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PROCEEDINGS of the FIFTH 
BERKELEY SYMPOSIUM ON 
MATHEMATICAL STATISTICS 
AND PROBABILITY 


Held at the Statistical Laboratory 
University of California 
June 21-July 18,1965 

__ and _ 

(^December 27,1965-Janua ry *7,1966 

with the support of 
University of California 
National Science Foundation 
National Institutes of Health 
Air Force Office of Scientific Research 
Army Research Office 
Office of Naval Research 

VOLUME IV 


Biology and Problems of Health 

Edited by LUCIEN M. LE CAM 
and JERZY NEYMAN 


UNIVERSITY OF CALIFORNIA PRESS 

BERKELEY AND LOS ANGELES 

1967 



CELL GENERATION TIMES: 
ANCESTRAL AND 
INTERNAL CONTROLS 

HERBERT E. KUBITSCHEK 
Argonne National Laboratory 


1. Introduction 

In order to develop adequate models for the kinetics of growth of cell popula¬ 
tions, it is necessary to know the generation time distribution for the individual 
cells and the degree to which the generation times of related individuals are 
associated. In essence, the generation time of a cell is that period between suc¬ 
cessive cell divisions, that is, the period between the birth of the cell by fission 
of its parent and the later instant at which its own fission occurs. In practice, 
the generation times of cells are measured by recording the passage of some 
particular state during or near fission as the individual cells are observed at 
successive times. This state must be chosen with some caution since any variabil¬ 
ity in its passage will cause a corresponding loss of resolution in the measured 
generation time distribution and will lead to an unduly negative correlation 
between successive generation times. 

Early models proposed to describe generation time distributions, by Rahn [1] 
and by Kendall [2], assumed that generation times of cells were independent of 
one another. We now know that the generation times of closely related bacterial 
cells are not independent since the generation times of sister bacterial cells are 
consistently observed to be positively correlated (Powell [3], [4], Kubitschek 
[5], Schaechter, Williamson, Hood, and Koch [6], Powell and Errington [7]), 
and associations continue to exist for at least three generations [7], [8]. Similar 
associations may be anticipated for other cells because of the similarity of their 
generation time distributions to those of bacteria [5], [9]. 

2. Correlations between the generation times of bacterial cells 

2.1. Lateral correlations. Powell measured generation times for six species 
of bacteria that were grown under almost constant conditions [3]. His criterion 
for cell division (or more precisely, termination of a generation) was cell separa¬ 
tion. He observed a significant positive correlation between the generation 
times of sister cells in all six species. Later, in a study of four of these species 
[4] he found in each a positive correlation between first cousins (that is, cells 

Studies performed under the auspices of the A.E.C. 

549 



550 


FIFTH BERKELEY SYMPOSIUM: KUBITSCHEK 


with a grandmother as their most recent common ancestor) and, in a more 
extensive study with Errington [7] in which more than 800 generation times 
were measured for each of three species grown on two different media, observed 
a positive correlation for second cousins (that is, cells with a greatgrandmother 
as their most recent common ancestor). These results provide evidence for 
lateral correlations between the cells in any given generation, for a period of at 
least three generations after their descendence from a common ancestor. In 
these experiments, however, there were no corresponding significant longitudinal 
correlations between mothers and daughters or between grandmothers and 
granddaughters. In addition, it should be pointed out that the distributions of 
generation times were skewed so that it was necessary to correct the product 
moment correlation coefficients according to a formula developed by Powell [4]. 

Although there appear to be no other reports of significant correlations 



Figure 1 

The generation rate distribution for E. coli B/r. 

The number of observations N is given for each microcolony. 

This distribution does not include potentially biased data: 
data from the first generation are excluded (see figure 10) since 
the cells in some microcolonies might not yet have achieved balanced growth; 
data are also excluded for all generations in which two or more cells 
failed to divide by the end of the photographic record. 

For two microcolonies only a single cell failed to divide in the final 
generation used. If these single cells had divided immediately after 
recording ended, they would have given the maximum rates indicated by <0; 
their true values are almost certainly smaller. 




CELL GENERATION TIMES 


551 


between first or second cousins, the conclusion that there are associations 
between the generation times of cells for some three generations is confirmed 
by my own studies [8]. This agreement is of special interest because I used a 
different criterion for termination and obtained a different frequency function 
for the distribution of generation times; the criterion for termination was the 
completion of the septum dividing the cell into two compartments. Cells of 
Escherichia coli, strain B/r, were grown on nutrient agar squares at 37°C on the 
stage of a phase microscope and photographed at a magnification of 1000, at 
six or ten second intervals. The time at which a septum was completed could be 
determined within a resolution of about two per cent. The data were obtained 



Colony M 2.1 2.2 2.3 



DIFFERENCE IN GENERATION RATE, per second 

Figure 2 

The distributions of the differences between the generations rates 
of mother-daughter and sister-sister cell pairs. 

Absolute values of the differences are plotted and fitted 
with the positive halves of normal distributions centered at zero; 
<x is standard deviation; N is number of pairs of differences. 




