J. theor. Biol. (1998) 195, 69–85Article No. jt980781

0022–5193/98/021069+17 $30.00/0 7 1998 Academic Press

Bifurcation Analysis of a Model of Mitotic Control in Frog Eggs

M T. B J J. T*

Department of Biology, Virginia Polytechnic Institute and State University,Blacksburg, VA 24061, U.S.A.

(Received on 13 March 1998, Accepted in revised form on 7 July 1998)

Novak and Tyson have proposed a realistic mathematical model of the biochemicalmechanism that regulates M-phase promoting factor (MPF), the major enzymatic activitycontrolling mitotic cycles in frog eggs, early embryos, and cell-free egg extracts. We usebifurcation theory and numerical methods (AUTO) to characterize the codimension-one and-two bifurcation sets in this model. Our primary bifurcation parameter is the rate constantfor cyclin synthesis, which can be manipulated experimentally by adding exogenouslysynthesized cyclin mRNA to extracts depleted of all endogenous mRNA molecules. For thesecondary bifurcation parameter we use the total amount of one of the principal regulatoryenzymes in the extract (APC, the enzyme complex that labels cyclin for degradation; Wee1,the kinase that inhibits MPF; or Cdc25, the phosphatase that activates MPF). We find a richarray of physiologically distinct behaviors exhibited by the model as these parameters arevaried around values that seem plausible for frog eggs and extracts. In addition to unique,stable steady states (cell cycle arrest) and limit cycle oscillations (autonomous, periodic celldivision), we find parameter combinations where the control system is bistable. For instance,an interphase-arrested state may coexist with a metaphase-arrested state, or two stable limitcycles of different amplitude and period may coexist. We suggest that such strange behavioris nearly unavoidable in a complex regulatory system like the cell cycle. Perhaps cells exploitsome of these exotic bifurcations for control purposes that are as yet unrecognized byphysiologists.

7 1998 Academic Press

1. Introduction

1.1.

The cell cycle is the sequence of events bywhich a growing cell duplicates all its com-ponents and then divides this material betweentwo daughter cells so that they can repeat theprocess. The development and reproduction ofall living organisms is based on this fundamentalability of a cell to replicate itself. Although thereare many unique features of cell proliferation indifferent organisms, the basic events of theeukaryotic division cycle are stereotypical: first,

the DNA molecule within each chromosome isfaithfully replicated during S phase (‘‘syn-thesis’’), and then one copy of each DNAmolecule is segregated to each sister cell duringM phase (‘‘mitosis’’). S and M phases alternatein time, in response to signals from a network ofenzymatic reactions that is highly conservedacross all eukaryotic lineages, from fungi andplants to insects and mammals. In this paper weshall concentrate on cell cycle regulation in frogeggs and early embryos, where the controlsystem is exceptionally simple.

Excellent summaries of the physiology andmolecular biology of the eukaryotic cell cycle can*Author to whom correspondence should be addressed.

Progesterone Fertilization

Interphasearrest

Metaphasearrest

Oocyte Meiosis I Meiosis II Fertilizedegg

Firstmitosis

Secondmitosis

MP

F a

ctiv

ity

. . . . 70

be found in Alberts et al. (1994), Lodish et al.(1995), Murray & Hunt (1993). These referencesmay be consulted for confirmation of the basicfacts gathered in this introduction.

1.2.

As frog eggs develop (Fig. 1), they grow toabout 1 mm diameter and arrest, with replicatedDNA, before the first meiotic division. In thisstate, they are known as immature oocytes.Exposure to hormone (progesterone) triggersimmature oocytes to undergo two meioticdivisions, generating mature eggs arrested atmetaphase of meiosis II. After fertilization, theegg nucleus completes meiosis II and fuses withthe sperm nucleus to create a zygote. The zygoteundergoes 12 rapid, synchronous mitotic cyclesto form a hollow ball of 4096 cells. At this stage(called the midblastula transition), cell divisionin the embryo slows down and becomesasynchronous, and major changes occur inmRNA expression.

We are concerned in this paper with regulationof the meiotic and mitotic divisions from egg

maturation to the midblastula transition. Thesedivisions are triggered by M-phase promotingfactor (MPF), a protein kinase consisting of twosubunits: Cdc2 (the catalytic subunit) and cyclinB (the regulatory subunit). (‘‘Cdc2’’ is alsoknown as ‘‘Cdk1’’.) When activated, MPFphosphorylates an array of proteins involved inchromosome condensation, nuclear envelopebreakdown, spindle formation, and other eventsof meiosis and mitosis. (In particular, MPF canbe assayed biochemically by its ability tophosphorylate histone H1, a DNA-bindingprotein abundant in chromosomes). As cells exitM phase, MPF activity drops and DNA ispermitted to replicate. Thus, during egg matu-ration and early embryonic cell divisions, thealternation of S and M phases is controlled bytemporal fluctuations in MPF activity.

1.3.

There are two primary modes of regulation ofMPF activity in frog eggs. (1) Although Cdc2 ispresent at constant level throughout the celldivision cycle, its partner fluctuates dramatically

F. 1. MPF fluctuations in the meiotic and mitotic cycles of frog eggs (adapted from Murray & Hunt, 1993, Fig. 2–6).The immature oocyte (far left) is arrested in G2 phase with low MPF activity and replicated chromosomes in homologouspairs (the X-shaped symbols represent one pair of homologous chromosomes: the two arms of each X represent identicalDNA molecules called sister chromatids). Progesterone-induced activation of MPF triggers meiosis I, during whichhomologous chromosomes line up on the spindle. The paired chromosomes are separated to the two poles of the spindleas MPF activity drops at the end of meiosis I. One set of chromosomes remains in the egg and the other set is discardedin a small polar body. As MPF activity rises again, the egg enters meiosis II, with its replicated chromosomes attachedagain to the spindle. The mature egg arrests in this state, awaiting fertilization. Sperm entry triggers destruction of MPFin the egg, coincident with separation of sister chromatids to the two poles of the spindle. One set of chromatids is discardedin a small polar body, and the other set combines with sperm chromatids to reconstitute the diploid state in the fertilizedegg (homologous pairs of unreplicated chromosomes, represented by the I-shaped symbols, one from the egg and one fromthe sperm). Shortly thereafter, the DNA in each chromosome is replicated (they become X-shaped, not shown) and about90 min after fertilization the egg is driven into mitosis 1 by rising MPF activity. Subsequent cycles of MPF activation andinactivation drive a series of rapid, synchronous mitotic divisions to produce a hollow ball of cells, called the blastula.

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due to changes in the rate of degradation ofcyclin B. Since Cdc2 is catalytically active onlywhen bound to cyclin, MPF activity comes andgoes with phases of net cyclin synthesis anddegradation. (2) MPF activity can be modifiedby phosphorylation of the Cdc2 subunit. To becatalytically active, Cdc2 must be phosphory-lated at a specific amino acid (threonine atposition 161), whereas phosphorylation of adifferent amino acid (tyrosine at position 15)inhibits Cdc2.

In the mature egg, arrested at metaphase ofmeiosis II, MPF activity is high because cyclindegradation is repressed. At fertilization, cyclindegradation is derepressed, and MPF activitydrops as cyclin is destroyed. The following 12mitotic cycles are driven by cytoplasmic reac-tions that periodically activate and inactivateMPF by cyclin turnover and Cdc2 phosphoryl-ation, resulting in MPF activity being low ininterphase (the period between subsequent Mphases, when DNA is being synthesized) andhigh in mitosis.

1.4.

These reactions can be studied conveniently inextracts prepared from mature eggs (Murray &Kirschner, 1989). After crushing the eggs, theircytoplasmic components are separated by low-speed centrifugation from lipids and membra-nous debris. Calcium ions, liberated fromorganelles during the extraction procedure,release the arrest on cyclin degradation, andspontaneous oscillations of MPF activity ensue.MPF oscillations in the extract can be observedby adding sperm nuclei, which undergo periodicmitoses whenever MPF activity is high. Theinter-mitotic period in extracts (about 60 min) islonger than the period in intact eggs (about30 min). In extracts, Cdc2 is extensively tyrosine-phosphorylated during interphase, whereas inintact eggs it is not (Ferrell et al., 1991).

Biochemical manipulation of the extract canbe used to explore properties of the underlyingmitotic control system (Murray & Kirschner,1989). For example, addition of cycloheximideblocks all protein synthesis and prevents MPFoscillations because cyclin cannot be made.Exogenously synthesized cyclin can be added tothe cycloheximide-blocked extract to reconsti-

tute MPF activity and drive indicator nuclei intomitosis. Another classic experimental protocol(Murray & Kirschner, 1989) is to destroy allendogenous mRNA in the extract (this killsMPF oscillations because the extract cannotsynthesize cyclin) and then add back exogen-ously produced mRNA for cyclin B. In this waythe rate of cyclin synthesis in the extract can becontrolled with some precision. If a sufficientquantity of cyclin mRNA is added back, MPFoscillations resume. This proves that cyclin Bsynthesis is sufficient to drive the early embry-onic cell cycle.

1.5.

Biochemical investigations of frog egg ex-tracts, supplemented by genetic observations inyeast, have led to a consensus picture of themolecular mechanism regulating MPF activity(Fig. 2). The mechanism consists of reactionsgoverning phosphorylation and dephosphoryla-tion of Cdc2 and activation of the AnaphasePromoting Complex (APC). The APC attachesubiquitin moieties to cyclin B, and polyubiquiti-nated cyclin B is then rapidly degraded. (TheAPC is also known as the ‘‘cyclosome’’ becauseof its involvement in cyclin degradation).

This mechanism has been converted into a setof nonlinear ordinary differential equations(Table 1) by Novak & Tyson (1993a), who usedthe model to explain many physiological andbiochemical characteristics of mitotic control inintact frog eggs and extracts. Novak & Tyson(1993a, b) found solutions relevant to fourobserved physiological states:

1. steady state with low MPF activity (cf.immature oocyte);

2. steady state with high MPF activity (cf.mature egg);

3. limit cycle with extensive tyrosine phos-phorylation of Cdc2 in interphase (cf. oscillatingextract);

4. limit cycle with little tyrosine phosphoryl-ation of Cdc2 (cf. early embryo).

Naturally we might ask whether there areother types of solutions to the underlyingdifferential equations which correspond to otherphysiological states as yet undiscovered. In thispaper we use bifurcation theory to characterize

P P

PP

f

Wee1

e

ch

g d

b

a

CAK

Wee1

Cdc25

Active MPF

Y T

P P

Pre-MPF

ActiveMPF

P

Y T P

P

APC

Y T

ActiveMPF

PT

P

PP CAK

3

2 1

Mitosis&

celldivision

Y T

P Wee1

Cdc25

Y T Y T Y T

Aminoacids

(a) The phosphorylation states of MPF

(b) Positive feedback loops (c) Negative feedback loop

Cdc25 P

Cdc25

Wee1

IE IE

APC*

+

+

+

Y PTY

. . . . 72

F. 2. A model of the cell cycle engine in frog eggs (adapted from Novak & Tyson, 1993a, Fig. 1). (a) The dimer box.Oval=cyclin, rectangle=Cdc2, Y=tyrosine-15, T= threonine-161. Weel and Cdc2 are the kinase and phosphatase thatoperate on tyrosine-15. CAK is the threonine kinase; its opposing phosphatase (PP) is unknown at this time. Active MPFis the dimer in the upper right corner of the box. (b) Wee1 is inactivated and Cdc25 is activated by phosphorylation. ActiveMPF promotes these phosphorylation steps directly or indirectly. (c) Cyclin degradation is initiated by the anaphasepromoting complex (APC). MPF activates APC indirectly, as represented by phosphorylation of an intermediary enzyme(IE). IE could be the cdc20 gene product, whose role in cyclin degradation is currently being delineated.

the qualitatively different solutions of theNovak–Tyson model, in order to determine therange of physiologically distinct behaviorsavailable to the control system. In addition todetermining how solutions 1–4 are related toeach other, we find new solutions that predictunusual responses never before observed in eggsor extracts. The theory suggests many novelexperimental tests of the underlying mechanism.

A preliminary version of this work, with moremathematical details, can be found in the firstauthor’s dissertation (Borisuk, 1997).

2. One-parameter Bifurcation Diagrams

The differential equations (DEs) in Table 1determine the temporal behavior of the MPFregulatory network in Fig. 2. An instantaneousstate of the network can be thought of as a pointin multidimensional phase space (one coordinatefor each time-dependent variable in the model),and a solution of the DEs can be represented bya trajectory of state points traced through phasespace as time proceeds (Odell, 1980; Segel, 1984;Edelstein-Keshet, 1988; Kaplan & Glass, 1995).A collection of representative solution trajec-

73

tories in phase space sketches out the phaseportrait of the system. As parameter values ofthe model (Table 2) are changed, the phaseportrait may undergo a dramatic qualitativechange. For instance, a steady-state solution (asingle point in phase space, corresponding to anarrested extract) may lose stability and bereplaced by a stable limit cycle solution (a closedcurve in phase space, corresponding to periodicactivation of MPF). In such cases, we say thesystem has undergone a bifurcation. (For readerswho are unfamiliar with bifurcation theory, weprovide a primer of the basic ideas and jargon inthe Appendix.) Our intent is to characterize thekinds of qualitative changes that may occur inphase portraits of the Novak–Tyson model asrepresentative biochemical parameters arechanged.

To this end we first construct a one-parameterbifurcation diagram (Fig. 3), choosing MPFactivity to represent the behavior of the systemand k1, the rate constant for cyclin synthesis, asa particularly important, experimentally ad-justable parameter. [The diagram is created by apowerful computer program, AUTO, written byE. Doedel (Doedel & Wang, 1995).] In the figurewe plot representative values of MPF activity ask1 is varied. For k1 greater than about 0.2 min−1,there exists a single stable steady-state solution,and all initial conditions lead eventually (ast : a) to this steady state. The activity of MPFat the steady state is plotted as a solid line fork1 q 0.2 (hereafter it is understood that k1 carriesunits min−1). For k1 somewhat less than 0.2, thissteady-state solution has lost stability (nowrepresented by a dashed line), and the only stable

T 1Novak–Tyson model of MPF regulation in frog eggs

d[w]dt = k1[amino acids]− k2[w]− k3[w][q]

d[[Dimer]]dt= kPP[[Dimer]P]− (kwee + kCAK + k2)[[Dimer]]+ k25[P[Dimer]]+ k3[w][q]

d[P[Dimer]]dt = kwee[[Dimer]]− (k25 + kCAK + k2)[P[Dimer]]+ kPP[P[Dimer]P]

d[P[Dimer]P]dt = kwee[[Dimer]P]− (kPP + k25 + k2)[P[Dimer]P]+ kCAK[P[Dimer]]

d[[Dimer]P]dt = kCAK[[Dimer]]− (kPP + kwee + k2)[[Dimer]P]+ k25[P[Dimer]P]

d[Cdc25P]dt = ka[MPF][total Cdc25]− [Cdc25P])

Ka +[total Cdc25]− [Cdc25P] − kb[PPase][Cdc25P]Kb +[Cdc25P]

d[Wee1P]dt = ke[MPF][total Wee1]− [Wee1P])

Ke +[total Wee1]− [Wee1P] − kf[PPase][Wee1P]Kf +[Wee1P]

d[IEP]dt = kg[MPF]([total IE]− [IEP])

Kg +[total IE]− [IEP] − kh[PPase][IEP]Kh +[IEP]

d[APC*]dt = kc[IEP]([total APC]− [APC*])

Kc +[total APC]− [APC*] − kd[anti-IE][APC*]Kd +[APC*]

k25 =V25'([total Cdc25]− [Cdc25P])+ V250[Cdc25P]

kwee =VWee'[Wee1P]+VWee0([total Wee1]− [Wee1P])

k2 =V2'([total APC]− [APC*])+ V20[APC*]

Notes: We write a differential equation for the concentration (or relative activity) of eachof the nine regulatory proteins in Fig. 2. The right hand side of each DE has the formsynthesis–degradation+ activation–inhibition. The kis are rate constants and the Kjs areMichaelis constants. The total concentrations of Cdc2, Cdc25, Wee1, APC and IE are all takento be constant. Vi' (resp. Vi0) is the turnover number for the less active (more active) form ofeach enzyme.

(a)0.35

122

0.15

0.25

0.05

10–210–3 10–1 1

Rate constant for cyclin synthesis

MP

F a

ctiv

ity

60 38

35

(b) 270

0.15

0.25

0.05

0.00350.0025 0.0045 0.0055

Rate constant for cyclin synthesis

MP

F a

ctiv

ity

180

106

8

. . . . 74

T 2Parameter values used in the Novak–Tyson

model1993 1998

These parameters are dimensionlessKa/[total Cdc25] 0.1 0.1Kb/[total Cdc25] 0.1 1.0Kc/[total APC] 0.01 0.01Kd/[total APC] 0.01 1.0Ke/[total Wee1] 0.3 0.1Kf/[total Wee1] 0.3 1.0Kg/[total IE] 0.01 0.01Kh/[total IE] 0.01 0.01These parameters have units min−1

k1[amino acids]/[total Cdc2] 0.1 0.01k3[total Cdc2] 1.0 0.5V2'[total APC] 0.015 0.005V20[total APC] 1.0 0.25V25'[total Cdc25] 0.1 0.017V250[total Cdc25] 2.0 0.17Vwee'[total Wee1] 0.1 0.01Vwee0[total Wee1] 1.0 1.0kCAK 0.25 0.64kPP 0.025 0.004ka[total Cdc2]/[total Cdc25] 1.0 2.0kb[PPase]/[total Cdc25] 0.125 0.1kc[total IE]/[total APC] 0.1 0.13kd[anti IE]/[total APC] 0.095 0.13ke[total Cdc2]/[total Wee1] 1.33 2.0kf[PPase]/[total Wee1] 0.1 0.1kg[total Cdc2]/[total IE] 0.65 2.0kh[PPase]/[total IE] 0.087 0.15

Notes: The ‘‘1993’’ parameter set is from Novak &Tyson (1993a), the ‘‘1998’’ set is from Marlovits et al.(1998).

Fig. 3(b)]. This point has the characteristics of a‘‘saddle-node on an invariant circle’’ bifurcation:as k1 decreases through kSNIC

1 , a stable periodicsolution of finite amplitude disappears becauseits period diverges to infinity, and it is replacedby a pair of steady states, a stable node [the lowersolid curve in Fig. 3(b)] and an unstable saddlepoint (the dashed curve). Following this curve ofsaddle points, we find that it folds into a differentcurve of nodes at kSN

1 =0.0029864.Examining the upper branch of steady-state

solutions in Fig. 3(b), we find two additionalHopf bifurcations at kH'

1 = 0.0029870 andkH0

1 =0.0041859. Between these limits the stablesteady state coexists (in part) with unstable limitcycles. The larger branch of unstable limit cycles

F. 3. One-parameter bifurcation diagram. Rateconstant for cyclin synthesis= k1. (—— stable steady state,(– – –) unstable steady state, (W) stable limit cycle, (w)unstable limit cycle, (Q) Hopf bifurcation points. For eachlimit cycle, the upper (lower) circle represents the maximum(minimum) value of MPF observed during an oscillation.To interpret this diagram, you must think of k1 fixed at acertain value and MPF activity varying periodically withtime between the two limits on the figure. The encirclednumbers indicate the period of oscillation at different valuesof k1. In (a), notice the logarithmic scale for k1; (b) is ablow-up of the region of multiple stable solutions.

self-perpetuating solution of the DEs is anoscillatory state, for which MPF activityfluctuates periodically between maximum andminimum values represented by the filled circleson the diagram.

The value of k1 where the steady state losesstability and is replaced by a stable limit cycleoscillation is called a supercritical Hopf bifur-cation point, kH

1 =0.18226. As k1 is decreasedslightly below the bifurcation point, the period ofoscillation stays nearly constant and the ampli-tude of oscillation (max MPF activity–min MPFactivity) increases in proportion to zkH

1 − k1.As the rate of cyclin synthesis is decreased

further, we find that the stable periodic solutionpersists to very small values of k1. Its period isroughly constant at 35–40 min until k1 dropsbelow about 0.03, after which the period ofoscillation increases rapidly and the oscillatorysolution disappears at kSNIC

1 =0.0040975 [see

75

derives from a subcritical Hopf bifurcation at kH01

and disappears at a saddle-loop bifurcation atkSL

1 =0.0032912. The tiny branch of unstablelimit cycles bifurcates from the subcritical Hopfbifurcation at kH'

1 and disappears almost immedi-ately at a saddle-loop bifurcation.

