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Adhesion Induced by Mobile Stickers: A List of Scenarios P.-G. de Gennes,*,†,‡ P.-H. Puech,‡ and F. Brochard-Wyart‡ Colle` ge de France, 75231 Paris Cedex 05, France, and Laboratoire Physico-chimie Curie, UMR 168, Institut Curie, 11 rue P. et M. Curie, 75231, Paris, France Received March 12, 2003. In Final Form: May 5, 2003 We consider a vesicle carrying on its surface a small number of mobile “stickers”. When facing a wall with suitable receptors for the stickers, the vesicle builds up an adhesion patch. Our aim is to analyze the patch growth, allowing for a variety of physical situations. (a) Sometimes the process is facilitated by a nonspecific adhesion process (e.g., van der Waals). But with suitably protected surfaces, this is absent: we can then use heavy vesicles, who fall by their weight on the surface but remain at a certain distance from it. (b) For certain model systems, the stickers diffuse freely. But for other systems, they may be stuck as soon as they reach their complement. (c) With some cellular adhesion molecules, we may find a long reaction time τ for fixation. Our discussion includes all these possibilities and ends up with a list of possible scenarios, leading to different growth laws.
1. Moving Stickers The problem to be discussed here is summarized in Figure 1. We start with a lipid vesicle covered by a dilute population of “binders” or “stickers” (Γ0 stickers per unit area). By some means (to be discussed later), we impose a contact between the vesicle and a solid surface. On this surface, the stickers can find a bonding partner and gain an energy U per sticker: we call this specific adhesion. The adhesion patch is an attractive region for the stickers: they will migrate toward the patch. What are the laws for the patch growth? A number of experimental and theoretical papers1-4 have been devoted to this problem: there is, in fact, a vast catalog of possibilities. Our aim in the present text is to list and discuss the main scenarios, covering both the statics and the dynamics of the contact zone. Our analysis is very crude, limited to the level of scaling laws. 1.1. The Role of Osmotic Pressures. The starting point is the role of the osmotic pressure Π(Γ) in the twodimensional fluid of stickers. This osmotic pressure is related to the separation energy W, that is, the work required for separating a unit area of vesicle from the wall in very slow processes, where thermodynamics holds. The work W is in general a superposition of nonspecific (Wn) and specific (Ws) adhesion W ) Ws + Wn, where Wn can be due, for instance, to van der Waals forces or to a combination of attractive and repulsive forces, while Ws is controlled by the stickers. At first sight, we might be tempted to write simply Ws ) ΓiU where Γi is the sticker concentration inside the plaque. But this is not correct, as pointed out first by Bell, Dembo, and Bongrand long ago.5 Actually
Ws ) Π(Γi) - Π(Γout)
(1)
is the difference between the osmotic pressures inside and outside the border (the contact line surrounding the patch). † ‡
Colle`ge de France. Institut Curie.
(1) Dustin, M.; Fergawov, M.; Chan, P.; Springer, T.; Golan, D. J. Cell Biol. 1996, 132, 465. (2) Ra¨dler, J.; Sackmann, E. J. Phys. France 1993, II.3, 727. (3) Boulbitch, A.; Guthnberg, Z.; Sackmann, E. Biophys. J. 2001, 81, 2743.
Figure 1. A vesicle with stickers approaching a solid surface. The solid is densely packed with receptor sites for the stickers. But the stickers themselves are originally dilute (on the membrane surface).
It is instructive to check eq 1 by considering the balance of horizontal forces at the contact line, that is, the Young equation. If γ0 is the surface tension in the absence of stickers, the surface tension outside of the contact zone is γ ) γ0 - Π(Γout). Assuming that Wn ) 0, the surface tension inside is γ0 - Π(Γi) ) γ - Π(Γi) + Π(Γout). The balance of forces is
γ - Π(Γi) + Π(Γout) ) γ cos θe
(2)
θe being the equilibrium contact angle. Equation 2 is equivalent to Ws ) γ(1 - cos θe), a form which we shall often use. If we have Wn * 0, this is shifted to
Ws + Wn ) γ(1 - cos θe)
(3)
Some remarks are important at this point. (a) Very often, the tension γ has been imposed by the preparation of the vesicle: by an internal solute additive, or by the very establishment of the contact, as pointed out in ref 6. In most of the present paper, we assume that γ remains constant during the whole growth of the adhesion patch. This should hold for small θ. The opposite case is discussed in Appendix B. For typical conditions, we find there that γ ) const for θ < 0.2. (b) In many cases (but not all), the osmotic pressure outside Π(Γout) is negligible when compared to Π(Γi). (4) Bruinsma, R.; Behrisch, A.; Sackmann, E. Phys. Rev. E 1999, 61, 4253. (5) Bell, G.; Dembo, M.; Bongrand, P. Biophys. J. 1985, 45, 1051. (6) Ra¨dler, J.; Feder, T.; Strex, H.; Sackmann, E. Phys. Rev. E 1995, 51, 4526.
