  The Turing patterning mechanism is believed to underly the formation of
repetitive structures in development, such as zebrafish stripes and mammalian
digits, but it has proved difficult to isolate the specific biochemical species
responsible for pattern formation. Meanwhile, synthetic biologists have
designed Turing systems for implementation in cell colonies, but none have yet
led to visible patterns in the laboratory. In both cases, the relationship
between underlying chemistry and emergent biology remains mysterious. To help
resolve the mystery, this article asks the question: what kinds of biochemical
systems can generate Turing patterns? We find general conditions for Turing
pattern inception -- the ability to generate unstable patterns from random
noise -- which may lead to the ultimate formation of stable patterns, depending
on biochemical non-linearities. We find that a wide variety of systems can
generate stable Turing patterns, including several which are currently unknown,
such as two-species systems composed of two self-activators, and systems
composed of a short-range inhibitor and a long-range activator. We furthermore
find that systems which are widely believed to generate stable patterns may in
fact only generate unstable patterns, which ultimately converge to
spatially-homogeneous concentrations. Our results suggest that a much wider
variety of systems than is commonly believed could be responsible for observed
patterns in development, or could be good candidates for synthetic patterning
networks.
