# High Cooperativity in Negative Feedback can Amplify Noisy Gene Expression

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## Abstract

Burst-like synthesis of protein is a significant source of cell-to-cell variability in protein levels. Negative feedback is a common example of a regulatory mechanism by which such stochasticity can be controlled. Here we consider a specific kind of negative feedback, which makes bursts smaller in the excess of protein. Increasing the strength of the feedback may lead to dramatically different outcomes depending on a key parameter, the noise load, which is defined as the squared coefficient of variation the protein exhibits in the absence of feedback. Combining stochastic simulation with asymptotic analysis, we identify a critical value of noise load: for noise loads smaller than critical, the coefficient of variation remains bounded with increasing feedback strength; contrastingly, if the noise load is larger than critical, the coefficient of variation diverges to infinity in the limit of ever greater feedback strengths. Interestingly, feedbacks with lower cooperativities have higher critical noise loads, suggesting that they can be preferable for controlling noisy proteins.

## Keywords

Stochastic gene expression Protein bursting Negative feedback Delayed production Asymptotic expansions## Mathematics Subject Classification

92C40 60K40 41A60## Notes

### Acknowledgements

We thank an anonymous referee for useful comments and important insights, in particular those leading to the analysis of “Appendix A”.

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