Download Cell Cycle Dynamics Inferred from the Static

Journal of General Microbiology (1982), 128, 2877-2892.
Printed in Great Britain
2877
Cell Cycle Dynamics Inferred from the Static Properties of Cells in
Balanced Growth
By A. L. KOCH’* A N D M . L. H I G G I N S 2
Program on Microbiology, Department of Biology, Indiana University, Bloomington,
Indiana 47405, U.S.A.
Department of Microbiology and Immunology, Temple University School of Medicine,
Philadelphia, Pennsylvania 19140, U . S . A .
(Received 15 April 1981 ;revised 9 April 1982)
The duration of a morphological phase of the cell cycle is reflected in the steady state
distribution of the sizes of cells in that phase. Relationships presented here provide a method for
estimating the timing and variability of any cell cycle phase. It is shown that the mean size of
cells initiating and finishing any phase can be estimated from (1) the frequency of cells
exhibiting the distinguishing morphological or autoradiographic features of the phase ; (2) the
mean size of cells in the phase; and (3) their coefficient of variation. The calculations are based
on a submodel of the Koch-Schaechter Growth Controlled Model which assumes that (i) the
distribution of division sizes is Gaussian; (ii) there is no correlation in division sizes between
successive generations; and (iii) every cell division gives rise to two daughter cells of equal size.
The calculations should be useful for a wider range of models, however, because the extrapolation factors are not sensitive to the chosen model. Criteria are proposed to allow the user to
check the method’s applicability for any experimental case.
The method also provides a more efficient test of the dependence of growth on cell size than
does the Collins-Richmond method. This is because the method uses the mean and coefficient
of variation of the size of the total population, in conjunction with those of the cells in a final
phase of the cell cycle, to test potential growth laws. For Escherichia coli populations studied by
electron microscopy, an exponential growth model provided much better agreement than did a
linear growth model.
The computer simulations were used to generate rules for three types of cell phases : those that
end at cell division, those that start at cell division, and those totally contained within a single
cell cycle. For the last type, additional criteria are proposed to establish if the phase is well
enough contained for the formulae and graphs to be used. The most useful rule emerging from
these computer studies is that the fraction of the cell cycle time occupied by a phase is the
product of the frequency of the phase and the ratio of the mean size of cells in that phase to the
mean size of all cells in the population.
A further advantage of the techniques presented here is that they use the ‘extant’ distributions
that were actually measured, and not hypothesized distributions nor the special distributions
needed for the Collins-Richmond method that can only be calculated from the observed
distributions of dividing or newborn cells on the basis of an assumed growth law.
INTRODUCTION
Charting the events of the cell cycle is especially difficult with prokaryotes. This is largely
because light microscopy has insufficient resolution and electron microscopy (with or without
autoradiography) precludes following a single cell in time. Synchronization methods devised so
far have poor resolution and are subject to severe criticism. An alternative approach, proposed
here, is to study fixed populations of cells taken from steady state cultures in balanced growth
0022-1287/82/0000-9910 $02.00 01982 SGM
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2878
A. L . KOCH A N D M . L . HIGGINS
Cell mass, rn
2
2
E l
0
0
On’25
0.50
0.75
1.0
Age, a
Fig. 1. The canonical size distribution and the analysis of an idealized cellular phase. The cell size (rn)
on a linear scale with the birth size chosen at 1 and a division size at 2 is shown at the top. A second scale
for cell age (a) is given as the abscissa; it is related to the upper scale by the relationship, rn = 2“, where
a goes from 0 to 1. The ordinate is the probability that the cell size will be in the range rn to rn
drn. The
cross-hatched region is a hypothetical phase (designated by X because it is entirely internal to the cycle)
that initiates at size x, = 1.7, corresponding to age 0.765, and finishes at size x,- = 1.9, corresponding to
age 0.926. The cumulative frequency of the entire distribution is 1 and that of phase Xis E = 0.124. For
the Xphase, X = 1.796and qx = 0.0321. From these values of E, Y,q, together with the mean size of the
entire population,t = 1.386, x, and xfcould be deduced from the relationships: xf - x, = (In 2)S(qs,Z
1 ) X 2 / i and In x,-/x, = (In 2)EY/li:
+
+
and make careful measurement of the dimensions, morphology, and autoradiographic patterns
amongst a large number of cells. If a phase can be defined on the basis of morphology or
labelling, then the end points of the phase can be calculated. Thus, from (1) the percentage of
cells showing a certain character, (2) their mean, and (3) the coefficient of variation of the
distribution in dimensions of cells belonging to this phase, the distribution of cell dimensions as
cells enter or leave the phase can be inferred. These size relationships may be converted into
time relationships, if desired.
The simplest possible case is shown in Fig. 1. It would apply if all cells arose at division with
the same size, grew exponentially in mass and all divided in exactly the same time at exactly the
same size. For this ‘canonical’case, size (m)and age (a)are directly related by m = 2a, and a scale
for each is indicated on the abscissa. If a particular phase of the cell cycle began at a particular
age (or size) and ended at given age (or size) (for example, as indicated by the cross-hatched
region of Fig. l), then the age when cells begin and end this phase could be inferred from the
extremes of sizes of cells exhibiting the characteristics of the phase. The same endpoints of the
phase can also be deduced from (1) the percentage of cells exhibiting the phase (the area of crosshatching), (2) their mean size, and (3) the coefficient of variation. For the hypothetical phase
corresponding to the cross-hatched area of Fig. 1, such a calculation is given in the legend.
Evidently, the latter method is preferable to searching for extremes of range because of
biological and statistical artefacts. We propose here methods to allow the same type of
interpretation to be made for the less regular cell cycle of actual bacteria. Because in the actual
case, as contrasted with the ‘canonical’ case of Fig. 1, there is variability in the size at cell
division, the size distribution of the total population and any particular phase is broader.
As a second example, consider two histograms of the volumes of rod-shaped bacteria taken
from a single population of bacteria in balanced growth; one concerns cells with a central
constriction, the other concerns all cells in the sample population. Many examples of this kind
have been published by the Amsterdam workers (Woldringh et al., 1977; Koppes et al., 1978;
Trueba, 1982). Evidently, constriction is a criterion for a cell cycle phase that precedes and
terminates with the actual cell division event. From data about the cell lengths or volumes of
those cells showing constriction, we might wish to calculate the distribution of lengths or
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
Charting the prokaryotic cell cycle
2879
volumes of cells in the final act of cell division; i.e., to calculate the extant distribution of mother
cells [t$,(m), in the terminology of Painter & Marr (1968)l. The fraction of cells actually dividing
at any instant is infinitesimal, while typically, 10-20% of cells are classified as showing
constriction, so the actually observed phase of constricting cells ends with division, but does
occupy a significant portion of the cell cycle. The actual critical event of leaving the constriction
phase (i.e., dividing) defining the 4e(m)distribution occurs with a larger mean size, but a smaller
standard deviation (and coefficient of variation) than characterizes the population of cells
showing constrict ion.
This problem of calculating the variability of size at division from the variability of size of any
observable phase which ends the cell cycle is related to a common problem treated in elementary
statistics. The problem arises when data are grouped into classes. By grouping, the computation
of means and statistical measure of variations is greatly simplified : however, corrections must
be applied (see, for example, Sokal & Rohlf, 1969).The most often used is Sheppard’s correction
for the second moment, applied by the formula:
s =
[sg2
- h2/12]’
where s is the corrected standard deviation, sg is the standard deviation of the grouped
measurements, and h is the grouping interval. If the class measure was the midpoint, the mean
resulting from the grouped data is an unbiased estimate of the population mean and may be very
accurate if enough measurements have been taken. If the class interval designates one or the
other boundary of the group, then the mean must be corrected by 1/2 the grouping interval.
We may draw the analogy between the grouping interval and the range of sizes of a phase of
the cell cycle. The frequencies of such a group, and the means and variabilities of size within the
group, are experimentally measurable quantities. What we wish to know about a cell cycle phase
is the mean size for entry and exit from that phase and its variability. Because the class intervals
that we need to consider are sometimes broad and the distributions far from normal, it is not at
all evident that a Sheppard-like correction would be appropriate. However, the analogy with
Sheppard’s problem of grouping data turns out to be an appropriate one. It will be shown below
that a formula like equation (1) can be used to calculate the standard deviation of the size of cells
as they move through a phase boundary, the only change necessary being to replace the usual
factor 12 by a factor, a, whose numerical values are given for different cases below. By computer
modelling, we have found that the coefficient of variation of phase size, together with the value
of a suitable for the position in the cycle where the phase occurs in the cycle, can lead to an
estimate of the standard deviation of the size at transition into or out of phase with little error.
The results of computer calculation also yield factors needed to find the initiating mean size of
the phase, and the finishing mean size of the phase. These correction factors (and a) vary
depending on whether the phase is near the beginning, middle, or the end of the cell division
cycle.
ANALYSIS OF POPULATION DISTRIBUTION DATA
Nomenclature of cell cycle phases
Basically, the method presented here was developed to apply when synthesis of protoplasm is
continuous, as it probably is for most bacteria. The method will be presented, therefore, in the
terminology used in prokaryote biology. Although as explicitly presented, exponential growth is
assumed, growth laws between the limits of linear growth and exponential growth could be
handled quite similarly. If there are quiescent periods or periods of net degradation in the cycle
as can occur in eukaryotic cells, our method would require considerable revisions.
In this paper, we have adopted a set of symbols as consistent as possible with those of our
previous work (Koch & Schaechter, 1962; Koch, 1977)and of others. The symbols are defined in
Table 1. Briefly, capital letters are used to identify both the phase and the duration of that phase,
lower case letters for cell sizes, and capital Greek letters for the frequency of cells in a given
phase. Superscript bars indicate means of phases, and the subscripts i andfindicate the mean
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2880
A
L . KOCH A N D M . L . H I G G I N S
Table I Notation for cell cycle parameters
Symbols
Capital letter
Small letters
Capital Greek letters
Subscripts
i
f
Statistical measures
Superscript bar
Yl3
Bz
41-
Meaning
Name & duration of cell phase
Size of cell
Fraction of total population
in extant phase
Mean initiating size of coefficient
of variation of phase
Mean finishing size of coefficient
of variation of phase
Examples
S, C, E, T, X
s, c, e, t , .Y
z, r, H ,
T, 3
XI? 9x1
-yr,4 x 1
Mean
Skewness, kurtosis of total extant
population
Coefficient of variation of extant
X-phase cells
Phases of the cell cycle
Cell
division
Cell
division
Normalization of
the total cycle
B+C+D=T= 1
3 =b = t = l
1
1
1
e, = d, = t, = 2
P+T+d=T=l
Special definitions
Current size of a cell = rn
Age of cell = a
4s =qe =4
Sheppard’s correction factor = OL
Mean phase sizes relative to mean
population size = R = ?/i
initiating and finishing sizes of phases. The statistical measures are designated by q for
coefficient of variation, y1 for measure of skewness, and /I2 for the measure of kurtosis. Note that
the value of f12 for a normal population is 3.
Certain established phases of the cell cycle are indicated in Table 1. These include B, the gap
between cell division and the initiation of DNA synthesis, C, the period of DNA synthesis, and
D, the gap between the termination of DNA synthesisand cell division. In addition, the total cell
cycle is indicated by T. Without lack of generality, ?and 6have been set to 1 and $to 2. We also
use S for phases that start with cell division and E for phases that end with cell division.
The symbol X is used to denote any phase of the cell cycle of interest. Consequently, this
symbol will change its meaning in the course of the paper. At times, it will be set equal to well
established prokaryote phases, B, C or D and at others to that phase where the cells show
terminal constriction. But this list is not inclusive; the whole point of the paper is that the
method of data analysis developed here may be applied to measurements based on any
classification scheme, even those that have not been yet discovered.
The correction factors presented here have been calculated on the assumption that cell
division splits the cytoplasm precisely in two. This is a great simplification from the
mathematical and numerical point of view. In a number of bacteria, the ratio of a daughter’s size
to its mother’s size has a coefficient of variation less than 10% (e.g., Trueba, 1982). A special
symbol q, bearing no subscript, is defined for the coefficient of variation of the size of cell
distribution at division; i.e., the coefficient of variation of the #,(m)distribution. Therefore, the
coefficient of variation of the division size distribution and the birth size are equal and qsi = qef
- qk = q4f = q.
The caveat of the current paper is that the rules and graphs presented here only apply to those
phases that lie entirely within a single T-period and do not extend into earlier or later cell cycles.
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
Charting the prokaryotic cell cycle
288 1
The Koch-Schaechter method for cell cycle statistics
It can be assumed that cell division is the normal response of the cell when it has succeeded in
growing sufficiently. It is more controversial, but not unreasonable, to assume that under
constant conditions the mass of an individual cell grows approximately exponentially with time ;
i.e., the rate of growth of an individual cell is directly proportional to its own mass or its content
of ribosomes. If a cell divided precisely when it achieved a critical size, then the frequency
distribution of cell sizes in a population of cells in asynchronous balanced growth would be as
indicated in Fig. 1. There are four times as many cells in the smallest size class as in the largest
for the canonical distribution. One factor of twofold arises simply because one cell divides into
two cells. The second factor of two arises because the population distribution represents a census
of the culture at a given instant of time, but classified on a basis of cell size. Obviously it takes
twice as long for a small, newborn cell to increase its size a unit amount than it does a large cell
that is just about to divide and has twice the mass, number of ribosomes, etc., needed to carry out
the necessary anabolism. This means that the smallest cells will spend twice as long in a size
category and will, therefore, be twice as highly represented as the largest cells in the population
distribution. Because the birth size has been defined as 1 and the division size as 2, this
distribution can be described by:
8(m) = 2/m2 dm; 1
< rn < 2
Of course, actual populations of prokaryotes do show a small degree of variation in the size
that they attain as they divide, This variability causes a further widening of the population size
distribution above that shown in Fig. 1 (see Fig. 3, later). The degree of broadening of the
distribution depends on variability of the size at cell division.
For the calculations presented below, we have assumed a particular sub-model of the general
growth controlled (GC) model of Koch & Schaechter (1962). This specific model is designated
SGC (standard growth control). The program computes a cell size distribution by summing
many distributions each like Fig. 1. These sub-distributions differ in their minimum and
maximum size. Figure 2 is a flow chart for the program. First, the computer successively chooses
birth sizes varying systematically from three standard deviations below the mean to three
standard deviations above the mean. For each choice, it makes weighted contributions to the
population distribution with the corresponding probability value calculated from the normal
distribution. For each choice of birth size, the computer considers a range of division sizes from
three standard deviations below the mean to three standard deviations above the population
mean and weights them as it did the birth sizes; i.e., according to the value drawn from a
Gaussian table of probabilities. In this way the computer has taken into account the variation of
birth and division sizes. For each combination of birth and division size, there are bacteria that
arose at various times before the instant of sampling and have grown to some intermediate size.
Their contribution to the frequency distribution depends on the inverse square law of equation
(2), as depicted in Fig, 1. The computer adds these contributions into appropriate memories
indicated on the right-hand side of Fig. 2 to form the cell size histogram. In this way, the
distribution shown by a solid line in Fig. 3 was generated. These calculations were laborious and
involved consideration of 601 birth sizes, 601 division sizes and 300 potential sizes of cells,
giving rise to a total of about lo8 contributions for each distribution.
The distribution of cell sizes if increase in cell size occurs according to a linear growth law
instead of an exponential law, is shown by a dashed line. The computer program in this case was
operated as indicated above, except that the inverse square weighting factor was replaced with a
negative exponential weighting factor. For linear growth, size and age are interchangeable. It is
well known that for the canonical case, the age distribution is a negative exponential (Koch &
Schaechter, 1962). This relationship was originally proposed by Euler in 1775 (Koch &
Blumberg, 1976). The calculations presented in this paper supersede earlier calculations and
have increased precision because finer intervals of birth and division sizes were used than in
previous work (Koch, 1966, 1977). In both this paper and Koch (1977), a range of sizes
extending three standard deviations below and above the mean were considered.
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2882
A . L . KOCH A N D M . L . H I G G I N S
0
size
size
prob.
cell
prob..
(Gaussian)
prob.
(Gaussian)
mf-0
NO
-
1
m
m
Fig. 2. Computer flow diagram. For each combination of birth size, division size, and intermediate
extant size, a contribution is calculated as the triple product of the probability of that combination
occurring. The contribution is added to a position in an array corresponding to that value of m.Two
such arrays are shown at the bottom of the figure. The contribution is assigned to one or the other
contingent on the basis of whether the cells under consideration correspond to belonging to the phase
(right-hand distribution) or failing to belong (left-hand distribution).
h
5
rn
2
1
Cell size, m
Fig. 3. Size distribution for steady state growth controlled cells growing either exponentially (solid line)
or linearly (dashed line). Both cases were calculated from a model that assumed that the division size
and birth size of extant cells was Gaussian and uncorrelated with q = 0.1, and that cell division
partitions cell contents evenly.
Value
Quantity
Symbol
Mean size*
Coefficient of variation
Skewness
Kurtosis
*
t
q
Y1
P.
Exponential
model
Linear
model
1-3776
0.2253
1.4452
0.2202
0.3814
2-4267
0.5685
2.6650
Mean birth size, t,, is equal to 1.
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2883
Charting the prokaryotic cell cycle
This method of treating the cell cycle is consistent with the Collins & Richmond (1962)
method. The two approaches are the inverse of each other. As usually used, both methods are
deterministic and assume that a cell of a certain size will grow at the same rate as any other cell of
the same size, independently of how long an interval has passed since the last cell division, or the
interval that will pass until the next cell division.
Relationship to the Collins-Richmond treatment
Collins & Richmond (1962) derived an equation for calculating the growth rate as a function
of cell size from the measurement of size distributions of cells taken from balanced,
asynchronous cultures. Their equation (Painter & Marr, 1968) depends on knowing three distributions : the size distribution of the extant total population, the size distribution of a sample of
mothers, and the size distribution of a sample of babies. In practice, the first distribution is
measured and the other two are assumed. The approximation has usually been to assume apriori
that both are narrow normal distributions. It has been generally assumed that both the
distributions of division and birth sizes have the same coefficient of variation. This requires that
division creates equal sized daughters. These approximations are needed because the required
division and birth distributions have not been measured. For many applications, the exact shape
of these distributions is not critical. Examples where this was the case were presented by Harvey
et al. (1967). The Collins-Richmond technique yields the average velocity of size increase as a
function of cell size. An early study reported that the velocity of size increase was very low for
both extremely small cells and extremely large cells (Harvey et al., 1967), but substantially
proportional to cell volume for the bulk of the cells.
In their critical review of the mathematics of microbial populations, Painter & Marr (1 968)
consider several types of frequency functions for different phases of the cell cycle (see their Table
1). It is to be noted that all the distributions formulated here correspond to those classified as
extant size distributions. Powell (1964) referred to them as 'realized' distributions. This choice
means that there is a direct correspondence between the experimentally observed size
distributions of cells in E phases, S phase, the total population, and the limiting distributions of
cells that are just dividing, +,(m), or are newly arisen, +&n). It is necessary to point out that the
distributions needed for the Collins-Richmond technique are quite different from those used
here. For this technique to be used on a completely experimental footing, these must be
computed from the extant distributions by assuming a growth law. Even if an exponential
growth law is assumed, the conversion is quite complicated [see Equation (27) of Powell (1964)l.
Otherwise the appropriate sample of cells must be followed in time until they divide and a
distribution prepared from the sizes observed just prior to division, +(m),and immediately after
division, has occurred, +(rn).
Choice of growth law
The application of the Collins-Richmond technique has led to a variety of interpretations
(Collins & Richmond, 1962; Harvey et al., 1967;Zusman et al., 1971;Cullum & Vincente, 1978;
Trueba, 1982). However, we suggest that the published evidence is consistent with exponential
growth of individual cells, but also indicates that a low percentage of cells are abnormal and
grow much more slowly than the remainder. The abnormal cells are usually either very large or
very small (Koch, 1980). When only the mid-range of cell sizes are considered, there is little
support for a linear model of growth.
For the purposes of analysing cell cycle phases, however, the choice of a growth law is of
relatively little importance. Even if the growth in cell size were linear, instead of exponential,
little effect can be expected on the magnitude of the correction factors presented below. This, as
noted by several authors previously, is because there is very little difference in cell mass, rn,
between, for example, a linear and an exponential mathematical function when both functions
are constrained to go from m of 1 to rn of 2 as the independent variable (the age of the cell), a,
goes from 0 to 1. Thus the greatest difference between the lines m = 1 a and m = 2" in the
region of 0 < a < 1 is only slightly greater than 6%. Consequently, the course of cell
enlargement is not much different in the two cases. From the distributions for cell sizes shown in
+
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2884
A . L. KOCH A N D M . L . HIGGINS
*-O
*-.
r
I
0.5
Age, a
1.0
Fig. 4. Linear, bilinear, and exponential growth. Cellular growth curves are shown for the linear and
exponential (bold line) case and for those two bilinear cases deviating least from the exponential mode
of growth. The residual sums of squares of differences (RSS) between the exponential case and all
possible bilinear modes of growth is plotted versus the relative cell age at which the growth rate of the
bilinear model doubles. Linear growth is the special case where this doubling occurs at a = 0 or a = 1.
The lines marked 0.28 and 0.85 are the bilinear growth curves, in which the growth rates double at these
fractions of the cell cycle; these are the cases that most closely approach the case of exponential growth.
Fig. 3, it can be seen that there is really very little difference between the linear and the
exponential model. This impression is reinforced from inspection of the statistical parameters of
the distributions given in the legend to Fig. 3. The difference between the two models is that it
takes a constant time for cells to pass through a size interval for linear growth, while this time is
inversely proportional to cell size for exponential growth.
The differences in the statistical parameters of the size distributions provide a test for the
linear model versus the exponential one. Although the mean values differ, this is only of value in
distinguishing between the growth laws experimentally where there are accurate independent
measurements of the mean birth or division size. This is an important point and is a critical
advantage of the present method (see below). The small difference in y1 and p2 requires accurate
data, unbiased by truncation or special properties of a few abnormal cells. The coefficient of
variation, qr, which is the most accurately measurable and accessible quantity, shows almost no
difference between linear and exponential growth.
Today, the linear model in the original form, in which the growth rate is assumed to double at
the instant of cell division, has been revised to assume that the growth rate doubles at a fixed age
in the cell cycle (Kubitschek, 1970, 1981). Such ‘bilinear’ models may approach the simple
exponential much more closely than does the linear (Fig. 4). This figure shows a linear (thin
line), an exponential (bold line), and two bilinear growth curves. The ordinate values on the
curved heavy line marked RSS (residual sums of squares) are the sums of the squares of the
differences between cell sizes at the same ages for exponential growth and bilinear growth. Each
point on this curve corresponds to the sum for ages ranging from 0 to 1, but for different bilinear
models in which the doubling point for the growth rate was shifted systematically from a = 0 to
a = 1 . This curve goes through two minima corresponding to the two bilinear growth curves
shown (where the doubling point occurs at a = 0.28 and at a = 0.85). In a number of cases in the
literature where bilinear growth has been suggested, the data were consistent with growth in
which the break occurs at about the age of the second minimum of the RSS curve. Consequently,
it is felt that the methods developed below should apply with an extremely small error even if
volume growth takes place by a bilinear mechanism.
PROCEDURE
The following operational approach is proposed. (1) Measure sufficient cells (at least 500)
from a population in a well defined state of balanced growth in order to define accurately the
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2885
Charting the prokaryotic cell cycle
1-39 I
'
T
--
0.6
s
x
h
v
m
v)
C
3
1.37
0.5
.-v)
La
0
*
Lf
m
-
1-36 1
0.20 1
2'
I
I
0.05
I
I
1
0.10
4
1
0.15
Fig. 5. Statistical parameters of the cell size distribution of a total population. The curves are plotted
from size distributions calculated for the SGC model. The values for the statistical parameters for the
first four moments are plotted as a function of 4, the coefficient of variation of the extant distribution of
sizes of dividing (mother) cells. As a more accurate alternative to reading values off the graph, the
following polynomials are provided. They were obtained by regression through the computer
simulation values for q = 0.05, 0.075, 0.1, 0.125, and 0.15.
+ 3.5617q2 - 35.66q3 + 106q4
+ 6.8629' - 33.48q3 + 78-99'
== 0.5113 - 1.0347q + 31.53q' - 190*9q3+ 362.7q'
= 2-1375 - 2.3963q + 136.71q' - 7169' + 11636.
Tit, =
1.38688 - 0.1978539
qr = 0.20959 - 0.27319
pz
population size distribution. (2) Compute the first four moments of the distribution. (3) Measure
the size distribution of the population in any phase of the cell cycle that ends with cell division
(e.g. septum of the cells exhibiting a cell constriction, etc.). (4) Calculate the mean size and
coefficient of variation of the size at division from this distribution using the methods presented
below. (5) From these two parameters, read off the expected moments of the population of total
cells from Fig. 5 , or calculate them from the formulae given in the figure legend. (6) Compare
expected parameters of total cell population from step 5 with observed values of step 2. If the
agreement is good, then the SGC assumptions (of independent sizes at successive divisions, of a
Gaussian division distribution, of exponential growth of individual cells, and of the division of
the mother cell into equal daughters) are a satisfactory representation of the cell populations
under study. (7) Apply the graphs and formulae presented below to the study of another phase or
phases of interest. We have prepared a program for the Hewlett-Packard HP41C pocket
calculators that carries out all the necessary arithmetic. The listing or magnetic strips can be
obtained from the authors. An example of the calculation is given below.
