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Cell division in Escherichia coli after changes in the velocity of DNA replication
J. theor. Biol. (1983) 102,101-120
Cell Division in Escherichia cob after Changes in the Velocity of DNA Replication H. BREMER AND L. CHUANG Biology (Received
Programs, The University of Texas at Dallas, 688, Richardson, Texas 75080, U.S.A. 20 August
P.O. Box
1981, and in final form 1 November
1982)
A method of computer analysis was developed to evaluate the kinetic changes in the rate of cell division in non-synchronous cultures of E. coli resulting from changes in the velocity or initiation of chromosome replication. This method takes into account that the cell division pathway in E. coli includes a reaction of indeterminate length described by a probability function that applies to the cell population. The analysis yields a hypothetical cell number kinetics as it would be observed if the stochastic element in the division pathway were absent. Since this derived cell number curve responds to experimentally induced perturbations of replication at defined times whereas the actual cell number curve reflects these perturbations only in a blurred fashion, replication and division events can be precisely correlated with this method. The method was applied to the evaluation of thymine starvation experiments with two YYhy- derivatives of E. coli B/r; one of the strains has a mutationally altered (60% increased) cell mass at initiation of chromosome replication. In both strains, the stochastic phase of the cell cycle had the same half-life value of 10 min and began 18 min after each termination of replication. This suggests that the time of cell division is linked to replication, not to cell mass or length. This interpretation is supported by results of experiments in which the rate of cell growth was altered at the time of thymine starvation.
Introduction In order to study causal relationships between chromosome replication events (initiation, termination) and cell division in bacteria, two experimental approaches have been used in the past; observation of replication and division in synchronous or age-fractionated bacterial cultures (e.g., Helmstetter & Cooper, 1968), or perturbation experiments with exponential cultures, where replication parameters were suddenly altered and the ensuing changes in the rate of division were observed (e.g., Meacock & Pritchard, 1975). A quantitative interpretation of either type of experiment is hampered, however, because the cell division pathway includes a reaction 101
0022-5193/83/090101+20$03.00/0
@ 1983 Academic Press Inc. (London) Ltd.
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of indeterminate length that blurs the relationship under study (Bremer & Chuang, 1981a,b). Partly because of this difficulty, it is still not clear to what extent the control of cell division in bacteria is coupled to DNA replication (see review by Helmstetter et al., 1978). In this work, a method was developed that employs a computer analysis of cell number kinetics observed after perturbations of exponential cultures. With this analysis, replication and division times can be accurately correlated without the need for synchronization of growth, which is never complete, and without the need for radioactive DNA labeling. The utility of this method is illustrated by analysing several thymine starvation experiments, using thymine-requiring derivatives of E. co/i B/r. Materials (A)
BACTERIAL
STRAINS
and Methods AND
GROWTH
MEDIA
The strains used were TJK16, a thyA deoB dnaA derivative of E. coli that has an increased “initiation mass” relative to E. coli Bfr due to the mutation in the dnaA gene; LEE130, a phe(Ap) thyA deoB derivative of E. cofi B/r obtained from LEE103. Origin and properties of TJK16 and LEE103 have been described (Choung, Estiva & Bremer, 1981). The bacteria were grown exponentially at 37°C in Medium C (Helmstetter, 1967) supplemented with 0.2% glucose, 10 P.g/ml of thymine, and (only for Lee 130) 20 pg/ml of phenylalanine. “Conditioned” medium was prepared by growing E. coli B/r A in Medium C plus glucose to an ODdeO of O-5. The bacteria were removed by filtration (Millipore Corp., 0.45 km pore size); the filtrate was diluted five-fold with the same medium, and the diluted medium was again filtered through a membrane filter of 0.2 +rn pore size. B/r
tB1
THYMINE
STARVATION
Thymine starvation was brought about by a 300-fold dilution of the cell culture (ODAeo = 0.4) into thymine-free, pre-warmed, conditioned medium. Thus, the final thymine concentration in this medium was 0.3 +g/ml. Identical results and the same final slope of the cell number curve were obtained when the initial concentration of thymine in the undiluted culture was 20, 10, 5, or 2.5 pg/ml (final concentrations after 300-fold dilution were between 0.6 and 0.075 kg/ml). This suggests, therefore, that the rate of DNA synthesis after dilution was not limited by residual exogenous thymine but rather by residual endogeneous synthesis due to leakiness of
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the thyA mutations used. This was especially true for LEE130 in which the thyA mutation was found to be temperature-sensitive (see below). The diluted cultures were incubated in a shaker waterbath. In control experiments in which the shaker was turned to high speed or was completely turned off, it was found that the shaker speed did not seem to matter in the diluted culture, but it was used at medium speed in all experiments. (0
CELL
COUNTJNG
Samples (7.5 ml) of diluted culture were added to an equal volume of counting solution (36 g NaCl, 8 ml 15% HCHO per liter of distilled water) which had been repeatedly filtered through two membrane filters (0.2 km pore size). These samples were counted with a particle counter (Coulter Electronics, 20 km aperture) equipped with a loo-channel particle size analyser. The statistical error was always less than 1%. Theory (A)
VARIABILITY
OF
THE
D-PERIOD
We have previously observed (Bremer & Chuang, 1981a, 6) that the D-period, or time between termination of a round of chromosome replication and the following cell division (Helmstetter & Cooper, 1968) varies within a population of E. coli cells. Cells that have just terminated a round of replication do not divide for a period of 17 to 18 min; this period is designated as the “minimum D-period”, D,. Then the cells start dividing at a maximum rate that immediately begins to decrease exponentially with a half-life, h of 5 to 10 min, as illustrated in Fig. l(a). Why the D-period varies in this manner is not known although it presumably reflects the involvement of a random element in the division pathway, perhaps in the initiation of septation. The curve in Fig. 1 (a) has not been observed directly because a population of bacteria that has synchronously terminated a round of replication cannot be experimentally obtained. However, the same information that is contained in the hypothetical division rate curve of Fig. l(a) can be extracted from kinetics of residual cell divisions observed when an exponential culture is subjected to an inhibition of DNA replication, e.g. by thymine starvation (Fig. l(b)). During exponential growth prior to starvation, the numbers of chromosome termini and of cells increase with the same doubling time, r; the cell number curve is displaced from the terminus curve by a length of time that defines the D-period (Fig.l(b), for t < 0; Bremer & Churchward,
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-
(mln)
1. Theoretical, idealized relationships between termination of chromosome replication and cell division. (a) Rate of cell division in a subpopulation of cells synchronously terminating a round of replication at zerotime. D, = minimum D-period = 18 min; h = half-life = 10 min = time at which the rate has decreased to l/2 of the maximum value at I = D,. The area under the curve is equal to the number of cells present in the subpopulation at t = 0; when all cells have divided at t + 00, the number is twice as great. (b) Number of chromosome termini and of cells in a hypothetical culture that was growing and dividing exponentially until zero time, when the culture was subjected to thymine starvation such that the replication velocity immediately dropped to zero. D = time distance between the terminus curve (Ter) and the cell number curve (Cells); this distance defines the D-period. D, = constant (minimum) portion, D, = variabfe portion of the D-period; d = hypothetical curve representing the number of cells that have finished D, and have entered D,. do, dl = value of dat t = 0 and at t = I,; t, = time at which the d-curve changes its slope; the slope is given by its doubling time, 7. (c) Same as (b) except that the replication velocity does not drop to zero (incomplete thymine starvation); the final cell number doubling time is 72 = 300 min. The cell number curves were calculated (see Appendix), using f0 = r1 = 43 min, D, = 18 min. h = 10 min. - - -, a different d-curve, assuming a reduced value of D,, = 4 min after tr (see text): this would result in a higher cell number curve not shown. D2 = final D-period. FIG.
