Johnson-Kendall-Roberts Theory Applied to Living Cells
Yeh-Shiu Chu,1Sylvie Dufour,1Jean Paul Thiery,1Eric Perez,2and Frederic Pincet2,*
1UMR144,Centre National de la Recherche Scientifique et Institut Curie,26rue d’Ulm,75248Paris Cedex05,France 2Laboratoire de Physique Statistique de l’Ecole Normale Supe´rieure,UMR8550,
Centre National de la Recherche Scientifique et Universite´s Paris6et7,24rue Lhomond,75251Paris Cedex05,France
(Received19December2003;published18January2005)
Johnson-Kendall-Roberts(JKR)theory is an accurate model for strong adhesion energies of soft
slightly deformable material.Little is known about the validity of this theory on complex systems such as
living cells.We have addresd this problem using a depletion controlled cell adhesion and measured the
force necessary to parate the cells with a micropipette technique.We show that the cytoskeleton can
provide the cells with a3D structure that is sufficiently elastic and has a sufficiently low deformability for
JKR theory to be valid.When the cytoskeleton is disrupted,JKR theory is no longer applicable.
DOI:10.1103/PhysRevLett.94.028102PACS numbers:87.17.–d,68.35.Np,82.35.Lr,87.80.Fe
A quantitative understanding of the adhesion of living cells is not often possible,and the study reported here is one of the rare exceptions.In contrast,the adhesion of solid elastic bodies has been extensively studied in the past,and a complete mathematical description has been derived[1]. In general,when the contacting surfaces adhere only weakly and deform little,the Derjaguin-Muller-Toporov approach[2]allows prediction of the behavior of the system.At higher adhesion and deformability,when ad-hering surfaces are subject to a parating force,there is a finite,nonzero contact area at paration.In this ca, Johnson-Kendall-Roberts(JKR)theory[3]gives the rela-tion between the pull off force F s and the adhesion energy W adh via the radii of curvature of the materials.For solid, homogeneous spheres,
W adh 2F s= 3 R m ;(1) where R m is the harmonic mean of the radii of the two spheres.
Many experimental studies on simple elastic materials have verified this description[4].Similarly,the relation between F s and W adh has been recently derived for spheri-cal shells[5]:
W adh F s= R m :(2) However,the adhesion of soft bodies such as cells is much more difficult to characterize.Several attempts to probe the adhesion strength of two biological cells have been made using techniques including shearflow or centrifugation[6]. Adhesion experiments using micromanipulation were con-ducted more than a decade ago[7,8]using red blood cells, which have well-defined membrane elasticity and a rela-tively simple,liquid interior.In contrast,it is much more difficult to extract quantitative results from adhesion mea-surements involving nucleated cells,which are often char-acterized by an irregular surface with folds and wrinkles and who interior exhibits a complex rheology.Chien’s group has developed a model inspired by Evans’s results [9]involving the mechanical equilibrium of the cell mem-brane.Using this model,they measured the adhesion be-tween cytolytic T cells and target cells[10,11].Treating the paration of the cells as a peeling process,they ana-lyzed their experiments in terms of adhesion energies and junction avidity.
The prent study involves living cells that do not spon-taneously adhere.We cau them to adhere t
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hrough a depletion effect in the suspending medium.We show that,when the cytoskeleton of the cells has a complete 3D structure that maintains a slightly deformable spherical shape,JKR theory is applicable to relate the paration force to the adhesion energy.It gives an elastic modulus coherent with the one independently measured with a surface force apparatus(SFA)and with tho found in the literature[12].In this ca,where the3D cytoskeleton is responsible for their spherical shape,the cells do not behave like shells but like elastic spheres.
The general principle of our approach consists of micromanipulating two murine sarcoma S180cells[13] with micropipettes,making them adhere in a highly con-centrated dextran solution and balancing the depletion-induced adhesion by the aspiration pressure in a micro-pipette.
It is well documented that nonadsorbing,water-soluble polymers can induce an attraction of phospholipid bilayers [14,15].The adhesion energy W adh induced by the deple-tion of dextran has been measured experimentally on lipid vesicles[16]and analyzed theoretically[17].de Gennes has derived the expression of W adh as a function of the volume fraction of polymers :
W adh k B T=a2 1:5;(3) where k B T is the thermal energy and a the size of a monomer.
For this study,we ud a protocol similar to that ud by Chien’s group[11].It is described in Fig.1.Bef
ore ana-lyzing the adhesion behavior,we establish that the adhe-sion obrved in the polymer solution is due only to this depletion effect.It was already known that S180cells are devoid of intrinsic intercellular adhesion properties[18]
becau they do not express cell-cell adhesion receptors at their surface.This is consistent with our obrvation that S180cells brought to clo contact do not adhere without dextran.In contrast,in the prence of dextran,S180cells do adhere when they are mechanically pushed together with the micropipettes.Equilibrium under zero compres-sion is reached after this mechanical constraint is removed (after less than a cond).Further,the obrvation that adhering cells parated immediately after transfer in a dextran-free chamber shows that no receptor was activated during the adhesion pha.This indicates that the adhesion of S180cells obrved here was purely a depletion effect. During paration,the cells appear elastic and slightly deformable(e Fig.1)and the contact area at paration remainsfinite.Therefore,it is interesting to analyze the paration process with JKR theory and with the spherical shell model.
As shown by Yeung and Evans[19],the cells may dis-play viscoelastic behaviors that could induce force gra-dients.To avoid any artifact due to this problem,we have checked that the aspiration force in the pipette equals the force transmitted to the contact zone:we ud a direct method of probing thi
s transmitted force by aspirating a cell in a4–5 m micropipette with a gentle suction and placing the opposite side of the cell on a spring(a micro-needle with a known stiffness),the results of the force experiments indicate that,in the range of force,time,and velocity ud,the measured force equals exactly the aspi-ration one.
Thus,it is possible to test the validity of JKR and spherical shell theories on the cells.The paration force F s is clo to the average of the aspiration forces of the penultimate cycle nÿ1and the last cycle n:
F s P nÿ1 P n R2p=2;(4) where R p is the pipette inner radius, P i being the aspira-tion during cycle i.
The adhesion energy predicted by JKR and spherical shells theories can be calculated from the measurements of F s and the radii of the cells.The values can be compared (Fig.2)to the theoretical[17]expression for the energy due to the depletion effects[Eq.(3)]and the experimental measurements[16]of that energy.Figure3shows a
very 0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
00.050.10.150.20.250.3
dextran volume fraction
a
d
h
e
s
i
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n
e
n
e
r
g
y
(
J
/
m
²
)
FIG.2.Adhesion energy,as deduced from JKR theory[dia-monds,Eq.(1)]and spherical shells[cross,Eq.(2)]as a function of the volume fraction of dextran.Two sizes of dextran molecules(4:6 105and2 106M W)were ud.The results can be compared to the theoretical ones given by de Gennes[17] (line)and to experimental ones obtained by Evans by contact angle measurements on lipid vesicles[16](squares).
a
b
c
e d
d b
FIG.1.(a),(b)Two cells,held under weak aspiration by micro-pipettes,are placed in contact and1s later became adherent. Separation process(c),(d):One cell is held by the right micro-pipette under strong aspiration.The aspiration applied to the other cell is incread and the right micropipette displaced away. Either the cell leaves the left micropipette(c)or both cells parate(d).In thefirst ca,the cell is reized by the left micropipette(b),the aspiration incremented,and the right micro-pipette displaced again.This cycle is repeated until the cells parate and the paration force is deduced from the last aspiration pressures.During the measurements,the pipettes were moved at a velocity of about20 m=s.The whole process of paration lasted1min at most.The aspiration level on pressure employed in each cycle was monitored continuously.
宝宝牙刷good agreement with JKR theory while spherical shells theory does not em to be suitable.
