Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
Mathematical modeling of the budding yeast cell cycle
During a
G1
division cycle, a
n
isio
div
l
l
e
c
Sta
rt
cell must
S
M
replicate all of
its components
Finish
(telophase)
and divide them
into two nearly
identical
G2
M
(anaphase)
daughter cells.
Since DNA
M
(metaphase)
The cell division cycle
stores the genetic
information, it
must be
accurately replicated during S phase and then the two copies precisely
segregated to the daughter cell during mitosis (M phase). Furthermore, it is
crucial that S phase and M phase occur in the right order: S phase precedes
mitosis and that a strict alternation of S and M phase be maintained.
These requirements are enforced by a special family of protein kinases named
cyclin-dependent kinases (Cdk’s for short). These kinases are heterodimers
consisting of a catalytic kinase subunit and a regulatory subunit called cyclin.
Cyclin binding is essential for the enzyme activity. In other words, Cdk is active
only in complex with its cyclin partner. For this reason, these protein kinases are
called cyclin dependent protein kinases.
1
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
In simple eukaryotes,
G1
Cdk
ion
vis
l di
l
e
c
Cln
like budding yeast, the
Sta
rt
role of maintaining the
Enemies
S
M
Finish
(telophase)
correct order and strict
alternation of S and M
Cdk
phases is fulfilled by a
CycB
single Cdk/Cyclin-B
G2
c20
Cd
M
(anaphase)
+A
PC
complex. Cdk/CycB
activity is required for
M
the cell to undergo the
(metaphase)
The role of Cdk/CycB in cell cycle regulation.
START transition to
initiate S phase.
Furthermore, its activity has to be removed for the cell to undergo the FINISH
transition to leave mitosis and divide. That is, Cdk/CycB activity needs to be
high from Start to Finish (which is during S/G2/M phase of the cycle) and low
from Finish to next Start (called G1 phase of the cycle). The Cdk/CycB activity
is controlled by interaction with its enemies (there are several of them, and I
will describe in more detail later). These enemies all have an antagonistic
relationship with the Cdk/CycB complex in such a way that they inhibit Cdk
activity, but Cdk’s can fight back and inactivate them.
Because of their mutual antagonism, Cdk/CycB and its enemies cannot coexist.
That is, the control system can exist in one of the two alternative steady states,
as you will see it later. In G1 phase, cells do not have any Cdk/CycB activity,
because enemies win, they are present in high levels to down-regulate the
Cdk/CycB activity. Whereas in S/G2/M phase, the Cdk’s win, kinase activity is
high and enemies level low.
2
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
For the cell cycle to pass from one phase to another, helper molecules are
needed: Cdk/Cln kinases are the helpers to the Cdk/CycB kinases for the Start
transition and Cdc20 are the helpers to the enemies for the Finish transition.
Cln is another kind of cyclin, different from cyclin B. Cdk/Cln is not inactivated
by the enemies, rather it helps Cdk/CycB to win against the enemies.
Cdc20 is part of a huge enzyme complex called Anaphase Promoting
Complex or APC, involved in protein degradation. The Cdc20/APC complex
has two important functions, one is to remove the cohesion proteins that hold
siter chromatids together and the other is to help the enemies to destroy
Cdk/CycB activity as cells exit from mitosis.
There are two enemies of the Cdk/CycB complex in budding yeast. One is
Cdh1, which is another component of the APC. Like Cdc20, Cdh1 specifically
targets CycB to the APC
core and promotes its
Details about antagonism and helper molecules
hand, Cdk/CycB
G1
h1
Cd
+APC
CKI
Enemies
degradation. On the other
phosphorylates Cdh1 and
prevents its association
c20
Cd
Cdk
+APC
Cln
with the APC core, thereby
turns off Cdk1 activity.
The other enemy of
Cdk
CycB
S/G2/M
unreplicated DNA
unaligned X’s
Cdk/CycB complex is a
3
small cell mass
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
stochiometric Cdk inhibitor (CKI for short) named Sic1 in budding yeast, which
binds to the Cdk/cycB complex and inhibits its kinase activity. However Cdk
phosphorylates the CKI and promotes its degradation.
As described earlier, because of these antagonistic relationships, the control
system has two alternative steady states. Since the steady states are selfmaintaining, helper molecules (Cdk/Cln and Cdc20/APC) are needed to turn the
cell cycle engine from one state to another. The Cdk/Cln kinases, because they
are not inhibited by CKI, nor degraded by Cdh1, can help Cdk/CycB and trigger
the Start transition by phosphorylating and eliminating the CKI (one of the
enemies). This happens abruptly as the synthesis of Cln’s is activated
autocatalytically.
The Finish transition is pushed by Cdc20/APC, which activates Cdh1 (the other
enemy of the Cdk/CycB). Cdc20 is able to drive the transition because it is not
inhibited by the CycB kinase.
These helper molecules (and therefore the transitions) are regulated by
checkpoint mechanisms. The Start transition is controlled by cell mass: the
positive feedback loop for Cln synthesis is not turned on if cells are smaller than
a critical size. The activity Cdc20 (the helper for the Finish transition) is also
regulated by a surveillance mechanism: if DNA is not fully replicated or if
chromosomes are not aligned on the metaphase plate, Cdc20 is kept inactive.
For cells to proliferate, to make a repetitive sequence of properly controlled
Start and Finish transitions, the helper molecules must be removed after they
have done their jobs. For instance, once Start is accomplished, Cln kinases
4
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
must disappear,
The negative feedback loops on helper molecules
otherwise it will
Enemies
h
Cd
1
+APC
work against the
CKI
G1
Finish transition.
For this reason the
c20
Cd
Cdk
+APC
Cln
helpers are regulated
by negative feedback
S/G2/M
Cdk
CycB
loops.
The Cln’s disappear
because the synthesis of the Cln’s is inhibited by Cdk/CycB complex. That is a
negative feedback loop: minus times minus times minus is minus.
Similarly, Cdc20 disappears after the Finish transition because its synthesis is
dependent on the Cdk/CycB complex. That is also a negative feedback loop:
plus times minus times plus is minus.
In the previous slide, only the skeleton of the eukaryotic cell cycle engine is
shown but you can already see how complicated the regulation is. One cannot
understand fully the operation of such kind of complicated network by intuitive
reasoning alone.
A natural way to make the connection from the wiring diagram to cell
physiology is to convert the diagram into mathematical equations (e.g.
differential equations) and then solve them numerically. The solution of these
equations will describe the outcome of the diagram precisely and can be
compared rigorously with the physiology of the cell.
5
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
Numerical simulation
Differential
equations
Network
Physiology
cycle engine has two
S/G2/M
characteristic stable
steady states with
Start
Finish
Cdk/CycB activity
confirms that the cell
different Cdk/CycB
activity: the low Cdk
G1
activity state
[ Cln ]
[ Cdc 20]
Hysteresis
corresponds to the G1
state, whereas the high
Cdk activity state the S/G2/M state of the cycle. If the Cln/Cdc20 ratio is very
small, the system is in the G1 state; if the ratio is very high then it is in S/G2/M
state. For certain intermediate ratios, the two states coexists and which state is
achieved depends on the previous history, an effect called “hysteresis”. Which
state the control system is at depends on two factors: one is the ratio of the
helper molecules, Cln versus Cdc20, and the other is the history of the system.
If the cell is in G1 phase, as the level of Clns is increasing, the control system is
pushed through Start into the S/G2/M state. Notice that, after the Start
transition, the level of Cln is decreasing, but the system stays in the S/G2/M
state. It does not flip back to the G1 state, because the CycB kinase is enough
now to keep Cdh1 and CKI level low without further help from the Clns. The
irreversibility of the Start transition is the consequence of the bi-stable steady
states, and it is intimately related to the phenomenon of hysteresis.
Similar situation exist for the Finish transition. Later in the cell cycle, when
DNA replication is complete and chromosomes is aligned, Cdc20 gets activated,
that pushes the control system through Finish transition, back to G1. CycB is
6
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
degraded, Cdc20 level drops, the cell goes back to its original state and the
cycle repeats itself.
Simulation of the budding yeast cell cycle
2
This graph shows
the numerical
mass
1
1 .0
simulation of the
budding yeast cell
CKI
Cln
cycle. There are two
0 .5
characteristic cell
0 .0
G1
1 .5
Cdh1
1 .0
S/M
CycB
0 .5
the enemies of
0 .5
Cdc20
0 .0
0 .0
0
50
cycle states: in G1,
100
150
Time (min)
Cdk/cyclin-B are
there and Cdk/CycB
activity is low; and the reverse is true in S/M state. The regulatory system
switches from G1 to S/M state with the help of Cln’s and it changes back with
the help of Cdc20. Observe that after each transition the helper molecules go
away, but the control system does not move back the original state. These
transitions are irreversible and show the characteristic effect of hysteresis.
One main advantage of such quantitative model is that it is very easy to test and
uncover the role of each component. For example, wild type cells undergo Start
transition about 70min after birth (top panel), when they inactivate the enemies
of the Cdk/CycB complex with the help of starter kinases Cdk/Cln. If we take
the starter kinase out of the model, as in cln- deletion mutants, then cells cannot
pass Start and arrest in G1 with low CycB activity (middle panel). However, if
we also eliminate CKI, as in a cln- cki1- double mutant, then cells can pass Start
at almost normal time as wild type (bottom panel). This clearly shows that the
major role of the Cln starter kinases is to eliminate the CKI.
7
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
It also suggests that Cln’s
and CKI are not essential
components of the cell cycle
regulation of this double
mutant might be very close
Cd
Cd
h1
+APC
CKI
engine. We postulate that the
Cdk
+APC
c20
Cln
to the mechanism for the
Cdk
CycB
most primitive eukaryotic
cell, which might be driven
by the antagonism between
Cdk/CycB and Cdh1/APC.
Simulation of the budding yeast cell cycle mutants.
Our aim is to make the connections between molecular control system and cell
physiology. To do so requires that we look at a simple reality from three
different point of view: the molecular network of the control mechanism, its
transformation into differential equations and its analysis by dynamical system
theory. These three
Three views of the same reality
Cd
h1
+APC
points of views
CKI
Cd
complement each
Cdk
c20 +APC
Cln
other, and together
Cdk
CycB
Molecular network
d CDK
= k1 - (v2 ’ + v2” . Cdh1 ) . CDK
dt
d Cdh1 (k3’ + k3 ” . Cdc20A) (1 - Cdh1) (k4’ + k4” . CDK . M) Cdh1
=
dt
J3 + 1 - Cdh1
J4 + Cdh1
d IEP
= k9 . CDK . M . (1 – IEP ) – k10 . IEP
dt
d Cdc20T
(CDK . M)4
= k5’ + k5” 4
- k6 . Cdc20T
dt
J5 + (CDK . M)4
.
.
d Cdc20A k7 IEP (Cdc20T - Cdc20A) k8 MAD Cdc20A
=
- k6 . Cdc20T
dt
J7 + Cdc20T - Cdc20A
J8 + Cdc20A
a in-depth
understanding of
Differential equations
S/G2/M
Start
the dynamics of the
Finish
Cdk/CycB activity
they would give us
network and how it
G1
[Cln]
[Cdc20]
really works the
Dynamical System Theory
way manifested in
the physiology of the cell. The molecular network is the natural view of the
8
Bela Novak:
Talk at the HHMI Meeting of the International Research Scholars,
HHMI ,Chevy Chase, June 2000
molecular geneticists. Ideas from the theory of dynamical systems, like
bistability and hysteresis are the natural language of modelers. The differential
equations provide a machine-readable form of these ideas, allowing both
experimentalist and theoreticians to explore the relations between their
hypothetical molecular mechanisms and the actual behavior of living cells.
The work I presented is published in the January 1, 2000 issue of the journal
Molecular Biology of the Cell. It is the collaborative work of my lab and John
Tyson’s lab in Virginia Tech. Attila and Bela are two graduate students from
my lab and Kathy Chen is a post-doc in Tyson’s lab.
9