Tile automaton in the well-mixed medium

By introducing a mean-field version of the tile automaton model introduced in earlier works, growth of molecules through chemical reaction networks is studied with explicit consideration for molecule shape as a “tile”. Tiles are picked up randomly to collide, and with a certain rule they react to form new tiles. A non-trivial growth pattern, called joint growth is found, with which tiles grow by combining tiles successively. This joint growth leads to a power-law distribution of tile sizes, by forming a positive feedback process for reproduction of tiles through cooperative relationship among large tiles. This effective growth is achieved by spontaneous differentiation of time scales: quick process for an autocatalytic network and a slower process with joint growth. We also discuss the relevance of the present results to the origin of life as a loose set of reproducting chemicals.