THEORETICAL ANALYSIS OF A METHOD FOR DETERMINING THE ... · macromolecular synthesis and...

J. Cell Sd. 62, 187-207 (1983) 187Printed in Great Britain © The Company of Biologists Limited 1983

THEORETICAL ANALYSIS OF A METHOD FOR

DETERMINING THE PATTERN OF

MACROMOLECULAR SYNTHESIS DURING THE CELL

CYCLE

R. S. S. FRASER AND A. BARNES*Biochemistry and Statistics Sections, National Vegetable Research Station,Wellesbourne, Warwick CV35 9EF, U.K.

SUMMARY

The dual-labelling centrifugal-elutriation method has been extensively used to study patterns ofmacromolecular synthesis and accumulation in the yeast cell cycle. Cells are long-term labelled witha radioactive precursor for about 1 • 5 cycles (a measure of macromolecular mass), then pulse-labelledwith a precursor containing a different radioactive isotope for the final CM cycle (a measure of rateof synthesis). Harvested cells are fractionated into cell cycle stages by centrifugal elutriation, andchanges in pulse: long-term labelling during the cycle determined. This pattern of change is com-pared with theoretical changes in rate: mass calculated for various patterns of synthesis. Using thismethod, it has been suggested that rates of synthesis and accumulation of several types of RNA andprotein all increase exponentially through the cycle.

In contrast, experiments using synchronous cultures or zonal centrifugation for cell cycle analysishave suggested other synthetic patterns, including periodic doubling in rate in each cycle. In thispaper we analyse whether the dual-labelling, centrifugal-elutriation method is capable ofdiscriminating between exponential and periodic rate-doubling patterns. Three possible sources ofimprecision in the method and its application are examined.

(1) Theoretical rate: mass curves have been simulated for macromolecules with a wider range ofproperties than previously considered. It is shown that for some important classes, such as messen-ger RNAs, with turnover rates in the range measured experimentally, and proteins, differencesbetween rate: mass curves for exponential and periodic rate-doubling models are considerablysmaller than previously suggested.

(2) Long-term labelling is shown to be an accurate measure of macromolecular mass, but pulse-labelling can be inadequate as a measure of rate of synthesis. The error is greater with RNAs withfaster turnover, and again reduces the ability of the method to discriminate exponential and periodicrate-doubling models of synthesis.

(3) Imperfections in cell cycle fractionation by centrifugal elutriation are examined, and bycomputer simulation it is shown that these also reduce the ability of the method to distinguishbetween the two models of synthesis.

The three sources of imprecision are cumulative. It is concluded from simulation analyses thatthe differences between exponential and periodic rate-doubling patterns analysed by the methodwould be so small as to be almost impossible to establish against the background of error in experi-mental measurement. We therefore suggest that in practice, the dual-labelling centrifugal-elutriation method is unable to discriminate between the exponentially increasing and periodic rate-doubling models.

The mathematical treatment developed in this paper should be applicable to analysis of othermethods and cell cycle events.

•Present address: Statistical Adviser, Faculty of Science, University of Queensland, St Lucia,Queensland, Australia 4067.

188 R. S. S. Fraser and A. Barnes

INTRODUCTION

Changes in rates of macromolecular synthesis during the cell cycle have beenmeasured to gain an understanding of the mechanisms involved in control and co-ordination of cell growth. Such studies generally involve either preparation of syn-chronously dividing cultures (Mitchison & Vincent, 1965) or fractionation of anasynchronous, exponentially growing population into different cell cycle stages bytechniques such as zonal centrifugation (Carter, Sebastian & Halvorson, 1971) orcentrifugal elutriation (Gordon & Elliot, 1977).

In attempting to determine the pattern of synthesis of a particular macromoleculeduring the cycle, it is essential to establish whether the method of cell cycle analysisused is in fact capable of discriminating between different patterns of synthesis.However, with a few exceptions (e.g., see Mitchison & Creanor, 1969), such testshave not been performed. Our objective in this paper is to take one widely usedmethod of cell cycle analysis, and study theoretically whether it is capable of givingunequivocal information about synthetic patterns.

The method to be analysed involves three experimental stages. An asynchronous,exponentially growing culture is labelled for about 1-5 times the mean duration of thecell cycle with a radioactive precursor of the macromolecule under examination (the'long-term' label). It is then 'pulse-labelled' for a final short period, generally about0-1 cycle, with a precursor containing a different radioactive isotope. The cells arechilled to stop further synthesis, and fractionated by centrifugal elutriation. Thisseparates on the basis of cell size and thus approximately by position in the cell cycle.The radioactive macromolecule is extracted and purified, and radioactivities incor-porated in the long-term and pulse-labellings determined by dual isotope counting.Pulse-labelling is taken as a measure of rate of synthesis, and long-term labelling torepresent the mass of the macromolecule. This allows calculation of a 'rate': 'mass'value free from errors arising from incomplete recovery of the macromolecule duringpurification.

It is possible to calculate changes in theoretical rate: mass values during the cellcycle for different cell cycle patterns of synthesis and accumulation, and to comparethese with the experimental data.

Using this and related types of cell cycle analysis with the budding yeast Sac-charomyces cerevisiae, it has been suggested that while DNA synthesis is periodic(Elliot & McLaughlin, 1978), rates of synthesis and accumulation of ribosomal RNA(rRNA) (Sogin, Carter & Halvorson, 1974; Elliot & McLaughlin, 1979a),polyadenylated messenger RNA (poly(A) mRNA) and transfer RNA (tRNA) (Elliot& McLaughlin, 1979a), ribosomal proteins (Elliot, Warner & McLaughlin, 1979) anda large number of other proteins (Elliot & McLaughlin, 1978, 19796), all increaseexponentially during the cell cycle.

In contrast, in experiments using different methods of cell cycle analysis withbudding yeast and the fission yeast Schizosaccharomyces pombe, others have sugges-ted that rates of poly(A) mRNA and rRNA synthesis may double in a periodic 'step'once in each cycle (Fraser & Carter, 1976; Fraser & Moreno, 1976; Fraser & Nurse,

Cell cycle analysis 189

1978). It has also been suggested that several individual enzymes in S. pombe mayaccumulate at a linear rate that doubles at a particular stage in each cell cycle (Mitch-ison & Creanor, 1969) and that the rate of synthesis of ribosomal proteins increasesperiodically during the cell cycle (Wain & Staatz, 1973). The rate of total proteinsynthesis has been reported to increase discontinuously in the cell cycles of fissionyeast (Creanor & Mitchison, 1982) and budding yeast (Gull^v, Friis & Bonven,1981).

In this paper we analyse whether the dual-labelling centrifugal-elutriation methodof cell cycle analysis is capable of distinguishing between exponential and discon-tinuous patterns of increase in the rate of macromolecular synthesis during the cellcycle. Our analysis involves further simulations of theoretical rate: mass changes forvarious patterns of synthesis. We then examine the extent to which the pulse and long-term labellings are adequate measures of rate and mass, respectively, and the extentto which imperfections in cell cycle fractionation by centrifugal elutriation may maskpredicted changes in rate: mass values.

ANALYSIS

Simulation of rate: mass patterns for different cell cycle patterns of macromolecularsynthesis

Elliot & McLaughlin (1978) simulated theoretical rate: mass curves for three pat-terns of macromolecular synthesis: for completeness these are shown in Fig. 1. Wheresynthetic rate and mass increase exponentially throughout the cycle, rate: mass isconstant (Fig. 1A). With periodic synthesis, as that of DNA or the putative 'step'enzymes (Halvorson, Carter & Tauro, 1971), rate: mass is zero for much of the cycle,but at the start of the synthetic period it rises sharply, then falls gradually beforereturning to zero at the end of the period of synthesis (Fig. 1B). For models where thesynthesis is continuous through the cell cycle, but doubles in rate at a discrete pointin each cycle, rate: mass falls during the cycle; doubles instantaneously when syn-thetic rate doubles; then decreases over the remainder of the cycle (Fig. lc).

In trying to distinguish between different patterns, we are interested not only in theshape of the rate: mass curve during the cell cycle, but in the extent of fluctuationabout the mean value. Elliot & McLaughlin (1978) calculated a 'percent standarddeviation' (%S.D.) [(standard deviation/mean) X 100%], as a measure of this fluc-tuation. For an 'ideal' cell population, and in the absence of experimental error inmeasurements, the%s.D. is 0 for the exponential model; for the rate-doubling modelit is 22, and for periodic synthesis it is very much higher, the actual value dependingon the finite cell cycle time occupied by the synthetic period. To allow comparisons,we have computed similar % S.D. values for our simulations, but will later commenton the suitability of this measure of fluctuation.

Elliot & McLaughlin (1978) compared experimentally measured pulse: long-termlabelling patterns, and % S . D . values, with these theoretical models, for synthesis ofa variety of macromolecules including rRNA, total poly(A) mRNA and individual


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Cell cycle

0-8 10

Fig. 1. Rate: mass curves for different patterns of macromolecular synthesis during thecell cycle, A. Exponential increase and accumulation, B. Periodic synthesis between (Hand 0-4 of the cycle, with step accumulation of a stable product, c. Instantaneous doublingin rate of synthesis at 0-3 of the cycle.

proteins. However, the patterns shown in Fig. 1 do not adequately represent severalmodels of biological interest, including mRNA and individual proteins.

Fig. 2 shows the effects of RNA half-life on theoretical rate: mass curves throughthe cell cycle. For all these curves, it is assumed that synthetic rate doubles instan-taneously at a fixed point in the cycle, and that cell cycle fractionation is perfect. Themathematical derivation of the curves is described in Appendix IA.

Very stable RNA, such as rRNA, shows a rate: mass curve (Fig. 2) conforming tothe Elliot & McLaughlin (1978) model (Fig. lc). However, with unstable RNAs,such as mRNAs, with half-lives a fraction of the cell cycle, the shape of the rate: masscurve changes progressively with decreasing half-life, and the %s.D. drops. Therange of mRNA half-lives chosen for the calculations in Fig. 2 includes the actualrange measured in budding yeast by Chia & McLaughlin (1979).

Altered RNA turnover rate has no effect on the calculated rate: mass pattern in the


0-4 0-6 0-8 10

Cell cycle

Fig. 2. Simulated rate: mass curves for mRNAs with various half-life times (tj), withan instantaneous doubling in rate of synthesis at 0-3 of the cycle. ( • ) t\ = 10 (20-4%);(O) * i=l -0 (20-5%); (A) *j = 0-3 (20-3%); ( • ) <j = 0-2 (19-8%); (A) <j = 0-05(14-9%). Half-lives are measured in fractions of the cell cycle. Figures in parenthesesare the percentage standard deviations, computed on the basis of 100 points per

exponential model: the ratio remains constant at all half-lives. This may beestablished by the proof in Appendix IB. However, it is clear that with the rate-doubling model, the rate: mass pattern and % S.D. values come closer to those of theexponential model with faster RNA turnover. Thus this method of cell cycle analysisis less able to discriminate between exponential and rate-doubling models for RNAswith half-lives similar to those measured experimentally for poly(A) mRNA.

For measurements of total messenger RNA (Elliot &McLaughlin, 1979a), it seemsunreasonable to limit the rate-doubling model by assuming that the rate of synthesisof all individual messenger species doubles synchronously and instantaneously at afixed point in the cell cycle. Another model has been proposed in which the rate ofmRNA synthesis is dependent on DNA replication (Fraser & Carter, 1976). A simplemodel to encompass this would be for the period during which rate of total mRNAsynthesis doubles to reflect the duration of DNA replication. Derivation of rate: massratios is described in Appendix IC. Where DNA replication (Williamson, 1965) istaken to occupy 0-3 of the cycle, the simulated rate: mass curves for RNA (Fig. 3A)are smoother, and with smaller % S.D. values, than for instantaneous doubling (Fig.2). Fig. 3A also shows that the shorter the half-life of the RNA, the closer the rate:mass pattern is to the exponential model.

Calculation of rate: mass curves for synthesis of individual proteins is complicatedby the presence of an intermediate step involving mRNA; for the rate-doubling casethe model and calculations of Elliot & McLaughlin (1978) (Fig. lc) cannot be applied


directly. Consider the case of a protein synthesized by translation of an mRNA, thesynthesis of which doubles instantaneously at a fixed point in the cycle. It is assumedthat the protein is stable, and that the rate of protein synthesis at any time dependson the concentration of its mRNA at that instant. Fig. 3B shows simulated rate: masscurves for proteins synthesized on messengers having various half-lives. The deriva-tion of these curves is described in Appendix ID. It is clear that the longer the mRNAhalf-life, the less fluctuation there is in the rate: mass curve. For mRNAs with half-lives similar to those measured experimentally (Chia & McLaughlin, 1979), the% S . D . of all protein rate: mass curves was considerably lower than the 22 predictedby the basic Elliot & McLaughlin (1978) model, and at longer half-lives was closer tothe 0 value of the exponential model.

In summary, these theoretical considerations show that for important classes ofmacromolecule, such as mRNA and proteins, the differences between rate: masspatterns for rate-doubling and exponential models of synthesis are very much less thanoriginally calculated and applied (Elliot & McLaughlin, 1978, 1979a; Elliot et al.

0-4 0-6

Cell cycle

0-8 10

Fig. 3. A. Simulated rate: mass curves for mRNAs with various half-life times, with alinear doubling in rate of synthesis occurring between 0-1 and 0-4 of the cycle (shown bythe bar). The half-lives, and in parentheses the %S.D. values, are: ( • ) t\= 10 (14-0%);(O) / 4 = 1-0 (13-8%); (A) i» = 0-3 (12-9%); (D) tt = 0-2 (12-1%); (A) ft = 005(6-1 %). B. Simulated rate: mass for a stable protein, synthesized by translation of mRNAswith various half-lives, which double their rate of synthesis instantaneously at 0-3 of thecycle. The mRNA half-lives, and in parentheses the % S.D. values of the protein rate: masscurves, are: ( • ) /j = 10 (2-0%); ( • ) tt = 0-5 (5-1 %) ; (A) tt = 0-3 (7-0%); (V) n = 0-1(13-0%). All %s.D. values were computed on the basis of 100 points per curve.


1979). This inevitably diminishes the ability of the dual-label centrifugal-elutriationmethod to discriminate between different models.

Adequacy of the pulse and long-term labellings as measures of synthetic rate and mass

In the preceding section, synthetic rate and mass were the values of these parametersat the instant of harvest. We have also calculated the effects of using pulse-labelling

0 0-2 0-4 06 0-8Cell cycle

Fig. 4. The effects of using pulse-labelling (0-1 cycle) as a measure of rate of synthesis,and long-term labelling (1-5 cycle) as a measure of mass. The simulated curves are formRNAs doubling their rates of synthesis instantaneously at 0-3 of the cycle, and with half-life times of: A, 10 cycles; B, 0-3 cycles; and c, 0-05 cycles. (•) True rate: mass; (A)pulse-labelling: mass; and ( • ) pulse-labelling: long-term labelling. The%s.D. values foreach curve, computed on the basis of 100 points per curve, are: A ( D ) 2 0 - 4 % , (A) 17-5%,(A) 17-2%; B ( D ) 20-3%, (A) 14-7%, (A) 14-6%; and c (•) 14-9%, (A) 3-7%, (A)3-7%.

10


as a measure of rate, and long-term labelling as a measure of mass. Appendix IIdescribes the general mathematical derivation. For brevity, we consider here only thecase of RNAs with different turnover rates, with true synthetic rate doubling instan-taneously once per cycle. The general conclusions apply to other models of discon-tinuously increasing synthesis.

