Introduction to cell cycle modeling Attila Csikász-Nagy
G1 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 DNA replication M (mitosis) G2
G1 S cell cycle engine DNA replication M (mitosis) G2
G1 Cyclindependent kinase S Cyclin DNA replication B-type cyclin M (mitosis) G2
Cyclin dependent protein kinase cyclin protein + A P P P protein + A P P P cyclin A P P P
G1 S Cyclin DNA replication M (mitosis) G2/M G2
chromosome cycle - DNA replication - mitosis (precise doubling and halving) growth cycle - cell growth - cell division (approx. doubling and halving)
Cell cycle engine T C T D Cytoplasmic growth 4 Size control 3 T C > T D Cytoplasmic mass 2 1 T C = T D exponential balanced growth T C < T D 0 0 1 2 3 4
Uncoupling of the growth and chromosomal cycles oogenesis embryogenesis
growing (somatic) cells early embryos
Modeling cell cycle regulation I. Embryonic cycles
Yoshio Masui
Cell cycle remodeling begins at the MBT. rapid cleavage divisions no gap phases no checkpoints no apoptosis apoptotic checkpoint gap phases cell cycle inhibitors DNA replication and damage checkpoints
The negative feedback loop Cdc20 d = k 1 (k 2 + k 2. Cdc20 A ) IE 9 10 IE-P 7 8 d IE-P = k 9.. 1 - IE-P J 9 + 1 - IE-P - k 10. IE-P J 10 + IE-P 2 +APC d Cdc20 A = k 7. IE-P. 1 - Cdc20 A J 7 + 1 - Cdc20 A k 8. Cdc20 A J 8 + Cdc20 A 1 AA
Early embryonic cell cycles 0.5 1 / IE-P IE-P & Cdc20 0.0 0 50 100 0
John J. Tyson Béla Novak
MPF is regulated by positive and negative feedback loops P Wee1 + IE + IEP P Cdc20 OFF P Cyclin P Wee1 Cdc25 Cyclin ON + degraded cyclin Cdc25
Direct negative feedback: NO oscillations Delayed negative feedback: sinusoid oscillations Positive & negative feedbacks: relaxation oscillations
1 3 2 1 Cyclin k Cyclin k k d Cyclin = 1 ) ( 3 25 2 Cyclin k PYT k YT k k k MPF k d YT cak wee pp + + + + = ) ( 2 25 PYTP k PYT k k k YT k PYT d pp cak wee + + + = ) ( 2 25 PYT k PYTP k k k MPF k PYTP d cak pp wee + + + = ) ( 25 2 PYTP k MPF k k k YT k MPF d wee pp cak + + + = 25 25 25 25 ) 25 25 ( 25 P Cdc K P Cdc PPase k P Cdc Cdc total K P Cdc total Cdc MPF k P Cdc d b b a a + + = 1 1 1 1 ) 1 1 ( 1 P Wee K P Wee PPase k P Wee total Wee K P Wee totalwee MPF k P d Wee f f e e + + = ) ( IEP K IEP PPase k IEP IE total K IEP IE total MPF k IEP d h h g g + + = * * * *) ( * APC K APC IE Anti k APC APC total K APC APC total IEP k APC d d d c c + + = 25 ) 25 25 ( " 25 ' 25 25 P Cdc V P Cdc Cdc total V k + = ) 1 1 ( 1 " ' P Wee total Wee V P Wee V k wee wee wee + = * *) ( " 2 ' 2 2 APC V APC APC total V k + = Differential Equations in Novak-Tyson Model
Mathematical simulations generate oscillations in the model of Xenopus extracts 30 total Cyclin 20 P Cyclin Cyclin 10 0 0 60 120 180 time (min)
Cyclin threshold to induce mitosis Marc Solomon
Prediction: The threshold concentration of cyclin B required to activate MPF is higher than the threshold concentration required to inactivate MPF. hysteretic non-hysteretic MPF activity MPF activity T i T a T cyclin level cyclin level
Jill Sible Joe Pomerening
+Cdc2AF +Wee1-OP11 Joe Pomerening
Modeling cell cycle regulation II. Somatic cycles
G1 S Cyclin DNA replication M (mitosis) G2/M G2
The Cell Division Cycle 2 mass growth 1 2 DNA nucleus - - division 1 G1 S G2 M
Stabilizing the G1 phase AA 1 2 Mass +APC 4 3 positive feedback (double-negative) Cdh1 d d Cdh1 = k 1. M - ( k 2 + k 2. Cdh1 ). = k 3. (1 - Cdh1) J 3 + 1 - Cdh1 - (k 4 + k 4. ) Cdh1 J 4 + Cdh1
G1 d = k. 1 M - ( k 2 + k 2. Cdh1 ). d Cdh1 = k 3. (1 - Cdh1) J 3 + 1 - Cdh1 - (k 4 + k 4. ) Cdh1 J 4 + Cdh1 Cdh1 S/M
Nullclines G1 nullcline: k. 1 M = v 2 ' + v 2 ". Cdh1 Cdh1 Cdh1 nullcline: = k 3' (1 - Cdh1). (J 4 + Cdh1) (J 3 + 1 - Cdh1) k 4 ". Cdh1 - k 4' k 4 " S/M
G1 Cdh1 Cdh1 starter kinase cell growth S/M S/M Cdh1 S/M
Turning on/off the - Cdh1 switch AA mass +APC Cdc14 TF-P mass TF Cln Cdh1 degraded Cln
d Cdh1 = (k 3 ' + k 3 ". Cdc20 A ) 1 - Cdh1 J 3 + 1 - Cdh1 - (k 4 '. SK + k 4. ) Cdh1 J 4 + Cdh1 d SK = k 13 ' + k 13 ". TF - k 14. SK d TF = k 15 '. M 1 - TF J 15 + 1 - TF - (k 16 ' + k 16. ) TF J 16 + TF AA 1 +APC 2 4 Cdc14 3 TF-P mass 15 16 TF 13 Cln 14 Cdh1 degraded Cln
IE 9 10 5 6 IE-P Cdc20 7 8 negative feedback AA 1 2 +APC d Cdc20 T = k 5 ' + k 5". 4 J 5 4 + 4 - k 6. Cdc20 T dcdc20 A = k 7. IE-P Cdc20 T -Cdc20 A J 7 + Cdc20 T - Cdc20 A - k 8. Cdc20 A J 8 + Cdc20 A - k 6. Cdc20 A d IE-P = k 9. 1 - IE-P J 9 + 1 - IEP - k 10. IE-P J 10 + IE-P
d = k. 1 M - (k 2 ' + k 2 ". Cdh1 + k 2 '". Cdc20 A ) d Cdh1 = (k 3 ' + k 3 ". Cdc20 A ) 1-Cdh1 J 3 +1-Cdh1 - (k 4 '. SK + k 4. ) Cdh1 J 4 + Cdh1 d SK = k 13 ' + k 13 ". TF - k 14. SK d TF = k 15 '. M 1 - TF J 15 + 1 - TF - (k 16 ' + k 16. ) TF J 16 + TF d Cdc20 T = k 5 ' + k 5". 4 J 5 4 + 4 - k 6. Cdc20 T dcdc20 A = k 7. IE-P Cdc20 T -Cdc20 A J 7 + Cdc20 T - Cdc20 A - k 8. Cdc20 A J 8 + Cdc20 A - k 6. Cdc20 A d IE-P = k 9. 1 - IE-P J 9 + 1 - IE-P - k 10. IE-P J 10 + IE-P
1.0 mass 0.5 2 Cdc20 total 0.5 IE-P 1 0 0.0 1 0.5 Cln 0.0 0 0 100 200 300 time (min)
Cdc20 IE IE-P Budding yeast +APC Cdh1 AA CKI CKI P TF-P mass TF SK Cln
WT CKI SK 2 0.5 1 sk cln - - 0.0 00.5 0 110 000 CKI 0 5 00.0 sk cln - - cki - - 0 1001 0 1 Cdh1 CKI 0.5 0.0 0 110 000 time (min) 0 Cln
What makes us believe that our set of equations and parameters are close to life? Test the model! Compare the results to life: What can be easy to measure? Length of different cell cycle phases Cell size at different transitions Not only wild type, but many mutants! One wild type parameter set. To simulate mutants use this as a basal parameter set. Current Budding yeast model 131 mutants Current Fission yeast model 59 mutants
Hysteresis in budding yeast cell cycle? CKI Cdh1 Cdc14 Clb Cln2 CKI Cdh1 Cln
S/G2/M Clb2/ activity Start Finish G1 A/B A + Cln2 B+Cdc14 Cln2 Cdc14 time
S/G2/M Clb2/ activity Neutral G1 A/B A + Cln2 B+Cdc14
Protocol to demonstrate hysteresis Genotype: cln1 cln2 cln3 GAL-CLN3 cdc14 ts Fred Cross Knockout all the G1cyclins Turn on CLN3 with galactose; turn off with glucose Temperaturesensitive allele of CDC14: on at 23 o C, off at 37 o C. Neutral conditions: glucose at 37 o C (no Cln s, no Cdc14)
Start with all cells in G1 R = raffinose G = galactose Make some Cln Standard for protein loading G1 cells Shift to neutral S/G2/M cells
P G2/M P V awee V i25 Wee1 Wee1 Cdc25 Cdc25 V iwee V a25 Schizosaccharomyces pombe P k wee TF-P k 25 mass AA 15 11 16 TF CKI 13 12 IE SK 10 9 5 6 Cdc20 +APC 2 4 3 Cdh1 CKI 14 IE-P P 7 8 G1/S Mitosis
2 cell mass 1 P 1 1 Wee1 0 0.5 M CKI G1 S/G2 M 0 0.05 CKI Cdh1 0 0 0 50 100 150 time (min) Fission yeast
2 cell mass 1 P Cdc2 Cdc13 1 1 Cdc2 Cdc13 Cdc2 Cdc13 Wee1 0 0.5 M Rum1 G1 S/G2 M 0 0.05 Rum1 Ste9 0 0 0 50 100 150 time (min) Fission yeast
Bifurcation diagram for cell cycle regulation (Signal response curve) Bifurcation parameter (signal): cell mass (reports environmental conditions) State variable (response): main controller of the system =
2 cell mass abscissa 1 P 1 ordinate 1 Wee1 0 0.5 M CKI G1 S/G2 M 0 0.05 CKI Cdh1 0 0 0 50 100 150 time (min) Fission yeast
P G2/M d T = k. 1 cell mass AA - (k 2 ' + k 2 ". Cdh1 +k 2 AA '". Cdc20 A ). Cdc20 T dpmpf = Pk Wee1. wee ( T - pmpf) - k. 25 pmpf - (k 2 ' + k 2 ". Cdh1 + k 2 '". Cdc20 A ). pmpf die delay = k 9. MPF. 1- IE J 9 +1- IE - k 10. IE J 10 + IE dcdc20 = k. 7 IE. 1 - Cdc20 J 7 + 1 - Cdc20 - k 8. Cdc20 J 8 + Cdc20 - k. 6 Cdc20 dcdh1 Wee1 = k 3 '. 1 - Cdh1 J +APC 3 + 1 - Cdh1 - (k 4'. SK + k. 4 MPF). Cdh1 J 4 + Cdh1 dcki T = k 11 Cdc25 - (k 12 + Pk 12 '. SK + k 12 ". MPF). CKI T dsk = k 13 ' + k 13 ". M - k. 14 SK dm = µ. M Cdc25 2 T CKI T Trimer = Cdh1 T + CKI T + K -1 eq + ( T + CKI T + K -1 eq ) 2-4 T CKI T CKI k wee = k wee ' + (k wee " - k wee '). GK(V awee, V iwee ' + V iwee ". MPF, J awee, J iwee ) k 25 = k 25 ' + (k 25 " - k 25 '). GK(V a25 ' + V a25 ". MPF, V i25, J a25, J i25 ) AA MPF = ( T - pmpf). ( T - Trimer) T CKI CKI P Start Finish
/ P G2/M 10 0 10-1 10-2 S/G2 P Wee1 Wee1 Cdc25 Cdc25 cell mass AA P CKI 0.1 2.0 cell mass M AA / CKI / 1 AA delay 10-5 0.1 10-1 10-2 10-3 10-4 +APC CKI Cdc20 cell mass 0.1 2.0 G1 Cdh1 P M M Start Finish 0.025 0.600 cell mass
2 mass / nuclei 1.6 1.5 stable s.s. unstable s.s. min/max 1 1 M G1 S/G2 M 1 / 0.5 0.4 0.3 M 0 0 0 50 100 150 time (min) 0.2 S/G2 0.1 G1 0 0 1 2 cell mass
Fission yeast wild type max 1e+0 unstable steady state (M) / 1e-1 1e-2 1e-3 stable steady state (S-G2) stable oscillation 1e-4 min 1e-5 stable steady state (G1) 0 1 2 3 4 cell mass
Wild type wee1 - mutant Paul Nurse Wee1 Wee1 CKI Cdh1 CKI Cdh1 SK SK
1e+0 wild type / 1e-1 1e-2 1e-3 1e-4 M 1e-5 0 1 2 3 4 wee1 - G2 Wee1 / 1e-1 1e-2 1e-3 G1 CKI Cdh1 1e-4 1e-5 0.0 0.5 1.0 1.5 2.0 cell mass
Two dimensional bifurcation diagram SN HB SN 1.0 genetics Wee1 activity 0.8 0.6 0.4 0.2 wee1 - wild-type <30 60 90 120 150 180 210 240 270 300 300< period (min) 0.0 0 1 2 3 4 5 6 cell mass cell physiology
spindle defect / 1.6 1.2 0.4 0.8 0.4 Metaphase G1 G2 checkpoint checkpoint M S/G2 G1 G1 cell mass M S/G2 S/G2 G1 0.0 0 1 2 3 4 5 M G1 CKI Cdh1 G2 Wee1 pheromone SK DNA damage Unreplicated DNA
Yeast homologues in our models
generic cell cycle model 1 Cdc20 2 APC APC-P Cdc14 Cyc D, E,A,B 4 3 TFB Cdc14 5 Wee1 Cdc25 P CKI P Wee1 Cdc25 Cdh1 CKI 8 7 6 TFI CKI Cyc D, E,A,B 9 CKI CycE TFE CKI CycA 12 Cyc E,A,B CycE 10 Cyc D,A,E CycA Cyc A,B 11 13
This model Budding yeast Fission yeast Clb2 Cdc13 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 Mammalian cells
Mammalian cells 10 1 S/G2/M 10 0 Active 10-1 10-2 G1 10-3 0 1 2 3 4 cell mass
Cell-matrix attachment Cell-Cell attachment Growth factors Survival factors Cell division Mitosis Mammalian cell cycle engine DNA synthesis DNA damage Apoptosis Death factors
The Dynamical Perspective Molecular Mechanism??? Physiological Properties
The Dynamical Perspective Molecular Mechanism Kinetic Equations Vector Field d T = k. 1 M - (k 2 ' + k 2 ". Ste9 +k2'". Slp1 A ). T dste9 = k 3 '. 1 - Ste9 J 3 + 1 - Ste9 - (k 4'. SK + k. 4 ). Ste9 J 4 + Ste9 d Rum1 T = k 11 - (k 12 + k 12 '. SK + k 12 ". ). RUM1 T dslp1 A = k. 7 IE. Slp1 T - Slp1 A - k J 7 + Slp1 T - Slp1. Slp1 A 8 - k A J 8 + Slp1. 6 Slp1 A A dm = µ. M Stable Attractors Physiological Properties
References:
Acknowledgements: Budapest University of Technology and Economics John J. Tyson Béla Novak Kathy Chen Andrea Ciliberto Ákos Sveiczer