Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, Hebrew University, Faculty of Medicine, PO Box 12272, Jerusalem 91120, Israel1
Section of Molecular Cytology, Institute for Molecular Cell Biology, University of Amsterdam, BioCentrum, Kruislaan 316, 1098 SM Amsterdam, The Netherlands2
Author for correspondence: N. B. Grover. Tel: +972 2 675 8360. Fax: +972 2 643 6890. e-mail: norman{at}md.huji.ac.il
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ABSTRACT |
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Keywords: cell shape during growth cycle, correlation between cell dimensions, diameter changes with cell age, length versus volume regulation, models of individual cell growth
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INTRODUCTION |
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In the early work on bacterial shape and form, cell diameter was at the limit of optical resolution and so measurements concentrated on length. Cell length was seen to double during the cell cycle, whilst the diameter appeared to remain constant (Schaechter et al., 1958 , 1962
). Later, electron-microscopic studies confirmed the essential findings and reported only small, apparently random fluctuations in diameter (Harvey & Marr, 1966
). Trueba & Woldringh (1980)
observed that the average diameter of slow-growing cells decreased gradually as a function of cell length prior to constriction, rising again during the constriction process; this was later corroborated by Vardi & Grover (1992)
in an elaborate analysis of more than 55000 cells. Trueba & Woldringh (1980)
inferred from their findings that the diameter of individual cells decreases systematically during their elongation, possibly because of the maintenance of a constant surface-to-volume ratio during the cell cycle.
This interpretation of the observed negative correlation between cell diameter and cell length was challenged by Cooper (1989) , who proposed instead that elongating E. coli bacteria maintain a constant diameter throughout the cell cycle: because the cell wall serves as a constraining hoop, the cells can only change their diameter slowly, over the course of several generations. In this view, individual cells are assumed to initiate the various cell-cycle events at essentially the same volume and to accommodate any variation in width by a compensating change in length. Thus cells that happen to be wider than average will initiate DNA replication or cell division when they are relatively shorter, whereas narrower cells will do so when they are longer. It is this reciprocal relationship at fixed cell volumes that produces the overall negative correlation between cell diameter and cell length even though actual cell diameter at the level of the individual bacterium does not change during the cell cycle (Cooper, 1991
).
These two approaches give rise to two opposing models of shape regulation that make very different predictions about the variations in cell dimensions at a regulated cell-cycle event and about correlations between the various dimensions. The models can be described as (i) a Length model (Fig. 1, solid lines), in which cell length is the regulating dimension and so displays only minimal variation at any particular cell-cycle event (cell birth, for example: narrow distribution in Fig. 1a
but broad distribution in Fig. 1b
), and (ii) a Volume model (Fig. 1
, dashed lines), in which volume is the regulating dimension and so displays minimal variation at cell-cycle events (narrow distribution in Fig. 1b
but broad distribution in Fig. 1a
).
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In the Volume model, which could be termed a constrained hoop model, it is assumed that during growth the individual cells maintain a constant diameter imposed by a rigid cell wall (dashed line in Fig. 1f) and show no correlation between diameter and volume at any particular cell-cycle event (dashed line in Fig. 1d
). Fluctuations in cell diameter are therefore accompanied by compensatory changes in length, causing a strong negative correlation between the two (dashed line in Fig. 1c
).
Clearly, the Length model and the Volume model make very different predictions about the correlations between the various dimensions of the cell and about the variations in these dimensions at all regulated cell-cycle events. We have therefore examined both previously published (Vardi & Grover, 1992 ) and newly acquired data in the light of these two models and carried out extensive simulation studies in an attempt to distinguish between them.
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METHODS |
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Growth conditions.
Cells were grown in batch cultures using either MOPS medium (MOPS supplemented with 0·1% L-alanine; Sigma) to obtain a doubling time of 113 min, or in glucose minimal medium (Helmstetter & Cooper, 1968 ) to obtain a doubling time of 42 min. Steady-state growth was reached after culturing the cells for at least 10 generations at the same growth rate whilst maintaining cell concentration below 107 cells ml-1 by periodic dilutions. Cells were deemed to have attained steady state when either average cell size, as measured by an electronic particle counter, or their length distributions, as measured by image cytometry, no longer changed with time.
