J. theor. Biol. (2001) 210, 249}263doi:10.1006/jtbi.2001.2293, available online at http://www.idealibrary.com on

Regulation of the Eukaryotic Cell Cycle: Molecular Antagonism, Hysteresis,and Irreversible Transitions

JOHN J. TYSON*- AND BELA NOVAK?

*Department of Biology,

FIG. 1. The cell cycle. Outer ring illustrates the chromosome cycle. The nucleus of a newborn cell contains unreplicatedchromosomes (represented by a single bar). At Start, the cell enters S phase and replicates its DNA (signi"ed by replicationbubbles on the &&chromosome''). At the end of S phase, each chromosome consists of a pair of sister chromatids (X) heldtogether by tethering proteins. After a gap (G2 phase), the cell enters mitosis (M phase), when the replicated chromosomes arealigned on the metaphase spindle, with sister chromatids attached by microtubules to opposite poles of the spindle. At Finish,the tether proteins are removed so that the sister chromatids can be segregated to opposite sides of the cell (anaphase). Shortlythereafter the cell divides to produce two daughter cells in G1 phase. The inner icons represent the fundamental molecularmachinery governing these transitions. Start is triggered by a protein kinase, Cdk, whose activity depends on association witha cyclin subunit. Cdk activity drives the cell through S phase, G2 phase, and up to metaphase. Finish is accomplished byproteolytic machinery, APC, which destroys the tethers and cyclin molecules. In G1 phase, APC is active and Cdk inactive,because it lacks a cyclin partner. At Start, the APC must be turned o! so that cyclins may accumulate. Cdk and APC areantagonistic proteins: APC destroys Cdk activity by degrading cyclin, and cyclin/Cdk dimers inactivate APC by phos-phorylating one of its subunits.

250 J. J. TYSON AND B. NOVAK

nucleocytoplasmic ratio of the cell is maintainedwithin advantageous limits. Of course, there areexceptions to this rule (Murray & Hunt, 1993),such as oocytes, which grow very large withoutdividing, and fertilized eggs, which divide rapidlyin the absence of growth. Nonetheless, the long-term viability of a cell line depends on balancedgrowth and division.

START AND FINISH

The chromosome cycle is usually subdividedinto four phases (G1, S, G2, M), but it is better tothink of it as two alternative &&states'' (G1 andS}G2}M) separated by two transitions (Start andFinish), as in Fig. 1 (Nasmyth, 1995, 1996; Novaket al. 1998a, b). In G1, chromosomes are unrep-licated and the cell is uncommitted to the replica-tion}division process. At Start (the transition

from G1 to S phase), a cell con"rms that internaland external conditions are favorable for a newround of DNA synthesis and division, and com-mits itself to the process. The decision is irrevers-ible: once DNA synthesis commences, it goes tocompletion.

During the process of DNA replication, sisterchromatids are tethered together by speci"c pro-teins, called cohesins. As the mitotic spindleforms in M phase, microtubules from the spindlepoles attach to chromosomes and pull them intoalignment at the center of the spindle (meta-phase). When DNA replication is complete andall chromosomes are aligned, the second irrevers-ible transition of the cycle (Finish) is triggered.The cohesins are destroyed, allowing sisterchromatids to be pulled to opposite poles of thespindle (anaphase). Shortly thereafter, daughternuclei form around the segregated chromatids

EUKARYOTIC CELL CYCLE 251

(telophase), and the incipient daughter cellsseparate.

These major events of the cell cycle must betightly regulated. For instance, balanced growthand division is achieved in most cells by a sizerequirement for the Start transition. That is, cellsmust grow to a critical size before they can com-mit to chromosome replication and division. Ifthis requirement is compromised by mutation,cells may become morbidly large or small(Moreno & Nurse, 1994). A second crucial regu-latory constraint is to hold o! the Finishtransition if there have been any problems withDNA replication or chromosome alignment.Were anaphase to commence under such condi-tions, then daughter nuclei would not receivea full complement of chromosomes, which isusually a fatal mistake (Murray, 1995).

MOLECULAR CONTROLS

Cell cycle events are controlled by a network ofmolecular signals, whose central components arecyclin-dependent protein kinases (Cdks). In theG1 state, Cdk activity is low, because its obligatecyclin partners are missing, because cyclinmRNA synthesis is inhibited and cyclin protein israpidly degraded. At Start, cyclin synthesis isinduced and cyclin degradation is inhibited, caus-ing a dramatic rise in Cdk activity, which persiststhroughout S, G2 and M (see Fig. 1). High Cdkactivity is needed for DNA replication, chromo-some condensation, and spindle assembly.

At Finish, a group of proteins, making up theanaphase-promoting complex (APC), is activated(Zachariae & Nasmyth, 1999). The APC attachesa &&destruction label'' to speci"c target proteins,which are subsequently degraded by the cell'sproteolysis machinery. The APC consists ofa core complex of about a dozen polypeptidesplus two auxiliary proteins, Cdc20 and Cdh1,whose apparent roles (when active) are to recog-nize speci"c target proteins and present them tothe core complex for labeling (Visintin et al.,1997; Zachariae & Nasmyth, 1999). Activation ofCdc20 at Finish is necessary for degradation ofcohesins at anaphase, and for activation of Cdh1.Together, Cdc20 and Cdh1 label cyclins fordegradation at telophase, allowing the controlsystem to return to G1. We must distinguish

between these two di!erent auxiliary proteins,because Cdc20 and Cdh1 are controlled di!er-ently by cyclin/Cdk, which activates Cdc20 andinhibits Cdh1.

