[PDF] Adhesion modulates cell morphology and migration within dense

43559_72020_jpcm_adhesion.pdf

Journal of Physics: Condensed Matter

J. Phys.: Condens. Matter32(2020) 314001 (12pp)https://doi.org/10.1088/1361-648X/ab7c17

Adhesion modulates cell morphology and

migration within dense fibrous networks

Maurício Moreira-Soares

1,4, Susana P Cunha2, José Rafael Bordin3

and Rui D M Travasso1

1CFisUC, Department of Physics, University of Coimbra, Rua Larga, 3004-516 Coimbra, Portugal

2CQC, Department of Chemistry, University of Coimbra, Rua Larga, 3004-535 Coimbra, Portugal

3Department of Physics, Institute of Physics and Mathematics, Federal University of Pelotas, Rua dos

Ipês, Capão do Leão, RS, 96050-500, Brazil

E-mail:

mmsoares@uc.ptandjrbordin@ufpel.edu.br Received 11 November 2019, revised 21 February 2020

Accepted for publication 3 March 2020

Published 6 May 2020

Abstract

One of the most fundamental abilities required for the sustainability of complex life forms is active cell migration, since it is essential in diverse processes from morphogenesis to leukocyte chemotaxis in immune response. The movement of a cell is the result of intricate mechanisms, that involve the coordination between mechanical forces, biochemical regulatory pathways and environmental cues. In particular, epithelial cancer cells have to employ mechanical strategies in order to migrate through the tissue"s basement membrane and in?ltrate the bloodstream during the invasion stage of metastasis. In this work we explore how mechanical interactions such as spatial restriction and adhesion affect migration of a self-propelled droplet in dense ?brous media. We have performed a systematic analysis using a phase-?eld model and we propose a novel approach to simulate cell migration with dissipative particle dynamics modelling. With this purpose we have measured in our simulation the cell"s velocity and quanti?ed its morphology as a function of the ?bre density and of its adhesiveness to the matrix ?bres. Furthermore, wehave compared our results to a previousin vitromigration assay of ?brosarcoma cells in ?brous matrices. The results show good agreement between the two methodologies and experiments in the literature, which indicates that these minimalist descriptions are able to capture the main features of the system. Our results indicate that adhesiveness is critical for cellmigration, by modulating cell morphology in crowded environments and by enhancing cell velocity. In addition, our analysis suggests that matrix metalloproteinases (MMPs) play an important role as adhesiveness modulators. We propose that new assays should be carried outto address the role of adhesion and the effect of different MMPs in cell migration under con?ned conditions. Keywords: adhesion, cancer metastasis, invasion, phase-?eld, dissipative particle dynamics 4 Author to whom any correspondence must be addressed. SSupplementary material for this article is availableonline (Some ?gures may appear in colour only in the online journal)

1. Introduction

Cancer progression is a multistep process. During metastasis in solid carcinomas, tumour cells leave the primary colony, invade the circulatory system, reach distant organs and form

new cancercell colonies. This movementof cancerouscells toother organs and tissues is the main cause of death in onco-logical patients [

1,2]. The existence of metastasis is one of

the markers that indicate poor prognosis [

3,4]. This intri-

cate mechanism can often be divided into distinct stages: invasion, intravasation, extravasation and colonisation[ 1,5]. In this work we focus on the invasion stage, described by the migratory event that takes place after the detachment

1361-648X/20/314001+12$33.001©2020 IOP Publishing Ltd Printed in the UK

J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al of a tumour cell from the primary colony. As the tumour cells leave the primary tumour and start their journey in the direction of a blood vessel there are signalling cascades and mechanical regulation events that modulate cellular migration [

6-8]. In the last decades, due to the advance of microscopy

and computational modelling, experimentalists and computa- tional biologists have been studying the mechanical aspects of cellular behaviour with the aim of better understanding the complex process that is metastasis [

9-14]. However, the

current state of the art is still far from the full comprehen- sion of the mechanics behind cancer cell migration in tissues in vivo. One of the main regulators of cell function is the microen- vironment, and in particular the extracellular matrix (ECM). The ECM is a rich network of macromolecules that occupies the interstitial space between cells and provides biochemical and mechanical support [

15]. The ECM is composed of water,

proteoglycansand?brousproteinssuch as collagen,as wellas signalling molecules and growth factors [

6,16]. This struc-

ture can be remodelled and degraded by cells through their mechanical action or by chemical reactions [

17,18]. Enzymes

present in this environment, among other functions, regulate and/or promote the proteolytic activity within the ECM while reshaping the medium [

19,20]. The biochemical content and

the mechanical properties of the ECM may vary from tissue to tissue, affecting the behaviour of the cells that are depen- dentoftheseproperties[

16].Inparticular,themechanicalcon-

straintsimposedbytheECMmayaffectdirectlycellmigration [

21,22]. For instance, high density collagen matrices induce

the cells to employ new strategies in order to enhance migra- tion within the restricted space environment [

23-25]. There

are several experimental observationswhich suggest that cells in con?ned space migrate preferentially along ?bronectin ?bres and preexisting paths in the ECM [

26,27], searching

for the direction of least resistance [

28-31] and deforming

their shape in order to follow these channel-like structures [

32,33].

The cell"s capability of adhering to their surrounding is mediated by transmembrane proteins in the cell membrane. These matrix receptors bind proteins present in the ECM to form cell-matrix junctions [

34]. The main group of cell adhe-

sion molecules (CAMs) responsible for linking the cell to the matrix components is the integrin family. Integrins are able to form anchoring junctions linking actin ?laments and interme- diate ?laments in the cell interior to the collagen, ?bronectin andlamininfromtheECM[

35].Toincreaseadhesionataloca-

tion, integrins cluster at that region of the membrane strength- ening locally the anchorage of the membrane to the ECM ?laments. There are different kinds of adhesion structures with speci?c functions: focal adhesions (FAs), responsible for strong adherent sites during cell migration; ?brillar adhe- sions, that are originated from the continuous applicationof forcesthroughFAs,dependingonactomyosincontractilityand matrix bending rigidity [

36]; podosomes/invadopodia, which

are actin rich structures and distinguishable from FAs due to

their shorter lifespan of a few minutes and their additionalfunction in promoting invasion in the ECM [

37,38], among

others. The matrix metalloproteinases (MMPs) belong to a family of enzymes that catalyse the cleavage of proteins [

39] and ini-

tiallyemergedas themaincollagen-degradingenzymes.Since then, further studies have demonstratedtheir proteolyticin?u- ence over other components of the ECM [

