Michaelis-Menten kinetics

Michaelis-Menten kinetics describes the kinetics of 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).

Determination of constants

To determine the maximum rate of an enzyme mediated reaction, the substrate concentration (^*) 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.

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 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) 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.

Note: K_M can only be used to describe an enzyme's affinity for substrate when product formation is rate-limiting, i.e., when k₂ << k_-1 and K_M becomes k_-1/k₁. Often, k₂ >> k_-1, or k₂ 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 steady state approximation:

$\frac{d = k_1$ ^*" target="_blank" >- k_{-1}^* = 0

$= \frac{k_1$ ^*}{k_{-1} + k_2}

Let's define:

$K_m = \frac{k_{-1} + k_2}{k_1}$

Therefore:

$= \frac{$ ^*}{K_m} (1)

The rate (or velocity) of the reaction is:

$\frac{d = k_2[ES$ (2)

The total concentration of enzyme is:

$=$ ^*

Hence:

$=$ ^* (3)

Substituting (3) into (1) gives:

$= \frac{($ ^*)" target="_blank" >[S}{K_m}

Rearranging gives:

$\frac{K_m}{$ ^*" target="_blank" >- [ES

$+ \frac{K_m}{$ ^*

$=$ ^*}} (4)

Substituting (4) in (2) and multiplying numerator and denominator by ^*:

$\frac{d = k_2$ ^*}{K_m" target="_blank" >+ ^*}{K_m" target="_blank" >+ [S}

This equation may be analyzed experimentally with a Lineweaver-Burk 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^*/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^* 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, ^*)" target="_blank" >approaches 1. Therefore, the rate of product formation is equal to k₂[E₀ in this case.

When equals K_m, ^*)" target="_blank" >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 ^*.

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

Enzyme kinetics Chemical kinetics Ordinary differential equations