For age selection, cells in steady-state growth (doubling time 42 min) were poured over a nitrocellulose filter for the elution of baby cells according to the membrane elution technique (Helmstetter & Cooper, 1968 ).
Sample preparation.
For electron microscopy, cells were fixed by adding OsO4 to the growth medium at a final concentration of 0·1%, and prepared using the agar filtration method (Woldringh et al., 1977 ). For phase-contrast microscopy, samples from the eluted suspension of baby cells and from the original, steady-state culture were fixed by adding formaldehyde to the growth medium at a final concentration of 0·25%. The cells were then sent from Florida to Amsterdam where they were postfixed by adding OsO4 at a final concentration of 0·1%; the time between fixations was 6 d. Finally, the cells were prepared in their hydrated state on a thin slab of 1·5% agarose as described previously (Van Helvoort et al., 1998
). Sample size for phase-contrast microscopy was 488 for membrane-eluted cells and 1141 for steady-state cells.
Microscopy and data acquisition.
The agar filters were photographed in focus on 70 mm film (SD-281; Kodak) using a transmission electron microscope (EM300; Philips) at 60 kV with a final magnification of x3600; exposure time was 1·5 s. The electron micrographs were input to an image processing system (System 100; Liacom) controlled by a computer (PDP-11/23; Digital Equipment), with a spatial resolution of 0·004 µm pixel-1 and 256 grey levels. The system (Vardi, 1990 ) recognizes the cells on the micrographs and detects their boundaries automatically by means of a multi-thresholding method that locates edges along the maximum optical density gradient. The cells are characterized using a syntactical approach and their morphological properties identified, on the basis of which the various cell dimensions are then calculated. These calculations require no geometrical assumptions other than that the cell be locally radially symmetrical along its midline; thus, the dimensions of a cell reflect its actual shape rather than the usual idealizations.
A total of 55269 cells were recorded in this way, with a mean diameter±SD of 0·447±0·034 µm, a mean length of 1·36±0·29 µm and a mean volume of 0·186±0·051 µm3. (SD refers to the standard deviation; a complete list of symbols used appears in Table 1). Dividing and newborn cells were obtained from the deeply constricted cells and from their prospective daughters, respectively, chosen so that the width of the constricted region
did not exceed 50% of the mean width of the prospective daughter cells. (This threshold, 0·50, was based on the smallest value that still gave a reasonable sample size, 637 constricting cells or 1274 prospective daughters; below that level, the number of constricted cells falls off rapidly.) Constriction was taken to begin at a relative
of 0·79 (chosen for the abrupt increase in cell length beyond that point), resulting in 50796 nonconstricting cells (92%); all cells with relative
in the inclusive range 0·650·78 were considered to be in constriction onset. (This interval was selected so as to be as narrow as possible consistent with providing an adequate number of cells, 688.)
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RESULTS |
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Computation by simulation
To obtain a more quantitative comparison between the two models, simulations on populations of 106 cells were carried out as described in the Appendix, with parameter values based on actual data (Vardi & Grover, 1992 ) wherever possible; the results are summarized in Table 4
. The choices for cvD were derived from the experimental observations: Vardi & Grover (1992)
found 7·6% and Trueba & Woldringh (1980)
reported 10·7%.
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The change in rDL with CVD in the Volume model is not unexpected. As the coefficient of variation of cell diameter CVD decreases, the lengthvolume correlation increases until, for constant diameter D, it approaches unity. This follows directly from the assumed relationship between length and volume: V(a)=D2[
(a)-
D]. (It will always be somewhat less than 1, however, because rLV is actually the correlation between the volume of a cell V(a) and its length L(a), not the mean length at that age
(a), and L(a) is taken to be randomly distributed about
(a) to allow for experimental error, as described in the Appendix.) Since
is the proportion of variance in L accounted for by V, as CVD decreases there is less and less unique variance left to be attributed to the diameter and eventually, with D constant and rLV at its maximum (and rDV set at 0, as required by the Volume model), rDL approaches zero. Similar arguments apply to rDV in the Length model.
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DISCUSSION |
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A regulating dimension is defined here as one that presents a structural or biochemical constraint to an underlying control mechanism performing an essential step in the cell cycle. Stating that a particular dimension (length or volume, depending on the model) regulates cell-cycle events in steady-state growth is insufficient by itself to allow quantitative predictions of cell shape. More information is required.
