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tissues
Fadia Bekkal Brikci, Jean Clairambault, Benjamin Ribba, Benoît Perthame
To cite this version:
Fadia Bekkal Brikci, Jean Clairambault, Benjamin Ribba, Benoît Perthame. An age-and-cyclin- structured cell population model with proliferation and quiescence for healthy and tumoral tissues.
[Research Report] RR-5941, INRIA. 2006. �inria-00081301v2�
inria-00081301, version 2 - 29 Jun 2006
a p p o r t
d e r e c h e r c h e
N0249-6399ISRNINRIA/RR--5941--FR+ENG
Thème NUM
An age-and-cyclin-structured cell population model with proliferation and quiescence for healthy and
tumoral tissues
Fadia Bekkal Brikci, Jean Clairambault, Benjamin Ribba and Benoît Perthame
N° 5941
Juin 2006
Unité de recherche INRIA Rocquencourt
Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)
Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30
healthy and tumoral tissues
Fadia Bekkal Briki
∗
, Jean Clairambault
†
, Benjamin Ribba
‡
and
Benoît Perthame
§
Thème NUM Systèmes numériques
Projet BANG
Rapportde reherhe n° 5941 Juin 2006 25pages
Abstrat: We present a nonlinear model of the dynamis of a ell popula-
tiondividedinaproliferativeandaquiesentompartments. Theproliferative
phaserepresentstheompleteelldivisionyle(G1−S−G2−M)ofapopula-
tionommittedtodivideatitsend. Themodelisstruturedbythetimespent
by a ellinthe proliferativephase, and by the amountof ylin D/(CDK4 or
6) omplexes. Cellsan transit fromone ompartment tothe other, following
transitionrules whih dieraording tothe tissue state: healthy or tumoral.
The asymptoti behaviour of solutions of the nonlinear model is analysed in
∗
Projet BANG, UR Roquenourt, Institut de Reherhe en Informatique et en Au-
tomatique,BP105,F78153Roquenourt,Frane,Fadia.Bekkal_Brikiinria.fr
†
Projet BANG, UR Roquenourt, Institut de Reherhe en Informatique et en Au-
tomatique, BP 105, F78153 Roquenourt and INSERM U 776 Rythmes biologiques et
aner, Hpital Paul-Brousse, 14, Av. Paul-Vaillant-Couturier, F94807 Villejuif, Frane,
Jean.Clairambaultinria.fr
‡
Institut de Médeine Théorique, Département de Pharmaologie Clinique EA3736,
FaultédeMédeineR.-T.-H.Laënne,UniversitéLyon1,RueParadin,F69376LyonCedex
08,Frane,brupl.univ-lyon1.fr
§
DépartementdeMathématiquesAppliquées,ÉoleNormaleSupérieure,45,rued'Ulm,
F75005ParisandProjetBANG,URRoquenourt,InstitutdeReherheenInformatique
etenAutomatique,BP105,F78153Roquenourt,Frane, Benoit.Perthameens.fr
both ases, exhibitingtissue homeostasis ortumour exponentialgrowth. The
model is simulated by numerial solutions whih onrm its theoretial pre-
ditions.
Key-words: Cellyle, Populationdynamis,PartialDierentialEquations,
Cylins, Tumour growth,Caner.
quiesene pour des tissus sains et tumoraux
Résumé : Nous présentons un modèle non linéaire de la dynamique d'une
populationellulairediviséeenunompartimentproliférantetunompartiment
quiesent. Laphasede proliférationreprésentel'ensembledu yle de division
ellulaire(G1−S−G2−M) d'unepopulationdeellulesappelées àsediviser
ennde yle. Lemodèleest struturéparletempspasséparuneelluledans
laphasede proliférationetpar laquantité de omplexesylineD/(CDK4 ou
6). Lesellulespeuventpasserd'unompartimentàl'autre,ensuivantdes lois
detransitionquidièrentsuivantqueletissuqu'ellesonstituentestdenature
saine ou tumorale. Le omportement asymptotique des solutions du modèle
nonlinéaireest analysédanslesdeuxas, etmontreunehoméostasietissulaire
danslepremier,etuneroissaneexponentielle(tumorale)dansleseond. Des
simulationsnumériques du modèle onrment ses préditions théoriques.
Mots-lés : Cyle ellulaire, Dynamique des populations, Équations aux
dérivées partielles,Cylines, Croissane tumorale, Caner.
