Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Bimodality in gene expression without feedback: from Gaussian white noise to log-normal coloured noise

1 Department of Microbial Sciences, Faculty of Health and Medical Sciences, University of Surrey, GU2 7XH Guildford, United Kingdom
2 Department of Physics, Faculty of Engineering and Physical Sciences, University of Surrey, GU2 7XH Guildford, United Kingdom

Special Issues: Cells as dynamical systems

Extrinsic noise-induced transitions to bimodal dynamics have been largely investigated in a variety of chemical, physical, and biological systems. In the standard approach in physical and chemical systems, the key properties that make these systems mathematically tractable are that the noise appears linearly in the dynamical equations, and it is assumed Gaussian and white. In biology, the Gaussian approximation has been successful in specific systems, but the relevant noise being usually non-Gaussian, non-white, and nonlinear poses serious limitations to its general applicability. Here we revisit the fundamental features of linear Gaussian noise, pinpoint its limitations, and review recent new approaches based on nonlinear bounded noises, which highlight novel mechanisms to account for transitions to bimodal behaviour. We do this by considering a simple but fundamental gene expression model, the repressed gene, which is characterized by linear and nonlinear dependencies on external parameters. We then review a general methodology introduced recently, so-called nonlinear noise filtering, which allows the investigation of linear, nonlinear, Gaussian and non-Gaussian noises. We also present a derivation of it, which highlights its dynamical origin. Testing the methodology on the repressed gene confirms that the emergence of noise-induced transitions appears to be strongly dependent on the type of noise adopted, and on the degree of nonlinearity present in the system.
  Article Metrics

Keywords noise-induced transitions; Langevin equation; log-normal noise; white and coloured noise; gene regulation

Citation: Gerardo Aquino, Andrea Rocco. Bimodality in gene expression without feedback: from Gaussian white noise to log-normal coloured noise. Mathematical Biosciences and Engineering, 2020, 17(6): 6993-7017. doi: 10.3934/mbe.2020361


  • 1. J. R. Pomerening, Uncovering mechanisms of bistability in biological systems, Curr. Opin. Biotechnol., 19 (2008), 381-388.
  • 2. M. C. A. Leite, Y. Wang, Multistability, oscillations and bifurcations in feedback loops, Math. Biosci. Eng., 7 (2010), 83-97.
  • 3. G. Balázsi, A. van Oudenaarden, J. J. Collins, Cellular decision-making and biological noise: From microbes to mammals, Cell, 144 (2011), 910-925.
  • 4. L. Wang, M. C. Romano, F. A. Davidson, Translational control of gene expression via interacting feedback loops, Phys. Rev. E, 100 (2019), 050402.
  • 5. M. Santillan, M. C. Mackey, E. S. Zeron, Origin of bistability in the lac operon, Biophys. J., 92 (2007), 3830-3842 .
  • 6. J. W. Williams, X. Cui, A. Levchenko, A. M. Stevens, Robust and sensitive control of a quorum-sensing circuit by two interlocked feedback loops, Mol. Sys. Biol., 4 (2008), 234.
  • 7. P. Melke, P. Sahlin, A. Levchenko, H. Jonsson, A cell-based model for quorum sensing in heterogeneous bacterial colonies, PLoS Comp. Biol., 6 (2010), e1000819.
  • 8. J. E. Ferrell, Bistability, bifurcations, and Waddington's epigenetic landscape, Curr. Biol., 22 (2012), R458-R466.
  • 9. M. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.
  • 10. A. P. Gasch, F. B. Yu, J. Hose, L. E. Escalante, M. Place, R. Bacher, et al., Single-cell RNA sequencing reveals intrinsic and extrinsic regulatory heterogeneity in yeast responding to stress, PLOS Biol., 15 (2017), e2004050.
  • 11. H. Ochiai, T. Hayashi, M. Umeda, M. Yoshimura, A. Harada, Y. Shimizu, et al., Genome-wide analysis of transcriptional bursting-induced noise in mammalian cells, preprint, BioRxiv, 2019: 736207.
  • 12. J. M. Raser, E. K. O'Shea, Noise in gene expression: Origins, consequences, and control, Science, 309 (2005), 2010-2013.
  • 13. A. Eldar, M. B. Elowitz, Functional roles for noise in genetic circuits, Nature, 467 (2010), 167-173.
  • 14. L. S. Tsimring, Noise in biology, Rep. Prog. Phys., 77 (2014), 026601.
  • 15. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, 2007.
  • 16. J. Elf, M. Ehrenberg, Fast evaluation of fluctuations in biochemical networks with the linear noise approximation, Genome Res., 13 (2003), 2475-2484.
  • 17. D. T. Gillespie, Exact stochastic simulations of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361.
  • 18. D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, J. Chem. Phys., 115 (2001), 1716-1733.
  • 19. P. S. Swain, M. B. Elowitz, E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Natl. Acad. Sci., 99 (2002), 12795-12800.
  • 20. J. Dattani, M. Barahona, Stochastic models of gene transcription with upstream drives: Exact solution and sample path characterisation, J. R. Soc. Interface, 14 (2017), 20160833.
  • 21. P. C. Bressloff, E. Levien, Propagation of extrinsic fluctuations in biochemical birth-death processes, Bull. Math. Biol., 81 (2019), 800-829.
  • 22. W. Horsthemke, R. Lefever, Noise-Induced Transitions, Springer, 1984.
  • 23. C. W. Gardiner, Handbook of Stochastic Methods, Springer, 1985.
  • 24. J. Garcia-Ojalvo, J. Sancho, Noise in Spatially Extended Systems, Springer, 1999.
  • 25. C. Zeng, C. Zhang, J. Zeng, H. Luo, D. Tian, H. Zhang, et al., Noises-induced regime shifts and -enhanced stability under a model of lake approaching eutrophication, Ecol. Complexity, 22 (2015), 102-108.
  • 26. J. Jungeilgesa, T. Ryazanova, Noise-induced transitions in a stochastic Goodwin-type business cycle model, Struct. Change and Econ. Dyn. 40 (2017), 103-115.
  • 27. M. Samoilov, S. Plyasunov, A. P. Arkin, Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations, Proc. Natl. Acad. Sci., 102 (2005), 2310-2315.
  • 28. M. Samoilov, A. P. Arkin, Deviant effects in molecular reaction pathways, Nat. Biotechnol., 24 (2006), 1235-1240.
  • 29. A. Rocco, Stochastic control of metabolic pathways, Phys. Biol., 6 (2009), 016002.
  • 30. E. Pujadas, A. P. Feinberg, A role for regulated noise in the epigenetic landscape of development and disease, Cell, 148 (2012), 1123-1131.
  • 31. M. Weber, J. Buceta, Dynamics of the quorum sensing switch: stochastic and non-stationary effects, BMC Syst. Biol., 7 (2013), 6.
  • 32. M. Weber, J. Buceta, Stochastic stabilization of phenotypic states: The genetic bistable switch as a case study, PLOS One, 8 (2013), e73487.
  • 33. A. Ochab-Marcinek, M. Tabaka, Bimodal gene expression in noncooperative regulatory systems Proc. Natl. Acad. Sci., 107 (2010), 22096-22101.
  • 34. A. Ochab-Marcinek, J. Jedrak, M. Tabaka, Hill Kinetics as a noise filter: The role of transcription factor autoregulation in gene cascades, Phys. Chem. Chem. Phys., 19 (2017), 22580-22591.
  • 35. M. R. Birtwistle, J. Rauch, A. Kiyatkin, E. Aksamitiene, M. Dobrzyński, J. B. Hoek, et al., Emergence of bimodal cell population responses from the interplay between analog single-cell signaling and protein expression noise, BMC Syst. Biol., 6 (2012), 109.
  • 36. K. H. Kim, H. M. Sauro, In search of noise-induced bimodality, BMC Biol., 10 (2012), 89.
  • 37. M. Dobrzyński, L. K. Nguyen, M. R. Birtwistle, A. von Kriegsheim, A. B. Fernàndez, A. Cheong, et al., Nonlinear signalling networks and cell- to-cell variability transform external signals into broadly distributed or bimodal responses, J. R. Soc. Interface, 11 (2014), 20140383.
  • 38. N. Q. Balaban, J. Merrin, R. Chait, L. Kowalik, S. Leibler, Bacterial persistence as a phenotypic switch, Science, 305 (2004), 1622-1625.
  • 39. C. Lou, Z. Li, Q. Ouyang, A molecular model for persister in E. coli, J. Theo. Biol., 255 (2008), 205-209
  • 40. S. M. Hingley-Wilson, N. Ma, Y. Hu, R. Casey, A. Bramming, R. J. Curry, et al., Loss of phenotypic inheritance associated with ydcI mutation leads to increased frequency of small, slow persisters in escherichia coli, Proc. Natl. Acad. Sci., 117 (2020), 4152-4157.
  • 41. A. Rocco, A. Kierzek, J. McFadden, Slow protein fluctuations explain the emergence of growth phenotypes and persistence in clonal bacterial populations, PloS One, 8 (2013), e54272.
  • 42. A. Rocco, A. Kierzek, J. McFadden, Systems Biology of Tuberculosis, Springer, 2013.
  • 43. L. Arnold, W. Horsthemke, R. Lefever, White and coloured external noise and transition phenomena in nonlinear systems, Z. Phys. B Condens. Matter, 29 (1978), 367-373.
  • 44. L. E. Reichl, W. C. Schieve, Instabilities, Bifurcation, and Fluctuations in Chemical Systems, University of Texas Press, Austin, 1982.
  • 45. N. Rosenfeld, J. W. Young, U. Alon, P. S. Swain, M. B. Elowitz, Gene regulation at the single-cell level, Science, 307 (2005), 1962-1965.
  • 46. B. B. Kaufmann, Q. Yang, J. T. Mettetal, A. van Oudenaarden, Heritable stochastic switching revealed by single-cell genealogy, PLoS Biol., 5 (2007), 1973-1980.
  • 47. P. Jung, P. Hänggi, Dynamical systems: A unified colored-noise approximation, Phys. Rev. A, 35 (1987), 4464-4466.
  • 48. P. Grigolini, L. A. Lugiato, R. Mannella, P. V. E. McClintock, M. Merri, M. Pernigo, Fokker-pLanck description of stochastic processes with colored noise, Phys Rev. A, 38 (1988), 1966.
  • 49. J. Holehouse, A. Gupta, R. Grima, Steady-state fluctuations of a genetic feedback loop with fluctuating rate parameters using the unified colored noise approximation, J. Phys. A: Math. Theor., Forthcoming.
  • 50. L. Borland, Itô-Langevin equations within generalized thermostatistics, Phys. Lett. A, 245 (1998), 67-72.
  • 51. R. V. Bobryk, A. Chrzeszczyk, Transitions induced by bounded noise, Phys. A, 358 (2005), 263-272.
  • 52. S. de Franciscis, G. Caravagna, A. d'Onofrio, Bounded noises as a natural tool to model extrinsic fluctuations in biomolecular networks, Nat. Comput. 13 (2014), 297-307.
  • 53. H. S. Wio, R. Toral, Effect of non-Gaussian noise sources in a noise induced transition, Phys. D, 193 (2004), 161-168.
  • 54. A. d'Onofrio, Bounded-noise-induced transitions in a tumor-immune system interplay, Phys. Rev. E, 81 (2010), 021923.
  • 55. A. d'Onofrio, A. Gandolfi, Resistance to antitumor chemotherapy due to bounded-noise-induced transitions, Phys. Rev. E, 82 (2010), 061901.
  • 56. L. Cai, N. Friedman, X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362.
  • 57. M. Bengtsson, A. Stahlberg, P. Rorsman, M. Kubista, Gene expression profiling in single cells from the pancreatic islets of langerhans reveals lognormal distribution of mRNA levels, Genome Res., 5 (2005), 1388-1392.
  • 58. L. Ham, R. D. Brackston, M. P. H. Stumpf, Extrinsic noise and heavy-tailed laws in gene expression, Phys. Rev. Lett., 124 (2020), 108101.
  • 59. D. T. Gillespie, A rigorous derivation of the chemical master equation, Phys. A, 188 (1992), 404-425.
  • 60. D. T. Gillespie, The chemical Lengevin equation, J. Chem. Phys., 113 (2000), 297.
  • 61. F. Wong, M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36 (1965), 1560-1564.
  • 62. V. Shahrezaei, J. F. Ollivier, P. S. Swain, Colored extrinsic fluctuations and stochastic gene expression, Mol. Sys. Bio., 4 (2008), 196.
  • 63. N. Rosenfeld, M. B. Elowitz, U. Alon, Negative autoregulation speeds the response times of transcription networks, J. Mol. Biol., 323 (2002), 785-793.
  • 64. M Thattai, A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci., 98 (2001), 8614.
  • 65. B. H. Toyama, M. W. Hetzer, Protein homeostasis: Live long, won't prosper, Nat. Rev. Mol. Cell. Biol., 14 (2013), 55-61.
  • 66. A. Loinger, A. Lipshtat, N. Q. Balaban, O. Biham, Stochastic simulations of genetic switch systems, Phys Rev., 75 (2007), 021904.
  • 67. G. Q. Cai, Y. K. Lin, Generation of non-Gaussian stationary stochastic processes, Phys. Rev. E, 54 (1996), 299-303.
  • 68. M. Kessler, M. Sørensen, Estimating equations based on eigenfunctions for a discretely observed diffusion process, Bernoulli, 5 (1999), 299-314.
  • 69. N. Friedman, L. Cai, X. S. Xie, Linking stochastic dynamics to population distribution: An analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302.
  • 70. V. Shahrezaei, P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci., 105 (2008), 17256-17261.
  • 71. Y. Taniguchi, P. J. Choi, G. W. Li, H. Chen, M. Babu, J. Hearn, et al., Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells, Science, 329 (2010), 533-538.
  • 72. H. Salman, N. Brenner, C. K. Tung, N. Elyahu, E. Stolovicki, L. Moore, et al., Universal protein fluctuations in populations of microorganisms, Phys. Rev. Lett., 108 (2012), 238105.
  • 73. N. Brenner, C. M. Newman, D. Osmanovic, Y. Rabin, H. Salman, D. L. Stein, Universal protein distributions in a model of cell growth and division, Phys. Rev. E, 92 (2015), 042713.
  • 74. A. Sigal, R. Milo, A. Cohen, N. Geva-Zatorsky, Y. Klein, Y. Liron, et al., Variability and memory of protein levels in human cells, Nature, 444 (2006), 643-646.
  • 75. J. Lei, Stochasticity in single gene expression with both intrinsic noise and fluctuation in kinetic parameters, J. Theo. Biol., 256 (2009), 485-492.
  • 76. F. Maleki, A. Becskei, An open-loop approach to calculate noise-induced transitions, J. Theo. Biol., 415 (2017), 145-157.
  • 77. D. Angeli, J. E. Ferrell Jr, E.D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Natl. Acad.. Sci., 101 (2004), 1822-1827.
  • 78. C. Hsu, V. Jaquet, F. Maleki, A. Becskei, Contribution of bistability and noise to cell fate tansitions determined by feedback opening, J. Mol. Biol., 428 (2016), 4115-4128.
  • 79. R. Kupferman, G. A. Pavliotis, A. M. Stuart, Itô versus Stratonovich white noise limits for systems with inertia and colored multiplicative noise, Phys. Rev. E, 70 (2004), 036120.
  • 80. G. A. Pavliotis, A. M. Stuart, Analysis of white noise limits for stochastic systems with two fast relaxation times, Multiscale Model. Simul., 4 (2005), 1.
  • 81. P. Thomas, R. Grima, A. V. Straube, Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators, Phys. Rev. E, 86 (2012), 041110.
  • 82. B. Bravi, K. J. Rubin, P. Sollich, Systematic model reduction captures the dynamics of extrinsic noise inbiochemical subnetworks, preprint, arXiv:2003.08704.
  • 83. R. Perez-Carrasco, P. Guerrero, J. Briscoe, K. M. Page, Intrinsic noise profoundly alters the dynamics and steady state of morphogen-controlled bistable genetic switches, PLoS Comput. Biol., 12 (2016), e1005154.
  • 84. M. Pájaro, O. Otero-Muras, C. Vázquez, A. A. Alonso, Transient hysteresis and inherent stochasticity in gene regulatroy networks, Nat. Commun., 10 (2019), 4581.
  • 85. A. Brock, H. Chang, S. Huang, Non-genetic heterogeneity-a mutation-independent driving force for the somatic evolution of tumours, Nat. Rev. Gen., 10 (2009), 336-342.
  • 86. A Sanchez, S. Choubey, J. Kondev, Regulation of noise in gene expression, Annu. Rev. Biophys., 42 (2013), 469-491.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved