Michaelis–Menten kinetics on arvau

Wikipedia / Michaelis–Menten kinetics

In biochemistry, Michaelis–Menten kinetics, named after Leonor Michaelis and Maud Menten, is the simplest case of enzyme kinetics, applied to enzyme-catalysed reactions involving the transformation of one substrate into one product. In 1913, Michaelis and Menten expanded on Victor Henri's fundamental equation of enzyme kinetics, which was established in 1902. It takes the form of a differential equation describing the reaction rate (rate of formation of product, the concentration of the substrateA (using the symbols recommended by the IUBMB). The formula below is given by the Michaelis–Menten equation:

, represents the limiting rate approached by the system at saturating substrate concentration for a given enzyme concentration. The Michaelis constant. 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

has often been called a "Michaelis–Menten plot", even recently, but this terminology is historically misleading, as Michaelis and Menten. Michaelis and Menten.

, but it remains valuable to compare the properties of several enzymes across a broad range of substrate concentrations - such as isoenzymes. 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), spanning a 2500-fold range. A conventional (linear) plot would compromise on readability for the high-affinity isoenzyme graphs, but a semi-logarithmic plot allows to read off the kinetic parameters for all isoenzymes.

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. 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. In 1913, they proposed a mathematical model of the reaction. 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. This may be represented schematically as

(catalytic rate constant) denote the rate constants, 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

is the initial enzyme concentration. The reaction order (first-order kinetics).), asymptotically, 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.

is not affected by the concentration or purity of an enzyme. 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. 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. 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, including alveolar clearance of dusts, the richness of species pools, clearance of blood alcohol, the photosynthesis-irradiance relationship, and bacterial phage infection.

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

Specificity

The specificity constant (also known as the catalytic efficiency. Diffusion limited enzymes, such as fumarase, work at the theoretical upper limit of , limited by diffusion of substrate into the active site.

as follows:

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

when the substrate concentration is high:

, then the two rates when both are present simultaneously are as follows:

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

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, 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:

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

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

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:

is the dissociation constant of the enzyme-substrate complex. Hence the rate equation is the Michaelis–Menten equation,

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. As their approach is never used today it is sufficient to give their final rate equation:

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, 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.. The resulting rate equation is as follows:

where

This is the generalized definition of the Michaelis constant.

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. 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. In practice, therefore, treating the movement of substrates in terms of diffusion is not likely to produce major errors. Nonetheless, Schnell and Turner consider it more appropriate to model the cytoplasm as a fractal, in order to capture its limited-mobility kinetics.

Estimation of Michaelis–Menten parameters

Graphical methods

Determining the parameters of the Michaelis–Menten equation typically involves running a series of enzyme assays, 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. 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, the Hanes plot, and the Lineweaver–Burk plot (also known as the double-reciprocal plot. Of these, is subject to errors with uniform standard deviation. occupies a finite range of ordinate scale, making it impossible to choose axes that conceal a poor experimental design.

(Propagation of uncertainty. This was well understood by Lineweaver and Burk, who had consulted the eminent statistician W. Edwards Deming before analysing their data., before deciding on the appropriate weights. This aspect of the work of Lineweaver and Burk received virtually no attention at the time, and was subsequently forgotten.

The direct linear plot.

Weighting

Many authors, for example Greco and Hakala, 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. 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. alone can represent.

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. 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. Namely,

where W is the Lambert W function and

The above equation, known nowadays as the Schnell-Mendoza equation, from time course data.

Reactions with more than one substrate

held constant. Then it is convenient to reorganize the equation as follows:

This has exactly the form of the Michaelis–Menten equation

defined as follows:

Linear inhibition

is the competitive inhibition is the uncompetitive inhibition constant. This equation includes the other types of inhibition as special cases:

and the resulting behaviour is competitive inhibition.
and the resulting behaviour is uncompetitive inhibition.
are finite the behaviour is mixed inhibition.
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. 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: