Force-Induced De-Adhesion of Specifically Bound Vesicles: Strong

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Force-Induced De-Adhesion of Specifically Bound Vesicles: Strong Adhesion in Competition with Tether Extraction Ana-Suncˇana Smith*,†,‡ and Udo Seifert‡ E22 Institut fu¨ r Biophysik, Technische Universita¨ t Mu¨ nchen, D-85748 Garching, Germany, and II. Institut fu¨ r Theoretische Physik, Universita¨ t Stuttgart, D-70550 Stuttgart, Germany Received May 16, 2005. In Final Form: July 20, 2005

A theoretical study of the thermodynamic equilibrium between force-induced tether formation and the adhesion of vesicles mediated by specific ligand-receptor interactions has been performed. The formation of bonds between mobile ligands in the vesicle and immobile receptors on the substrate is examined within a thermodynamic approximation. The shape of a vesicle pulled with a point force is calculated within a continuous approach. The two approaches are merged self-consistently by the use of the effective adhesion potential produced by the collective action of the bonds. As a result, the shapes of the vesicle and the tether, as well as the number of formed bonds in the contact zone, are determined as a function of the force, and approximate analytic expressions for them are provided. The de-adhesion process is characterized by the construction of a phase diagram that is a function of the density of the ligands in the vesicle, the surface coverage by receptors, the ligand-receptor binding affinity, and the reduced volume of the vesicle. In all cases, the phase diagram contains three regions separated by two nonintersecting lines of critical forces. The first is the line of onset forces associated with a second-order shape transition from a spherical cap to a tethered vesicle. The second line is attributed to the detachment forces at which a first-order unbinding transition from a tethered shape to a free vesicle occurs.

Introduction Of the many fascinating aspects of the living cell, those associated with its mechanical properties are among the most fundamental as well as the most amenable to physical analyses. Along the road to obtaining a better understanding of such properties in complex cellular systems, an extremely successful approach has been to investigate them in simpler, more controlled environments such as vesicles. In particular, the study of vesicles adhering to flat substrates has yielded many important insights into the process of cell adhesion.1 In the first instance, the adhesion occurring in such systems may be nonspecific in nature. A more realistic elaboration occurs, however, when a particular type of ligand molecule is embedded into the vesicle membrane. These ligands are chosen in such a way that they are subject to a complementary interaction with appropriate receptors, which are distributed on the surface of the flat substrate. The adhesion of the vesicle to the substrate in such a system is thus known as ligandreceptor-mediated or specific adhesion.2 Naturally, a model system consisting of a ligandcontaining vesicle and a receptor-coated substrate is an ideal context in which one can better understand the fundamental aspects of specific adhesion. In addition to this, such a system is also a convenient means to access information concerning the response of specifically adhered vesicles to external perturbations, such as the application of force. The latter control mechanism is known * To whom correspondence should be addressed. E-mail: [email protected]. † Technische Universita ¨ t Mu¨nchen. ‡ Universita ¨ t Stuttgart. (1) Sackmann, E. Physical basis of self-organization and function of membranes: Physics of vesicle. In Handbook of Biological Physics; Lipowsky, R., E. Sackmann, E., Eds.; Elsevier Science: Amsterdam, 1995, Chapter 5. (2) Bongrand, P. Rep. Prog. Phys. 1999, 62, 921-968.

to be important in a cellular context, for example, in connection with the rolling of leukocytes.3 The simplest example of a “specific” type of adhesion being subjected to an external force is the case in which a force load is applied to a single ligand-receptor “bond”. The physics in this scenario can be understood by simply applying the Arrhenius law of chemical kinetics,4,5 with the caveat that the bonds appear to be more resistant to higher force rates; therefore, rapid, force-induced unbinding is more difficult than the slower loading alternative.6-8 In addition, geometrically constraining ligands and receptors binding to a two-dimensional (2D) surface can significantly decrease their binding affinity in comparison to their affinity in solution. The 2D binding affinity, as well as the forward and reverse rate constants, could be measured by the use of micropipet-related techniques for the force spectroscopy of either a single bond or a very low number of bonds9-11 and could be understood on the basis of a stochastic theory.10 In contrast, the understanding of the resistance of ensembles of ligand-receptor pairs under force is still rudimentary. The first to address the problem of a force acting on membrane-confined ligand-receptor bonds were Bell,12 Dembo and co-workers,13 as well as (3) Springer, T. A. Cell 1994, 76, 301-314. (4) Merkel, R. Phys. Rep. 2001, 346, 344-385. (5) Leckband, D.; Israelachvili, J. Q. Rev. Biophys. 2001, 34, 105267. (6) Evans, E.; Ritchie, K. Biophys. J. 1997, 72, 1541-1555. (7) Alon, R.; Hammer, D. A.; Springer, T. A. Nature 1995, 374, 539542. (8) Merkel, R.; Nassoy, P.; Leung, A.; Ritchie, K.; Evans, E. Nature 1999, 397, 50-53. (9) Piper, J. W.; Swerlick, R. A.; Zhu, C. Biophys. J. 1998, 74, 492513. (10) Chesla, S. E.; Selvaraj, P.; Zhu, C. Biophys. J. 1998, 75, 15531572. (11) Nguyen-Duong, M.; Koch, K. W.; Merkel, R. Europhys. Lett. 2003, 61, 845-851. (12) Bell, G. I. Science 1978, 200, 618-627. (13) Dembo, M.; Torney, D. C.; Saxman, K.; Hammer, D. Proc. R. Soc. London, Ser. B 1988, 234, 55-83.

10.1021/la051303f CCC: $30.25 © 2005 American Chemical Society Published on Web 10/18/2005

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Evans.14 When the bonding components are restricted to surfaces, the complex keeps the surfaces in close proximity and thus allows for broken bonds to reform.12,15 Depending on the manner in which the force is transduced, the bonds in the complex of numerous bonds may respond as though they are either in series, in which only molecules in the outer rim feel the load, or in parallel, where the load is shared by all the bonds in the complex.16,17 A model for the unbinding of a specifically adhered vesicle by the application of a force load to the bonds in series was subsequently suggested.18 Sometime later, a stochastic theory for the unbinding of a complex of parallel bonds under constant force19,20 and a force load21 was developed. From an experimental point of view, the dissociation of multiple bonds that formed between a cell and a vesicle under a force load was studied with the micropipet aspiration technique.22 Although good agreement could be obtained with the theoretical predictions,16 no direct observation of events in the contact zone between the vesicle and the cell could be obtained. Similar experiments were recently performed with red blood cells adhering to a substrate but under constant force.23 By the use of an interferometric technique, a more detailed analysis of the dissociation of bonds under force in the contact zone could be obtained.24 However, in this case, the deformation of the vesicle shape, which occurs as a response to the force but also mediates the force to the bonds, remained somewhat obscure because only a small portion of the shape could be reconstructed. The response of a vesicle to the induction of a pointlike force strongly depends on whether the vesicle is weakly or strongly adhered.25 In the case of weakly adhered vesicles, elliptical deformations that use excess free surface area have been theoretically predicted26 and concomitantly observed by combining the fluorescent and interferometric approaches.27 The detachment of such vesicles is associated with a first-order unbinding transition, wherein the adhesion in the contact zone fails to provide adequate counteraction to pulling, and the vesicle disengages from the substrate while still in possession of a finite contact zone. This type of detachment has also been observed in experiments on nonspecifically and weakly adhered vesicles pulled with micropipets.23 On the other hand, ellipsoidal deformations are energetically unfavorable in the case in which the vesicle is a tense spherical cap. In this case, as has been demonstrated in several different pulling experiments,28-30 the (14) Evans, E. A. Biophys. J. 1985, 48, 175-183; 1985, 48, 185-192. (15) Bell, G. I.; Dembo, M.; Bongrand, P. Biophys. J. 1984, 45, 10511064. (16) Seifert, U. Phys. Rev. Lett. 2000, 84, 2750-2753. (17) Seifert, U. Europhys. Lett. 2002, 58, 792-798. (18) Boulbitch, A. Eur. Biophys. J. 2003, 31, 637-642. (19) Erdmann, T.; Schwarz, U. S. Phys. Rev. Lett. 2003, 92, 108102. (20) Erdmann, T.; Schwarz, U. S. J. Chem. Phys. 2004, 121, 89979017. (21) Erdmann, T.; Schwarz, U. S. Europhys. Lett. 2004, 66, 603609. (22) Prechtel, K.; Bausch, A.; Marchi-Artzner, V.; Kantlehner, M.; Kessler, H.; Merkel, R. Phys. Rev. Lett. 2002, 89, 028101. (23) Pierrat, S.; Brochard-Wyart, F.; Nassoy, P. Biophys. J. 2004, 87, 2855-2869. (24) Guttenberg, Z.; Bausch, A.; Hu, B.; Bruinsma, R.; Moroder, L.; Sackmann, E. Langmuir 2000, 16, 8984-8993. (25) Seifert, U.; Lipowsky, R. Phys. Rev. A 1990, 42, 4768-4771. (26) Smith, A.-S.; Sackmann, E.; Seifert, U. Europhys. Lett. 2003, 64, 281-287. (27) Smith, A.-S.; Goennenwein, S.; Lorz B.; Seifert, U.; Sackmann E. 2004, unpublished work. (28) Evans, E.; Yeung, A. Chem. Phys. Lipids 1994, 73, 39-56. (29) Heinrich, V.; Bozˇicˇ, B.; Svetina, S.; Zˇ eksˇ, B. Biophys. J. 1999, 76, 2056-2071. (30) Waugh, R. E. Biophys. J. 1982, 38, 19-27; 1982, 38, 29-37.

Smith and Seifert

vesicle undergoes a shape transition toward a tethered shape. Such experiments have been used to measure a wide variety of membrane mechanical properties.31 Recently, tether formation has also been used as a technique for probing a small number of ligand-receptor bonds.32 If the vesicle is strongly adhered, it will adopt a shape of a spherical cap. Because of the geometrical constraints, the spherical cap is uniquely defined by the reduced volume of the vesicle only. The excess surface area of a vesicle, obtained in the process of volume reduction (equilibration of the osmotic pressures of the inner and outer vesicle solutions), is entirely used for the formation of the largest possible contact zone. Consequently, no excess area is left in the free part of the vesicle, and the membrane fluctuations are reduced to very small oscillations around the stationary shape and can therefore be omitted. The simplest way to model adhesion is by the use of a nonspecific contact potential.25 It is also the only model that is capable of directly accounting for the vesicle shape. In this model, the spherical cap is a solution of a limit when WRs2κ . 1 (W is the adhesion energy, Rs is the radius of a sphere with equivalent area and volume, and κ is the bending rigidity). Furthermore, it is in this limit that Young-Dupre´’s theory applies, and an effective strength of adhesion can be quantified by means of a droplet wetting theory.25,33 Pulling on such a nonspecifically adhered vesicle with a point force indeed produces a tether as a part of a first-order shape transition. The detachment process resulting from that model culminates in a secondorder unbinding transition at finite forces.34 Many more sophisticated models have been developed over time concentrating on the specificity of vesicle adhesion.35-39 Within the same physical frame as that used previously,12,13,35 a simple thermodynamic model for vesicle adhesion mediated by numerous ligand-receptor bonds was recently developed.40 In that model, explicit consideration has been given to the following factors: (i) the enthalpy of binding, (ii) the mobility of the ligands in a vesicle through a contribution to the mixing entropy, (iii) the bending energy of the vesicle shape, (iv) the finite number of ligands contained in the vesicle, and (v) the constant density of receptors on the substrate. Several important outcomes have emerged from the model. First, the explicit calculation of the bending energy has been used to show that the determination of the vesicle shape can be decoupled from the equilibration of ligand-receptor binding in the contact zone. Second, the density of the formed bonds has been found to never decrease in response to a reduction in the size of the contact zone. Finally, and perhaps most importantly, the formation of bonds has been found to result in a quantifiable effective adhesion strength, which acts as a vesicle spreading pressure. In this work, we propose a mechanism for vesicle detachment in which the vesicle is first strongly adhered (31) Dai, J.; Sheetz, M. P. Laser Tweezers in Cell Biology; Sheetz, M. P., Ed.; Accademic Press: San Diego, 1998; see also references therein. (32) Heinrich, V.; Leung, A.; Evans E. Biophys. J. 2005, 88, 22992308. (33) Sackmann, E.; Bruinsma, R. F. ChemPhysChem 2002, 3, 262269. (34) Smith A.-S.; Sackmann, E.; Seifert, U. Phys. Rev. Lett. 2004, 92, 208101. (35) Dembo, M.; Bell, G. I. Curr. Top. Membr. Transp. 1987, 29, 71-89. (36) Zuckerman, D. M.; Bruinsma, R. F. Phys. Rev. Lett. 1995, 74, 3900-3903. (37) Lipowsky, R. Phys. Rev. Lett. 1996, 77, 1652-1655. (38) Weikl, T. R.; Andelman, D.; Komura, S.; Lipowsky R. Eur. Phys. J. E 2002, 8, 59-66. (39) De Gennes, P. G.; Brochard-Wyart, F. Langmuir 2003, 19, 71127119. (40) Smith A.-S.; Seifert, U. Phys. Rev. E. 2005, 71, 061902.

Strong Adhesion Versus Tether Extraction

to the substrate by numerous ligand-receptor bonds and then pulled, in a vertical direction, away from the substrate with a constant force. The equilibrium approach undertaken should apply as long as both the diffusion of the ligands and the kinetics of the bond formation are fast in comparison to the force load. Our aim is to blend the continuous approach to macroscopic vesicle shapes with a more elaborate model that is capable of accounting for the microscopic nature of vesicle adhesion mediated by the formation of ligand-receptor bonds. In particular, the thermodynamic approach40 is combined with the calculations of tethered shapes in a continuous model.34 The tether formation induced by the force is in competition with adhesion because the constraints on the total area and volume impose that the material from the tether must come from the contact zone between the vesicle and the substrate. Furthermore, the force induces the failure of ligand-receptor bonds. The consequence of the application of the force is thus de-adhesion seen as a decrease of the vesicle-substrate contact zone. On the other hand, the vesicle opposes de-adhesion by increasing the density of bonds in the remainder of the contact zone, leading to an increased spreading pressure of the vesicle. The balance between the force and the spreading pressure typically provides a thermodynamic equilibrium where the contact zone is smaller than the initial one. Within a contact zone that is thus constrained to a particular size, the equilibrium number of formed bonds can be determined by the use of the thermodynamic model, which also provides the appropriate effective adhesion strength. The latter can be used as an input parameter for continuous models that result in vesicle shapes. Consequently, a self-consistent description of both the tethered vesicle shape and the average number of ligand-receptor bonds emerges from the procedure. Construction of the Model Notation. The vesicle surface is separated into a region that is parallel to the substrate that forms the contact zone and a region that consists of the remaining (in the absence of force, spherical) part of the vesicle. The interaction of the ligands with the receptors occurs within the contact zone (see Figure 1). Nevertheless, the regions are able to exchange ligands and area. In particular, during the force-induced detachment process, the area from the contact zone will be gradually used for the formation of the extending tether. The notation used is generally in reduced units, in which the relevant length scale is the radius of the sphere with the equivalent area and volume Rs. However, sometimes the real values will be calculated. To distinguish the reduced from the real values, the first letter of the real variable will be capitalized (e.g., lT T LT). A particular superscript may be added to indicate a critical value of a given parameter when necessary. Generally used notation includes: κ - bending rigidity of the membrane. kB - Boltzmann constant. Rs - radius of a sphere with the area equivalent to this of a vesicle. v - reduced volume of the vesicle (v ) 3V/4πRs3), 0 e v e 1. For the purpose of describing the vesicle shape, the following parameters will be used: hv - height of the spherical cap, 0 e hv e 2. rv - radius of the spherical cap. lT - tether length. rT - tether radius.

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Figure 1. Vertical pulling on a specifically adhered vesicle results in the formation of a tether at the north pole of the vesicle. Only a fraction of the substrate and the vesicle surface is covered by receptors and ligands, respectively. Furthermore, it is possible that there will be both ligands and receptors in the contact zone that are not participating in the formation of bonds (see text for notation).

To account for the discrete nature of the ligand-receptor binding, the vesicle surface is divided into sites of area R. The area of the site is associated with the ligand gyration area. The following notation will be used for the description of the contact zone: St - total number of sites in the vesicle (St ) 4πRs2/R, in which R is the area of a single ligand). sc - fraction of the total vesicle area forming the contact zone (sc ) A*/4πRc2 ) Sc/St, in which A* ) ScR and is the area of the contact zone, and Sc is the total number of sites in the zone41). Fr - fraction of the substrate surface covered by receptors in the adhesion zone. Ea - ligand-receptor binding energy (in units of kBT). This is the enthalpy change for the chemical reaction ligand + receptor T complex, in which all of the contributors in the reaction are already under the conditions of the experiments (in terms of temperature, pH, and surface confinement). This enthalpy does not include the change in enthalpy between standard conditions and the conditions of the experiment and is independent of the constituent concentrations. Generally, it is a forcedependent quantity. nt - total surface concentration of ligands in a vesicle (e.g., for nt ) 0.5, 50% of the vesicle surface is covered by ligands). nb - fraction of total ligands that are in the contact zone and bound to receptors (e.g., for nb ) 0.5, 50% of ligands in a vesicle are bound to the substrate). nf - fraction of total ligands that are in the contact zone and free. (41) The area of the contact zone should not be confused with the area associated with tight adhesion complexes that are often seen with reflection interference contrast microscopy. The area of tight adhesion is, to the first-order approximation, proportional to the number of formed bonds, whereas the area of the contact zone is the entire area in the vicinity of the substrate (the closest 100 nm).

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Table 1. A Set of Expressions Characterizing the Onset of the Tether, the Detachment Process, and the Conditions for the Vesicle Unbinding reduced units

geometrical first-order transition for the tether appearance

f ) FRs/πκ

xhov =

2

f lT ) LT/Rs rT ) RT/Rs σ ) ∑Rs2/κ p ) PRs3/κ

o

2

2

+ 16soc + 4soc hov w - hov

-1

po = 0.5(4w - f o2)hov hov ) hov(v)

sc ) Sc /(4R2s π)

rov soc

) )

2 0.5hov(4soc + hov ) 2 o sc (v) ) 0.5 - 0.125hov

f

f d = 2v1/3w

lT = (1 - v2/3)f rT = 2f-1 σ ≈ 0.125f 2 -

ldT = 2(v1/3 - v)w rdT = 2v2/3w-1 σd = 0.5v2/3w2 pd = -v1/3w2

rv ) 0.5hv(4sc + hv2)

hdv = 2v1/3 rdv = v1/3 0

-

All of the variables that are expressed as fractions adopt values between 0 and 1. The only exception is the fraction of the vesicle area in the contact zone (sc), which has its maximum value geometrically constrained to 0.5. Pulling on Vesicles Strongly Adhered in a Contact Potential. To calculate the macroscopic shape of a tethered vesicle, the proven continuum approach for the calculation of shapes has been undertaken. As mentioned previously, such a model has been presented elsewhere in reasonable detail34 for the case of nonspecifically bound vesicles. However, in the following section, we will summarize the results of these calculations, which are important for the construction of a model for the detachment of specifically adhered vesicles. Modeling the detachment of an adhered vesicle by means of a point force can be achieved by finding stationary solutions of the appropriate free energy by means of variation calculus. Most generally, such a free energy can be written as:

F ) Fel + PV + ΣA + F(Z - Z0) + Fadh

(1)

The first term in eq 1 is the bending contribution of the shape. The following two terms are responsible for maintaining the volume (V) and the area (A) of the vesicle shape at constant values. This is achieved by appropriate adjustments to the osmotic pressure (P) and tension (Σ), respectively. The fourth term is proportional to the force (F) and the displacement of the vesicle’s north pole from its position in the absence of force (Z0). The last term in eq 1 accounts for the interaction between the vesicle and the substrate. The vesicle is adhered in a nonspecific contact potential Fadh ) -WA*, in which W > 0 and is the a strength of a contact potential. The free energy emerging from eq 1 can be written, by the use of reduced variables, in units of πκ as:

f )

second-order unbinding transition

2soc

0 0 σo = 0.125 f o2

hv ) Hv/Rs rv ) Rv/Rs

tether extraction

lT + pv + σ + f(lT + hv) - 4wsc rT

(2)

The first term in eq 2 is the bending energy of a tube presented in standard reduced units. The variation of the bending energy of a spherical cap is at least one order of magnitude smaller and is hence omitted from the calculations. The subsequent two terms represent the constraints for the total volume and area of the shape, in which the reduced volume and the area of a shape are given by 8v ) 12sc + hv3 + 0.5lTrT2 and a ) 2sc + 0.25hv2 + 0.5lTrT ) 1, respectively. The fourth term describes the action of the force. The last term is the adhesion energy

in which the conversion of the contact potential is given by w ) WRs2/κ. For other variables, the transformation from real to reduced units is given in the first column of Table 1. The detachment process of a vesicle adhered in a constant contact potential can be summarized as follows:34 At zero force, the thermodynamic equilibrium shape of a vesicle is a spherical cap. Because of the area and volume constraint, the membrane material for the (tubular) deformation of the shape must arise from the contact zone. Thus, at least partial reduction of the contact zone must be achieved to form a tether. As there is only one spherical cap for a given reduced volume, there is a range of nonzero forces in which the initial shape of the vesicle will not be influenced. The increase of force, in this regime, serves mainly to build up the tension and the pressure in the vesicle. There is, however, a critical force that is sufficient to overcome the adhesive potential. It is at this force that a geometrical first-order transition from a spherical cap to a tethered shape occurs and the detachment process of the vesicle begins. (When this onset of tether formation is referenced, all variables will have an additional superscript o, as shown in the second column of Table 1.) The increase in the force above the onset results in a detachment process in which the tether grows (sometimes to lengths much larger than the vesicle height) until the contact zone is gradually spent. At a critical force, this detachment process culminates in a secondorder unbinding transition, in which the vesicle separates from the substrate. The characteristic values of the system parameters when this transition occurs are characterized by a superscript d and are presented in the last column of Table 1. The onset height and contact area are nontrivial functions of the reduced volume and can be calculated from the geometrical constraints when rT ) lT ) 0. The pressure, the area, and the height of the vesicle in the detachment process could not be determined approximately. However, it is possible to obtain them directly from the minimization of eq 2 for any force. Modeling Ligand-Receptor-Mediated Adhesion of Vesicles. It was shown in our previous study that the adhesion of a vesicle to a substrate by the formation of ligand-receptor bonds results in an effective adhesion strength.40 This effective adhesion strength acts as a spreading pressure of the vesicle and, in the case of a force applied to the membrane, is expected to provide counteraction. Furthermore, a direct correspondence was established between this effective adhesion strength and the contact potential used in the previous section for the calculation of tethered shapes. Because this former work40


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ligands in both the contact zone and the free part of the vesicle. Consequently, stochastic distributions of ligands should be expected in both these regions. This level of approximation is clearly unable to account for the bond aggregation that is often observed in the contact zone.33 However, such an approach should provide good estimates for the average number of formed bonds as long as all ligand-ligand interactions remain small (no clustering prior to the adhesion). Fractions of Ligands in the Contact Zone. The free energy from eq 3 is a function of three unknown parameters: nb, nf, and sc. Minimization of this free energy can be performed in two steps. First, the minimization with respect to ligand density variables (nb and nf) results in: nb )

Figure 2. The density of bound ligands (nb/sc) in the contact zone for different ligand-receptor binding strengths Ea. Independent of the density of the system constituents and their binding strengths, the density of the ligands never increases with the increasing size of the contact zone.

contains an overview of the relevant literature as well as a thorough characterization of the behavior of the effective adhesion strength as a function of the ligand and receptor densities and their binding strength, we will not reproduce this material in full herein. We will, however, outline, in some detail, the elements of the model pertinent to the unbinding of specifically adhered vesicles by the application of a point force. The Free Energy. In accordance with eq 1, the free energy of adhesion (in units of kBT, in which T is the temperature) can be written as follows:

F ) Fel(v,sc) - nbntStEa - ln Ω

(3)

in which

Ω)

(

)(

)(

(1 - sc)St sc(1 - Fr)St scFrSt nbntSt nfntSt (1 - nf - nb)ntSt

)

The first term in eq 3 is the bending energy of the vesicle shape at the given size of the contact zone for a vesicle with fixed reduced volume v. The evaluation of this term shows that the bending energy diverges in a manner that is inversely proportional to the size of the contact zone as the maximum size of the contact zone obtained for the shape of the spherical cap42 is approached. However, such calculations are equivalent to the calculations of shapes in a given contact potential with an adjustable contact zone size.25 In the present case, the contact potential resumes the role of a Lagrange multiplier. For the details of these shapes and their calculations, see the relevant literature. The second term in eq 3 is the enthalpy of binding and is proportional to the absolute number of formed bonds (nbntSt) and the binding strength of the ligand-receptor pair Ea (in units of kBT). The negative sign indicates that bond formation is favorable in terms of the total free energy. The last term is the mixing entropy of ligands having the freedom of occupying receptors in the contact zone and can be evaluated approximately by the use of the Stirling formula. This term assumes ideal mixing of (42) Smith, A.-S. Ph.D. Thesis, Technische Universita¨t Mu¨nchen, Germany, 2004; http://tumb1.biblio.tu-muenchen.de/publ/diss/ph/2004/ smith.html.