552 FIFTH BERKELEY SYMPOSIUM: KUBITSCHEK 

from only four microcolonies; fewer than 200 values were obtained, and gave a 
skewed distribution of generation times. However, it was found that the recip¬ 
rocals of the generation times, termed generation rates for convenience, have a 
distribution that is approximately normal, as shown in figure 1. The correlation 
between the generation rates of sister cells was positive and significant; the 
value of the product moment correlation coefficient was 0.45. For sister cells 
the standard deviation a of the differences in generation rates was reduced to 
about 80 per cent of the values found for unrelated cells or between mothers 
and daughters, figure 2. There was no significant correlation between the gen¬ 
eration times of mother cells and their daughters except when daughter cells 
had unusually short generation times. Then the generation times of their 
mothers were significantly longer than the mean, figure 3. 

2.2. Variance of the sum of successive generation times. A recent reexamination 
[8] of the data from my experiments was stimulated by Koch’s finding that the 



GENERATION RATE OF DAUGHTER CELLS, per second 


Figure 3 

Grouped values of mother cell generation rates 
as a function of the generation rates of their daughters. 

The eleven daughter cells in the two most rapidly dividing groups 
have come from mother cells with a mean generation rate 
that is significantly less than the mean for all cells. 




CELL GENERATION TIMES 


553 


variance of the sum of successive generation times is far more constant than 
would be expected if generation times were independent [10]. Koch’s approach 
permits an estimate of the extent to which generation times of cells are cor¬ 
related with those of their ancestors. To see this, let T h • • • , T n represent 
the successive generation times in a progeny line of n generations, and define 
the sum 

(2.1) S n = 75 + T 2 + • • • + T„. 

Then, if these generation times are independent, 

(2.2) Var S n = Var T, + Var 7’ 2 + • • • + Var T n . 

We can ask what the behavior of Var S n would be for either of two extremes, 
independence of generation times or complete dependence. However, we shall 
be interested only in cultures for which the distributions of generation times 
does not vary from generation to generation. In order to obtain cultures of this 
kind it is necessary to maintain balanced growth. In balanced growth every 
extensive property of the culture increases at the same rate. For such cultures 
the distributions of cell number, mass, and composition remains unchanged in 
successive generations. 

At the first extreme, if generation times are independent, or if they are merely 
uncorrelated, then 

(2.3) Var T, = Var T 2 = • • • = Var T n = Var T, 

where Var T is the constant variance contributed by any generation. In this 
case, 

(2.4) Var S n = n Var T. 

This linear dependence of the variance on n is shown by line A in figure 4. 

At the other extreme, if we consider the particular case where fluctuations in 
generation time only occur just before or just after cell division and if the 
fluctuations occurring just after division completely nullify those occurring just 
before, then Var S n cannot increase, but must remain constant at the value it 
would have had for a single complete generation 

(2.5) Var S n = Var T. 

This kind of constancy is shown by line B in figure 4. 

Values of Var S n were calculated for those generations in my experiments in 
which it was possible to obtain measurements of the interdivision times of all 
cells or all but one; for example, for n = 2, all mother-daughter pairs were 
included from the mother cell in the initial generation to the daughter cells in 
the final generation. The dependence of Var S n upon the number of elapsed 
generations, shown by the experimental points in figure 4, does not agree with 
either extreme dependence, line A or line B. Instead, the variance increases 
rather uniformly for the first three generations, then remains relatively constant 
thereafter. These results show that the generation times of cells in a progeny line 



554 


FIFTH BERKELEY SYMPOSIUM I KUBITSCHEK 


are essentially uncorrelated with those of their immediate ancestors, but they 
cannot be independent since Var S n approaches constancy after three genera¬ 
tions. Furthermore, these results are in agreement with the absence of longitu¬ 
dinal correlations discussed earlier, and they support the finding of Powell and 



Figure 4 

The dependence of Var S n , 

the variance of the sum of successive generation times, 
upon the number of generations. Estimates of the errors of 
the variances [8] are too small to portray conveniently. 


Errington that associations exist between related cells over a period of three 
generations [7]. 

2.3. Longitudinal correlations. Since the distribution of generation rates for 
the bacterial cells in my study was approximately normal, it was possible to use 
a simple sign test for the presence of correlations between cells in progeny lines. 
The deviation of the generation rate from the mean was recorded for each cell 





CELL GENERATION TIMES 


555 


as positive (+) or negative (—). Deviations of related pairs were compared to 
see if signs were the same (+,+ or —) or alternate (+,— or —the 
frequency of alternate signs was compared to that expected by chance, and the 
significance of the result was assessed by t test. Further comparisons were 
made between the deviations of single cells (of arbitrary generation g ) and the 
sums of the deviations of two or more cells in successive generations in the same 
progeny line (such as granddaughter and greatgranddaughter in generations 
g + 2 and g + 3). The results presented in table I show that there are signif- 


TABLE I 

Correlation Between Generation Rates in Ancestral and Progeny Lines 


Generation 


Ancestral 

Progeny 

N 

A 

t 

P 

g 

g + 1 

160 

81 

.159 

NS 

g 

g + 2 

144 

76 

.667 

NS 

g 

0+3 

112 

61 

.941 

NS 

g 

0+4 

64 

28 

1.000 

NS 

g 

0 + 1) 0 + 2 

144 

74 

.334 

NS 

g 

0 + 2, 0 +3 

112 

72 

3.015 

.01 

g 

0 + 3, 0 + 4 

64 

31 

.250 

NS 

g 

0 + l»0+2, 0+3 

112 

67 

2.072 

.05 

g 

0+2, 0+3, 0+4 

64 

35 

.750 

NS 

g 

0 + 1, 0+2, 0 +3, 0+4 

64 

31 

.250 

NS 

g - 1, g — 2 

0 

144 

83 

1.834 

NS 

g - 2, g - 3 

0 

112 

71 

2.825 

.01 

g - 3, g - 4 

0 

64 

32 

0 

NS 

0-1, 0-2, 0-3 

0 

112 

73 

3.203 

.01 

0-2, 0-3, 0-4 

0 

64 

34 

.501 

NS 

- 1, g - 2, g - 3, g - 4 

0 

64 

32 

0 

NS 

g, g + i 

0 + 2, 0 + 3 

112 

74 

3.390 

.01 

g, g + 1 

0+3,0+4 

64 

41 

2.250 

.05 

g, g + 1 

0+2, 0 + 3, 0+4 

64 

38 

1.501 

NS 

g, g + b g + 2 

0 + 3, 0 + 4 

64 

44 

3.000 

.01 


icant negative correlations between the generation rates of single cells and the 
mean deviation of some pairs or triplets of successive cells in their ancestral 
and progeny lines: significant correlations exist with the means of greatgrand¬ 
mothers and grandmothers, with the means of all three preceding generations, 
with the means of triplets of successive cells in the three following generations, 
and with the means of granddaughters and greatgranddaughters in the same 
line (that is, with the means of g — 3, g — 2, of g — 3, g — 2, g — 1, with 
g + 1, g -I- 2, g +- 3, and with g + 2, g + 3). 

Deviations from the mean cell generation rate were compared for each cell of 
an arbitrary generation g and its progeny in later generations g +- i, or its 
ancestors in earlier generations g — i, where i — 1, 2, 3, or 4. A comparison 






550 FIFTH BERKELEY SYMPOSIUM: KUBITSCHEK 

was also made with the average rate for sums of generations in ancestral or 
progeny lines. In the table, N is the total number of pairs of values, and A is 
the number of pairs of values with deviations of alternate sign. These values 
were compared by t test. The probability P that the observed frequency of 
alternate signs would have occurred by chance is given in the final column; NS 
indicates that the difference was not significant. 

It is instructive to plot the experimental values for the frequencies of alternate 
signs, figure 5. This figure shows that the frequencies of alternate signs increase 
to a maximum value between 2 and 2.5 generations. 



GENERATION 

Figure 5 

Observed frequencies 

for pairs of generation time values with alternate signs. 

The deviation of the generation rate of the reference cell 
(generation 0) from the mean generation rate was compared with 
the deviation of related single cells in ancestral or progeny lines, 
and with the mean value of successive pairs or triplets in these lines. 
Frequencies of alternate deviations are plotted 
at the average generation number. 

Filled circles indicate pairs of single cells; 
cross indicates reference cell and two successive generations; 
open circles indicate reference cell and three successive generations. 




CELL GENERATION TIMES 


557 


These correlations are in agreement with the variance analysis, and again 
confirm earlier results for the absence of significant longitudinal correlations 
between single pairs of related cells in a progeny line. In addition, they extend 
those results, demonstrating that significant correlations arise when average 
generation rates over two or three generations are considered. 

Finally, in these studies it has also been possible to relate the generation 
times to cell size [8]. From the original photographs, it was possible to measure 
the relative lengths of these rod shaped bacteria just after formation of the 
septum in the parental cell (“birth”) and their lengths in turn at the instant of 
formation of their own septa. The distributions of these lengths are shown in 
figure 6. As generation rates are increased, average birth lengths first decrease to 
a minimum value, and then increase again. Final cell lengths appear to go 
through a greater change and to reach a minimum value at a greater generation 



Figure 6 

Distributions of relative lengths of cells at birth and 
final lengths at division as a function of generation rate. 
Lengths of each cell at birth and 
at division are connected with a vertical line. 

The two curved lines are visual approximations. 







































558 


FIFTH BERKELEY SYMPOSIUM! KUBITSCHEK 


rate. Thus, for almost all birth lengths there are two different average values of 
generation rate, and the cells in each of the two classes grow to different final 
average lengths at division. Similarly, for many final cell lengths there are two 
different classes of length at birth. It follows that the generation rates of cells 
cannot be determined solely by their lengths at birth or by their final lengths. 
The latter possibility was assumed in a deterministic model for the kinetics of 
bacterial division suggested by Koch and Schaechter [11]. 

