CONTROL OF CELL SIZE AND CYCLE TIME IN SCHIZOSACCHAROMYCES POMBE

Transcription

1 J. Cell Set. 24, (1977) 51 Printed in Great Britain CONTROL OF CELL SIZE AND CYCLE TIME IN SCHIZOSACCHAROMYCES POMBE P. A. FANTES Department of Zoology, University of Edinburgh, West Mains Road, Edinburgh, EHg 2JT, Scotland SUMMARY Steady-state and perturbed cells of Schizosaccttaromyces pombe have been observed through several division cycles by time-lapse photomicrography. Perturbed cells were produced by the use of a conditional cell division cycle mutant in which nuclear division is reversibly blocked at high temperature. These experiments show that in both populations cell length at division and cell cycle duration are homeostatically controlled, probably by a primary size-control mechanism. Cycle time is indirectly controlled, as cells which have an extended cycle are on average larger at division, so that daughters of such cells need to grow by a smaller amount and for a shorter period, before dividing again. In general, deviations from the mean are corrected within a single cycle, but in the case of very long cells the control breaks down because the cycle cannot be shortened by more than a quarter under the conditions used. These cells take more than one cycle to return to normal. INTRODUCTION It is a familar concept that cells of a given type growing exponentially under constant cultural conditions maintain a constant mean size (Maaloe & Kjeldgaard, 1966; Wehr & Parks, 1969; Yen et al. 1975). Cell size at any stage of the cell cycle, such as cell division, also has a constant mean value, and the variation around this mean does not increase from generation to generation (Schaechter, Williamson, Hood & Koch, 1962; Prescott, 1956; Anderson, Bell, Petersen & Tobey, 1969). Cell size is related to cell age in the cycle by the pattern of growth, and the control of timing of cell division might therefore be expected to be associated with the control of cell size: a mechanism which controls one parameter will probably control the other. Investigation of how constancy of cell size and cell cycle duration is maintained has in the past concentrated mainly on the control of cycle duration (Schaechter et al. 1962; Sudbery & Grant, 1975; Froese, 1964; Miyamoto, Zeuthen & Rasmussen, 1973; Dawson, Madoc-Jones & Field, 1965). In a few cases only has cell size also been considered (Yen et al. 1975; Lovlie, 1963; Killander & Zetterberg, 1965; James, Hemond, Czer & Bohman, 1975; Prescott, 1956). More detailed investigation of the relation between size and cycle time in eukaryotes has not been made because of the difficulty of estimating the size of most types of individual living cells. The fission yeast Schizosaccharomyces pombe is a eukaryote in which the size of individual cells may be be determined with relative ease: cells are cylindrical, with approximately hemispherical ends, and growth occurs primarily by length extension 4-2

