7475096

the Michaelis Menten equation describes the rates of irreversible enzymatic reactions by relating reaction rate to the concentration of the substrate. ... The validity of the following derivation rests on the reaction scheme given below and two key assumptions: that the total enzyme concentration and the concentration of the intermediate complex do not change over time. The most convenient derivation of the Michaelis Menten equation, described by Briggs and Haldane, is obtained as follows (Note that often the experimental parameter kcat is used but in this simple case it is equal to the kinetic parameter k2): The enzymatic reaction is assumed to be irreversible, and the product does not bind to the enzyme. E + S \overset{k_1}\underset{k_{-1}}{\begin{smallmatrix}\displaystyle\longrightarrow \\ \displaystyle\longleftarrow \end{smallmatrix}} ES \overset{k_2} {\longrightarrow} E + P \qquad \qquad (1) The first key assumption in this derivation is the quasi-steady-state assumption (or pseudo-steady-state hypothesis), namely that the concentration of the substrate-bound enzyme ([ES]) changes much more slowly than those of the product ([P]) and substrate ([S]). source: http://en.wikipedia.org/wiki/Michaelis-Menten_equation

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <apply> <and/> <apply> <eq/> <ci>v</ci> <apply> <divide/> <apply> <times/> <ci>d</ci> <list> <ci>P</ci> </list> </apply> <apply> <times/> <ci>d</ci> <ci>t</ci> </apply> </apply> </apply> <apply> <eq/> <apply> <divide/> <apply> <times/> <ci>d</ci> <list> <ci>P</ci> </list> </apply> <apply> <times/> <ci>d</ci> <ci>t</ci> </apply> </apply> <apply> <divide/> <apply> <times/> <ci> <msub> <mi>V</mi> <mi>max</mi> </msub> </ci> <list> <ci>S</ci> </list> </apply> <apply> <plus/> <ci> <msub> <mi>K</mi> <mi>m</mi> </msub> </ci> <list> <ci>S</ci> </list> </apply> </apply> </apply> </apply> </math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mo>&InvisibleTimes;</mo> <mfenced close="]" open="["> <mi>P</mi> </mfenced> </mrow> <mrow> <mi>d</mi> <mo>&InvisibleTimes;</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mi>max</mi> </msub> <mo>&InvisibleTimes;</mo> <mfenced close="]" open="["> <mi>S</mi> </mfenced> </mrow> <mrow> <msub> <mi>K</mi> <mi>m</mi> </msub> <mo>+</mo> <mfenced close="]" open="["> <mi>S</mi> </mfenced> </mrow> </mfrac> </mrow> </math>

v = \frac{d [P]}{d t} = \frac{ V_\max {[S]}}{K_m + [S]}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mfenced close="]" open="["> <mi>P</mi> </mfenced> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mi>max</mi> </msub> <mfenced close="]" open="["> <mi>S</mi> </mfenced> </mrow> <mrow> <msub> <mi>K</mi> <mi>m</mi> </msub> <mo>+</mo> <mfenced close="]" open="["> <mi>S</mi> </mfenced> </mrow> </mfrac> </mrow> <annotation encoding="SnuggleTeX">\[ v = \frac{d [P]}{d t} = \frac{ V_\max {[S]}}{K_m + [S]} \]</annotation> </semantics> </math>