Michaelis-Menten kinetics
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2008-07-14T03:24:20Z
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'''Michaelis-Menten kinetics''' (occasionally also referred to as '''Michaelis-Menten-Henri kinetics''') describes the [[enzyme kinetics|kinetics]] of many [[enzyme]]s. It is named after [[Leonor Michaelis]] and [[Maud Menten]]. This kinetic model is relevant to situations where the concentration of enzyme is much lower than the concentration of substrate (i.e. where enzyme concentration is the limiting factor), and when the enzyme is not [[Allosteric regulation|allosteric]].
==History==
The modern relationship between substrate and enzyme concentration was proposed in [[1903]] by Victor Henri.<ref>Victor Henri. Lois Générales de l'Action des Diastases. Paris, Hermann, 1903.</ref> A microscopic interpretation was thereafter proposed in [[1913]] by [[Leonor Michaelis]] and [[Maud Menten]], following earlier work by [[Archibald Vivian Hill]].<ref>Leonor Michaelis, Maud Menten (1913). Die Kinetik der Invertinwirkung, Biochem. Z. 49:333-369.</ref> 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 the concentrations of the intermediate complexes remain constant) was proposed by Briggs and Haldane.<ref>G. E. Briggs and J. B. S. Haldane (1925) A note on the kinetics of enzyme action, Biochem. J., 19, 339-339.</ref>
==Determination of constants==
[[Image:Michaelis-Menten saturation curve of an enzyme reaction.svg|thumb|300px|right|Saturation curve for an enzyme showing the relation between the concentration of substrate and rate.]]
To determine the maximum rate of an enzyme mediated reaction, a series of experiments is carried out where the [[substrate (biochemistry)|substrate]] concentration (''[S]'') is increased until a constant initial rate of product formation is achieved. This is the ''maximum velocity'' (''V''<sub>max</sub>) of the enzyme under the conditions of the experiment. In this state, enzyme active sites are saturated with substrate.
==Reaction rate/velocity ''V''==
The reaction rate ''V'' is the number of reactions per second catalyzed per mole of the enzyme. The reaction rate increases with increasing substrate concentration [S], [[asymptote|asymptotically]] approaching the maximum rate ''V<sub>max</sub>''. There is therefore no clearly-defined substrate concentration at which the enzyme can be said to be saturated with substrate. A more appropriate measure to characterise an enzyme is the substrate concentration at which the reaction rate reaches half of its maximum value (''V<sub>max</sub>/2''). This concentration can be shown to be equal to the Michaelis constant (''K<sub>M</sub>'').
==Michaelis constant '''K''<sub>m</sub>'==
For enzymatic reactions which exhibit simple Michaelis-Menten kinetics and in which product formation is the rate-limiting step (i.e., when ''k''<sub>2</sub> << ''k''<sub>-1</sub>) ''K''<sub>m</sub>≈''k''<sub>-1</sub>/''k''<sub>1</sub>=''K''<sub>d</sub>, where ''K''<sub>d</sub> is the [[dissociation constant]] ([[affinity]] for [[Substrate (biochemistry)|substrate]]) of the [[enzyme]]-[[Substrate (biochemistry)|substrate]] (ES) complex. However, often ''k''<sub>2</sub> >> ''k''<sub>-1</sub>, or ''k''<sub>2</sub> and ''k''<sub>-1</sub> are comparable, in which case nothing can be said about the enzyme affinity from the Michaelis constant alone.<ref>Nelson, DL., Cox, MM. (2000) Lehninger Principles of Biochemistry, 3rd Ed., Worth Publishers, USA</ref>
The Michaelis constant can be defined as:
<math>K_m = \frac{k_{\textrm{-}1} + k_2}{k_1}</math>
==Equation==
The most convenient derivation of the Michaelis-Menten equation, described by Briggs and [[J. B. S. Haldane|Haldane]], is obtained as follows:
The enzymatic reaction is assumed to be irreversible, and the product does not bind to the enzyme.
<math>
E + S
\begin{matrix}
k_1 \\
\longrightarrow \\
\longleftarrow \\
k_{\textrm{-}1}
\end{matrix}
ES
\begin{matrix}
k_2 \\
\longrightarrow\\
\
\end{matrix}
E + P
</math>
The rate of production of the product, <math>d[P]/dt</math> is referred to as the reaction rate, V in enzyme kinetics. It is dependent on the conversion rate constant, k<sub>2</sub> (often referred to as the catalytic constant, k<sub>cat</sub>) and <math>[ES]</math>, the concentration of enzyme that is bound to substrate. As <math>[ES]</math> is not usually measurable, it must be expressed in terms of the known parameters of the system, namely the concentration of enzyme and substrate originally added.