10.1021/la034421g CCC: $25.00 © 2003 American Chemical Society Published on Web 07/10/2003
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(c) An interesting limit is obtained at low concentrations, where Π(Γi) ) kTΓi. This then gives Ws ) kT(Γi - Γout) = kTΓi. In this regime, Ws can be much smaller than UΓi. (d) The opposite limit is obtained with dense stickers. Here Ws returns to the naive value Ws ) UΓi. This can be understood by equating the chemical potentials in and out,
∫ΓΓ
i
out
1 dΠ ) U Γ
(4)
When Γout is close to Γi (for a compact system of stickers), this reduces to
Ws ≡ Π(Γi) - Π(Γout) ) ΓiU The buildup of the adhesion patch when nonspecific adhesion helps to start the process (Wn * 0) has been discussed by us in a recent theoretical paper.7 Here we shall only summarize the conclusions briefly: in a first stage, nonspecific adhesion is expected to dominate, and the patch should grow by conventional wetting processes. This should take place in a short time (seconds). Then, later, the specific stickers begin to gather and dominate the patch growth, which lasts typically for 1 h. 1.2. Cases without Primary Adhesion. In the present paper, we shall mostly concentrate our attention on cases where all the adhesion is due to the stickers (Wn ) 0). This is often achieved in practice, through the use of protective layers (grafted poly(ethylene glycol) (PEG) on vesicles, or a natural glycocalix for biological cells) or through the use of suitable spacers. When Wn ) 0, if we want to bring a vesicle into contact with a horizontal surface, we must use an external driving force. The most common one is gravity. We load the vesicle with suitable (sugar) additives inside, making it heavier than the surrounding water; then it will fall on the surface. This situation appears simple at first sight. In fact it is rather complex, for two reasons. (a) The hydrodynamics of approach includes the formation of a “dimple”, which is eliminated very slowly. It often happens that the adhesion patch nucleates not at the center but at some point on a circle of closest approach.9 We shall not discuss this “dissymmetric” nucleation here: we assume that the vesicle has reached a full hydrostatic equilibrium before building a patch. (b) Because of entropic effects, the membrane is repelled from the solid (Figure 2). Thus, in equilibrium, we expect a “cushion” of water between vesicle and wall. It is sometimes assumed that the cushion thickness results from a balance between the weight and the entropic forces invented by W. Helfrich. However, the Helfrich calculation was restricted to vesicles with 0 surface tension. As pointed out by Ra¨dler et al.,6 the (very weak but finite) tension γ reached in usual experiments modifies the picture for the cushion very seriously. This is summarized in section 2. In section 3, we use the same ideas to construct the membrane profile around the first successful sticker, and also around a multisticker patch. 1.3. Mobilities. (a) When the population of stickers is initially rather dense, they do not have to move. This can occur in model experiments. It leads to a rather trivial situation, where specific adhesion is not dynamically different from nonspecific adhesion. (7) Brochard-Wyart, F.; de Gennes, P.-G. Proc. Natl. Acad. Sci. U.S.A. 2001, 99, 7854. (8) Helfrich, W.; Servuss, R. Nuevo Cimento 1984, 3D, 137. (9) Martchi-Artzner, V.; Lorz, B.; Gosse, C.; Jullien, L.; Merkel, R.; Kessler, H.; Sackmann, E. Langmuir 2003, 19, 835.
Figure 2. A vesicle with no interaction with the substrate (e.g., protected by a PEG brush). Here, the vesicle is made heavy (by incorporation of suitable sugars) and it falls toward the substrate. The thickness hc of the remaining water cushion is a compromise between weight and fluctuation forces, discussed by Ra¨dler and Sackmann (ref 2) and summarized in section 1.2.
