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A Cellular Potts Model of single cell migration in presence of durotaxis

Rachele Allena, Marco Scianna, Luigi Preziosi

To cite this version:

Rachele Allena, Marco Scianna, Luigi Preziosi. A Cellular Potts Model of single cell migration in

presence of durotaxis. Mathematical Biosciences, 2016, pp.57-70. �hal-02375906�

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A Cellular Potts Model of single cell migration in presence of durotaxis

R. Allena

a,

, M. Scianna

b

, L. Preziosi

b

aArts et Metiers ParisTech, LBM/Institut de Biomecanique Humaine Georges Charpak, 151 bd de l’Hopital, 75013 Paris, France

bDipartimento di Scienze Mathematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Keywords:

Cell migration Durotaxis Cell polarity Anisotropy CPM

a b s t r a c t

Cellmigrationisafundamentalbiologicalphenomenonduringwhichcellssensetheirsurroundingsand respondtodifferenttypesofsignals.Inpresenceofdurotaxis,cellspreferentiallycrawlfromsofttostiff substratesbyreorganizingtheircytoskeletonfromanisotropictoananisotropicdistributionofactinfil- aments.Inthepresentpaper,weproposeaCellularPottsModeltosimulatesinglecellmigrationover flatsubstrateswithvariablestiffness.Wehavetestedfiveconfigurations:(i)asubstrateincludingasoft andastiff region,(ii)asoftsubstrateincludingtwoparallelstiff stripes,(iii)asubstratemadeofsucces- sivestripeswithincreasingstiffnesstocreate agradientand (iv)astiff substratewithfourembedded softsquares.Foreachsimulation,wehave evaluatedthemorphologyofthecell,thedistancecovered, thespreadingareaandthe migrationspeed.Wehavethencomparedthenumericalresultstospecific experimentalobservationsshowingaconsistentagreement.

1. Introduction

Cellmigrationisacriticalphenomenonoccurringinseveralbi- ologicalprocesses, such asmorphogenesis [1], woundhealing [2]

andtumorogenesis[3].Ittakesplaceinsuccessiveandcyclicsteps [4] and it is triggered by specific interactions with the extracel- lular matrix (ECM).Actually, cell migration mayoccur inthe ab- sence of external signals thereby typically resulting in a random walk. However, in most situations, cells are able to sense their surroundingenvironmentandtorespondforinstancetochemical (i.e., chemotaxis) [5], electrical (i.e., electrotaxis) [6]or mechani- cal (i.e.,mechanotaxis)[7]fieldsoryet tostiffnessgradients(i.e., durotaxis) [8,9]. Thelatter mechanismconsistsofthe cell prefer- ential crawlingfromsoftmatrixsubstratestostifferones,evenin the absenceofanyadditionaldirectional cues[10,11].Byforming local protrusions (i.e., pseudopodia), the cells are in fact able to probe the mechanicalpropertiesof thesurrounding environment andto morestrongly adhereover stiff regions. Additionally,such behavior results in a substantial reorganization of the intracellu- lar cytoskeleton. In fact, over soft substrates cells typically show anunstableandisotropicdistributionofactinfilaments,whichare poorlyextendedandradiallyoriented,whereasoverstiff substrates cell morphology ismore stableandexhibitssignificant spreading andoftenanisotropicarrangementsofactinfilamentsinthedirec- tionofmigration(i.e.,polarization)[12–16].

Corresponding author. Tel.: +33 1 44 24 61 18; fax: +33 1 44 24 63 66.

E-mail address: rachele.allena@ensam.eu (R. Allena).

Althoughseveralcomputationalmodelshavebeenproposed in literature to investigate single cell migration, only few of them deal with durotaxis. Among others,it is worth to cite the work by Moreoet al.[17] who proposed a continuum approach based onanextensionoftheHill’smodelforskeletalmusclebehaviorto investigatecellresponseontwo-dimensional(2D)substrates.They showed,in agreement with experimental observations, that cells seemtohavethesamebehaviorwhencrawlingonstiffersubstrate andon pre-strained substrates. Harlandet al.[18] instead repre- sentedacellasacollectionofstressfibersundergoingcontraction andbirth/deathprocessesandshowedthatonstiff substratescells exhibitdurotaxisandstressfiberssignificantlyelongate.Dokukina andGracheva[19] developeda 2D discrete modelof aviscoelas- ticfibroblastcellusingaDelaunaytriangulation.Ateachnodethe balance of the forces was determined by the contributions i) of the frictions betweenthe cell and the substrate, ii)of a passive viscoelasticforce and iii)of an intrinsicactive force. The authors thenevaluated cell behavior overa substratewitha rigidity step ingoodagreementwithspecificexperimentalobservations.Infact, they found that the cell preferentially moves on the stiffer sub- strateand turnsaway fromthe softsubstrateasreported by[8]. Stefanoni etal. [20] proposed a finite element approach able to accountforthelocalmechanicalpropertiesoftheunderneathsub- strateandtoanalyzeselectedcellmigratorydeterminantsontwo distinct configurations:an isotropic substrateandabiphasicsub- strate (which consistsoftwo adjacent isotropic regions withdif- ferentmechanicalproperties).Trichet etal.[14]employed instead the active gel theory to demonstratethat cells preferentially mi- grateoverstiff substratesfoundinganoptimalrangeofrigidityfor

http://dx.doi.org/10.1016/j.mbs.2016.02.011

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a maximal efficiency of cell migration. Further, in [21] a vertex- basedapproach(i.e.,theso-calledSubcellularElementModel,SCE) wassettorepresentintracellularcytoskeletal elementsaswell as their mechanical properties. In particular, the dynamics of such subcellulardomainswere describedby Langevinequations,which account for a weak stochastic component (i.e., that mimic cyto- plasmicfluctuations)and elasticresponses (i.e., modeled by gen- eralized Morse potentials) to both intracellular and intercellular biomechanicalforces. The same methodwas successfullyapplied in[22]formodeling substrate-drivenbacterialocomotion. Finally, inAllenaandAubry[23]a2Dmechanicalmodelwasproposedto simulatecellmigrationover anheterogeneoussubstrateincluding slippingregionsandtoshowthatoversofterregionsthecellslows downandislessefficient.

In thepresentwork, wedescribe aCellular PottsModel(CPM, developedin[24,25]andreviewedin[25–29]),whichisalattice- basedstochasticapproachemployinganenergyminimizationphi- losophy,toreproducesinglecellmigrationoverflatsubstrateswith different rigidity. In particular, we test four configurations: (i) a substrateincluding a soft anda stiff region, (ii) a soft substrate includingtwoparallelstiff stripes,(iii)asubstratemadeofsucces- sivestripes withincreasing stiffnesstocreatea gradient and(iv) a stiff substrate with four embeddedsoft squares. For each sce- nario,we analyzecell behavior interms ofmorphology, distance covered,spreading/adhesivearea andmigrationspeed inorderto capturetheessentialmechanismsofdurotaxis.Thecomputational outcomesarethen comparedwithspecific experimental observa- tionstakenfromtheexistingliterature.

The rest of this paper is organized as follows. In Section 2, we clarifythe assumptions onwhich our approach is based and present the model components. The simulation results are then shown in Section 3. Finally, a justification of our model choices aswell asa discussion on possibleimprovements is proposed in Section 4.Additionally, thearticle isequipped withan Appendix thatdealswithstatisticsandparameterestimates.

2. Mathematicalmodel

The cell-substratesystemisrepresentedusinga CPMenviron- ment[24,25].The simulationdomain isa three-dimensional(3D) regular lattice

R3 constituted by identical closed grid sites, whichareidentified bytheir centerxR3 andlabeled byan in- tegernumber

σ

(x)N(whichcanbeinterpretedasadegenerate spin)[30,31].Theboundaryofagenericsitex,oneofitsneighbors anditsoverallneighborhoodaredefinedas

x,xand

x,respec- tively.Subdomainswithidenticallabel

σ

formdiscreteobjects

σ (withborder

σ),which havean associatedtype

τ

(

σ).In the caseofourinterest,

τ

=Mstandsfor themedium,

τ

=C for the cellsand

τ

=Si for the ith type of substrate. In this respect,we anticipatethat each type ofmatrixregion willdiffer forstiffness andthereforeforadhesiveaffinitywithmovingindividuals.

Celldynamicsresultfroman iterativeandstochasticreduction of the energy of the overall system, given by a Hamiltonian H (units: kgm2/s2), whose expression will be clarified below. The employed algorithm is a modification of the Metropolis method for Monte Carlo–Boltzmann dynamics [24,32], which is particu- larlysuitabletosimulatetheexploratorybehaviorofbiologicalin- dividuals as cells. Procedurally, at each time step t of the algo- rithm, calledMonte Carlo Step (MCS), a randomly chosen lattice sitexsource belonging to a cell tries to allocate its spin

σ

(xsource)

tooneofitsunlike neighborsxtarget

x,whichisalsorandomly selected.Then,thenetenergydifference

Hduetotheproposed changeofsystemconfigurationiscalculatedas

H

|

σ (xsource)σ

(

xtarget

)

=H(afterspin copy)H(be f orespincopy) (1)

The trial spin update is finally validated by a Boltzmann-like probabilityfunctiondefinedas

P[

σ (

xsource

)

σ (

xtarget

)

]

(

t

)

=min

1,eTCH

(2)

wheretistheactualMCSandTCR+isaBoltzmanntemperature, thathasbeeninterpretedinseveralwaysbyCPMauthors(see[33]

foracommentonthisaspect).However,wehereopttogiveTCthe sense ofa cellintrinsicmotility(i.e.,agitationrate),followingthe approachin [25].Finally,it isusefultounderline that thematrix substratesareconsideredfixedandimmutable.

As seen, the simulated system evolves to iteratively and stochasticallyreduceitsfree energy,whichisdefinedby aHamil- tonianfunctionHwhich,foranygiventimestept,reads

H

(

t

)

=Hadhesion

(

t

)

+Hshape

(

t

)

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Hadhesion(t)isdeducedfromtheSteinberg’sDifferentialAdhesion Hypothesis(DAH)[24,34]andisduetotheadhesionbetweencells andextracellularcomponents(i.e.,themediumoragiventypeof substrate).Inparticular,itreads

Hadhesion

(

t

)

=Hadhesion

(

t

)

= (∂xσ)

(

xσ

)

Jτ

(

σ (x)

)

,τ (σ(x)) (4)

with x and x two neighboring sites and

σ and

σ two neighboring objects (with borders

σ and

σ, respectively).

Jτ (

σ (x)),τ (σ(x))R+ are constant and homogeneous binding forcesperunit area.Theyaresymmetricwithrespecttotheirin- dicesandcanbespecifiedasfollows:

- JC,M is the adhesive strength between the cells and the col- lagenous medium which is constituted by a mixture of sol- uble adhesive ligands (i.e., carbohydrate polymers and non- proteoglycanpolysaccharides)andwatersolvent;

- JC,S

i givestheadhesivestrengthbetweenthecellsandithtype ofsubstrate.RecallingtheminimizationtheoryoftheCPM,we assume that the stiffer the substrate i, the lower the corre- spondingvalue JC,S

i (i.e.,the higherthe adhesionbetweenthe cellsandtheithtype of substrate).This isa pivotalhypothe- sis of our approach: it is consistent since it has been widely demonstratedintheexperimentalliteraturethatcellsgenerate highertractionforcesandgeneratemorestablefocaladhesion pointswhenmigratingoverstiffersubstrates[16,35–38]. Hshape(t) defines the geometrical attributes of each cell

σ, whicharewrittenaselasticpotentialsasitfollows:

Hshape

(

t

)

=Hvolume

(

t

)

+Hsur f ace

(

t

)

= σ

[

κ

σ

( v

σ

(

t

)

VC

)

2+

ν

σ

(

t

) (

sσ

(

t

)

SC

)

2] (5)

where

v

σ

(

t

)

andsσ

(

t

)

aretheactualvolumeandsurfaceofthe cell

σ, whereas VC and SC the corresponding cell characteristic measures in the initial resting condition.

κ

σ and

ν

σ

(

t

)

are in- stead two mechanical moduli in units of energy. The former is linked to volume changesand, assuming that cellsdo not signif- icantlygrowduringmigration,isconsideredconstant withahigh value (i.e.,

κ

σ =

κ

C1) foranyindividual

σ. Thelatter refers totherigidity ofacell. Aswe willexplainindetails lateron, for each cell

σ,

ν

σ isassumedtodependon theunderneath type ofsubstrate.Inparticular,eachcelldecreasesitsinitiallyhigh(i.e., 1) rigidity, thereby being more able to deform, if it comes in contactwith a stiff substrate.This assumption isconsistent with experimental observations on the fact that cell contactwith stiff matrixregions activatedownstreamintracellularpathwaysresult- inginacto-myosindynamicsandthereforeincytoskeletalremod- eling [8,39]. More specifically, it seems that certain cells have a binarysensor attheir membrane junctionsites that allows them

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Fig. 1. Cell behavior is determined by a modified Metropolis algorithm, which is based on a iterative and stochastic minimization of the cell–matrix system energy, defined by a Hamiltonian functional H. In particular, it includes energetic contributions for cell geometrical attributes and cell-substrate adhesive affinity. A Boltzmann-like law finally controls the likelihood of the acceptance of domain configuration updates, which is further biased by the intrinsic cell motility, established by parameter T C.

Table 1

Main parameters of the model.

