Enzyme Kinetics
...he "velocity" of the reaction and is symbolized by v. By means of the above mechanism and ignoring the back reaction, the velocity can then be stated by the equation below: The next manipulation to proceed is to find an expression for ES in terms of enzyme and substrate concentrations. This can be done by using a “steady state approximation” for the complex ES. This assumes that the net rate of change of ES is very small when compared to the rates of formation and destruction of ES. From this approximation the following statement is true: = 0 For laboratory purposes, an expression for [ES] in terms of the concentration of S and E needs to be found, since these are the species that we can "control" in the laboratory. We can then substitute this expression into our velocity equation and thus obtain an expression for the (measured) velocity of reaction in terms of the substrate and enzyme concentrations. In order to do this the differential equation must be solved. To solve the above differential equation two assumptions must be made. The first assumption to be looked at is termed the “equilibrium assumption” (described in detail later). This is the assumption that k-1 >> k2 . If this is true, then the reversible step in the mechanism does achieve equilibrium and we can write the law of chemical equilibrium for this reversible step as follows: Ks represents the dissociation constant, and when the above equation is solved for [ES] in terms of [E] and [S], we get: Substituting this into our velocity expression, we arrive at: The enzyme-substrate (ES) complex is known as the Michaelis complex (dept.physics.upenn.edu). The second assumption made is the “steady state assumption”. This is the assumption that [S] » [E], or that substrate concentration is in much greater excess then enzyme concentration. Except for initial build-up of [ES] at the beginning of the reaction (a few milliseconds) the [ES] remains approximately constant until substrate is nearly all used up. Hence, d[ES]/dt is approximately zero, or the intermediate doesn’t change with respect to time. Below is an example plot depicting this (dept.physics.upenn.edu): Hence, = 0 All of the concentrations above are "instaneous" concentrations. One can then solve the above expression for [ES] as a function of [E] and [S]. The total enzyme concentration [E]T which is the initial amount of enzyme introduced to the system, can then be solved for. From conservation of mass (at constant volume), it is easy to see that: The expression [ES] = [E]T - [ES] is then substituted into our steady-state expression above to yield: Then solving for [ES] gives: Multiplying the right-hand side of the above equation by "1" in the form of k-1 /k1gives: The ratio of (k-1 + k2)/ k1 is known as the Michaelis Constant shown below: With both of the above assumptions made, the rate constants can then be determined. The above expression is even a better approximation to the measured velocity of reaction (v) if v is measured "early" in the reaction, or after [ES] reaches its constant value but before the production of P becomes reversible. Thus, the above expression is commonly written in terms of the initial velocity (v0) as: The use of the initial velocity (v0) usually defined as the measured velocity before more than about 10 percent of the substrate (S) has been converted to product (P) minimizes complicating factors such as the reversible reaction mentioned above as well as inhibition of the enzyme by product and progressive inactivation of the enzyme. As the [S] becomes very large, the velocity of the reaction will not increase indefinitely, but for a fixed amount of [E]T will reach a limiting value termed Vmax, the maximal velocity (dept.physics.upenn.edu). . Substituting Vmax for k2[E]T in the above expression Michaelis-Menten equation: k2 in the above equation is termed the turnover number, or number of sucrose molecules hydrolyzed per second per molecule of enzyme when the enzyme is saturated with substrate (Shoemaker). To properly measure a k2[E]T and a KM value two plots are common plots are used as to give a linear interpretation. Fist a plot of 1/v0 vs. 1/[S0] is made and the intercept deter...