the form of the Michaelis–Menten equation with apparent constants, as can be seen by writing the equation above as follows:

[math]\displaystyle{ v = \dfrac{\dfrac{V}{1 + i/K_\mathrm{iu}} \cdot a} {\dfrac{K_\mathrm{m}(1 + i/K_\mathrm{ic})} {1 + i/K_\mathrm{iu}} +a} = \frac{V^\mathrm{app} a}{K^\mathrm{app}_\mathrm{m} + a} }[/math]

with apparent values [math]\displaystyle{ V^\mathrm{app} }[/math] and [math]\displaystyle{ K^\mathrm{app}_\mathrm{m} }[/math] defined as follows:

[math]\displaystyle{ V^\mathrm{app} = \dfrac{V}{1 + i/K_\mathrm{iu}} }[/math]
[math]\displaystyle{ K^\mathrm{app}_\mathrm{m} = \dfrac{K_\mathrm{m}(1 + i/K_\mathrm{ic})}{1 + i/K_\mathrm{iu}} }[/math]

See also

References

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  2. ↑ "Symbolism and terminology in enzyme kinetics. Recommendations 1981". Arch. Biochem. Biophys. 234 (2): 732–740. 1983. doi:10.1016/0003-9861(83)90262-X. 
  3. ↑ "Symbolism and terminology in enzyme kinetics. Recommendations 1981". Biochem. J. 213 (3): 561–571. 1982. doi:10.1042/bj2130561. PMID 6615450. 
  4. ↑ Cornish-Bowden, A. (2014). "Current IUBMB recommendations on enzyme nomenclature and kinetics". Perspectives in Science 1 (1–6): 74–87. doi:10.1016/j.pisc.2014.02.006. Bibcode: 2014PerSc...1...74C. 
  5. ↑ The subscript max and term "maximum rate" (or "maximum velocity") often used are not strictly appropriate because this is not a maximum in the mathematical sense.
  6. ↑ 6.0 6.1 Cornish-Bowden, Athel (2012). Fundamentals of Enzyme Kinetics (4th ed.). Wiley-Blackwell, Weinheim. pp. 25–75. ISBN 978-3-527-33074-4. 
  7. ↑ Busch, T.; Petersen, M. (2021). "Identification and biochemical characterisation of tyrosine aminotransferase from Anthoceros agrestis unveils the conceivable entry point into rosmarinic acid biosynthesis in hornworts". Planta 253 (5): 98. doi:10.1007/s00425-021-03623-2. PMID 33844079. 
  8. ↑ M. A. Chrisman; M. J. Goldcamp; A. N. Rhodes; J. Riffle (2023). "Exploring Michaelis–Menten kinetics and the inhibition of catalysis in a synthetic mimic of catechol oxidase: an experiment for the inorganic chemistry or biochemistry laboratory". J. Chem. Educ. 100 (2): 893–899. doi:10.1021/acs.jchemed.9b01146. Bibcode: 2023JChEd.100..893C. 
  9. ↑ Huang, Y. Y.; Condict, L.; Richardson, S. J.; Brennan, C. S.; Kasapis, S. (2023). "Exploring the inhibitory mechanism of p-coumaric acid on α-amylase via multi-spectroscopic analysis, enzymatic inhibition assay and molecular docking". Food Hydrocolloids 139: 19)08524. doi:10.1016/j.foodhyd.2023.108524. 
  10. ↑ Cárdenas, M. L.; Cornish-Bowden, A.; Ureta, T. (1998). "Evolution and regulatory role of the hexokinases". Biochim. Biophys. Acta 1401 (3): 242–264. doi:10.1016/S0167-4889(97)00150-X. PMID 9540816. 
  11. ↑ Henri, Victor (1903). Lois Générales de l'Action des Diastases. Paris: Hermann. https://archive.org/details/b28114024. 
  12. ↑ "Victor Henri". Whonamedit?. http://www.whonamedit.com/doctor.cfm/2881.html. 
  13. ↑ 13.0 13.1 Michaelis, L.; Menten, M.L. (1913). "Die Kinetik der Invertinwirkung". Biochem Z 49: 333–369.  (recent translation, and an older partial translation)
  14. ↑ 14.0 14.1 Chen, W.W.; Neipel, M.; Sorger, P.K. (2010). "Classic and contemporary approaches to modeling biochemical reactions". Genes Dev 24 (17): 1861–1875. doi:10.1101/gad.1945410. PMID 20810646. 
  15. ↑ 15.0 15.1 Laidler K.J. and Meiser J.H. Physical Chemistry (Benjamin/Cummings 1982) p.430 ISBN:0-8053-5682-7
  16. ↑ Ninfa, Alexander; Ballou, David P. (1998). Fundamental laboratory approaches for biochemistry and biotechnology. Bethesda, Md.: Fitzgerald Science Press. ISBN 978-1-891786-00-6. OCLC 38325074. 
  17. ↑ Lehninger, A.L.; Nelson, D.L.; Cox, M.M. (2005). Lehninger principles of biochemistry. New York: W.H. Freeman. ISBN 978-0-7167-4339-2. https://archive.org/details/lehningerprincip00lehn_0. 
  18. ↑ 18.0 18.1 Chakraborty, S. (23 Dec 2009). Microfluidics and Microfabrication (1 ed.). Springer. ISBN 978-1-4419-1542-9. 
  19. ↑ Yu, R.C.; Rappaport, S.M. (1997). "A lung retention model based on Michaelis–Menten-like kinetics". Environ Health Perspect 105 (5): 496–503. doi:10.1289/ehp.97105496. PMID 9222134. 
  20. ↑ Keating, K.A.; Quinn, J.F. (1998). "Estimating species richness: the Michaelis–Menten model revisited". Oikos 81 (2): 411–416. doi:10.2307/3547060. 
  21. ↑ Jones, A.W. (2010). "Evidence-based survey of the elimination rates of ethanol from blood with applications in forensic casework". Forensic Sci Int 200 (1–3): 1–20. doi:10.1016/j.forsciint.2010.02.021. PMID 20304569. 
  22. ↑ Abedon, S.T. (2009). "Kinetics of phage-mediated biocontrol of bacteria". Foodborne Pathog Dis 6 (7): 807–15. doi:10.1089/fpd.2008.0242. PMID 19459758. 
  23. ↑ Ding, Shinghua; Sachs, Frederick (1999). "Single Channel Properties of P2X2 Purinoceptors". The Journal of General Physiology 113 (5): 695–720. doi:10.1085/jgp.113.5.695. PMID 10228183. 
  24. ↑ Dugdale, RCJ (1967). "Nutrient limitation in the sea: Dynamics, identification, and significance". Limnology and Oceanography 12 (4): 685–695. doi:10.4319/lo.1967.12.4.0685. Bibcode: 1967LimOc..12..685D. 
  25. ↑ Stroppolo, M.E.; Falconi, M.; Caccuri, A.M.; Desideri, A. (Sep 2001). "Superefficient enzymes". Cell Mol Life Sci 58 (10): 1451–60. doi:10.1007/PL00000788. PMID 11693526. 
  26. ↑ Deichmann, U.; Schuster, S.; Mazat, J.-P.; Cornish-Bowden, A. (2013). "Commemorating the 1913 Michaelis–Menten paper Die Kinetik der Invertinwirkung: three perspectives". FEBS J. 281 (2): 435–463. doi:10.1111/febs.12598. PMID 24180270. 
  27. ↑ Mathews, C.K.; van Holde, K.E.; Ahern, K.G. (10 Dec 1999). Biochemistry (3 ed.). Prentice Hall. ISBN 978-0-8053-3066-3. http://www.pearsonhighered.com/mathews/. 
  28. ↑ 28.0 28.1 28.2 Keener, J.; Sneyd, J. (2008). Mathematical Physiology: I: Cellular Physiology (2 ed.). Springer. ISBN 978-0-387-75846-6. 
  29. ↑ Van Slyke, D. D.; Cullen, G. E. (1914). "The mode of action of urease and of enzymes in general". J. Biol. Chem. 19 (2): 141–180. doi:10.1016/S0021-9258(18)88300-4. 
  30. ↑ Briggs, G.E.; Haldane, J.B.S. (1925). "A note on the kinetics of enzyme action". Biochem J 19 (2): 338–339. doi:10.1042/bj0190338. PMID 16743508. 
  31. ↑ Laidler, Keith J. (1978). Physical Chemistry with Biological Applications. Benjamin/Cummings. pp. 428–430. ISBN 0-8053-5680-0. 
  32. ↑ In advanced work this is known as the quasi-steady-state assumption or pseudo-steady-state-hypothesis, but in elementary treatments the steady-state assumption is sufficient.
  33. ↑ Murray, J.D. (2002). Mathematical Biology: I. An Introduction (3 ed.). Springer. ISBN 978-0-387-95223-9. 
  34. ↑ Zhou, H.X.; Rivas, G.; Minton, A.P. (2008). "Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences". Annu Rev Biophys 37 (1): 375–97. doi:10.1146/annurev.biophys.37.032807.125817. PMID 18573087. 
  35. ↑ Mastro, A. M.; Babich, M. A.; Taylor, W. D.; Keith, A. D. (1984). "Diffusion of a small molecule in the cytoplasm of mammalian cells". Proc. Natl. Acad. Sci. USA 81 (11): 3414–3418. doi:10.1073/pnas.81.11.3414. PMID 6328515. Bibcode: 1984PNAS...81.3414M. 
  36. ↑ Schnell, S.; Turner, T.E. (2004). "Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws". Prog Biophys Mol Biol 85 (2–3): 235–60. doi:10.1016/j.pbiomolbio.2004.01.012. PMID 15142746. 
  37. ↑ Segel, L.A.; Slemrod, M. (1989). "The quasi-steady-state assumption: A case study in perturbation". SIAM Review 31 (3): 446–477. doi:10.1137/1031091. https://zenodo.org/record/1059052. 
  38. ↑ Eadie, G. S. (1942). "The inhibition of cholinesterase by physostigmine and prostigmine". J. Biol. Chem. 146 (1): 85–93. doi:10.1016/S0021-9258(18)72452-6. 
  39. ↑ Hofstee, B. H. J. (1953). "Specificity of esterases". J. Biol. Chem. 199 (1): 357–364. doi:10.1016/S0021-9258(18)44843-0. 
