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Michaelis-Menten kinetics

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'''Michaelis-Menten kinetics''' describe the rate of enzyme mediated reactions for many enzymes. It is named for Leonor Michaelis and Maud Menten. This kinetic model is valid only when the concentration of enzyme is much less than the concentration of substrate (i.e., enzyme concentration is the limiting factor), and in the particular case of steady-state, where the concentration of the enzyme-substrate complex is constant --> d[ES]/dt ~ 0 '''in comparison to''' d[S]/dt and d[P]/dt. This is an example of the application of the quasi steady-state assumption which is really just assuming that d[ES]/dt<< (d[S]/dt and d[P]/dt)

Determination of constants
To determine the maximum rate of an enzyme mediated reaction, the substrate (biochemistry) substrate concentration (''[S]'') is increased until a constant rate of product formation is achieved. This is the ''maximum velocity'' (''V''_max) of the enzyme. In this state, enzyme active sites are saturated with substrate. Note that at the ''maximum velocity'', the factors that affect the rate of enzyme mediated reactions (ie. pH, temperature, etc) are at optimal values. image:Michaelis-Menten.png
''Diagram of reaction speed and Michaelis-Menten constant.''

Reaction rate/velocity 'V'
The speed ''V'' means the number of reactions per second that are catalyzed by an enzyme. With increasing substrate concentration [S], the enzyme is asymptotically approaching its maximum speed ''V''_max, but never actually reaching it. Because of that, no [S] for ''V''_max can be given. Instead, the characteristic value for the enzyme is defined by the substrate concentration at its half-maximum speed (''V''_max''/2''). This K_M value is also called Michaelis-Menten constant.

Michaelis constant '''K''_M'
Since the substrate concentration at ''V''_max cannot be measured exactly, enzymes must be characterized by the substrate concentration at which the rate of reaction is half its maximum. This substrate concentration is called the Michaelis-Menten constant (''K''_M) a.k.a. Michaelis constant. This represents (for enzyme reactions exhibiting simple Michaelis-Menten kinetics) the dissociation constant (affinity for Substrate (biochemistry) substrate) of the enzyme-Substrate (biochemistry) substrate (ES) complex. Low values indicate that the ES complex is held together very tightly and rarely dissociates without the substrate first reacting to form product. Note: ''K''_M can only be used to describe an enzyme's affinity for substrate when ''k''₂ is rate-limiting, i.e. ''k''₂ << ''k''₁ and ''K''_M becomes ''k''_-1/''k''₁. Often, ''k''₂ >> ''k''₁, or ''k''₂ and ''k''₁ are comparable. ''Nelson, DL., Cox, MM. (2000) Lehninger Principles of Biochemistry, 3rd Ed., Worth Publishers, USA''

Equation
This derivation of "Michaelis-Menten" was actually described by Briggs and J. B. S. Haldane Haldane. It is obtained as follows: The enzymatic reaction is supposed to be irreversible, and the product does not rebind the enzyme. $E\; +\; S\; \backslash begin\{matrix\}\; k\_1\; \backslash \backslash \; \backslash longrightarrow\; \backslash \backslash \; \backslash longleftarrow\; \backslash \backslash \; k\_\{-1\}\; \backslash end\{matrix\}\; ES\; \backslash begin\{matrix\}\; k\_2\; \backslash \backslash \; \backslash longrightarrow\backslash \backslash \; \backslash \; \backslash end\{matrix\}\; E\; +\; P$ Because we follow the Steady state (chemistry) steady state approximation: $\backslash frac\{d[ES]\}\{dt\}\; =\; k\_1[E][S]\; -\; k\_\{-1\}[ES]\; -\; k\_2[ES]\; =\; 0$ $[ES]\; =\; \backslash frac\{k\_1[E][S]\}\{k\_\{-1\}\; +\; k\_2\}$ Let's define: $K\_m\; =\; \backslash frac\{k\_\{-1\}\; +\; k\_2\}\{k\_1\}$ Therefore: $[ES]\; =\; \backslash frac\{[E][S]\}\{K\_m\}$ (1) The rate (or velocity) of the reaction is: $\backslash frac\{d[P]\}\{dt\}\; =\; k\_2[ES]$ (2) The total concentration of enzyme is: $[E\_0]\; =\; [E]\; +\; [ES]$ Hence: $[E]\; =\; [E\_0]\; -\; [ES]$ (3) Substituting (3) into (1) gives: $[ES]\; =\; \backslash frac\{([E\_0]\; -\; [ES])\; [S]\}\{K\_m\}$ Rearranging gives: $[ES]\; \backslash frac\{K\_m\}\{[S]\}\; =\; [E\_0]\; -\; [ES]$ $[ES](1\; +\; \backslash frac\{K\_m\}\{[S]\})\; =\; [E\_0]$ $[ES]\; =\; [E\_0]\backslash frac\{1\}\{1+\backslash frac\{K\_m\}\{[S]\}\}$ (4) Substituting (4) in (2) and multiplying numerator and denominator by $[S]$: $\backslash frac\{d[P]\}\{dt\}\; =\; k\_2[E\_0]\backslash frac\{[S]\}\{K\_m\; +\; [S]\}\; =\; V\_\{max\}\backslash frac\{[S]\}\{K\_m\; +\; [S]\}$ This equation may be analyzed experimentally with a Lineweaver-Burke diagram. *E₀ is the total or starting amount of enzyme. It is not practical to measure the amount of the enzyme substrate complex during the reaction, so the reaction must be written in terms of the total (starting) amount of enzyme, a known quantity. *d[P]/dt a.k.a. V₀ a.k.a. ''reaction velocity'' a.k.a. ''reaction rate'' is the rate of production of the product. Note that the term ''reaction velocity'' is misleading and ''reaction rate'' is preferred. *k₂[E₀] a.k.a. V_max is the ''maximum velocity'' or ''maximum rate''. k₂ is often called k_cat. Notice that if [S] is large compared to K_m, [S]/(K_m + [S]) approaches 1. Therefore, the rate of product formation is equal to k₂[E₀] in this case. When [S] equals K_m, [S]/(K_m + [S]) equals 0.5. In this case, the rate of product formation is half of the maximum rate (1/2 V_max). By plotting V₀ against [S], one can easily determine V_max and K_m. Note that this requires a series of experiments at constant E₀ and different substrate concentration [S].

History
The relationship between substrate concentration and enzyme concentration was proposed in 1913 by Leonor Michaelis and Maud Menten, following earlier work by Archibald Vivian Hill. Leonor Michaelis, Maud Menten (1913). Die Kinetik der Invertinwirkung, Biochem. Z. 49:333-369. The current derivation has been proposed by Briggs and Haldane. G. E. Briggs and J. B. S. Haldane (1925) A note on the kinetics of enzyme action, Biochem. J., 19, 339-339.

Sources

- Catalysis chapter from the ''Biochemistry'' textbook released under the GFDL Category:Enzyme kinetics Category: Chemical kinetics Category:Ordinary differential equations de:Michaelis-Menten-Theorie es:Cinética de Michaelis-Menten fr:Équation de Michaelis Menten nl:Michaelis-Menten

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