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Network Dynamics andNetwork Dynamics andCell PhysiologyCell Physiology
John J. Tyson John J. Tyson
Department of Biological Sciences Department of Biological Sciences
& Virginia Bioinformatics Institute& Virginia Bioinformatics Institute
Outline
1. Cell Signaling: Physiology
2. Cell Signaling: Molecular Biology
3. Chemical Kinetics
4. Sniffers, Buzzers & Toggles
5. Bistability & Oscillations in Frog Eggs
6. Dynamical Perspective
7. Example: Fission Yeast Cell Cycle
nutrients
repellants
damage
hormones
heat shock
growth & division
movement
geneexpression
death
BacteriaGlucose
Lactose
lactosemetabolizing
enzymes
1
0 0
Fission Yeast
14 m
7 m
Wild type Mutant (wee1)
Fibroblast
Growth Factor
PROLIFERATION
Extracellular Matrix
Cell-Cell Contact
Fibroblast
ProgrammedCell Death
Suprachiasmatic Nucleus12hL:12hD
ActivityBody temp
Outline
1. Cell Signaling: Physiology
2. Cell Signaling: Molecular Biology
3. Chemical Kinetics
4. Sniffers, Buzzers & Toggles
5. Bistability & Oscillations in Frog Eggs
6. Dynamical Perspective
7. Example: Fission Yeast Cell Cycle
Hanahan & Weinberg (2000)
Signal Transduction Network
Cdk
C K
I
Cdk
Cyclin
C K
I
Cdk
Cyclin
Cdk
Cyclin
P
Cyclin
Cdk
Each icon represents a chemical species. Each arrow represents a chemical reaction that occurs at a certain rate.
CyclinMPF =
M-phase Promoting Factor
X(t) = [cyclin]
1 0
0 1
, (0)
( )
dXk X X
dtX t X k t
Time (min)
Cyclin(nM)
20
40
4020 60
Interphase arrestedFelix et al. (1990)Nature 346:379, Fig. 1
Metaphase releasedTang et al. (1993)EMBO J 12:3427, Fig. 2
1. Synthesis
Estimate k1 from the “red” data:
2
2 1/ 2
2 1/ 2
2 1/ 2
2 0
0
1/ 2 0 0 0
1/ 22 2
, (0)
( )
1 1( )
2
ln 2 0.72 , or
k t
k tk t
k t
dXk X X X
dt
X t X e
X t X X e Xe
e tk k
2. Degradation
Estimate k2 from the “blue” and “green” data above.
How can it be that cyclin has different half-lives in
different phases of the cell cycle?
3 3 0 0
0 03 0 0
0 0
( ) ( ) ( )( ), (0) 0
(1 )( ) , where ( )
t
t
dMk C t X t k C M X M M
dt
C X eM t k C X
C X e
3. Dimerization
X(t) = [cyclin], C(t) = [Cdc2], M(t) = [dimer],
Estimate k3 from the data below, given that C0 = 100 nM.
Time (min)
Dimers(nM)
20
40
105 15
Kumagai & Dunphy (1995)Mol Biol Cell 6:199, Fig. 3B
2
1 2
1
2
1
2
, (0) 0,
( ) 1
Note: as , ( ) (stable steady state)
k t
dXk k X X
dt
kX t e
k
kt X t
k
From your previous estimates of k1 and k2, estimate the steady state concentrations of cyclin in interphase and late anaphase (end of mitosis).
4. Synthesis and Degradation
Phase k1 k2 Xss
Interphase
Anaphase
This case is unusual in that one can write down an “exact” solution of the differential equation in terms of elementary functions. When an exact solution is not available, one can always take other approaches…
Numerical1 2
1 2
( ) ( )( )
( ) ( ) ( ( ))
X t t X tk k X t
tX t t X t k k X t t
This always works, but doesn’t provide much insight.
Graphical
+
dX/dt
X
k1
k1/ k2
dX/dt = 0 at X = k1/k2, called a “steady state” solution
X(t) approaches k1/k2 for t large (“stable” steady state)
Outline
1. Cell Signaling: Physiology
2. Cell Signaling: Molecular Biology
3. Chemical Kinetics
4. Sniffers, Buzzers & Toggles
5. Bistability & Oscillations in Frog Eggs
6. Dynamical Perspective
7. Example: Fission Yeast Cell Cycle
R
S
0
0.5
0 1 2 3
res
po
ns
e (
R)
signal (S)
linear
0
5
0 0.5 1
S=1
R
rate
(d
R/d
t)
rate of degradation
rate of synthesis
S=2
S=3
Gene Expression
Signal-ResponseCurve1 2
d,
d
Rk S k R
t 1
ss2
k SR
k
R
Kinase
RP
ATP ADP
H2OPi
Protein Phosphorylation
0
1
2
0 0.5 1
RP
rate
(d
RP
/dt)
0.25
0.5
1
1.5
2
Phosphatase 0
0.5
1
0 1 2 3
res
po
ns
e (
RP
)
Signal (Kinase)
1 R 0
R
S
EP E0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.5R
rate
(d
R/d
t) S=0
S=8
S=16
0
0.5
0 10
res
po
ns
e (
R)
signal (S)
Protein Synthesis:Positive Feedback
Example: Fuse
0
0.5
0 10
resp
on
se (
R)
signal (S)
dying
Apoptosis(Programmed Cell Death)
living
Outline
1. Cell Signaling: Physiology
2. Cell Signaling: Molecular Biology
3. Chemical Kinetics
4. Sniffers, Buzzers & Toggles
5. Bistability & Oscillations in Frog Eggs
6. Dynamical Perspective
7. Example: Fission Yeast Cell Cycle
0
0.5
1
0 1 2
resp
on
se (
MP
F)
signal (cyclin)
MPF
Cdc25-PCdc25
MPF-P
Wee1
(inactive)
0
0.5
1
0 0.5 1 1.5
MPF
Cd
c2
5-P
0
0.5
1
0 1 2 3
Cd
c2
5-P
MPF
S = Total Cyclin
centrifuge
Solomon’s protocol for cyclin-induced activation of MPF
cytoplasmic extract
pellet
Ca2+ M
Cyclin
Cyclo-heximide
Cdk1Wee1
Cdc25
Cyclin
Cdk1
Cell 63:1013 (1990)
Threshold
0
20
40
60
80
100
120
0 10 20 30
Cyclin (nM)
CD
K a
ctiv
ity Solomon et al. (1990)
Cell 63:1013.
Novak & Tyson (1993) J. Cell Sci. 106:1153
Pomerening et al., Nature Cell Biology 5:346-351 (2003)
Sha et al., PNAS 100:975-980 (2003)
Testing activation threshold for Mitosis I
Interphase
Mitosis I
90Cyclin B1 and 100 µg/ml CHX
Testing Thresholds in Cycling Extracts
Testing inactivation threshold for Mitosis I
Interphase Interphase
Mitosis I
90Cyclin B1
100 µg/ml CHX
MPFactivity
time
16 24 32 40 090 cyclin B (nM) :
90 min
0 min
60 min
140 min
090 cyclin B (nM) : 16 3224 40
M
M M M
The activation threshold for Mitosis I is between 32 and 40 nM.
The inactivation threshold for Mitosis I is between 16 and 24 nM.
0
0.5
1
0 1 2
MP
F
cyclin
MPF
Cdc25-PCdc25
MPF-P(inactive)
cyclin synthesis
cyclin degradationAPC
If knock-out positive feedback loop, then oscillations become faster and smaller amplitude…
Figure 4. Pomerening, Kim and Ferrell
With + feedback Without + feedback
• Tyson, Chen & Novak, “Network dynamics and cell physiology,” Nature Rev. Molec. Cell Biol. 2:908 (2001).
• Tyson, Csikasz-Nagy & Novak, “The dynamics of cell cycle regulation,” BioEssays 24:1095 (2002).
• Tyson, Chen & Novak, “Sniffers, buzzers, toggles and blinkers,” Curr. Opin. Cell Biol. 15:221 (2003).
• Csikasz-Nagy et al., “Analysis of a generic model of eukaryotic cell-cycle regulation,” Biophys. J. 90:4361 (2006).
References
Outline
1. Cell Signaling: Physiology
2. Cell Signaling: Molecular Biology
3. Chemical Kinetics
4. Sniffers, Buzzers & Toggles
5. Bistability & Oscillations in Frog Eggs
6. Dynamical Perspective
7. Example: Fission Yeast Cell Cycle
Cdk
C K
I
Cdk
Cyclin
C K
I
Cdk
Cyclin
Cdk
Cyclin
P
Cyclin
Cdk
Wee1
Cdc25
= k1 - (kwee + k2) * MPF + k25 (cyclin - MPF)
= k1 - k2 * cyclin
d MPFdt
d cyclindt
MPF
Cyclin
Phase Plane dx/dt=f(x,y)dy/dt=g(x,y)
(xo,yo)
x=f(xo,yo) t
y=g(xo,yo) t
One-parameter bifurcation diagram
parameter
variable
stable steady state
unstable steady state
saddle-nodesaddle-node
Signal Response
t t
p x
OFF
ON
(signal)
(response)
x
y
One-parameter bifurcation diagram
parameter
variable
stable steady state
unstable steady state
saddle-nodesaddle-node Hopf
(signal)
(response)
MPF
Cyclin
Phase Plane dx/dt=f(x,y)dy/dt=g(x,y)
MPF
Cyclin
Phase Plane dx/dt=f(x,y)dy/dt=g(x,y)
MPF
Cyclin
Phase Plane dx/dt=f(x,y)dy/dt=g(x,y)
Hopf BifurcationHopf Bifurcation
x2
p1
stable limit cycle
sss
uss
slc max
min
Hopf BifurcationHopf Bifurcation
x2
p1
sssuss
slc
parameter(signal)
variable(response)
Hopf
Second Parameter
subcritical
Second Parameter
CF
parameter(signal)
variable(response)
SNIC
Second Parameter
SL
SNIC BifurcationSNIC Bifurcation
Invariant Circle
Limit Cycle
x2
p1
node
saddle
Saddle-Node on anInvariant Circle
max
min
max
SNIC
Signal-Response Curve = One-parameter Bifurcation Diagram
•Saddle-Node•Supercritical Hopf•Subcritical Hopf•Cyclic Fold•Saddle-Node Invariant Circle
Outline
1. Cell Signaling: Physiology
2. Cell Signaling: Molecular Biology
3. Chemical Kinetics
4. Sniffers, Buzzers & Toggles
5. Bistability & Oscillations in Frog Eggs
6. Dynamical Perspective
7. Example: Fission Yeast Cell Cycle
S
G1
DNAreplication
G2Mmitosis
cell division
1) Alternation ofS phase and M phase.
2) Balanced growth anddivision.
3) Checkpoints
P
Cdc25
Wee1
Wee1P
Cdc25
CycB
PCdc20
Cdc20
Cdh1
CK
I
CycB
CycBCK
I
CK
I
CycA
CycA
APC-PAPCTFBI
TFBA
CycE
CycD
TFEA
TFEI
Cyc E,A,B
CycE
TFIA
TFII
Cdc20
CK
I
CycE
Cdc14
Cdc14
Cdc14
CycA
CycA
CycB
CycD
Cdh1CycD
0 50 100 150 200 250 300
0
1
2
3
4
5
mass/nucleus
P Cdk1
CycB
Cdk1
CycB
CKI
Cdh1
Cdc20
Wee1
Cdc25
Time (min)
S G2 MG1 S G2 MG1 S
Gene Viable? Traitcdc2
No block in G2cdc13
No block in G2rum1
Yes sterileste9
Yes sterileslp1
Yeswee1
Yes smallcdc25
No block in G2
cdc2 OP Yes wtcdc13 OP Yes wtrum1 OP No endoreplic.ste9 OP Yes wtwee1 OP Yes largecdc25 OP Yes small
wee1 rum1 No extremely small
wee1 cdc25 Yes quantized cycles
wee1 cdc25OP No cut
wee1 OP cdc25 No block in G2
Mutants in Fission YeastMutants in Fission Yeast
P
Cdc25
Wee1
Wee1P
Cdc25
CycB
PCdc20
Cdc20
Cdh1
CK
I
CycB
CycBCK
I
CK
I
CycA
CycA
APC-PAPCTFBI
TFBA
CycE
CycD
TFEA
TFEI
Cyc E,A,B
CycE
TFIA
TFII
Cdc20
CK
I
CycE
Cdc14
Cdc14
Cdc14
CycA
CycA
CycB
CycD
Cdh1CycD
mass/ DNA
0.0 0.5 1.0 1.5
Cdc2
/Cdc1
3
10-5
10-4
10-3
10-2
10-1
100
G1
Mmass/ DNA
0 1 2
Cdc2
/Cdc1
3
10-3
10-2
10-1
100
S/G2
M
mass/nucleus
mass/ DNA
0.0 0.5 1.0 1.5 2.0
Cdc2
/Cdc1
3
0.1
1
M
0 1 2 3 4 5
0
0.4
0.8
3.0
mass/nucleus
Cd
k1:C
ycB
G1S/G2
M
SNICHopf SN1SN2SN3
P
Cdc25
Wee1
Wee1P
Cdc25
CycB
PCdc20
Cdc20
Cdh1
CK
I
CycB
CycBCK
I
CK
I
CycA
CycA
APC-PAPCTFBI
TFBA
CycE
CycD
TFEA
TFEI
Cyc E,A,B
CycE
TFIA
TFII
Cdc20
CK
I
CycE
Cdc14
Cdc14
Cdc14
CycA
CycA
CycB
CycD
Cdh1CycD
mass/nucleus
wee1
mass/nucleus
Cd
k1:C
ycB
0 1 2 3 4 5
0
0.4
0.8
1.2
G1
S/G2
M
P
Cdc25
Wee1
Wee1P
CycB
PCdc20
Cdc20
Cdh1
CK
I
CycB
CycBCK
I
CK
I
CycA
CycA
APC-PAPCTFBI
TFBA
CycE
CycD
TFEA
TFEI
Cyc E,A,B
CycE
TFIA
TFII
Cdc20
CK
I
CycE
Cdc14
Cdc14
Cdc14
CycA
CycA
CycB
CycD
Cdh1CycD
Cdc25
mass/nucleus
mass/nucleus
Cd
k1:C
ycB
G1S/G2
M
0 1 2 3 4 5
0
0.4
0.8
3.0 cki
The Start module is not required during mitotic cyclesThe Start module is not required during mitotic cycles
P
Cdc25
Wee1
Wee1P
Cdc25
CycB
PCdc20
Cdc20
Cdh1
CK
I
CycBCK
I
CK
I
CycA
CycA
APC-PAPCTFBI
TFBA
CycE
CycD
TFEA
TFEI
Cyc E,A,B
CycE
TFIA
TFII
Cdc20
CK
I
CycE
Cdc14
Cdc14
Cdc14
CycA
CycA
CycB
CycD
Cdh1CycD
CycB
0
0.4
0.8
2.0
0 1 2 3 4 5
G1
S/G2
M
cki wee1ts
mass/nucleus
Cd
k1:C
ycB
Unbalanced Growthand Division …
is Lethal !
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