2 52 P. A. Fantes (Mitchison, 1957), though there is some evidence for an increase in width in successive generations (Johnson & Lu, 1975). The mean diameter of haploid cells is 3-5 /.cm (Mitchison, 1970), and the length increases from about 8 to 14/tm during the cell cycle under normal conditions. Cell extension does not occur at a constant rate: several authors (Mitchison, 1957; Streiblova & Wolf, 1972; Johnson, 1965) have reported that the extension rate increases for about three quarters of the cell cycle, at which time extension stops. Nuclear division occurs at about this time, and about o-i of a cycle later, a cell plate is laid down across the cell centrally, or very nearly so. The final event of the cycle is cleavage of the cell plate, followed by the new daughter cells rounding off. A number of cell division cycle (cdc) mutants have been isolated in S. pombe (Nurse, Thuriaux & Nasmyth, 1976). They divide normally at 25 C, but on shifting to 35 C block at specific stages of the cell cycle, and do not divide further. Studies of these mutants, and previous studies using inhibitors (Mitchison & Creanor, 1971; Herring, 1974) have shown that cell plate formation and cell division are dependent on prior nuclear division. One of the cdc mutants has been used in this work to produce specific inhibition of nuclear division and hence cell division. The purpose of this work is to gain a better insight into the homeostatic mechanisms by which individual cell sizes and cycle times are controlled in a eukaryote. The approach taken has been to follow, by means of time-lapse photomicrography, the growth and division of individual cells. Steady-state populations of cells have been examined in this way, and in an attempt to define the operational limits of the control, abnormally long cells were produced by blocking nuclear division, and their recovery followed after release of the block. MATERIALS AND METHODS Organisms Two strains of Schizosaccharomyces pombe were used: the wild-type haploid 972 with Ir mating type (Gutz, Heslot, Leupold & Loprieno, 1974), and a cell division cycle mutant cdc 2-33 derived from 972 by mutagenesis. Some properties of this mutant are described in the text: further information about its isolation and characterization are published elsewhere (Nurse et al. 1976). Routine handling of the organism was according to Mitchison (1970) and Gutz et al. (1974). Media For time-lapse studies, the medium used was a modification of EMM2 as described by Nurse (1975), with 05 % Ion agar (Oxoid No. 2). In most experiments, yeast extract (Difco) was also included at 0-5 %, which leads to a faster rate of growth (generation time reduced from 4-3 to 2-9 h at 25 C): it is stated in the text when minimal medium alone was used. Time-lapse photomicrography Cells were observed and photographed using a low-power (16 x) objective with dark-field optics on a Zeiss Photomicroscope. Photographs were taken every 20 min. To control the growth temperature, a special growth chamber was constructed. This consisted of a glass microscope slide (38 x 76 mm) with a hollow aluminium annulus (internal

3 Control of cell size and cycle time 53 diameter 20 mm; external diameter 35mm; height 5 mm) attached to the surface with Araldite. Water from a thermostatted bath was pumped through the hollow channel in the annulus. On the inside of the annulus was a ledge (1 mm high; 3 mm wide) in contact with the glass slide: this served as support for a 16-mm circular coverslip. The ledge was formed as an integral part of the annulus. For use, aflatdisk of agar was made by putting a drop of molten agar on the slide, and lowering a coverslip on top. The agar was allowed to set, the coverslip removed, and the slide placed in position on the microscope stage with water circulating to allow temperature equilibration. A drop of cell culture was placed on the agar, the cells allowed to settle for a few minutes, and a fresh coverslip placed on top and sealed to the ledge with silicone grease to prevent the agar from drying out. A suitable field for observation ( cells) was then selected. The temperature on the agar was checked with a contact thermistor in trial experiments. Temperature shifts were carried out by moving the water inlet tube from one waterbath to another: the new temperature was reached within 2 min. Aeration was not found necessary for exponential growth (see below). Scoring of cell length and cell division time The processed negative film was viewed by projection on to a screen. A typical cell had a projected length of 15 mm at division, equivalent to a real value of 13-2 fim. Measurements were made to the nearest mm. A cell was scored as having divided at the first frame when the daughter cells showed any rounding off at their adjacent ends: cell length at division was measured in the preceding frame. As the length of cells does not increase much, if at all, near the end of the cycle (Mitchison, 1970), the error in. length determination because of growth in this period is small. Cell length at birth was not routinely measured: measurement of a number of cells of different length at division showed that cells divide symmetrically, or very nearly so. The lengths of daughter cells differed by o or 1 mm (0-9 /tm). Taking the mean of the 2 values would therefore produce no errors larger than 0-5 mm (04 ftm), which would have little effect on the distribution of cell length at birth of the population. The sum of the birth lengths of 2 daughter cells was found in a number of cases to be a constant amount (3 mm or 2-6 fim) greater than the length of the mother at division, the difference being due to the rounding off of the daughter cells. Cell size at birth was therefore derived by adding the mother cell's division length to the correction factor for rounding off, and dividing by 2. RESULTS Two basic types of experiment were analysed in detail. The first experiment was the observation of wild-type cells growing at 25 C in steady state. Cells of strain 972 were transferred from a shaken exponential-phase liquid culture to the temperature-controlled slide, and 88 cell lines followed for 2 complete cycles, that is, 3 divisions. Two further cells were initially included, but these failed to divide at all and appeared dead. As a check that conditions on the temperature-controlled slide allowed steady-state growth, the following tests were made, (i) The increase in cell number was nearly exponential (Fig. 1), with a doubling time of 3-1 h, compared with the value of 2-9 h in liquid culture on the same medium. Mean cycle times for the first and second cycles are shown in Table 1. The observation that the first cycle is on average slightly longer than the second is paralleled by similar findings by Powell (1955) using bacteria. To reduce any effect this might have on the interpretation of results, the steady-state measurements shown in Figs. 2-6 refer to the second cycle only, (ii) The mean cell lengths at the first, second and third divisions are shown in Table 1. A small decrease at successive divisions occurred, but the change

