Introduction to Introduction to cell cycle modelingcell cycle modeling

Attila Attila CsikásCsikászz--NNaagygy

S

G1

DNAreplication

G2M(mitosis)

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

...

S

G1

DNAreplication

G2M(mitosis)

cell cycleengine

Cyclin-dependent

kinase

B-typecyclin

Cdk

Cyclin

S

G1

DNAreplication

G2M(mitosis)

cyclin Cdk

Cyclin dependent protein kinase

cyclin

Cdk

A P P P+

A P P

+

P

A P P P

protein protein

Cdk

Cyclin

S

G2

G1

DNAreplication

G2/MM

(mitosis)

chromosome cycle growth cycle

- DNA replication- mitosis(precise doubling

and halving)

- cell growth- cell division(approx. doubling

and halving)

0 1 2 3 40

1

2

3

4

TC > TD

TC < TD

Cyt

opla

smic

mas

s

TC = TD

exponential exponential balanced balanced growthgrowth

Size control

TDTC

Cell cycleengine

Cytoplasmicgrowth

oogenesis embryogenesis

Uncoupling of the growth and chromosomal cycles

growing (somatic) cells early embryos

Modeling cell cycle regulationModeling cell cycle regulation

I. Embryonic cyclesI. Embryonic cycles

Yoshio Masui

rapid cleavage divisionsno gap phases no checkpointsno apoptosis

apoptotic checkpointgap phasescell cycle inhibitorsDNA replication and damage

checkpoints

Cell cycle remodeling begins at the MBT.

AA

The negative feedback loop

Cdc20

IE IE-P

+APCCdk

CycBCdk

d CycBdt = k1 – (k2’ + k2”’

. Cdc20A) CycB

d IE-P

dt = k9 . CycB .

1 - IE-PJ9 + 1 - IE-P - k10

. IE-P

J10 + IE-P

d Cdc20A

dt = k7 . IE-P .

1 - Cdc20AJ7 + 1 - Cdc20A

– k8 .

Cdc20AJ8 + Cdc20A

1

9

10

87

2

0 50 1000.0

0.5

0

1

Early embryonic cell cycles

Cdk

CycB

IE-P IE-P

& C

dc2

0

Cd

k/C

ycB

Béla NovakJohn J. Tyson

Wee1

Wee1

Cdc25

P

degradedcyclin

Cdk

Cyclin Cdk

OFF

ONCdk

Cyclin

+

P

Cdc20

+

����

Cdc25P

����P

IE IEP

+

MPF is regulated by positive and negative feedback loops

Direct negative feedback:

NO oscillations

Delayed negative feedback:

sinusoid oscillations

Positive & negative feedbacks:

relaxation oscillations

]][1[][][

321 CyclinCdkkCyclinkkdt

Cyclind −−=

]][1[][])[(][][

3252 CyclinCdkkPYTkYTkkkMPFkdt

YTdcakweepp ++++−=

][])[(][][

225 PYTPkPYTkkkYTkdt

PYTdppcakwee +++−=

][])[(][][

225 PYTkPYTPkkkMPFkdt

PYTPdcakppwee +++−=

][])[(][][

252 PYTPkMPFkkkYTkdt

MPFdweeppcak +++−=

]25[

]25][[

]25[]25[

])25[]25]([[]25[PCdcK

PCdcPPasek

PCdcCdctotalK

PCdcCdctotalMPFk

dt

PCdcd

b

b

a

a

+−

−+−=

]1[

]1][[

]1[]1[

])1[]1]([[]1[

PWeeK

PWeePPasek

PWeeWeetotalK

PWeeWeetotalMPFk

dt

PWeed

f

f

e

e

+−

−+−=

][

]][[

][][

])[]]([[][

IEPK

IEPPPasek

IEPIEtotalK

IEPIEtotalMPFk

dt

IEPd

h

h

g

g

+−

−+−

=

*][

*]][[

*][][

*])[]]([[*][

APCK

APCIEAntik

APCAPCtotalK

APCAPCtotalIEPk

dt

APCd

d

d

c

c

+−

−+−=

]25[])25[]25([ "25

'2525 PCdcVPCdcCdctotalVk +−=

])1[]1([]1[ "' PWeeWeetotalVPWeeVk weeweewee −+=

*][*])[]([ "2

'22 APCVAPCAPCtotalVk +−=

Differential Equations in Novak-Tyson Model

time (min)

0 60 120 1800

10

20

30

Mathematical simulations generate oscillations in the model of Xenopus extracts

total

Cdk

Cyclin

P

Cyclin

Cdk

Cyclin

Cyclin threshold to induce mitosis

Marc Solomon

Prediction: The threshold concentration of cyclin B required to activate MPF is higher than the thresho ld

concentration required to inactivate MPF.

Ti Ta Tcyclin level cyclin level

MP

F ac

tivity

MP

F ac

tivity

hysteretic non-hysteretic

Jill Sible

Joe Pomerening

+Cdc2AF+Wee1-OP11

Joe Pomerening

Modeling cell cycle regulationModeling cell cycle regulation

II. Somatic cyclesII. Somatic cycles

Cdk

Cyclin

S

G2

G1

DNAreplication

G2/MM

(mitosis)

1

2

2

1division

growth

DNAnucleus

mass

The Cell Division Cycle

--

S G2 MG1

Cdh1

Stabilizing the G1 phase

AACdk

CycBCdk

posi

tive

feed

back

(dou

ble-

nega

tive)

d CycB dt = k1

. M - ( k2’ + k2” . Cdh1 ) . CycB

d Cdh1dt =

k3’ . (1 - Cdh1) J3 + 1 - Cdh1 -

(k4’ + k4” . CycB) Cdh1 J4 + Cdh1

+APC

1 2

4

3

Mass

d CycB dt = k1

. M - ( k2’ + k2” . Cdh1 ) . CycB

d Cdh1

dt = k3’

. (1 - Cdh1) J3 + 1 - Cdh1 -

(k4’ + k4” . CycB) Cdh1

J4 + Cdh1

CycB

Cdh

1G1

S/M

CycB nullcline:

CycB = k1

. M v2' + v2"

. Cdh1

Cdh1 nullcline:

CycB = k3' (1 - Cdh1) . (J4 + Cdh1)(J3 + 1 - Cdh1) k4"

. Cdh1 - k4'k4"

CycB

Cdh

1

Nullclines

G1

S/M

Cdh

1

CycB

S/M

Cdh

1

CycB

cell growth

G1

S/M

starter kinase

CycB

S/M

Cdh

1

+APC

Cdh1

Cdc

14

AACdkCycB

Cdk

CdkCln

TF-P

mass

TF

Cdk

degradedCln

Turning on/off the Cdk - Cdh1 switch

mass

+APC

Cdh1

Cdc

14

AACdkCycB

Cdk

CdkCln

TFTF-P

mass

Cdk

degradedCln

d Cdh1dt = (k3' + k3"

. Cdc20A) 1 - Cdh1

J3 + 1 - Cdh1 - (k4' . SK + k4

. CycB) Cdh1

J4 + Cdh1

d SK

dt = k13' + k13" . TF - k14

. SK

d TF

dt = k15' . M

1 - TFJ15 + 1 - TF - (k16' + k16”

. CycB) TF

J16 + TF

1

2

4

3

13

1415

16

+APCAA

CdkCycB

Cdk

Cdc20

IE IE-P

nega

tive

feed

back

d Cdc20T

dt = k5' + k5" . CycB4

J54 + CycB4 - k6

. Cdc20T

dCdc20A

dt = k7 . IE-P Cdc20T-Cdc20A

J7 + Cdc20T - Cdc20A - k8 .

Cdc20A

J8 + Cdc20A - k6 . Cdc20A

d IE-Pdt = k9

. CycB 1 - IE-P

J9 + 1 - IEP - k10 . IE-P

J10 + IE-P

9

108

7

5 6

1 2

dCycBdt = k1

. M - (k2' + k2" . Cdh1 + k2'"

. Cdc20A) CycB

d Cdh1

dt = (k3' + k3" . Cdc20A)

1-Cdh1J3+1-Cdh1 - (k4'

. SK + k4 . CycB)

Cdh1J4 + Cdh1

d SK

dt = k13' + k13" . TF - k14

. SK

d TF

dt = k15' . M

1 - TFJ15 + 1 - TF - (k16' + k16”

. CycB) TF

J16 + TF

d Cdc20T

dt = k5' + k5"

. CycB4

J54 + CycB4 - k6

. Cdc20T

dCdc20A

dt = k7 . IE-P

Cdc20T-Cdc20A

J7 + Cdc20T - Cdc20A - k8

. Cdc20A

J8 + Cdc20A - k6

. Cdc20A

d IE-P

dt = k9 . CycB

1 - IE-P J9 + 1 - IE-P - k10

. IE-P

J10 + IE-P

0 100 200 3000.0

0.5

0

1

0

1

2

0.0

0.5

0.5

1.0

Cdk

CycB

mass

IE-P

Cdc20total

time (min)

Cdk

CycB

CdkCln

Cdc20

Cdh1

P

CK

I

CK

I

AA

IE IE-P

SK

TF-P TF

+APCCdk

CycBCdk

mass

Budding yeast

Cdk

CycB

CdkCln

Cdk

CycB

Cdh1CKI

0 1000 .0

0 .5

0

1

2

0 1000 .0

0 .5

0

5

0 1000.0

0.5

0

1

Cdk

CycB

CK

IWT

CK

I

sk-

sk- cki-

Cdk

CycB

Cdk

CycB

SK

0 1000 .0

0 .5

0

1

2

0 1000 .0

0 .5

0

5

0 1000.0

0.5

0

1

Cdk

CycB

CK

IWT

CK

I

sk-

sk- cki-

Cdk

CycB

Cdk

CycB

SK

0 1000 .0

0 .5

0

1

2

0 1000.0

0.5

0

5

0 1000.0

0.5

0

1

Cdk

CycB

CK

IWT

CK

I

cln-

cln- cki-

Cdk

CycB

Cdk

CycB

SK

time (min)

CdkCln

What makes us believe that our set of equations and What makes us believe that our set of equations and

parameters are close to life?parameters are close to life?

What can be easy to measure?What can be easy to measure?•Length of different cell cycle phases

•Cell size at different transitions

Current Budding yeast model 131 mutantsCurrent Fission yeast model 59 mutants

Not only wild type, but many mutants!Not only wild type, but many mutants!One wild type parameter set. To simulate mutants use this as a basal

parameter set.

Test the model! Compare the results to life:

Cdk

CycB

CKICdh1

Cln2

Cdc14

Hysteresis in budding yeast cell cycle?

Cdk

Clb

CKICdh1

CdkCln

Clb2/Cdkactivity

A + Cln2B+Cdc14

A/B

G1

S/G2/M

Start

Finish

time

Cln2 Cdc14

Clb2/Cdkactivity

A + Cln2B+Cdc14

A/B

G1

S/G2/M

“Neutral”

Protocol to demonstrate hysteresis

Genotype: cln1∆ cln2∆ cln3∆ GAL-CLN3 cdc14ts

Knockout all the G1cyclins

Turn on CLN3with galactose; turn off with glucose

Temperature-sensitive allele of CDC14: on at 23oC, off at 37oC.

“Neutral” conditions: glucose at 37oC (no Cln’s, no Cdc14)

Fred Cross

R = raffinoseG = galactose

Standard for protein loading

G1 cells

S/G2/M cells

Start with allcells in G1

Make some Cln

Shift toneutral

P

Wee1P

Cdc25

Cdc25

Wee1

G2/

M

Cdk

CycBP

Cdc20

Cdh1

PC

KI

CK

I

AA

IE IE-P

SK

TF-P TF

G1/

SM

itosi

s

+APC

Schizosaccharomycespombe

Cdk

CycBCdk

2

kwee

k25

5 6

9

10 7

8

4 3

1112

VaweeViwee

Vi25

Va25

mass13

1415

16

Cdk

CycB

CKICdh1

Wee1

Fission yeast

0

1

0

1P CdkCycB

CdkCycB

1

2

cell mass

time (min)0 50 100 150

0

0.5

0

0.05

CK

I

S/G2G1M M

Cdc2

Cdc13

Rum1Ste9

Wee1

Fission yeast

0

1

0

1P Cdc2Cdc13

Cdc2Cdc13

1

2

cell mass

time (min)0 50 100 150

0

0.5

0

0.05

Rum

1

S/G2G1M M

Cdk

CycB==main controller of the systemmain controller of the system

State variable (response):State variable (response):

Bifurcation parameterBifurcation parameter (signal):(signal):

cell masscell mass(reports environmental conditions)(reports environmental conditions)

Bifurcation diagram for cell cycle Bifurcation diagram for cell cycle regulationregulation((Signal Signal –– response curveresponse curve ))

0

1

0

1P CdkCycB

CdkCycB

1

2

cell mass

time (min)

Cdk

CycB

CKICdh1

Wee1

Fission yeast0 50 100 150

0

0.5

0

0.05

CK

I

S/G2G1M M

abscissa

ordinate

P

Wee1P

Cdc25

Cdc25

Wee1

G2/

M

Cdk

CycBP

Cdc20

AA

Sta

rtF

inis

h

+APCCdk

CycBCdk

Cdk

CycBCK

I

AAAA

Cdk

CycB

Cdh1

PC

KI

CK

I

delay

dCycB T

dt = k1 . cell mass - (k2' + k2" . Cdh1 +k 2'" . Cdc20 A) . CycB T

dpMPFdt = kwee . (CycB T - pMPF) - k25

. pMPF - (k 2' + k2" . Cdh1 + k 2'"

. Cdc20 A) . pMPF

dIEdt = k9

. MPF . 1- IE

J9 +1- IE - k10 .

IE J10 + IE

dCdc20dt = k7

. IE . 1 - Cdc20

J7 + 1 - Cdc20 - k8 . Cdc20

J8 + Cdc20 - k6 . Cdc20

dCdh1dt = k3'

. 1 - Cdh1

J3 + 1 - Cdh1 - (k4' . SK + k 4

. MPF) . Cdh1

J4 + Cdh1

dCKIT

dt = k11 - (k12 + k12' . SK + k 12" . MPF) . CKIT

dSKdt = k13' + k 13" . M - k14 . SK

dMdt = µµµµ . M

[Trimer] = 2 [CycB T] [CKI T]

[CycB T] + [CKI T] + Keq-1 + ([CycB T] + [CKI T] + Keq

-1) 2 - 4 [CycB T] [CKI T]

kwee = kwee' + (kwee" - k wee') . GK(Vawee, Viwee ' + V iwee" . MPF, Jawee, J iwee) k25 = k25' + (k25" - k 25')

. GK(Va25' + Va25" . MPF, Vi25, Ja25, J i25)

MPF = (CycB T - pMPF) . (CycB T - Trimer)

CycB T

P

Wee1P

Cdc25

Cdc25

Wee1

G2/

M

Cdk

CycBP

Cdc20

AA

Sta

rtF

inis

h

+APCCdk

CycBCdk

Cdk

CycBCK

I

AAAA

Cdk

CycBcell mass

Cdh1

PC

KI

CK

I

delay

0.1 2.0

0.1

1

M

cell mass

Cdk

/Cyc

B

0.025 0.60010-5

10-4

10-3

10-2

10-1

G1

M

cell mass

Cdk

/Cyc

B

0.1 2.0

10-2

10-1

100

S/G2

M

cell mass

Cdk

/Cyc

B

0 50 100 1500

1

0

1

CdkCycB

1

2

mass / nuclei

0 1 20

0.1

0.2

0.3

0.4

0.5

1.5

1.6

G1

S/G2

M

stable s.s.unstable s.s.min/max����

cell masstime (min)

Cdk

/Cyc

B

S/G2G1M M

0 1 2 3 41e-5

1e-4

1e-3

1e-2

1e-1

1e+0

Fission yeast – wild type

stable steady state (G1)

stable steady state (S-G2)

unstable steady state (M)

stab

le o

scill

atio

n

max

min

cell mass

Cdk

/Cyc

B

Cdk

CycB

SK

CKICdh1

Wee1Cdk

CycB

SK

CKICdh1

Wee1

Wild type wee1- mutant

Paul Nurse

0 1 2 3 41e-5

1e-4

1e-3

1e-2

1e-1

1e+0 wild type

0.0 0.5 1.0 1.5 2.0

1e-5

1e-4

1e-3

1e-2

1e-1

wee1-

cell mass

Cdk

/Cyc

BC

dk/C

ycB

Cdk

CycB

CKICdh1

Wee1

G1

G2

M

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

cell mass

Wee

1 ac

tivity

wildwild--typetype

wee1wee1--

<30 60 90

120 150 180 210 240 270 300

period(min)

300<

cell physiologycell physiology

gene

tics

gene

tics

Two dimensional bifurcation diagram

HBSN SN

Cdk

CycB

CKICdh1

Wee1

SK

G1

G2

M

pheromone

Cdk

/Cyc

B

cell mass

G1S/G2

M0.4

G1 checkpoint

0.00 2 431 5

DNA damageUnreplicated DNA

0.0G1

S/G2

M0.4

G2 checkpoint

0 2 431 5

Cdk

/Cyc

B

cell mass

spindle defect

G1 S/G2

M

0.4

0.0

0.8

1.2

1.6

0 2 431 5

Metaphase checkpoint

Cdk

/Cyc

B

cell mass

Yeast homologues in our models

11

55

CK

ITFI

TFE

CycECyc E,A,B

CycA

CK

I

CycE CK

I

CycA

Cyc D,A,E Cyc A,B

Cyc D, E,A,B

Cdc20

CycB

APC-PAPC

TFB

Cdc14

Cdh1

Cdc14

CycB P

Cdc25Wee1

CycB

CK

I

PWee1

Cdc25

33

22

44

77 6688

1010

99

1111 1313

1212

Cyc D, E,A,B

CK

I

CycB

generic cell cycle model

This model Budding yeast

Fission yeast

Mammalian cells

CycB Clb2 Cdc13 CycB

CycA Clb5 Cig2 CycA

CycE Cln2 - CycE

CycD Cln3 Puc1 CycD

CKI Sic1 Rum1 Kip1

Cdh1 Cdh1 Ste9 Cdh1

Wee1 Swe1 Wee1 Wee1

Cdc25 Mih1 Cdc25 Cdc25c

Cdc20 Cdc20 Slp1 Cdc20

Cdc14 Cdc14 Clp1/Flp1 Cdc14

TFB Mcm1 - Mcm

TFE Swi4/Swi6 Cdc10 E2F

TFI Swi5 - -

IE APC APC APC

0 1 2 3 4

10 -3

10 -2

10 -1

10 0

10 1

G1

S/G2/M

cell mass

Act

ive

Cyc

B

Mammalian cells

Mammaliancell cycleengine

Cell-Cell attachment

Cell-matrix attachment

Growth factors

DNA damage

Apoptosis

DNA synthesis

Mitosis

Cell division

Survival factors

Death factors

The Dynamical PerspectiveThe Dynamical Perspective

Molecular MechanismMolecular Mechanism

??

??

??

Physiological PropertiesPhysiological Properties

The Dynamical PerspectiveThe Dynamical Perspective

Molecular MechanismMolecular Mechanism

Kinetic EquationsKinetic Equations

Vector FieldVector Field

Stable AttractorsStable Attractors

Physiological PropertiesPhysiological Properties

d CycBT

dt = k1 . M - (k2' + k2"

. Ste9 +k2'" . Slp1A) . CycBT

dSte9

dt = k3' .

1 - Ste9J3 + 1 - Ste9 - (k4'

. SK + k4 . CycB) .

Ste9J4 + Ste9

d Rum1T

dt = k11 - (k12 + k12' . SK + k12"

. CycB) . RUM1T

dSlp1A

dt = k7 . IE .

Slp1T - Slp1A

J7 + Slp1T - Slp1A - k8

. Slp1A

J8 + Slp1A - k6

. Slp1A

dMdt = µµµµ . M

References:References:

Béla Novak

Kathy Chen

Budapest University of Technology and Economics

John J. Tyson

Andrea Ciliberto Ákos Sveiczer

Acknowledgements:Acknowledgements: