Skip to main content
SpringerLink
Log in

Stationary Distributions and Metastable Behaviour for Self-regulating Proteins with General Lifetime Distributions

  • Conference paper
  • First Online:
Computational Methods in Systems Biology (CMSB 2020)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 12314))

Included in the following conference series:

Abstract

Regulatory molecules such as transcription factors are often present at relatively small copy numbers in living cells. The copy number of a particular molecule fluctuates in time due to the random occurrence of production and degradation reactions. Here we consider a stochastic model for a self-regulating transcription factor whose lifespan (or time till degradation) follows a general distribution modelled as per a multi-dimensional phase-type process. We show that at steady state the protein copy-number distribution is the same as in a one-dimensional model with exponentially distributed lifetimes. This invariance result holds only if molecules are produced one at a time: we provide explicit counterexamples in the bursty production regime. Additionally, we consider the case of a bistable genetic switch constituted by a positively autoregulating transcription factor. The switch alternately resides in states of up- and downregulation and generates bimodal protein distributions. In the context of our invariance result, we investigate how the choice of lifetime distribution affects the rates of metastable transitions between the two modes of the distribution. The phase-type model, being non-linear and multi-dimensional whilst possessing an explicit stationary distribution, provides a valuable test example for exploring dynamics in complex biological systems.

CÇ is supported by the Comenius University grant for doctoral students Nos. UK/201/2019 and UK/106/2020. PB is supported by the Slovak Research and Development Agency under the contract No. APVV-18-0308, by the VEGA grant 1/0347/18, and the EraCoSysMed project 4D-Healing. AS is supported by the National Science Foundation grant ECCS-1711548 and ARO W911NF-19-1-0243.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abel, J.H., Drawert, B., Hellander, A., Petzold, L.R.: GillesPy: a python package for stochastic model building and simulation. IEEE Life Sci. Lett. 2(3), 35–38 (2017)

    Article  Google Scholar 

  2. Alon, U.: An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman & Hall/CRC (2007)

    Google Scholar 

  3. Andreychenko, A., Bortolussi, L., Grima, R., Thomas, P., Wolf, V.: Distribution approximations for the chemical master equation: comparison of the method of moments and the system size expansion. In: Graw, F., Matthäus, F., Pahle, J. (eds.) Modeling Cellular Systems. CMCS, vol. 11, pp. 39–66. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-45833-5_2

    Chapter  Google Scholar 

  4. Backenköhler, M., Bortolussi, L., Wolf, V.: Control variates for stochastic simulation of chemical reaction networks. In: Bortolussi, L., Sanguinetti, G. (eds.) CMSB 2019. LNCS, vol. 11773, pp. 42–59. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-31304-3_3

    Chapter  Google Scholar 

  5. Becskei, A., Séraphin, B., Serrano, L.: Positive feedback in eukaryotic gene networks: cell differentiation by graded to binary response conversion. EMBO J. 20, 2528–2535 (2001)

    Article  Google Scholar 

  6. Blake, W., Kaern, M., Cantor, C., Collins, J.: Noise in eukaryotic gene expression. Nature 422, 633–637 (2003)

    Article  Google Scholar 

  7. Bokes, P., Lin, Y., Singh, A.: High cooperativity in negative feedback can amplify noisy gene expression. Bull. Math. Biol. 80, 1871–1899 (2018)

    Article  MathSciNet  Google Scholar 

  8. Bokes, P.: Postponing production exponentially enhances the molecular memory of a stochastic switch. BioRxiv (2020). https://doi.org/10.1101/2020.06.19.160754

    Article  Google Scholar 

  9. Bokes, P., Borri, A., Palumbo, P., Singh, A.: Mixture with delayed distributions in a stochastic gene expression model feedback: a WKB approximation approach. J. Math. Biol. 81(1), 343–367 (2020). https://doi.org/10.1007/s00285-020-01512-y

    Article  MathSciNet  MATH  Google Scholar 

  10. Bokes, P., Hojcka, M., Singh, A.: Buffering gene expression noise by MicroRNA based feedforward regulation. In: Češka, M., Šafránek, D. (eds.) CMSB 2018. LNCS, vol. 11095, pp. 129–145. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99429-1_8

    Chapter  Google Scholar 

  11. Bokes, P., King, J.R., Wood, A.T., Loose, M.: Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression. J. Math. Biol. 64, 829–854 (2012). https://doi.org/10.1007/s00285-011-0433-5

    Article  MathSciNet  MATH  Google Scholar 

  12. Bokes, P., Singh, A.: Controlling noisy expression through auto regulation of burst frequency and protein stability. In: Češka, M., Paoletti, N. (eds.) HSB 2019. LNCS, vol. 11705, pp. 80–97. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-28042-0_6

    Chapter  Google Scholar 

  13. Bortolussi, L., Lanciani, R., Nenzi, L.: Model checking Markov population models by stochastic approximations. Inf. Comput. 262, 189–220 (2018)

    Article  MathSciNet  Google Scholar 

  14. Bratsun, D., Volfson, D., Tsimring, L.S., Hasty, J.: Delay-induced stochastic oscillations in gene regulation. Proc. Natl. Acad. Sci. U.S.A. 102(41), 14593–14598 (2005)

    Article  Google Scholar 

  15. Cai, L., Friedman, N., Xie, X.: Stochastic protein expression in individual cells at the single molecule level. Nature 440, 358–362 (2006)

    Article  Google Scholar 

  16. Cinquemani, E.: Identifiability and reconstruction of biochemical reaction networks from population snapshot data. Processes 6(9), 136 (2018)

    Article  Google Scholar 

  17. Cinquemani, E.: Stochastic reaction networks with input processes: analysis and application to gene expression inference. Automatica 101, 150–156 (2019)

    Article  MathSciNet  Google Scholar 

  18. Dar, R.D., Razooky, B.S., Singh, A., Trimeloni, T.V., McCollum, J.M., Cox, C.D., Simpson, M.L., Weinberger, L.S.: Transcriptional burst frequency and burst size are equally modulated across the human genome. Proc. Natl. Acad. Sci. U.S.A. 109, 17454–17459 (2012)

    Article  Google Scholar 

  19. Deneke, C., Lipowsky, R., Valleriani, A.: Complex degradation processes lead to non-exponential decay patterns and age-dependent decay rates of messenger RNA. PLoS ONE 8(2), e55442 (2013)

    Article  Google Scholar 

  20. Escudero, C., Kamenev, A.: Switching rates of multistep reactions. Phys. Rev. E 79(4), 041149 (2009)

    Article  Google Scholar 

  21. Friedman, N., Cai, L., Xie, X.: Linking stochastic dynamics to population distribution: an analytical framework of gene expression. Phys. Rev. Lett. 97, 168302 (2006)

    Article  Google Scholar 

  22. Griffith, J.: Mathematics of cellular control processes II. Positive feedback to one gene. J. Theor. Biol. 20(2), 209–216 (1968)

    Article  Google Scholar 

  23. Gross, D.: Fundamentals of Queueing Theory. Wiley, Hoboken (2008)

    Book  Google Scholar 

  24. Guet, C., Henzinger, T.A., Igler, C., Petrov, T., Sezgin, A.: Transient memory in gene regulation. In: Bortolussi, L., Sanguinetti, G. (eds.) CMSB 2019. LNCS, vol. 11773, pp. 155–187. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-31304-3_9

    Chapter  MATH  Google Scholar 

  25. Hanggi, P., Grabert, H., Talkner, P., Thomas, H.: Bistable systems: master equation versus Fokker-Planck modeling. Phys. Rev. A 29(1), 371 (1984)

    Article  MathSciNet  Google Scholar 

  26. Hinch, R., Chapman, S.J.: Exponentially slow transitions on a Markov chain: the frequency of calcium sparks. Eur. J. Appl. Math. 16(04), 427–446 (2005)

    Article  MathSciNet  Google Scholar 

  27. Innocentini, G.C.P., Antoneli, F., Hodgkinson, A., Radulescu, O.: Effective computational methods for hybrid stochastic gene networks. In: Bortolussi, L., Sanguinetti, G. (eds.) CMSB 2019. LNCS, vol. 11773, pp. 60–77. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-31304-3_4

    Chapter  MATH  Google Scholar 

  28. Innocentini, G.C., Hodgkinson, A., Radulescu, O.: Time dependent stochastic mRNA and protein synthesis in piecewise-deterministic models of gene networks. Front. Phys. 6, 46 (2018)

    Article  Google Scholar 

  29. Jackson, J.R.: Jobshop-like queueing systems. Manage. Sci. 10(1), 131–142 (1963)

    Article  Google Scholar 

  30. Jia, T., Kulkarni, R.: Intrinsic noise in stochastic models of gene expression with molecular memory and bursting. Phys. Rev. Lett. 106(5), 58102 (2011)

    Article  Google Scholar 

  31. Johnson, N., Kotz, S., Kemp, A.: Univariate Discrete Distributions, 3rd edn. Wiley, Hoboken (2005)

    Book  Google Scholar 

  32. van Kampen, N.: Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  33. Kelly, F.P.: Reversibility and Stochastic Networks. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  34. Kendall, D.: Stochastic processes and population growth. J. Roy. Stat. Soc. B 11, 230–282 (1949)

    MathSciNet  MATH  Google Scholar 

  35. Kurasov, P., Lück, A., Mugnolo, D., Wolf, V.: Stochastic hybrid models of gene regulatory networks - a PDE approach. Math. Biosci. 305, 170–177 (2018)

    Article  MathSciNet  Google Scholar 

  36. Lagershausen, S.: Performance Analysis of Closed Queueing Networks, vol. 663. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32214-3

    Book  Google Scholar 

  37. Lestas, I., Paulsson, J., Ross, N., Vinnicombe, G.: Noise in gene regulatory networks. IEEE Trans. Circ. I 53(1), 189–200 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Liu, L., Kashyap, B., Templeton, J.: On the \({GI}^{X}/{G}/\infty \) system. J. Appl. Probab. 27(3), 671–683 (1990)

    Article  MathSciNet  Google Scholar 

  39. McShane, E., Sin, C., Zauber, H., Wells, J.N., Donnelly, N., Wang, X., Hou, J., Chen, W., Storchova, Z., Marsh, J.A., et al.: Kinetic analysis of protein stability reveals age-dependent degradation. Cell 167(3), 803–815 (2016)

    Article  Google Scholar 

  40. Miȩkisz, J., Poleszczuk, J., Bodnar, M., Foryś, U.: Stochastic models of gene expression with delayed degradation. Bull. Math. Biol. 73(9), 2231–2247 (2011)

    Article  MathSciNet  Google Scholar 

  41. Michaelides, M., Hillston, J., Sanguinetti, G.: Geometric fluid approximation for general continuous-time Markov chains. Proc. Roy. Soc. A 475(2229), 20190100 (2019)

    Article  MathSciNet  Google Scholar 

  42. Michaelides, M., Hillston, J., Sanguinetti, G.: Statistical abstraction for multi-scale spatio-temporal systems. ACM Trans. Model. Comput. Simul. 29(4), 1–29 (2019)

    Article  MathSciNet  Google Scholar 

  43. Newby, J., Chapman, J.: Metastable behavior in Markov processes with internal states. Journal of Mathematical Biology 69(4), 941–976 (2013). https://doi.org/10.1007/s00285-013-0723-1

    Article  MathSciNet  MATH  Google Scholar 

  44. Norris, J.R.: Markov Chains. Cambridge Univ Press, Cambridge (1998)

    MATH  Google Scholar 

  45. Prajapat, M.K., Ribeiro, A.S.: Added value of autoregulation and multi-step kinetics of transcription initiation. R. Soc. Open Sci. 5(11), 181170 (2018)

    Article  Google Scholar 

  46. Ross, S.M.: Introduction to probability models. Academic Press, Cambridge (2014)

    MATH  Google Scholar 

  47. Samal, S.S., Krishnan, J., Esfahani, A.H., Lüders, C., Weber, A., Radulescu, O.: Metastable Regimes and Tipping Points of Biochemical Networks with Potential Applications in Precision Medicine. In: Liò, P., Zuliani, P. (eds.) Automated Reasoning for Systems Biology and Medicine. CB, vol. 30, pp. 269–295. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17297-8_10

    Chapter  Google Scholar 

  48. Schnoerr, D., Sanguinetti, G., Grima, R.: Approximation and inference methods for stochastic biochemical kinetics-a tutorial review. J. Phys. A: Math. Theor. 50(9), 093001 (2017)

    Article  MathSciNet  Google Scholar 

  49. Soltani, M., Vargas-Garcia, C.A., Antunes, D., Singh, A.: Intercellular variability in protein levels from stochastic expression and noisy cell cycle processes. PLoS Comput. Biol. 12(8), e1004972 (2016)

    Article  Google Scholar 

  50. Taniguchi, Y., Choi, P., Li, G., Chen, H., Babu, M., Hearn, J., Emili, A., Xie, X.: Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells. Science 329, 533–538 (2010)

    Article  Google Scholar 

  51. Thattai, M., van Oudenaarden, A.: Intrinsic noise in gene regulatory networks. Proc. Natl. Acad. Sci. U.S.A. 98(15), 8614–8619 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Comenius University, 84248, Bratislava, Slovakia

    Candan Çelik & Pavol Bokes

  2. Mathematical Institute, Slovak Academy of Sciences, 81473, Bratislava, Slovakia

    Pavol Bokes

  3. Department of Electrical and Computer Engineering, University of Delaware, Newark, Delaware, 19716, USA

    Abhyudai Singh

Authors
  1. Candan Çelik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavol Bokes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abhyudai Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavol Bokes .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Alessandro Abate

  2. Department of Computer and Information Science, University of Konstanz, Konstanz, Germany

    Tatjana Petrov

  3. Department of Computer Science, Saarland University, Saarbrücken, Germany

    Verena Wolf

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Çelik, C., Bokes, P., Singh, A. (2020). Stationary Distributions and Metastable Behaviour for Self-regulating Proteins with General Lifetime Distributions. In: Abate, A., Petrov, T., Wolf, V. (eds) Computational Methods in Systems Biology. CMSB 2020. Lecture Notes in Computer Science(), vol 12314. Springer, Cham. https://doi.org/10.1007/978-3-030-60327-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-60327-4_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-60326-7

  • Online ISBN: 978-3-030-60327-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation