Nutrient Shielding in Clusters of Cells

Cellular nutrient consumption is influenced by both the nutrient uptake kinetics of an individual cell and the cells' spatial arrangement. Large cell clusters or colonies have inhibited growth at the cluster's center due to the shielding of nutrients by the cells closer to the surface. We develop an effective medium theory that predicts a thickness $\ell$ of the outer shell of cells in the cluster that receives enough nutrient to grow. The cells are treated as partially absorbing identical spherical nutrient sinks, and we identify a dimensionless parameter $\nu$ that characterizes the absorption strength of each cell. The parameter $\nu$ can vary over many orders of magnitude between different cell types, ranging from bacteria and yeast to human tissue. The thickness $\ell$ decreases with increasing $\nu$, increasing cell volume fraction $\phi$, and decreasing ambient nutrient concentration $\psi_{\infty}$. The theoretical results are compared with numerical simulations and experiments. In the latter studies, colonies of budding yeast, \textit{Saccharomyces cerevisiae}, are grown on glucose media and imaged under a confocal microscope. We measure the growth inside the colonies via a fluorescent protein reporter and compare the experimental and theoretical results for the thickness $\ell$.