Problem 1
You have purchased a binding protein against a cell surface protein, and perform a cell surface titration to estimate that Kd = 3 nM. After labeling your cells at 500 nM of the binding protein, you wash and resuspend them in protein-free buffer at room temperature, and leave for lunch. Estimate how many minutes will pass before half of the bound antibody has dissociated from the cells.

Notes/Hints:
i) A titration refers to an experiment where you label one species (cells here) with varying concentrations of the other species. Often the objective is to find the fraction of the first species that is bound by the second species (y) as a function of concentration. This data can then be used to estimate the equilibrium dissociation constant by non-linear regression.
ii) What is the fraction of cell surface protein that is bound to the binding protein under the experimental conditions?
iii) What happens when you wash and re-suspend the cells in protein-free buffer? Will the protein start dissociating? Can you make any assumptions about the likelihood of the dissociated protein binding? For instance, what happens if the buffer volume is reasonably large relative to volume occupied by the cells?
iv) Do you have all the kinetic parameters necessary? Can you make assumptions about the on rate? Can you obtain an estimate for the off rate?

Problem 2
Consider a protein/ligand interaction with Kd =10 pM and kon = 105 M−1s−1. If a ligand titration of [P]o =
10 pM of this protein is performed with [L]o =1, 3, 10, 30, and 100 pM, what time is required to reach 95% of the equilibrium complex concentration for each of the values of [L]o? Calculate the fractional complex formation if equilibration is allowed to proceed for only one hour. Fit a value of Kd to this curve; what is the magnitude of the error in the estimate?

Notes/Hints:
i) A titration refers to an experiment where you label one species (P here) with varying concentrations of the other species (L here). Often the objective is to find the fraction of the first species that is bound by the second species (y) as a function of concentration. This data can then be used to estimate the equilibrium dissociation constant by non-linear regression.
ii) Does the amount of time required to reach equilibrium depend on the concentrations? Do you have all the kinetic parameters required to find the amount of time? Can you use the relationship between the equilibrium constant and the on/off rates to estimate any kinetic parameters required?
iii) You can calculate y at equilibrium for the different values of concentration. Now, you can also calculate the y-actual when you wait for only one hour. Thus you can have a plot of y-actual vs. L0. From this data set (y-actual vs. L), you can estimate a value of Kd by non-linear regression. This is the Kd that is experimentally determined. Compare this with the value provided (10 pM). One of the goals of this problem is to emphasize the effect of kinetics on equilibrium measurements.

Problem 1 You have purchased a binding protein against a cell su...