Self-Consistent Examination of Donachie's Constant Initiation Size at the Single-Cell Level.

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.

1. Introduction

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., 1958). This is one of the first quantitative principles in bacterial physiology. Another important quantitative principle is the bacterial cell cycle model, whose two cornerstone assumptions are (i) in balanced growth the duration of replication (C period) of Escherichia coli chromosome is constant independent of the growth condition and (ii) cell divides after a constant time (C + D period) has elapsed since replication initiation (Cooper and Helmstetter, 1968; Helmstetter, 1968; Cooper, 1969).

In an important work, Donachie studied the consequences of the growth law and the cell cycle model together (Donachie, 1968). He concluded that, if both models are correct, the size of the cell per origin at the moment replication is initiated should be constant for all growth conditions. Furthermore, if the two models are correct, then the growth law can be expressed using the measured C + D as where m is the average cell size and T is average cell doubling time. In other words, Donachie was able to make experimentally testable predictions by self-consistently examining the relationship between two different assumptions. Furthermore, conversely, the predicted relationship can be used to estimate C + D using size m and the average doubling time T, which can be measured and tested independently. In Appendix A, we present another example of self-consistency check, i.e., by self-consistently combining the initiator model (Cooper, 1969; Helmstetter, 1969) and the cell cycle model, we can show that the growth law emerges.

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. (2015b) and discussions therein]. Single-cell data reveal information about fluctuations, heterogeneity and correlations between measurable parameters, which are masked in population measurements. In particular, we and others have shown that bacteria employ an “adder” principle to maintain size homeostasis during steady-state growth (Campos et al., 2014; Taheri-Araghi et al., 2015a). That is, cells grow by a constant size from birth to division, irrespective of the birth size. This automatically ensures that deviations in cell-size are corrected within a few generations. The adder principle however raises an important issue of when during the cell cycle size control is imposed.

This work presents a single-cell version of Donachie's analysis to our data in Taheri-Araghi et al. (2015a). We assume that C + D is constant for all cells. Using this assumption, we retrace C + D minutes backward in time from each cell division to extract a hypothetical initiation size of individual cells. We then ask if these assumptions lead to constant initiation size at the single-cell level. We found that, if the C + D period is indeed constant for all cells, the constant initiation size is consistent with the adder principle at the single-cell level. Another prediction of our self-consistent analysis is that a cell can initiate two rounds of replication between birth and division. These predictions can be tested experimentally to verify the validity of the assumptions.

2. Materials and methods

2.1. Experimental data on growth and division of E. coli

We used experimental data of cell length vs. time for seven different growth conditions for E. coli reported in Taheri-Araghi et al. (2015a). The media, average generation time, and average newborn size of cells are listed in Table 1. For the details of the experiments and growth media see Taheri-Araghi et al. (2015a) and its Supplementary Material. For the details of the single-cell growth experiment see Wang et al. (2010).

Table 1

Name of the growth conditions, average generation time, and average cell size at birth.

Name of growth media	Generation time (minutes)	Size at birth (μm3)
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

2.2. Retracing length vs. time data to infer initiation size

We apply the cell cycle model by Helmstetter and Cooper (Cooper and Helmstetter, 1968; Helmstetter, 1968; Cooper, 1969) to infer the initiation size. That is, we assume that individual cells initiate replication C + D minutes prior to cell division (Figure 1A). We estimate C + D self-consistently by fitting the population average size vs. growth data from Taheri-Araghi et al. (2015a) to Equation (1). The fitting outcome is that C + D = 69 min.

Figure 1

(A) Retracing cell size C + D minutes prior to cell divisions to infer size at initiation of replication. Constancy of C + D predicts some cells have two initiations in one division cell cycle. (B) Fractions of initiations that occur in generations with double initiations. (C) Distributions of hypothetical initiation size can be bimodal. Each panel refers to one growth condition where filled area show the distribution of double initiations and solid lines show the distribution of all hypothetical initiation sizes. (D) The peaks of the distributions in C collapse onto each other. (E) Inferred initiation size per origin of replication from various growth condition collapse onto each other. (F) Mother-daughter correlations of Δs/#ori (growth per origin of replication), based on inferred initiation moments. Panel (A) is reproduced from Taheri-Araghi et al. (2015a) with permission from Elsevier.

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 A2).

