Tight coupling of cell width to nucleoid structure in Escherichia coli

Abstract

Cell dimensions of rod-shaped bacteria such as Escherichia coli are connected to mass growth and chromosome replication. During their interdivision cycle (τ min), cells enlarge by elongation only, but at faster growth in richer media, they are also wider. Changes in width W upon nutritional shift-up (shortening τ) occur during the division process. The elusive signal directing the mechanism for W determination is likely related to the tightly linked duplications of the nucleoid (DNA) and the sacculus (peptidoglycan), the only two structures (macromolecules) existing in a single copy that are coupled, temporally and spatially. Six known parameters related to the nucleoid structure and replication are reasonable candidates to convey such a signal, all simple functions of the key number of replication positions $n(=C/\tau )$, the ratio between the rates of growth $\left({\tau }^{-1}\right)$ and of replication $\left(C^{-1}\right)$. The current analysis of available literature-recorded data discovered that, of these, nucleoid complexity $NC\left[=\left(2^n-1\right)/\left(n\times \ln 2\right)\right]$ is by far the most likely parameter affecting cell width W. The exceedingly high correlations found between these two seemingly unrelated measures (NC and W) indicate that coupling between them is of major importance to the species’ survival. As an exciting corollary, to the best of our knowledge, a new, indirect approach to estimate DNA replication rate is revealed. Potential involvement of DNA topoisomerases in W determination is also proposed and discussed.

Significance

The analysis described in this article concludes that, under dynamic steady states of exponential growth, there is a biologically unusual low coefficient of variation between bacterial cell width W and chromosome structure, so-called nucleoid complexity $NC\left[=\left(2^n-1\right)/\left(n\times \ln 2\right)\right]$, where $n\left[=C/\tau \right]$ is the ratio between the time to replicate the chromosome C and that to duplicate the cell τ. In addition, an innovative, indirect method is described to estimate C from (τ, W) data sets. DNA topoisomerase may be involved in W determination.

Introduction

The Copenhagen School

Bacterial Physiology was established as a Quantitative Biology field in Copenhagen during the 1950s and intensively overviewed then (1) and again lately (2). Its major outlook is the coordination between bacterial growth and the cell cycle in rigorous molecular biology and physics terms (3). Elucidating the underlying principles of interactions between macromolecules, metabolites, and structures clarifies some macroscopic and microscopic observations. Cell dimensions, however, are yet to be better explained, as described here (and see (4)).

Bacterial growth, dimensions, and the cell cycle (and see Appendix)

Cells of bacillary (rod-shaped) bacteria such as Escherichia coli grow by elongation and divide perpendicularly at mid-point to two identical daughters. Under dynamic steady state (5), cell mass (and volume) increases exponentially during its individual division cycle and in its culture mean size (6)—with a population doubling time (τ min) that depends on medium composition (7). At 37°C, with a moderate to fast mass growth rates (with τ_m < $\approx$ 70 min) (8), a cell divides $D\approx 20$ min after termination of a round of replication that initiated $\approx 40$ min (C) earlier. Replication of the single, circular chromosome, bidirectionally from oriC to terC, is a linear process, initiating at a strain-specific constant cell mass per oriC, Mi, or $2^{\mathrm{n}}$-multiple (here, n = 1, 2, 3, etc.) of Mi (9). Under conditions of $\tau <\left(C+D\right)$, replication-initiation occurs in the mother cell, and when $\tau <C$ initiation happens before the previous replisome (Appendix) terminates its ongoing round, hence the multiforked replicating chromosome contains more DNA. This model generates a larger mean cell $M\left(=\ln 2\times Mi\times 2^{\left(C+D\right)/\tau }\right)$ containing more DNA (in genome equivalents) $G\left(=\left(\tau /C\ln 2\right)\times \left(2^{\left(C+D\right)/\tau }-2^{D/\tau }\right)\right)$, although smaller DNA concentration $\left[G/M=\left(1-2^{-n}\right)/\left(Mi\times n\times \ln 2\right)\right]$, where $n\left[=C/\tau \right]$, when growing faster in richer media (8). Such cells are not only longer by default, but surprisingly also wider (7,10). The question whether the larger G in faster growing cells requires wider cells for proper segregation and partition before division is still moot.