3. Experimental Predictions

As we have pointed out, the rate of cyclinsynthesis (k1) can be controlled experimentally infrog egg extracts by enzymatic degradation ofendogenous mRNAs and then addition ofexogenously synthesized cyclin mRNA (Murray& Kirschner, 1989). The one-parameter bifur-cation diagram described in Fig. 3 makes someremarkable predictions about the behavior of asequence of extracts prepared with increasingamounts of cyclin mRNA. Let k1 =0.015represent the standard rate of cyclin synthesis inoscillating extracts (period1 60 min). At a20-fold higher rate (k1 =0.3), the extract shouldarrest in a mitotic state, with high MPF activity;and at a 10-fold lower rate (k1 =0.0015), theextract should arrest in interphase with low MPFactivity. By setting up a series of mRNA-ablatedegg extracts supplemented with increasingamounts of exogenous cyclin mRNA, one couldsearch for the bifurcation points that separatethese qualitatively distinct behaviors.

These bifurcation points will have distinct andeasily recognized properties, according to Fig. 3.At high levels of cyclin mRNA, one shouldobserve a supercritical Hopf bifurcation. Justbelow the bifurcation point, extracts wouldexhibit MPF oscillations of small amplitude(perhaps MPF activity does not drop lowenough for nuclei to go into interphase). Theperiod of these oscillations should be short(135 min) and show little dependence onmRNA level. In addition, maximum MPFactivity in this series of extracts increasesdramatically as cyclin mRNA level decreases!This prediction is surprising because one wouldexpect that less cyclin mRNA would generateless cyclin protein and, consequently, less MPFactivity. However, because of subtle interactionsbetween MPF and APC, a slower rate of cyclinsynthesis reduces the extent of APC activation

and permits a larger amplitude of MPFoscillations.

At low mRNA levels, the model predicts aSNIC bifurcation: the oscillatory solution haslarge amplitude (i.e. it drives nuclei in and out ofmitosis), and the period becomes very long as thelevel of cyclin mRNA approaches the bifurcationpoint [Fig. 3(b)]. Just below the bifurcationpoint, the extract will arrest in interphase, withlow MPF activity. Notice, however, that justbelow the SNIC bifurcation (for k1 between0.0029864 and 0.0040975) the model exhibits twosteady states, with low and high MPF activities,respectively. It would be striking to confirm thisfeature of the model experimentally, but ourcalculations (not shown) indicate that thedomain of attraction of the stable metaphase-ar-rested steady state is too small to be found bysimple perturbations (e.g. by adding active MPFto the interphase-arrested extract).

In Fig. 3(b) there is a tiny region,0.0041282Q k1 Q 0.0042567, where two stableoscillatory solutions coexist. Although it wouldbe impossible to confirm this prediction exper-imentally, it illustrates the complexity ofbehavior possible in this control system.

4. Two-parameter Bifurcation Diagrams

Are the predictions made on the basis ofFig. 3 reliable when we acknowledge that theparameter values estimated by Novak & Tyson(1993a) are quite uncertain? What becomes ofour one-parameter bifurcation diagram if westart to vary a second, third, fourth parameter?Are the predictions robust? Do they hold up inthe face of uncertainty about the otherparameters in the model?

We approach this question by characterizingthe bifurcation diagram of the model when twoparameters are varied simultaneously. First wewill construct a two-parameter bifurcationdiagram based on k1 and V20 (the rate constantfor cyclin degradation when the APC is active).This diagram gives a planar section through thefull 26-dimensional parameter space. Then wewill examine two other planar sections: (1) k1 and[Wee1]total (the total amount of Wee1 in theextract), and (2) k1 and [Cdc25]total. We will showthat all two-parameter bifurcation diagrams

. . . . 76

have certain generic features that are subject toexperimental verification.

A two-parameter bifurcation diagram isconstructed as follows. Let us start in Fig. 3(a)at the SN bifuration at k1 =0.0029864. We askAUTO to do the following: change k1 a little bit(the system moves off the point of SNbifurcation), then change V20 just enough tobring the system back to a SN bifurcation point.By repeating this process many times, AUTOcan trace out a locus of SN bifurcation points inthe (k1, V20) parameter plane [the red, inverted Vin Fig. 4(a)]. We see that there are two loci of SNbifurcations, which come together at a cusppoint. The left-hand branch contains the SNbifurcation at k1 =0.0029864 in Fig. 3(a), andthe right-hand branch contains the SN foldwhich is a component of the SNIC bifurcation atk1 =0.0040975 in Fig. 3(a). Inside the V, thesystem exhibits three steady states (two nodesand a saddle point). Outside the V, the systemhas only one steady-state solution.

Each branch of the V is a locus of‘‘codimension-one’’ bifurcation points, becausewe need vary only one parameter (either k1 orV20) to cross the bifurcation. The cusp is a‘‘codimension-two’’ bifurcation point, becausewe must specify the values of two parameters(both k1 and V20) in order to locate this point.Bifurcation theory tells us that the V-shape ofthese SN bifurcations is ‘‘generic’’, i.e. it is nota freakish property of our specific choice ofparameters, but rather V-shaped loci of SNbifurcations can be expected in any planarsection through parameter space.

Next, we return to the Hopf bifurcation atk1 =0.18226 in Fig. 3(a), and we ask AUTO tofollow it. AUTO returns with a closed curve[blue in Fig. 4(a)] of Hopf bifurcation points.Inside this closed curve we always find anunstable steady state accompanied by a stableperiodic solution; outside, the steady state isstable. It should be obvious that the Hopfbifurcations at 0.18226 and 0.0041859 in Fig. 3belong to the blue curve in Fig. 4(a), but theHopf bifurcation at 0.0029870 in Fig. 3(b) doesnot. Asking AUTO to follow the latter point, weget the light blue curve of Hopf bifurcations inFig. 4(b), which seems to attach in two places tothe red curve of SN bifurcations. Consulting anystandard text on bifurcation theory, we find thatsuch attachment points are typical of acodimension-two bifurcation first described byTakens & Bogdanov (TB). At a TB bifurcationthere must also exist a third locus of codimen-sion-one bifurcation points, called saddle-loop(SL) bifurcations [the green curves in Fig. 4(b)].At an SL bifurcation, a limit cycle is annihilatedby a saddle point; consequently the period ofoscillation diverges (: a) at the bifurcationpoint. The SL loci emanating from the TBbifurcation points attach elsewhere to the SNcurve at two SNL points (saddle-node-loopbifurcations, codimension two). Between the twoSNLs, we have a locus of codimension-one SNICbifurcations.

The basic structure of Fig. 4 is familiar toexperts in chemical dynamics. The cusp-shapedregion of multiple steady states with a loop ofHopf bifurcations attached at two TBs is a

F. 4. Two-parameter bifurcation diagram (incomplete). Rate constant for cyclin degradation= V20. Dashed horizontalline at V20 =1 represents the one-parameter cut in Fig. 3. (a) Within the red V-shaped region, bounded by saddle-node (SN)bifurcations, there exist three steady-state solutions; outside only one. Within the closed blue curve of Hopf (H) bifurcations,there exist an unstable steady state and a stable limit cycle; (b) a second locus of Hopf bifurcations (light blue) attaches tothe SN loci (red) at two Takens–Bogdanov bifurcation points (TB), along with a locus of saddle-loop (SL) bifurcations (green).The SL loci also attach to SN loci at two saddle-node-loop bifurcation points (SNL). Between the SNLs the bifurcation curveis a SNIC (saddle-node on an invariant circle).

F. 5. Two-parameter bifurcation diagram (complete). Key: red=saddle-node (SN), blue=Hopf (H), green=saddle-loop (SL), black=cyclic fold (CF). The phase portraits in each region of parameter space are described by icons: s= stablenode, u=unstable node, x= saddle point, solid circle= stable limit cycle, dashed circle=unstable limit cycle.

F. 7. Stable limit cycle oscillations: (a) the thick colored curves follow limit cycles of fixed period (20 min, 25 min, etc.)across the two-parameter plane; (b) the region of existence of at least one stable limit cycle is delineated by pieces of variousbifurcation sets that compose the outer edge of this bounded domain. Within the domain there are several subdomains withat least one other stable solution.

77

generic feature of reaction mechanisms withpositive feedback loops (autocatalysis) (Gucken-heimer, 1986), and the closed loop of Hopfbifurcations (dark blue) is typical of negativefeedback oscillators (Goldbeter & Guilmot,1996).

The two-parameter bifurcation diagram(Fig. 4) is incomplete because we have yet toaccount for places where periodic solutionschange their stability. Hopf bifurcations changefrom subcritical to supercritical at codimension-two, degenerate Hopf (DH) bifurcation points;and saddle-loops change from stable to unstableat codimension-two, neutral saddle-loop (NSL)bifurcation points. For our model, AUTO findssix DH points and two NSL points. At each ofthese eight codimension-two bifurcation pointsthere appears a branch of cyclic fold (CF)bifurcations: codimension-one bifurcationswhere stable and unstable limit cycles coalesceand disappear. There are four CF branches thatconnect the DH and NSL points in pairs: DH1

to NSL2, DH2 to DH5, DH3 to DH4, and DH6 toNSL1 (see Figs 5 and 6).

This completes our description of the two-par-ameter bifurcation diagram for the Novak–Tyson model, using the parameter set in their1993 paper. Full details of the diagram aregiven in Figs 5 and 6, along with icons thatpresent all the qualitatively different phaseportraits predicted by the model.

5. Biological Implications

Figure 5 gives a comprehensive view of thedifferent types of behavior implicit in therelatively simple molecular network controllingMPF activity in frog egg extracts. We can thinkof all the ‘‘action’’ occurring in a restricteddomain of parameter space. On the outskirts ofthis domain, the control system is arrested at aunique stable steady state. Within the domain,the system exhibits stable oscillatory solutions,multiple steady states, and bi-rythmicity (coexist-ing stable limit cycles). The largest subdomain,bounded in part by the dark blue Hopf curve inFig. 5, has a simple phase portrait: a singleunstable steady state accompanied by a stable

limit cycle, with period 20–60 min. This subdo-main corresponds to the well-known oscillatorybehavior of Murray–Kirschner (Murray &Kirschner, 1989) cycling extracts. It is reassuringto notice that this behavior is very robust: theoscillations persist over a broad range of rates ofcyclin synthesis and degradation. As we shallshow later, these oscillations are cruciallydependent on the negative feedback loop ofMPF on its own destruction, through IE andAPC [Fig. 2(c)].

A second subdomain of simple oscillations(stable limit cycle and a single unstable steadystate) exists at large values of V20 (bounded by thelight blue curve in Fig. 5). These oscillations arediven by the positive feedback loops: MPFactivates its friend (Cdc25) and inhibits its enemy(Wee1).

Stable limit cycle solutions exist only within arestricted region of the two-parameter plane.Figure 7(a) shows how the period of oscillationdepends on position within this region. Figure7(b) shows that the oscillatory domain isbounded by bits and pieces of the bifurcationsets in Fig. 5, and delineates subdomains whereother stable solutions coexist with a stable limitcycle. Figure 7(b) demonstrates that the tinyslice of bi-rhythmicity in Fig. 3 opens up into alarger stretch of k1 values at larger values of V20.There is even a tiny region of tri-stability inparameter space. Although it would be difficultto confirm these regions of multi-stability, weshould remember that the MPF regulatorysystem has the potential for such complexbehavior.

As we get into the region of multiple steadystates, within the red cusp in Fig. 5, the behaviorof the control system becomes very complicatedindeed. This complexity of solutions is not afreakish property of the Novak–Tyson model;rather it is exactly what can be expected of adynamical system that admits both limit cycleoscillations and multiple steady states. Where thetwo types of solutions collide, they typicallygenerate this sort of hodge-podge of solutions(Guckenheimer, 1986).

Might the complex dynamical properties ofthe MPF regulatory system play important rolesin control of the cell cycle? It is too early toanswer this question. There is no experimental

(a) (b)

S

S,X,US,X,S

S,X,

S,U,

S

S

S

U,

U

U

X

X

S ,U

U,U

X

XX

S,UX

SS,X

S

SS

S,X

CF

C1DH1

Hopf

Hopf

S

S

S

S

S

S,X, S

S,X, S

SN

SN

SN

SL

SL

SN

SNIC

SNIC

SNL4

SNL1

NSL1

TB2DH6

NSL2

SNL3

SNL2

SN

SNIC

CF

CF

HopfSL

50.0

[APC]total

[Wee1]total [Cdc25]total20.0

40.0

30.0

10.0

0.000.500.00 1.00 1.50 2.00

Relative level of expression

Per

iod

of

osc

illa

tio

ns

(min

)

. . . . 78

F. 6. Schematic representations of the most complicated regions of the two-parameter bifurcation diagram.

evidence for any behavior subtler than a stablesteady state (cell cycle arrest) or a simpleoscillation (mitotic cycles in early embryos). Onthe other hand, no one has ever thought to lookfor evidence of these more complex phaseportraits in the physiology and biochemistry ofthe eukaryotic cell division cycle.

6. How Robust are these Results?

6.1.

Many rate constants are proportional to totalenzyme concentration and thus may vary bytwo-fold between homozygous and heterozygousindividuals, if the heterozygote carries a non-functional copy of the gene. In general we wouldexpect a two-fold change in parameter value tocause a significant change in the period of a limitcycle oscillation, and it might eliminate the limitcycle altogether (by driving the system across abifurcation set). However, our model of MPFoscillations has the unexpected virtue that theperiod of the limit cycle is quite insensitive to thetotal concentration of key regulatory enzymes,over a broad range of concentrations (seeFig. 8).

6.2. -

Recently, Marlovits et al. (1998) have re-esti-mated the parameters of the Novak–Tysonmodel (Table 2), based on careful kineticmeasurements by Kumagai & Dunphy (1995).Figure 9(a) shows that this presumably better setof rate constants predicts a two-parameterbifurcation diagram very similar to Fig. 5,

F. 8. Period of oscillation as a function of the amountsof the major regulatory enzymes. The parameter setin Table 2 is normalized so that [Wee1]total =[Cdc25]total =[APC]total =1. If we were to change the level ofexpression of one of these genes, then the total proteinconcentration would deviate from 1. This figure demon-strates that the period of limit cycle oscillations is quiteinsensitive to expression level over at least a four-fold range.

Fig. 4

Fig. 5

Fig. 7

79

computed for the original parameter set. Thisobservation suggests that the basic features ofthe two-parameter bifurcation diagram are notsensitive to precise values of the other par-ameters in the model.

6.3.

Figure 9(a) shows one planar slice through a26-dimensional parameter space. Are the strangefeatures of this diagram unique to how we havesliced the object? In Fig. 9(b,c) we show twoadditional slices: k1 vs. [Wee1]total and k1 vs.[Cdc25]total. Although these slices look quitedifferent from each other and from the k1 vs. V20

slice, they are all composed of the same genericelements: a V-shaped domain of multiple steadystates, two families of Hopf bifurcations, and along stretch of SNIC bifurations (where the redand green curves overlap). Loosely speaking, theuniverse of different types of behavior in26-dimensional parameter space looks equallystrange no matter how you slice it!

6.4.

Finally, we can use AUTO to determine towhat degree the full Novak–Tyson model (withnine variables) can be reduced without losingvital qualitative features of the solution set. In anearlier paper, Novak & Tyson (1993b) usedsingular perturbation theory to simplify theirmodel to just two DEs: for active MPF and totalcyclin. Although the two-component version stillshows oscillations and arrested states, it does nothave the wealth of complex bifurcations ob-served in the nine-component model [seeFig. 10(a)]. The two-component model has aV-shaped region containing multiple steadystates, and a loop of Hopf bifurcationsconnected to the V at TB points, but it lacks thelarge region of oscillatory solutions enclosed bythe dark blue curve of Hopf bifurcations inFig. 4.

Following the lead of that paper, we havesimplified the model in stages and recomputedthe two-parameter bifurcation diagram to seehow the behavior of the model depends on itscomplexity. If we let k3 : a (fast dimerizationof Cdc2 and cyclin) and kCAK : a (fast

phosphorylation of threonine-161), we simplifythe system from nine variables to six, because wecan eliminate cyclin monomers and express theconcentrations of T161 unphosphorylateddimers in terms of T161 phosphorylated dimers.Nonetheless, there are no significant changes inthe two-parameter bifurcation diagram [see Fig.10(b)]. In addition, we can assume rapidphosphorylation and dephosphorylation ofWee1 or Cdc25, reducing the model from sixvariables to four, and still retain all thecharacteristic features of the two-parameterbifurcation diagram [Fig. 10(c)]. However, if weassume rapid phosphorylation and dephospho-rylation of IE, the model loses the large regionof limit cycle oscillations and the diagram lookslike Fig. 10(a). Since the role of IE is to introducea time delay in the activation of APC by MPF,we conclude that the region of oscillations withinthe dark blue curve in Fig. 4 is attributable to thenegative feedback loop in the model. Oscillationswithin the light blue loop, on the other hand, aregenerated by the positive feedback loops,whereby MPF activates Cdc25 and inhibitsWee1.

7. Discussion

Using bifurcation theory, we have uncoveredan unexpectedly complex collection of physio-logically distinct behaviors exhibited by thebiochemical network that controls MPF activityin frog eggs. Some solutions of the modelequations correspond to well-known physiologi-cal states of the egg: interphase arrest with lowMPF activity (immature oocyte), M-phase arrestwith high MPF activity (mature egg), andspontaneous limit cycle oscillations (auton-omous mitotic cycles of the early embryo)(Tyson, 1991). Other sets of solution are morebizarre and have never been recognized exper-imentally: for example, coexisting states ofinterphase and M-phase arrest, or two stableoscillatory solutions of different amplitude andperiod.

7.1.

Our theoretical analysis of the MPF controlsystem makes some dramatic predictions aboutwhere to look for unusual behavior. The basic

Fig. 9 Fig. 10

F. 9. Two-parameter bifurcation diagrams for the 1998 parameter set: (a) all the major conclusions we have drawn fromFig. 5 (the corresponding diagram for the 1993 parameter set) are still valid; (b) and (c) bifurcation diagrams using [Cdc25]total

and [Wee1]total as the second parameter. We use the same color scheme as before (Figs 4 and 5) for the bifurcation curves.Developmental paths: IO= immature oocyte, ME=mature egg, EE=early embryo, LE= late embryo.

F. 10. Bifurcation diagrams for reduced versions of the Novak–Tyson model. Basal parameters: 1993 set. RCCD=rateconstant for cyclin degradation: (a) the two-variable model [active MPF and total cyclin only; see Novak & Tyson (1993a)];(b) six-variable model (active MPF, total cyclin, Wee1, Cdc25, IE, APC); (c) four-variable model (active MPF, total cyclin,IE, APC).

. . . . 80

protocol calls for Xenopus egg extracts depletedof maternal mRNA and supplemented withcontrolled quantities of cyclin B mRNA (Murray& Kirschner, 1989).

1. At low levels of cyclin mRNA, extractsshould arrest in interphase, at intermediate levelsthey should exhibit mitotic oscillations, and athigh levels they should arrest in M phase.

2. As cyclin mRNA level is adjusted carefullybetween these three regimes, the extracts shouldundergo characteristic transitions (bifurcations).As mRNA content is reduced from high levels,the mitotic-arrested state should lose stability bya Hopf bifurcation, meaning that MPF oscil-lations, at first, have small amplitude. MPFactivity might not fluctuate enough to drivesperm nuclei in and out of mitosis (the easy wayto assay MPF fluctuations in extracts), butoscillations could be detected by histone H1kinase assays. As cyclin mRNA level is reducedfurther, the model predicts that maximum cyclinprotein content (and MPF activity) in the extractwould increase abruptly, but the period ofoscillation would increase only slightly: both ofthese predictions are counterintuitive. At stilllower levels, the period of oscillation wouldabruptly lengthen, and the extract would arrestin interphase, with plenty of cyclin and low MPFactivity. The model predicts that, at high levelsof cyclin mRNA, the oscillatory state should belost as the amplitude of oscillation goes to zerowith the period of oscillation holding constant,a signature of Hopf bifurcation. On theother hand, at low levels of cyclin mRNA,oscillations should be lost as the period goes toinfinity with the amplitude holding constant, acharacteristic of SNIC bifurcation.

3. For cyclin mRNA levels just below theSNIC bifurcation, the extract should exhibitbistable behavior. Sperm nuclei in the extractcould arrest either in interphase or in metaphase,depending on subtle details of how the extract isprepared.

4. Also in the vicinity of the SNIC bifur-cation, we might find more exotic bistablebehavior, e.g. stable MPF oscillations coexistingwith a stable arrested state.

In designing such tests, it would be advisableto have experimental control over a second

parameter in the model, in order to exploitinformation in two-parameter bifurcation dia-grams. Figure 9 should be especially useful inthis regard, because [Wee1]total and [Cdc25]total areparameters that can be manipulated experimen-tally.

7.2.

Tyson (1991) pointed out that frog eggs seemto use the bifurcation properties of the MPFcontrol system to control cell division duringdevelopment. By regulating the expression ofgenes for cyclin B, Wee1 and Cdc25, as well asother genes that interact with the MPF network,the egg can select different regions of parameterspace with different physiological properties. Forinstance, it is well known that, after exposure toprogesterone, frog eggs synthesize the mosprotein, which then interferes with cyclindegradation (Sagata et al., 1989; Hunt, 1989). InFig. 9(a) we show that a large decrease in the rateconstant for cyclin degradation, combined witha modest increase in the rate constant for cyclinsynthesis, can move an egg from an interphase-arrested steady state to a metaphase-arrestedsteady state (‘‘developmental path’’ IO : ME).The fertilized egg must then move across theHopf bifurcation into the region of sustainedoscillations [ME : EE in Fig. 9(a)], which mightbe attributable to the degradation of Mos afterfertilization (Watanabe et al., 1989).

