Module: Cell Cycle
Authors: Jill C. Sible and John J. Tyson
I. Abstract
DNA synthesis and mitosis in frog eggs is controlled by periodic synthesis and
degradation of the cyclin subunit of a cyclin-dependent protein kinase, and also
by periodic phosphorylation and dephosphorylation of the kinase subunit. We
construct a differential-equation model of these reactions and show that, under
reasonable conditions, the model can exhibit bistability (arrest in interphase or in
mitosis) or spontaneous oscillations (periodic replication and division).
II. Physiology
The cell division cycle is the sequence of events by which a growing cell
replicates its DNA and divides the products of replication (the ‘sister chromatids’)
equally between to daughter cells. In eukaryotes, the cell cycle consists of four
sequential phases: G1 (first gap: DNA unreplicated), S (DNA synthesis), G2
(second gap: DNA replicated), and M (mitosis). Progression through these
phases is unidirectional. Checkpoints within the system ensure that one phase of
the cell cycle is complete before the next phase begins. Checkpoints also
monitor the cell for DNA damage and delay cell cycle progression to allow for
repair of the damage. In animal cells the restriction point (R) regulates entry into
the cell cycle in the presence of growth factors. In the absence of growth factors,
the cell withdraws to a state of quiescence called G0.
Regulation of even the simplest cell cycles (such as those of frog egg extracts,
described here) is a complex affair that is difficult to understand in depth by
intuitive (verbal) arguments alone. Why is the cell cycle unidirectional? Once a
cell initiates mitosis, why does it never slip back into S or G2? What controls the
timing of cell cycles, which can range in length from eight minutes (in fly
embryos) to more than 24 hours (in adult mammals)? Mathematical models of
cell cycle controls offer a systems-level view that can answer these questions by
revealing fundamental regulatory properties of the system. In this tutorial, we will
explore (1) the concept of hysteresis, which explains the irreversible switch-like
behavior of the cell cycle, (2) the concepts of lag times and “critical slowing
down”, which contribute to regulation of cell cycle timing, and (3) the feedback
loops that generate autonomous oscillations through the cell cycle (Sible and
Tyson, 2007).
In the exercises, we will explore several specific behaviors that were discovered
by experimentation and can be better understood in the context of a
mathematical model. First, we will consider the observation by Murray and
Kirschner that synthesis and degradation of cyclin is all that is needed to drive
cell cycle oscillations in frog egg extracts (Murray and Kirschner, 1989; Murray et
al., 1989). Second, we will investigate the observation by Solomon that a
threshold amount of cyclin is required to drive an extract into mitosis (Solomon et
al., 1990). We will uncover the role of positive feedback in cell cycle progression,
the bistable nature of the molecular control network, and the effect of
unreplicated DNA on cell cycle progression.
Figure 1. The eukaryotic cell cycle.
Is the mitotic spindle assembled?
Are chromosomes properly aligned on the spindle?
M
G0
G2
G1
Is DNA damaged?
Is DNA replication complete?
restriction point
S
Is DNA damaged?
Is cell large enough?
adapted from: Alberts et al.
Molecular Biology of the Cell, 3rd ed.
III. Molecular Biology
Progression through the eukaryotic cell cycle is driven by a family of enzymes
called cyclin - dependent kinases (Cdks). The role of a Cdk is to attach a
phosphate group to serine or threonine residues on target proteins. By controlling
the phosphorylation state of these target proteins, the Cdks control the timing of
cell cycle events. In order to be catalytically active, a Cdk subunit must bind to a
cyclin subunit. Cyclin molecules are generally unstable proteins that are
synthesized and degraded periodically during the cell cycle. Different cyclin-Cdk
complexes are responsible for progression through specific phases of the cell
cycle. For example, cyclin E-Cdk2 drives the cell into S phase by phosphorylating
components of the DNA origin replication complex to trigger origin firing and
subsequently to block rereplication of DNA (Furstenthal et al., 2001). A simplified
view of the cyclin-Cdk complexes that drive progression through the eukaryotic
cell cycle is depicted in Figure 2.
Figure 2. Cyclin-Cdk complexes in eukaryotic cells.
MPF
cyclin B
cyclin A
Cdk1
Cdk1
cyclin D
Cdk6
M
G2
cyclin D
G1
cyclin A
Cdk4
S
Cdk2
cyclin E
Cdk2
adapted from: Alberts et al.
Molecular Biology of the Cell 3rd ed.
In this exercise, we focus on cyclin B-Cdk1, which catalyzes the G2/M - phase
transition. This Cdk complex is also known as MPF (M-phase promoting factor)
for historical reasons. Like most Cdk complexes, cyclin B-Cdk1 activity is
regulated by multiple mechanisms. (1) Cyclin synthesis and degradation. Cyclin
levels accumulate as the cell cycle progresses, peaking at M-phase, coincident
with the peak in Cdk activity. Then, cyclin is rapidly degraded and the cell exits
mitosis. Cyclin degradation is stimulated indirectly by cyclin B-Cdk1 itself, forming
a time delayed negative feedback loop. (2) Stoichiometric inhibitors. A family of
proteins, called cyclin-dependent kinase inhibitors (CKIs), bind to cyclin-Cdk
complexes and repress kinase activity. Cyclin B-Cdk1 is susceptible to these
inhibitors, but as they are not present in the frog egg extracts, they will not be
introduced into our model. (3) Activating phosphoryation. Phosphorylation of the
catalytic subunit by Cdk-activating kinase (CAK) is required for activation
(Solomon et al., 1992). Because this phosphorylation event does not appear to
be regulated in frog egg extracts, it will be ignored in our model. (4) Inhibitory
phosphoryations. When cyclin B-Cdk1 is phosphorylated on threonine 14 and
tyrosine 15 by Wee1/Myt1 (Mueller et al., 1995a; Mueller et al., 1995b), its kinase
activity is inhibited. The opposing (activating) phosphatases are members of the
Cdc25 family (Gautier et al., 1991; Kumagai and Dunphy, 1991). These two
groups of enzymes, which regulate the phosphorylation states of Thr 14 and Tyr
15, play a central role in our mathematical model as they are engaged in
feedback loops with cyclin B-Cdk1.
k1
amino acids
Cyclin
k3
kb
P
Cdk1
Cdc25
Cdc25
Cdk1
Cyclin
P
k25
Cdk1
Cyclin
kwee
Wee1
kh
kd
IE
ka
P
kc
IE
kg
OFF
APC
P
ke
C
ON
Cdk1
k2
Wee1
kf
AP
P
P
degraded
cyclin
Figure 3. Wiring diagram for the regulation of cyclin B-Cdk1 activity in frog egg
extracts. Solid lines represent biochemical reactions; dashed lines represent
catalytic effects. IE = intermediate enzyme (hypothetical). APC = anaphase
promoting complex, which tags cyclin B for degradation.
IV. Model
The biochemical network regulating mitosis (Figure 3) can be described by a set
of ordinary differential equations (ODEs; Figure 4). The model centers on cyclin
B-Cdk1, the Cdk complex that drives entry into mitosis. MPF refers to the active
(unphoshorylated) form of cyclin B-Cdk1, and preMPF to the inactive
(phosphorylated) form. Wee1 represents the collective kinase activity (a
combination of Wee1 and Myt1 activities) that phosphorylates cyclin B-Cdk1 on
Thr 14 and Tyr 15. Cdc25 represents the collective opposing phosphatase
activity, (a combination of Cdc25A, Cdc25C and possibly other isoforms). Cyclin
B-Cdk1 phosphorylates and activates Cdc25 (resulting in a positive feedback
loop) and phosphorylates and inactivates Wee1 (resulting in a double-negative
feedback loop). Cyclin B-Cdk1 phosphorylates an intermediate enzyme, which
activates the APC to tag cyclin B for degradation (resulting in a time-delayed,
negative feedback loop).
1.
d
[Cycli n] = k1 k2[Cycli n]  k3 [Cycli n] [Cdk]
dt
2.
d
[MPF] = k3 [Cycli n] [Cdk]  k2[MPF]  kwee[MPF]  k25 [preMPF]
dt
3.
d
[preMPF] =  k2[preMPF]  kwee[MPF]  k25 [preMPF]
dt
4.
d
ka[MPF]([total Cdc25] - [Cdc25P]) kb[PPase][Cdc25P]
[Cdc25P] =

dt
Ka + [total Cdc25] - [Cdc25P]
Kb + [Cdc25P]
5.
d
ke[MPF]([total Wee1] - [Wee1P]) kf[PPase][Wee1P]
[Wee1P] =

dt 
Ke + [total Wee1] - [Wee1P]
Kf +[Wee1P]
6.
d
kg[MPF]([total IE] - [IEP]) kh[PPase][IEP]
[IEP] =

dt 
Kg + [total IE] - [IEP]
Kh +[IEP]
7.
d
kc[MPF]([total APC] - [APC]) kd[PPase][APC]
[APC] =

dt
Kc + [total APC] - [APC]
Kd +[APC]

8. [Cdk] = [Total Cdk]  [MPF]  [preMPF]
9. k25= V25' ([Total Cdc25]  [Cdc25P]) + V25" [Cdc25P]
10. kwee = Vwee' [Wee1P] + Vwee" ([Total Wee1]  [Wee1P])
11. k2 = V2' ([Total APC]  [APC*]) + V2" [APC*]
Figure 4. Mathematical model of the cell cycle control network in frog egg
extracts (Novak and Tyson, 1993). MPF = active form of cyclin B–Cdk1. Pre-MPF
= inactive (phosphorylated) form of cyclin B-Cdk1. Other protein names
appended with a “P” refer to the phosphorylated form of these proteins.
V. Exercises
Open MPF.ode in XPP or WinPP. A set of parameter values and initial conditions
have been provided.
1. Using the Xi vs. T function, plot preMPF and MPF versus time. Also plot cyclin
and totalcyclin versus time.
a. What biological behaviors regarding the cell cycle in frog egg extracts are
reproduced by these simulations?
b. In what order do the peaks in preMPF and MPF occur in each cell cycle?
c. What happens to the oscillations in MPF activity if you change the rate of
cyclin synthesis to 1? to 0.2? to 0.05? What are the bounds on the rate of cyclin
synthesis to produce sustained oscillations in MPF activity?
2. In 1990, Solomon et al. published a study in which extracts were treated with
cycloheximide (to block protein synthesis) and supplemented with fixed amounts
of a mutant, non-degradable form of cyclin B (cyclin B). Simulate this
experiment.
a. What parameter values did you change? Why?
b. What is the minimal threshold concentration of cyclin required to activate
MPF?
c. Why do you think this cyclin threshold exists?
d. What happens if you raise the concentration of cyclin incrementally just above
the activation threshold?
3. Solomon et al. measured a cyclin threshold for MPF activation. Now let’s use
the model to predict a new behavior: a cyclin threshold for MPF inactivation. As
in Exercise 2, set k1 = V2’ = V2” = 0, and set the following initial conditions: cyclin
= 0, preMPF = 0, Cdc25P = 1, Wee1P = 1, IEP = 0, APC = 0, MPF = 20, 15, 10,
… In this case, you are simulating an extract in which all the cyclin is initially in
the form of active MPF. What is the cyclin threshold for MPF inactivation?
a. How does the lag-time for MPF inactivation depend on total cyclin
concentration?
b. Using your results in Exercises 2 and 3, plot the steady state concentration of
MPF as a function of total cyclin concentration for the two cases: (i) initial MPF =
0 and total_cyclin increasing, and (ii) initial MPF = total_cyclin decreasing. Notice
that the system has two stable steady states for total_cyclin between 8 and 16.
4. Using the original set of parameter values, plot MPF vs. total cyclin. What is
the interpretation of the oval-shaped curve you will see?
a. Compare this oval to your graph of steady states in Exercise 3b.
b. How are MPF oscillations related to the activation and inactivation thresholds
investigated in Exercises 2 and 3?
c. What advantages does bistability confer to the physiology of mitosis?
5. In 2005, Pomerening et al. published a study in which a frog egg extract was
supplemented with a recombinant form of Cdk1 in which Thr 14 and Tyr 15 were
mutated to Ala (A) and Phe (F), respectively (Pomerening et al., 2005). (We will
refer to this mutant protein as Cdk1AF; for historical reasons, Pomerening et al.
call it Cdc2AF.)
a. Assume that endogenous Cdk1 was removed from the extract and precisely
replaced by Cdk1AF. What parameter value(s) would you change to simulate
these conditions? Why?
b. Plot MPF vs. time under these conditions. Describe the simulations.
c. What does this experiment tell us about the feedback loops that affect MPF
phosphorylation?
d. How would you simulate the original conditions of the experiment, in which
endogenous Cdk1 is supplemented with an equal amount of Cdk1AF?
6. In the presence of unreplicated DNA, a cell cycle checkpoint is activated. A
protein kinase called Chk1 is activated. Chk1 phosphorylates both Wee1 and
Cdc25 (on residues distinct from the MPF phosphorylation site), resulting in
activation of Wee1 and inhibition of Cdc25.
a. Adjust parameters to represent the effect of unreplicated DNA on the mitotic
control system. Describe the simulation of MPF vs. time.
b. By how much must Vwee” be raised to engage the checkpoint?
c. To engage a checkpoint, is it sufficient for Chk1 to phosphorylate only Wee1 or
only Cdc25?
d. If replicated chromosomes cannot be properly aligned on the mitotic spindle,
then the cell engages a different checkpoint that prevents activation of the APC.
How might you model cell cycle arrest at this checkpoint?
VI. Highlights
 Built a mathematical model of MPF activation in frog egg extracts.
 Verified spontaneous MPF oscillations driven by periodic cyclin
degradation.
 Observed thresholds for MPF activation and inactivation in the case of no
cyclin synthesis or degradation.
