Electronic Journal of Statistics

Cox Markov models for estimating single cell growth

Federico Bassetti, Ilenia Epifani, and Lucia Ladelli

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Recent experimental techniques produce thousands of data of single cell growth, consequently stochastic models of growth can be validated on true data and used to understand the main mechanisms that control the cell cycle. A sequence of growing cells is usually modeled by a suitable Markov chain. In this framework, the most interesting goal is to infer the distribution of the doubling time (or of the added size) of a cell given its initial size and its elongation rate. In the literature, these distributions are described in terms of the corresponding conditional hazard function, referred as division hazard rate. In this work we propose a simple but effective way to estimate the division hazard by using extended Cox modeling. We investigate the convergence to the stationary distribution of the Markov chain describing the sequence of growing cells and we prove that, under reasonable conditions, the proposed estimators of the division hazard rates are asymptotically consistent. Finally, we apply our model to study some published datasets of E-Coli cells.

Article information

Electron. J. Statist. Volume 11, Number 2 (2017), 2931-2977.

Received: September 2016
First available in Project Euclid: 11 August 2017

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Digital Object Identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 62N02: Estimation 62P10: Applications to biology and medical sciences
Secondary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation

Asymptotic consistency cell size growth in bacteria Cox partial likelihood division hazard rate extended Cox model positive Harris recurrent Markov chains

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Bassetti, Federico; Epifani, Ilenia; Ladelli, Lucia. Cox Markov models for estimating single cell growth. Electron. J. Statist. 11 (2017), no. 2, 2931--2977. doi:10.1214/17-EJS1306. https://projecteuclid.org/euclid.ejs/1502416820

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  • [1] Amir, A. 2014. Cell size regulation in bacteria., Physical Review Letters, 112, 208102_1-208102_5.
  • [2] Andersen, P., & Gill R. 1982. Cox’s regression model for counting processes: a large sample study., Annals of Statistics, 10, 1100–1120.
  • [3] Goldie, C. M., & Maller, R. A. 2000. Stability of perpetuities., Annals of Probability, 28, 1195–1218.
  • [4] Kennard, A. S., Osella, M. C., Javer, A., Grilli, J., Nghe, P., Tans, S., Cicuta, P., & Cosentino Lagomarsino, M. C. 2015. Individuality and universality in the growth-division laws of single E. coli cells., Physical Review E, 93, 012408
  • [5] Klein, J. P., & Moeschberger, M. L. 2003., Survival analysis: Techniques for censored and truncated data. Springer, New York.
  • [6] Liese, F., & Miescke, K. J. 2008., Statistical decision theory. Springer, New York.
  • [7] Marantan, A. & Amir, A. 2016. Stochastic modeling of cell growth with symmetric or asymmetric division., arXiv Preprint arXiv:1602.01848 [q-bio].
  • [8] Meyn, S., & Tweedie, R. L. 1993., Markov Chains and Stochastic Stability. Springer-Verlag, London.
  • [9] Osella, M., Nugent, E., & Cosentino Lagomarsino, M. 2014. Concerted control of Escherichia coli cell division., PNAS DOI: 10.1073/pnas.1313715111.
  • [10] Osella M., Tans, S. J. & Cosentino Lagomarsino, M. 2017. Step by Step, Cell by Cell: Quantification of the Bacterial Cell Cycle., Trends in Microbiology, 25, 250–256.
  • [11] Soifer, I., Robert, L., & Amir, A. 2016. Single-Cell Analysis of Growth in Budding Yeast and Bacteria Reveals a Common Size Regulation Strategy., Current Biology, 26, 356–361.
  • [12] Taheri-Araghi, S., Bradde, S., Sauls, J., Hill, N. S., Levin, P. A., Paulsson, J., Vergassola, M., & Jun, S. 2015. Cell-size control and homeostasis in bacteria., Current Biology, 25, 385–391.
  • [13] Tierney, L. 1994. Markov Chains for Exploring Posterior Distributions., Annals of Statistics, 22, 1701–1762.
  • [14] Turner, J. J., Ewald, J. C., & Skotheim, J. M. 2012. Cell size control in yeast., Current Biology, 22, R350–R359.
  • [15] Tyson, J. J., & Diekmann, O. 1986. Sloppy size control of the cell division cycle., Journal of Theoretical Biology, 118, 405–426.
  • [16] Venables, W. N., & Ripley, B. D. 2002., Modern Applied Statistics with S, Fourth Edition Springer, New York.
  • [17] Wang, P., Robert, L., Pelletier, J., Dang, W. L., Taddei, F., Wright, A. & Jun, S. 2010. Robust growth of Escherichia coli., Current Biology, 20, 1099–1103.
  • [18] Wilcox, R. R. 2009. Nonparametric Estimation in, Handbook of Computational Econometrics (eds D. A. Belsley and E. J. Kontoghiorghes), John Wiley & Sons, Ltd, Chichester.
  • [19] Zucker, D. M. 1998. Restricted mean life with covariates: modification and extension of a useful survival analysis method., Journal of the American Statistical Association, 93, 702–709.