Cell growth kinetics, division asymmetry and volume control at division in
the marine dinoflagellate Gonyaulax polyedra: a model of circadian clock
control of the cell cycle
K. HOMMAf and J. WOODLAND HASTINGS*
Department of Cellular and Developmental Biology, Biological Laboratories, Harvard University, 16 Divinity Ave, Cambridge, MA 021JS, USA
•Author for correspondence
•(•Present address: Tonen Fundamental Research Laboratory, 1-3-1 Nishi-Tsurugaoka, Ohi-machi, Iruma-gun, Saitama 354, Japan
Summary
A new method of determining the dependence of
cell growth on the initial cell volume in the absence
of cell division is presented. The assumptions are
that volume in a certain period of time is either
increasing or decreasing, but not both, and is
independent of the history of cells. Applying this
method to Gonyaulax polyedra in a 12 h light-12h
dark cycle, growth in volume between the 3rd and
12th hours of the light period is found to be more
exponential-like than linear. The magnitude of
growth in the time period is determined solely by
cell volume and environmental conditions, not by
cell age. All cells decrease in volume slightly in the
dark from the 12th to 23rd hour, and then increase a
little from the 23rd to 3rd hour of the following day.
Cell division in this species is significantly asymmetric, and the extent of asymmetry is estimated
mathematically. Simulations based on the growth
patterns and the asymmetric division reveal that
cell division must at least partly depend on the
volume of cells.
The dependence of conditional cell division probability on cell volume is then experimentally determined. The probability is zero up to a certain cell
volume, and then it gradually increases to a plateau
level, which is less than unity. Neither the strict size
control model nor the transition probability model
is fully consistent with the observed shape of the
conditional probability function. A hybrid model
postulating a 'sloppy' critical volume with a constant probability of division above that volume
adequately accounts for the conditional probability. With the use of the observed volume growth
law, cell division dependence on volume, and the
extent of asymmetry in cell division, cell volume
distributions are successfully simulated for cells
growing in a 12 h light-12h dark cycle. Another
simulation reveals that the true coefficient of variation in generation time is 33%. On the basis of
these findings, a model of the cell cycle is presented
that incorporates the circadian clock as a cyclic Gt
phase. According to this scheme, cells satisfying the
minimum cell volume requirement between the
12th and the 18th hour probably exit to the replication/segregation sequence ending in division, and
re-enter the cyclic portion after a fixed time interval.
Introduction
cells change in shape, from the difficulties in measuring
motile cells, and from problems in measuring a large
number of cells.
Such problems are avoided if the volumes of individual
cells in a population are measured electronically, and
volume growth rates are determined using peak cell
volumes at a series of time points. By this method,
Sinclair & Ross (1969) found that, except for one cell
line, the growth in volume of Chinese hamster cells is
consistent with both exponential and linear growth. The
resolution of volume growth laws can be improved with
The simplest way of determining the growth law is to
measure the changes in size of individual cells. In this
way, Schaechter et al. (1962) reported that the increase in
length of the bacteria Salmonella typhimurium and
Escherichia coli is roughly exponential, while Mitchison
& Nurse (1985) concluded that the growth in Schizosaccharomyces pombe consists of two linear segments, with
the rate change in the middle of the cell cycle. This 'size'
method suffers from measurement inaccuracies when
Journal of Cell Science 92, 303-318 (1989)
Printed in Great Britain © The Company of Biologists Limited 1989
Key words: cell growth law, cell cycle, cell division, circadian
rhythm, dinoflagellates.
303
methods that determine the volume growth rate based on
the volumes of all cells; but this is not easy with random
cell division times, where one is obliged to consider both
increase in cell volume and the occurrence of cell
division, in order to distinguish volume growth from
changes in cell volume distribution. However, assumptions with regard to division and cell volume distributions
may be made, and applications of this method to Chinese
hamster cells (Anderson et al. 1969), Bacillus cereus
(Collins & Richmond, 1962), E. coli (Harvey et al. 1967;
Cullum & Vicente, 1978), Azotobacter agilis (Harvey et
al. 1967), Myxococcus xanthus (Zusman et al. 1971),
and Euglena gracilis (Kempner & Marr, 1970) all led to
results showing exponential-like growth in volume, with
some deviations.
Koch & Higgins (1982), whose method was based on
the Koch & Schaechter (1962) growth-controlled model,
found for procaryotes that growth in volume is close to
exponential. They assumed that the distribution of
division volumes is Gaussian, that there is no correlation
in volumes at division of successive generations, and that
every cell division gives rise to two equal-volume
daughters. But, although procaryotic cell division is
generally assumed to give equal-sized daughter cells
(Sargent, 1979), there are many reports to the contrary,
including Caulobacter crescentus (Terrana & Newton,
1975), Anabaena catemda (Mitchison & Wilcox, 1972),
and even E. coli (Trueba, 1982). Using this method,
moreover, one can consider only a few volume growth
laws and judge which of the predicted cell volume
distributions fits the actual distribution best.
With linear growth in volume, the growth rate does not
depend on initial cell volume, while with exponential
growth in volume it does. Brooks & Shields (1985) used a
cell sorter to select the largest and smallest 10 % of Swiss
3T3 cells. Comparing the growths in volume of the
unsorted cells and the two subpopulations, they concluded that it is neither linear nor exponential.
In fact, we know of no published case of either strictly
linear or exponential growth in size: what is reported as
linear is actually bilinear with a change in growth rate
within the cell cycle (e.g. see Mitchison & Nurse, 1985)
and what is largely exponential always has deviations at
either small or large cell sizes (e.g. see Anderson et al.
1969; Kempner & Marr, 1970). In practical terms, the
difference in cell volume growth in bilinear and exponential volume growth modes is almost too small to be
detected. Since what really matters is how much increase
in the standard deviation of cell volume is expected in a
cell cycle, more measurements of volume growth are
needed. However, care must be taken not to make
unverified assumptions, such as that of symmetric division.
A consequence of the circadian control of cell division
in G. polyedra is that it is phased (Sweeney & Hastings,
1958; Horama & Hastings, unpublished): cell division is
restricted to a certain time period, making the separation
of volume increase and cell division quite easy. To be
specific, changes in cell volume distributions between the
3rd and 23rd hour in 12 h light-12 h dark are almost
entirely due to growth, while changes between the 23rd
304
K. Homma and J. VI". Hastings
and 3rd hour of the following day are mostly due to cell
division. Exploiting this fortuitous property, we show
that the increase in volume of G. polyedra is also neither
exponential nor linear.
The extent of asymmetry in cell division is important,
because the exact amount of variability in cell volume
introduced by division is critical in determining the
stability of cell volume distribution. Cell division of many
eukaryotes, especially that of dinoflagellates, is clearly
unequal (Jagadish et al. 1977; Thomas et al. 1980;
Duspiva, 1977). On the basis of measurements of the
volume distributions of & population of dividing cells, we
find that G. polyedra cell divisions are in general significantly unequal, and quantify the extent of asymmetry.
Of the several proposed cell cycle models, the simplest
class may be deterministic: cells go through an obligatory
sequence of processes that ends in cell division. A strict
deterministic model cycle, however, has difficulty in
explaining the spread in the sizes and generation times of
newly born cells. The coefficient of variation (c.v.,
standard deviation divided by the mean) in the sizes of
newly born cells is of the order of 10%, while that in
generation time is 20% (Mitchison, 1977; Fantes &
Nurse, 1981). If the cell cycle involves a fixed sequence of
events, then the deviations should not be so large.
Transition probability models (see Brooks, 1981) postulate that the event that commits cells to divide is
essentially random, instead of deterministic, not dependent on the volume, age or past history of the cell.
Variation in the cell cycle is generated by the probabilistic
nature of this event. In the original model (Smith &
Martin, 1973), the cell cycle is divided into two phases, A
and B. The A phase is indeterminate in length, and
includes most of Gi, while the B phase has a constant
length and ends in mitosis. (Since the model was
constructed for rapidly proliferating mammalian cell
lines, Go was not included.) The transition from A to B is
postulated to occur probabilistically, and the probability
may depend on environmental conditions, but not on the
cell's age, volume or past history.
