How a system of genetically identical biological cells organizes into spatially heterogeneous tissues is a central question in biology. Even when the molecular and genetic underpinnings of cell-cell interactions are known, how these lead to multicellular patterns is often poorly understood. Of particular interest are dynamic patterns such as traveling waves, which confer spatiotemporal control over key developmental processes such as differentiation, segmentation and cell division. Theoretical approaches based on mathematical descriptions of underlying physical and chemical processes provide a promising avenue to explore biological pattern formation. In particular, theoretical models connect processes on the molecular scale to biological function on the tissue level and may provide mechanistic descriptions of how patterns are generated and maintained.

Cells form spatial patterns by coordinating their gene expressions. How a group of mesoscopic numbers (hundreds to thousands) of cells, without pre-existing morphogen gradients and spatial organization, self-organizes spatial patterns remains poorly understood. Of particular importance are dynamic spatial patterns such as spiral waves that perpetually move and transmit information. We developed an open-source software for simulating a field of cells that communicate by secreting any number of molecules. With this software and a theory, we identified all possible “cellular dialogues”—ways of communicating with two diffusing molecules—that yield diverse dynamic spatial patterns. These patterns emerge despite widely varying responses of cells to the molecules, gene-expression noise, spatial arrangements, and cell movements. A three-stage, “order-fluctuate-settle” process forms dynamic spatial patterns: cells form long-lived whirlpools of wavelets that, following erratic dynamics, settle into a dynamic spatial pattern. Our work helps in identifying gene-regulatory networks that underlie dynamic pattern formations. Dang et al. developed a software and a theoretical framework to discover and classify all moving spatial patterns (e.g., waves) that cells can form by secreting two diffusible molecules that control their gene expressions. They identified all gene regulations that the molecules can have for forming moving patterns, which self-organize through a three-stage, “order-fluctuate-settle” dynamic.

Communicating cells can coordinate their gene expressions to form spatial patterns, generating order from disorder. Ubiquitous “secrete-and-sense cells” secrete and sense the same molecule to do so. Here we present a modeling framework—based on cellular automata and mimicking approaches of statistical mechanics—for understanding how secrete-and-sense cells with bistable gene expression, from disordered beginnings, can become spatially ordered by communicating through rapidly diffusing molecules. Classifying lattices of cells by two “macrostate” variables—“spatial index,” measuring degree of order, and average gene-expression level—reveals a conceptual picture: a group of cells behaves as a single particle, in an abstract space, that rolls down on an adhesive “pseudo-energy landscape” whose shape is determined by cell-cell communication and an intracellular gene-regulatory circuit. Particles rolling down the landscape represent cells becoming more spatially ordered. We show how to extend this framework to more complex forms of cellular communication.