Michaelis-Menten kinetics

Overview
Michaelis-Menten kinetics describes the kinetics of many enzymes. It is named after 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), and when the enzyme is not allosteric.

History
The modern relationship between substrate and enzyme concentration was proposed in 1903 by Victor Henri. . A microscopic interpretation was thereafter proposed in 1913 by Leonor Michaelis and Maud Menten, following earlier work by Archibald Vivian Hill. . 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.

Determination of constants
To determine the maximum rate of an enzyme mediated reaction, the substrate concentration ([S]) is increased until a constant rate of product formation is achieved. This is the maximum velocity (Vmax) 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 [S], the enzyme is asymptotically approaching its maximum speed Vmax, but never actually reaching it. Because of that, no [S] for Vmax can be given. Instead, the characteristic value for the enzyme is defined by the substrate concentration at its half-maximum speed (Vmax/2). This KM value is also called the Michaelis-Menten constant.

Michaelis constant 'KM'
Since Vmax cannot be reached at any substrate concentration (because of its asymptotic behaviour, V keeps growing at any [S], 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 (KM) 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 KM 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 KM becomes k-1/k1. Often, k2 >> k-1, or k2 and k-1 are comparable.

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:

$$\frac{d[ES]}{dt} = k_1[E][S] - k_{-1}[ES] - k_2[ES] = 0$$

$$[ES] = \frac{k_1[E][S]}{k_{-1} + k_2}$$

Let's define the Michaelis constant:

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

This simplifies the form of the equation:

$$[ES] = \frac{[E][S]}{K_m}$$ (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:

$$[E_0] = [E] + [ES]$$

$$[E] = [E_0] - [ES]$$ (2)

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

$$[ES] = \frac{([E_0] - [ES]) [S]}{K_m}$$

Rearranging gives:

$$[ES] \frac{K_m}{[S]} = [E_0] - [ES]$$

$$[ES]\left(1 + \frac{K_m}{[S]}\right) = [E_0]$$

$$[ES] = [E_0]\frac{1}{1+\frac{K_m}{[S]}}$$ (3)

The rate (or velocity) of the reaction is:

$$V = \frac{d[P]}{dt} = k_2[ES]$$ (4)

Substituting (3) in (4) and multiplying numerator and denominator by $$[S]$$:

$$\frac{d[P]}{dt} = k_2[E_0]\frac{[S]}{K_m + [S]} = V_{max}\frac{[S]}{K_m + [S]} $$

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[P]/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[E0] a.k.a. Vmax is the maximum velocity or maximum rate. k2 is often called kcat.

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

When [S] equals Km, [S]/(Km + [S]) equals 0.5. In this case, the rate of product formation is half of the maximum rate (1/2 Vmax). By plotting V0 against [S], one can easily determine Vmax and Km. 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 law of mass action derived from the assumptions of free (Fickian) diffusion and 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/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 ). 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 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 . In general physics and chemistry, limited mobility-derived kinetics have been successfully described by the fractal-like kinetics.  R. Kopelman, M.A. Savageau , and S. Schnell pioneered the “fractal enzymology”, which is further developed by other researchers (see, e.g. Feng Xu ).