regulation of the mammalian cell cycle: a model of the g...

Regulation of the mammalian cell cycle:a model of the G1-to-S transition

ZHILIN QU, JAMES N. WEISS, AND W. ROBB MACLELLANCardiovascular Research Laboratory, Departments of Medicine (Cardiology)and Physiology, University of California, Los Angeles, California 90095Submitted 14 February 2002; accepted in final form 19 September 2002

Qu, Zhilin, James N. Weiss, and W. Robb MacLellan.Regulation of the mammalian cell cycle: a model of theG1-to-S transition. Am J Physiol Cell Physiol 284:C349–C364, 2003. First published October 9, 2002; 10.1152/ajpcell.00066.2002.—We have formulated a mathematicalmodel for regulation of the G1-to-S transition of the mamma-lian cell cycle. This mathematical model incorporates the keymolecules and interactions that have been identified experi-mentally. By subdividing these critical molecules into mod-ules, we have been able to systematically analyze the contri-bution of each to dynamics of the G1-to-S transition. Theprimary module, which includes the interactions betweencyclin E (CycE), cyclin-dependent kinase 2 (CDK2), and pro-tein phosphatase CDC25A, exhibits dynamics such as limitcycle, bistability, and excitable transient. The positive feed-back between CycE and transcription factor E2F causesbistability, provided that the total E2F is constant and theretinoblastoma protein (Rb) can be hyperphosphorylated.The positive feedback between active CDK2 and cyclin-de-pendent kinase inhibitor (CKI) generates a limit cycle. Whencombined with the primary module, the E2F/Rb and CKImodules potentiate or attenuate the dynamics generated bythe primary module. In addition, we found that multisitephosphorylation of CDC25A, Rb, and CKI was critical for thegeneration of dynamics required for cell cycle progression.

positive feedback; phosphorylation; nonlinear dynamics; bi-furcation; simulation

THE EUKARYOTIC CELL CYCLE regulating cell division isclassically divided into four phases: G1, S, G2, and M(35, 38). Quiescent cells reside in the G0 phase and areinduced to reenter the cell cycle by mitogenic stimula-tion. In the S phase, the cell replicates its DNA, and atthe end of the G2-to-M transition the cell divides intotwo daughter cells, which then begin a new cycle ofdivision. This critical biological process is orchestratedby the expression and activation of cell cycle genes,which form a complex and highly integrated network(24). In this network, activating and inhibitory signal-ing molecules interact, forming positive- and negative-feedback loops, which ultimately control the dynamicsof the cell cycle. Although many of the key cell cycleregulatory molecules have been cloned and identified,

the dynamics of this complicated network are too com-plex to be understood by intuition alone.

Over the last decade, mathematical models havebeen developed that provide insights into dynamicalmechanisms underlying the cell cycle (1, 2, 12, 13, 16,23, 36, 37, 39, 57, 59, 61, 62). Cell cycle dynamics havebeen modeled as limit cycles (13, 16, 36, 39), cell mass-regulated bistable systems (37, 60, 61), bistable andexcitable systems (57, 59), and transient processes (1,2, 23). Whereas each of these approaches has led tomodels that nominally replicate the dynamics of thecell cycle under specific conditions, no unified theory ofcell cycle dynamics has emerged.

Given the complexity of the cell cycle, a logicalapproach is to model its individual components (e.g.,the G1-to-S transition and the G2-to-M transition)before linking them together. The biological supportfor this approach comes from the existence of twostrong checkpoints, one at the beginning of eachtransition (10). In this study, we mathematicallymodel the G1-to-S transition as a first step in thisprocess. Mathematical models of regulation of theG1-to-S transition have been previously proposedand simulated (2, 16, 23, 39, 40), with the model ofAguda and Tang (2) being the most detailed. The keyregulators in control of the G1-to-S transition arecyclin D (CycD), cyclin E (CycE), cyclin-dependentkinases (CDK2, CDK4, and CDK6), protein phospha-tase CDC25A, transcription factor E2F, the retino-blastoma protein (Rb), and CDK inhibitors (CKIs),particularly p21 and p27. Many interactions amongthese regulators have been outlined experimentally,and these experiments provide guidelines for devel-oping a physiologically realistic model. To investi-gate the dynamics systematically, we have taken theapproach of breaking down the regulatory network ofthe G1-to-S transition into individual simplified sig-naling modules, with the primary component involv-ing CycE, CDK2, and CDC25A. After analyzing thedynamical properties of this primary signaling mod-ule, we systematically add E2F/Rb, CKI, and CycD ina stepwise fashion and explore the features they addto dynamics of the G1-to-S transition.

Address for reprint requests and other correspondence: Z. Qu,Dept. of Medicine (Cardiology), 47-123 CHS, 10833 Le Conte Ave.,University of California, Los Angeles, Los Angeles, CA 90095(E-mail: [email protected]).

The costs of publication of this article were defrayed in part by thepayment of page charges. The article must therefore be herebymarked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734solely to indicate this fact.

Am J Physiol Cell Physiol 284: C349–C364, 2003.First published October 9, 2002; 10.1152/ajpcell.00066.2002.

0363-6143/03 $5.00 Copyright © 2003 the American Physiological Societyhttp://www.ajpcell.org C349

MATHEMATICAL MODELING

Figure 1 summarizes the key interactions betweenregulators of the G1-to-S transition in the mammaliancell cycle that have been identified experimentally overthe last two decades and have been incorporated intoour model.

CycE and CDK2 Regulation

Increased CDK2 activity marks the transition fromthe G1 to the S phase. CDK2 activity is regulated by atleast two mechanisms, including 1) transcriptional reg-ulation of its catalytic partner, CycE, and 2) posttrans-lational modification of the CycE-CDK2 complex itself.For the purposes of our model, we will assume thatCycE transcription is primarily regulated by two mech-anisms (step 1). Mitogenic stimulation promotes Myc-dependent transcription of CycE (4, 34, 45), which weassume occurs at a constant rate (k1). In addition, E2Fis also an important transcription factor for CycE syn-thesis (4, 9). We assume that it induces CycE synthesisat a rate proportional to E2F concentration (k1ee in Eq.1a). Thus the total CycE synthesis rate is k1 � k1ee. ForCycE and CDK2 binding and activation, we adoptedthe scheme proposed by Solomon et al. (51, 52). CycEand CDK2 bind together, forming an inactive CycE-CDK2 complex (step 3), with CDK2 phosphorylated atThr14, Tyr15, and Thr160. CDC25A dephosphorylatesThr14 and Tyr15 and activates the kinase (step 5) (18,47, 51, 52). The complex is inactivated by ubiquitin-mediated degradation of CycE in its free form (step 2)or in the active bound form (CycE-CDK2, step 7) (66,67). We assume that the rates of degradation throughboth pathways are proportional to their concentrations(k2y for free CycE and k7x for active CycE-CDK2 inEq. 1a).

CDC25A Regulation

CDC25A is a phosphatase that dephosphorylatesand activates CDK2 by removing inhibitory phos-phates from Thr14 and Tyr15. CDC25A is a transcrip-tional target of Myc (28) and E2F (64). We assume thatMyc induces CDC25A synthesis at a constant rate (k8)and that E2F induces CDC25A synthesis in proportionto E2F concentration (k8ee in Eq. 1a). CDC25A is alsodegraded through ubiquitination, which we assume isproportional to its concentration (k9z0 in Eq. 1a). ForCDC25A to become active, it must itself be phosphor-ylated. This phosphorylation is catalyzed by the activeCycE-CDK2 complex (18), which forms a key positive-feedback loop in CycE-CDK2 regulation. It has beenshown that CDC25C is highly phosphorylated at theG2-to-M transition and has five serine/threonine-pro-line sites: Thr48, Thr67, Ser122, Thr130, and Ser214 (17,25, 32). The number of functionally important phos-phorylation sites on CDC25A has not been determinedexperimentally, but we assume that CDC25A has a

total of L phosphorylation sites and that its multisitephosphorylation occurs sequentially, with each phos-phorylation step catalyzed by CycE-CDK2 (Fig. 1B).We also assume that highly phosphorylated CDC25A isdegraded through ubiquitination (step 10).

E2F/Rb Regulation

E2F is a transcription factor for a number of cellcycle genes that are critical for G1-to-S transition inmammalian cells (49, 68). This family of transcriptionfactors binds to and is inactivated by a second family ofproteins known as pocket proteins, the prototypicalmember being the Rb gene product. In G0 or early G1,E2F is complexed to Rb and is inactive. E2F is freed byRb phosphorylation, which occurs sequentially first byCycD-CDK4/6 and subsequently by CycE-CDK2, form-ing another positive-feedback loop. E2F is also synthe-sized de novo as the cell progresses from G0 to G1 andautocatalyzes its own production (9, 27). E2F can beinactivated by binding to dephosphorylated Rb and bydegradation through ubiquitination after phosphoryla-tion by CycE-CDK2 (9). In the model, we assume thatE2F is synthesized at a constant rate k11 and a raterelated to free E2F concentration [k11eg(e) in Eq. 1b]and degraded at a rate proportional to its concentra-tion (k12e in Eq. 1b) and CycE-CDK2 (k12xe in Eq. 1b).Although it is known that Rb has 16 phosphorylationsites (14), the number of functionally important sites isunknown. Therefore, we assume that Rb has a total ofM� phosphorylation sites and that E2F is dissociatedfrom Rb when a certain number (M) of sites are phos-phorylated and Rb is hyperphosphorylated by phos-phorylation of the other phosphorylation sites (M� �M) on Rb. In our model, we vary M and M� to study theeffects of multisite phosphorylation.

CycD and CDK4/6 Regulation

Mitogenic stimulation of cells in the G0 phase trig-gers synthesis of CycD (28). CycD interacts with CDK4or CDK6 to form a catalytically active CycD-CDK4/6complex, which phosphorylates Rb to free active E2F(9, 49). It also potentiates CDK2 activity indirectly bytitrating away inhibitory CKIs from CycE-CDK2 (50).

CKI Regulation

CKIs, such as p21 or p27, bind to CycD-CDK4/6 orCycE-CDK2 to form trimeric complexes. CKI activity ishigh during the G0 phase but decreases during the cellcycle (49). Because factors regulating CKI synthesisare not well understood, we assume that CKI is syn-thesized at a constant synthesis rate (k18) and de-graded at a rate proportional to its concentration (k19iin Eq. 1d), which is low enough to ensure a high CKIlevel at the beginning of the G1-to-S transition. It hasbeen shown that p27 binds to CycE-CDK2 and has to

C350 MODELING THE MAMMALIAN CELL CYCLE

AJP-Cell Physiol • VOL 284 • FEBRUARY 2003 • www.ajpcell.org

Fig. 1. Schematic model of molecular sig-naling involved in regulation of the G1-to-S transition. A: signaling network di-vided into functional modules for the G1-to-S transition. Cyclin E (CycE), cyclin-dependent kinase (CDK) 2 (CDK2), andprotein phosphatase CDC25A form mod-ule I (red box), retinoblastoma protein (Rb)and transcription factor E2F form moduleII (green box), cyclin D (CycD) and CDK4/6form module III (orange box), and CDKinhibitor (CKI) forms module IV (bluebox). B: reaction scheme for CDC25A mul-tisite phosphorylation catalyzed by activeCycE-CDK2 complex. C: E2F-Rb pathway,in which multisite phosphorylation of Rbis catalyzed by CycD-CDK4/6 and activeCycE-CDK2. When Rb in the Rb-E2F com-plex is phosphorylated at M sites, E2F isfreed. Free Rb can then be phosphorylatedat the rest of its phosphorylation sites(M� � M). D: multisite phosphorylation ofthe trimeric complex CycE-CDK2-CKI iscatalyzed by active CycE-CDK2. WhenCKI has N sites phosphorylated, it bindsto F-box protein for ubiquitination anddegradation (step 25), and CycE-CDK2 isfreed. See Table 1 for rate constants.

C351MODELING THE MAMMALIAN CELL CYCLE


be phosphorylated on Thr187 by active CycE-CDK2 forubiquitination and degradation (31, 48, 65). Ishida etal. (19) showed that p27 is phosphorylated on manysites, including Ser10, Ser178, and Thr187, and showedthat Ser10 was phosphorylated-dephosphorylated in acell cycle-dependent manner and contributed to p27stability. Although it is known that Thr187 phosphory-lation is required for p27 ubiquitination, it is not clearwhether phosphorylation and dephosphorylation ofany other sites are also needed for p27 ubiquitinationin the mammalian cell cycle. A recent study (33) inyeast has shown that Sic1, a homolog of p27, has ninephosphorylation sites and that six of these sites have tobe phosphorylated by Cln-CDC28 to bind to F-boxproteins for ubiquitination and degradation. Becausethere is no explicit information on how many phosphor-ylation sites are required for CKI ubiquitination anddegradation in the mammalian cell cycle and no infor-mation on what kinase and phosphatase are responsi-ble for phosphorylation and dephosphorylation of thephosphorylation sites, except Thr187, in this model, wegenerally assume that N sites on CKI have to bephosphorylated by active CycE-CDK2 for CKI ubiquiti-nation to occur. We vary N from 1 to a certain numberto study the effects of phosphorylation. After CKI isubiquitinated and degraded, active CycE-CDK2 isfreed from the trimeric complex. This forms a positive-feedback loop between CycE-CDK2 and CKI. CKI alsobinds to CycD-CDK4/6 to form a trimeric complex (49,50) and is modeled analogously.

On the basis of the model outlined in Fig. 1, weconstructed differential equations (see Table 1 for def-initions of symbols and parameters), which are listedseparately for each module as follows.

For module I (CycE � CDK2 � CDC25A)

y � k1 � k1ee � k2 y � k3 y � k4x1

x1 � k3 y � k4x1 � �k5 � f�z��x1 � k6x

x � �k5 � f�z��x1 � k6x � k7x � k23xi � k24ix0 � k25ixN

z0 � k8 � k8ee � k9z0 � kz�z1 � kz

�z0(1a)

�

zl � kz�zl � 1 � kz

�zl � kz�zl � 1 � kz

�zl, l � 1, L � 1

�

zL � kz�zL � 1 � kz

�zL � k10zL

where f(z) � ¥l � 1L lzl represents the catalyzing

strength of CDC25A on CDK2 and l is a weighingparameter. L is the total number of phosphorylationsites of CDC25A. kz

� � bz � czx is the rate constant forCycE-CDK2-catalyzed phosphorylation of CDC25A,and kz

� � az is the rate constant for dephosphorylationof CDC25A. In Eq. 1a, we assumed that CDK2 concen-tration was much higher than cyclin concentration (3)and, thus, set CDK2 concentration to be constant 1.

Table 1. Definitions of variables in Eq. 1 and defaultparameter set

Variable Definition

x Active CycE-CDK2y Free CycEx1 Inactive CycE-CDK2zl l-Site-phosphorylated CDC25A (l � 0, L)e Free E2FPm m-Site-phosphorylated Rb-E2F (m � 0, M)rm m-Site-phosphorylated Rb (m � 0, M�)d Free CycDd4 CycD-CDK 4/6i Free CKIixn n-Site-phosphorylated CycE-CDK2-CKI (n � 0, N)id CycD-CDK4/6-CKI

Parameter Value

Rate constants for reaction steps in Fig. 1Ak1 200k1e 1k2 1k3 50k4 50k5 0.1k6 1k7 9k8 100k8e 0k9 1k10 0k11 0k11e 3k12 0.1k12x 0.01k13 0k14 1k15 1k16 1k17 1k18 100k19 1k20 0.1k21 1k22 0.1k23 0.4k24 1k25 5

Constant for CDC25A, Rb, and CKI phosphorylationand dephosphorylation in Fig. 1, B-D

az 100bz 1cz 1ar 50br 1cr 1ai 50bi 1ci 1

Rate constant for E2F and Rb affinity and dissociation in Fig. 1Ckm

� 0.01

(m�0,M�1)km

� 0.5kM

� 10kM

� 0.01

Total Rb concentrationR0 100

Parameters served as default set. Any parameter not given in corre-sponding figure legend is set to default value. CycE and CycD, cyclins Eand D; CDK, cyclin-dependent kinase; Rb, retinoblastoma protein; CKI,CDK inhibitor.



For module II (Rb � E2F)

e � k11 � k11e g�e� � �k12 � k12x x�e

� m�0

M

km�pm �

m�0

M

km�erm

p0 � kr�p1 � kr

� p0 � k0�p0 � k0

�er0

�

pm � kr�pm�1 � kr

�pm � kr�pm � kr

� pm�1 � km� pm

� km�erm, m � 1, M � 1

�

pM � kr�pM�1 � kr

�pM � kM� pM � kM

�erM

r0 � k0�p0 � k0

�er0 � kr�r0 � kr

�r1 (1b)

�

rm � km� pm � km

�erm � kr�rm�1 � kr

�rm � kr�rm

� kr�rm�1, m � 1, M

�

rm� � kr�rm� � 1 � kr

�rm� � kr�rm�

� kr�rm��1, m� � M � 1, M� � 1

�

rM� � R0 � m�0

M

pm � m�0

M��1

rm

where g(e) is a function representing E2F synthesisby E2F itself and will be defined later. M is thenumber of phosphorylation sites on Rb needed fordissociation of E2F from Rb. M� is the total numberof phosphorylation sites on Rb. km

� and km� (m � 0,M)

are the rate constants for E2F dissociation fromm-site-phosphorylated Rb and for association withm-site-phosphorylated Rb, respectively. kr

� � br �crx � crd4 is the combined rate constant for CycD-CDK4/6- and CycE-CDK2-catalyzed phosphorylationof Rb, and kr

� � ar is the rate constant for Rbdephosphorylation. For simplicity, here we assumedthat CycD-CDK4/6 and CycE-CDK2 have the samecatalytic effects on Rb and that the phosphorylationand dephosphorylation are not state dependent; i.e.,we used the same rate constant for each phosphory-lation or dephosphorylation step. R0 is the total Rbconcentration.

For module III (CycD � CDK4/6)

d � k13 � k14d � k15d � k16d4(1c)

d4 � k15d � k16d4 � k17d4 � k20d4i � k21id

For module IV (CKI)

i � k18 � k19i � k20d4i � k21id � k22id � k23xi � k24ix0

id � k20id4 � k21id � k22id

ix0 � k23xi � k24ix0 � ki�ix0 � ki

�ix1

� (1d)

ixn � ki�ixn�1 � ki

�ixn � ki�ixn � ki

�ixn�1, n � 1, N � 1

�

ixN � ki�ixN�1 � ki

�ixN � k25ixN

where ki� � bi � cix is the rate constant for CycE and

CDK2-catalyzed CKI phosphorylation and ki� � ai is

the rate constant for dephosphorylation. N is the totalnumber of phosphorylation sites on CKI. We also as-sumed that the rate constants are the same for eachphosphorylation step.

When we simulate one module or a combination ofsome modules, we indicate that we remove all theinteraction terms in Eq. 1 from the modules that arenot involved. In numerical simulation, we used thefourth-order Runge-Kutta method to integrate Eq. 1.The time step we used in simulation was �t � 0.005.

RESULTS

Dynamics of Module I (CycE � CDK2 � CDC25A)

The full model shown in Fig. 1 is too complex for acomplete analysis of its dynamical properties. Wetherefore divided the major components of the G1-to-Stransition into modules and examined their individualbehaviors before reintroducing them into the full modelto determine their combined effects. We consider theprimary module (module I in Fig. 1A) as comprised ofCycE, CDK2, and CDC25A, with CDC25A driving apositive-feedback loop catalyzing active CycE-CDK2production.

Figure 2 shows the case for two functional phosphor-ylation sites on CDC25A (L � 2) and f(z) � zL (hereaf-ter defined as the default conditions for module I,unless otherwise indicated), in which the steady stateof active CycE-CDK2 is plotted as a function of theCycE synthesis rate k1. Figure 2, C and D, also showsthe corresponding bifurcations vs. k1. Depending onthe stability of the steady state and the other param-eter choices, the system exhibits the following dynam-ical regimes.

Regime 1: monotonic stable steady-state solution. Forany CycE synthesis rate k1, there is only one steady-state solution, and it is stable (Fig. 2A).

Regime 2: bistable steady-state solutions. There arethree steady-state solutions over a range of k1 (k1 �418–612 in Fig. 2B): two are stable, and one is a saddle(unstable). As k1 is increased from low to high, CycE-CDK2 remains low until k1 exceeds a critical value, atwhich point there is a sudden increase in CycE-CDK2(upward arrow in Fig. 2B). When k1 decreases fromhigh to low, however, the transition occurs at a differ-



ent k1 (downward arrow in Fig. 2B), forming a hyster-esis loop.

Regime 3: limit cycle solution. The steady state in theparameter window (k1 � 90–290 in Fig. 2C) is anunstable focus, and CycE and CycE-CDK2 oscillatespontaneously (Fig. 2E, with k1 � 150). The transitionfrom the stable steady state to the limit cycle is viaHopf bifurcation.

Regime 4: multiple steady-state and limit cycle solu-tions. There are three steady-state solutions over arange of k1 (k1 � 192–245 in Fig. 2D): a stable node, asaddle, and an unstable focus. For a range of k1 justbeyond the triple steady-state solution (k1 � 245–320in Fig. 2D), a limit cycle solution exists. In this case,the system undergoes a saddle-loop (or homoclinic)bifurcation (54).

Regime 5: excitable transient. Suprathreshold stim-ulation causes a large excursion that gradually returnsto the stable steady state.

Role of CDC25A

The dynamics shown in Fig. 2 are critically depen-dent on CDC25A phosphorylation. Figure 3 shows

steady-state fully phosphorylated CDC25A (Fig. 3A)and total CDC25A (Fig. 3B) vs. active CycE-CDK2. Asthe number of phosphorylation sites increases, thefully phosphorylated CDC25A increases more steeplyand at a higher threshold as active CycE-CDK2 in-creases. This steep change in CDC25A is critical forinstability, leading to limit cycle, bistability, and otherdynamical behaviors. When CDC25A had only onephosphorylation site (L � 1), the steady state wasalways stable, regardless of other parameter choices(Fig. 4, A and B). It can be demonstrated analyticallythat the steady state can become unstable and lead tointeresting dynamics only when CDC25A has morethan one phosphorylation site and requires phosphor-ylation of both sites to become active (see APPENDIX). InFig. 2, we show various dynamics for L � 2; onlybiphosphorylated CDC25A is active. In Fig. 4, A and B,we also show bifurcations for L � 3. When only triphos-phorylated CDC25A is active, i.e., f(z) � z3 in Eq. 1a,the range of the limit cycle is k1 � 300–520 (Fig. 4A)and the bistability occurs at k1 � 675–2,385 (dashed-dotted line in Fig. 4B), which is a much higher range ofk1 than for L � 2. If we assume that bi- and triphos-

Fig. 2. Steady-state solutions and bi-furcations for module I. Parameters aredefault values in Table 1, unless other-wise indicated. A: steady state of activeCycE-CDK2 vs. CycE synthesis rate(k1), with k5 � 1. B: steady state ofactive CycE-CDK2 vs. k1, with k2 � 5.SN, saddle-node bifurcation. Dashedline, unstable steady state, which is asaddle. As k1 increases from small tolarge, a transition from lower to highersteady state occurs at SN (upward ar-row). If k1 decreases from large tosmall, a transition from the higher tothe lower steady state occurs at theother SN (downward arrow), forming ahysteresis loop. C: steady state and bi-furcation of active CycE-CDK2 vs. k1,with k2 � 0.5. H, Hopf bifurcationpoint, at which a stable focus becomesan unstable focus and oscillation be-gins; solid lines, stable steady state;dashed-dotted line, unstable steadystate; circles, maximum and minimumvalues of active CycE-CDK2. Whensteady state is stable, maximum andminimum values of active CycE-CDK2are equal and the same as the steady-state value. When the steady state isan unstable focus, active CycE-CDK2oscillates as a limit cycle, and its max-imum and minimum differ (E). All bi-furcation diagrams for limit cycleshave been plotted as maximum andminimum active CycE-CDK2 vs. k1. D:same as C, except k2 � 1.25 and k3 �0.02. E: free CycE and active CycE-CDK2 vs. time for a limit cycle in C atk1 � 150. F: free CycE and active CycE-CDK2 vs. time for an excitable case inD for k1 � 210. At t � 50 (arrow), weheld active CycE-CDK2 at 6 for a dura-tion of 0.02 to stimulate the large ex-cursion.



phorylated CDC25A are equally active, i.e., f(z) � (z2 �z3)/2 in Eq. 1a, the limit cycle (open circle in Fig. 4A)and the bistability (dotted line in Fig. 4B) occur atmuch lower k1, very close to the case of L � 2.

Figure 4, C and D, shows the effects of CDC25Asynthesis rate on limit cycle and bistability. Decreas-ing the synthesis rate of CDC25A shifts the limit cycleand bistable regions to higher but wider k1 ranges. Forexample, when k8 � 25, the limit cycle occurred at k1 �195–480 (Fig. 4C, cf. k1 � 90–290 in Fig. 2C), andbistability occurred at k1 � 860–1,400 (Fig. 4D, cf. k1 �418–612 in Fig. 2B). In Figs. 2 and 4, we assumed nodegradation of phosphorylated CDC25A; i.e., k10 � 0. Ifwe assume that fully phosphorylated CDC25A is de-graded at a certain rate (k10 � 0), the limit cycle regionis widened and the bistable region is shifted to a higherrange of k1. For example, the range of the limit cycleregime shown in Fig. 2C was k1 � 90–365 and therange of bistability in Fig. 2B was k1 � 475–640 afterwe set k10 � 5.

In Figs. 2 and 4, we assumed that the CDC25Aphosphorylation and dephosphorylation rates werefast. If these rates were slow, the steady state didnot change and bistable behavior was not altered, butlimit cycle behavior was affected. Slowing the phos-phorylation and dephosphorylation rates caused nar-rowing of the k1 range over which limit cycle behavioroccurs, and eventually limit cycle behavior disappeared(Fig. 4E).

Role of E2F

The results presented thus far show that the pri-mary module of the G1-to-S transition by itself exhibitsmultiple dynamical regimens. We now examine howRb and E2F (module II in Fig. 1), known to play crucialroles in mammalian cell cycle progression, regulate thedynamics of G1-to-S transition.

With total E2F constant. If there is no E2F synthesisand degradation, i.e., steps 11 and 12 are absent inmodule II, then the total E2F is constant (for simplic-ity, we set it equal to total Rb). Figure 5A shows thesteady state for free E2F concentration vs. active CycE-CDK2 or CycD-CDK4/6. As the number of Rb phos-

phorylation sites M required to free E2F, as well as thetotal number of phosphorylation sites M�, increases,the threshold for E2F dissociation and the steepness ofthe response increased.

A steep sigmoidal function due to multisite phos-phorylation of Rb as shown in Fig. 5A, coupled with thepositive-feedback loop between CycE and E2F, mightbe predicted to generate instability. To test this, weremoved the CDC25A from module I and simulated Eq.1, a–c, together. We assumed that dephosphorylationof Thr14 and Tyr15 was carried out by a phosphatase(possibly CDC25A) at a constant rate and, thus, setk5 � 1 and f(z) � 0 in Eq. 1a. In this system, we foundthat bistability could be generated, but no other dy-namics such as limit cycle occurred. Figure 5B showsan example of a bistable steady state of active CycE-CDK2 vs. CycE synthesis rate k1, with (k13 � 50) orwithout (k13 � 0) CycD for M � 2 and M� � 16. Thepresence of CycD caused the bistability to occur atlower k1, because CycD-CDK4/6 phosphorylates Rband frees E2F, which can then promote the E2F-depen-dent synthesis of CycE. In Fig. 5C, we show the condi-tions in the M � M� space under which bistabilityoccurs. This demonstrates that multisite phosphoryla-tion of Rb is critical for the CycE-E2F positive-feedbackloop to generate bistability.

We then added the CDC25A function back to moduleI and simulated Eq. 1, a–c, to study how E2F modu-lates the dynamics of module I. With module I in thelimit cycle regimen (Fig. 2C), Fig. 5D shows two bifur-cations with (k13 � 50) and without (k13 � 0) CycD.Without CycD, the limit cycle range occurred at k1 �90–250 (compared with k1 � 90–290 in Fig. 2C) and aperiod 2 oscillation occurred at around k1 � 230 (Fig.5D, inset). With CycD, the range decreased to k1 �75–220. We then set module I in the bistability regi-men as in Fig. 2B. Without CycD (Fig. 5E), the range ofthe bistable region was k1 � 405–610 (compared withk1 � 420–612 in Fig. 2B); with a low CycD level (k13 �50 in Fig. 5E), the range was k1 � 385–598; with a highCycD level (k13 � 200 in Fig. 5E), the range was k1 �320–514. Because in our model the CycD module (mod-

Fig. 3. A: steady state of fully phos-phorylated CDC25A (zL) vs. activeCycE-CDK2 (x) for different numbersof total phosphorylation sites (L � 1, 2,and 5). B: total CDC25A vs. activeCycE-CDK2. Results were obtained bysimulating the CDC25A phosphoryla-tion module in B and using x as acontrol parameter.



ule III) does not include any feedback, it did not lead toany novel dynamics. CycD-CDK4/6 is simply propor-tional to CycD. The major effect of CycD-CDK4/6 is tophosphorylate Rb and, thus, produce more free E2F.Greater free E2F promotes more CycE synthesis,which causes the Hopf or saddle-node (SN) bifurcationsat lower k1. Another effect of CycD-CDK4/6 shown inFig. 5D is the removal of period 2 oscillation. This canbe explained as follows: because CycD-CDK4/6 frees acertain amount of E2F from the Rb-E2F complex, theavailability of Rb-E2F for active CycE-CDK2 to phos-phorylate is reduced. This makes the steady-state re-sponse curve of free E2F vs. CycE-CDK2 less steepthan is the case without CycD. The reduction of steep-ness of the free E2F response to active CycE-CDK2thus causes the period 2 behavior to disappear. In fact,

even in the case of no CycD, if we reduce the hyper-phosphorylation sites (M� � M) of Rb, this period 2 willalso disappear because of the reduction of steepness ofthe response of free E2F to active CycE-CDK2.

With E2F synthesis and degradation. If we introduceE2F synthesis and degradation (steps 11 and 12) intothe E2F-Rb regulation network, then the steady-stateconcentration of E2F as a function of active CycE-CDK2 (x) is simply determined by steps 11 and 12 andsatisfies the following equation

k11 � k11e g�e0� � �k12 � k12x x�e0 � 0 (2)

If g(e) � e, then e0 � k11/(k12 � k12xx � k11e). If k11e �k12, no steady-state solution of E2F exists when x issmall. If k11e k12, the steady state is always stable forany x and will always decrease as x decreases. For

Fig. 4. Effects of CDC25A phosphorylation andsynthesis rate on stability of steady states andbifurcations of module I. A: bifurcation in thelimit cycle regime in Fig. 2C. Parameters are thesame as in Fig. 2C, except for total number ofphosphorylation sites. Solid line, 1 phosphoryla-tion site on CDC25A (1p). F and dashed line, 3phosphorylation sites on CDC25A; only the fullyphosphorylated site is active (3p1). E and dottedline, 3 phosphorylation sites on CDC25A; 2- and3-site phosphorylation sites are active (3p2). B:bistability (dashed-dotted line), as in the regimeshown in Fig. 2B, for different phosphorylationsites on CDC25A. CDC25A phosphorylation islabeled as in A. C and D: CDC25A synthesis rateon limit cycle and bistability. Parameters are thesame as in Fig. 2, B and C, except for k8. E: speedof CDC25A phosphorylation and dephosphoryla-tion on limit cycle stability. Parameters are thesame as in Fig. 2C, except az, bz, and cz weredivided by �.



g(e) � e/(� � e), the situation is similar. This does notagree with the experimental observation that free E2Fis low in G0 or early G1 but high during the G1-to-Stransition. With g(e) � e2/(� � e2), Eq. 2 results in abistable solution. Figure 6A shows two bistable solu-tions for k11 � 0.02 and 0.1. Symbols in Fig. 6, A and B,are values of free E2F and the total Rb-E2F complexfor k11 � 0.02 by simulating module II and settingactive CycE-CDK2 (x) as a control parameter. In Fig. 6,A and B, x is shown changing from high to low and from

low to high, with E2F being initially set on the upperbranch. Even at very low free E2F, the complexedRb-E2F (Fig. 6B) is high when active CycE-CDK2 (x) islow. With the bistability feature of free E2F, the obser-vation that free E2F is low in G0 and early G1 but highduring the G1-to-S transition can be explained as fol-lows: in G0 or early G1, free E2F is at the lower branchof the bistable curve and most of the E2F is stored asthe Rb-E2F complex. As CycD increases, E2F freedfrom Rb-E2F brings free E2F into the upper branch.

Fig. 5. Effects of E2F on dynamicswhen total E2F is constant (steps 11and 12 set to 0 in module II). Total E2Fwas set equal to total Rb (R0) at 100. A:steady-state free E2F vs. active CycE-CDK2 for different M and M�. B: bist-ability generated by E2F and CycEfeedback loop. Simulation was donewith modules I and II, with f(z) � 0,k2 � 0.5, k5 � 1, k7 � 1, M � 2, and M�� 16. Solid line, k13 � 0; dashed line,k13 � 50. C: phase diagram showingbistability and monostability for differ-ent M and M� combinations. F, M andM� combinations for which bistabilityoccurs; when combinations are in theregion marked “monostability,” bi-stability is absent. Region is markedM � M� by definition, because M � M�has to be satisfied. Parameters are thesame as in B, and k13 � 0. D: E2Feffects on limit cycle bifurcations fork13 � 0 (F) and k13 � 50 (E), with M �2 and M� � 16. Parameters for moduleI were the same as in Fig. 2C. Inset:active CycE-CDK2 vs. time for k1 �230 and k13 � 0. E: E2F effects onbistability, with M � 2 and M� � 16.Parameters for module I are the sameas in Fig. 2B. No CycD, k13 � 0; lowCycD, k13 � 50; high CycD, k13 � 200.



Alternatively, increasing the E2F synthesis (k11) couldalso shift the bistable region into a higher x range(k11 � 0.1 in Fig. 6A), causing E2F transit to the upperbranch at low x.

Except for the bistability generated by the positivefeedback of E2F on its own transcription rate, thepositive feedback between CycE and E2F does notgenerate any new dynamics without the CDC25A feed-back loop in module I. When the CDC25A feedbackloop is present, high E2F increases the transcription ofCycE, which causes limit cycle and bistability of mod-ule I at lower k1. Figure 6, C and D, shows activeCycE-CDK2 and free E2F vs. time for k1 � 75 andk11 � 0.1 and for k1 � 100 and k11 � 0.1, respectively,with module I in the limit cycle regime as in Fig. 1C. Atk1 � 75, the oscillation is slower and the maximumE2F is much higher than the steady-state value shownin Fig. 6A. At k1 � 100, the oscillation becomes muchfaster and E2F decays to a much lower level, indicatingthat the dynamics are governed more by module I.

Role of CKI

Experiments in yeast have shown (33) that six of itsnine phosphorylation sites have to be phosphorylatedfor Sic1 ubiquitination. Although it has been shownthat phosphorylation of p27 on Thr187 by CycE-CDK2is required for p27 ubiquitination in the mammaliancell cycle (31, 48, 65), whether additional phosphoryla-tion sites may also be important for p27 stability isunknown (19). To assess the possible dynamics causedby multisite phosphorylation, we assume that a certainnumber of sites have to be phosphorylated (or dephos-phorylated) by active CycE-CDK2 for ubiquitination

(module IV, Fig. 1, A and D). We first simulated thesteady-state responses of fully phosphorylated CKIand total CKI vs. active CycE-CDK2 (x). As more phos-phorylation sites were assigned to CKI, the fully phos-phorylated CKI had a steeper response to x and at ahigher threshold (Fig. 7A). The total CKI increasedfirst to a maximum and then decreased to a very lowlevel as x increased (Fig. 7B). This indicates that CKIand CycE-CDK2 were first buffered in the incompletelyphosphorylated CKI states. As x increased beyond thethreshold, positive feedback caused the sharp decreaseof total CKI, which may cause instability and lead tointeresting dynamics.

Dynamics caused by CKI phosphorylation by CycE-CDK2. To study the dynamics caused by the positive-feedback loop between CycE-CDK2 and CKI alone, wesimulated Eq. 1, a and d, with CDC25A removed frommodule I [f(z) � 0 in Eq. 1a]. When CKI had only onephosphorylation site (N � 1), the steady state wasstable for any k1 (Fig. 7C). When CKI had more thanone site (N � 1), the steady state became unstable andled to a limit cycle over a range of k1. At larger N, thelimit cycle occurred at a higher k1 threshold and had alarger oscillation amplitude. In a previous study,Thron (58) showed that the positive-feedback loop be-tween CycE-CDK2 and CKI caused bistability. In ourpresent model, there is no bistability, because thesteady state of the active CycE-CDK2 is simply propor-tional to k1. The difference between our model andThron’s model is that in the latter the total CycE-CDK2 remained constant, whereas in our model itvaried.

Fig. 6. Role of E2F when total E2F isnot constant. g(e) � e2/(50 � e2). A:steady-state free E2F vs. active CycE-CDK2 of module II. Dashed line, k11 �0.02; solid line, k11 � 0.1. Circles, sim-ulation results of module II, with x ascontrol parameter: F, x from small tolarge; E, x from large to small. B: totalRb-E2F vs. x when module II was sim-ulated as described in A. C: activeCycE-CDK2 and free E2F vs. time.Simulation was done with modules Iand II, with k11 � 0.1. Parameters formodule I are the same as in Fig. 2C,with k1 � 75. D: same as C, with k1 �100.



CKI modulation of the dynamics of module I. Wenext investigated how CKI modulates the dynamics ofmodule I by simulating Eq. 1, a and d, with CDC25A inmodule I. Because the CKI module does not change thesteady state of active CycE-CDK2, we studied only thecase for module I in the limit cycle regime (Fig. 2C).Figure 7D shows bifurcations for different numbers oftotal phosphorylation sites on CKI. With one site (N �1), the limit cycle occurred at k1 � 97–300 (comparek1 � 90–290 in Fig. 2C), showing that CKI had a littleeffect on the dynamics. When N � 2, the limit cycleoccurred at k1 � 112–326, and when N � 6, k1 �268–350. Thus increasing the number of phosphoryla-

tion sites of CKI caused the instability to occur at ahigher k1 threshold and narrowed the range of theinstability.

CKI mutation. Finally, we examined the effects of asimulated mutation of CKI on the dynamics of theG1-to-S transition. We assume that CKI is mutated sothat it cannot bind to F-box protein for degradation bysetting k25 � 0 in our simulation. Figure 7E shows abifurcation diagram for N � 0, 1, and 5 at high and lowCKI expression (k18 � 100 and 25, respectively). Athigh CKI, the steady state was always stable for any N(solid line in Fig. 7E). In other words, the limit cycledynamics generated by module I were blocked by CKI

Fig. 7. Effects of CKIs on stability andbifurcations. A: steady-state fully phos-phorylated CKI vs. active CycE-CDK2 forN � 1 (1p), 2 (2p), and 6 (6p). Results wereobtained from module IV, with x as a con-trol parameter. B: total CKI vs. activeCycE-CDK2. C: limit cycle bifurcationgenerated by CKI and CycE-CDK2 feed-back loop. Simulations used modules I andIV with f(z) � 0, k2 � 0.5, k5 � 1, and k7 �1. D: modulation of dynamics of module Iby CKI. Simulations used modules I andIV. Parameters for module I are the sameas in Fig. 2C. E: effects of mutating CKI ondynamics of module I. Simulations weredone as in D, but with k25 � 0. Low CKI,k18 � 25; high CKI, k18 � 100.



mutation. However, if CKI was low, the limit cycle stilloccurred for N � 0, 1, and 5, but the range wasnarrower with more phosphorylation sites. Contrary tothe control case shown in Fig. 7D, the mutation hadlittle effect on the k1 threshold of instability.

DISCUSSION

We have presented a detailed mathematical model ofregulation of the G1-to-S transition of the mammaliancell cycle. Our approach was to divide the full G1-to-Stransition model into individual signaling modules (15)and then analyze the dynamics in a stepwise fashion.Our major findings are as follows. 1) Multisite phos-phorylation of cell cycle proteins is critical for instabil-ity and dynamics. 2) The positive feedback betweenCycE-CDK2 and CDC25A in the primary module gen-erates limit cycle, bistable, and excitable transientdynamics. 3) The positive feedback between CycE andE2F can generate bistability, provided total E2F isconstant and Rb is phosphorylated at multiple sites. 4)The positive feedback between CKI and CycE-CDK2can generate limit cycle behavior. 5) E2F and CKImodulate the dynamics of the primary module.

Although all the dynamical regimes manifested bythe primary module in this study have been describedin previous models, there are several important advan-tages to the present formulation. 1) The full G1-to-Stransition model is reasonably complete with respect toincorporating the state of knowledge about experimen-tally determined physiological details. 2) All the rela-tionships between components were modeled accordingto biologically realistic reaction schemes, rather thanphenomenological representations, as in many priormodels. 3) Despite its complexity, we achieved a rea-sonably complete description of the dynamics of the fullmodel by breaking it down into modules and system-atically examining the dynamical consequences of re-combining the individual modules. 4) The G1-to-S tran-sition model exhibits a wide range of dynamicalbehaviors, depending on the parameter choices. This isa powerful aspect, since it provides a wide degree offlexibility for fitting the model to experimental obser-vations. Specifically, experimental perturbations thatalter G1-to-S transition features may correspond totransitions between dynamical regimes.

The dynamics responsible for the checkpoint and cellcycle progression at the G1-to-S transition or duringthe entire cell cycle are not clearly understood (61). AHopf or an SN bifurcation can mimic the G1-to-S tran-sition or other checkpoint transitions. Here we cannotdistinguish unequivocally the dynamics responsible forthe G1-to-S transition, so we consider both dynamics aspossible candidates and discuss the biological implica-tions of our modeling results.

CycE Expression and Degradation

Proper CycE regulation is important for normal cellcycle control. Insufficient CycE results in cell arrest inthe G1 phase, whereas overexpression of CycE leads topremature entry into the S phase (41, 42, 44), genomic

instability (53), and tumorigenesis (8, 21). In our model(Figs. 3 and 4), insufficient CycE expression keepsCycE-CDK2 activity very low, and the cell remains inthe G1 phase. As CycE expression increases, CycE-CDK2 moves into the bistable or limit cycle regimen.As CycE expression further increases, CycE-CDK2stays stably high. However, CycE-CDK2 has to bedownregulated for stable DNA replication (43). There-fore, the stable high CycE-CDK2 caused by overexpres-sion of CycE might be the cause of genomic instabilityand tumorigenesis.

Our simulations show that high degradation of freeCycE and CycE bound to CDK2 makes active CycE-CDK2 very low, whereas a low degradation rate keepsCycE-CDK2 stably elevated. Recent studies (22, 30, 55)showed that the failure to degrade CycE stabilizedCycE-CDK2 activity and was tumorigenic, similar tooverexpression of CycE.

CDC25A

CDC25A is a key regulator of the G1-to-S transitionand is highly expressed in several types of cancers (6,7). Overexpression of CDC25A accelerates the G1-to-Stransition (5). It is a target of E2F and is required forE2F-induced S phase (64). It is also the key regulator ofthe G1 checkpoint for recognizing DNA damage (4, 11,29). CDC25A is downregulated and, thus, delays theG1-to-S transition. In our modeling study, CDC25A isrequired for limit cycle and bistability. At low CDC25Aexpression levels, these dynamics require a high CycEsynthesis rate (Fig. 4C). In other words, overexpressionof CDC25A makes the dynamics occur at a lower CycEsynthesis rate or stabilizes CycE-CDK2 at a high level.These results may explain the experimental observa-tions described above.

Role of CKI

Overexpression of CKIs, such as p27, causes G1 cellcycle arrest (50). According to our model, the presenceof CKIs makes the limit cycle occur at higher CycEsynthesis rates, which agrees with the observation thatG1 cell cycle arrest by p27 can be reversed by overex-pression of CycE (26). Mutation of CKI so that it cannotbe degraded is predicted in the model to prevent thelimit cycle regime in the CycE-CDK2-CDC25A networkand to maintain active CycE-CDK2 at a low level. Thisagrees with the experimental observation that overex-pression of nondegradable CKI permanently arrestscells in G1 (33, 46, 63).

E2F-Rb Pathway

Rb was the first tumor suppressor identified. Block-ing Rb’s action shortens the G1 phase, reduces cell size,and decreases, but does not eliminate, the cell’s re-quirement for mitogens (49). According to our simula-tions, if total E2F is conserved, E2F-Rb has little effecton the threshold for limit cycle and bistability whenCycD-CDK4/6 is absent. However, when CycD-CDK4/6is present, the threshold for these regimens shifts to alower CycE synthesis rate (k1). If we increase Rb syn-



thesis or reduce E2F, these bifurcations occur at higherk1, and vice versa if Rb synthesis is reduced or E2Fsynthesis is increased. These simulation results agreewith the observation that overexpressing E2F acceler-ates the G1-to-S transition, whereas overexpression ofRb delays or blocks the G1-to-S transition. One impor-tant consequence is that overexpressing E2F or delet-ing Rb causes CycE-CDK2 to remain stably high,which may promote tumorigenesis.

Importance of Multisite Phosphorylation

Multisite phosphorylation is common in cell cycleproteins. Multisite phosphorylation has two effects: itsets a high biological threshold and causes a steepresponse (33). A steep response is well known to becritical for instability and dynamics in many systems,including cell cycle models, and here we identify mul-tisite phosphorylation as the biological counterpart re-sponsible for this key feature. In our G1-to-S transitionmodel, the elements involved in feedback loops,CDC25A, Rb, and CKI, had to be phosphorylated attwo or more sites to become active. For Rb, an evengreater number was required. A caveat for CDC25A isthat, in our model, we assumed that the dephosphory-lations of Thr14 and Tyr15 occurred simultaneously.If we assume that these dephosphorylations occursequentially, then first-order phosphorylation ofCDC25A may be enough to generate the dynamics.Nevertheless, experiments have shown that CDC25 ishighly phosphorylated during the cell cycle and hasmultiple phosphorylation sites (17, 25, 32). To ourknowledge, the point that multisite phosphorylationmay be the biological mechanism critical for cell cycledynamics has not been explicitly appreciated.

Summary and Implications

Although without additional experimental proof wecannot identify the specific dynamical regime(s) in-volved in the G1-to-S transition, the model described inour study provides a means to approach this goalsystematically. The model is consistent with most ofthe available experimental observations about the G1-to-S transition, including the checkpoint dynamics reg-ulating the G1-to-S transition under physiological con-ditions and the loss of checkpoint under certainpathophysiological conditions, and the cyclical changesin G1-to-S transition cell cycle proteins. In concert withexperimental approaches to define more precisely thedynamical regimes under which this physiologicallydetailed G1-to-S transition model operates, the nextmajor goal is to develop analogously detailed modelsfor the G2-to-M and other transitions. These modelscan then be coupled to reconstruct a complete formu-lation of the mammalian cell cycle as a modular sig-naling network, the underlying dynamical behavior ofwhich has been thoroughly investigated.

Limitations

In this study, we constructed a network model de-scribing regulation of the G1-to-S transition and ana-

lyzed its dynamics and the biological implications.However, there are some limitations in our study. Theparameters were chosen arbitrarily to investigate pos-sible dynamical behaviors of the model and were notbased on any experimental data. There are so manyparameters in this model that it is impossible for us toanalyze the model completely, and this may prevent usfrom identifying other dynamics, such as high period-icity and chaos. There are other regulatory interac-tions, such as wee1 phosphorylation (52) by active CDKand CDC25 phosphorylation by enzymes other thanactive CDK (20), which we did not incorporate into ourmodel. These interactions may have new consequencesto the dynamics of the G1-to-S transition. Anothercaveat of this study is that we assumed that cell cycleproteins were distributed uniformly throughout thecell, but actually they distribute nonuniformly anddynamically inside the cell (56), which should be ad-dressed in future studies.

APPENDIX

If we assume that steps 3 and 4 occur very fast, i.e., k3 andk4 are very large, we can remove x1 from Eq. 1a. In addition,if we assume that phosphorylation and dephosphorylation ofCDC25A is also very fast, we can treat CDC25A as a functionof active CycE-CDK2 (x). We have the following two-variablesimple model for module I

x � �k5 � f�x�� y � k6x � k7x(A1)

y � k1 � �k5 � f�x�� y � k6 x � k2 y

where f(x) represents the catalytic effect of phosphorylatedCDC25A. If we assume that k10 � 0, then from Eq. 1a

f�x� � l � 1

L

l zl � l � 1

L

l

�bz � cz x�l k8

azl k9

(A2)

By setting x � 0 and y � 0, the steady-state solution (x0,y0)of Eq. A1 satisfies

y0 ��k6 � k7�x0

k5 � f �x0�, k1 � �k2 � k5 � f �x0��y0 � k6x0 (A3)

Following the standard method of linear stability analysis(54), we have for the steady state of Eq. A1

�1 � [�� 2 � 4��]/2(A4)

�2 � [�� a2 � 4��]/2

where �1 and �2 are the two eigenvalues from the linearstability analysis and � and � are

� � k2 � k5 � k6 � k7 � f�x0� � fx y0(A5)

� � k2�k6 � k7 � fx y0� � k7�k5 � f�x0��

fx � df(x)/dx�x�x0. Three types of steady-state behaviors de-pend on the values of � and �:

1) (�2 � 4�) 0. The eigenvalues are a pair of conjugatecomplex numbers. The steady state is a focus. When � � 0, itis a stable focus. When � 0, it is an unstable focus.

2) (�2 � 4�) � 0 but � � 0, the steady state is a node. When� � 0, it is a stable node. When � 0, it is an unstable node.

3) (�2 � 4�) � 0, but � 0, for whatever �, one of the twoeigenvalues is positive. Then the steady state is a saddle.

For f(x) � ax or ax/(b � x), (k6 � k7 � fxy0) in Eq. A5becomes k5(k6 � k7)/[k5 � f(x0)] or (k6 � k7)[k5 � ax0

2/(b �



Fig. 8. Phase diagrams in various parameterspaces for simplified model in the APPENDIX. BS,region in which 2 of the triple steady-state solu-tions are stable; TS, region of triple steady-statesolutions in which 1 is a stable node, 1 is asaddle, and 1 is an unstable focus; LC, region oflimit cycle. Unmarked regions have only steady-state solution that is stable. Solid lines, SN bi-furcation; dashed lines, H bifurcation. In regionbetween 2 dashed lines, steady state or 1 of thesteady states is an unstable focus. In regionbetween 2 solid lines, triple steady-state solu-tions exist. A: k6 � 1, k7 � 9. B: k5 � 0.1, k6 � 1.C: k6 � 1, k7 � 9. D: k2 � 0.5, k7 � 9.



x02)]/[k5 � f(x0)], which implies � � 0 and � � 0 for any x and

any parameters a and b in Eq. A4; therefore, the steady-statesolution is always stable. For f(x) � ax2, the term (k6 � k7 �fxy0) in Eq. A5 becomes (k6 � k7)(k5 � ax0

2)/[k5 � f(x0)], whichcan make � or � negative in a certain range of x if k6 and k7

are large and k5 is small compared with the other parame-ters. Therefore, to cause instability in Eq. A1, a higher orderof f(x) is required, or CDC25A has to have at least twophosphorylation sites to create a higher-order response inmodule I. To give an overview of the dynamics generatedfrom Eq. A1, we show in Fig. 8 the phase diagram for variousparameters of the rate constants and for f(x) � [(bz � czx)2/az

2]k8/k9. Limit cycle occurred at small k2 and large k7,whereas bistability occurred at large k2. This can intuitivelybe understood as follows. If we let z � x � y and combine thetwo equations in Eq. A1, we have z � k1 � (k7 � k2)x � k2z.The nullcline for this equation is z � k1/k2 � (1 � k7/k2)x;another nullcline from Eq. A1 is z � (k6 � k7)x/[k5 � f(x)] �x. The second nullcline is a typical N-shaped curve; the firstis a straight line. For these curves to have three intersec-tions, k1/k2 and k7/k2 have to be in a certain range.

This study was supported by funds from the University of Cali-fornia, Los Angeles, Department of Medicine and by the Kawata andLaubisch Endowments.

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