Postulating sloppiness in deterministic models is
equivalent to the incorporation of a probabilistic component. But the probabilistic models must assume some
degree of volume dependence of cell division in order to
account for the constancy of the standard deviation in cell
volume in exponential growth in volume. Also, when
volume growth is somewhere between exponential and
linear, one is often forced to assume cell volume dependency of cell division to keep the standard deviation in
cell volume constant in the face of rather large asymmetry
in cell division. As most positively skewed distributions
in cell age can give apparently exponential alpha curves
(e.g. see Castor, 1980), pure probabilistic models are not
the only explanation for exponentially decreasing alpha
curves. On the basis of such considerations, various
hybrid models have been proposed. Shilo et al. (1976)
suggested that the traverse of start requires growth of the
cell to a critical size, followed by a random transition that
starts the cell cycle. Wheals (1982) and Tyson & Diekmann (1986) proposed sloppy size control models that
postulate a transition probability that increases with cell
Materials and methods
G. polyedra (strain 70) was obtained from the ProvasoliGuillard culture collection (Bigelow Laboratories for Ocean
Sciences, West Boothbay Harbor, Maine 04575, USA). A clone
of this strain was obtained by placing an individual cell in a
capillary tube, with transfers to larger culture volumes as the
cells proliferated, finally in 2-8-1 Fernbach flasks containing
1500 ml of f/2 medium (Guillard & Ryther, 1962), omitting the
silicate and adding 0-5% autoclaved extract of soil. In some
experiments, cells were resuspended in conditioned medium,
which was obtained from a 2-day-old culture by filtration.
Cultures were grown photoautotrophically in a cycle of 12 h of
light and 12 h of darkness (LD 12:12) with light from cool
white fluorescent bulbs (light intensity = 166 ± 9 microeinsteinsm~ 2 s~') at 22(±3)°C (referred to as 'the standard
conditions'). Circadian time (CT) specifies the phase of the
circadian clock in constant conditions and is defined by dividing
the free running period into 24 equal portions, with CT 0 taken
as the beginning of the subjective light period. In an LD 12:12
light cycle, the period is equal to exactly 24 h and time is
First filtration ,.
larger cells
^
retained
Cell
divisions —
—
D
/STB
12
Day 1
LTD 23, Day 1
A
.
B
y
\
_3
Second nitration
smaller cells
/
retained
D
LDT
Cel
size. For an excellent review of these models see
Tyson (1987).
It is possible that the deterministic and the probabilistic models are two extreme cases of a reality that
involves both. However, there are not many measurements of cell division dependence on cell size that are
precise enough to evaluate their respective contributions.
The fact that size control acts on cell division was shown
in the budding yeasts, Sac'charomyces cerevisiae (Johnston et al. 1977), Candida utilis (Thomas et al. 1980),
and C. albicans (Bedell et al. 1980), and the fission yeast
S.pombe (Fantes, 1977; Miyataef al. 1978). In addition,
Sa. cerevisae has been shown to have a size control point
in Gi (Johnston et al. 1977), and S. pombe is reported to
exhibit size control over the initiation of DNA synthesis
(Nurse, 1975; Nurse & Thuriaux, 1977; Fantes & Nurse,
1978). As Fantes (1977) has demonstrated, however,
such data can be interpreted by both the deterministic
and the probabilistic models.
Heath & Spencer (1985) proposed a model of the cell
cycle in a marine diatom in which a temperaturecompensated and light-dependent period is followed by a
temperature-dependent and light-independent period.
There was no description, however, of the volume
dependence of growth and division of such cells. Donnan
& John (1983) demonstrated that both time- and sizedependent controls function in Chlamydomonas in the
ultradian mode of growth.
Wheals (1982) presented data concerning how cell
division depends on size in Sa. cerevisiae, and concluded
that two hybrid models describe the observations equally
well. Species whose cell division is controlled by the
circadian clock, such as the alga Gonyaulaxpolyedra, are
especially suitable for modelling the cell cycle because
growth in volume and cell division are naturally separated
in time. In this paper, we report determinations of the
cell division dependence on volume, and then proceed to
simulate the cell volume distributions based on the actual
data. On the basis of these results, a model of the cell
cycle of G. polyedra is presented.
B
Unfiltered
After first
\
filtration,
_\ large cells
\ X retained
Day 2
LTD 3, Day 2
A"»— Large cells after
r\
division round
/ 1
1 1
1 1 After second
/ l^- filtration, small
j
Cell volume
Fig. 1. Cell synchronization to the beginning of Gi by two
sequential filtrations. The protocol described in the text is
presented schematically at the top. Panels below show cell
volume distributions at about L D T 23 (A) for unfiltered and
filtered cells (left) and for the large cells after a division
round at L D T 3 (B) with and without a second filtration
step.
designated by L D T , with L D T 0 defined as the beginning of
the light period.
Cell filtrations were done with a Nitex screen (Tetko Inc.,
Elmsford, NY, USA). The fabric number and mesh size used
are specified in each case. Cell densities and cell volume
distributions were measured with a Coulter counter model ZB1
coupled to a 100-channel pulse height analyser (channelyser)
(Coulter Electronics, Hialeah, FL, USA). The cells are very
close to being spherical and, by virtue of their rigid cellulose
theca, are not subject to deformation during passage through
the aperture (140 ftm). Computations on cell sizes were carried
out with a PDPll/10 computer (Digital Equipment Corp.,
Maynard, MA, USA) and cell size distribution data were stored
on disks. The proportionality factor for cell volume to channel
numbers for the settings used throughout was determined to be
589 /im3 per channel, by calibration with beads of known
diameter supplied by the manufacturer; the relationship between channel number and cell volume was accurately linear.
For counting cell numbers, the lower limit was set at channel
17, and the upper limit at 98.
As shown some years ago (Sweeney & Hastings, 1958), cell
division in G. polyedra is phased with reference to the 24 h
light-dark cycle, so that divisions are restricted to a time
spanning the end of the dark and the beginning of the light
period (LDT 23-3). Such cells are phased but not synchronized with respect to cell division. To synchronize cells we
selected newly born cells by two sequential filtrations (Homma
& Hastings, 1988; and Fig. 1, top). In the first filtration, large
cells were selected between L D T 8 and L D T 10 of day 1,
resuspended in fresh medium, and allowed to proceed through
the next division about 12 h later. Those cells that did divide
were then selected after L D T 3 of day 2 as small cells, namely
those that passed through the same screen and collected in the
filtrate. Such a culture is referred to as one with cells synchronized to the beginning of Gi. Cell volume distributions at A and
B are shown in Fig. 1.
Growth and division control in Gonyaulax
305
5-89r 100
-2-95
|
O
2
U
3
4
Time (days)
5
Fig. 2. Maximum and minimum cell volumes (left) during
growth (cell densities, right) measured each day at LDT 3
with a cloned culture grown in LD 12:12 at a light intensity
of 120 ± 17microeinsteinsm~ / s ~ . Cells were collected by
filtration using Nitex screen with openings of 37 ftm (fabric
HC3-37) and resuspended in fresh f/2 medium to an
approximate density of 400 cells ml"'. The maximum cell
volume is defined as the channel above which fewer than
0-36% of the total cells fall in a single channel; the minimum
is specified by the channel with minimum counts between
small debris and G. polyedra cells (see Fig. 3).
2-95
Results
Cell volume distribution in the exponential phase of
growth
The maximum and minimum cell volumes were estimated for four consecutive days and are plotted along
with cell density (Fig. 2). As expected, there is no
dispersion in cell volume distribution in the exponential
phase, i.e. the standard deviation in cell volume stays
constant, generation after generation, while cells are
increasing exponentially. This represents evidence for
the stability in the cell volume distributions for newly
born cells and for dividing cells.
Coefficient of variation in the volume of newly born
cells
To determine the coefficient of variation in the volume of
newly born cells, it is first necessary to find the cell
volume distributions of dividing and newly born cells.
This can be accomplished by comparing the cell volume
distributions immediately before (at L D T 23) and after
(LDT 3 of the following day) a division round (Homma
& Hastings, 1988; and Fig. 3A). Since the amount of
growth in volume between L D T 23 and L D T 3 of the
following day is rather small (approximately 4 channels),
the difference between the two distributions is almost
entirely due to cell division. To determine the difference,
a program was written to subtract the cell volume
distribution at L D T 3 from that at L D T 23 of the
previous day, after small corrections for the volume
growth during the time period. The subtraction
(Fig. 3B) gives a negative part representing cells that
divided, and a positive part, corresponding to newly born
cells. The mean channel number of the newly born cells
is 27-9 ± 0 8 (n = 5), while its coefficient of variation is
16-1 ± 0 - 5 % (« = 5).
306
K. Homma and J. W. Hastings
Fig. 3. Determination of cell volume distribution of cultures
before and after a division round. A. Typical cell volume
distributions at LDT 23 and L D T 3 of the following day in
LD 12:12, at a light intensity of 120microeinsteinsm~ s" 1
The graphs shown were normalized so that the area under
each curve is equal to the number of cells per ml.
B. Difference between the cell volume distributions at L D T 3
and L D T 23 of the previous day, determined channel by
channel. The negative part of the graph corresponds to cells
that divided, and the positive part to the newly born cells.
Because cells in fresh medium grow about four channels
between L D T 0 and L D T 3, and the mean time of division is
around LDT 1 (Sweeney & Hastings, 1958), the actual values
of volume growth before and after cell division used for
cultures in fresh medium are one and three channels,
respectively.
One possible problem is that there is an overlapping
region of dividing and newly born cells that cannot be
represented after subtraction: if some cells that underwent division were as small as 34 channels in Fig. 3A,
then a predominance of newly born cells of the same
volume more than compensates for them, and prevents
the small cells destined to divide from showing up in
Fig. 3B. Plausible distributions of dividing and newly
born cells around the overlapping region in Fig. 3B can
be estimated.
The dependence of conditional cell division probability
on cell volume
Shelving for the time being the discrepancy due to the
overlap, let us consider the dependence of cell division on
cell volume, as determined by the measurements of
Fig. 3. We define P(x) as the probability of cells at
volume x at the time of a cell division round undergoing
cell division. P(x) is determined simply by dividing the
Original volume
distribution (1/6)
10
Assumed cell
_ Expected dividing
division probability ~ cell distribution
Underestimated
(+5%)
Corrected P(x)
a.
.D
O
S 0-5
Overestimated
(+18%)
Optimal
(+12%)
Channel no.
Cell volume
(xlO4/im3)
20
1-18
4-71
Fig. 4. Dependence of conditional cell division probability
on initial cell volume. The apparent number of dividing cells
at each channel in Fig. 3B was dividing by the corresponding
number of cells immediately before the division round, i.e.
the number of cells at one channel more than the channel
number at L D T 23. The resultant values are plotted against
channel numbers (D, uncorrected P(x)). Next, the estimated
actual distribution of dividing cells was substituted for the
apparent distribution, and the actual cell division probability
curve is indicated by the dotted line ( • , corrected P(x)).
number of dividing cells for each channel by the corresponding cell number immediately before cell division,
which is obtained by introducing the small volume
growth correction to the cell volume distribution at
L D T 23. A program was written for this calculation; the
results give the probability of cell division as a function of
cell volume (Fig. 4, uncorrected). This illustrates, for
example, the fact that cells at channel 20 a moment before
the cell division round had zero probability of dividing,
while cells at channel 60 had better than a 90 % chance of
undergoing cell division.
Overlapping region
Before proceeding to the determination of the actual
probability of cell division, including cells in the region of
overlap, we observe that the (apparent) probability
dependence on cell volumes can be divided into three
regions: the first with zero probability, then a linear-like
region of increase, and finally a high plateau (Fig. 4).
Thus the curve can be approximately represented by
three parameters: the beginning (XQ) and the end (X\)
channels of the linear-like increase, and the final plateau
level (Pmax). Notice that the overlap mentioned above
mainly affects the lower end of the non-zero portion of
the probability curve, but not so much the upper end. If
it is assumed that the increase is linear-like for the actual
probability curve, including the overlap region, then Xo
needs to be adjusted in order to estimate the actual
probability dependence curve. Varying Xo (Fig. 5)
changes the predicted cell density and volume distribution after the division round: decreasing XQ means that
some small cells that previously had no chance of division
20 40 60 80
Channel no.
20 40 60
80
Fig. 5. Illustration of the procedures used to determine the
actual dependence of conditional cell division probability on
cell volume. Under-, over- and optimal estimation of the
overlapping region (A, B and C, respectively). By channelby-channel multiplications of the cell volume distribution just
before the division round (left) and the uncorrected division
probability parametrized by Xo, X\ and P m n x (middle), the
expected number of dividing cells are obtained (right).
are allowed to divide, and therefore increases the predicted cell density, and adds some small newly born cells,
and vice versa (Fig. 2). To identify the optimal value of
XQ, the expected fractional increases were calculated for
various values of XQ, with X\ and P m a x fixed, and
compared with the actual increases.
The corrected probability curve is plotted in Fig. 4 for
that case; in any event the correction in Ao is less than or
equal to four channels, a rather small amount.
New method for calculating volume growth rate
dependence on cell volume
Since most cell division takes place between L D T 23 and
L D T 3 of the following day, volume growth rate over this
time period can be readily calculated. The only assumptions made were that cell volume either increases or
decreases monotonically, and that the rate of volume
growth is independent of the cells' history. It follows that
growth in volume does not alter the ordering: if cell A is
larger than cell B at time t, then it remains so at any later
time in the absence of cell division. Examinations of the
growth rates of a population and its subgroup showed
that exactly equivalent cells will be found at precisely
equivalent profile positions in samples taken at different
times (Homma, 1987). This observation justifies the
above assumptions.
If the ordering of cell volume remains unchanged, then
cells at the mode at time to (Vmodo) will grow to the mode
at a later time tx (V modl ), the smallest cells (Vm;no) to the
smallest cells (Vmini)> and the largest cells (Vmaxo) to the
largest cells (V,liaxl) (see Fig. 6A). This is because the
Growth and division control in Gonyaulax
307
maximum cell volume Vmaxo, then the final volume must
satisfy the equation:
J
After growth in
volume (LDT 12)
C v>
V"o
n(to)dU=
«(
I'minO n(t\)
,
where n(to) and
are theJ Vnumbers
of the cells at
volume t / a t times to and t\, respectively. The number of
cells smaller than Vo at time to (the shaded fraction in
Fig. 6A) must be the same as the number of cells smaller
than V\ at time t\. Knowing the number of cells at
channel x, it is easy to find Vi satisfying the above
equation for each VQ value smaller than Vmodo. Similarly,
if the initial volume VQ falls somewhere between Vmodo
and ^maxO) then V\ can be found:
min
6
Cell volume
V
K
^minO
118
•
min
2-36
J
Linear growth
Fig. 6. Method for calculating volume growth rate. A. The
cell volume distributions before and after growth in volume.
The cell volume distributions of a cloned culture at LDT 3
and L D T 12 of the same day in LD 12: 12 were measured
with a Coulter counter and plotted. For other initial volumes,
interpolations were used based on the idea that the shaded
fractions represent the same groups of cells, and that V'o in
the figure corresponds to V]. B. The dependence of volume
growth rate on the initial volume. The absolute amount of
volume increase, V\ — Vo, divided by the time interval (9h in
this case) was calculated for each V'o with the use of the
PDPll/lO computer. Expected shapes of volume growth
curves in genuine linear and exponential growth in volume
are shown (broken lines). The actual volume growth curve
can be approximated by the minimum rate (Rmin), the least
squares fit straight line for the exponential-like part, and the
maximum rate {Rmax). The volumes where the exponentiallike part starts and ends are designated UQ and U\,
respectively.
cells whose volumes are between Vmjno and Vmod0 must
grow to cells whose volumes fall between Vmini and
^modii ar) d those between V"mod0 and Vmaxo must become
cells between VmOdi and V max i. With the correspondence
principle, the volume growth rate of all the other cells can
be calculated by interpolation: if the initial cell volume VQ
is between the minimum cell volume Kmin0 and the
308
K. Homma andj. W. Hastings
Vo
n(t0) dU
- J"
The results are expressed as the increase in the number
of channels divided by the number of hours in between,
i.e. {Vl-rV0)/(t1—t0). The volume growth rate dependence on the initial channel number VQ is presented in
Fig. 6B. Linear growth in volume gives a straight horizontal line because the absolute amount of growth in
volume is constant and does not depend on the initial cell
volume Vo- An exponential growth in volume, on the
other hand, is represented as a straight line going through
volume zero, if the time interval used is small compared
to mass doubling time. As shown, the actual volume
growth of G. polyedra cells is more exponential-like,
although there is a linear-like portion at small initial
volumes, and the extrapolation intersects with the x axis
at a positive channel number, rather than at the origin.
Dependence of volume growth on cell age
For both experimental and theoretical analyses, the
independence of cell growth in volume on the age of a cell
is often tacitly assumed, i.e. it is assumed that cells of a
certain volume all increase in volume at the same rate,
regardless of age. This assumption needs to be tested,
especially in a species like G. polyedra, whose inexact cell
volume control leads to cells whose volumes are the same
but whose ages are quite different. For example, cells at
channel 49 at L D T 12 in LD 12: 12 (see Fig. 6A) consist
of 5-, 4-, 3-, 2- and 1-day-old cells whose volumes at birth
are calculated (see below) to have been at channels 17, 22,
27, 33 and 41, respectively. The volume growth curve in
LD 12:12 for the period from L D T 12 to L D T 20
(during D) was calculated. The results (data not shown)
indicate that cells stop growing in volume, and actually
shrink by approximately two channels during the night in
LD 12:12(Homma, 1987).
To examine the question of age dependence of volume
growth, an experiment with a synchronized culture with
daily resuspension in the conditioned medium was carried out. It was considered necessary to keep cells in the
same medium every day in order to compare daily volume
growth rates, because cell growth in volume slows down
in old medium (data not shown). The newly born cells
were collected by the two sequential filtrations technique
at L D T 3 of day 2, and the synchronized culture was kept
in LD 12: 12 with daily resuspension in the conditioned
owth rate
From LTD 3 to LTD 12 in LD 12:12
Day
10 -
i
Da y 5
"5
c
Volume
char
00
J
ture. The functional form presented above is only
applicable under the specific conditions described above.
1
\
\X
'
0-5 -
\ Day 4
ay 2
i
Channel no 10
Cel volume
0-589
1
1
1
1
20
30
40
50
1-18
1-78
2-36
2-98
1
Fig. 7. Age dependence of volume growth rate. The
experimental procedures are given in the legend to Table 1.
The daily volume growth curves were calculated from the cell
volume distribution data.
medium at L D T 10. Volume growth curves from L D T 3
to L D T 12 of days 2 to 6 were determined from cell
volume distributions; they are very similar (Fig. 7).
As the first generation cells grow to larger volumes, the
second generation cells, i.e. divided cells, appear first at
the smaller volumes. A computer simulation based on the
volume growth rate and cell division probability data
obtained from this experiment (see below) shows that all
the cells are in the first generation on day 2, while only
channels above 35 correspond to genuine first generation
cells on day 3, those above 43 on day 4, and those above
53 on day 5. The nearly identical volume growth curves
for the entire range of Vo values shows that the volume
growth rates are the same for days 2-5 for the first
generation cells. Thus the volume growth rates are
independent of cell age.
In order to facilitate mathematical treatment of the
volume growth rate function to be used in the modelling
presented below, a linear approximation was made for the
middle range of volume growth between initial volumes
UQ and U\ (see Fig. 6B). The linear portion is expressed
as (A)(VQ) + B, where A is the slope in h" 1 , andfi is they
intercept in channelsh" 1 . Below Uo and above U\,
volume growth rates stay roughly constant, and the
constant levels near the lower and the upper limits are
defined as Rm\n and fimax, respectively (Fig. 6B).
Although a constant volume growth rate is reproducibly
observed at the lower end, levelling off at the upper end is
not always observed (Fig. 7). When there is no plateau
level at around Vmax, i? max is simply taken to be the
volume growth rate of the largest cells. These parameters, as well as the expected densities normalized for
dilutions at the time of resuspension, appear in Table 1.
From this, the mean parameters of growth in volume
were calculated and found to be:
Cell growth in volume between LDT 23 and 3 in
LD 12:12
Thus far, cell growth in volume has been considered
between L D T 3 and L D T 23, i.e. outside the time
period when cell divisions occur. It is important, however, to estimate the magnitude of cell growth in volume
between L D T 23 and L D T 3, for both dividing and nondividing cells, because the method of separating out
dividing and newly born fractions from cell volume
distributions presented above rests on the assumption
that cell growth in volume in the time interval is small.
To determine the volume growth of non-dividing cells,
the data from a rerun of the experiment described in the
legend to Table 1 were used. Identifying a non-dividing
fraction in the cell volume distributions at L D T 22 of day
2 and L D T 3 of day 3, we calculated volume growth rate.
There is apparently a uniform growth in volume of about
four channels: the cell volume increase is small, and
independent of the initial cell volume (Homma, 1987).
To determine the volume growth rate of newly born
cells in the light, we selected relatively large cells by
filtration at L D T 10 of day 1, resuspended the culture in
a conditioned medium (the conditioned medium was
used in order to find the average behaviour of cells in the
exponential phase, and not the behaviour immediately
after inoculation in a fresh medium), split the culture into
two, placed one of them in LD 12: 12, while the other was
kept in constant darkness (DD) from L D T 12 of day 1.
We then measured the volume distributions of the
cultures at hour 3 of day 2. The treatment of the two
groups differed only by the 3 h of light on day 2, because
the group in LD 12:12 is also in the dark from L D T 12
to L D T 24 of day 1. Since more than half of the cells
divide after such selection, and most of the cell division of
that round is complete by L D T 3 of day 2 (even in
darkness), a comparison of the cell volume distributions
of the two (Fig. 8A) must reveal the effects of light on
newly born cells (Homma, 1987). The volume growth
was calculated using the previously explained method,
with the group in DD taken as the distribution before
growth in volume and the group in LD as that after
growth in volume. The results demonstrated that newly
born cells grow by four channels right after birth.
Therefore, all cells grow by four channels from L D T 23
to L D T 3 of the following day, irrespective of the initial
cell volume and of whether they are non-dividing or
newly born cells.
Extent of unequal cell division
To determine the extent of asymmetry in cell division, we
started with a population of dividing and newly born cells
(Fig. 3B). The basic idea is to start from the volume
Growth rate in volume (in channel h )
distribution
of the dividing cells and simulate that of the
= 0-33 (when Vo < channel 28)
newly
born
cells,
assuming some degree of asymmetry,
= 0-036 Vo - 0-55 (when channel 28 =S VQ =S channel 50)
and to compare the result with the actual distribution of
= 0-86 (V o > channel 50).
newly born cells. The degree of asymmetry that predicts
the actual distribution of newly born cells best is defined
These parameters are quite susceptible to environmento be the actual extent of unequal cell division. For
tal changes, such as medium, light intensity or temperaGrowth and division control in Gonyaulax
309
Table 1. Age dependence of volume growth rates
Day
Day
Day
Day
2
3
4
5
Mean ±s.E.M.
Slope (A)
(h- 1 )
v intercept (B)
(channel h" 1 )
Min. growth
(channel h" 1 )
Max. growth
(channel h~')
0-029
0-046
0-030
0-040
0036 ±0-003
(" = 4)
-0-43
-0-86
-0-24
-0-68
-0-55 ±0-12
(» = 4)
0-33
0-44
0-33
0-33
0-36 ±0-02
(» = 4)
0-88
111
0-88
1-00
0-9710-02
(n = 4)
A C. polyedra culture was synchronized to the beginning of G! at LDT 3 of day 2 by the two sequential filtrations technique (see Fig. 1 and
Materials and methods). After the second filtration, the synchronized culture was returned to LD 12: 12 at 22(±2)°C at a light intensity of
112± 12microeinsteinsm~ 2 s~' during L. At LDT 10 of days 2, 3, 4 and S, the culture was resuspended to comparable densities in the same
medium. The resultant volume growth rates between LDT 3 and LDT 12 on days 2, 3, 4, 5 and 6 were computed from the cell volume
distribution data. The four parameters, that is, the slope (A) and the y intercept (B) of the linear part of dx/dt, and minimum (Rmm) and
maximum (Rmax) volume growth rates, were estimated from the plots shown in Fig. 7.
Expected distribution
assuming equal division
Table 2. The variation in cell volume introduced by
cell division
Best-fit values
Case
Observed distribution
Expected distribution
assuming asymmetric
division
I
80
4-71
Fig. 8. Calculation of the extent of asymmetry in cell
division. The difference of cell volume distributions at
L D T 3 and L D T 23 of the previous day as shown in Fig. 3B
was duplicated (thick line). The positive part of the graph
corresponds to newly born cells and the negative part to
dividing cells. The cell volume distribution of the newly born
cells was simulated assuming both equal (broken line) and
asymmetric (continuous thin line) cell division. The better
simulation was obtained assuming asymmetric division, with
epsilon (see the text) equal to 0-10.
simulations it was assumed that a dividing cell with
volume x gives k daughter cells whose volumes are
distributed normally with the mean cell volume of alpha x
and with the standard deviation proportional to the mean
cell volume of the daughters. That the number of
daughters per mother was not set equal to 2 is because of
the existence of an overlapping region of dividing and
newly born cells: if one lets the overlapping area be A,
and the actual number per ml of dividing cells be D, then
the actual number per ml of newly born cells must be 2D.
Taking the overlapping region into account, the area of
the negative fraction is given by (D—A), while that of the
positive fraction (2D—A). Thus the apparent number of
daughters per mother per ml (designated k) is given by
310
K. Homma andjf. W. Hastings
1
2
3
4
5
Mean±s.E.M.
a0-591
0-595
0-507
0-476
0-516
0-54 ±0-02
(« = S)
e
2-35
2-22
2-18
2-04
3-80
0-100
0-128
0078
0-074
0-126
2-5 ±0-3
(« = 5)
0 1 0 ± 0-01
(" = 5)
The cell volume distribution of newly born cells at LDT 3 was
simulated from that of dividing cells with the number of daughter
cells per mother (k), the ratio of the mean cell volume of the newly
born cells to that of the dividing cells (alpha), and the proportionality
factor of variation introduced at cell division (epsilon) taken as
variables. The three parameters were optimized to produce the best
fitting volume distribution of newly born cells. Five cases with the
optima values are tabulated together with the mean value ± S.E.M. of
each parameter.
(2D-A)/(D-A),
which is a little larger than 2 (see
Table 2). The distribution of daughter cell volumes must
be symmetric around the mean value because each pair
must be distributed symmetrically, and it is reasonable to
assume it to be Gaussian. We assumed that the standard
deviation of each Gaussian distribution of daughters is
proportional to the mean, and the proportionality constant is designated as epsilon.
A computer program was written to simulate the cell
volume distribution of newly born cells using the parameters, alpha (the ratio of the mean values of daughters
to those of mothers), k and epsilon, and to calculate the
fitness parameter that is equal to the sum of the square of
the difference of the actual number of cells and the
simulated number for each channel. The strategy
adopted for optimization was to find the value of alpha
that gives the minimum fitness parameter, then to get the
best values for k and epsilon, in that order, and to repeat
the optimization round once more if necessary. Video
display was used to see what a simulated volume distribution actually looks like, in order to avoid ending up
with local minima of the fitness parameter that are larger
than the absolute minimum.
When no asymmetry is assumed, i.e. when epsilon is
set equal to zero after the optimizations for alpha and k,
the expected volume distribution of newly born cells is
narrower than the actual one (Fig. 8). Evidently some
degree of asymmetry that spreads the volume distribution
of newly born cells must be present. When cell division
was allowed to be unequal, i.e. when epsilon was allowed
to vary, then much better simulation was obtained. Five
cases were analysed in this way and the optimal value of
epsilon was found to be 0-10 ±0-01 (n = 5) (Table 2).
This means that, for example, sister cells whose mean
channel number is 30 are distributed normally with the
standard deviation of 3 (30x0-10) channels. This is
certainly a small but significant variation between sister
cell volumes.
Simulations of cell volume distribution without volume
control at cell division
Since cell volume distribution was found to be stable in
this species, a natural question to ask is how such
homeostasis of cell volume is maintained. As pointed out
by Fantes & Nurse (1981), in the absence of cell volume
control at division linear growth in volume reduces
variation in volume, while exponential growth remains
constant if cell division is symmetric. If cell division is
not completely symmetric, however, the variation in
volume will increase generation after generation in exponential growth, and thus cell volume distribution is
unstable. (The same conclusion was reached mathematically by Tyson & Hannsgen (1985)). It can also be
shown that a large degree of asymmetry at division
destabilizes cell volume distribution when growth in
volume is between linear and exponential.
The actual growth law of G. polyedra is neither purely
exponential nor linear, but contains some elements of
both. Since there is also significant asymmetry in cell
division, it is necessary to carry out simulations in order
to examine whether or not there is cell volume control to
account for the observed stability in cell volume distribution. With the knowledge of both the volume growth
law and the extent of asymmetry, the steady state of cell
volume distribution, if any, in the absence of generationtime dependence on initial cell volume, can be examined.
Since the actual mean generation time is approximately 3
days in the light and temperature conditions in which the
volume growth equation is determined, the same generation time was assumed for cells of all initial volumes for
simulation.
The increase in cell volume between L D T 3 and
L D T 12 can be calculated for each initial volume from
the volume growth equation presented above. The
amounts of growth assumed in the following will be
double-checked by the success in simulating cell volume
distributions later in the paper. For cells with initial
volumes less than or equal to channel 26, 27 and 29, 30
and 32, 33 and 35, 36 and 38, and greater than or equal to
channel 39, the growths in volume were by 3,-4, 5, 6, 7
and 8 channels, respectively. (To facilitate simulation,
the volume increases were rounded to integral channel
numbers.) Using these values and the observations that
cells shrink uniformly by two channels from L D T 12 to
Theoretical
for doubling '
Actual
50
50
Initial cell volume (channel no.)
Fig. 9. Final volumes of cells initially at different volumes
after 3 days of growth in volume without division. The
volumes of cells at' the end of day 3 starting from various
volumes at the beginning of day 1 were computed, assuming
that there is no cell division in the meantime. Shrinkage and
swelling of cells from L D T 12 to LDT 23, and from LDT 23
to L D T 3 of the following day, respectively, were included in
the calculation. The calculated final volumes are plotted by
thick lines and the final volumes, assuming that they are
simply twice the initial volumes, are shown by thin lines.
L D T 23, but grow uniformly by four channels from
L D T 23 to L D T 3 of the following day, the amount of
volume growth for each initial cell volume from the
beginning of day 3 to the end of day 5 (LDT 23) can be
calculated. (The 'initial' cell volumes were expressed as
those before the growth in volume in the first 3 h on day 3,
i.e. as volumes of newly born cells at L D T 3 less four
channels.) Fig. 9 shows the dependence of the expected
final volumes on the initial volumes.
Cell division was assumed to be asymmetric, with the
standard deviation equal to 0* 10 times the mean volume
of sister cells. The ratio of the mean cell volume of
daughters to that of mothers, alpha, is taken to be 0-54,
i.e. the mean value of the simulation carried out above.
To calculate the lower limit of a volume distribution after
division, observe that the smallest mothers account for it.
If cell division is equal, and the volume of the smallest
mothers is channel 50, then the lower limit of the
daughters is 50 times 0-54, or channel 27. With the extent
of asymmetry assumed, the standard deviation of the
daughters around the mean value is 27 times 0-10, or 2-7
channels.
Starting with hypothetical volume distributions of
newly born cells, the expected cell volume distributions
for all subsequent generations can be simulated with the
volume growth and cell division parameters described
above. Fig. 10 shows that cell volume distributions all
converge to a distribution whose lower and upper limits
are 11 and 55 channels: if the initial volume distribution
Growth and division control in Gonyaulax
311
Initial
distribution
•of B
Stable
upper limit
60
c
Stable
distribution =
Initial
distribution
of A
I §40
3. •«
„ o
20
Stable
i lower limit
0
1 2
3
Generation no.
4
5
Fig. 10. Simulations of cell volume distributions for several
generations assuming independence of generation times on
initial cell volume. The upper and lower limits in the
beginning of generations only are shown for three different
initial cell volume distributions. A represents an initial
distribution that coincides with the steady state, while B and
C represent initially broader and narrower distributions,
respectively.
ranges exactly from channel 11 to channel 55 (case A) as
shown to the right of generation 5, then the distribution
remains unchanged generation after generation. An initially narrower distribution of cell volume (case C, which
is shown to the right of generation 0) spreads out with
generation, and reaches the steady-state distribution (A)
after a few generations (shown to the right of generation
5). In case the initial volume distribution is broader than
that of A (case B, graphed to the left of generation 0),
then the distribution shrinks down to distribution A after
a few rounds. It can be proven mathematically that after a
large enough number of generations, any initial cell
volume distribution converges to the above steady-state
distribution. In contrast, adding the plateau levels tips
the balance toward linear volume growth law, and
permits the existence of a steady state.
The actual minimum cell volume of newly born cells is
clearly larger than the predicted steady-state value of
channel 11, while the actual maximum cell volume of
newly born cells is smaller than channel 55 (Fig. 3B).
This discrepancy is attributable to the initial assumption
that generation times are independent of initial cell
volume: actually, small newly born cells must spend
more than 3 days before division, and therefore grow in
volume more than those in the simulated case above. As a
result, the minimum cell volume of newly born cells in
steady state exceeds 11 channels. By contrast, in actual
cases, initially large cells spend less than 3 days before
division, making the actual maximum volume of newly
born cells smaller than 55 channels. There must therefore
be some dependence of generation times on initial cell
volumes.
Looked at in another way, cells whose volumes immediately after division are less than average do not
312
K. Homma andjf. W. Hastings
divide within 3 days, but take longer and undergo extra
growth in volume. The added volume growth will at least
partially compensate for the smallness of initial volumes,
and makes the cell volume variation at division less than
predicted by the above simulation. Newly born cells that
are larger than average divide sooner and therefore grow
less in volume before division. Thus, a relatively tight cell
volume distribution for dividing cells is observed as a
result of the dependence of generation times on initial cell
volumes. This conclusion is consistent with the observation presented earlier that conditional cell division
probability depends on cell volume.
Observed coefficients of variation (c.v.) in generation
times
A volume control model of cell division must be able to
simulate the observed c.v. in generation time. In order to
test the division control model presented later, we first
measured the observed c.v. in generation time. For these
experiments, we made use of the two sequential filtrations technique to select newly born, thus cell-cyclesynchronized, cells. These were placed in LD 12:12 at
various light intensities, and were thus also circadianentrained. Cell densities and volumes were measured
each day after the completion of cell division; the
generation times were then calculated from the daily
increases in cell density. In analysing data, it was
assumed that all the increase in cell density, up to the
overall doubling, is due to first generation cell division.
This is not strictly correct, since some large secondgeneration cells may divide before some small firstgeneration cells. Since simulations to be described later
remedy this defect, let us just state that the apparent
value of c.v. is an underestimate of the real value.
Table 3 shows the distributions of generation times
with the light intensities used along with the calculated
c.v. for each case. The mean c.v. in generation time is
25-1 ± 0 - 9 % (K = 4 ) , which is somewhat larger than
published values of 10-20% (Mitchison, 1977).
Average division probability dependences on cell
volume in LD 12:12 and LL
The calculation of cell division probability dependence
on cell volume, P(x), described above, was made for 11
cases of cell division, and averaged for cultures in
LD 12:12 and in LL in fresh f/2 medium. (Cells in a
depleted medium tend to divide at smaller volumes, and
therefore cannot be treated similarly.) For each probability curve, P(x), the three parameters, A'0) X\ and
fnM, were determined, and then averaged separately for
the cases in LD 12: 12 and in LL. The values of Xo, A',
and P m a x in LD 12:12 are channels 45-4 ± 1 - 3 ,
56-4 ±1-9 and 0-87 ± 0-05 (» = 7), respectively; the
corresponding values in LL are 40-0 ±0-7, 49-0 ± 0-9
and 0-91 ± 0-03 (n = 4). These results are illustrated in
Fig. 11.
The differences inXo and A'i are statistically significant
(at P less than 0-02). Cells continue to grow in volume
during the subjective night in LL (data not shown) while
in LD 12:12 a small decrease in volume is observed
between L D T 12 and L D T 23 (see above). Since the
Table 3. The calculation of coefficients of variation in generation times
Light intensity
(microeinsteins
% of cells whose generation time is equal to:
1 day
2 days
3 days
4 days
5 days
Mean generation
time
(days)
8-2
4-9
4-6
2-6
72-0
20-9
19-8
0-0
19-8
57-6
48-4
3-6
0
16-6
27-2
30-7
0
0
0
63-2
2-12
2-86
2-98
4-44
—
—
m-V)
118 + 6
112 ± 12
78 ± 4
37 + 4
Coefficient
of variation
(%)
24-5
26-0
27-2
22-6
Cultures of C. polyedra were grown and kept throughout in LD 12:12 at light intensities ranging from 37 to 118 microeinsteinsm 2 s ', at
22(±3)°C. Newly born cells were selected between LDT 4 and LDT 6 on day 2 by the two sequential filtration procedure. The cell densities
and cell volume distributions were measured after LDT 6 of subsequent days, and the generation times were calculated from the data. The table
shows the simulated distribution of generation times of the first generation cells, the mean generation time, and the c.v. in generation times.
p
/T
' max
-
c
.2
So ^_^
IS
" ••=
«J X>
In LL '
0-50
/
/
'
O X)
5 2
•§*•
0
J
-
o
i In LD 12:12
/
^
/
Xo/
•
Channel no. 20
Cell volume i io
(xl04Mm3)
J/_ j /
40
2-36
60
3-53
Fig. 11. Average cell division probability dependence on cell
volume in LD 12: 12 and LL. Cell division probability
dependence on cell volume, P(x), was calculated for seven
cases of cell division in LD and four cases in LL (light
intensity in L: 140 ± 20 microeinsteins m~ s~ .temperature:
22(±3)°C), as described in the text.
trigger event of cell division is located before L D T 18
(Homma & Hastings, 1989), the amount of growth in
volume between the trigger event and division in LL
must be larger than that in LD 12:12. Therefore, if the
volume dependence of cell division at the time of the
trigger event is the same in both cases, then the apparent
probability at the time of division in LL will be at a larger
volume than in LD 12:12, i.e. P(x) at each x value will
be smaller in LL than that in LD. As the ordering is just
the opposite, we are led to conclude that cell division
probability at the trigger event is higher in LL than that
in LD.
Plateau level of P(x)
In LD 12:12 the level of the plateau, P m a x , was found to
be 0-87 ±0-05 (w = 7), and its departure from unity is
barely significant (significant at P less than 0-10 but not at
P less than 0-01). Also, the corresponding value in LL
(0-91 ± 0-03 (n = 4)) is not significantly different from 1.
P m a x values less than unity mean that, no matter how
large cells are, some of them fail to divide at least by
L D T 3.
We examined the question of whether large undivided
cells at L D T 3, i.e. the cells above^i (Fig. 11) at L D T 3
in LD 12:12, are the cells whose divisions were merely
delayed for a few hours or they are the cells that remain
undivided till the time of next division round. If they do
not divide between L D T 3 and L D T 12, then they
merely grow in volume. Since the cells larger than
channel 38 grow by the constant volume of eight channels, and X\ is larger than channel 38, the large cells
(>X\) will grow by eight channels in the 9-h period.
Thus the right side of the volume distribution at L D T 12
is predicted to be the same as the corresponding side at
L D T 3, except that it is shifted to the right by eight
channels. The predicted cell volume distribution for this
case, i.e. assumingP max = 0-67, is plotted in Fig. 12A. If,
on the other hand, all the cells above X\ (=channel 54 in
this case) divide between L D T 3 and L D T 12, i.e. if
PmnK= 1> then there will not be any cells above channel
62 (=54+8) at L D T 12, as is also shown in Fig. 12A.
The actual volume distribution at L D T 12 (thick line)
agrees well with the predicted distribution, assuming
P m a x = 0-67; i.e. assuming no further division between
L D T 3 and L D T 12.
Fig. 12B represents similarly predicted volume distributions at L D T 23 with the two alternative assumptions
(thin lines), and the actual distribution at L D T 23 (thick
line). The figure also supports the assumption that there
is no further division between L D T 3 and L D T 23. The
experimental evidence is thus in agreement with the idea
that no cells above X\ that remain undivided at L D T 3
undergo division until L D T 23. We thus conclude that
the plateau level of P(x) is actually less than unity in
LD 12:12; i.e. cells larger than the critical volume have
less than 100 % chance of undergoing division in the next
division round.
Age dependence o/P(x)
The use of cell volume only as a parameter for the
probability of cell division assumes that the event does
not depend on cell age. This is not self-evident, because
some factor like a mitogen may accumulate with time
after cells' birth, giving those that remained undivided
for a longer time a higher mitogen level and thus a higher
probability of division. We must therefore justify this
assumption by demonstrating that cell division probability dependence on cell volume, P(x), of the first day
in the cell cycle is not dissimilar to that of the second day,
and so forth.
