# High Cooperativity in Negative Feedback can Amplify Noisy Gene Expression

## 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”.

## References

- Abramowitz M, Stegun I (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Washington, DCMATHGoogle Scholar
- Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2002) Molecular biology of the cell. Garland Science, New YorkGoogle Scholar
- Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
- Barrio M, Burrage K, Leier A, Tian T (2006) Oscillatory regulation of hes1: discrete stochastic delay modelling and simulation. PLoS Comput Biol 2:e117CrossRefGoogle Scholar
- Becskei A, Serrano L (2000) Engineering stability in gene networks by autoregulation. Nature 405:590–593CrossRefGoogle Scholar
- Be’er S, Assaf M (2016) Rare events in stochastic populations under bursty reproduction. J Stat Mech Theory Exp 2016:113501MathSciNetCrossRefGoogle Scholar
- Biancalani T, Assaf M (2015) Genetic toggle switch in the absence of cooperative binding: exact results. Phys Rev Lett 115:208101CrossRefGoogle Scholar
- Blake W, Kaern M, Cantor C, Collins J (2003) Noise in eukaryotic gene expression. Nature 422:633–637CrossRefGoogle Scholar
- Bokes P, Singh A (2015) Protein copy number distributions for a self-regulating gene in the presence of decoy binding sites. PLoS ONE 10:e0120555CrossRefGoogle Scholar
- Bokes P, Singh A (2017) Gene expression noise is affected differentially by feedback in burst frequency and burst size. J Math Biol 74:1483–1509MathSciNetCrossRefMATHGoogle Scholar
- Bokes P, King J, Wood A, Loose M (2013) Transcriptional bursting diversifies the behaviour of a toggle switch: hybrid simulation of stochastic gene expression. Bull Math Biol 75:351–371MathSciNetCrossRefMATHGoogle Scholar
- Bruna M, Chapman SJ, Smith MJ (2014) Model reduction for slow-fast stochastic systems with metastable behaviour. J Chem Phys 140:174107CrossRefGoogle Scholar
- Cai L, Friedman N, Xie X (2006) Stochastic protein expression in individual cells at the single molecule level. Nature 440:358–362CrossRefGoogle Scholar
- Cao Y, Terebus A, Liang J (2016) State space truncation with quantified errors for accurate solutions to discrete chemical master equation. Bull Math Biol 78:617–661MathSciNetCrossRefMATHGoogle Scholar
- Dar RD, Razooky BS, Singh A, Trimeloni TV, McCollum JM, Cox CD, Simpson ML, Weinberger LS (2012) Transcriptional burst frequency and burst size are equally modulated across the human genome. Proc Natl Acad Sci USA 109:17454–17459CrossRefGoogle Scholar
- Dattani J, Barahona M (2017) Stochastic models of gene transcription with upstream drives: exact solution and sample path characterization. J R Soc Interface 14:20160833CrossRefGoogle Scholar
- Dessalles R, Fromion V, Robert P (2017) A stochastic analysis of autoregulation of gene expression. J Math Biol. https://doi.org/10.1007/s00285-017-1116-7 MathSciNetMATHGoogle Scholar
- Elf J, Ehrenberg M (2003) Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res 13:2475–2484CrossRefGoogle Scholar
- Elowitz M, Levine A, Siggia E, Swain P (2002) Stochastic gene expression in a single cell. Science 297:1183–1186CrossRefGoogle Scholar
- Friedman N, Cai L, Xie X (2006) Linking stochastic dynamics to population distribution: an analytical framework of gene expression. Phys Rev Lett 97:168302CrossRefGoogle Scholar
- Gillespie D (1976) A general method for numerically simulating stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434MathSciNetCrossRefGoogle Scholar
- Golding I, Paulsson J, Zawilski S, Cox E (2005) Real-time kinetics of gene activity in individual bacteria. Cell 123:1025–1036CrossRefGoogle Scholar