4 54 P. A. Fantes was smaller than that in cycle time and does not have an important effect on any conclusions made. In view of these results, cultural conditions on the slide were considered an adequate approximation to those in liquid steady-state culture u Time, h 10 Fig. i. Abscissa, time, h;ordinate,cell number in the microscope field. Increase in cell number of a population of strain 972 growing on the temperature controlled slide at 2S C. Table 1. Mean cell lengths at division and mean cycle times of steady state and perturbed cells Parameter measured Length at division, /im Cycle time, h Experimental condition -MC2-33; 25 c ffa/c2-33; 25 C after 3 h pulse at 35 C 972;25 C tfcrfc2-33; 25 C after 3 h pulse at 25 C C 1st Mean ± S.D ± ± ± 2'09 3-i4±o ±0-57 Cycle Number of cells before 1st division: #,! B8;f, 2 S ;tt,62. The number of cells before the 2nd and 3rd divisions are 2 and 4 times the initial values. A 2nd Meant S.D ± ± ± ± ±050 3rd Meant S.D i i 113

5 Control of cell size and cycle time 55 The second type of experiment carried out involves the use of the conditional cell division cycle mutant cdcz-^t, (Nurse et al. 1976). This mutant grows at the same rate as wild-type at 25 C, the permissive temperature. Cell size under these conditions is slightly larger than that of the wild-type, as shown in Table 1. This is probably because the temperature-sensitive gene product is slightly defective at 25 C. On shift to the restrictive temperature, 35 C, cells of this strain continue to progress through the cell cycle up to a stage before nuclear division, where they are blocked: the nuclear morphology is typical of interphase cells. Growth continues at the restrictive temperature, bulk protein and RNA increasing in a similar way to wild-type for at least 5 h (Nurse et al. 1976), during which time viability remains above 95%. The inhibition of nuclear division while growth continues leads to the production of abnormally long cells. On shift back to 25 C, cells continue to elongate for a period, after which time they divide. This regimen of pulsing cells at the restrictive temperature is a powerful and specific method of delaying nuclear division and hence cell division. Existence of size homeostasis Two lines of evidence indicate the existence of size homeostasis: one from steadystate experiments and the other from an experiment where the recovery of cells of cdc2~22 was followed after holding at 35 C for 3 h. Mean lengths and cycle times are E 20 Z 16 U Cell length at division n. 24 Fig. 2. Abscissa, cell length at division n, /tm; ordinate, cell length at division n + i, /tm. Mean values, and bars showing the standard error of the mean for division n +1, are shown for each size class at division n. Results for n = 2 shown for 972 ( ) in steady state at 25 C C; results for n = 1 and n = 2 shown for cdcz-22 ( ) at 25 C after a period of 3 h at 35 C.