As the embryo approaches the mid-blastulatransition, cell cycle time lengthens and cellsfinally arrest in interphase. In Drosophila thistransition is thought to occur by depleting theegg of Cdc25 (Edgar & O’Farrell, 1990), whichsuggests that the control system crosses theSNIC bifurcation as indicated in Fig. 9(b)(EE : LE). Cells in this state (LE) are arrestedat an interphase checkpoint. In order to entermitosis, the checkpoint must be lifted; forexample, by crossing the SNIC curve into theoscillatory domain (by increasing [Cdc25]total )and then returning to the region of interphasearrest (by decreasing [Cdc25]total). In this way, celldivision can be suited to the differentiation ofvarious tissues in the developing embryo.

In frog eggs, the mid-blastula transition isassociated not with fluctuations in Cdc25 levelbut with a dramatic disappearance of cyclin E

81

(Hartley et al., 1996). Guadagno & Newport(1996) showed that, when Cdk2/cyclin E isblocked by a specific inhibitor, mitotic cycling offrog embryos halts and Cdc2/cyclin B accumu-lates in the inactive, tyrosine-phosphorylatedform, apparently because Cdk2/cyclin E isnecessary to keep Cdc25 partially phosphory-lated and activated. Therefore, loss of Cdk2/cy-clin E activity at the mid-blastula transitionmight decrease the maximal activity of Cdc25(changing the parameter V250 in our equations)and thereby create an interphase-arrested state.Release from this checkpoint in later cell cyclemay be driven by re-expression of cyclin E or byfluctuations in other cyclin-dependent kinases.

In any case, if embryonic cell cycles arecontrolled by limit cycles (early) and checkpoints(later), then embryos must adopt parametervalues close to all the ‘‘action’’ in our bifurcationdiagrams, because it is in this region that changesin gene expression can create dramatic changesin cell division. If so, then it will be hard for cellsto avoid the more complex bifurcations thatcome with the territory. Perhaps cells make largechanges in parameter values (as in the develop-mental paths in Fig. 9) so that they skip from onelarge region of parameter space to another(where the behavior is simple and robust) andthereby miss the smaller domains where compli-cated bifurcations occur. Although this scenarioseems likely to us, we recognize the possibilitythat cells exploit some of the more complicatedbifurcations for subtle physiological purposes.We can cite no examples, but that may say moreabout what physiologists recognize and namethan what frog eggs are capable of doing.

Pancreatic cells are known to generatecomplex ‘‘bursting’’ patterns of insulin secretionthat depend on crossing bifurcation sets similarto the ones we have discovered here (Sherman,1997). We propose that cell cycle regulation, likeplasma glucose regulation, is intimately tied tobifurcations in the dynamics of the underlyingmolecular mechanism. In this paper we haveshown what complexity is inherent to the MPFcontrol system. Later in development, at leastfour different kinase/cyclin pairs and theirancillary proteins govern the full cell cycle ofvertebrates. To understand the bifurcationproperties of such comprehensive mechanisms

and the role they play in the physiology of thecell cycle will be a major challenge for futuremodeling in this field.

7.3.

The current paradigm of cell cycle control ineukaryotes posits a linear sequence of threecheckpoints: at the G1/S transition, the G2/Mtransition, and the meta/anaphase transition(Alberts et al., 1994). Cells progress through thedivision cycle, according to this view, bytransiting from one checkpoint to the next.Before they may pass the G1 checkpoint andenter S phase, cells must grow large enough towarrant a new round of DNA synthesis and theymust acquire the proper ‘‘visas’’. (For example,mammalian cells generally require exposure tocirculating growth factors, whereas buddingyeast cells require that sex pheromones be absentfrom their external environment). Before passingthe G2 checkpoint and entering mitosis, cellsmake sure that their DNA is completely andcorrectly replicated, and before separating sisterchromatids at anaphase, cells make sure thatevery chromosome is properly aligned on themitotic spindle.

We contend that these checkpoints are stable,steady-state solutions of the underlying kineticequations governing cell cycle events. In thisview, checkpoints are imposed/lifted by bifur-cations that create/destroy stable steady states.In this paper we have illustrated several kinds ofbifurcations that could serve this role. Forinstance, a checkpoint could be lifted at a Hopfbifurcation, where a steady state loses stability,or at a saddle-node bifurcation, where a stablesteady state is annihilated by coalescing with anunstable saddle point. Both scenarios have beenproposed to explain cell cycle transitions ineukaryotes (Tyson, 1991; Novak & Tyson,1993a, 1997; Thron, 1997), and SNIC bifur-cations work equally well (Kaern & Hunding,1998). The best experimental system for testingthese theoretical ideas is the frog egg extract. Ifwe can show (as proposed above) that mitoticcycles are lost at low levels of cyclin mRNA bya SNIC or SL bifurcation (infinite period), wewill have direct evidence that cell cycletransitions are intimately tied to subtle bifur-cation processes.

. . . . 82

7.4. -

?

In analysing a model of time-delayed negativefeedback in cyclin degradation (without positivefeedback of MPF on Wee1 and Cdc25),Goldbeter & Guilmot (1996) found robustoscillations but no evidence of multiple steadystates. Our analysis of the Novak–Tyson modelshows that the negative feedback loop doesindeed play an essential role in generating robustMPF oscillations characteristic of frog eggextracts and embryos. If we remove the timedelay from the negative feedback loop (byassuming that IE responds rapidly to changes inMPF activity), then we lose the large domain ofrobust oscillations bounded by the dark bluecurve in Fig. 10(c). The only remainingoscillatory domain relies on the positive feed-back loop generating large fluctuations intyrosine-15 phosphorylation, which are notobserved in mitotic oscillations in intact embryos(Ferrell et al., 1991).

The essential features of mitotic control inXenopus (multiple steady states and robustoscillations) seem to depend on both the positiveand negative feedback loops known to exist inthe MPF regulatory system. We cannot ade-quately describe the behavior of frog egg cellcycles with simplified models, like those initiallyproposed by Goldbeter (1991), Norel & Agur(1991), Tyson (1991). Although these simplemodels may be useful in understanding certainfeatures of mitotic control, they do not tell thewhole story. Therefore we must have tools forstudying the dynamics of large sets of coupledDEs that govern realistic reaction mechanismswith many interacting components. We havefound that a combination of computational andanalytical tools works well. Singular-pertur-bation arguments can be used to simplifycomplex models, so that the intuitively appeal-ing, geometrical tools of phase plane analysis canbe applied to subsets of the full network. Thenaccurate numerical solutions of the DEs can becomputed, to support the predictions of sim-plified models and to provide quantitative datafor comparison to experiments. Finally, thetheory of codimension-one and -two bifurcationsets, in combination with powerful numerical

programs like AUTO, can be used to unravel theproperties of complex networks with manyinteracting components.