 Discovered the phenomenon of ‘critical slowing down’ (i.e., long time lags)
for cyclin concentrations close to the threshold levels.
 Observed that, between the two thresholds, the MPF control system is
bistable.
 Discovered the relationship between bistability and oscillations.
 Discovered that negative feedback alone is sufficient for small amplitude
oscillations that cannot drive robust cycling between DNA synthesis and
mitosis.
 Discovered how checkpoint mechanisms can convert stable oscillations
(cell cycle progression) into stable steady states (cell cycle arrest).
 All of these properties of the model are confirmed by experiments
documented in the papers by Solomon et al. (1990), Novak & Tyson
(1993), Sha et al. (2003), and Pomerening et al. (2003, 2005).
VII. Further Readings
Furstenthal, L., Kaiser, B., Swanson, C., and Jackson, P., 2001. Cyclin E uses Cdc6 as a
chromatin-associated receptor required for DNA replication. J. Cell Biol. 152,
1267-1278.
Gautier, J., Solomon, M.J., Booher, R.N., Bazan, J.F., and Kirschner, M.W., 1991. cdc25
is a specific tyrosine phosphatase that directly activates p34cdc2. Cell 67, 197-211.
Kumagai, A., and Dunphy, W.G., 1991. The cdc25 protein controls tyrosine
dephosphorylation of the cdc2 protein in a cell-free system. Cell 64, 903-914.
Mueller, P.R., Coleman, T.R., and Dunphy, W.G., 1995a. Cell cycle regulation of a
Xenopus Wee1-like kinase. Mol. Biol. Cell 6, 119-134.
Mueller, P.R., Coleman, T.R., Kumagai, A., and Dunphy, W.G., 1995b. Myt1: a
membrane-associated inhibitory kinase that phosphorylates Cdc2 on both
threonine-14 and tyrosine-15. Science 270, 86-90.
Murray, A.W., and Kirschner, M.W., 1989. Cyclin synthesis drives the early embryonic
cell cycle. Nature 339, 275-280.
Murray, A.W., Solomon, M.J., and Kirschner, M.W., 1989. The role of cyclin synthesis
and degradation in the control of maturation promoting factor activity. Nature
339, 280-286.
Novak, B., and Tyson, J.J., 1993. Numerical analysis of a comprehensive model of Mphase control in Xenopus oocyte extracts and intact embryos. J. Cell Sci. 106,
1153-1168.
Pomerening, J.R., Sontag, E.D., and Ferrell, J.E.Jr., 2003. Building a cell cycle oscillator:
hysteresis and bistability in the activation of Cdc2. Nat. Cell Biol. 5, 346-51.
Pomerening, J.R., Kim, S.Y., and Ferrell, J.E.Jr., 2005. Systems-level dissection of the
cell-cycle oscillator: bypassing positive feedback produces damped oscillations.
Cell 122, 565-578.
Sha, W., Moore, J., Chen, K., Lassaletta, A.D., Yi, C.S., Tyson, J.J., and Sible, J.C.,
2003. Hysteresis drives cell-cycle transition inXenopus laevis egg extracts. Proc.
Natl. Acad. Sci. USA 100, 975-80.
Sible, J.C., and Tyson, J.J., 2007. Mathematical modeling as a tool for investigating cell
cycle control networks. Methods 41, 238-47.
Solomon, M.J., Lee, T.H., and Kirschner, M.W., 1992. Role of phosphorylation in p34cdc2
activation: identification of an activating kinase. Mol. Cell. Biol. 3, 13-27.
Solomon, M.J., Glotzer, M., Lee, T.H., Phillippe, M., and Kirschner, M.W., 1990. Cyclin
activation of p34cdc2. Cell 63, 1013-1024.
VIII. Historical Notes
This model of frog egg extracts was first proposed by Novak & Tyson (1993),
based in large part on Solomon’s 1990 publication of the cyclin threshold for MPF
activation. Novak and Tyson made three predictions from their model: (1) There
must be a different (lower) cyclin threshold for MPF inactivation (corollary: the
control system is bistable between these two thresholds). (2) For cyclin levels
close to the thresholds, the time lag for changes in MPF activity should get long.
And (3) the unreplicated DNA checkpoint will operate by raising the cyclin
threshold for MF activation. These three predictions were not confirmed until
2003, in the elegant experiments of Sha et al. and Pomerening et al. In 2005,
Pomerening et al. published very careful measurements of MPF activity and total
cyclin concentrations in cycling egg extracts. They showed that limit cycle
oscillations encircles the hysteresis loop, as expected from the Novak-Tyson
model, and that, if the positive feedback loop is broken, then the negative
feedback oscillations are indeed faster and smaller amplitude. In this case, the
nuclei in the extract do not show clear rounds of DNA replication followed by
nuclear division.