In developing some of the elementary principles of the kinetics of enzyme reactions, we shall discuss an enzymatic reaction that has been suggested by Levine and LaCourse as part of a system that would reduce the size of an artificial kidney.8 The desired result is the production of an artificial kidney that could be worn by the patient and would incorporate a replaceable unit for the elimination of the body’s nitrogenous waste products, such as uric acid and creatinine. In the microencapsulation scheme proposed by Levine and LaCourse, the enzyme urease would be used in the removal of urea from the bloodstream. Here, the catalytic action of urease would cause urea to decompose into ammonia and carbon dioxide. The mechanism of the reaction is believed to proceed by the following sequence of elementary reactions:
Symbolically, the overall reaction is written as
We see that some of the enzyme added to the solution binds to the urea, and some of the enzyme remains unbound. Although we can easily measure the total concentration of enzyme, (Et), it is difficult to measure either the concentration of free enzyme, (E), or the concentration of the bound enzyme (E · S).
Letting E, S, W, E · S, and P represent the enzyme, substrate, water, the enzyme–substrate complex, and the reaction products, respectively, we can write Reactions (9-13), (9-14), and (9-15) symbolically in the forms
Here P = 2NH3 + CO2.
The corresponding rate laws for Reactions (9-16), (9-17), and (9-18) are
where the specific reaction rates are defined with respect to (E · S). The net rate of formation of product, rP, is
For the overall reaction
we know –rS = rP.
This rate law (Equation 9-19) is of not much use to us in making reaction engineering calculations because we cannot measure the concentration of the enzyme substrate complex (E · S). We will use the PSSH to express (E · S) in terms of measured variables.
The net rate of formation of the enzyme-substrate complex is
rE·S = r1E·S + r2E·S + r3E·S
Substituting the rate laws, we obtain
Using the PSSH, rE·S = 0, we can now solve Equation (9-20) for (E · S)
and substitute for (E · S) into [Equation (9-19)]
We still cannot use this rate law because we cannot measure the unbound enzyme concentration (E); however, we can measure the total enzyme concentration, Et.
In the absence of enzyme denaturation, the total concentration of the enzyme in the system, (Et), is constant and equal to the sum of the concentrations of the free or unbounded enzyme, (E), and the enzyme–substrate complex, (E · S):
Substituting for (E · S)
solving for (E)
substituting for (E) in Equation (9-22), the rate law for substrate consumption is
Note: Throughout the following text, Et ≡ (Et) = total concentration of enzyme with typical units such as (kmol/m3) or (g/dm3).
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