Internalization of bound receptors is a way of downregulating the response to extracellular ligand stimulation. Consider a cell with a surface receptor concentration, R0, when no ligand is present. When a ligand, L, is added at t=0, it binds reversibly with cell surface receptors to form receptorligand complexes, C. The complexes are internalized at a rate proportional (internalization rate constant, keCeC ) to the surface concentration of bound receptors (with no recycling). Assume that the ligand is present in concentrations much greater than the total receptor concentration such that the concentration of ligand may be treated as a constant. a) Draw a diagram of the /reaction scheme that represents this system and write the differential equation for the surface concentrations of free and bound receptors. b) Assume that the receptor-ligand binding reaction is very fast compared to the internalization rate (rapid equilibrium assumption). Solve for the surface concentration of bound receptors as a function of time. (Hint: use the rapid equilibrium condition to find the initial conditions for your approximate solution for C.) c) Simulate the complete system in part (a) (i.e., without the rapid equilibrium assumption in part (b)) using the following parameter values kf=1061/(Ms) forward reaction rate constant for ligand binding receptor ks=0.01(1/s) reverse reaction rate constant for ligand binding receptor kC=0.006(1/s) internalization rate constant R0=100,000 (\#/cell) total receptor concentration L=107M (Hint: make the total time for the simulation 500s) d) Compare the simulation with your result from (b) using the parameters in (c). Was the rapid equilibrium assumption for the receptor-ligand binding step a good assumption