40,41]. For a long

time the importance of the MMP family was restricted to this single role and consequently obscured its crucial positionin cell dynamics. In fact, MMPs act also by regulating cell func- tionsinabroadersensethroughthecleavageofothersproteins, which have consequences in cell adhesion, proliferation, sur- vival and migration. MMP activity is linked to the cell ability of adhering to the ECM due to its in?uence on speci?c CAMs [

42-44], and even to the regulation of actin polymerisation

[ 45].
Mathematical modelling has demonstrated its usefulness in the study of biological systems when coupled with exper- imental assays [

46,47]. In our perspective, the aim of

building a mathematical description for such intricate sys- tems is not to completely characterise the system in its full complexity. Instead, the challenge is to implement assertive assumptions and approximationsthat simplify the study while preservingthekeyfeaturesofthebiologicalsystem.Strikingly, recent works have adopted two or more different mathemat- ical approaches to tackle the same problem in order to test whether the common mechanistic assumptions of each model are relevant or not [

48,49]. When different models of a sin-

gle biologicalprocess yield similar behavioursthen the results are robust to the modelling implementation. In this work we employ two particular approaches for cell migration that are complementary to each other, the phase-?eld model (PFM) and the dissipative particle dynamics (DPD). PFMs have been extensively used to study biological sys- tems, such as in tumour growth [

50-53], vessel growth

[

54-57],cellmonolayers[58,59],axonaldevelopmentinneu-

rons[

60],immunesystemresponse[61,62]andcellularmotil-

ity [

63-68]. Since they are focused on the interface dynamics,

a major advantageof PFMs is the lower numberof parameters when compared to other established methods such as cellular potts models (CPMs) [

69,70], agent-based models (ABMs)

or mixture models [

71-73]. As a result, this low number of

parameters gives the possibility to build minimalist models to approach speci?c questions about the system of interest. In particular, in this work we model a self propelled cell as a droplet within a porous material composed of a ?bre net- work.The?brenetworkismeanttodescribecollagenscaffolds where cells are deposited forin vitroexperiments, mimick- ing the ECM structure. The cell maintains its shape due to the surface tension, and interacts with the ?bres by adhesion and depletion.We carry out a systematic study by varyingtwo parameters: the density of the ?bres and the adhesion strength between the cell and the ?bres. The effect of mechanical con- ?nement on cells has been explored with multiple models and at different scales: from the ?lopodia dynamics [

74], single

cell migration [

69] to multicellular systems [75], where some

aspects of the velocity dependencewith ECM properties were discussed. 2 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al Recently, the DPD approach has been used in the compu- tational biology ?eld to describe tissues and cells [

76-79].

TheDPD modelis aparticle-basedmethodintroducedin 1992 for hydrodynamics phenomena simulation which combines the algorithmic scheme of molecular dynamics (MD) and the time-stepping of lattice-gas automata (LGA) [

80]. These

features delivered a method faster than MD and more ?exi- ble than LGA, thus enabling the exploration of systems with larger spatial scale and for longer timescales. Similar to the aforementionedmethods,DPD uses Newtonianmechanicsfor calculating explicitly the interaction forces between particles. In this work we introduce a novel description for modelling the cell membrane and the ECM using DPD. Similarly to the phase-?eld model, our system is composed by the cell and the ?bres. The cell is described as a spherical distribution of monomers around a central monomer at a ?xed distance. The monomers at the cell surface are linked to the central monomer through a harmonic potential, mimicking an elas- tic membrane. On the other hand, the ?bres are modelled as chains of monomers connected via a harmonic potential and randomly distributed in space. In the following section, we present the two computational methods adopted and the mathematical details behind each approach. In the section Results and Discussion, we com- pare the results between the two models. We also compare our results with experimental data available in the literature and, ?nally in the Conclusions section, we draw some conclu- sions and introduce hypotheses about the biological processes modelled with these systems.

2. Methods

2.1. Phase-field model

Our simulation is based on the mathematical model for mul- ticellular systems introduced in reference [

81] and explored

furtherin [

82].First wede?nea free energyfunctionalF[φ,ψ]

that will determinethe system dynamics.Similarly to [

56],we

use two order parametersφ(r,t) andψ(r,t). The cell is iden- ti?ed in space by the regions whereφ≈1 and the ECM is de?ned by the domain whereψ≈1 (see ?gure

1). The inter-

stitial space is de?ned as the region where both functions are close to zero. This free energy functional is the sum of a func- tional related to the cell"s energyFcell[φ], with another func- tional describing the cell"s interaction with other biological entitiesFint[φ,ψ], such as the ECM: F[φ,ψ]=Fcell[φ(r,t)]+Fint[φ(r,t),ψ(r,t)].(1)

The ?rst component of this functional, written in

equation (

2), is given by the Ginzburg-Landau free energy

with a double well potential, with two stable solutions φ={0,1}, which correspond respectively to the exterior and interior of the cell (see ?gure 2), F cell[φ]=? Ω? ε2

2|?φ|2+14φ2(1-φ)2?

d 3r + αv

12?Vtarget-V[φ]?2, (2)

Figure 1.The ?eldsφandψare depicted in the image. The cell (red) is de?ned by the region whereφ=1 andψ=0, while the ?bres (cyan) are represented by the domains ofψ=1 andφ=0. The interstitial space is represented by both ?elds being zero simultaneously (white). Figure 2.The double-well potential is used in order to maintain the two stable phases for describing the cell interior and the exterior. The two minima are symmetrical and atφ=1 andφ=0 respectively [ 83].
whereΩdenotes the total volume of the system. The coef- ?cientεis a positive constant that is related to the width of the interface and to the surface tensionσ(see SI section S2 stacks.iop.org/JPhysCM/32/314001/mmedia). The squared gradient term acts as an energetic cost for maintaining the interface: more interface costs more energy. The coef?cient α vin the last term is the Lagrange multiplier for the volume V [φ]=? Ωh(φ)d3r, whereh(φ)=φ2(3-2φ),αvis the reinforcement coef?cient for the volume constraint andVtarget is the target volume of the cell: ifV[φ]>Vtargetthis term removes material from the surface and whenV[φ]81] for a detailed description). The interaction component of the free energy is given by F int[φ,ψ]=? Ω [η?h(φ)· ?ψ+γh(φ)h(ψ)]d3r.(3) 3 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al The ?rst term characterises the adhesion between the cell and the matrix (controlled by the parameterη), and the sec- ond penalises energetically the overlap between cells and matrix ?bres (controlled by the parameterγ). Given the free energy functional, the equation for the cell dynamics is derived as a gradient descent equation with an advective term. Therefore, in this case we consider theφchange rate to be proportionalto the gradient of the free energy functional [

83,84]

∂ tφ+v· ?φ=-δF

δφ.(4)

Taking the free-energy functional derivative, we ?nally have the equation for the cell dynamics ∂ tφ+v· ?φ=ε2?φ+φ(1-φ)?