There are additional complexities in the Cdknetwork (Mendenhall & Hodge, 1998). The con-trol systems found in budding and "ssion yeastsconsist of two di!erent families of cyclins, astoichiometric inhibitor of cyclin/Cdk complexes,kinases and phosphatases that modify the Cdksubunit, transcription factors that control theexpression of cell-cycle genes, and a carefullyregulated phosphatase that opposes cyclin/Cdkactivity at crucial control points in the network(Chen et al., 2000; Novak et al., 1999). The con-trol system in mammalian cells is more compli-cated still, with seven di!erent Cdks, seven familiesof cyclins, a dozen stoichiometric inhibitors, andhundreds of relevant interactions among thesecomponents (Bartek et al., 1996; Kohn, 1999). Inaddition, signal transduction pathways (&&surveil-lance mechanisms'') convey information from theinterior and exterior milieus to control progressthrough the cell cycle (Elledge, 1996; Hanahan& Weinberg, 2000).

A major challenge for theoretical molecularbiologists is to explain the physiology of cellproliferation in a variety of unicellular and multi-cellular eukaryotes in terms of their underlyingmolecular control systems. Of necessity, suchconnections will be made by ambitious computa-tional models that re#ect some of the inescapablecomplexity of real cell cycle controls (Chen et al.,2000; Novak & Tyson, 1993). In order to designsuch models and understand how they work, we"rst need a solid grasp of the basic control prin-ciples of the cell cycle.

The purpose of this paper is to draw attentionto a simple theme that runs through the morassof molecular details (Tyson et al., 1995; Aguda,1999; Thron, 1999): the irreversible transitions ofthe cell cycle (Start and Finish) are consequencesof the creation and destruction of stable steadystates of the molecular regulatory mechanism bydynamic bifurcations. At the core of the cell cycleis a hysteresis loop deriving from the funda-mental antagonism between Cdk and APC(Novak et al., 1998): the APC extinguishes Cdkactivity by destroying its cyclin partners, whereascyclin/Cdk dimers inhibit APC activity by

FIG. 2. Phase plane portrait for the pair of nonlinearODEs (1) and (2). Parameter values are given in Table 1.Curves are nullclines (see text) for A"0 and m"0.3 or 0.6.Arrows indicate direction "eld for m"0.3 only. Form"0.3, the control system has three steady states: a stablenode (G1) at ([CycB], [Cdh1])+(0.039, 0.97), a saddle pointnear (0.10, 0.36), and another stable node (S+G2+M) near(0.90, 0.0045). Suppose a newborn cell resides at G1 (Cdh1active and CycB missing). As the cell grows (m increases), theG1 steady state is lost by a saddle-node bifurcation (atm+0.53), and the control system is forced to the S+G2+Msteady state.

252 J. J. TYSON AND B. NOVAK

phosphorylating Cdh 1 (Fig. 1). This antagonismcreates two, alternative, stable steady statesof the control system: a G1 state, with highCdh1/APC activity and low cyclin/Cdk activity,and an S}G2}M state, with high cyclin/Cdkactivity and low Cdh1/APC activity. Transitionsbetween these two states are facilitated by&&helper'' molecules that are insensitive tothe antagonists.

In the following sections, we "rst construct andanalyse a simple model of this antagonism. Next,we add a cyclin-dependent kinase inhibitor (CKI)to the model, to create a simple yet accuratemodel of yeast cell cycle controls. Finally, wespeculate on how to extend the yeast model toa useful picture of cell cycle regulation in multi-cellular eukaryotes.

Our approach is a tribute to the spiritof Joel Keizer, who recognized that many inter-esting and important problems in molecular cellbiology can be formulated as physicochemicalprocesses in space and time and studied success-fully by modern tools of nonlinear dynamicalsystems.

Hysteresis in the Interaction of Cyclin B/Cdkand Cdh1/APC

The biochemical reactions in the center ofFig. 1 can be described by a pair of nonlinearordinary di!erential equations (ODEs):

d[CycB]dt

"k1!(k@

2#kA

2[Cdh1]) [CycB], (1)

d[Cdh1]dt

"(k@3#kA3A)(1![Cdh1])J3#1![Cdh1]

!k4m[CycB] [Cdh1]J4#[Cdh1] . (2)

In these equations, [CycB] and [Cdh1] are theaverage concentrations (grams of protein pergram of total cell mass) of cyclin B/Cdk dimersand active Cdh1/APC complexes, respectively.The k's are rate constants, the J's are Michaelisconstants, m represents cell &&mass'' (not to beconfused with M for &&mitosis''). The terms ineqn (1) represent synthesis and degradation ofCycB, with degradation rate dependent on Cdh1

activity (assuming that APC cores are in excess).We assume that cyclin B molecules combinerapidly with an excess of Cdk subunits, so that wedo not have to keep track of CycB and Cdkmonomers. The terms in eqn (2) represent activa-tion and inactivation of Cdh1, formulated asMichaelis}Menten rate laws. We assume that thetotal Cdh1 concentration is constant (scaled to 1),and that J

3and J

4are both ;1, so that Cdh1

behaves like a &&zero-order ultra-sensitive switch''(Goldbeter & Koshland, 1981). This switch-likebehavior of Cdh1 is crucial to hysteresis in ourmodel. Cdh1 is activated by a generic &&activator'':for now A is simply a parameter, but, in the nextsection, we will show how A relates to Cdc20.We assume that the inactivation of Cdh1 byCycB/Cdk1 takes place in the nucleus, whereCycB accumulates. Under this assumption, thee!ective concentration of CycB in the nucleuswill increase as the cell grows, so [CycB] is multi-plied by m in the second term in eqn (2). Thismass-dependence plays a crucial role in ourmodel by connecting the Start transition to cellgrowth.

The phase plane portrait for system (1)}(2) isillustrated in Fig. 2. The nullclines are described

FIG. 3. Bifurcation diagram for eqns (1) and (2). The steady-state concentration of CycB is plotted as a function of thebifurcation parameter, p"(k@

3#kA

3A)/k

4m. Other para-

meters: b"e"0.04. Saddle-node bifurcations occur atp1+0.0545 and p

2+0.261.