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1977). After inhibition of termination by thymine starvation, the terminus curve “plateaus” while the cell number continues to increase until it reaches the plateau of the terminus curve. The shape of this cell number curve was shown to reflect the parameter D, and h (Bremer & Chuang, 1981b). Fig. l(b) also shows a curve labeled “d” which reflects the cell number curve for the hypothetical case that all cells would divide exactly 18 min ( =D, min) after termination, i.e. in the absence of a stochastic element in the division pathway. In contrast to the actual cell number curve which shows no sharp breaks, the d-curve shows sudden changes in slope in response to sudden changes in replication parameters. (B)
EVALUATION
OF
REPLICATION
PERTURBATION
EXPERIMENTS
If the replication velocity is suddenly altered to a finite value (>O), e.g. by thymine deprivation, such that terminations continue after zerotime at a reduced rate (Fig. l(c)), the cell number curve becomes more complex to analyse, but also yields additional information. In this case, the minimum D-period might still be 18 min (Fig. ((c), or it might become shorter as in the hypothetical example of Fig. l(c). For the purpose of this analysis, the experiment is dissected into a number of adjacent intervals designated by the subscript i. For each interval, three parameters are defined: ri is the doubling time of the d-curve during the interval considered, hi is the halflife parameter, and ti is the interval endpoint, given either by a break in the slope of the d-curve or as the beginning or end of the experiment. The triplet TV, ho, to refers to the time of exponential growth before the experiment (from t = -co to t = 0). These intervals are illustrated in Fig. l(c) where the d-curve has three sections, each marked by a doubling time defining its slope in the semilog plot: r. before zerotime. r1 between zero and tl, and r2 after tl. In this hypothetical case, it was assumed that r1 = TV. Thus, the three time intervals reach from --cc to 0, from 0 to tl (here tl =D,), and from t1 to the end of the experiment at 80 min ( = t2). During each time interval i for the d-curve in Fig. l(c), the half-life parameter values are ho, hl, and hZ. These values are not a priori known and might be constant during a replication perturbation experiment, such that ho = hl = hz, or they might change. In principle, it should be possible to determine changes, if any, in h from an observed cell number kinetics since these kinetics would be affected by such changes. However, not only changes in h, but also changes in 7 have to be considered. Furthermore, the times at which these changes occur, i.e. the ti-values, are generally not known (except the &-values representing the beginning and end of the experiment, f. and f2 in Fig. (1~)).
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The method of analysis used here is based on a generalisation of a theory reported previously (Bremer & Chuang, 19816). An equation was derived (equation (Al)), that describes any cell number curve from the values of T,, hi, and tie In order to evaluate the results of a replication perturbation experiment, the values for these parameters are systematically varied and used to calculate theoretical expectations for the cell number curve with which the observed data are compared. When the calculated curve fits the observed data, the onset and halflife of the stochastic phase of the division process, as well as the terminus curve and the d-curve (Fig. l(b), (c); Fig. 8), can be obtained from the parameters used for the calculation. The theoretically infinite number of combinations of parameter values required to fit the calculated curve to observed data was minimized by applying two principles: (1) If a similar experiment had been analysed earlier which gave a consistent and presumably correct interpretation, then the principle of the “minimum number of changes from experiment to experiment” was applied. For example, in the experiment of Fig. 4 below, a repeat of the experiment of Fig. 3, theoretical considerations suggested a change in the value of h at zero time. Therefore, the same values as for the experiment of Fig. 3 were used, except the value for h was systematically varied. This strategy quickly led to a curve that fitted the observed data. (2) The principle of the “minimum number of changes during the course of one experiment” was applied. This means that, in one experiment, only a few stepwise changes in parameter values are assumed to occur. For example in a thymine starvation experiment (Fig. l(b), (c)), only one break in the d-curve is expected. Such stepwise changes (break in the d-curve or change in h) lead to a change in the second derivative of the observed cell number curve; therefore, the times of change, i.e. the ti-values, can be approximately estimated by visual inspection of the cell number curve. The fact that consistent results could be obtained when these principles were followed suggests that the interpretation of data (e.g. those in Table 1 below) was essentially correct.
FIG. 2. Calculated cell number curves in hypothetical thymine starvation experiments, illustrating the effects of changes in parameter values. Each panel shows one “control” kinetics (- - -), calculated from the same set of parameter values (similar as in the experiment of Fig. 3): q = q = 45 min, q = 2000 min, D, = fI = 18 min, ho = hI = h2 = 9 min. In each panel, only one parameter was varied, whereas the other parameters were kept constant as in the control. (a) Variation in 7s = 500,1000,2000 (control), and cc min. (b) Variation in tl = 15,18 (control) or 21 min. (c) Variation in rr = 30, 45 (control), or 60 min. (d) Variation in h = 6,9 (control), or 12 min.
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I
(0)
r2 = 500’
(cl
r, = 30’
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80
108
H. (C)
EFFECT
BREMER
OF
AND
INDIVIDUAL
THE
CELL
L.
CHUANG
PARAMETER NUMBER
VALUES
ON
KINETICS
To show how the above parameters affect the cell number kinetics, individual parameter values were systematically varied and the resulting cell number kinetics were calculated (Fig. 2). In every panel of Fig. 2, one curve (dashed) was obtained with the same fixed set of parameter values (from the experiment in Fig. 3 below). The effects of changing 72 (a), t, (b), 71 (c), and hi = h2 (d) are illustrated. The value of r2 affects the kinetics only after tl (18 min in this example). Different values of Q result from incomplete thymine starvation, when the replication velocity is reduced to some finite value greater than zero. Changes in tl produce parallel (up or down) shifts of the later, nonexponential phase (after ri), and increase or shorten the duration of the initial, exponential phase of the cell number curve (Fig. 2(b)). Changes in tl, as well as in 71 (see below), relate to changes in the minimum D-period. Logically, tl cannot exceed a certain value that would bring the d-curve above the terminus curve (Fig. l(c)). Any change in 7l (Fig. 2(c)) would indicate a gradual change in the minimum D-period. 71 cannot change without altering tl; therefore the curves in Fig. 2(c), where 71 changes but tl was held constant, cannot occur in reality. In the experiments shown below, changes in r1 have not been observed. A change in the half-life parameter h alters the initial slope of the kinetics (Fig. 2(d)). Therefore, when a sudden change in the rate of cell division is observed in a culture, a change in the value of h can be inferred (see below). Application (A)
EVALUATION
OF
THYMINE WITH
STRAIN
of Theory STARVATION
EXPERIMENTS
TJK16
Figure 3(a) shows the results of an experiment reported previously (Bremer & Chuang, 19816), in which the residual increase in cell number was followed in a culture of a Thy- derivative of E. coli B/r, TJK16, that was subjected to thymine starvation by a 300-fold dilution into thymine-free medium. The cell number values have been normalized to the zero-time value by applying a linear regression analysis to the logarithms of the cell numbers of the first ten data points. The doubling time during the initial exponential phase agreed with the mass doubling time before dilution (rO = 46 min). These data were previously evaluated by a “special plot” method (Bremer & Chuang, 1981b) that is illustrated here for comparison:
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0.5
Time
after
thy -starvation
(rn11-1)
FIG. 3. Evaluation of an observed thymine starvation experiment with strain TJK16. (a) Observed cell number kinetics, C,, normalized to the cell number value at zero-time which is set at 1.0. The points shown were observed, the curve is calculated (see Appendix) with the following parameter values: r0 = r1 = 46 min; r2 = 2000 min, ti = 18 min, ho = hi = h2 = 9 min. (b) Replot of the initial points from panel (a) as J2”4’ - C, versus time. The intersection of the final section with the base line corresponds to the time ri = 18 minutes at which the rate of cell division begins to decrease; this time is interpreted as D,. (c) Replot of later data points, after t = 18 minutes, showing the difference “final plateau minus C’,” normalized to the difference at 18 minutes (set at 1.0) uusus time. The slope of the curve in the semilog plot corresponds to a halflife value h = 9 min. (d), (e) Graphic evaluation, as in panels (b), (c) of the calculated data points.
