When a collective outcome triggers a rare individual event: a mode of metastatic process in a cell population

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When a collective outcome triggers a rare individual event: a mode of metastatic process in a cell population

Michel Malo, Amandine Cartier-Michaud, Elizabeth Fabre-Guillevin, Guillaume Hutzler, Franck Delaplace, Georgia Barlovatz-Meimon, Annick

Lesne

To cite this version:

Michel Malo, Amandine Cartier-Michaud, Elizabeth Fabre-Guillevin, Guillaume Hutzler, Franck De-

laplace, et al.. When a collective outcome triggers a rare individual event: a mode of metastatic

process in a cell population. 2009. �hal-00407546v2�

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event: a mode of metastati proess in a ell population

M.Malo

(a)

, A. Cartier-Mihaud

(a)

, E. Fabre-Guillevin

(a,b)

, G.Hutzler

(a)

, F. Delaplae

(a)

,

G. Barlovatz-Meimon

(a,c)

and A. Lesne

(d,e,∗)

July 27, 2009

(a)IBISCFRE3190CNRS-Université d'Evry,523Plae desTerrasses,91000Evry, Frane.

(b)HpitalGeorges Pompidou,20rue Leblan,75908 PARISCedex15,Frane.

()Université Paris12,61avenuedu GénéraldeGaulle,94010 Créteiledex,Frane.

(d)InstitutdesHautesÉtudesSientiques,35routedeChartres ,91440Bures-sur-Yvette,Frane.

(e)LPTMCUMR7600,Université PierreetMarieCurie,4plaeJussieu,75252Paris, Frane.

(*)Correspond ingauthor.

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of metastati proess in a ell population

Abstrat

Amodelofearlymetastatiproessisbasedontheexperimentallyassessedroleof

aprotein,PAI-1,whihathighenoughextraellularonentrationpromotesthetransi-

tionofanerellstoaphysiologialandmorphologialstatepronetomigration. This

transitionisdesribedatthesingleelllevelasabi-stableswithgeneriallyassoiate d

withasubritialbifuration. Inamultilevelreation-diusionsenario,themiroen-

vironment ofthetumoris modiedby theproliferatingellpopulationsoas topush

theonentration ofPAI-1abovethebifurationthreshold. Theformulationinterms

ofpartialdierentialequationsfailstoapturespatio-temporalheterogeneity.Cellular-

automataandagent-basedsimulationsofellpopulationssupportthehypothesisthat

arandomlyloaliz edaumulationof PAI-1anariseand triggertheesapeofafew

isolatedells. Awayfromtheprimarytumor,theseellsexperieneareversetransition

baktoaproliferativestateandouldgenerateaseondarytumor.Theproposedrole

ofPAI-1inontrollingthismetastatiylewouldpartlyexplainitswell-doumented

roleinanerprogression.

Runningtitle: Multilevelmetastatiesapemodel

Keywords: ell population, metastasti esape, multilevel modeling, reation-diusion,

agent-basedsimulation,multi-stability.

Abbreviation. PAI-1: Plasminogen-Ativator-Inhibitor-1(a protein,eitherintra-ellular

(subsript

i

^), ^soluble ⁱⁿ ^the extraellular spae (subsript

s

^), ^or matrix-bound (subsript

m

^)).

1 Introdution

Duetolethalonsequenesofmetastatispreadingofaner,understandingandontrolling

theproessesunderlyingtheformationofmetastasesisamajorhallenge,remaininglargely

open. Several modes of metastati spreading (letting aside surgial dissemination) were

identied: (i)transportinlymphatiirulation,(ii)transportinbloodirulation,and(iii)

amode involving aspei migration mehanism, the amoeboid migration (Friedl, 2004).

Thepresentpaperfousesonthislattermode. Takingplaeattheellsale,itappearsless

pervasive than the rst two ones, where irulation-failitated transport spans the whole

organism. However it is less dependent on the anatomialfeatures of the loation of the

tumorandisaandidatefortheearlyeventsofthemetastatispreading,beforemetastati

ellsreahthelymphatiorthebloodirulation. Itmightwellbeanessentialpreliminary

stepommontoallmetastatiproesses.

Adiultyomesfrom thefat thatearlyeventsinvolvedintheesapeofaanerell

fromtheprimary tumorarerareevents,toorareto beeasily observedor experimentedin

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tumors. Aordingly,experimentalprotoolsarerestritedtoindiretinvestigations,mainly

genetiandbiohemialanalysesofmetastatiellsomparedtothoseoftheprimarytumor

(Witz,2008),orstatistialtrakingofthenumber,loation,andgenetilineageofseondary

tumors(Albini,2008). Aninreasingnumberofexperimentsfousonthebiohemialanal-

ysisofthesurroundingmiroenvironment(Tayloretal.,2008),themorphologialsignature

of potentially metastati ells (Vinan et al., 2007) and the reprodution in vitro of the

epithelial-mesenhymatous transition andthe mesenhymatous-amoeboid transition whih

aetthemorphologyandtheproliferativeandmigratoryapaities(amoeboidmigration)

of ells of epithelial origin (Malo et al., 2006). These omplementary experiments have

shown that the metastati proess involvesjointly geneti determinants (aumulation of

spei mutations (Gerstung and Beerenwinkel in this issue)), biohemial fators (trig-

gering newpathways or swithing existing ones, leadingto modiationsin the ellstate

andmetabolism),andrequirementsaboutthestateandgeometryofthemiroenvironment

thatis,theextraellularspaeandmatrixofthetumorells. Howeverthereisnointe-

gratedunderstandingofthisproesssofar. Thediultyomesfromthedierentnatures,

loations,andtimesalesofthepotentialausalfators,andisstrenghtenedbytherarityof

metastatievents. Modelingisthenessentialtoartiulatethedierentpartialandindiret

experimentalresults,andtovalidatetheirinterpretationinanintegratedsenario,bridging

moleular,ellular,extraellular, andellpopulationlevels.

Frombiologialfatsaboutthemesenhymatous-amoeboidtransitionandamoeboidmi-

gration,wewish to explain how permissive onditionsfor themestastati esape of afew

aner ellsmight be olletivelyindued at the ell population level. Our laim is that

proliferation-indued modiations of the tumor miroenvironment ould produe a feed-

bak loalized in a few privileged individual ells, seleted by a omplex onjuntion of

stohastiand history-dependentmoleular events. Ourworkinghypothesis, supportedby

experimental results (Maloet al., 2006), is the entral role of the protein, PAI-1, synthe-

sized at high rate by the aner ells and released in their immediate environment. We

shall desribe how this moleuleould mediate an interplay betweenintra-ellular, extra-

ellular,andellpopulationfeatures,swithingafewellsintoastatepronetoesapeand

migration,thenswithing thembakinto aproliferativestateat adistane oftheprimary

tumor(thisistheproessofmetastasis). Thissenarioisrootedinageneridesriptionof

thesingleell statein bifuration theory, supported by in vitroexperiments. Its spatially

extended formulationattheellpopulationlevelis ahieved inareation-diusionmodel,

implementedeitherinthestandardframeworkofpartialdierentialequationsorintheone

ofellularautomata andagent-basedsimulations. The preditionsofourstudy motivated

andguidednewexperimentsprovingtheexisteneofthereversehangefromtheamoeboid

tothemesenhymatousstateandbakingupthedynaminatureofthistransition. Finally,

weresortto atastrophetheoryto suggestapossiblepath towardstheanerstage where

amoeboidstateandmigrationan beobserved.

2 Biologial Setting

Consideringepithelial ells, arst transition towards aanerousstate is observed, origi-

natingin aumulating mtuations and leadingto the so-alled mesenhymatous state(see

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thedestabilizationofepitheliumandpronetoproliferation. (right)Amoeboidstatehara-

terizedbyablebbingmorphologyand bymodiedadhesionleadingto aspeial migratory

ability. Themesenhymatous-amoeboid transition(as experienedbytherightmost ellin

theleft handsidepiture)islikelytoplayakeyroleinearlymetastatiesape.

Figure1left). Inthisstate,ell-ell juntions arenolonger establishedand theepithelium

isdestabilized. Thisstatehasmoreoverastrongproliferativeapaity,henethetransition

tothis mesenhymatousstateisgenerallyassoiatedwith theappearaneofawell-dened

tumor(Thiery,2002). Ininvasiveepithelialtumors,itisthedefaultstateoftheells(Gavert

andBen-Ze'ev,2008)and itwill bethedefaultstateoftheellpopulationinourmodel.

A seondtransition mayourtowardstheso-alledamoeboid state(seeFigure1right)

identied by a spei and persistent blebbing morphology (round shapewith dynami

atin ringsvisible atthe ellperiphery). Thismesenhymatous-amoeboidtransition is as-

soiatedwith ahangein adhesionproperties (more preisely, adhesionbeomes integrin-

independent, theatin ytoskeletonreorganizesandamodiationof oneof itsregulatory

pathways,RhoA-pathway,ours: itnowinvolvesanauxiliaryprotein,ROCK, whilethere

isnolongerproteolysisof theextraellularmatrix(Friedl, 2004;MCarthy,2009). Due to

itspeuliar features,ablebbingellanmovefastandprogressbyexploitingintersties of

thesubstratewithnoneedofmatrixproteolysis. Aordingly,amoeboidmigrationisavery

eientmodeofmigrationinatissue,enounteredinnormalonditionsduringsomedevel-

opmental stages (Thiery, 2002); in apathologialontext, itwas suggestedasaprivileged

modeofmetastatimigration(Friedl andWolf,2003;Berxetal.,2007).

In the mesenhymatous state, migration is quite ineient while proliferation is very

ative. Proliferation ismostly ontrolled byell density for obvioussteri reasons. Inthe

amoeboidstate,ellmigrationisveryeient. Roughly,prolif erati on oursathigh rate in

themesenhymatousstatewhereasmigrationoursmoreeientlyintheamoeboidstate,in

agreementwiththeurrentwisdomthatproliferationandmigrationaremutuallyexlusive

proessesin agivenell.

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ellmorphologyandthemesenhymatous-amoeboidtransition. Theproportionofblebbing

MDA-MB-231breastanerellsismeasuredatxedPAI-1onentration(values5,10,20,

40

µ

^g/m²^);

∗ p < 0, 05

^;

∗ ∗ p < 0, 01

^;

∗ ∗ ∗ p < 0, 001

^.

Reentobservationsinvivohintatakeyplayerinamoeboidmigration,metastases,and

moregenerally aner progression: the plasminogen-ativator-inhibitor protein of type 1,

heneforth termed by its aronym PAI-1. It is an ubiquitous speies involved in several

pathwaysandfuntions,among whih someaspetsarerelevantformetastati proess. It

is found in the surroundings of the most invasive tumors (Pedersen, 2005; Wilkins-Port

andHiggins, 2007;Wilkins-Portet al., 2007)and onsideredas amarkerof badprognosis

(Jänikeeal.,2001;Looketal.,2002;Castelloetal.,2007;Biermannetal.,2008). PAI-1is

enounteredunder severalforms: thenewlysynthesizedmoleulein theell( internal PAI-

1),asasolubleformintheextraellularmedium( solublePAI-1 ). Thislatterformaneither

diuseintheextraellularmedium, orbindtotheextraellularmatrix( matrix-boundPAI-

1),or betrappedontheellsurfaeanddeativated,orbeinternalizedanddegradedwith

nofurther known onsequeneonthestate ofthe ell. Onthe ontrary,internalization of

matrix-boundPAI-1oursthroughtheformationofatripartiteomplexwithamembrane

reeptor,uPAR,andamoleule,uPA.Itsroleinmodifyingtheellphysiology(speially,

inmodifyingtheativityofRhoApathway)isaknowledged(Chazaudet al.,2000). Strik-

ingly,whenanerellsareplaedonartiialsubstrateswithhighonentration

c

m

> c

^∗_m

of matrix-bound PAI-1, they experiene the above-mentioned mesenhymatous-amoeboid

transition(Maloet al.,2006). Inthisrespet,matrix-boundPAI-1anpromoteanerell

migration, at leastin vitro(Friedland Wolf, 2003). Moreover,these experimental results,

presentedonFigure2,indiatethatthemesenhymatous-amoeboidtransitionisnotdueto

somemutations butis ratheradynamitransition betweentwodierentstatesoftheell,

ontrolledbyitsenvironment.

Canerells synthesize morePAI-1 thannormalells do;PAI-1 moleules are thense-

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ary regions, where the matrix is not fully oupied by ells. Hene, onentration of the

matrix-boundPAI-1intheborderregionofthetumorisexpetedtobehigherthanaround

normalells. This was observedexperimentally(Look et al., 2002; Chazaud et al., 2002).

CanerellsalsoproduemoreuPAandhaveaninreasednumberofuPAmembranereep-

torsuPAR, diretly involvedin the internalizationof matrix-boundPAI-1. Thenetresult

isaninreasedinternalization ux

J

i ⁱⁿ âner êlls, ^heneânâmpliationôf^theênsuing

ellmetaboli andmorphologialhangesomparedto normalells.

Fromtheseexperimental fats,weshall startfrom thefat thatgivenanerell,with

regards to its metastati potentialities, an be in two dierent states, the mesenhyma-

tousandthe amoeboid one. Weadopt theleadingpattern aordingto whih theabrupt

mesenhymatous-amoeboid transition of a ell is ontrolled by its internalization ux of

matrix-boundPAI-1. Theoupleddynamisofseveralellsandextraellularmediumhave

then to be onsidered. Indeed, proliferation in mesenhymatous state turns a single ell

intoanaggregateolletivelyontributingtotheonentrationofmatrix-boundPAI-1. We

shallthereforeembedthedynamisofasingleellinaspatiallyextendedpopulationmodel,

fousing on the spatio-temporal varying internalization ux of the matrix-bound PAI-1,

onsideredasamarkerofthemetastatipotentialitiesoftheells.

3 Model

3.1 Challenges

Let us rst avoid a possible misunderstanding: what we term amodel is a way to hek

the onsisteny of several experimental fats and partial mehanisms (possibly ouring

at dierent levels), to put forward ontrol parameters, thresholds, swith behaviors and

quantitativehintson themajordeterminantsof thefate ofthe system. It should allowus

toderivetestableonsequenes ofthehypothesesand suggestnovel protools. A model is

aneessaryintermediarystepbetweenqualitativeunderstanding andfurther experimental

evideneandanbeseenasawayofhypothesistesting. .

Theguidelineof ourinvestigationsis themultileveland intriatemehanismsbywhih

PAI-1 ouldplayarolein thefate ofanerellsand inthemetastati proess. As shown

inSetion2,thesemehanismshavefuntional onsequenesatfourdierentlevels:

•

^at ^the ^moleular (intra-ellular) level : internalization of matrix-bound PAI-1. The internalization of solublePAI-1 hasno intra-ellularonsequenes and amountsto a

meredegradation;

•

^at ^the ^ellular ^{level :} mesenhymatous-amoeboid transition. The experimental fat

that the onentration

c

m ^of matrix-bound PAI-1 is a major determinant of the mesenhymatous-amoeboidtransition (Figure2)willbeformulatedin theframework

ofbifurationtheoryatthesingleelllevel,withaontrolparameterrelatedto

c

m^. ^The

moleularanalysisofintra-ellularpathwaysandmorphologialtransformationsshows

thatamorestraightforwarddeterminantofthemesenhymatous-amoeboidtransition

istheinternalizationux

J

iôf^theêll. Ôur^hypothesisîs^thatintra-ellulardynamis

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twomarkedlydierentmesenhymatousandamoeboidstates,atsomethresholdvalue

J

i

= J

_i^∗^.