During nutritional shift-up transition from poorer to richer medium (11), cells overshoot their length during a “rate maintenance” period of $\approx \left(C+D\right)$ before divisions are enhanced, and width W expands at the divisome to form temporarily “pear-shaped” cells (12). Thus, the spatial and temporal coupling between DNA structure/replication/segregation and peptidoglycan (PG) synthesis seems to be associated with determination of W during the division process itself. These two essential macromolecules (DNA and PG) and the structures they form (nucleoid and sacculus, respectively) are the only two existing in a single copy per cell each and must therefore coordinate their duplications for species’ survival. As the carrier of genetic information, it is reasonable for the nucleoid to “instruct” the PG synthetic machinery “when” and “where” to divide rather than the other way around. Much of the mechanism for the division process in the divisome (after replication-termination and at mid-cell) is known (e.g., (13)), but very little is known about “how” cell width W is determined. Here, we attempt to find out which parameter of chromosome content/replication/structure (14) is most likely, statistically, to affect W.

The bacterial chromosome

Six parameters of chromosome structure and reproduction are known ((14); Appendix, adapted from (4)):

• i)the key number of “replication positions” $n=\left(C/\tau \right)$, i.e., sets of replication “forks” synchronously initiating from all oriC sites existing at the time of replication-initiation, and the other five that depend on it, as follows:
• ii)the number of replicating forks $F\left(=\left(2^n-1\right)\right)$, i.e., replisomes,
• iii)the ratio between the frequency of oriC (site of replication-initiation) and that of terC (termination site), $o/t$ $(=2^n)$,
• iv)the amount of DNA in genome equivalents associated with a single terC, i.e., a chromosome (14,15), later termed “nucleoid complexity” (16), $NC\left(=\left(2^n-1\right)/\left(n\times \ln 2\right)\right)$,
• v)the cell’s “DNA concentration” $\left(G/M=\left(1-2^{-n}\right)/\left(Mi\times n\times \ln 2\right)\right)$ (17), and
• vi)the amount of DNA per cell G (8), the only parameter that depends on D as well.

Results and discussion

Choosing nucleoid complexity NC

Linear regression analysis of the relationships between cell width W and each of the six parameters in E. coli growing at seven moderate and fast rates (18), with the conventional C of 40 min (8,18), is displayed in Fig. 1. The linear equations (i.e., $W=\mathrm{b}\times P$, where P is the parameter value) and the R² values obtained are presented for each in the adjacent insets. The best fit, with R² = 0.9994, is obtained for nucleoid complexity NC, but the values for the other five parameters (ranging from 0.9584 to 0.9963) are not much lower numerically. Table 1 summarizes (using Excel worksheets) the relationships and the statistics between W and these parameters. Least-square fitting of the data is usually a very good way of presenting the issue. Here however, SDs of W vs. all parameters do not accentuate the extremely low values of the SD for the NC parameter. The differences are accentuated by examining the values of coefficients of variation (CVs) of the ratios of each, $\mathrm{b}=W/P$, where P is the parameter value. The remarkably low CV of 1.5% in the ratio $W/NC$ (=b; Table 1) is rare in biological systems hence reinforcing the suggestion (15,16,4) that it is not simply fortuitous. The outstandingly higher CV values (16.8–40.6%) of the ratios between W and each of the other five n-dependent parameters (items i–iii, v, and vi) are also consistent with the involvement of NC (iv) in W determination.

Figure 1.

Linear regression analysis of the relationships between cell width W and each of the six parameters in E. coli growing at seven moderate and fast rates, with the conventional C = 40 min.

Table 1.

Correlations between E. coli cell width W and six n-dependent parameters related to nucleoid structure and replication, assuming C = 40 min

	τ	W	n	F	o/t	NC	G/M	G	W/n	W/F	W/(o/t)	W/NC	W/(G/M)	W/G
	17.40	0.98	2.299	3.921	4.921	2.460	0.500	5.458	0.426	0.250	0.199	0.398	1.960	0.180
	22.22	0.80	1.800	2.483	3.483	1.990	0.571	3.713	0.444	0.322	0.230	0.402	1.400	0.215
	26.67	0.72	1.500	1.828	2.828	1.758	0.622	2.957	0.480	0.394	0.255	0.409	1.158	0.243
	30.77	0.67	1.300	1.462	2.462	1.623	0.659	2.546	0.515	0.458	0.272	0.413	1.017	0.263
	37.50	0.61	1.067	1.095	2.095	1.480	0.707	2.143	0.572	0.557	0.291	0.412	0.863	0.285
	50.85	0.55	0.787	0.725	1.725	1.330	0.771	1.746	0.699	0.759	0.319	0.414	0.714	0.315
	51.28	0.55	0.780	0.717	1.717	1.326	0.772	1.738	0.705	0.767	0.320	0.415	0.712	0.316
Mean									0.549	0.501	0.269	0.409	1.118	0.260
SD									0.115	0.204	0.045	0.006	0.446	0.051
CV									0.209	0.406	0.168	0.015	0.399	0.196

Values of cell width W were measured in steady-state cultures of E. coli (18) growing with the indicated doubling times τ (leftward block), calculated values of the six parameters (middle block, in bold; see below and in text), and the respective values of the ratios between W and each of the derived parameters (rightward block, in bold) of the chromosome structure: n (positions), F (forks), o/t (oriC/terC marker frequencies), NC (nucleoid complexity, in italics), G/M (DNA concentration), and G (DNA/cell). The ratios between W and each of the six n-dependent parameters and their statistics (underlined) were calculated using conventional values (8) of C (40 min) and D (20 min, for G only).

