Cardiovascular Research Laboratory, Departments of Medicine
(Cardiology) and Physiology, University of California, Los Angeles,
California 90095
positive feedback; phosphorylation; nonlinear dynamics; bifurcation; simulation
 |
INTRODUCTION |
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 negative-feedback 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 mass-regulated 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.
| |
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.
View larger version (27K):
[in this window]
[in a new window]
|
Fig. 1.
Schematic model of molecular signaling involved in regulation of
the G1-to-S transition. A: signaling network
divided into functional modules for the G1-to-S transition.
Cyclin E (CycE), cyclin-dependent 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.
|
|
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
Myc-dependent 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 CycE-CDK2 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
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 through
both pathways are proportional to their concentrations (k2y for free CycE and
k7x for active CycE-CDK2 in
Eq. 1a).
CDC25A 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
positive-feedback 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 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
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
[k11eg(e) in Eq. 1b] and degraded at a rate proportional to its
concentration (k12e in Eq. 1b) and CycE-CDK2
(k12xe 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.
CycD and CDK4/6 Regulation
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).
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 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 positive-feedback 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)
|
(1a)
|
where f(z) = 
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.
k
= bz + czx is the rate constant for CycE-CDK2-catalyzed
phosphorylation of CDC25A, and
k
= 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.
For module II (Rb + E2F)
|
(1b)
|
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. k
and k
(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
= br + crx + crd4 is the combined rate constant
for CycD-CDK4/6- and CycE-CDK2-catalyzed phosphorylation of Rb, and
k
= ar
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)
|
(1c)
|
For module IV (CKI)
|
(1d)
|
where k
= bi + cix is the rate
constant for CycE and CDK2-catalyzed CKI phosphorylation and
k
= 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 |
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.
View larger version (33K):
[in this window]
[in a new window]
|
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
steady-state 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 CycE-CDK2 vs. time for a limit cycle in C at
k1 = 150. F: free CycE and
active CycE-CDK2 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.
|
|
Regime 1: monotonic stable steady-state solution.
For any CycE synthesis rate k1, there is only
one steady-state 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, CycE-CDK2 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 different 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
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 (dashed-dotted line in
Fig. 4B), which is a much higher range of
k1 than for L = 2. If we assume that
bi- and triphosphorylated 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.
View larger version (17K):
[in this window]
[in a new window]
|
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.
|
|
View larger version (30K):
[in this window]
[in a new window]
|
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).  and dashed line, 3 phosphorylation sites on
CDC25A; only the fully phosphorylated site is active (3p1).
 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  .
|
|
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 CycE-CDK2 or CycD-CDK4/6. As the
number of Rb phosphorylation 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.
View larger version (31K):
[in this window]
[in a new window]
|
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 CycE-CDK2 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. , 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  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 () and k13 = 50 (), 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.
|
|
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 CycE-CDK2 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 (module 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, 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 CycE-CDK2
(x) is simply determined by steps 11 and
12 and satisfies the following equation
|
(2)
|
If g(e) = e, then
e0 = k11/(k12 + k12xx 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 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
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. 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.
View larger version (26K):
[in this window]
[in a new window]
|
Fig. 6.
Role of E2F when total E2F is not 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, simulation results of
module II, with x as control parameter:
, x from small to large; ,
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.
|
|
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.
View larger version (29K):
[in this window]
[in a new window]
|
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.
|
|
Dynamics caused by CKI phosphorylation by CycE-CDK2.
To study the dynamics caused by the positive-feedback 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 CycE-CDK2
remained constant, whereas in our model it varied.
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 phosphorylation 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 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 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,
CycE-CDK2 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 CycE-CDK2 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 synthesis
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 G1-to-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 analyzed 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.
TOP
ABSTRACT
INTRODUCTION
![<A><AC>x</AC><AC>˙</AC></A><SUB>1</SUB>=k<SUB>3</SUB>y−k<SUB>4</SUB>x<SUB>1</SUB>−[k<SUB>5</SUB>+f(z)]x<SUB>1</SUB>+k<SUB>6</SUB>x](/content/vol284/issue2/fulltext/C349/img006.gif)
![<A><AC>x</AC><AC>˙</AC></A>=[k<SUB>5</SUB>+f(z)]x<SUB>1</SUB>−k<SUB>6</SUB>x−k<SUB>7</SUB>x−k<SUB>23</SUB>xi+k<SUB>24</SUB>i<SUB>x0</SUB>+k<SUB>25</SUB>i<SUB>xN</SUB>](/content/vol284/issue2/fulltext/C349/img007.gif)
![<A><AC>x</AC><AC>˙</AC></A>=[k<SUB>5</SUB>+f(x)]y−k<SUB>6</SUB>x−k<SUB>7</SUB>x](/content/vol284/issue2/fulltext/C349/img045.gif)
![<A><AC>y</AC><AC>˙</AC></A>=k<SUB>1</SUB>−[k<SUB>5</SUB>+f(x)]y+k<SUB>6</SUB>x−k<SUB>2</SUB>y](/content/vol284/issue2/fulltext/C349/img046.gif)