φ-1

2-γh(ψ)

+η?2ψ+αV?Vtarget-V[φ]?? , (5) where the velocityvis written as the gradient of the chemical ?eldc,whichformetastaticcellscanbeassumedastheoxygen concentration in the tissue guiding the cell to a blood vessel during invasion: v=χ?c, (6) whereχis chemotactic parameter. In this study we assumed that thisvelocityis constantanddirectedalongthex-direction, which allowed us to isolate the in?uence of the mechanical constraints in the movement.The details regardingthe numer- ical implementationare in the supportinginformation(SI)and thesoftwareisavailableunderthe

DOI10.5281/zenodo.35342

[85,86].

2.2. Dissipative particle dynamics

We propose a model for the system described in the previous sections using a DPD approach.Moreover,the DPD allows us to exploredifferentperspectivesthatwouldbeextremelycom- plicated using only the PFM, such as the introduction of elas- tic ?bres. Brie?y stated, in DPD we solve Newton"s equation of motion based on the total force that acts on each particle i[

77,87]:

? i?=j F ij =? i?=j ?

FCij +FDij +FRij +FHij ?.(7)

As usual in DPD models, each bead is interpreted through a coarse-grained perspective, where distinct numbers of parti- cles are combined into a single bead. Each bead species has distinct properties, de?ned by their interaction potentials. The ?rstterminthisequationistheconservativeforce,whichmod- elstheinteractionbetweenthedifferentbeadtypes,asfunction of the bead"s types. Here, our conservative term is F C ij (rij )=aij ωCR(rij )ˆrij -bij ωCA(r)ˆrij , (8) where the ?rst term is the repulsive force, withaijbeing the maximum repulsion between the particlesiandj, and the second term the attractive force with maximum intensitybij.

In equation (

8)rij=ri-rjis the distance between the twoparticles, andωCR(r), known as the weight function, usually

assumes the form ω

CR(rij )=??????

1-rij

r1? , ifrij 0, otherwise, (9) withr1=1.25μm. This term is responsible for the repulsion due the excluded volume of the beads. To model the adhe- sion of cells to the ECM ?bre we added the second term of equation(

8). This attractiveterm in the conservativeforcewas

proposedrecentlyforthestudyofcelldeformationinasurface [

88]. The attractive weight function is given by:

ω

CA(rij )=??????

1-rij -r1

r2-r1? , ifr10, otherwise, (10) wherer2=2.50μm. This term acts only near the ?bre"s sur- face and describes the biochemical adhesion forces between the cell and the ?bres.

The dissipative force,FDij in equation (

7) represents the

effect of viscosity. It uses the relative velocity between two particles to slow down the motion with respect to each other and is given by F D ij =-γωD(rij )(ˆrij·vij )ˆrij, (11) whereγis a coef?cient,vij=vi-vjis the relative veloc- ity between the particles andωD(rij) is a weight function that depends on the distance between the particles: ω

D(rij )=??????

1-rij

r2? , ifrij 0, otherwise.(12) Finally, the random forceFRij represents the thermal or vibrational energy of the system and includes ?uctuation effects, F R ij =σωR(rij )ξij ˆrij , (13) whereσis a coef?cient,ωR(rij) the weight function andξija random constant obtained from a Gaussian distribution. The dissipative and random forces are related by their coef?cients and weight functions, ω

D(r)=?ωR(r)?2,σ2=2γkBT

m, (14) wherekBis the Boltzmann constant andTthe equilibrium temperature. These relations turn the DPD into a thermostat. Since the algorithmdependson the relative velocitiesbetween the beads and since all interactions are spherically symmetric, the DPD is an isotropic Galilean invariant thermostat which preserves the hydrodynamics[ 89].
Here,weproposea modeltostudytherelationbetweencell shapeanddynamics.Ourcell consists of452beadsdistributed around a sphere of radiusac=6μm, the cell radius. Cell sur- face beads c repel themselves by the conservative force with an intensityofacc=12.5×10-11N. These surfacecell beads 4 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al Figure 3.(a) Schematic depiction of the DPD cell model. (b) Cell in a bulk simulation showing the central ghost bead. (c) Same as(b), but with the surface beads with the correct size that is included in the simulation. Figure 4.Depiction of the DPD model for a ?exible network. The ?bres (cyan) are represented by a chain of monomers connected by a harmonic potential. The cell (red) is modelled as an approximately spherical set of beads bound by an elastic potential to a central bead inside the cell. The solvent is represented in yellow. For better visualisation the solvent beads are smaller than the other beads and only a few ?bre chains are shown. areattachedbya standardharmonicforceFHtoacentralghost beadg, as depicted in ?gure 3(a): F H cg=-kcg(rcg-ac)ˆrcg, (15) wherercgis the distancebetweena c beadandthe centralbead g, andkcg=30 pNμm-1. Figure

3(b) shows a snapshot of the

cell in a bulk simulation with small surface beads so the ghost bead can be seen. In ?gure

3(c) we show the same snapshot

but with the surfacebeads depictedin their regularsize. Aswe can see, the cell remains approximately spherical-obviously less spherical than the raspberry model [

90]-and, as we will

demonstrate, it can deform to pass through obstacles. The other two species in the DPD system are the solvent and the ?bre matrix. Solvent beads s have a number density ρ

N=4.0, consistent with previous works [

91,92], and are

repelled by c beads with a strengthacs=65×10-11N with a cut off radius of 2.5μm. The ?bres are modelled as polymers with beads of type f, and the interaction parameters betweenc andf beadsarethesameasbetweencands,acs=acf.Theonly attraction interaction in the system is between c and f beads. In this case, the attractiveparameterplays the role ofthe adhe- sion de?nedbythe relationbcf=ηDPDacf. HereηDPDprovides the strength of the adhesion between the cell and the polymer network. In the case of a rigid ?bre network,the polymersare placed randomly inside the simulation box and remain ?xed, i.e., the corresponding beads are not integrated in time. In the case of ?exible ?bre network only the beads placed at the ends of the polymeric chains are ?xed, and the rest of the polymer can Figure 5.Examples of cell deformations inside rigid (A) and ?exible (B) ?bres in DPD compared to phase-?eld model (C). oscillate similarly to a rope with ?xed ends. In this ?exible case, the beads in the same polymer are bound by the har- monicbond,equation(

15),withkff=kcg=30pNμm-1. The

case of rigid polymers can be imagined as the limit case of k ff→ ∞. This description is motivated by the ?brous struc- tures of collagen present in the ECM, which may have dif- ferent stiffness depending on the collagen concentration,its origin (skin, tendon, vasculature, organs, bone) and/or poly- merization process of collagen-based scaffolds forin vitro experiments [

93-95].

The cell is placed in the left buffer zone of the simulation boxandaconstantforceinthelongitudinaldirectionisapplied to the cell mimicking the chemical gradient. Then we analyse thesystem propertiesduringthe cell migrationin the ?bre net- work region. The simulation ends when the cell reaches the rightbuffer.TheSIpresentsadetaileddescriptionofthesetup.

We show in ?gure

4a schematic depiction of the simulation

box.

3. Results and discussion

As previously described in section

2, we start by simulating a

dropletmigratingtroughthe?brenetworkusingboththePFM and DPD model, and quantify the in?uence of the medium in its shape and movement. As expected, the properties of the ?brenetwork,particularlytheporecrosssection,adhesiveness and elasticity, affect cell migration and morphology. We quantify migration by measuring the mean velocity ?v?, the mean squared displacement (MSD) and the diffusion exponentβ(see SI section S3 for the de?nition). In addi- tion, morphology is quanti?ed by the cell"s radius of gyration R gand surface energy, which measure deformation and the surface roughness respectively. These properties are obtained as a function of the ?bre densityρ, de?ned by the quotient of occupied and total volume of the system, the mean pore cross section (see SI) and the adhesion strengthηbetween the 5 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al

Figure 6.Position of the cell center of mass as function of time duringits movement for ?bre densities (a)ρ=0.20, (b)ρ=0.55 and (c)

ρ=0.70 for the DPD model with rigid ?bres. The inset in (c) shows the curve for the ?exible matrix.

dropletsandthe?bres.Wecompareourresultstoexperimental data for cell migrationin vitro.

3.1. Cell movement and matrix density

For the cell to overcome the obstacles in dense ?bre net- works it needs to deform. In ?gure

5we show some snapshots

from the cell migration. Row (A) refers to cell modelled with DPD when the ?bres are rigid, while row (B) shows exam- ples for the ?exible matrix scenario. For better visualisation, the ?bres are smaller thanthe cell beads, andthe solventbeads are not shown. Comparing (A.I) and (B.I) snapshots, we can see that even for a simple obstacle, which is to bypass a sin- gle ?bre, the cell is more deformed by the rigid ?bre. The deformation becomes larger when the cell is blocked by more than one rigid ?bre, as the (A.II) and (A.III) snapshots show. On the other hand, the cell is only slightly deformed by the ?exible matrix, as the (B.II) and (B.III) snapshots show. We expect that the time needed by the cell to escape from a ?bre trap is directly related with the time necessary to deform its spherical shape and cross through the available space. The PFM presents similar results to the DPD with rigid ?bres, which we can see when comparing the rows (A) and (C) in ?gure 5. To understand the velocity dependence on the ?bre density ρfor rigid and ?exible networks we present in ?gure

6typical

examples of thexcomponent of the cell center of mass (CM) of the DPD cell as functionoftime forthreedifferentdensities ρ={0.20,0.55,0.70}.We presentsimilar resultsforthePFM in ?gure

2of section 3 in SI.

In the examples for the lowest ?bre density,the cell may be blocked a few times for both rigid and ?exible ?bre matrices, but it diffuses almost without obstacles for most of the time. In these low density matrices, the time required for the cell to leave the rigid or ?exible network is similar. However, as ρincreases the scenario changes. Forρ=0.55, ?gure 6(b), we can see that both curves have two trap regions, but the time spent by the cell to leave those regions is much longer in the rigid network. The difference in the stop time for the rigidand?exiblematrices becomesevenmorepronouncedfor

ρ=0.70, ?gure

6(c). Figure 7.Exponentβas function of the ?bre densityρ. For lower density of ?bres (0.1?ρ?0.5) the exponentβremains unaltered with a value near 2, which indicates a ballistic behaviour present in the three models. When the density is increased even furthera transition region appears for the PFM and for the DPD with ?xed ?bres, but not for the DPD with ?exible ?bres. Anotherinterestingfeaturein these systems is the diffusion regime. We quantify the diffusion by scaling the MSD and the time, namely?|r(t)-r(0)|2? ≈tβ. If the MSD increases lin- earlywithtime,β=1.0,thesystemisinthenormalorFickian regime.Ifβ <1.0the diffusionis anomalousandcalled subd- iffusive, while in the case ofβ >1.0 the regime is superdiffu- sive.Thelimitofβ=2.0iscalledballistic.Tounderstandhow the rigid or ?exible matrix can in?uence the diffusion regime we evaluated the exponentβas functionof the ?bre densityρ. The value ofβused here is the obtainedast→ ∞, and not the short time exponent.

We can observein ?gure

7that a superdiffusiveregimewas

observed at lower values ofρfor the three models, in agree- ment with experimental studies [

96]. However, at higher net-

work densities,ρ?0.50, we observe a change in the regime for the DPD model with rigid ?bres and for the PFM. There is a transition from a superdiffusive to a subdiffusive regime. This transition can be directly related to the longer times the cell gets blocked the ?bre matrix. At low densities, since there is force pullingthe cell, the movementis almost ballistic,with 6 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al

Figure 8.Migration velocity as a function of the density of ?bres withthree values of adhesion coef?cientη=0,0.5,1.0 for (a) PFM, (b)

DPD with rigid ?bres and (c) DPD with ?exible ?bres. The velocity starts to decrease forρ >0.4 and the cells stop moving (except for the

?exible ?bres) atρM≈0.7. occasional collisions with the matrix. However, as the density increases, the cell MSD is strongly affected,with long periods of nomobility.Onthe otherhand,the migrationinside?exible matrix is only slightly affected. The movement changes from a near-ballistic to a superdiffusive regime. Here, the change is small and even at high densities the time that the cell remains blocked is small, as we have showed in ?gure

6(c). Therefore,

the ?exibility of the ?bre network plays an important role not only in the velocity of migration, but also in cell deformation and in diffusion regime.

