Michaelis-menten Kinetics

HISTORY

The modern relationship between substrate and enzyme concentration was proposed in 1903 by Victor Henri. Victor Henri. Lois Générales de l'Action des Diastases. Paris, Hermann, 1903.. A microscopic interpretation was thereafter 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.. It postulated that enzyme (catalyst) and substrate (reactant) are in fast equilibrium with their complex, which then dissociates to yield product and free enzyme.

The current derivation, based on the quasi steady state approximation, that is the concentration(s) of intermediate complex(es) do(es) not change, 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.

DETERMINATION OF CONSTANTS

To determine the maximum rate of an enzyme mediated reaction, the Substrate concentration ('' {Link without Title} '') 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 other factors that affect the rate of reaction (ie. pH, temperature, etc) are at optimal values.

REACTION RATE/VELOCITY ''V''

The speed ''V'' means the number of reactions per second that are catalyzed by an enzyme. With increasing substrate concentration 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 the Michaelis-Menten constant.

MICHAELIS CONSTANT '''K''<SUB><SMALL>M</SMALL></SUB>'

Since ''V''_max cannot be reached at any substrate concentration (because of its Asymptotic behaviour, ''V'' keeps growing at any {Link without Title} , albeit ever more slowly), enzymes are usually 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 ) of the Enzyme - 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.

It is worth noting that ''K''_M can only be used to describe an enzyme's affinity for substrate when product formation is rate-limiting, i.e., when ''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 Haldane . It is obtained as follows:

The enzymatic reaction is supposed to be irreversible, and the product does not rebind the enzyme.

E + S
  \begin{matrix}
    k_1 \
    \longrightarrow \
    \longleftarrow  \
    k_{-1}
  \end{matrix}
 ES
  \begin{matrix}
    k_2 \
    \longrightarrow\
    \ 
  \end{matrix}
 E + P

Because we follow the quasi Steady State approximation. the concentrations of the intermediates are assumed to equillibrate much faster than those of the product and substrate, i.e. their time derivatives are zero:

rac{d = k_1[E - k_{-1}[ES - k_2[ES] = 0

[S]}{k_{-1} + k_2}

Let's define the Michaelis constant:

K_m = rac{k_{-1} + k_2}{k_1}

This simplifies the form of the equation:

[S]}{K_m}

(1)

The total (added) concentration of enzyme is a sum of that which is free in the solution and that which is bound to the substrate, and the free enzyme concentration is derived from this:

= [E + [ES]

= [E_0 - [ES]

(2)

Using this concentration (2), the bound enzyme concentration (1) can now be written:

= rac{([E_0 - [S}{K_m}

Rearranging gives:

[ES] rac{K_m}{[S]} = - [ES

+ rac{K_m}{[S}
ight) = [E_0]

rac{1}{1+ rac{K_m}{[S]}}

(3)

The rate (or velocity) of the reaction is:

rac{d = k_2[ES

(4)

Substituting (3) in (4) and multiplying numerator and denominator by

{Link without Title} :

rac{d = k_2[E_0 rac{+ [S} = V_{max}rac{+ [S}

This equation may be analyzed experimentally with a Lineweaver-Burk Diagram or a Hanes-Woolf Plot.

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 {Link without Title} /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 {Link without Title} a.k.a. V_max is the ''maximum velocity'' or ''maximum rate''. k₂ is often called k_cat.

Notice that if is large compared to K_m, [S /(K_m + approaches 1. Therefore, the rate of product formation is equal to k₂[E₀ in this case.

When equals K_m, [S /(K_m + 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].

Note that the Michaelis-Menten equation is a rate describing equation. A steady state solution for a chemical equilibrium modeled with Michaelis-Menten kinetics can be obtained with the Goldbeter-Koshland equation.

LIMITATIONS

Michaelis-Menten kinetics, like other classical biochemical kinetic theories, relies on the enzymatic reactions, such as those of membrane enzymes, molecular mobility of the enzyme or substrates can also be severely restricted, due to the immobilization or phase-separation of the reactants. For some homogenous enzymatic reactions, the mobility of the enzyme or substrate may also be limited, such as the case of DNA Polymerase where the enzyme moves along a chained substrate, rather than having a three-dimensional freedom. The limitation on molecular mobility (as well as other “non-ideal” conditions) demands modifications on the conventional mass-action laws, and Michaelis-Menten kinetics, to better reflect certain real world situations. Although it has been shown that the law of mass action can be valid in heterogenous environments (see, R. Grima and S. Schnell R. Grima, S. Schnell (2007). A systematic investigation of the rate laws valid in intracellular environments. Biophys. Chem., 124, 1-10.[http://dx.doi.org/10.1016/j.bpc.2006.04.019]. In general physics and chemistry, limited mobility-derived kinetics have been successfully described by the fractal-like kinetics. R. Kopelman R. Kopelman (1988) Fractal reaction kinetics, Science, 241, 1620–1626.[http://www.sciencemag.org/cgi/reprint/241/4873/1620.pdf], M.A. Savageau M.A. Savageau (1995) Michaelis-Menten mechanism reconsidered: Implications of fractal kinetics, J. Theor. Biol., 176, 115–124. and S. Schnell S. Schnell, T. E. Turner (2004) Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws. Prog. Biophys. Mol. Biol., 85, 235-260.[http://dx.doi.org/10.1016/j.pbiomolbio.2004.01.012 pioneered the “fractal enzymology”, which is further developed by other researchers (see, e.g. Feng Xu F. Xu and H. Ding (2007) A new kinetic model for heterogeneous (or spatially confined) enzymatic catalysis: Contributions from the fractal and jamming (overcrowding) effects” Appl. Catal. A Gen. 317, 70-81[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6TF5-4MD465X-1-T&_cdi=5217&_user=2009156&_orig=search&_coverDate=01%2F27%2F2007&_sk=996829998&view=c&wchp=dGLbVzz-zSkzV&md5=b1717d5bce455e248254870b420910e4&ie=/sdarticle.pdf]).

EXTERNAL LINKS

NIH guide , enzyme assay development and analysis

REFERENCES

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Michaelis-menten Kinetics