Note that although results are given numerically in three or four digits, no inferred accuracy is indicated since no measurement errors for the data are available.

A similar analysis was performed with data (7) from the closely related species Salmonella typhimurium. The slightly higher CV (3.2%) of $W/NC$ (Table 2) in this species than in E. coli may stem from at least two reasons: a smaller number of points (four rather than seven), and the degree of accuracy in the method of measuring W (7). As observed in E. coli, the much higher CV values (19.8–65.1%) of the other five ratios (Table 2) firmly support the working hypothesis of this report that a tight linear correlation exists between W and NC, at least in the Gram-negative species studied here, E. coli and S. typhimurium (Tables 1 and 2, respectively).

Table 2.

Correlations between Salmonella typhimurium cell width W and six n-dependent parameters related to nucleoid structure and replication, assuming C = 40 min

	τ	W	n	F	o/t	NC	G/M	G	W/n	W/F	W/(o/t)	W/NC	W/(G/M)	W/G
	22.00	1.43	1.818	2.526	3.526	2.005	0.568	3.764	0.787	0.566	0.406	0.713	2.516	0.380
	32.00	1.22	1.250	1.378	2.378	1.591	0.669	2.454	0.976	0.885	0.513	0.767	1.824	0.497
	60.00	0.93	0.667	0.587	1.587	1.271	0.801	1.602	1.395	1.583	0.586	0.732	1.161	0.581
	98.00	0.87	0.408	0.327	1.327	1.156	0.871	1.331	2.132	2.661	0.656	0.753	0.999	0.653
Mean									1.322	1.424	0.540	0.741	1.625	0.528
SD									0.596	0.928	0.107	0.024	0.693	0.117
CV									0.451	0.651	0.198	0.032	0.426	0.223

Values of cell width W were measured in steady-state cultures of Salmonella typhimurium (7) growing with the indicated doubling times τ (leftward block), calculated values of the six parameters (middle block, in bold; see below and in text), and the respective values of the ratios between W and each of the derived parameters (rightward block, in bold) of the chromosome structure: n (positions), F (forks), o/t (oriC/terC marker frequencies), NC (nucleoid complexity, in italics), G/M (DNA concentration) and G (DNA/cell). The ratios between W and each of the six n-dependent parameters and their statistics (underlined) were calculated using conventional values (8) of C (40 min) and D (20 min, for G only).

Note that although results are given numerically in three or four digits, no inferred accuracy is indicated since no measurement errors for the data are available.

These results may be biased by the fixed value (40 min) assigned for C. To probe whether other values of C bring about other conclusions, Fig. 2, A and B, displays CV values of each of the six W/P (i–vi) over a wide range (1–60 min) of C, extracted from Tables S1 and S2, respectively. The results confirm the conclusion of a potential NC→W signal: refined analyses (displayed in Fig. 3A) reveal CV minima (0.5% for E. coli and 2.9% for S. typhimurium) at slightly lower values of C, 37.7 and 37.5 min, respectively. These almost identical C values of ≈37.6 min are remarkably close to the 39 min found recently (19) using the accurate qPCR method, thus backing up, yet again, the NC as the decisive parameter for W determination.

Figure 2.

A and B: CV values of each of the six W/Parameter (i–vi) as a function of C, extracted from Tables S1 and S2, respectively.

Figure 3.

A–C: CV of the ratio W/NC as a function of C, extracted from Tables S1 and S2, respectively.

A new, indirect method to independently estimate C

Accurate and reproducible measurements of W are hampered by various methods of cell preparations and optical constraints, e.g., (20). Values of C have been estimated using many modes, e.g., (8,17,18,21,22), some reviewed in (16), only two of which are direct, in vivo (23,24). The uncertainties in values of C, the exceedingly low CVs of $W/NC$ (=b; Tables 1 and 2), and the small number of parameters (C and τ only) involved in calculating NC led us to accept the possibility to obtain C values using a method completely independent of the others: finding the minimal CV value of $W/NC$ from the available measured (τ, W) data sets (Tables 1, 2, and S1). Thus, if W depends linearly on NC $\left(W=\mathrm{b}\times NC\right)$, and we examine the change of the CV of the constant b $=\left(W/NC\right)$ for all cultures as a function of C, we get the best constant value of $W/NC$ $\left(=\mathrm{b}\right)$ from the lowest CVs (Fig. 2). This happens at $C=37.7$ min and $W/NC=0.42$ for E. coli (Table S1) and at $C=37.5$ min and $W/NC=0.76$ for S. typhimurium (Table S2).

Using this method for the other P values, it is seen (Fig. 2, A and B) that either no minimum is obtained or (for $W/o/t$ only) it is about 2% for $C=21$ min in both species, much too far from the realistic value of C. We cannot explain the reason for this apparently strong coupling at ∼20 min that is near the conventional D.

Additional existing examples

Three additional sets of (τ, W) data exist for other strains of E. coli than the K12 NCM3722 of (18). For B/r H266 (Fig. 3C in (15)), the actual values of W in each of the 11 cultures (covering a wide range of τ values) are not available (C.L. Woldringh, personal communication, 2023) and were cropped from the plot in (15). The analysis was performed (not shown) with these data points, resulting in $C=47$ min and CV of $W/NC=7.3\%$, somewhat higher values than those obtained with the two major examples analyzed here (Tables 1 and 2), but again the lowest of all six W/P values. Considering the lower accuracy in the mode of cropping the measured W points, this result supports our major conclusion about the relationship between NC and W. It goes without saying that changes of W with τ (and even C; see below) may differ among strains of the same species (e.g., Fig. 3), which therefore requires a separate treatment for each.

Similar analyses of available data, albeit with only two determinations of W, were performed. For E. coli strain 15 growing in media yielding 2 different τ values (40 and 60 min; Table 4 in (25)), the analysis (Fig. 3C) results in $C=46$ min with an extremely low CV of 0.05%. Likewise, a low CV of 0.2% was obtained during a nutritional shift-up of E. coli strain B/r H266 (Fig. 1B in (12)) from slow-growing culture $(\tau =72$ min, with $W=0.6$ μm) to threeold faster ($\tau =24$ min, with $W=0.94$ μm; Fig. 3B). The fourfold higher CV here (0.2% rather than 0.05%) likely stems from the difference, although small, in the real values of C in cells growing under such a large τ range (C.E. Helmstetter, personal communication, 2023).

Thymine limitation

Coupling W to the number of replication positions $n\left(=C/\tau \right)$ is supported also, at least qualitatively, by the dramatic changes in W (25) upon modulating chromosome replication rate $\left(C^{-1}\right)$ by thymine concentration [T] supplied to thyA auxotroph (17), i.e., thymine limitation, without changing mass growth rate $\left(=60/\tau \right)$. Quantifying this relationship is impeded by the lack of a real steady state: under thymine limitation at short τ values (<∼50 min), cell size M changes continuously for long periods, and at a rate that is inversely related to [T] (Fig. 7 in (25)). This change occurs firstly by increasing W up to a limit, then by branching (26). The W limit is consistent with an existing eclipse (27,28): a minimal distance from oriC needed for a replisome to reach before the next replisome can move forward to replicate the chromosome away from the new, temporarily abandoned initiating oriC. The recent finding that the rate of in vivo replication is not constant along the chromosome but rather oscillates, and in a still enigmatic manner (24), complicates matters further, more so in thymine-limited slow replication rates as needed for accurate extensions of the analyses such as performed here.

Perspectives

The option that the presumed signal for W determination is delivered at a specific time during the cell cycle (e.g., upon termination of replication) rather than by the weighted mean value of NC during the whole cell division cycle has not skipped our minds; it may need a modified mode of calculations to that used here. To test such analyses, additional results such as in, e.g., (29), are obviously needed. The observed weak asymmetry between the two polar caps of nonseptated steady-state growing cells (30) may reflect a slight asymmetry in replication rate of the two nascent sister (or daughter) chromosomes, possibly related to the strand-specific segregation of hyperstructures, which would include those containing nucleoid associated proteins, topoisomerases (Topos), and wall-synthesizing enzymes (31). Under such conditions, the NC values between the two emerging daughter cells would differ so that the one with larger NC displays a so-called pear shape, like that observed during nutritional shift-up (12). This observation supports the existence of an NC→W signal but cannot per se distinguish between the notions of weighted mean or temporary signal.