3.2. Adhesiveness as a key modulator in migration

We then explored how adhesion affects cell migration in both PFM and DPD models. The mean velocity is measured as the quotient of the distance traveled by the center of mass of the cell andits respectivetime interval.In ?gure

8we presenthow

the ?bre densityρaffects migration for three different values of adhesionη. The overall behaviour is similar and indepen- dent of the adhesion coef?cient: for lower ?bre densities we obtainthehighestvelocities,whileasthedensityincreases,the velocity becomes smaller the cell gets trapped in the matrix. Nonetheless, by comparing the migration velocity for differ- ent values of adhesion at the higher densitiesρ={0.6,0.7}, we observe that the velocity is higher forη=1 than for the lower valuesη={0.0,0.5}. This dynamic is anticipated by Figure 9.Migration velocity as a function of the adhesion coef?cient forin silicoresults and as a function of the ?bronectin coating concentration for thein vitroassay with ?broblasts [

97]. The

?bronectin concentration is directly related to the adhesiveness, higher concentrations of ?bronectin strengthen the adhesion between the cell and the ECM. The simulation results show clearly a peak in velocity aroundη=1.0. 7 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al experimental results of cell migration, but it is striking to observe this behaviour in this simple model. For further investigation concerning the dependence of the migration velocity on the adhesion we ?xed the density of ?bresatρ=0.6andransimulationsforseveralvaluesofadhe- sion. In ?gure

9the mean velocity is plotted as a function

of the adhesion coef?cient for our computational models and it includes thein vitroresults adapted from [

97] for ?brob-

last migration as a function of the ?bronectin concentration. We can see that the cell migration velocity increases with the adhesion coef?cient until it reaches a maximum, after which it decreases. This suggests that there is an optimal value of adhesionformigratoryevents[

98,99].Indeed,wehavesuper-

posed the experimental data for the cell velocity as a function of the ?bronectin concentration obtained by Maheshwari and colleagues [

97] and the same qualitative behaviour is veri-

?ed (see ?gure

9). Fibronectin appears as protagonist in cell

adhesion and its concentration is a proxy for altering adhe- siveness-more?bronectinpresentintheECMleadstohigher adhesiveness. Experimentally the adhesiveness is linked to the cell capa- bility to exert traction forces in the ?bres in order to pull its body forward. Thus this is the main reason by which the in vitroassay exhibits an increase in velocity with the adhe- siveness and then decreases after reaching a critical point where adhesionis so strongthat it suppressesmigration.How- ever, in our mathematical model we have a slightly differ- ent circumstance that leads to the same qualitative results. The ?rst main difference we should stress here is that adhe- sion is a passive attribute while in nature adhesion is an active dynamical event performed by the cell. The migration of our droplet is affected by adhesion in two different man- ners: (i) changing cell plasticity and (ii) introducing an arti?- cial pulling force. A higher adhesion increases the cell defor- mity (see ?gure

10) thus facilitating the migration through

small pores. Moreover, the adhesion acts as a pulling force due to the leading edge of the cell that feels the ?bres upfront and is attracted to them by the adhesive force. This effect mimics some of the features observed in real cells during migration. The study realised by Wolfet alused anin vitromodel of tumour cells and immune cells (T-cells and neutrophils) migrating in 3D collagen scaffolds of different porositiesto study the physical limits of migration [

9]. In order to compare

our results to their experimental data, calculations were per- formed to determinate the mean pore cross-sectionof the ?bre networks. Each matrix was sliced in several planes orthogo- nal to the migrationdirectionandfor eachplanethe connected component algorithm (CCA) was applied to obtain the area of the pores (see section 5 of SI). Then the procedure was repeated for several random generated matrices of the same densityandthemeanvaluesandtheirrespectiveunbiasedstan- dard deviation were obtained. On the other hand, for the DPD matrices we have used a method described in [

100] that esti-

mates the area of the pores by the effective space available [

101].Theporesizes is inverselyproportionaltothedensityof

?bres.Thecomputationaldatais depictedin?gure

11together

with experimental results adapted from [

9] (?gure2(b)). Theexperimentaldata was measuredforhuman?brosarcomacellsof the line HT1080 migrating in rat tail collagen matricestreated with an MMP inhibitor (GM6001) and for untreatedmatrices (with medium).

As shown in ?gure

11the cell velocity increases linearly

with the pore size for both experimental and computational models. For small pore sizes the cells start to have MMP- dependent migration, degrading the ?bres and opening space formigration.Inourmodelthecellbecomestrappedforhigher pore sizes than in the experimental setup due to the lack of this mechanism (?gure

11). For the DPD model with ?exible

?bres (?gure

11(b)) we observe a higher velocity than in the

rigid case, which is expected since matrix elasticity favours migration. In the experimental assay the authors had also repeated the migration study isolating the role of MMPs. For this purpose, the cells were treated with the potent broad spectrum MMP inhibitor GM6001, also known as ilomastat, which mitigates the in?uence of MMP activity during migration. In ?gure 12 we present the migration velocity as a function of the pore cross section for (a) PFM and (b) DPD results and com- pare with the GM6001 experimental data. As before, we can observe for the three cases a linear dependency with the pore cross section.

Comparing the two lines in ?gure

12(a) we can notice

that one is shifted in relation to another, as well when com- paring the DPD rigid in ?gure

12(b). This difference is the

result of a larger plasticity of the cancer cells used in the experiments and their capability of adhering to the ?bres, which facilitates migration, when compared with the capac- ity of the simulated cells to deform. The ?brosarcoma cells are able to migrate under extreme con?nement conditions as seen in the transwell ?lter assay where cells can pass through apertures of only 3μm wide (?gure 7(b) of [

9]). The fact

that the line presented in ?gure

12(b) for the DPD ?exible

is shifted to the left corroborate this hypothesis and reveals some level of equivalence between matrix elasticity and cell plasticity. We then increased the adhesiveness in the PFM and ran simulations forη=0.5 andη=1.0 in order to evaluate the contribution of adhesion to migration. We can see in ?gure 13 that increasing adhesion fromη=0 toη=0.5 the velocity dependency with the pore size rotates, becomes more simi- lar to the result obtained experimentally with MMP inhibi- tion (RT GM6001). When increasing even further the adhe- siveness (η=1.0) the PFM results became analogous to the experimentalresultswithouttheinhibitor(RTMedium).These results indicate that MMP inhibitors are able to affect signi?- cantly adhesiveness by suppressing the cleavage of molecules that act as adhesion promoters. Moreover this brings adhe- siveness as a key modulator of migration in crowded media. In addition, we suggest that more experimental assays should be carried to unveil how MMPs activity acts in different mechanisms of cell migration, using speci?c inhibitors to tar- get isolated members of the MMP family. Some MMP iso- forms are known to affect mainly collagen degradation and remodelling while others act regulation cell adhesion and migration. 8 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al

Figure 10.Surface energy (left) and radius of gyration (right) as function of the density of ?bres and the adhesion coef?cient. Thesurface

energy gives an estimative of surface roughness and deformation. The surface energy increases with the adhesion with ?bre density, which

indicates a higher cell plasticity.