The results of figure 6 again imply that the generation rates of daughter cells 
are strongly dependent upon those of their mothers, despite the absence of 
significant correlations in earlier tests. Cells with the smallest generation rates 
(longest generation times) reach the greatest final lengths, and their progeny 
are so large at birth that they fall into the class of birth lengths associated only 
with the most rapid generation rates. This relationship provides a more de¬ 
tailed accounting for the negative correlation shown earlier in figure 3. Further¬ 
more, this kind of dependence of the generation rates of daughter cells upon 
those of their mothers must occur for all such pairs: the terminal length of a 
mother cell establishes the permissible ranges of birth lengths of its daughters, 
and thereby, the permissible ranges of their generation times. Since these ranges 
are large and since there are two classes of generation time for most birth 
lengths, these relationships are usually concealed in tests of correlation. 

The results of figure 6 require the presence of at least two nongenetic factors 
controlling generation rates in daughter cells, transmitted from mother to 
daughter and tending to be compensatory [8]. One of these might account for 
the increase in generation rate as birth lengths are increased. The other factor 
would then have to account for the decrease in generation rate with increase in 
birth length that occurs at the smallest generation rates. 

2.4. Summary. To summarize this section, there is a great deal of evidence 
that the generation times of bacterial cells depend upon those of their ancestors, 
despite the absence of observable correlations between the generation times of 
individual cells. The range of permissible generation times of daughter cells is 
established by the lengths of their mothers through the transmission of factors 
that tend to compensate for each other. These factors give rise to the appearance 
of two classes of generation time for most cell lengths at birth, tending to obscure 
this dependence in simple correlation tests. Nevertheless, significant positive 
lateral correlations have been observed between sisters, between first cousins, 
and between second cousins, and significant negative longitudinal correlations 
have been observed between single cells and the mean values for several 
ancestral or progeny generations. These correlations are supported by a variance 
analysis of the time required for several successive cell divisions; Altogether 
they demonstrate the association of generation times over a period of three 
generations, 



CELL GENERATION TIMES 


559 


3. On the generality of the truncated normal distribution of generation rates 

3.1. Evidence supporting the distribution. The observation that the genera¬ 
tion rate distribution was approximately normal led to an examination of gen¬ 
eration time distributions published for other kinds of cells [5]. Generation 
rate distributions were constructed for the yeast Saccharomyces cerevisiae from 
the data of Burns [12], for the protozoan Tetrahymena geleii HS from the data 
of Prescott [13], for mammalian cells in culture, HeLa, from the data of Hsu 



0.6 0.8 1.0 1.2 1.4 1.6 

GENERATION RATE 
MEAN GENERATION RATE 

Figure 7 

Cumulative distributions of cell generation rates 
when control is relaxed. 

[14], and in a later communication [15] for another protozoan, Euglena gracilis, 
from the data of Cook and Cook [16]. The criterion of termination was different 
for each cell type: the initial appearance of a daughter bud was scored for yeast, 
cytoplasmic fission for the protozoa, and mitotic anaphase for HeLa cells. 
Despite these different criteria, each distribution of generation rate is an approx¬ 
imate agreement with a normal distribution. The agreement is evident even 
when the data consists of no more than 160 values, as in the cumulative dis¬ 
tributions shown in figure 7, and the agreement is improved for larger numbers, 



560 


FIFTH BERKELEY SYMPOSIUM: KUBITSCHEK 


figure 8 and figure 9. In addition, Sisken and Morasca [17] have recently ob¬ 
served that the generation rates of human amnion cells appear to be normally 
distributed in culture. 

In the more extensive experiments the agreement with the normal distribution 
of generation rates is so good that rather a large number of observations are 
required to give a good estimate of the deviation from normality. The data for 


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GENERATION RATE 
MEAN GENERATION RATE 


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Figure 8 

Cumulative distributions of cell generation rates 
when control is stringent. 


figure 9 led to no significant departure from normality for generation rates as 
small as three standard deviations below the mean and as large as two standard 
deviations above the mean [9]. This unusual agreement was unexpected because 
growth and division of cells are limited by a variety of processes and consequently 
a maximum generation rate must exist, and might have been expected to occur 
at smaller values. 

These results lead to the possibility that generation rate distributions are 
similarly normal for all or almost all cells dividing by simple regular fission. 
This point is of some importance because it implies that cell division might then 




CELL GENERATION TIMES 561 

be controlled in a generally similar fashion, and if true, it would then be desirable 
to construct a general model. 

3.2. Evidence against the normal distribution. Some published generation 
time distributions do not yield even approximately normal distributions of 
generation rate. We need not consider reports prior to Powell’s initial study 
[3] since the earlier work either was not sufficiently systematic, or there are 



Figure 9 

Cumulative distributions of cell generation rates 
for Euglena in two different growth media. 

ambiguities in the description of the measurements, and furthermore, the con¬ 
cept of balanced growth was not enunciated nor can we ascertain that it was 
intuitively understood. Some of the experiments done since can be discarded 
because they fail to satisfy one or more of the conditions that are required to 
obtain unbiased distributions. 

