Download analysis of the significance of a periodic, cell size

J. Cell Sci. 35, 41-51 (1979)
Printed in Great Britain © Company of Biologists Limited 1979
41
ANALYSIS OF THE SIGNIFICANCE OF A
PERIODIC, CELL SIZE-CONTROLLED
DOUBLING IN RATES OF MACROMOLECULAR
SYNTHESIS FOR THE CONTROL OF
BALANCED EXPONENTIAL GROWTH OF
FISSION YEAST CELLS
A. BARNES
National Vegetable Research Station, Wellesbourne, Warwick CK35 gEF, England,
PAUL NURSE AND R. S. S. FRASER*
University of Edinburgh, Department of Zoology, West Mains Road,
Edinburgh EHg T,JT, Scotland
SUMMARY
Mutant strains of the fission yeast Schizosaccharomyces pombe are available which divide at
smaller mean sizes than wild type. Earlier work by the present authors has shown that all these
strains double their rates of polyadenylated messenger RNA synthesis as a step once in each cell
cycle. The smaller the cell, the later in the cycle is the doubling in rate of synthesis. Strains of
all sizes, however, double their synthetic rate when at the same threshold size. We show here
that the differences in cell cycle stage of doubling in rate of polyadenylated messenger RNA
synthesis are enough to explain the reduced mean steady state polyadenylated messenger RNA
contents of the smaller strains. The cell size-related control over doubling in rate of synthesis is
also shown to maintain the mean polyadenylated messenger RNA content as a constant proportion of cell mass, irrespective of cell size. This control thus allows cells to maintain balanced
exponential growth, even when absolute growth rate per cell is altered by mutation. It is also
shown that the concentration of polyadenylated messenger RNA itself could act as a monitor of
the threshold size triggering the doubling in rate of synthesis in each cell cycle.
INTRODUCTION
During balanced exponential growth of cell cultures, individual parameters of
growth remain at a constant proportion of total cell mass. At the level of the individual
cell, the amount of each cell component must on average double during each cell
cycle, so that the daughter cells formed at division are identical to the parent at the same
stage in the preceding cycle. One mechanism which can account for the doubling in
amount of a component is a stepwise doubling in the rate of synthesis of that component at a fixed point in each cell cycle. In the fission yeast Schizosaccharomyces
pombe, there are periodic doublings once in each cell cycle in the rates of synthesis
of several enzymes (Mitchison & Creanor, 1969), ribosomal RNA (rRNA) (Wain
& Staatz, 1973; Fraser & Moreno, 1976) and polyadenylated messenger RNA
(poly (A)+mRNA) (Fraser & Moreno, 1976; Fraser & Nurse, 1978a).
• Present address: National Vegetable Research Station, Wellesbourne, Warwick CV35 9EF,
England.
42
A. Barnes, P. Nurse and R. S. S. Fraser
One approach to the problem of how individual growth parameters may be maintained at a constant mean proportion of cell mass has been made possible by the discovery of mutants of fission yeast which divide at different mean sizes from wild type
(Nurse, 1975). Fraser & Nurse (1978a, b) showed that the smaller the cell, the later in
the cell cycle was the stepwise doubling in rate of poly(A)+mRNA or rRNA synthesis.
In this paper we will show that the observed delay in the doubling in rate of
poly(A)+mRNA synthesis in the cell cycle is enough to account for the reduced
poly(A)+mRNA content of the smaller cells (Fraser & Nurse, 1978 a). Furthermore, if
the doublings in synthetic rate always occur in cells of the same size, then poly(A)+mRNA
content will be maintained at a constant proportion of cell mass irrespective of cell size
or absolute growth rate per cell. We also show that the concentration of poly(A)+mRNA
itself could act as a signal for the onset of doubling in rate of poly(A)+mRNA synthesis
in each cell cycle.
Table 1. Volumes at division and mean protein contents of 5 strains of
Schizosaccharomyces pombe
Mean cell vol.
at division,
Strain
Haploids
Wild type
wee 1-50
Diploids
Wild type/wild type
wee 1-50/wild type
toee 1-50/iuee 1-50
fim3
Mean protein
per cell,
Pg±S.E.M.
149
12-1 ±0-3
73
6-9 ±0-4
283
22-6 ±O'5
185 ±08
13-0 ±O-2
231
144
MATERIALS AND METHODS
Wild type cells were strain 972 h~ of Schizosaccharomyces pombe Lindner. The derived mutant
wee 1-50 (Nurse, 1975) has an alteration in the control over initiation of nuclear division, such
that it undergoes nuclear division and cell division at slightly more than half the size of wild
type. Cell cycle data were obtained for haploid wild type and wee 1-50 (Fraser & Nurse, 1978a)
and for 3 diploid strains: wild type and wee 1—50 homozygous diploids and the heterozygous
wee 1-50/wild type diploid (Fraser & Nurse, 19786). The 3 diploid and 2 haploid strains
together cover a size range from approximately o-6 to 1 -9, relative to the haploid wild type size of
1 0 (Table 1).
RESULTS AND DISCUSSION
Table 2 summarizes the relative poly(A)+mRNA contents of the 3 diploid and
2 haploid strains studied, and the times in their cell cycles when the rate of
poly(A)+mRNA synthesis underwent a stepwise doubling. It is clear that in both haploid
and diploid series, the smaller the cell, the lower the relative content of poly(A)+mRNA
per cell, and the later in the cell cycle the doubling in rate of poly(A)+mRNA synthesis.
We wish first to establish that the delay between doubling in rate of poly(A)+mRNA
synthesis in wild type and smaller mutants is all that is required to account for the
Exponential growth offissionyeast
43
reduced mean poly(A)+mRNA content of the smaller cells. This will be done by
calculating how much the poly(A)+mRNA content of the smaller cells would be, if the
delay in doubling of the synthetic rate were the only difference between the 2 strains,
and comparing this with the experimentally measured ratio.
Table 2. Mid-points of doublings in rate of poly(A)+mRNA synthesis, and
poly(A)+mRNA contents of 5 strains of Schizosaccharomyces pombe
Poly(A)+mRNA content per
cell as a percentage of
wild type of the same
ploidy level
Cell cycle stage
of mid-point in
doubling of rate
of poly(A)+mRNA
synthesis
Strain
Haploids
Wild type
wee 1-50
Diploids
Wild type/wild type
ivee 1-50/wild type
wee 1-50/wee 1-50
O-II ±004
o-8i ±0-06
0-70 ±007
Calculated
Measured
62 ±2
59 ±3
0-03 ±0-02
0-47 ±0-04
0-50 ±003
8o± 4
I-OI ±005
104 ±004
5° ±2
S7±2
Mid-points of doublings in rate of poly(A)+mRNA synthesis were measured in synchronous
cultures (data from Fraser & Nurse, 1978a, 6). / is the time, in cell cycle units, between the midpoint of doubling in a small mutant cell and in wild type of the same ploidy. Poly(A)+mRNA
content of small mutant cells was calculated as z~' of wild type content, as explained in the text.
All values are means ± standard errors.
1
1
2 -
y
/f
/
/
I
-
Rate
1 -
Time, t
Fig. r. The assumed patterns of rates of synthesis of poly(A)+mRNA in wild type
(
) and small mutant wee 1-50 (
) cell cycles. Wild type cells are assumed to
synthesize poly(A)+mRNA at a rate f(t) (
), and wee 1-50 cells at a rate i(t — l)
(
). ax and a, are the cell cycle stages of the mid-points of the doublings in rate of
poly(A)+mRNA synthesis in wild type and wee 1-50 cells respectively. / is the time
between the mid-points of doubling in rates of poly(A)+mRNA synthesis in wild
type and wee 1-50, expressed as a fraction of the cell cycle. One time equals a complete
cell cycle.
