J. theor. Biol. (1995) 173, 283–305

0022–5193/95/070283+23 $08.00/0 7 1995 Academic Press Limited

Quantitative Analysis of a Molecular Model of Mitotic Control in

Fission Yeast

B N† J J. T‡

Department of Biology, Virginia Polytechnic Institute and State University, Blacksburg, Virginia24061-0406, U.S.A.

(Received on 22 April 1994, Accepted in revised form on 24 October 1994)

A wealth of information has accumulated about the physiology, genetics and molecular biology of thecell cycle in the fission yeast, Schizosaccharomyces pombe. From this information we have constructeda detailed molecular mechanism of M-phase control based on the modification of M-phase promotingfactor (MPF) by a suite of protein kinases (e.g. Wee1 and Mik1 which inhibit MPF) and phosphatases(e.g. Cdc25 which activates MPF). In particular, we analyze the interphase checkpoint in S. pombe, wherethe wild-type cell confirms that S phase is complete and that the cell is large enough to finish the divisioncycle. In our model, incomplete DNA replication restrains the onset of M phase by inhibiting Cdc25 andactivating Mik1, whereas increasing size biases the cell towards mitosis by down-regulating Wee1. Bystandard mathematical methods of chemical kinetics, we show that our model gives a quantitativelyaccurate account of the effects of hydroxyurea treatments, nutritional shifts and other perturbations ofthe division cycle of wild-type and mutant cells.

Introduction

Our present understanding of mitotic control in theeukaryotic cell cycle has been derived from physiologi-cal experiments (monitoring the responses of intactcells to external disturbances such as nutritional shiftsor transient drug treatments), genetic studies (mutantisolation and characterization), andmolecular analysis(gene cloning and expression, in vitro reconstitution).The last stage has been spectacularly successful inidentifying the major molecular components of themitotic control system and sketching out the ‘‘wiringdiagram’’ of their interactions. From this informationwe should be able to explain the physiological behaviorof intact cells; however, the molecular mechanism ofM-phase control has become so elaborate andcomplicated that it is hard to understand the system indetail by biochemical intuition and casual verbalarguments. We have proposed that mathematical

modeling can be useful as a bridge between molecularbiology and cell physiology (Novak & Tyson, 1993a),and we have illustrated this approach with acomprehensive model of M-phase control in Xenopusoocyte extracts (Novak & Tyson, 1993b).

In this paper we extend our model to anotherorganism that has played a prominent role inunraveling the details of cell cycle control, the fissionyeast Schizosaccharomyces pombe (see Forsburg &Nurse, 1991, for a review). Although the majorcomponents of mitotic control are identical (and eveninterchangeable) in the two organisms and the wiringdiagrams appear to be similar, their respective cellcycles are quite different physiologically. Frog eggextracts and intact embryos execute spontaneousoscillations in M-phase controlling factors, to a largeextent independent of DNA synthesis and nuclearresponses, whereas the mitotic cycle of fission yeast isclosely coupled to the DNA replication cycle and tooverall mass increase of the cell (Murray & Kirschner,1989; Nurse, 1990). How can two so fundamentallydifferent division cycles be generated by nearly thesame underlying control system?

† Permanent address: Department of Agricultural and ChemicalTechnology, Technical University of Budapest, 1521 BudapestGellert Ter 4, Hungary.

‡ Author to whom correspondence should be addressed.

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F. 1. The cell cycle of S. pombe. Timing of events is taken fromMitchison (1989), Table II. The relative timing of events is measuredfrom one cell division to the next. S phase starts at phase=−0.05(+0.95 in the previous cycle) and ends at +0.05. M phasecommences about 3/4 of the way through the cycle and ends at 0.81.

1992). In Xenopus oocyte extracts, cyclin bindingfacilitates both these phosphorylation steps (Solomonet al., 1990, 1992), and we assume that the same is truein S. pombe. T167 is phosphorylated quickly aftercyclin binds to Cdc2, and this phosphorylationreaction does not seem to be regulated during the cellcycle (Gould et al., 1991). The major regulatory stepis Y15 phosphorylation and dephosphorylation[Fig. 2(a)], carried out by Wee1 (Featherstone &Russell, 1991) and Cdc25 (Millar et al., 1991a,b),respectively. There is a second tyrosine kinase,Mik1 (Lundgren et al., 1991), that can phosphorylateY15, and at least one backup phosphatase as well,Pyp3 (Millar et al., 1992a). MPF accumulates in G2 astyrosine-phosphorylated, inactive forms (primarily thedoubly phosphorylated form) (Gould et al., 1991), andit is activated just prior to mitosis by dephos-phorylation of the tyrosine residue of the doubly-phosphorylated form (Gould & Nurse, 1989; Morenoet al., 1989; Gould et al., 1990).

In vitro experiments with Xenopus extracts indicatethat the abrupt and autocatalytic activation of MPF isgenerated by two positive-feedback loops, wherebyMPF activates Cdc25 and inhibits Wee1 (Murray,1993). We assume the same feedback loops areoperative in fission yeast. In Fig. 2(b) these loops arerepresented as direct phosphorylations of Cdc25 andWee1 by MPF, but it is not particularly important toour model whether MPF acts directly on Cdc25 andWee1 or through intermediary kinases.

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Unreplicated DNA and small cell size are thought toinhibit M phase by favoring the tyrosine-phosphory-lated form of MPF. Enoch & Nurse (1990) concludedthat S–M coupling works through Cdc25 rather thanWee1 because wee1− cells have normal S–M couplingwhereas cdc25OP cells are defective in this regard. Inaddition, since wee1− Dcdc25 has normal S–Mcoupling (Enoch et al., 1992), we assume that Mik1 isa second target of unreplicated DNA. In ourmechanism [Fig. 2(b)], unreplicated DNA inactivatesCdc25 and activates Mik1. Because the mechanism ofS–M coupling is still uncertain, we present this schemeonly as a preliminary, simple, and reasonablepossibility.

A collection of hus− mutants (hydroxyurea sensitive)in fission yeast suggests that, between unreplicatedDNA and the tyrosine-modifying enzymes, there is along signal transduction pathway (Enoch et al., 1992),which we represent by a single switch-like variable W.The indicator for unreplicated DNA is species X,which accumulates after START and is degraded aftercompletion of S phase. Species X has properties similar

After reviewing some characteristic features of theS.pombe cell cycle, we elaborate the Xenopus model to fitthe situation in fission yeast. We discuss in detail thebehavior of the model in relation to wild-type yeast cellcycles, focusing on the fundamental physiologicalproperties of S–M coupling and size control. Then weinvestigate the division cycles in various mutants, forwhich S–M coupling and/or size control is broken. Weshow that all major features of the fission yeast cellcycle can be derived from a model of mitotic controlthat is almost identical to the Xenopus situation.

A Molecular Model of M-Phase Control in

Fission Yeast

The fission yeast cell cycle is diagrammed in Fig. 1.In defined medium (EMM3) at 29°C, cell numberdoubles every 180 min, which we take as our standardcycle time. Most of the cell cycle is spent in G2; S phaselasts only 18 min and M phase is even shorter. Cellabscission is considerably delayed from the end ofmitosis, so G1 and the onset of S actually occur beforethe mother cell divides. Wild-type cells divide at alength of about 14 mm. During G2, cells increase inlength nearly exponentially (Mitchison & Nurse,1985). Between mitosis and cell division there is hardlyany length extension, although there is still someincrease in cellmass during this time (Mitchison, 1957).

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We now know that M phase in eukaryotic cells istriggered by activation of a protein kinase called MPF,a dimer of Cdc2 (the catalytic subunit, named after itsgene cdc2 in fission yeast) and cyclin B (the regulatoryor targeting subunit) (Nurse, 1990). There are severalB-type cyclins in fission yeast, coded by cdc13 (Haganet al., 1988), cig1 (Bueno et al., 1991) and cig2 (Bueno& Russell, 1993). MPF activity is regulated byphosphorylation of Cdc2 at an inhibitory tyrosine site(Tyr-15) (Gould & Nurse, 1989) and an activatorythreonine site (Thr-167) (Gould et al., 1991; Fleig et al.,

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F.2.Amolecularmodelof themitoticcontrolsysteminS.pombe.(a) Cyclin monomers (ovals) are synthesized (step 1) and degraded(step 2), and they combine with Cdc2 monomers (step 3). The Cdc2subunit of cyclin/Cdc2 dimers is phosphorylated and dephosphory-lated by a suite of kinases andphosphatases. Tyr-15 phosphorylationis inhibitory (Y site on left of icon) and Thr-167 phosphorylation isactivatory (T site on right of icon). The active form of MPF [-qw-P]

stimulatesmitosis and cell division. (b) The kinases (Wee1 andMik1)and phosphatase (Cdc25) that control Tyr-15 phosphorylation areregulatory enzymes. Wee1 is inhibited by active MPF and by cell sizethroughNim1andPK.(ThedoublyphosphorylatedformofWee1hasbeen left off this diagram for the sake of simplicity.) Mik1 is activatedby unreplicated DNA. Cdc25 is activated by MPF and inhibited byunreplicated DNA. DNA synthesis is inhibited by Ta-P, a molecularsignal that M phase is in progress. (c) The ubiquitination pathway(UbE) by which cyclin is degraded is stimulated by MPF through anintermediaryenzyme(IE),butthestimulationwillbeblockedifcertaintarget proteins (Ta) are not successfully phosphorylated by MPFduring mitosis. All four dimers are degraded by this pathway.

to the cdc18 gene product (Kelly et al., 1993), which ishomologous to budding yeast CDC6 (Bueno &Russell, 1992).

Following a suggestion of Enoch & Nurse (1990), weassume that cell size operates on Wee1 through Nim1,whichphosphorylatesWee1on sitesdistinct fromMPFphosphorylation sites (Coleman et al., 1993; Parkeret al., 1993; Wu & Russell, 1993). Because nim1 is notan essential gene, we assume there is a second proteinkinase, PK, which mediates the size effect on Wee1.†Wedistinguish fourdifferent formsofWee1,dependingon whether the enzyme is phosphorylated on itsN-terminal domain (MPF sites) or its C-terminaldomain (Nim1andPKsites).Allphosphorylated formsare assumed to be less active (about 25-fold) than theunphosphorylated form (Coleman et al., 1993; Parkeret al., 1993; Tang et al., 1993; Wu & Russell, 1993).

