 Solution

Take $x=0$ at chamber A and $x=L$ at chamber B. At steady state, the diffusion equation with constant positive diffusion coefficient givesThe two Dirichlet boundary conditions give $A=c_0$ and $B=-c_0/L$. ThusThe concentration gradient is constant, $dc/dx=-c_0/L$. By Fick's first law, the diffusive flux is $J=-Ddc/dx=D{c}_0/L$, directed from A towards B. The profile is linear independently of $D$, while the flux and the time needed to approach it depend on $D$.

This result assumes no significant bulk flow, chemoattractant loss to the walls, degradation or cellular uptake. Experimentally, a uniform-concentration control with matched reservoir concentrations tests for directional cell migration caused by flow or apparatus asymmetry; reversing the reservoir concentrations tests whether the directional response follows the chemical gradient. Keeping both reservoirs well mixed and measuring the concentration profile checks the assumed boundary concentrations and whether uptake has distorted it. These controls distinguish the assumed passive diffusion field from other directional cues; the derived profile alone does not establish the mechanism of chemotaxis.