Enzymes are large proteins that biological systems use to increase the rate at which reactions occur. For example, food is usually composed of large molecules that are hard to digest; enzymes break down the large molecules into small nutrients as part of the digestive process. One such enzyme is amylase, contained in human saliva. It is commonly known that if you place a piece of uncooked pasta in your mouth its taste will change from paper-like to sweet as amylase breaks down the carbohydrates into sugars. Enzyme breakdown is often expressed by the following relation:
In this expression a substrate (S) interacts with an enzyme (E) to form a combined product (C) at a rate k1. The intermediate compound is reversible and gets disassociated at a rate k-1. Simultaneously some of the compound is transformed into the final product (P) at a rate k2. The kinetics describing this reaction are known as the Michaelis-Menten equations and consist of four nonlinear differential equations. However, under some conditions these equations can be simplified. Let E0 and S0 be the initial concentrations of enzyme and substrate, respectively. It is generally accepted that under some energetic conditions or when the enzyme concentration is very big (E0 >> S0), the kinetics for this reaction are given by
where the following constant terms are used (Schnell, 2004):
a. Assuming the initial conditions for the reaction are S(0) = S0; E(0) = E0; C(0) = P(0) = 0, find the Laplace transform expressions for S, C, and P: ℒ{S}; ℒ{C}, and ℒ{P}, respectively.
b. Use the final theorem to find S(∞); C(∞), and P(∞).
