Bistability, switches, and oscillators
Jim Ferrell
July 2006
Coping with the complexities of cell signaling
Sambrano,
Arkin,
Gilman
Positive feedback as a recurring motif
System
Loops
System
Loops
Mitotic
trigger
Cdc2 -> Cdc25 -> Cdc2
Cdc2 -| Wee1 -| Cdc2
Cdc2 -| Myt1 -| Cdc2
Platelet
activation
Activation -> 5-HT release
-> activation
Activation -> aggregation
-> activation
START
traversal
Cdc28 -| Sic1 -| Cdc28
Cdc28 -> Cln xcription ->
Cdc28
Ras
activation
Sos -> Ras -> Sos
RTK
signaling
EGFR -> ROS -| PTP1B -| EGFR
EGFR -> sheddases -> EGFR
[others proposed]
Action
potential
Depolarization -> Na influx
-> depolarization
Oocyte
maturation
Mos -> MAPK -> Mos
Myogenesis
MyoD -> Myogenin
Myogenin -> MyoD
Calcium
spikes
Ca cyt -> IP3R -> Ca cyt
Ca cyt -> PLC -> Ca cyt
IP3R -> ER dep -> SOC ->
Ca cyt -> IP3R
p53
regulation
p53 -> PTEN -| Akt -> MDM-2 -|
p53
p53 -> p21 -| Cdk2 -| MDM-2 -|
p53
Under certain circumstances, a positive
feedback loop can function as a bistable switch
Bistability is not an automatic consequence of
the pos feedback or double-neg feedback
topology
Monod & Jacob: the differentiated state is
actively maintained through self-sustaining
patterns of gene expression
Enzyme’s product inhibits
a repressor of the enzyme’s
transcription
Monod and Jacob, CSHSQB 26:389-401 (1961)
Bistability in MAPK activation
during Xenopus oocyte maturation
• All-or-none MAPK response in individual oocytes
• Blocking positive feedback changes the MAPK
response to a more graded one, and changes an
irreversible response to a reversible one
• Continuously variable, reversible signal
transducers
-> discrete, irreversible responses
– Ferrell & Machleder Science 1998
– Xiong & Ferrell Nature 2003
Some biological oscillators
• Circadian rhythms
• Pacemaker action potentials
• The cell cycle
• Repetitive calcium spikes
• p53/Mdm2
• NF-B localization
Periodic activation and
inactivation of Cdc2 drives the
embryonic cell cycle
Cdc2 H1K
activity
S
APC
activity
M
S
M
What are the design principles
of this oscillator?
One simple way the oscillator might work
A
B
C
• Negative feedback loop, limit cycle oscillations
• There must be at least three elements in the loop
• But the topology does not guarantee oscillations
Michael Elowitz actually built this
type of oscillator in E. coli
A
B
C
Elowitz & Leibler (2000) Nature
GFP
The basic wiring of the
Cdc2/APC oscillator
The role of neg feedback seems obvious.
What is the pos feedback doing?
Knock out the neg feedback loop (use nondestructable cyclin as stim); examine the
steady-state response of Cdc2 to cyclin
Is there hysteresis?
Experimental system... cytoplasmic
Xenopus egg extracts
lipid
EXTRACT
yolk &
pigment
Eggs
Packed
Extracts à la Murray and others
Spun a bit harder
CSF-arrested
(M-phase)
Ca2+
extract
Interphase
65 cyclin
H1 kinase activity
Experimental approach
Time
CSF-arrested
(M-phase)
Ca2+
extract
65 cyclin
Interphase
Mphase
65 cyclin
Ca2+
H1 kinase activity
Experimental approach
?
Time
CSF-arrested
(M-phase)
Ca2+
extract
65 cyclin
Interphase
Mphase
65 cyclin
Ca2+
H1 kinase activity
Experimental approach
?
Time
0
25
40
45
50
55
60
[65 cyclin B1] (nM)
0
25
40
45
50
55
60
M-phase
interphase
Steady state response is hysteretic
going up
coming down
Cdc25
Wee1
Erk2
Pomerening et al. Nat Cell Biol 200
Sha et al. PNAS 2003
0
25
40
45
50
55
60
[65 cyclin B1] (nM)
0
25
40
45
50
55
60
M-phase
interphase
Steady state response is hysteretic
going up
coming down
Cdc25
Wee1
Erk2
Pomerening et al. Nat Cell Biol 200
Sha et al. PNAS 2003
What is the
significance of
these steady-state
properties
[hysteresis and
bistability] of one
part of the system
for the dynamical
properties of the
whole system?
How to abrogate
positive feedback?
• Ideally: replace Cdc2 with Cdc2AF
• Supplement endogenous Cdc2 with
(modest concentrations of) Cdc2AF
Pooled H1K data, running averages
How to force the extract to run on Cdc2AF
and ignore the endogenous Cdc2?
Wee1
Kim et al. MCB (2005)
With 20 nM OP11 added to extracts,
endogenous Cdc2 is pretty much
out of the picture
Cdc2AF + OP11-Wee1 severely
compromises cycling
The cyclical destruction of cyclin is
similarly compromised
Conclusions so far
• Compromising positive feedback ->
– Less explosive Cdc2 activation
– Less abrupt and less complete cyclin
destruction
– Sustained oscillations become damped
• The Cdc2/APC oscillator relies on
positive feedback
Pomerening Kim & Ferrell Cell (2005)
What is special about running an
oscillator off a bistable trigger?
• Discrete states, suppressed chatter
• Another idea: both the Hodgkin-Huxley
oscillator and the cell cycle oscillator
need to have an adjustable frequency,
but the amplitude should stay approx.
constant when the frequency is adjusted.
How do the two types of oscillators
perform in this respect?
Vary ksynth for the
pos +neg feedback oscillator
And vary ksynth for the
neg feedback oscillator
Over a wider range of ksynth values…
Rationale for the constant
amplitude/adjustable frequency?
Could you change frequency without
changing amplitude by varying one of
the other 20 parameters in the neg
feedback model?
Try some other oscillators…
Try some other oscillators…
Neg feedback oscillators with subcritical
vs. supercritical Hopf bifurcations?
Summary
• The inhibition of Wee1 by Cdc2 is highly
ultrasensitive
• Pos feedback + ultrasensitivity can yield
bistability
• The steady-state response of the
Cdc2/Wee1/Cdc25 system to non-degradable
cyclin is bistable
• Positive feedback is essential for oscillations in
the whole Cdc2/APC system
• Oscillators with a bistable trigger can have an
adjustable frequency with a robust, constant
amplitude
Joe Pomerening
Sun Young Kim
Eric Machleder
Wen Xiong
Bill Dunphy
Helen Piwnica-Worms
Dave Morgan