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a plot of the square root of the difference “exponential curve minus observed cell number” (Fig. 3(b)) shows that the cell number began to deviate systematically from the exponential at 18 min, which gives the value for the minimum D-period, D,. Figure 3(c) shows the deviation of the observed cell numbers after 18 min from the final “plateau”; this plot gives the halflife of the stochastic phase, equal to 9 min. Assuming, as simplest interpretation, the following parameter values: ho = h, = h2 = 9 min, 7,) = TI= 46 min, 72 = 2000 min = observed “final” slope, and ti =D, = 18 min, a theoretical expectation for the cell number kinetics was calculated from equation (Al). The resulting curve shown in Fig. 3(a) is seen to agree with the observed data points. Similarly, if the calculated kinetics are evaluated by the special plot method, it yields the same parameter values that went into its calculation (Fig. 3(d), (e)). This agreement is not a necessity (see below). Immediately after a 300-fold dilution of the culture, the cell number was frequently observed to increase with a somewhat different doubling time as compared to the growth before dilution. An extreme example is illustrated in Fig. 4, which shows a repeat of the experiment in Fig. 3, but where the initial doubling time was 33 rather than 46 min. Evaluation by the special plot method gave reduced values of D,, and h (12 and 7 min rather than 18 and 9), and when these values were used to calculate an expectation, the theoretical curve did not agree with the observed data (Fig. 4(a), short dashes). It is implicit in the theory (equation (Al)), that sudden changes in the rate of cell division in a culture cannot occur unless the half-life parameter h changes suddenly. Thus, using the “best fit method”, the apparent shorter doubling time of 33 min in the experiment of Fig. 4 was found to be the result of a sudden reduction in the value of h from 10 min before the experiment to 5 min after the dilution at zero time. With this change in II, and using the same values for TV, 72 and tl as in the experiment of Fig. 2, the theoretical cell number curve (solid curve in Fig. 3(a)) agreed with the observed data points. Figures 3(d),(e) shows that the special plot method, when applied to the theoretical curve, gives in fact erroneous values for D, and h, which means that the “special plot” analysis cannot be used if the value of h changes during the course of an experiment. However, the “best fit method” is still applicable in such cases. (B)
EFFECT
OF THE
ALTERED TIME
REPLICATION OF
CELL
INITIATION
OF
DIVISION
Figure 5 shows the results of two thymine starvation experiments with strain LEE130 that differs from TJK16 used above in its control of
REPLICATION
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1
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30 after
thy-starvation
40
I 0
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1
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(mini
FIG. 4. Effect of culture dilution on the rate of cell division; repeat of thymine starvation experiment shown in Fig. 2; see text for details. Parameter values used for calculation were: (-) 70 = 71 = 46 min, ‘~2 = 2000 min; tI = 18 min; ho = 10 min, hl = hZ = 5 min. (- - -) 70 = +, = 33 min, T* = 2000 min, tl = 12 min, ho = h, = hz = 7 min (see text).
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I 8 16 14
Time
after
thy-starvotlon
(rn~n)
FIG. 5. Two thymine starvation experiments with LEE130. The curves calculated using the parameter values given in Table 1.
points
are observed,
the
replication initiation. Due to a mutation in the dnaA gene, TJK16 has a 60% greater initiation mass than LEE130 that has the wildtype initiation mass (Choung et al., 1981). Evaluation of the experiments in Fig. 5 by the best fit method showed that the values for D, and h are essentially the same in Lee130 and TJK16, but 1 to 2 min changes in the value of h TABLE
Evaluation
starvation experiments with TJK16 and LEE130 (same experiments as shown in Figs 3, 4 and 5).
Experiment
Strain
m
Fig. 3 Fig. 4
TJK16 TJK16
3.3 3.5 3.43$ 2.8 2.8 2.66$
Fig. 5(a) Fig. 5(b)
1
of thymine
LEE130 LEE130
46 46
9 10
46 46
9 5
2000 2000
9 5
18 18
30 31
42 43
11 10
42 43
10 12
1000 1000
10 12
18 18
32 31
11fi = average cell mass = OD46c units per lo9 cells. 9: r0 = mass doubling time observed before the beginning of thymine starvation. t rr, r2, ho, hr. hl: values in min, obtained from cell number kinetics observed after thymine starvation by comparison with calculated curves; see text. The subscripts 0, 1 and 2 refer to 3 different time intervals: -co to 0 ( = to); 0 to 18 (= tt); and 18 to 80 (= r2) minutes. 7 D, = minimum D-period = t 1. $0 D-period calculated from equation (A9). $ Cell mass averages from ten further experiments.
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occurred at zerotime (Table 1). The two experiments shown are representative for the reproducibility of the results, i.e. the experiment in Fig. 4 was exceptional, as mentioned. The final slope of the cell number curves for LEE130 (Q = 1000 min) is greater than for TJK16 (T* = 2000 min; Table 1). This difference is due to a leakiness of the thyA mutation in LEE130 (at 30”, LEE130 is thA’; at 37”, it does not form colonies in the absence of thymine). In these experiments, the value of Q obtainined by the best fit method is always somewhat greater than the “visually” determined value. This reflects the fact that the experiments ended at 80 min, when the final slope was being approached, but not reached. CC) EFFECT
OF
CELL
GROWTH
To see how the rate of cell growth, i.e. cell elongation, affects the rate of cell division in the absence of replication, a culture of TJK16 growing
CJ
2
16
:
14 E
E I2 E z IO 0
20 Time after
40 thy-starvation
60 (rmn)
80
FIG. 6. Effect of growth rate on therate of cell division during thymine starvation. (a) A culture of TJK16 grown to an OD46c = 0.5 was 1: 10 diluted into the same glucose minimal medium with 10 kg/ml of thymine @-A), or into medium containing 0.2% (w/v) a-methyl glucose instead of glucose (C-O), or into medium containing 0.2% glucose and 0.15% (w/v) casamino acids (O-9). The cultures were further incubated while the ODd6c was followed. (b) Same experiment as in (a) except that dilution was 1: 300 into thymine-free medium, and that the cell number was followed.
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exponentially in glucose medium (+ thymine) was diluted into thymine-free medium and divided into three portions (Fig. 6): one portion was left as a control, the second portion received a growth-inhibiting glucose analogue (a-methylglucose); the third portion was supplemented with a mixture of 20 amino acids to stimulate growth. The glucose analogue reduced the growth rate by 25%, the amino acids stimulated growth by 40% (Fig. 6(a)). Neither condition, however, had a significant effect on the rate of cell division during thymine starvation (Fig. 6(b)). Even a drastic (90%) reduction in the metabolic rate brought about by 10 mM sodium azide (Fig. 7(a)) did not affect the rate of cell division after thymine starvation (Fig. 7(b)). The slight reduction in the rate of division seen in the presence of amino acids (Fig. 6(b)) is probably a second-order phenomenon, since in this case, the cells became so big during the course of the experiment (six to eight-fold increase in cell mass) that they became difficult to count.
Tome after
oddltton
of ande
(ml”
1
~‘~~~
0
20 Time
FIG. 7. Effect of sodium in glucose minimal medium was added to a portion of OD4s0 of the main culture of it was diluted 1:300 into the cell number was followed
40 after
thy-starvctlon
60
80 (mln)
azide on growth and cell division. A culture of TJK16 was grown with 10 ug/ml of thymine. (a) At an OD46a = 0.21, 10 mM NaN, the culture, and the OD 460 was followed (0-O). (b) When the in (a) (without azide; C-0) had reached a value of 0.4, a portion prewarmed thymine-free medium, containing 10 mM NaNs, and (O-O). The “control” curve is a replot of the curve in Fig. 2(a).
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These experiments showed that the changes in the half-life parameter h that occurred on dilution of the culture into thymine-free medium are not due to changes in the rate of cell growth in the fresh medium. Further controls showed that neither the conditioned medium used for dilution nor traces of detergent on our glass ware were responsible for these effects. This was checked by comparing unconditioned medium with medium in which wild-type E. coli B/r had grown to stationary phase, and by studying the effects of detergent (Triton x-100, and another detergent used for washing our glass ware; concentrations up to 0.1%). Also the extent of aeration in the diluted culture had no effect on the rate of cell division. Discussion (A)
UTILITY
OF
THE
METHOD
A mathematical method was described that allows one to evaluate the kinetic changes in the rate of cell division in bacterial cultures after a complete or partial inhibition of DNA chain elongation, or after a change in the initiation of replication. The information obtained from such experiments is in the form of a set of parameter values which can be used to reconstruct the kinetic changes in the number of chromosome replication termini during the course of the experiment, and in the number of cells that have entered the stochastic phase of the cell cycle. The latter cells may divide at any moment, with a constant probability per unit time, given by a half-life parameter whose value is obtained from the analysis. These constructed kinetics of termination and division (d-curve, Fig. 8) accurately show the temporal relationship between an experimentally induced perturbation of replication and the ensuing change in cell division. The method uses computer-generated, theoretical cell number curves that are compared with an observed cell number curve; the set of parameter values used for the theoretical curve that gives the “best fit” with observed data is assumed to be the correct one describing the actual situation in the experimental culture. Obviously, the observed cell number curve has to be sufficiently complex and precise to contain the amount of information sought; without perturbations of replication, i.e. during exponential growth, this complexity is absent. The “best fit” method was compared with another analytical method developed earlier for a similar purpose (Bremer & Chuang, 1981b), that might be designated “special plot” method. The special plot method was designed for the analysis of experiments involving a complete inhibition of replication, whereas the new method can be applied to a variety of
116
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0
AND
20 lime
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L.