In order to obtain valid generation time, or rate, distributions three conditions 
must be met [3], [4], [5], [9]. 




562 


FIFTH BERKELEY SYMPOSIUM I KUBITSCHEK 


(1) Whatever the criterion chosen for the termination of a generation, it 
must permit good resolution: the terminal instant must be clearly distinguishable 
and have but small variability in the sequence of events composing the cell 
division cycle. 

(2) Cells must be in balanced growth. (That is, every extensive cellular 
property, such as mass, number, and composition, must increase at the same 
rate. Statistical fluctuations would occur for small numbers of cells, but these 
would be negligible if the culture were indefinitely large.) This requirement 
ensures that the generation time distribution remains unchanged from genera¬ 
tion to generation. 

(3) The distribution must not be calculated from data so selected that the 
distribution is biased. For example, selection (“cutoff” bias) would occur 
against long generation time values if every value in a complete photographic 
record was used to construct the distribution, since only cells with successively 
shorter generation times can be recorded as the end of the record (“cutoff”) 
approaches. Although corrections for bias of this kind are available [3], or the 
bias can be made negligible by properly discarding appropriate values [9], it is 
far more satisfying to avoid this bias by deciding beforehand to include data 
for all cells for only a fixed number of generations, as suggested by Powell [3]. 

These conditions must also be satisfied, of course, when the population under 
study contains inviable cells or subpopulations known to be under other kinds 
of selection. In addition, it is necessary in these cases to establish the constancy 
of this selection, generation by generation. There are as yet no reliable studies 
on populations of these kinds. In one study that included nonviable bacteria 
[18], the frequency of nonviable cells did not remain constant at the level found 
in early generations, and in addition, mean generation times decreased with 
successive divisions, failing to satisfy the second condition of balanced growth. 

Techniques for culturing mammalian cells have been developed only relatively 
recently, and it is frequently quite difficult to maintain constant growth condi¬ 
tions over a period of many generations. Even when precautions are taken to 
minimize variability in sera and other growth factors, some cells are strongly 
influenced by their local environment, which changes rapidly with the growth 
of the population. It is especially important to provide evidence for balanced 
growth in studies of these cells. No such evidence was reported for the two 
generations in Froese’s study of HeLa cells [19], although cell division rates 
were more rapid with human serum (28 values) than with fetal calf serum 
(about 200 values). The differences between the three experiments using fetal 
calf serum strongly suggest that growth conditions were not well controlled; 
in one of the experiments cells were so unusually motile that some 50 per cent 
of the divisions could not be followed. 

In a similar study of a strain of rat sarcoma cells [20], the constancy of the 
mean generation time was tested statistically for each clone and each passing 
generation. Except for one experiment, the means did not differ significantly 



CELL GENERATION TIMES 


563 


among the seven experiments comprising some 200 values. The data were also 
examined for possible cutoff bias, but since none could be detected, almost all 
of the data were used. In itself, this inability to obtain evidence for this bias is 
disturbing, since such bias must exist. (Actually, the reduced variance in the 
partial data for the fifth generation is evidence for this bias.) More serious, 
however, was the fact that the data were not complete even to the fourth 
generation (see, for example, their figure 4), presumably because of cell migra¬ 
tion. To make absence of bias convincing, these authors would need to establish 
that the unobtainable data were not in a selected subpopulation. 

The strongest evidence against a normal distribution of generation rates 
would appear to come from the studies of the bacterium E. coli by Schaechter, 
Williamson, Hood, and Koch [6] and the more extensive studies by Powell [3], 
[4] and Powell and Errington [7]. Schaechter, Williamson, Hood, and Koch 
observed a normal distribution of generation times, while Powell’s experiments 
led to a distribution skewed toward longer generation times (Pearson Type III, 
or similar). My own studies led to a distribution of generation times even more 
highly skewed toward longer generation times [5]. Furthermore, in all of these 
studies there was no evidence that growth was unbalanced, nor did it seem that 
the data were collected in a manner leading to cutoff bias. Thus, these studies 
would seem to have led to three different distributions of generation time for 
the same bacterial species. 

However, it should be noted that the criterion of termination in these other 
experiments was different from mine. They measured cell wall separation, the 
production of two essentially mechanically independent daughters (termed o- 
fission by Powell [4]), whereas in my studies the criterion was cytoplasmic 
separation, the development of a septum isolating the contents of a parental 
cell into two daughter units that remained tightly attached to each other 
(Powell’s p-fission). Powell anticipated the possibility of different distributions 
when he expected only a rough correspondence between these two generation 
times [4]. 

