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energy and even the force exerted by receptor-ligand pairs can be measured by interferometric analysis of the contour of soft shells close to the surface on the basis of elastic boundary condition as shown below.

6Measurement of adhesion strength by interferometric contour analysis

The contour of adhering soft shells close to the contact line (which defines the circumference of the contact disc) exhibits the general shape shown in Figure 5c. The membrane is slightly bent at the contact line and goes smoothly over into a straight line region and the contour is thus completely defined by a contact curvature Rc and a contact angle θc. The straight line region is a consequence of the membrane tension induced by the adhesion. In analogy to the well-known Young law on the balance of surface tensions (at the triple line of partially wetting fluid droplets) the contact angle is related to the adhesion strength, W , according to

W = σ(1 − cos θc)

(6.1)

Whas the dimension of a surface pressure (force per unit length) and is thus often called spreading pressure. Similarly, the contact curvature is determined by the balance of bending moments which provides a second relationship between the free adhesion energy and the contact curvature

W= κ/Rc2 [10, 15].

The two boundary conditions provide a powerful tool to determine the physical parameters W and σ through measurements of the geometric parameters θc and Rc (if the bending sti ness is measured in a separate experiment e.g. by flicker analysis [8]).

A more rigorous analysis of the membrane deformation close to the contact zone in terms of the boundary conditions provides a more general relationship for the contour in the direction, s, perpendicular to the contact line [6]

H (s) = θc (s − λ) + θcλ exp −	s	·	(6.2)
	λ

Where λ is the “capillary length” λ = κ/σ, which is related to Rc as Rc = θc · λ. By fitting this curve to the contour of the adhering shell (as determined by the microinterferometric technique illustrated in Fig. 5) Rc and θc and thus W and σ may be determined with high precision. Note that λ has a simple meaning. It is the length over which the contour is determined by the bending elasticity before it becomes tension dominated and is typically λ 1 µm.

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7 Switching on specific forces: Adhesion as localized dewetting process

The adhesion scenario changes dramatically when specific forces are switched-on. At low receptor densities of the order of cR ≤ 10 µm−2 (which are comparable to the situation in real cells) the adhesion area decomposes into domains of tight adhesion (formed by 2D assemblies of receptor ligand pairs) which are separated by areas of weak adhesions which may exhibit strong flickering. This phenomenon is reminiscent of the formation of adhesion plaques or focal adhesion contact sites of adhering cells and it is in both cases a consequence of adhesion-induced lateral receptor segregation. The adhesion is driven by the lowering of the short range (van der Waals) minimum of the double well potential by switching on the receptor-ligand interaction potential Uoo (where Uoo is of the order of 10 kBT for integrinmediated lock-and-key forces corresponding to an energy density of about 10−4 J/m2). It can thus be understood in terms of a first order dewetting transition of a two-dimensional fluid.

It is important to note that the adhesion domain formation is a transient (nucleation and growth) process and that the domains merge in the time course of hours. The coarsening is driven by the line tension arising due to the bending deformation at the rim of the domains exhibiting a width ξ = (κ/V )1/4. V is defined in equation (4.1). ξ, which is defined in Figure 7a, is called the correlation length [6, 11] and is of the order of ξ 10 nm (that is much smaller than the capillary length λ). As is well known from the dynamics of nucleation and growth, the merging of domains with time t is very slow and follows a t1/3-law. In biomembranes the merging of adhesion plaques is expected to be further slowed down by the attraction between receptors, their coupling to focal adhesion complexes and actin bundles. The domain structure is thus a quasi-static state on biological time scales.

8Measurement of unbinding forces, receptor-ligand leverage and a new role for stress fibers

The thermodynamic concept described above breaks down if the binding energy between receptor-ligand pairs becomes considerably larger than the thermal energy (w ≥ 10 kBT ). This becomes most clearly evident if we consider a situation of soft shells adhering through a small number of pinning centers. These can be exposed and studied if one pulls the shell at the top by a force F (applied through magnetic tweezers glued to the top pole of the shell; cf. Fig. 8 and [6]). The contact line exhibits relatively sharp protrusions of nearly triangular shape. A closer inspection also shows that the tip of the protrusion has rounded edges exhibiting a radius of

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Fig. 7. a) Adhesion leads invariably to lateral segregation of receptors and ligands resulting in the formation of tightly bound adhesion plaques (distance do 10 nm) separated by weakly adhering often flickering regions (with distance d1 30 nm) determined by repeller repulsion and undulation forces. The adhesion may be described as a localized dewetting transition. Note that membrane forms a dimple near adhesion plaques extending over a correlation length of ξ 20 nm. µL and µp are the chemical potentials of the repeller (xi molar fractions). b) RICM image of cell model adhering through tight adhesion plaques (encircled and indicated by arrows).

curvature ρc [17]. These edges can only exist if they are stabilized by a line force f mediated by the membrane tension (cf. [15], Sect. 12). The vertical component f which balances the force generated by the receptor-ligand pairs is related to the contour ζ (h) of the shell above the substrate (defined in Fig. 8a)

f = κ ζ + ρc−1ζ

(8.1)

(with ζ = ∂ζ/∂h). This equation follows from the boundary condition for the mechanical stability of an adhering plate with a curved contour and is derived in reference [15] (Sect. 12). It is a second order boundary condition since it depends on the third derivative of the vertical deflection of the shell and κ · ζ is the gradient of the bending moment. However, for small radii of curvatures of the contour (ρc) the force f becomes large and the second order boundary condition becomes important. The contour at distances r > λ from the edge of the pinning center is no longer a straight line as in the regions of weak adhesion but the slope decreases with the intermembrane distance h as ζ = F λ/2πσr. The components of the line force f perpendicular to the substrate can be determined by the following trick (illustrated in Fig. 8b). Since the membrane tension σ of fluid membranes is

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Fig. 8. a) Magnetic tweezer technique to expose pinning centers by pulling vesicle in vertical direction with force Fm. ζ is the distance between the substrate and the membrane. b) Exposure of pinning center along contact line by application of force at top vesicle with magnetic tweezer. The white lines parallel to the contour line mark contours of equal height above surface yielding local contact angle. Note that the contact line of the pinning center exhibits a triangular shape forming an angle with round tips of radius ρc. c) Plot of line force (force per unit length) along contact line from which receptor-ligand unbinding force can be measured.

isotropic it can be measured by analyzing the contour at any region of weak adhesion along the contact line. If θc (s) is measured along the contours the vertical line force is:

f = σ sin θc (s)			(8.2)
and is related to the applied force as F = σ			sin θc (s) ds. θc (s) can often
	the first and second interference
be measured by the distance ∆d (s) between

fringe yielding f (s). Figure 8b shows a plot of f (s) along the contour and the sharp peaks clearly define the pinning centers. By simultaneously measuring f and the radius of the pinning centers while F is increased, one can measure the line force where the bonds break and thus obtain the binding force per link if the receptor density is known.

A very surprising result is that bonds break at very weak forces ( 1 pN). It has been postulated [6] that this is caused by the leverage of the receptor-ligand bond due to an amplification of the unbinding force by the bending moment exerted by the external force on the site of the receptor-ligand pairs. This finding attributes a new role to the stabilization of sites of cell adhesion by focal adhesion complexes associated with stress fibers. By this trick, nature prevents the disrupture of cells subjected to large hydrodynamic shear stresses such as endothelial cells lining in the blood vessels.

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9An application: Modification of cellular adhesion strength by cytoskeletal mutations

The microoptical method opens new possibilities to quantify the cell adhesion strength in order to explore the e ect of mutations or deceases on cell-adhesion. Very often cytoskeletal mutations do not a ect the phenomenological behavior of cells substantially, and quantitative methods of characterization are necessary [18]. On the other side, specific mutations provide a valuable tool to induce distinct structural alterations of the composite membrane for systematic studies of the correlations between membrane structure and adhesion. Thus, the e ect of the coupling strength of the actin cortex to the lipid/protein bilayer has been studied by removal of the actin-membrane coupling protein talin (cf. Fig. 2a).

In order to measure the work of adhesion, the membrane tension and the bending modulus simultaneously, the changes of the surface profile by hydrodynamic shear fields were measured. As shown in Table 1 the membrane bending sti ness of the talin-deficient mutant is decreased by a factor of 20 and the adhesion strength W by a factor of 5. In fact, the bending sti ness of the mutant agrees well with that of the pure bilayer containing 50 mole% cholesterol. Two major messages of this experiment are: (i) talin is indeed essential for the coupling of the actin cortex to the membrane and (ii) the adhesion strength is closely related to membrane bending sti ness.

Table 1. E ect of knock-out of talin on the bending sti ness of the cellular envelope and on adhesion strength (according to Simson et al. [18]).

Cell type	κ [kBT ]	Wad µJ/m2
wild type	1000	20
mutant	50	5

10 Conclusions

Insight into the physical basis of cell adhesion is gained by interferometric studies of vesicles or cells adhering on substrates which mimick the role of cell or tissue surfaces and adhesion forces can be measured locally by analysis of the contour of adhering shell in terms of the equilibrium of elastic stresses close to the adhesion zone.

The adhesion process is controlled by interplay of short range specific forces between repeller-ligand pairs and long range repulsion between cell

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surface molecules forming the glycocalix. This competition can lead to spontaneous segregation of receptors resulting in the formation of tight adhesion domains similar to the generation of focal contact sites of adhering cells. The adhesion process shares common features with a first order dewetting transition which explains why a rather small number of receptors can drive cell adhesion. The merging of the adhesion plaques is very slow (typical for coarsening processes) and could be further slowed down by the coupling of cell receptors to the actin cortex. The adhesion strength is closely related to the membrane elasticity and thus to the receptor-cytoskeleton coupling and its measurement can yield valuable insight into modification of the actin cortex by mutations or cell damaging agents.

Our model membrane studies attribute an important role to the glycocalix, namely the maintenance of mechanical equilibrium of adhering cells. Mobile repeller molecules (but also non-adherent receptors) in the nonadhering part of adhering soft shells exerts a two-dimensional osmotic pressure on the adhering membrane fraction which relaxes strong receptor-ligand forces. Surprisingly weak forces on adhering shells (e.g. hydrodynamic shear forces) can disrupt receptor-ligand pairs due to leverage. This could be the reason for the strategy of cells to stabilize adhesion plaques by stress fibers.

We greatly appreciate the help of Nikita Ter-Oganessian with the preparation of the manuscript.

A Appendix: Generic interfacial forces

Below we summarize the essential generic interfacial forces governing the adhesion of soft shells. The modification of these forces by steric repulsion associated with membrane undulations (or the dynamic surface roughness) are discussed at the end. The dominant forces are the ubiquitous Van der Waals and electrostatic forces, short range steric or solvation induced forces and steric repulsion mediated by repellers of the glycocalix and/or of the target tissue.

The Van der Waals attraction between a membrane of thickness δ ( 2 nm) and planar surface can be expressed as

	H	1			1
VVdW ≈ −				−			(A.1)
	12π		h2		(h + δ)2

where H is the Hamaer constant and is of the order of kBT . The adhesion energy for a distance of h ≈ 10 nm (the distance enforced typically by

repellers) is VVdW 10−6 J/m2. It is by an order of magnitude smaller for two interacting bilayers. For bare membranes d is about 2 nm (cf. Evans

et al.) and VVdW 10−5 J/m2. The electrostatic interaction between

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two di erently charged membranes in the presence of electrolyte exhibiting charge densities (per unit area) σ1 and σ2, respectively, is best expressed in terms of the disjoining pressure P = ∂V /∂h [19].

P (h) =	8π			σ12 + σ22 − 2 |σ1| σ2 cos κh			κh
	ε			(2 sin κh)2	e−			(A.2)

where ε is the dielectric constant (ε ≈ 80) and κ is the Debye screening length which depends on the salt concentration

κD2 =	8πe2c	·	(A.3)
	εkBT

(Note that for room temperature the screening length (in units of nm) is

κ−D1 = 3.08√c−1 where the salt concentration c is measured in units of mole/l).

If the two membranes exhibit the same sign of the surface charge the electrostatic force is repulsive and counteracts the Van der Waals attraction. An extensive discussion of adhesion mediated by attractive electrostatic forces between oppositely charged receptor-ligand pairs can be found in [14].

There are several contributions to the short range steric repulsion. A contribution which becomes only relevant at distances of 0.2 nm is the dehydration force associated with the removal of hydration water between the interfaces. The potential decays exponentially: Vhyd = V0 exp {−h/λ} with a screening length of λ 0.3 nm [20]. A steric short range repulsive force arises for the case of the adhesion of cells or cell models on biofunctionalized solid supports due to the repulsion exerted by the surface protein layer. It dominates the short range repulsion at full hydration and can also

be expressed in terms of an exponential law Vrep = K/h exp {−h/τ }, where K is the compression modulus of the protein cover and τ the thickness of

the biofunctional film [12].

The third and under practical conditions most important short range force is the repulsion mediated by the glycocalix. It depends critically on the surface concentration cR and the mobility of repeller molecules (besides the Van der Waals radius of the headgroup). An approximate expression for the interaction potential can be given for the model systems where lipids with polyethyleneoxide head groups are used as repellers (cf. Bruinsma et al. [6]). The size of the head group is determined by the Flory radius of the flexible macromolecule. For the specific (saturation) situation that the concentration of the repellers is adjusted in such a way that the head groups (of radius Rg) start to overlap (cRRg2 ≈ 1) the repulsion potential

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can be expressed as (cf. Bruinsma et al. [6])

Vrep (h) = 6		kBT cR	hg			exp −1.5	Rg			·	(A.4)
	π2			R	2			h	2

For cells the situation is more complicated but it is still possible to define an e ective Flory radius of the semiflexible head groups such as chains of IgG-like domains and a similar exponential expression is expected to hold.

Much more important than the detailed structure of the membrane proteins forming the glycocalix is the fact that the repellers are in general mobile and can exchange between the adhesion zone and the bulk of the non-adherent membrane fraction. The repeller molecules in the nonadherent (bulk) membrane fraction provide a reservoir of fixed chemical

potential µR,bulk = kBT ln c◦R/Rg2 which modifies the repulsive interaction potential. In equilibrium µR,bulk must be equal to the repeller chemical potential, µR, in the adhesion zone. The e ective thermodynamic po-

tential of the repellers in the adhesion zone is determined by the sum of the bare repulsion potential Vrep (Eq. (A.4)) and the translational entropy (kBT ·ln cR (h) /Rg2) where the concentration of the repellers depends on the interfacial distance h. The e ective potential which depends now on c◦R is

VR,e (h) = kBT 1 − exp − 6

hg

exp −1.5 Rg

·

(A.5)

π2

R

2

h

2

It is now helpful to consider two limiting situations:

For large interfacial distances h Rg one obtains

Vrep (h) ≈ kBT cR◦ hg exp

−2

Rg

(A.6)

R

3

h

2

which corresponds to the interaction potential at fixed concentration c◦R of the repeller.

For h Rg one obtains

Vrep (h) = kBT cR◦

(A.7)

which is the maximum value of Vrep (h) and corresponds to the two-dimen- sional osmotic pressure of the repeller.

The interplay between Van der Waals attraction and the thermodynamic repulsive interaction potential of the repellers has two important consequences

•The interaction potential can exhibit two minima corresponding to a state of strong adhesion (the Van der Waals minimum) and a state of

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weak adhesion. On physically or chemically rough surfaces (exhibiting lateral inhomogeneities of the interaction potential VVdW) this can lead to a decomposition of the adhesion zones into domains of tight adhesion separated by regions of weak coupling;

•With increasing repeller concentration the depth of the Van der Waals minimum is shifted to higher energies and reaches the value of the

shallow minimum at c◦RRg2 ≈ 0.6. Since the two minima are separated by an activation barrier this leads to a first order unbinding transition of the adhering shell driven by the chemical potential of the repellers;

•The binding strength of the tight adhesion is strongly reduced by the lateral osmotic pressure exerted by the repeller molecules in the non-adherent membrane fraction, a point discussed in the main text (cf. Ref. [21] for an experimental verification).

Consider now the e ect of undulation forces. This beautiful concept was introduced by Helfrich to explain the spontaneous swelling of lipid multilayers. It is an entropic force associated with the gradual freezing-in of long wavelength bending excitations if a flickering membrane approaches a surface. The decrease in entropy gives rise to a repulsive disjoining pressure Pund which can be estimated as follows [22]. The bending excitations are statistically excited local events and local deflections U (r, t) decay with a correlation length ζp [22, 23]. Because ζp ( 1 µm) is small compared to the size of vesicles or cells (erythrocytes) we can consider the soft shell to be composed of small segments (cushions) of dimension ζp · ζp which exhibit independent Brownian motions in the normal direction. In the Helfrich image a disjoining pressure arises due to the local collisions of the membrane segments with the wall, similar to the pressure generated by an ideal gas through the momentum transfer onto the wall. Since equipartition theory predicts an energy transfer of kBT per collision the undulation pressure is

Pund =	kBT	(A.8)
	2ζp2 · h

where h is the average distance between the two interfaces.

The correlation length ζp is proportional to the average distance h according to ζp ≈ κ/kBT h . This relationship follows from the conditions that the mean square amplitude (the so-called roughness) u2L of a membrane segment L L is u2L = (kBT /κ) L2. This relationship follows by integrating over all bending modes of amplitude u2q = kBT /κq4L2 from q = π/L to q = π/∆, with ∆ the membrane thickness. The roughness of a

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membrane at average distance h must be u2L ≤ h 2. It therefore follows

Pund = c	kBT	·	(A.9)
	h3

For one-component membranes the constant c has been more rigorously calculated by renormalization group calculations [23] and by Monte Carlo simulations to c = 0.1 and is considered as a universal number. The situation may be quite di erent for mixed membranes due to concentration fluctuations induced by the interaction of soft membranes with surface (S. Marx et al., unpublished data of authors laboratory).

References

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[6]Z. Guttenberg, A. Bausch, B. Hu, L. Moroder, R. Bruinsma and E. Sackmann, Langmuir 16 (2000) 8984-8993; G. Wiegand, K. Neumaier and E. Sackmann, Appl. Opt. 37 (1998) 6892-6905.

[7]Bin Hu et al., Biochemistry 39 (2000) 12284-12294.

[8]W. Helfrich, Z. Naturf, 33A (1978) 305-312; U. Seifert, Phys. Rev. Lett. 74 (1995) 5060-5063; J. R¨adler, T. Feder, H. Strey and E. Sackmann, Phys. Rev. E 51 (1995) 4526-4531.

[9]A. Zilker, M. Ziegler and E. Sackmann, Phys. Rev. A 46 (1992) 7998-8001.

[10]R. Lipowsky and U. Seifert, Langmuir 7 (1991) 1867.

[11]P.S. Swain and D. Andelmann, Langmuir 15 (1999) 8902.

[12]G.I. Bell, M. Dembo and P. Bongrand, Biophys. J. 45 (1984) 1051; P. Bongrand,

Rep. Progr. Phys. 62 (1999) 921.

[13]If we start from a spherical shell with contour U (r) the deformed shape of the adhering shell is U (r) + ζ (r) (where ζ (r) is the local deformation). One can then

write σ∆O = 12 {grad ζ (r)}2 dxdy and H = ∆ζ (r)

[14]J. Nardi, R. Bruinsma and E. Sackmann, Phys. Rev. E 58 (1998) 6340-6354.

[15]L.D. Landau and E.M. Lifschitz, Theory of Elasticity (Pergamon Press, New York, 1986).

[16]R. Bruinsma and E. Sackmann, Comptes Rendus de l’Acad´emie des Sciences – Ser. IV – Phys. 2 (2001) 803-815.

[17]From the apex angle φ of the triangular shape one can determine the tangential

force f according to f = (κ · λ)1/2 θc2 (π/2 − φ) (cf. Albersd¨orfer et al. Europhys. Lett. 42 (1998) 227 for details).