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1
http//www.icmb.ed.ac.uk/research/earnshaw/mitosis
GrL.gif
Mathematical Modeling of the Cell Cycle
Baltazar D. Aguda, PhD Dept Genetics
Genomics Boston Univ Sch Medicine Boston, MA
02118 USA http//gg.bu.edu
Tutorial Lecture, 21 March 2004 APS Meeting,
Montreal, Canada
http//www.icmb.ed.ac.uk/research/earnshaw/Mitosis
.htm
2
Phases of the Eukaryotic Cell Cycle
Slide from John Tyson
3
Phases of the Eukaryotic Cell Cycle (hrs shown
for typical mammalian cells)
R
http//ntri.tamuk.edu/cell/celcycl.gif
4
S phase
M phase
The Cell Division Cycle
alternating S M phases balanced growth
division
growth cycle
chromosome cycle
slide from B. Novak
5
KEY IDEAS/PHENOMENA FOR MODELLING
3 irreversible transitions during normal mitotic
cycle (1) commitment to DNA replication at
START (2) commitment to mitosis at G2/M
transition (3) cell exits from mitosis at
FINISH
At least 3 requirements for a successful cell
cycle (1) S phase precedes M phase (2)
must be responsive to checkpoint controls
(3) balanced growth and division (cell cycle
must be sensitive to the
cytoplasmic mass per DNA ratio)
6
WHAT YOU WILL GET FROM THIS TUTORIAL

• An overview of the molecular machinery of the
  cell cycle
• engine (mainly the work of the
  Tyson/Novak group on
• budding yeast)

(2) Review of essential nonlinear biochemical
kinetics and how they are applied to cell
cycle modeling. (application of nonlinear
dynamical systems theory)
(3) Modeling of the Restriction Point in the
mammalian cell cycle. (Demonstrates how
qualitative network stability analysis leads
to a robust switch model of the G1/S transition)
7
cyclins cyclin-dependent kinases (CDKs)
8
Cyclins CDKs work together to phosphorylate
proteins
http//hykim.chungbuk.ac.kr/lectures/cellbio/13/18
_4.jpg
9
cyclin/CDK
ATP
Protein
P
Protein
ADP
10
Cyclin and MPF levels during early mitotic cycles
cyclin B/cdc2 (cdc2cdk1)
( MPF mitosis/maturation promoting factor )
Will talk about the kinetic modeling of these
oscillations.
11
The oscillations in cyclin/CDK activities
orchestrate the sequence of cell cycle phases.
Autonomous oscillations in early embryonic cell
cycles in yeast .
12
Background on Biochemical Kinetics
1. Cyclic enzyme reactions
2. Bistability hysteresis
3. Hysteretic oscillation
4. Coupled cyclic enzyme reactions
13
Cyclic enzyme reactions
Mass-action kinetics v1
k1X (uncatalyzed) v2 k2Y
k1 k1E1
(catalyzed)
k2 k2E2
14
Zeroth-order Ultrasensitivity Goldbeter
Koshland (1981) PNAS 786840.
Michaelis-Menten ( Km ? 0 )
Michaelis-Menten (Km large)
mass-action
15
Bistability Hysteresis
Tyson et al. (1995) Prog. Cell Cycle Res vol. 1,
pp. 1-8.
E
3
4
Cln/CDK
A
1
v1 k1EY/(Km1Y) k1bAY/(Km1bY)
inactive SBF
active SBF
v2 k2CX/(Km2X)
Y
X
2
v3 k3X k0
C
v4 k4E
dX/dt v1 v2 dY/dt v2 v1 dE/dt v3 v4
s
s
u
k00.0001, k30.04, k40.1, k2C0.4, Km20.001,
k1k1bA1.5, Km1Km1b0.01 (XY)total0.7
s
16
Hysteretic Oscillations
coupled () and (-) feedback loops
nullclines
Fig 2b of Tyson JJ, Chen KC, Novak B (2003)
Curr. Opin. Cell Biol. 15 221-231.
17
Coupled cyclic enzyme reactions


BD Aguda (1999) Oncogene 18 2846.
18
Transcritical Bifurcation in Positively Coupled
Cycles
Y2ss
s
u
s
0
E2
Y2ss
0
E1
Y1 Y2 turn ON only if E1E2 gt (k1r/k1f)(k2r/k2f)
Ei Xi Yi
Y1
v1f k1fX1Y2 v1r k1rY1
ss
v2f k2fY1X2 v2r k2rY2
Y2
0
E1
19
The budding yeast cell cycle
Chen KC et al. (2000) Kinetic analysis of a
molecular model of the budding yeast cell
cycle, Molecular Biology of the Cell 11 369-391.
20
slide from Novak/Tyson group
21
mass
CDK APC dont like to coexist !
Positive Feedback Loops
slide from Novak/Tyson group
22
Negative Feedback Loops
slide from Novak/Tyson group
23
slide from Novak/Tyson group
24
Fig 2 of Chen et al. (2000)
25
dClb5T/dt (k kMBF) mass - Vd Clb5T
Vd kd kdCdc20
dmass/dt m mass
26
Vi ki kiCln3 eiCln2 eiClb5
eiClb2
27
Fig3 of Chen et al. (2000)
28
The G1 checkpoint in the mammalian cell cycle
29
Fig. M1
30
INITIATION OF S PHASE Regulation of G1/S
transition
GFs
Cyclin-D/cdk4
Cyclin-A/cdk2
TK, DHFR
p16, p27
pRb
ORC
Cdc6
Cdc45
E2F
DP
MCMs
GFs
Cdc7/Dbf4
Cdk2/Cyclin-E
Cdk2/Cyclin-E
Cdc25A
p27
Myc
Max
Fig. M2 Aguda Tang (1999)
GFs
31
Cyclin-D/cdk4
pRb
E2F
DP
Cdk2/Cyclin-E
Cdc25A
p27
Model from a Subnetwork
Fig. M3
32
CELL CYCLE TRANSITIONS CHECKPOINTS
A typical temporal profile of the major kinase
activities as the mammalian cell cycle is
traversed is shown in Fig. M1. The modeling
presented in this section adopts the major
assumption that the activities of Cyclin E/CDK2
and Cyclin B/CDK1 are the markers for the G1-S
and G2-M transitions, respectively. As one can
see in the figure, these Cyclin/CDKs are
correlated with the entries into S and M phases,
respectively. Cyclin A/CDK2 is necessary for
the maintenance of S phase and will not be
explicitly included in the models here.
G1/S the Restriction Point
The molecular regulation of entry into S phase
(or the G1/S transition) is complex, as shown in
Fig. M2 which is a qualitative network that
attempts to summarize the essential pathways
involved. Details can be found in the
following references Aguda (2001), Aguda Tang
(1999), and Qu, Weiss MacLellan (2003). As
discussed in the previous section, we will look
for feedback cycles that affect the stability of
the network and suggest a model for the switching
behavior associated with the Restriction Point
(R-point) in the mammalian cell cycle.
The R-point is a commitment point for entry into
S phase. Quiescent cells have to be exposed to
mitogenic growth factors (GFs) up until the
R-point in order to commit once this checkpoint
is crossed, GFs can be withdrawn with- out
affecting cell cycle progression. The importance
of studying the regulation of the R-point
mechanism comes from the claim that virtually all
known human cancers have aberrations affecting
genes involved in the R-point.
We will now sketch the major pathways presented
in Fig. M2 more details will be given in the
lectures. A family of E2F transcription factors
induce expression of various S-phase genes
including cyclins E and A, and the phospha- tase
Cdc25A. The E2Fs are inhibited by the
retinoblastoma protein (pRb) in quiescent cells.
Growth factors stimu- late various intracellular
signaling pathways that eventually induce cyclin
D synthesis and consequently activate CKD4 or
CDK6. These kinases start the ball rolling they
phosphorylate pRb, thus initiating pRbs
deactivation and derepression of the E2Fs. In
addition, a CDK inhibitor called p27Kip1 is known
to have high protein levels in quiescent cells
and need to be downregulated in order to enter S
phase.
33
The model network that was considered in the
computer simulations presented in Aguda Tang
(1999) is shown in Fig. M3. Note that there are
three positive feedback loops shown the
pRb-E2F-Cdk2 loop, the Cdc25A-Cdk2 loop, and the
p27Kip1-Cdk2 loop.
In order to carry out kinetic simulations, the
model qualitative network has to be translated
into a mechanistic model. The detailed model
considered by Aguda Tang (1999) is shown in
Fig. M4. The mechanistic details have
been gathered from the literature and are
summarized in the aforementioned reference. You
will get to know more of these details in the
computer lab exercise. The symbols iCycE/CDK2
and aCycE/CDK2 refer to the inactive and active
forms of Cyclin E/CDK2, respecive- ly. The
positive feedback loop involving Cdc25A is shown
in more detail in Fig. M5. As you can see, it is
an example of the positively coupled cyclic
enzyme reactions we discussed in the previous
section. This is a key observation which is
really the basis of the Aguda-Tang model of the
R-point. The positive feedback loop (mutual
antagonism) between p27Kip1 and CDK2 occurs in
the following way ini- tially, high levels of
p27Kip1 binds and inhibits whatever Cyclin/CDKs
are present. However, once these CDKs get
activated (due to growth factor-stimulated
signals that induce expression of the activating
cyclins via the pRb-E2F-CDK positive loops) the
inhibitor p27Kip1 gets phosphorylated thereby
rendering it a target for the ubiquitin-mediated
pathway of protein degradation.
Shown in Figures M6 M7 are some results of the
integration of the set of differential equations
of the R-point model. Fig. M6 shows that p27Kip1
levels are reduced prior to the sharp increase in
Cyclin E/CDK2 activity. A simulation of the
R-point, i.e. cutting off growth-factor signaling
and showing a commitment point (in time) for
the activation of Cyclin E/CDK2, is shown in Fig.
M7. These figures will be discussed in more
detail in the lectures.
34
p27/CycE/CDK2
CycD/CDK4/p27
.
.
.
.
.
.
iCycE/CDK2
p27
aCycE/CDK2
CycD/CDK4
.
.
.
.
Cdc25A
.
pRB-P
.
pRB/E2F
E2F
pRB
Kinetic Model
Fig. M4
35
p27/CycE/CDK2
CycD/CDK4/p27
.
.
.
.
.
iCycE/CDK2
p27
aCycE/CDK2
CycD/CDK4
.
.
.
.
aCdc25A
iCdc25A
E2F
Coupled PD cycles between Cdc25A and CDK2
Fig. M5
36
dp27/dt (v7 v20 v10 ) - (v19 v9 v8
v22 ) where v7 k7 v20
k20CycD/cdk4/p27 v10 k10p27/CycE/cdk2
v19 k19p27CycD/cdk4 v9
k9p27aCycE/cdk2 v8
k8p27aCycE/cdk2 v22 k22p27
Fig. III.6
Sample differential equation in the Aguda-Tang
R-point model
Aguda III-8
37
Effect of p27Kip1 level on E2F and CycE/cdk2
Fig. M6
E/cdk2
p27
E2F
BD Aguda Y. Tang (1999) Cell Prolif. 32 321
38
Simulation of the Restriction Point (CDK2
activation)
1 sustained 2 t_off 80 3 t_off 50 4
t_off 30 5 t_off 29 6 t_off 28
time
GFs
Cyclin D/CDK4
BD Aguda Y Tang (1999) Cell Prolif. 32 321.
39
References Other Readings Aguda BD. (1999a)
Instabilities in phosphorylation-dephosphorylatio
n cascades and cell cycle checkpoints,
Oncogene 18 2846-2851. Aguda BD. (1999b) A
quantitative kinetic analysis of the G2 DNA
damage checkpoint system, Proc. Natl.
Acad. Sci. USA 96 11352-11357. Aguda BD. (2001)
Kick-starting the cell cycle From growth-factor
stimulation to initiation of DNA replication,
Chaos 11 269-276. Aguda BD and Tang Y. (1999)
The kinetic origins of the Restriction Point in
the mammalian cell cycle, Cell
Proliferation 32 321-335. Chen KC et al. (2000)
Kinetic analysis of a molecular model of the
budding yeast cell cycle, Molecular Biology of
the Cell 11 369-391. Goldbeter A Koshland DE
(1981) Proc. Natl. Acad. Sci. USA 78 6840 Pines,
J. (1999) Nature Cell Bio 1 E73. Qu Z., Weiss JN
and MacLellan WR. (2003)Regulation of the
mammalian cell cycle a model of the G1-to-S
transition, Am J Physiol Cell Physiol 284
C349-C364. Tyson JJ et al. (1995). Prog Cell
Cycle Res 1 1-8.
40
http//www-micro.msb.le.ac.uk/3035/kalmakoff/bacul
o/pics/Apoptosis.gif
41
APOPTOSIS
In this section, we learn some details of the
pathways for activating the caspases the set of
proteases that are involved in cutting down
proteins during the programmed cell death
process. We will also study in detail
the kinetic model proposed by Fussenegger, Bailey
Varner (2000). I think the most important
feature of this model is what I call the
mitochondria-as-amplifiers hypothesis (Kumar
Vaux, 2002 Aguda Algar, 2003).
Routes of caspase activation
An overview of the caspase cascade is shown in
Fig. IV.1. Two major pathways are shown, one
initiating from the plasma membrane through the
formation of the complex called DISC
(Death-Inducing Signaling Complex) and
activation of caspase-8, and the other initiating
with the release of cytochrome c from
mitochondria followed by the formation of the
apoptosome a complex of Apaf-1, cytochrome c
and procaspase-9 and then the activa- tion of
caspase-9. Caspase-8 and -9, called the
initiator caspases, will then activate the
executioner caspases (cas- pase-3, -7) which
carry out the death sentence to the cell. An
excellent way to get to know the intimate details
of the caspase cascade is to study the
Fussenegger-Bailey-Varner model which we turn to
next.
The Fussenegger-Bailey-Varner (FBV) Model
FAS death domain clustering R L
RL (1) RL ? RL (2) where R
Fas receptor, L Fas Ligand, RL
unclustered Fas/Fas Ligand complex, RL
clustered Fas/Fas Ligand complex
42
Fig. IV.1. Pathways of Caspase activation
(from Zheng Flavell, 2000)
43
FADD Binding RL F
RL.F (3) RL.F F RL.F2 (4)
where F FADD
Procaspase-8 activation C8z RL.F2
(RL.F2).C8z (5)
(RL.F2).C8z C8z (RL.F2).(C8z)2 (6)
(RL.F2).(C8z)2 ? RL.F2
2C8a (7) I8 RL.F2
(RL.F2).I8 (8) I8 (RL.F2).C8z
I8.(RL.F2).C8z (9) C8z C8z ?
2C8a (10)
where C8z procaspase-8, C8a active
caspase-8, I8 decoy protein that competes
against procaspase-8 for binding with
RL.F2
Formation of Apoptosome Cc
A1 A1.Cc (11)
A1 bx A1.bx (12) bx
be bx.be (13) where Cc
cytochrome c, A1 Apaf-1, bx
anti-apoptotic Bcl-2 family member,
be proapoptotic family member
Procaspase-9 activation C9z A1.Cc
(A1.Cc).C9z (14)
(A1.Cc).C9z C9z (A1.Cc).(C9z)2 (15)
(A1.Cc).(C9z)2 ? A1.Cc
2C9a (16) I9 A1.Cc
(A1.Cc).I9 (17) I9 (A1.Cc).C9z
I8.(A1.Cc).C9z (18) C9z C9z ?
2C9a (19) where
C9z procaspase-9, C9a active caspase-9,
I9 decoy protein that competes
against procaspase-9 for binding with A1.Cc
44
Executioner caspase activation C8a CEz
C8a.CEz (20) C8a.CEz ? C8a
CEa (21) C9a
CEz C9a.CEz (22)
C9a.CEz ? C9a CEa (23)
IEa C8a C8a.IEa (24) IEa
C9a C9a.IEa (25) where CEz
executioner procaspase, CEa active
executioner caspase, IEa decoy
proteins
Executioner caspase inactivation CEa
IAPs CEa.IAPs (26) where
IAPs inhibitors of apoptosis.
Cytochrome c release The
rate, rc , of cytochrome c release from
mitochondria was modelled by FBV in an ad hoc
kind of way by assigning a threshold for the
ratio of CEa (executioner caspase) to that of
bx (antiapoptotic Bcl-2 family member). I am
not happy with this. I therefore substituted the
following expression which takes into account the
observations that both active caspase-8 (C8a) and
executioner caspases (CEa) induce cytochrome
c release. rc kc0 kc1C8a
kc2CEa (27) The ks are constants.
Note that because rc is proportional to CEa, we
say that there is a positive feedback loop
between caspase-9 and the executioner caspases.
45
COORDINATION BETWEEN THE CELL CYCLE APOPTOSIS
This section is speculative. I will tell you
about some ideas we have proposed recently (Aguda
Algar, 2003) regarding the links among
signaling pathways, cell cycle initiation, and
apoptosis trigger. The work is based on two
major premises one is that the network is
separable into modules, and the other is that it
is the interaction between these modules that are
essential in the coordination between the
initiation of the cell cycle (or entry into S
phase) and of apoptosis. If I manage to ignite
your interest on these matters, you are quite
welcome to discuss them with me outside class.
Control nodes between the cell cycle and apoptosis
I will assume that apoptosis is directly
associated with activation of the executioner
caspases (caspase-3,-7) and that entry into the
cell cycle is marked by the activation of S-phase
CDKs (CDK2). One of the most convincing
evidence that the control of apoptosis is closed
linked with that of the cell cycle (G1/S) is the
recent report that transcription factors (e.g.
E2F-1 and c-Myc) which induce expression of
S-phase genes are also found to induce expression
of caspase genes (Nahle et al., 2002). Also, it
has also been observed that if you overexpress
E2F-1 or c-Myc, apoptosis is triggered.
I claim that signaling pathways that impinge (or
affect) the cell cycle necessarily affect the
regulation of apoptosis. Thus, nodes must exist
between the cell cycle and apoptosis, and these
nodes are the relevant targets of
signaling pathways. A classification of these
nodes and associated signaling pathways is shown
in Fig. V.1.
Interactions among signaling, cell cycle
apoptosis modules
The modules can interact according to Fig. V.2-A.
Look at Fig. III.2 and see if you can figure out
the interactions Shown as a and b in Fig.
V.2-A. The inhibition b is based on certain
reports that Ras is inhibited by a pathway that
emanates from the pRb/E2F pathway. The
interaction c is based on certain reports that
CDK1 induces apoptosis.
46
Fig. V.1. Signaling pathways and nodes linking
the cell cycle and apoptosis. Case I could be
exemplified by mitogenic signaling pathways that
activate transcription factors such as E2F-1,
c-Myc, c-Fos, and others. Case II involves
oncogenic signaling pathways such as those
involving the protein kinase Akt which acts both
as a proliferative and a survival (i.e.
apoptosis-inhibiting) factor. The MAPK cascade
is another example of a signaling pathway that
has both proliferative and anti-apoptotic
function. Case II may also be active in normal
cells since apoptosis can be induced under serum
starvation. Cell cycle checkpoints would trigger
sig- nals belonging to case III or case IV in
cells destined to quiescence. A good example of
case III involves the tumor-suppressor protein
p53. An example for the node in case IV would be
the CDK inhibitor p21Cip1 which can arrest the
cell cycle as well as inhibit apoptosis. (Figure
is from Aguda Algar, 2003)
47
details of Case I
Myc
E2F
Rb
cycD/cdk4
p53
Mdm2
ARF
Bax
Bcl-2
Bad
p27
Cdc25A
cycE/cdk2
DISC
S phase
Apoptosome
Exec. Caspases
Apoptosis
48
A
B
vs
vms
signal
signal
b
v1
S1
S2
b
vm1
v2
vm2
G2
a
c
v4
v3
die
A1
A2
C1
C2
cycle
vm4
vm3
Fig. V.2. (A) Interactions among signaling,
cell cycle and apoptosis (die) modules. Arrows
mean activate hammerheads mean
inhibit. (B) An explicit implementation of
the modular network in (A).
49
Computer simulations using the kinetic model in
Fig. V.2-B are shown in Fig. V.3. The rate
expressions and parameters are given in Table 1
except for the values of ksd2 which are shown on
the top right corner of each of the four panels
in Fig. V.3. The time course of the signal
(sig), C2 and A2 are generated from the numerical
integration of the differential equations given
in Table 1. C2 corresponds to an active cell
cycle factor, and A2 corresponds to an active
apoptosis factor.
A series of computer simulations for increasing
signal strengths (by decreasing a cell
cycle-dependent signal- degradation parameter,
ksd2, involved in the rate vms) is shown in Fig.
V.3. As ksd2 is decreased from 0.15 to 0.01,
the peak of the signal increases (curves labeled
sig refer to signal in Table 1). No cell
cycle oscillations (curves labeled C2
referring to C2 in Table 1) are observed at first
and then a certain range of ksd2 gives sustained
oscillations (as exemplified by the second panel
where ksd20.081). Apoptosis is eventually
triggered when the signal gets sufficiently high
as exemplified by curve A2 (referring to A2 in
Table 1) for ksd20.01 (lowermost panel). Note
that the value of km2a is nonzero if it were set
to zero, sustained oscillations can still be
achieved by increasing ksd2 (simulations not
shown for example, if km2a0, oscillations occur
for ksd20.091 oscillations suddenly disappear
for ksd20.09). We have interpreted the
oscillations here to represent CDK oscillations
involved in the cell cycle.
50
Table 1. Reaction rate expressions and kinetic
equations for the model shown in Fig.
V.2-B
Rate
Expressions
Kinetic (differential) Equations
Base Parameter Values
Initial Conditions



0
.
1


0
.
1


0
.
1
Eta
Etc
Ets



0001
.
0
2


01
.
0
1


01
.
0
ksd
ksd
ks



02
.
0
1


0
.
1
1



0
.
1
1
KM
Vm
k



01
0.
2



0
.
1
2


02
.
0
1
a
k
k
KMr



0
.
1
3


001
.
0
2


01
.
0
2
k
a
km
km



02
.
0
3


02
.
0
3


0
.
1
3
KMr
KM
Vm



0
.
1
4


005
.
0
4


5
.
0
4
Vm
a
k
k


02
.
0
4


02
.
0
4
KMr
KM


51
Fig. V.3. Computer simulations using the model
in Fig. V.2-B and the kinetic expressions and
parameters given in Table 1.
52
References Aguda BD. (1999a) Instabilities in
phosphorylation-dephosphorylation cascades and
cell cycle checkpoints, Oncogene 18
2846-2851. Aguda BD. (1999b) A quantitative
kinetic analysis of the G2 DNA damage checkpoint
system, Proc. Natl. Acad. Sci. USA 96
11352-11357. Aguda BD. (2001) Kick-starting the
cell cycle From growth-factor stimulation to
initiation of DNA replication, Chaos 11
269-276. Aguda BD and Tang Y. (1999) The kinetic
origins of the Restriction Point in the mammalian
cell cycle, Cell Proliferation 32
321-335. Clarke BL. (1980) Stability of Complex
Reaction Networks Advances in Chemical Physics
43 1-215. Fussenegger M, Bailey JE and Varner J.
(2000) A mathematical model of caspase function
in apoptosis, Nature Biotech 18
768-774. Goldbeter A Koshland DE (1981) Proc.
Natl. Acad. Sci. USA 78 6840 Kumar S and Vaux
DL(2002) A cinderella caspase takes center
stage, Science 297 1290-1291. Nahle Z et al.
(2002) Direct coupling of the cell cycle and
cell death machinery by E2F, Nature Cell
Biology 4 859-864. Pines, J. (1999) Nature Cell
Bio 1 E73. Qu Z., Weiss JN and MacLellan WR.
(2003)Regulation of the mammalian cell cycle a
model of the G1-to-S transition, Am J
Physiol Cell Physiol 284 C349-C364. Tyson JJ et
al. (1995). Prog Cell Cycle Res 1 1-8. Zheng TS
and Flavell RA(2000) Death by numbers, Nature
Biotechnology 18 717-718.