44
A. Barnes, P. Nurse and R. S. S. Fraser
Derivation of an expression for the average poly{A)+mRNA
compared to wild type
content of a mutant small cell
Consider first the rates of poly(A)+mRNA synthesis in synchronously dividing populations of wild type and small mutant wee 1-50 cells. The cell cycle commences with
the same absolute rate of poly(A)+mRNA synthesis per cell in each strain (Fraser &
Nurse, 1978 a). The rate of synthesis in wild type cells increases as a function of time
i(t), so that at the end of the cell cycle it is exactly double the initial rate (Fig. 1). In
wee 1-50 cells, the rate of synthesis of poly(A) + mRNA follows the same pattern as
in wild type, except for a delay of a fraction / of a cell cycle before the rate of
poly(A) + mRNA synthesis doubles. Cell cycle patterns [of poly(A) + mRNA synthesis
such as shown in Fig. 1 have been demonstrated experimentally for wild type and wee
1-50 cells(Fraser& Nurse, 1978 a). However, the following proof is valid for any pattern
of increase during the cell cycle which is the same for wild type and wee 1-50.
Let R' and R'm be the amounts of poly(A)+mRNA in synchronously dividing populations of wild type and wee 1-50 cells; R' and R'm depend on time t, which is measured
in cell cycle units (one cycle or mean generation time = i-o cell cycle unit). The degradation of poly(A)+mRNA has been shown to follow approximately first-order kinetics
(Fraser, 1975); we assume that the half-life is the same in both strains. Thus the net
rates of poly(A)+mRNA accumulation are given by:
^ - = N(t)i(t)-AR'
J
for wild type
n'
?^
=N(t)f(t-l)-AR'm
for wee 1-50,
(1)
where A is the degradation rate constant, N(t) is the number of cells in the population
at any time t, and f (t) is the rate of synthesis of poly(A)+mRNA in wild type cells;
i(t — I) is the rate in wee 1-50 cells (Fig. 1).
In addition, as the populations double in each cell cycle unit of time, but at any time
consist of cells identical to those of the previous generation at the same stage in the
cell cycle, we may write
2f(«-l).
(2)
Now consider asynchronous, exponentially growing populations of wild type and
wee 1-50 cells, with the same numbers of cells per ml, and growing with the same mean
generation time (Nurse, 1975). R and Rm are the amounts of poly(A)+mRNA in the
wild type and mutant populations respectively. From the canonical cell age distribution equation (Cook & James, 1964) the fraction of cells aged a to a + 8a is given by
lnz.z1-"^
(3)
at all times. From Fig. 1 it is clear that the rates of synthesis of poly(A)+mRNA for
cells aged a are given by f(a) and {(a —I) for wild type and wee 1-50 respectively. By
Exponential growth offissionyeast
45
integrating these rates of synthesis, weighted for cell age, over all possible ages, we
obtain the average rate of synthesis of poly(A)+mRNA in asynchronous population as
f1
f (a). In 2.21~a da in wild type
Jo
f(a-/).ln2.2 1 ~ a da in wee 1-50.
and
Jo
As the number of cells in each population is increasing exponentially, the total
number of cells may be written as N = No efct for both populations, where iV0 and k
are constants. The total rates of synthesis of poly(A)+mRNA in each population are
given by the total number of cells multiplied by the average rates of synthesis.
We may now write analogous equations to (1) for rates of accumulation of
poly(A)+mRNA in asynchronous, exponentially-growing populations:
for wild type
(5a)
^ = C T r t a ^ 1 - da-AR
at
Jo
and for wee 1-50
da-ARm,
(56)
where C = No efct.ln 2.
Equation (5 b) may be written as
1
f(a-/)2^+'d(a-/)-Ai?m
(6)
and since f is a periodic function integrated over a whole period, (6) may be rewritten
as
Therefore, writing
B = f1f(a)21-«da
Jo
equations (5) may be rewritten as
~dt=
and
'
—r^ = C .B .z^—XR,,,.
As the numbers of cells in the 2 populations are increasing at the same exponential
rate and as the age distribution of the population remains fixed, the amount of
poly(A)+mRNA in the 2 populations must also increase at the same exponential rate.
So
where q is a constant.