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40 dllutton
60
80
(mln)
FIG. 8. Evaluation of the experiments in Figs 3(a), 4(b), and 5(a). calculated ceil number kinetics that describe the observed data; same curves as shown in Figs. 3, 4 and 51aL - - - d-curve = cell number curve as it would occur in the absence of a stoachastic reaction in the division pathway (see text to Figs l(b),(c). number of replication termini. All curves were calculated using the equations given in the Appendix and the parameter values of Table 1.
perturbations (see above), including sequential perturbations such as thymine starvation restart experiments (removal and later readdition of thymine to Thy bacteria). Thus, the new method is more versatile than the one previously reported. Further, if the value of h changes during the course of an experiment the special plot method is not applicable (Fig. 4) while the new methou MII still be used. (B)
SIGNIFICANCE
OF
EXPERIMENTAL
RESULTS
The new method of analysis was applied to four thymine starvation experiments with two Thy- derivatives of E. coli B/r, TJK16 with increased initiation mass and LEE130 with wildtype initiation mass. The average D-period, D, and the minimum D-period, D,, were 31 f 1 and 18 min,
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respectively; the halflife parameter during exponential growth before starvation, ho, was 10 f 1 min for both strains (Table 1). These values agree with previously determined values of D, and h (Bremer & Chuang, 19816) and confirm the existence of a stochastic phase during the D-period of the cell cycle (Bremer & Chuang, 1981~). In addition, the results suggest that cell division is linked to chromosome replication rather than cell mass or cell length since a delayed initiation of replication in the mutant strain TJK16 resulted in an equally delayed time of cell division. Similarly, the results of the experiments in Figs 6 and 7 indicated that cell growth (rate of mass or length increase) has very little, if any, effect on the timing of cell division. The parameter values obtained by the best fit analysis are quite reproducible (Table 1). The greatest variation was observed in the changes in h at the beginning of starvation, ranging between -5 and +2 min. Further experiments (not illustrated) indicated that the -5-min value ( = 50% change) was exceptional; the average change is about lo%, i.e. barely detectable, given the 1 to 2% accuracy of the cell number counts. We have not been able to find a cause for the sudden changes in the value of h at the beginning of thymine starvation. The phenomenon may be related to a previous observation that the average cell size of E. coli B/r varies from culture to culture under what appears to be identical conditions of growth, and these variations were mainly the result of variations, of a few minutes, in the length of the D-period (Choung et al., 1981). In wildtype (Thy+) E. coli B/r, it was previously found from an analysis of synchronous cultures that the halflife parameter varied between 4 and 6 min (Bremer & Chuang, 1981a). Further, the D-period was a few minutes longer and the average cell size accordingly increased in several Thy- strains in comparison to their Thy’ parental strains (Choung et al., 1981). This suggests that the value of the halflife parameter is affected, i.e. increased by about 5 min, by the Thy- mutation. The 18 min duration of the minimum D-period includes the time required for septum formation and for the reactions that lead to the final separation of two daughter cells; this time is called T-period. If the T-period would last 10 min and were invariant within a population of cells it would mean that the earliest septa would begin to form 8 min after termination such that the earliest divisions occur 18 min after termination, as observed. In that case the initiation of septum formation would be the stochastic process in the cell cycle. Alternatively, septum formation might begin a constant time after termination of replication but the time for completion of the septum and cell separation, i.e. the T-period, might be the variable period in the cell cycle. Currently, the answer to this question is not known.
118
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The variability of the D-period is not the result of “stickiness” of already completed daughter cells that are then separated randomly by some external mechanical force. This was shown previously by sonicating an exponential culture with an energy sufficient to lyse 10% of the cells; this treatment did not reduce the number of the largest cell couples that were shortly before separation (Bremer & Chuang, 1981a). It was concluded that the final cell separation involves an active metabolic process. Hence the variability of the D-period and the change in the parameter h after dilution (experiment of Fig. 4) does not reflect trivial mechanical effects like changes in the shaker speed. This work was supported by a grant from the National Institutes of Health. We thank R. Little for helpful discussions and S. Rahn for her expert typing. REFERENCES BREMER,H.& CHUANG,L. (1971a). J. theor.Biol.88,47. BREMER, H.& CHUANG,L (1981 b).J. theor.Bid.93,909. BREMER, H. & CHURCHWARD, G. (1977). 1 theor.Biol.69,645. CHOUNG,K.,ESTIVA,E.& BREMER,H.(~~~~).J. Bacterioi: 1451239. HELMSTE~R,~. (1967).J. mol. BioL24,417. HELMSTETTER,~. &COOPER, S. (1968j.J. mol. BioL 31,507. HELMSTE~R,~., PIERUCCI, 0.. WEINBERGER,M., HOLMF.S,M. & TANG,E. 11978). In: The Bacteria Vol. VII. (Sokatch, J. R. & Omston, L. N. eds). New York: Academic Press. MEACOCK, P. & PRITCHARD, R. (1975). J. Bncteriol. 122,931.
APPENDIX Calculation of Cell Number Kinetics The kinetics of the cell number (C, = cell number at time r, normalized to the zero-time value) were calculated from the values of two parameters, 7 ( = doubling time of the d-curve, see Fig. l(b)) and h ( = half-life of the division rate curve, see Fig. l(a)). The values for these parameters were assumed to change stepwise during a given experiment at times which define different adjacent time intervals, e.g. from to to tl, from tl to r7, etc., in general from t, to ti( j = i + 1). The kinetic curve was calculated between zero (to = 0) and somemaximum time value (tmax= 80 min) corresponding to the beginning and end of an actual or hypothetical experiment. Before zero-time, the cells were assumedto be in steady-state exponential growth, meaning that h and T had constant values, ho and 7. at least until to = 0. Thus the first time interval reaches from t = --COto r = 0. During the next interval from t = 0 to f = tl, 7 and h have the values 71 and h,, etc.
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In a thymine starvation experiment, three intervals have to be considered, given by three value triplets: 70, ho, to; TV, hl, tl and r2, h2, t2 where f. = 0, tl =D, (see Fig. l(b)), and t2 = rmax. The cell number at time t, C,, is a function of this set of “constants” and of the independent vsriable t. This function (based on equation (A13) of Bremer & Chuang, 19816) can be written as follows: C~=Aj(2”‘~-1)+Bj(1-2P”h’)+Ci
for tiItItj
(Al)
where q=t-ti
Aj =
fOrti5tStj
di
1 +(hj +7j)
for ti < t < tj
forti
Bj=Aj-Ci
L42)
(A31 (A4)
Ci = C, at tj
(A5)
Co=l-0
VW
dj = di x 2(‘,-‘h
(A7)
do = 1+ (ho/~o)
(A8)
Note that di is the value of the d-curve (Fig. l(b)) at t = ti. The first two terms of equation (Al) give the increase in cell number since c = ti ; the third term is the cell number Ci that had been reached at t = tie Evidently it is not possible to use the formula to calculate the cell number for any value of t greater than ti without having first calculated the cell number at ti. Thus, the calculation must start from the beginning, using Co = 1.0 (equation (A6)), and then proceed to C1 at t = tl and so forth. The first term in equation (Al), which has the constant A and the positive time exponent, represents the increase due to cells newly entering the variable portion of the D-period; the term with the constant B and the negative time exponent represents the increase due to dividing cells that were in the D,-period already at the beginning of the time interval considered. With increasing time after ti; this B-term becomes a constant contribution to the cell number (Bi = Ai - Ci), which means that it no longer contributes to the rate of cell division. The D-period was calculated (Table 1) from equation (B17) of Bremer & Chuang (1981b): D =D,
+z
{In [l + (hO/~O)]}.
(A9)
120
H.
RREMER
The number of replication the following equations:
AND
termini
T = 2’fCD”T~
L.
CHUANG
(T) relative to C, was calculated for t 5 0.
T = 2 D’i~~x 2 f’72 for 0 5 t < t,
from (A101 (All,
where t,y is the thymine starvation time (starvation from t = 0 to t = t, 1; and (A121 for t 2 t, T=2 D/r,, x 2’,172x 2(~-fh
Cell Division in Escherichia cob after Changes in the Velocity of DNA Replication H. BREMER AND L. CHUANG Biology (Received
Programs, The University of Texas at Dallas, 688, Richardson, Texas 75080, U.S.A. 20 August
P.O. Box
1981, and in final form 1 November
1982)
A method of computer analysis was developed to evaluate the kinetic changes in the rate of cell division in non-synchronous cultures of E. coli resulting from changes in the velocity or initiation of chromosome replication. This method takes into account that the cell division pathway in E. coli includes a reaction of indeterminate length described by a probability function that applies to the cell population. The analysis yields a hypothetical cell number kinetics as it would be observed if the stochastic element in the division pathway were absent. Since this derived cell number curve responds to experimentally induced perturbations of replication at defined times whereas the actual cell number curve reflects these perturbations only in a blurred fashion, replication and division events can be precisely correlated with this method. The method was applied to the evaluation of thymine starvation experiments with two YYhy- derivatives of E. coli B/r; one of the strains has a mutationally altered (60% increased) cell mass at initiation of chromosome replication. In both strains, the stochastic phase of the cell cycle had the same half-life value of 10 min and began 18 min after each termination of replication. This suggests that the time of cell division is linked to replication, not to cell mass or length. This interpretation is supported by results of experiments in which the rate of cell growth was altered at the time of thymine starvation.
Introduction In order to study causal relationships between chromosome replication events (initiation, termination) and cell division in bacteria, two experimental approaches have been used in the past; observation of replication and division in synchronous or age-fractionated bacterial cultures (e.g., Helmstetter & Cooper, 1968), or perturbation experiments with exponential cultures, where replication parameters were suddenly altered and the ensuing changes in the rate of division were observed (e.g., Meacock & Pritchard, 1975). A quantitative interpretation of either type of experiment is hampered, however, because the cell division pathway includes a reaction 101
0022-5193/83/090101+20$03.00/0
@ 1983 Academic Press Inc. (London) Ltd.
102
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BREMER
AND
L.
CHUANG
of indeterminate length that blurs the relationship under study (Bremer & Chuang, 1981a,b). Partly because of this difficulty, it is still not clear to what extent the control of cell division in bacteria is coupled to DNA replication (see review by Helmstetter et al., 1978). In this work, a method was developed that employs a computer analysis of cell number kinetics observed after perturbations of exponential cultures. With this analysis, replication and division times can be accurately correlated without the need for synchronization of growth, which is never complete, and without the need for radioactive DNA labeling. The utility of this method is illustrated by analysing several thymine starvation experiments, using thymine-requiring derivatives of E. co/i B/r. Materials (A)
BACTERIAL
STRAINS
and Methods AND
GROWTH
MEDIA
The strains used were TJK16, a thyA deoB dnaA derivative of E. coli that has an increased “initiation mass” relative to E. coli Bfr due to the mutation in the dnaA gene; LEE130, a phe(Ap) thyA deoB derivative of E. cofi B/r obtained from LEE103. Origin and properties of TJK16 and LEE103 have been described (Choung, Estiva & Bremer, 1981). The bacteria were grown exponentially at 37°C in Medium C (Helmstetter, 1967) supplemented with 0.2% glucose, 10 P.g/ml of thymine, and (only for Lee 130) 20 pg/ml of phenylalanine. “Conditioned” medium was prepared by growing E. coli B/r A in Medium C plus glucose to an ODdeO of O-5. The bacteria were removed by filtration (Millipore Corp., 0.45 km pore size); the filtrate was diluted five-fold with the same medium, and the diluted medium was again filtered through a membrane filter of 0.2 +rn pore size. B/r
tB1
THYMINE
STARVATION
Thymine starvation was brought about by a 300-fold dilution of the cell culture (ODAeo = 0.4) into thymine-free, pre-warmed, conditioned medium. Thus, the final thymine concentration in this medium was 0.3 +g/ml. Identical results and the same final slope of the cell number curve were obtained when the initial concentration of thymine in the undiluted culture was 20, 10, 5, or 2.5 pg/ml (final concentrations after 300-fold dilution were between 0.6 and 0.075 kg/ml). This suggests, therefore, that the rate of DNA synthesis after dilution was not limited by residual exogenous thymine but rather by residual endogeneous synthesis due to leakiness of
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the thyA mutations used. This was especially true for LEE130 in which the thyA mutation was found to be temperature-sensitive (see below). The diluted cultures were incubated in a shaker waterbath. In control experiments in which the shaker was turned to high speed or was completely turned off, it was found that the shaker speed did not seem to matter in the diluted culture, but it was used at medium speed in all experiments. (0
CELL
COUNTJNG
Samples (7.5 ml) of diluted culture were added to an equal volume of counting solution (36 g NaCl, 8 ml 15% HCHO per liter of distilled water) which had been repeatedly filtered through two membrane filters (0.2 km pore size). These samples were counted with a particle counter (Coulter Electronics, 20 km aperture) equipped with a loo-channel particle size analyser. The statistical error was always less than 1%. Theory (A)
VARIABILITY
OF
THE
D-PERIOD
We have previously observed (Bremer & Chuang, 1981a, 6) that the D-period, or time between termination of a round of chromosome replication and the following cell division (Helmstetter & Cooper, 1968) varies within a population of E. coli cells. Cells that have just terminated a round of replication do not divide for a period of 17 to 18 min; this period is designated as the “minimum D-period”, D,. Then the cells start dividing at a maximum rate that immediately begins to decrease exponentially with a half-life, h of 5 to 10 min, as illustrated in Fig. l(a). Why the D-period varies in this manner is not known although it presumably reflects the involvement of a random element in the division pathway, perhaps in the initiation of septation. The curve in Fig. 1 (a) has not been observed directly because a population of bacteria that has synchronously terminated a round of replication cannot be experimentally obtained. However, the same information that is contained in the hypothetical division rate curve of Fig. l(a) can be extracted from kinetics of residual cell divisions observed when an exponential culture is subjected to an inhibition of DNA replication, e.g. by thymine starvation (Fig. l(b)). During exponential growth prior to starvation, the numbers of chromosome termini and of cells increase with the same doubling time, r; the cell number curve is displaced from the terminus curve by a length of time that defines the D-period (Fig.l(b), for t < 0; Bremer & Churchward,
104
H.
BREMER Ttme
0 QJ zL
IO-
AND after
30 1
e-D,,--
:055 a
CHUANG
termlnotlon
1
’
L.
(mln) I
I
60 I
I
_
h
(a) 0 20
I
I
1
I
Time
I
after
I
thy-starvation
1
1
I
I
(b)
-
Cc)
-
(mln)
1. Theoretical, idealized relationships between termination of chromosome replication and cell division. (a) Rate of cell division in a subpopulation of cells synchronously terminating a round of replication at zerotime. D, = minimum D-period = 18 min; h = half-life = 10 min = time at which the rate has decreased to l/2 of the maximum value at I = D,. The area under the curve is equal to the number of cells present in the subpopulation at t = 0; when all cells have divided at t + 00, the number is twice as great. (b) Number of chromosome termini and of cells in a hypothetical culture that was growing and dividing exponentially until zero time, when the culture was subjected to thymine starvation such that the replication velocity immediately dropped to zero. D = time distance between the terminus curve (Ter) and the cell number curve (Cells); this distance defines the D-period. D, = constant (minimum) portion, D, = variabfe portion of the D-period; d = hypothetical curve representing the number of cells that have finished D, and have entered D,. do, dl = value of dat t = 0 and at t = I,; t, = time at which the d-curve changes its slope; the slope is given by its doubling time, 7. (c) Same as (b) except that the replication velocity does not drop to zero (incomplete thymine starvation); the final cell number doubling time is 72 = 300 min. The cell number curves were calculated (see Appendix), using f0 = r1 = 43 min, D, = 18 min. h = 10 min. - - -, a different d-curve, assuming a reduced value of D,, = 4 min after tr (see text): this would result in a higher cell number curve not shown. D2 = final D-period. FIG.