We shall not specify what the cell measures as its‘‘size’’ (no one knows). We only presume that thereis some size variable that increases exponentially‡throughout the cycle. We assign this variable the value1 at the end of mitosis in wild type cells. This impliesthat wild-type cells divide at size=1.14 and are bornat size=0.57. When comparing the model toexperiments, we use protein content per cell, whenavailable, or cell length at division.

We assume that the nutritional status of the cellimpinges on the G2 sizer through Nim1: faster growthrates decrease the activity of Nim1, so the cell mustgrow to a larger size before it can enter mitosis (Fantes& Nurse, 1977).

Cyclin (cdc13) is a periodic protein in S. pombe(Booher et al., 1989), being degraded rapidly afterM phase (Moreno et al., 1989). As in the Xenopusmodel, we assume that cyclin degradation [Fig. 2(c)] isstimulated by active MPF through an intermediaryenzyme (IE). Unlike Xenopus oocyte extracts and earlyembryos, S. pombe has a ‘‘checkpoint’’ at metaphasewhich does not permit cyclin degradation and exit fromM phase until all components of the mitotic machineryare in place (Moreno et al., 1989). The mitoticmachinery is represented by a generic ‘‘Target’’ protein

† As nim1 is allelic with cdr1 (Feilotter et al., 1991), perhaps PK isthe product of cdr2 (Young & Fantes, 1987).

‡ Exponential growth is a useful first approximation. Morecomplicated growth laws could be implemented without changingthe basic features of the model.

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F. 3. An automaton model of START control in S. pombe. Between exit from mitosis and entry into S phase, a yeast cell must carryout a number of G1-specific processes. Because the molecular basis of these processes is not yet known, we model G1 by a set of logicalinstructions. This flow chart together with the molecular mechanism in Fig. 2 provides a mathematically complete model of the fission yeastcell cycle (see Appendix).

(Ta) that must be phosphorylated by MPF; only afterTa is phosphorylated can the cyclin degradationmachinery be turned on and the cell leave M phase.(We define M phase as the period when more than 50%of Ta is phosphorylated.) Furthermore, we assumethat the mitotic state is incompatible with DNAsynthesis [Fig. 2(b)]: if MPF is activated while S phaseis still in progress, as in cases of mitotic catastrophe,then Ta-P slows the rate of DNA synthesis (see, forexample, Walker et al., 1992).

Figure 2 represents a molecular mechanism thatdescribes the S. pombe cell cycle from sometime afterthe onset of DNA synthesis to shortly after the exitfrom mitosis. To finish the model we need somedescription of how cells proceed through G1 and into Sphase. Unfortunately, not much is known aboutthe molecular mechanism of this part of the fissionyeast cell cycle, so we cannot represent it by a setof biochemical reactions but must settle for a set

of logical rules—an ‘‘automaton’’ (Fig. 3)—based onphysiological observations. When cells exit M phase,they start degrading Cdc25 at an elevated rate(Ducommun et al., 1990; Moreno et al., 1990; Millaret al., 1991a) for 20 min in themodel, but they continuecyclin synthesis for 10 min. (When cells exit M phase,we reduce cell mass two-fold; i.e. our size variablerefers to size per nucleus.) Ten minutes after mitosis,cells consult a size monitor (Nurse & Fantes, 1981) todetermine if they are large enough to pass START(sizeq0.35, a value we calculate by comparing theprotein content per cell in wild-type and wee1− cellsNurse, 1975; Nurse & Thuriaux, 1977; Nasmyth et al.,1979). If not, they discontinue cyclin synthesis† andgrow until they satisfy the minimum size requirementfor START. When cells pass START, they resumecyclin synthesis (which puts them on the road to Mphase) and they become committed to DNA synthesis(which commences after a 15 min preparation period).The details of this ‘‘automaton’’ are not particularlyimportant; it is intended only as a ‘‘crutch’’ for theM-phase control system until a satisfactory molecularmechanism of START control in fission yeast isavailable.

The model in Figs 2 and 3 can be translated into aset of differential equations for the concentrations ofthe various forms of MPF and its regulatory enzymes.

† Our assumption that the rate of cyclin synthesis is reduced if G1is long is based on observations in budding yeast (Nasmyth, 1993)that mitotic cyclins are not resynthesized until cells past START.Unfortunately, the situation in fission yeast in this regard is stillmurky.

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F. 4. Biochemical changes during the cell cycle in wild-type yeastcells. (a) Time-courses for total cyclin (monomer+dimers), activeMPF [-qw-P], and inactive MPF [P-qw-P], relative to total Cdc2. The

dotted lines indicate the cyclin level and MPF activity that would bemeasured in a partially synchronous culture, with a degree ofsynchrony comparable to that in the experiments of Creanor &Mitchison (1993). (b) Each hatched region represents the phase inthe cycle when the indicated component exceeds 50% of its maximalconcentration. M phase, by our convention, is defined as the timeperiod when Ta is more than 50% phosphorylated.

type fission yeast. The cycle time is 180 min. During Sphase, and for some time thereafter, X, W and Mik1activities are high [Fig. 4(b)], thus S phase inhibitsprogress into M phase by forcing Cdc2/cyclin dimersto accumulate in the inactive, doubly phosphorylatedform [Fig. 4(a)]. When S phase is completed and Wdrops below 50% maximal activity, about 40 min aftercell division, the brake that S phase was exerting on thepathway toward mitosis is released, and this shows upas a noticeable change in the rate of appearance ofMPF activity. About 80 min after cell division, the sizerequirement for mitosis is lifted (Nim1 inactivatesWee1). MPF dimers are dephosphorylated in aself-accelerating fashion (because of the positivefeedback loops whereby active MPF activates Cdc25and inhibits Wee1), and the cell moves into mitosis at135 min. This sort of pattern of MPF activation hasbeen observed recently in careful measurements byCreanor & Mitchison (1993), cf. the dashed curve inour Fig. 4(a) with their Fig. 2.

To understand the fission yeast cycle ‘‘from theinside’’, we employ the same graphical ‘‘portraiture’’that we introduced in the Xenopus model (Novak &Tyson, 1993b). The state of the mitotic control systemis envisioned as a point in the plane spanned by totalcyclin concentration (free monomers+heterodimers)on the vertical axis and MPF activity on the horizontalaxis.As cyclin is synthesized and degraded and asMPFchanges its phosphorylation state, the point specifyingthe state of the control system moves around on thecyclin/MPF plane (Novak & Tyson, 1993a). Forinstance, the oscillation in Fig. 4(a) appears as a closedtrajectory in the cyclin/MPF plane [Fig. 5(a)]. Stationsalong this cycle are identified by relative phase (celldivision=0 or 1, S phase from 0.95 to 0.05, M phasefrom 0.73 to 0.81).

As we have explained in earlier papers (Novak& Tyson, 1993a, b), the motion of the state-pointis governed to a great extent by two curves: thecyclin balance curve (along which the rate of cyclinsynthesis=the rate of cyclin degradation) and thedimer equilibriumcurve (alongwhich the four differentphosphorylated forms of MPF are in equilibrium witheach other). In Fig. 5(b) we graph these curves on thecyclin/MPF plane at one point in the fission yeast cellcycle (phase=0.3). Cyclin and MPF concentrationsare expressed relative to the total concentration ofCdc2 subunits, which remains constant throughout thecell cycle (Simanis & Nurse, 1986). The cyclin balancecurve is a horizontal line at [total cyclin]=1 because(we assume) the steady-state cyclin level in G2 phaseis roughly equimolar to Cdc2. This is a significant

These equations are collected in the Appendix, alongwith the numerical values we have selected for thevarious kinetic parameters. The simulations presentedin this paper were carried out by numerical solution ofthese differential equations with the basal parametervalues suitably modified to reflect the experimentalsituation. This model does not take into accountcompartmentalization of any of the reactions.

Analysis of the Model for the Wild-type Cell Cycle

In Fig. 4 we illustrate the behavior of the modelunder conditions that simulate the cell cycle in wild

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departure from the situation in frog eggs and embryos,where Cdc2 is always in great excess over cyclin, butrepresentative of the situation in somatic cells(Minshull et al., 1989b).

The state point at phase 0.3 (the black square)lies below the cyclin balance curve, so synthesis isgreater than degradation and the state point is movingupward toward the cyclin balance curve. As cyclinslowly accumulates, it combines with free Cdc2monomers, and the phosphorylation and dephospho-rylation reactions at Y15 and T167 keep the state pointclose to the dimer equilibrium curve. The location ofthe state point close to the dimer equilibrium curve tells

us roughly how the dimers are divided among activeand inactive forms. At phase 0.3, [active MPF]=0.11and [total cyclin]=[active MPF]+[inactiveMPF]=0.88, so about 13% of the MPF dimers areactive and 87% inactive (mostly the doublyphosphorylated form). At this point in the cell cycle,the state point is moving upward toward the steadystate at the intersection of the cyclin balance and dimerequilibrium curves. If these curves were to stay put, thestate of the control system would come to rest at thisintersection point.

However, as the cell proceeds through this portionof the cell cycle, the dimer equilibrium curve is slowlydropping because, as cell mass increases, Nim1 and PKtotal activities increase (total activity=specific ac-tivity×cell mass), and the distribution of Wee1 formsis shifted toward the inactive, C-terminalphosphorylated forms. This growth-induced shiftin Wee1 favors the active form of MPF, whichis manifested in the cyclin/MPF plane as a steadyerosion of the hump in the dimer equilibrium curve, aswe shall see.

In Fig. 6 we follow the evolution of the mitoticcontrol system throughout the cell cycle of S. pombe.In S phase and early G2 [phase 0–0.2 in Fig. 6(a)],the dimer equilibrium curve is a sharply increasing linebecause the Tyr-kinase to Tyr-phosphatase ratio ishigh (Wee1 is active, as the cell is small; Mik1 is activeand Cdc25 is inactive due to the braking action ofunreplicated DNA; and the total amount of Cdc25 islow because it was degraded after M-phase).Consequently, as total cyclin increases, dimersaccumulate in inactive, tyrosine-phosphorylatedforms. The control system is moving toward a steadystate (at the intersection of the cyclin balance and thedimer equilibrium curves) with lots of cyclin andinactive dimers [Fig. 6(a), phases 0–0.2]. We call thissteady state, on the left branch of the dimer equilibriumcurve, an interphase checkpoint.

In early G2 (phase 0.05–0.2) the state point movesslowly to the right as the S-phase brake on mitoticprogression is gradually released (X and W activitiesdecrease; Mik1-activating and Cdc25-inactivatingphosphatases turn off), and hence more active MPFappears. In our model, even after the S-phase brake iscompletely disengaged (phase 0.3), Wee1 maintains asecond brake on mitotic progression. As the cell growsin size, Wee1 activity decreases and the hump in thedimer equilibrium curve drops [Fig. 6(a), phase 0.3–0.5].

Midway through the cell cycle (phase=0.5) theinterphase checkpoint disappears as the local

F. 5. An alternative picture of the wild-type cell cycle. (a) Thecurves for total cyclin and active MPF in Fig. 4(a) are used to plota closed curve in the cyclin/MPF plane, with time as a parameter(time is expressed as a fraction of the division cycle period). (b) Thecyclin balance curve and dimer equilibrium curve at phase 0.3. Thestate of the system at this phase in the cell cycle is located by the blacksquare (Q), The lowest curve is the line [total cyclin]=[active MPF];it is not straight because the coordinate system is semilogarithmic.The intersection of the cyclin balance and dimer equilibrium curves,marked by W, is a stable steady state corresponding to G2 arrest(cyclin level high, MPF activity low).

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F. 6. Dynamical portraits of the wild-type cell cycle. As in Fig. 5(b) we plot the cyclin balance and dimer equilibrium curves, and useblack squares to locate the state point at indicated phases of the cell cycle. The dark lines are parts of the cell cycle trajectory in Fig. 5(a).(a) Early G2 (phase 0–0.5). The dimer equilibrium curve falls precipitously as the S-phase brake is released (0–0.2) and then slowly as thecell grows (0.3–0.5). (b) Late G2 (phase 0.5–0.7). The interphase checkpoint has disappeared and the state point moves toward a mitoticcheckpoint. (c) M (phase 0.72–0.78). As the cell moves successfully through mitosis, cyclin degradation is switched on and the cyclin balancecurve drops precipitously. (d) Late M and early G1 (phase 0.78–0.82). As the state point moves into the G1 region of the plane (cyclin levellow, MPF activity low), the cyclin balance and dimer equilibrium curves move back into position for the start of a new cycle.

maximum of the dimer equilibrium curve drops belowthe cyclin balance curve. There no longer exists a stablesteady state with low MPF activity. The only steadystate now lies on the right-hand branch of the dimerequilibrium curve—it is a state of high MPF activity (amitotic state)—and the state point must now move inthat direction.†

The movement of the dimer equilibrium curveduring early G2 provides a conceptual bridge betweenthe molecular mechanism of M-phase initiation andthe physiological notions of checkpoints, S–Mcoupling, and size control. The interphase checkpointis a stable steady state of high cyclin level but low MPFactivity (because the cyclin/Cdc2 dimers are mostly

sequestered as tyrosine-phosphorylated forms). In thecyclin/MPF plane, this steady state is located at theintersection of the cyclin balance curve and a steeplyrising dimer equilibrium curve. The cell cannot pass theinterphase checkpoint until DNA synthesis is completeand cell size is sufficient. Satisfaction of theserequirements is reflected in a two-stage process bywhich the dimer equilibrium curve collapses and theinterphase checkpoint disappears. As the S-phasebrake disengages, the dimer equilibrium curve dropsfrom ‘‘steeply rising’’ to ‘‘N-shaped’’, and then, as thecell grows, the local maximum of the N-shaped curvedrops until it finally dips below the cyclin balancecurve. This last event corresponds to ‘‘pulling thetrigger of the size control mechanism’’. However,‘‘trigger’’ is hardly a suitable metaphor. The cell sizerequirement is met half-way through the cycle, in thesense that, if we could halt cell growth withoutany other effects on the mitotic control system, the cellwould eventually enter M phase; but there isa considerable time lag beforeMPF is activated and thecell enters M phase irrevocably. As we shall

† In simulations, the state point actually begins its motion towardthe mitotic state at phase 0.42, 15 min earlier than expected. Thisdiscrepancy occurs because (i) the two-dimensional dynamicalportrait in Fig. 6(a) is only a rough approximation of the full18-dimensional set of differential equations in the appendix, and(ii) the state point always lies some distance from the steady state,so its motion is not predicted exactly by the properties of the steadystate.

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see [Fig. 9(b)], cells which have satisfied the sizerequirement and are on their way toward mitosis canbe recalled to the interphase checkpoint.

After the interphase checkpoint disappears, thecontrol system proceeds toward the mitotic state bydephosphorylating the Tyr-15 residues of inactiveMPF dimers [Fig. 6(b)]. There is a considerable lag(45 min) before the cell enters mitosis because, evenafter the interphase checkpoint is erased, progresstoward mitosis is slow in the region where the twobalance curves used to intersect (phase 0.5–0.6). As thestate point moves out of this stagnant zone, itaccelerates—as the positive feedback loops throughCdc25 and Wee1 engage—and approaches the mitoticcheckpoint (phase 0.6–0.7).

If the cell enters mitosis successfully (Ta4 Ta-P,in the model), then the cyclin degradation pathway isset in motion, the cyclin balance curve collapses, andcyclin level drops [Fig. 6(c), phase 0.72–0.76].

If for one reason or another MPF does notsuccessfully phosphorylate Ta, then the control systemsticks at the mitotic checkpoint because the cyclinbalance curve does not collapse. This resembles thesituation in nda3 mutants (Hiraoka et al., 1984), whichare defective in b-tubulin and arrest in mid-mitosiswith high H1 kinase activity (Moreno et al., 1989).ln simulations of this situation (not shown), MPFactivity increases and stays high together withcyclin, keeping Cdc25 in its active form and Wee1inactive.

At phase 0.81, the target protein becomes less than50% phosphorylated (which we take to be the end ofM phase), and the cyclin balance curve starts movingback to its original position as cyclin degradation turnsoff [Fig. 5(d)]. At this time Cdc25 begins to be degradedrapidly. The remaining MPF dimers are phosphory-lated on Tyr-15, and the resulting loss of MPF activityshifts Wee1 back to its active form and any remainingCdc25 to its inactive form. The dimer equilibriumcurve regains its characteristic N-shape [phase 0.82 inFig. 6(d)] and becomes even more steeply increasing asS-phase brakes are re-established [phase 0 in Fig. 6(a)].

All other cell cycles which we shall discuss (thoseperturbed by nutritional shifts or drug treatment orthose rearranged by mutation) can be understood interms of modifications in the dynamical portraits(Fig. 6) that underlie the wild-type cycle.

Perturbations of the Wild-type Cycle

In the wild-type cell cycle of fission yeast, twoessential requirements must be met before the cell can

proceed into mitosis (Nurse, 1991): DNA synthesismust be complete, and the cell must be large enough.Both signals operate through tyrosine phosphoryl-ation of theCdc2 subunit ofMPFdimers. In ourmodelof the fission yeast cell cycle, unreplicated DNAinactivates MPF by activating a tyrosine kinase (Mik1)and inactivating the primary tyrosine phosphatase(Cdc25). Size control operates on a second tyrosinekinase (Wee1). If one or the other kinase is operative,entry into M phase is blocked because MPF dimers areinactive. Once both requirements are met, the cellproceeds into mitosis in a self-accelerating fashion,because increasing MPF activity feeds back positivelyon Cdc25 and Wee1 activities. To look at these controlsignals in more detail we examine three classicalphysiological studies of wild-type S. pombe.

Wild-type fission yeast cells have an effectiveS–M coupling mechanism: if S phase is blocked bychemicals (e.g. hydroxyurea) or mutation, cells will notenter mitosis (Mitchison & Creanor, 1971; Nurse et al.,1976). Miyata et al. (1978) further studied the effectsof pulses of hydroxyurea on the subsequent cell cycle.If S phase was delayed less than 1/3 of a cycle (undertheir experimental conditions, cycle time(CT)=110 min, sowe express their timemeasurementsas fractions of CT), then entry into M phase was notdelayed at all (i.e. G2 can be shortened up to 0.33 CT).But, if S phase was delayed more than 0.33 CT, thenentry into M phase was delayed by an amount=S-phase delay−0.33 CT [see Fig. 7(b), open squares].

We simulate these experiments in Fig. 7. Notice[Fig. 7(a)] that in all cases MPF shows an initial rise inactivity about 30 min after S phase ends (about 50 minafter hydroxyurea is removed): this is the time it takesto remove the S-phase brake on MPF. If MPF activitystarts to rise before 80 min post cell division, then itsrise is immediately damped by the still-operative sizecontrol (cells grownormally in hydroxyurea, so the sizerequirement is not met until about 80 min after celldivision). After the 80 min point there is no block dueto insufficient size, and the cell proceeds along thedephosphorylation pathway [Fig. 6(b)] as soon as theDNA replication brake is released.

Figure 7(b) summarizes the effects of S-phase delayon timing of mitosis. Note that wild-type cells have acompensation mechanism in G2: if S phase is delayedless than 0.33CT, this delay can be accommodated by ashortened G2 so that M phase occurs on schedule. Butlonger S-phase delays cannot shorten G2 any further,so M phase is pushed back. However, wee1− mutant

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cells do not have this compensation mechanism(Fantes & Nurse, 1978), as we shall discuss later.

