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

http://www.youtube.com/watch?v=I_xh-bkiv_c&NR=1

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 !