Programming cell growth into different cluster shapes using diffusible signals

Recent advances in genetic engineering technologies has made it possible to construct artificial genetic circuits and use them to control how cells respond to their surroundings. This has been used to generate spatial patterns of differential gene expression. In addition to the spatial arrangement of different cell types, another important aspect of spatial structure lies in the overall shape of the group of cells. However, the question of how cells can be programmed, and how complex the rules need to be, to achieve a desired tissue morphology has received less attention. In this paper, we attempt to address these questions by developing a mathematical model to study how cells can use diffusion-mediated local rules to grow into clusters with different shapes. Within our model, cells are allowed to secrete diffusible chemicals which can either directly regulate the growth rate of cells ( growth regulator), or indirectly affect growth by changing the secretion rate or the effect of other growth regulators. We find that (1) a single growth inhibitor can be used to grow a rod-like structure, (2) multiple growth regulators are required to grow multiple protrusions, and (3) the length and shape of each protrusion can be controlled using growth-threshold regulators. Based on these regulatory schemes, we also postulate how the range of achievable structures scales with the number of signals: (A) the maximum possible number of protrusions increases exponentially with the number of growth inhibitors involved, and (B) to control the growth of each set of protrusions, it is necessary to have an independent threshold regulator. Together, these experimentally-testable findings illustrate how our approach can be used to guide the design of regulatory circuits for achieving a desired target structure.