Question 2: Enzyme Kinetics
Enzymes are catalysts that help convert molecules that we will call substrates into other
molecules that we will products. They themselves are not changed by the reaction. Within
cells, enzymes are typically proteins. They can speed up biological reactions, sometimes by
up to millions of times. They are also regulated by a very complex set of positive and
negative feedback systems. Computational biologists are painstakingly mapping out this
complex set of reactions. In this problem, we will model and simulate a simplified enzyme
An enzyme E converts the substrate S into the product P through a two-step process. First,
E forms a complex with S to form an intermediate species ES in a reversible manner at the
forward rate k1 and reverse rate k2. The intermediate ES then breaks down into the product
Pat a rate k3, thereby releasing E. Schematically, we write
8.1. Using the law of mass action, write down four equations for the rate of changes of the
four species, E, S, ES, and P.
8.2. Write a code to numerically solve these four equations using the fourth-order Runge-
Kutta method. For this exercise, assume that the initial concentration of E is 1 uM, the initial
concentration of S is 10 uM, and the initial concentrations of ES and Pare both 0. The rate
constants are: K1=100/uM/min, K2=600/min, k3=150/min.
8.3. We define the velocity, V, of the enzymatic reaction to be the rate of change of the
product P. Plot the velocity V as a function of the concentration of the substrate S. You
should find that. when the concentrations of S are small. the velocity V increases
approximately linearly. At large concentrations of S, however, the velocity V saturates to a
maximum value, Vm. Find this value Vm from your plot.