Michaelis-Menten theory

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The Michaelis-Menten kinetics describes the enzyme kinetics as follows vereinfachendem mechanism : The free enzyme first binds reversibly to its substrate . In the bound state (enzyme-substrate complex) the substrate is converted and the reaction product is released from the enzyme. If the breakdown of the complex in enzyme and substrate dominates over the formation of the product, the law of mass action and the Michaelis-Menten equation for the speed of the overall reaction (substrate consumption and product formation) depending on the substrate concentration apply for the reversible reaction after the steady state has been established and other parameters. This can be used to explain, for example, the saturation of the production rate of products in enzymatic reactions.

The Michaelis-Menten kinetics is by Leonor Michaelis and Menten Maud named the 1913 improved experimental and demonstrated methods of analysis for the enzyme kinetics. Adolphe Wurtz had published the hypothesis for the mechanism with the complex as an intermediate as early as 1880. In 1902 Victor Henri derived the Michaelis-Menten equation from this.

Theoretical background

Simple description of an enzymatic reaction

As biocatalysts, enzymes E form a complex ES (enzyme-substrate complex) with their substrate S , from which the reaction to product P takes place:

k 1 and k ' 1 are the rate constants for the association (agglomeration) of E and S or the dissociation of the enzyme-substrate complex ES. k 2 and k ' 2 are the corresponding constants for the reaction to the product and the back reaction to the substrate. k 3 and k ' 3 describe the dissociation or association of an enzyme-product complex. This reverse reaction is negligible under the conditions of the enzyme kinetics (small concentration [P]). Also, k ' 2 is usually much smaller than k 3 , so that the following simplification is justified:

This system can generally be described by a system of ordinary differential equations which, under one of the conditions [E] >> [S] or [S] >> [E], can be solved approximately analytically, otherwise numerically . The Michaelis-Menten equation applies under the further assumption of steady state.

Steady state

In general, enzymes are able to compensate for fluctuating substrate concentrations , i.e. H. very quickly a steady set ( "steady state") in that they adjust their activities to the offer. This means that the concentration of the enzyme-substrate complex remains constant on the slower time scale that is valid for the process of product formation. So the following applies: This steady state assumption was developed by GE Briggs and John Burdon Sanderson Haldane . The Michaelis-Menten kinetics are only valid assuming this steady state with a constant [ES].

The Michaelis-Menten equation

The formulas at the Graz University of Technology .

The Michaelis-Menten kinetics derived from the reaction equation can generally be represented as:

Here, v 0 indicates the initial reaction rate at a certain substrate concentration [S]. v max is the maximum response speed.

The Michaelis constant  K m is a parameter for an enzymatic reaction . It depends on the particular enzymatic reaction. K m indicates the substrate concentration at which the conversion rate is half the maximum ( v  = ½ · v max ), which is therefore present at half saturation. It turns out to be

for the case that k 2 cannot be neglected compared to k 1 (Briggs-Haldane situation). A special case ("Michaelis-Menten case") is given if k 2  <<  k ' 1 . Here, K m is simplified to:

This corresponds to the dissociation constant of the enzyme-substrate complex. In this case, K m can be viewed as a measure of the affinity of the enzyme for the substrate.

Another important variable is the turnover number , also called molecular activity or "turnover number". This is the rate constant of the rate-determining step of the reaction and is denoted by k cat . If, as in the case mentioned above, the second step determines the rate, the definition of the reaction rate shows that

and thus

.

Simplified derivation of the Michaelis-Menten equation

The assumed steady state enables a formal derivation of the Michaelis-Menten equation from a suitable formulation of the law of mass action (which in turn is based on kinetic considerations). Sufficient requirements for the derivation are:

  • the formulation of the steady state;
  • the context ; where the concentration of the enzyme assumed to be constant is the total (i.e. with or without bound substrate);
  • the proportionality .

The procedure not only saves (like the source mentioned in the previous section) the solution of differential equations, but also the explicit consideration of the individual rate constants . Furthermore, the above formulation of the steady state makes understandable without further calculation,

  • why a small constant means a high affinity of the enzyme for the substrate (the value of the fraction falls for a given numerator as the denominator increases ), and
  • why the constant has the dimension of a concentration.
Mathematical derivation of the Michaelis-Menten equation  

and in results:

Ruptures fall ;
(Michaelis-Menten) .

Saturation of the enzymatic reaction

In contrast to the kinetics of uncatalyzed reactions, there is the phenomenon of saturation in enzyme kinetics : at very high substrate concentrations, the conversion rate  v cannot be increased further, that is, a value v max is reached.

K m corresponds to the concentration for which v = ½  v max applies

The saturation function of a "Michaelis-Menten enzyme" can be formulated as follows using the parameters K m and v max :

This Michaelis-Menten relationship is the equation of a hyperbola .

Calculation for the classification of the relationship as hyperbola  

The hyperbola is given.

  • Shift by -K m in [S] direction results
  • (The following) reflection at the [S] axis results in:
  • (The following) shift by + v max in the v-direction results in:

since the Michaelis-Menten relationship can be generated graphically by chaining congruence maps from a hyperbolic equation, it is itself a hyperbolic equation.

It shows the following properties (see figure):

  • The v value of the horizontal asymptote corresponds to v max .
Calculation to determine the asymptote  

The limit value can be determined by factoring out and reducing :

  • If the substrate concentration [S] corresponds to the K m value, then half of the enzyme E originally present is in the form of the enzyme-substrate complex ES, the other half is free: [ES] = [E] = ½ [E] 0 .
Generalization: If the substrate concentration [S] is -fold of K m , then the rate of turnover is -fold of ; further then [ES] is -fold of [E] 0 , and -fold of [E] 0 is free enzyme (concentration [E]). With follow for [S] = K M in the section "The Michaelis-Menten equation" called half-maximum conversion rate as well as "[ES] = [E] = ½ [E] 0 ".
Calculation to derive the named relationships  

Substituting in gives:

;

For the following transformations, relationships from the section "Simplified derivation of the Michaelis-Menten equation" are used. - Equate with

results in:

;

Inserting in gives:

;
  • Since the saturation is approximated asymptotically, this requires substrate concentrations that are more than ten times the K m value. Conversely, if a saturation hyperbola has been measured for an enzyme, i. H. If the turnover rate v is determined as a function of the substrate concentration [S], v max (the activity) and K m (the reciprocal affinity ) can be derived from this. A relatively new, simple and yet precise method for this purpose is direct linear plotting (see enzyme kinetics and S-system ).

Inhibitors and their influence on Michaelis-Menten kinetics

Inhibitors , including important drugs and poisons, change the properties of enzymes and inhibit the enzymatic reaction. Inhibitors can be divided into different classes (see: enzyme inhibition ). Depending on the mode of action of the inhibitor, it has a different influence on the Michaelis-Menten equation:

  • “Competitive” inhibitors increase the K m value, but do not change v max .
  • “Uncompetitive” inhibitors (rarely found) bind specifically to the enzyme-substrate complex. They lower v max and the apparent K m value.
  • Mixed-type inhibitors increase the K m value and decrease v max
  • The “non-competitive” inhibitor, which only lowers the v max value and leaves the K m value unchanged, is a special case of the mixed type . This type does not occur with single-substrate enzymes.

literature

  • Andrés Illanes: Enzyme biocatalysis: principles and applications. Springer, Dordrecht 2008, ISBN 978-1-4020-8360-0 .
  • David L. Nelson, Michael M. Cox: Lehninger Biochemie. 4th edition. Springer, Berlin / Heidelberg 2009, ISBN 978-3-540-68637-8 . Chapter: Enzymes .

Individual evidence

  1. Athel Cornish-Bowden: One hundred years of Michaelis-Menten kinetics . In: Perspectives in Science . tape 4 , March 2015, ISSN  2213-0209 , p. 3–9 , doi : 10.1016 / j.pisc.2014.12.002 (free full text).
  2. ^ Entry on Michaelis – Menten kinetics . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.M03892 Version: 2.3.1.
  3. ^ Entry on Michaelis – Menten mechanism . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.M03893 Version: 2.3.1.
  4. Chen WW, Niepel M, Sorger PK: Classic and contemporary approaches to modeling biochemical reactions . In: Genes Dev. . 24, No. 17, September 2010, pp. 1861-75. doi : 10.1101 / gad.1945410 . PMID 20810646 .
  5. For the derivation see Enzyme Kinetics (PDF).
  6. Entry on Michaelis constant . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.M03891 Version: 2.3.1.