A key assumption in this derivation is the quasi [[Steady state (chemistry)|steady state]] approximation, namely that the concentration of the substrate-bound enzyme (and hence also the unbound enzyme) change much more slowly than those of the product and substrate. Starting from the quasi [[Steady state (chemistry)|steady state]] approximation, this allows us to express the relationship between the substrate concentration and the bound and unbound enzyme concentrations in terms of the various rate constants:
<math> \frac{d[ES]}{dt} = 0 = k_1[E][S] - [ES] (k_{\textrm{-}1} + k_2) </math>
Which can be rearranged to:
<math>[ES] = \frac{k_1[E][S]}{k_{\textrm{-}1} + k_2}</math>
To simplify the equation, we define the Michaelis constant as:
<math>K_m = \frac{k_{\textrm{-}1} + k_2}{k_1}</math>
yielding:
<math>[ES] = \frac{[E][S]}{K_m}</math> (1)
The total concentration of enzyme (<math>[E_0]</math>) is the sum of the free enzyme in solution (<math>[E]</math>) and that which is bound to the substrate (<math>[ES]</math>), allowing us to derive the free enzyme concentration from (1):
<math>[E_0] = [E] + [ES]</math>
<math>[E] = [E_0] - [ES]</math> (2)
Using this concentration (2), the bound enzyme concentration (1) can now be written:
<math>[ES] = \frac{([E_0] - [ES]) [S]}{K_m}</math>
Rearranging gives:
<math>[ES] \frac{K_m}{[S]} = [E_0] - [ES]</math>
<math>[ES]\left(1 + \frac{K_m}{[S]}\right) = [E_0]</math>
<math>[ES] = [E_0]\frac{1}{1+\frac{K_m}{[S]}}</math> (3)
The reaction rate is:
<math>V = \frac{d[P]}{dt} = k_2[ES]</math> (4)
Substituting (3) in (4) and multiplying the numerator and denominator by <math>[S]</math> gives:
<math>\frac{d[P]}{dt} = k_2[E_0]\frac{[S]}{K_m + [S]} = V_{\mbox{max}}\frac{[S]}{K_m + [S]} </math>
Because the concentration of substrate changes as the reaction takes place, the initial reaction rate (V<sub>0</sub>) is used to simplify analysis, taking the initial concentration of substrate as <math>[S]</math>.
[[Image:Lineweaver-Burke plot.svg|thumb|400px|[[Lineweaver-Burk plot]]]]
This equation may be represented by a [[Lineweaver-Burk plot]] or a [[Hanes-Woolf plot]].
If [S] is large compared to K<sub>m</sub>, [S]/(K<sub>m</sub> + [S]) approaches 1. Therefore, the rate of product formation is equal to k<sub>2</sub>[E<sub>0</sub>] in this case.
When [S] equals K<sub>m</sub>, [S]/(K<sub>m</sub> + [S]) equals 0.5. In this case, the rate of product formation is half of the maximum rate (1/2 V<sub>max</sub>). By plotting V<sub>0</sub> against [S], one can easily determine V<sub>max</sub> and K<sub>m</sub>. The most common way of generating this data is with a series of experiments at constant E<sub>0</sub> and different substrate concentration [S].
The Michaelis-Menten equation describes the rates of irreversible reactions. A steady state solution for a chemical equilibrium modeled with Michaelis-Menten kinetics can be obtained with the [[Goldbeter–Koshland kinetics|Goldbeter-Koshland]] equation.
==Limitations==
Michaelis-Menten kinetics, like other classical [[Biochemistry|biochemical]] kinetic theories, relies on the [[law of mass action]] derived from the assumptions of free ([[Fickian]]) [[diffusion]] and [[Thermodynamics|thermodynamically]]-driven random collision. However, many biochemical or cellular processes deviate significantly from such conditions. For example, the [[cytoplasm]] inside a cell behaves more like a gel than a freely flowable or watery [[liquid]], due to the very high concentration of protein (up to ~400 mg/mL) and other “solutes”, which can severely limit molecular movements (diffusion or collision) (see e.g. Olsen <ref>S. Olsen (2006) Applications of isothermal titration calorimetry to measure enzyme kinetics and activity in complex solutions, Thermochim. Acta 448, 12–18.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THV-4K96S9K-1-M&_cdi=5292&_user=2009156&_orig=browse&_coverDate=09%2F01%2F2006&_sk=995519998&view=c&wchp=dGLzVlz-zSkzV&md5=2a11b736f4850bddb62370c5175da47d&ie=/sdarticle.pdf]</ref>).
