## ABSTRACT

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, 19780).

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 (1978*a, 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.

## 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, 1978*a*) and for 3 diploid strains: wild type and *wee* 1—50 homozygous diploids and the heterozygous *wee* 1–50/wild type diploid (Fraser & Nurse, 1978*b*). The 3 diploid and 2 haploid strains together cover a size range from approximately 0· 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 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.

### Derivation of an expression for the average poly (A)^{+}mRNA content of a mutant small cell compared to wild type

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 f(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 Z 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 validforany pattern of increase during the cell cycle which is the same for wild type and *wee* 1–50.

*R’*and be the amounts of poly(A)+mRNA in synchronously dividing populations of wild type and

*wee*1–50 cells;

*R’*and depend on time

*t*, which is measured in cell cycle units (one cycle or mean generation time = 1 ·0 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: where λ 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; f(

*t*—

*l*) is the rate in

*wee*1–50 cells (Fig. 1).

*wee*1–50 cells, with the same numbers of cells per ml, and growing with the same mean generation time (Nurse, 1975).

*R*and

*R*

_{m}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 α to α + δ α is given by at all times. From Fig. 1 it is clear that the rates of synthesis of poly(A)+mRNA for cells aged α are given by f(α) and f(α —

*l*) for wild type and

*wee*1–50 respectively. By 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 As the number of cells in each population is increasing exponentially, the total number of cells may be written as

*N = N*

_{o}

*e*

^{kt}for both populations, where

*N*

_{o}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.

*b)*may be written as and since f is a periodic function integrated over a whole period, (6) may be rewritten as Therefore, writing equations (5) may be rewritten as 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. where

*q*is a constant.

*wee*1–50 containing the same numbers of cells and growing at the same exponential rate. The delay, of a fraction

*I*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

*l*, 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

*t —I*. In the intervening interval both populations would have increased in cell number by a factor

*2*

^{1}. Thus the average amount of poly(A)+mRNA per cell in the

*wee*1–50 population at any time will be

*2*

^{− 1}of that in the average cell of the wild type population.

Using this expression, and the values for *I* measured in synchronous cultures for the haploid and diploid series of *S*.*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(AfimRNA synthesis in small mutant cells keeps the ratio of the average poly(AfimRNA 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, 1978*b*). 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.

_{m}(α) be the value of the growth parameter (be it protein, mass or volume) of cells aged α, of wild type and small mutants respectively. Both

*V*(α) and V

_{m}(α) increase exponentially with the same specific rate of increase. This rate being such that the cell sizes at α = 1 must be twice the sizes at α = 0, we may write: and Let the cell age at the mid-point of the doubling in rate of poly(A)+mRNA synthesis be α

_{1}in wild type cells and α

_{2}in small mutant cells; so that α

_{2}=

*α*

_{1}

*—l*(Fig. 1). The property of cell size parameters assumed for the proof implies that Substituting α

_{1}in (11α) and α

_{2}in

*(*11

*b)*and making use of (12) it follows that Using (13) to eliminate U(0) and

*V*

_{m}(0) from (11) leads to In asynchronous, exponentially growing populations, cell age distribution remains 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

^{2}larger than in small mutant populations. From equation (10) the average amount of poly(A)+mRNA in wild type cells is also 2

^{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 size-related 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 synthesis in each cell cycle

The 3 diploid strains of different mean sizes all double their rate of poly(A)+mRNA synthesis at points in their cell cycles when they have very similar protein contents per cell (Fraser & Nurse, 1978 6). 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 of 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 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 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)+mRNA 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).

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.

## 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 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.

## ACKNOWLEDGEMENTS

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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