Figure 11.Migration velocity as a function of the pore cross-sectionin silicoandin vitrofor untreated rat-tail collagen [9]. The results for

(a) the phase-?eld model and (b) the DPD model are similar. The slopes andR2are the following: 0.010(0.87) (RT medium), 0.017(0.97)

(PFM), 0.016(0.98) (DPD-rigid) and 0.017(0.97) (DPD-?exible). The error bars are the unbiased standarddeviation of the mean for several

randomly generated matrices.

4. Conclusion

Inthis articlewe presentedtwo minimalistmathematicalmod- els for droplets: the DPD and the PFM. We carried out a systematic study for the movement of a droplet in ?brous media,measuringitsmigrationvelocityandmorphologywhile varying the ?bre density and adhesiveness. We observed that cell migration is strongly modulated by spatial condi- tions, and specially by the adhesion between the cell and the substrate. Our results indicate that adhesion is critical for cell migra- tion [

102], by modulating cell morphology [103-106] in

crowded environments and enhancing cell velocity [ 107].
We have compared our results to experimental data [

9,96].

The theoretical results for the velocity and the experimental data shown the same qualitative behaviour as a function of the matrice"s pore cross section and adhesion strength. The results are model independent and have shown good agree- ment between the two methodologies and experiments in the literature, which indicates that these minimalist descriptions

are able to capture the main features and the basic mechanicalnature of cell migration. In addition, the two approaches are

complementary to each other, which helps to ful?l some gaps due to model assumptions. Our analysissuggests that theMMP family playsan impor- tant role as adhesion regulator, since our results reproduce the effect of MMP inhibition by the only mean of varying the cell-ECM adhesion: a stronger adhesion in our model gives similarresults to themigrationassay withoutMMP inhibition; while a weaker adhesion describes the experimental data with the GM6001 inhibitor. In fact, in the recent years the MMPs have been gathering attention by its many faceted action on modulatingcellmigrationandinvasiveness,pushingitsimpor- tance beyond a collagen degrading enzyme [45,108,109]. For example, the MT1-MMP binds more than one hundred of cell surface partners[

110,111],including:tetraspaninsfamily

[

112], known for regulating cancer progression [110]; inte-

grins and CD44 [

113], which are protagonists in cell adhesion

[ 114].
The DPD model is a promising novel approach to model cells in a ?exible substrate. In this work we described the 9 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al

Figure 12.Migration velocity as a function of the pore cross-sectionin silicoandin vitro[9]. The DPD ?exible ?tting is perfectly

superposed with the RT GM6001 experimental line. The slopesandR2are the following: 0.017(0.99) (RT GM6001), 0.017(0.97) (PFM),

0.016(0.98) (DPD-rigid) and 0.017(0.97) (DPD-?exible). The error bars are the unbiased standarddeviation of the mean for several

randomly generated matrices. Figure 13.Migration velocity as a function of the pore cross-section for three values of adhesion coef?cientin silicoη={0,0.5,1}and in vitro?brosarcoma cells in rat-tail collagen [

9]. The slopes andR2

for the simulation results from lower to higher adhesions are

0.017(0.97), 0.014(0.92), 0.011(0.92) and 0.010(0.87) (RT

medium) and 0.017(0.99) (RT GM6001) for the experimental data. ?bres as polymeric chains under a simple harmonic poten- tial, but the DPD implementation provides a natural method to introduce nonlinear elasticity. The elastic ?bres model allows to explore different problems in cellular systems, such as how cell-to-cell mechanical communication happens through force propagation via the network and its in?u- ence on migration. For these reasons further investigation will be carried in order to address the effect of coopera- tive migration in the tumour microenvironment of metastasis (TMEM).These results help to elucidate the behaviour of soft cells diffusing in complex media. Mainly, it has shown that adhe- sion is a key modulator for cellular plasticity and migra- tion in con?ned conditions. In addition we showed that there is a strong correlation between migration velocity and the cell deformation. We propose that new assays should be car- ried to address the role of adhesion and the effect of dif- ferent MMPs in cell migration under con?ned conditions. In particular, we suggestin vitroexperiments with 3D col- lagen scaffolds using MMP speci?c inhibitors [

115] and

adhesion modulators, such as integrins inhibitors/promoters (RGD/S peptides) [

116]. In addition, recent con?nement

micro?uidic assays may be helpful to explore a different perspective [ 117].

Acknowledgments

The simulations were done in the Satolep Cluster of the Federal University of Pelotas and in the Centaurus cluster of the University of Coimbra. This work was funded by FEDER funds through the Operational Programme Compet- itiveness Factors-COMPETE and by national funds by FCT- Foundation for Science and Technology under the strategic projects UID/FIS/04564/2016 and POCI-01-0145-FEDER-

031743-PTDC/BIA-CEL/31743/2017 (M M-S, R T). M M-S

acknowledges the support of the National Council of Techno- logical and Scienti?c Development (CNPq) under the Grant

235101/2014-1 and thanks Pedro V P Cunha for the fruit-

ful discussions. The authors would like to thank Beatriz Costa-Gomes for constructive criticism of the manuscript.

J R B acknowledges CNPq and FAPERGS for ?nancial

support.

ORCID iDs

Maurício Moreira-Soareshttps://orcid.org/0000-0001-

9928-7256

JoséRafael Bordinhttps://orcid.org/0000-0002-8025-6529 10 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al

References

[1] Lambert A W, Pattabiraman D R and Weinberg R A 2017Cell

168670-91

[2] Taketo M M 2011Cancer Prev. Res.4324-8 [3] Steeg P S 2016Nat. Rev. Cancer16201 [4] Mehlen P and Puisieux A 2006Nat. Rev. Cancer6449 [5] Fidler I J 2003Nat. Rev. Cancer3453 [6] Venning F A, Wullkopf L and Erler J T 2015Frontiers Oncol.5 224
[7] LamouilleS, XuJand Derynck R2014Nat. Rev. Mol. Cell Biol. 15178
[8] Chiodoni C, Colombo M P and Sangaletti S 2010Cancer

Metastasis Rev.

29295-307

[9] Wolf K, Te Lindert M, Krause M, Alexander S, Te Riet J, Willis A L, Hoffman R M, Figdor C G, WeissSJ and Friedl P2013

J. Cell Biol.

2011069-84

[10] Brodland G W and Veldhuis J H 2012PLoS One7e44281 [11] Malandrino A, Kamm R D and Moeendarbary E 2017ACS

Biomater. Sci. Eng.

4294-301

[12] KatiraP,Bonnecaze R TandZaman MH2013FrontiersOncol. 3145
[13] Wise S M, Lowengrub J S, Frieboes H B and Cristini V 2008J.

