Bespoke Turing systems

摘要: Reaction–diffusion systems are an intensively studied form of partial differential equation, frequently used to produce spatially heterogeneous patterned states from homogeneous symmetry breaking via the Turing instability. Although there many prototypical “Turing systems” available, determining their parameters, functional forms, and general appropriateness for a given application is often difficult. Here, we consider reverse problem. Namely, suppose know parameter region associated with reaction kinetics in which patterning required—we present constructive framework identifying that will exhibit instability within this region, whilst addition allowing selection desired features, such as spots, or stripes. In particular, show how build system two populations governed by polynomial morphogen the: domain (in any spatial dimension), phases even type resulting pattern up dimensions) can all be determined. Finally, employing temporal heterogeneity, demonstrate mixed mode patterns (spots, stripes, complex prepatterns) also possible, one arbitrarily complicated landscapes. Such employed pedagogically, variety contemporary applications designing synthetic chemical biological systems. We discuss implications freedom design has on using reaction–diffusion modelling suggest stronger constraints needed when linking theory experiment, simple easily generated choose kinetics.