  40. ↑ Hanes, C.S. (1932). "Studies on plant amylases. I. The effect of starch concentration upon the velocity of hydrolysis by the amylase of germinated barley". Biochem. J. 26 (2): 1406–1421. doi:10.1042/bj0261406. PMID 16744959. 
  41. ↑ 41.0 41.1 Lineweaver, H.; Burk, D. (1934). "The Determination of Enzyme Dissociation Constants" (in en). Journal of the American Chemical Society 56 (3): 658–666. doi:10.1021/ja01318a036. https://pubs.acs.org/doi/abs/10.1021/ja01318a036. 
  42. ↑ The name of Barnet Woolf is often coupled with that of Hanes, but not with the other two. However, Haldane and Stern attributed all three to Woolf in their book Allgemeine Chemie der Enzyme in 1932, about the same time as Hanes and clearly earlier than the others.
  43. ↑ This is not necessarily the case!
  44. ↑ "The dissociation constant of nitrogen-nitrogenase in Azobacter". J. Amer. Chem. Soc. 56: 225–230. 1934. doi:10.1021/ja01316a071. 
  45. ↑ 45.0 45.1 Burk, D.. "Nitrogenase". Ergebnisse der Enzymforschung 3: 23–56. 
  46. ↑ Eisenthal, R.; Cornish-Bowden, A. (1974). "The direct linear plot: a new graphical procedure for estimating enzyme kinetic parameters". Biochem. J. 139 (3): 715–720. doi:10.1042/bj1390715. PMID 4854723. 
  47. ↑ Greco, W.R.; Hakala, M.T. (1979). "Evaluation of methods for estimating the dissociation constant of tight binding enzyme inhibitors". J Biol Chem 254 (23): 12104–12109. doi:10.1016/S0021-9258(19)86435-9. PMID 500698. 
  48. ↑ Storer, A. C.; Darlison, M. G.; Cornish-Bowden, A. (1975). "The nature of experimental error in enzyme kinetic measurements". Biochem. J. 151 (2): 361–367. doi:10.1042/bj1510361. PMID 1218083. 
  49. ↑ Askelöf, P; Korsfeldt, M; Mannervik, B (1975). "Error structure of enzyme kinetic experiments: Implications for weighting in regression-analysis of experimental-data". Eur. J. Biochem. 69 (1): 61–67. doi:10.1111/j.1432-1033.1976.tb10858.x. PMID 991863. 
  50. ↑ Mannervik, B.; Jakobson, I.; Warholm, M. (1986). "Error structure as a function of substrate and inhibitor concentration in enzyme kinetic experiments". Biochem. J. 235 (3): 797–804. doi:10.1042/bj2350797. PMID 3753447. 
  51. ↑ Cornish-Bowden, A; Endrenyi, L. (1986). "Robust regression of enzyme kinetic data". Biochem. J. 234 (1): 21–29. doi:10.1042/bj2340021. PMID 3707541. 
  52. ↑ Schnell, S.; Mendoza, C. (1997). "A closed form solution for time-dependent enzyme kinetics". Journal of Theoretical Biology 187 (2): 207–212. doi:10.1006/jtbi.1997.0425. Bibcode: 1997JThBi.187..207S. 
  53. ↑ Olp, M.D.; Kalous, K.S.; Smith, B.C. (2020). "ICEKAT: an interactive online tool for calculating initial rates from continuous enzyme kinetic traces". BMC Bioinformatics 21 (1): 186. doi:10.1186/s12859-020-3513-y. PMID 32410570. 
  54. ↑ Goudar, C. T.; Sonnad, J. R.; Duggleby, R. G. (1999). "Parameter estimation using a direct solution of the integrated Michaelis–Menten equation". Biochimica et Biophysica Acta (BBA) - Protein Structure and Molecular Enzymology 1429 (2): 377–383. doi:10.1016/s0167-4838(98)00247-7. PMID 9989222. 
  55. ↑ Goudar, C. T.; Harris, S. K.; McInerney, M. J.; Suflita, J. M. (2004). "Progress curve analysis for enzyme and microbial kinetic reactions using explicit solutions based on the Lambert W function". Journal of Microbiological Methods 59 (3): 317–326. doi:10.1016/j.mimet.2004.06.013. PMID 15488275. 
  56. ↑ According to the IUBMB Recommendations inhibition is classified operationally, i.e. in terms of what is observed, not in terms of its interpretation.
  57. ↑ Cleland, W. W. (1963). "The kinetics of enzyme-catalyzed reactions with two or more substrates or products: II. Inhibition: Nomenclature and theory". Biochim. Biophys. Acta 67 (2): 173–187. doi:10.1016/0926-6569(63)90226-8. PMID 14021668. 

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