Cooper (1989) augmented his original specification of control by cell volume, inspired by Koch & Schaechter (1962)
, with that of a constant diameter imposed on a growing cell by a rigid wall, the constraining hoops of the peptidoglycan chains. Trueba & Woldringh (1980)
added to control by cell length their empirical observation that diameter decreases with cell length up to constriction; instead of restraining peptidoglycan molecules, they suggested that shape was perhaps determined by a constant surface-to-volume ratio. Such a constraint will of course provide the requisite negative correlation between diameter and length unfortunately, it also predicts a strong negative diametervolume correlation, just the opposite of what is found experimentally (Table 4
). In the present Length model, therefore, we just assume a weak negative linear diameterlength correlation without specifying its origin. The implication of this is that here, in sharp contrast to the assumption in the Volume model, an individual cell can and does change its diameter during the cell cycle.
To represent the entire population of nonconstricting cells, the contributions from all the subpopulations have to be summed. For the Length model, this leads to a positive correlation between diameter and volume (solid line in Fig. 1f) because each individual component is positive (solid line in Fig. 1d
); the overall diameterlength correlation is, of course, negative as specified (solid line in Fig. 1e
). The Volume model predicts a negative correlation between diameter and length (dashed line in Fig. 1e
) due to the summing of the individual contributions, which are all negative (dashed line in Fig. 1c
); as expected, there is no correlation between diameter and volume (dashed lines in Fig. 1d
and f
).
Selection by dimension, any dimension, is not informative. In cells with the same length, the diametervolume correlation will always be large and positive whereas in cells with the same volume, the diameterlength correlation will always be large and negative. This is a geometrical imperative, quite independent of any model or growth law, and applies to all shapes in which diameter and length are both necessary and sufficient to determine volume.
A curious point should be noted here. The Volume model states that the diameter of an individual cell remains constant during its cell cycle in steady-state growth. Nonetheless, the strong negative correlation at any particular event in the cycle between diameter and length that is inherent in this model gives rise to an apparent decrease in diameter with cell age when expressed in terms of length. Looked at differently, a negative rDL does not necessarily imply that cell diameter decreases during the cell cycle. This effect of combining the contributions of separate subpopulations was first pointed out by Cooper (1989) and termed correlated variables. The same holds true in the Length model: the positive correlation at particular cell-cycle events between diameter and volume gives rise to an apparent increase in diameter with cell age when expressed in terms of volume.
The superior fit of the Length model as regards both variation (Table 2) and correlation (Table 3
), indicates that cells at particular events during their cycle exhibit constant length rather than constant volume. Why length? We have suggested previously (Woldringh et al., 1990
) that constant length could reflect the amount of DNA in a non-replicating chromosome that self-organizes into a nucleoid of fixed linear dimension. During DNA replication and cell elongation, the daughter chromosomes are segregated and eventually organized into two daughter nucleoids in a cell twice the original length.
Regardless of the actual mechanism, however, what we have shown here is that cell diameter is more likely to vary systematically with cell age than to conform to the constraints of a rigid hoop. Such behaviour would be consistent with the observed ability of cell diameter to change after a nutritional shift-up (Woldringh et al., 1980 ) or shift-down (Zaritsky et al., 1993
).
Although it seems highly unlikely that any major cell-cycle event would be regulated directly by a single dimensional parameter, the magnitude of the variations in cell volume or in cell length observed at such events could shed light on the underlying biological control system. Thus, a particularly small variation in volume at initiation of DNA replication, Boye et al. (1996) report less than 9%, might indicate a biochemical sensing mechanism for the concentration of an initiator protein (Hansen et al., 1991
), whereas a small variation in length at a particular event, onset of constriction for example, could reflect a constraint on the displacement of the nucleoid along the cell axis (Woldringh et al., 1994
).
Looked at in that way, it is not unreasonable to consider a composite model: volume regulation at initiation of chromosome replication, say, and length regulation at initiation of cell constriction. We have not been able to test this experimentally because our cells do not contain external physical markers for initiation of chromosome replication, as they do for constriction and division, and so we cannot rule it out at this stage.
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APPENDIX |
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A step-by-step description of the execution of the Length model, the more complicated of the two, is given, accompanied by an annotated flow chart (Fig. 3); the simulation of the Volume model is presented more concisely, without a figure. The Appendix concludes with a summary of our test for robustness.
