Read the Manfrin et al$.,$ paper, and consider the opposing gradients of BMP$4$ and NOGGIN. Source: Manfrin, Andrea, et al$.$ "Engineered signaling centers for the spatially controlled patterning of human pluripotent stem cells." Nature methods $16\mathrm{.}7(2019)$: $640-648.$

a$.$ Define the boundary conditions for each molecule, BMP$4$ and NOGGIN. Assume the molecule is washed away at the outlet.

b$.$ Look up the molecular weights of each molecule and convert it to a hydrodynamic radius $($www$.$

fluidic.com$/$resources$/$Toolkithydrodynamic$-$radius$-$Converter$/).$ Treat each as a model sphere, and calculate the diffusion coefficients at $37C.$

c$.$ Assuming a rectangular chamber, derive the concentration profiles for each molecule. Assume steady, $1$D flow with no reactions. What is the approximate concentration of each molecule at the peak of cell response $($refer to Figure $6$D$).$