The mitotic size control in fission yeast ensuresthat cells large at birth will have a shorter than averagecycle time, and vice versa for small cells. Fantes

F. 8. Cycle time (min) and birth length (relative to wild type).(Q) Data from (Fantes, 1977). (——) our simulations.

F. 7. Simulation of hydroxyurea pulse treatments. (a)Hydroxyurea (HU) was added to wild-type cells sometime in G1(time Q−9 min; cell division occurs at time 0) and removed at thetime indicated by the black arrow. This is the time at which DNAsynthesis starts in each treated cell, and the number attached to eacharrow gives the S-phase delay (as a fraction of the normal cycle time,180 min) induced by each HU treatment. The curves track simulatedchanges in MPF activity after the block-and-release (the thickercurve is the untreated control). Cells released after a short time inHUgo intomitosis nearly synchronouslywith the control, indicatinga G2-shortening mechanism that can compensate for brief delays inthe completion of S phase. Cells released later are delayed in entryin mitosis. (b) The results in part (a) generate the curve (Q) labeledWT. Delays are plotted as fractions of the normal cycle time. Theopen squares are data of Miyata et al. (1978) (see their figure 2). Wecalculate the S-phase delay from their data by assuming that themidpoint of septum formation ((0.85+1)/2=0.925) overlaps withstart of S phase (0.95). The same simulation, repeated forwee1− cells,generates the curve marked by closed triangles.

(1977) demonstrated this negative correlation betweencycle time and birth length in an elegant experiment(see Fig. 8, closed squares). To extend the range ofobservable birth lengths, Fantes used a cdc2ts mutantstrain which he held for 3 hr at the restrictivetemperature; during this time the cells could grow butnot divide. On return to the permissive temperature,these cultures produced abnormally large newborncells which then displayed very short interdivisiontimes, down to aminimumcycle timeof about 110 min.Our simulations of these experiments (Fig. 8) are inperfect agreement with Fantes’ data. The minimumcycle time exhibited by very large cells can be brokendown into these steps: G1= 25 min, S=18 min,remove DNA brake=33 min, remove sizebrake=0 min, proceed into M=20 min, M=14 min.

Another classical set of experiments on size controlin S. pombe was the nutritional-shift experimentsof Fantes & Nurse (1977). In Fig. 9 we simulate anutritional shift-up. At time 0 we abruptly increasem (the rate constant for mass increase) by 30%, whichcorresponds to a change from proline (medium 6) toglutamate (medium 3) in the Fantes & Nurse (1977)paper (see their Fig. 1). The observed increase in sizeat division suggests that the critical size for division isaffected by nutritional conditions (Fantes & Nurse,1977). To account for this effect in our model, weassume that the 30% increase in m causes a 50%decrease in the specific activity of Nim1. The dropin Nim1 activity causes Wee1 activity to increase,which raises the local maximum of the dimerequilibrium curve [Fig. 9(b)]. This shift introduces apronounced delay in the progress of G2 cells toward M

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F. 9. Simulation of a nutritional shift-up experiment. (a) At t=0an asynchronous population of cells is shifted from poor growthmedium to a richer medium. (The two lines show the increase inprotein content of the shifted culture and the unshifted control.) Celldivision persists for about 40 min at the small cell size characteristicof the poor growth medium, then there is a 1-hr hiatus in cell division(cell number plateaus) before cells start dividing again at the largercell size characteristic of the richermedium.Compare this simulationto figure 2 of Fantes & Nurse (1977). (b) Consider a pair of cellswhich are at phases 0.6 and 0.7 in the cell cycle when the cellpopulation is shifted to the richer growth medium. The state points(Q) for these cells are plotted in the cyclin/MPF plane. Both cellsunder consideration are proceeding toward mitosis at the time of theshift. The dimer equilibrium curve at phase 0.65 is plotted bothbefore and after the shift. After the shift the cell at phase 0.7 can stillproceed towardmitosis, but the cell at phase 0.6 is called back towarda newly created interphase checkpoint; the open squares locate them20 min after the shift. The earlier cell cannot enter mitosis until itgrows to the larger size necessary to destabilize the interphasecheckpoint at the higher growth rate.

there is a burst of new cell divisions at a much largersize, and then the cell number curve asymptoticallyapproaches the curve characteristic of the new growthrate. Our Fig. 9 is in good agreement with Fig. 2of Fantes & Nurse (1977).

Inverse changes (acceleration into mitosis) happenduring a nutritional shift-down experiment, andthe model gives predictions consistent with Fantes &Nurse (1977) experiments (simulations not shown).During a shift down, Wee1 activity is down-regulated(because Nim1 gets more active), so the situation isanalogous to a shift of a temperature-sensitivewee1 mutant from ‘‘permissive’’ to ‘‘restrictive’’temperature (see next subsection).

The Cell Cycle in Various Mutant Strains of

Fission Yeast

wee1−

A major breakthrough in genetic investigationsof the eukaryotic cell cycle was Nurse’s (1975)discovery of wee mutants of fission yeast. The firstof these mutant cells (originally called cdc9.50,then renamed wee1.50) carries a temperature-sensitiveallele: it grows and divides nearly normally at 25°C butdivides at about half-normal size at 35°C. It isimportant to remember that these mutant cells areperfectly viable (unlike classical cdc mutations whichare, by definition, temperature sensitive lethals), andthat they grow and divide just as well as the wild type.The only difference is that the mitotic control systemhas been ‘‘reprogrammed’’ so that cells divide at asmall size (Nurse, 1991).

To simulate the effects of shifting wee1ts mutant cellsto the restrictive temperature, we assume that the shiftcauses the catalytic activity of the active form of Wee1to drop 15-fold. Figure 10(a) shows the expected MPFactivity of cells which are at different stages in the cyclewhen Wee1 is knocked out. Knocking out Wee1completely eliminates the size requirement at theinterphase checkpoint: notice in Fig. 10(b) (dashedcurves) that there is no longer a hump in the dimerequilibrium curve after the S-phase brake comes off(phase 0.25). So when the temperature shift is donelater than about 45 min (0.25 CT) into the cycle, cellsimmediately start to dephosphorylate inactive MPF;these wee1− cells are advanced into mitosis and divideat an abnormally small size. (In our simulation ofwee1.50 at the restrictive temperature, we have notbothered to change the growth rate to reflect the highertemperature.) Cells from phase 0.25 to 0.65 arrive atmitosis at the same time, about 10 min after the shift,and 45 min later they divide synchronously. Cells that

phase, to the extent that some cells, which had fulfilledthe old size requirement and were headed for themitotic checkpoint, are forced to return to theinterphase checkpoint [see Fig. 9(b)] and grow tothe larger size dictated by the higher growth rate.This effect shows up as a distinct plateau in cellnumber, as a function of time, and a hiatus in celldivision [Fig. 9(a)]. After a delay of about one hour

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F. 10. Simulation of the effects of shifting a wee1ts mutant cellto the restrictive temperature at different phases in the cell cycle. (a)The phase of the shift is indicated by the arow. At any phase, the shiftaccelerates cells into mitosis. (b) In the cyclin/MPF plane we plot thecyclin balance curve (thin solid line), the wild-type cell cycle (darksolid curve), which locates the cell’s mitotic control system justbefore the shift, and the dimer equilibrium curves (dashed) just afterthe shift (from phase 0.05 to phase 0.3 in intervals of 0.05 CT). Cellsshifted early in the cycle are held up by the S-phase brake, but whenit is released they proceed quickly into mitosis because there is nooperative size control in wee1− cells. Cells shifted later in the cyclego immediately in mitosis because, after the shift, there is nofunctional tyrosine kinase to oppose Cdc25.

The simulation we have just described agreesquite well with the experiments of Thuriaux et al.(1978) except that cells carrying the temperature-sensitive wee1.50 allele do not have a wild-type sizeat the permissive temperature, rather they aresemi-wee (Nurse, 1975; Thuriaux et al., 1978) withslightly delayed S phase (Nasmyth et al., 1979). Oursimulations of this case (not shown) are in perfectagreement with Fig. 2 of Thuriaux et al. (1978).

If we allow wee1− cells to achieve a state of balancedgrowth and division, they will express a cell cycle verydifferent from wild type [Fig. 11(a)]. S phase in the weecycle is shifted to a later position (phase of mid-S=0.4), followed by a short G2.† G1 phase, on the otherhand, is greatly extended in wee cells because they arebornat a relatively small size (0.292, i.e. 51.2%of the sizeof wild-type cells at birth) and do not achieve the criticalsize for START (0.35) until phase 0.26 of their cycle.

F. 11. The wee1− cell cycle. (a) Cyclin and MPF changes asfunctions of time. (b) Cell cycle changes in the cyclin/MPF plane.Thick solid curve: state points (Q) along the cell cycle (cell divisionat phase 0). Thin solid curves: the cyclin balance curve (upper) fromphase 0.26 to phase 0.87 and (lower) from phase 0.87 to 0.26 in thenext cycle. Dashed curves: dimer equilibrium curves (from upper tolower) at phases 0.45, 0.57, 0.59, 0.61, 0.63, and 0.80.

are earlier in the cycle (0–0.2) at the time of thetemperature shift find their unreplicated-DNA brakestill in operation, so they cannot divide along with thesynchronous subpopulation.

† The timingof events inour simulationof thewee1− cell cycledoesnotagreepreciselywithobservations.Forinstance, thephaseofmid-Sin our model (0.4) is 0.11 later than experimentally determined (0.29)for wee1.50 at 35°C (Nurse, 1975). We attribute this discrepancy tothe fact that the gap between M phase and cell division is longer inwee1− cells than in wild-type cells (Creanor & Mitchison. 1993). Thisshortens the observed gap between cell division and mid-S, althoughthe duration of G1 is the same in the model and in the mutant cell.Taking intoaccount thisdiscrepancy in the timingof cell division,ourcalculated time for execution of START (phase 0.26) fits nicely withexperimental data (0.2–0.24) (Fantes & Nurse, 1978; Singer &Johnston, 1985) on the transition point of the cdc10.129 STARTmutation in wee1− cells (cdc10ts wee1− double mutant).

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F. 12. Simulations of cell size and tyrosine phosphorylation levels in mutants at the wee1 and cdc25 loci. ‘‘Cell size’’ (at cell division)and ‘‘Y15P’’ (the maximum concentration of doubly phosphorylated cyclin/Cdc2 dimers during a complete cycle) are expressed relative towild-type values. We simulate both increased levels and decreased activities of both gene products (relative to wild type). The closed trianglesrepresent experimental data points for size: 1 (Russell & Nurse, 1987a, 2 (Russell & Nurse, 1986), 3 (Ducommun et al., 1990), 4 (Nurse, 1975).

When plotted on the cyclin/MPF plane, the weecell cycle looks very different from the wild-type cellcycle [compare Figs. 11(b) and 5(a)]. In particular,there is a G1 checkpoint in the wee cycle (phase0.87–0.26), when cyclin synthesis is turned off and thecell is growing to meet the size requirement forSTART. After START is executed (0.26), cyclinsynthesis recommences, the cyclin balance curveshifts from the low to the high position, and MPFdimers accumulate in tyrosine-phosphorylated formsbecause the S-phase brake is engaged. In thecyclin/MPF plane, the brake shows up as a steeplyincreasing dimer equilibrium curve around the timeof DNA synthesis (phase 0.35–0.45). About 25 minafter S phase is completed, the brake disengages andthe dimer equilibrium curve shifts abruptly to theright (phase 0.57–0.61). Inactive dimers are rapidlydephosphorylated (phase 0.6–0.7) and the cell entersmitosis. G2 phase is short because in wee1− cells thereis no size requirement to be met in late G2. After Sphase is completed, Mik1 is down-regulated, and,there being no other active tyrosine kinase in thewee1− mutant, Cdc25 drives the cell directly into

mitosis. Deletion of cdc25 in a wee1− cell is not lethalbecause there is a back-up tyrosine phosphatase, Pyp3(Millar et al., 1992a), which can drive cells lackingWee1 into mitosis, even though Pyp3, compared toCdc25, is not a very active phosphatase and possiblylacks positive feedback from MPF. If mitosis proceedssuccessfully, cyclin degradation is initiated, and themutant cell exits mitosis exactly as wild-type cells do.

In the model one can decrease Wee1 activity in stepsand see cell size drop abruptly to about half wild-typesize (Fig. 12, lower left) with little change in Y15P levelof Cdc2 (Gould et al., 1990; Enoch et al., 1991; Fleig& Gould, 1991). These changes are observed, perhaps,in certain wee1− alleles which seem to reduce onlypartially the turnover number of Wee1: for example,the wee1.302 allele exhibits a partial-wee phenotype(division size=75% of wild type) (Fantes, 1981), andthewee1.50 temperature-sensitive allele is a partial-weeat the permissive temperature (division size=81% ofwild type) (Nasmyth et al., 1979).

Because loss of Wee1 activity does not interfere withthe signal from unreplicated DNA, wee1− cells haveperfect S–M coupling, as observed in experiments

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(Enoch & Nurse, 1990) and in our simulations. Thereis no compensation for S-phase delays in wee1− cells[Fig. 7(b)] because the duration of G2 is constant(Fantes, 1984; Fantes & Nurse, 1978): the time it takesto remove the brake exerted by S phase plus the timeit takes to dephosphorylate inactiveMPFand carry thecell into mitosis.

Singer & Johnston (1985) grew wee1− cells inthe continuous presence of low concentrations ofhydroxyurea, which lengthened the duration of Sphase without blocking DNA replication completely.They observed that longer S phase causes a shorteningof G1, as evidenced by shifting START to an earlierpoint in the cycle. Our model behaves similarly(simulations not shown). Furthermore, the modelpredicts that, as the concentration of hydroxyureaincreases, START-control goes cryptic, balancedgrowth is no longer maintained, and cells should getlarger each cycle until they become inviable.

In our model, as in experiments (Fantes & Nurse,1978), wee1− cells do not respond at mitosis tonutritional shifts, because mitotic nutritional controloperates through Wee1.

wee1OP

If Wee1 is overproduced, either by increasing thegene dosage of wee1+ or by putting the wee1+ gene ona plasmid behind a good promoter, cell size at divisionincreases (see Fig. 12, upper right quadrant), exactly asobserved experimentally (Russell & Nurse, 1987a;Millar et al., 1992a, b). Even though Wee1 protein isproduced in excess, maximum level of tyrosinephosphorylation remains constant (Fig. 12, upperright, open squares).† This prediction of the model hasyet to be verified experimentally. Massive overexpres-sion of Wee1 leads to cell cycle arrest in the model andin experiment (Millar et al., 1992a,b).

cdc25OP

Another way to produce the wee phenotype is tooverexpressCdc25 (Fig. 12, lower right).Cell size dropsquickly to about half wild-type size (Russell & Nurse,1986). (In the model the drop is more abrupt thanexperiments warrant.) In contrast to the case of Wee1reduction, Cdc25 overexpression causes a dramaticdrop in the level of tyrosine phosphorylation (Gouldet al., 1990). Because S–M coupling depends on thephosphorylation of Tyr-15, cdc25OP mutants are

subject to abnormal dependency between S phase andM phase, as we shall see in the next section.Interestingly, in cdc25OP cells, thepool ofCdc25proteindoes not get heavily phosphorylated before M phaseprobablybecause the great excess of substrate saturatesthe protein kinase. The cell goes into and out of mitosisby activating only a small fraction of the Cdc25 pool(simulations not shown).On an SDSgel therewould beno significant shift of the Cdc25 band to the lowermobility (heavily phosphorylated) form. Only if cellswere blocked in mitosis (say, cdc25OP nda3ts) wouldthere be enough time for active MPF to phosphorylatemost of the Cdc25 pool and cause a noticeable gel shift(P. Russell, personal communication).

cdc25−

If Cdc25 activity is reduced (Fig. 12, upper left),cell size increases and Y15P levels remain constant.Larger size depresses Wee1 activity so that the tyrosinephosphorylation level remains about the same in spiteof reduced Cdc25 activity. Experimentally, Cdc25activity can be reduced in steps by mutating subsets ofits potentially regulatory phosphorylation sites or bytruncating the N-terminus of the protein (P. Russell,private communication). If we reduce Cdc25 activitysufficiently, cells become blocked in G2, which is thecharacteristic phenotype of cdc25− alleles.

hus−

Next we turn from mutations that interfere with sizecontrol to mutations that disrupt S–M coupling. Anumber of non-allelic hydroxyurea-sensitive (hus)mutants have been isolated in fission yeast (Enoch &Nurse, 1990). These cells are perfectly normal exceptthat, in the presence of hydroxyurea, they enterM phase with unreplicated DNA, which is a lethalmistake. We model this mutation by eliminatingcomponent W from Fig. 2(b). Although unreplicatedDNA no longer exerts a brake on MPF, the sizerequirement is still operative, so G2 is not shortened(cells are not advanced into mitosis). Under normalconditions, DNA synthesis is completed long beforecells grow large enough to pass the interphasecheckpoint. But, in the presence of hydroxyurea, cellgrowth eliminates the interphase checkpoint (i.e. pullsthe dimer equilibrium curve below the cyclin balancecurve, see Fig. 13) on schedule with untreated cells,so the treated cells enter mitosis with catastrophicresults.Enoch et al. (1992) call this the ‘‘cut’’ phenotypebecause, in the presence of hydroxyurea, cell platesform prematurely and cut the unreplicated, undividednucleus with disastrous results. We call the phenotype‘‘conditional mitotic catastrophe’’ (cond MC) becauseit arises only in the presence of hydroxyurea.

† Tyr-15 phosphorylation level of MPF is independent of Wee1level, either reduced or overproduced (Fig. 12), because the controlsystem automatically regulates Wee1 activity to maintain the correctMPF activity. If Wee1 level is reduced, cell size decreases so the totalactivity of Nim1 is smaller, and Wee1 is more active. On the otherhand, if Wee1 is overproduced, cell size increases, Nim1 activity islarger, and Wee1 is down-regulated to a total activity comparableto wild-type cells.

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F. 13. The cell cycle in hus− cells. As before, the dark solid curveis the cell cycle trajectory, the thin solid curve is the cyclin balancecurve, and the dashed curves are dimer equilibrium curves fromphase 0.10 (upper) to phase 0.70 (lower) at intervals of 0.15 CT.There is no S-phase brake on progress toward mitosis: compare thedimer equilibrium curve at phase 0.1 in this figure with that in Fig.6(a). As the cell grows, the dimer equilibrium curve is brought downuntil the G2 arrest point disappears and the cell proceeds towardmitosis. The same sequence of events happens in the presence ofhydroxyurea, resulting in a cut phenotype (Enoch & Nurse, 1990).

cdc25OP wee1ts

The phenotype of these cells depends on tempera-ture and the extent of overproduction of Cdc25.Consider first the situation at the permissivetemperature, i.e. cdc25OP wee1+. As we increase theextent of overproduction of Cdc25, cell size decreasesby about 50%, Y15P levels drop (Fig. 12), and G2shortens to a plateau of about 40 min (Fig. 15). Foroverexpression less than about ten-fold, the S–Mcoupling mechanism is intact, because the addition ofhydroxyurea blocks cells from entering mitosis(Fig. 15). Larger overexpression leads to conditionalmitotic catastrophe: cells are viable in the absence ofhydroxyurea, but theywill divide catastrophically in its

F. 14. Unconditional mitotic catastrophe in hus− wee1ts cells. (a)Cells are shifted to the restrictive temperature at different phases inthe cell cycle (time=0 at cell division). MPF activity rises abruptlyas all available cyclin/Cdc2 dimers are rapidly dephosphorylated. (b)In the cyclin/MPF plane we plot the hus− cycle (from Fig. 13) tolocate the control system at various stages in the cycle at the timeof the temperature shift. The curve that runs close to the diagonal,[total cyclin]=[active MPF], is the dimer equilibrium curve after theshift. The length of each arrow represents the movement of the statepoint over 10 min after the shift. The cell shifted in G1 (phase 0.9)enters mitosis 10–20 min after the shift, long before it can completeDNA replication.

Unconditional Mitotic Catastrophe

If we burden fission yeast with mutations in both theS–M coupling pathway and the size controlmechanism, the results are fatal (Enoch et al., 1992).

hus− wee1ts

On raising such cells to the restrictive tempera-ture, they start immediately toward M phase, nomatter their size or state of DNA replication[Fig. 14(a)]. For cells late in the cycle, this is no problem(at first), but cells which are in G1 when shifted willenter mitosis with unreplicated DNA, withcatastrophic results. Of course, cells that complete thefirst mitosis normally will undergo this catastrophe intheir second cycle. This is a lethal phenotype, calledunconditional mitotic catastrophe. Clearly there aretwo and only two requirements to be met by fissionyeast at the interphase checkpoint: completion ofDNA synthesis and growth to a minimal size.Knocking out both controls is a lethal combination ofmutations.

In Fig. 14(b) we show what is happening in thecyclin/MPF plane. On shifting to the restrictivetemperature the dimer equilibrium curve collapsesalmost to the diagonal line [total cyclin]=[activeMPF]. As cyclin is synthesized, it is converted directlyinto active MPF; there are no active tyrosine kinasesto inactivate the dimers.

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F. 15. Mitotic catastrophe in cdc25OP strains. We plot theduration of G2 as a function of Cdc25 overproduction (wild-typelevel=1). By ‘‘G2’’ in the presence of hydroxyurea (+HU) we meanthe period from the end of S phase in the control (−HU) to the startof mitosis in the treated (+HU) culture. G2Q0 in the absence of HUrepresents an unconditional MC, and G2Qa in the presence of HUrepresents a conditional MC.

presence (Fig. 15). Notice that, between 20- and27-fold overexpression of Cdc25, cells enter mitosis atexactly the same time with or without hydroxyurea inthe growth medium.† Finally, if Cdc25 is over-expressed sufficiently (q27-fold) in the model, the cellsenter M phase before finishing S (G2 duration Q0),which is unconditional mitotic catastrophe. This lattereffect has never been observed, perhaps because noknown promoters can drive Cdc25 to such high levelsor because Tyr-15 reactions are compartmentalized inthe cell (see the discussion).

At the restrictive temperature, i.e. cdc25OP wee1−, wehave a similar scenario (Fig. 15), just moved to lowerlevels of Cdc25. In particular, for a 15-foldoverexpression of Cdc25, cdc25OP wee1+ is aconditional MC but cdc25OP wee1− is an unconditionalMC (Russell & Nurse, 1986). That is to say, the effectsof wee1 mutation and cdc25 overexpression areadditive: either change alone produces viable wee cells,but both changes in concert produce inviable MC cells(Russell & Nurse, 1986, 1987a).

Other Mutations

In Table 1 we summarize the results of oursimulations of many single- and multiple-locus

† Enoch & Nurse (1990) and Enoch et al. (1992) have observedthis behavior in the cdc2.3w mutant, which behaves similarly tocdc25OP in many ways. Perhaps Cdc2.3w remains active even whenphosphorylated on Tyr-15 (Enoch et al., 1991). We do not try tomodel the dominant cdc2-wee mutations, cdc2.1w (Nurse &Thuriaux, 1980) and cdc2.3w (Fantes, 1981), because there is nowidely accepted explanationof themolecular basis of this phenotype.

T 1Properties of various mutant strains of fission yeast, as determined by the model

Strain Y15P Size G1 S G2 +HU Parameter

WT 100 1.00 0.14a 0.10a 0.68a cdcb see tablewee1− 82c 0.51d 0.54 0.11 0.27 cdcb V0wee=0.06×Dwee1 77c 0.51 0.54 0.12 0.26 cdc Weetotal=0wee1OP† 98 2.53e 0.14 0.10 0.68 cdc Weetotal=7cdc25OP† 45c 0.50f 0.57 0.14 0.22 cutb ksyn=15×cdc2OP 103 0.95 0.14 0.10 0.68 cdc Cdc2total=30cdc13OP 128 0.96g 0.14 0.10 0.62 cdc k1=10×Dmik1 99 0.94h 0.14 0.11 0.68 delayed cuti Miktotal=0mik1OP 109 1.89h 0.14 0.10 0.68 cdc Miktotal=8hus− 98 1.00j 0.14 0.11 0.68 cutj kh=0Dnim1 99 1.27k 0.14 0.11 0.68 cdc Nim1=0nim1OP 87 0.52k 0.52 0.11 0.29 cdcb Nim1=15×nim1OP wee1− 81 0.51k 0.54 0.11 0.27 cdc Nim1=15×, V0wee=0.06×cdc25OP Dmik1 30 0.47h 0.66 0.18 0.07 cut ksyn=15×, Miktotal=0hus− wee1− 18 unconditional mitotic catastrophe j kh=0, V0wee=0.06×cdc25OP wee1− 25 unconditional mitotic catastrophe f ksyn=15×, V0wee=0.06×cdc25OP nim1OP 28 unconditional mitotic catastrophe l ksyn=15×, Nim1=15×Dmik1 wee1− 28 unconditional mitotic catastropheh Miktotal=0, V0wee=0.06×cdc25OPcdc13OP 52 unconditional mitotic catastrophe ksyn=15×, k1=2×cdc25OPwee1OP 109 1.13 0.14 0.10 0.68 cdcj ksyn=15×, Weetotal=15×cdc25OPwee1−mik1OP 74 0.51h 0.55 0.12 0.25 cdch ksyn=15×, V0wee=0.06×

Miktotal=8

The differential equations in theAppendix are solved using the parameter values given there, except for the change(s) listed in the last column.The notion ‘‘0.06×’’ means 0.06 times the wild-type value for this parameter.

Y15P: the maximum in doubly phosporylated form.Size: cell mass or length relative to WT.G1, S, G2: phase duration as fraction of cycle time; M-phase duration is 0.08 in all cases except cdc13OP, for which it is 0.14.† For more data see Figs 12 and 15.References: a(Mitchison, 1989), b(Enoch & Nurse, 1990), c(Gould & Nurse, 1990; Enoch et al., 1991), d(Nurse, 1975), e(Russell & Nurse,

1987a), f(Russell & Nurse, 1986), g(Booher & Beach, 1988), h(Lundgren et al., 1991), i(Rowley et al., 1992), j(Enoch et al., 1992), k(Russell& Nurse, 1987b), l(Wu & Russell, 1993).

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mutations in elements of the cell cycle control systemin S. pombe. The model is in complete agreementwith experiments.

cdc13OP

Cyclin overproduction has no effect on the timing ofmitosis in fission yeast (Booher & Beach, 1988),in contrast to the situation in oocytes where additionalcyclin mRNA can induce premature activation ofMPF (Minshull et al., 1989a; Novak & Tyson 1993b;Swenson et al., 1986). Oocytes have a considerableexcess of Cdc2 subunits over cyclin subunits, soadditional cyclin can generate more MPF dimers. Forfission yeast we presume that the two subunits are innearly equimolar quantities in G2, so additional cyclinaccumulates harmlessly as unbound subunits.

cdc2OP

Cdc2 overproduction has very little effect on size atdivision (Durkacz et al., 1986). However, a cdc2OP cellshould behave like an oocyte in the sense that cyclinoverproduction should advance cdc2OP cells intomitosis (i.e. shorten G2) even to the extent of anunconditional MC.

cdc25OP cdc13OP

Overproduction of Cdc25 produces wee cells(Russell & Nurse, 1986) that are controlled by sizecontrol at START, followed by a ‘‘cyclin timer’’.As cells synthesize cyclin after START, they go intomitosis as soon as they accumulate enough T167-phos-phorylated dimers. If completion of S phase is delayedby hydroxyurea, the cells exhibit conditional mitoticcatastrophe (Enoch&Nurse, 1990). If the rate of cyclinsynthesis is increased by multiple copies of cdc13, thesecells can be forced into MC in the absence ofhydroxyurea. To our knowledge, this genotype has notbeen described in the literature.

cdc25OP wee1OP

Overexpression of both cdc25+ and wee1+ re-establishes normal cell size at division and normal S–Mcoupling.

Dmik1

Because Mik1 activity is down-regulated in late G2,Mik1 does not play any role in late G2 and Dmik1mutants divide at roughly wild type size. In themodel, Dmik1 cells should show delayed conditionalmitotic catastrophe because in hydroxyurea they

will eventually grow large enough to down-regulateWee1 and enter mitosis. Although Dmik1 cells wereoriginally reported not to show cut phenotype inhydroxyurea (Lundgren et al., 1991), a later paperreports that Dmik1 cells move into septation anddie ahead of wild-type cells (Rowley et al., 1992).Similarly, just as reported (Lundgren et al., 1991),Dmik1cdc25OP is not catastrophic, although the modelpredicts that higher overexpression of Cdc25 shoulddrive Dmik1 cells into mitotic catastrophe.In summary, deletion of mik1 has much differentconsequences than wee1 mutation, exactly as observedexperimentally.

mik1OP

Overproduction of Mik1 makes cells larger and alsorescues the wee1− cdc25OP mitotic catastrophe.†

Dnim1

Deletion of Nim1 increases size by 27%, as reported(Russell & Nurse, 1987b), because cells must grow toa larger size to inactivate Wee1 sufficiently to enter Mphase. The nim1 locus is identical to cdr1, and it is wellknown that when a cdr1− culture is shifted tonitrogen-starvation medium, many cells block in G2(Young & Fantes, 1987) instead of piling up at the G1checkpoint, as normal cells do. This happens becausethe nutritional shift stops cell growth but does notdown-regulate Wee1 (the signal is not relayed bydefective Nim1), so all cells between the G1 and G2checkpoints block in G2.

nim1OP

Nim1 overproduction has the same effect asknocking out Wee1 (Russell & Nurse, 1987b). SinceNim1 works only on Wee1 and does not down-regulateMik1 as suggested by Wu & Russell(1993), overproduction of Nim1 is not additive withwee1 mutation: nim1OP wee1− are just wee, notcatastrophic (Russell & Nurse, 1987b).