6 56 P. A. Fantes shown in Table 1. Fig. 2 shows cell lengths at successive divisions plotted against one another in both steady-state and perturbed situations. In neither case is there a definite relation between cell lengths at successive divisions, except where the length at one division is greater than about 18-5 /tm. In all but these very long cells, therefore, there is no tendency for a cell longer (or shorter) than average at division to produce daughters which divide at either a larger or a smaller size than average. Cell size is therefore not 'inherited', nor is there any evidence for 'overshoot', which would \ M C u 10 Cell length at birth, Fig. 3. Abscissa, cell length at birth, /tm; ordinate, length extension during the cycle, /tm. Means and standard error bars are shown for each size class. Symbols and experimental details as in Fig. 2: only second cycle results are included for 972. The lines shown are calculated regression lines with slopes 076 for the steady state cells, and 082 and +002 for the 2 lines for the perturbed cells. result in a negative correlation between mother and daughter division sizes. Some control must operate such that deviations in length at one division are compensated, with most of the compensation occurring in a single cycle. In the steady-state situation, this has the effect of maintaining a constant size at successive divisions: where the initial cells are abnormally long, the mean length at division decreases at successive divisions, being close to the steady-state value after 3 divisions. Another way of demonstrating the compensatory mechanism is shown in Fig. 3, where the amount of growth or extension during the cycle (length at division minus length at birth) is plotted against length at birth. This shows a strong negative correlation between the variables, except for cells longer than 10-5 /tm at birth, discussed below. The slopes of the regression lines fitted to the data are 0-76 for the steady-

7 Control of cell size and cycle time 57 state case, and 0-82 for the perturbed situation. These values are close to 1, which value would indicate that all the compensation occurred within a single cycle. The best regression line for cells longer than 10-5 /jm at birth has a slope of and is also shown. Mode of action of size homeostasis There are 2 ways in which the size homeostasis described in the previous section might be attained: (i) the duration of individual cycles might be independent of cell Cell length at birth, //m E a. 20 I 3 a Cell length at division, /im Fig. 4. A: abscissa, cell length at birth, /im; ordinate, mean, extension rate, /im h" 1. B: abscissa, cell length at division, /im; ordinate, mean extension rate, /im h" 1. Symbols and experimental details as for Fig. 3.

8 58 P. A. Fantes length at birth, and the homeostasis maintained by alteration of growth rate inversely with cell size; or (ii) the duration of the cycle might vary inversely with cell length at birth, and the growth rate would not decrease with increasing cell length at birth. Fig. 4 A shows that alteration of extension rate (extension during the cycle/cycle time) plays no part in maintaining constant cell length at division. The extension rate through any cycle is independent of cell length at the start of that cycle: in fact Fig. 4B shows that extension rate is strongly positively correlated with cell length at the end of the cycle. Thus the overall relation between cell size and extension rate is a positive one, large cells growing more rapidly than small ones. The effect of this is to increase rather than decrease the variation in cell length at division, thus working against homeostasis. Yri % OS Relative cell age 10 Fig. 5. Abscissa, relative cell age; ordinate, relative cell length. Idealized growth curves for individual S. pombe cells after Mitchison (1957). Heavy lines are growth curves for: (i) a cell of average birth length and division length (abc); (ii) a cell with short birth length and average division length (xabc); (iii) a cell with average birth length and long division length (abyz). Light lines are straight lines representing the mean extension rates of type (i) cells (ac), type (ii) cells (xac), and type (iii) cells (as). The discrepancy between the relations birth length/extension rate and division length/extension rate may be explicable as a consequence of the shape of the growth curve of individual cells. As described by several authors (Mitchison, 1957; Streiblova & Wolf, 1972; Johnson, 1965), growth is not constant through the cycle; the growth rate increases from birth for about three quarters of the cycle, then growth stops during a constant volume period. An idealized growth curve for a cell of average birth

