a drug is diffusing from the surface of a spherical pill through a radial shell with thickness d1. the radius of the pill is r1. the drug concentration at the pill surface is c1 and the concentration outside the shell is c2. the diffusion coefficient of the solute is D. from first principles, ficks laws, derive a steady state expresssion for the drug concentration c(r) and flux J(r) inside the shell

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Proteins and electrolytes can be transported into cells in two ways: either through diffusion through the plasma membrane, or facilitated diffusion through a channel. Some proteins or electrolytes can be transported through both mechanisms. We can make a simplified model of this by modeling two membranes in parallel with each other, and assuming 1D diffusion through both. Assume that protein A can be transported through the membrane in either method (i.e., direct diffusion or facilitated diffusion) (NOTE: this picture doesnât really describe the system well. Imagine the red circles can move through the membrane AND/OR through the channel). Outside of the cell, the concentration of protein A is 100 nM. Inside the cell, the concentration is 10 nM. Assume the thickness of the plasma membrane is 10 nm. The diffusion constant for direct diffusion of Protein A in the membrane is 1x10^-10 cm2 /s, and the diffusion constant for facilitated diffusion is 5x10^-10 cm2 /s. The partition coefficient Î¦ for Protein A in the plasma membrane is 0.01, and the partition coefficient Î¦ for Protein A in the channel is 0.05.

A. Assume that initially the channel is closed, so Protein A can only get into the cell via direct diffusion through the membrane. Calculate the steady-state flux of Protein A across the membrane. Assume 1-D diffusion, steady state diffusion, and no convection.

B. Assume that the channel is now open, and acting in parallel to direct diffusion. Assume 1-D diffusion through both the membrane and the channel, assume steady state diffusion, and assume no convection.

C. Determine a general expression for the effective diffusion coefficient through the system in Part B, and show how these two pathways behave like resistors in parallel.

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