EUKARYOTIC CELL CYCLE 253

by simple algebraic equations:

[CycB]" bJ2#[Cdh1] (CycB nullcline),

[CycB]"p (1![Cdh1]) (J4#[Cdh1])[Cdh1] (J

3#1![Cdh1])

(Cdh1 nullcline),

where b"k1/k

2, J

2"k@

2/kA

2and p"(k@

3#

kA3A)/(k

4m). The CycB nullcline is a simple hyper-

bola. For J3"J

4;1, the Cdh1 nullcline is a sig-

moidal curve passing through [CycB]"p at[Cdh1]"1

2.

The Cdh1 nullcline can be rewritten for[Cdh1] as a function of [CycB], [Cdh1]"G (p, [CycB], J

3, J

4), where G is the Gold-

beter}Koshland function (Goldbeter & Kosh-land, 1981):

G(<a, <

i, J

a, J

i)" 2c

b#Jb2!4ac, a"<

i!<

a,

b"<i!<

a#<

aJi#<

iJa,c"<

aJi.

This function will come in handy later.The control system has steady-state solutions

wherever the nullclines intersect. The number ofintersections depend on the value of p (Fig. 3).For p

1(p(p

2, the ODE system (1)}(2) has

three steady states: two stable nodes separated bya saddle point. The stable nodes we refer to as G1(Cdh1 active, CycB low) and S+G2+M (Cdh1inactive, CycB high); in bold face to distinguishthe theoretician's stable steady state from theexperimentalist's cell cycle phase. It can be shownthat, when J

2"J

3"J

4,e;1, the saddle-

node bifurcations occur at

p1+b M1#2 Je#O (e3@2)N

[Cdh1]+1!Je#e

[CycB]+b (1#Je!e), and

p2+b/(4e)

[Cdh1]+e [CycB]+b/(2e).

Progress through the cell cycle can be thoughtof as a tour around the hysteresis loop in Fig. 3.For a small, newborn cell in G1 phase (withA+0 and p+k@

3/k

4m'p

1), the CycB}Cdh1

control system is attracted to the stable G1steady state. As the cell grows, m increases andp decreases. Eventually, p drops below p

1, and the

G1 steady state disappears, forcing the controlsystem to jump irreversibly to the S+G2+Msteady state. High CycB/Cdk activity initiates theprocesses of DNA synthesis and mitosis, as thecell continues to grow. We assume that, whenDNA replication is complete and the chromo-somes are properly aligned on the mitoticspindle, the parameter A increases abruptly, for-cing p to increase above p

2. Consequently, the

S+G2+M steady state is lost by a saddle-nodebifurcation, and the control system jumps irre-versibly back to the G1 state. The cell divides(mPm/2), A decreases back to 0, and the controlsystem returns to its starting condition.

In this simple model, the irreversibletransitions of the cell cycle (Start and Finish) arethe abrupt jumps of the hysteresis loop, at thesaddle-node bifurcation points. The G1PS+G2+M transition is driven by cell growth, and

FIG. 4. Phase plane portrait for eqns (1) and (3). [Cdh1] iscomputed from [CycB] by solving for the steadystate of eqn (2). Parameter values given in Table 1.(a) m"0.5; (b) m"1. Solid curve: CycB nullcline.Dotted curve: Cdc20 nullcline. Dashed curve: trajectory.The control system undergoes a saddle-node-loop bifurca-tion at m+0.8.

254 J. J. TYSON AND B. NOVAK

the reverse transition is driven by chromosomealignment on the mitotic spindle.

Activation of Cdh1/APC at Anaphase

To "ll out the picture in the previous section,we must identify the activator of Cdh1/APC anddescribe why A increases abruptly at the meta-phasePanaphase transition and decreases in G1phase. The form of eqn (2) suggests that A isa phosphatase, removing from Cdh1 the inhibit-ory phosphate groups placed there by CycB/Cdk.Recent experimental evidence in budding yeast(Jasperson et al., 1999; Visintin et al., 1998) identi-"es A as the product of the CDC14 gene. At themetaphasePanaphase transition, Cdc14 phos-phatase is activated indirectly by Cdc20/APC,which destroys an inhibitor of Cdc14. To keepour model as simple as possible, we assume thatAJ[Cdc14]J[Cdc20] and write a di!erentialequation for the production of Cdc20:

d[Cdc20T]

dt"k@

5#kA

5

([CycB]m/J5)n

1#([CycB]m/J5)n

!k6

[Cdc20T]. (3)

Because Cdc20 is synthesized only in S+G2+Mphase of the budding yeast cell cycle (Shirayamaet al., 1998; Zachariae & Nasmyth, 1999), we haveassumed that its transcription factor is turned onby CycB/Cdk according to a Hill function withparameters n and J

5. (The signi"cance of the

subscript ¹ will become clear shortly.)By supposing that Cdh1 activity responds

rapidly to changes in [CycB] and [Cdc20T], we

can solve eqn (2) for [Cdh1]"G(k@3#kA

3[Cdc20

T], k

4m [CycB], J

3, J

4), where G is the

Goldbeter}Koshland function described earlier.With this assumption, our control system is stillrepresentable by a pair of ODEs, eqns (1) and (3),and by phase plane portraits (Fig. 4). Considera newborn cell in G1 phase [Fig. 4(a)]. As the cellgrows, the CycB nullcline moves to the right andthe control system undergoes a saddle-node-loopbifurcation at a critical value of m (m

crit+0.8).

