Michaelis-Menten kinetics (idea)

Michaelis-Menten Kinetics is a model for describing enzymatic reactions, and obtaining an expression for the reaction rate.

An enzyme is a type of protein that catalyzes chemical reactions in living organisms. These chemical reactions are usually highly specific, which means that the enzyme catalyzes only a few or one reactants into a few or one product. Thus, enzymes are specific to one biochemical reaction, and facilitate reactions to occur at low temperature (i.e. body temperature). An example of an enzyme is urease, that converts urea into ammonia and carbon dioxide. In more general terms: an enzyme converts a substrate to one or more products.

Enzymatic reactions can be quite difficult in their dynamics; the actual chemical conversion can occur in many reaction steps. However, in many enzymatic reactions where the enzyme and substrate are (water) soluble, the mechanism of the reaction can be simplified to a few elementary reactions:

1. The enzyme (E) reacts reversibly to the substrate (S) to form an enzyme-substrate complex (ES):

   E + S -> ES

2. The complex can decompose back to the enzyme and substrate:

   ES -> E + S

3. The complex can decompose irreversibly to form a product (P), and free the enzyme:

   ES -> P + E

Thus, the overall reaction is that substrate (S) is converted to product (P). The enzyme participates in the reaction, but is recovered in reaction 3. It is then ready to be used for another catalytic cycle, which is usually called a turnover.

Substrate is converted to complex in reaction 1. This is an elementary reaction, which means that it is a linear function of both the enzyme and substrate concentrations:

     r1, S = k1[E][S]

where k denotes the rate constant for this reaction step. However, substrate is also generated by reaction 2.

     r2, S = k2[ES]

Thus, the net rate of disappearance, given by -r_s) (negative because it is a reactant) is a function of the rates of reactions 1 and 2:

     -rS = k1[E][S] - k2[ES]                   (1)

Similarly, the rate of formation of the enzyme-substrate complex (ES) cam be written as a function of the rates of the three elementary steps:

     rES = k1[E][S] - k2[ES] - k3[ES]

Now an important assumption is made: it is assumed that reaction 3 is the limiting reaction step. Thus, the enzyme-substrate complex is formed instantaneously, and its concentration doesn't change during the course of the reaction. This assumption is called the Quasi-Steady State Approximation (QSSA) or Pseudo-Steady State Hypothesis (PSSH). Because the concentration of Enzyme-Substrate complex is assumed constant, its rate of formation is equal to zero:

     0 = k1[E][S] - k2[ES] - k3[ES]            (2)

The enzyme is present as free enzyme, or as complex. However since the enzyme is not consumed, its total concentration remains constant:

     [ET] = [E] + [ES]

     [E] = [ET] - [ES]                         (3)

Substituting (3) into (2) and solving for [ES] yields:

     [ES] = k1[ET][S] / (k1[S] + k2 + k3)       (4)

Combining (1) and (2) yields:

     -rS = k3[ES]                               (5)

Substituting (4) into (5) yields:

     -rS = k1k3[ET][S] / (k1[S] + k2 + k3)

Now we replace apply the following substitutions:

     Km = (k3 + k2) / k1

     Vmax = k3[ET]

and we obtain the common form of the Michaelis-Menten Equation:

     -rS = Vmax[S] / (Km + [S])

where V_max is the maximum rate of reaction for a given total enzyme conversion, and K_m is the Michaelis constant. It can be shown that the Michaelis constant is equal to the substrate concentration at which the reaction rate is equal to one-half of V_max.

The Michaelis-Menten equation is important because it shows that the rate of substrate conversion can be described by only two kinetic parameters, and the substrate concentration.

At low substrate concentration ([S] << K_m), the rate becomes proportional to the substrate concentration:

     -rS ≅ Vmax[S] / Km

At high substrate concentration, ([S] >> K_m), the rate becomes independent of the substrate concentration:

     -rS ≅ Vmax

The values for V_max, and K_m need to be determined experimentally. This is typically done by measuring the change in substrate concentration over time, and calculating the rate as a function of substrate concentration. Usually, the Michaelis-Menten Equation is inverted:

     1/-rS = (1/Vmax) + (Km/Vmax[S])

A plot of the reciprocal reaction rate versus the reciprocal substrate concentration yields a straight line, with intercept 1/V_max and slope K_m/V_max. This plot is called a double reciprocal plot, or a Lineweaver-Burk plot.