Patch growth occurs then by transformation of the energy ΓU into viscous losses inside the edge (of angle θ) near the contact line: this is similar to the early regime of growth described in ref 7. The patch reaches a final radius θeRv (where Rv is the overall vesicle radius) in a time ∼ ηRv/γθe2, where η is the water viscosity and θe is the equilibrium angle derived from eq 3. (b) We shall focus our attention here on the opposite limit: very dilute stickers, which must move to build up a significant adhesion patch. In the unbound parts of the membrane, they will then have a certain diffusion coefficient D (in the range 10-10 to 10-7 cm2/s). What happens inside the patch is more delicate; we can think of two limits: The first limit is a “free” case where both key and lock are mobile. This means that we still have diffusion in the patch after bonding, with a diffusion coefficient Db comparable to D. The second limit is a “bound” case where the key is mobile but the lock is not (Db , D). An example of this case is the system arginin glycin aspartate (RGD)/integrin.10 In section 4, we shall explore both cases. To our surprise, it turns out that the distinction between free and bound is not dramatic for the growth problem: we retain the same scaling laws, provided that the key reacts fast with the lock. The opposite case is discussed below. 1.4. Delayed Sticking. When we think of the sophisticated cellular adhesion molecules (CAMs) which are of interest, such as integrins or cadherins, we realize that the assembly of a sticker to a receptor may be in itself a slow process. The pathway for assembly is often complex. (Also, it may be that the sticker reacts only when it is in one special orientation. When they are dilute, the stickers might arrive into utterly wrong orientations.) Thus we may expect, in some cases, that there is a long time τ required for fixation. We shall incorporate this possibility in section 5. For growth times t < τ, we then expect completely different laws of growth. 1.5. Quasi-Stactic Regimes. In experiments with dilute stickers, patch growth is often a slow process, taking place over hours. Thus it is tempting to assume that the contact angle θ (at the border of the patch) is equal to the local equilibrium value θe derived from the Young equation (eq 3).We call this the quasi-static regime. This regime does indeed apply when there is no cushion, as discussed in ref 7. However, the quasi-static approximation fails when we start from the cushion discussed in section 2. The adhesion patch grows inside the cushion (10) Kloboucek, A.; Behrisch, A.; Faix, J.; Sackmann, E. Biophys. J. 1999, 77, 2311.
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by dewetting the water film between vesicle and support, and the dynamic angle θ is not equal to θe (but is roughly θe/x2). This will be discussed in section 4. 1.6. Elimination of Special Complications. (a) All of our discussion will be restricted to neutral stickers. As shown in ref 11 if the stickers are charged, and if the two charge densities (on the wall and on the membrane) are different, the membrane develops blisters, due to counterions, and the physics is more complex. In practice, we often find some deviations from neutrality, but we assume here that they are small. (b) We also assume that we are dealing with a single type of stickers: when we have two populations, with different sizes, various conflicts occur and often result in a two-dimensional phase separation.12 (c) On the other hand, a single type of stickers may spontaneously show a phase separation, the simplest case being from a two-dimensional gas to a two-dimensional liquid. An interesting mechanism for this has been proposed in ref 4. Thus we shall retain the possibility of a coexistence plateau between a dilute and a dense phase of stickers. This has novel effects if a cushion is present. (d) Ultimately the patch will reach an equilibrium size. In this final regime, the concentration out has gone down, Γ0 f Γ0f (where f stands for final) and the concentration in has reached a final value Γif. A relation between these two concentrations is imposed by the equality of chemical potentials: for the simplest case (ideal gas for the osmotic pressures), this reduces to Γ0f/Γif ) exp(-U/kT). In our discussion of the dynamics, we are mainly concerned with growth processes at times t much smaller than the overall equilibration time τeq. This time is discussed in more detail in Appendix A. 2. Heavy Vesicles on a Flat Support: The Cushion 2.1. Vesicle without Surface Tension (γ ) 0). Here the basic feature is the entropic repulsion first imagined by W. Helfrich.8 A fluctuating membrane approaching a solid surface is hindered and loses entropy. The resulting energy per unit area is
EH(h) ) A/2h2
(5)
A = (kT)2/K
(6)
Here h is the cushion thickness, kT is the thermal energy, and K is the curvature rigidity of the membrane (typically K ∼ 10kT). The coefficient in the equation for A is approximate, but sufficient for our purposes. The cushion is compressed by a hydrostatic pressure
p ) ∆Fg(2Rv)
(7)
where ∆F is the difference of density between the inside and the outside of the vesicle (typically 10 kg/m2); g is the gravitational acceleration, and 2Rv is the vesicle diameter. Balancing p against the force per unit area derived from eq 5, we would arrive at a large cushion thickness, hH ∼ 0.5 µm. But, as we shall see in section 2.2, this situation does not hold in practice. 2.2. Role of the Vesicle Tension γ. In most practical experiments, the vesicle is not completely floppy, but its manipulation has imposed a finite surface tension γ. γ is (11) Nardi, J.; Bruinsma, R.; Sackmann, E. Phys. Rev. E 1998, 58, 6340. (12) Bruisnma, R.; Sackmann, E. C. R. Acad. Sci. Paris 2001, 2, 803.
Figure 3. Qualitative plot of the interaction energy, E, versus thickness, h, for a water cushion separating a fluctuating membrane from a solid, repulsive wall. Below a critical thickness h0, the strong Helfrich repulsion prevails. At h > h0, the interaction falls dramatically. For a simple explanation of this plot, see ref 11.
Figure 4. Deformation of the membrane from a flat shape (cushion) when a single sticker is bound at point 0.
very small (of order 1 µJ/m2), but it does limit seriously the membrane fluctuations. For a lucid and simple discussion of this limitation, see the review by U. Seifert.13 The Helfrich interactions are killed beyond a thickness h0:
h0 )
(kTγ )
1/2
(8)
Typically, h0 ∼ 0.1 µm is somewhat smaller than the thickness discussed in section 2.1. A qualitative plot of the energy E(h) is shown in Figure 3. At thickness h > h0, E(h) falls out very fast (exponentially). The resulting cushion has been discussed by Ra¨dler and Sackmann.6 The overall conclusion is that the interactions behave nearly like a hard wall standing at h ) h0. The cushion thickness is expected to be h0 and is nearly independent of the hydrostatic pressure p (except for logarithmic factors). The horizontal radius Rc of the cushion zone is obtained from classical capillarity arguments14 and scales according to Rc = κRv2 where κ-1 is the Laplace length, defined by κ2 ) ∆Fg/γ. In all our later discussions of an adhesion patch, we shall assume that the patch is smaller than the cushion R < Rc (in the opposite limit we would return to the growth problem without cushion discussed in ref 7). 3. Pinning through the Cushion 3.1. A Single Sticker. When, through some fluctuation, one sticker has penetrated the cushion, reached the binding surface, and achieved its binding, we expect to reach a membrane profile h(r) shown qualitatively on Figure 4. (13) Seifert, U. Adv. Phys. 1997, 46, 13. (14) de Gennes, P.-G.; Brochard-Wyart, F.; Que´re´, D. Gouttes, bulles, perles et ondes; Belin: Paris, 2002.
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These profiles have been analyzed in detail by Menes and Safran15 for the case of “thick cushions” (γ ) 0). But the discussion has to be transposed here for “thin cushions” (γ * 0, hc = h0). We must distinguish two regions in the profile, obtained by comparing r with a characteristic length
λ)
(Kγ )
1/2
(9)
The length λ describes the competition between rigidity, K, and surface tension, γ. It is typically of order 0.1 µm. The length λ is well-known in discussion of vesicles in close contact with a wall. (a) At distances r smaller than λ, the energy stored in the profile is essentially the sum of a Helfrich term and a rigidity term,
F )
∫2πr dr[21K(∇2h)2 + 21A/h2]
(10)
where ∇2h ) C is a two-dimensional Laplacian defining the local curvature of the membrane. The equilibrium equation obtained by minimizing F is simple,
K∇2C - A/h3 ) 0
(11)
and there is an exact solution with a conical shape
h ) Ψr Ψ ≡ (kT/K)
(12) 1/2
(13)
(typically kT/K ∼ 0.1 and Ψ ∼ 20 °). (b) At distances r > λ, we reach height h comparable to the cushion thickness h0, and the profile reaches hc exponentially in the simplest models. The central result is given by eq 12 and deserves some comments. (i) The potential energy stored in the cone is
Epot =
∫rλ
min
2πr dr A/(2h2)
(14)
where rmin is a cutoff at small distance (the lateral size of the sticker ) a few nanometers). This gives -2
Epot ∼ πAΨ
ln(λ/rmin) ∼ 10kT
(15)
This result is doubled if we add the curvature energy. Thus there is a large potential barrier opposing the entry of the first sticker. In practice, however, the nucleation process will often be catalyzed by a dust particle or some other defect (such as a cluster of dye molecules or of the sticker itself). (ii) At each point of the cone, we have a repulsive force (due to Helfrich interactions) and an attractive force (tending to decrease h) due to curvature effects, and they essentially add up to 0. But each of them is large. The repulsive force, for instance, is
∫
Ψ A fREP ) r 2πr dr 3 = K min r h min λ
(16)
But this should not be misinterpreted: the overall force acting on the sticker is much smaller than fREP. It can be estimated through the outer region (r > λ) and is f ∼ γ2πλΨ ∼ γh0 ∼ kT/h0. This is a very small force. (15) Menes, R.; Safran, S. A. Phys. Rev. E 1997, 56, 1891.
Figure 5. Qualitative profile of the membrane when two stickers are bound at points S1 and S2.
Figure 6. Structure near the contact line of an adhesion patch. The patch radius R is assumed to be much smaller than the cushion radius Rc: then the height far from the contact line approaches the cushion thickness hc. The angle near the line is equal to Ψ (defined in eq 12) except for weak logarithmic corrections.
3.2. Two Stickers (Figure 5). For two stickers, we have made only a guess on the profiles and energies involved. Our proposed scaling form for the energy V(r12) of two stickers separated by a distance r12 < λ is
V(r12) = E1 + 2πkT ln
( ) r12 rmin
(17)
where E1 ) 2Epot is the energy of the membrane with one sticker only (see eq 15). Equation 17 ensures that V f E1 when the two stickers are adjacent (r12 ∼ rmin) and V f 2E1 when they are separated by the characteristic distance λ. The intersticker force f12 resulting from the interaction V is attractive and is simply
f12(r) ) -
kT dV = -2π dr r12
(18)
This is a weak force. We have estimated the time required for two stickers to meet, under this force after starting from a distance r: this time remains comparable to a diffusion time r2/D. 3.3. A Full Adhesion Patch (Figure 6). We now consider the profile near the contact line (marked by point L on Figure 6). Our observation point M is at a distance x from the line. For x < λ, the free energy is still given by eq 12, but the equilibrium equation is now one-dimensional,
d4h A )0 dx4 h3
(19)
K
and the solution is nearly linear:
h(x) ) xΨ
(
)
x 1 ln 2 xmin
1/4
= Ψx
(20)
(here xmin should be the distance between stickers in the patch) where Ψ is always given by eq 14. Thus the tilt angle is essentially the same. The most interesting
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(we must remember that θe(t) usually varies with time, because the sticker concentration in the patch does so). The horizontal size of the rim l is fixed by volume conservation,
πR2h0 = 2nRl2θ × const Figure 7. Growth of a patch under a cushion of water. The water taken out from the patch region (r , R(t)) is collected into a rim, with a dynamic wedge angle θ. In the crude model of section 4, θ differs from the equilibrium contact angle θe only by a numerical factor θ ∼ 0.7θe.
l2 ∼ Rh0/θ Whenever R . h0/θ, we have l , R and the assumption VA ) VB is valid. Then, returning to eq (22, 23) we find
consequence of this calculation is the scaling form of the line tension ul (energy per unit length of the contact line L. We expect (ignoring coefficients)
∫
θ ) θe/x2 Collecting these results, we arrive at
∞
A kT A ul = x dx ∼ ∼ min 2 h 2ψ2xmin xmin
(21) VA )
For very small patches, this line tension would modify the balance of forces in the Young equation (eq 2). But this is not important for our purposes, as can be seen from the following estimate. The relative shift in contact angle due to line tension is derived from
γ(1 - cos θe) ) Π(Γi) - ul/R giving
ul ∆θe 1 1 ∼ ∼ ∼ θe 2Π(Γi)R ΓiRxmin Np where Np is the number of stickers at the periphery of the patch (within a distance xmin from the contact line). The total number of stickers in the patch is N ∼ Np2. If we take N ) 104, the relative angular correction is only 10-2. 4. Growth of an Adhesive Patch inside a Cushion Let us assume that our heavy vesicle sits on an equilibrated cushion on the supporting surface. At one point, the adhesive patch has nucleated. The patch then grows. We consider the regime where the patch radius R(t) is smaller than the cushion radius. 4.1. The Dewetting Model. This scenario is described in Figure 7. The water which was initially in the central part of the cushion is collected into a rim of horizontal size l. We describe this rim with the simplest language used for dewetting processes.14 The dissipation is concentrated in the wedges near points A and B. Thus, at a crude level, we can assume that the pressure is constant in the fat region of the rim: then, according to Laplace’s law, the curvature of the rim profile is constant, and the profile is a portion of a circle. The wedge angle θ at both ends (A and B) is then the same. The velocities VA ()dR/dt) and VB are nearly the same and are given by the scaling laws
γ η V = γ(1 - cos θ) = θ2 θ B 2 γ η V = γ(cos θ - cos θe) = (θe2 - θ2) θ A 2
(22) (23)
(
)
Π(Γi) - Π(Γout) dR = V*θe3 = V* dt γ
3/2
(25)
where V* ) γ/η. We must finally specify the law for the osmotic pressure Π(Γ). Here we shall focus on the perfect gas limit, because (as we shall see) Γi is not much larger than Γout. 4.2. Dewetting of a Cushion: The Case of Freely Diffusing Stickers. We assume here that the stickers remain mobile inside the patch; then they achieve a flat concentration Γ ) Γi inside. Just out of the patch, we have a very small concentration Γ+ < Γ0. Far from the patch (r , R), we return to the initial concentration Γ0. The inward diffusion current near the patch is
Γ0 - Γ+ J ) kD lD
(26)
Γ+ is negligible; k is a numerical constant of order unity. The length lD is the size of the outer diffusion region. In our early study on growth initiated by primary adhesion,7 we had a patch radius R much smaller than (Dt)1/2, and we assumed
lD ) (Dt)1/2
(27)
But in the present problem, the situation is quite different: the dewetting process for a thin film is fast, and it turns out that R . (Dt)1/2. Then the size lD is reduced and is given by the classical formula:
D V
lD )
(28)
where V = dR/dt is the dewetting velocity (eq 25). The number of stickers inside the patch is ΓiπR2, and it grows from the current J,