To check that JKR theory is indeed valid,we have measured the variation of the contact area R c during the paration process and deduced the elastic modulus K from the relation [3]:
R c 3
R m
2K
ÿF 3 R m W adh ÿ6 R m W adh F 3 R m W adh 2q ;
(5)
where F is the (positive)pulling force.The results are plotted in Fig.3and give K 3500 1500Pa .To check this value,we have conducted SFA [20]experiments be-tween two layers of cells in which the reduction of the two layers of thickness with the load is measured (Fig.4).The measurements give K 4200 1000Pa ,which is in excellent agreement with values obtained by micro-manipulation and with values from the literature [12](1–5kPa).As a final proof of the validity of the JKR theory,the ratio between the contact radius at paration
and the one under zero load was measured.The obtained value is 0:65 0:12,again in excellent agreement with the expected one,0.63.Therefore,the main features of JKR theory are verified here.This result may em surprising as living cells in general display very complex mechanical behaviors and JKR should obviously not be valid for all types of cells.In the prent ca,the cytoskeleton is responsible for the shape of the cells and its 3D elastic properties.We have verified by imaging actin,tubulin,and vimentin filaments that the S180cells have an extended 3D cytoskeleton (data not shown).However,elasticity is ex-pected of only the behavior of the cytoskeleton for shape c
hanges that are sufficiently rapid that there is no time for the cytoskeleton to exhibit plastic flow during the detach-ment.This is the ca here.The experiments lasted a few tens of conds,whereas the time taken by a cell to regain its spherical shape after it has been expelled from a pipette was a few
minutes.
2000
4000
60008000
10000
12000
(δ3R)1/2 (µm 2)
f (µN )
FIG.4.Force between two layers of S180cells deposited on mica surfaces in a SFA function of the parameter 3R 1=2where is the reduction of the two cell layers thickness under com-pression and R the radius of the substrate.For smaller than the cell size,the slope gives the elastic modulus
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[3].
00.050.10.150.20.250.30.350.40
20
40
6080100120
R c 3 (µm 3)
A (p J )
FIG.3(color online).Parameter A R m 2 ÿF 3 R m W adh
ÿ6 R m W adh F 3 R m W adh 2p plotted as a function of R 3c .As indicated in Eq.(5),in the ca of JKR theory,the slope gives the elastic modulus.The large error bars are due to the low accuracy in the measurement of the contact radius in optical microscopy.The points are taken from three different experiments at various dextran volume fractions.
a
0.0002
0.00040.00060.00080.0010.0012
00.10.20.3
dextran volume fraction
E n e r g y f r o m J K R t h e o r y (J /m ²)
c b
FIG.5.(a),(b)Morphology of the cells treated with Lat B during the paration process.Note the difference from Fig.1.(c)Adhesion energy as it would be obtained through JKR theory [Eq.(1)]as a function of the dextran volume fraction in the prence of 0:1 M (filled diamonds)or 1:5 M (empty dia-monds)latrunculine B.The solid line is the expected value deduced from the applied depletion force [17].
To confirm the assumption that the cytoskeleton is re-sponsible for the elastic behavior of the cell,the same micromanipulation experiments were done in the prence of0.1or1:5 M of Latrunculin B(Lat)which inhibits actin polymerization and questers actin monomers [21,22].When the cell is made more deformable by alter-ation(0:1 M Lat)or disruption(1:5 M Lat)of the actin cytoskeleton network,there is a drastic change in the adhesion measurements as shown in Fig.5.In thefirst concentration,JKR theory ems to work correctly at low dextran concentrations(weak forces)while it is not appli-cable at higher ones.In1:5 M Lat,the measured apparent adhesion is weak and independent of the dextran concen-tration.In the cas,the cells prent a much larger deformation and take a long time(up to veral minutes) to recover their initial shape,and it is meaningless to try to deduce adhesion energies with the approach prented here.The actin cytoskeleton is mostly cortical in round cells in suspension and allows the mechanical connec-tion of the membrane to the tridimensional elastic structure of the rest of the cell.It is therefore not surprising that in this ca JKR and spherical shell theories are not valid anymore.
The measurements show that JKR theory can reason-ably be applied to predict the adhesion energy of the cells.Micropipette experiments are ideal to measure such an adhesion as the aspiration pressure gives a good mea-surement of the paration force.Whether such measure-men
ts are valid for cells of other kinds remains open.The applicability of JKR theory to the adhesion of other living cells could be checked directly using depletion forces,as here.However,the results suggest that the deformation of the cell during the detachment process is a good indi-cator of whether JKR or spherical shells theories are ap-plicable:if the cells prent a small deformation with a finite contact area at paration,this suggests a nearly elastic behavior of the cytoplasm and therefore the like-lihood that the theories will be
applicable.
*Corresponding author.
Electronic address:Frederic.s.fr
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