Michaelis-Menten equation

The usual equation to describe the conversion of a substrate in an enzymatic reaction. Let $ S ( t ) $ be the concentration of some substrate which is converted by an enzyme $ E $ into a product $ P $. The reaction rate is proportional to $ S ( t ) $ for small values of $ S ( t ) $, but there is a maximum rate, which is not surpassed even for large $ S ( t ) $. These observations can be expressed by the equation

$$ { \frac{d S ( t ) }{dt } } = - { \frac{k _ {a} \cdot S ( t ) }{K + S ( t ) } } . $$

In this case the first reaction parameter, $ k _ {a} $, describes the maximal conversion speed, and the second reaction parameter, $ K $, is equal to the substrate concentration at which the reaction speed is exactly $ { {k _ {a} } / 2 } $. For small values of $ S ( t ) $, the reaction rate is $ k _ {a} \cdot K ^ {- 1 } \cdot S ( t ) $.

The equation can be derived from the chemical equilibrium reactions between the substrate $ S $ and the enzyme $ E $, which combine to a compound $ ES $. This compound is rearranged in an equilibrium reaction into a compound $ EP $, which dissociates into the enzyme $ E $ and product $ P $. In a formula:

$$ S + E \rightleftarrows _ \beta ^ \alpha SE \rightleftarrows _ \mu ^ \gamma EP \rightleftarrows _ \sigma ^ \lambda E + P. $$

In this reaction, the step from $ SE $ to $ EP $ is usually far slower than the other reactions, and if the reaction from $ EP $ to $ E + P $ is irreversible, or if the product $ P $ is removed by some transport mechanism, then the two reaction constants $ \mu $ and $ \sigma $ can be discarded, and the rearrangement coefficient $ \gamma $ determines a simplified reaction equation:

$$ S + E \rightleftarrows _ \beta ^ \alpha SE { \mathop \rightarrow \limits ^ \gamma } E + P. $$

Putting $ B = SE $, the following system of differential equations emerges from the reactions:

$$ { \frac{dS ( t ) }{dt } } = \beta \cdot B - \alpha \cdot S ( t ) \cdot E, $$

$$ { \frac{dE }{dt } } = ( \beta + \gamma ) \cdot B - \alpha \cdot S ( t ) \cdot E, $$

$$ { \frac{dB }{dt } } = \alpha \cdot S ( t ) \cdot E + ( \beta + \gamma ) \cdot B. $$

These equations imply that $ E + B = E _ {0} $ is a constant, representing the maximal amount of either $ E $ or $ B $. Usually, $ S ( t ) $ is large with respect to fluctuations in $ E _ {0} $ and so the reaction enters a steady state, in which the concentrations of $ E $ and $ B $ remain almost constant over a large time interval. In that case the proportion $ E \simeq B $ equals $ \beta + \gamma \simeq \alpha \cdot S ( t ) $, which yields

$$ B = { \frac{\alpha \cdot S ( t ) \cdot E _ {0} }{\gamma + \beta + \alpha \cdot S ( t ) } } , $$

and

$$ { \frac{dS ( t ) }{dt } } = \beta \cdot B - \alpha \cdot S ( t ) \cdot E = $$

$$ = - \gamma \cdot B = - { \frac{\alpha \cdot \gamma \cdot E _ {0} \cdot S ( t ) }{\gamma + \beta + \alpha \cdot S ( t ) } } . $$

The values of $ K $ and $ k _ {a} $ can be inferred from the differential equation. This gives:

$$ K = { \frac{\gamma + \beta } \alpha } , k _ {a} = \gamma \cdot E _ {0} , $$

and so

$$ { \frac{dS ( t ) }{dt } } = - { \frac{k _ {a} \cdot S ( t ) }{K + S ( t ) } } . $$

The values can be fitted from observations of $ S ( t ) $, but it is impossible to also find the parameters $ \alpha $, $ \beta $, $ \gamma $, and $ E _ {0} $ from the curve of $ S ( t ) $.

The equation was derived by L. Michaelis and M.L. Menten [a1].

References

[a1]	L. Michaelis, M.L. Menten, "Die Kinetik der Invertinwirkung" Biochem. Zeitschrift , 2 (1913) pp. 333–369