Assignment: Introduction to Biomolecular Engineering

1. The Michaelis Menten equation quantifies the relationship between reaction rate r, substrate concentration [S], total enzyme concentration [E]_tot, the turnover rate k_cat, and the Michaelis Menten constant K_m. This equation is:

r = (k_cat [E]_tot [S]) / (K_m + [S])

A member of your team has performed a series of experiments to measure enzyme reaction rates. They have added 10 µM of hexokinase and excess ATP to varying concentrations of D-glucose and measured the resulting production rate of D-glucose 6- phosphate. The data from these experiments is listed below. Using this experimental data, determine the k_cat of hexokinase and the K_m of glucose in this reaction. You may use either graphing techniques (e.g. a Line-Weaver Burke plot) or non-linear regression. You must show all work, including any graphs, calculations, or necessary code.

2. Consider an enzyme that catalyzes the conversion of a single substrate, S, to a product, P, while also being inhibited by an inhibitor I. The inhibitor binds to either the enzyme (E) or the enzyme-substrate complex (ES). In the first case, the inhibitor forms an enzyme-inhibitor complex (EI). In the second case, the inhibitor forms an enzyme-substrate-inhibitor complex (ESI). This is called non-competitive inhibition.

Derive an expression that relates the enzyme-catalyzed reaction rate r to the substrate concentration [S], the inhibitor concentration [I], the total enzyme concentration [E]_tot, the enzyme turnover number k_cat, the substrate's Michaelis Menten constant K_m, and the inhibitor's equilibrium disassociation constant, K_i. The K_i has units of mM. You must also define the kinetic constants that appear in your definition of K_m and K_i.

A. Begin your derivation by writing down all chemical reactions occurring in the system.

B. Then employ mass action kinetics to derive a system of ordinary differential equations that describes how the concentrations of each molecular species change over time.

C. Write down a mole conservation equation for the free enzyme and enzyme complexes.

D. Then solve for the enzyme complex concentrations by assuming that their concentrations have reached steady-state.

E. Finally, employ these expressions to determine the reaction rate r, which is equal to the production rate of the product species, in terms of the substrate concentration [S], the inhibitor concentration [I], the total enzyme concentration [E]_tot, the enzyme turnover number k_cat, the substrate's Michaelis Menten constant K_m, and the inhibitor's equilibrium disassociation constant, K_i.