REPLICATION
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IN
105
BACTERIA
1977). After inhibition of termination by thymine starvation, the terminus curve “plateaus” while the cell number continues to increase until it reaches the plateau of the terminus curve. The shape of this cell number curve was shown to reflect the parameter D, and h (Bremer & Chuang, 1981b). Fig. l(b) also shows a curve labeled “d” which reflects the cell number curve for the hypothetical case that all cells would divide exactly 18 min ( =D, min) after termination, i.e. in the absence of a stochastic element in the division pathway. In contrast to the actual cell number curve which shows no sharp breaks, the d-curve shows sudden changes in slope in response to sudden changes in replication parameters. (B)
EVALUATION
OF
REPLICATION
PERTURBATION
EXPERIMENTS
If the replication velocity is suddenly altered to a finite value (>O), e.g. by thymine deprivation, such that terminations continue after zerotime at a reduced rate (Fig. l(c)), the cell number curve becomes more complex to analyse, but also yields additional information. In this case, the minimum D-period might still be 18 min (Fig. ((c), or it might become shorter as in the hypothetical example of Fig. l(c). For the purpose of this analysis, the experiment is dissected into a number of adjacent intervals designated by the subscript i. For each interval, three parameters are defined: ri is the doubling time of the d-curve during the interval considered, hi is the halflife parameter, and ti is the interval endpoint, given either by a break in the slope of the d-curve or as the beginning or end of the experiment. The triplet TV, ho, to refers to the time of exponential growth before the experiment (from t = -co to t = 0). These intervals are illustrated in Fig. l(c) where the d-curve has three sections, each marked by a doubling time defining its slope in the semilog plot: r. before zerotime. r1 between zero and tl, and r2 after tl. In this hypothetical case, it was assumed that r1 = TV. Thus, the three time intervals reach from --cc to 0, from 0 to tl (here tl =D,), and from t1 to the end of the experiment at 80 min ( = t2). During each time interval i for the d-curve in Fig. l(c), the half-life parameter values are ho, hl, and hZ. These values are not a priori known and might be constant during a replication perturbation experiment, such that ho = hl = hz, or they might change. In principle, it should be possible to determine changes, if any, in h from an observed cell number kinetics since these kinetics would be affected by such changes. However, not only changes in h, but also changes in 7 have to be considered. Furthermore, the times at which these changes occur, i.e. the ti-values, are generally not known (except the &-values representing the beginning and end of the experiment, f. and f2 in Fig. (1~)).
106
H. BREMER
AND
L. CHUANG
The method of analysis used here is based on a generalisation of a theory reported previously (Bremer & Chuang, 19816). An equation was derived (equation (Al)), that describes any cell number curve from the values of T,, hi, and tie In order to evaluate the results of a replication perturbation experiment, the values for these parameters are systematically varied and used to calculate theoretical expectations for the cell number curve with which the observed data are compared. When the calculated curve fits the observed data, the onset and halflife of the stochastic phase of the division process, as well as the terminus curve and the d-curve (Fig. l(b), (c); Fig. 8), can be obtained from the parameters used for the calculation. The theoretically infinite number of combinations of parameter values required to fit the calculated curve to observed data was minimized by applying two principles: (1) If a similar experiment had been analysed earlier which gave a consistent and presumably correct interpretation, then the principle of the “minimum number of changes from experiment to experiment” was applied. For example, in the experiment of Fig. 4 below, a repeat of the experiment of Fig. 3, theoretical considerations suggested a change in the value of h at zero time. Therefore, the same values as for the experiment of Fig. 3 were used, except the value for h was systematically varied. This strategy quickly led to a curve that fitted the observed data. (2) The principle of the “minimum number of changes during the course of one experiment” was applied. This means that, in one experiment, only a few stepwise changes in parameter values are assumed to occur. For example in a thymine starvation experiment (Fig. l(b), (c)), only one break in the d-curve is expected. Such stepwise changes (break in the d-curve or change in h) lead to a change in the second derivative of the observed cell number curve; therefore, the times of change, i.e. the ti-values, can be approximately estimated by visual inspection of the cell number curve. The fact that consistent results could be obtained when these principles were followed suggests that the interpretation of data (e.g. those in Table 1 below) was essentially correct.
FIG. 2. Calculated cell number curves in hypothetical thymine starvation experiments, illustrating the effects of changes in parameter values. Each panel shows one “control” kinetics (- - -), calculated from the same set of parameter values (similar as in the experiment of Fig. 3): q = q = 45 min, q = 2000 min, D, = fI = 18 min, ho = hI = h2 = 9 min. In each panel, only one parameter was varied, whereas the other parameters were kept constant as in the control. (a) Variation in 7s = 500,1000,2000 (control), and cc min. (b) Variation in tl = 15,18 (control) or 21 min. (c) Variation in rr = 30, 45 (control), or 60 min. (d) Variation in h = 6,9 (control), or 12 min.
REPLICATION
AND
I -
I
I
DIVISION I
IN I
BACTERIA
I
107
I
(0)
r2 = 500’
(cl
r, = 30’
I
-
16-
l-6
-
I.8
-
l-6
-
I.6 ‘.6
-
0
-__-----
Cdl
4
h=6’
20 Time
after
40 thy-starvatmn
6C (mn)
80
108
H. (C)
EFFECT
BREMER
OF
AND
INDIVIDUAL
THE
CELL
L.
CHUANG
PARAMETER NUMBER
VALUES
ON
KINETICS
To show how the above parameters affect the cell number kinetics, individual parameter values were systematically varied and the resulting cell number kinetics were calculated (Fig. 2). In every panel of Fig. 2, one curve (dashed) was obtained with the same fixed set of parameter values (from the experiment in Fig. 3 below). The effects of changing 72 (a), t, (b), 71 (c), and hi = h2 (d) are illustrated. The value of r2 affects the kinetics only after tl (18 min in this example). Different values of Q result from incomplete thymine starvation, when the replication velocity is reduced to some finite value greater than zero. Changes in tl produce parallel (up or down) shifts of the later, nonexponential phase (after ri), and increase or shorten the duration of the initial, exponential phase of the cell number curve (Fig. 2(b)). Changes in tl, as well as in 71 (see below), relate to changes in the minimum D-period. Logically, tl cannot exceed a certain value that would bring the d-curve above the terminus curve (Fig. l(c)). Any change in 7l (Fig. 2(c)) would indicate a gradual change in the minimum D-period. 71 cannot change without altering tl; therefore the curves in Fig. 2(c), where 71 changes but tl was held constant, cannot occur in reality. In the experiments shown below, changes in r1 have not been observed. A change in the half-life parameter h alters the initial slope of the kinetics (Fig. 2(d)). Therefore, when a sudden change in the rate of cell division is observed in a culture, a change in the value of h can be inferred (see below). Application (A)
EVALUATION
OF
THYMINE WITH
STRAIN
of Theory STARVATION
EXPERIMENTS
TJK16
Figure 3(a) shows the results of an experiment reported previously (Bremer & Chuang, 19816), in which the residual increase in cell number was followed in a culture of a Thy- derivative of E. coli B/r, TJK16, that was subjected to thymine starvation by a 300-fold dilution into thymine-free medium. The cell number values have been normalized to the zero-time value by applying a linear regression analysis to the logarithms of the cell numbers of the first ten data points. The doubling time during the initial exponential phase agreed with the mass doubling time before dilution (rO = 46 min). These data were previously evaluated by a “special plot” method (Bremer & Chuang, 1981b) that is illustrated here for comparison:
REPLICATION
AND
DIVISION
IN
BACTERIA
109
0.5
Time
after
thy -starvation
(rn11-1)
FIG. 3. Evaluation of an observed thymine starvation experiment with strain TJK16. (a) Observed cell number kinetics, C,, normalized to the cell number value at zero-time which is set at 1.0. The points shown were observed, the curve is calculated (see Appendix) with the following parameter values: r0 = r1 = 46 min; r2 = 2000 min, ti = 18 min, ho = hi = h2 = 9 min. (b) Replot of the initial points from panel (a) as J2”4’ - C, versus time. The intersection of the final section with the base line corresponds to the time ri = 18 minutes at which the rate of cell division begins to decrease; this time is interpreted as D,. (c) Replot of later data points, after t = 18 minutes, showing the difference “final plateau minus C’,” normalized to the difference at 18 minutes (set at 1.0) uusus time. The slope of the curve in the semilog plot corresponds to a halflife value h = 9 min. (d), (e) Graphic evaluation, as in panels (b), (c) of the calculated data points.
110
H.
BREMER
AND
L.
CHUANG
a plot of the square root of the difference “exponential curve minus observed cell number” (Fig. 3(b)) shows that the cell number began to deviate systematically from the exponential at 18 min, which gives the value for the minimum D-period, D,. Figure 3(c) shows the deviation of the observed cell numbers after 18 min from the final “plateau”; this plot gives the halflife of the stochastic phase, equal to 9 min. Assuming, as simplest interpretation, the following parameter values: ho = h, = h2 = 9 min, 7,) = TI= 46 min, 72 = 2000 min = observed “final” slope, and ti =D, = 18 min, a theoretical expectation for the cell number kinetics was calculated from equation (Al). The resulting curve shown in Fig. 3(a) is seen to agree with the observed data points. Similarly, if the calculated kinetics are evaluated by the special plot method, it yields the same parameter values that went into its calculation (Fig. 3(d), (e)). This agreement is not a necessity (see below). Immediately after a 300-fold dilution of the culture, the cell number was frequently observed to increase with a somewhat different doubling time as compared to the growth before dilution. An extreme example is illustrated in Fig. 4, which shows a repeat of the experiment in Fig. 3, but where the initial doubling time was 33 rather than 46 min. Evaluation by the special plot method gave reduced values of D,, and h (12 and 7 min rather than 18 and 9), and when these values were used to calculate an expectation, the theoretical curve did not agree with the observed data (Fig. 4(a), short dashes). It is implicit in the theory (equation (Al)), that sudden changes in the rate of cell division in a culture cannot occur unless the half-life parameter h changes suddenly. Thus, using the “best fit method”, the apparent shorter doubling time of 33 min in the experiment of Fig. 4 was found to be the result of a sudden reduction in the value of h from 10 min before the experiment to 5 min after the dilution at zero time. With this change in II, and using the same values for TV, 72 and tl as in the experiment of Fig. 2, the theoretical cell number curve (solid curve in Fig. 3(a)) agreed with the observed data points. Figures 3(d),(e) shows that the special plot method, when applied to the theoretical curve, gives in fact erroneous values for D, and h, which means that the “special plot” analysis cannot be used if the value of h changes during the course of an experiment. However, the “best fit method” is still applicable in such cases. (B)
EFFECT
OF THE
ALTERED TIME
REPLICATION OF
CELL
INITIATION
OF
DIVISION
Figure 5 shows the results of two thymine starvation experiments with strain LEE130 that differs from TJK16 used above in its control of
REPLICATION
AND
0
081
DIVISION
40
20
1
IN
I
I
60 ,
I,(
111
BACTERIA
80 )-
Cc) h=?’
0:
i-
02
;i
0 0
0
/ 0
0 0
0.8'
3 ; ,'
fi
z,
I-C I-
0.:
i-
0
0.1 0
IO
20 Time
30 after
thy-starvation
40
I 0
I
1
20
(mini
FIG. 4. Effect of culture dilution on the rate of cell division; repeat of thymine starvation experiment shown in Fig. 2; see text for details. Parameter values used for calculation were: (-) 70 = 71 = 46 min, ‘~2 = 2000 min; tI = 18 min; ho = 10 min, hl = hZ = 5 min. (- - -) 70 = +, = 33 min, T* = 2000 min, tl = 12 min, ho = h, = hz = 7 min (see text).
112
H.
BREMER
AND
L. CHUANG
I 8 16 14
Time
after
thy-starvotlon
(rn~n)
FIG. 5. Two thymine starvation experiments with LEE130. The curves calculated using the parameter values given in Table 1.
points
are observed,
the
replication initiation. Due to a mutation in the dnaA gene, TJK16 has a 60% greater initiation mass than LEE130 that has the wildtype initiation mass (Choung et al., 1981). Evaluation of the experiments in Fig. 5 by the best fit method showed that the values for D, and h are essentially the same in Lee130 and TJK16, but 1 to 2 min changes in the value of h TABLE
Evaluation
starvation experiments with TJK16 and LEE130 (same experiments as shown in Figs 3, 4 and 5).
Experiment
Strain
m
Fig. 3 Fig. 4
TJK16 TJK16
3.3 3.5 3.43$ 2.8 2.8 2.66$
Fig. 5(a) Fig. 5(b)
1
of thymine
LEE130 LEE130
46 46
9 10
46 46
9 5
2000 2000
9 5
18 18
30 31
42 43
11 10
42 43
10 12
1000 1000
10 12
18 18
32 31
11fi = average cell mass = OD46c units per lo9 cells. 9: r0 = mass doubling time observed before the beginning of thymine starvation. t rr, r2, ho, hr. hl: values in min, obtained from cell number kinetics observed after thymine starvation by comparison with calculated curves; see text. The subscripts 0, 1 and 2 refer to 3 different time intervals: -co to 0 ( = to); 0 to 18 (= tt); and 18 to 80 (= r2) minutes. 7 D, = minimum D-period = t 1. $0 D-period calculated from equation (A9). $ Cell mass averages from ten further experiments.
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occurred at zerotime (Table 1). The two experiments shown are representative for the reproducibility of the results, i.e. the experiment in Fig. 4 was exceptional, as mentioned. The final slope of the cell number curves for LEE130 (Q = 1000 min) is greater than for TJK16 (T* = 2000 min; Table 1). This difference is due to a leakiness of the thyA mutation in LEE130 (at 30”, LEE130 is thA’; at 37”, it does not form colonies in the absence of thymine). In these experiments, the value of Q obtainined by the best fit method is always somewhat greater than the “visually” determined value. This reflects the fact that the experiments ended at 80 min, when the final slope was being approached, but not reached. CC) EFFECT
OF
CELL
GROWTH
To see how the rate of cell growth, i.e. cell elongation, affects the rate of cell division in the absence of replication, a culture of TJK16 growing
CJ
2
16
:
14 E
E I2 E z IO 0
20 Time after
40 thy-starvation
60 (rmn)
80
FIG. 6. Effect of growth rate on therate of cell division during thymine starvation. (a) A culture of TJK16 grown to an OD46c = 0.5 was 1: 10 diluted into the same glucose minimal medium with 10 kg/ml of thymine @-A), or into medium containing 0.2% (w/v) a-methyl glucose instead of glucose (C-O), or into medium containing 0.2% glucose and 0.15% (w/v) casamino acids (O-9). The cultures were further incubated while the ODd6c was followed. (b) Same experiment as in (a) except that dilution was 1: 300 into thymine-free medium, and that the cell number was followed.
114
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exponentially in glucose medium (+ thymine) was diluted into thymine-free medium and divided into three portions (Fig. 6): one portion was left as a control, the second portion received a growth-inhibiting glucose analogue (a-methylglucose); the third portion was supplemented with a mixture of 20 amino acids to stimulate growth. The glucose analogue reduced the growth rate by 25%, the amino acids stimulated growth by 40% (Fig. 6(a)). Neither condition, however, had a significant effect on the rate of cell division during thymine starvation (Fig. 6(b)). Even a drastic (90%) reduction in the metabolic rate brought about by 10 mM sodium azide (Fig. 7(a)) did not affect the rate of cell division after thymine starvation (Fig. 7(b)). The slight reduction in the rate of division seen in the presence of amino acids (Fig. 6(b)) is probably a second-order phenomenon, since in this case, the cells became so big during the course of the experiment (six to eight-fold increase in cell mass) that they became difficult to count.
Tome after
oddltton
of ande
(ml”
1
~‘~~~
0
20 Time
FIG. 7. Effect of sodium in glucose minimal medium was added to a portion of OD4s0 of the main culture of it was diluted 1:300 into the cell number was followed
40 after
thy-starvctlon
60
80 (mln)
azide on growth and cell division. A culture of TJK16 was grown with 10 ug/ml of thymine. (a) At an OD46a = 0.21, 10 mM NaN, the culture, and the OD 460 was followed (0-O). (b) When the in (a) (without azide; C-0) had reached a value of 0.4, a portion prewarmed thymine-free medium, containing 10 mM NaNs, and (O-O). The “control” curve is a replot of the curve in Fig. 2(a).
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These experiments showed that the changes in the half-life parameter h that occurred on dilution of the culture into thymine-free medium are not due to changes in the rate of cell growth in the fresh medium. Further controls showed that neither the conditioned medium used for dilution nor traces of detergent on our glass ware were responsible for these effects. This was checked by comparing unconditioned medium with medium in which wild-type E. coli B/r had grown to stationary phase, and by studying the effects of detergent (Triton x-100, and another detergent used for washing our glass ware; concentrations up to 0.1%). Also the extent of aeration in the diluted culture had no effect on the rate of cell division. Discussion (A)
UTILITY
OF
THE
METHOD
A mathematical method was described that allows one to evaluate the kinetic changes in the rate of cell division in bacterial cultures after a complete or partial inhibition of DNA chain elongation, or after a change in the initiation of replication. The information obtained from such experiments is in the form of a set of parameter values which can be used to reconstruct the kinetic changes in the number of chromosome replication termini during the course of the experiment, and in the number of cells that have entered the stochastic phase of the cell cycle. The latter cells may divide at any moment, with a constant probability per unit time, given by a half-life parameter whose value is obtained from the analysis. These constructed kinetics of termination and division (d-curve, Fig. 8) accurately show the temporal relationship between an experimentally induced perturbation of replication and the ensuing change in cell division. The method uses computer-generated, theoretical cell number curves that are compared with an observed cell number curve; the set of parameter values used for the theoretical curve that gives the “best fit” with observed data is assumed to be the correct one describing the actual situation in the experimental culture. Obviously, the observed cell number curve has to be sufficiently complex and precise to contain the amount of information sought; without perturbations of replication, i.e. during exponential growth, this complexity is absent. The “best fit” method was compared with another analytical method developed earlier for a similar purpose (Bremer & Chuang, 1981b), that might be designated “special plot” method. The special plot method was designed for the analysis of experiments involving a complete inhibition of replication, whereas the new method can be applied to a variety of
116
H.
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0
AND
20 lime
after
L.
CHUANG
40 dllutton
60
80
(mln)
FIG. 8. Evaluation of the experiments in Figs 3(a), 4(b), and 5(a). calculated ceil number kinetics that describe the observed data; same curves as shown in Figs. 3, 4 and 51aL - - - d-curve = cell number curve as it would occur in the absence of a stoachastic reaction in the division pathway (see text to Figs l(b),(c). number of replication termini. All curves were calculated using the equations given in the Appendix and the parameter values of Table 1.
perturbations (see above), including sequential perturbations such as thymine starvation restart experiments (removal and later readdition of thymine to Thy bacteria). Thus, the new method is more versatile than the one previously reported. Further, if the value of h changes during the course of an experiment the special plot method is not applicable (Fig. 4) while the new methou MII still be used. (B)
SIGNIFICANCE
OF
EXPERIMENTAL
RESULTS
The new method of analysis was applied to four thymine starvation experiments with two Thy- derivatives of E. coli B/r, TJK16 with increased initiation mass and LEE130 with wildtype initiation mass. The average D-period, D, and the minimum D-period, D,, were 31 f 1 and 18 min,
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respectively; the halflife parameter during exponential growth before starvation, ho, was 10 f 1 min for both strains (Table 1). These values agree with previously determined values of D, and h (Bremer & Chuang, 19816) and confirm the existence of a stochastic phase during the D-period of the cell cycle (Bremer & Chuang, 1981~). In addition, the results suggest that cell division is linked to chromosome replication rather than cell mass or cell length since a delayed initiation of replication in the mutant strain TJK16 resulted in an equally delayed time of cell division. Similarly, the results of the experiments in Figs 6 and 7 indicated that cell growth (rate of mass or length increase) has very little, if any, effect on the timing of cell division. The parameter values obtained by the best fit analysis are quite reproducible (Table 1). The greatest variation was observed in the changes in h at the beginning of starvation, ranging between -5 and +2 min. Further experiments (not illustrated) indicated that the -5-min value ( = 50% change) was exceptional; the average change is about lo%, i.e. barely detectable, given the 1 to 2% accuracy of the cell number counts. We have not been able to find a cause for the sudden changes in the value of h at the beginning of thymine starvation. The phenomenon may be related to a previous observation that the average cell size of E. coli B/r varies from culture to culture under what appears to be identical conditions of growth, and these variations were mainly the result of variations, of a few minutes, in the length of the D-period (Choung et al., 1981). In wildtype (Thy+) E. coli B/r, it was previously found from an analysis of synchronous cultures that the halflife parameter varied between 4 and 6 min (Bremer & Chuang, 1981a). Further, the D-period was a few minutes longer and the average cell size accordingly increased in several Thy- strains in comparison to their Thy’ parental strains (Choung et al., 1981). This suggests that the value of the halflife parameter is affected, i.e. increased by about 5 min, by the Thy- mutation. The 18 min duration of the minimum D-period includes the time required for septum formation and for the reactions that lead to the final separation of two daughter cells; this time is called T-period. If the T-period would last 10 min and were invariant within a population of cells it would mean that the earliest septa would begin to form 8 min after termination such that the earliest divisions occur 18 min after termination, as observed. In that case the initiation of septum formation would be the stochastic process in the cell cycle. Alternatively, septum formation might begin a constant time after termination of replication but the time for completion of the septum and cell separation, i.e. the T-period, might be the variable period in the cell cycle. Currently, the answer to this question is not known.
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The variability of the D-period is not the result of “stickiness” of already completed daughter cells that are then separated randomly by some external mechanical force. This was shown previously by sonicating an exponential culture with an energy sufficient to lyse 10% of the cells; this treatment did not reduce the number of the largest cell couples that were shortly before separation (Bremer & Chuang, 1981a). It was concluded that the final cell separation involves an active metabolic process. Hence the variability of the D-period and the change in the parameter h after dilution (experiment of Fig. 4) does not reflect trivial mechanical effects like changes in the shaker speed. This work was supported by a grant from the National Institutes of Health. We thank R. Little for helpful discussions and S. Rahn for her expert typing. REFERENCES BREMER,H.& CHUANG,L. (1971a). J. theor.Biol.88,47. BREMER, H.& CHUANG,L (1981 b).J. theor.Bid.93,909. BREMER, H. & CHURCHWARD, G. (1977). 1 theor.Biol.69,645. CHOUNG,K.,ESTIVA,E.& BREMER,H.(~~~~).J. Bacterioi: 1451239. HELMSTE~R,~. (1967).J. mol. BioL24,417. HELMSTETTER,~. &COOPER, S. (1968j.J. mol. BioL 31,507. HELMSTE~R,~., PIERUCCI, 0.. WEINBERGER,M., HOLMF.S,M. & TANG,E. 11978). In: The Bacteria Vol. VII. (Sokatch, J. R. & Omston, L. N. eds). New York: Academic Press. MEACOCK, P. & PRITCHARD, R. (1975). J. Bncteriol. 122,931.
APPENDIX Calculation of Cell Number Kinetics The kinetics of the cell number (C, = cell number at time r, normalized to the zero-time value) were calculated from the values of two parameters, 7 ( = doubling time of the d-curve, see Fig. l(b)) and h ( = half-life of the division rate curve, see Fig. l(a)). The values for these parameters were assumed to change stepwise during a given experiment at times which define different adjacent time intervals, e.g. from to to tl, from tl to r7, etc., in general from t, to ti( j = i + 1). The kinetic curve was calculated between zero (to = 0) and somemaximum time value (tmax= 80 min) corresponding to the beginning and end of an actual or hypothetical experiment. Before zero-time, the cells were assumedto be in steady-state exponential growth, meaning that h and T had constant values, ho and 7. at least until to = 0. Thus the first time interval reaches from t = --COto r = 0. During the next interval from t = 0 to f = tl, 7 and h have the values 71 and h,, etc.
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In a thymine starvation experiment, three intervals have to be considered, given by three value triplets: 70, ho, to; TV, hl, tl and r2, h2, t2 where f. = 0, tl =D, (see Fig. l(b)), and t2 = rmax. The cell number at time t, C,, is a function of this set of “constants” and of the independent vsriable t. This function (based on equation (A13) of Bremer & Chuang, 19816) can be written as follows: C~=Aj(2”‘~-1)+Bj(1-2P”h’)+Ci
for tiItItj
(Al)
where q=t-ti
Aj =
fOrti5tStj
di
1 +(hj +7j)
for ti < t < tj
forti
Bj=Aj-Ci
L42)
(A31 (A4)
Ci = C, at tj
(A5)
Co=l-0
VW
dj = di x 2(‘,-‘h
(A7)
do = 1+ (ho/~o)
(A8)
Note that di is the value of the d-curve (Fig. l(b)) at t = ti. The first two terms of equation (Al) give the increase in cell number since c = ti ; the third term is the cell number Ci that had been reached at t = tie Evidently it is not possible to use the formula to calculate the cell number for any value of t greater than ti without having first calculated the cell number at ti. Thus, the calculation must start from the beginning, using Co = 1.0 (equation (A6)), and then proceed to C1 at t = tl and so forth. The first term in equation (Al), which has the constant A and the positive time exponent, represents the increase due to cells newly entering the variable portion of the D-period; the term with the constant B and the negative time exponent represents the increase due to dividing cells that were in the D,-period already at the beginning of the time interval considered. With increasing time after ti; this B-term becomes a constant contribution to the cell number (Bi = Ai - Ci), which means that it no longer contributes to the rate of cell division. The D-period was calculated (Table 1) from equation (B17) of Bremer & Chuang (1981b): D =D,
+z
{In [l + (hO/~O)]}.
(A9)
120
H.
RREMER
The number of replication the following equations:
AND
termini
T = 2’fCD”T~
L.
CHUANG
(T) relative to C, was calculated for t 5 0.
T = 2 D’i~~x 2 f’72 for 0 5 t < t,
from (A101 (All,
where t,y is the thymine starvation time (starvation from t = 0 to t = t, 1; and (A121 for t 2 t, T=2 D/r,, x 2’,172x 2(~-fh