Edited by: Jaan Männik, University of Tennessee, USA
Reviewed by: Vic Norris, University of Rouen, France; Christine Jacobs-Wagner, The Howard Hughes Medical Institute, USA; Marco Cosentino Lagomarsino, University Pierre et Marie Curie, France
*Correspondence: Sattar Taheri-Araghi
This article was submitted to Microbial Physiology and Metabolism, a section of the journal Frontiers in Microbiology
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How growth, the cell cycle, and cell size are coordinated is a fundamental question in biology. Recently, we and others have shown that bacterial cells grow by a constant added size per generation, irrespective of the birth size, to maintain size homeostasis. This “adder” principle raises a question as to when during the cell cycle size control is imposed. Inspired by this question, we examined our single-cell data for initiation size by employing a self-consistency approach originally used by Donachie. Specifically, we assumed that individual cells divide after constant C + D minutes have elapsed since initiation, independent of the growth rate. By applying this assumption to the cell length vs. time trajectories from individual cells, we were able to extract theoretical probability distribution functions for initiation size for all growth conditions. We found that the probability of replication initiation shows peaks whenever the cell size is a multiple of a constant unit size, consistent with the Donachie's original analysis at the population level. Our self-consistent examination of the single-cell data made experimentally testable predictions, e.g., two consecutive replication cycles can be initiated during a single cell-division cycle.
The coordination between growth and the cell cycle is a fundamental aspect of cellular physiology. The classic work of Schaechter, Maaløe and Kjelgaard established the “growth law,” which states that the average size of bacterial cells in steady-state growth condition scales exponentially with the respective average growth rate (Schaechter et al.,
In an important work, Donachie studied the consequences of the growth law and the cell cycle model together (Donachie,
In recent years, single-cell experiments have significantly improved our understanding of growth and cell-size control in bacteria [For a review see Taheri-Araghi et al. (
This work presents a single-cell version of Donachie's analysis to our data in Taheri-Araghi et al. (
We used experimental data of cell length vs. time for seven different growth conditions for
TSB | 17.1 | 2.73 |
Synthetic Rich | 22.5 | 1.64 |
Glucose+12 a.a. | 26.7 | 1.04 |
Glucose+6 a.a. | 30.2 | 0.80 |
Glucose | 37.7 | 0.59 |
Sorbitol | 50.8 | 0.46 |
Glycerol | 51.3 | 0.42 |
We apply the cell cycle model by Helmstetter and Cooper (Cooper and Helmstetter,
Since we do not have direct experimental data on the actual fluctuations of C and D periods, we cannot quantify the error arising from the retracing method. However, we can add noise to C + D extracted by fitting data to Equation (1), and use it to check robustness of our conclusions. In Appendix B, we present a detailed discussion on the effect of noise in C + D. We find that the predictions of our analysis are robust to random fluctuations in the C and D periods, unless the added noise is larger than ≳ 20% of the generation time (Figure
We provide a final self-consistency check that our single-cell analysis agrees with the population level results in Appendix C.
We computed distributions of hypothetical initiation size by retracing the single-cell length vs. time data for seven different growth conditions (Figure
Indeed, we found that the peaks of the inferred initiation size distributions collapse onto each other, with the peak positions increase in exponent of 2 from the position of the left-most peak. We then calculated inferred initiation size per replication origin (Figure
A prediction of our self-consistent analysis is the possibility of double initiations. For significant fractions of subpopulations of cells, retracing by constant C + D predicted two initiations separated by growth of a constant size per origin between them within a single generation (Figures
Another important question is whether the Helmstetter-Cooper model based on constant C + D is consistent with the adder principle. The organized pattern of inferred initiation size in Figures
Growth by a constant size per origin is consistent with the classic initiator model by Helmstetter and Cooper, stating that chromosome replication starts once the accumulation of initiators reach a critical threshold level (Cooper,
While the initiator model seems plausible for the coordination of cell size and the replication cycle, there are experimental data that cannot be explained by the initiator model. For example, it has been shown that both an ectopic origin and the original wildtype origin initiate simultaneously without significant changes in growth kinetics (Wang et al.,
In this work, we applied Donachie's self-consistent analysis to the single cell data we reported recently. With the assumption that C + D is constant for individual cells, our analysis makes two predictions that can be directly tested experimentally in the future work: (i) double initiations of chromosome replication in one division cycle, and (ii) growth by a constant size between two consecutive replication initiations. Single-cell level test of these predictions will clarify whether our assumption of constancy of C + D is valid. Cell-size dependency or large fluctuations of C + D can change these predictions.
Several recent models discussed various size control routes in bacteria (Amir,
Finally, while adder principle appears general for all bacterial organisms tested so far, eukaryotes are not perfect adder (Jun and Taheri-Araghi,
This work was supported by Paul G. Allen Foundation, the Pew Charitable Trusts, and the National Science Foundation CAREER Award (MCB-1253843) to Suckjoon Jun.
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This work was accomplished under guidance of Suckjoon Jun. The theoretical tools and analysis were developed through discussions with members of the Jun Lab in the University of California, San Diego. We thank Massimo Vergassola, Terrence Hwa, Steven D. Brown, and John T. Sauls for critical reading of the manuscript and stimulating discussions.
In this appendix we model a nutrient shift experiment where the timing of the initiation of chromosome replication and, thus, cell divisions are calculated based on the initiator model proposed and tested by Helmstetter and Cooper (Cooper,
Below is the list of the assumptions of the model:
Chromosome replication initiates once the initiators accumulate to a critical level. When a round of replication starts, initiators get destroyed.
There is a constant time gap,
During steady-state and nutrient-shift the timing of initiations of replication are given by:
In steady-state: period of initiators accumulation up to the threshold is During nutrient shift from doubling times
The cell grows exponentially and the size increase rate changes promptly upon a nutrient shift.
To begin with, consider a cell growing in steady-state condition with doubling time
We find the timing of initiations of chromosome replication, both before nutrient shift and after nutrient shift.
From the timing of replication initiations, we find timing of cell divisions, assuming every initiation results in a cell division after
From the timing of cell division we calculate cell size considering that size increases exponentially with a rate instantaneously proportional to nutrient condition.
Since multiple cell cycles can overlap in bacteria, we assume at the time of nutrient shift,
Since there is a one-to-one correspondence between initiation of replication and cell division, the time gap between the
Let's consider the next chromosome replication (the first after nutrient shift) starts at
The time
Each initiation of chromosome replication leads to a cell division after a time gap of (
Let's consider
From
Substituting
Equation (A7) denotes that if cells grow in any steady-state condition with generation time
This is the growth law (Schaechter et al.,
Consider that cells growing in a steady-state condition with generation time
Since 2
One may question the effect of noise in C + D in retracing analysis and the extent it influences the distributions reported in Figure
There are two different points that should not be mixed:
The actual fluctuations in C and D periods.
The noise that can be possibly added to retracing time, C + D.
The reading error, defined as the difference between the actual moment of initiation and the inferred moment, is a result of the combination of (i) and (ii). Regarding the point (i), we believe that there are fluctuations in C and D periods. However, to date, we do not have any experimental data measuring and addressing these fluctuations at a single-cell level. Regarding the point (ii), retracing is an indirect method of estimating initiation time and size. Adding noise to retracing time can test the extent the outcome is robust with respect to noise.
Let's consider the actual fluctuations of C + D periods (combined) have the standard deviation of σ
If both actual fluctuations of C + D and added noise are Gaussian, we can calculate the standard deviation of the reading error. If the actual initiation points happen (C + D + δ
Since δ
Therefore, to minimize the error in retracing analysis, we should minimize the noise in C + D + δ
To visualize the effect of noise in the analysis, we added Gaussian noise in retracing time and tracked the changes in the distributions of the inferred initiation sizes. As expected, the noise widens the distributions and beyond certain points, it influences the bimodal shape of the distributions. Various levels of noise is tested with standard deviations, σ
Figure
In Figure
In conclusion, for the retracing analysis used in this work, a constant value for retracing time C + D minimizes the analysis error. Our test on the distributions of the inferred initiation cell size shows that the shape of distributions are more or less conserved if the noise in the retracing time is up to ~20% of the generation time. Beyond this threshold the shape of the distributions start to change.
In this appendix, we reproduce the Donachie's analysis on the
Figure
Based on inferred moments of initiation, one can mathematically calculate the number of origins during the cell cycle. A graphical example is presented in Figure
To examine correlation between #
1As a consistency check we can also extract