Evolution of gene auto-regulation in the presence of noise

Auto-regulatory negative feedback loops, where the protein expressed from a gene inhibits its own expression are common gene network motifs within cells. We investigate when will introducing a negative feedback mechanism be beneficial in terms of increasing a fitness function that is given by the probability of maintaining protein numbers above a critical threshold. Our results show the existence of a trade-off as introducing feedback decreases the average number of protein molecules driving this number closer to the critical threshold (which decreases fitness) but also reduces stochastic fluctuations around the mean (which increases fitness). We provide analytical conditions under which a negative feedback mechanism can evolve, that is, introducing feedback will increase the above fitness. Analyses of these conditions show that negative feedbacks are more likely to evolve when (i) the source of noise in the protein population is extrinsic (i. e. noise is caused by fluctuations in exogenous signals driving the gene network) and not intrinsic (i. e. the randomness associated with mRNA/protein expression and degradation); (ii) the dynamics of the exogenous signal causing extrinsic noise is slower than the protein dynamics; and (iii) the critical threshold level for the protein number is low. We also show that mRNA/protein degradation rates are critical factors in determining whether transcription or translational negative feedback should evolve. In particular, when the mRNA half-life is much shorter than the protein's half-life, then a transcriptional negative feedback mechanism is more likely to evolve. On the other hand, a translational negative feedback mechanism is preferred with more stable mRNAs that have long half-lifes.