In making the comparisons under step 6, the skewness and the kurtosis measure can be
expected to be slightly higher than the calculated values because of a small fraction of defective
cells of extreme size. We believe that for many bacteria under diverse growth conditions, the
population does contain a significant but small percentage of cells that are defective or
pathological, even during balanced growth. Such cells will tend to be either very small
(minicells) or very large (filaments), but at either extreme they are not engaged in protein
synthesis or volume increase as actively as would have been inferred for their size from the
behaviour of the predominant kind. Because they grow slowly or not at all, either sub-population
will have gross effects on the measurement of skewness and kurtosis of the total cell size
distribution (Koch, 1980). This is why we strongly recommend autoradiography of balanced
cultures, pulse-labelled with a precursor of protein, e.g., [35S]methionine. In this way, the
proportion of slow-growing or non-growing cells and their size distribution can be estimated and
individual cells identified. Even in the absence of objective data, the corrections would work
well for the bulk of the cells, but some subjective decisions to eliminate abnormally growing cells
must be made (unless pulse autoradiography has been done to make such correction
objective). A final caveat is that the phase under study must be contained entirely within a single
cell cycle to employ the correction factors of Figs 7-9 (see later).
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2886
A . L . KOCH A N D M . L . HIGGINS
CORRECTION FACTORS FOR CELL CYCLE PHASES
The computer program used in the present study is a revision of one originally constructed to
study the phase of DNA synthesis (Koch, 1977). For the more general purposes of the present
paper, the program was reworked into several variants, spanning a range of assumptions as to
criteria for entry and exit of cells from the phase of interest. The modified programs calculate, as
outlined above, the contributions of cells from birth and division size classes for each
intermediary size. But in addition (Fig. 2), the computer now checks each contribution to see if it
belongs to the phase under consideration. If a contribution [i.e. the triple product of the
probability of the birth size, the division size, and the inverse square probability from equation
(2)] belongs to a cohort of cells that satisfy the criteria for inclusion within the phase, the
contribution is added into the register for that size of cell in the memory bank indicated on the
right-hand side of Fig. 2. In the opposite case, when the cells do not belong to the phase in
question, the contribution is added to the registers indicated on the left of Fig. 2. When all
combinations have been made and totalled, the computer then calculates a variety of items
concerning the distribution of cells within the phase. These include mean, standard deviation,
etc., but also include items allowing the generalizations presented in the sections below to be
applied.
Phases ending with cell division, E phases
The phases ending the cell cycle will be considered first, since morphological changes are
generally observable in cells about to divide and therefore the method of this section is always
applicable. E-phases are also considered first because their treatment is a prelude to predicting
the properties of the entire population. This permits an essential step in the method-the
comparison between the predicted and observed total size distributions in order to decide
whether the method is appropriate. If so, then the method can be applied to other kinds of
phases.
Distributions of cell sizes were calculated as indicated above for five choices of the coefficient
of variation for the dividing population (i.e., q = 0~05,0~07,0-09,0~11
and 0.13). This covers the
range of variability in size of division seen in experimental studies with many prokaryotes. For
each value of q, the computer was asked to calculate the distribution of the population of Ephase cells assuming that the duration of Econstituted the last 5, 10, 15,20, or 25% of the mean
cell cycle time, F.In so doing, we have tacitly assumed that no matter whether a cell is larger or
smaller than average at division, it will take just as long for the cell to pass through the terminal
phases of the cell cycle.
For each of these 25 cases, data from computer simulation were used to compare the fraction
of the total population in the terminal phase, (N),
the coefficient of variation of sizes in the
phase (qe), and in the total population (4). It was found empirically that:
q
=
(end of cycle)
(qe2 - H2/14-4)4
(3)
This expression can be accurately used in calculating q over the entire range of durations of the
E-phase and over the range of q values. Thus, an equation of the form of equation (l), but with
14.4 replacing the usual 12, can be used to estimate the coefficient of variation of the division
size distribution.
Correcting the mean of the class to the size of the dividing cell is not quite as simple, but Fig. 6
gives a graph of appropriate factors. More precisely, the ratio of the mean size at the finishing of
an E-phase to the mean size of the phase is given by:
eiZ = 0.9942
+ 0.168q + 0-489H
-
OelqH
(end of cycle)
(4)
-
0.05qH (end of cycle)
(5)
The mean size for initiating an E-phase is given by:
e,/Z
=
0.9922
+ 0.1648q
-
0-431H
As is evident from the system of nomenclature (Table l), D and Tare also phases that end the
cell cycle. Therefore, ef = d, = tp since all are symbols for the mean size at cell division. The
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2887
Charting the prokaryotic cell cycle
End of cycle . ,
,
;’
/
”
.-
Y
.-u
.-L
.-
A
W
0-9
I
I
0
1
I
I
\,
’l . ,
.
J
10
20
30
2, Cells in class (96)
Fig. 6. Factors for estimating the mean initiating and finishing sizes of cell phases that either end or
start with a cell division event. These factors are to be multiplied with the mean size of cells exhibiting a
morphological or autoradiographic phase. The lines in the upper half give correction factors to compute
the mean finishing size of phases that either start or end with cell division. Those in the lower half give
correction factors for the mean initiating size for phases that either start or end with cell division. The
correction factors for phases terminating with cell division are partially dependent on q (dashed line,
q = 0.05; dotted line, q = 0.1 1) and linear interpolation is necessary and sufficient. For phases starting
with cell division, the value of q is much less critical and no interpolation is necessary. The equations
(4),(9,(7), and (8) (see text) can be used more accurately than the figure for calculating mean initiating
and finishing sizes when q is in the range 0.05 to 0.1 1. When q is larger (0.15@25), different regressions
are appropriate; these can be obtained from the authors.
mean initiating size of an E-phase, ei, may or may not have any intrinsic relationship to other
phases of the cell cycle.
Phases starting with cell division, S phases
In eukaryotic systems, the phase extending from the end of mitosis to the beginning of DNA
synthesis is designated G1. It appears to be the important phase in the regulation of the cell cycle
(see, e.g. Stubblefield, 1981). Many hormones and growth factors seem to act on cells in this
phase. In slow-growing bacterial cells, there can be an interval between cell division and the
initiation of DNA synthesis, designated B. In many circumstances, B may be negative and vary
widely for individual cells within the same population. There is at least one morphological phase
of short duration that seems to be initiated at cell division, or very close to it. This is the
band splitting in Streptococcus faeciurn, which in slow growing cultures occurs in most cells
immediately after cell fission (M. L. Higgins and A. L. Koch, unpublished observation).