(

1 1 Fs + + 2 2nt r c

)

1 - xe2Ea(nt - Frsc)2 - 2eEa[nt2 + Fr2sc2 - (nt + Frsc)] + (nt + Frsc - 1)2 (eEa - 1)

(4a) sc(1 - Fr) 1 - scFr

nf ) (1 - nb)

(4b)

Both nb and nf in eq 4, parts a and b, respectively, are still functions of the size of the contact zone. If both parts of eq 4 are divided by sc, the density of the ligands in the contact zone is obtained (see Figure 2 for the density of the bound ligands). The dependence of the density of the formed bonds on the size of the contact zone clearly demonstrates the interplay between the entropic contribution to the free energy and the total enthalpy of binding (Figure 2). When Ea ) 0, the density of bound ligands in the contact zone becomes independent of the size of the contact zone and results simply from an entropically driven equipartition of ligands over the entire vesicle surface (the bond is formed whenever the ligand is positioned above the receptor). For small binding strengths (e.g., Ea ) 1), the entropy still dominates the thermodynamic equilibrium, and the effective adhesion strength is very weakly dependent on the size of the contact zone. When Ea f +∞, the enthalpy completely dominates the free energy, and the number of bonds is maximized. When there are more ligands in the vesicle than receptors on the substrate, the density of bound ligands saturates to a constant value because all of the receptors on the substrate are bound (see Figure 3 when Ea ) 10, for small sc). This is a case of a so-called receptor-dominated type of equilibrium in which the absolute number of bound ligands decreases linearly with the size of the contact zone. If there are more receptors in the contact zone than ligands in the vesicle, when Ea f +∞, all of the ligands will be bound, and a ligand-dominated type of equilibrium (see Figure 2 when Ea ) 10, for large sc) will result. This will lead to an increase in the density of formed bonds as the contact zone becomes smaller, although the total number of bonds remains basically constant. The crossover from a ligand- to receptor-dominated type of equilibrium arises at the critical size of the contact zone s/c ) nt/Fr, at which the number of receptors on the substrate is the same as the number of ligands in the vesicle. For binding strengths in the range of 0 to ∞, the density of bonds remains a monotonically increasing function as the size of the contact zone decreases, indicating the reorganization of the ligand-receptor bonds. This fact will become crucial when attempting to decrease the

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Figure 3. The free energy per site (F/St) comprising the bending contribution of a vesicle with the reduced volume v ) 0.85 (thin line) and the free energy with no bending contribution (dashed line) as a function of the size of the contact zone (sc). The bending energy term induces a stable boundary minimum at sc ) smax , c the value at which the shape of the vesicle is a spherical cap. Otherwise, the bending term does not influence the total free energy. Inset: dependence of the size of the contact zone associated with the spherical cap on the reduced volume of the vesicle.

contact zone by force because it implies that doing so should actually lead to an increased binding of ligand-receptor pairs. Decoupling of the Vesicle Shape. Substituting eq 4 into eq 3 results in an expression for the free energy that depends only on the size of the contact zone, an example of which can be seen in Figure 3. An inspection of Figure 3 shows that the free energy is a decreasing function of sc, leading to a boundary minimum at sc ) smax with c respect to the same variable induced by the divergence of the bending energy. To reach the thermodynamic equilibrium, the vesicle will thus maximize its area of contact with the substrate. However, the size of the contact zone is restricted by the volume and area constraints and the bending energy of the vesicle. Consequently, in the thermodynamic equilibrium, the vesicle shape is always that of a spherical cap, with the optimum number of bound and free ligands in the contact zone given by eq 4. The spherical cap is uniquely defined by the reduced volume. Furthermore, apart from inducing a minimum, the bending energy does not influence the free energy at other values of sc < smax . This is because the magnitude of the c bending energy term is generally much smaller than that of the other terms in the free energy. Therefore, the bending energy term can be omitted from the calculcations in the absence of force. In the presence of force, the existence of the boundary minimum has important implications. The model presented for tethering vesicles that are adhered in a constant contact potential has shown that the force is of the same order of magnitude as that of the bending term. Hence, the force term will be able to influence the position of the boundary minimum (inducing a detachment) without influencing the free energy curve. As a result, a detachment process will do nothing other than move the boundary minimum along the free energy curve (as shown in Figure 3 as a dashed line) toward smaller values for the size of the contact zone. Our task is thus reduced to that of attributing a particular free energy or a particular size of the contact zone to the appropriate force. The Effective Adhesion Strength. The effective adhesion strength is the work required to exchange the area of contact of the vesicle with the substrate for one surface unit (one site) at a constant receptor surface

Smith and Seifert

Figure 4. The effective adhesion strength (ω) as a function of the size of the contact zone (sc) for different binding strengths (Ea) of the ligand-receptor pair. Independent of the chosen system parameters, the effective adhesion strength decreases monotonically from its maximum value (at sc ) 0).

coverage. Therefore, the vesicle spreading pressure is determined as a function of the size of the contact zone. Expressed in units of kBT/R (R is the area of the site), the following expression40 is developed:

ω≡-

(

)

Fr Frsc - nb ∂F ) Fr ln - ln ∂sc 1 - Fr (1 - Fr)sc - nf

(5)

Here, nb and nf are given by eq 4 and should be evaluated at the appropriate size of the contact zone. Importantly, the spreading pressure always increases monotonically as the size of the contact zone decreases. This can be confirmed by an inspection of Figure 4 in which the effective adhesion strength is presented for a set of parameters identical to those used for determining the density of the formed bonds in Figure 2. The fastest changes of the effective adhesion strength are obtained in the vicinity of s/c ) nt/Fr. However, the position of s/c can be influenced by changing the composition of the system. Furthermore, the effective adhesion strength reaches its maximum value for the vanishing size of the contact zone. This value can be determined exactly and is given by the following expression:

ωd ) Fr ln[(eEa - 1)nt + 1]

(6)

This spreading pressure needs to be overcome to detach a vesicle (hence, it is characterized by a superscript d). Even when the density within the contact zone has saturated to its maximum value, the effective adhesion strength is found to increase in response to a contact-zone reduction. This can be seen by a comparison of Figures 2 and 4, which were derived from the same system parameters. Hence, it should be expected that successful pulling will always become progressively more difficult as the contact zone becomes smaller. The Conversion Rule. It is relatively simple to establish a direct mapping of the effective adhesion strength ω (calculated in the previous section) and the contact potential w used in the continuous model.34 The


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Figure 5. Phase diagram as a function of the fraction of ligands in the vesicle nt (left), ligand-receptor binding strength Ea (middle), and the surface coverage by receptors Fr (right). A region associated with a spherical cap, a tethered shape, and a free vesicle can be seen in each graph. The regions are separated by lines of critical forces. Solid lines indicate the onset force (f o) and are associated with the first-order shape transition. Dashed lines correspond to the second-order unbinding transition obtained for the detachment force f d.

obtained forces and other variables could, however, be several orders of magnitude different if different conversion factors had been used.

direct comparison of units provides:

w ) Ξω, in which Ξ )

Rs2 kBT R κ

(7)

Importantly, the conversion factor Ξ can adopt values ranging over several orders of magnitudes, even under standard laboratory conditions. For example, a membrane with a bending rigidity of 10 kBT that belongs to a vesicle for which Rs ) 10 µm, which contains ligands with a gyration radius of 5 Å, will lead to Ξ ≈ 108. If a vesicle of the same size contains more cholesterol, which can increase the bending stiffness up to 100 kBT, and a larger ligand (with a gyration radius of 1 nm), the conversion factor will be considerably smaller (Ξ ≈ 106). If the same ligand is embedded into a smaller vesicle (Rs ) 1 µm) comprised of the same type of membrane, the conversion factor will be even more markedly affected (Ξ ≈ 104). On the other hand, vesicles of different size with very different compositions may produce the same conversion factor. Although the w and ω are clearly potentials of different magnitude scales, there is a range of parameters for which such mapping is sensible. This is because vesicles can sustain relatively large adhesive potentials before undergoing lysis. Furthermore, the more the volume of a vesicle is reduced, the greater the adhesive potential required to obtain strong adhesion becomes. This leads to different orders of magnitude of the potential strength required for vesicles with reduced volumes close to that of a sphere (e.g., v ) 0.99), in comparison to strongly deflated ones (e.g., v ) 0.8). In conclusion, mapping between the continuous and the thermodynamic models requires the determination of Ξ. However, once Ξ is set, all other conversion rules (presented in the first column of Table 1) are defined as well. For this reason, it is not convenient to present the mapping in a general way. In the following sections, we will present the results for the detachment of a specifically adhered vesicle for a chosen conversion factor. The

Consequences of the Model - Results By using the conversion rule, it is possible to determine the “contact potential” required for obtaining a particular shape for any given composition of the vesicle and the substrate. The shape results in a particular size of the contact zone that, in turn, provides a new effective adhesion strength. The unbinding process that occurs between the onset and the detachment parameter values can be evaluated by calculating the force required to achieve a particular size of the contact zone associated with a particular value of the effective adhesion strength. The Phase Diagram. A particular family of solutions for eq 1 that is associated with given forces can be recognized as a phase. Different phases can be found as a function of the system parameters. For tethering nonspecifically bound vesicles, the system parameters are the strength of the contact potential and the reduced volume of the vesicle.34 For the case in which adhesion is mediated by specific binders, the reduced volume is still an important system parameter but the adhesion strength becomes a function of three parameters: the fraction of ligands in the vesicle nt, the surface coverage Fr, and the binding strength Ea. The most transparent manner in which to present the phase diagram is to calculate sections of the diagram by varying one of these three parameters while holding the two others constant (Figure 5). In a fashion similar to that used in the continuum model (for which the effective adhesion strength is constant during detachment), the phase diagram was found to consist of three regions. The first region belongs to the shape of a spherical cap. In the second region, the equilibrium shape is a tethered vesicle. These two parts of the phase diagram are separated by a line associated with the onset force f o, which is where the first-order shape transition takes place. The region of tethered shapes is separated from the third

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region that belongs to free vesicles. A line separating these two regions is a line of the second-order unbinding transition and is associated with the detachment force f d. At zero force, the shape of the vesicle is uniquely defined by the reduced volume v, which provides the equilibrium and the height (hv) of the size of the contact zone smax c is determined, choosing a ligand and a vesicle. Once smax c compatible receptor (at concentrations nt and Fr, respectively) that bind with a specific binding strength Ea makes it possible to evaluate the equilibrium fractions of ligands in the contact zone nb and nf (using eq 4). The fractions nb and nf (for smax ) provide an effective adhesion strength c ω that is calculated from eq 5. According to the continuous calculations, forces below the onset force are not sufficient to compete with the effective adhesion strength of the vesicle, and thus no changes in the size of the contact zone, and consequently in ω, will occur for forces below , and f o. Hence, the onset size of the contact zone soc ) smax c the calculated ω can be transformed into wo in accordance with the appropriate conversion rule (eq 7). Inserting of wo and smax into the formula presented in the first row of c the second column of Table 1 results in the onset force f o for a given composition of the system and the particular binding strength of the ligand-receptor pair. Inspection of Figure 5 shows that the onset force f o varies over several orders of magnitude and is generally an increasing function of the parameter chosen for presentation. The only exception to this rule is the onset force when presented as a function of the receptor surface coverage (right panel in Figure 5). In this particular case, f o is an increasing function until the coverage reaches the value Fr ) nt/smax . Below this value, s/c > smax , and the c c system is in a receptor-dominated type of equilibrium. Above this value of coverage, there are more receptors in the contact zone than there are ligands in the vesicle, and the system relaxes in a ligand-dominated type of equilibrium. The transition between these two types of equilibria can be traced in a diagram in which the control parameter is nt (left panel in Figure 5). Specifically, at nt ) Frsmax , a kink suddenly appears in f o, indicating a c change in the type of equilibrium. When the binding affinity Ea is the control parameter, a saturation of the onset force f o at large ligand-receptor binding strengths can be observed. This is only true because the concentrations of the constituents have been chosen in such a way that the system is in a liganddominated equilibrium (nt/Fr < smax , for any Ea). For these c equilibria, the effective adhesion strength saturates to a finite value as Ea f ∞.40 If, for another choice of parameters, the system would be in a receptor-dominated equilibrium, then f o would continue to increase until xEa is reached. This is a consequence of the linear dependence of the effective adhesion strength on the binding strength of the ligand-receptor pair (ω ≈ Ea) when Ea f ∞ for receptor-dominated equilibria.40 The system parameters Fr, nt, and Ea determine the maximum value of the effective adhesion strength ωd in correspondence with eq 6. However, this value is achieved at exactly sc ) sdc ) 0. Consequently, ωd is the effective adhesion strength that needs to be overcome to detach the vesicle. Introducing the equivalent wd together with the chosen v into the formula found in the first row of the last column in Table 1 provides the detachment force f d. By using the remaining expressions in the last column of Table 1, this is sufficient to determine all remaining parameters, such as the detachment length and width of the tether and the tension and pressure in the vesicle.

Smith and Seifert

Figure 6. Phase diagram as a function of the fraction of the reduced volume of a vesicle (v) for a fixed composition of the system (nt, Ea, and Fr are constant, as indicated in the legend of the graph; for notation, see Figure 5). Inset: dependence of the onset and detachment adhesion strengths (wo and wd, respectively) as a function of the reduced volume v.

The behavior of the detachment force can be understood by analyzing the maximum value of the effective adhesion strength (ωd from eq 6). Because f d ≈ ωd, then f d ≈ Fr. Similarly, f d ≈ Ea at large binding strengths, whereas, when Ea f 0, the detachment force tends to zero logarithmically. The same logarithmic tendency of f d can be observed when nt f 0 [the f(nt) diagram in Figure 5 is a double logarithmic plot]. On the other hand, when the fraction of ligands in the vesicle becomes very large (nt f 1), then f d f FrEa. The remaining system parameter to be explored is the reduced volume of the vesicle. To understand its influence, a phase diagram is constructed for a fixed composition of the system, including a chosen binding strength of the ligand-receptor pair (see Figure 6). The reduced volume can be easily controlled by changing the osmotic conditions by adding salts, sugars, or water into the buffer surrounding the vesicle. Because the detachment adhesion strength wd is independent of the reduced volume of the vesicle (inset of Figure 6), the line of the second-order unbinding transition is simply proportional to v1/3. This dependency is exactly the same as that predicted by the continuum model (see the second row of the last column in Table 1). The line of the first-order shape transition is, on the other hand, a function of ho and soc , which are explicit functions of v, and wo, which is an implicit function of v (see inset in Figure 6). Hence, f o varies strongly with the reduced volume. Moreover, as the vesicle becomes more and more spherical (v f 1), the onset and detachment adhesion strengths become very similar and, at v ) 1, ultimately coincide. In this regime, the onset force rapidly increases with the reduced volume to values several orders of magnitude larger than those of the onset forces for substantially deflated vesicles.


The Extraction of Tethers. If both the diffusion of the ligands on the vesicle surface and the ligand-receptor binding are fast in comparison to the force load, the detachment of the vesicle can be regarded as a quasistatic process. In principle, the force load must also be slower than the typical unbinding time. This may appear to be difficult because high bond enthalpies usually correspond to very low dissociation rates of the bonds at zero force. However, the dissociation rate exponentially decreases with the force. In addition, the enthalpy of binding may also be decreased by the force. Hence, a quasistatic situation does become accessible in experiments with constant force, even if the unloaded bond enthalpies are high. Similarly, detachment can be induced by the successive application of forces of increasing strength on a vesicle that is allowed to reach a new equilibrium under every given force. In both of these cases, the thermodynamic approach presented herein should provide a good description of the vesicle unbinding. To accomplish the unbinding of a specifically adhered vesicle with a contact zone at a given value sc, forces higher than those that would be required in the case of a constant effective adhesion strength (typical for nonspecific adhesion) must be applied. This is due to the increase in the effective adhesion strength that takes place in response to the detachment of specifically bound vesicles. The pulling force under the conditions of variable w remains of the same order of magnitude as that for the case of constant w until the final stages of the detachment process. When the remaining contact zone becomes small (below 10% of the total vesicle area), the resistance to pulling is considerably increased. In the final stages of detachment (sc < 0.01), the resistance to pulling is further enhanced. This effect is independent of the strength of the potential providing the resistance. However, because the detachment force is proportional to wd, if the effective adhesion strength from the onset to detachment changes by an order of magnitude, the detachment force will do so as well. Thus, in this last stage, the specific and the nonspecific force curves can differ by up to several orders of magnitude. The above-described behavior can be clearly seen in Figure 7 in which several detachment curves are shown. The detachment curves presented here are for parameters identical to those used to generate Figures 2 and 4, in which the appropriate density of bound ligands and the effective adhesion strength, respectively, are shown as functions of the size of the contact zone. If ω from Figure 4 is in the range of 0-1, w will adopt values in the range of 0 to the order of magnitude of the conversion factor. Henceforth, for the given choice of nt and Fr, at Ea ) 1, the w is basically constant, and the detachment curve is very similar to one that would correspond to the nonspecific adhesion. For larger binding strengths (Ea ) 5 and Ea ) 10), w changes by more than 1 order of magnitude, and although the onset forces for these two binding strengths are almost the same, the detachment forces differ considerably. Faster changes in the force curve can be seen around s/c (the point at which the transition from the ligand- to the receptor-dominated equilibria occurs), particularly if the relevant effective adhesion strength curve (Figure 4) experiences similar features. Furthermore, if s/c is small (below 0.1), the force increase due to changes in w will be combined with the intrinsic enhancement of the force at small contact zone sizes so that even stronger differences will arise between the detachments of specifically and nonspecifically adhered vesicles.

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Figure 7. For a given composition of the vesicle and the substrate and the indicated reduced volume, the force associated with the thermodynamic equilibrium as a function of the size of the contact zone is calculated for several ligand-receptor binding strengths. The inset contains a double logarithmic plot of the region close to sc ) 0.

The calculation presented in Figure 7 is performed under the assumption of large numbers of both sites and ligands. Such an approximation imposes a thermodynamic limit that will be reached for very small contact zone sizes (below 100 sites). However, because a vesicle typically possesses 106-109 sites, the contact zone sizes that are amenable to the thermodynamic approach are sc ) 10-4-10-7 and hence will have no impact on Figure 7. Other parameters of the shape, such as the tether length or width, are approximated to be simple functions of the force (see the third column in Table 1). Thus, the behavior of the force can be directly projected onto these parameters. Consequently, considerably longer and thinner tethers should be pulled in the case of specifically adhered vesicles in comparison to those pulled under the influence of nonspecific adhesion. However, the radius of a tether is directly proportional to the inverse of the force and does not provide any information about the specificity of the binding, the adhesion strength, or the reduced volume of the vesicle. Accordingly, the tether length does not explicitly depend on the adhesion strength but is linear with force (see the third row of the second column in Table 1). Consequently, linear slopes (dependent only on the reduced volume) that are the same as those produced in the detachment process of a nonspecifically adhered vesicle should be expected. Last, the tension in the vesicle, according to the continuous calculations, is quadratic in force. Hence, pulling on a specifically adhered vesicle will induce much larger tensions than will pulling on a nonspecifically adhered vesicle. In some cases, particularly when forces approach the critical detachment force, the instability of the vesicle shape due to the lysis under tension can be

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expected to occur rather than the detachment. Specifically, for slow loads, which are also assumed in this model, the typical lysis tension is measured to be on the order of 1-10 mJ m-2.43,44 For the vesicles presented in Figures 5-7, in accordance with the conversion from reduced to real units (given in the fist column of Table 1), the expected lysis force should be ∼103 pN. Discussion We have developed a thermodynamic model for the unbinding of vesicles that are specifically adhered to the substrate by exerting a point force in a direction opposite to the substrate. Numerous ligand-receptor bonds are formed to adhere the vesicle. The ligands are capable of diffusion because they are constrained to the vesicle membrane, whereas the receptors are immobilized on the substrate. The effects of the finite vesicle size are taken into account. By the determination of the effective adhesion strength produced by the formed bonds, the continuous calculations of the vesicle shape adhered in a contact potential are self-consistently merged with the thermodynamic approach to adhesion, which is mediated by discrete ligand-receptor interactions. The tether emerges as part of a second-order shape transition when a critical onset force, sufficient to overcome the adhesion, is applied. A quasi-static detachment process is predicted to follow at forces higher than the onset force that culminates in a continuous unbinding transition occurring at a critical detachment force. Finally, the number of formed bonds, the effective adhesion strength opposing the force, and all shape parameters, including the tether length and width, the size of the contact zone, and the tension in the vesicle, are obtained for all stages of the detachment process. One of the most important advantages of the presented model is that the equilibration with respect to the formation of bonds is decoupled from the equilibration with respect to the size of the contact zone and the shape. Because the bending contribution to the free energy is several orders of magnitude smaller than the contributions associated with the ligand-mixing entropy and the enthalpy of binding, the force serves as a control mechanism for maintaining a particular size of the contact zone. It opposes the effective adhesion strength produced by the formation of bonds. However, the bending energy serves to define the shape of the tether but not the shape of the body of the vesicle, which is controlled by adhesion and force. As a part of the self-consistent calculation, a rule for the conversion of the effective adhesion strength produced by the bonds into a contact adhesion potential is established. Interestingly, the obtained rule implies that vesicles from the same preparation adhering to the same substrate will experience different effective adhesion strengths if they are of different sizes. In the context of the thermodynamic model used to describe the adhesion, it is shown that when the size of the contact zone is reduced, the equilibrium density of the formed bonds increases until all available receptors are occupied, and then it remains constant (or it continues increasing until all of the contact zone is lost; see Figure 2). Consequently, increased binding in the contact zone is expected in response to an attempt to unbind the vesicle. This is an important prediction that enables one to understand the thermodynamic equilibrium for adhesion mediated by ligand-receptor bonds under force, which (43) Evans, E.; Needham, D. J. Phys. Chem. 1987, 91, 4219-4228. (44) Hatagen, A.; Law, R.; Kahn, S.; Discher, D. E. Biophys. J. 2003, 85, 2746-2759.

Smith and Seifert

was already observed experimentally in a similar system.24 It is interesting to note the behavior observed in the system constructed by Piper and collaborators (a cell adhering to receptor-coated wells pulled away by centrifugation) is completely consistent with that predicted by our adhesion model. Although their mechanism for detachment is somewhat different from that assumed in our work, those experiments support our conclusion that the physics driving the balance between the entropy and the binding enthalpy is of the same origin, even in the thermodynamic limit of our model, when the number of bonds becomes small. An important result emerging from the presented calculations is the range of forces that is capable of influencing a strongly adhered vesicle. Forces on the order of 10 pN are sufficient to induce the unbinding of numerous ligand-receptor bonds (as presented for typical experimental conditions in Figure 7). However, breaking all of the bonds to completely disengage the vesicle from the substrate appears to be a task that is considerably more resilient to the force. It is shown that critical detachment forces are several orders of magnitude larger than the forces characterizing the onset of the detachment process. In addition, the densities of ligands and receptors as well as their binding strengths considerably influence the detachment forces. This result could possibly be used to understand why the beads that are specifically adhered to vesicles by a low number of bonds provide such a strong and stable resistance mechanism to the force commonly used in micropipet experiments.29,45 Somewhat more surprising is the effect of the reduced volume on the onset force. The extraction of tethers in vesicles with a volume that has been reduced to some degree appears to be much easier than that in almost spherical vesicles. The typical forces obtained in the presented calculations for the onset of tether extraction are in good agreement with the forces reported in the literature,28,30,45 in which at least slightly deflated vesicles are generally used. Conclusions The model presented herein provides a description of the equilibrium for the competition between membrane deformations and specific adhesion. Certainly for living cells, such a simple approach should provide basic physical principles that govern the response of an adhered cell to the force. However, the complexity of the living system should not be neglected. In the context of tether extraction, membrane-cytoskeleton adhesion is a very important contribution to the counteraction of the tether-expelling force. Nevertheless, tether extraction in liposomes that are adhered to cells is a very important technique used for understanding the adhesion mechanisms that are acting in vivo.22 The presented model is simple and well-defined enough to make it a good starting point for equivalent experimental studies. Although some of the expressions that arise from the calculations (such as the tether radius and tension relation with the force) are well-known and commonly used in experiments with micropipets,45-47 direct measurements by which the remainder of this model could be tested have not yet been performed. It appears that producing a setup in which a controlled pulling (45) Heinrich, V.; Waugh, R. E. Ann. Biomed. Eng. 1996, 24, 595605. (46) Bukman, D. J.; Yao, J. H.; Wortis, M. Phys. Rev. E 1996, 54, 5463-5468. (47) Waugh, R. E.; Hochmuth R. M. Biophys. J. 1987, 52, 391-400.


mechanism that can be monitored over large vertical distances is combined with the detailed characterization of the contact zone is a laborious task. Nevertheless, the present study will provide a tool for the analysis of such an experiment should it be achieved. Furthermore, the results herein can still be used advantageously, even if such sophisticated apparati are not available. For example, because the difference between the responses of specifically and nonspecifically adhered vesicles to the application of force is so significant, the model could be used to test which type of binding is operative between the vesicle and the substrate. Finally, this comprehensive model of the thermodynamic equilibrium is a very well-defined

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starting point for a dynamical study of vesicle detachment. This latter task will hopefully be undertaken in the near future. Acknowledgment. We thank E. Sackmann for motivating this work and for numerous helpful discussions. A.-S.S. is grateful to the Hochschul- und Wissenschaftsprogramm (HWP II) for financial assistance, as well as the Sonderforschungsbereich 465/C4 for support. A.-S.S. thanks Ron Clarke of the University of Sydney, Australia, for hospitality during the preparation of this manuscript. LA051303F

Force-Induced De-Adhesion of Specifically Bound Vesicles: Strong

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