Parameter Description Value Unit Reference(s)

V C Initial/target cell volume 16 ×10 3 μm 3 [46]

S C Initial/target cell surface 3.8 ×10 3 μm 2 [46]

T C Motility of the cell 50 ×10 −27 kg m 2/s 2 Sensitivity analysis (see Fig. 10 ) κC Compressibility of cell volume 25 ×10 −9 kg/s 2m 4 Sensitivity analysis in [80,89]

νC Cell intrinsic rigidity 25 ×10 −3 kg/s 2m 2 Sensitivity analysis in [57,68,80,89]

νt Threshold value of cell rigidity 10 ×10 −3 kg/s 2m 2 Parameter analysis ( Appendix A.3 ) J C,M Cell-medium adhesive strength 25 ×10 −15 kg/s 2 Parameter analysis ( Appendix A.3 ) J soft Adhesive strength between cells and the softest substrate 25 ×10 −15 kg/s 2 Sensitivity analysis (see Fig. 10 ) J stiff Adhesive strength between cells and the stiffest substrate 1 ×10 −15 kg/s 2 Parameter analysis ( Appendix A.3 )

toswitchfromarelaxedandroundedmorphology,whenthesub- strate is softer than the cell’s elastic modulus [39–43], to a fan- shaped morphology with abundant stress fibers, when the sub- strateisstifferorasstiff asthecell itself[39].Further,it hasbeen shown that cellstend to isotropically and poorly spread on soft substrates, whereas they formpseudopodia randomly distributed along the membraneon stiff substrates, resultingin a significant anisotropicspreading[16].Inthisrespect,accordingtoseveralex- perimental observations [16,35–38,44], there exists a linear rela- tionship betweenthe adhesion forces exerted by the cell on the substrate and the spreading area of the cell. More specifically, the larger the contact area between the cell and the substrate, the higher the number of focal adhesion points that can be es- tablished.Nonetheless,the sequenceofeventsisstillunclear and two main processes may occur when a cell is seeded on a stiff substrate[45]:

(i) the cell adheres because of the stiffness of the substrate, thenitsignificantlyspreads;

(ii) the cell spreads because of the stiffness of the substrate, thenitmorestronglyadheres.

Such uncertaintyisthereasonwhyinthepresentmodelboth the adhesive parameters and the cell rigidity directly depend on thesubstratestiffness,butareindependentfromeachother.

Themaincomponentsandthescalesinvolvedintheproposed modelaresummarizedinthediagraminFig.1.Finally,allthepa- rametersofthesimulationsarereportedinTable1,whiletheAp- pendixprovidesacarefulexplanationofhowtheyhavebeenesti- mated.

3. Numericalsimulations

The characteristic size of each lattice site is 4

μ

m and

the geometrical domain

is a 70×70×30 regular grid (280

μ

m×280

μ

m×120

μ

m)withno-fluxboundaryconditionsin

alldirections.Thischoice mimics thesituationofa delimitedex- perimentaldevice,wherecellsarenotabletoovercomethephysi- calbarriers.AllourCPMcellsareinitiallyahemisphereofaradius of20

μ

m,whoseinitialpositionwillbespecifiedforeachsimula-

tionsetting.AMCSissettocorrespondto2sofactualunitoftime (see the Appendix for a commenton thisaspect), which results in simulations covering time intervals between 16min to 5.5h.

Thischoiceenablescellstomigrateoversufficientlylongpathsin orderto compare numericalresults andproper experimental ob- servations.Wehavetestedseveralcell–matrixsettings,whichare presented in the followings. The resulting simulations were per- formed on a modified version of the open source package Com- puCell3D(downloadableatwww.compucell3d.org).Inparticular,a Phytonscript wasspecificallydevelopedto accountforsubstrate- dependentcellrigidity.

3.1. Cellspreferentiallycrawloverstiff substrates

Wefirstconsiderasubstratesplitintoasoftandastiff region, i.e.,

τ

=S1 and

τ

=S2 (see Fig. 2a). A cell

1 is then seededat thecenterofthesubstrateanditisallowedtomovefor500MCS (approximately16min).Therigidityof

1,

ν

1,hasinitiallyahigh value

ν

1=

ν

C= 25×103kg/s2m2.However,itisallowedtode- crease,of103kg/s2m2forMCSuntilathresholdvalue

ν

tequalto

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Fig. 2. Snapshots of the tested substrate configurations: (a) soft (red: τS1with J C,S1= 25 ×10 −15kg/s 2) and stiff (yellow: τS2with J C,S2= 1 ×10 −15kg/s 2) substrates, (b) soft (red:

τS1with J C,S1= 25 ×10 −15kg/s 2) substrate with two stiff (yellow: τS2with J C,S2= 1 ×10 −15kg/s 2) stripes, (c) sequence of stripes with different stiffness (red: τS1 with kg/s J C,S1= 25 ×10 −15, dark orange: τS2with J C,S2= 20 ×10 −15kg/s 2, orange: τS3with J C,S3= 15 ×10 −15kg/s 2, light orange: τS4with J C,S4= 10 ×10 −15kg/s 2, dark yellow: τS5with J C,S5= 5 ×10 −15kg/s 2, yellow: τS6with J C,S6= 1 ×10 −15kg/s 2), (d) stiff (yellow: τS2with J C,S2= 1 ×10 −15kg/s 2) substrate with embedded soft (red: τS1with J C,S1= 25 ×10 −15kg/s 2) squares. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

102kg/s2m2,whilethecellisincontactwiththestiff region S2, therebyleadingtoa flatteningofthe initiallyrigidcellularhemi- sphere.Inmathematicalterms,weindeedhavethat

ν

1

(

t

)

=

max

( ν

1

(

t−1

)

−103;

ν

t

)

if

(

x,x

x

)

:x

1 andxS2;

ν

1

(

t−1

)

else,

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foreachMCS.

We then study how cell behavior is affected by variations in the ratio between the adhesive affinity of the cell with either thesoftor thestiff substrate region.In particular,we keep fixed JC,S

1= Jso f t=25×1015kg/s2 while decreasing the value ofJC,S

2

from25×10−15kg/s2to1×10−15kg/s2(whichisequaltoJstiff,the lowestvalue consistentwiththecaseofourinterest,seethe Ap- pendix).AssummarizedinFig.3c,when JC,S

2 decreases,thecellis biasedtocrawltowardthestiff domain, asitisconfirmedby the plotof the trajectories of its center ofmass deriving frominde- pendentsimulations.Infact,overaperiodof500MCS(≈16min), thecellrandomlymovesaroundthesubstratecenterwhen JJC,S1

C,S2=1

(Fig.3a)while,when JJC,S1

C,S2=25,thecelltrajectoriesdramaticallyshift overthestiff partofthesubstrate(Fig.3b). Ournumericalresults are sustained and consistent with the experimental observations accordingtowhichcells(i.e.,fibroblasts,smoothmusclecells,Mes- enchymalStemCells(MSCs))crawlfromsoft(1–5kPa)tostiff (34–

80kPa) substrates (i.e., gels or polyacrylamide sheets) [9–11,46]. Notablyduringmotiontowardthestiff substrate, ourCPM cell is alsoallowed toincrease itsremodelingability, asits rigidity

ν

1

progressivelydecreases uponcontactwithsubstrateS2,according to Eq. (6). In this respect, a further set of simulations evaluates cellmorphologicaldifferencesduetotheunderneathtypeofsub- strate. Keeping the samedomain asin Fig.2a, two cells, i.e.,