When the G1 steady state is destroyed by coales-cence with the saddle point, [CycB] starts toincrease, see the dotted trajectory in Fig. 4(b)

CycB-dependent kinase activity drives the cellinto S phase and mitosis, and it turns on syn-thesis of Cdc20. Notice that the S+G2+M steadystate in Fig. 4(b) is unstable: as Cdc20 accumu-lates, the control system loops around the unsta-ble steady state. Cdh1/APC is activated byCdc20, CycB is destroyed, and the cell exits mito-sis. At cell division, m is reduced two-fold, and thenullclines readopt the con"guration in Fig. 4(a).The control system is captured by the stable G1steady state, until cell size, m (t), once more in-creases to m

crit.

In this picture, cells exit from mitosis &&auto-matically'' a certain time after Start (the timerequired to make enough Cdc20 to activateAPC); there is no connection between alignmentof replicated chromosomes on the metaphaseplate and the transition to anaphase. In buddingyeast, the connection is established through fur-ther controls on Cdc20 (see bottom part ofFig. 6). Newly synthesized Cdc20 is inactive.A Cdc20-activating signal derives indirectly fromCycB/Cdk; some intermediate steps betweenCycB synthesis and Cdc20 activation assurea minimum time lag for DNA synthesis and chro-mosome alignment to be completed beforeanaphase commences. If they are not completedon time, a Cdc20-inactivating signal is imposedby the MAD-family of spindle checkpoint genes(Hwang et al., 1998). To take these additional

FIG. 5. Simulation of eqns (1)}(6), with parameter valuesin Table 1. Middle panel: CycB

T(solid curve) scale to the

right. Cell division occurs when [CycBT] crosses 0.1 from

above.

EUKARYOTIC CELL CYCLE 255

feature into account, we write

d[Cdc20A]

dt"k7 [IEP]([Cdc20T]![Cdc20A])

J7#[Cdc20

T]![Cdc20

A]

!k8[Mad] ) [Cdc20A]J8#[Cdc20

A]

!k6[Cdc20

A], (4)

d[IEP]dt

"k9m[CycB](1![IEP])!k

10[IEP].

(5)

Here, [Cdc20A] is the concentration of &&active''

Cdc20, and [Cdc20T] is the total concentration

of both active and inactive forms. From now on,we set A"[Cdc20

A] in eqn (2). We treat [Mad]

as a parameter; [Mad]"1, if chromosome align-ment is completed on schedule, and"some largenumber, if not. [IEP] is the concentration of theactive form of a hypothetical &&intermediary en-zyme'', whose total concentration is scaled to 1.IEP is put in the model to create a time lag(as observed) between the rise in CycB/Cdk acti-vity and the activation of Cdc20. Because themolecular basis of this time lag has yet to beidenti"ed, we must be content with this "ctionalcomponent.

To complete this primitive model of cell cyclecontrols, we provide a di!erential equation forcell growth:

dmdt

"km A1!mm

*B , (6)

where m*

is the maximum size to which a cellmay grow if it does not divide, and k is thespeci"c growth rate when m;m

*. Our model

consists of eqns (1)}(6), with the proviso thatmPm/2, whenever the cell divides (i.e. when[CycB] drops below some threshold level, takento be 0.1). A typical simulation is presented inFig. 5.

This simple model ful"lls all the requirementsof a functional, eukaryotic cell cycle, with twoirreversible transitions: Start (dependent on cellgrowth) and Finish (dependent on chromosomealignment) (Nasmyth, 1995). Although this pic-ture may represent the primitive control system

in the earliest eukaryotes, as they evolved fromprokaryotic progenitors (Novak et al., 1998a, b),all present-day organisms that have been studiedin detail have more complicated mechanisms ofcell cycle regulation.

Cell Cycle Controls in Yeast

To get a reasonable model of cell cycle controlsin modern yeasts, we need to add a cyclin-depen-dent kinase inhibitor (CKI) to the picture (Fig. 6).CKI binds to CycB/Cdk to form inactive trimers.The existence of trimers changes slightly the in-terpretation of eqn (1):

d[CycBT]

dt"k

1!(k@

2#kA

2[Cdh1]

#k@@@2

[Cdc20A]) [CycB

T], (1@)

256 J. J. TYSON AND B. NOVAK

where [CycBT]"[CycB]#[Trimer]. We also

need a kinetic equation for total CKI:

d[CKIT]

dt"k

11!(k@

12#kA

12[SK]

#k@@@12

m[CycB]) [CKIT]. (7)

In eqn (7), the rate of CKI degradation dependson CycB/Cdk activity, because CycB-dependentphosphorylation of CKI renders it unstable. (Inbudding yeast, this CKI is called Sic1; its kineticproperties are described in Mendenhall &Hodge, 1998). Thus, CKI and CycB/Cdk are mu-tual antagonists. The model (Fig. 6) postulatesa &&starter'' kinase (SK) that phosphorylates CKI inthe absence of CycB/CdK activity. (In buddingyeast, the starter-kinase role is played by Cln-de-pendent kinases; see Mendenhall & Hodge.) Noticealso, in eqn (1@), that we have given Cdc20 someability to degrade cyclin B, as indicated by experi-ments (Irniger et al., 1995; Visintin et al., 1997).

We assume that CKI/CycB/Cdk trimers arealways in equilibrium with CKI monomers andCycB/Cdk dimers: [Trimer]"K

eq[CycB]

[CKI]"Keq

([CycBT]![Trimer]) ([CKI

T]!

[Trimer]), or

[Trimer]" 2 [CycBT] [CKIT][CycB

T]#[CKI

T]#K~1

eq#J([CycB

T]#[CKI

T]#K~1

eq)2!4[CycB

T][CKI

T].

To understand how the CKI part of the con-trol system works, let us consider the[CycB

T]}[CKI

T] phase plane (Fig. 7) de"ned by

eqn (1@) and (7). In these equations,[Cdh1]"G (k@

3#kA

3[Cdc20

A], k

4m [CycB],

J3, J

4), [CycB]" [CycB

T]![Trimer], with

[Trimer] given by the equation directly above,and [SK], [Cdc20

A] and m are treated as para-

meters. The CKIT-nullcline is N-shaped because

of the antagonism between CycB and CKI, andthe CycB

T-nullcline is N-shaped because of the

antagonism between CycB/Cdk and Cdh1/APC.For proper choice of parameters (Table 1), the

control system exhibits bistability, with a stableG1 state (CycB low, Cdh1 active, CKI abundant)and a stable S+G2+M state (CycB high, Cdh1

inactive, CKI missing). Exit from mitosis(S+G2+MPG1) is carried out by Cdc20, asdescribed in the previous section. To start thechromosome cycle (G1PS+G2+M), we assumethat synthesis of the &&starter kinase'' is turned onwhen the cell reaches a characteristic size. As[SK] increases, the local maximum of the CKI

T-

nullcline is depressed (Fig. 7), destroying the G1steady state by a saddle-node bifurcation.

To complete the picture, we need a dynamicalequation for the time course of [SK]. Again, wetake our hint from budding yeast, where twocyclins, Cln1 and Cln2, in combination withCdc28 (the catalytic subunit) phosphorylate CKIat Start, permitting B-type cyclins (Clb1}6) toaccumulate and drive the cell through S andM phases. Synthesis of Cln1}2 is controlled bya transcription factor (TF; called SBF in buddingyeast) that is activated by Cln/Cdc28 and inhib-ited by Clb/Cdc28, so we write

d[SK]dt

"k13

[TF]!k14

[SK],

[TF]"G (k@15

m#kA15

[SK], k@16

#kA16

m[CycB], J15

, J16

), (8)

where G( . . . ) is the Goldbeter}Koshland func-tion, as usual. Size control at Start enters thismodel through the term k@

15m in the "rst argu-

ment of G; when the cell gets su$ciently large,k@15

m+k@16

, it begins to synthesize SK. Increas-ing [SK] activates its own transcription, eqn (8),and destroys CKI, eqn (7). We also assume thatSK phosphorylates Cdh1, although not ase$ciently as CycB/Cdc28; that is, in eqn (2)we replace k

4m[CycB][Cdh1] by (k@

4[SK]#

k4m [CycB])[Cdh1], with k@

4;k

4.

With these changes, the system of equations(1@), (2)}(8) accounts for many characteristic fea-tures of wild type and mutant budding yeast cells.Wild-type cells have a long G1 (unbuddedperiod) and short S}G2}M (budded period); see

FIG. 6. The basic cell cycle engine in eukaryotic cells. Thegeneric components in this mechanism correspond to speci-"c gene products in well-studied organisms (see Table 2).Dynamical properties of this mechanism are determined bya set of kinetic equations (1@), (2)}(7), with A"[Cdc20

A].

A basal set of parameter values, suitable for yeast cells, isgiven in Table 1.

FIG. 7. Phase plane portrait for eqns (1@) and (7). [Cdh1]is computed from [CycB] by solving for the steady state ofeqn (2). Parameter values in Table 1, plus [Cdc20

A]"0,

m"1, [SK] adjustable. Dotted curve: CycB nullcline. Solidcurves: CKI nullclines, for [SK]"0.02, 0.1 and 1.d"stable steady state, L"unstable steady state. The con-trol system has a saddle-node bifurcation at [SK]+0.9.

EUKARYOTIC CELL CYCLE 257

Fig. 8(a). Throughout G1, SK (Cln) level steadilyrises, causing CKI (Sic1) level to fall. In late G1,several events occur in close succession. First, TFis activated and [SK] rises sharply, causing rapid

TABLParameter

Component Rate constants (min~1)

CycB k1"0.04, k@

2"0.04, kA

2"1, k@@@

2"1

Cdh1 k@3"1, kA

3"10, k@

4"2, k

4"35

Cdc20T

k@5"0.005, kA

5"0.2, k

6"0.1

Cdc20A

k7"1, k

8"0.5

IE k9"0.1, k

10"0.02

CKI k11"1, k@

12"0.2, kA

12"50, k@@@

12"100

SK k13"1, k

14"1, k@

15"1.5, kA

15"0.05, k@

16"

M k"0.01

destruction of the remaining CKI. When [CKI]drops below [CycB

T], active CycB/Cdc28 dimers

make their appearance, helping SK to inactivateCdh1. As Cdh1 activity drops o! rapidly, CycBlevel rises further. Newly produced CycB/Cdc28initiates DNA synthesis at about the same timethat Cln/Cdc28 initiates a new bud. Meanwhile,CycB/Cdc28 turns o! SK production and SKlevel drops. CKI and Cdh1 do not make a come-back because CycB/Cdc28 keeps these proteinsphosphorylated. Persistent CycB/Cdc28 activitydrives the cell into M phase. After a delay ofabout 45 min [during which IE is activated*notshown in Fig. 8(a)], Cdc20 is activated and

E 1values

Dimensionlessconstants

[CycB]threshold

"0.1J3"0.04, J

4"0.04

J5"0.3, n"4

J7"10~3, J

8"10~3, [Mad]"1

Keq"103

1, kA16"3 J

15"0.01, J

16"0.01

m*"10

FIG. 8. The budding yeast cell cycle. (a) Wild-type cells:simulation of eqns (1@), (2)}(8), with parameter values inTable 1, except k"0.005 min~1. (b) Mutant cells lackingSK (k

13"0) block in G1 with copious CKI and active APC.

(c) Mutant cells lacking SK and CKI (k11"k

13"0) are

viable.

258 J. J. TYSON AND B. NOVAK

destroys enough CycB to allow Cdh1 to reacti-vate and CKI to reaccumulate. When CycBdrops low enough, the cell divides. [In Fig. 8(a),we assume symmetric division for simplicity,

TABLCell cycle regulatory protein

Component Budding Fissioyeast yeas

Cdk Cdc28 CdcCycB Clb1}6 Cdc1Cdh1 Cdh1 Ste9Cdc20 Cdc20 Slp1IE Cdc5? Plo1CKI Sic1 RumSK Cln1}2 Cig

although budding yeast cells typically divideasymmetrically.]

Mutant cells lacking SK (k13"0) block in G1

with abundant CKI and active Cdh1 [Fig. 8(b)],which is the phenotype of cells in which allCln-cyclins are deleted (cln1* cln2* cln3*)(Richardson et al., 1989). (Although Cln3 playsa di!erent role than Cln1}2, it can serve as theirbackup; so it too must be deleted to see theexpected phenotype.) Because the only essentialjob of the Cln-cyclins is to remove Sic l (CKI), thequadruple-deletion mutant cln1* cln2* cln3*sic1* is viable (Tyers, 1996); see Fig. 8(c). Fora more thorough analysis of the budding yeastcell cycle, consult Chen et al. (2000).

Properly interpreted (Table 2), eqns (l@), (2)}(8)also provide a reasonable description of cell cyclecontrols in "ssion yeast cells lacking Wee1 (wee1*mutants; see (Novak et al., 1998a, b). To describewild-type "ssion yeast, we must add a new level ofcontrol to Fig. 6, involving tyrosine phosphoryla-tion and dephosphorylation to Cdk subunits, asin Novak & Tyson (1995) and Sveiczer et al.(2000).

Checkpoints and Surveillance Mechanisms

The job of the basic cell cycle engine (Fig. 6) isto coordinate the events of DNA synthesis andmitosis with overall cell growth. We have seenhow cell size (m) might feed into the engine toensure balanced growth and division. If cells aretoo small, the engine stops at the stable steadystate (G1 in Fig. 7). Only when m exceedssome critical value is this stable steady state lost

E 2s in yeasts and vertebrates

n Frog Mammaliant egg cell

2 Cdc2 Cdk13 Cyclin B Cyclin B

Fizzy-related Cdh1Fizzy p55cdc

? Plx1? Plk1?1 Xic1 p27K*1-2 Cyclin E? Cyclin D, E

EUKARYOTIC CELL CYCLE 259

by coalescence with an unstable steady state(saddle-node bifurcation). (In some cases, thestable G1 state might be lost by a change ofstability*a Hopf bifurcation; e.g. Fig. 3 inNovak & Tyson, 1997.) When the G1 attractor islost, the cell can proceed into S phase. In thisway, Start can be controlled by cell size.In some species (e.g. "ssion yeast), size controloperates at the G2PM transition by the sameprinciple: a stable G2 steady state is lost whencells grown beyond a critical size (Novak &Tyson, 1995).

We believe that this is a general principle of cellcycle control. A checkpoint corresponds toa stable steady state of the cell cycle engine (nofurther progress). The checkpoint is lifted bychanges in crucial parameters, carrying thecontrol system across a bifurcation. The crucialparameters are controlled by surveillance mecha-nisms that monitor the internal and externalmilieus of the cell. For instance, if DNA synthesisstalls for any reason, an inhibitory signal sup-presses mitosis until the genome is fully rep-licated. If DNA is damaged in G1 or G2 phases,other surveillance mechanisms suppress entryinto S phase or M phase, respectively. If chromo-some alignment on the metaphase plate is de-layed for any reason, a signal inactivates Cdc20and blocks progression from metaphase toanaphase.

Spontaneous Oscillations in Fertilized Eggs

Although the proliferation of most eukaryoticcells is controlled by checkpoints, as described,cell division during early embryogenesis is not.The fertilized frog egg, for example, undergoes 12rapid, synchronous, mitotic cell divisions, whichare unconstrained by the usual requirements forgrowth and DNA integrity (Murray & Hunt,1993). In the fertilized egg, the checkpoints (stablesteady states) are missing, and the cell cycleengine exhibits its capacity for free-runningoscillation.

The cyclin/Cdk network controlling cell divis-ions in early embryos is a stripped-down versionof Fig. 6, lacking CKI and Cdh1. (With CKImissing, SK has no role to play in the model.) Theonly remaining components are CycB, IEP and

Cdc20, described by the following equations:

d[CycB]dt

"k1!(k@

2#k@@@

2[Cdc20

A])[CycB], (9)

d[IEP]dt

"k9[CycB](1![IEP])!k

10[IEP],

(10)

d[Cdc20A]

dt"k7[IEP]([Cdc20T]![Cdc20A])

J7#[Cdc20

T]![Cdc20

A]

!k8[Mad][Cdc20A]J8#[Cdc20

A]

. (11)

For simplicity, we assume that Cdc20 ("zzy) isa stable protein in the early embryo and set[Cdc20

T]"1 in eqn (11). Furthermore, [Mad]

"1, because the spindle assembly checkpoint isinoperative. Cell size (m) does not appear in theseequations, because the embryo is not growing.

The system of equations (9)}(11) is a classicalnegative-feedback oscillator; see Goldbeter(1991). As shown in Fig. 9, it has limit-cyclesolutions, corresponding to spontaneous oscilla-tions in activity of CycB/Cdk (usually calledMPF in the frog-egg literature). After 12 rapid,synchronous divisions, the frog egg undergoes anabrupt reorganization of the cell cycle (called themidblastula transition). Expression of zygoticgenes provides the missing components of the cellcycle checkpoints. Consequently, the pace of celldivision slows, as the checkpoint controls comeinto play.

Discussion

Kohn (1999) has recently summarized ourknowledge of the molecular signals controllingthe cell cycle in mammals. The &&wiring'' diagramextends in "ne print over four journal pages, andmost people would agree that we are only begin-ning to unravel the details. How are we to makesense of a control system of such complexity?Where casual verbal arguments will not su$ce,reliable quantitative tools are needed. To thisend, we have been developing analytical andcomputational methods to study the molecularmachinery of cell cycle regulation.