d (πR2(Γi - Γ0)) ) 2πRJ dt
(29)
Here θe is the local equilibrium angle, derived from Young’s equation (eq 2),
(this scaling formula ignores logarithmic factors). Since Γ+ is much smaller than Γ0, we may simplify this equation, obtaining
γ Π(Γi) - Π(Γout) ) θe2(t) 2
(Γi - Γ0)2πR dR ) 2πJ dt
(24)
(30)
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Assembling eqs 26, 28, and 30, we arrive at a remarkable result:
Γ i - Γ0 )k Γ0
(31)
The internal concentration Γi is then expected to be constant, Γi ) Γ0(1 + k), and to be rather small. Returning to eq 25, we see that θe is constant, and thus the dewetting velocity must also be constant. From eq 26, we get
R ) 3/2V*t
(32)
where we have introduced (following ref 7) the dimensionless parameter
2kTΓ0 ,1 γ
)
(33)
Equation 32 predicts a rapid growth (R . (Dt)1/2) which does not allow for a strong migration of the stickers. 4.3. Immobile Stickers inside the Adhesion Patch. Here we assume that as soon as a sticker reaches the adhesion patch, it cannot move any more. The diffusion balance of eq 30 still holds, and we again arrive at a constant (small) internal concentration Γi (eq 32). But there is an interesting modification in the adhesion energy W (first pointed out to us by F. Amblard). The Bell argument (eq 1) does not hold for immobilized stickers, and we must return to the naive formula W ) ΓiU. Combining this with eqs 26, 28, and 30, we find a growth law similar to eq 32 but with a different coefficient:
R ) ˜ 3/2V*t
2UΓi γ
with ˜ )
(34)
˜ is typically 4 times larger than . 5. Cases of Slow Reaction 5.1. General Principles. As pointed out in section 1.4, it may happen that a sticker, when entering the adhesion plaque, does not immediately react with a receptor of the supporting solid. We must then distinguish two sticker concentrations in the plaque: the overall concentration Γi and the reacted, or bound, concentration Γb. In a first discussion (section 5.2), we shall assume that the profiles of Γi and Γb inside the patch are essentially flat. This will turn out to be correct if the bound stickers are still diffusing fast enough inside the patch. In section 5.3, we discuss the opposite limit (slow diffusion). 5.1.1. Chemical Kinetics. We shall assume a rate equation for binding of the form
dΓb 1 ) (fΓi - Γb) dt τ
(35)
Here τ is a reaction time for binding, and f is the bound fraction at equilibrium (for a fixed Γi). In what follows, we shall focus our attention on cases of strong fixation f ∼ 1. Equation 35 leads to two regimes: (a) for t > τ we should return to Γb = fΓi = Γi and our previous analysis (section 4) should hold; (b) for t < τ we may assume Γb , Γi and write
dΓb = Γi/τ dt
(36)
5.1.2. Osmotic Pressures. What happens to the law of Bell, Dembo, and Bongrand5 when we have two coexisting populations of stickers (bound ) Γb, unbound ) Γi - Γb) inside the patch? Answer: The osmotic pressure participating in the Young force (eq 3) is associated with those stickers which cannot cross the osmotic filter (here, the contact line); this is the osmotic pressure of the bound fraction Π(Γb). Thus we now write, instead of eq 3,
θe2 ) Π(Γb) ) kTΓb 2
γ(1 - cos θe) = γ
(37)
Here we have gone directly to an ideal gas behavior, because the bound fraction is dilute in the early stages. We must now discuss the growth of the bound fraction. 5.2. Fast Diffusion inside the Patch. In the early stages of interest here, we have simply Γi ) Γ0 (the patch concentration has not yet had time to grow) and eq 36 then gives
texp Γb ) Γ 0 τ
(38)
where texp is the time interval during which our stickers have been exposed to the support: this exposure time is 0 at the contact line (where exposure has just begun), in the center texp is equal to the full time t elapsed after nucleation. The average exposure time ht is thus of the form kt where k is a numerical constant (the averaging is performed through diffusion inside the patch; we come back to this question in section 5.3). Then the scaling law is simply
t Γb ∼ Γ0 τ
(39)
We shall now apply these ideas to our main examples: (1) a heavy vesicle eliminating a cushion of water; (2) a sticky vesicle, with no cushion, as in ref 7. 5.2.1. Heavy Vesicles. Here the velocity is given by eq 25 provided that we replace Π(Γi) by Π(Γb). The result is
t R ) V*t τ
( )
3/2
(40)
5.2.2. Sticky Vesicles. Here, after the fast growth of a nonspecific patch, we expect an increasing contact angle θe(t) corresponding to a quasi-static Young equilibrium described by eq 37. Then the patch radius is expected to scale as
t R ) Rvθe(t) = Rv τ
( )
1/2
(41)
5.3. Slow Reaction plus Slow Diffusion. We assumed that the bound sticker concentration Γb(r) inside the patch (r < R) had a nearly flat profile. This will clearly hold if the bound stickers diffuse fast. However, we may ask what happens if the diffusion coefficient Db of the bound stickers is small. The full transport equation for Γb(r, t) has the form
∂Γb 1 ) Db∇2Γb + Γi ∂t τ
(42)
and, as argued above, we may replace Γi by Γ0 in the regime t , τ.
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de Gennes et al. Table 1. Summary of the Different Situations
vesicles sticky
reaction rapid slow
heavy
Figure 8. Concentration profiles Γb for the bound stickers inside the adhesion patch. (a) Dotted line: when the mobility is zero, the stickers have a concentration proportional to the exposure time. On the figure, we assume a constant velocity R˙ . Then the exposure time is (R - r)/R˙ . (b) Continuous line: with a small but nonzero diffusion constant Db for the bound species, the profile is smoothed out on lengths comparable to the diffusion length (Dt)1/2.
rapid slow
osmotic pressure
growth law R ∼ tn n ) ...
ideal gas plateau ideal gas
1/4 0 1/2
ideal gas
1 5/2
For sticky vesicles, using the first equality in eq 41, we arrive at
Dbt2 2 R τ v
R4 )
(48)
that is, a radius R(t) ∼ t1/2. 6. Concentrated Stickers
The condition at the moving boundary r ) R(t) is
dΓb dR | ) Γb -Db dr R dt
(43)
It may be expressed in terms of an extrapolation length
b ) Γb/
( )
dΓb Dbt ∼ dr R
(44)
(a) Fast internal diffusion (by “internal”, we refer to the patch zone) is obtained when Dbt > R2 and imposes b . R. Then the profiles are flat and discussion of section 5.2 holds. Consider as an example the case of a sticky vesicle, where eq 41 holds. Using eqs 44 and 41, we see that b > R when Db > Rv2/τ. A similar discussion can be given for heavy vesicles, using eq 40 instead of eq 41. (b) Slow internal diffusion occurs when Dbt , R2. Here the profile is qualitatively shown in Figure 8 where we have chozen a pedagogical example: we assume that the growth velocity dR/dt is constant. Then, if we consider first the 0 diffusion case (Db ) 0) the profile is triangular; Γb is maximum at the center, where the exposure time was long, and vanishes at the contact line. In this gedanke experiment, there is no osmotic force at the border Π(Γb) ) 0 and no motion. If we now switch on a small diffusion constant Db, the profile becomes smoothed out slightly, and the concentration at r ) R is of order
Γb(R) =
Dbt t b Γb(0) = 2 Γ0 R τ R
(45)
The equilibrium contact angle θe is defined by
θe2 ) Π(Γb)r)R γ 2
(46)
Here we assume that (since Db is small but finite) we may still use the Bell formula (eq 1) for W.
Dbt2
θe2 =
R2τ
(47)
This special situation was mentioned in section 1.3. Here the stickers are dense from the start, with a high concentration Γ. Migration effects outside of the patch are not important. 1. If the bridging reaction is fast (t > τ in the notations of section 5), we return to a classical problem of dynamical capillarity: putting a drop (with a very low surface tension) on a surface.14 The drop (assumed viscous) reaches its equilibrium contact angle in a time τfast = Rvη/Ws, where Ws is the thermodynamic adhesion energy (and Ws ) ΓU in the dense limit). 2. If the bridging reaction is slow (t < τ), we can find either of the two regimes discussed in section 5.3: (a) fast diffusion when Dbt > R2; (b) slow diffusion when Dbt < R2. 7. Recapitulation of the Growth Laws Expected with Dilute Stickers 7.1. List of Assumptions. We have discussed many possible situations: a summary is clearly needed. Let us first recall the main assumptions used: (a) The surface tension γ is nonvanishing and does not vary during growth. The limits of the regime are discussed in Appendix B. (b) The stickers diffuse freely on the membrane outside of the patch. (c) On the other hand, in the patch region, diffusion may be fast or slow. (d) We have considered the possibility of delayed bridging in section 5. (e) For the osmotic pressures of a two-dimensional fluid of stickers, we have kept mainly two simple limits: (e-i) ideal gas behavior; (e-ii) plateau behavior when the stickers undergo a phase transition. (f) We constantly assumed that the pool of stickers initially present on the surface was not exhausted at time t. This implies t < Rv2/D. The opposite limit is discussed in Appendix A. 7.2. Growth Laws: Comparison with Data. Our predictions are summarized in Table 1. There is mainly one set of careful data which can be compared to them. Boulbitch et al.3 used vesicles protected by a PEG brush, as an analogue to the glycocalix of a real cell. The stickers were lipid-anchored peptides carrying the specific group RGD. The corresponding receptor was an integrin, physically adsorbed on a glass surface. These vesicles were made heavy: having a sugar solution inside after preparation, they are resuspended in an iso-osmotic salt solution. This ensures that they are slightly floppy. The adhesion here is relatively strong. The stickers (RGD
Adhesion Induced by Mobile Stickers
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peptide attached to a lipid) diffuse freely outside of the contact zone, but they are stuck as soon as they reach their immobile partner (integrin). From our point of view, looking at Table 1, we would expect a growth law R ∼ t in the most usual case (t . τ), far from what the authors find for small Γ0 (n ) 1/2). The experiments of Albersdorfer et al.16 are similar in spirit: the sticking pair is streptavidin/biotin. There is a protective PEG layer preventing nonspecific adhesion. On the supported bilayer, the streptavidin molecules are stuck (and probably crystallize) as shown by fluorescence recovery after photobleaching (FRAP) experiments: thus bound stickers cannot move. The authors of ref 16 quote one case where they monitor the growth of the adhesion patch and find (for t j 300 s) an exponent n ) 1/2. At longer times, growth slows down. Many effects can explain such a slowing down: (i) raise in θ beyond 0.2 when the surface tension rises; (ii) exhaustion of the reservoir of stickers. In the experiments of Kloboucek et al.,10 using an homophilic adhesion molecule (csA) linked to a lipid, we are still dealing with specific adhesion. Diffusion of bound doublets inside the patch may be fast, but we do not have much information on the growth law. It may be that this is a case showing a phase transistion, in which one we would expect n ) 1 or n ) 5/2 depending on the regime. Reference 3 contains a theoretical model leading to n ) 1/2. The basic assumption here is that the concentration Γi of the stickers inside the patch is large, corresponding to full saturation of a dense sheet of receptors. A simplified form of the argument can be given as follows: assume a regime of slow growth (R < (Dt)1/2); then a small patch reaps the stickers from a large area Dt and the diffusion balance is (ignoring logs)
πR2(Γi - Γ0) ) (const)ΓoutDt
Appendix A: Approach to the Final Equilibrium As mentioned in section 1.6d, the growth process ends by an equilibrium situation, with final concentrations Γif and Γof in and out of the patch (related by one equilibrium condition). The final patch radius R corresponds to
πRf2Γif + (4πRv2 - πRf2)Γof ) 4πRv2Γ0 Near this final state, we may write a diffusion balance similar to eq 35,
d (πR2Γi) = 2πRD(Γof - Γ+) dt where Γ+ ) Γi exp(-U/kT), if we have a rapid equilibrium at the contact line. Taking R ) Rf ) constant, we get a linear relaxation equation for Γi, with a certain characteristic time τeq,
D 1 ) (const) 2 exp(-U/kT) τeq R f
All of our discussion in the text assumes that we are still far from the final equilibrium, that is, that the time under study, t, is much smaller than τeq. But we require even more: the “diffusion hole”, of size (Dt)1/2, must be smaller than the vesicle size Rv. Appendix B: Increase in Tension Due to the Flattening of the Vesicle The relative increase in area due to a contact with an angle θ is
∆A θ4 ) A 16
(49)
Thus, at fixed Γi, R2 is proportional to t. This assumption of saturated Γi may be excellent in a certain regime of long times. But it is not acceptable for our discussion for the following reason: fixing Γi fixes a certain osmotic pressure Π(Γi), and this reacts on the growth. For instance, when we are dealing with a cushion below a heavy vesicle, the radius of the patch should increase according to eq 25.
dR ∼ [Π(Γi)]3/2 dt We must ensure self-consistence between capillarity and transport. To summarize: the model in ref 3 may be adequate in certain regimes of long times but cannot describe the full buildup of the patch. Many more experiments, monitoring the control parameters and the initial size of the cushion, will be required to clarify these points. On the whole, our paper is very tentative. It may be that unexpected complications show up in the experiments. But we have, at least, produced a list of relatively simple regimes, which may possibly be seen in the future. (16) Alberdorfer, A.; Feder, T.; Sackmann, E. Biophys. J. 1997, 73, 245.
(θ , 1)
This induces a change in surface tension (γ f γ0). In most usual situations, where fluctuations effects are dominant, we then have (as explained in ref 7)
()
kT γ ∆A ) ln A 8πKb γ0
We may define a crossover angle θc by imposing γ = 2.7γ0,
θc )
( ) 2kT πKb
1/4
For θ < θc, γ is constant (γ = γ0): this is the regime considered in ref 7 and in this present paper. But if θ > θc, γ/γ0 increases fast, and this blocks the growth. For instance, in the discussion of sticky vesicles, we have
θ ∼ t1/4 θ ∼ (ln t)1/4
(θ , θc) (θ . θc)
In practice, this means that θ may saturate around θc. LA034421G