For [[heterogeneous]] 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 homogeneous 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 heterogeneous environments (see, R. Grima and S. Schnell <ref>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]</ref>). In general physics and chemistry, limited mobility-derived kinetics have been successfully described by the fractal-like kinetics. R. Kopelman <ref>R. Kopelman (1988) Fractal reaction kinetics, Science, 241, 1620–1626.[http://www.sciencemag.org/cgi/reprint/241/4873/1620.pdf]</ref>, M.A. Savageau <ref> M.A. Savageau (1995) Michaelis-Menten mechanism reconsidered: Implications of fractal kinetics, J. Theor. Biol., 176, 115–124.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6T2K-3WH53P4-2-5P&_cdi=4921&_user=2009156&_orig=search&_coverDate=06%2F07%2F1998&_sk=999529998&view=c&wchp=dGLbVtb-zSkWA&md5=98198068127ec65569c44f2ba3daa223&ie=/sdarticle.pdf]</ref>, and S. Schnell <ref> 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]</ref> pioneered the “fractal enzymology”, which has been further developed by other researchers.<ref> 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]</ref>
== Equation Optimization ==
The Michaelis-Menten equation is expressed here as:
:<math> {v} = \frac{V_{max} [S]}{K_{m} + [S]}</math>
where K<sub>m</sub> = Michaelis-Menten rate constant, [S] = substrate concentration, v =
initial rate of production of the product, and V<sub>max</sub> =
maximum initial rate of production of the product .
The Michaelis-Menten equation can be optimized by linear regression and nonlinear regression methods.
Commonly used linear regression methods are: Lineweaver-Burk, Eadie-Hofstee, Scatchard, and Hanes-Woolf.
The double reciprocal of the Michaelis-Menten equation yields the Lineweaver-Burk equation:
:<math> \frac{1}{v} = \frac{1}{V_{max}} + \frac{K_m}{V_{max}[S]}</math>
A plot of (1/v) versus (1/[S]) yields a slope = K<sub>m</sub>/V<sub>max</sub> and an intercept = 1/V<sub>max</sub>.
The Lineweaver-Burk regression is very sensitive to data error and it is strongly biased toward fitting the
data in the low concentration range. It was proposed in 1934. Another common linear form of the Michaelis-Menten equation
is the Eadie-Hofstee equation:
:<math> v = V_{max} - \frac{v K_m}{[S]}</math>
A plot of (v) versus (v/[S]) yields a slope = -K<sub>m</sub> and an intercept = V<sub>max</sub>.
The Eadie-Hofstee regression has some bias toward fitting the data in the low concentration range.
It was proposed in 1942 and 1952.
Note that if you invert the ''x'' and ''y'' axes, then this regression would convert into the Scatchard regression:
:<math> \frac{v}{[S]} = \frac{V_{max}}{K_m} - \frac{v}{K_m}</math>
A plot of (v/[S]) versus (v) yields a slope = -1/K<sub>m</sub> and an intercept = V<sub>max</sub>/K<sub>m</sub>.
The Scatchard regression is biased toward fitting the data in the high concentration range.
It was proposed in 1949.
Note that if you invert the ''x'' and ''y'' axes, then this regression would convert into the Eadie-Hofstee regression
discussed earlier. The last linear regression commonly used is the Hanes-Woolf linear regression:
:<math> \frac{[S]}{v} = \frac{[S]}{V_{max}} + \frac{K_m}{V_{max}}</math>
A plot of ([S]/v) versus ([S]) yields a slope = K<sub>m</sub>/V<sub>max</sub> and an intercept = 1/V<sub>max</sub>.
The Hanes-Woolf regression was proposed in 1932 and 1957. The Hanes-Woolf regression has very little sensitivity to data error. It has some bias toward fitting the data in the middle and high concentration range.
There are two kinds of nonlinear least squares (NLLS) regression techniques that can be used to optimize the Michaelis-Menten equation. They differ only on how the goodness-of-fit is defined. In the v-NLLS regression method, the best goodness-of-fit is defined as the curve with the smallest ''vertical'' error between the optimized curve and the data. In the n-NLLS regression method, the best goodness-of-fit is defined as the curve with the smallest ''normal'' error between the optimized curve and the data. Using the vertical error is the most common form of NLLS regression criteria. Definitions based on the normal error are less common. The normal error is the error of the datum point to the nearest point on the optimized curve. It is called the normal error because the trajectory is normal (that is, perpendicular) to the curve.
It is a common misconception to think that NLLS regression methods are free of bias. However, it is important to note that the v-NLLS regression method is biased toward the data with low [S] values. This is because the Michaelis-Menten equation has a sharp rise at low concentration values, which results in a large vertical error if the regression does not optimize this region of the graph well. Conversely, the n-NLLS regression method does not have any significant bias toward any region of the saturation curve.
Whereas linear regressions are relatively easy to pursue with simple programs, such as excel or hand-held calculators, the nonlinear regressions are much more difficult to solve. The NLLS regressions are best pursued with any of various computer programs.
==References==
<references/>
==Further reading==
*{{Wikibooks-inline|Biochemistry/Catalysis}}
== External links ==
*[http://www.ncgc.nih.gov/guidance/section4.html#inhibition-constant NIH guide], enzyme assay development and analysis
* LMMpro, a Michaelis-Menten optimization software @ http://www.alfisol.com [http://www.alfisol.com/IFS/IFS-003/LMMpro-Michaelis-Menten.php Website]
{{Enzymes}}
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