For a highly stable RNA, the pulse-labelling: mass curve shows some smoothing incomparison with the true rate: mass curve (Fig. 4A). The pulse: long-term labellingcurve is very similar to the pulse-labelling: mass curve.

For more unstable RNAs, with half-lives of 0-2 (Fig. 4B) or 0-05 cycle (Fig. 4c) thepulse-labelling: mass curves show considerable smoothing in comparison with thetrue rate: mass curves. Again the pulse: long-term labelling curves are very similar tothe pulse-labelling: mass curves. The extent of the smoothing in each case is measuredby the % S.D. for each curve, given in the legend to Fig. 4.

These calculations show that long-term labelling does give a good representationof mass. In contrast, pulse-labelling can be an inadequate measure of synthetic rate.The discrepancy between true rate and 'rate' measured by pulse-labelling increasesmarkedly with decreasing RNA half-life.

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Fig. 5. A. Relative numbers, and proportions of the cell cycle occupied by four micro-scopically recognizable stages of the yeast cell cycle. U, unbudded; B, budded; ND,nuclear division; P, binucleate pairs. The histogram shows data derived from Fig. 1 ofElliot & McLaughlin (1978); ( ) the canonical cell age frequency distribution foran exponentially growing culture; and (V — V) the calculated median cell age in eachfraction. B. Cell cycle fractionation by centrifugal elutriation. Data recalculated from Fig.1 of Elliot & McLaughlin (1978). ( • ) Total cell number in each flow-rate fraction; ( • )unbudded cells; (A) budded cells; (O) nuclear division; ( • ) binucleate pairs. ( V - - - V )The mean cell age in each fraction.


We conclude that for mRNAs with half-lives within the range measured experi-mentally (Chia & McLaughlin, 1979), and presumed to double in rate of synthesis ata fixed time in each cycle, the inadequacy of pulse-labelling as a measure of rate wouldtend to distort the 'rate': mass curve towards the exponential form. In contrast, theinadequacy of pulse-labelling as a measure of true rate does not affect the calculatedconstant rate: mass pattern of the exponential model. Therefore, this source of errorwill tend to decrease the ability of the dual-label method to discriminate the rate-doubling and exponential increase models.

Effects of imperfections in cell cycle fractionation

For this analysis, we used the experimental cell cycle fractionation data in Fig. 1of Elliot & McLaughlin (1978). For fractions taken from the elutriating rotor atdifferent flow rates, the authors present total cell number, and the percentages of cells

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in each of four microscopically recognizable cell cycle stages: unbudded (U); budded(B); cells with migrating nuclei (nuclear division, ND); and binucleate pairs of cells(P). From the total number of each stage recovered, the total of all cells recovered,and the canonical cell age frequency distribution (Cook & James, 1964), the number,and proportion of the cell cycle occupied by each stage may be calculated (Fig. 5A).This would be the distribution of cells recovered from the elutriating rotor if a perfectcell cycle fractionation had occurred.

Fig. 5B shows the real data, recalculated from Elliot & McLaughlin (1978). It isclear that a cell cycle fractionation has occurred, in that early cycle stages arerecovered preferentially at low flow rates, and later stages at high flow rates. Thefractionation at the beginning of the cycle is better than at the end. There are twomajor types of imperfection. First, it is clear that fractionation of different cell cyclestages is not complete, especially at the end of the cycle. Thus in the 23 ml/minfraction, the number of unbudded cells (presumed to be close to the start of the cycle)actually exceeds the numbers of cells in nuclear division or having completed nucleardivision. Secondly, and as a consequence of the first point, the rotor fractions on aflow-rate basis do not represent a complete fractionation of cell cycle time, as shownby the calculated curve for mean cell age in each fraction (Fig. 5B). These datatherefore imply that cell cycle fractionation by elutriation is less than perfect, asituation partly analogous to imperfect synchrony in a synchronized culture.

Exponentially growing cultures of S. cerevisiae do not comply fully with the idealcanonical cell age distribution, and do not show an ideal relationship between cellvolume and cell cycle stage. For example, newly released buds and young mothercells, both at the start of the cell cycle, are of different sizes, and the size of the mothercell increases with the number of buds it has borne (Hartwell & Unger, 1977; Lord& Wheals, 1981). More generally, there is the problem of 'momentary variation' in cellsize for any given point in the cycle (Scherbaum & Rasch, 1957), which increases thetrue range of volumes over the cell cycle from twofold to a higher value. The inverseof this effect is that cells of a fixed size in the population represent a range of ages. Wehave made no attempt in our analysis to allow for these imperfections in the 5.cerevisiae cell cycle; they would undoubtedly have a further smoothing effect on anypattern of rate: mass, additional to the effects of imperfections in experimental cellcycle fractionation.

For a macromolecule whose rate of synthesis increases exponentially through thecell cycle, imperfections in fractionation of the cell cycle would not alter the constantrate: mass during the cycle. Indeed, a method of cycle fractionation that completelyfailed to give any fractionation based on cell age would still give constant rate: massin all fractions.

With other patterns of macromolecular synthesis, such as periodic rate doubling,imperfections in cell cycle fractionation would tend to alter the pattern of rate: mass.To estimate the extent of this effect, we have calculated rate: mass curves for variouspatterns of synthesis, with the imperfections in cycle fractionation that are presentexperimentally. The simulated curves also take account of imperfections in the pulseand long-term labellings as measures of synthetic rate and macromolecular mass.


In Appendix III, an equation is derived that gives the ratio of amounts of pulse andlong-term labelling of macromolecules in a cell population with any given agedistribution. The equation is appropriate to any lengths of pulse and long-term label-ling, to stable molecules or to unstable molecules with first-order decay kinetics. Itcan handle any form of doubling of synthetic rate over the course of one cell cycle.

In order to simulate the expected patterns of fluctuation in pulse to long-termlabelling over different flow-rate fractions recovered from the elutriating rotor, it isnecessary to estimate the cell age distributions of the sub-populations recovered at

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Fig. 6. Relative numbers, and proportions of the cell cycle occupied by four micro-scopically recognizable cell cycle stages, in two representative flow-rate fractions from theelutriating rotor, A. 15mlmin~'; B, 21 mlmin"1. U, unbudded; B, budded; ND, nucleardivision; P, binucleate pairs. Histogram data derived from Fig. 1 of Elliot & McLaughlin(1978). The continuous lines show the smoothed age distributions fitted to the experi-mental data.


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Fig. 7. Comparison of the canonical cell age distribution for an exponentially growingculture ( ) with the whole-population age distribution (O) calculated from thesmoothed age distributions fitted to all flow-rate fractions from the elutriating rotor, asexemplified in Fig. 6, and weighted for contribution to the whole population.

each flow rate. Histograms of age distribution can be plotted, but as there are only fourage classes, and one dominates, it is necessary to attempt some smoothing to constructa more reliable estimate of the population composition. Fig. 6 shows examples ofhistograms and the smoothed curves fitted for two representative flow rates. A partialcheck of the accuracy of the smoothed age distribution can be made by summing therelative numbers of cells at each age, weighted for contribution to the whole popula-tion. This should generate an age distribution close to the (approximately) canonicalage distribution of an exponentially growing population. The agreement shown inFig. 7 is quite satisfactory.

The ratios of pulse to long-term labelling for several models of synthesis have beensimulated. Firstly, and as a further test of the accuracy of the smoothing, we simulateda labelling ratio pattern for DNA, assuming that this is replicated periodically, at aconstant rate, between 0-1 and 0-4 of the cycle (Williamson, 1965). The long-term andpulse-labelling times were those used by Elliot & McLaughlin (1978). Fig. 8A showsthat the simulated pattern was quite close to the experimentally measured pattern(reproduced from the data of Elliot & McLaughlin, 1978), except for the pointrepresenting the lowest flow rate. The low ratio measured experimentally for this flowrate may have reflected a substantial proportion of the population being prematurelyreleased buds, or daughter cells that are known to be smaller than their parents, andrequire a period of growth before reaching the 'start' point in the cycle (Hartwell &Unger, 1977; Lord & Wheals, 1981). Neither of these classes of cells would have


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Fig. 8. Simulated pulse: long-term labelling curves for different cell cycle patterns ofmacromolecular synthesis, for an exponentially growing yeast culture fractionated into cellcycle stages by centrifugal elutriation. Simulated data are shown (O O). A. DNA,synthesized from 0-1-0-4 of the cycle, B. mRNA with t\ of 0-3 cycle, with a rate of synthesisthat doubles linearly between 0-1 and 0-4 of the cycle, c. mRNA with ij of 0-3 cycle, anda rate of synthesis doubling instantaneously at 0-4 of the cycle, D. Stable protein,synthesized by translation of an mRNA with t{ of 0-3 cycle, the rate of synthesis of whichdoubles instantaneously at 0-1 of the cycle. For the simulations, the distributions ofdifferent cell ages over the different flow rates were calculated from Fig. 1 of Elliot &McLaughlin (1978), and the effects on patterns of macromolecular pulse and long-termlabelling computed, as explained in the text and Appendix III . Durations of the pulse andlong-term labellings used in the simulations were the same as those used experimentallyby Elliot & McLaughlin (1978) for their experimental data. ( • • ) Pulse: long-term labelling for DNA measured experimentally by Elliot & McLaughlin (1978). The%s.D. values for each curve, calculated on the basis of eight points per curve, are: B, 5-2;c, 5-9; andD, 2 1 .

started DNA synthesis, but neither is allowed for in our simulation. Finally, it mustbe stressed that the cells in the lowest flow-rate fraction account for only 0-3 % of thetotal population.

We therefore consider the agreement between experimental and simulated labelling


ratios for DNA to be quite satisfactory and an indication that the smoothed agedistribution used is an adequately accurate representation.

Fig. 8 also shows representative simulations of 'rate': 'mass' for various models ofRNA and protein synthesis. For mRNA, with a synthetic rate doubling over 0"3 ofa cycle (Fig. 8B) or instantaneous (Fig. 8c), the curves are very flat, with %S.D.values of less than 6. Curves for RNAs with other half-lives, including stable RNA,are also very flat (not shown). A curve for a stable protein synthesized by translationof an unstable mRNA that has an instantaneously doubling rate of synthesis showseven less deviation from a constant value (Fig. 8D) with a % S.D. of only 2. Overall,the extent of smoothing caused by the additive effects of imperfect cell cycle fractiona-tion and shortcomings of the pulse label as a true measure of rate is shown by com-parison of the families of curves in Figs 3, 4 and 8. Further simulations, similar tothose in Fig. 8, for RNAs with other half-life times, all showed very flat rate: masscurves with small % S . D . values.