1

and

2,areinitiallyseededinthemiddleofthesoftandthestiff regions, respectively. The rigidity of the two cells is then regu- lated by Eq.(6). As reproduced in Fig. 4 (in particular, panel (a) represents the final cell morphologies as resulted from a single representativesimulation,whereas panel (b)givesthemeanfinal cell morphologies,asthe plainellipsoids derive froman interpo- lationprocedureofthe celladhesiveareascomingfromindepen- dentsimulations,seethe Appendixforfurtherdetails),bothindi- vidualsdonotsignificantlymoveacrossthedomainduringatime lapseof 500MCS (approximately16min). However, the adhesive area of thecell located over the softregion is almost 30% lower than the adhesive area ofthe cell that crawls over the stiff sub- strate(Fig.4c).Such acellbehaviorisconsistentwiththeexperi- mentaldataby Loandco-workerson 3T3fibroblastscultured on flexible polyacrylamidesheetscoatedwithtype Icollagen,where a transition in rigidity was introduced by a discontinuity of the bis-acrylamidecross-linker, thatresulted intwosubstrateregions withYoung’s modulus equal toeither 14kPa and 30kPa [46].In particular,ononehand,thevalueoftheadhesiveareaofourCPM cellseededonthesoftsubstrateisnot surprisinglysimilartothe correspondingdatabyLoandco-workers[46],sinceweusedsuch anexperimentalquantificationforourparameterestimate(seethe Appendix). Onthe otherhand, theadhesive area oftheCPM cell seededonthestiff regionisinsteadacompletelyindependentand

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Fig. 3. Simulation for a substrate with soft (red) and stiff (yellow) regions. As the ratioJJC,SC,S,,12, increases, the cells are typically biased to migrate toward the stiff region (c).

This is also confirmed by the trajectories of the cell center of mass, which are relatively close to the center of the substrate whenJJC,SC,S,,12=1 (a), whereas they are substantially shifted on the stiff region whenJJC,SC,S,,12=25 (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

self-emerging modeloutcome: thereforeits consistency withthe measurements by Lo andcolleagues [46] is relevantpoint ofour work.

3.2. Stiff versus soft substrate in the presence of an external cue

For the second series of simulations we consider againa do- mainsplitintoasoft(

τ

=S1suchasJC,S

1=Jso f t=25×1015kg/s2) andastiff (

τ

=S2suchasJC,S2=Jsti f f=1×10−15kg/s2)region,but an additional external potential is introduced. This results in an imposedartificialbiasinthespinflipratethatisabletoaffectthe directionofcellmigration.Enteringmoreindetails,theexpression oftheHamiltonianfunction presentedinEq.(3)ismodifiedasit follows

H

(

t

)

=Hadhesion

(

t

)

+Hshape

(

t

)

+Hpotential (7)

whereHpotential=−

v

ext

(

xtargetxsource

)

andvextisavectorwhose components determine the direction of the potential and whose modulus gives the relative importance in the overall system en- ergy. Forthe sake of simplicity,we assume that the potential is constantintimeandhomogeneousthroughouttheentiredomain.

Aswewillseelater,itisinfactanartificialtermthatsimplyhelps cellstomaintainasustaineddirectionalmovement.Inthisrespect, what isrelevantisonly itsmodulus, i.e.,|vext|. We thentest two configurations:

(a) a cell

1 placed atthe south-east cornerandthe external potentialdirectedtowardthenorth-westcorner;

(b) thesamecell

1placedatthesouth-westcornerofthesub- strateandtheexternalpotentialdirectedtowardthenorth- eastcorner.

Inboth cases,we set

| v

ext

|

=7×1021 kgm/s2, whichresults in plausible cell velocities (see later) and the simulations last 10,000 MCS (approximately 5.5h). Further, cell rigidity is again regulatedbyEq.(6).

Insystemconfiguration (a) (see Movie1 andFig. 5a), theex- ternalcueguidesthecelltowardthenorth-westcornerofthedo- main.Inparticular,whenapartofthecellcomesintocontactwith thestiffersubstrate,itbecomestheleadingedge.Further,themov- ingindividual clearlyacceleratesassoon asit crossesthe bound- ary between the two matrix regions (3.6

μ

m/s versus 4.5

μ

m/s,

Fig.5c),asexperimentallyobservedin[8]forfibroblastscrawling overpolyacrylamide sheets.Anincrementof theadhesive area is observedaswellwhenthecellshiftsoverthestiff region.

Inthe case (b),the external potential forces the cell tomove toward the north-east corner of the domain (see Movie 2 and Fig. 5b). However, assoon asthe individual approaches the soft region, it changes orientation, and starts moving and elongating paralleltotheboundarybetweenthetwosubstrateregions.These outcomesmaybecomparedto theexperimentalobservationsob- tainedbyLoetal.in[46],whoculturedfibroblastsonthealready describedsubstrate system, i.e., characterized by two areas with differentYoung’s modulus.Inparticular,Loandcolleaguesseeded cellsatlow densitytominimize the effectsofintercellular inter- actionsandtoavoidthatpullingorpushingforcesfromneighbors individuals may alter cell substrate probing processes (thereby

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Fig. 4. Two cells are initially seeded on a soft (red) and a stiff (yellow) substrate, respectively. (a) Simulation snapshot of the final positions (i.e., at MCS = 500 corresponding to nearly 16 min) of the two cells. (b) Initial (dashed) and final (plain) contour shapes give an idea of the position and the morphology of the two cells. (c) Cell adhesive area as a function of the type of substrate. The area is about 30% higher in the case of the cell seeded over the stiff substrate, due to the specific constitutive law given to cell rigidity (i.e., Eq. (6) ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Simulation for a substrate with soft (red) and stiff (yellow) subdomains. The trajectories of the cell center of mass as well as the initial (dashed) and the final (plain) cell contours are traced respectively for (a) a cell initially seeded at the south-east corner and an external potential introduced toward the north-west corner and (b) a cell initially seeded at the south-west corner and an external potential directed toward the north-east corner. (c) Cell average velocity over either the stiff and soft substrate.

(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

impedingcellstofreelymoveacrossthesoftandthestiff regions).

Then,cell migrationwasrecordedover atime spanof10h. Sim- ilarlyto ournumerical outcomes, theauthors found that ascells move toward a stiffer substrate, new lamellipodia are formed in thedirectionofmigration,therebyresultinginthedominantfront end of the individuals. On the opposite, local retractions occur whencellsapproacha softregion,inducing thereforea changeof direction. In a second series of experiments, Lo and co-workers showed that mechanical inputs triggered by substrate deforma- tionsmight alsocontrol formationandretraction oflamellipodia.

Inparticular,theyexternally pulledorpushedthesubstrateaway ortowardthecellscentertofindthat,duetothecentripetalforces exerted by the 3T3fibroblasts on the substrate [46],in the first case less motion is produced, since cells experience a softening ofthesubstrate,whereas inthesecond casetheoverallmotionis increased,since cellsperceivethe substrateasstiffer.IntheCPM model proposed here, the matrix substrates are not deformable,

therefore the numerical simulations are unable to capture the experimentalobservationscomingfromthissecondsetofassays.

3.3. Twostiff stripesembeddedinasoftsubstrate

The third configuration that has been tested includes a soft substrate (again

τ

=S1 with JC,S

1=Jso f t=25 ×1015kg/s2) with two embedded stiff stripes (again

τ

=S2 with JC,S2=Jsti f f=1

×1015kg/s2),whichare both28

μ

m-wide(Fig.2b).Acell

1 is initiallyseededatthesouth-westcorner,whoserigidityisallowed todecreasefollowingtheconstitutivelaw(Eq.(6)).Anexternalpo- tentialisthenintroduced towardthenorth-eastcornerofthedo- main: its intensity|vext| is allowed tovary froma minimal value of7×1021kgm/s2 toa maximal value of28×1021kgm/s2.All simulationslast 10,000MCS,whichcorrespondto nearly5.5h.In thecaseofa low|vext| =7×1021kgm/s2 (see Fig.6a andMovie 3),thecelltypicallymigratestowardthefirststiff substratestripe:

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Fig. 6. Configuration with a soft (red) substrate with two embedded stiff stripes (yellow). (a) and (c) Simulations with |vext|= 7 ×10 −21kg m/s 2 and |vext|= 28 ×10 −21kg m/s 2respectively. Representative cell trajectories are plotted together with the initial (dashed) and the final (plain) cell contours. (b) Relative cell frequency as function of | v ext|. (d) Cell average velocity over the different substrate regions in the case of |vext|= 28 ×10 −21kg m/s 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

then it remains stuck over it and goes on migrating along such a matrix region.Furthermore, its morphology,due tothe depen- dencyofitselasticityontheunderneathtypeofsubstrate,changes asthecrawlingindividualacquiresanelongatedshape.Suchabe- havior is due to the fact that the external potential is too low to overcome the adhesive interactions between the cell and the stiffest substrate: in particular, the individual has not energetic benefits (deriving fromthe external bias) to move further inthe domain,i.e.,topassthefirststiff stipe.TheoutcomesofourCPM areconsistenttothatobservedforcells(i.e.,endothelialcellsorfi- brosarcomacells)seededon2Dsubstrates(i.e.,maleicacidcopoly- mersurfaces)structuredwithfibronectinstripeswhichorienttheir actinfibersalongthestripedirection[47–49].

Ontheother hand,ifthemodulusoftheexternalpotentialin- creases,wehaveahigherpercentageofcellsthatareabletocross the entire domain(Fig. 6b). Inparticular, when |vext| is maximal (i.e.,28×10−21kgm/s2,Fig. 6candMovie4), thecellsconstantly migrate at the north-eastcorner of the domain passing also the second stiff stripe.Inthiscase,the cellaveragevelocityincreases over the stiff stripes (about4.4

μ

m/s)whereas itvaries between

3.6

μ

m/s and3.9

μ

m/s over the soft regions (Fig. 6d). With the

maximalexternalpotential,cellmorphologydoesnotsignificantly vary,asthemovingindividualstypicallymaintainanalmosthemi- sphericshape,withoutsubstantialelongationorincrementsinthe adhesive area during the entire motion. They in fact behave as translating rigidbodies, subjected toan external highforce. This interesting behavior is the consequence ofthe fact that the cells do notneed toreorganize (nor haveenough time todo it)to be

ableto crawl,astheir motionis mainlydueto theexternal bias:

the specific substrate regions are only able to further accelerate (orpartiallyslowdown)cellmovement,aspreviouslycommented.

Thenumericaloutcomesinthecaseofloworintermediatevalues of|vext| can be comparedto those experimentally testedby Choi etal.[50]andVincentetal.[51],wheredifferentcellphenotypes wereseededon micropatternedhydrogelswithstiffnessgradient.

Although no external bias was introduced in such experimental configurations, a similar behavior may be observed. In the for- merwork [50],the authorsproposed twomechanically-patterned hydrogels:one constitutedby 100

μ

m stiff (10kPa) and 500

μ

m

soft(1kPa) stripesand one containing500

μ

mstiff (10kPa)and

100

μ

msoft (1kPa).First, Adipose-derivedStem Cells(ASCs)and C2C12 myoblasts were allowed to adhere and both were able to sense the stiffness gradient and to migrate toward the stiffer stripes(i.e.,durotaxis)[46].Suchbehaviorwasalsoobservedwhen cells were far away from the stripe interface (around 250

μ

m).

Nevertheless,sincecellsonlydetectstiffnessdifferencesovershort distances(aroundsomemicrons)[52],inthiscasetheauthorsim- pliedthat the phenomenon wasmostly dueto random walk to- ward the interface rather than to durotaxis itself. Regarding the morphology ofthecells, both ASCSandC2C12myoblasts aligned inthedirectionofthelongaxisofthestripeasweobserveinour numericalsimulations(Fig.5aandMovie3)inthecaseoflowin- tensityoftheexternalpotential.Second,lesscontractilecellssuch as neurons were seeded on the hydrogels, which did not show anypreferentialadhesionconfirmingpreviousexperimentalobser- vationsaccordingtowhichtheypreferasofterniche[53].

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Fig. 7. Results for the simulation with a soft to stiff gradient ( Section 3.4 ). (a) Relative cell frequency as | v ext| increases. (b) Average cell velocity over the different substrate regions in case of |vext|= 28 ×10 −21kg m/s 2.

In the latterwork [51],the authors developed three types of polyacrylamide(PA)hydrogelsystemsofstiffnessgradients:physi- ological(1Pa/

μ

m),pathological(10Pa/

μ

m)andstep(100Pa/

μ

m).

Thestep stiffnessgradient, which isthe configuration ofinterest for the simulations presented above in this section, was consti- tutedby500

μ

mwideregionsofsoftPAalternatedwith∼100

μ

m

widestripesofstiff hydrogelproducingastrippedstiffnessprofile.

MSCs were plated and they spread and attached independently of the gradient strength or the stiffness within hours after the seeding, whereas after 3days they started to migrate toward stifferregions. Additionally,cellscrawledat18±0.7

μ

m/h,which

is approximately sixfold faster than on the other gradient con- figurations discussed in the same paper (i.e. physiological and pathological)andconfirmsthatdurotaxisvelocityisinfluencedby gradientstrength[11].

3.4.Stiffnessgradient

In thissection,we presenttheresultsfora simulationinvolv- ing a substrate made of sixsuccessive stripes (i.e.,

τ

=Si where i=1, …, 6, each 46

μ

m-wide) which are organized to obtain a

soft-to-stiff gradient from the left to the right side of the do- main(fromthered totheyellowsubdomains).Suchsubstratere- gionsarecharacterizedbydifferentcelladhesiveaffinity,i.e., JC,S

i, whichvaryfrom JC,S

1 =Jso f t= 25×1015kg/s2 to JC,S

6=Jsti f f= 1×10−15kg/s2,respectively(seeFig.2candthecorrespondingcap- tion forthe specific details). A cell

1 is initially seeded at the south-westcornerandanexternal potential isintroduced toward thenorth-eastcornerofthedomain,whosemagnitude|vext|isvar- iedagainfroma minimal valueof 7×10−21kgm/s2 (Movie5) to amaximalvalue of28×1021kgm/s2 (Movie6).The rigidity

ν

1

of

1 is allowed to decrease (from the usual initial high value of

ν

C=25×10−3kg/s2m2) withalawanalogouswithEq.(6),but whichtakesintoaccountofthepresenceofdifferenttypesofsub- strates,i.e.,

ν

1

(

t

)

=

max

( ν

1

(

t1

)

v

i;

ν

t

)

if

(

x,x

x

)

:x

1 andxSi;

ν

1

(

t1

)

else, (8)

wheret istheactual MCS,

ν

t is theusual thresholdvalue (equal

to10−2kg/s2m2)andi=2,…,6.Inthisrespect,

ν

i= 0.05×10−3, 0.06×103,0.1×103,0.2×103,1×103kg/s2m2 whilethe cell isincontactwith substrateS2,S3, S4, S5,S6,respectively.

ν

1 re- mainsindeedconstant andequalto

ν

C ifthecell islocatedover thesoftest substrate S1.All the resulting simulations last 10,000 MCS(5.5h). AsreproducedinFig.7a,thepercentageofcellsable

toreachthenorth-eastcornerincreasesconcomitantlywithincre- mentsof|vext|.Moreover,byfixing

| v

ext

|

=28×1021kgm/s2,itis possible to observe that the cell average velocity increases from 3.6

μ

m/sto4.4

μ

m/sastheymovefromsoftertostiffersubstrates

(Fig. 7b).This resultiscoherentwiththe modeloutcomesofthe previoussetofsimulations(i.e.,seeFig.6candd),wherewehave observedthat inthecaseofveryhighexternalpotential cellsac- celerate while crossing on stiffer matrixregions, even if they do notsignificantlyundergomorphologicaltransitions.

A similar configuration wasexperimentally proposed by Che- ungetal.[10]who,usinga microfluidics-basedlithographytech- nique, fabricated a micropatterned cell-adhesive substrate made of a series of PEG-fibrinogen hydrogels with uniform stiffness ranging from 0.7 to 50kPa. Human Foreskin Fibroblasts (HFFs) were then plated and their migratory trajectories were analyzed over 22h. The authors found that the cells that were initially seededona stiffnessfrontiertendedtomigratetowardthestiffer region, while cells plated on uniform stiffness spread in both directions.

3.5. Roleofthecharacteristicdimensionofthegradientstiffness

Theexternal potential introducedin mostof theprevious sets ofsimulations,is anartificialtermthat isincludedinthe Hamil- tonian to bias and sustain cell movement across the entire ma- trix substrate. In experimental assays, the directional component in cell motion is typically established by geometrical cues, such asmicrotracksandmicrochannels [54,55],orgradients ofsoluble or insoluble chemical substances (chemotaxis and haptotaxis, re- spectively) or, in the case of our interest, gradients of substrate stiffness[10,51].However,wehaveobservedfromoursimulations thatthesequenceofdifferenttypesofsubstratestripesemployed intheprevious section doesnotsufficetodetermine apersistent cell movement across theentirematrix, sincea high enoughex- ternalpotential hastobeincludedtoallowcellsreachthenorth- east corner of the domain (see the plot in Fig. 7a). The reason ofthisdiscrepancybetweencomputational andexperimental out- comesreliesinthefactthat “realcells”,onceestablisheda direc- tionofmovement,are abletodramaticallyorienttheir cytoskele- ton(viathepolarizationofactinfilaments)and,eventually,starta persistentshape-dependentlocomotion. Thisway,realindividuals areabletocrossalsolargeportions ofsubstrateswithoutslowing down orchanging direction. Such a cell behavior cannot be cap- tured in our approach since we do not include a proper model componentreproducingselected intracellularcytoskeletal dynam- ics (in this respect, the interested reader may refer to [56,57],

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Fig. 8. Results for the simulations with a soft to stiff gradient ( Section 3.5 ). (a) | v ext| necessary to allow cells reach the opposite border of the domain versus width of substrate stripes. (b) Average cell velocity over the different substrate regions for different widths of the matrix stripes in case of a stiffness gradient sufficient fine-grained to have a cell persistent movement even in the absence of an external potential.

where polarization processes and the subsequent cell persistent movementsaresimulatedinCPMseitherbyintroducinganasym- metric correction to the Boltzmann probability law orby adding a further inertialtermin theHamiltonian). TheCPM cellsofour modelareonly abletoisotropicallyspread(due todecrementsin their rigidity upon contact with stiff substrates) or elongate fol- lowing thegeometryof theunderlying matrixregion in orderto maximize their adhesive interactionswiththe stiffer areasofthe domain(butonlywhentheexternalpotentialissubstantiallylow, see Fig. 6a and c). However, the model presented in this paper can beusedtopredict ifa sustainedcellmotioncanbe achieved by onlyvaryingthegeometrical characteristicsofthematrixsub- strate. With this purpose in mind, we employ the same type of domainasinSection3.4,butweprogressivelydecreasethewidth ofthesubstratestripes.Wethenevaluatetheminimal magnitude of the external potential neededby cells to reach the border of thedomainoppositetotheirinitialposition(againthesouth-west corner). Cell rigidity follows the law in Eq. (8) and the simula- tionslast10,000MCS(5.5h).Assummarizedinpanel(a)ofFig.8, we canobserveatri-phasicbehavior. Forsufficientlywidestripes (i.e.,>45

μ

m), a cellsustained movementresults onlywithvery

highexternalpotentials(i.e.,>25×10−21kgm/s2).Then,forlower stripewidths(i.e.,intherangeof35–45

μ

m)thecriticalvalue of

the external potential modulus decreases almost linearly. Finally, forlow enough stripewidths (i.e.,<35

μ

m), thepotential neces-

sary to have a sustainedcell movement significantly drops, until becoming negligible for stripe widths lower than 35

μ

m (Movie

7).Summingup,wecanstatethatthecharacteristicdimensionof the stiffness gradient (here determined by the widthof the ma- trixstripes),whichallowsapersistentcellmovementwithoutthe artificialhelp ofan externalbias,is lowerthan themeancell di- ameter (i.e., that in our simulations is around 40–45

μ

m). From

a computational viewpoint, the rationale of this behavior is that when aCPMcell islocatedon agivensubstratestripeitishow- ever ableto wandering its closeproximity (dueto the stochastic Metropolis algorithm) which, if the stripe width is low enough, includes the neighboring matrix region. In thisrespect, the CPM cell simultaneously experiences the adhesive affinity with a cou- pleofneighboringsubstratestripesandthenitmovestowardthe stifferone,therebyadvancingacrossthedomain.Suchaprocessis reiterated forall pairs of substratestripes, thereby resultingin a sustained directionalmovement. These resultscan be interpreted from an experimental viewpoint asa prediction on the fact that cellsmayexhibitapersistentmotionalsowithoutanintracellular polarization, i.e.,by onlymaintainanamoeboidmovement,ifthe substratestiffnessgradientissufficientlyfine-grained.

