Tunable oscillations and chaotic dynamics in systems with localized synthesis.

Biological systems contain biochemical control networks that reside within a remarkable spatial structure. We present a theoretical study of a biological system in which two chemically coupled species, an activating species and an inhibiting species forming a negative feedback, are synthesized at unique sites and interact with each other through diffusion. The dynamical behaviors in these systems depend on the spatial locations of these synthetic sites. In a negative feedback system with two sites, we find two dynamical modes: fixed point and stable oscillations whose frequency can be tuned by varying the distance between the sites. When there are multiple synthetic sites, we find more diverse dynamics, including chaos, quasiperiodicity, and bistability. Based on this theoretical analysis, it should be possible to create in the laboratory synthetic circuits displaying these dynamics. This study illustrates the concept of "spatial switching," in which bifurcations in the dynamics occur as a function of the geometry of the system.