9 Control of cell size and cycle time 59 length and division length is shown in Fig. 5 by line abc. The mean extension rate determined in this study is the slope of the straight line ac. The duration of the constant volume stage does not vary much with the length of the cell at that stage (Fantes, unpublished results; James et al. 1975), and for clarity will here be considered independent of cell length. Furthermore, the shape of the growth curve will be taken as identical for cells of all lengths. (These assumptions are not critical to the argument which follows, but merely modify the predicted relationships slightly.) Consider a cell E a. Cycle time, h Fig. 6. Abscissa, cycle time, h; ordinate, length extension during the cycle, /im. Experimental details as for Fig. 3 : mean extension ± standard error is shown for the various cycle time classes. which is shorter than average at birth, which starts its cycle at point x. Such a cell has a mean extension rate given by the slope of line xc, which passes very close to point a, and its mean extension rate is therefore very close to that of an average cell. The same argument applies to a cell larger than average at birth, and therefore cell length at birth has little effect on mean extension rate. Now consider a cell of average birth length which is longer than average at division, represented by the growth curve abyz. The mean extension rate of such a cell is the slope of line az, which is greater than that of ac. A similar argument applies to cells shorter than average at division, and thus mean extension rate is positively correlated with division length. Fig. 6 shows that the amount of extension during the cycle is strongly positively correlated with cycle time, and that therefore the inverse relation between birth size and extension of Fig. 3 should be reflected in a similar inverse relation between birth

10 6o P. A. Fantes size and cycle time, that is, the longer the cell, the shorter the cycle time. This is confirmed by the results in Fig. 7, at least for cells shorter than 10-5/tm at birth, which includes the whole of the normal length range. Cells longer than this show no such relationship: their cycle time is about 2-3 h and varies little with birth size. This period of 2-3 h presumably reflects the time which the cell needs in order to complete the metabolic and related processes necessary for cell division, and this requirement overrides the normal control mechanism for division of the cell. The breakdown of control is also reflected in the abnormal behaviour of very long cells shown in Figs. 2, 3. Cells \, E 3 (J 10 Cell length at birth, Fig. 7. Abscissa, cell length at birth, fim; ordinate, cycle time, h. Experimental details as for Fig. 3. which at the start of a cycle are very long must undergo a minimum cycle and therefore must extend by a certain minimum amount in that cycle; in general such cells will still be longer than normal at the subsequent division. These cells therefore take more than a single cycle to return to steady state. The length of the minimum cycle time appears to be independent of growth rate, as a similar experiment done with cells growing on minimal medium (doubling time 4-3 h) also showed that very long cells undergo a minimum cycle of between 2 and 2-5 h. Cycle time homeostasis The relationships described in the previous section suggest a mechanism by which constancy of cycle time might be maintained in steady-state cultures. Consider a cell of average length at birth. If, because of random variation, the cell undergoes a cycle longer than average, then from Fig. 6 its length at division will be above average.

11 Control of cell size and cycle time 61 When it divides, its daughters will be longer than average, and such cells have shorter than average cycles (Fig. 7). Thus a long interdivision time in one cycle should result in the next cycle being shorter than average. To test this prediction directly, the durations of the first and second cycles in the steady-state experiment were plotted against one another (Fig. 8). There is a statistically significant negative correlation between successive cycle times (r = 0-16: P < 0-02), but even so the relation is not as strong as might have been expected. The reason for this is not clear; probably there is an underlying relationship but this is masked by random variation. o c o e Duration of cycle n, h Fig. 8. Abscissa, duration of cycle n, h; ordinate, duration of cycle n + 1, h. Results for n = 1 shown for 972 ( ) in steady state at 25 C; results for n = 1 and n = 2 shown for cdc2-33 ( ) at 25 C, with a pulse of 1-3 h at 35 C during cycle 1. By increasing the experimental variation relative to the random variation, it should be possible directly to demonstrate cycle time homeostasis, that is, longer cycles should be followed by shorter compensatory ones. An asynchronous population of strain cdcz-yy growing at 25 C was pulsed for 1-3 h at 35 C under the microscope. The effect on cell number is shown in Fig. 9: about 0-7 h from the start of the pulse cell division stops, and cell number remains almost constant until 1-3 h after the end of the pulse. After this, cell number rapidly increases as the cells recover from the nuclear division block, then the rate of division falls to a value which is still faster than in the steady state, and finally returns to the original rate. The duration of the delayed cycle is plotted against the duration of the following cycle in Fig. 8. The mean durations of the 2 cycles are 4-00 and 2*52 h respectively, and the value for the third cycle is 3-37 h, near to the steady-state value of 3-07 h. Fig. 8 shows that the greater the delay in one cycle, the more advanced is the division in the following cycle.

12 62 P. A. Fantes u I I 6 8 Time, h Fig. 9. Abscissa, time, h; ordinate, cell number in the microscope field. Increase in a population of strain cd.c2.-22 growing at 25 C: a i'3-h pulse at 35 C was applied at the time indicated. DISCUSSION Relevance to other systems The main conclusion to be drawn from the experiments described here is that both cell size at division and cycle time show homeostasis, and that deviations from the population mean value are corrected within a small number of cycles, in most cases within a single cycle. Elements of these relationships have been reported in other systems; Prescott (1956) found a negative correlation between birth weight and generation time in Amoeba proteus, and a similar relation was found for Tetrahymena pyriformis by Lovlie (1963). A negative correlation between cycle time and cell size at mitosis in human lymphoid cells has been reported recently by Yen et al. (1975), but this result is not analogous to any obtained in this study, as the relation found was between size at mitosis and the length of the preceding cycle. It should be pointed out that in some studies on mammalian cells, no effect of cell size on cycle duration was observed (Fox & Pardee, 1970; Fournier & Pardee, 1975). Studies on Physarum polycephalum (Sudbery & Grant, 1975; Devi, Guttes & Guttes, 1968; Sachsenmaier, Donges, Rupff & Czihak, 1970) have shown that cycle time is under homeostatic control, and that deviations are corrected within a single cycle. Long delays to mitosis are followed by a minimum cycle equal to about three quarters of the normal cycle on rich medium, and the length of the minimum cycle is independent of growth rate (Sudbery & Grant, 1975). Although size is not such a

13 Control of cell size and cycle time 63 familiar concept in Physarum, the amount of cytoplasm per nucleus is analogous, and also appears to be controlled (Sudbery & Grant, 1975). A detailed analysis of the normal S. pombe cell cycle has been made by James and co-workers (1975), and although the parameters determined were slightly^ different from those measured in this study, the basic conclusion that cells which are larger than average at birth have shorter than average cycles is common to both studies. These authors found that growth rate was negatively correlated with cell size at birth, though the correlation was weak, whereas I found no such correlation. Their estimate of growth rate did not include the period at the constant volume stage, whereas the mean extension rate referred to here does include this period. However, it can be seen from Fig. 5 that the exclusion of this period should tend to decrease any negative correlation between birth size and growth rate: the slopes of lines ac and xc are equal (includes constant volume stage), but the slope of the straight line ab is greater than that of xb (excludes constant volume stage). Thus exclusion of the constant volume stage in this case should produce a. positive correlation between birth size and growth rate. Therefore differences in the method of estimating growth rate are not responsible for the discrepancy between the 2 series of observations, and this discrepancy must remain unexplained. Possible control models Cell size and cell age are linked through growth: long cells are old cells, and length at division is positively correlated with cycle time. It is therefore possible that the primary control acts either on cycle time or on cell size, and accordingly 2 different types of model can be proposed (Fantes et al. 1975). A simple clock or timer model in which there is a constant time interval between some early event in the cycle and the next division can maintain long-term constancy of cell size only if the mass doubling time is exactly equal to the cell cycle time. Even if this were the case, a cell which because of random drift was too large or too small would produce daughters which were also abnormal in size, and thus cell size would be heritable, contrary to observation in S. pombe. In any case, such a model cannot explain the correction within one cycle of deviations from the mean. These arguments apply to models where only time is considered, and not size, such as the limit cycle theory of Kauffman & Wille (1975), and the transition probability theory of Smith & Martin (1974). There are several modifications which can be made to timer models in order to introduce an element of size control. One could propose that the rate of progress of the timer is proportional to growth rate, which means that the parameter measured by the cell is amount of growth since the last division rather than time, and this will predict long-term constancy of cell size. This is because a cell which is too large at one division will transmit half its extra size to its daughters, and after the constant amount, of growth required for division, these daughters will divide at a size halfway between the mother division size and the population mean value (assuming constancy of growth rate). This model predicts that every cell should extend by a constant amount irrespective of initial size, between successive divisions. The plot of extension against