We finally conclude thissection by analyzing how cell veloc- ityisaffectedbythewideofthesubstratestripes,intherangeof valuessufficientlylowtohaveasustainedcellcrawlingintheab- senceofanexternalpotential(i.e.,<35

μ

m).Asitispossibletosee

inpanel(b)ofFig.8,lowerwidthsofthesubstrateregions(which means, as previously seen, more fine-grained stiffness gradients) resultsin incrementsincell average velocity.From thecomputa- tionalviewpoint,thisisduetothefactthatthemorethedifferent stripesofthematrixaresmall,themorethepreviouslydescribed cellprobing mechanismisfacilitatedandaccelerated, thereby re- sultinginhighercellaveragevelocities.

3.6.Softsquaresembeddedinastiff substrate

As a final simulation, we test the substrate configuration in Fig. 2d, where four soft squares (

τ

=S1 with JC,S

1=Jso f t= 25×10−15kg/s2) are embedded in a stiff substrate (

τ

=S2 with

JC,S

2=Jsti f f= 1×1015kg/s2) at its three corners (north-west, north-east and south-east) and at the center. A cell

1 is ini- tially seeded at the south-west corner and an external poten- tial is introduced toward the north-east corner of the domain, whose magnitude |vext| has been set equal to an intermediate 14×10−21kgm/s2.Asusual,cellrigidityisallowedtodecreaseac- cording to Eq. (6) and the observation time is 10,000 MCS (i.e., nearly 5.5h). The cell startsmoving in the directiondetermined bythepotentialwithatrajectoryofapproximately45°but,assoon asitencountersthecentralsoftsquare,thecellavoidsandcircum- ventsit.Astheoriginalpathisrecovered,thecellneedstosqueeze betweenthenorth-eastsquare andthesubstratefrontierinorder to achieve thetarget corner ofthe domain (Movie8 andFig. 9).

Thechoiceofthemigrationtrackmaydependontheinitialposi- tionofthecell.Inthepresentsimulation,thecellisseededalong the substratediagonal, thus the probability ofcircumventing the centralsoftsquarecounterclockwise(asithappenshere)orclock- wise are the same.However, ifthe cell is seededslightly down- wardand/or right, it will mostcertainly employa counter clock- wisetrajectory,whereasifitisplatedupward,itwillprobablyfol- lowaclockwisepath.Itisusefultonoticethatwithasignificantly highermodulusoftheexternalpotentialthecellwouldhavebeen able to pass across the soft regions, without deforming to avoid them,coherentlywiththesimulationsproposedinFig.6c.

Thisconfigurationissimilartothatproposedin[23]wherethe cellmustavoidstwoslippingregionsinordertoreachtheexternal cue placed at 45°. Although the employed numerical approaches are substantially different, taken together the outcomes confirm

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Fig. 9. Snapshots from a representative simulation dealing with a domain with four soft squares (red) embedded in a stiff substrate (yellow). The cell is initially seeded at the south-west corner and migrates in the direction of an external potential ( |vext|= 14 ×10 −21kg m/s 2), i.e., toward the north-east corner. Snapshots are taken at 2 min (a), 30 min (b), 1.5 h (c), 2 h (d), 2.5 h (e), 3.5 h (f), 4.5 h (g) and 5.5 h (g). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

thetendency of the cell to migrateover stiffer substrates where thehigheradhesionforcesmaybedeveloped.

4. Conclusions

In thispaper we haveproposed a three-dimensionalCPM ap- proachto simulatesingle cell migration over matrix domains in whichsoftandstiff regionsarecombined.

The CPM method is becoming an increasingly common tech- niqueforthemathematicalmodelingofawiderangeofbiological phenomena, includingavascular andvascular tumor growth[58–

61],gastrulation[62],skinpigmentation[63],yeastcolonygrowth [64], stem cell differentiation [65], fruiting body formation of Dictyostelium discoideum [66], epidermal formation [67], hydra regeneration [66], retinal patterning [68], wound healing [69,70], biofilms[71],chick limb-budgrowth [72–74],cellular differentia- tionandgrowthoftissues,bloodflowandthrombusdevelopment [75–77], angiogenesis [70,78–81], dynamics of vascular cells [82–85],cell scattering[86],cell migration onand within matrix environments[56,57,87]. Notably, in [88] the authors introduced acompartmentalizedapproachtosubdivide aMyxococcusxanthus into strings of subcellular domains with different rigidity, this in order to give the bacterium a particular geometry and to control its overall length. Further, in [89] a keratocyte has been represented with a set of undifferentiated hexagonal subunits, whichhasallowedtoreproduceitspolarization duringmotion.In thisrespect,itis usefultounderline that, ascommented in[25], althoughtheseapproachesarecorrect, thefactthat theproposed subcellularcompartmentsdonothaveanimmediateordirectcor- respondencewithrealsubcellularelements,haslimitedthepracti- calityandtheusefulnessoftherelativemodels.Themostaccurate wayof realistically reproducing different and extremely complex cell morphologies is to compartmentalize them according to the compartmentalization“suggestedinnature”,andthustoexplicitly represent for instance the plasmamembrane (PM), the cytosolic

region, thenucleus, and other intracellular organelles (e.g.,mito- chondria,ribosomes,Golgiapparatus,andsecretorygranules).This wayisinfactpossible,forexample,tolocalizewithin theproper cell compartment selected biochemicalpathwaysand/or tostudy the role play by the nucleusin cell movement, given its higher rigiditywithrespecttothesurroundingcytoplasm[56,57,87].

Keybenefits oftheCPM energeticformalism are itssimplicity andextensibility:almost anybiological mechanismcaninfact be included inthe model,simply by addingan appropriate general- izedpotentialtermintheHamiltonianfunctional.Forinstance,itis possibletoeasilycomprehendtheimportanceofeachmechanism involvedin thesimulated phenomenonby only alteringtherela- tive Potts parameter, so that the other terms inthe Hamiltonian scaleaccordingly.Inparticular, byequatingall theother termsto zero,itispossibletounderstandwhetheramechanismisindivid- uallycapableofproducingthephenomenonofinterestorwhether it requires cooperative processes. Further critical features of the CPM(comparedtoalternativecell-basedmodelingapproachesthat represent biological individuals as point particles, such as Inter- acting Particle Systems or purely discrete models, or fixed-sized spheresorellipsoids,suchasCellularAutomata)isthati)itdiffer- entiatesbetweenbound andunbound regions ofcell membranes andii)morphologicalchangescanbeeasilyandrealisticallyrepro- duced. Thesecharacteristicshavebeenfundamentalinourchoice ofusingaCPMto describethephenomenon ofourinterestsince they are particularly suitable to implement our two main model assumptions,drawnaccordingtotheexperimentalobservationsre- ported in the literature: i) the adhesiveness of cells changes ac- cordingtothesubstratestiffness,thatmodelsthefactthathigher traction forcesandmore stablefocal pointsare generatedover a stiffersubstrate[16,35–38]andii)eachcell adaptsitsmorphology asa function ofthe substratestiffnessso that over a softregion itmaintainsaroundedshape,whereasoverastifferdomainasig- nificantspreadingoccurs[39–43].Theconsiderationsabovearein remarkableagreementwiththescholarlydissertationproposedby

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Voss-Böhmeintheconclusivesectionofherarticle[33].Sheinfact argued that the applicationofCPMs is reasonablewhenthe bio- logicalproblemofinterestinvolves“considerablevariabilityincell sizesandshapes”,whichisthecaseofthecellmorphologicaltran- sitions dueto contact withsoft/stiff substrates. On the opposite, when“essentiallyisotropic,non-polarizedcellsofuniformsizeare considered”,itwouldbepreferabletheuseofmorecoarse-grained modeling approaches, like the alreadycited Cellular Automataor InteractingParticleSystems,whicharebetteranalyzedbothmech- anisticallyandanalytically.

Further,we haveoptedfora3D settingsincetheadhesivein- teractionsbetweencellsandmatrixsubstratesoccurunderthecell body (i.e., they are localized over the contact area between the cellsthemselvesandthe underneathsubstrate).Inbi-dimensional CPMscell–matrixinteractionsinsteadoccuronly“laterally”,asthe cellsdonotmove onsubstratesbutwithinthesameplane asthe matrix.Indeed,athree-dimensionaldomainismoreappropriateto reproduceanadhesive-drivencellmigration.

We have then used our CPM-based approach to test cell be- haviorindifferentdomainconfigurations,wheresoftandstiff sub- stratescoexisted.Inparticular,thenumericaloutcomeshavebeen consistently compared to specific experimental data, in terms of cell morphology, distance covered, spreading/adhesive area and migrationspeed.Inthisrespect,followingthedichotomyproposed inthealreadycitedworkbyVoss-Böhme[33],wehaveinterpreted ourCPMasaphenomenologicalmethod.Inparticular,theresulting remarkableagreement (notonly qualitative butalsoquantitative) between in vitro and in silico data has allowed us to conclude that our approach,although stronglysimplified, wasable tocap- ture themain mechanisms underlying cell migrationin presence ofdurotaxis.Wehavefinallyturnedtouseourmodelinapredic- tive manner, with the aimto analyze how the external potential andthe criticaldimensionsofa substratestiffnessgradient(here representedby thewidthofthedifferenttypesofmatrixstripes) affectcellmovement.Inthisrespect,wehavefoundthatcellsare abletoachieveasustainedcellmigrationintheabsenceofanex- ternalbias(andintheabsenceofintracellular polarizationmech- anisms) wheretheunderneath matrixis characterizedby asuffi- cientfine-grainedgradientofrigidity.

However, our approach is not free of some serious short- comings. First, it does not reproduce the active and continuous reorganization of the cytoskeleton, which provides the support forcellsandmediatestheir coordinatedanddirected movements, mainlyinresponsetomechanicaltensionsandstressesexchanged with the underneath substrate. In this respect, selected geomet- ricalandmechanicalpropertiesofthecells,such their elongation andelasticity,shouldevolveaccordingtoamodelofactinfilament dynamics, which are powered, for example, by ATP (adenosine triphosphate) hydrolysis and controlled by inside-out signaling mechanisms transmitted from and by the extracellular matrix via focal adhesion points. Further, in our model, the substrates are not deformable and therefore it has not been possible to account how the matrix reacts to the probing processes exerted bycrawlingcells. Finally,itisusefultounderlinethatourspecific CPMapplicationdoesnotsufferofthelimitationthatVoss-Böhme proved to characterize most CPMs (see again [33]), i.e., cellsdie outinthelong-runduetomodificationsintheoriginalMetropolis algorithm.Wehaveinfact focusedonrelativelyshortobservation times:ourmodelhasindeedworkedinawell-behavedparameter regimewherethetemporalevolutionofthesimulatedsystemhas been still directed toward the minimization of the Hamiltonian functional and the non-controlled, voter-like part of the lattice updateshasbeennegligible.

Movie1Simulation ofcell migrationoverastiff-soft substrate (yellow=stiff region,red=softregion)inpresenceofanexternal

potentialdirectedtowardthenorth-westcorner(Section 3.2). The cellisinitiallyseededatthesouth-eastcorner.

Acknowledgments

Thisworkwasinitiatedandpartiallycompleted whileRachele Allenawasa visitor to the Dipartimento diScienze Matematiche ofthePolitecnicodiTorinofundedbytheGruppoNazionaleperla FisicaMatematica(GNFM).

Appendix

A.1. Morphologicalandmigratorydeterminants

The positionof a cell wasestablished by thecoordinate ofits centerofmass(CM).Inparticular,acellwasassumedtobelocated ona giventype ofsubstrateifits center ofmasswaslocated on thatmatrixregion.Inthisrespect,themigratorytrajectoryofacell wasgeneratedbytrackingthepositionofitscenterofmassateach timestep(i.e.,ateachMCS).

Theadhesive areaofa cell wasdefinedasthe extensionofits surfaceincontactwiththesubstrateofinterestatthefinalobser- vationtime.

Theaveragevelocityofanindividualonagiventypeofsubstrate wasmeasuredastheratiobetweenthewidthofthesubstratere- gionitself (whichisclarified foreachsimulationsetting) andthe time neededby thecell to crossit.In thisrespect, toobtain the amount oftime spent by a cell to pass a given matrix region it issufficienttomultiplythecorresponding averagevelocityforthe widthofthesubstrateofinterest.

A.2.Statistics

In the plots, we represented cell trajectories coming from 10 independent and randomly chosen simulations. A number of 10 was chosen since we observed that it was sufficient to have a correctinterpretation ofthesimulation outcomesbutit wasalso low enoughto havean acceptable graphical quality,astoomany cell paths overlappedone toeach other, thereby resultingundis- tinguishable.

Cellaverage velocity and adhesive area were instead given in the corresponding graphs asmean ± sd over 100 independent simulations.

Inthe plotsrepresentingthe cell final distributionon thedif- ferenttypesofsubstrate, therelative frequencywasgivenby the number of individuals that, over 100 independent simulations, werelocatedovereachmatrixregionattheendoftheobservation time. Indeed, the sumof the relative frequencies is, in all cases, equalto100.

Finally, dashed and plain ellipsoids representing, respectively, initial and final cell morphologies in a given simulation setting were established by interpolating the cell adhesive areas coming from10independentsimulations(typicallytheonesusedtotrack thecelltrajectoriesforthesamesimulationsetting).Obviously,the initial cell position wasthe constant for each simulation setting, whereas the initial cell shape was the same for all cases (i.e., a hemisphereof20

μ

mofradius).

A.3.Parameterestimates

GiventheenergeticnatureoftheCPM,adirectone-to-onecor- respondence betweenmodel parameters and experimental quan- tities is not straightforward (as commented also in [27] and in [90]).Inparticular,asexplainedindetailsin[33],theCPMparam- eterscanbesubdividedini)directlyinterpretableandmeasurable

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