Recent results of my own support Powell’s expectation. Cultures of E. coli 
15 THU - (requiring thymine, histidine, and uracil), kindly supplied by S. S. 
Cohen, were grown in a minimal medium (M9 salts, 2 jug/ml thymidine, 20 Mg/ml 
histidine, 20 jug/ml uracil, and 0.1 per cent glucose) and transferred daily to 
fresh medium for several w r eeks. In the final culture, exponential growth was 
permitted for only about six generations, and the cells w ere then observed by 
phase microscopy. Few, if any, long filamentous forms were seen. However, 
there were many pairs of joined cells that appeared to be about twice the usual 
cell length. These pairs usually were constricted at their attachment, with 
diameters frequently reduced from only slightly to about half the usual cell 
diameter. Of 1872 cells examined, at least 15 per cent were pairs of this kind. 
As a control, cells of a similar strain, B/r/Tl,try - , from a continuous (chemostat) 
culture were also examined. In this culture not more than 1.5 per cent were 



564 


FIFTH BERKELEY SYMPOSIUM: KITBITSCHEK 


joined pairs. Since constriction is concomitant with septum formation or soon 
follows it, these observations show that cell wall separation was unusually 
delayed in the 15 THU - strain. 

The observation that cell separation can be unusually delayed provides a 
possible explanation for the three different kinds of generation time distribu¬ 
tions observed for E. coli. If the criterion of termination is cell wall separation, 
then cells that are unusually tardy in separating after their septa are formed 
will give unusually large values for their generation times. However, if this 
unusual delay is absent at the next or later divisions, then two or more progeny 
cells will appear to have unusually short generation times. Thus, an unusually 
variable interval between septum formation and cell wall separation would lead 
predominantly to an apparent increase in the frequency of short generation 
times (and also to an increased value of the coefficient of variation CV for 
generation rates). The distortion of the distribution would increase with both 
the magnitude of this delay and the frequency with which it occurs. For these 
reasons, cell separation would seem to be a poor criterion of termination, failing 
to satisfy the first condition of providing good resolution. Septum formation 
appears to afford greater resolution. 

If these conclusions are correct, the studies of Powell, of Powell and Errington, 
and of Schaechter, Williamson, Hood, and Koch present no strong evidence 
against the generality of the truncated normal distribution of generation rates. 

3.3. Estimation of mean generation rates. In order to obtain a valid genera- 



Figure 10 

Growth curve from a single cell of E. coli B/r. 



CELL GENERATION TIMES 


565 


tion time (or rate) distribution, the culture must be in balanced growth with 
reproducible mean rates of cell division. Mean generation times are not very 
accurate when they are computed from clonal histories comprised of 50 to 60 
cell divisions, and minor deviations from balanced growth can escape detection, 
leading nevertheless to biased distributions. A more accurate procedure is to 
plot the growth curve for each microcolony arising from a single cell, as in 
figure 10, and to make a visual estimate of the line best fitting these data. 
These estimates have a variability that is three to four times smaller than the 
mean generation rate computed from the average of individual values (see, for 
example, table II in [5]). This increased accuracy can be attributed to the 
added information of the order in which cell divisions occurred, contained 
implicitly in the growth curve [5]. It would be valuable to have a mathematical 
procedure for fitting the best straight line to data with cyclic fluctuations of 
this kind; a major problem is that such fitting could not be allowed to depend 
unduly strongly upon the part of the cycle at which data collecting was dis¬ 
continued. 

3.4. The advantage of the generation rate distribution. Generation rate dis¬ 
tributions have at least one theoretical advantage over generation time distribu¬ 
tions when inviable cells are present. Even one such cell would cause the mean 
generation time to become infinite. However, such a cell would add only a 
single value to the zero class of a generation rate distribution, and the mean 
generation rate could be correspondingly reduced, usually only slightly. 

4. Discrete distributions of generation rate 

4.1. Evidence for two groups. When the coefficient of variation CV was cal¬ 
culated for the generation rate distributions of cells growing in complex (organic) 
media, a striking relationship emerged for these cultures growing at or nearly at 
their maximum growth rates: only two classes of values of CV were observed 
[9]. That is, the widths of these two distributions did not appear to be distributed 
at random, but fell into two groups. The presence of the two groups is easily 
seen by comparing the slopes of the cumulative distributions in figures 7 and 8, 
since the values of CV are proportional to these slopes. Considering each figure 
separately, the slopes are not significantly different from one another. However, 
the slopes in figure 8 are all about twice as great as those in figure 7. The broader 
distributions in figure 7 have values of CV of about 20 per cent. We may inter¬ 
pret these distributions as reflecting an intrinsic control by each cell, and we may 
say that the control of generation rate is relaxed for the cultures in figure 7. 
Correspondingly, the narrower distributions in figure 8 have values of CV of 
about 10 per cent, and for these control is stringent. The distribution for Euglena 
grown in an organic medium (figure 9) also has a value of about 10 per cent 
for CV, and therefore these cells were also under stringent control. 

This grouping of values of CV is lost when cells are not grown near or at 
their maximum rates of division. When Euglena was grown in a salts minimal 



566 


FIFTH BERKELEY SYMPOSIUM! KUBITSCHEK 


medium (figure 9) the value for CV was increased to 16.0 per cent (SE: 0.4 
per cent), intermediate to the two groups [16]. Another intermediate value, 
13.6 per cent, was observed by Sisken and Morasca for human amnion cells 
cultured in a salts medium supplemented with amino acids, vitamins, horse 
serum and antibiotics [17]. Furthermore, in each of the three species of bacteria 
that Powell and Errington studied [7], growth in a chemically defined medium 
gave markedly larger values for CV than did growth in a complex organic 
medium. 

