  Negative and positive transcriptional feedback loops are present in natural
and synthetic genetic oscillators. A single gene with negative transcriptional
feedback needs a time delay and sufficiently strong nonlinearity in the
transmission of the feedback signal in order to produce biochemical rhythms. A
single gene with only positive transcriptional feedback does not produce
oscillations. Here, we demonstrate that this single-gene network in conjunction
with a simple negative interaction can also easily produce rhythms. We examine
a model comprised of two well-differentiated parts. The first is a positive
feedback created by a protein that binds to the promoter of its own gene and
activates the transcription. The second is a negative interaction in which a
repressor molecule prevents this protein from binding to its promoter. A
stochastic study shows that the system is robust to noise. A deterministic
study identifies that the dynamics of the oscillator are mainly driven by two
types of biomolecules: the protein, and the complex formed by the repressor and
this protein. The main conclusion of this paper is that a simple and usual
negative interaction, such as degradation, sequestration or inhibition, acting
on the positive transcriptional feedback of a single gene is a sufficient
condition to produce reliable oscillations. One gene is enough and the positive
transcriptional feedback signal does not need to activate a second repressor
gene. This means that at the genetic level an explicit negative feedback loop
is not necessary. The model needs neither cooperative binding reactions nor the
formation of protein multimers. Therefore, our findings could help to clarify
the design principles of cellular clocks and constitute a new efficient tool
for engineering synthetic genetic oscillators.