The biochemical mechanism governing W in the model Gram-positive rod species Bacillus subtilis is operated by the opposing action of the two main cell walls’ synthetic systems (32). Similar correlation between W and directional MreB filament density/curvature in E. coli rod mutants suggests that this model may be generalized to species that elongate via the rod complex (33), but the presumed primary signal affecting W by changing growth rate through the medium composition, let alone by thymine limitation, is still elusive.

Deciphering the mechanism of the presumed signal NC→W, particularly activities of DNA topoisomerases (Topos) preceding cell division, may enlighten the cellular activities during the D period (34) and lead to design of a new family of effective antibacterial drugs. Such a “thought experiment” follows the potential of, e.g., cell size and timing of chromosome replication-initiation in the 1960s (8,9) to bring about far-reaching understanding of the function of DnaA in regulating initiation in the 1970s. The “elusive role” of E. coli’s Topo I (topA) and Topo III (topB) seem (35,36,37) to “prevent unregulated replication in the Ter region.” These enzymes reduce “the topological entanglement between daughter chromosomes to exactly zero” for each cell division. Specific, partial inhibition of the Topo(s) that decatenate the two daughter (sister) chromosomes upon replication-termination would delay the signal while NC is large and hence expand W to form pear-shaped cells as found during nutritional upshift (12) or ovoidal cells and nucleoids (pictures in Figures 4–8 in (38)). To this end, growth of temperature-sensitive mutants at intermediate temperatures may expose the mechanistic details of this signal. Anticipated backup activities of several other Topos in the vicinity of the Ter region will no doubt hamper clear-cut conclusions, but such difficulties will be solved by wise and knowledgeable molecular biophysicists. The genes encoding dispensable backup enzymes can be knocked out, for which end a unified conventional nomenclature for the numerous existing Topos, their known functions and encoding genes must be implemented. An analogous nomenclature that was developed for bacterial-derived insecticidal proteins (39) has been highly successful in advancing that field.

Most importantly perhaps, the robust results of these analyses, exceedingly low values of CV in each set of studied data, imply that coupling of W to NC is of major importance, not yet understood, to survival of (at least) the species studied so far.

Author contributions

A.Z. conceived and developed the project, gleaned data sets from the literature, wrote the manuscript, and obtained the references. C.B.H. produced the concept of using Excel to work out lowest CV for W/Parameters, built Excel tables and produced figures from data, and helped in editing the manuscript. G.Y. helped in all stages throughout: contributing ideas for the project, producing figures from data, and editing the manuscript. A.R. analyzed the results of the data and their significance, and helped to edit the manuscript.

Editor: Alexander Fletcher.

Appendix: Glossary—parameters and definitions of field-specific terms used (the cell cycle and dimensions; adapted from (4,28))

Cell growth and cycle parameters (all averages in a population)

τ, doubling time of cell or culture growing exponentially under dynamic steady state with an age a distribution function $f\left(a\right)=\ln 2\times 2^a$ (40), where $a=t/\tau$, and t is time since birth by dividing mother cell.

τ_m, doubling time of mass, equals to τ at steady state of exponential growth.

C, replication time, taken to duplicate the entire chromosome, from origin oriC to terminus terC.

D, time between replication-termination and subsequent cell division.

E, eclipse, minimal possible distance along the chromosome needed for a replisome to be away from oriC before a succeeding replisome can initiate a next replication round there.

Mi, initiation mass, cell mass per number of oriC copies at the time of replication-initiation.

n, number of “replisome positions,” sets of replication “forks,” synchronously initiating from all oriC sites existing at the time of replication-initiation, equals to $C/\tau$.

NC, nucleoid complexity, amount of DNA in genome equivalents associated with a single terC, equals to $\frac{\left(2^n-1\right)}{\left(n\times \ln 2\right)}\text{.}$

F, number of replication forks (i.e., replisomes), equal to $\left(2^n-1\right)\text{.}$

$o/t$, the ratio between the frequency of oriC (site of replication-initiation) and that of terC (termination site), equals $2^{\mathrm{n}}$.

Cell dimensions and composition

M, average cell mass in a steady-state culture growing in batch, equal to $\ln 2\times Mi\times 2^{\left(C+D\right)/\tau }$.

V, cell volume.

W, cell width.

G, amount of DNA per cell in genome equivalents.

$G/M$, DNA concentration, i.e., amount of DNA (in genome equivalents) per cell mass (or volume).

PG, peptidoglycan.

[T], concentration of thymine supplied to growth media of thyA mutants.

Hyperstructures and processes

Divisome: a contractile ring of polypeptides involved in bacterial cell division.

Replisome: a matrix of enzymes that is the site of DNA replication.

Topos: DNA topoisomerase enzymes.