14 64 P. A. Fantes birth length in Fig. 3 is in disagreement with this prediction, and the model may therefore be ruled out. The simplest interpretation, though not necessarily the only possible one, of the control operating within a single cycle, is that cells have a means of monitoring their absolute size, and that division, or commitment to divide, can occur when a critical size has been attained. The primary control would then act on this event, which might be followed by a fairly constant period of time up to division. A cell which by random variation divided a little late would be larger than average at division, and its daughters would have to grow by a reduced amount before attaining the critical size. This would take a shorter time than normal. Thus cycle time would be controlled indirectly by growth and the existence of a size control. In this context it is interesting that while size homeostasis is easy to demonstrate in steady-state cells, the negative correlation between successive cycle times is weak, suggesting that size, the event proposed as primary control, is more precisely regulated than cycle time. Using a different experimental approach, other results from this laboratory (Nurse, 1975; Fantes & Nurse, in preparation) point strongly to the existence of a size control acting at early nuclear division in 5. pombe. Minimum cycle times Cells longer than about 10-5 /am at birth do not show the expected homeostatic control, but instead, undergo a cycle of mean duration 2-3 h, both on minimal and complete medium, where the mass doubling times are different (4-3 and 3-1 h respectively). The period of 2-3 h is presumably the minimum time in which the cell can carry out all the processes necessary for division. Thus the cycle may include an incompressible period: this is probably largely G 2, as DNA is normally replicated early in the cycle. In this context it is of interest that of cell cycle mutants arresting at nuclear division, some have transition points as early as 0-4 (Nurse et al. 1976). It may be that this early function is concerned with events in the incompressible period. Size control and transition probability models Smith & Martin (1974) have proposed a transition probability model of the cell cycle. In this model, the first part of the cycle, including most of G lt consists of the A phase, which is of indeterminate length. Each cell in the A phase has a constant probability (P) of leaving this phase in any time interval, when it enters the B phase. The duration of the B phase (T 6 ) is essentially constant, ending in mitosis. This model predicts a particular distribution of cycle times, and no correlation between cell size and cycle time, or between successive cycle times. Although the model appears unnecessary to explain my observations on 5. pombe, and is in fact in conflict with certain aspects, it is of interest to see whether some of its predictions hold true. Fig. 10 shows the distribution of interdivision times of steady-state cells on the a plot used by Smith & Martin; the proportion of undivided cells is plotted against cell age. The length of the B phase (T b ) can be estimated at 2-5 h, and the transition probability (P) at 0-95 h" 1, although these estimates are based on only a proportion of the data, as the linear part of the curve only fits the points for the half of the cells whose cycle times

15 Control of cell size and cycle time 65 are greater than 2-7 h. Sister cell cycle times were positively correlated (r = 0-39; P<o-i %), and a /? plot (Smith & Martin; 1974) was constructed (Fig. 10). This shows the cumulative difference between the cycle times of sister cells. The slope of the straight part of the plot is 0-97 h" 1, close to the estimate of P from the a plot, which is consistent with the transition probability model. The straight line section of the /? plot does not include the first (100%) point: this may be due to statistical bias as rather few classes are present, but if not, the data become inconsistent with the transition probability model I Time, h Fig. 10. Abscissa, cell age, h for a; age difference of sister cells at division for /9; ordinate, a (% cells at the age specified which are still undivided, A), and /? (% of sister cell pairs whose ages at division differ by at least the time specified, A). The population considered is the second cycle of the steady-state culture. 5 CEL 24

16 66 P. A. Fantes Thus, if the whole population is considered and no account taken of cell size, my results are consistent with the transition probability model. However, the model in its simplest form has no way of explaining the relation between cycle time and birth size or the fact that deviations are corrected within a single cycle. It may be possible to modify the model by inserting a size control, which must be outside the indeterminate A phase in order to maintain the random element central to the model. Insertion of a size control before nuclear division into the transition probability model makes the model consistent with the results presented here and with further results from this laboratory (Nurse, 1975; Fantes & Nurse, unpublished). Another possibility is that re-entry into the A phase after nuclear division might be dependent on the attainment of a critical cell size. Thus the existence of a size control and the occurrence of random transitions can be reconciled at least in principle. Whether such an integration is necessary is much less clear, as the results presented here are capable of interpretation without proposing any random events. I would like to thark Dr P. Nurse and Professor J. M. Mitchison for helpful discussion during the course of this work and for critical reading of the manuscript. I was in receipt of a Research Fellowship from the Royal Commission for the Exhibition of 1851 during the period of this work. REFERENCES ANDERSON, E. C, BELL, G. I., PETERSEN, D. F. & TOBEY, R. A. (1969). Cell growth and division. IV. Determination of volume growth rate and division probability. Biophys. J. 9, DAWSON, K. B., MADOC-JONES, H. & FrELD, E. O. (1965). Variations in the generation times of a strain of rat sarcoma cells in culture. Expl Cell Res. 38, DEVI, V. R., GUTTES, E. & GUTTES, S. (1968). Effects of ultraviolet light on mitosis in Physarum polycephalum. Expl Cell Res. 50, FANTES, P. A., GRANT, W. D., PRITCHARD, R. H., SUDBERY, P. E. & WHEALS, A. E. (1975). The regulation of cell size and the control of mitosis. J. theor. Biol. 50, FOURNIER, R. E. & PARDEE, A. B. (1975). Cell cycle studies of mononucleate and Cytochalasin- B-induced binucleate fibroblasts. Proc. natn. Acad. Sci. U.S.A. 72, Fox, T. O. & PARDEE, A. B. (1970). Animal cells: noncorrelation of length of G 1 phase with size after mitosis. Science, N. Y. 167, FROESE, G. (1964). The distribution and interdependence of generation times of HeLa cells. Expl Cell Res. 35, GUTZ, H., HESLOT, H., LEUPOLD, U. & LOPRIENO, N. (1974). Schizosaccharomyces pombe. In Handbook of Genetics, vol. 1 (ed. R. C. King), pp New York: Plenum Press. HERRING, A. (1974). A Study of Induced Delay in the Division of the Yeast Schizosaccharomyces pombe. Ph.D. Thesis, University of Edinburgh. JAMES, T. W., HEMOND, P., CZER, G. & BOHMAN, R. (1975). Parametric analysis of volume distributions of Schizosaccharomyces pombe and other cells. Expl Cell Res. 94, JOHNSON, B. F. (1965). Autoradiographic analysis of regional wall growth of yeast, Schizosaccharomyces pombe. Expl Cell Res. 39, JOHNSON, B. F. & Lu, C. (1975). Morphometric analysis of yeast cells. IV. Increase of the cylindrical diameter of Schizosaccharomyces pombe during the cell cycle. Expl Cell Res. 95, IS4-IS8. KAUFFMAN, S. & WILLE, J. J. (1975). The mitotic oscillator in Physarum polycephalum. J. theor. Biol. 55,

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