Why doesn’t the amount of ligand needed to reach a fraction-bound value of 0.5 increase as the starting protein concentration increases?

In a Kd binding curve of fraction-bound on the y-axis and ligand concentration on the x-axis, it’s known that the Kd is equal to the ligand concentration [L] needed for [PL]/[Ptotal] =0.5. However, I’m confused as to how the ligand concentration needed to result in that fraction-bound value does not increase as you up the starting concentration of protein in solution. Shouldn’t the amount of ligand needed to bind more protein be larger as well? How can you put in the same concentration of ligand and still get an equal fraction-bound value even if you doubled the starting concentration of your protein in solution?

• The assumption is that Ptot << Ltot, so that the change in L concentration due to binding to the protein is negligible and [L]free ~ [L]tot. That remains true even if we double the protein concentration. If [P] gets high enough that the assumption does not hold, then we would need to plot using the concentration of free L rather than total L because they would not be approx equal Nov 9 '21 at 13:15
• I see, thank you. That clarified it :). Nov 15 '21 at 20:08