How to Color a French Flag--Biologically Inspired Algorithms for Scale-Invariant Patterning

TL;DR

This paper explores biologically inspired algorithms for the French flag problem, demonstrating that message passing between simple agents can effectively produce complex, scale-invariant patterns in distributed systems, unlike concentration-based models.

Contribution

It introduces a message-passing approach for pattern formation that overcomes limitations of concentration-based models in 2D, using simple state machines for agents.

Findings

Message passing enables 2D French flag patterning with simple agents.

Concentration models struggle to produce accurate patterns without positional info.

Message-based models are promising for nano-robot patterning applications.

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 that relies on morphogens (diffusing proteins acting as chemical signals) and a more abstract version based on reliable message passing between cellular agents. Much of developmental biology research has focused on concentration-based approaches using morphogens, since morphogen gradients are thought to be an underlying mechanism in tissue patterning. We show that both our model types 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 that our insights may lay some groundwork for what kind of message passing abstractions or guarantees, if any, may be 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.