Theoretically, it would be possible to attempt to distinguish a predicted pattern oflabelling ratio with such a small fluctuation (% S.D. of 2-6) from the 0 % predictedby the exponential model. However, our theoretical simulations and calculations of% S . D . have all assumed zero measurement error. In practice, experimentallymeasured points will also have a finite measurement error. If the measurement S.D.is (say) 10%, and S.D. due to the periodic nature of rate: mass is 5 % for the rate-doubling model, then the expected overall % S.D. for the exponential model is 10, andfor the instantaneous rate-doubling model is V102+52— 11. It would be almostimpossible to distinguish between these two values in practice.

It is perhaps noteworthy that in their examination of rate: mass patterns for 111individual proteins, Elliot & McLaughlin (1978) found an average % S.D. of 11. Thisis larger than the 0% predicted by the exponential increase model, and the 2—6%predicted by our various simulations based on instantaneous doubling in rate ofmessenger synthesis.

Two further comments on the suitability of % S.D. as a measure of periodic fluctua-tion in rate: mass are relevant. Firstly, the method of calculation gives undue weightto very low and very high flow-rate fractions, which contain comparatively low num-bers of cells. Calculation of population-weighted % S.D. values might be preferable.For simulations such as those shown in Fig. 8, population-weighted % S.D. valueswere generally lower than unweighted % S.D. values. Secondly, for any given patternof rate: mass, the % S.D. may depend on the number of points taken, and for somepatterns of synthesis the variation can be quite marked. This makes it impossible touse % S . D . directly for comparison of curves with markedly different numbers ofpoints.

CONCLUSION

Our calculations suggest that it is most unlikely that the dual-labelling centrifugal-elutriation method can distinguish between exponential and periodic rate-doublingpatterns of macromolecular synthesis through the cell cycle. The disagreement


between reports in the literature claiming exponentially increasing or discontinuouslyincreasing patterns of synthesis of various RNAs and proteins therefore remainsunresolved.

Our analysis has not dealt extensively with the model for periodic synthesis, forexample, of the putative 'step' enzymes. The dual-labelling centrifugal-elutriationmethod undoubtedly allows greater discrimination between this model and either ofthe two involving continuous synthesis, although this ability will decrease with in-creasing cell cycle duration of the 'step'. Our simulation of the effects of imperfect cellcycle fractionation on the pattern of DNA labelling ratio shows the extent of degrada-tion that may occur. This particular step occurs early in the cycle, where cell fractiona-tion is good. Steps occurring later in the cycle, where fractionation is poor, mightpresent problems of analysis.

This paper has dealt with a single method of cell cycle analysis in order to highlightsome theoretical and practical problems. The approaches adopted, and the mathemat-ical treatment developed, should be applicable to analysis of other cell cycle situa-tions.

We are grateful to Professor C. S. McLaughlin for permission to use his data in our simulations.We thank Dr G. H. Freeman for painstaking criticism and comment on the mathematics, andProfessor J. M. Mitchison and Dr P. M. Nurse for helpful discussions. The computer programsU3ed in this work are available on application.

REFERENCES

CARTER, B. L. A., SEBASTIAN, J. & HALVORSON, H. O. (1971). The regulation of the synthesisof arginine-catabolizing enzymes during the cell cycle of Saccharvmyces cerevisiae. Adv. enzymeReguln 9, 253-266.

CHIA, L. & MCLAUGHLIN, C. S. (1979). The half life of mRNA in Saccharvmyces cerevisiae.Molec.gen. Genet. 170, 137-144.

COOK, J. R. & JAMES, T. W. (1964). Age distribution of cells in logarithmically growing cellpopulations. In Synchrony in Cell Division and Growth (ed. E. Zeuthen), pp. 485-495. NewYork: Interscience.

CREANOR, J. & MITCHISON, J. M., with a statistical appendix by WILLIAMS, D. A. (1982). Patternsof protein synthesis during the cell cycle of the fission yeast Schizosaccharomyces pombe.J. CellSci. 58, 263-285.

ELLIOT, S. G. & MCLAUGHLIN, C. S. (1978). Rate of macromolecular synthesis through the cellcycle of the yeast Saccharvmyces cerevisiae. Proc. natn. Acad. Sci. U.SA. 75, 4384—4388.

ELLIOT, S. G. & MCLAUGHLIN, C. S. (1979a). Regulation of RNA synthesis in yeast. III. Molec.gen. Genet. 169, 237-243.

ELLIOT, S. G. & MCLAUGHLIN, C. S. (19796). Synthesis and modification of proteins during thecell cycle of the yeast Saccharotnyces cerevisiae. J. Bact. 137, 1185-1190.

ELLIOT, S. G., WARNER, J. R. & MCLAUGHLIN, C. S. (1979). Synthesis of ribosomal proteinsduring the cell cycle of the yeast Saccharvmyces cerevisiae. j . Bact. 137, 1048-1050.

FRASER, R. S. S. & CARTER, B. L. A. (1976). Synthesis of polyadenylated messenger RNA duringthe cell cycle of Saccharvmyces cerevisiae. jf. molec. Biol. 104, 223-242.

FRASER, R. S. S. & MORENO, F. (1976). Rates of synthesis of polyadenylated messenger RNA andribosomal RNA during the cell cycle of Schizosaccharomyces pombe, with an appendix: calcula-tion of the pattern of protein accumulation from observed changes in the rate of messenger RNAsynthesis. J . Cell Sci. 21, 497-521.

