Stochastic delays suppress noise in a genetic circuit with negative feedback

We consider a mechanistic stochastic model of an autoregulatory genetic circuit with time delays. More specifically, a protein is expressed in random bursts from its corresponding gene. The synthesized protein is initially inactive and becomes active after a time delay. Rather than considering a deterministic delay, a key aspect of this work is to incorporate stochastic time delays, where delay is an independent and identically distributed random variable. The active protein inhibits its own production creating a negative feedback loop. Our analysis reveals that for an exponentially-distributed time delay, the noise in the protein levels decreases to the Poisson limit with increasing mean time delay. Interesting, for a gamma-distributed time delay contrasting noise behaviors emerge based on the negative feedback strength. At low feedback strengths the protein noise levels monotonically decreases to the Poisson limit with increasing average delay. At intermediate feedback strengths, the noise levels first increase to reach a maximum, and then decease back to the Poisson limit with increasing average delay. Finally, for strong feedbacks the protein noise levels monotonically increase with the average delay. For each of these scenarios we provide approximate analytical formulas for the protein mean and noises levels, and validate these results by performing exact Monte Carlo simulations. In conclusion, our results uncover a counter intuitive feature where inclusion of stochastic delays in a negative feedback circuit can play a beneficial role in buffering deleterious fluctuations in the level of a protein.