Problem $3$: Morphogen Diffusion, Adapted from Deen $3-13(30$ PoINTs$)$

A morphogen is a ligand whose binding to a receptor triggers the differentiation of a target cell in a multi$-$

cellular organism. The variation in its concentration as a function of distance from where it is secreted helps

to determine how structures are formed during development. The model of morphogen secretion$,$ diffusion,

and binding in this problem is a simplified version of one applied by Goentoro et al$.(2006)$ to fly oogenesis.

Assume that that two layers of cells of indefinite extent are separated by a fluid space of height $h,$ as shown in

Figure $2.$ Suppose that the morphogen is secreted from a circular area of radius $a$ in the lower layer $(z=0)$

at a constant flux $N_0.$ Elsewhere on the lower surface, the flux is zero. Assume that binding to receptors on

target cells at the upper surface $(z=h)$ follows first$-$order kinetics. The flux there is $kC(r,h),$ where $k$ is

the rate constant and $C(r,z)$ is the local concentration. Binding may occur anywhere on the upper surface.

$($a$)$ State the governing equations and boundary conditions for $C(r,z).($Hint: A second order PDE requires

two boundary conditions per coordinate.$)$

$($b$)$ Using order of magnitude analysis, under what conditions is $\frac{C}{C}1,$ where $C$ is the variation in

$C$ across the height of the liquid?

$($c$)$ Assuming the conditions in $($b$)$ are true, the model can be lumped in the $z$ direction and the PDE

for $C(r,z)$ can be converted to an ODE for the vertically$-$averaged concentration $\frac{?}{\text{b}\text{a}\text{r}}(C)(r)=\frac{1}{h}{\displaystyle \underset{0}{\overset{h}{}}}dzC(r,z).$

Determine the governing equation and boundary conditions for $\frac{?}{\text{b}\text{a}\text{r}}(C)(r).($Hint: Take the vertical average of

the entire PDE.$)$

Figure $2$: Model for morphogen diffusion. Molecules secreted from a circular area on the bottom surface may bind

anywhere on the upper surface.