dont use chat gpt

Synthetic biology is transforming how we engineer cells to make custom products. An exciting

new area in the field is in cell$-$free synthesis whereby the cell wall is removed or an artificial cell

wall is introduced. Here, we will consider a new cell wall for plant cells that is highly permeable

to paclitaxel $($cancer therapeutic$).$ Our hypothesis is that the high permeability will enable

continuous extraction$/$production of the drug at a higher rate.

For this problem, assume that the artificial cell will be $100$m in diameter with a polymer cell

wall of $5$ um$.$ The diffusion coefficient of paclitaxel is $5\mathrm{.}2\backslash$times $10-6$ cm$2/$s through the polymer wall.The cell is well mixed and always producing the secondary metabolite such that the

concentration within the artificial cell remains at $11$ nM$.$ The cell culture media surrounding the

sphere is able to instantly wash away the drug once it makes it to the outside.

a$)$ Draw the system in a simplified diagram. Be sure to include the coordinate $($origin$),$

length scale$($s$),$ and direction of flux.

b$)$ What is the domain over which mass transfer is occurring?

c$)$ Write out the GMB in the correct coordinate system, list assumptions and simplify the

expression to a solvable ODE.

d$)$ What is the initial condition? If it$$s not needed, state why.

e$)$ What are the boundary conditions?

f$)$ Solve for the concentration of paclitaxel within the domain.

g$)$ What is the production rate of one cell?

h$)$ What volume of reactor would you need to produce $1000$ doses per day if the typical dose

is $6$ mg$?$