I have suggested that these findings might be explained in the following way 
[9]. The generation time of each cell is determined by the time required to 
complete a large number of reactions, any one of which might be made rate 
limiting. Some of these reactions can be omitted if these cells are exposed to an 
exogenous supply of compounds that they would otherwise be required to 
synthesize. The elimination of these reactions would reduce the time necessary 
for cellular replication. In addition, the elimination of these reactions would 
also eliminate the fluctuations arising from them, thereby decreasing the value 
for CV. 

4.2. Alternate states of control of cell division. Since only a relatively small 
number of studies of generation rate distributions have been made for cultures 
grown in complex media, there was the possibility that the grouping of values 
of CV was accidental, and therefore of no further significance. Fortunately, it 
was possible to test for the occurrence of discrete groups by a different approach, 
suggested by the results of Burns’ studies [12]. For diploid Saccharomyces, 
generation rates were under stringent control at 20°C and 30°C (figure 8) with 
a value of CV of about 10 per cent, but control was relaxed at 38°C (figure 7) 
with a value of about 20 per cent. Burns had also observed a loss of control 
manifested by the failure of mother and daughter cells to bud simultaneously 
at 38°C, although they do so at the two lower temperatures. Further charac¬ 
teristics of these cells and related experiments are described by Tobias [21]. 

This loss of control led us to inquire about the dependence of CV at inter¬ 
mediate temperatures [22]. If the observed grouping of values were accidental, 
then we expected to obtain intermediate values of CV as the temperature was 
raised from 30°C to 38°C. On the other hand, if discrete states of control of 
cell division exist, then we expected to observe an abrupt transition from 
stringent to relaxed control at some intermediate temperature. When the ex¬ 
periment was performed, neither expectation was fulfilled, but the results did 
confirm the existence for two discrete states of control of cell division in diploid 
Saccharomyces. 

We obtained generation rate distributions for this strain of yeast at a number 
of different temperatures between 27°C and 35°C, using Burns’ criterion of 
budding for termination. Our experiments were not as accurate as those of 
Burns, since we did not separate cells by micromanipulation as he did. Instead, 
because we anticipated the need for measuring a much larger number of gen¬ 
eration times (2052 values were obtained), we decided to forego micromanipula- 



CELL GENERATION TIMES 


567 


tion and accept any consequent increase in CV that might occur from unbalanced 
growth arising through variations in local growth conditions in the growing 
microcolonies. As a result, it will be seen that our values of CV are consistently 
about a third higher than those he obtained. 

At temperatures below 32.5°C, the values of CV were essentially the same 
in each experiment (figure 11). Above this temperature, however, we obtained 



Figure 11 

Values of CV observed in daily experiments with Saccharomyces. 

The standard errors of CV are represented 
by the vertical bars above and below the circles. 

The square at 33.5°C indicates the value that was observed 
when the temperature of the culture was deliberately increased at 0.5°C 
for a period of 20 minutes at the beginning of the recording period. 

an unexpected result: although some values of CV were about the same as in 
the earlier experiments, many values were about twice as large. That is, the 
region from 32.5 to 34.5°C is a transition region in which the cells in these 
cultures can be either under stringent or relaxed control at any temperature in 
this region. In figure 11, it can also be seen that the frequency of large values 
of CV increases with temperature in the transition region. 

Since multiple values of CV implied some inadvertent failure to maintain 
completely constant conditions, we reexamined the experimental records and 
found that large values of CV were correlated with rather small inadvertent 
decreases in temperature, about 0.5 to 1°C, which occurred at the beginning of 





568 


FIFTH BERKELEY SYMPOSIUM: KUBITSCHEK 


the recording period in some of these experiments. This correlation suggested 
that the control of the interdivision period is unstable in the transition region, 
and that a short, slight temperature shift in either direction might trigger the 
onset of relaxed control in some cells. An instability of this kind would also 
explain the increase in frequency of large values of CV with increasing tem¬ 
perature in the transition region: the instability would be expected to increase 
with temperature. To test this possibility, at the beginning of a final experiment 
the temperature was increased 0.5°C above the steady state temperature for a 
period of 20 minutes. A large value was again observed for CV; it is represented 
by the square in figure 11. 

Although the standard errors of the values in the transition region were small 
enough to make it highly unlikely that we had sampled from a single population 
in these experiments, the variability of the larger values of CV suggested that 
not all parental cells were triggered to relaxed control. The data had been 
recorded as clonal histories of cell pairs (cells cannot be separated until budding 
begins [12]) allowed to divide for two generations, making possible a determina¬ 
tion of the value of CV for the twelve generation rate values obtained for each 



Figure 12 


The distribution of clonal values of CV. 

Clonal values of CV were calculated for the 12 cell generation rates 
observed for each clone that grew from its parental cell doublet. 
The shaded line encloses that portion of this bimodal distribution 
for which the average value of CV is 26.3 per cent. 




CELL GENERATION TIMES 


569 


clone. (Inviable cells were not observed.) The distribution of the clonal values 
of CV (figure 12) was clearly bimodal, and was fitted by a graphical analysis of 
the cumulative data. The results of this analysis are also shown in figure 12; 
the shaded boundary encloses the estimated distribution of those clones with 
the larger value of CV. The parameters are: stringent control, 154 clones, with 
a mean value of 13.7 per cent for CV; relaxed control, 37 clones, with a mean 
value of 26.3 per cent. Slightly better estimates can be calculated for these mean 
values of CV and their standard errors [22]. 

Along with these shifts between states of control of the interdivision period, 
we also observed a corresponding decrease in the mean generation rate of these 
cultures, figure 13. Our estimate of the average decrease, about 10 per cent, is 



TEMPERATURE, # C 

Figure 13 


Cumulative daily generation rates as a function of temperature. 

Open circles indicate experiments for which the value of 
CV was less than 20 per cent; filled circles indicate 
experiments for which the value of CV was greater 
than 20 per cent; square indicates the experiment 
for which the temperature was increased for 20 minutes. 

necessarily crude, and it is entirely possible that actual decreases in generation 
rate might be dependent upon the generation rate of the unperturbed culture. 

In a further attempt to examine these changes in control, cultures in diploid 
Saccharomyces in exponential (balanced) growth in suspension in an organic 
medium (malt extract broth, Difco Laboratories) were divided into two parts, 




570 


FIFTH BERKELEY SYMPOSIUM: KUBITSCHEK 


one of which was briefly exposed to an increased temperature. At first, these 
experiments showed no consistent change in cell division rate. Later it was 
realized that good results might require triggering all of the parental cells to 
relaxed control, and therefore, larger temperature shifts might be required. 
The results of a more successful experiment of this kind are shown in figure 14. 



Change in cell division rate of a culture of diploid yeast exposed 
to a temperature increase of 9°C for a period of two minutes. 

In this experiment the temperature was raised 9°C for a period of two minutes. 
Cell division rates were determined using a Coulter electronic cell counter. 
After the temperature perturbation, the cell division rate was reduced by about 
20 per cent in the shifted culture. This reduction in mean generation rate agrees 
with those from our earlier cell studies. 

4.3. Control “memory systems Since the state of control of cell division, 
relaxed or stringent, depends in these experiments upon the history of the 
culture, it follows that a “memory system” is involved. Cox [23] described 
temperature dependent hysteresis effects in macromolecular RNA that should 
permit the storage of information, and Katchalsky, Oplatka, and Litan [24] 




CELL GENERATION TIMES 


571 


have reviewed the evidence that cells contain systems capable of a “memory” 
of this kind. Macromolecular complexes would appear to be the most likely 
candidates for regulating cellular reactions, and thereby affecting cell generation 
times. Nevertheless, at present these temperature dependent hysteresis effects 
seem insufficient for a detailed explanation of alternative states of control, 
since either a temporary increase or a temporary decrease can lead to relaxed 
control [22]. 

5. Concluding and summarizing remarks 

If the results of the experimental studies and the interpretations that I have 
presented are essentially correct, then we have entered a new phase in our 
knowledge of the processes controlling generation times. Clearly, generation 
times are not entirely independent, as was assumed in the earlier models; nor can 
it be supposed that generation times of progeny continue indefinitely to be 
significantly dependent upon their remote ancestors. Rather, there are associa¬ 
tions between the generation times of cells for an intermediate period, some 
three generations in bacterial cultures, as demonstrated both by lateral and by 
longitudinal correlations between related cells. 

The dependence of the generation times of daughter cells upon those of their 
mothers is usually not revealed in correlation tests, but this dependence becomes 
evident when the sizes of cells at division are considered. These results show 
that generation times of progeny are influenced by nongenetic factors trans¬ 
mitted from their ancestors, and furthermore, that at least two such factors 
are necessary to account for the absence of correlation between mother and 
daughter, with opposite effects upon the generation times of daughter cells. 

Moreover, if evidence continues to support the generality of the truncated 
normal distribution of generation rates for cells growing at or near maximal 
growth rates, then it is probable that similar kinds of dependences will be found 
in other kinds of cells. In this case, too, a general model predicting this distribu¬ 
tion would be broadly useful. 

However, such a model must take into account the cellular transmittal of 
compensatory factors leading to the correlations observed between related cell 
pairs, although it might ignore the existence of discrete states of control of the 
interdivision period. Such a model must also take into account the results of 
the excellent study by Sisken and Morasca [17], which showed that an early 
part of the interdivision cycle before onset of DNA synthesis (G 1) also gave a 
normal distribution for the reciprocals of the times required for the completion 
of this period, and furthermore, that this period appeared to be even more 
variable than the total cycle. 


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572 FIFTH BERKELEY SYMPOSIUM: KUBITSCHEK 

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