Am J Physiol Cell Physiol 284: C349–C364, 2003.
First published October 9, 2002; 10.1152/ajpcell.00066.2002.
Regulation of the mammalian cell cycle:
a model of the G1-to-S transition
ZHILIN QU, JAMES N. WEISS, AND W. ROBB MACLELLAN
Cardiovascular Research Laboratory, Departments of Medicine (Cardiology)
and Physiology, University of California, Los Angeles, California 90095
Submitted 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 the
G1-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 mathematical
model for regulation of the G1-to-S transition of the mammalian cell cycle. This mathematical model incorporates the key
molecules and interactions that have been identified experimentally. By subdividing these critical molecules into modules, we have been able to systematically analyze the contribution of each to dynamics of the G1-to-S transition. The
primary module, which includes the interactions between
cyclin E (CycE), cyclin-dependent kinase 2 (CDK2), and protein phosphatase CDC25A, exhibits dynamics such as limit
cycle, bistability, and excitable transient. The positive feedback between CycE and transcription factor E2F causes
bistability, provided that the total E2F is constant and the
retinoblastoma protein (Rb) can be hyperphosphorylated.
The positive feedback between active CDK2 and cyclin-dependent kinase inhibitor (CKI) generates a limit cycle. When
combined with the primary module, the E2F/Rb and CKI
modules potentiate or attenuate the dynamics generated by
the primary module. In addition, we found that multisite
phosphorylation of CDC25A, Rb, and CKI was critical for the
generation of dynamics required for cell cycle progression.
THE EUKARYOTIC CELL CYCLE regulating cell division is
classically divided into four phases: G1, S, G2, and M
(35, 38). Quiescent cells reside in the G0 phase and are
induced to reenter the cell cycle by mitogenic stimulation. In the S phase, the cell replicates its DNA, and at
the end of the G2-to-M transition the cell divides into
two daughter cells, which then begin a new cycle of
division. This critical biological process is orchestrated
by the expression and activation of cell cycle genes,
which form a complex and highly integrated network
(24). In this network, activating and inhibitory signaling molecules interact, forming positive- and negativefeedback loops, which ultimately control the dynamics
of the cell cycle. Although many of the key cell cycle
regulatory molecules have been cloned and identified,
the dynamics of this complicated network are too complex to be understood by intuition alone.
Over the last decade, mathematical models have
been developed that provide insights into dynamical
mechanisms underlying the cell cycle (1, 2, 12, 13, 16,
23, 36, 37, 39, 57, 59, 61, 62). Cell cycle dynamics have
been modeled as limit cycles (13, 16, 36, 39), cell massregulated bistable systems (37, 60, 61), bistable and
excitable systems (57, 59), and transient processes (1,
2, 23). Whereas each of these approaches has led to
models that nominally replicate the dynamics of the
cell cycle under specific conditions, no unified theory of
cell cycle dynamics has emerged.
Given the complexity of the cell cycle, a logical
approach 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 support
for this approach comes from the existence of two
strong checkpoints, one at the beginning of each
transition (10). In this study, we mathematically
model the G1-to-S transition as a first step in this
process. Mathematical models of regulation of the
G1-to-S transition have been previously proposed
and simulated (2, 16, 23, 39, 40), with the model of
Aguda and Tang (2) being the most detailed. The key
regulators in control of the G1-to-S transition are
cyclin D (CycD), cyclin E (CycE), cyclin-dependent
kinases (CDK2, CDK4, and CDK6), protein phosphatase CDC25A, transcription factor E2F, the retinoblastoma protein (Rb), and CDK inhibitors (CKIs),
particularly p21 and p27. Many interactions among
these regulators have been outlined experimentally,
and these experiments provide guidelines for developing a physiologically realistic model. To investigate the dynamics systematically, we have taken the
approach of breaking down the regulatory network of
the G1-to-S transition into individual simplified signaling modules, with the primary component involving CycE, CDK2, and CDC25A. After analyzing the
dynamical properties of this primary signaling module, we systematically add E2F/Rb, CKI, and CycD in
a stepwise fashion and explore the features they add
to 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 the
payment of page charges. The article must therefore be hereby
marked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734
solely to indicate this fact.
positive feedback; phosphorylation; nonlinear dynamics; bifurcation; simulation
http://www.ajpcell.org
0363-6143/03 $5.00 Copyright © 2003 the American Physiological Society
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MATHEMATICAL MODELING
Figure 1 summarizes the key interactions between
regulators of the G1-to-S transition in the mammalian
cell cycle that have been identified experimentally over
the last two decades and have been incorporated into
our model.
total of L phosphorylation sites and that its multisite
phosphorylation occurs sequentially, with each phosphorylation step catalyzed by CycE-CDK2 (Fig. 1B).
We also assume that highly phosphorylated CDC25A is
degraded through ubiquitination (step 10).
E2F/Rb Regulation
CycE and CDK2 Regulation
Increased CDK2 activity marks the transition from
the G1 to the S phase. CDK2 activity is regulated by at
least two mechanisms, including 1) transcriptional regulation of its catalytic partner, CycE, and 2) posttranslational modification of the CycE-CDK2 complex itself.
For the purposes of our model, we will assume that
CycE transcription is primarily regulated by two mechanisms (step 1). Mitogenic stimulation promotes Mycdependent transcription of CycE (4, 34, 45), which we
assume occurs at a constant rate (k1). In addition, E2F
is also an important transcription factor for CycE synthesis (4, 9). We assume that it induces CycE synthesis
at a rate proportional to E2F concentration (k1ee in Eq.
1a). Thus the total CycE synthesis rate is k1 ⫹ k1ee. For
CycE and CDK2 binding and activation, we adopted
the scheme proposed by Solomon et al. (51, 52). CycE
and CDK2 bind together, forming an inactive CycECDK2 complex (step 3), with CDK2 phosphorylated at
Thr14, Tyr15, and Thr160. CDC25A dephosphorylates
Thr14 and Tyr15 and activates the kinase (step 5) (18,
47, 51, 52). The complex is inactivated by ubiquitinmediated 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 through
both pathways are proportional to their concentrations
(k2 y for free CycE and k7x for active CycE-CDK2 in
Eq. 1a).
E2F is a transcription factor for a number of cell
cycle genes that are critical for G1-to-S transition in
mammalian cells (49, 68). This family of transcription
factors binds to and is inactivated by a second family of
proteins known as pocket proteins, the prototypical
member being the Rb gene product. In G0 or early G1,
E2F is complexed to Rb and is inactive. E2F is freed by
Rb phosphorylation, which occurs sequentially first by
CycD-CDK4/6 and subsequently by CycE-CDK2, forming another positive-feedback loop. E2F is also synthesized de novo as the cell progresses from G0 to G1 and
autocatalyzes its own production (9, 27). E2F can be
inactivated by binding to dephosphorylated Rb and by
degradation through ubiquitination after phosphorylation by CycE-CDK2 (9). In the model, we assume that
E2F is synthesized at a constant rate k11 and a rate
related to free E2F concentration [k11e g(e) in Eq. 1b]
and degraded at a rate proportional to its concentration (k12e in Eq. 1b) and CycE-CDK2 (k12x e in Eq. 1b).
Although it is known that Rb has 16 phosphorylation
sites (14), the number of functionally important sites is
unknown. Therefore, we assume that Rb has a total of
M⬘ phosphorylation sites and that E2F is dissociated
from Rb when a certain number (M) of sites are phosphorylated and Rb is hyperphosphorylated by phosphorylation of the other phosphorylation sites (M⬘ ⫺
M) on Rb. In our model, we vary M and M⬘ to study the
effects of multisite phosphorylation.
CDC25A Regulation
CycD and CDK4/6 Regulation
CDC25A is a phosphatase that dephosphorylates
and activates CDK2 by removing inhibitory phosphates from Thr14 and Tyr15. CDC25A is a transcriptional target of Myc (28) and E2F (64). We assume that
Myc induces CDC25A synthesis at a constant rate (k8)
and that E2F induces CDC25A synthesis in proportion
to E2F concentration (k8ee in Eq. 1a). CDC25A is also
degraded through ubiquitination, which we assume is
proportional to its concentration (k9z0 in Eq. 1a). For
CDC25A to become active, it must itself be phosphorylated. This phosphorylation is catalyzed by the active
CycE-CDK2 complex (18), which forms a key positivefeedback loop in CycE-CDK2 regulation. It has been
shown that CDC25C is highly phosphorylated at the
G2-to-M transition and has five serine/threonine-proline sites: Thr48, Thr67, Ser122, Thr130, and Ser214 (17,
25, 32). The number of functionally important phosphorylation sites on CDC25A has not been determined
experimentally, but we assume that CDC25A has a
Mitogenic stimulation of cells in the G0 phase triggers synthesis of CycD (28). CycD interacts with CDK4
or CDK6 to form a catalytically active CycD-CDK4/6
complex, which phosphorylates Rb to free active E2F
(9, 49). It also potentiates CDK2 activity indirectly by
titrating away inhibitory CKIs from CycE-CDK2 (50).
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CKI Regulation
CKIs, such as p21 or p27, bind to CycD-CDK4/6 or
CycE-CDK2 to form trimeric complexes. CKI activity is
high during the G0 phase but decreases during the cell
cycle (49). Because factors regulating CKI synthesis
are not well understood, we assume that CKI is synthesized at a constant synthesis rate (k18) and degraded at a rate proportional to its concentration (k19i
in Eq. 1d), which is low enough to ensure a high CKI
level at the beginning of the G1-to-S transition. It has
been shown that p27 binds to CycE-CDK2 and has to
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MODELING THE MAMMALIAN CELL CYCLE
Fig. 1. Schematic model of molecular signaling involved in regulation of the G1to-S transition. A: signaling network divided into functional modules for the G1to-S transition. Cyclin E (CycE), cyclindependent kinase (CDK) 2 (CDK2), and
protein phosphatase CDC25A form module I (red box), retinoblastoma protein (Rb)
and transcription factor E2F form module
II (green box), cyclin D (CycD) and CDK4/6
form module III (orange box), and CDK
inhibitor (CKI) forms module IV (blue
box). B: reaction scheme for CDC25A multisite phosphorylation catalyzed by active
CycE-CDK2 complex. C: E2F-Rb pathway,
in which multisite phosphorylation of Rb
is catalyzed by CycD-CDK4/6 and active
CycE-CDK2. When Rb in the Rb-E2F complex is phosphorylated at M sites, E2F is
freed. Free Rb can then be phosphorylated
at the rest of its phosphorylation sites
(M⬘ ⫺ M). D: multisite phosphorylation of
the trimeric complex CycE-CDK2-CKI is
catalyzed by active CycE-CDK2. When
CKI has N sites phosphorylated, it binds
to F-box protein for ubiquitination and
degradation (step 25), and CycE-CDK2 is
freed. See Table 1 for rate constants.
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MODELING THE MAMMALIAN CELL CYCLE
be phosphorylated on Thr187 by active CycE-CDK2 for
ubiquitination and degradation (31, 48, 65). Ishida et
al. (19) showed that p27 is phosphorylated on many
sites, including Ser10, Ser178, and Thr187, and showed
that Ser10 was phosphorylated-dephosphorylated in a
cell cycle-dependent manner and contributed to p27
stability. Although it is known that Thr187 phosphorylation is required for p27 ubiquitination, it is not clear
whether phosphorylation and dephosphorylation of
any other sites are also needed for p27 ubiquitination
in the mammalian cell cycle. A recent study (33) in
yeast has shown that Sic1, a homolog of p27, has nine
phosphorylation sites and that six of these sites have to
be phosphorylated by Cln-CDC28 to bind to F-box
proteins for ubiquitination and degradation. Because
there is no explicit information on how many phosphorylation sites are required for CKI ubiquitination and
degradation in the mammalian cell cycle and no information on what kinase and phosphatase are responsible for phosphorylation and dephosphorylation of the
phosphorylation sites, except Thr187, in this model, we
generally assume that N sites on CKI have to be
phosphorylated by active CycE-CDK2 for CKI ubiquitination to occur. We vary N from 1 to a certain number
to study the effects of phosphorylation. After CKI is
ubiquitinated and degraded, active CycE-CDK2 is
freed from the trimeric complex. This forms a positivefeedback loop between CycE-CDK2 and CKI. CKI also
binds 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, we
constructed differential equations (see Table 1 for definitions of symbols and parameters), which are listed
separately for each module as follows.
For module I (CycE ⫹ CDK2 ⫹ CDC25A)
ẏ ⫽ k 1 ⫹ k 1e e ⫺ k 2 y ⫺ k 3 y ⫹ k 4 x 1
ẋ 1 ⫽ k 3 y ⫺ k 4 x 1 ⫺ 关k 5 ⫹ f共z兲兴x 1 ⫹ k 6 x
Table 1. Definitions of variables in Eq. 1 and default
parameter set
Variable
x
y
x1
zl
e
Pm
rm
d
d4
i
ixn
id
(1a)
⯗
ż l ⫽ k z⫹zl ⫺ 1 ⫺ kz⫺zl ⫹ kz⫺zl ⫹ 1 ⫺ kz⫹zl,
l ⫽ 1, L ⫺ 1
⯗
ż L ⫽ k z⫹zL ⫺ 1 ⫺ kz⫺zL ⫺ k10 zL
where f(z) ⫽ ¥lL⫽ 1␴lzl represents the catalyzing
strength of CDC25A on CDK2 and ␴l is a weighing
parameter. L is the total number of phosphorylation
sites of CDC25A. kz⫹ ⫽ bz ⫹ czx is the rate constant for
CycE-CDK2-catalyzed phosphorylation of CDC25A,
and kz⫺ ⫽ az is the rate constant for dephosphorylation
of CDC25A. In Eq. 1a, we assumed that CDK2 concentration was much higher than cyclin concentration (3)
and, thus, set CDK2 concentration to be constant 1.
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Active CycE-CDK2
Free CycE
Inactive CycE-CDK2
l-Site-phosphorylated CDC25A (l ⫽ 0, L)
Free E2F
m-Site-phosphorylated Rb-E2F (m ⫽ 0, M)
m-Site-phosphorylated Rb (m ⫽ 0, M⬘)
Free CycD
CycD-CDK 4/6
Free CKI
n-Site-phosphorylated CycE-CDK2-CKI (n ⫽ 0, N)
CycD-CDK4/6-CKI
Parameter
k1
k1e
k2
k3
k4
k5
k6
k7
k8
k8e
k9
k10
k11
k11e
k12
k12x
k13
k14
k15
k16
k17
k18
k19
k20
k21
k22
k23
k24
k25
ẋ ⫽ 关k 5 ⫹ f共z兲兴x 1 ⫺ k 6 x ⫺ k 7 x ⫺ k 23 xi ⫹ k 24 i x0 ⫹ k 25 i xN
ż 0 ⫽ k 8 ⫹ k 8e e ⫺ k 9 z 0 ⫹ k z⫺z1 ⫺ kz⫹z0
Definition
az
bz
cz
ar
br
cr
ai
bi
ci
Value
Rate constants for reaction steps in Fig. 1A
200
1
1
50
50
0.1
1
9
100
0
1
0
0
3
0.1
0.01
0
1
1
1
1
100
1
0.1
1
0.1
0.4
1
5
Constant for CDC25A, Rb, and CKI phosphorylation
and dephosphorylation in Fig. 1, B-D
100
1
1
50
1
1
50
1
1
Rate constant for E2F and Rb affinity and dissociation in Fig. 1C
⫹
km
0.01
⫺
km
0.5
⫹
kM
10 (m ⫽ 0, M ⫺ 1)
⫺
kM
0.01
Total Rb concentration
R0
100
Parameters served as default set. Any parameter not given in corresponding figure legend is set to default value. CycE and CycD, cyclins E
and D; CDK, cyclin-dependent kinase; Rb, retinoblastoma protein; CKI,
CDK inhibitor.
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For module II (Rb ⫹ E2F)
For module IV (CKI)
i̇ ⫽ k 18 ⫺ k 19 i ⫺ k 20 d 4 i ⫹ k 21 i d ⫹ k 22 i d ⫺ k 23 xi ⫹ k 24 i x0
ė ⫽ k 11 ⫹ k 11e g共e兲 ⫺ 共k 12 ⫹ k 12x x兲e
M
⫹
兺
M
⫹
km
pm ⫺
m⫽0
兺
⫺
km
erm
m⫽0
i̇ d ⫽ k 20 id 4 ⫺ k 21 i d ⫺ k 22 i d
i̇ x0 ⫽ k 23 xi ⫺ k 24 i x0 ⫺ k i⫹ix0 ⫹ ki⫺ix1
⯗
ṗ 0 ⫽ k r⫺p1 ⫺ kr⫹ p0 ⫺ k0⫹p0 ⫹ k0⫺er0
(1d)
⫹
i xn⫺1
i̇ xn ⫽ k i
⯗
m ⫽ 1, M ⫺ 1
⯗
⫹
⫺
ṗ M ⫽ k r⫹pM⫺1 ⫺ kr⫺pM ⫺ kM
pM ⫹ kM
erM
ṙ 0 ⫽ k 0⫹p0 ⫺ k0⫺er0 ⫺ kr⫹r0 ⫹ kr⫺r1
(1b)
⯗
⫹
⫺
pm ⫺ km
erm ⫹ kr⫹rm⫺1 ⫺ kr⫺rm ⫺ kr⫹rm
ṙ m ⫽ k m
⫹ kr⫺rm⫹1,
m ⫽ 1, M
⯗
ṙ m⬘ ⫽ k r⫹rm⬘ ⫺ 1 ⫺ kr⫺rm⬘ ⫺ kr⫹rm⬘
m⬘ ⫽ M ⫹ 1, M⬘ ⫺ 1
⯗
r M⬘ ⫽ R 0 ⫺
兺p
m
m⫽0
M⬘⫺1
⫺
⫺
i xn⫹1
⫺k i ⫺k i ⫹k i
,
n ⫽ 1, N ⫺ 1
i̇ xN ⫽ k i⫹ixN⫺1 ⫺ ki⫺ixN ⫺ k25ixN
where k⫹
i ⫽ bi ⫹ cix is the rate constant for CycE and
CDK2-catalyzed CKI phosphorylation and k⫺
i ⫽ ai is
the rate constant for dephosphorylation. N is the total
number of phosphorylation sites on CKI. We also assumed that the rate constants are the same for each
phosphorylation step.
When we simulate one module or a combination of
some modules, we indicate that we remove all the
interaction terms in Eq. 1 from the modules that are
not involved. In numerical simulation, we used the
fourth-order Runge-Kutta method to integrate Eq. 1.
The time step we used in simulation was ⌬t ⫽ 0.005.
RESULTS
⫹ kr⫺rm⬘⫹1,
M
⫹
i xn
⯗
⫹
ṗ m ⫽ k r⫹pm⫺1 ⫺ kr⫺pm ⫺ kr⫹pm ⫹ kr⫺ pm⫹1 ⫺ km
pm
⫺
⫹ km
erm,
⫺
i xn
兺r
m
m⫽0
where g(e) is a function representing E2F synthesis
by E2F itself and will be defined later. M is the
number of phosphorylation sites on Rb needed for
dissociation of E2F from Rb. M⬘ is the total number
⫹
⫺
of phosphorylation sites on Rb. km
and km
(m ⫽ 0,M)
are the rate constants for E2F dissociation from
m-site-phosphorylated Rb and for association with
m-site-phosphorylated Rb, respectively. k⫹
r ⫽ br ⫹
cr x ⫹ crd4 is the combined rate constant for CycDCDK4/6- and CycE-CDK2-catalyzed phosphorylation
of Rb, and k⫺
r ⫽ a r is the rate constant for Rb
dephosphorylation. For simplicity, here we assumed
that CycD-CDK4/6 and CycE-CDK2 have the same
catalytic effects on Rb and that the phosphorylation
and dephosphorylation are not state dependent; i.e.,
we used the same rate constant for each phosphorylation or dephosphorylation step. R0 is the total Rb
concentration.
For module III (CycD ⫹ CDK4/6)
ḋ ⫽ k 13 ⫺ k 14 d ⫺ k 15 d ⫹ k 16 d 4
(1c)
ḋ 4 ⫽ k 15 d ⫺ k 16 d 4 ⫺ k 17 d 4 ⫺ k 20 d 4 i ⫹ k 21 i d
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Dynamics of Module I (CycE ⫹ CDK2 ⫹ CDC25A)
The full model shown in Fig. 1 is too complex for a
complete analysis of its dynamical properties. We
therefore divided the major components of the G1-to-S
transition into modules and examined their individual
behaviors before reintroducing them into the full model
to determine their combined effects. We consider the
primary module (module I in Fig. 1A) as comprised of
CycE, CDK2, and CDC25A, with CDC25A driving a
positive-feedback loop catalyzing active CycE-CDK2
production.
Figure 2 shows the case for two functional phosphorylation sites on CDC25A (L ⫽ 2) and f(z) ⫽ zL (hereafter defined as the default conditions for module I,
unless otherwise indicated), in which the steady state
of active CycE-CDK2 is plotted as a function of the
CycE synthesis rate k1. Figure 2, C and D, also shows
the corresponding bifurcations vs. k1. Depending on
the stability of the steady state and the other parameter choices, the system exhibits the following dynamical regimes.
Regime 1: monotonic stable steady-state solution. For
any CycE synthesis rate k1, there is only one steadystate solution, and it is stable (Fig. 2A).
Regime 2: bistable steady-state solutions. There are
three 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, CycECDK2 remains low until k1 exceeds a critical value, at
which point there is a sudden increase in CycE-CDK2
(upward arrow in Fig. 2B). When k1 decreases from
high to low, however, the transition occurs at a differ-
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Fig. 2. Steady-state solutions and bifurcations for module I. Parameters are
default values in Table 1, unless otherwise indicated. A: steady state of active
CycE-CDK2 vs. CycE synthesis rate
(k1), with k5 ⫽ 1. B: steady state of
active CycE-CDK2 vs. k1, with k2 ⫽ 5.
SN, saddle-node bifurcation. Dashed
line, unstable steady state, which is a
saddle. As k1 increases from small to
large, a transition from lower to higher
steady state occurs at SN (upward arrow). If k1 decreases from large to
small, a transition from the higher to
the lower steady state occurs at the
other SN (downward arrow), forming a
hysteresis loop. C: steady state and bifurcation of active CycE-CDK2 vs. k1,
with k2 ⫽ 0.5. H, Hopf bifurcation
point, at which a stable focus becomes
an unstable focus and oscillation begins; solid lines, stable steady state;
dashed-dotted line, unstable steady
state; circles, maximum and minimum
values of active CycE-CDK2. When
steady state is stable, maximum and
minimum values of active CycE-CDK2
are equal and the same as the steadystate value. When the steady state is
an unstable focus, active CycE-CDK2
oscillates as a limit cycle, and its maximum and minimum differ (E). All bifurcation diagrams for limit cycles
have been plotted as maximum and
minimum active CycE-CDK2 vs. k1. D:
same as C, except k2 ⫽ 1.25 and k3 ⫽
0.02. E: free CycE and active CycECDK2 vs. time for a limit cycle in C at
k1 ⫽ 150. F: free CycE and active CycECDK2 vs. time for an excitable case in
D for k1 ⫽ 210. At t ⫽ 50 (arrow), we
held active CycE-CDK2 at 6 for a duration of 0.02 to stimulate the large excursion.
ent k1 (downward arrow in Fig. 2B), forming a hysteresis loop.
Regime 3: limit cycle solution. The steady state in the
parameter window (k1 ⫽ 90–290 in Fig. 2C) is an
unstable focus, and CycE and CycE-CDK2 oscillate
spontaneously (Fig. 2E, with k1 ⫽ 150). The transition
from the stable steady state to the limit cycle is via
Hopf bifurcation.
Regime 4: multiple steady-state and limit cycle solutions. There are three steady-state solutions over a
range of k1 (k1 ⫽ 192–245 in Fig. 2D): a stable node, a
saddle, and an unstable focus. For a range of k1 just
beyond the triple steady-state solution (k1 ⫽ 245–320
in 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 stimulation causes a large excursion that gradually returns
to the stable steady state.
Role of CDC25A
The dynamics shown in Fig. 2 are critically dependent on CDC25A phosphorylation. Figure 3 shows
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steady-state fully phosphorylated CDC25A (Fig. 3A)
and total CDC25A (Fig. 3B) vs. active CycE-CDK2. As
the number of phosphorylation sites increases, the
fully phosphorylated CDC25A increases more steeply
and at a higher threshold as active CycE-CDK2 increases. This steep change in CDC25A is critical for
instability, leading to limit cycle, bistability, and other
dynamical behaviors. When CDC25A had only one
phosphorylation site (L ⫽ 1), the steady state was
always stable, regardless of other parameter choices
(Fig. 4, A and B). It can be demonstrated analytically
that the steady state can become unstable and lead to
interesting dynamics only when CDC25A has more
than one phosphorylation site and requires phosphorylation of both sites to become active (see APPENDIX). In
Fig. 2, we show various dynamics for L ⫽ 2; only
biphosphorylated CDC25A is active. In Fig. 4, A and B,
we also show bifurcations for L ⫽ 3. When only triphosphorylated 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 (dasheddotted line in Fig. 4B), which is a much higher range of
k1 than for L ⫽ 2. If we assume that bi- and triphos-
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Fig. 3. A: steady state of fully phosphorylated CDC25A (zL) vs. active
CycE-CDK2 (x) for different numbers
of total phosphorylation sites (L ⫽ 1, 2,
and 5). B: total CDC25A vs. active
CycE-CDK2. Results were obtained by
simulating the CDC25A phosphorylation module in B and using x as a
control parameter.
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 at
much lower k1, very close to the case of L ⫽ 2.
Figure 4, C and D, shows the effects of CDC25A
synthesis rate on limit cycle and bistability. Decreasing the synthesis rate of CDC25A shifts the limit cycle
and bistable regions to higher but wider k1 ranges. For
example, when k8 ⫽ 25, the limit cycle occurred at k1 ⫽
195–480 (Fig. 4C, cf. k1 ⫽ 90–290 in Fig. 2C), and
bistability occurred at k1 ⫽ 860–1,400 (Fig. 4D, cf. k1 ⫽
418–612 in Fig. 2B). In Figs. 2 and 4, we assumed no
degradation of phosphorylated CDC25A; i.e., k10 ⫽ 0. If
we assume that fully phosphorylated CDC25A is degraded at a certain rate (k10 ⬎ 0), the limit cycle region
is widened and the bistable region is shifted to a higher
range of k1. For example, the range of the limit cycle
regime shown in Fig. 2C was k1 ⫽ 90–365 and the
range of bistability in Fig. 2B was k1 ⫽ 475–640 after
we set k10 ⫽ 5.
In Figs. 2 and 4, we assumed that the CDC25A
phosphorylation and dephosphorylation rates were
fast. If these rates were slow, the steady state did
not change and bistable behavior was not altered, but
limit cycle behavior was affected. Slowing the phosphorylation and dephosphorylation rates caused narrowing of the k1 range over which limit cycle behavior
occurs, and eventually limit cycle behavior disappeared
(Fig. 4E).
Role of E2F
The results presented thus far show that the primary module of the G1-to-S transition by itself exhibits
multiple dynamical regimens. We now examine how
Rb and E2F (module II in Fig. 1), known to play crucial
roles in mammalian cell cycle progression, regulate the
dynamics of G1-to-S transition.
With total E2F constant. If there is no E2F synthesis
and degradation, i.e., steps 11 and 12 are absent in
module II, then the total E2F is constant (for simplicity, we set it equal to total Rb). Figure 5A shows the
steady state for free E2F concentration vs. active CycECDK2 or CycD-CDK4/6. As the number of Rb phosAJP-Cell Physiol • VOL
phorylation sites M required to free E2F, as well as the
total number of phosphorylation sites M⬘, increases,
the threshold for E2F dissociation and the steepness of
the response increased.
A steep sigmoidal function due to multisite phosphorylation of Rb as shown in Fig. 5A, coupled with the
positive-feedback loop between CycE and E2F, might
be predicted to generate instability. To test this, we
removed the CDC25A from module I and simulated Eq.
1, a–c, together. We assumed that dephosphorylation
of Thr14 and Tyr15 was carried out by a phosphatase
(possibly CDC25A) at a constant rate and, thus, set
k5 ⫽ 1 and f(z) ⫽ 0 in Eq. 1a. In this system, we found
that bistability could be generated, but no other dynamics such as limit cycle occurred. Figure 5B shows
an example of a bistable steady state of active CycECDK2 vs. CycE synthesis rate k1, with (k13 ⫽ 50) or
without (k13 ⫽ 0) CycD for M ⫽ 2 and M⬘ ⫽ 16. The
presence of CycD caused the bistability to occur at
lower k1, because CycD-CDK4/6 phosphorylates Rb
and frees E2F, which can then promote the E2F-dependent synthesis of CycE. In Fig. 5C, we show the conditions in the M ⫺ M⬘ space under which bistability
occurs. This demonstrates that multisite phosphorylation of Rb is critical for the CycE-E2F positive-feedback
loop to generate bistability.
We then added the CDC25A function back to module
I and simulated Eq. 1, a–c, to study how E2F modulates the dynamics of module I. With module I in the
limit cycle regimen (Fig. 2C), Fig. 5D shows two bifurcations 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 a
period 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 regimen as in Fig. 2B. Without CycD (Fig. 5E), the range of
the bistable region was k1 ⫽ 405–610 (compared with
k1 ⫽ 420–612 in Fig. 2B); with a low CycD level (k13 ⫽
50 in Fig. 5E), the range was k1 ⫽ 385–598; with a high
CycD level (k13 ⫽ 200 in Fig. 5E), the range was k1 ⫽
320–514. Because in our model the CycD module (mod-
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Fig. 4. Effects of CDC25A phosphorylation and
synthesis rate on stability of steady states and
bifurcations of module I. A: bifurcation in the
limit cycle regime in Fig. 2C. Parameters are the
same as in Fig. 2C, except for total number of
phosphorylation sites. Solid line, 1 phosphorylation site on CDC25A (1p). F and dashed line, 3
phosphorylation sites on CDC25A; only the fully
phosphorylated site is active (3p1). E and dotted
line, 3 phosphorylation sites on CDC25A; 2- and
3-site phosphorylation sites are active (3p2). B:
bistability (dashed-dotted line), as in the regime
shown in Fig. 2B, for different phosphorylation
sites on CDC25A. CDC25A phosphorylation is
labeled as in A. C and D: CDC25A synthesis rate
on limit cycle and bistability. Parameters are the
same as in Fig. 2, B and C, except for k8. E: speed
of CDC25A phosphorylation and dephosphorylation on limit cycle stability. Parameters are the
same as in Fig. 2C, except az, bz, and cz were
divided by ␤.
ule III) does not include any feedback, it did not lead to
any novel dynamics. CycD-CDK4/6 is simply proportional to CycD. The major effect of CycD-CDK4/6 is to
phosphorylate Rb and, thus, produce more free E2F.
Greater free E2F promotes more CycE synthesis,
which causes the Hopf or saddle-node (SN) bifurcations
at lower k1. Another effect of CycD-CDK4/6 shown in
Fig. 5D is the removal of period 2 oscillation. This can
be explained as follows: because CycD-CDK4/6 frees a
certain amount of E2F from the Rb-E2F complex, the
availability of Rb-E2F for active CycE-CDK2 to phosphorylate is reduced. This makes the steady-state response curve of free E2F vs. CycE-CDK2 less steep
than is the case without CycD. The reduction of steepness of the free E2F response to active CycE-CDK2
thus causes the period 2 behavior to disappear. In fact,
AJP-Cell Physiol • VOL
even in the case of no CycD, if we reduce the hyperphosphorylation sites (M⬘ ⫺ M) of Rb, this period 2 will
also disappear because of the reduction of steepness of
the response of free E2F to active CycE-CDK2.
With E2F synthesis and degradation. If we introduce
E2F synthesis and degradation (steps 11 and 12) into
the E2F-Rb regulation network, then the steady-state
concentration of E2F as a function of active CycECDK2 (x) is simply determined by steps 11 and 12 and
satisfies the following equation
k 11 ⫹ k 11e g共e 0 兲 ⫺ 共k 12 ⫹ k 12x x兲e 0 ⫽ 0
(2)
If g(e) ⫽ e, then e0 ⫽ k11/(k12 ⫹ k12x x ⫺ k11e). If k11e ⬎
k12, no steady-state solution of E2F exists when x is
small. If k11e ⬍ k12, the steady state is always stable for
any x and will always decrease as x decreases. For
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Fig. 5. Effects of E2F on dynamics
when total E2F is constant (steps 11
and 12 set to 0 in module II). Total E2F
was set equal to total Rb (R0) at 100. A:
steady-state free E2F vs. active CycECDK2 for different M and M⬘. B: bistability generated by E2F and CycE
feedback loop. Simulation was done
with 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 showing
bistability and monostability for different M and M⬘ combinations. F, M and
M⬘ combinations for which bistability
occurs; when combinations are in the
region marked “monostability,” bistability is absent. Region is marked
M ⬎ M⬘ by definition, because M ⱕ M⬘
has to be satisfied. Parameters are the
same as in B, and k13 ⫽ 0. D: E2F
effects on limit cycle bifurcations for
k13 ⫽ 0 (F) and k13 ⫽ 50 (E), with M ⫽
2 and M⬘ ⫽ 16. Parameters for module
I were the same as in Fig. 2C. Inset:
active CycE-CDK2 vs. time for k1 ⫽
230 and k13 ⫽ 0. E: E2F effects on
bistability, with M ⫽ 2 and M⬘ ⫽ 16.
Parameters for module I are the same
as in Fig. 2B. No CycD, k13 ⫽ 0; low
CycD, k13 ⫽ 50; high CycD, k13 ⫽ 200.
g(e) ⫽ e/(␣ ⫹ e), the situation is similar. This does not
agree with the experimental observation that free E2F
is low in G0 or early G1 but high during the G1-to-S
transition. With g(e) ⫽ e2/(␣ ⫹ e2), Eq. 2 results in a
bistable solution. Figure 6A shows two bistable solutions for k11 ⫽ 0.02 and 0.1. Symbols in Fig. 6, A and B,
are values of free E2F and the total Rb-E2F complex
for k11 ⫽ 0.02 by simulating module II and setting
active CycE-CDK2 (x) as a control parameter. In Fig. 6,
A and B, x is shown changing from high to low and from
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low to high, with E2F being initially set on the upper
branch. Even at very low free E2F, the complexed
Rb-E2F (Fig. 6B) is high when active CycE-CDK2 (x) is
low. With the bistability feature of free E2F, the observation that free E2F is low in G0 and early G1 but high
during the G1-to-S transition can be explained as follows: in G0 or early G1, free E2F is at the lower branch
of the bistable curve and most of the E2F is stored as
the Rb-E2F complex. As CycD increases, E2F freed
from Rb-E2F brings free E2F into the upper branch.
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Alternatively, increasing the E2F synthesis (k11) could
also shift the bistable region into a higher x range
(k11 ⫽ 0.1 in Fig. 6A), causing E2F transit to the upper
branch at low x.
Except for the bistability generated by the positive
feedback of E2F on its own transcription rate, the
positive feedback between CycE and E2F does not
generate any new dynamics without the CDC25A feedback loop in module I. When the CDC25A feedback
loop is present, high E2F increases the transcription of
CycE, which causes limit cycle and bistability of module I at lower k1. Figure 6, C and D, shows active
CycE-CDK2 and free E2F vs. time for k1 ⫽ 75 and
k11 ⫽ 0.1 and for k1 ⫽ 100 and k11 ⫽ 0.1, respectively,
with module I in the limit cycle regime as in Fig. 1C. At
k1 ⫽ 75, the oscillation is slower and the maximum
E2F is much higher than the steady-state value shown
in Fig. 6A. At k1 ⫽ 100, the oscillation becomes much
faster and E2F decays to a much lower level, indicating
that the dynamics are governed more by module I.
Role of CKI
Experiments in yeast have shown (33) that six of its
nine phosphorylation sites have to be phosphorylated
for Sic1 ubiquitination. Although it has been shown
that phosphorylation of p27 on Thr187 by CycE-CDK2
is required for p27 ubiquitination in the mammalian
cell cycle (31, 48, 65), whether additional phosphorylation sites may also be important for p27 stability is
unknown (19). To assess the possible dynamics caused
by multisite phosphorylation, we assume that a certain
number of sites have to be phosphorylated (or dephosphorylated) by active CycE-CDK2 for ubiquitination
(module IV, Fig. 1, A and D). We first simulated the
steady-state responses of fully phosphorylated CKI
and total CKI vs. active CycE-CDK2 (x). As more phosphorylation sites were assigned to CKI, the fully phosphorylated CKI had a steeper response to x and at a
higher threshold (Fig. 7A). The total CKI increased
first to a maximum and then decreased to a very low
level as x increased (Fig. 7B). This indicates that CKI
and CycE-CDK2 were first buffered in the incompletely
phosphorylated CKI states. As x increased beyond the
threshold, positive feedback caused the sharp decrease
of total CKI, which may cause instability and lead to
interesting dynamics.
Dynamics caused by CKI phosphorylation by CycECDK2. To study the dynamics caused by the positivefeedback loop between CycE-CDK2 and CKI alone, we
simulated Eq. 1, a and d, with CDC25A removed from
module I [f(z) ⫽ 0 in Eq. 1a]. When CKI had only one
phosphorylation site (N ⫽ 1), the steady state was
stable for any k1 (Fig. 7C). When CKI had more than
one site (N ⬎ 1), the steady state became unstable and
led to a limit cycle over a range of k1. At larger N, the
limit cycle occurred at a higher k1 threshold and had a
larger oscillation amplitude. In a previous study,
Thron (58) showed that the positive-feedback loop between CycE-CDK2 and CKI caused bistability. In our
present model, there is no bistability, because the
steady state of the active CycE-CDK2 is simply proportional to k1. The difference between our model and
Thron’s model is that in the latter the total CycECDK2 remained constant, whereas in our model it
varied.
Fig. 6. Role of E2F when total E2F is
not constant. g(e) ⫽ e2/(50 ⫹ e2). A:
steady-state free E2F vs. active CycECDK2 of module II. Dashed line, k11 ⫽
0.02; solid line, k11 ⫽ 0.1. Circles, simulation results of module II, with x as
control parameter: F, x from small to
large; E, x from large to small. B: total
Rb-E2F vs. x when module II was simulated as described in A. C: active
CycE-CDK2 and free E2F vs. time.
Simulation was done with modules I
and II, with k11 ⫽ 0.1. Parameters for
module I are the same as in Fig. 2C,
with k1 ⫽ 75. D: same as C, with k1 ⫽
100.
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CKI modulation of the dynamics of module I. We
next investigated how CKI modulates the dynamics of
module I by simulating Eq. 1, a and d, with CDC25A in
module I. Because the CKI module does not change the
steady state of active CycE-CDK2, we studied only the
case for module I in the limit cycle regime (Fig. 2C).
Figure 7D shows bifurcations for different numbers of
total phosphorylation sites on CKI. With one site (N ⫽
1), the limit cycle occurred at k1 ⫽ 97–300 (compare
k1 ⫽ 90–290 in Fig. 2C), showing that CKI had a little
effect on the dynamics. When N ⫽ 2, the limit cycle
occurred 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 a
higher k1 threshold and narrowed the range of the
instability.
CKI mutation. Finally, we examined the effects of a
simulated mutation of CKI on the dynamics of the
G1-to-S transition. We assume that CKI is mutated so
that it cannot bind to F-box protein for degradation by
setting k25 ⫽ 0 in our simulation. Figure 7E shows a
bifurcation diagram for N ⫽ 0, 1, and 5 at high and low
CKI expression (k18 ⫽ 100 and 25, respectively). At
high CKI, the steady state was always stable for any N
(solid line in Fig. 7E). In other words, the limit cycle
dynamics generated by module I were blocked by CKI
Fig. 7. Effects of CKIs on stability and
bifurcations. A: steady-state fully phosphorylated CKI vs. active CycE-CDK2 for
N ⫽ 1 (1p), 2 (2p), and 6 (6p). Results were
obtained from module IV, with x as a control parameter. B: total CKI vs. active
CycE-CDK2. C: limit cycle bifurcation
generated by CKI and CycE-CDK2 feedback loop. Simulations used modules I and
IV with f(z) ⫽ 0, k2 ⫽ 0.5, k5 ⫽ 1, and k7 ⫽
1. D: modulation of dynamics of module I
by CKI. Simulations used modules I and
IV. Parameters for module I are the same
as in Fig. 2C. E: effects of mutating CKI on
dynamics of module I. Simulations were
done as in D, but with k25 ⫽ 0. Low CKI,
k18 ⫽ 25; high CKI, k18 ⫽ 100.
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mutation. However, if CKI was low, the limit cycle still
occurred for N ⫽ 0, 1, and 5, but the range was
narrower with more phosphorylation sites. Contrary to
the control case shown in Fig. 7D, the mutation had
little effect on the k1 threshold of instability.
DISCUSSION
We have presented a detailed mathematical model of
regulation of the G1-to-S transition of the mammalian
cell cycle. Our approach was to divide the full G1-to-S
transition model into individual signaling modules (15)
and then analyze the dynamics in a stepwise fashion.
Our major findings are as follows. 1) Multisite phosphorylation of cell cycle proteins is critical for instability and dynamics. 2) The positive feedback between
CycE-CDK2 and CDC25A in the primary module generates limit cycle, bistable, and excitable transient
dynamics. 3) The positive feedback between CycE and
E2F can generate bistability, provided total E2F is
constant and Rb is phosphorylated at multiple sites. 4)
The positive feedback between CKI and CycE-CDK2
can generate limit cycle behavior. 5) E2F and CKI
modulate the dynamics of the primary module.
Although all the dynamical regimes manifested by
the primary module in this study have been described
in previous models, there are several important advantages to the present formulation. 1) The full G1-to-S
transition model is reasonably complete with respect to
incorporating the state of knowledge about experimentally determined physiological details. 2) All the relationships between components were modeled according
to biologically realistic reaction schemes, rather than
phenomenological representations, as in many prior
models. 3) Despite its complexity, we achieved a reasonably complete description of the dynamics of the full
model by breaking it down into modules and systematically examining the dynamical consequences of recombining the individual modules. 4) The G1-to-S transition model exhibits a wide range of dynamical
behaviors, depending on the parameter choices. This is
a powerful aspect, since it provides a wide degree of
flexibility for fitting the model to experimental observations. Specifically, experimental perturbations that
alter G1-to-S transition features may correspond to
transitions between dynamical regimes.
The dynamics responsible for the checkpoint and cell
cycle progression at the G1-to-S transition or during
the entire cell cycle are not clearly understood (61). A
Hopf or an SN bifurcation can mimic the G1-to-S transition or other checkpoint transitions. Here we cannot
distinguish unequivocally the dynamics responsible for
the G1-to-S transition, so we consider both dynamics as
possible candidates and discuss the biological implications of our modeling results.
CycE Expression and Degradation
Proper CycE regulation is important for normal cell
cycle control. Insufficient CycE results in cell arrest in
the G1 phase, whereas overexpression of CycE leads to
premature entry into the S phase (41, 42, 44), genomic
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instability (53), and tumorigenesis (8, 21). In our model
(Figs. 3 and 4), insufficient CycE expression keeps
CycE-CDK2 activity very low, and the cell remains in
the G1 phase. As CycE expression increases, CycECDK2 moves into the bistable or limit cycle regimen.
As CycE expression further increases, CycE-CDK2
stays stably high. However, CycE-CDK2 has to be
downregulated for stable DNA replication (43). Therefore, the stable high CycE-CDK2 caused by overexpression of CycE might be the cause of genomic instability
and tumorigenesis.
Our simulations show that high degradation of free
CycE and CycE bound to CDK2 makes active CycECDK2 very low, whereas a low degradation rate keeps
CycE-CDK2 stably elevated. Recent studies (22, 30, 55)
showed that the failure to degrade CycE stabilized
CycE-CDK2 activity and was tumorigenic, similar to
overexpression of CycE.
CDC25A
CDC25A is a key regulator of the G1-to-S transition
and is highly expressed in several types of cancers (6,
7). Overexpression of CDC25A accelerates the G1-to-S
transition (5). It is a target of E2F and is required for
E2F-induced S phase (64). It is also the key regulator of
the G1 checkpoint for recognizing DNA damage (4, 11,
29). CDC25A is downregulated and, thus, delays the
G1-to-S transition. In our modeling study, CDC25A is
required for limit cycle and bistability. At low CDC25A
expression levels, these dynamics require a high CycE
synthesis rate (Fig. 4C). In other words, overexpression
of CDC25A makes the dynamics occur at a lower CycE
synthesis rate or stabilizes CycE-CDK2 at a high level.
These results may explain the experimental observations described above.
Role of CKI
Overexpression of CKIs, such as p27, causes G1 cell
cycle arrest (50). According to our model, the presence
of CKIs makes the limit cycle occur at higher CycE
synthesis rates, which agrees with the observation that
G1 cell cycle arrest by p27 can be reversed by overexpression of CycE (26). Mutation of CKI so that it cannot
be degraded is predicted in the model to prevent the
limit cycle regime in the CycE-CDK2-CDC25A network
and to maintain active CycE-CDK2 at a low level. This
agrees with the experimental observation that overexpression of nondegradable CKI permanently arrests
cells in G1 (33, 46, 63).
E2F-Rb Pathway
Rb was the first tumor suppressor identified. Blocking Rb’s action shortens the G1 phase, reduces cell size,
and decreases, but does not eliminate, the cell’s requirement for mitogens (49). According to our simulations, if total E2F is conserved, E2F-Rb has little effect
on the threshold for limit cycle and bistability when
CycD-CDK4/6 is absent. However, when CycD-CDK4/6
is present, the threshold for these regimens shifts to a
lower CycE synthesis rate (k1). If we increase Rb syn-
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thesis or reduce E2F, these bifurcations occur at higher
k1, and vice versa if Rb synthesis is reduced or E2F
synthesis is increased. These simulation results agree
with the observation that overexpressing E2F accelerates the G1-to-S transition, whereas overexpression of
Rb delays or blocks the G1-to-S transition. One important consequence is that overexpressing E2F or deleting Rb causes CycE-CDK2 to remain stably high,
which may promote tumorigenesis.
Importance of Multisite Phosphorylation
Multisite phosphorylation is common in cell cycle
proteins. Multisite phosphorylation has two effects: it
sets a high biological threshold and causes a steep
response (33). A steep response is well known to be
critical for instability and dynamics in many systems,
including cell cycle models, and here we identify multisite phosphorylation as the biological counterpart responsible for this key feature. In our G1-to-S transition
model, the elements involved in feedback loops,
CDC25A, Rb, and CKI, had to be phosphorylated at
two or more sites to become active. For Rb, an even
greater number was required. A caveat for CDC25A is
that, in our model, we assumed that the dephosphorylations of Thr14 and Tyr15 occurred simultaneously.
If we assume that these dephosphorylations occur
sequentially, then first-order phosphorylation of
CDC25A may be enough to generate the dynamics.
Nevertheless, experiments have shown that CDC25 is
highly phosphorylated during the cell cycle and has
multiple phosphorylation sites (17, 25, 32). To our
knowledge, the point that multisite phosphorylation
may be the biological mechanism critical for cell cycle
dynamics has not been explicitly appreciated.
Summary and Implications
Although without additional experimental proof we
cannot identify the specific dynamical regime(s) involved in the G1-to-S transition, the model described in
our study provides a means to approach this goal
systematically. The model is consistent with most of
the available experimental observations about the G1to-S transition, including the checkpoint dynamics regulating the G1-to-S transition under physiological conditions and the loss of checkpoint under certain
pathophysiological conditions, and the cyclical changes
in G1-to-S transition cell cycle proteins. In concert with
experimental approaches to define more precisely the
dynamical regimes under which this physiologically
detailed G1-to-S transition model operates, the next
major goal is to develop analogously detailed models
for the G2-to-M and other transitions. These models
can then be coupled to reconstruct a complete formulation of the mammalian cell cycle as a modular signaling network, the underlying dynamical behavior of
which has been thoroughly investigated.
Limitations
In this study, we constructed a network model describing regulation of the G1-to-S transition and anaAJP-Cell Physiol • VOL
lyzed its dynamics and the biological implications.
However, there are some limitations in our study. The
parameters were chosen arbitrarily to investigate possible dynamical behaviors of the model and were not
based on any experimental data. There are so many
parameters in this model that it is impossible for us to
analyze the model completely, and this may prevent us
from identifying other dynamics, such as high periodicity and chaos. There are other regulatory interactions, such as wee1 phosphorylation (52) by active CDK
and CDC25 phosphorylation by enzymes other than
active CDK (20), which we did not incorporate into our
model. These interactions may have new consequences
to the dynamics of the G1-to-S transition. Another
caveat of this study is that we assumed that cell cycle
proteins were distributed uniformly throughout the
cell, but actually they distribute nonuniformly and
dynamically inside the cell (56), which should be addressed in future studies.
APPENDIX
If we assume that steps 3 and 4 occur very fast, i.e., k3 and
k4 are very large, we can remove x1 from Eq. 1a. In addition,
if we assume that phosphorylation and dephosphorylation of
CDC25A is also very fast, we can treat CDC25A as a function
of active CycE-CDK2 (x). We have the following two-variable
simple model for module I
ẋ ⫽ 关k 5 ⫹ f共x兲兴 y ⫺ k 6 x ⫺ k 7 x
ẏ ⫽ k 1 ⫺ 关k 5 ⫹ f共x兲兴 y ⫹ k 6 x ⫺ k 2 y
(A1)
where f(x) represents the catalytic effect of phosphorylated
CDC25A. If we assume that k10 ⫽ 0, then from Eq. 1a
L
f共x兲 ⫽
兺
L
␴l zl ⫽
l⫽1
兺
l⫽1
␴l
共b z ⫹ c z x兲 l k 8
a zl k 9
(A2)
By setting ẋ ⫽ 0 and ẏ ⫽ 0, the steady-state solution (x0, y0)
of Eq. A1 satisfies
y0 ⫽
共k 6 ⫹ k7兲x0
,
k5 ⫹ f 共x0兲
k1 ⫽ 关k2 ⫹ k5 ⫹ f 共x0兲兴 y0 ⫺ k6 x0
(A3)
Following the standard method of linear stability analysis
(54), we have for the steady state of Eq. A1
␭ 1 ⫽ [⫺␣ ⫹ 冑共␣2 ⫺ 4␤兲]/2
␭ 2 ⫽ [⫺␣ ⫺ 冑共a2 ⫺ 4␤兲]/2
(A4)
where ␭1 and ␭2 are the two eigenvalues from the linear
stability analysis and ␣ and ␤ are
␣ ⫽ k 2 ⫹ k 5 ⫹ k 6 ⫹ k 7 ⫹ f共x 0 兲 ⫺ f x y 0
␤ ⫽ k 2 共k 6 ⫹ k 7 ⫺ f x y 0 兲 ⫹ k 7 关k 5 ⫹ f共x 0 兲兴
(A5)
fx ⫽ df(x)/dx兩x⫽x0. Three types of steady-state behaviors depend on the values of ␣ and ␤:
1) (␣2 ⫺ 4␤) ⬍ 0. The eigenvalues are a pair of conjugate
complex numbers. The steady state is a focus. When ␣ ⬎ 0, it
is 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 two
eigenvalues is positive. Then the steady state is a saddle.
For f(x) ⫽ ax or ax/(b ⫹ x), (k6 ⫹ k7 ⫺ fxy0) in Eq. A5
becomes k5(k6 ⫹ k7)/[k5 ⫹ f(x0)] or (k6 ⫹ k7)[k5 ⫹ ax02/(b ⫹
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MODELING THE MAMMALIAN CELL CYCLE
Fig. 8. Phase diagrams in various parameter
spaces for simplified model in the APPENDIX. BS,
region in which 2 of the triple steady-state solutions are stable; TS, region of triple steady-state
solutions in which 1 is a stable node, 1 is a
saddle, and 1 is an unstable focus; LC, region of
limit cycle. Unmarked regions have only steadystate solution that is stable. Solid lines, SN bifurcation; dashed lines, H bifurcation. In region
between 2 dashed lines, steady state or 1 of the
steady states is an unstable focus. In region
between 2 solid lines, triple steady-state solutions exist. A: k6 ⫽ 1, k7 ⫽ 9. B: k5 ⫽ 0.1, k6 ⫽ 1.
C: k6 ⫽ 1, k7 ⫽ 9. D: k2 ⫽ 0.5, k7 ⫽ 9.
AJP-Cell Physiol • VOL
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MODELING THE MAMMALIAN CELL CYCLE
x02)]/[k5 ⫹ f(x0)], which implies ␣ ⬎ 0 and ␤ ⬎ 0 for any x and
any parameters a and b in Eq. A4; therefore, the steady-state
solution is always stable. For f(x) ⫽ ax2, the term (k6 ⫹ k7 ⫺
fxy0) in Eq. A5 becomes (k6 ⫹ k7)(k5 ⫺ ax02)/[k5 ⫹ f(x0)], which
can make ␣ or ␤ negative in a certain range of x if k6 and k7
are large and k5 is small compared with the other parameters. Therefore, to cause instability in Eq. A1, a higher order
of f(x) is required, or CDC25A has to have at least two
phosphorylation sites to create a higher-order response in
module I. To give an overview of the dynamics generated
from Eq. A1, we show in Fig. 8 the phase diagram for various
parameters of the rate constants and for f(x) ⫽ [(bz ⫹ czx)2/
az2]k8/k9. Limit cycle occurred at small k2 and large k7,
whereas bistability occurred at large k2. This can intuitively
be understood as follows. If we let z ⫽ x ⫹ y and combine the
two equations in Eq. A1, we have ż ⫽ k1 ⫺ (k7 ⫺ k2)x ⫺ k2 z.
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 first
is a straight line. For these curves to have three intersections, k1/k2 and k7/k2 have to be in a certain range.
This study was supported by funds from the University of California, Los Angeles, Department of Medicine and by the Kawata and
Laubisch Endowments.
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