- Griffith J (1968) Mathematics of cellular control processes I. Negative feedback to one gene. J Theor Biol 20:202–208CrossRefGoogle Scholar
- Grönlund A, Lötstedt P, Elf J (2013) Transcription factor binding kinetics constrain noise suppression via negative feedback. Nat Commun 4:1864CrossRefGoogle Scholar
- Hinch EJ (1991) Perturbation methods. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
- Innocentini GC, Forger M, Radulescu O, Antoneli F (2016) Protein synthesis driven by dynamical stochastic transcription. Bull Math Biol 78:110–131MathSciNetCrossRefMATHGoogle Scholar
- Jedrak J, Ochab-Marcinek A (2016a) Influence of gene copy number on self-regulated gene expression. J Theor Biol 408:222–236MathSciNetCrossRefMATHGoogle Scholar
- Jedrak J, Ochab-Marcinek A (2016b) Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound poisson process. Phys Rev E 94:032401CrossRefMATHGoogle Scholar
- Johnson R, Munsky B (2017) The finite state projection approach to analyze dynamics of heterogeneous populations. Phys Biol 14:035002CrossRefGoogle Scholar
- Komorowski M, Miekisz J, Stumpf MP (2013) Decomposing noise in biochemical signaling systems highlights the role of protein degradation. Biophys J 104:1783–1793CrossRefGoogle Scholar
- Kumar N, Platini T, Kulkarni RV (2014) Exact distributions for stochastic gene expression models with bursting and feedback. Phys Rev Lett 113:268105CrossRefGoogle Scholar
- Lafuerza L, Toral R (2011) Role of delay in the stochastic creation process. Phys Rev E 84:021128CrossRefGoogle Scholar
- Leier A, Barrio M, Marquez-Lago TT (2014) Exact model reduction with delays: closed-form distributions and extensions to fully bi-directional monomolecular reactions. J R Soc Interface 11:20140108CrossRefGoogle Scholar
- Lester C, Baker RE, Giles MB, Yates CA (2016) Extending the multi-level method for the simulation of stochastic biological systems. Bull Math Biol 78:1640–1677MathSciNetCrossRefMATHGoogle Scholar
- Lin YT, Doering CR (2016) Gene expression dynamics with stochastic bursts: construction and exact results for a coarse-grained model. Phys Rev E 93:022409MathSciNetCrossRefGoogle Scholar
- Lin YT, Galla T (2016) Bursting noise in gene expression dynamics: linking microscopic and mesoscopic models. J R Soc Interface 13:20150772CrossRefGoogle Scholar
- Maarleveld TR, Olivier BG, Bruggeman FJ (2013) Stochpy: a comprehensive, user-friendly tool for simulating stochastic biological processes. PLoS ONE 8:e79345CrossRefGoogle Scholar
- McAdams H, Arkin A (1997) Stochastic mechanisms in gene expression. Proc Natl Acad Sci USA 94:814–819CrossRefGoogle Scholar
- Monk N (2003) Oscillatory expression of hes1, p53, and nf-\(\kappa \)b driven by transcriptional time delays. Curr Biol 13:1409–1413CrossRefGoogle Scholar
- Munsky B, Neuert G, Van Oudenaarden A (2012) Using gene expression noise to understand gene regulation. Science 336:183–187MathSciNetCrossRefMATHGoogle Scholar
- Murray J (2003) Mathematical biology: I introduction. Springer, New YorkGoogle Scholar
- Newby J (2015) Bistable switching asymptotics for the self regulating gene. J Phys A Math Theor 48:185001Google Scholar
- Ochab-Marcinek A, Tabaka M (2010) Bimodal gene expression in noncooperative regulatory systems. Proc Natl Acad Sci USA 107:22096–22101CrossRefGoogle Scholar
- Ochab-Marcinek A, Tabaka M (2015) Transcriptional leakage versus noise: a simple mechanism of conversion between binary and graded response in autoregulated genes. Phys Rev E 91:012704CrossRefGoogle Scholar
- Ozbudak EM, Thattai M, Kurtser I, Grossman AD, Van Oudenaarden A (2002) Regulation of noise in the expression of a single gene. Nat Genet 31:69–73CrossRefGoogle Scholar