Theor. Biol.

253524-43

[14] Byrne H and Chaplain M A 1996Math. Comput. Modelling24 1-17 [15] Hay E D 2013Cell Biology of Extracellular Matrix(Berlin:

Springer)

[16] Frantz C, Stewart K M and Weaver V M 2010J. Cell Sci. 123

4195-200

[17] Friedl P and Alexander S 2011Cell147992-1009 [18] Visse R and Nagase H 2003Circ. Res.92827-39 [19] Kessenbrock K, Plaks V and Werb Z 2010Cell14152-67 [20] Wolf K and Friedl P 2011Trends Cell Biol.21736-44 [21] KhatiwalaCB,PeytonSR andPutnamAJ2006Am.J.Physiol.

Cell Physiol.

290C1640-50

[22] GoetzJG,MinguetS,Navarro-LéridaI,LazcanoJJ,Samaniego R, Calvo E, Tello M, Osteso-Ibáñez T, Pellinen T and

Echarri Aet al2011Cell

146148-63

[23] Wullkopf L, West A K V, Leijnse N, Cox T R, Madsen C D,

Oddershede L B and Erler J T 2018Mol. Biol. Cell

29

2378-85

[24] Gaggioli C, Hooper S, Hidalgo-Carcedo C, Grosse R, Marshall

J F, Harrington K and Sahai E 2007Nat. Cell Biol.

91392
[25] Lugassy C, Zadran S, Bentolila L A, Wadehra M, Prakash R, Carmichael S T, Kleinman H K, Péault B, Larue L and

Barnhill R L 2014Canc. Microenviron.

7139-52

[26] Wang W, Wyckoff J B, Goswami S, Wang Y, Sidani M, Segall

J E and Condeelis J S 2007Canc. Res.

673505-11

[27] Sahai E, Wyckoff J, Philippar U, Segall J E, Gertler F and

Condeelis J 2005BMC Biotechnol.

514
[28] Yamauchi K, Yang M, Jiang P, Yamamoto N, Xu M, Amoh Y, Tsuji K, Bouvet M, Tsuchiya H and Tomita Ket al2005

Canc. Res.

654246-52

[29] Weigelin B, Bakker G J and Friedl P 2012IntraVital132-43 [30] Provenzano PP,EliceiriKW,Campbell JM,Inman DR,White

J G and Keely P J 2006BMC Med.

438
[31] Paul C D, Mistriotis P and Konstantopoulos K 2017Nat. Rev. Canc. 17131
[32] Friedl P and Wolf K 2010J. Cell Biol.18811-9 [33] Friedl P 2004Curr. Opin. Cell Biol.1614-23 [34] Costa P and Parsons M 2010 New insights into the dynamicsof cell adhesionsInternational Review of Cell and Molecular

Biologyvol 283 (Amsterdam: Elsevier) pp 57-91

[35] Alberts B, Bray D, Lewis J, Raff M, Roberts K and Watson J

2002 (London: Garland Science)

[36] Zaidel-BarR,CohenM,AddadiLandGeigerB2004Hierarchi- cal assembly of cell-matrix adhesion complexesBiochem.

Soc. Trans.

32416-20

[37] Destaing O, Saltel F, Géminard J C, Jurdic P and Bard F 2003

Mol. Biol. Cell

14407-16

[38] Oikawa T and Takenawa T 2009Cell Adhes. Migrat.3195-7 [39] Jabło´nska-Trypu´c A, Matejczyk M and Rosochacki S 2016J.

Enzym. Inhib. Med. Chem.

31177-83

[40] Gross J and Lapiere C M 1962Proc. Natl Acad. Sci. USA48 1014
[41] Rodríguez D, Morrison C J and Overall C M 2010Biochim.

Biophys. Acta, Mol. Cell Res.

180339-54

[42] Jiao Y, Feng X, Zhan Y, Wang R, Zheng S, Liu W and Zeng X

2012PloS One

7e41591

[43] Conant K, Lim S T, Randall B and Maguire-Zeiss K A 2012

Curr. HIV Res.

10384-91

[44] Butler G S, Dean R A, Tam E M and Overall C M 2008Mol.

Cell Biol.

284896-914

[45] Ferrari R, Martin G, Tagit O, Guichard A, Cambi A, Voituriez R, Vassilopoulos S and Chavrier P 2019Nat. Commun. 10 1-5 [46] Ubezio B, Blanco R A, Geudens I, Stanchi F, Mathivet T, Jones

M L, Ragab A, Bentley K and Gerhardt H 2016Elife

5 e12167 [47] Li X, Padhan N, Sjöström E O, Roche F P, Testini C, Honkura N, Sáinz-Jaspeado M, Gordon E, Bentley K and Philippides

Aet al2016Nat. Commun.

711017

[48] Osborne J M, Fletcher A G, Pitt-Francis J M, Maini P K and

Gavaghan D J 2017PLoS Comput. Biol.

13e1005387

[49] LakatosD,SomfaiE,MéhesEandCzirókA2018J.Theor.Biol.

456261-78

[50] Frieboes H B, JinF, Chuang Y L, WiseS M, Lowengrub JS and

Cristini V 2010J. Theor. Biol.

2641254-78

[51] Lima E, Oden J and Almeida R 2014Math. Models Methods

Appl. Sci.

242569-99

[52] Lorenzo G, Hughes T J, Dominguez-Frojan P, Reali A and

Gomez H 2019Proc. Natl Acad. Sci. USA

1161152-61

[53] Travasso R D, Castro M and Oliveira J C 2011Phil. Mag.91

183-206

[54] Milde F, Bergdorf M and Koumoutsakos P 2008Biophys. J.95

3146-60

[55] Santos-Oliveira P, Correia A, Rodrigues T, Ribeiro-Rodrigues T M, Matafome P, Rodríguez-Manzaneque J C, Seic¸a R, Girão H and Travasso R D M 2015PLoS Comput. Biol. 11 e1004436 [56] Moreira-Soares M, Coimbra R, Rebelo L, Carvalho J and

Travasso R D M 2018Sci. Rep.

81-2
[57] Vilanova G,Colominas I and Gomez H 2017J. R. Soc. Interface

1420160918

[58] Palmieri B, Bresler Y, Wirtz D and Grant M 2015Sci. Rep.5 11745
[59] Najem S and Grant M 2016Phys. Rev.E93052405 [60] Najem S and Grant M 2015Soft Matter114476-80 [61] Najem S and Grant M 2014Soft Matter109715-20 [62] Najem S and Grant M 2013Phys. Rev.E88034702 [63] Shao D, Levine H and Rappel W J 2012Proc. Natl Acad. Sci. USA

1096851-6

[64] Marth W and Voigt A 2014J. Math. Biol.6991-112 [65] Marth W, Praetorius S and Voigt A 2015J. R. Soc. Interface12

20150161

[66] KulawiakDA,CamleyBAandRappelWJ2016PLoSComput. Biol.