Figure A2

(A) To test the effect of noise in the analysis, a Gaussian noise δ with standard deviation of standard deviation of σ is added to the retracing time. (B) Various levels of noise is tested with standard deviations, σ, chosen at 0, 2.5, 5, 7.5, and 10% of C + D. (C) Solid lines refer to the distribution from whole cell population and filled area refer to cells with double initiations in a division cycle. Standard deviation, σ, in each row is the same as the one in the corresponding row in (B). The ratio of the standard deviations, σ, to the average doubling time, T, is noted on the left side of each sub panel. (D) Mother-daughter correlations of Δs/#ori with noise added in retracing.

We provide a final self-consistency check that our single-cell analysis agrees with the population level results in Appendix C.

3. Results and discussion

3.1. Distribution of inferred initiation size shows distinct peaks, consistent with donachie's constant initiation size model

We computed distributions of hypothetical initiation size by retracing the single-cell length vs. time data for seven different growth conditions (Figure 1B). All distributions showed peaks. An obvious question is whether these peaks are multiples of constant initiation size as Donachie inferred from population data. To answer this question, we overlaid the distributions (Figure 1D).

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 1E). Distributions from various growth conditions collapse on each other in the form of single-peak distributions. This is consistent with the model that replication initiates whenever the cell size per origin reaches a constant critical size, regardless of the growth condition (Donachie, 1968; Pritchard, 1968). (In Appendix D we show how the number of replication origins is calculated.)

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 1B–E). This is not what is expected from the basic assumptions of the cell cycle control, which requires one-to-one correspondence between replication cycle and division cycle (Mitchison, 1971). Since this prediction seemingly violates a basic assumption, direct experimental test at the single-cell level will be important.

3.2. Conditions for consistency of constant (C + D) model with adder principle

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 1D,E can support such consistency. Unfortunately, with our current data we cannot answer whether replication starts at a critical size or after the cell grows for a constant size per origin from previous initiation. However, we can test if the constant C + D assumption and the adder principle are consistent by mother-daughter correlations. In Figure 1F, we show that there are no significant correlations between the mother and the daughter cells in terms of added size per origin (Δs/#ori), as expected by the adder principle. That is, growth of the daughter cell by a constant Δs/#ori between initiation events is independent of the mother. Since Δs/#ori has been estimated by the constant C + D assumption, our analysis suggest that the two assumptions are mutually consistent.

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, 1969; Helmstetter, 1969). A feedback mechanism was proposed by Sompayrac and Maaloe (1973) to maintain initiator level proportional to cell size. We showed in Appendix A how the initiator and the cell cycle model by Helmstetter and Cooper can lead to the growth law.

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., 2011). Another example is synchronous replication of minichromosomes that carry similar origin of replication in cells (Messer et al., 1978; Leonard and Helmstetter, 1986). In these examples, the relationship between size and number of origins do not follow the wild-type. At this point, we do not have sufficient experimental evidence to confirm the initiator model and the critical size for initiation and its link to the adder principle. Nevertheless, one way to reconcile a consistency between adder and constant C + D is to have an adder-like behavior for cell size at the initiation of chromosome replication.

3.3. Future work

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, 2014; Campos et al., 2014; Iyer-Biswas et al., 2014a,b; Kennard et al., 2014; Osella et al., 2014; Taheri-Araghi et al., 2015a). An interesting, unresolved question is how size control principles align with the cell cycle control. For a conclusive answer, we need direct experimental data on the progression of cell cycle in individual cells.

Finally, while adder principle appears general for all bacterial organisms tested so far, eukaryotes are not perfect adder (Jun and Taheri-Araghi, 2015). Further insights on the molecular mechanism of the adder principle can be gained through experimental tests in which we can perturb the perfect adder. Previously, perturbation of cell division machinery has been experimentally linked to variations of cell size (Weart et al., 2007; Chien et al., 2012; Hill et al., 2013). The timing of replication initiation was also linked to cell size, where E. coli mutants of smaller size delay initiation until they reach the appropriate initiation size (Hill et al., 2012). Interestingly, a modest over expression of DnaA-ATP can recover the replication initiation timing. We believe experiments on wild-type or size mutants in which the rate of accumulation of possible initiators can be temporarily decoupled from cell size (with overexpression or inhibition of their expression) will reveal valuable information on the regulation of cell size and the coordination of the cell cycle with cell size.

Funding

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.

Conflict of interest statement

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.