- Pájaro M, Alonso AA, Otero-Muras I, Vázquez C (2017) Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting. J Theor Biol 421:51–70MathSciNetCrossRefMATHGoogle Scholar
- Platini T, Jia T, Kulkarni RV (2011) Regulation by small rnas via coupled degradation: mean-field and variational approaches. Phys Rev E 84:021928CrossRefGoogle Scholar
- Popovic N, Marr C, Swain PS (2016) A geometric analysis of fast-slow models for stochastic gene expression. J Math Biol 72:87–122MathSciNetCrossRefMATHGoogle Scholar
- Roberts E, Be’er S, Bohrer C, Sharma R, Assaf M (2015) Dynamics of simple gene-network motifs subject to extrinsic fluctuations. Phys Rev E 92:062717CrossRefGoogle Scholar
- Rosenfeld N, Elowitz MB, Alon U (2002) Negative autoregulation speeds the response times of transcription networks. J Mol Biol 323:785–793CrossRefGoogle Scholar
- Schikora-Tamarit MA, Toscano-Ochoa C, Espinos JD, Espinar L, Carey LB (2016) A synthetic gene circuit for measuring autoregulatory feedback control. Integr Biol 8:546–555CrossRefGoogle Scholar
- Schuss Z (2009) Theory and applications of stochastic processes: an analytical approach. Springer Science & Business Media, BerlinMATHGoogle Scholar
- Scott M, Hwa T, Ingalls B (2007) Deterministic characterization of stochastic genetic circuits. Proc Natl Acad Sci USA 104(18):7402–7407CrossRefGoogle Scholar
- Shahrezaei V, Swain P (2008) The stochastic nature of biochemical networks. Curr Opin Biotechnol 19:369–374CrossRefGoogle Scholar
- Singh A (2011) Negative feedback through mrna provides the best control of gene-expression noise. IEEE Trans Nanobiosci 10:194–200CrossRefGoogle Scholar
- Singh A, Hespanha JP (2009) Optimal feedback strength for noise suppression in autoregulatory gene networks. Biophys J 96:4013–4023CrossRefGoogle Scholar
- Smith S, Shahrezaei V (2015) General transient solution of the one-step master equation in one dimension. Phys Rev E 91(6):062119MathSciNetCrossRefGoogle Scholar
- Soltani M, Bokes P, Fox Z, Singh A (2015) Nonspecific transcription factor binding can reduce noise in the expression of downstream proteins. Phys Biol 12(055):002Google Scholar
- Stekel DJ, Jenkins DJ (2008) Strong negative self regulation of prokaryotic transcription factors increases the intrinsic noise of protein expression. Bmc Syst Biol 2:6CrossRefGoogle Scholar
- Suter DM, Molina N, Gatfield D, Schneider K, Schibler U, Naef F (2011) Mammalian genes are transcribed with widely different bursting kinetics. Science 332:472–474CrossRefGoogle Scholar
- Swain PS (2004) Efficient attenuation of stochasticity in gene expression through post-transcriptional control. J Mol Biol 344:965–976CrossRefGoogle Scholar
- Taniguchi Y, Choi P, Li G, Chen H, Babu M, Hearn J, Emili A, Xie X (2010) Quantifying
*E. coli*proteome and transcriptome with single-molecule sensitivity in single cells. Science 329:533–538CrossRefGoogle Scholar - Thattai M, van Oudenaarden A (2001) Intrinsic noise in gene regulatory networks. Proc Natl Acad Sci USA 98:151588598CrossRefGoogle Scholar
- Wang J, Lefranc M, Thommen Q (2014) Stochastic oscillations induced by intrinsic fluctuations in a self-repressing gene. Biophys J 107(10):2403–2416CrossRefGoogle Scholar
- Yang X, Wu Y, Yuan Z (2017) Characteristics of mrna dynamics in a multi-on model of stochastic transcription with regulation. Chin. J Phys 55:508–518CrossRefGoogle Scholar
- Yates JL, Nomura M (1981) Feedback regulation of ribosomal protein synthesis in
*E. coli*: localization of the mrna target sites for repressor action of ribosomal protein l1. Cell 24:243–249CrossRefGoogle Scholar