From the model in Fig. 2 it is evident that manyprotein phosphatases play crucial roles in mitoticcontrol in fission yeast (Yanagida et al., 1992),especially the two that activate Wee1 and the one(or more) that inactivates Cdc25. These phosphatasesare negative regulators of M phase: knocking them outadvances cells into mitosis and overexpressing themdelays mitosis (Kinoshita et al., 1990; Millar et al.,1992b; Ottilie et al., 1992; Kinoshita et al., 1993). Ourmodel is in general agreement with these results(simulations not shown), but not enough is known yetabout these phosphatases to associate them withspecific steps in the mechanism.

† Lundgren et al. (1991) discovered mik1 by looking foroverexpressed genes that could rescue cdc2.3w wee1− from mitoticcatastrophe. Because cdc2.3w behaves similarly to cdc25OP, wepresume that mik1OP cdc25OP wee1− is a viable genotype.

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Discussion

The cell cycle of fission yeast is organized aroundthree checkpoints: an interphase (G2) checkpointwhere the cell makes sure that DNA synthesis iscomplete and that cell size is sufficient, an M-phasecheckpoint where the cell makes sure that itschromosomes are successfully aligned on themetaphase plate, and START (a G1 checkpoint) wherethe cell verifies that it is large enough to begin anotherround of DNA replication. We model the interphasecheckpoint with a molecular mechanism for MPFactivation and inactivation with regulatory signalsfrom unreplicated DNA and cell size (Fig. 2). Mostof our modeling and analysis concerns theinterphase checkpoint because here we findenough molecular details to build a reasonablemodel and enough physiological and geneticinformation to test it thoroughly. The M-phasecheckpoint is related to the initiation of cyclindegradation, but little else is known about it.Wemodelit by a primitive requirement for phosphorylation of a‘‘target’’ protein [Fig. 2(c)]. START is a secondplace in the yeast cycle where size control is exerted.Because molecular details about START are stillmeagre, we model G1 progression by a logicalflow-chart (Fig. 3).

We constructed the model based on facts andideas distributed through many experimental paperson the fission yeast cell cycle, and it is a reasonablycomplete and accurate picture of the control of celldivision in S. pombe, as best we know at the presenttime. Clearly the model, in its entirety, is toocomplicated to be studied reliably by the intuitivearguments that have been used in the literature todescribe its individual pieces. Therefore, we con-verted the model in Figs 2 and 3 into a set ofdifferential equations and studied these equations bycomputer simulation and by graphical analysis ofdynamical portraits. The model is successful inquantitative detail in accounting for all the majorphysiological and genetic properties of the fissionyeast cell cycle in terms of the underlying mechanismof molecular controls.

Our model of the fission yeast cell cycle is verysimilar to an earlier model proposed for Xenopus

oocyte extracts and intact embryos (Novak & Tyson,1993b); even the parameter values are similar. It isremarkable that essentially the same mechanism canaccount for two very different types of divisioncycles: frog eggs show rapid MPF oscillationswithout growth and largely independent of periodicDNA replication and nuclear division, whereasfission yeast cycles are much slower, with G1 and G2phases, with strict dependencies between S and Mphases, and with two checkpoints for cell size.Nonetheless, essentially the same mechanism forMPF activation and inactivation can describenumerous subtle details of the entry into mitosis inboth cell types. Not only are the basic componentsof M-phase control the same in frog eggs andfission yeast, but also they appear to be wiredtogether in the same way despite drastic differencesin the physiology of the cell cycle in these twoorganisms.

We suggest that these differences can be ‘‘dynami-cal’’ in origin (Tyson, 1991; Novak & Tyson, 1993a).That is, the behavior of the control system (e.g.autonomous oscillations or stable rest states) dependsin subtle ways on the values of the parameters (rateconstants, enzyme concentrations, etc) that specify thesystem. ‘‘Bifurcation theory’’, which describes thedependence of the behavior of a dynamical systemon its parameters, is a natural mathematical toolto derive the diversity of cell cycle physiology froma fundamentally universal mechanism of cell cyclecontrol.†

Other authors in earlier work have pointed outthat many features of cell cycle control are genericproperties of dynamical systems (see Hyver & LeGuyader, 1990; Goldbeter, 1991, 1993; Norel &Agur, 1991; Thron, 1991; Tyson, 1991; Obeyeskereet al., 1992; Busenberg & Tang, 1994). These authorsmodel the cell cycle with small systems of nonlinearordinary differential equations (two or three ODEs):capturing cell cycle control in broad strokes andachieving rough agreement with major qualitativefeatures of the cell division cycle. We are pushing thisapproach to the next level: solving large systems ofODEs that describe the molecular control system insufficient detail to achieve quantitative agreementwith a broad range of experimental observations. Toretain some of the spirit and intuitive appeal of thetwo-component models, we analyze our numericalresults by projecting the multi-dimensional statespace onto two principal coordinates, total cyclin andactive MPF. By introducing dimer-equilibrium andcyclin-balance curves we can envision the forces thatdrive the cell cycle in wild-type and mutant fissionyeast.

† Biologists with a solid understanding of differential calculus canfind intelligible descriptions of bifurcation theory in a review byOdell (1980) and a textbook by Edelstein-Keshet (1988).

. . . 300

?

The model suggests that, in wild-type fission yeast,there is no event in G2 that can be called a mitotic‘‘trigger’’. The final requirement for mitosis (namelycell size large enough) is met in mid-cycle, not in lateG2. It initiates a Tyr-15 dephosphorylation reactionthat is agonizingly slow at first, not a fast, explosiveentry into mitosis. The commitment to Tyr-15dephosphorylation is not irrevocable; for instance, anutritional shift-up can recall ‘‘committed’’ cells to theinterphase checkpoint they had left [Fig. 9(b)].Only thelast stage of entry into mitosis, when the MPF/Cdc25positive feedback loop is fully engaged, looks anythinglike an irrevocable commitment. But the decision to setthis process in motion was made long before the finalexplosive activation of MPF. The mitotic controlsystem is not so much like squeezing the trigger of apistol as it is like lighting the fuse on a stick ofdynamite. There is a considerable time lag after theinitial decision, during which you may change yourmind, clip the fuse and abort the explosion.

Normal progress through the cell cycle can bedisrupted by treatments with metabolic inhibitorsor by mutation. For instance, mutation of wee1removes the size requirement in late G2, so wee1− cellsenter mitosis soon after the S-phase brake is removed(Fig. 11). These cells divide at an abnormally small size,determined by the G1 size requirement (at START)followed by an {S+minimum G2} timer (Fantes &Nurse, 1978) enforced by the S–M couplingmechanism. Abnormally small cells can also be createdby overexpressing Cdc25, but, unlike the wee1− cycle,the cdc25OP cell cycle is determined by size control atSTART followed by a cyclin timer (the time necessaryto synthesize enough cyclin to drive MPF activity highenough to initiate mitosis). There is no interphasecheckpoint at all in cdc25OP cells: they lack both S–Mcoupling and G2 size control. This is apparent from thefact that cdc25OP cells show a conditional mitoticcatastrophe whereas wee1− cells have perfect S–Mcoupling.

The double mutant cdc25OP wee1− exhibitsunconditional mitotic catastrophe (going into M phasewhile the cell is still replicating DNA even in theabsence of hydroxyurea). The fact that cdc25OP andwee1− are additive has been taken to indicate thatWee1 and Cdc25 catalyze sequential reactions ratherthan opposing reactions, but our model demonstratesthat the widely accepted notion of Wee1 and Cdc25 asa kinase/phosphatase pair in opposition is consistentwith the additive effects of mutations in these genes.

Unconditional mitotic catastrophes (MC) arise inour model only if several requirements are met.First, mutations must abolish both pathways forTyr-15 phosphorylation (the unreplicated-DNA path-way through Mik1 and Cdc25 and the size-controlpathway through Wee1) so there is no interphasecheckpoint in the cycle. Second, the cell mustsynthesize cyclin, which means that it must executeSTART. Third, it must make cyclin fast, the cyclinmust combine avidly with Cdc2 monomers, thecyclin/Cdc2 dimers must be good substrates for CAK,and the Thr-167 phosphorylated dimer must havenearly wild-type MPF activity. Even if the interphasecheckpoint is knocked out, the second and thirdrequirements may prevent mitoses from beingunconditionally catastrophic. For instance, theengineeredmutant ofCdc2, inwhichTyr-15 is replacedby phenylalanine, certainly permits no G2-arrestedstate, and yet it is a viable mutation (Gould & Nurse,1989), presumably because the aberrant Cdc2associates less readily with cyclin or has lower MPFactivity as a dimer. By tinkering with the rate of cyclinB synthesis, it should be possible to rescue certainlethal MC genotypes—as does Dcig2 (Bueno &Russell, 1993)—or to push borderline genotypes overthe edge—for example, cdc13OP should drive cdc25OP

into lethal MC. Our second requirement, that cellsmust pass START to go into MC, is consistent with allknown MC phenotypes except wee1− Dmik1 (Lund-gren et al., 1991).

Many physiological and genetic experimentsdemonstrate the importance of cell size in determiningthe timing of M phase in the S. pombe cell cycle (Nurse,1975; Fantes, 1977; Fantes & Nurse, 1977; Nurse &Fantes, 1981; Russell & Nurse, 1987a). In this modelwe have attributed G2 size control to Nim1 and someother protein kinase (PK) that inactivate Wee1 byphosphorylating its C-terminus. We assume that theactivities of these two protein kinases increase inproportion to cell size; for instance, they could be madein step with all the other proteins in the cell (increasingexponentially in total amount as the cell grows) andthen collected in some constant-volume compartmentof the cell where they phosphorylate Wee1. Anotherscenario might have size operating through thephosphatases that oppose Nim1 and PK in theinactivation of Wee1: if cell growth ‘‘diluted out’’ thesephosphatases, it would shift the balance towardinactive forms of Wee1, and this alternative modelwould behave similarly to the one we have presented.A third possibility, which we described earlier (Novak& Tyson, 1993a), is that size control operates through

301

Cdc25 rather thanWee1.At present there is insufficientevidence to decide confidently for one of thesepossibilities or some other that may come to mind.

If increasing cell size down-regulates Wee1, then avery large wee1+ cell should be similar to a wee1− cell.However, wee1+cdc25− cells, which grow quite large,are inviable whereas wee1−cdc25− cells are viable(Fantes, 1979). On the other hand, if size controloperates through Cdc25, then a wee1−cdc25+ cell inhydroxyurea should eventually grow large enough toinitiate mitosis, but conditional mitotic catastrophesare not observed in wee1− strains (Enoch & Nurse,1990). Although we can circumvent either of theseproblems by a suitable choice of parameters (so thatreal cells never grow large enough to generate themitotic signal), we want to point out that similarobjections can be raised to both scenarios.

If size control acts through Wee1, then wee1− strainsshould have a G2 phase of fixed duration, independentof cell size. But if size acts through Cdc25, then thelength of G2 phase in large wee1− cells should beslightly shorter than that in smaller cells. Experimentsby Fantes & Nurse (1978) suggest that G2 phase inwee1− is of fixed duration, independent of size, so weprefer the Wee1 scenario, although the whole question,in our minds, is still open.

The model we have explored ignores compart-mentalization of M-phase control for two reasons.First, there is no widely accepted view of where inthe cell various reactions take place, so any model thatrelies heavily on transport processes between or-ganelles and cytoplasm would be highly speculative.Second, there is no convincing evidence thatcompartmentalization plays an essential role in themechanism, and we wanted to see to what extenta non-compartmental model could account for theobserved properties of the fission yeast cell cycle.Frankly, we were surprised at how successful such amodel is.

If compartmentalization of the mechanism turns outto be important, it can be introduced into the model inlater versions. For instance, cyclin targets Cdc2 to thenucleus (Booher et al., 1989), and there is somesuspicion that Wee1 is located primarily in the nucleusand Cdc25 in the cytoplasm. To dephosphorylatepreMPF in late G2, either Cdc25 must be transportedinto the nucleus or cyclin/Cdc2 dimersmust be shuttledto the cytoplasm and back into the nucleus. If thesetransport processes are rate-limiting, compartmental-ization would be important and would magnify theadditive effects of cdc25OP and wee1−. But to pursuemodeling in this direction seems premature at present.

The model makes a number of predictions about thefission yeast cell cycle that have not yet beenestablished experimentally:

, Tyrosine phosphorylation in wee1OP and cdc25-truncated mutants should remain very close tothe wild-type level, even though size at divisionincreases dramatically.

, Sublethal doses of hydroxyurea to wee1− cells(slowing down DNA synthesis without stopping italtogether) should lead to unbalanced growth,with cells getting larger each cycle until they die.

, Although cyclin overproduction does not advancewild-type cells into mitosis, it should advancecdc2OP mutants. This experiment would test ourassumption that cyclin B and Cdc2 are present inroughly equimolar amounts in G2 phase ofwild-type cells.

, Cyclin overproduction should drive cdc25OP

mutants into unconditional mitotic catastrophe.In Fig. 2 we propose how the pieces of the

M-phase-control puzzle need to be assembled in orderto explain the observed physiology and genetics ofthe fission yeast cell cycle. Is there a crucial experimentthat could establish or demolish the entire edifice? Themost crucial and still controversial part of our modelis the N-shape of the dimer equilibrium curve, whicharises from the positive feedback loops that activateMPF. All of the subtle behavior of the model can betraced ultimately to deformations of this N-shapedcurve, as we have tried to show with our dynamicalportraits in the cyclin/MPF plane. And yet we haveonly indirect evidence for the N shape, for example therecall of ‘‘committed’’ cells by a nutritional shift-up[Fig. 9(b)]. In our earlier paper on the Xenopus cellcycle (Novak & Tyson, 1993b), we proposed a simpleanddirect testof theN-shapeddimerequilibriumcurve.

Our model is a collection of many assumptionsabout the molecular mechanism and logical organiz-ation of the eukaryotic cell cycle, and some of theseassumptions may prove faulty. As our picture of cellcycle controls is refined and elaborated over time, thetechniques of model-building and mathematicalanalysis illustrated in this paper will remain valid anduseful tools for correlating the physiology, geneticsand molecular biology of the cell cycle.

We acknowledge support from NSF grants DMS-9123674, MCB-9207160, and INT-9212471, and from theHungarian Academy of Sciences, grant OTKA 5-376. We aregrateful to Paul Russell and Murdoch Mitchison for sharingwith us their unpublished data and for commenting on anearly draft of this paper. We also thank Kathy Chen for hergenerous advice at all stages of this research.

. . . 302

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APPENDIX

The model outlined in Figs 2 and 3 is formally defined in terms of the following differential equations and rateconstant definitions:

ddt

[w]=k1 [AA]−k2 [w]−k3 [w] [q]

ddt

[-qw-]=kINH [-qw-P]−0 kwee

KM+[-qw-]+kMik+kCAK+k21[-qw-]+k25 [P-qw-]+k3 [w] [q]

ddt

[P-qw-]=kwee [-qw-]KM+[-qw-]+kMik [-qw-]−(k25+kCAK+k2)[P-qw-]+kINH [P-qw-P]

. . . 304

ddt

[P-qw-P]=kwee [-qw-P]KM+[-qw-P]+kMik [-qw-P]−(kINH+k25+k2)[P-qw-P]+kCAK [P-qw-]

ddt

[-qw-P]=kCAK [-qw-]−0kINH+kwee

KM+[-qw-P]+kMik+k21[-qw-P]+k25 [P-qw-P]

ddt

[q]=k20[-qw-]+[P-qw-]+[P-qw-P]+[-qw-P]1−k3 [w] [q]

ddt

[cdc25-P]=ka [-qw-P]([total cdc25]−[cdc25-P])

Ka+[total cdc25]−[cdc25-P]−

(kbs+Fb [W])[cdc25-P]Kb+[cdc25-P]

ddt

[Wee1]=kf1 [Wee1-P]−ke1 [-qw-P] [Wee1]+kf2 [P-wee1]

Kfs2+[P-wee1]−

ke2 [Wee1]Ke2+[Wee1]

([PK]+[Nim1])mass

ddt

[Wee1-P]=ke1 [-qw-P] [Wee1]−kf1 [Wee1-P]+kf2 [P-Wee1-P]

Kf2+[P-Wee1-P]−

ke2 [Wee1-P]Ke2+[Wee1-P]

([PK]+[Nim1])mass

ddt

[P-Wee1]=kf1 [P-Wee1-P]−ke1 [-qw-P] [P-Wee1]−kf2 [P-Wee1]

Kf2+[P-Wee1]+

ke2 [Wee1]Ke2+[Wee1]

([PK]+[Nim1])mass

[P-Wee1-P]=[total Wee1]−[Wee1]−[Wee1-P]−[P-Wee1]

ddt

[X]=kt−kx [X]

ddt

[W]=kh [X]([total W]−[W])Kh+[total W]−[W]

−kg [W]

Kg+[W]

ddt

[Mik1]=kp [W]([total Mik1]−[Mik1])−ko [Mik1]

ddt

[IE-P]=ki [-qw-P]([total IE]−[IE-P])

Ki+[total IE]−[IE-P]−

kj([total Ta]−[Ta-P])[IE-P]Kj+[IE-P]

ddt

[UbE]=kc [IE-P]([total UbE]−[UbE])

Kc+[total UbE]−[UbE]−

kd [UbE]Kd+[UbE]

ddt

[Ta-P]=kk [-qw-P]([total Ta]−[Ta-P])

Kk+[total Ta]−[Ta-P]−

kl [Ta-P]Kl+[Ta-P]

ddt

[total Cdc25]=ksyn−kdeg [total Cdc25]

ddt

[DNA]=V

1+F[Ta-P], [DNA]=0 at onset of S-phase, =1 at the end of S-phase

ddt

[mass]=m[mass], [mass]=1 at the end of M-phase in WT cells

k25=V'25([total Cdc25]−[Cdc25-P])+V025 [Cdc25-P]

kwee=V'wee([total Wee1]−[Wee1])+V0wee [Wee1]

k2=V'2 ([total UbE]−[Ube])+V02 [UbE]

kmik=V'mik([total Mik1]−[Mik1])+V0mik [Mik1]

305

To model the wild-type cell cycle of fission yeast, we use the following parameter values, listed in order as theyappear in the differential equations. The units of all rate constants are min−1.

k1 [AA]=0.025 (cyclin synthesis on), =0.003 (cyclin synthesis off)

k2=0.025 ([total UbE]−[UbE])+0.5 [UbE], [total UbE]=1

k3=10, kINH=0.01, kCAK=1, KM=1

k25=0.045 ([total Cdc25]−[Cdc25-P])+0.45 [Cdc25-P], [total Cdc25]=1

kwee=0.0333 ([total Wee1]−[Wee1])+0.75 [Wee1], [total Wee1]=1

kmik=0.02 ([total Mik]−[Mik])+0.2 [Mik], [total Mik]=1

ka=0.5, Ka=0.1, kbs=0.2, Fb=2, Kb=0.1

ke1=1, kf1=1, ke2=1, Ke2=0.15, kf2=1, Kf2=0.15

[PK]=0.965, [Nim1]=0.258

kt=0.1 (when cell is in post-start G1 and S-phase), =0 (otherwise), kx=0.1

kh=2, Kh=0.05, kg=0.2, Kg=0.05, [total W]=1

kp=2, ko=0.2,

ki=0.2, Ki=0.01, kj=0.2, Kj=0.01, [total IE]=1

kc=0.1, Kc=0.01, kd=0.05, Kd=0.01

kk=1, Kk=0.02, kl=0.5, Kl=0.02, [total Ta]=1

ksyn=0.04 G, G=cdc25 gene dosage

kdeg=0.04 (during most of the cycle), =0.2 (for 20 min after M)

V=0.0555, F=15, m=0.00385