cell cycle modeling - unitrento · bifurcation diagram for cell cycle bifurcation diagram for cell...
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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: