dynamical models for the cell cycle - concordia university

Dynamical modelsfor the cell cycle

W.Govaerts,C.Sonck

Introduction

Regulation of the cellcycle

The toy model ofTyson and Novak

Description

Equilibriumbifurcation diagram(k4 = 35)


HSN, HHS and NCHorbits

The budding yeastmodel of Tysonand Novak

Description

Bifurcation diagrams


The dynamic masscell model

A slow-fast system

Growth rate versusmass increase inS-G2-M phase

Robustness of themodel

Dynamical models for the cell cycle

W. Govaerts C. Sonck

Department of Applied Mathematics and Computer ScienceGhent University

Belgium

BAA Workshop, Montreal 2010


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Outline

IntroductionRegulation of the cell cycle

The toy model of Tyson and NovakDescriptionEquilibrium bifurcation diagram (k4 = 35)Equilibrium bifurcation diagram (k4 = 18)HSN, HHS and NCH orbits

The budding yeast model of Tyson and NovakDescriptionBifurcation diagramsHSN, HHS and NCH orbits

The dynamic mass cell modelA slow-fast systemGrowth rate versus mass increase in S-G2-M phaseRobustness of the model


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



The basic mechanisms of the cell cycle

I Cell cycle: 4 phases (G1, S, G2, M)

I Irreversible transitions (Start and Finish)

I Need for tight regulation


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



I Antagonistic relationship between cyclin/Cdk andCdh1/APC proteins, central components in the cellcycle.

I 2 stable steady states:I G1 state with high Cdh1/APC activity and low

cyclin/Cdk activityI S-G2-M state with high cyclin/Cdk activity and low

Cdh1/APC activity


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Models for the cell cycle

I 2 of the models proposed by J.J. Tyson and B. NovakI the toy modelI the budding yeast model

I ReferencesI Tyson J.J. and Novak B. (2001) Regulation of the

Eukaryotic Cell Cycle: Molecular Antagonism,Hysteresis, and Irreversible Transitions. J. theor. Biol210, 249-263.

I Fall C.P., Marland E.S., Wagner J.M. and Tyson J.J.,ed. (2002) Computational Cell Biology, Springer,Chapter 10.

I Computational results: MatCont


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



The toy model of Tyson and Novak

I Basic model for a fixed cell mass m:

dX

dt= k1 − (k ′

2 + k ′′2 Y )X ,

dY

dt=

(k ′3 + k ′′

3 A)(1− Y )

J3 + 1− Y− k4mXY

J4 + Y,

dA

dt= k ′

5 + k ′′5

(mX )n

Jn5 + (mX )n

− k6A

with X = [cyclin/Cdk], Y = [active Cdh1/APC] andA = [activator of Cdh1/APC at Finish]

I For growing mass: equationdm

dt= µm added

I µ growth parameter


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Equilibrium bifurcation diagram for the toymodel (k4 = 35)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

m

X

LP1

LP2

H

Figure: Equilibrium bifurcation diagram for the toy model (parametersk1 = 0.04, k ′2 = 0.04, k ′′2 = 1, k ′3 = 1, k ′′3 = 10, k4 = 35, J3 = 0.04,J4 = 0.04, k ′5 = 0.005, k ′′5 = 0.2, k6 = 0.1, J5 = 0.3, n = 4)


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



I 2 disjoint stable parts: before LP1, between LP2 and H

I Stable periodic orbits coming from the right andtending to a HSN orbit

I Time in S-G2-M phase tends to infinity as µ tends to 0

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

m

X

LP1

LP2

H

00.2

0.40.6

0.8

0

0.2

0.4

0.6

0.8

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X

LP2

H

Y

LP1

m

Figure: Intersection of the equilibrium bifurcation curve with the HSNorbit for the toy model in the (m,X )-plane and in the (X ,Y ,m)-space(for k4 = 35).


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Equilibrium bifurcation diagram for the toymodel (k4 = 18)

I In domain relevant for interpretationI stable equilibria before Hopf point H1I other parts unstable

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

m

X

H1 LP1

LP2

H2

Figure: Equilibrium bifurcation diagram for the toy model (sameparametervalues as before, except k4 = 18)


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



I Stable periodic orbits coming form the right andtending to a HHS orbit

I Time in S-G2-M phase is a priori limited (independentof µ)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

m

X

H1LP1

LP2

H2

1.61 1.612 1.614 1.616 1.618 1.62 1.622 1.6240.045

0.0455

0.046

0.0465

0.047

0.0475

0.048

0.0485

0.049

0.0495

0.05

m

XH1

LP1

Figure: Intersection of the equilibrium bifurcation curve with the HHSorbit for the toy model in the (X ,A)-plane (for k4 = 18).


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



HSN, HHS and NCH orbitsI Large stable periodic orbits can end either in a HHS or

a HSN orbitI Transition: NCH orbit

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

X

A

NS

NCH

0.04 0.05 0.06 0.07 0.08 0.09 0.10.05

0.052

0.054

0.056

0.058

0.06

0.062

0.064

0.066

0.068

0.07

X

A

NS

NCH

Figure: A curve of HHS orbits tending to a NCH orbit for the toymodel in the (X ,A)-plane.


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



The budding yeast model of Tyson and Novak

d [CycB]T

dt= k1 − (k′2 + k′′2 [Cdh1] + k′′′2 [Cdc20]A)[CycB]T ,

d [Cdh1]

dt=

(k′3 + k′′3 [Cdc20]A)(1− [Cdh1])

J3 + 1− [Cdh1]−

(k4m[CycB] + k′4[SK ])[Cdh1]

J4 + [Cdh1],

d [Cdc20]T

dt= k′5 + k′′5

(m[CycB])n

Jn5 + (m[CycB])n

− k6[Cdc20]T ,

d [Cdc20]A

dt=

k7[IEP]([Cdc20]T − [Cdc20]A)

J7 + [Cdc20]T − [Cdc20]A−

k8[Mad ][Cdc20]A

J8 + [Cdc20]A− k6[Cdc20]A,

d [IEP]

dt= k9m[CycB](1− [IEP])− k10[IEP],

d [CKI ]T

dt= k11 − (k′12 + k′′12[SK ] + k′′′12 m[CycB])[CKI ]T ,

d [SK ]

dt= k′13 + k′′13[TF ]− k14[SK ],

d [TF ]

dt=

(k′15m + k′′15[SK ])(1− [TF ])

J15 + 1− [TF ]−

(k′16 + k′′16m[CycB])[TF ]

J16 + [TF ],

[CycB] = [CycB]T − [Trimer ],

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

Σ +√

Σ2 − 4[CycB]T [CKI ]T,

Σ = K−1eq + [CycB]T + [CKI ]T .


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Equilibrium bifurcation diagram

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m

Cyc

BT

H1

LP1

LP2

H2

H3

I supercritical Hopf points H1, H2 and H3

I 3 disjoint stable parts: before H1, between LP2 and H2,beyond H3


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Periodic orbit bifurcation diagram

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m

CycBTLPC

NS

LPC

LPC

H1

LP

LP

H2

H3

I Stable periodic orbits born at H1 are really short-livedand die at HHS orbit

I Stable periodic orbits born at H2 lose stability at LPCand return as unstable periodic orbits

I Stable periodic orbits born at H3I turn and become unstable at LPC1I turn again and become stable again at LPC2I die eventually at HHS orbit


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



The dynamic mass cell model

I Equationdm

dt= µm added to the budding yeast model

I With growing mass, different stages:I lower left branch of stable equilibria (growth in G1

phase)I loses stability at H1 for m = mREP = 0.6546307 (start

of DNA replication)I attracted by stable periodic orbits born at H3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m

CycB

T

LPC

LPC

H1

LP

LP

H2

H3

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.850

0.1

0.2

0.3

0.4

0.5

0.6

0.7

m

CycB

T


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



HSN, HHS and NCH orbits

I The large limit cycles (born at H3) can end either in aHHS or a HSN orbit dependent on parameter values

I analogue to the case of the toy model

I Time in S-G2-M phaseI either limited independently of growth rate µI either arbitrary large for µ sufficiently small


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



The slow-fast system

I For a large range of initial values of m and theconcentration variables: same behaviour of the initialsegment of the orbit

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

m

CycB

T

m(0)=0.428

m(0)=0.478

m(0)=0.528

Figure: Orbits starting from m = 0.428, 0.478 and 0.528.

I Slow-fast system with m as slow variable andconcentration variables as fast ones

I Computation of the cell cycle as a fixed point of a mappossible


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Growth rate versus mass increase in S-G2-Mphase

I Effect of change of growth rate µ on orbits

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

m

CycB

T

µ=0.005

µ=0.01

µ=0.0025

Figure: Growth of a cell for three different values of µ.

I all orbits first converge to quasi-steady state solutionsI when m > mREP : system starts oscillating with a

damped amplitude


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Growth rate versus mass increase in S-G2-Mphase

I mNEW : mass of a newborn cell.

I mREP : mass at onset of DNA replication.

I mDIV = value of m at which [CycB]T reaches its firstminimum value after mREP .

µ mREP mDIVmDIV−mREP

µ

0.02 0.6546307 1.5933184 46.93440.01 0.6546307 1.0724438 41.7813

0.005 0.6546307 0.8564426 40.36240.0025 0.6546307 0.7557679 40.4549

0.00125 0.6546307 0.7053451 40.57150.000625 0.6546307 0.6797931 40.2599

0.0003125 0.6546307 0.6674274 40.94970.00015625 0.6546307 0.6644945 63.1282

Table: Values of the growth rate µ with corresponding valuesmREP and mDIV , and ratio mDIV−mREP

µ .


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



I For values of µ between 0.0003125 and 0.01I ratio mDIV−mREP

µ nearly constant ≈ C = 40.73.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

µ

m1−m

0

Figure: Graphical interpretation of Table 1.


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Interpretation for mDIV −mREP ≈ C µ [1]

0.65 0.7 0.75 0.8 0.85 0.90

10

20

30

40

50

60

70

m

mT

(m)

Figure: Representation of mT (m) versus m for the large stable orbitsborn at H3.

I m in the range of the stable periodic orbits born at H3

I T (m) is the period of the large stable periodic orbit atm


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system




I mT (m) ≈ C1 ≈ 30 in this range of m except for m veryclose to the homoclinic orbit

Put

I ∆t = time between birth of cell and cell divisionI ∆t = ∆1t + ∆2t, with

I ∆1t = time between birth of cell and onset ofDNA-replication (time in G1 phase)

I ∆2t = time between onset of DNA-replication and celldivision (time in S-G2-M phase)


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Let the periodic orbits in the fast manifold during theS-G2-M phase be parameterized by a phase variableφ ∈ [0, 1]. Orbits in the neighborhood can be parameterizedaccordingly by identifying points on an isochron.

isochron

periodic orbit

orbit

t0

t1

t2

During a time dt a fraction dtT (m(t)) is traversed by the

isochron that contains the fast state variable. Let ρ be thetotal fraction of a periodic orbit traversed during S-G2-Mphase.


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system




We then have ∫ ∆2t

0

dt

T (m(t))= ρ∫ ∆2t

0m(t)

dt

m(t)T (m(t))= ρ

Taking into account that m(t)T (m(t)) ≈ C1, we conclude∫ ∆2t

0mREPeµtdt ≈ C1ρ

1

µmREP(eµ∆2t − 1) ≈ C1ρ

mDIV −mREP

µ≈ C1ρ


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system




I C ≈ C1ρ

I Total fraction of a periodic orbit traversed duringS-G2-M phase: ρ ≈ C

C1≈ 1.33 is constant, i.e.

independent of µ.I Argument fails if

I µ is very small: important part of the orbit in regionclose to homoclinic orbit, where mT (m) is large

I µ is large: important part of the orbit in region whereno periodic orbits exists


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Duration of the phases of the cell cycle

Put

I mNEW = mDIV /ν

I mDIV = mNEW eµ∆t = νmNEW

I eµ∆t = ν or ∆t =lnν

µ

I mDIV = mREPeµ∆2t ≈ mREP + Cµ

I ∆2t ≈1

µln

(1 +

Cµ

mREP

)I ∆1t ≈

lnν

µ− 1

µln

(1 +

Cµ

mREP

)


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



Robustness of the model

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m

CycBTLPC

NS

LPC

LPC

H1

LP

LP

H2

H3

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m

CycB

T

LPC

H1LP

LP

H2

H3

Figure: Bifurcation diagram for resp. k ′13 = 0 (as before) and k ′13 = 0.2.


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



I Different behaviour for change of parameter values

I 2-parameter study of the Hopf points and the LPCs

0.5 0.6 0.7 0.8 0.9 1 1.10

0.05

0.1

0.15

0.2

0.25

0.3

m

k13’

GH

BT

Figure: Continuation of resp. H2 (green curve), H1 (browncurve), LPC corresponding to H2 and first LPC corresponding toH3 (blue dotted line), second LPC corresponding to H3 (light bluedotted line) and H3 (red curve) in (m, k ′13)-space.

I H1 (brown curve): loss of stability of G1 phase


W.Govaerts,C.Sonck

Introduction



Description





Description




A slow-fast system



I For rising value of k ′13: domain of bistability of periodic

orbits becomes biggerI For example for k ′13 = 0.2: “small” stable periodic

orbits between green and red curve and “large” stableperiodic orbits to the left of dotted light blue curve

I Example of orbit that is attracted to “small” stableperiodic orbit for k ′13 = 0.2 and m = 0.65 (value slightlylarger than that of H1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

Cdh1

CycB

T

I Robustness of the model?

dynamical models for the cell cycle - concordia university

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