4
CEL 33
46
A. Barnes, P. Nurse and R. S. S. Fraser
Combining equations (7) and (8) by eliminating — and . '" and rearranging gives
= C.B
and
(9)
from which it follows directly that
(IO)
R
This result may be more readily appreciated from the following simpler explanation,
which is not, however, a formal mathematical proof. Consider populations of wild
Time
Fig. 2. Poly(A)+mRNA contents of asynchronous, exponentially growing populations
of wild type (upper curve) and wee 1-50 cells (lower curve). The 2 populations contain
the same number of cells at any time. The fixed time shift, of magnitude / between the
curves, is a consequence of the delay, of the same magnitude, between the doubling in
rate of poly(A)+mRNA synthesis in the wild type and wee 1-50 cell cycles.
type and wee 1-50 containing the same numbers of cells and growing at the same
exponential rate. The delay, of a fraction / of the cell cycle, in the doubling of the rate
of synthesis of poly(A)+mRNA in wee 1-50 cells (Fig. 1) implies that the amount of
poly(A)+mRNA in the wee 1-50 population will lag behind that of the wild type
population by a time interval /, as shown in Fig. 2. Thus if the wee 1-50 population
contained a particular amount of poly(A)+mRNA at a particular time t, the wild type
population would have contained the same amount at time / —/. In the intervening
interval both populations would have increased in cell number by a factor 2'. Thus the
average amount of poly(A)+mRNA per cell in the wee 1-50 population at any time will
be 2~* of that in the average cell of the wild type population.
Exponential growth offissionyeast
47
Using this expression, and the values for / measured in synchronous cultures for the
haploid and diploid series oiS.pombe, we have calculated mean relative poly(A)+mRNA
contents for the small mutant cells of each series as a fraction of wild type. Table 2
(p. 43) shows that these calculated values are in good agreement with the values determined experimentally in asynchronous, exponentially growing cultures.
We conclude that delaying the doubling in rate of poly(A)+mRNA synthesis in the
cell cycle is alone sufficient to account for the reduced poly(A)+mRNA content of
those cells forced by mutation to be of smaller mean size than wild type.
Delayed doubling in the rate of poly(A)+mRNA synthesis in small mutant cells keeps the
ratio of the average poly(A)+mRNA content to average total cell mass the same as in wild
type
It has been shown experimentally for the haploid and diploid series that at the time
of the mid-point of doubling in rate of poly(A)+mRNA synthesis, the members of each
series have similar protein contents per cell (Fraser & Nurse, 1978b). We have taken
protein content per cell as a measure of cell size, as it is easy to measure accurately.
Protein increases close to exponentially through the cell cycle (Stebbing, 1971) and it is
likely that other parameters of growth such as total cell mass or volume will also increase
close to exponentially through the cell cycle. We shall refer to any of these parameters
representing overall cell size as V, and will demonstrate that if the value of V is the
same for wild type and small mutant cells at the time of the mid-point in doubling of
the rate of poly(A)+mRNA synthesis, then the ratio of the average poly(A)+mRNA
content to average V in asynchronous, exponentially growing populations must be the
same for strains of all sizes.
Let V(a) and Vm(a) be the value of the growth parameter (be it protein, mass or
volume) of cells aged a, of wild type and small mutants respectively. Both V(a) and
Vm(a) increase exponentially with the same specific rate of increase. This rate being
such that the cell sizes at a = 1 must be twice the sizes at a = o, we may write:
V(a) = V(o)za
(11 a)
VJai) = VJp)z*.
(nb)
and
Let the cell age at the mid-point of the doubling in rate of poly(A)+mRNA synthesis
be <xx in wild type cells and Oj, in small mutant cells; so that <x2 = <tx — l (Fig. 1). The
property of cell size parameters assumed for the proof implies that
(12)
Substituting a± in (11 a) and c^ in (11b) and making use of (12) it follows that
V(o) = Vm(o)z'
(13)
Using (13) to eliminate V(o) and Vm(o) from (11) leads to
V(a) = VJ*W.
(14)
In asynchronous, exponentially growing populations, cell age distribution remains
48
A. Barnes, P. Nurse and R. S. S. Fraser
fixed and is the same for wild type and small mutant populations. As equation (14)
implies that the wild type cells are always 2' larger than the mutant of corresponding
age, it follows directly that the average cell size in a wild type population will always be
2' larger than in small mutant populations. From equation (10) the average amount of
poly(A)+mRNA in wild type cells is also 2' greater than in mutant cells at all times.
Therefore, it follows directly that the ratio of average poly(A)+mRNA content to
average cell size must be the same for wild type and small mutant strains.
From this proof, we can conclude that the control of the cell cycle stage of doubling
in the rate of poly(A)+mRNA synthesis by a threshold cell size which is the same for
mutants of a range of mean sizes (Fraser & Nurse, 1978 a, b) will maintain average
poly(A)+mRNA content at a constant proportion of total cell protein, mass or volume
during growth, irrespective of cell size or absolute growth rate per cell. This control
therefore acts homeostatically: when the growth of the cell is distorted by the presence
of the wee 1 mutation, leading to an altered absolute growth rate per cell, the sizerelated control is nonetheless able to maintain balanced growth and keep
poly(A)+mRNA content in line with total cell growth. A further property of the cellsize control over doubling in rate of poly(A)+mRNA synthesis is that it enables cells to
compensate for variation in gene concentration. For example, the 3 diploid strains
have different gene concentrations but the size control over doubling in rate of RNA
synthesis allows them to grow with the same mean RNA concentration (Fraser &
Nurse, 19786).
The concentration of poly(A)+mRNA could itself act as the trigger initiating the doubling
in rate of poly(A)+mRNA synt/iesis in each cell cycle
The 3 diploid strains of different mean sizes all double their rate of poIy(A)+mRNA
synthesis at points in their cell cycles when they have very similar protein contents per
cell (Fraser & Nurse, 19786). Similarly, the 2 haploid strains have similar protein
contents per cell when they double their rates of poly(A)+mRNA synthesis (Fraser &
Nurse, 1978 a). All 5 strains double their rate of poly(A)+mRNA synthesis when they
have similar protein contents per haploid genome per cell, suggesting that the timing
of the doubling in rate of synthesis involves monitoring of some aspect of cell size. We
wish now to establish that the concentration of poly(A)+mRNA itself could act as a
size-monitoring mechanism.
As poly(A)+mRNA is unstable, if it is synthesized at a constant rate per cell the
content per cell will approach a steady state. Doubling the rate of synthesis as a
discrete step once per cell cycle will lead to an increase in the poly(A)+mRNA content,
which will again tend to a steady state at a higher level (Fig. 3). We assume that cell
volume increases exponentially, and that the half-life of poly(A)+mRNA is 0-275 °f
a generation time (Fraser, 1975) in all strains. As the cells must exactly double their
poly(A)+mRNA content in one cell cycle, knowledge of the half-life is sufficient to
determine the relative initial rates of synthesis of poly(A)+mRNA. From these figures
and the times of doubling in rates of synthesis in the cell cycle, the cell content of
poly(A)+mRNA can be calculated for any stage in the cell cycle. Dividing the
Exponential growth offissionyeast
49
+
poly(A) mRNA content by the cell volume at that stage gives the poly(A)+mRNA
concentration.
Using experimentally measured values for total protein per cell as a measure of cell
mass or volume, Fig. 4 shows calculated changes in poly(A)+mRNA concentration
during the cell cycles of the 3 strains for which we have the most extensive cell cycle
1
1
1
r
1
20
s
2
cc
s
/ '
synthesi!
"o
-
1-6
o
^11 volume anc1 comtent
_
,«*
E 1-8
/
/
-
1-4
/
_
1-2
w
I
.{•'
•'/
-
10
|
0
0-2
I
|
0-4
0-6
Cell cycle time
1
1
0-8
1-0
Fig. 3. Changes in rate of poly(A)+mRNA synthesis (- -) and in calculated
poly(A)+mRNA content per cell (
) during the cell cycle of ideal cells. The
example shown is haploid wild type with a doubling in rate of poly(A)+mRNA
synthesis at o-n of the cell cycle (Fraser & Nurse, 1978a). The rate doubling is
assumed to be instantaneous.