12e1005239

[67] Camley B A, Zhao Y, Li B, Levine H and Rappel W J 2017

Phys. Rev.E

95012401

[68] Moure A and Gomez H 2017Comput. Methods Appl. Mech. Eng.

320162-97

[69] Scianna M, Preziosi L and Wolf C 2013Math. Biosci. Eng.10

235-61

[70] Kumar S,Kapoor A,Desai S, Inamdar MMand SenS 2016Sci. Rep.

619905

[71] Reinhardt J W and Gooch K J 2014J. Biomech. Eng.136

021024

11 J. Phys.: Condens. Matter32(2020) 314001M Moreira-Soareset al [72] Ahmadzadeh H, Webster M R, Behera R, Valencia A M J, Wirtz D, Weeraratna A T and Shenoy V B 2017Proc. Natl Acad. Sci.

114E1617-26

[73] Cao X, Ban E, Baker B M, Lin Y, Burdick J A, Chen C S and

Shenoy V B 2017Proc. Natl Acad. Sci.

114E4549-55

[74] Kim M C, Whisler J, Silberberg Y R, Kamm R D and Asada

H H 2015PLoS Comput. Biol.

111-29

[75] Macnamara C K, Caiazzo A, Ramis-Conde I and Chaplain M A

2020J. Comput. Sci.

40101067

[76] Groot R D and Rabone K L 2001Biophys. J.81725-36 [77] Yamamoto S,Maruyama Y and Hyodo SA 2002J. Chem. Phys.

1165842-9

[78] Basan M, Prost J, Joanny J F and Elgeti J 2011Phys. Biol.8

026014

[79] Ye T, Phan-Thien N and Lim C T 2016J. Biomech.492255-66 [80] Hoogerbrugge P and Koelman J 1992Europhys. Lett.19155 [81] Nonomura M 2012PLoS One71-9 [82] Akiyama M, Nonomura M, TeroA and Kobayashi R 2018Phys. Biol.

16016005

[83] Bray A 1994Adv. Phys.51481-587 [84] Bray A J, Cavagna A and Travasso R D 2001Phys. Rev.E65

016104

[85] Moreira-Soares M 2019Software Package for Cell Research (Spiccato) https://doi.org/10.5281/zenodo.3534284 [86] Moreira-Soares M 2019 Zenodohttps://doi.org/10.5281/ zenodo.3537983 [87] LiuM,LiuG,ZhouLandChangJ2015Arch.Comput. Methods Eng.

22529-56

[88] Macis M, Lugli F and Zerbetto F 2017ACS Appl. Mater.

Interfaces

919552-61

[89] Allen M P and Schmid F 2007Mol. Simul.3321-6 [90] Lobaskin V and Dünweb B 2004New J. Phys.654 [91] Blumers A L, Tang Y H, Li Z, Li X and Karniadakis G E 2017

Comput. Phys. Commun.

217171-9

[92] Gatsonis N A, Potami R and Yang J 2014J. Comput. Phys.256

441-64

[93] Mason B N, Starchenko A, Williams R M, Bonassar L J and

Reinhart-King C A 2013Acta Biomater.

94635-44

[94] Kniazeva E and Putnam A J 2009Am. J. Physiol. Cell Physiol.

297C179-87

[95] Kraehenbuehl T P, Zammaretti P, Van der Vlies A J, Schoen- makers R G, Lutolf M P, Jaconi M E and Hubbell J A 2008

Biomaterials

292757-66

[96] Dieterich P, Klages R, Preuss R and Schwab A 2008Proc. Natl

Acad. Sci. USA

105459-63

[97] Maheshwari G, Wells A, Grif?th L G and Lauffenburger D A

1999Biophys. J.

762814-23

[98] Kelley L C, Lohmer L L, Hagedorn E J and Sherwood D R 2014 Traversing the basement membranein vivo: a diversity of strategiesJ. Cell. Bio.

204291-302

[99] Maheshwari G, Brown G, Lauffenburger D A, Wells A and

Grif?th L G 2000J. Cell Sci.1131677-86

[100] Bordin J R 2018Phys.A

495215-24

[101] Bordin J R 2019Fluid Phase Equilib.499112251 [102] Lock J G, Wehrle-Haller B and Strömblad S 2008 Cell-matrix adhesion complexes: master control machinery of cell migrationSemin. Cancer Biol.

1865-76

[103] Natale C, Lafaurie-Janvore J, Ventre M, Babataheri A and

Barakat A 2019J. R. Soc. Interface

1620190263

[104] Oakes P W, Banerjee S, Marchetti M C and Gardel M L 2014

Biophys. J.

107825-33

[105] Bischofs I B, Schmidt S S and Schwarz U S 2009Phys. Rev. Lett.

103048101

[106] Banerjee S and Marchetti M C 2013New J. Phys.15035015 [107] Tantivejkul K,Vucenik I and Shamsuddin A M 2003Anticancer

Res.233681-9

[108] Takino T, Watanabe Y, Matsui M, Miyamori H, Kudo T, Seiki

M and Sato H 2006Exp. Cell Res.

3121381-9

[109] Itoh Y 2006IUBMB Life58589-96[110] Tomari T, Koshikawa N, Uematsu T, Shinkawa T, Hoshino

D, Egawa N, Isobe T and Seiki M 2009Canc. Sci.

100

1284-90

[111] Knapinska A M and Fields G B 2019Pharmaceuticals12 [112] SchröderH,HoffmannS,HeckerM,KorffTandLudwigT2013

Int. J. Biochem. Cell Biol.

451133-44

[113] Gálvez B G, Matías-Román S, Yáñez-Mó M, Sánchez-Madrid

F and Arroyo A G 2002J. Cell Biol.

159509-21

[114] Senbanjo L T and Chellaiah M A 2017Frontiers in Cell and

Developmental Biology

518
[115] Vincenti M P 2001 The matrix metalloproteinase (mmp) and tissue inhibitor of metalloproteinase (timp) genesMatrix

Metalloproteinase Protocols(Berlin: Springer) pp

121-48

[116] Hersel U, Dahmen C and Kessler H 2003Biomaterials24

4385-415

[117] Brückner D B, Fink A, Schreiber C, Röttgermann P J, Rädler

J O and Broedersz C P 2019Nat. Phys.

15595-601

12

Politique de confidentialité -Privacy policy