Living organisms are complex, and are typically composed
of many interacting subsystems. Recently a few subsystems
on the genetic level, namely, genetic repressilators or
oscillators, and bi-stable gene circuits, have been constructed
and manipulated. In the former case, while mathematical model
predicts simple oscillations, it has been observed that period
varies from one oscillation to another considerably. Realizing
that laboratory biochemical experiments take place in space in
a distributed way, we study coupled repressilators. Several
different types of coupling are considered. We find that
synchronized oscillations may occur for a narrow range of coupling
strength between nearly matched oscillators. In the remaining wide
range of coupling strength, chaos occurs, both via period doubling
and quasi-periodicity. This work thus naturally explains why the
periods observed in the experiments vary so considerably. Further
implications of the work is considered, including how oscillators
in the genetic level may interact to generate macroscopic orderly
behavior, why there might be a need to re-interpret the variations
in the expression level among difference experiments.
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