Michaelis-Menten Kinetics (Enzyme Kinetics)

Insert the substrate concentration [S] into the equation v = V_max·[S]/(K_m + [S]). V_max is the maximum rate at substrate saturation, K_m the Michaelis constant, which states at which substrate concentration half the maximum rate is reached. Example: with V_max = 10 µmol/(L·s), K_m = 2 mmol/L and [S] = K_m = 2 mmol/L you get v = 10·0.002/(0.002 + 0.002) = 5 µmol/(L·s), that is exactly half of V_max. At very large [S], far above K_m, v approaches the value V_max. Watch for consistent concentration units for [S] and K_m.

The Michaelis constant K_m is the substrate concentration at which the reaction rate reaches exactly half of the maximum rate V_max. This follows directly from the equation: inserting [S] = K_m gives v = V_max/2. K_m therefore has the unit of a concentration, usually mol/L, and is a measure of the affinity of the enzyme for the substrate. A small K_m means high affinity, the enzyme works efficiently already at low substrate concentration. A large K_m means low affinity, it needs a lot of substrate for high rate. A common mistake is to confuse K_m with the maximum rate; K_m is a concentration, V_max a rate.

Because the enzyme has a limited number of active sites, which are almost constantly occupied at high substrate concentration. This is called saturation. In the equation v = V_max·[S]/(K_m + [S]) the term [S] dominates over K_m at very large [S], so numerator and denominator almost cancel and v tends towards V_max. Further substrate addition can then barely increase the rate, because all enzymes are already working. At low [S], by contrast, the curve is nearly linear, because free active sites are available. This transition from linear rise to a saturation plateau is the characteristic hallmark of enzyme kinetics and distinguishes it from a simple first-order reaction.

The equation assumes a simple enzyme kinetics with a quasi-steady enzyme-substrate complex. It does not apply to allosteric enzymes, whose activity is regulated through binding at other sites and which often show a sigmoidal rather than hyperbolic curve. It also fails under substrate inhibition, where too much substrate slows the reaction. Further limits are multiple substrates, reversible product formation in both directions or very fast reactions where the quasi-steady-state assumption no longer holds. For many classic enzymes with one substrate and without complex regulation, however, the equation describes the kinetics very well and is the standard model of biochemistry.

Directly from the hyperbolic curve V_max and K_m are hard to read, because the rate approaches the maximum only asymptotically. Therefore one linearizes the equation, classically with the Lineweaver-Burk plot, in which 1/v is plotted against 1/[S]. A straight line results, from whose intercepts V_max and K_m can be read: the y-intercept gives 1/V_max, the x-intercept −1/K_m. Alternatively one uses modern nonlinear regression, which fits the parameters directly to the measured points and is more accurate. For this one measures the initial rate at different substrate concentrations. These characteristic values describe an enzyme and allow comparisons between enzymes or the study of inhibitors.