Some patients with prostate cancer have elevated levels of a protein called prostate-specific antigen (PSA) in the blood. A concentration above 2 ng/mL often leads to further evaluation, although many factors, including PSA dynamics, must be considered. You aim to develop a diagnostic test to quantify PSA concentration from blood samples. You have a ligand that binds to PSA with 0.5 nM affinity. The association rate constant of this ligand for PSA is 3 10^5 M^1 s^1. You are characterizing a pilot version of your blood test using a patients blood sample. You adsorb 10 femtomoles of ligand inside a test tube. 500 L of blood is added to the tube, and a long wait allows association and dissociation of the PSA-ligand complex to near equilibrium. You rinse the tube to remove unbound PSA, while PSA- ligand complex remains adsorbed. An enzymatic detection scheme enables you to quantify that 500 million PSA molecules are present in the tube.

a. Is this patients PSA level above the 2 ng/mL limit? b. What fraction of ligand binding sites would be complexed if the blood concentration were 2 ng/mL? c. You needed to make a quick estimate of the bound ligand fraction at the normal limit, so you assumed that the PSA was present in large excess relative to ligand. How much does your estimate of the bound fraction deviate from the true value?

d. What should be done to reduce the error to 1%?