Contents
1 Introdution 5
2 Moleular mehanisms involved in the G1 phase 6
3 Physiologially strutured model 7
3.1 Unequal division . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Transitionontrolbetween proliferationand quiesene . . . . . 10
4 Analysis and qualitative behaviour 12
4.1 Linear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Healthy tissue: Non-extintion (a prioriboundfrom below) . . . 14
4.3 Healthy tissue: Limited growth (apriori bound from above) . . 15
4.4 Tumoraltissue: Unlimitedgrowth . . . . . . . . . . . . . . . . . 16
4.5 Steady state forhealthy tissue . . . . . . . . . . . . . . . . . . . 17
5 Numerial simulations 19
6 Disussion and onlusion 21
1 Introdution
Living tissues, subjet to renewal, are onstituted of two dierent ategories
of ells: the proliferating ells (p) and the quiesent ells (q). Proliferating ells grow and divide, givingbirth at the end of the ellyle tonew ells, or
else transit tothe quiesent ompartment(often referred toas the G0 phase),
whereas quiesent ells donot grownor divide but eithertransit tothe prolif-
erative ompartment orelse stay in G0 and eventually dierentiate aording tothe tissuetype.
Inatumourellpopulationthe numberofproliferatingellsinreases on-
tinuously aslong as itis malignantand ative,whereas in a normal(healthy)
ell population, the size of the proliferative ompartment remains bounded
sine the total number of ells, proliferating and quiesent, remains onstant
(atleast inthe mean,e.g. by averagingover24hours)soastomaintaintissue
homeostasis.
Duringtherstphase(oftenreferredtoasG1)oftheproliferatingellyle,
until the restrition point (R) in late G1 has been reahed, proliferating ells may enter the quiesent G0 phase and stopproliferation. Indeed, experiments
byZetterbergandLarsson[10,34℄showedthattherestritionpoint(R)divides
the G1 phase into two parts: before R, ells may enter the quiesent phase,
but one it has been passed, they are ommitted to proeed along the other
phases(S,G
2,M, whih willnot be onsideredhere as suh) untilelldivision.
The swithing of ells between quiesene and proliferation depends on
extraellularenvironmentalonditions suhasgrowth and antigrowth fators,
and is regulated dierently in normal and tumour ells, due to dierenes in
the expression of the involved genes.
Cellpopulationmodelswithproliferativeandquiesentompartmentshave
beeninvestigatedbyauthorswhostudiedtheasynhronousexponentialgrowth
property [3, 14, 15, 28℄. Our goal here is to design a generi ell population
modelappliable toboth anerand normaltissue growth.
Unlimitedtumourgrowth, byoppositiontohealthytissuehomeostasis,an
beseen inpartiularasaderegulationof transitionsbetweenproliferativeand
quiesentompartments. Furthermore,reentmeasurements[16℄indiatethat
ylins are the most determinant ontrolmoleules forphase transitions.
For these reasons, and sine we are interested in studying in parallel the
behaviourofhealthy and tumourells, westruture ourellpopulationmodel
inage andylin ontent, aproess whihwedesribestep by stepinsetions
2and3. Insetion4,weanalysethetheoretialpropertiesofthe model,whih
we illustrate by numerial simulations in setion 5. Finally, some omments
and future prospets are briey developed insetion 6.
2 Moleular mehanisms involved in the G1 phase
A variety of proteins are produed during the proliferative ell yle. The
progressionof aellthrough the yle is ontrolled by omplexesomposed of
two proteins: a ylin (strutural protein) and a ylin dependent kinase (or
CDK),anenzymewhihisneededforthe ylintoativate. Eahphaseofthe
ellyle has spei ylin/CDK omplexes. In partiular, ylin D/(CDK4
or 6) and ylin E/CDK2 ativate duringthe G1 phase. Cylin D isthe rst
ylin whihis synthesized at the beginning of the ellyle. The level of y-
linD isontrolledbythe extraellularenvironment. Thus, ylin D synthesis
isindued by spei growth fators (GFs)[5℄, and its level deays when ells
are deprived of GFs. GFs bind to spei reeptors onthe external ytoplas-
mimembrane,stimulatinganintraellularsignallingpathway(Ras/Raf/Map
kinase) by means of whih ylin D is eventually synthesized (see [2, 4, 30℄,
for more details). Experiments reported in [17, 35℄ show the important role
of ylin D as a regulator of the transition between G1 and G0. They show
that a redued exit from G1 to G0 ours when ylin D is overexpressed, whereas non overexpressing ells remain in G0. Progression through the re-
strition point (R) is essentially related to ylin D level inasmuh as when thereisasuient amountofylin D,ells pass therestrition pointand are
ommitted to proeed through the rest of the ell yle. Moreover, ylin D
makesomplexeswitheitherCDK4 orCDK6 kinasesandthese omplexesare
able to phosphorylate other proteins whih are importantfor ellprogression
in the G1 phase through the restrition point and further for the rest of the
elldivision yle: DNA repliation, mitosisand elldivision.