FIG. 9. Spontaneous oscillations of CycB/Cdk activity inearly embryonic cells. Simulation of eqns (9)}(11), with para-meter values in Table 1. Period"41 min.

260 J. J. TYSON AND B. NOVAK

Following the reductionist approach of experi-mentalists, we have broken down the theoreticalproblem into manageable pieces. By understand-ing the dynamical properties of restricted parts ofthe network "rst, we are able to put the piecestogether into ever more comprehensive,computational models of intact control systemsin speci"c organisms (Marlovits et al., 1998;Novak et al., 1998a, b; Novak & Tyson, 1997).In this paper we summarize what we havelearned over the years about bistability and oscil-lations in the control system, and their relationsto checkpoints and surveillance mechanisms inliving cells.

THE CELL CYCLE ENGINE AS A HYSTERESIS LOOP

We subscribe to the opinion of Nasmyth (1996)that the eukaryotic cell cycle is an alternationbetween two self-maintaining states: G1 (unrep-licated DNA) and S}G2}M (DNA replication andmitosis). The cell switches between these statesby the production and destruction of cyclin-dependent kinase activities. In G1, the cyclindegradation machinery is active and cyclins arescarce. At &&Start,'' the cyclin degradation machin-ery is turned o!, and cyclins accumulate, drivingthe cell from G1 into S}G2}M. At &&Finish,'' thecyclin degradation machinery is turned back on,cyclins disappear, the cell divides and its progenyenter G1.

From a dynamical point of view, these twoself-maintaining states of the cell cycle are stablesteady states of the kinetic equations describingthe interactions between cyclin-dependentkinases (Cdk) and the cyclin degradation machin-ery (APC). The control system is bistable becauseof a fundamental antagonism: APC destroys Cdkactivity (by labeling its cyclin partner forproteolysis), and Cdk turns o! the APC (byinactivating one of its components, Cdh1). Asexpected for a dynamical system of this sort,bistability is observed only within a restrictedregion of parameter space; the boundaries of thisregion are parameter values where saddle-nodebifurcations occur (e.g. Fig. 3). The control sys-tem can be driven from one state to the other byparameter changes that carry the system acrossthe boundary of saddle-node bifurcation points.Because the stable state initially occupied by thecell (G1) is lost at the saddle-node bifurcation, thecell is forced to make an irreversible transition(Start) to the other stable state (S+G2+M). Ingeneral, the opposite transition (Finish) can onlybe induced by parameter changes that carry thesystem across a di!erent boundary, where theS+G2+M state is lost and the system jumps irre-versibly to G1. When traced out in a diagram likeFig. 3, these parameter changes and statetransitions create a &&hysteresis loop''. Our lineof reasoning suggests that the irreversibletransitions of the cell cycle are intimately connec-ted to the molecular antagonism of Cdk andAPC. The connection is established by classicalideas from dynamical systems theory: bistability,saddle-node bifurcation, and hysteresis.

The phenomenon of hysteresis, upon whichour whole notion of cell cycle control depends, isnot a special feature of eqns (1) and (2) but rathera general property of dynamical systems withantagonism. Indeed, the establishment of G1phase is not a simple matter of antagonism be-tween Cdk and APC. On the contrary, in mostcells, proteins called cyclin-dependent kinase in-hibitors (CKIs) play signi"cant roles in stabiliz-ing the G1 state. Any realistic model of cell cycletransitions must take these CKIs into account.However, because CKI and Cdk are also antag-onistic proteins, the generic property of hysteresisis maintained in more comprehensive modelswith more state variables and parameters.

EUKARYOTIC CELL CYCLE 261

Figures 2 and 3 provide convenient illustrationsof the basic ideas that we believe will be generallyapplicable to cell cycle models of any complexity.

The &&parameter'' changes that drive cellsthrough Start and Finish are carried out by addi-tional components of the cell cycle control sys-tem, which we have called &&helper'' molecules.The role of &&starter kinases'' is to destroy CKIs sothat the cell can leave G1, and the role of Cdc20 isto activate Cdh1 and stabilize CKIs so that thecell can reenter G1. The helpers do not partici-pate in the antagonistic interactions: starterkinases are not inhibited by CKIs and notdegraded by APC, and Cdc20 is not inhibited byCdk. Helper activity is only transient: it rises toinduce a transition, but then falls back down inpreparation for the reverse transition. Were thehelper activity to stay high, it would impede thereverse transition. Mutations that interfere withthe rise or fall of helper proteins are usuallyinviable or severely compromised in progressthrough the cell cycle. For a thorough discussionof such mutants in budding yeast, and a presenta-tion of the experimental evidence for multiplestable steady states in the yeast cell cycle engine,see Chen et al. (2000).

IS THE CELL CYCLE ENGINE A

LIMIT-CYCLE OSCILLATOR?

We have been arguing that progress throughthe cell cycle consists of a series of irreversibletransitions from one stable steady state to an-other of the underlying molecular regulatorysystem; cell reproduction is a periodic processbecause these steady states are connected ina ring. In the simplest case (budding yeast), thereare two stable steady states (G1 and S+G2+M)and the cell #ips back and forth between them atStart and Finish. In higher eukaryotes, there areat least three checkpoints, in G1, S}G2, andM (Alberts, 1994), and the cell cycles throughthem in order. Instead of this &&domino-like'' the-ory of the cell cycle (Murray & Kirschner, 1989b),would not it be simpler, more elegant, andperhaps more accurate to describe periodic celldivision by a limit-cycle oscillator (LCO)?Indeed, many theoreticians have put forwardLCO models of cell cycle regulation (Norel &Agur, 1991; Obeyesekere, 1999; Hatzimanikatis

et al., 1999), assuming that interdivision time"period of an LCO. We do not think that LCOs,in general, provide a useful picture of cell cycleregulation, for the following reasons.

Generally speaking, the period of an LCO isa complicated function of all the kinetic para-meters in the di!erential equations. However, theperiod of the cell division cycle depends on onlyone kinetic property of the cell: its mass-doublingtime. That is, the period of the chromosome cycle(DNA synthesis, mitosis and cell division) is al-ways equal to the period of the growth cycle (thetime it takes for the cell to duplicate all its othercomponents: total protein, membranes, etc.). Ifthe chromosome cycle ran faster than the growthcycle, cells would get smaller at each division andeventually die. If the chromosome cycle ranslower than the growth cycle, cells would becomelarger at each division, which is also eventuallyfatal. Hence, in the long run, growth and divisionmust be balanced, and the period of the celldivision cycle must depend only on the growthrate of the culture [the parameter k in eqn (6) inthis paper]. For this reason, cell cycle period isinsensitive to large changes in all other kineticparameters of the control system. For example,by mutating cyclin genes or APC-related genes,one can modify the rate constants for cyclin syn-thesis or degradation many-fold without alteringthe balance between cell division time and mass-doubling time (see the literature citations andsimulations in Chen et al., 2000). Our models areconsistent with this fundamental property of thecell cycle because they take into account the roleplayed by cell growth in driving the Starttransition of the cell cycle. (In wild-type "ssionyeast cells, size control operates at the G2PMtransition, but the basic principle is the same.) Inour opinion, to understand balanced growth anddivision, it is more pro"table to think of progressthrough the cell cycle as a domino-like sequenceof irreversible transitions from one checkpoint tothe next than as a clock-like sequence of eventsdriven by an underlying LCO.

Nonetheless, there is an LCO &&hidden'' inthe cell cycle regulatory mechanism, and it isrevealed under speci"c circumstances. For in-stance, it is possible, by mutation, to remove allsize controls from the "ssion yeast cell cycle (thewee1~ rum1* mutant of Moreno & Nurse, 1994),

262 J. J. TYSON AND B. NOVAK

in which case the Cdk-regulatory system exhibitsits underlying, free-running, oscillatory mode.These cells divide more rapidly than they cangrow, and they soon die. In other words, theLCO mode of control is indeed possible for yeastcells, but it is a lethal mistake! The free-runningoscillations must be restrained by checkpointmechanisms (especially size control) in orderto create viable, balanced cycles of growth anddivision.

There is an important circumstance in whichgrowth and division are unbalanced, namely, therapid division of the fertilized egg to producea multicellular blastula. These division cyclesseem to be driven by an LCO in the Cdk-regula-tory system (Goldbeter, 1991; Novak & Tyson,1993; Borisuk & Tyson, 1998). All checkpointsare inoperative, and the period of the oscillator isnoticeably dependent on kinetic parameters, suchas the rate of cyclin synthesis (Murray &Kirschner, 1989a). At the midblastula transition,checkpoints are re-established in the cell cycle,rapid division ceases, and cell division eventuallybecomes size-regulated again.

Finally, the LCO hypothesis seem inappropri-ate for cell cycle control because anotherhallmark of LCOs (Type-1 phase resettingin response to small perturbations) was not evi-dent in the unique experimental setting where itcould have been observed, namely, plasmodialfusion experimental in Physarum polycephalum(Tyson & Sachsenmaier, 1978; Winfree, 1980,pp. 444}448).

CELL CYCLE CONTROLS IN METAZOANS

Our analysis of cell cycle control shows thatthe complex molecular regulatory system can beunderstood in terms of some basic buildingblocks, whose dynamical features are most evi-dent in the proliferation of yeast cells and otherunicellular eukaryotes. Within the complex wir-ing diagram of mammalian cell cycle controls(Kohn, 1999), we can easily "nd the same basicbuilding blocks (Novak & Tyson, in prepara-tion). In multicellular organisms, unlike yeast,cell growth and division are under additional&&social'' constraints, because most somatic cells,though they "nd themselves bathed in a richlynutritious medium, are restrained from prolif-

erating. Only if they receive speci"c &&permission''from the body as a whole may these cells growand divide. The permission slips include growthfactors (small polypeptides secreted into theblood stream or interstitial #uids), and signalsthat re#ect cell}cell contacts and adhesion to theextracellular matrix (Alberts et al., 1994). Surveil-lance mechanisms monitor these signals and holdthe cell in a resting state (alive but not proliferat-ing) until conditions permit cell growth anddivision. If these surveillance mechanismsbecome mutated so that a cell loses crucial socialconstraints, it becomes transformed, in stages, toan invasive cancer, whose uncontrolled prolifer-ation eventually interferes with some vital func-tion and kills the organism (Hanahan & Wein-berg, 2000). By modeling the dynamical proper-ties of these surveillance mechanisms and consid-ering how they might plug into the cell cycleengine (Fig. 6), we hope eventually to constructa realistic model of mammalian cell cyclecontrols that will faithfully describe thephysiology of normal and cancerous cells andaccurately predict the results of pharmaceuticalinterventions.

This work was supported by grants from theNational Sciences Foundations of the USA (DBI-9724085) and Hungary (T-022182), and by theHoward Hughes Medical Institute (75195-542501).Our understanding of cell cycle controls has beenin#uenced greatly by conversations with Paul Nurse,Kim Nasmyth, Jill Sible, Kathy Chen, Attila Csikasz-Nagy, Bela Gyor!y, and Attila Toth.

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IntroductionFIGURE 1

Hysteresis in the Interaction of Cyclin B/Cdk and Cdh1/APCFIGURE 2FIGURE 3

Activation of Cdh1/APC at AnaphaseFIGURE 4FIGURE 5

Cell Cycle Controls in YeastFIGURE 6FIGURE 7TABLE 1FIGURE 8

Checkpoints and Surveillance MechanismsTABLE 2

Spontaneous Oscillations in Fertilized EggsDiscussionFIGURE 9

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