Switches, Oscillators, and
the Cell Cycle
What to notice so far
•  There are two ways to design a regulatory
cell network:
•  (1) protein-protein interactions (mutual
phosphorylation, etc etc) (time scale: secmin)
•  (2) gene networks (time scale: hrs day) Gene circuits
Protein circuits
Protein circuits
Kinase
M
phosphatase
Mp
Other things to notice
•  By building up feedback interactions it is
possible to obtain new dynamics :
•  (1) Simple decay to steady state
•  (2) Switch (bistability)
•  (3) Oscillator (stable cycles) No feedback
x
Decay to a single
stable steady
state
No feedback
x
Decay to a single
stable steady
state
A
Positive feedback
x
Bistability and
switch-like
behaviour
possible
A
Add negative feedback
x
y
Stable cycles
possible
Example: Phosphorylation cycle
Kinase
Inactive
M
phosphatase
Mp
Active
Decay to a single
stable steady
state
Add positive feedback to kinase
Kinase
M
phosphatase
Mp
Bistability and
switch-like
behaviour
possible
Add further negative feedback Kinase
M
phosphatase
Mp
Stable cycles
possible
Simple mathematical example
x
A
A switch (Generic bistability)
The parameter A
controls the
switch
A controls the switch
“Switch” (Generic bistability)
y
The “parameter”
y controls the
switch
The curve dx/dt=0
y
y controls the switch
y
As y varies, we can go around the
hysteresis loop
y
Add negative feedback to the switch
x
y
Now y is dynamic
]
x
y
Switch becomes an oscillator
]
Example:
This is the Fitzhugh
Nagumo model
“Switch” (Generic bistability)
Bifurcation
diagram
y
“Switch” (Generic bistability)
Let us flip it
over
y
“Switch” (Generic bistability)
y
y
The xy phase plane
The curve dy/dt=0
The curve dx/dt=0
Oscillator
a=0.7, b=0.8, c=3, j=0.35
Get an oscillator
Application to the Cell Cycle
•  Work by John Tyson (Virginia Tech):
•  The control of the cell division is
maintained by an intricate web of signaling
pathways, that incorporates many signals to
decide when to divide.
•  The cycle has “checkpoints” at which
decisions are made.
G1
Slide by John
Tyson
cell division
1) Alternation of
S phase and M phase.
2) Balanced growth and
division.
M
mitosis
S
DNA
replication
G2
G1
Slide by John
Tyson
cell division
The cell cycle is the
sequence of events whereby
a growing cell replicates all
its components and divides
them more-or-less evenly
between two daughter cells
...
M
mitosis
S
DNA
replication
G2
G1
Slide by John
Tyson
cell division
S
Cyclin-dependent
kinase
Cyclin B
P
M
mitosis
DNA
replication
Cdk1
CycB
P
G2
P Wee1
Cdc20
Cdc14
Wee1
CycB
Cdc25
TFBI
APC
APC-P
Cdc14
P
CycB
Cdc14
CKI
Cdc25
CycD
CycB
CycE
TFII
Cdh1
TFIA
CKI
CKI
CycA CycD
CKI
P
TFBA
CycE
CycA
Cyc E,A,B
CycE
CycA
TFEA
CycB
CycD
CycA
TFEI
Checkpoints
in phase
G1 there is
low Cdk
and low
cyclin
buildup of
cyclin/Cdk
Cdk1
CycB
APC is activated, leading to
destruction of cyclin and loss of CdK
activity.
Cyclin is produced and degraded
APC is inactivated by phosphorylation
Cyclin
Active form
APC
(no phosphate)
pi
Inactive
form
APC is inactivated by phosphorylation
Cyclin
APC
This will be modeled by a typical
equation that we have already seen.
pi
Schematic
Cyclin
Cyclin
APC
pi
Negative feedback
Cyclin
APC
pi
APC and Cyclin mutually antagonistic
Cyclin
APC
pi
Model
Model
Model
Bistable switch
Cyclin
Cell Mass
Cell mass is the parameter that
flips the switch
Cyclin
Cell Mass
P Wee1
Cdc20
Cdc14
Wee1
CycB
TFBI
APC
APC-P
Cdc14
P
Cdc25
CycB
Cdc14
Cdc25
CKI
Bistable switch
CycD
CycB
CycE
M
TFIA
CKI
CycA CycD
10-1
CKI
Cdc2/Cdc13
CKI
S/G2
10-2
CycE
CycA
Cyc E,A,B
10-3
CycE
0
1
[cyclin]
mass/DNA
TFEA
CycA
2
CycB
CycD
CycA
TFII
Cdh1
100
[kinase]
P
TFBA
TFEI
Activation of APC by Cdc20 (“A”)
cYclin A
A= Cdc20. It increases sharply during metaphase and activates APC
A is turned on by cyclin (sigmoidally)
cYclin A
Activation of APC by Cdc20 (“A”)
cYclin A
A= Cdc20. It increases sharply during metaphase and activates APC
Three variable model:
Now we get a cell cycle.