In this paper, we are interested in the moleular interations that are re-
latedto the ativityof the ylin D/(CDK4or 6) omplexes. Several authors
[24,25,26, 32℄havedesribed andsimulated,underspei assumptions,part
of these omplex reations. Here, we give a simple modelto desribe the a-
tivity of ylin D/(CDK4 or 6) indued by growth fators.
Let x be the amount of the omplexes ylin D/(CDK4 or 6) and w an
aggregatedvariablerepresenting the amountofthe various moleulesinvolved
inthesynthesisofylinDsuhasRas/Raf. Weonsiderxandwasregulating
variablesinasimplenonlinearsystemofordinarydierentialequations(ODEs)
with respet to age a in the G1 phase. Synthesis of x ours at a rate c1 and
its degradation at a rate c2; we assume that the synthesis of w is indued by
growthfators (GFs)ataonstantratec3,itsdegradationourring atarate
c4. The ODE modelan thus be writtenas follows:
dx
da =c1 x
1 +xw−c2x, x(0) =x0 >0, dw
da =c3−c4w, w(0) =w0 >0.
(1)
Theonlynonlinearityofthe modelisloatedinthe termc1
x
1 +x representing a positively autoregulating oeient with saturation for x under the linear
inuene of the lumped variable w. Substituting the solution of the seond equationof (1), we an redue(1) toone equation inx:
dx da =c1
x 1 +x
c3
c4
+e−c4a(w0−c3
c4
)
−c2x, x(0) =x0. (2)
This holds only for the G1 phase sine we assume that ylin amount x and
agearemainonstantinG0 phase. Anaturalquantityarisesinthequalitative analysis of (2), the x-nullline:
X(a) = c1
c2
c3
c4
+e−c4a(w0− c3
c4
)
−1.
We assume that w0 ≤ c3
c4
and c1c3 > c2c4 whih is a way to express that the
lumped variable w is inreasing from its initial to its asymptoti value, and
that in the early G1 phase the overall synthesis of the hemials involved in
the progression of the G1 phase overomes their degradation. Therefore, a fundamental property of equation (2) is that the ylin onentration x is
limitedby:
xmax= c1c3
c2c4
−1>0. (3)
We keep this simple model for our next purpose whih is to desribe a
populationof ells, in proliferativeor quiesent state.
3 Physiologially strutured model
We onsider here only twophases: a quiesent one (physiologiallyG0)and a
proliferativeone (physiologially G1−S−G2−M). The ellpopulations we study are rstly strutured by the time spent inside the proliferative phase.
Thisphaserepresentsherethe ompleteelldivisionyle sineellbirth,and
thistimeinthephasewillhereafterbereferredtoasa,forphysiologialage in the yle. As proposed in[6, 33℄, we alsostruture the model by the amount
of ylin D/(CDK4 or 6) omplexes, denoted by variable x. Indeed, as men-
tioned earlier, this biologial quantity is the most important determinant of
progression up tothe restrition pointR inthe late G1 phase.
Letp(t, a, x)andq(t, a, x)berespetivelythe densitiesofproliferatingand quiesent ells with age a and ontent x inylin D/(CDK4 or 6) omplexes
attime t.
We also onsider a total weighted population, i.e., an eetive population
density, N dened by:
N(t) =
+∞
Z
0 +∞
Z
0
ϕ∗(a, x)p(t, a, x) +ψ∗(a, x)q(t, a, x)
dadx. (4)
Heretheweightsϕ∗andψ∗representenvironmentalfatorssuhasgrowthand anti-growth fators ating on the populations of proliferating and quiesent
ells, respetively. N is the density of the fration of the total population
onsisting inthe ells that are sensitivetothese fators and are thusqualied
toinuene,e.g. by amehanism apparented todensity inhibition,the G0/G1
transition. This exludes forinstane apoptotior pre-apoptotiells.
Exitsfromthe quiesent ompartmentare dueeithertoapoptosis (physio-
logialelldeath)ataratedortotransitiontotheproliferativephaseaord- ingtoa reruitment orgetting in funtion G, whih isassumed todepend
onthe total weighted populationN. We alsoassume that ells may leave the
proliferative ompartment for the quiesent one aording to a demobilisa-
tion or leak funtion L(a, x). These funtions L and G, whih represent
the ore mehanism of exhange between proliferation and quiesene in our
model, willbe desribed insetion 3.2. The modelmay thus be writtenas:
∂
∂tp(t, a, x) + ∂
∂a(Γ0p(t, a, x)) + ∂
∂x(Γ1(a, x)p(t, a, x)) =
−(L(a, x) +F(a, x) +d1)p(t, a, x) +G(N(t))q(t, a, x),
∂
∂tq(t, a, x) =L(a, x)p(t, a, x)−(G(N(t)) +d2)q(t, a, x).
(5)
Quiesentellsareassumedtobehaltedintheirindividualphysiologialevolu-
tion,inthesensethatoneaellbeomesquiesent,itsageandylin ontent
are xed at their last values as belongingto a proliferative ell. In this way,
quiesent ells donot age and donot hange their ylin ontent.
The parameter Γ0 denotes the evolution speed of physiologial age a with
respettotimet,whihisassumedtobeonstantinthismodel;ifforexample Γ0 = 0.5,itmeansthat physiologialagea evolves twieasslowlyasreal time t. Similarly,thefuntionΓ1 representstheevolutionspeedofylin D/(CDK4
or 6) with respet to time, i.e., Γ0 times the speed
dx
da of x with respet to
physiologialage a,whih isgiven by equation(2), with w1 =w0− c3
c4 <0: dx
da = Γ1(a, x) Γ0
=c1
x 1 +x
c3 c4
+e−c4aw1
−c2x.
The parameters d1, d2 are apoptosis rates for proliferatingand quiesent ells respetively, and F(a, x) is the fration of ells whih leave the proliferative populationtodivide aording toa proess whih willbedesribed later.
To ompletethe desription ofthe model(5), wespeify initialonditions:
p(0, a, x) =pi(a, x), a≥ 0, x≥0, (6)
and
q(0, a, x) =qi(a, x), a≥0, x≥0, (7)
where pi and qi are nonnegative funtions.
Inthe followingsetion,wedesribeaonditionforenteringthe proliferat-
ingphase(physiologiallyinG1)atagea= 0,butnotethat nosuhondition
is needed at x= 0, sine ylin level x= 0 is never reahed in the proess
desribed by (2) beause Γ1 vanishes at x= 0.
3.1 Unequal division
The distributionof the ellularmaterialbetween daughter ells isassumed to
beunequal. Duetovariabilityinylinontentbetweenthetwodaughterells
when division ours (see [18℄, and [31℄ for a relationwith ageing), some ells
mayinheritabiggeramountofertainproteinsasylins,whereasothersstart
the yle with a smaller amount of the same proteins. We onsider that the
distributionoftheamountofylin D/(CDK4or 6)betweenthetwodaughter
ellsis given by aonditionaldensity f(a, x, y)suh thatthe probabilityfora daughter ell, born from a mother ell with ontent y in ylin D/(CDK4 or
6) with x1 ≤y≤x2, tohave itselfontent x inylin D is:
x2
Z
x1
f(a, y, x)dy
+∞
Z
0
f(a, y, x)dy .
We also onsider that all newborn ells are at birth in the proliferative om-
partment. Then we have the following ondition atthe boundarya = 0, p(t,0, x) = 2
Γ0 +∞
Z
0 +∞
Z
0
f(a, x, y)p(t, a, y)dady. (8)
The following onditions followfromthe above interpretation:
(1)The ylinamountofadaughter ellissmallerthan thatofitsmotherell
atthe timeof division:
f(a, x, y) = 0 if x > y.
(2)Theylin amounty ofthe motherellisexatlyonserved andshared by
the two daughters
f(a, x, y) = f(a, y−x, y)
and
Z +∞ 0
f(a, x, y)dx=F (a, y),
whereF(a, y) isthe fration of ellswhihat age a andylin ontent y leave
the proliferatingphaseto undergo elldivision. Theseells disappear and are
replaed by two daughter ells whih immediately restart in the proliferative
phase fortheir own part.
Wehoose for F astandard Hill funtion:
F (a, y) = k1yγ1
k2γ1 +yγ11l[A∗,+∞[(a),
where k1 is the maximum eet of ylin D on elldivision , k2 is the ylin
ontent yielding its half-maximum eet, γ1 is the Hill oeient tuning the
steepnessofthe swithat y =k2 between 0andk1 fortheeet, andA∗ isthe
minimalellyle duration;we alsoonsider thatylin repartitionisuniform
afterdivision:
f(a, x, y) = F (a, y)
y 1l[0,y](x).
3.2 Transition ontrol between proliferation and quies-
ene
Lynh [19℄ has studied the eet of a transription fator that inhibits the
proliferationof humanolon anerells by reduingylin D geneexpression
and hene induing an aumulation of ells in G0. Deprivation of growth fators (GFs) inthe early G1 phase alsoleads to a low ylin D level in ells,
when ylin D/CDK4 is the only ylin/CDK omplex present, and the low
level of ylin D is suh that ells exitG1 toenter the G0 phase.
