Table Of ContentForce localization in contracting cell layers
Carina M. Edwards1 and Ulrich S. Schwarz1,2,∗
1University of Heidelberg, Bioquant, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany
2University of Heidelberg, Institute for Theoretical Physics, Philosophenweg 19, 69120 Heidelberg, Germany
(Dated: January 16, 2012)
Epithelialcelllayersonsoftelasticsubstratesorpillararraysarecommonlyusedasmodelsystems
forinvestigatingtheroleofforceintissuegrowth,maintenanceandrepair. Hereweshowanalytically
that the experimentally observed localization of traction forces to the periphery of the cell layers
doesnotnecessarilyimplyincreasedlocalcellactivity,butfollowsnaturallyfromtheelasticproblem
of a finite-sized contractile layer coupled to an elastic foundation. For homogeneous contractility,
theforcelocalizationisdeterminedbyonedimensionlessparameterinterpolatingbetweenlinearand
2
exponential force profiles for the extreme cases of very soft and very stiff substrates, respectively.
1
If contractility is sufficiently increased at the periphery, outward directed displacements can occur
0
at intermediate positions, although the edge itself still retracts. We also show that anisotropic
2
extracellular stiffness leads to force localization in thestiffer direction, as observed experimentally.
n
a PACSnumbers: 87.10.+e,87.16.Ln,87.17.Rt
J
3
1 Actively generated forces have emerged as a central
element in the way tissue cells interact with their en-
]
O vironment, for example during development and tissue
maintenance [1]. It is becoming increasingly clear that
T
cells use physical force to probe the mechanical prop-
.
o erties of their environments and to create a structurally
i
b coherentstatewithinthecellandtissue. Experimentally,
- however, it is challenging to measure cellular forces in a
q FIG. 1. (a) Scanning electron micrographs of an epithelial
physiologicalcontext. Thereforetheireffectisusuallyas-
[ cell layer on an array of microfabricated elastomeric pillars
sessed in an indirect manner, for example by mechanical
(image reproduced from [5].) (b) A schematic representation
1
relaxation after laser cutting [2]. of thepillar assay. The cell layer contracts and is resisted by
v
9 In order to directly measure the forces that cells ex- a distribution of linear springs on the surface.
7 ertontheirenvironments,twocomplementarytechniques
7 have been established over the last decade. Embedding
2 homogeneous contraction, we show that the problem is
marker beads into soft elastic gels enables the measure-
. determined by one dimensionless parameter which inter-
1 ment of the displacement field generatedby cell traction
polates between the extreme cases of soft pillars with a
0
forces applied to the surface of the gel, which can then
2 linear force profile and stiff pillars with an exponential
be estimated using elasticity theory [3]. Alternatively,
1 forceprofile. Wealsoshowhowourresultschangeforin-
: microfabrication techniques are used to create an array creasedcontractile activity at the rim and for substrates
v
ofelastomericpillars,whichforsmalldisplacementseach
i with anisotropic stiffness.
X act as a linear spring [4].
Elastic model. Microstructured surface assays consist
r Inordertoaddresstheroleofforcesforlargercellclus- of a regular array of flexible elastomeric pillars with de-
a
ters, both soft elastic substrates and pillar assays have fined surface chemistry, see Fig. 1. For small displace-
been used for confluent layers of epithelial cells. Plac- ments and slender pillars, the relation between the trac-
ing such cells on a pillar array, it was found that cellu- tion applied to the pillars T and the displacement u fol-
lar traction forces are localized to the edge of the cell lows from linear elasticity theory [9],
layer [5, 6]. A qualitatively similar effect has also been
3π r4
observed for cell sheets migrating over soft elastic sub- T=ku = E u, (1)
strates [7], although in this case the length scales over 4 pL3
(cid:18) (cid:19)
which the stresses decayed were greater. It is tempting where r and L are the radius and height of the pillar,
to assume that force localization to the edge reflects in- respectively,andE is the Young’s modulus. The pillars
p
creased mechanical activity of the cells at the edge, for thus behaveas simplelinearspringswitheffective spring
exampleofputative leadercells[8]. However,a morede- constant k.
tailed analysis requires a mechanical analysis of the cell Intheassaysofinterestthesubstrateisusuallycoated
sheet coupled to the elastic foundation. with adhesion-promoting molecules so that the cells
In this Letter, we analytically solve the problem of a spread thinly and form a confluent layer. We thus as-
contracting cell layer coupled to a layer of springs. For sume the layer behaves as a continuous elastic solid. In
2
addition, as the typical lateral extension of the sheet is a b 9
much largerthan the layer thickness h, we have the con- 0.12
ditions for plane-stress in the layer and can use a two-
dimensional model for the cells. The force balance then 0 m) 6
reads u|/l0.08 µ/
| N
div·F−kNu=0, (2) 0.04 F (n 3
whereFisthetwo-dimensionalin-planestresstensorob-
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
tained by averaging the layer stress over the thickness, x/l0 x/l0
andN is the number density ofpillars. Fora continuous
elastic substrate, the restorative force applied to the cell FIG. 2. (Color online) Plot of (a) displacement |u| and (b)
layerwouldnottakethissimpleform. Moreovermorede- stress F within the cell layer for the stripe geometry with
tailedassumptionswouldberequiredtodefinetheelastic P0 = 0.7, Ec = 10kPa, h = 1µm, ν = 0.45 and γ =5, 6, 8,
10, 15 (from top tobottom in (a) and from bottom totop in
problem of the two coupled elastic layers. Nevertheless,
(b)).
the linear traction force term from Eq. 2 can be also
considered as a first order approximation for soft elastic
substrates [10, 11]. introduced a new length scale, the localization length
The cell layer is mechanically active, with cells simul-
taneously repositioning and contracting. Here we con- hE 4hE L3
c c
sider only the final configuration of a contractile cell l =skN(1−ν2) =s3π(1−ν2)EpNr4, (5)
layer. Then the problem of modelling layer contraction
is similar to that of thermoelasticity, where temperature with the second expression following from Eq. (1).
changes generate expansion or contraction in the mate- Assuming that the cell layer contracts uniformly, P
rial [10]. In thermoelasticity one typically introduces an is constant. For the stripe case, it is convenient to set
expansiontermproportionaltothetargetvolumechange P =−2P /(1+ν)whereP isthenthetargetcontraction
0 0
into the constitutive relation [9]. In analogy we take inthe x-direction. We thenfindthatEq.(4)issolvedby
hE ν hE u sinh(γx/l )
F = c e + e δ − c Pδ . (3) =−P 0 (6)
ij 1+ν ij 1−ν kk ij 2(1−ν) ij l 0 γcoshγ
(cid:18) (cid:19) 0
The firsttermis theusualplane-stresslinearelasticcon- where we havedefined the dimensionless localizationpa-
stitutive relation with Ec the cellular Young’s modulus, rameter γ = l0/l, the ratio of layer dimension to the
ν the cellular Poisson’s ratio, and e the strain tensor. localization length l. The internal stress F(x) can be
ij
ThesecondterminEq.(3)capturescellularcontractility, obtainedby substituting Eq.(6) intothe constitutive re-
where wehaveassumedthatthe contractionis isotropic. lation Eq. (3), see SI [12].
If the layer is under no stress, F = 0 and thus P = e In Fig. 2 the displacement u and internal stress F are
ij ii
(summation convention applies), and so P is the local plotted for different values of γ. We see that a constant
target volume change of a material element in the cell cellularcontractionleadstonon-constantpillardisplace-
layer. ment, with greatest deflections at the edge. The non-
At the layer boundary no stress is applied, giving the dimensional parameter γ controls the profile of the solu-
boundary condition F·n = 0 on Γ. Integrating Eq. (2) tion and the rate of decrease of observed tractions away
over the cell area D and applying the zero-stressbound- fromthelayeredge. Whenγislarge,thepillarsdominate
aryconditiongives kNu=0. Thusmomentumisnot the system and restrict the displacements to a small re-
D
generatedinthesystem,asexpectedforaclosedsystem. gionnearthe rim. When γ is small, the substrate resists
Solution for a coRntracting stripe. It is instructive to less and the tractions decay slower. Note that both dis-
firstsolveourelasticprobleminonedimension,although placement and stress scale linearly with cell contraction
experimentally even very anisotropic cell layers always P0.
contract in both lateral directions. We consider a rect- Solution for a contracting disc. We now consider the
angularsheetwith e =0,suchthat the layercontracts two-dimensional problem of a circular cell layer with
yy
only in the x-direction over the stripe width l . Substi- isotropiccontraction,whichisdirectlycomparabletoex-
0
tuting Eq. (3) into Eq. (2) we find that u(x) satisfies periments with finite-sized and isotropically contracting
celllayers. Fromsymmetryargumentsu=u(r)e ,where
r
d2u 1 1 dP r is the radialdistance from the centre of the plate. The
− u = (1+ν) (4)
dx2 l2 2 dx radially symmetric equivalent to Eq. (4) reads
with the zero stress boundary condition du/dx = (1+ d2u du r2 1 dP
r2 +r − 1+ u= (1+ν)r2 . (7)
ν)P/2atx=±l0 andwithu=0atx=0. Herewehave dr2 dr l2 2 dr
(cid:18) (cid:19)
3
The boundary conditions are now u = 0 at r = 0, while a b
atCro=nsrid0ewrienghaavgeaiPnt=he2c(dasue/odfrc+onus/tar)n/t(c1e+lluνla).rcontrac- u/r00 u/r00
tion, P =−P0, the solution of Eq. (7) with the specified −0.02 −0.02
boundary conditions can be calculated in terms of mod-
−0.04
ified Bessel functions: −0.04
u 1 −0.06
r0 =−2γP0(1+ν)A(γ)I1(γr/r0), (8) 0 0.2 0.4 0.6 0.8x/r01 −0.060.4 0.6 0.8 x/r01
ν−1 −1
A(γ)= I (γ)+ I (γ) . (9) FIG. 3. (Color online) Plot of displacement u in a disc
0 1
γ
(cid:18) (cid:19) with contractile rim for (a) P0/P1 =0.1, 0.3, 0.5, 0.7, 1 with
Again γ = r /l is the ratio of sample dimension and r1/r0 =0.9 (from top to bottom), (b) r1/r0 =0.75, 0.8, 0.85,
0
0.9, with P0/P1 = 0.5 (from top to bottom). In both plots
localization length, i.e. the decay length scale for the
γ =8, P1 =0.7, ν =0.45.
traction forces l. From Eq. (8) the radial stresses F
rr
and hoop stresses F in the layer can be calculated, see
θθ
SI [12], which show a similar behaviour as in the one- tion. We now apply our model for the contractile cell
dimensional case from Fig. 2(b), except that the hoop layer to two further situations of large practical inter-
stress does not vanish at the boundary. The plot for est, namely non-homogeneous contraction in the layer
displacement essentially looks like the one for the stripe and anisotropic layer stiffness. We first consider what
case,Fig.2(a),seeSI[12]. Ourresultsareconsistentwith thedisplacementswouldlooklikeifthe layerhadamore
the simulations in [13], where a uniformly contracting mechanically active rim. We consider that the contrac-
networkof actinfibres on a discrete pillar arraywas also tion jumps up at the rim:
observed to exhibit localisation of force to the edges of
the network. −P x<x or r ≤r
P = 0 1 1 , (11)
Regarding the comparison with experiments, typical −P x <x<l or r <r <r
1 1 0 1 0
parameter values are N = 0.25µm−2 for the pillar den- (cid:26)
sity, r = 50µm for layer dimension, h = 1µm for layer for both the contracting stripe and disc. Eqs. (4) and
0
thickness, E =10kPafor cellular Young’s modulus and (7) are unchanged, as are the boundary conditions ap-
c
ν =0.45forPoisson’sratio. With pillarstiffness ranging plied at both the origin (u = 0) and edge of the layer
over1-200nNµm−1[6],thisgivesatypicalrangeforthe (zero normal stress). At the interface between the two
dimensionless parameter γ ∼ 7−100. The clear decay contractile regions (i.e. at x = x1 and r = r1, for the
of the traction profile reported in [6] agrees nicely with stripe and disc case, respectively), however, we impose
our results. A more spread-out traction profile has been the additional conditions that the displacement u and
reportedin[7],whichmaybe attributableto alow value stresses F (for stripes) and Frr (for discs) are continu-
of γ. However, as this study focused on the dynamics in ousacrosstheinterface. Withtheseadditionalboundary
a migrating sheet, direct comparison is difficult in this conditionsit is possible to obtainanalyticalsolutions for
case. the displacement u, see SI [12].
To further examine the role of γ in determining the For the disc geometry, the resulting displacement is
profile of the solution, we consider the limiting cases of plotted in Fig. 3(a). Note that the plot of the displace-
γ small and large. When γ is small, the pillars are very ment u has a kink at the point of transition between the
weak and the layer will be able to contract by a large two contractilities,and that as the inner region becomes
amount. When γ ≪ 1, we obtain to lowest order a lin- weakerand P0/P1 decreases,this feature is accentuated.
ear profile u ∼ −P0r/2, as one can infer either directly Equally, as P0/P1 → 1, this feature disappears and we
from the differential equation or from the Bessel func- recover the solution for constant contractility. The dis-
tion. Alternatively, in the case γ ≫ 1, when the springs placements for the contracting stripe solution are quali-
dominate the behaviour of the sheet, the displacement tatively the same.
localizes strongly to the edge and we get an exponential For a sufficiently inactive inner region, there is the
profile, possibility that contraction of the outer rim dominates
to such an extent that it can pull the inner region to-
P r (1+ν)
u∼− 0 0 e−γ(1−r/r0). (10) wards it, resulting in positive displacements at interme-
2γ
diate positions, although the layer as a whole retracts
Theexactprofileofthesolutionisthusclearlycontrolled its edge. This effect depends sensitively on the spatial
by the non-dimensional parameter, γ, which quantifies extent of the contracile rim. For the stripe case with
the relative strengths of pillars versus cells. x /x = 1 − ǫ and ǫ ≪ 1, positive displacements are
1 0
Effect of a contractile rim. Up to now, we have possible only when P ∼ ǫ2γ2P /2, see SI [12]. This de-
0 1
assumed that the layer undergoes a uniform contrac- pendence on the extent of the more contractile region
4
gatedinthedirectionoflargerstiffness(x-direction). We
also see that the traction forces f = k(θ)u are concen-
tratedatthepointedendsofthelayer. Wedonotexpect
this effect to be sufficientto accountfor the shape of the
elongated tissue islands reported in [6, 14], but it is im-
portant to note that an anisotropy in tissue shape and
forceprofilecanbeinducedpurelythroughthesubstrate
properties. Indeed this result of uniform cellular tension
generating anisotropic internal stresses could be used by
cellularassembliestosensethedirectionofmaximalstiff-
ness and to polarize accordingly.
FIG. 4. (Color online) Plot of (a) displacement |u|, arrows
indicate displacement directions and magnitude, (b) traction
forcef/Kr0 exertedon pillars. Inbothplotsγ =5,P0 =0.8
and ν = 0.45, ǫ = 3 (major axis in x-direction), the dotted ACKNOWLEDGMENTS
black line indicates theinitial shape of thelayer.
CMEwassupportedbyapostdoctoralfellowshipfrom
is also seen in Fig. 3(b). In summary, the existence of the Center for Modelling and Simulation in the Bio-
kinks and positive displacements in experiments would sciences (BIOMS) at Heidelberg. USS is a member of
be a clear signature of increasedcontractilityat the rim. the Heidelberg cluster of excellence CellNetworks.
Solution for anisotropic pillars. In a natural setting,
cellsareoftenconfrontedwithhighlyanisotropicenviron-
ments, which can have a strong effect on their response.
Thisimportanteffectcanbe investigatedexperimentally
∗
[email protected]
by using pillars with elliptical cross-sections [4, 6, 14].
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