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Cell interdivision time
follows a Pearson type III distribution, CV=20%
Proportion of nonconstricting cells=90%
Mean cell diameter
=0·440 µm, CV=5%
For the Length model
Cell length determines the timing of cell-cycle events
Mean cell length at age a,
(a), increases exponentially with cell age; L(a) is normally distributed about
(a), CV at controlled cell-cycle event=7·5%,
=1·32 µm
Mean cell volume at age a,
(a), is determined by cell geometry; V(a) is normally distributed about
(a), CV due to experimental error=10%
Mean cell diameter decreases linearly with L; diameterlength correlation
=-0·12, D is normally distributed about
For the Volume model
Cell volume determines the timing of cell-cycle events
Mean cell volume at age a,
(a), increases exponentially with cell age; V(a) is normally distributed about
(a), CV at controlled cell-cycle event=10%,
=0·181 µm3
Mean cell length at age a,
(a), is determined by cell geometry; L(a) is normally distributed about
(a), CV due to experimental error=7·5%
Mean cell diameter is independent of V; D is normally distributed about
Implementation
The simulations were carried out on a population of 106 cells.
The Length model.
To be able to set the diameterlength correlation in advance of the simulation it is necessary to determine the value of the coefficient of variation of cell length CVL: once this is known, one can ensure that will indeed be as required (see below). The value of CVL is computed in a preliminary simulation loop, as illustrated on the left-hand side of Fig. 3
, which we now describe.
(1) The simulation begins by a decision as to the model. The properties of the Length model can be quantified as follows.
Cell length L determines the timing of all cell-cycle events and its growth is exponential:
(a)=L(0)2a, with a in units of the mean interdivision time
Cell diameter D decreases with L (
<0, here we use the value -0·12 given by Vardi & Grover, 1992
) up to constriction. The model does not specify the behaviour of constricting cells nor is there much experimental data in that range, and we make no attempt to simulate cell growth beyond constriction.
Cell volume V is determined by cell geometry (right circular cylinder with hemispherical polar caps): V=
D2(L-
D)
The distribution of interdivision times f(), and so also the steady-state age distribution
(a), are Pearson type III:
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(2) The simulations were carried out with parameter values based on actual data wherever possible: CVDa=0·05, CVLa=0·075, CVVa=0·10, CV=0·20,
=0·440 µm,
=1·32 µm, where CVXa denotes the coefficient of variation of dimension X in cells of age a, and
refers to the nonconstricting subpopulation only.
(3) At this point, a decision is taken as to the subpopulation to simulate. The various subpopulations were defined to correspond to those observed empirically (Vardi & Grover, 1992 ), as follows.
Dividing cells: experimentally, obtained from the deeply constricted cells, chosen so that the width of the constricted region
did not exceed 0·50 of the mean width of the prospective daughter cells, sample size 637 out of a total of 55269; in simulation, oldest 1·15% of the population
Newborn cells: experimentally, the prospective daughters of the dividing cells, sample size 1274; in simulation, youngest 2·31% of the population
Nonconstricting cells: experimentally, relative
>0·78, sample size 50796; in simulation, youngest 91·9% of the population
Onset of constriction: experimentally, relative
in the inclusive range 0·650·78, sample size 688; in simulation, youngest 1·24% of the constricting cell population
The age of the youngest cell within a particular subpopulation is labelled a1 and the oldest a2, and these were determined from
(a)da=proportion of cells younger than those of the subpopulation, and
(a)da=proportion of cells within the subpopulation. L(0) and V(0) are given by
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where ac is the value of a2 in the subpopulation of nonconstricting cells (in which, of course, a1=0).
(4) A random number u is drawn from a standard uniform distribution. Cell age a is computed from
(a)da=u. If it turns out that the cell is not a member of the subpopulation being simulated (that is, if a<a1 or a>a2), we reject it and repeat the draw. If it does belong to the appropriate subpopulation, we proceed by computing the mean cell length at that age
(a) from
(a)=L(0)2a and then drawing a second random number z1, this time from a standard normal distribution. We have assumed that the length of a particular cell at any age L(a) is normally distributed about its mean at that age
(a), and so L(a)=
(a)[1+z1CVLa], where CVLa is the coefficient of variation of cell length at age a, a fixed value, part of the input data, and set at 0·075.
(5) Two more random numbers z2 and z3 are drawn from the standard normal distribution and discarded; this ensures that the main loop (right-hand side of Fig. 3) produces the same cell sample as the preliminary one (left-hand side). Various counters are updated, such as
L(a) and
L2(a), that will be used to compute CVL. When the total number of cells examined, including those not part of the subpopulation being studied, reaches one million, we exit the preliminary loop and compute CVL, the coefficient of variation of length in the entire sample. This is to be distinguished from CVLa which, of necessity, is always smaller than CVL. We are now ready for the main simulation process.
(6) After resetting the various counters, we compute an interim variable '=
CVL/CVLa. This is done to ensure that the overall correlation between D and L is
(see below). As stated previously, the parameter values used are based on actual data wherever possible. Nonetheless, in order to test the robustness of our conclusions, we repeated the simulations for a large range of parameter values, as listed under the subheading Robustness towards the end of this Appendix. In particular, we also tested CVLa=0, and the extra decision boxes in the main loop were inserted in order to cover that condition. Apart from this, the main loop is very similar to the preliminary loop down to the calculation of L(a). At that stage we draw a third random number z2 (or compute it, in the case CVLa=0) and define D' as D'=z1
'+z2
.
(7) Because z1 and z2 have mean 0 and variance 1 and are mutually independent, D' also has mean 0 and variance 1. Furthermore, the correlation between D' and z1, and hence between D' and L(a), is '. Since
'=
CVL/CVLa, the correlation between D' and the overall L is simply
, as required. Because D is assumed to be normally distributed about
with a coefficient of variation CVDa independent of cell age, D is given by D=
[1+D'CVDa]; this assures that D has a mean of
, a coefficient of variation of CVDa and a correlation with L of
via D'. The mean volume at that age
(a) is computed from the assumed cell shape of a right circular cylinder with hemispherical polar caps:
(a)=
D2[L(a)-
D]. Since V(a) is taken to be normally distributed about
(a) with a fixed coefficient of variation of CVVa, a fourth random number z3 is used to compute it: V(a)=
(a)[1+z3CVVa].
At this stage we update the various counters of the main loop, D,
D2, sample size n,
L(a),
L2(a),
V(a) and
V2(a), and check to see whether the total number of cells examined in this loop, including those not part of the subpopulation being studied, has reached one million. When it has, we exit the main loop and compute the final output data: the coefficients of variation CVD, CVL and CVV, and the correlation coefficients rDL, rDV and rLV.
The Volume model.
The Volume model, as mentioned above, is much simpler. In particular, there is no preliminary loop and no need for '. A random number u is drawn from a standard uniform distribution and cell age a determined as in the Length model. For cells belonging to the subpopulation being studied,
(a) is computed from
(a)=V(0)2a. A second random number z1, this time drawn from a standard normal distribution, is used to compute V(a): V(a)=
(a)[1+z1CVVa]. A third random number z2, again from a standard normal distribution, is drawn and D computed from D=
[1+z2CVDa].
(a) is then obtained from the assumed cell geometry:
(a)=4V(a)/
D2+
D. Finally, a fourth random number z3, once again from a standard normal distribution, is used to get L(a): L(a)=
(a)[1+z3CVLa]. The various counters are updated, as in the case of the Length model, and when the total number of cells drawn has reached one million, we exit the loop and compute the three coefficients of variation and the three correlation coefficients.
Robustness
The results reported here were obtained with the parameters and assumptions listed above, but many other combinations were tested. In no case were the implications or inferences affected, although the actual numerical output of the simulations did differ somewhat with each combination, usually by less than 1%.
Proportion of constricting cells: 8095%
CV of interdivision time: 025%
CV of cell diameter: 2·510%
CV at controlled cell-cycle event (V for the Volume model, L for the Length model): 020%
CV due to experimental error (L in the Volume model, V in the Length model): 020%
Cell shape: right circular cylinder without polar caps
Cell growth [
(a) in the Volume model,
(a) in the Length model]: increases linearly with cell age
Cell interdivision time: follows a normal distribution
Sample size: 105107 cells
Random number generation: three different seeds
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ACKNOWLEDGEMENTS |
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REFERENCES |
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Received 5 July 2000;
revised 29 August 2000;
accepted 8 September 2000.