Kathy Chen helped us considerably in preparingthis material for an audience of cell and molecularbiologists. Garry Odell gave us many valuablesuggestions about applying our analysis to frog eggextracts. John Guckenheimer advised us in the earlystages of pinning down Fig. 5. This work wassupported by National Science Foundation GrantsMCB-9600536 and DBI-9724085.

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B, M. T. (1997). Bifurcation analysis of a model ofthe frog egg cell cycle. Ph.D. Thesis. Virginia PolytechnicInstitute & State University, Blacksburg, VA. http://scholar.lib.vt.edutheses/etd-search.html

D, E. J. & W, X. J. (1995). AUTO94: software forcontinuation and bifurcation problems in ordinarydifferential equations. Pasadena, CA: Center for Researchon Parallel Computing, California Institute of Technol-ogy.

E-K, L. (1988). Mathematical Models inBiology. New York: Random House.

E, B. A. & O’F, P. H. (1990). The threepostblastoderm cell cycles of Drosophila embryogenesisare regulated in G2 by string. Cell 62, 469–480.

F, J., J. E., W, M., G, J. C. & M, G. S.(1991). Cell cycle tyrosine phosphorylation of p34cdc2 anda microtubule-associated protein kinase homologin Xenopus oocytes and eggs. Mol. Cell. Biol. 11,1965–1971.

G, A. (1991). A minimal cascade model for themitotic oscillator involving cyclin and cdc2 kinase. Proc.Nat. Acad. Sci. U.S.A. 88, 9107–9111.

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K, A. & D, W. G. (1995). Control of theCdc2/Cyclin B complex in Xenopus egg extracts arrestedat a G2/M checkpoint with DNA synthesis inhibitors.Mol. Biol. Cell 6, 199–213.

(a)0.8

0.4

0.0

–0.4

–0.8

–0.8–1.2 –0.4 0.0 0.4 0.8

x

y

(b)0.8

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0.0

–0.4

–0.6

–1.0 –0.6 –0.2 0.2 0.6 1.0

x

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K, Y. A. (1995). Elements of Applied BifurcationTheory. New York: Springer-Verlag.

L, H., B, D., B, A., Z, S. L.,M, P. & D, J. (1995). Molecular CellBiology, 3rd Edn, Chap. 25. New York: W. H. FreemanCo.

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M, A. W. & K, M. W. (1989). Cyclinsynthesis drives the early embryonic cell cycle. Nature339, 275–280.

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APPENDIX

Primer on Bifurcation Theory

Fuller introductions to bifurcation theory canbe found in Strogatz (1994), Borisuk (1997).Industrial strength treatises that we consulted areNayfeh & Balachandran (1985), Guckenheimer(1986), Kuznetsov (1995).

A kinetic model of a biochemical controlsystem, like Fig. 2 and Table 1, consists ofdifferential equations (DEs), one for eachtime-varying component (‘‘variable’’) in thereaction network. The right-hand-side of eachDE is a sum of terms, each one representing areaction in the network. Each term is precededby a + or − sign depending on whether itsreaction increases or decreases the concentration(or relative activity) of the variable. Each

F. A1. Phase plane portraits for eqn (*) in the Appendix. Each curve (trajectory) is a locus of points traced out bya solution x(t) and y(t), as t varies from 0 to a. Parameter values: b=0.5, c=0.1, p=1: (a) a=−0.8; (b) a=−0.55.

x

y

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–1.5–0.50–1.00 0.00 0.50 1.00

a

x(b)

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–1.5–0.25 0.25–0.75 0.75

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x(a)

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. . . . 84

F. A2. Hopf bifurcation in eqn (*). As a increases (at fixed b=5, c=0.1 and p=1), the steady state loses stabilityand a family of small amplitude, stable limit cycle solutions arises: (a) the envelope of these closed orbits traces out aparaboloid surface in three-dimensional space (x,y,a); (b) the amplitude of the limit cycle (xmin and xmax) is plotted as afunction of a. Supercritical Hopf bifurcations occur at a=−0.5905 and +0.5494.

F. A3. Saddle-node bifurcation in eqn (*): (a) phase portrait (a=−5.0, b=−0.36229, c=0.1, p=1); (b) bifurcationdiagram. As b increases (at fixed a, c and p), a new stable node is created, seemingly ex nihilo, in combination with a saddlepoint. As b increases further, the saddle point moves over and annihilates the original stable node.

F. A4. Saddle-node on an invariant circle: (a) phase portrait (a=−0.7, b=0.81979, c=0.53, p=1); (b) bifurcationdiagram. As b increases (at fixed a, c and p), a saddle and node coalesce and are replaced by a stable limit cycle.

85

reaction term is an algebraic function ofvariables and ‘‘parameters’’, those rate constantsand Michaelis constants that are necessary todescribe how fast biochemical reactions proceed.

For simplicity, let us suppose we have a systemdescribed by two variables (x and y) and fourparameters (a, b, c and p):

dxdt

= f(x,y;p)= p[x(1− x2)− y]

dydt

= g(x,y;a,b,c)= (x− a)(b− y)− c(*)

We have chosen specific functions, f and g, forillustrative purposes only. They do not representa biochemical control system because thevariables and parameters are allowed to take onnegative values (i.e. they are not concentrationsand rate constants). For a given set ofparameters values, if we specify initial con-ditions, x(0)= x0 and y(0)= y0, then we cansolve the DEs numerically to obtain twofunctions, x(t) and y(t), that describe exactlyhow the system evolves in time. For instance, thedynamical system may approach a stable steadystate: x(t) : x* and y(t) : y*, as t : a (here,x* and y* are constants). Or it may approach astable (limit cycle) oscillation: x(t) : xper(t) andy(t) : yper(t), as t : a [here, xper(t) and yper(t)are periodic functions of minimal period T].These ideas are illustrted in Fig. A1, where we

plot solutions of eqn (*) as trajectories in statespace (also called the ‘‘phase’’ plane).

The qualitative appearance of trajectories inthe phase plane is called the ‘‘phase portrait’’ ofthe system. In Fig. A1 all trajectories are suckedinto a single stable steady state for a small;whereas, for larger values of a, trajectories leavean unstable steady state and are attracted to astable limit cycle. At some intermediate value ofa (called the bifurcation point), the system mustmake a smooth transition from one phaseportrait to the other. The bifurcation isillustrated schematically in Fig. A2(a). In eqn (*),as a increases through aH, the steady state losesstability and a family of small amplitude, stablelimit cycles is born [see Fig. A2(b)]. This is calleda supercritical Hopf bifurcation. At a subcriticalHopf bifuration (not shown), an unstable steadystate gains stability and a family of smallamplitude, unstable limit cycles is born.

Equations (*) can also illustrate a saddle-nodebifurcation (Fig. A3) and a SNIC bifurcation(Fig. A4). In the latter case, a saddle-nodebifurcation occurs on an invariant circle[Fig. A4(a)], which is a closed trajectory thatproceeds out of the saddle-node and then swingsaround to come back into the saddle-node in adifferent direction as t : a. Beyond the SNICbifurcation, the system has a stable limit cycle oflong period [Fig. A4(b)].