For phases starting with cell division, the same range of values of q was examined as in the
previous section. As before, the computer output where the duration of S constituted 5 , 10, 15,
20,25% of the mean cell cycle time, was examined to develop an empirical relationship for the
interpretation of the simulation data. Over these ranges, the analogue of the Sheppard’s
correction formula is:
r,
q = [qs‘
+ C2/44I7
1
(start of cycle)
(6)
Because the Sheppard’s correction factor for S-phase is so large, unless the phase under
consideration includes a very large fraction of the population, q will be very nearly equal to qs.
Thus, the observed variation in the size of a cell in a starting phase is almost equal to the
variation in birth size. To compute the mean values at the end of an S-phase, the solid line in the
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2888
A . L . KOCH AND M . L . H I G G I N S
upper part of Fig. 6 can be consulted or, more accurately, the following regression can be
employed.
sfl?
=
1.00035
+ 0-24526C + 0-1233Z2
(start of cycle)
(7)
The mean size at initiation of an S-phase can be calculated from the solid line in the lower part of
Fig. 6 or is given by:
si/? = 0.99974 - 0-24612C
+ 0.0253C2
(start of cycle)
(8)
Of course, in this case, si is an estimate of bi or one-half of df.
Phases contained entirely within the cell cycle, X phases
A quite different problem concerns phases of the cell cycle that neither originate nor
terminate with cell division. The phase of D N A synthesis has generally been assumed to be an
example of such an X-phase in slow-growing cells. For this class of problem, the computer was
given parameters concerned with the assumed duration and position of the phase of interest in
the cell cycle, at various fixed times relative to either the start or the end of the cell cycle; the
computer was then asked to calculate the distribution of cells in phase X . We carried out
computations under two different assumptions : (1) that the duration of the interval between the
end of Xand the end of the cell cycle has no variability, and (2) that the interval between division
and the beginning of the X phase has no variability. Except for some caveats described below,
these two extremes give rise to nearly the same rules for finding the mean initiating and finishing
sizes x,and xf, the variability of these sizes (qsxiand qxf)as well as the duration of the phase, X .
For the case in which the X phase is identified with the D N A synthesis phase, these two
assumptions correspond, respectively, to the assumption that D has no variability and that B has
no variability (see Koch, 1977). In these cases, all the variability must reside in the other phase.
These two cases, of course, do not exhaust the possibilities. Other reasonable possibilities
include cases where there is variation in the duration of the X phase and where the phase starts
in each cell at random at a certain mean size with a certain coefficient of variation,
independently of the size of that cell at the start or end of its cell cycle.
A condition that must be met to use the graphs and regressions presented in this section is that
the phase must not extend outside the duration of a single cell cycle. For example, the method as
described would not give accurate results in those cases where the variability in the start of D N A
synthesis is so large that D N A synthesis in part of the population starts in a different cell
generation from that in which it is completed. The method would be inapplicable even if in most
cell cycles the C phase is contained within a single cell cycle.
Consequently, the computational method given below is more likely to be successfully applied
if the phase is short and occurs near the middle of the cycle. A rough rule is that Xi?, the mean size
of the phase relative to the mean size of cells of the total population, should be somewhere
between 0.9 and 1.25.
Figures 7 and 8 show the value of the Sheppard’s correction factor, a, as a function of R = x / t .
As above, the value of a from the each simulation was calculated from:
a = P/(q,*
-
qxi?)
(internal to cycle)
(9)
It can be seen that the factor ranges between the limiting values of 44 and 14.4 presented in
previous sections, appropriate to the phases that start or end with a cell division event. It is also
apparent that a is dependent almost solely on Fir. Thus, for those cases in which the duration of
Xis assumed to be the same for all cells in the population, the user can calculate the coefficient of
variation and the size of initiating and finishing of the phase from three experimentally
accessible quantities: q x , q, and Z/T.
The duration of an Xphase can be calculated from the following empirical relationship, which
was discovered by inspection of the computer output: X = E l f . This relationship appears to
have no simple mathematical basis. That it is quite accurate, however, can be seen from Fig. 9.
The deviation from this simple relationship is less than & 10% as q is varied, as Xis varied from
0 to 0-6,or as the interval between X and either flanking cell division event is varied. The graph
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2889
Charting the prokaryotic cell cycle
26
8
c1
E
g
.c
8
5
18
I
1
I
I
I
I
50
40
30
20
cd
a
a
2
16
10
v)
14
0.8
d
0.9
1.0
1.1
1.2
I
0.7 0.8
1-3
I
I
I
I
I
0.9
1.0
1.1
1.2
1.3
I
1.4
R = XI7
R = X/7
Fig. 7
Fig. 8
Fig. 7. Sheppard’s correction factor (a)for phases interior to the cell division cycle. Calculations are for
q = 0.1 (solid lines); q = 0.05 (dashed lines). The values of OL are calculated, except for one line, on the
assumption that there is no variation in the interval from the end of variable phase X to cell division.
This exception (dotted line) applies when q = 0.05 and there is no variation in the time from cell
division to the initiation of phase X.
Fig. 8. Sheppard’s correction factor for short phases. The value of a is given for a phase which can occur
at any point in the cell cycle but maintains accurate relationship to the next division (continuous line) or
to the previous division (dotted line). The calculations are for q = 0.05 and X = 0.1. For very short
phases, R = T//fcan vary from 0.693 to 1.386. The predictions of the model both where the phase X
times with no error the next division, and where the previous division times the initiation of the Xphase
are in agreement for a considerable part of the range of .Y/i.They differ at lower values of R. This
corresponds to cases where there is a long gap before the next cell division. Then in an appreciable
fraction of cells, the phase must be started in the previous cell cycle for the cases where the X-phase
times the next division.
0.8
I
I
I
I
I
0.9
1.0
1.1
1.2
1.3
1.4
R = .?/i
Fig. 9. Error in the approximation X = ZUji. The ordinate is a correction factor which, if multiplied by
B.Y//T, yields the duration of the phase under study in units of fractions of a mean generation time.
Factors are given for q = 0.05 (dashed line) and 0.1 (solid line) and for X = 0.1 (thin line) and 0.5 (thick
line). These calculations are based on the assumption that the X phase times cell division. The origin of
each line corresponds to the situation in which the phase terminates with cell division. The lines end at
arrow heads when the interval from the end of the phase to cell division is sufficiently long so as to push
IO%of the phase Xcells into the preceding cell cycle. Consequently, the factors given here should apply
to data if the distribution of cells within the phase versus cell size is monomodal.
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2890
A: L . K O C H A N D
M. L . HIGGINS
can be employed when higher accuracy is required, but in most cases no correction factor is
needed.
A P P L I C A T I O N O F T H E M E T H O D T O Escherichiu coh'
In recent years, extensive and careful electron microscope studies of the properties of
balanced populations of E. coli have been reported by the group in Amsterdam. These studies
have been based on the measurements of tens of thousands of images of bacteria. At this point,
these published data can be used to justify the application of the method to a variety of strains of
E. coli grown under a variety of conditions. In two papers (Woldringh et ul., 1977; Koppes et al.,
1978), data are presented which are adequate to test the applicability of the procedure. These
workers measured the distribution of sizes of the sub-population of cells possessing constrictions
as well as that of the total population of cells. The data base consists of 14 cell samples where
between 632 and 3163 cells were measured and the first two moments of the total and
constricting populations were reported. In these two papers, 21 279 cells in total were measured.
While the mean size at division varied considerably with the growth conditions and the strain,
the coefficient of variation of the distribution of constricting cells was quite constant with a
mean value of 0.0853 and with a standard deviation of 0-0197.
As a numerical example of our method, consider the set of 1549 cells of E. coli B/rA growing
with a 109 min doubling time reported on the first line of Table 2 of the paper of Koppes et al.
(1978). For cells showing constriction, the coefficient of variation, qe, 'was reported as 0.084 and
the fraction of cells in the constriction phase, H , as 0.105. When these values were substituted
into equation (3), q was found to be 0.0793, slightly less than qe. By substituting q, H , and the
reported mean size of constricting cells, Z, of 2.0 pm, into equation (4), the mean size of dividing
cells, er, was found to be 2.1 16 pm. Then, substituting q into the equations in the legend to Fig. 5,
the moments of the entire population were predicted. These values are: i / t , = 1.380; qr =
0.2175; y1 = 0.5467; p2 = 2.4962. Consequently,i = 1.380 x 2.1 16 pm/2 = 1.47 pm which can
be compared with '7 reported by Koppes et al. (1978) of 1.47 pm. Similarly, the value of qt
calculated from distribution of constricting cells, 0.2175, can be compared with the value of qr
they calculated directly from the distribution of the total population of 0.226.
Similarly, for the remaining 13 cases, we computed q from equation (3) and Zffrom equation
(4). From the values of q and Zr, the mean size e,of the total cell population, t,was calculated
using Fig. 5. On average, the observed mean cell sizes of the 14 populations were only a little
smaller (2.3%) than the values calculated from the constricting population. Another way to
report the same finding is that the ratio of the observed to calculated mean sizes was 0.097 &
0.03.
A very large, unpublished, data sample was kindly given to us by Frank Trueba. It
consisted of 14 populations with a total of 14000 cells. What was of special importance about
this set of populations is that the extant distribution of the lengths of constricting cells and the
lengths of incipient daughter cells were measured. In this case, the ratio of the observed to
calculated mean size was 0.970 f 0.044 or 0.971 & 0.040, depending on whether the dividing
cell distribution or newly arisen cell distribution was used in the calculation.
The coefficientof variation for the total population, qt, was also calculated from Fig. 5 for the
three sets of populations. For the Woldringh et al. (1977) paper, the ratio of the observed to
calculated values of qt was 1.122 0.039; for the Koppes et al. (1978) paper the same ratio was
1.057 f 0.040; and for the Trueba unpublished data, the ratio was 1.045 f 0.0505. Thus, the
observed mean of the sizes in the total population is consistently a very little smaller than the
calculated mean, and the actual width of the population size distribution is slightly larger than
predicted from the constricting subpopulation.
These findings are close enough to support the SGC model for these strains of E . coli and the
application of the methods presented here to these strains and culture conditions. The slight
discrepancies for both the mean and width of the total population distribution are in a direction
such that they would be further enlarged if the calculations were repeated under the assumption
of a linear rate of volume increase. This argues against linear growth. However, the deviations
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
Charting the prokaryotic cell cycle
289 1
are consistent with the assumption that a few cells in the population are large and have a
decreased ability to carry out macromolecular biosynthesis, or that cells increase their volume
growth rate slightly above the exponential model in the last phases of the cell cycle.
The third and fourth moments of two population distributions of cell sizes of E. coli B/rA
cultures in balanced growth at doubling times of 125 and 109 min have been kindly provided by
Conrad Woldringh (personal communication). The values of y l , 0.74 and 0.77 from the total
population are larger than the values of 0.56 and 0.55 calculated from the observed distribution
of constricting cells. The values of kurtotic statistics are 2.89 and 3.15, respectively, larger than
the predicted 2.80 and 2-49. Again, the observed deviations from the expected values are
consistent with a few, abnormal, slowly growing cells that are predominantly larger.
DISCUSSION
The earliest studies of the cell cycle of prokaryotes were the microscopic studies of
interdivision times (Rahn, 1931-1932). His observations, which have been often confirmed,
were that within a single culture under constant conditions Tis highly variable. Typically, T has
a coefficient of variation of 20-25%. We first noted (Schaechter et a / . , 1962) that size at division
was much better controlled and typically had half this coefficient of variation. This means that
relating phases of the cycle to cell size had much greater potential for identifying critical events
than relating them to age.
It is curious that 20 years later, cell size as an index of progress through the cell cycle is only
now attaining ascendancy over cell age. This is largely because of advances in electron
microscopy, in electron microscopic autoradiography, and in computer techniques that
facilitate measurement of cellular dimensions from microphotographs. These methods, of
course, provide data primarily about cell size and not cell age.
The GC model (Koch & Schaechter, 1962) had as its basic tenet that growth in terms of
production of cytoplasm must ultimately control growth. It was proposed there, and further
stressed later (Koch, 1977), that a search for the least variable phase of the cell cycle could
identify the stage that responds to the cell’s success in synthesizing cytoplasm. This still remains
a working hypothesis, since in no single case has a controlling mechanism been dissected out or
defined.
An obvious candidate for a phase to be controlled by cell mass is the initiation of DNA
synthesis (Cooper & Helmstetter, 1968; Donachie, 1968), but the evidence is clear in certain
cases that DNA initiation has a broader variability than does cell division (Koch, 1977). In such
cases, it is therefore impossible that chromosome replication has the critical role in the division
cycle, although it may have a veto role. This, we believe, is a fair conclusion, even though
analysis of the published data of Myxococcus xanthus (Koch, 1977) and the analysis of the data of
Woldringh et al. (1977), and Koppes et al. (1978) are consistent with DNA synthesis having qa
values similar to, but not smaller than, the population q. Certainly, the size at initiation of DNA
synthesis does not seem to have a smaller coefficient of variation than does the size at cell
division; i.e., qa is not less than q.
The method developed here provides a highly critical way to test the growth laws of different
organisms under different conditions. Although the population distribution contains the
required information, the distribution must be very precisely defined (Koch, 1966) to make the
errors on the third and fourth moments small enough. We now point out that a comparison of the
mean of the total population calculated from the total distribution and the mean calculated from
an E phase provide a refined test. Although the sizes of cells vary widely, the standard error of
the mean can be quite small since many cells can be measured. Although there are usually fewer
E-phase cells, because the variation in size is smaller, the standard error of the calculated mean
size of the total population is in most cases comparable. Thus, with adequate sampling, both can
be small compared with the 4% difference corresponding to exponential versus linear growth
when q = 0.10 (see legend to Fig. 3). Thus the observed mean size of each of the E . coli
populations is not significantly different from the mean size calculated from the dividing
population when exponential growth is assumed. However, the difference for all populations is
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24
2892
A . L. KOCH A N D M. L. HIGGINS
highly significant when linear growth is assumed. Consequently, we conclude that E. coli grows
exponentially and that our new approach can exceed the power of tests based on total population
distributions alone.
The purpose of this paper has been to propose a method to aid in the definition of phases of the
cell cycle; it should aid in the search for the most precisely regulated phase as well as establishing
size and temporal relation in the cycle. Of course, the method should not be applied when the
standard growth controlled model (SGC) does not predict the population dynamics adequately,
nor if the phase under study overlaps generations. The criteria proposed must be satisfied if one
is to use the computational methods, but for those cases where it can be applied, we trust that
these graphs and formulae will serve as a Rosetta stone to translate pictures of fixed cells into an
understanding of the dynamic processes of the cell cycle.
We thank the consulting staff of the Wrubel Computing Center for aid during the development of the programs
presented here. The work was supported by the Indiana University Computing Fund. We thank Dr Lolita DaneoMoore for suggesting this problem, and Drs Conrad Woldringh and Frank Trueba for providing us with their
unpublished data. Experimental work in our laboratories is supported by NSF PCM 79 11241 (A.L.’K.) and NIH
A1 10971 (M.L.H.).
REFERENCES
COLLINS,J . F. & RICHMOND,
M. H. (1962). Rate of
growth of Bacillus cereus between divisions. Journal
of General Microbiology 28, 15-33.
COOPER,S. & HELMSTETTER,
C. E. (1968). Chromosome replication and the division cycle of Escherichia
coli Bjr. Journal of Molecular Biology 31, 5 19-540.
CULLUM,
J . & VINCENTE,
M. (1978). Cell growth and
length distribution in Escherichia coli. Journal of
Bacteriology 134, 330-337.
DONACHIE,
W. D. (1968). Relationship between cell
size and the time of initiation of DNA replication.
Nature, London 219, 1077-1079.
HARVEY,
R. J., MARR,A. G . & PAINTER,
P. R. (1967).
Kinetics of growth of individual cells of Escherichia
coli and Azotobacter agilis. Journal of Bacteriology 93,
605-6 17.
KOCH,A. L. (1966). Distribution of cell size in growing
cultures of bacteria and the applicability of the
Collins-Richmond principle. Journal of General
Microbiology 45, 409-41 7.
KOCH,A. L. (1977). Does the initiation of chromosome
replication regulate cell division? Adcances in
Microbial Physiology 16, 49-98.
KOCH, A. L. (1980). Does the variability of the cell
cycle result from one or many chance events? Nature,
London 286, 80-82,
KOCH,A. L. & BLUMBERG,
G. (1976). Distribution of
bacteria in the velocity gradient centrifuge.
Biophysical Journal 16, 389-405.
M . (1 962). A model for the
KOCH,A. L. & SCHAECHTER,
statistics of the cell division process. Journal uf’
General Microbiology 29, 435-454.
KOPPES,L. J. H., WOLDRINGH,
C. L. & NANNINGA,
N.
(1978). Size variation and correlation of different cell
cycle events in slow-growing Escherichia coli. Journal
of’ Bacteriology 134, 423433.
KUBITSCHEK,
H. E. (1970). Evidence for the generality
of linear cell growth. Journal of Theoretical Biology
28, 15-29.
KUBITSCHEK,
H. E. (1981). Bilinear cell growth of
Escherichia coli. Journal of Bacteriology 148,730-733.
PAINTER,
P. R. & MARR,A. G . (1968). Mathematics of
microbial populations. Annual Review of Microbiology 22, 5 19-548.
POWELL,E. 0. (1964). A note on Koch & Schaechter’s
hypothesis about growth and fission of bacteria.
Journal of General Microbiology 37, 23 1-249.
RAHN,0. (1931-1932). A chemical explanation of the
variability of growth rate. Journal of General
Phj.sio1og.v 15, 257-277.
SCHAECHTER,
M., WILLIAMSON,
J . P., HOOD,J . R. &
KOCH,A. L. (1962). Growth, cell and nuclear division in some bacteria. Journal of General Microbiology 29, 42 1-454.
SOKAL,R. R. & ROHLF,F. J. (1969). Biometry. San
Francisco : W . H. Freeman.
STUBBLEFIELD,
E. (1981). Cell cycle. In Advanced Cell
Biologji, pp, 1005-1025. Edited by L. M. Schwartz &
M. M. Azar. New York: Van Nostrand Reinhold.
TRUEBA,
F. J . (1982). On the precision and accuracy
achieved by Eschrrichia coli cells at fission about
their middle. Archit!es qf’ Microbiology 131. 55-59.
WOLDRINGH, c. L., DEJONG,N . A., VAN DEN BERG,
W . & KOPPES. L. J. H. (1977). Morphological
analysis of the division cycle of two Escherichia coli
substrains during slow growth. Journal of Bacteriology 131, 270-279.
ZUSMAN,
D., GOTTLIEB,
P. & ROSENBERG,
E. (1971).
Division cycle of My.~ococcusxanthus. 111. Kinetics
of cell growth and protein synthesis. Journal of Bacteriology 105, 81 1-819.
Downloaded from www.microbiologyresearch.org by
IP: 88.99.165.207
On: Sun, 18 Jun 2017 22:08:24