For this purpose, we selected newly born cells and
Growth and division control in Gonyaulax
313
2nd Experiment
Day 3
1st Experiment
Day 3
LDT12
Observed distribution
Predicted distribution
assuming PmBJi = 0-67
Predicted
distribution
assuming
LDT23
Predicted distribution
assuming Pmax = 0-67
6
Observed distribution
Predicted
distribution
assuming
/•„,„ = i-o v
Channel no. 20
Cell volume 1-18
(xl0 4 jun 3 )
60
3-53
100
5-89
20
118
60
3-53
100
5-89
%
60
70
80
Channel no.
90
Fig. 12. Simulations of cell volume distributions at L D T 12
and L D T 23 for different values of P m a x . A G. polyedra
culture grown in the standard conditions was synchronized to
the beginning of Gj at L D T 3 of day 2 by the two sequential
filtration technique with the light conditions kept as LD
12:12 on day 1, but with the light intensity decreased to
112 ± 12microeinsteinsm~ s~ from L D T 0 of day 2. After
the second filtration, the synchronized culture was returned
to LD 12:12 at 22(±2)°C at a light intensity of
112 ± 12microeinsteinsm~ z s~' during L. At LDT 10 of days
2, 3, 4 and 5, the culture was resuspended to comparable
densities in the same medium. The cell volume distributions
of day 5 were used.
determined P(x) after each cell division round for 4 days.
T h e resultant parameters, Xo, Xx and Pmax, of each day
for two essentially identical experiments are plotted in
Fig. 13. T h e parameters A'o, Xi and Pmax, which determine the character of the function, fluctuate somewhat
from day to day. However, the magnitudes of the
fluctuations are small, and the fluctuations are different
for the two experiments. These facts support the idea that
the fluctuations are not due to age dependence of P(x),
but are of statistical origin. We thus conclude that P(x) is
not strictly dependent on cell age.
Simulations of cell volume
distribution
By using the information gathered concerning growth in
volume and cell division, it is possible to simulate the
time-course of the cell volume distribution and the
densities in L D 12:12 for as many days as desired. A
comparison of the simulated cell volume distribution and
314
K. Homma and jf. W. Hastings
Fig. 13. Age dependence of conditional cell division
probability. The same experimental data whose protocol
appears in the legend to Fig. 4 were used. The cell division
probabilities for days 3, 4, 5 and 6 were calculated from the
cell volume distributions at L D T 23 of the previous day and
at L D T 3 with overlapping regions being taken into account.
The probability of each day as described by the values of A'o,
Xi and P m a x are plotted against cell volume in the left column
(1st Experiment). In addition, identical analysis was made of
the data of an experiment carried out with identical
procedures except that the cell densities were
500-900 cells ml" 1 and that the cell volume distributions
before division were measured at L D T 22 instead of at
L D T 23. The results are plotted similarly in the right column
(2nd Experiment).
density with the actual data provides a good test of the
adequacy of the description of a cell's behaviour whose
various parameters were independently determined. In
addition, considering the difficulty in experimentally
distinguishing the generations of cells, a simulation of
generation times of a culture initially synchronized to the
beginning of Gi provides a supplementary tool for
estimating the c.v. in generation times.
For the simulations, it is assumed that the following
steps occur: from L D T 3 to L D T 12, a volume growth
rate that depends on environmental conditions but not on
cell age; from L D T 12 to L D T 24, a decrease in volume
of all cells (by two channels); at L D T 0, asymmetric cell
division with probability, P(x), which is also age-independent but in some cases dependent on days of measurements; from L D T 0 to L D T 3, growth of all cells (by
four channels). T h e volume growth rate between L D T 3
and L D T 12 is expressed by the slope (A) andjy intercept
(B) of the linear part, and minimum and maximum
Table 4. Simulated distribution of generation times starting from a group of selected or non-selected newly born
cells
Cell division parameters used
% of cells whose generation time is equal to:
Q1 fY\ 111€tt1 f\ 1n
OHIlLlldLltJJ 11
number
1
Initial ppll \7oliimp
l l l l U l u l l^Cll VU1U-111C
distribution
Selected culture
xQ
(channel)
(day 1)
45
(day 2)
37
(day 3)
38
2
Selected culture
3
Non-selected culture
(day 4)
36
(all 4 days)
39
(all 4 days)
39
v
(channel)
P
1
max
50
1-00
54
1-00
51
1-00
40
1-00
49
49
1 day
2 days
Mean
C.v.
3 days
4 days
5 days
time (days)
6-5
18-9
50-5
23-3
0-7
2-93
28-8
100
9-2
17-5
52-4
19-6
1-3
2-86
30-8
1-00
9-8
41-8
36-8
11-3
0-3
2-51
33-3
The distribution of generation times of the first generation cells, the mean generation time, and the c.v. in generation times in each simulation
are tabulated. Simulations 1 and 2: cell volume and density data of the selected population of newly born cells were taken at LDT 3 of day 2.
1
1
fi dparameters
l 0 0used
3 6 1 are/l
( h ' ) =B
0 1 ); /? m i n = 033
' #
86
The specific
0-0361 (h~'); hB= -0-554
(channelh"
(channelh~');
(channelh" 1 );
) and the
values of A'o, A'i and Pmax are as shown in the table. Simulation 3: the two sequential protocols used to synchronize cells to the beginning of G\
select a biased group of newly born cells; to correct for this, the difference of the cell volume distributions at LDT 22 and LDT 3 of the
following day was used to estimate the actual distribution of unselected newly born cells. Taking this as the cell volume distribution at LDT 0
on day 2, the simulation program was used with the following parameters: A = 0-0361 h" 1 ; B = —0-554channelh""1; Rm-,n = 0-33channelh"*';
; and the values of AQ, A'I and P n l a x are as entered in the table.
volume growth rates (Rm[n and .Rmax)- The cell division
probability, P(x), is described by the channel at which
the ascending portion starts (XQ), the channel at which it
ends (Xi), and the plateau level (P max ) (see Fig. 4). The
sister cell volumes are assumed to be normally distributed
with the standard deviation proportional to 0-10 times the
mean volume.
Table 4 is a computer simulation of the experiment of
Fig. 13. The specific parameters used are given in the
legend; the volume growth parameters are the mean
values of those for the first 4 days (Table 1), and the cell
division probability parameters were taken from Fig. 13.
The simulated and measured cell volume distribution
curves at L D T 12 of days 2, 3, 4 and 5 are shown in
Fig. 14A-D.
Most of the features of cell volume distribution could
apparently be simulated by this program. The distribution of the generation times of the first generation of
cells after synchronization can be calculated from the cell
number changes (Table 4). For simulation no. 1, the
mean generation time was 2-93 days, and the c.v. 28-8 %,
a bit larger than the experimentally determined value of
25%. One reason for this difference is that a significant
fraction of second generation cells divide after 3 days:
experimentally, all the increase in cell density was attributed to cell division of first generation cells, and thus
the fractions of cells with generation times equal to 3 and
4 days are overestimated. Also, no cells are registered
experimentally as having a generation time of 5 days,
while there is a small fraction of such cells according to
the simulation (Table 4).
When the mean values of cell division parameters were
used for all four days (simulation no. 2 in Table 4), the
coefficient of variation in generation times was 30-8%,
2 % larger than that obtained above using different cell
division probability functions for different days. Since
the difference is small, it is likely that the average
conditional probability describes cell division adequately,
i.e. cell division has no significant age dependence.
It is then possible to simulate the time course of cell
density and volume distribution of a collection of unselected newly born cells (Table 4, 'simulation no. 3').
The c.v. in generation times for the unselected group is
33-3%, a little larger than that for the selected group.
This difference is attributable to the larger c.v. in the
initial distribution of unselected newly born cells as
compared to that of newly born cells selected by double
filtration. Since the standard deviation in cell volume is
amplified in quasi-exponential volume growth, in the
presence of cell volume control of division, the standard
deviation in generation times is larger if the c.v. of the cell
volume is larger at the outset. Therefore, the c.v. in
generation times is unusually large for unselected newly
born cells, and is due to a large c.v. in the cell volume of
newly born cells combined with quasi-exponential volume growth.
