- Physical Review Journals
All JournalsPhysics Magazine

PRX Life

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Cyclo-Stationary Distributions of mRNA and Protein Counts for Random Cell Division Times

Syed Yunus Ali1,*, Aditya Saran1, Ashok Prasad2,†, Abhyudai Singh3,‡, and Dibyendu Das1,4,§

  • *Contact author: syed_24@iitb.ac.in
  • †Contact author: ashok.prasad@colostate.edu
  • ‡Contact author: absingh@udel.edu
  • §Contact author: dibyendudas@iitb.ac.in
PDF

PRX Life 3, 043002 – Published 7 October, 2025

DOI: https://doi.org/10.1103/ysj3-gkrr

Abstract

There is a long history of using experimental and computational approaches to study noise in single-cell levels of mRNA and proteins. The noise originates from myriad factors: intrinsic processes of gene expression, partitioning errors during division, and extrinsic effects, such as random cell-cycle times. Although theoretical methods are well developed to analytically understand the statistics of copy numbers for fixed or Erlang distributed cell cycle times, the general problem of random division times is still open. For any random (but uncorrelated) division time distribution, we present a method to address this challenging problem and obtain exact series representations of the copy number distributions in the cyclo-stationary state. We provide explicit cell age-specific and age-averaged results, and analyze the relative contribution to noise from intrinsic and extrinsic sources. Our analytical approach will aid the analysis of single-cell expression data and help in disentangling the impact of variability in division times.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (101)

  1. A. Raj, P. van den Bogaard, S. A. Rifkin, A. van Oudenaarden, and S. Tyagi, Imaging individual mRNA molecules using multiple singly labeled probes, Nat. Methods 5, 877 (2008).
  2. A. Raj and A. Van Oudenaarden, Nature, nurture, or chance: Stochastic gene expression and its consequences, Cell 135, 216 (2008).
  3. D. Lovatt, B. K. Ruble, J. Lee, H. Dueck, T. K. Kim, S. Fisher, C. Francis, J. M. Spaethling, J. A. Wolf, M. S. Grady et al., Transcriptome in vivo analysis (TIVA) of spatially defined single cells in live tissue, Nat. Methods 11, 190 (2014).
  4. J. Cao, J. S. Packer, V. Ramani et al., Comprehensive single-cell transcriptional profiling of a multicellular organism, Science 357, 661 (2017).
  5. D. Mahdessian, A. J. Cesnik, C. Gnann et al., Spatiotemporal dissection of the cell cycle with single-cell proteogenomics, Nature (London) 590, 649 (2021).
  6. T. K. Suen, B. Al, and K. Placek, Cell-to-cell proteome variability: Life in a cycle, Sig. Transduct. Target Ther. 6, 229 (2021).
  7. A. Deloupy, V. Sauveplane, J. Robert, S. Aymerich, M. Jules, and L. Robert, Extrinsic noise prevents the independent tuning of gene expression noise and protein mean abundance in bacteria, Sci. Adv. 6, eabc3478 (2020).
  8. B. Munsky, G. Neuert, and A. van Oudenaarden, Using gene expression noise to understand gene regulation, Science 336, 183 (2012).
  9. O. Padovan-Merhar and A. Raj, Using variability in gene expression as a tool for studying gene regulation, WIREs Syst. Biol. Med. 5, 751 (2013).
  10. M. M. Saint-Antoine and A. Singh, Network inference in systems biology: Recent developments, challenges, and applications, Curr. Opin. Biotechnol. 63, 89 (2020).
  11. S. M. Shaffer, M. C. Dunagin, S. R. Torborg et al., Rare cell variability and drug-induced reprogramming as a mode of cancer drug resistance, Nature (London) 546, 431 (2017).
  12. K. S. Farquhar, D. A. Charlebois, M. Szenk, J. Cohen, D. Nevozhay, and G. Balázsi, Role of network-mediated stochasticity in mammalian drug resistance, Nat. Commun. 10, 2766 (2019).
  13. I. El Meouche, P. Jain, M. K. Jolly, and J.-P. Capp, Drug tolerance and persistence in bacteria, fungi and cancer cells: Role of non-genetic heterogeneity, Transl. Oncol. 49, 102069 (2024).
  14. C. A. Chang, J. Jen, S. Jiang et al., Ontogeny and vulnerabilities of drug-tolerant persisters in HER2+ breast cancer, Cancer Discov. 12, 1022 (2022).
  15. I. El Meouche, Y. Siu, and M. J. Dunlop, Stochastic expression of a multiple antibiotic resistance activator confers transient resistance in single cells, Sci. Rep. 6, 19538 (2016).
  16. A. Singh and L. S. Weinberger, Stochastic gene expression as a molecular switch for viral latency, Curr. Opin. Microbiol. 12, 460 (2009).
  17. P. S. Swain, M. B. Elowitz, and E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Natl. Acad. Sci. USA 99, 12795 (2002).
  18. J. M. Raser and E. K. O'Shea, Control of stochasticity in eukaryotic gene expression, Science 304, 1811 (2004).
  19. A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas, and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol. 4, e309 (2006).
  20. L. Cai, N. Friedman, and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature (London) 440, 358 (2006).
  21. Y. Taniguchi, P. J. Choi, G.-W. Li, H. Chen, M. Babu, J. Hearn, A. Emili, and X. S. Xie, Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells, Science 329, 533 (2010).
  22. A. Sanchez and I. Golding, Genetic determinants and cellular constraints in noisy gene expression, Science 342, 1188 (2013).
  23. G. Storz, J. A. Opdyke, and A. Zhang, Controlling mRNA stability and translation with small, noncoding rnas, Curr. Opin. Microbiol. 7, 140 (2004).
  24. M. Thattai and A. Van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. USA 98, 8614 (2001).
  25. J. Paulsson, Models of stochastic gene expression, Phys. Life Rev. 2, 157 (2005).
  26. A. Sanchez, H. G. Garcia, D. Jones, R. Phillips, and J. Kondev, Effect of promoter architecture on the cell-to-cell variability in gene expression, PLoS Comp. Biol. 7, e1001100 (2011).
  27. V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. USA 105, 17256 (2008).
  28. P. Bokes, J. R. King, A. T. Wood, and M. Loose, Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression, J. Math. Biol. 64, 829 (2012).
  29. A. Singh, C. A. Vargas, and R. Karmakar, Stochastic analysis and inference of a two-state genetic promoter model, in 2013 American Control Conference (IEEE, New York, 2013), pp. 4563–4568.
  30. A. Singh, B. S. Razooky, R. D. Dar, and L. S. Weinberger, Dynamics of protein noise can distinguish between alternate sources of gene-expression variability, Mol. Syst. Biol. 8, 607 (2012).
  31. Y. Wang, Z. Yu, R. Grima, and Z. Cao, Exact solution of a three-stage model of stochastic gene expression including cell-cycle dynamics, J. Chem. Phys. 159, 224102 (2023).
  32. E. D. Hawkins, J. F. Markham, L. P. McGuinness, and P. D. Hodgkin, A single-cell pedigree analysis of alternative stochastic lymphocyte fates, Proc. Natl. Acad. Sci. USA 106, 13457 (2009).
  33. R. Tsukanov, G. Reshes, G. Carmon, E. Fischer-Friedrich, N. S. Gov, I. Fishov, and M. Feingold, Timing of Z-ring localization in Escherichia coli, Phys. Biol. 8, 066003 (2011).
  34. G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, Timing the start of division in E. coli: A single-cell study, Phys. Biol. 5, 046001 (2008).
  35. A. H. K. Roeder, V. Chickarmane, A. Cunha, B. Obara, B. S. Manjunath, and E. M. Meyerowitz, Variability in the control of cell division underlies sepal epidermal patterning in Arabidopsis thaliana, PLoS Biol. 8, e1000367 (2010).
  36. E. B. Stukalin, I. Aifuwa, J. S. Kim, D. Wirtz, and S. X. Sun, Age-dependent stochastic models for understanding population fluctuations in continuously cultured cells, J. R. Soc. Interface 10, 20130325 (2013).
  37. C. A. Yates, M. J. Ford, and R. L. Mort, A multi-stage representation of cell proliferation as a Markov process, Bull. Math. Biol. 79, 2905 (2017).
  38. P. Thomas, Intrinsic and extrinsic noise of gene expression in lineage trees, Sci. Rep. 9, 474 (2019).
  39. D. Huh and J. Paulsson, Non-genetic heterogeneity from stochastic partitioning at cell division, Nat. Genet. 43, 95 (2011).
  40. D. Huh and J. Paulsson, Random partitioning of molecules at cell division, Proc. Natl. Acad. Sci. USA 108, 15004 (2011).
  41. C. J. Zopf, K. Quinn, J. Zeidman, and N. Maheshri, Cell-cycle dependence of transcription dominates noise in gene expression, PLoS Comput. Biol. 9, e1003161 (2013).
  42. N. A. Cookson, S. W. Cookson, L. S. Tsimring, and J. Hasty, Cell cycle-dependent variations in protein concentration, Nucleic Acids Res. 38, 2676 (2010).
  43. O. G. Berg, A model for the statistical fluctuations of protein numbers in a microbial population, J. Theor. Biol. 71, 587 (1978).
  44. D. R. Rigney, Stochastic model of constitutive protein levels in growing and dividing bacterial cells, J. Theor. Biol. 76, 453 (1979).
  45. F. Bertaux, S. Marguerat, and V. Shahrezaei, Division rate, cell size and proteome allocation: Impact on gene expression noise and implications for the dynamics of genetic circuits, R. Soc. Open Sci. 5, 172234 (2018).
  46. R. Dessalles, V. Fromion, and P. Robert, Models of protein production along the cell cycle: An investigation of possible sources of noise, PLoS ONE 15, e0226016 (2020).
  47. M. Soltani and A. Singh, Effects of cell-cycle-dependent expression on random fluctuations in protein levels, R. Soc. Open Sci. 3, 160578 (2016).
  48. C. H. L. Beentjes, R. Perez-Carrasco, and R. Grima, Exact solution of stochastic gene expression models with bursting, cell cycle and replication dynamics, Phys. Rev. E 101, 032403 (2020).
  49. Z. Cao and R. Grima, Analytical distributions for detailed models of stochastic gene expression in eukaryotic cells, Proc. Natl. Acad. Sci. USA 117, 4682 (2020).
  50. C. Jia, A. Singh, and R. Grima, Concentration fluctuations in growing and dividing cells: Insights into the emergence of concentration homeostasis, PLoS Comput. Biol. 18, 1 (2022).
  51. P. Fantes and P. Nurse, Control of cell size at division in fission yeast by a growth-modulated size control over nuclear division, Exp. Cell Res. 107, 377 (1977).
  52. I. G. Johnston and N. S. Jones, Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions, Proc. R. Soc. A: Math. Phys. Eng. Sci. 471, 20150050 (2015).
  53. J. Männik, P. Kar, C. Amarasinghe, A. Amir, and J. Männik, Determining the rate-limiting processes for cell division in Escherichia coli, Nat. Commun. 15, 9948 (2024).
  54. C. A. Vargas-Garcia, K. R. Ghusinga, and A. Singh, Cell size control and gene expression homeostasis in single-cells, Curr. Opin. Syst. Biol. 8, 109 (2018).
  55. G. Reshes, S. Vanounou, I. Fishov, and M. Feingold, Cell shape dynamics in Escherichia coli, Biophys. J. 94, 251 (2008).
  56. A. Zilman, V. V. Ganusov, and A. S. Perelson, Stochastic models of lymphocyte proliferation and death, PLoS ONE 5, e12775 (2010).
  57. X. Liu, J. Yan, and M. W. Kirschner, Cell size homeostasis is tightly controlled throughout the cell cycle, PLoS Biol. 22, e3002453 (2024).
  58. C. Nieto, C. A. Vargas-García, J. M. Pedraza, and A. Singh, Mechanisms of cell size regulation in slow-growing Escherichia coli cells: Discriminating models beyond the adder, npj Syst. Biol. Appl. 10, 61 (2024).
  59. M. Campos, I. V. Surovtsev, S. Kato, A. Paintdakhi, B. Beltran, S. E. Ebmeier, and C. Jacobs-Wagner, A constant size extension drives bacterial cell size homeostasis, Cell 159, 1433 (2014).
  60. C. Cadart, S. Monnier, J. Grilli, P. J. Sáez, N. Srivastava, R. Attia, E. Terriac, B. Baum, M. Cosentino-Lagomarsino, and M. Piel, Size control in mammalian cells involves modulation of both growth rate and cell cycle duration, Nat. Commun. 9, 3275 (2018).
  61. S. Taheri-Araghi, S. Bradde, J. T. Sauls, N. S. Hill, P. A. Levin, J. Paulsson, M. Vergassola, and S. Jun, Cell-size control and homeostasis in bacteria, Curr. Biol. 25, 385 (2015).
  62. C. Nieto, C. A. Vargas-García, and A. Singh, A generalized adder for cell size homeostasis: Effects on stochastic clonal proliferation, Biophys. J. 124, 1376 (2025).
  63. M. Soltani, C. A. Vargas-Garcia, D. Antunes, and A. Singh, Intercellular variability in protein levels from stochastic expression and noisy cell cycle processes, PLoS Comput. Biol. 12, e1004972 (2016).
  64. R. Perez-Carrasco, C. Beentjes, and R. Grima, Effects of cell cycle variability on lineage and population measurements of messenger RNA abundance, J. R. Soc. Interface. 17, 20200360 (2020).
  65. C. Jia and R. Grima, Frequency domain analysis of fluctuations of mRNA and protein copy numbers within a cell lineage: Theory and experimental validation, Phys. Rev. X 11, 021032 (2021).
  66. C. Nieto, C. Augusto Vargas-Garcia, and A. Singh, A moments-based analytical approach for cell size homeostasis, IEEE Control Syst. Lett. 8, 2205 (2024).
  67. C. W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer Series in Synergetics, Vol. 115 (Springer, Berlin, Heidelberg, 1985).
  68. K. R. Ghusinga, C. A. Vargas-Garcia, and A. Singh, A mechanistic stochastic framework for regulating bacterial cell division, Sci. Rep. 6, 30229 (2016).
  69. S. Iyer-Biswas, G. E. Crooks, N. F. Scherer, and A. R. Dinner, Universality in stochastic exponential growth, Phys. Rev. Lett. 113, 028101 (2014).
  70. S. Iyer-Biswas, C. S. Wright, J. T. Henry, K. Lo, S. Burov, Y. Lin, G. E. Crooks, S. Crosson, A. R. Dinner, and N. F. Scherer, Scaling laws governing stochastic growth and division of single bacterial cells, Proc. Natl. Acad. Sci. USA 111, 15912 (2014).
  71. L. Luo, Y. Bai, and X. Fu, Stochastic threshold in cell size control, Phys. Rev. Res. 5, 013173 (2023).
  72. K. Biswas and N. Brenner, Universality of phenotypic distributions in bacteria, Phys. Rev. Res. 6, L022043 (2024).
  73. D. Antunes and A. Singh, Quantifying gene expression variability arising from randomness in cell division times, J. Math. Biol. 71, 437 (2015).
  74. J. Jędrak, M. Kwiatkowski, and A. Ochab-Marcinek, Exactly solvable model of gene expression in a proliferating bacterial cell population with stochastic protein bursts and protein partitioning, Phys. Rev. E 99, 042416 (2019).
  75. N. Friedman, L. Cai, and X. S. Xie, Linking stochastic dynamics to population distribution: An analytical framework of gene expression, Phys. Rev. Lett. 97, 168302 (2006).
  76. I. Golding, J. Paulsson, S. M. Zawilski, and E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell 123, 1025 (2005).
  77. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ysj3-gkrr for derivation of the analytical cyclo-stationary distributions of mRNA and protein counts and the corresponding cumulants, a supporting simulation data, description of computational methods of series summation and kinetic Monte-Carlo simulations, and analytical results on age-specific distribution and deterministic limits of partitioning and gene expression.
  78. A. D. Bird, M. J. Wall, and M. J. Richardson, Bayesian inference of synaptic quantal parameters from correlated vesicle release, Front. Comput. Neurosci. 10, 116 (2016).
  79. E. G. Krächan, A. U. Fischer, J. Franke, and E. Friauf, Synaptic reliability and temporal precision are achieved via high quantal content and effective replenishment: Auditory brainstem versus hippocampus, J. Physiol. 595, 839 (2017).
  80. K. Rijal, N. I. C. Müller, E. Friauf, A. Singh, A. Prasad, and D. Das, Exact distribution of the quantal content in synaptic transmission, Phys. Rev. Lett. 132, 228401 (2024).
  81. Z. Vahdat, O. Gambrell, J. Fisch, E. Friauf, and A. Singh, Inferring synaptic transmission from the stochastic dynamics of the quantal content: An analytical approach, PLoS Comput. Biol. 21, e1013067 (2025).
  82. J. Yu, J. Xiao, X. Ren, K. Lao, and X. S. Xie, Probing gene expression in live cells, one protein molecule at a time, Science 311, 1600 (2006).
  83. Interestingly, the first passage times of threshold crossing of bacterial population size undergoing autocatalytic growth follows the Beta exponential distribution, as was shown by Dellbruck years before [101].
  84. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).
  85. E. O. Powell, Growth rate and generation time of bacteria, with special reference to continuous culture, Microbiology 15, 492 (1956).
  86. F. Jafarpour, C. S. Wright, H. Gudjonson, J. Riebling, E. Dawson, K. Lo, A. Fiebig, S. Crosson, A. R. Dinner, and S. Iyer-Biswas, Bridging the timescales of single-cell and population dynamics, Phys. Rev. X 8, 021007 (2018).
  87. P. Thomas, Making sense of snapshot data: Ergodic principle for clonal cell populations, J. R. Soc. Interface 14, 20170467 (2017).
  88. S. Chakrabarti, A. L. Paek, J. Reyes, K. A. Lasick, G. Lahav, and F. Michor, Hidden heterogeneity and circadian-controlled cell fate inferred from single cell lineages, Nat. Commun. 9, 5372 (2018).
  89. S. Jun, F. Si, R. Pugatch, and M. Scott, Fundamental principles in bacterial physiology—History, recent progress, and the future with focus on cell size control: A review, Rep. Prog. Phys. 81, 056601 (2018).
  90. A. Genthon, Analytical cell size distribution: Lineage-population bias and parameter inference, J. R. Soc. Interface 19, 20220405 (2022).
  91. Z. Zhang, I. Zabaikina, C. Nieto, Z. Vahdat, P. Bokes, and A. Singh, Stochastic gene expression in proliferating cells: Differing noise intensity in single-cell and population perspectives, PLoS Comput. Biol. 21, e1013014 (2025).
  92. K. Rijal, A. Prasad, A. Singh, and D. Das, Exact distribution of threshold crossing times for protein concentrations: Implication for biological timekeeping, Phys. Rev. Lett. 128, 048101 (2022).
  93. O. Padovan-Merhar, G. P. Nair, A. G. Biaesch et al., Single mammalian cells compensate for differences in cellular volume and DNA copy number through independent global transcriptional mechanisms, Mol. Cell. 58, 339 (2015).
  94. R. Losick and C. Desplan, Stochasticity and cell fate, Science 320, 65 (2008).
  95. K. Carniol, P. Eichenberger, and R. Losick, A threshold mechanism governing activation of the developmental regulatory protein σF in Bacillus subtilis, J. Biol. Chem. 279, 14860 (2004).
  96. R. Miyazaki, M. Minoia, N. Pradervand, S. Sulser, F. Reinhard, and J. R. van der Meer, Cellular variability of RpoS expression underlies subpopulation activation of an integrative and conjugative element, PLoS Genet. 8, e1002818 (2012).
  97. J. W. Little and C. B. Michalowski, Stability and instability in the lysogenic state of phage lambda, J. Bacteriol. 192, 6064 (2010).
  98. N. Q. Balaban, J. Merrin, R. Chait, L. Kowalik, and S. Leibler, Bacterial persistence as a phenotypic switch, Science 305, 1622 (2004).
  99. L. Schuh, M. Saint-Antoine, E. M. Sanford et al., Gene networks with transcriptional bursting recapitulate rare transient coordinated high expression states in cancer, Cell Syst. 10, 363 (2020).
  100. S. M. Shaffer, B. L. Emert, R. A. Reyes Hueros et al., Memory sequencing reveals heritable single-cell gene expression programs associated with distinct cellular behaviors, Cell 182, 947 (2020).
  101. M. Delbrück, Statistical fluctuations in autocatalytic reactions, J. Chem. Phys 8, 120 (1940).

Outline

Information

PRX Life 3, 043002– Published 7 October, 2025
  • Received 6 June 2025
  • Accepted 12 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation