Stochastic processes in cells are associated with fluctuations in mRNA1, protein production and degradation2,3, noisy partition of cellular components at division4, and other cell processes. Variability within a clonal population of cells originates from such stochastic processes, which may be amplified or reduced by deterministic factors5. Cell-to-cell variability, such as that seen in the heterogeneous response of bacteria to antibiotics, or of cancer cells to treatment, is understood as the inevitable consequence of stochasticity. Variability in cell-cycle duration was observed long ago; however, its sources are still unknown. A central question is whether the variance of the observed distribution originates from stochastic processes, or whether it arises mostly from a deterministic process that only appears to be random. A surprising feature of cell-cycle-duration inheritance is that it seems to be lost within one generation but to be still present in the next generation, generating poor correlation between mother and daughter cells but high correlation between cousin cells6. This observation suggests the existence of underlying deterministic factors that determine the main part of cell-to-cell variability. We developed an experimental system that precisely measures the cell-cycle duration of thousands of mammalian cells along several generations and a mathematical framework that allows discrimination between stochastic and deterministic processes in lineages of cells. We show that the inter- and intra-generation correlations reveal complex inheritance of the cell-cycle duration. Finally, we build a deterministic nonlinear toy model for cell-cycle inheritance that reproduces the main features of our data. Our approach constitutes a general method to identify deterministic variability in lineages of cells or organisms, which may help to predict and, eventually, reduce cell-to-cell heterogeneity in various systems, such as cancer cells under treatment.
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We thank H. Miyoshi at the Riken Tsukuba for the Fucci markers, N. Barkai, N. Shoresh, J. Theiler, S. Kadener, L. Glass, G. Asher, A. W. Murray and J. Paulsson for discussions, and M. Gorfine and R. Heller for advice on statistical analysis. We thank Q. Yang and A. van Oudenaarden for the Cyanobacteria data sets, and the authors of ref. 24 for making their published data available online. This work was supported by the ISF (grants no. 592/10 (N.Q.B.); no. 567/10(I.S.); and no. 9/09, 302/14 (O.A.)) and the ERC Starting Grant no. 281306 (I.S.), no. 260871 (N.Q.B.), the Chief Scientist Office of the Israel Ministry of Health and the Weinkselbaum family medical research fund (I.S.). I.S. thanks the USAID’s ASHA Program for the upgrading of the FACS laboratory. S.P.M. is supported by the Clore Foundation.
The authors declare no competing financial interests.
Extended data figures and tables
Cycle durations were binned into eight bins according to time elapsed from the beginning of the experiment to the midpoint of the cell cycles (Tmid). For each bin the mean and s.d. were calculated. Both fluorescence and bright-light exposure were minimized to prevent lengthening of the cell-cycle duration. n = 526; error bars represent the s.d.
Bar graphs of the measured correlations coefficients for Ttot of pairs of mother–daughter (red), sister (blue) and cousin (purple) cells, in different cell lines: two different clones of L1210 Fucci (n = 423, 432) and a wild-type (WT) L1210 (n = 283). The cousin–mother inequality is observed in all cases. Data are mean and s.d. of 100 independent random samples from the data set, as described in the online methods.
a, Lineage correlations in microfluidic devices (clone2) (Methods). Cells are monitored under constant media flow, keeping the sample in constant conditions while washing away potential by-products of the cells’ metabolism. Correlation between cousins remains higher than mother–daughter correlation, ruling out micro-environment bias (ρm-d = 0.3 ± 0.03; ρs-s = 0.64 ± 0.01; ρc-c = 0.57 ± 0.05; n = 381; standard deviations represent 100 independent random samples from the data set, as described in the online methods). b, c, Cycle-duration differences within quadruplet cousins. b, A schematic representation of a cousin quadruplet. c, Cell-cycle duration differences in near and far cousin pairs. No significant difference is found. (n = 66; one-tailed Wilcoxon signed rank test, α = 0.05; P > 0.15).
a, Computed dimension versus DE for a random series (red); L1210 cell-cycle duration data (blue); and the kicked cell cycle model (equations 1–4 (see Supplementary Information); green). b, Correlation dimension for surrogate data obtained from random shuffling of the cell-cycle durations. The mean and s.d. of 59 randomly shuffled data are shown (green), showing that the saturation in the correlation dimension observed for the experimental data (blue) is due to a deterministic process27. c, Correlation dimension for random data (red) and data from a simulation of the additive model developed by Cowan and Staudte17 (green).
The fraction of points closer than a distance r, C(r), is plotted versus r, on a log–log scale. for small values of r. Therefore, the correlation dimension dcorr is the saturation of the slope of log(C(r)) versus log(r), with increasing embedding dimension DE. a, Experimental data; b, Simulation data of the kicked cell cycle model. Note the saturation of the slope at DE = 3 and 4. c, Normally distributed random data with same mean and s.d. as experimental data, n = 236.
a, b, Simulation results of the kicked cell cycle model (equations 1–4). a, Pearson correlation of mother and daughter cells (ρm-d) is predicted to vary between negative and positive values, depending on k and T0. b, Grandmother–granddaughter correlations (ρgm-gd) follow a similar pattern, namely correlations that are close to zero, but can vary between slightly negative and positive values, as observed experimentally. The simulations were run on 300 lineages c, Correlation plot of experimentally observed total cell-cycle duration Ttot of grandmother and granddaughter cells (clone2) s.d. of 100 independent random samples from the data set, as described in the online methods is 0.06. Spearman; ρgm-gd = 0.22 ± 0.06, n = 380.
Extended Data Figure 7 Total cell-cycle duration (Ttot) measurements and circadian phase at birth in Cyanobacteria.
Data of circadian phase and cell cycles of single Synechococcus elongatus cells from 29 lineages was obtained from Yang et al.24 a, Mother–daughter correlation (ρm-d); b, Sister–sister correlation (ρs-s); c, Cousin–cousin correlation (ρc-c). Asterisks denote significant (P < 0.002) Spearman correlations. d, Cell-cycle duration versus circadian phase. Each dot represents a cell’s circadian phase at birth and its cell-cycle duration. Experimental data (blue) and simulation (red). The black line denotes the expected trend of the model (equation 5, see Supplementary Information). Parameters used for the simulation are shown on the right.
Extended Data Figure 8 Simulation results of the kicked cell cycle model of G1 and G2 durations fit the experimental observations.
a–c, Pearson correlations of simulated data of the total (Ttot), G1 (TG1) and G2 (TG2) durations between mother-daughter cells (red), sisters (blue) and cousins (purple) (equations 6–8, see Supplementary Information). Parameters as in Extended Data Table 1 ; β = 0.07. d, Typical plot of the simulated data for TG1 versus TG2 (in units of Tosc). The low correlation (−0.18; P = 0.07) despite the deterministic inheritance is consistent with the experiments: the correlation coefficients from 4 independent experiments range from –0.4 to 0.17.
This file contains Supplementary Text and Data, which relates to the Grassberger-Procaccia algorithm and theoretical model. (PDF 293 kb)
The video shows time lapse microscopy of dividing L1210 cells and their expression of the Fucci markers (clone 2). Cells were grown in PDMS chambers under the microscope and imaged with x20 magnification. Here a composite of phase-contrast, red fluorescence and green fluorescence is shown with time interval of 23 min. Phase contrast images were acquired at a faster rate. (AVI 7717 kb)
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Sandler, O., Mizrahi, S., Weiss, N. et al. Lineage correlations of single cell division time as a probe of cell-cycle dynamics. Nature 519, 468–471 (2015). https://doi.org/10.1038/nature14318
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