•

^at ^the ^ell populatio n level : the mutualized seretionof PAI-1, feeding extraellular PAI-1speies. Cellgrowthanddivision areonsideredatthislevel;

•

^at^themiroenvironmentlevel : solublePAI-1diusionintheextraellularmediumand bindingontheell-free matrix,thusturninginto matrix-boundPAI-1.

Thepointis toexplainhowtheinterplaybetweenthevariousformsof PAI-1 andthevari-

ouslevelsatwhihtheyareprodued,ontrolled,orused,antriggerthemesenhymatous-

amoeboidtransitionandmoregenerallyanexplainthemetastatiproess. Complementary

modelingapproahesareessentiallyneededtoapturethemultileveldeterminantsandmeh-

anisms at work. In order to bring out a robustexplanatory senario, we devisethe most

parsimonious model . Suh modelsoftenneed olletivevariables andeetiveparameters,

aountinginabottom-upandalreadyintegratedwayofawealthofelementarymehanisms

(here, the useof averagedensities and pseudo-rstorder kinetis, see below). Inomplex

systems,theyalsoinvolveeetiveinputsorboundaryonditions,aountinginatop-down

wayoftheinueneofthesystememergentfeaturesorstruturesandsurroundingsonele-

mentarypartsandmehanisms(Lesne,2008b). Themain qualityofaparsimoniousmodel

isthe robustnessof itspreditionswith respet to smallhanges in themirosopi ingre-

dients, beause theywill only slightlyaet thevalue of theeetive parameterswithout

modifyingthe generalform ofthe model (Lesne,2008a). For instane, abifuration (here

themesenhymatous-amoeboidtransition)willstillbeobservedwithpossiblyonlyashiftof

thebifurationloation. Theonfrontationwithexperimental observationswould validate

theleadingpriniplesandsenario. Itisthenanotherpartofthework,involvingingeneral

dierentdataandomputationaltools,tosubstantiatetheminimalmodelwithunderlying

mehanisms and expliit ingredients in order to derive the exat numerial values of the

eetiveparameters,interationsandreations. Theuseofaminimalmodelishereallthe

moreessentialthat noexperimental aesstothevaluesofe.g. kinetiparametersistoday

possible,noradiretexperimental investigationofthemetastatiproess.

3.2 Reation-Diusion Model

Werstonsider adesription(termed mean-eld-like desriptio n instatistial physisfor

interatingmany-body systems)in whih solublePAI-1,internal PAI-1, andmatrix-bound

PAI-1speiesaredesribedbymeansofsmoothdeterministionentrations. Theellpop-

ulationisdesribed by asmoothdeterministielldensity. Theexperimental observations

areformalizedin termsof hemialkinetis,diusion, andgrowth,butthedisrete nature

ofellsandmoleules,andthestohastiityoftheelementaryproessesarenopartofthe

mean-elddesription. Thesetofessentialvariablesinludestheonentrations

c

m

(~r, t)

^of

matrix-bound PAI-1,

c

s

(~r, t)

ôf ^soluble ^PÂI-1, ând

c

i

(~r, t)

ôf înternal ^PAI-1 ât ^time

t

^and

loation

~r

^, ^and ^a ^smooth ^variable

σ(~r, t)

âounting ^for ^the ^presene ôf êlls ât ^the ôn-

sideredposition

~r

^. ^Aording^to ^the^standardontinuous-medium approximation(Landau and Lifshitz, 1959), the element

d~r

^has ^to ^be ^large ênough ^to ôntain â ^large ^number ôf

moleules,sothat onentrationsaresmoothanddeterministi, butnottoolargeso as to

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ruledbyhemialkinetiequations(massationlaw)anddiusionequation(Fiklaw). Ina

similarspirit,thevariable

σ

^is^anhomogenizedversionoftheBooleanfuntion

σ

0

(~r, t)

^with

σ

0

(~r, t) = 1

îfâêllîs^presentⁱⁿ

~r

^at^time

t

ând⁰ôtherwise. ^Thereîs^need^neither^toôunt

ellsnortoaremuhfortheboundariesoftheellpopulationinadisretesetting,andthe

resultingelldensity

σ(~r, t)

îsâôntinuousêld^with

0 ≤ σ ≤ 1

^. ^In^partiular,^ell^growth

andelldivision anbetreatedsimilarly,bothproduingaspreadingofthesupportanda

loal inreaseof theeld

σ(~r, t)

^. ^This ^desription^is mean-eld-likeinsofaras orrelations betweentheutuating numbersofmoleules and ellsat various loations andtimes are

negletedandonlytheirloal averagesareonsidered(Lesne,2007). Theoveralldynamis

isdesribedinaspatiallyextendedkinetimodel,asillustratedonFigure3,aountingfor:

•

^ell^{divisi on}^and^growth,ontinuouslyenlargingtheregionoupiedbyells(theregion

where

σ > 0

^). ^Thisêxpansionôf ^theêll^populationîs ^measured^byâ^rate

k

g ^and^a

unimodal kernel

Γ(.)

ôf ^nite ^range: êll ^growth ând ^division îndue â ôntinuous

spreadingoftheontinuousregionswithshort-rangeinrementsweightedwith

Γ

^. ^As

mentioned,thereisnoneedtoonsiderseparateontributionsforgrowthanddivision.

This kernel is isotropi (

Γ(~r)

^depends ^only ^on

r

^), time-independent, normalized by settingitsintegraloverthewholespaeequalto1. Itswidthorrespondstypiallyto

theellradius;

•

â^soure ^term ^desribing ^the^synthesis ôf ^PAI-1 înside ^the êll. Ît ^desribes^the ^net

resultoftheproteinsynthesisfollowinggeneexpression,itsdegradationrightafterfor-

mation,andpossiblyanegativeself-regulatoryeetontheexpressionofPAI-1gene;

thisisaountedforbyaneetiveterm

f (c

i

)

^suh^that

f (0) > 0

^andmonotonously dereasingto0astheonentration

c

i ôfînternal^PAI-1înreases. ^Presribingâ^more

detailedformfor

f

îs^not^possible^given^the^limited^biologial^knowledgeândît^would

givean illusorypreision;wethus limitourselvesto well-assessedgeneralfeatures of

f

^,^that^will ^appear^to^be^suient^for^derivingqualitativeandrobustonlusions.

•

^the^releaseôf^soluble^PAI-1^whenêllsâre^present,^feedingôn^theirôntentⁱⁿînternal

PAI-1. Itisdesribedbyapseudo-rst-orderkinetis,aountingonlyforthespeiesof

interestandthesimpleproportionalityoftheseretedamountwithrespetto

c

i^. ^The

inuene of possibleadditional fatorsand speies other than PAI-1 (whose expliit

desriptionwouldobsurethedominantsenariothatwewanttoexplore)isimpliitly

takenintoaountin theeetiverate

k

s^:

•

^soluble ^PAI-1 ^then ^diuses ^with ^a ^diusion ^oeient

D

^. ^The ^fat ^that ^it ^diuses

onlyintheextraellularmediumisaountedforbyusingaspae-dependentdiusion

oeient

(1 − σ(~r, t))D

^whih^vanishes^at^maximal^ell^density

σ = 1

^;

•

^the^xationôf^soluble^PAI-1ôn^the^matrix^when^noêllîs^present,^produing^matrix-

boundPAI-1witharate

(1 − σ(~r, t))k

m^whih^vanishes^at^maximal^ell^density

σ = 1

^;

•

^thedeativation ofsoluble PAI-1or itsinternalization; thisproessis muh dierent fromtheinternalizationofmatrix-boundPAI-1withregardstoellphysiology: ithas

nosignalingroleanddoesnottriggeranypathway,havingnallynoonsequeneonthe

overallstateoftheell;heneitshouldnotbetakenintoaountintheinternalization