How to Color a French Flag

Abstract

In the French flag problem, initially uncolored cells on a grid must differentiate to become blue, white or red. The goal is for the cells to color the grid as a French flag, i.e., a three-colored triband, in a distributed manner. To solve a generalized version of the problem in a distributed computational setting, we consider two models: a biologically-inspired version relying on morphogens (gradients of chemicals acting as signals) and a more abstract version based on reliable message passing between cellular agents. We show that both models easily achieve a French ribbon - a French flag in the 1D case. However, extending the ribbon to the 2D flag in the concentration model is somewhat difficult unless each agent has additional positional information. Assuming that cells are are identical, it is impossible to achieve a French flag or even a close approximation. In contrast, using a message-based approach in the 2D case only requires assuming that agents can be represented as constant size state machines. We hope our insights may lay some groundwork for what kind of message passing abstractions or guarantees are useful in analogy to cells communicating at long and short distances to solve patterning problems. In addition, we hope that our models and findings may be of interest in the design of nano-robots.