, change in cell volume (or mass, or protein)
assumed to increase exponentially during the cell cycle.
data; all strains studied show the same basic pattern. In each curve, there is initially a
decline in concentration, then at the time in the cell cycle when the rate of
poly(A)+mRNA synthesis doubles, the concentration of poly(A)+mRNA begins to
rise. The minimum concentrations of poly(A)+mRNA reached in the different strains
are very similar.
In addition to this experimentally based evidence, it can be shown theoretically that
if exponentially growing cells from any 2 strains have the same volume at the time of
doubling in rate of poly(A)+raRNA synthesis, and if the rate of poly(A)+mRNA
synthesis follows the generalized pattern indicated in Fig. 1, then the minimum concentrations of poly(A)+mRNA must be the same and must occur at the time when the
rate of poly(A)+mRNA synthesis doubles. This result may be proved by integrating
equations (1) and making use of the property in equation (2).
50
A. Barnes, P. Nurse and R. S. S. Fraser
This experimental and theoretical evidence therefore indicates that the concentration of poly(A)+mRNA itself would be capable of acting in the mechanism which
triggers the doubling in rate of poly(A)+mRNA synthesis in each cycle. As the concentration of poly(A)+mRNA in the cell depends not only on the rate of synthesis of
poly(A)+mRNA but also on cell growth, it follows that this mechanism is essentially
a cell-size monitoring mechanism. A doubled rate of synthesis of poly(A)+mRNA is
switched on when the cell has grown to a sufficient size to reduce the concentration of
poly(A)+mRNA below a threshold level. Furthermore, as the trigger responds to a
concentration and not to an absolute amount per cell, the mechanism would be able to
operate in both haploid and diploid cells without further elaboration.
0-2
0-4
0-6
Cell cycle time
0-8
10
Fig. 4. Changes during the cell cycle in poly(A)+mRNA concentration in 3 strains
of S. pontbe. Changes in cell content of poly(A)+mRNA were calculated from the
observed times of doubling of rate of poly(A)+mRNA synthesis in the cell cycle
(Fraser & Nurse, 1978a, b) as shown in Fig. 3. Poly(A)+mRNA concentration was
calculated by dividing poly(A)+mRNA content by the cell size parameter (based on
total protein estimations) at each stage of the cell cycle.
, wild type haploid;
, wee 1-50 haploid;
, wild type/wee 1-50 heterozygous diploid.
CONCLUSION
In this paper we have analysed the behaviour of a component whose rate of synthesis doubles as a step during the cell cycle. We have shown that the component will
be maintained on average at a constant proportion of mass in cells of different sizes at
division, provided that the rate of synthesis per cell before the step doubling is the same
in the different cells, and that the cell size at the time of the step doubling is the same
in the different cells. We have used as an example the rate of synthesis of
poly(A)+mRNA, which doubles as a step. However, the proof will also apply to stable
molecules such as rRNA, and to other components regardless of their actual patterns
of increase through the cell cycle. As long as the rate of synthesis per cell and cell size
are the same at any particular point on the pattern, the component will be maintained
on average as a constant proportion of cell mass irrespective of actual cell mass.
Other cell components whose rates of synthesis are dependent upon a component
Exponential growth offissionyeast
51
regulated by a control of the type we have described will also be maintained at a constant average proportion of cell mass. An example of this may be provided in S. pombe
by 3 enzymes (Mitchison & Creanor, 1969) the accumulation of which may be
dependent on mRNA content (Fraser & Moreno, 1976). Therefore a regulatory
mechanism involving cell size control over rates of synthesis could be of widespread
significance in the control of balanced exponential growth of cells.
This work was supported by the Agricultural, Science and Medical Research Councils. We
thank Dr G. H. Freeman, Professor. J. K. A. Bleasdale and Professor J. M. Mitchison,
F.R.S. for useful comments.
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{Received 8 May 1978)