Homogeneous time constants promote oscillations in negative feedback loops

Biological oscillators are present in nearly all self-regulating systems, from individual cells to entire organisms. In any oscillator structure, a negative feedback loop is necessary, but not sufficient to guarantee the emergence of periodic behaviors. The likelihood of oscillations can be improved by careful tuning of the system time constants and by increasing the loop gain, yet it is unclear whether there is any general relationship between optimal time constants and loop gain. This issue is particularly relevant in genetic oscillators resulting from a chain of different subsequent biochemical events, each with distinct (and uncertain) kinetics. Using two families of genetic oscillators as model examples, we show that the loop gain required for oscillations is minimum when all elements in the loop have the same time constant. On the contrary, we show that homeostasis is ensured if a single element is considerably slower than the others.