FRASER, R. S. S. & NURSE, P. (1978). Novel cell cycle control of RNA synthesis in yeast. Nature,bond, in, 726-730.


GORDON, C. N. & ELLIOT, S. G. (1977). Fractionation of Saccharvmyces cerevisiae cell popula-tions by centrifugal elutriation.7. Bact. 129, 97-100.

GULL0V, K., FRIIS, J. & BONVEN, B. (1981). Rates of protein synthesis through the cell cycle ofSaccharomyces cerevisiae. Expl Cell Res. 136, 295-304.

HALVORSON, H. O., CARTER, B. L. A. & TAURO, P. (1971). Synthesis of enzymes during the cellcycle. Adv. microb. Physiol. 6, 47-106.

HARTWELL, L. H. & UNGER, M. W. (1977). Unequal division in Saccharomyces cerevisiae and itsimplication for the control of cell division.,7. CellBiol. 75, 422-435.

LORD, P. G. & WHEALS, A. E. (1981). Variability in individual cell cycles of Saccharomycescerevisiae. J. Cell Sri. 50, 361-376.

MITCHISON, J. M. & CREANOR, J. (1969). Linear synthesis of sucrase and phosphatases during thecell cycle of Schizosaccharomyces pombe. J. Cell Sri. 5, 373-391.

MITCHISON, J. M. & VINCENT, W. S. (1965). Preparation of synchronous cell cultures by sedi-mentation. Nature, Land. 205, 987-989.

SCHERBAUM, O. & RASCH, G. (1957). Cell size distribution and single cell growth in Tetrahymenapyriformis GL. Ada path, microbiol. scand. 41, 161-182.

SOGIN, S. J., CARTER, B. L. A. & HALVORSON, H. O. (1974). Changes in the rate of ribosomalRNA synthesis during the cell cycle of Saccharomyces cerevisiae. Expl Cell Res. 89, 127-138.

WAIN, W. H. & STAATZ, W. D. (1973). Rates of synthesis of ribosomal protein and total nucleicacid through the cell cycle of the fission yeast Schizosaccharomyces pombe. Expl Cell Res. 81,269-278.

WILLIAMSON, D. H. (1965). The timing of deoxyribonucleic acid synthesis in the cell cycle ofSaccharomyces cerevisiae. J. Cell Biol. 25, 517-528.

APPENDIX I

In this Appendix the synthesis rate: mass ratios for different patterns ofmacromolecular synthesis during the cell cycle are calculated.

A. Synthesis rate: mass for a macromolecule that doubles its rate of synthesis instan-taneously at a fixed time within each cell cycle.

For the purpose of calculation, and without loss of generality, it may be supposedthat the rate of synthesis doubles at a time 1-0, on a time-scale in which 1 unitrepresents 1 cell cycle. Then, with M as mass, t as time and A as the rate constant forbreakdown of molecules, the net rate of accumulation is given by:

^ for all 0-0 =Sf< 1-0, (IA1)

which implies that:

M(t) = t~h ( P e^ck + M(0))Jo

Now, as t approaches 1-0, M{t) approaches 2M(0) and, for convenience this conditionmay be written asM(l) = 2M(0), so thatM(0) = K/k, whereK= ( l-e"A)/(2-e"A),giving

Cell cycle analysts 203

•j(\ + {K-I)e-k'). (IA2)

As the gross rate of synthesis is 1 throughout 0 < t < 1-0,

Synthesis rate 1

B. Synthesis rate: mass for a macromolecule that has a synthesis rate that increasesexponentially throughout the cell cycle. The net rate of accumulation is given by:

for all 0-0 s=t< 1-0

AMat

which is constant whatever the degradation or synthesis rate.

C. Synthesis rate: mass for a macromolecule that has a synthesis rate that doubleslinearly over a fraction 1 —1\ of a cell cycle. Using the same time scale as above, it maybe supposed that doubling occurs between t = ti and t = 1-0. So that:

forO^t^f i (IC1)

and ^-=\ + l-f^--XM for *i < e < 1 -0. (IC2)at 1 — t\

From equation (IC1), M(t) = \(1 - e"A') + e~A( M(0), 0 ^ t =S ti (IC3)

and from equation (IC2), M(f) = i ! ~ 2 f ' | (1 - e~^'-h))(1 — t\)A

(l-U)X[ ' A A 1 ; J

e-A('-' '». ti<t<\. (IC4)

From equation (IC3), M{t\) = \(l - e^'1) + e"A'1 M(0). (IC5)A

Substituting (IC5) into (IC4) and usingM(l) = 2M(0) gives:

(IC6)

T̂o {2-2,1-I-(l-,1-I)


Making use of equations (IC5) and (IC6) in (IC3) and (IC4) gives equations forM(t) in terms of parameters t\ and A only.

Synthesis rate . . , 1—* is then given by -rr—,

mass M(t)where M(t) is given by equation (IC3) for 0 ̂ t ^ t\, and by

M(t)

where M(t) is given by equation (IC4) for t\ < t < 1.Fig. 3A (see the main text) was constructed using these results with rate doubling

occurring between 0-1 and 0-4 of the cell cycle.

