Ricardo Borges, Àngel Calsina, Sílvia CuadradoRicardo Borges, Àngel Calsina, Sílvia CuadradoUniversitat Autònoma de BarcelonaUniversitat Autònoma de Barcelona

On a cyclin content structured cell population

model

OutlineOutline

IntroductionIntroduction The modelThe model Stationary solutions Stationary solutions Asymtoptic behavior:Asymtoptic behavior:

xx-independent solutions-independent solutions Numerical simulationsNumerical simulations

M. Gyllenberg, G. Webb M. Gyllenberg, G. Webb Age-size structure in Age-size structure in populations with quiescencepopulations with quiescence Math. Biosc. (1987) Math. Biosc. (1987)

““it is hypothesized that growth arrest states in the G1 it is hypothesized that growth arrest states in the G1 phase of the cell cycle are closely related to an phase of the cell cycle are closely related to an integrated control of cell prolliferation and integrated control of cell prolliferation and differentiation, and that cancer may result form defects differentiation, and that cancer may result form defects that uncouple this integrated system”that uncouple this integrated system”

M. Gyllenberg, G. Webb M. Gyllenberg, G. Webb A nonlinear structured population A nonlinear structured population model of tumor growth with quiescence.model of tumor growth with quiescence.

J. Math. Biol. (1990)J. Math. Biol. (1990)

““A realistic description of proliferation and quiescence in A realistic description of proliferation and quiescence in tumors require a structured model”tumors require a structured model”

In 2001, Leland H. Hartwell, R. Timothy Hunt, and Paul In 2001, Leland H. Hartwell, R. Timothy Hunt, and Paul M. Nurse won the Nobel Prize in Medicine and M. Nurse won the Nobel Prize in Medicine and Physiology for Physiology for their discoveries regarding cell cycle their discoveries regarding cell cycle regulation by cyclin and cyclin-dependent kinases.regulation by cyclin and cyclin-dependent kinases.

•F.B. Bricki, J. Clairambault, B. Ribba, B. Perthame, An age-and-cyclin-structured population model for healthy and tumoral tissues. J. Math. Biol. (2008)

“Unlimited tumor growth can be seen in particular as a deregulation of transitions between proliferative and quiescent compartments. Furthermore, recent measurements indicate that cyclins are the most determinant control molecules for phase transitions”

•M. Doumic, Analysis of a Population Model Structured by the cells molecular content. MMNP (2007)

•R. Borges, A. C., S. Cuadrado, Equilibria of a cyclin structured cell population model. To appear in DCDS

Cell cycleCell cycle

cyclin regulation of the transition to G0 phase

Q(x, t)

G1-S-G2-M

cell division

apoptosisapoptosis

P(x, t)

The modelThe model

evolution speed of cyclin content in a particular cell

)(xdtdx

cell division rate of cells with x

cyclin content production of cells with x cyclin content per unit time:

mx

xdyytP

yyF ),()(2

Cyclin dependent

“demobilisation” function

Density dependent recruitment function

dxxtQxxtPxtN mx 0 21 ),()(),()()(

),())((),()(),(

,),()(2),())((),()()(

)),()((),(

2

1

xtQdtNGxtPxLxtQt

dyytPyyFxtQtNGxtPdxFxL

xtPxx

xtPt

mx

x

The model:

),()()()(0

,)()(2)()()()()())()((0

2

1

xQdNGxPxL

dyyPyyFxQNGxPdxFxLxPx

dxd mx

x

dxxQxxPxN mx 0 21 )()()()(

Steady states: P(x) and Q(x) such that

In abstract form:In abstract form:

GGdPxLPGNG ˆ))ˆˆ)(,ˆ(,()ˆ(

2

mxx

g

G

dyyPyyFxKP

xPdxFxLgd

dxPxdxdxPA

g

PKA

)()(2:)(

)()()())()((:)(

, positivefor where

,0ˆ)(

12

2

ˆ

))1,0((ˆ 1LP such that G in the range of function G andFind

and

)(:)( KAsgs g is a simple dominant eigenvalue with a corresponding positive eigenvector and it is the only eigenvalue with positive eigenvector. Moreover it is an increasing function of g.

Steady states:

GGdPxLPG ˆ))ˆˆ)(,ˆ(,(

2

))1,0((ˆ 1LP such that G

,0ˆ)( ˆ PKAG

in the range of G andFind

Under good hypothesis

Let us further assume

0)()0( 0 KAss

and healthy tissue!

amounts to population decrease when there is no

recruitment from the Q-stage

)()0(0 )0( KAss G amounts to population

increase when the population number is small

and

0)()ˆ(ˆ! ˆ KAsGsGG

that such

GNGN ˆ)ˆ(such that ˆ!

Then there is a unique (nontrivial) steady state given by

KAG ˆ

where P^ is the (positive) eigenvector corresponding to the 0 eigenvalue of the operator normalized in such a way that

NdxxPGd

xLxxmx ˆ)(ˆˆ

)()()(02

21

GdPxLP ˆ2

ˆ)(,ˆ

Asymptotic behaviorAsymptotic behaviorA particular case with A particular case with x-x-independent solutions: normalize independent solutions: normalize xxmm to 1 and taketo 1 and take

),,())((),(),(

),,())((),(34),(

32

)),()1((),(

20

1

10

xtQdtNGxtPLxtQ

xtQtNGdyytPxtPdxL

xtPxxxtP

t

x

xt

dxxtQxxtPxtN 10 21 ),()(),()()(

)(),( tQtP

10 2

10 120

10

)()()()()(),()))((()()(

)())(()()3/1()(

tQdxxtPdxxtNtQtNGdtPLtQ

tQtNGtPdLtP

which satisfy the o.d.e. system

xxFLxLxxx32)(,)(),1()( 0

has solutions

Phase portrait for the Phase portrait for the x-x-independent independent solutionssolutions

021 )/)0(1)(3/1( LdGd extinction

)/)0(1)(3/1(3/1 2101 dGdLd a unique nontrivial steady state

stable, but…. Hopf bifurcation

Phase portrait for the Phase portrait for the x-x-independent independent solutionssolutions

10 3/1 dL unbounded solutions

Phase portrait for the Phase portrait for the x-x-independent independent solutionssolutions

Numerical simulations in the general Numerical simulations in the general casecaseT. Kostova, T. Kostova, An explicit third-order numerical method for size-An explicit third-order numerical method for size-structured population equationsstructured population equations, Num. Methods in PDE (2003), Num. Methods in PDE (2003)

we have non local terms and moreover, the principal part Pt+(ΓP)x is such that Γ (x) vanishes at the ends of the domain.

An explicit numerical method based on integrationalong characteristics lines. A non uniform rectangular grid with constant time step.(xi,tj) and (xi+1,tj+1) located on the same characteristic curve:

Extinction:Extinction:extinction.gifextinction.gif

Numerical simulations in the Numerical simulations in the general casegeneral case

stabilizationstabilization::

Numerical simulations in the Numerical simulations in the general casegeneral case

OscillationsOscillations::

Numerical simulations in the Numerical simulations in the general casegeneral case

Unbounded solutions:Unbounded solutions:UNBOUNDED.gifUNBOUNDED.gif

Numerical simulations in the Numerical simulations in the general casegeneral case

ConclusionsConclusions A simplification of a pre-existing model of two stages cell A simplification of a pre-existing model of two stages cell

tissue growth has been consideredtissue growth has been considered

An analytical proof of existence and uniqueness of steady An analytical proof of existence and uniqueness of steady state for suitable density dependent transition rate from state for suitable density dependent transition rate from quiescent to proliferating stagequiescent to proliferating stage

Convergence to the steady state; but also selfsustained Convergence to the steady state; but also selfsustained oscillations of the populations (even in the case of healthy oscillations of the populations (even in the case of healthy tissue) if the transition rate is large and very sensitive to tissue) if the transition rate is large and very sensitive to changes in the populationschanges in the populations

Numerical simulations corroborating the resultsNumerical simulations corroborating the results