  Using agar plates hosting a 2D cell population stimulated with signaling
molecules is crucial for experiments such as gene regulation and drug discovery
in a wide range of biological studies. In this paper, a biophysical model is
proposed that incorporates droplet soaking, diffusion of molecules within agar,
cell growth over an agar surface, and absorption of signaling molecules by
cells. The proposed model describes the channel response and provides valuable
insights for designing experiments more efficiently and accurately. The
molecule release rate due to droplet soaking into agar, which is characterized
and modeled as the source term for the diffusion model, is derived.
Furthermore, cell growth is considered over the surface, which dictates the
dynamics of signaling molecule reactions and leads to a variable boundary
condition. As a case study, genetically-modified $E.\,coli$ bacteria are spread
over the surface of agar and Isopropyl-beta-D thiogalactopyranoside (IPTG) is
considered as a signaling molecule. IPTG droplets are dropped onto the
bacteria-covered agar surface. The parameters for the IPTG molecule release
rate as a diffusion source into the agar are estimated from this experiment.
Then, a particle-based simulator is used to obtain the spatio-temporal profile
of the signaling molecules received by the surface bacteria. The results
indicate that the number of molecules reacting with or absorbed by bacteria at
different locations on the surface could be widely different, which highlights
the importance of taking this variation into account for biological inferences.