Discussion
The method for calculating the dependence of growth
rate on initial volume presented above permits a detailed
examination of the growth law obeyed. This simple
method can be applied to other systems so long as: (1) all
cells either increase or decrease in volume, (2) there is no
history of dependence of growth in cell volume, and
(3) there is no significant cell division in a given time
interval. The first two requirements may be met by most
species, and one way to satisfy the last condition is to stop
cell division by adding an inhibitor. The possibility that
growth rate is affected by such a treatment can be tested
Growth and division contml in Gonyaulax
315
by administering the inhibitor to a culture that does not
divide even in the absence of the chemical, and comparing the resultant growth rate with that of a control
culture.
The probabilities of division for cells once they attain a
certain cell volume are different for different models: the
most rigid size control model predicts that no cells divide
below a critical volume, while all larger cells do so. In
effect, the conditional probability must be zero up to the
critical size and then increase abruptly to unity. According to the original transition probability model, the
conditional probability function at a large enough volume
is constant but is less than unity, and also includes an age
dependence: cells are assumed to divide with a constant
probability subsequent to a fixed time after cell formation, and sufficiently large cells must all be old enough to
satisfy this requirement. A synchronized culture, on the
other hand, initially has no probability of division irrespective of cell size, because no cells meet the age
requirement. Once a culture becomes old enough to
satisfy the time requirement, then all cells have the same
probability of division. Thus the conditional probability
is either zero for the entire range or a constant (which is
less than unity), depending on the time after synchronization when conditional probability is determined.
The observations that the plateau level (.Pmax) is less
than unity and that there is not a sharp step to the plateau
in the conditional probability (Fig. 4) do not support the
strict size control model. Although the transition probability model is consistent with the observed conditional
probability, it is inconsistent with lack of age dependence
(Fig. 13). Indeed, it was shown that without some kind
of cell volume control at division, the volume distribution
of newly born cells cannot be simulated. Thus, the
conditional probability is most adequately explained by a
hybrid model with a 'sloppy' size control combined with a
transition probability: before being allowed to divide
with a constant probability (P max ), cells must pass a
critical volume (from A'o to A']). This hybrid model
predicts no age dependence of conditional probability,
because cell volume is assumed to be the key factor for
cell division.
It must also be kept in mind that division is restricted
to a specific time of the day, controlled by the circadian
clock. Is the large c.v. in generation times (25-1 %) due to
the consequent quantization of generation times to integral multiples of days? If cells are allowed to divide
continuously, then the mean generation time will be 0-5
day shorter, and the standard deviation will be less, and is
actually expressed as (s2—(1/12))* (Sheppard's correction), where s is the standard deviation in the quantized
case. If one makes use of the definition, c.v. =.j/(mean
generation time), and the fact that the value of c.v. in the
quantized case is 25 %, it can be mathematically shown
that c.v. in the continuous case is larger than 25 % as long
as the mean generation time is longer than 1-6 days. Since
the mean generation times of G. polyedra under the above
experimental conditions all exceed 1-6 days, the c.v. in
generation time in the continuous case is expected to be
larger than that in the quantized case. That is, without
quantization, the generation c.v. in time would be even
316
K. Homma and J. W. Hastings
10
c
u
Day 0
LTD 12
Simulated
distribution
LTD 12, Day 2 B
Simulated
distribution
LTD 12
Actual
distribution
LTD 12, Day 1 C
LTD 12, Day 3 D
Actual
^/distribution
Actual
istribution
I
Channel no.
Cell volume
Actual
distribution
50
100
2-98
5-95
/
\\
Simulated
distribution
•U. W.,..
50
100
2-98
5-95
Fig. 14. Computer simulations of volume growth and cell
division in L D 12: 12. T h e cell volume distribution of a
synchronized culture was taken from the experiment
described in the legend to Fig. 4 with the cell density
normalized to 100 (drawn with a thick line in A). Taking this
as the starting distribution at L D T 3 of day 2, a computer
program simulated cell volume distributions and densities to
the beginning of day 7. T h e simulated cell volume
distributions at L D T 12 of days 2, 3, 4 and 5 are represented
by thin lines in A, B, C and D, respectively. The actual
volume distributions at corresponding times were provided by
the same experiment, and are shown by thick lines in the
corresponding figures.
Circadian
control?
Fig. IS. Model of the cell cycle and the circadian clock that
explains reported observations. Since the generation times are
quantized to multiples of days, the cell cycle is depicted as
Gi subcycles with a branch leading to cell division. To
explain the observation that there is a discrete D N A synthesis
phase just prior to division, the S phase must be located not
on the subcycle, but on the branch.
larger than 25-1 %. Hence quantization does not account
for the large c.v. in generation time observed.
Making use of both present and previously reported
results, we propose the following model of the cell cycle
and its relationship to the circadian clock of G. polyedra
(Fig. 15). The cell cycle has a cyclical part corresponding
to the traditional Gi phase, and a branch corresponding
to the S, G2 and M phases. The circular part represents
the circadian clock, therefore the duration of one cycle is
temperature-compensated, and phase can be shifted by
visible light. The entrance to the S-G2-M sequence
depends on cell volume, and can take place any time
between L D T (or CT) 12 and 18. Passage takes about
11 h but, in contrast to the circadian part, its duration is
not temperature-compensated. Cells proceeding through
the S-G2-M sequence maintain the same phase of the
circadian clock as those cycling in G.
Since transition to the branch is restricted to the 6-h
(circadian) interval, the phasing of the S phase with
respect to the circadian clock (Homma & Hastings,
unpublished) is explained. In addition, since it takes a
fixed time for the committed cells to proceed along the
branch to undergo cell division, cell division is correctly
predicted to be phased with respect to the circadian clock.
This single circadian control point is at variance with
Sweeney's (1982) observation that in Pyrocystis a circadian oscillator controls not only the transition to DNA
replication but also transitions in G] and G2. Also, the
assumption that cells on the replication-segregation
sequence are at all times in synchrony with non-dividing
cells explains the fact that cell division does not alter the
phase (Homma et al. 1989). Since it is postulated that the
entrance to the branch depends on cell volume, and all
entering cells are assumed to divide, cell division must
depend on cell volume, as was observed. The feature is to
be contrasted with the behaviour of Chlamydomonas cells
in the ultradian mode, which, upon attainment of a
critical volume, all enter the replicative sequence, with no
apparent circadian control (John, 1987).
The cyclical part leads to multimodal generation times
whose spacings are equal to one circadian day, and are
independent of temperature, is observed generally in
species in which cell division is controlled by the circadian clock (see Edmunds, 1984). On the other hand in
Chlamydomonas, a species whose cell division time can
be controlled by the circadian clock, the sequence,
S-G2-M is temperature-dependent (Donnan & John,
1983). The model is similar to the one proposed by Heath
& Spencer (1985) for a marine diatom, in that a temperature-compensated phase is followed by a temperaturedependent one. However, our model differs from theirs
in that no light dependence or independence of the two
phases is postulated, and that the first phase is cyclical.
The present model differs in two respects from the model
of quantal subcycles (G q ) proposed to explain quantized
generation time in mammalian cells (Klevecz, 1976).
First, our model assumes that the duration of the cyclic
part is one circadian day, while a G q cycle is 3-4 h. Since
the generation time of a G. polyedra is quantized to
integral multiples of days, and not of 3-4 h, ours is an
adequate model for this species. Second, our model
correctly predicts that cells return to the subcycle at the
right phase, as is found, while Klevecz's model leads to
the conclusion that divided cells will differ in phase from
undivided cells. Chisholm & Costello (1980) suggested a
model of the cell cycle of the Bacillariophyceae, Thalassiosirafliwiatilis, modified from Klevecz's in that a larger
value for G q was used. However, even if G q is allowed to
vary, Klevecz's model differs from ours on the second
point.
If the model is correct, one intriguing possibility would
be to identify what substance(s) are responsible for the
transition to the S-G2-M sequence. Since these putative
substance(s) should be element(s) of or something very
important in molecular mechanisms of the circadian
clock, this approach may prove fruitful for the elucidation of the circadian clock. Also, since these molecule^) trigger the sequence leading to cell division, their
identification will be useful in uncovering the molecular
mechanisms of cell division in cells that exhibit circadian
rhythmicity.
This research was supported in part by a grant from the
National Institutes of Health (GM 19536). The authors thank
Dr Richard Krasnow for the important ideas and generous
assistance that he contributed to this research, and Dr Till
Roenneberg for constructive criticism and suggestions.
Taken in part from a thesis of K.H. presented to the
Committee on Higher Degrees in Biophysics of Harvard
University in partial fulfilment of the requirement for the PhD.
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