3.2. Enzyme Kinetics

3.2.1. Kinetics of Enzyme-Catalyzed Reactions

Reaction Velocity. The time course of an enzymatic reaction permits one to deduce the substrate affinity, the catalytic mechanism in the active center, and the efficiency of the enzyme (maximum rate, turnover number). The rate of an enzyme-catalyzed single reactant reaction depends on the concentration of substrate and product, respectively. The velocity of the reaction V is:

where first term is the rate of dissapearance of substrate S and second term is the rate of appearance of product P (both S and P are in concentration).

Behavior of Initial Rates. The initial rate (Vo) is determined by extrapolating the slope of the time course of the substrate or product concentration to time zero (Fig. 3.5). The dependence of Vo on the substrate concentration, S (at constant enzyme concentration), is shown in Fig. 3.6. It reflects the typical substrate saturation. At first, Vo increases proportionally to the amount of substrate. Upon further enhancement of substrate concentration Vo levels off. The curve asymptotically approaches a maximum value, Vmax. When this plateau is reached, a change of S does not lead to a measurable change of Vo: the enzyme is saturated by substrate and has thus reached the limit of its efficiency.

Micahaelis-Menten Kinetics. These kinetics result from the fast and reversible formation of an enzyme-substrate complex, ES, which dissociates in a second, slower reaction under liberation of the product, P (Fig. 3.7):

E + S ES E + P (3)

Because the second reaction is rate-limiting, at very high substrate concentration almost all enzyme is present as enzyme-substrate com-plex. Under these conditions a steady state is reached in which the enzyme is steadily saturated by substrate and the initial rate is at a maximum (Vmax). This relation between substrate concentration and reaction rate may be described by the Michaelis-Menten equation:

where KM is the Michaelis constant of the enzyme for the given sub-strate. KM may also be described by:

Meaning of KM The relevance of KM becomes evident at S = KM. Then Vo = Vmax /2, i.e., KM is the substrate concentration at which the reaction rate is half maximum (Fig. 3.6). The KM value characterizes the affinity between the substrate and the enzyme. At known KM and Vmax, Vo can be calculated for each value of substrate concentration. A low KM value reflects high affinity. At substrate concentrations S << KM, the reaction rate is directly proportional to the substrate concentration (first order reaction); at high substrate concentration (S >> KM) the reaction is zero order and is no longer dependent on the substrate concentration but only on the enzyme activity.

Fig. 3.5. Determination of initial rates at different substrate concentrations.

Fig. 3.6. A plot of Vo vs. substrate concentration S.

Enzyme Kinetics

Rate of reaction:

Net formation rate of ES:

During reaction, the # of active sites occupied by S is constant:

Total enzyme sites = occupied sites + free sites:

Therefore,

Fig. 3.17. Enzyme kinetics.

Lineweaver-Burk Plot. To calculate KM and Vmax (and inhibitor constants) it is advantageous to transform the Michaelis-Menten relation so as to obtain linear relationships between S and Vo that can be evaluated graphically. An example is the Lineweaver-Burk equation, containing the reciprocal values of Vo and S:

An example of Lineweaver-Burk plot is shown in Fig. 3.8.