D. Synthesis rate: mass for a stable protein, P, synthesized by translation of anunstable messenger RNA, the synthesis rate of which doubles instantaneously onceper cell cycle.

The synthesis rate of the messenger RNA is again taken to double at a time, t = 1-0.The rate equations are given by:

and ^ = cR 0 ^ * < l - 0 , (ID2)

where R and P are mRNA and protein contents, respectively, A is the breakdown rateof mRNA and c is a proportionality constant. Boundary conditions are given by-2R(0)=R(t) and 2P(0) =P{t) as t approaches 1-0.

From equation (ID1), R(t)=\[\ + (K- l)e-A/],A

where K= ( l -e- A ) / (2- e - A ) , and from (ID2),

„ protein synthesis rate 1 + (K— l)e Xl

protein mass

APPENDIX II

Calculation of the apparent synthesis rate: mass ratio when the amounts of RNAradioactively labelled during labelling periods of length '/>' and 'q' cycles are used asmeasures of synthetic rate and mass, respectively. Only an RNA undergoing a discretedoubling in synthesis rate once per cell cycle is considered here. The derivation formore complex patterns of synthesis would be similar but more involved.


The RNA is assumed to double at times, t = .. . — 2, — 1, 0, 1. Where the time scaleis such that 1 unit represents a cell cycle. The rate of accumulation of labelled RNA,N, is then given by:

^ cXN for all*, (III)en

where c is the synthesis rate and may be taken to be given by c — 2 , where / equalsthe integer part of t. A boundary condition appropriate to radioactive labelling isN(t0) = 0, where t0 is the time of the start of the radioactive labelling.

To consider the effect of a short-term labelling of length/) (assumed less than onecycle), ending at time t, as an indication of the rate of synthesis, it is necessary tocalculate the ratio of accumulation over/) cycles to time t and the mass at time t. Ift >p, the whole period of accumulation occurs within one cycle and may be deter-mined from:

(dN(x))/(dx) = \-?

with a boundary condition that N(t-p) = 0 for 0=£f < 1-0. So for t>p:

|(l-e"^). (112)

For t </), synthesis takes place at two different rates during the labelling period oflength/), i.e.:

f o rx<0

and ^&=l-*N(x) forx^O.

A boundary condition for this system isN(t—p) = 0. These equations lead to:

(113)

forThe short-term labelling: mass ratio is then given by:

Accumulation over/) cycles N(t)Mass at the end of the labelling period M(t)'

where N(t) is given by equation (112) if p<t < 1-0 and by equation (113) if/>, and M{t) is given by equation (IA2) of Appendix IA.

For a longer labelling period between one and two cycles in length, q cycles say, asimilar analysis to the above leads to equations for the amount of labelled RNA at theend of the labelling period, * (assumed to be 0 < < < 1-0). \iq-\^t< 10, labellingwill take place at two rates and the amount of RNA labelled, say L(t), is given by:


(114)

It 0<t<q— 1, labelling will take place over three cycles and synthesis will occur atthree rates during the labelling period. This leads to:

+'' - l ^ o ^ O + O - L £*~^Q\ v /TT C\

o accumulation over p cycles _N(t)accumulation over q cycles L(t)

where N(t) is given by equation (113) for 0 s£ t =£/>,N(t) is given by (112) lorp<t< 1-0,L(t) is given by (115) for 0 « * =£<?-l-0,L(t) is given by (114) f o r ^ - l - 0 < < < 1-0.

APPENDIX III

Calculation of the effects of imperfections of cell cycle fractionation by centrifugalelutriation on patterns of rate: mass.

Let/(f) be the rate of synthesis of a macromolecule at time t in a population of cellsthat initially consisted of one cell aged 0 at time t = 0. Also, letM(/,a) be the amountof the macromolecule at time t in a population of cells that initially consisted of onecell aged a at time, t = 0.

Then, because of the periodically changing nature of the rate of synthesis, the rateof synthesis oiM(t, a) is given by f{t+a). The net rate of accumulation is then given by:

where A is the breakdown rate constant.If ML(t,a) is the amount of the macromolecule radioactively labelled during the

period (0,t), ML (t,a) is given by integrating equation (III1) leading to:

f(x)eXxdx, foral l0s=a<l-0. (1112)

At time t there will be I(t,a) cells aged /}= (a+t-J), where I(t,a) = 2? and j is theinteger part of t+a. At time t, these cells will have an average amount of labelledmacromolecule per cell of C/.(f,/3) =M/.(/,a)//(i,a), which, making use of (III2), isgiven by:

-t+j

(note, an alternative form of jf, used in solving these equations, is.7 = the integer partof t-0+1).

Now, in a population of cells with age distribution g(J3) at time t, the averageamount of labelled macromolecule per cell, Kt(t) say, is given by:


Jo J 0 - <

If a second radioactive labelling is given during the time interval t\ to £, where t\>0(i.e. equivalent to the pulse label), then, following similar arguments to those above,it can be shown that the average amount of labelled macromolecule per cell with thistype of label is:

Jo

The ratio of the two types of label, pulse to long-term, is then BL(t,ti)/Ki(t) for along-term labelling of t cell cycles and a short-term labelling of t—t\ cell cycles, andfor a fraction of the cell population with an age distribution given by g(P). Thisequation was solved by computer simulation after specification of the forms oif(x)

and the values of the parameters X,t,ti.

(Received JO December 1982 -Accepted 4 January 1983)

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