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Joint Distribution of Protein Concentration and Cell Volume Coupled by Feedback in Dilution

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Computational Methods in Systems Biology (CMSB 2023)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 14137))

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Abstract

We consider a protein that negatively regulates the rate with which a cell grows. Since less growth means less protein dilution, this mechanism forms a positive feedback loop on the protein concentration. We couple the feedback model with a simple description of the cell cycle, in which a division event is triggered when the cell volume reaches a critical threshold. Following the division we either track only one of the daughter cells (single cell framework) or both cells (population framework). For both frameworks, we find an exact time-independent distribution of protein concentration and cell volume. We explore the consequences of dilution feedback on ergodicity, population growth rate, and the bias of the population distribution towards faster growing cells with less protein.

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References

  1. Albayrak, C., et al.: Digital quantification of proteins and mRNA in single mammalian cells. Mol. Cell 61(6), 914–924 (2016). https://doi.org/10.1016/j.molcel.2016.02.030

    Article  CAS  PubMed  Google Scholar 

  2. Athreya, K.B., Ney, P.E., Ney, P.: Branching processes. Courier Corporation (2004)

    Google Scholar 

  3. Backenköhler, M., Bortolussi, L., Großmann, G., Wolf, V.: Abstraction-guided truncations for stationary distributions of Markov population models. In: Quantitative Evaluation of Systems: 18th International Conference, QEST 2021, Paris, France, August 23–27, 2021, Proceedings 18, pp. 351–371. Springer (2021). https://doi.org/10.1007/978-3-030-85172-9_19

  4. Backenköhler, M., Bortolussi, L., Wolf, V.: Variance reduction in stochastic reaction networks using control variates. In: Principles of Systems Design: Essays Dedicated to Thomas A. Henzinger on the Occasion of His 60th Birthday, pp. 456–474. Springer (2022)

    Google Scholar 

  5. Beentjes, C.H., Perez-Carrasco, R., Grima, R.: Exact solution of stochastic gene expression models with bursting, cell cycle and replication dynamics. Phys. Rev. E 101(3), 032403 (2020). https://doi.org/10.1103/PhysRevE.101.032403

    Article  CAS  PubMed  Google Scholar 

  6. Bokes, P.: Heavy-tailed distributions in a stochastic gene autoregulation model. J. Stat. Mech: Theory Exp. 2021(11), 113403 (2021). https://doi.org/10.1088/1742-5468/ac2edb

    Article  CAS  Google Scholar 

  7. Bokes, P.: Stationary and time-dependent molecular distributions in slow-fast feedback circuits. SIAM J. Appl. Dyn. Syst. 21(2), 903–931 (2022). https://doi.org/10.1137/21M1404338

    Article  Google Scholar 

  8. Bokes, P., Singh, A.: Protein copy number distributions for a self-regulating gene in the presence of decoy binding sites. PLoS ONE 10(3), e0120555 (2015). https://doi.org/10.1371/journal.pone.0120555

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Bokes, P., Singh, A.: Cell volume distributions in exponentially growing populations. In: Computational Methods in Systems Biology: 17th International Conference, CMSB 2019, Trieste, Italy, September 18–20, 2019, Proceedings 17. pp. 140–154. Springer (2019). https://doi.org/10.1007/978-3-030-31304-3_8

  10. Bokes, P., Singh, A.: Controlling noisy expression through auto regulation of burst frequency and protein stability. In: Hybrid Systems Biology: 6th International Workshop, HSB 2019, Prague, Czech Republic, April 6–7, 2019, Selected Papers 6, pp. 80–97. Springer (2019). https://doi.org/10.1007/978-3-030-28042-0_6

  11. Cai, L., Friedman, N., Xie, X.S.: Stochastic protein expression in individual cells at the single molecule level. Nature 440(7082), 358–362 (2006). https://doi.org/10.1038/nature04599

    Article  CAS  PubMed  Google Scholar 

  12. Çelik, C., Bokes, P., Singh, A.: Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions. In: Computational Methods in Systems Biology: 18th International Conference, CMSB 2020, Konstanz, Germany, September 23–25, 2020, Proceedings, pp. 27–43. Springer (2020). DOI: https://doi.org/10.1007/978-3-030-60327-4_2

  13. Çelik, C., Bokes, P., Singh, A.: Protein noise and distribution in a two-stage gene-expression model extended by an mRNA inactivation loop. In: Computational Methods in Systems Biology: 19th International Conference, CMSB 2021, Bordeaux, France, September 22–24, 2021, Proceedings 19, pp. 215–229. Springer (2021). https://doi.org/10.1007/978-3-030-85633-5_13

  14. Dawson, D.A., Maisonneuve, B., Spencer, J., Dawson, D.: Measure-valued Markov processes. Springer (1993)

    Google Scholar 

  15. De Jong, H., et al.: Mathematical modelling of microbes: metabolism, gene expression and growth. J. R. Soc. Interface 14(136), 20170502 (2017). https://doi.org/10.1098/rsif.2017.0502

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Dekel, E., Alon, U.: Optimality and evolutionary tuning of the expression level of a protein. Nature 436(7050), 588–592 (2005). https://doi.org/10.1038/nature03842

    Article  CAS  PubMed  Google Scholar 

  17. Desoeuvres, A., Szmolyan, P., Radulescu, O.: Qualitative dynamics of chemical reaction networks: an investigation using partial tropical equilibrations. In: Computational Methods in Systems Biology: 20th International Conference, CMSB 2022, Bucharest, Romania, September 14–16, 2022, Proceedings, pp. 61–85. Springer (2022). https://doi.org/10.1007/978-3-031-15034-0_4

  18. Doumic, M., Hoffmann, M.: Individual and population approaches for calibrating division rates in population dynamics: Application to the bacterial cell cycle. arXiv preprint arXiv:2108.13155 (2021).

  19. Duso, L., Zechner, C.: Stochastic reaction networks in dynamic compartment populations. Proc. Natl. Acad. Sci. 117(37), 22674–22683 (2020). https://doi.org/10.1073/pnas.2003734117

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  20. Facchetti, G., Chang, F., Howard, M.: Controlling cell size through sizer mechanisms. Curr. Opinion Syst. Biol. 5, 86–92 (2017). https://doi.org/10.1016/j.coisb.2017.08.010

    Article  Google Scholar 

  21. Fraser, L.C., Dikdan, R.J., Dey, S., Singh, A., Tyagi, S.: Reduction in gene expression noise by targeted increase in accessibility at gene loci. Proc. Natl. Acad. Sci. 118(42), e2018640118 (2021). https://doi.org/10.1073/pnas.2018640118

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  22. Friedman, N., Cai, L., Xie, X.S.: Linking stochastic dynamics to population distribution: an analytical framework of gene expression. Phys. Rev. Lett. 97(16), 168302 (2006). https://doi.org/10.1103/PhysRevLett.97.168302

    Article  CAS  PubMed  Google Scholar 

  23. Genthon, A.: Analytical cell size distribution: lineage-population bias and parameter inference. J. R. Soc. Interface 19(196), 20220405 (2022). https://doi.org/10.1098/rsif.2022.0405

    Article  PubMed  Google Scholar 

  24. Holehouse, J., Cao, Z., Grima, R.: Stochastic modeling of autoregulatory genetic feedback loops: a review and comparative study. Biophys. J . 118(7), 1517–1525 (2020). https://doi.org/10.1016/j.bpj.2020.02.016

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Huang, G.R., Saakian, D.B., Rozanova, O., Yu, J.L., Hu, C.K.: Exact solution of master equation with Gaussian and compound Poisson noises. J. Stat. Mech: Theor. Exp. 2014(11), P11033 (2014). https://doi.org/10.1088/1742-5468/2014/11/P11033

    Article  Google Scholar 

  26. Janson, S.: Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110(2), 177–245 (2004). https://doi.org/10.1016/j.spa.2003.12.002

    Article  Google Scholar 

  27. Jędrak, J., Kwiatkowski, M., Ochab-Marcinek, A.: Exactly solvable model of gene expression in a proliferating bacterial cell population with stochastic protein bursts and protein partitioning. Phys. Rev. E 99(4), 042416 (2019). https://doi.org/10.1103/PhysRevE.99.042416

    Article  PubMed  Google Scholar 

  28. Jędrak, J., Ochab-Marcinek, A.: 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(3), 032401 (2016). https://doi.org/10.1103/PhysRevE.94.032401

    Article  CAS  PubMed  Google Scholar 

  29. Jia, C., Grima, R.: Coupling gene expression dynamics to cell size dynamics and cell cycle events: exact and approximate solutions of the extended telegraph model. Iscience 26(1), 105746 (2023). https://doi.org/10.1016/j.isci.2022.105746

    Article  CAS  PubMed  Google Scholar 

  30. Jia, C., Singh, A., Grima, R.: Cell size distribution of lineage data: analytic results and parameter inference. Iscience 24(3), 102220 (2021). https://doi.org/10.1016/j.isci.2021.102220

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  31. Jia, C., Singh, A., Grima, R.: Concentration fluctuations in growing and dividing cells: insights into the emergence of concentration homeostasis. PLoS Comput. Biol. 18(10), e1010574 (2022). https://doi.org/10.1371/journal.pcbi.1010574

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  32. Jia, C., Xie, P., Chen, M., Zhang, M.Q.: Stochastic fluctuations can reveal the feedback signs of gene regulatory networks at the single-molecule level. Sci. Rep. 7(1), 1–9 (2017). https://doi.org/10.1038/s41598-017-15464-9

    Article  CAS  Google Scholar 

  33. Jia, C., Zhang, M.Q., Qian, H.: Emergent Lévy behavior in single-cell stochastic gene expression. Phys. Rev. E 96(4), 040402 (2017). https://doi.org/10.1103/PhysRevE.96.040402

    Article  PubMed  Google Scholar 

  34. Kumar, N., Platini, T., Kulkarni, R.V.: Exact distributions for stochastic gene expression models with bursting and feedback. Phys. Rev. Lett. 113(26), 268105 (2014). https://doi.org/10.1103/PhysRevLett.113.268105

    Article  CAS  PubMed  Google Scholar 

  35. Kumar, N., Singh, A., Kulkarni, R.V.: Transcriptional bursting in gene expression: analytical results for general stochastic models. PLoS Comput. Biol. 11(10), e1004292 (2015). https://doi.org/10.1371/journal.pcbi.1004292

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  36. Mackey, M.C., Tyran-Kaminska, M., Yvinec, R.: Dynamic behavior of stochastic gene expression models in the presence of bursting. SIAM J. Appl. Math. 73(5), 1830–1852 (2013). https://doi.org/10.1137/12090229X

    Article  Google Scholar 

  37. Molenaar, D., Van Berlo, R., De Ridder, D., Teusink, B.: Shifts in growth strategies reflect tradeoffs in cellular economics. Mol. Syst. Biol. 5(1), 323 (2009). https://doi.org/10.1038/msb.2009.82

    Article  PubMed  PubMed Central  Google Scholar 

  38. Nieto-Acuña, C., Arias-Castro, J.C., Vargas-García, C., Sánchez, C., Pedraza, J.M.: Correlation between protein concentration and bacterial cell size can reveal mechanisms of gene expression. Phys. Biol. 17(4), 045002 (2020). https://doi.org/10.1088/1478-3975/ab891c

    Article  CAS  PubMed  Google Scholar 

  39. Ochab-Marcinek, A., Tabaka, M.: Bimodal gene expression in noncooperative regulatory systems. Proc. Natl. Acad. Sci. 107(51), 22096–22101 (2010). https://doi.org/10.1073/pnas.1008965107

    Article  PubMed  PubMed Central  Google Scholar 

  40. Padovan-Merhar, O., et al.: Single mammalian cells compensate for differences in cellular volume and DNA copy number through independent global transcriptional mechanisms. Mol. Cell 58(2), 339–352 (2015). https://doi.org/10.1016/j.molcel.2015.03.005

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  41. Patange, O., et al.: Escherichia coli can survive stress by noisy growth modulation. Nat. Commun. 9(1), 5333 (2018). https://doi.org/10.1038/s41467-018-07702-z

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  42. Romanel, A., Jensen, L.J., Cardelli, L., Csikász-Nagy, A.: Transcriptional regulation is a major controller of cell cycle transition dynamics. PLoS ONE 7(1), e29716 (2012). https://doi.org/10.1371/journal.pone.0029716

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  43. Rotem, E., et al.: Regulation of phenotypic variability by a threshold-based mechanism underlies bacterial persistence. Proc. Natl. Acad. Sci. 107(28), 12541–12546 (2010). https://doi.org/10.1073/pnas.1004333107

    Article  PubMed  PubMed Central  Google Scholar 

  44. Shaffer, S.M., et al.: Rare cell variability and drug-induced reprogramming as a mode of cancer drug resistance. Nature 546(7658), 431–435 (2017). https://doi.org/10.1038/nature22794

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  45. Shahrezaei, V., Swain, P.S.: Analytical distributions for stochastic gene expression. Proc. Natl. Acad. Sci. 105(45), 17256–17261 (2008). https://doi.org/10.1073/pnas.0803850105

    Article  PubMed  PubMed Central  Google Scholar 

  46. Taheri-Araghi, S., et al.: Cell-size control and homeostasis in bacteria. Curr. Biol. 25(3), 385–391 (2015). https://doi.org/10.1016/j.cub.2014.12.009

    Article  CAS  PubMed  Google Scholar 

  47. Tan, C., Marguet, P., You, L.: Emergent bistability by a growth-modulating positive feedback circuit. Nat. Chem. Biol. 5(11), 842–848 (2009). https://doi.org/10.1038/nchembio.218

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  48. Thattai, M., Van Oudenaarden, A.: Intrinsic noise in gene regulatory networks. Proc. Natl. Acad. Sci. 98(15), 8614–8619 (2001). https://doi.org/10.1073/pnas.151588598

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  49. Thomas, P., Shahrezaei, V.: Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations. J. R. Soc. Interface 18(178), 20210274 (2021). https://doi.org/10.1098/rsif.2021.0274

    Article  PubMed  PubMed Central  Google Scholar 

  50. Turpin, B., Bijman, E.Y., Kaltenbach, H.M., Stelling, J.: Population design for synthetic gene circuits. In: Computational Methods in Systems Biology: 19th International Conference, CMSB 2021, Bordeaux, France, September 22–24, 2021, Proceedings 19, pp. 181–197. Springer (2021). https://doi.org/10.1007/978-3-030-85633-5_11

  51. Vadia, S., Levin, P.A.: Growth rate and cell size: a re-examination of the growth law. Curr. Opin. Microbiol. 24, 96–103 (2015). https://doi.org/10.1016/j.mib.2015.01.011

    Article  PubMed  PubMed Central  Google Scholar 

  52. Van Heerden, J.H., Kempe, H., Doerr, A., Maarleveld, T., Nordholt, N., Bruggeman, F.J.: Statistics and simulation of growth of single bacterial cells: illustrations with B. subtilis and E. coli. Sci. Reports 7(1), 16094 (2017). https://doi.org/10.1038/s41598-017-15895-4

  53. Vargas-Garcia, C.A., Ghusinga, K.R., Singh, A.: Cell size control and gene expression homeostasis in single-cells. Curr. Opinion Syst. Biol. 8, 109–116 (2018). https://doi.org/10.1016/j.coisb.2018.01.002

    Article  Google Scholar 

  54. Xia, M., Greenman, C.D., Chou, T.: PDE models of adder mechanisms in cellular proliferation. SIAM J. Appl. Math. 80(3), 1307–1335 (2020). https://doi.org/10.1137/19M1246754

    Article  PubMed  PubMed Central  Google Scholar 

  55. Zabaikina, I., Zhang, Z., Nieto, C., Bokes, P., Singh, A.: Quantifying noise modulation from coupling of stochastic expression to cellular growth: An analytical approach. bioRxiv, pp. 2022–10 (2022). https://doi.org/10.1101/2022.10.03.510723

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Acknowledgements

PB was supported by the Slovak Research and Development Agency under contract no. APVV-18-0308, and VEGA grants 1/0339/21 and 1/0755/22.

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Authors and Affiliations

  1. Comenius University, Bratislava, 84248, Slovakia

    Iryna Zabaikina & Pavol Bokes

  2. University of Delaware, Newark, DE, 19716, USA

    Abhyudai Singh

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  1. Iryna Zabaikina
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  2. Pavol Bokes
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  3. Abhyudai Singh
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Correspondence to Iryna Zabaikina .

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  1. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Jun Pang

  2. Inria Lille, Villeneuve d’Ascq, France

    Joachim Niehren

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Zabaikina, I., Bokes, P., Singh, A. (2023). Joint Distribution of Protein Concentration and Cell Volume Coupled by Feedback in Dilution. In: Pang, J., Niehren, J. (eds) Computational Methods in Systems Biology. CMSB 2023. Lecture Notes in Computer Science(), vol 14137. Springer, Cham. https://doi.org/10.1007/978-3-031-42697-1_17

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