Adhesion Kinetics between a Membrane and a Flat Substrate. An

J. Phys. Chem. B 2010, 114, 15495–15505

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Adhesion Kinetics between a Membrane and a Flat Substrate. An Ideal Upper Bound to the Spreading Rate of an Adhesive Patch Antonio Raudino* and Martina Pannuzzo Dipartimento di Scienze Chimiche, UniVersita` di Catania, Viale A. Doria 6-95125, Catania, Italy ReceiVed: July 20, 2010; ReVised Manuscript ReceiVed: October 18, 2010

A semiquantitative theory to describe the adhesion mechanism between an elastic membrane and a solid substrate (or another membrane) was developed. Since the membrane bending deformation requires a relatively small energy cost, thermally excited fluctuations may give rise to a local protrusion connecting the membrane to the substrate. This transient adhesion site is stabilized by short-range adhesion forces and it is destabilized by repulsion and elastic deformation energy. Above a critical radius of the contact site, adhesion forces prevail, enabling the contact site to expand until complete membrane-substrate adhesion is attained. This represents a typical nucleation mechanism involving both growth and dissolution processes. However, here we prove that also in the barrierless region, well beyond the critical radius, the spreading rate of a membrane still remains rather small, even under the favorable assumption of strong, sudden, and irreversible membrane-substrate adhesion. A detailed analysis of the membrane vibrational behavior near the adhesion patch rim suggests a reasonable mechanism for the spreading rate that has been analyzed by nonequilibrium statistical mechanics approaches. In relevant limiting cases, the model yields simple analytical formulas. Approximate relationships between the spreading rate and parameters like membrane elastic bending modulus, membrane-substrate interaction, temperature, and solvent viscosity have been found. 1. Introduction One of the most intriguing features in the field of membrane physics is the adhesion process between a soft membrane and a rigid substrate (or between two juxtaposed membranes). This process is ruled by a subtle interplay between generic interfacial forces, membrane bending elasticity, and specific short-range ligand-receptor interactions. In recent years, attention has been paid in understanding the spreading rate of an adhesion membrane patch onto a sticking substrate induced by ligandreceptors binding1-24 as shown in Figure 1A. A surprising finding common to most of the above works was the slowness of the expansion rate of the adhesion site. Explanations based on the slow diffusion of membraneembedded ligands moving toward the adhesion site have been proposed and thoroughly modeled; this point will be discussed shortly later. In this paper, we face the problem from a different standpoint: here we assume extremely large ligand-receptor binding energy and instantaneous bridging kinetics. Opposing a naı¨ve interpretation of the linear relationship between fluxes and conjugated forces, we anticipate that even in such ideal conditions the membrane spreading rate remains finite and it is not too far from the measured values. More precisely, the aim of this work is to calculate the upper limit to the spreading rate of an adhesive patch and to explore its dependence on membrane, substrate and solvent properties. Consider the flat region of a large deformable vesicle brought at the equilibrium distance η0 from a planar rigid substrate. We assume that the energy-distance curve exhibits two minima: a shallow long-distance minimum (the so-called secondary minimum) at η ) η0 and a deep short-distance well at η ) 0 (primary minimum) as shown in Figure 2. * To whom correspondence should be addressed. Phone: 0039957385078. Fax: 003995580138. E-mail: [email protected].

Figure 1. (A) Time evolution of a membrane adhesion site between a vesicle and a sticking solid substrate (for the sake of compactness, only the vesicle’s region facing the solid substrate has been drawn). The transition from an initial shallow long-distance energy minimum (left, t ) 0) to a tight short-distance energy minimum occurs through the nucleation and growth of a local tight adhesion patch of radius a. (B) Qualitative variation of the total energy (au) against the adhesion patch radius a calculated for different values of the membrane bending rigidity modulus KM (from KM ) 10kT (left) to KM ) 40kT (right)).

The energy-distance curve arises from different contributions. Classical DLVO theory25 considers a combination of repulsive electrostatic forces and attractive van der Waals interactions. Under certain conditions, they give rise to a doublewell potential. In biological membranes, other forces must be accounted for. The main short-range contribution comes from the overwhelming repulsive forces due to water ordering at the membrane-water interfaces.26-30 Altogether, repulsive forces are often stronger than the dispersion ones, so the energy-distance curve of either charged or neutral lipid membranes does not exhibit the short-range potential well predicted by the DLVO

10.1021/jp106722w  2010 American Chemical Society Published on Web 11/09/2010

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Figure 2. Energy variation against the solid-membrane distance η along the perpendicular z axis. The energy is described through a combination of two parabolas as shown by eq 2. The dashed line measures the energy barrier for the long-distance to the short-distance transition. Position and height of the barrier are defined in section 2.3.

theory. Hydration forces, however, can be overcame upon the formation of tight ligand-receptor couples (ranging from simple multivalent ions or membrane proteins); when this happens, the energy profile may exhibit a sharp minimum at η f 0.31 Next, let us introduce the contribution of the thermally excited membrane bending fluctuations. A well-known result is that the averaged membrane-substrate distance becomes larger than the equilibrium one. This effect is related to the pressure originating from time- and space-averaged membrane bending fluctuations (undulation forces32-35). The above mean-field picture is correct in the statistical limit of large surface areas and long times (it properly describes phenomena like enhanced membrane repulsion32-35 and unbinding transitions36), but it turns out to be wrong on smaller scales. Consider indeed the fate of a single bending fluctuation selected among those of the largest amplitude. A few fluctuations may locally reach amplitudes comparable with the distance from the sticky substrate. This unlikely event is strongly localized, so the average membrane spacing is still of order 〈η0〉 ≈ η0. Afterward, the large-amplitude fluctuation “trapped” inside the short-distance minimum evolves through different scenarios: (a) When the short-range binding energy is smaller than the membrane deformation energy the fluctuation decays. Typically, this happens when the adhesion area is too small. (b) When the adhesion area becomes large enough to overcome the membrane deformation energy cost, the local dimple spontaneously grows in size, until a complete transition from the long-distance to the short-distance adhesion is attained. A cartoon of the adhesion process is given in Figure 1A where we report the time evolution from loosely bound (left) to tight bound (right) vesicle via the formation of a localized dimple of radius a(t). For simplicity, only the adhesion region has been drawn. The lateral size L (the thick curve at t ) 0) of weakly bound states depends on the vesicle-substrate adhesion energy and vesicle radius;37 in any case L is much greater than the dimple radius a(t), so throughout the paper we will set L f ∞. In Figure 1B we report the qualitative variation of the total energy as a function of the patch radius a, evidencing a maximum at a ) acrit. Accordingly, we expect a restrained patch growth below the critical barrier (a e acrit, nucleation process) and a fast growth when a > acrit (spreading). In a recent paper,38 we calculated the nucleation barrier of an adhesion site (a e acrit) that, as expected, is small and depends on the ligand-receptor adhesion, membrane interaction, and bending energy. The ideas sketched before have been growing in these years among the researchers. While the notion of a double potential well as a key modulator of the membrane adhesion static and dynamics dates back to the pioneering Sakmann works on receptorsmediated vesicles adhesions onto solid substrates,3 the notion of correlated fluctuations of the membrane-substrate distance

Raudino and Pannuzzo is more recent. Starting from previous works on the statistical mechanics of membrane fluctuations,36,39 in a very recent paper Seifert and his co-workers developed a self-consistent theory to describe the evolution of a flat membrane lying onto a sticky substrate. They investigate the time evolution from the initial flat state into adhering (short distance) and nonadhering (long distances) coexisting patches.23 Since the reference state is the planar membrane set at an equilibrium distance from the substrate, Seifert’s choice seems to be the best reference suitable to investigate the nucleation step of the adhesion sites. On the contrary, the present model explores the late stages of adhesion (the barrierless region a > acrit defined in Figure 1B), and therefore, it is natural to assume as the reference state the boundary region between the adhering and nonadhering regions of a substrate-bound membrane (see Figure 1A). Around this boundary (the profile of which is calculated by energy minimization), we superimposed the membrane fluctuations that are assumed to be much faster than the growth rate of the boundary. The choice of this reference system makes the approach more difficult, but it does not crucially depend any longer on the correlation among the fluctuations that throughout the paper are assumed to be uncorrelated. At variance with the slow nucleation stage, the barrierless spreading kinetics should be fast. However, several experimental data evidence just the opposite behavior. Some authors ascribe the slowness of the spreading to the ligand-receptor encounter kinetics which is the main step for the adhesive patches growth.5-10,16,18 Indeed, for growth of the adhesion zone, it is necessary for the zone to recruit binders to its advancing edge, so that their density immediately behind the front can be increased to a value much higher than that found on the membrane surface. In the diffusion-controlled regime and floppy vesicles (like those of the present study, different results are obtained for tense vesicles), the time dependence of the adhesion patch spreading ranges between5 a(t) ≈ t1/2 and10 a(t) ≈ t5/8. A similar behavior occurs also during the biologically relevant hemifusion process where two lipid bilayers, brought at close contact, merge into a single bilayer. It was observed some years ago that the expansion rate of the hemifused disk is very slow,40 and calculations of hemifusion kinetics done by Hed and Safran41 et al.42 support these findings. The paper is organized as follows. In section 2.1 we describe the formation energy of an adhesive patch between a membrane and a solid substrate. In section 2.2 we calculate the adhesion growth rate as a function of the membrane fluctuations dynamics near the patch periphery. Through a Fokker-Planck equation, we relate the spreading rate to membrane and solvent properties. Calculations are performed under two drastic assumptions: (a) Ligands and receptors are uniformly distributed, and the pairs energy W is large. (b) The ligand-receptor pairing rate is instantaneous. These conditions ensure an extremely fast spreading rate of the adhesion patch that, according to well-founded theories4-7,10 and recent measurements,8,24 the spreading rate is proportional to the adhesion energy rate ≈|W|λ (λ being a model-dependent parameter, e.g., λ ) 3/2 for floppy vesicles and immobile binders10); then the spreading rate should tend to infinity in the limit |W| f ∞. This claim obviously is wrong; we will prove that (i) the spreading rate remains finite even for strong ligand-receptor interactions; (ii) within a restricted but meaningful range of parameters, the spreading rate could attain rather small values, sometimes comparable with the observed ones. Numerical calculations have been performed as described in section 2.3. The main predictions of the theory are summarized

Adhesion between a Membrane and a Flat Substrate

J. Phys. Chem. B, Vol. 114, No. 47, 2010 15497

and discussed in section 3, while possible evolutions of the model are sketched in section 4. 2. Theory 2.1. Formation Energy of an Adhesion Patch. Let us consider a planar membrane set an equilibrium distance η0 apart from a planar rigid substrate. Next we introduce a local deformation of the membrane planar shape. Let η ≡ η(r b) be the local membrane-substrate distance; any deformation requires an energy cost given to a good approximation by

E)

∫S [ 21 KM(∇2η) + G(η)] dS

(1) η1 ) η(0)

where the integration spans over the whole membrane surface S. The first term is the Helfrich bending deformation energy (with ∇2η the local membrane curvature and KM the bending rigidity per unit area; typically KM ≈ 10-19 J, a value strongly depending on thickness, area per molecule, surface charge, temperature, and composition43-53). The second term describes the membrane-surface interaction. As mentioned in the Introduction and shown in Figure 2, G(η) can be described by an asymmetric bistable potential with a shallow minimum at a distance η ) η0 from the substrate, and a ligand-receptor induced sharp minimum at the solid surface η ) 0. Usually, an asymmetric bistable potential is written as G(η) ≈ Aη2 + Bη3 + Cη4. Such a procedure is not convenient for our purposes, so, borrowing an idea widely used in theories like the solventassisted electron transfer, we describe G(η) by a combination of two parabolas centered at η ) 0 and η ) η0, respectively (see Figure 2)

{

1 -|W| + F1η2 2 G(η) ≈ 1 F (η - η0)2 2 0

S < S0

(2)

S > S0

where |W| is the membrane-substrate binding energy per unit area, while S0 and S are the surface area of the membrane regions inside and outside the adhesion patch. The usefulness of the piecewise functional (2) will be evident later when performing the limit F1 f ∞ (inextensible ligand-receptor binding). Let us calculate the formation energy of an adhering membrane as shown in Figure 1A. It is convenient to separately discuss the energy behavior inside and outside the adhesion patch. Inside the patch (S < S0) the membrane lies parallel to the solid surface at a still unknown, but presumably small, distance η ) η1. In this region, the membrane is flat and the bending deformation energy must vanish: E|SS0 ) ∫S>S0[(1/2)KM(∇2η)2 + (1/2)F0(η - η0)2] dS. Adding together the inner and outer contributions, we obtain a compact expression for the total free energy

1 1 E ) -|W| + F0(η1 - η0)2 + F1η21 S0 + 2 2 1 1 K (∇2η)2 + F0(η - η0)2 dS (3a) S>S0 2 M 2

(

)

∫

[

are derived from (3a) in the case of large adhesive patches. Let x be the radial coordinate measured from the patch rim (x ≡ r - a, with r > a). We may approximate ∇2η ≈ ∂2η/∂x2 (so doing, we neglect a term proportional to r-1∂η/∂r ) (a + x)-1∂η/∂x; since we are investigating patch radii well beyond the critical nucleation size acrit, this is a safe approximation). A model describing the adhesion energy at smaller a values has been recently developed by us.38 Because the membrane-substrate distance at the boundary x ) 0 between the adhering and nonadhering regions must be identical, we must impose in (3a)

(3b)

Recalling that the area increment dS is dS ) 2πa dx, eventually we rewrite (3a) as

1 1 E ) -|W| + F0(η(0) - η0)2 + F1η2(0) πa2 + 2 2 L-a ∂2η 2 πa 0 KM 2 + F0(η - η0)2 dx (3c) ∂x

(

∫

[ ( )

)

]

L - a ≈ L f ∞ being the lateral width of the membrane. The great advantage of the energy functional (3c) is the presence of quadratic terms alone that, after minimization, lead to a linear differential equation. On the contrary, a complete and correct representation of a bistable potential requires additional cubic and quartic terms to the integrand of (3c). However, one must bear in mind that (3c) holds only in the case of an extremely localized potential well at contact distance η ) 0 (viz., F1 . F0; see Figure 2); such a surface localization transforms the double-minimum problem into a simpler single minimum + surface term problem (employing large but finite F1 values could be a tool to describe elastically bound receptors, a situation found, e.g., in simulating cytoskeleton forces). To make a step forward, we partition η as

η(b, r t) ) η¯ (x) + ε(s(b), r t)

(4)

η j (x) and ε(s(r b),t) being the static and dynamic components of the membrane deformation, respectively. The symbol s(r b) indicates that the fluctuations are calculated in a local reference frame tangent to the equilibrium membrane surface. The static component η j (x) does depend on the radial coordinate x alone, while the dynamical part ε(s(r b),t) depends both on spatial and time coordinates. We first calculate the minimum energy profile η j≡η j (x) as follows.Consider a generic functional

E[η(x)] ) γ0(η(x0), ηx(x0)) + γ1(η(x1), ηx(x1)) +

∫xx f(ηxx(x), ηx(x), η(x)) dx 1

0

(5)

]

We may assume a circular shape for the adhesion patch, so S0 ) πa2, with a the patch radius. Particularly simple expressions

with ηx ≡ ∂η/∂x and ηxx ≡ ∂2η/∂x2. Let η˜ (x) be the function minimizing (5) and η(x) ) η˜ (x) + RV(x) a function close to η˜ (x). Hence

15498


(

)

Raudino and Pannuzzo

[(

∂f ∂f d2 ∂f d ∂f + 2 V dx + 0 ∂η dx ∂ηx ∂ηx dx ∂ηxx ∂γ0 ∂γ0 x1 d ∂f ∂f V+ Vx + V(x0) + V (x ) + dx ∂ηxx ∂ηxx x0 ∂η(x0) ∂ηx(x0) x 0

dE ) dR

∫xx

)

1

]

∂γ1 ∂γ1 V(x ) + V (x ) ∂η(x1) 1 ∂ηx(x1) x 1

(

-

-

∂f ∂ηxx

|

+

lim

xf∞

(7)

∂3η¯ )0 ∂x3

(10b)

Solution to (9) subjected to the boundary conditions (eqs 10a,b) is

(

( ) ( )) µx µx cos √2 √2

(11)

j a result similar to that obtained by other authors.54-56 Thus, η j ) η0 as x f ∞. Inserting (11) into ) (1/2)η0µx as x f 0 and η (3c) gives a compact expression for the optimized energy of the adhesion patch

1 1/4 EOPT ) -|W| + Fη20 πa2 + 2-1/2πaη20F3/4KM 2

(

)

(12a)

)| (

∂f d ∂f ∂ηx dx ∂ηxx

∂2η¯ )0 ∂x2

η¯ ) η0 1 - exp -

together with four boundary conditions

-

xf∞

(6)

The condition (dE/dR)0 ) 0 then gives the standard EulerLagrange equation

∂E ∂2 ∂E ∂ ∂E + 2 )0 ∂η ∂x ∂ηx ∂x ∂ηxx

lim

∂γ0 + ) 0; ∂η(x0)

x)x0

∂f d ∂f ∂ηx dx ∂ηxx

∂γ0 ) 0; ∂ηx(x0)

)|

-

+

x)x1

∂f ∂ηxx

x)x0

∂γ1 ) 0 (8a) ∂η(x1)

|

+

∂γ1 )0 ∂ηx(x1)

x)x1

(8b)

evidencing a balance between sticking and deformation energies. The use of the boundary condition57 ∂η j /∂x|x)0 ) 0 instead of j /∂x2|x)0 ) 0 (eq 10a leads to the same result, the only ∂2η difference being a numerical factor of 2 in the second term of the right-hand side of (12a). The membrane profile, however, j ≈ x as in (11)). The is different (η j ≈ x2 when xf0 insted of η plot of (12a) against the adhesion patch radius a exhibits a maximum at a ) acrit (Figure 1B): this represents the nucleation step of the adhesion kinetics previously described by us,38 while in this paper we investigate the following growth step a . acrit. Looking at (12a) we notice that, for stable adhesion to occur, the energy difference between planar and adhering configuration must be negative, and so the following inequality must be fulfilled

In our problem

1 γ0(η(x0), ηx(x0)) ) πa -|W| + F0(η(0) - η0)2 + 2 1 F η2(0) , γ1(η(x1), ηx(x1)) ) 0 2 1

(

2

)

[ ( )

f(ηxx, ηx, η) ) πa KM

∂2η ∂x2

2

+ F0(η - η0)2

]

Inserting this latter expression into (7) we obtain

∂4η¯ + µ4(η¯ - η0) ) 0 4 ∂x

(9)

µ ≡ (F0/KM)1/4 being a measure of the decay length of the membrane deformation. The boundary conditions to (9) at x ) j/ 0 are calculated from (8a); after simple algebra one finds KM(∂3η j (0) ) 0 and -KM(∂2η j /∂x2)|x)0 ) 0. Since our ∂x3)|x)0 + 2aF1η model is valid in the limit F1 f ∞ alone, we get simpler expressions

η¯ (0) ) 0

∂2η¯ ∂x2

|

x)0

)0

(10a)

Analogously, far from the adhesion patch, x f ∞, we obtain

(|W| - 21 Fη )a > 2 2 0

-1/2 2 3/4 1/4 η0F KM

(12b)

Since we are considering large |W| values and patch radii a . acrit, inequality (12b) is always satisfied. 2.2. Growth Rate of a Membrane Adhesion Patch. The calculation of the adhesion kinetics is a rather difficult task; a cartoon for the time evolution of the membrane-substrate adhesion patch is given in Figure 3. Let η be the local distance of the membrane from the flat substrate. We postulated that the adhesive forces are effective only within a small layer (the white layer of Figure 3) of thickness h f 0 set on the top of an undeformable planar substrate, while at shorter distances the adhesion energy suddenly drops to zero. Thus, once the amplitude of a membrane oscillation becomes comparable with the gap width (defined by the plane z ) 0 in Figure 3), short-range adhesion forces “trap” the vibrating membrane inside a deep short-distance minimum. This event widens the adhesion patch radius. Since the membrane-substrate distance at the adhesion rim (x ) 0) goes to zero, the adhesion rate should be instantaneous. Things, however, are more intricate because also the bending oscillations amplitude of a clamped membrane goes to zero at x ) 0 (see the forthcoming equations). In order to solve this puzzle, we must perform a careful investigation of both the oscillation amplitudes and the membrane-substrate equilibrium distance in a narrow region near x ) 0. Looking at the fluctuations-assisted adhesion mechanism reported in Figure 4, we partition the bending fluctuations into


Figure 3. A cartoon for the time evolution of an adhesion patch according to a bending fluctuations-assisted adhesion mechanism. The dashed region of the substrate represents its repulsive core, while the white thin layer of thickness h describes a cushion of receptors. Sudden adhesion occurs only when the membrane-substrate distance z is zero. From top to bottom: (a) Shape of the adhering membrane site at a generic time t in its minimum energy (equilibrium) configuration. Here the width of the adhesion patch is a. (b) Thermally excited bending fluctuations (dashed line) about the equilibrium configuration defined in (a). (c) Equilibrium shape of the adhering membrane at time t + dt: notice the increase of the adhesion size from a to a + dx on going from step (a) to step (c). Steps (a-c) self-replicate in time until full membrane-substrate adhesion is attained.

Figure 4. A more detailed view of the mechanism described in Figure 3. Membrane bending fluctuations around equilibrium position (thick line) calculated near the contact site rim set at x ) 0. Long-wavelength fluctuations (panel A) never reach the reactive plane z ) 0 where membrane and substrate irreversibly adhere (subcritical case). On the contrary, short-wavelength fluctuations (panel B) easily overlap the plane z ) 0, provided their amplitude is greater than a critical value Ycrit (supercritical case). When this happens, the adhesive patch widens. Lastly, when both the amplitude and wavelength of the bending fluctuations further increase, multiple encounters with the sticking plane could be observed (Figure 4, panel C). This means that the formation of multiple adhesion sites separated by nonadhering regions cannot be ruled out. The probability for the occurrence of these irregular patterns, however,isverysmallbecauseoftherapidgrowthofthemembrane-substrate spacing on leaving the adhesion patch rim. This mechanism could explain the observed irregular advancing boundaries when membranes are spreading onto substrates. Different hypotheses and detailed calculations dealing with the formation of irregular fronts of adhesion sites (blisters) have been proposed.1 This very unlikely event is not considered in this paper.

two classes: the nonreactive ones, whose amplitude will never reach the sticking plane z ) 0 (Figure 4, panel A) and the


Figure 5. Geometrical parameters used to describe the bending fluctuations dynamics. In the inset we report the membrane shape near an adhesion patch. The encircled part of the inset is enlarged and shown with more details in the main figure. εq(x′) (blue line) describes a bending fluctuation near the adhesion patch rim. The x axis is parallel to the solid surface, while the x′ axis is tangent to the equilibrium membrane surface η j (x).

reactive ones, which do overlap the sticking plane (Figure 4, panel B) contributing to its widening and determining the overall adhesion kinetics. The number of a reactive fluctuation crucially depends on fluctuation amplitude and wavelength. The calculations are made simpler noticing that the membrane-substrate distance rapidly grows on leaving the adhesion rim (see eqs 8a,8b), attaining a constant value η0 at large distance (η0 ≈ (3-4) × 10-9 m, a value far larger than the mean amplitude of membrane oscillations), and therefore, fluctuations responsible for the patch growth are localized into a narrow strip at the periphery of the adhesion patch. In the next section we will perform a reasonable calculation of the “flux” of reactive fluctuations reaching a sticking substrate. 2.2.1. Bending Fluctuations Dynamics. In order to describe the dynamics of bending fluctuations, we introduced in (4) the deviation ε(s(r b),t) of the instantaneous position from the equilibrium geometry η j (x). Near the membrane-solid adhesion site, x ) 0, we may rectify the curvilinear coordinate s(r b) tangent to the membrane equilibrium configuration (Figure 5) as s(r b) ≈ x′ and express ε(s(r b),t) through a normal modes expansion

ε(x′, t) )

∑ (Aoq(t) sin(qx′) + Boq(t) cos(qx′)) q

(13) where Aoq(t) and Boq(t) are coefficients to be determined At the patch periphery x ) 0, the amplitude of the bending fluctuations must vanish: ε(x′,t)|x′)0 ) 0, such a condition is satisfied setting Boq(t) ) 0. By maintaining constant the patch area a (the time evolution of the adhesive patch (Figure 1A) is much slower than the bending deformations ε ) ε(x′,t) of the adhering membrane), the time evolution of ε ) ε(x′,t) is obtained by locally balancing the mechanical force -δE/δε with the solvent viscous dissipation. In the simplest approximation, the dissipation behaves as ≈σΓ∂ε/∂t, where Γ (J · s · m-2) is a frictional drag coefficient per unit area proportional to solvent viscosity and σ (m-2) is the surface membrane density

15500


σΓ

∂ε δE )+ σR(x′, t) ∂t δε


(14)

Yf0

In (14) we added a random force R(x′,t) related to the heat bath fluctuations. We assume R(x′,t) to be a Gaussian white noise satisfying the time-averaging property 〈R(x′,t)〉 ) 0. We shall see that the thermal noise plays a key role in determining the membrane spreading kinetics. By expanding the noise term as R(x′,t) ) ∑qRq(t) sin (qx′) and using (13) we obtain from (14) a Langevin equation for each q mode

dAoq(t) ω2q 1 + A (t) ) Rq(t) dt Γ oq Γ

∑ Aoq(t) sin(qx′)

(16)

q

where Aoq(t) satisfies (15). In a straightforward attempt to solve the stochastic differential equation (15), we treat Aoq(t) as an independent variable, Aoq(t) ≡ Y, and introduce the probability Pq ≡ Pq(Y) for finding a prescribed amplitude Y. Following a standard procedure,58 we derive from the Langevin equation (15) a Fokker-Planck equation

(

)

∂Pq ∂Pq ω2q ∂ ) D(q) + YP ∂t ∂Y ∂Y kT q

(17)

the diffusion coefficient D(q) (m2 · s-1) being related to the viscous damping. The stationary (∂Pq/∂t ) 0) first integral of (17) is

(

)

∂Pq ω2q D(q) + YP ) -Iq ∂Y kT q

(

Pq ) exp -

)[

ω2q 2 Y -Iq 2kT

lim Pq ) 0

( ) ω2

]

(19)

The constants Iq and Cq will be determined by applying proper boundary conditions shown in the next subsection. 2.2.2. Calculation of the Adhesion Rate in the Strong Adhesion Limit. In order to calculate the first boundary condition, we impose that for small oscillations (Y f 0) the nonequilibrium distribution Pq approaches the equilibrium Gaussian distribution: Pq ≈ Nq exp(-ωq2Y2/2kT). The normalization constant Nq is obtained by imposing that the mass of a fluctuating membrane must be identical, after space averaging, to its mass M at rest: ∫VPqF dV ) M (M ) FSl, with F the 3D membrane mass density, S its surface, and l the thickness). Since in a fluctuating lamina dV ≈ S dY, we obtain the first boundary condition

(20)

(21)

where Ycrit is the still unknown critical amplitude. In order to apply the boundary condition (21), an estimate of Ycrit is required, and this goal is reached as follows. An oscillation is said to be reactive when its local amplitude εq(x′) equates (or is greater than) the equilibrium membrane-substrate distance. Let z-x be an axis frame with the x axis parallel to the solid substrate and z′-x′ be a rotated frame with the x′ axis parallel to the membrane near the adhesion rim x ) 0 as shown in Figure 5. The angle γ measures the equilibrium slope of the membrane near the adhesion patch (the behavior at large distances is irrelevant). Bending vibrations are defined in the z′-x′ frame; their local amplitude varies as εq(x′) ) Aoq(t) sin(qx′) ≡ Y sin(qx′). Looking at Figure 5, an oscillation reaches the substrate when its amplitude (projected onto the z axis perpendicular to the substrate) equates the equilibrium distance η j (x)

εq(x′) cos γ g η¯ (x)

(22)

Inserting εq(x′) ) Y sin (qx′) into (22) and noticing that x′ ) x/cos γ, we get

(18)

∫0Y D-1(q) exp 2kTq Y′2 dY′+Cq

(2πkT)1/2

YfYcrit

Ycos γ · sin

the integration constant -Iq (m · s-1) can be identified as the steady flux of oscillations reaching a given amplitude Y. A further integration gives

lωq

Let us calculate the second boundary condition. Looking at Figure 4 we notice that, when a bending mode (thin line) about equilibrium (thick line) reaches the sticking plane z ) 0, the membrane irreversibly binds to the substrate through tight shortrange adhesion forces (Figure 4, panel B), otherwise the fluctuation is ineffective to the binding (Figure 4, panel A). This behavior is well described by a sink condition

(15)

with ωq2 ≡ (1/σ)(F + KMq4). Thus, the bending fluctuations (13) become

ε(x′, t) )

lim Pq ) Nq )

( cosqxγ ) g η¯ (x)

(23)

Performing the limit x f 0, we derive from (23) the condition for having a reactive fluctuation near the adhesion patch rim

(

x Yq -

)

1 η0µ g 0 √2

(24)

Therefore, fluctuations do contribute to the spreading kinetics only when their maximum amplitude Y reaches a critical value

Ycrit ) η0

µ √2q

(25)

while when Y < Ycrit vibrations are ineffective. Let us pause for a moment to investigate the meaning of Ycrit. The q-dependence of Ycrit is related to the different behavior of the membranesubstrate equilibrium spacing η j (x) and that of the bending fluctuations amplitudes near x f 0. Indeed, η j (x) behaves as (1/2)η0µx, while the bending fluctuation of a clamped membrane varies as Yqx, and therefore long-wave oscillations (q f 0) are less effective in inducing adhesion than the shortwave ones despite their larger amplitudes.



Coming back to eq 19, introducing the boundary conditions (20) and (21), and using (25) to define Ycrit, we may calculate the integration constants

Cq ) Nq Iq )

(26a)

{

NqD(q)

( )

ω2q 2 Ycrit exp Y dY 0 2kT 2 ω3qYcrit ω2qYcrit l D(q) exp 2kT (2π)1/2 (kT)3/2 ≈ ω q l D(q) 1/2 Y (2πkT) crit

{

the Fourier components of the amplitudes Aoq(t) satisfy eq 15. Standard procedure yields D(q) ) kT/σΓ, where Γ is a q-dependent friction coefficient. For an oscillating membrane embedded in a viscous fluid and lying a mean distance η j above a flat substrate, a calculation based on the Navier-Stokes equation gives Γ(and then D(q)) in a close form59-61

∫

(

η¯ 3 2 qη¯ , 1 q 12ω D(q) ≈ 1 σkT qη¯ . 1 4ωq σkT

)

2 ω2qYcrit .1 2kT 2 ω2qYcrit ,1 2kT

(26b)

where in deriving (26b) we exploited the formulas: Z exp(-(1/ 2)Z2)∫0Z exp((1/2)z2) dz ) 1 + Z-2 + ... when Z . 1, and ∫0Z exp((1/2)z2) dz ) Z + (1/6)Z3 + ... when Z , 1. It can be easily shown that only the first of (26b) is meaningful for realistic values of the different parameters (see section 2.3). Equation 26b gives the steady flux Iq of those vibrations reaching the critical amplitude Ycrit near the adhesion patch rim. Its knowledge provides an estimate of the patch spreading rate

Vq ≡

da ) RIq dt

(27)

where R is a dimensionless numerical constant. Equation 27 applies to the spreading rate that is induced by a single vibrational mode of wavelength q. The effective rate 〈Vq〉 is just the average over all the bending modes. Assuming an uniform wavelengths distribution Gq in the range 0 < q < qMAX, we write

f 〈Vq〉 ) 2 ∑ GqVq Lf∞ ∫ 2qmax 0 q 1

1

qmax

Vq dq

ω being the solvent viscosity. An experimental test of (30) has been recently reported.62 Equation 30 applies when the membrane-substrate mean distance η j is constant. In our problem, the distance varies in the range 0 < η j (x) < η0 on going from x ) 0 to x f ∞, and therefore, the above figures are only a crude estimate. Because of the small membrane-substrate distance near the rim x ) 0, we used the first of (30) and set η j ≈ 〈η j (x)〉 ≈ (1/2)η0; thus, D(y) ≈ D0y2. As we shall see shortly, these uncertainties in the analytical expression for the diffusion coefficient do not substantially alter our conclusions because D(q) enters only as a prefactor to an exponential behavior. Once an estimate of D(q) has been provided, eq 29 can be calculated either numerically or by an asymptotic analysis accomplished as follows. By setting in (29) D(y) ≈ D0y2, we may conveniently rewrite 〈Vq〉 as

〈Vq〉 )

∫01 H(y) exp(-G(y)) dy

〈Vq〉 ≈ H(y*) exp(G(y*))

(

∫ y exp 21 ∂ ∂yG(y) | 1

0

2

2

y)y*(y

)

- y*)2 dy

(32)

Since the greatest contribution to the integral comes from the region y ≈ y*, we may replace the integration limits by (∞. Integration is straightforward

-1 4 3/2 -2 4 〈Vq〉 ≈ N ∫0 D(y)y (1 + Ay ) exp(-My (1 + Ay )) dy 1

〈Vq〉 ≈ ND0

(29) where

(31)

where H(y) ≡ ND0y(1 + Ay4)3/2 and G(y) ≡ (M/y2)(1 + Ay4). G(y) is a strongly peaked function around a critical value y ) y* calculated by seting ∂G(y*)/∂y ) 0, and this leads to y* ) A-1/4. Series expansion of G(y) about y* transforms (31) into

(28)

The multiplicative factor 1/2 has been introduced because only the fluctuations in the lower half-space facing the substrate can induce adhesion. Replacing in eqs 27 and 28 Iq by its analytical expression (26b), and introducing the dimensionless wave vector y ≡ q/qMAX, eventually we find

(30)

(2π)1/2 exp(-2MA1/2) 3/4 1/2 A M

(33)

provided MA1/2 . 1. After simple rearrangement of (33), we find a compact formula for the spreading rate 〈Vq〉

M≡

N≡

2 F3/2 0 η0

1

1/2 2 4σkTKM qmax

〈Vq〉 ≈ ω Z(∆, KM) exp(-∆/kT)

( )( )

πRlη0 F0 4qmax πσkT

3/2

F0 KM

1/4

4 /F0 . 1. In order to apply (29), one needs an and A ≡ KMqMAX estimate of the diffusion coefficient D(q) for the bending fluctuationsdynamics.Itcanbecalculatedfromthevelocity-velocity correlation function: D(q) ) ∫0∞ 〈A˙oq(t)A˙oq(t + t′)〉 dt′, where

where

∆≡ and

F0η20 2σ

(34)

15502


Raudino and Pannuzzo 1/4 qMAX), and A, M, and N are where G ≡ (F05/4η0)/(21/2σkTKM defined in (29). The integral (37) has been evaluated numerically as reported in section 3. A rough estimate by the steepest-descent technique gives

3 lF7/4 0 η0 R Z(∆, KM) ≡ 3/4 3 · 211/2 KM qMAX

This analytical result highlights an exponential decrease of 〈Vq〉 on an energy barrier ∆; this point will be thoroughly investigated in section 3. Our model leads to a simple expression for the adhesion patch spreading rate 〈Vq〉 as a function of different substrate and membrane parameters. As expected, 〈Vq〉 contains terms related both to viscous and elastic forces. This kind of activated process, which appears in a seemingly barrierless situation (spreading above the critical radius), somehow resembles the behavior of activated transport processes like the hopping diffusion of Brownian particles in condensed phases;63 its nature, however, is totally different. 2.2.3. Calculation of the Adhesion Rate in the Weak Adhesion Limit. The equations developed so far apply to the case of sudden and irreversible binding of the membrane when it reaches the sticky substrate. When short-distance adhesion forces are weak and/or the ligand-receptor pairing kinetics is slow, considerable deviations from the ideal behavior described by (34) are expected. In order to model the nonideal case, the sink condition (21) must be modified. This goal can be attained by imposing that the flux Iq of fluctuations reaching the critical amplitude for sticking is proportional to the fluctuation density Pq (the so-called radiation boundary condition widely used in diffusion problems). Thus, from (18)

(

∂Pq ω2q Iq ) lim D(q) + YP YfYcrit ∂Y σkT q

)|

Y)Ycrit

) -βPq | Y)Ycrit

(35) β (m · s-1) being a phenomenological parameter measuring the sticking efficiency. When β f ∞, all the fluctuations reaching the substrate contribute to the patch growth through irreversible binding. In that limit, the spreading rate takes its largest value and we recover the sink condition (21). On the other hand, when β f 0 the number of reactive fluctuations goes to zero and the adhesion patch does not expand any longer. This behavior suggests β to be related to the membrane-substrate adhesion energy W defined in (2). The specific relationship between W and β is not addressed in this paper; anyway, the relevant limit limWf∞β-1 ) 0 must hold, and thus β ≈ Ws with s an unknown coefficient. Calculations follow the same procedure employed in the previous section, enabling us to calculate the flux of reactive vibrations

Iq )

(

NqD(q)

ω2q

)

D(q) exp Y2 + β 2kT crit

∫0Y

crit

( )

exp

ω2q 2 Y dY 2kT

(36)

Combining (25) and (36), we obtain after simple algebra the nonideal spreading rate

〈Vq〉 ≈ N ∫0

1

D(y)y-1(1 + Ay4)3/2 1 1 + GD(y)y-1(1 + Ay4) β exp(-My-2(1 + Ay4)) dy (37)

〈Vq〉 ≈

〈Vq〉max 1+

1 K ωβ

(38)

〈Vq〉MAX being the maximum spreading rate of the adhesive patch (34) calculated in ideal conditions (β f ∞) and K ≡ [1/(3 · 29/ 3/2 4 1/2 2)](F0 η0)/(KM ). Making use of the qualitative result derived above, β ≈ Ws, we find from (38) a direct relationship between 〈Vq〉 and W, a behavior confirmed both theoretically and experimentally.4-10,24 2.3. Numerical Calculations. The asymptotic formulas developed so far yield simple relationships between the expansion rate of the adhesion site 〈Vq〉 and the physical properties of the lipid membranes. Numerical results were calculated using the following parameters: the bending elasticity modulus KM ) 0.4 × 10-19 J ≈ 10kT; the thermal energy kT ≈ 0.4 × 10-20 J at T ) 298 K; the water viscosity ω ) 10-3 J · m-3 · s; the equilibrium membrane-substrate distance η0 ) 4 × 10-9 m; the two-dimensional lipid membrane density σ ) 1.7 × 1018 m-2; and the bilayer thickness l ) 5 × 10-9 m; the shortest bending fluctuations wavelength is the inverse of the lipid-lipid distance d: qMAX ) 2π/d ≈ πσ1/2 ≈ 3 × 109 m-1. Finally, the derivative of the membrane pressure F0 was estimated as follows. Consider a planar membrane approaching to a flat rigid substrate along its perpendicular axis. Let η0 be the equilibrium position of the long-distance minimum and η ) 0 the position of the short-distance minimum shown in Figure 2. The bistable potential is given by (2), and in the limit of a narrow shortdistance well (F1/F0 . 1) it can be easily seen that the energy barrier (the dotted line in Figure 2) occurs at η* ≈ [(2|W| + F0η02)/(F1)]1/2 ≈ 0, while its height f ) f(η*) amounts to be

1 f ≈ F0η20 2

(39)

enabling one to estimate F0 once f is known. In neutral membranes, f is mainly determined by short-range hydration forces, while electrostatic and van der Waals forces play a role in determining position, η0, and depth of the shallow secondary minimum. Inside the primary minimum and at physiological salt concentration, the electrostatic and van der Waals forces contributions turn out to be about 10-20% of the hydration term, even at high surface potentials.25 Therefore, they can be neglected, while at long distance their effect is implicitly considered within the empirical distance η0. The hydration pressure behaves as PH(z) ≈ PH(0) exp(-z/λ) (typically26-30 PH(0) ≈ 108 J · m-3, λ ≈ 2 × 10-10 m), therefore: f ) -∫0η0PH(z) dz ≈ 10-2 J · m-2. Setting η0 ≈ 4 × 10-9 m, eventually we get from (31) F0 ≈ 1015 J · m-4. Such an estimate applies to simple lipids. A likely consistent reduction of the above figures could be due to the sugar residues attached to the outer cell surface.64 3. Results and Discussion The role of different parameters on the spreading rate 〈Vq〉 of an adhesion patch onto a rigid substrate has been obtained in close form. Extension to the case of membrane spreading onto soft surfaces (e.g., another membrane) is straightforward;



however, the adhesion between soft interfaces might give rise to instability effects.65 Purposely, we have introduced two extreme conditions: (a) the short-range ligand-receptor adhesion energy W per unit surface tends to the infinity; (b) the ligand-receptor pairing rate is instantaneous. Even in this favorable case the spreading rate is high but finite, its general expression being

Z

〈Vq〉 ≈ ω exp(-∆/kT)

(40)

where the pre-exponential term Z/ω (ω being the medium viscosity) is a frequency factor and ∆ an energy barrier. The frequency factor Z/ω measures the flux of vibrations reaching the sticking plane. Since at the adhesion rim, x f 0, the membrane-substrate distance goes to zero, one may conjecture extremely large Z/ω values (much larger than 1 m · s-1) and then a fast-spreading kinetics. What is the origin of the unexpected exponential factor? We have shown that, near the patch rim, x f 0, the equilibrium distance from the sticking surface behaves as (1/2)η0µx, while the bending fluctuation amplitude of a clamped membrane varies as Yqx, and thus, both the equilibrium distance and the bending fluctuation amplitude vanish in the limit x f 0. A correct analysis requires a deeper study of both the equilibrium distance and fluctuation in the small x region before to perform the limit x f 0. We had shown in section 2.2.2 that, in order to stick onto a substrate, the membrane fluctuations amplitude must be equal to (or greater) than a critical value Ycrit (see Figure 4). Since Ycrit ≈ const/q (eq 25), it follows that thermally excited long-wave oscillations (q f 0) are less effective than the short-wave ones in inducing adhesion, despite their larger amplitudes. The proposed mechanism picks up the reactive bending fluctuations (coming from the high-q region) and disregards the nonreactive ones (q f 0); this selection determines the exponential fall of the spreading rate. Theories like the celebrated transition state theory for chemical kinetics, the many nucleation theories, the Eyring model of condensed phases viscosity, and so on are always determined by various mechanisms that select events not reaching a critical threshold. All these theories give rise to an exponential behavior similar to that described by (40). Other relevant outcomes of the model are the following: (a) Since the spreading rate of the adhesion site 〈Vq〉 does not depend on time, we may conclude that the patch radius growth law scales as a(t) ≈ t. This result qualitatively agrees with that obtained in the case of immobile binders distributed over floppy vesicles.10 (b) The spreading rate of the adhesion site 〈Vq〉 sharply decreases with the membrane-substrate interaction parameter F0. We found 〈Vq〉 ≈ F07/4 exp(-|C|F0). High F0 are likely to occur in charged membranes, but also neutral phosphatidylcholine membranes (the most abundant lipid in cells) exhibit largeF0 because of overwhelming hydration forces.26-30 Some representative curves describing the behavior of the spreading rate 〈Vq〉 against F0 have been calculated by numerical integration of (29) and are shown in Figure 6. As expected, the absolute figures of 〈Vq〉 are higher than the experimental ones (of order 10-6m · s -1, see, e.g., refs 5 and 24). This discrepancy arises from the idealized nature of our model which imposes a complete and sudden binding when the membrane reaches the surface of the substrate. Therefore, our figures fix just a theoretical upper limit to the spreading rate. Surprisingly, the differences between theory and experiments

Figure 6. Variation of the logarithm of the spreading rate 〈Vq〉 (m · s-1) against the membrane-substrate interaction parameter F0 (J · m-4). Figures have been calculated at different values of the membrane bending rigidity modulus KM. From top to bottom: KM ) 10kT, 20kT, and 30kT.

are rather small because F0 values as large as ≈1016 J · m-4 are enough to bring the calculated rates 〈Vq〉 within the range of the experimental findings. These are not unreasonable figures because F0 values lying in the range 1015-1016 J · m-4 have been observed in bunches of lipid lamellas30 at low water content. This is perhaps the main result of this work, which shows that the slowing down of the adhesion kinetics is, to a large extent, related to the basic physics of the spreading mechanism. Other experimental data seem to support our conjecture. In concentrated negatively charged lamellas, the cation-induced collapse into closely packed arrays of tightly bound lamellas (for instance, the phosphatidylserine-Ca2+ system) follows extremely slow kinetics66-68 (of order of months). A likely cause is the slow diffusion of the cations in the intermembrane space; another explanation, however, could be related to the extremely slow spreading rate predicted by us in the limit of high F0 values. No conclusive answers are available at moment. (c) In eq 12a we had shown that the energy required to form an adhesion patch depends on the membrane bending rigidity KM. Therefore, one may hypothesize that also the adhesion patch spreading rate 〈Vq〉 should exponentially decrease with KM, resembling the behavior of 〈Vq〉 with F0. On the contrary, our theory foresees a weaker dependence of 〈Vq〉 on KM contained -3/4 . The rationale in the pre-exponential factor alone: 〈Vq〉 ≈ KM for this behavior naturally emerges noticing that the membranesubstrate equilibrium distance near the adhesive patch rim is inversely related to the membrane rigidity (from eq 11, η j (x) ) (1/2)η0(F0/KM)1/4x). Therefore, in soft membranes (low KM), the bending fluctuations must run over longer distances in order to reach the sticking substrate. This effect is balanced by the larger fluctuations amplitudes typical of soft membranes. Experimentally, this issue could be investigated because KM changes with temperature,69-74 by varying chain length or unsaturation of the lipid hydrocarbon chains,44,75,76 or by adding a second component to the bilayer (cholesterol, for instance, increases the membrane rigidity77-80). (d) There are direct and indirect thermal effects on the spreading rate, these latter being related to the thermal variation of different parameters. The direct effect predicts an increase of the spreading rate with temperature: rate ≈ exp(-|C1|/T). Among the indirect effects particularly relevant is the lowering of the membrane rigidity especially near the thermotropic gel f fluid phase transition.69-74 While a few papers deal with thermal effects on geometrical and energetic aspects of vesicle adsorption,81 to our knowledge the temperature behavior of the spreading rate of vesicles has not been investigated yet. (e) A pressure P perpendicularly applied to the adhering membrane (e.g., osmotic forces induced by water-soluble polymers sterically excluded from the interstitial region82 or by

15504



Figure 7. Panel A: variation of the mean spreading rate 〈Vq〉 (m · s -1 × 105) against the applied pressure P (N · m-2) calculated at high values of the membrane-substrate repulsion parameter F0 (from bottom to top: F0 ) 0.8, 1.0, 1.2 × 1016 J · m-4; 1 atm ≈ 105 N · m-2). Panel B: associated variation of the membrane-substrate equilibrium distance with the applied pressure P (η0 ) 4 × 10-9 m when P ) 0).

4. Perspectives and Outlook

Figure 8. Variation of the spreading rate 〈Vq〉 (m · s -1 × 105) against the membrane binding efficiency parameter β (m · s -1) calculated at different values of the membrane-substrate interaction parameter F0 (from the bottom to the top: F0 ) 1.2, 1.1, 1.0, 0.9 × 1016 J · m-4).

a gravitational field)22,83 reduces the membrane-substrate distance η0 as experimentally tested.84 In the harmonic approximation the variation of the lamellar spacing has been calculated from (2), augmented by a pressure-related term, -P(η - η0), where η is the new distance and η0 is the distance at P ) 0. Energy minimization leads to a shorter pressure-depending equilibrium distance: η0eff(P) ) η0 - P/F0. According to (34) the spreading rate 〈Vq〉 strongly depends on the distance η0eff(P): 3 2 eff rate ≈ (ηeff 0 (P)) exp(-const · (η0 (P)) ), a formula that explains the pressure effect on 〈Vq〉. A few numerical results are shown in Figure 7A, where we report the spreading rate against the applied pressure P at different values of the membrane-substrate interaction parameter F0. For the sake of comparison, in Figure 7B the variation of the distance ηeff 0 with P is also reported (notice that the pressurerelated enhancement of the spreading rate 〈Vq〉 is larger when F0 is small). The pressure behavior of cells adhesion is a wellknown and biologically relevant effect and different mechanisms have been reported in the literature.18 (f) The spreading rate of adhesive patches decreases on lowering the membrane-substrate binding efficiency parameter β. Useful limits are rate ≈ β when β f 0 and rate ≈ rateMAX when β f ∞. A few numerical examples are given in Figure 8. Qualitatively, this result is confirmed by data taken from the literature which report a decrease of the spreading rate on the binding efficiency4-10,24 obtained by lowering the ligand concentration or the binding adhesion energy per sticker. (g) The spreading rate is inversely related to the solvent viscosity ω through the diffusion coefficient D(q): rate ≈ ω-1. On decreasing the membrane-substrate binding efficiency parameter β, the viscosity effect becomes weaker: rate ≈ ((1/ β)const + ω)-1. This behavior could be experimentally investigated.

The approximate theory we have developed emphasizes the relevant role of combined energetic and dynamic effects in determining the spreading rate of a membrane onto a solid substrate. The model deals with the barrierless late stages of the spreading, where the adhesive patch size lies well beyond the critical nucleation radius a* (Figure 1A,B). We drew attention to the superfast spreading regime where the shortrange ligand-receptor adhesion energy W tends to infinity and the pairing rate is instantaneous. We proved that in the limit W f ∞ the expansion rate of the patch radius is linear in time and it levels up to an asymptotic value that depends on several factors (membrane-substrate potential, bending rigidity, temperature applied pressure), the most important one being the membrane-substrate potential as shown in the previous section. For a reasonable choice of the different parameters, the asymptotic rate is obviously greater than that observed in typical experiments; the differences, however, are smaller than expected. The model is simple. It, however, contains several shortcomings that could be eliminated in the future. A natural evolution of the theory would consider the following points: (a) Our model is tensionless. This seems to be a good approximation in the early stages of large vesicles adhesion where the adhesion-induced tension is low37,85 (experimentally, it is well-known that the developing tension of tightly adsorbed vesicles may induce lysis86). Tension influences both the equilibrium shape near the adhesion rim and the bending fluctuations dynamics, bringing substantial variations to the calculated spreading rate. Altogether, the growing tension with the adhesion patch radius a should reduce the spreading rate, especially at large a values: this is called the saturation regime of the spreading kinetics. (b) The adhesion rate decreases with medium viscosity through a phenomenological parameter Γ. In principle, Γ is related, through the Navier-Stokes equation, to the fluctuationsinduced hydrodynamics near the adhesion rim. This problem is relatively easy to solve at constant membrane-substrate j (x) varies spacing.59,60 Since in the present problem the distance η in the range 0 < η j (x) < η0, more specific and difficult hydrodynamic calculations are required. Development along this way has been recently reported in the literature.61 (c) We have extended in section 2.2.3 the model to the case of partial binding at the adhesive plane. Deviations from the ideal behavior are contained into a single model-dependent parameter β which measures the frequency of the effective membrane-substrate encounters. We have found that the spreading rate increases with β (eq 38). Analytical relationships between β and parameters like the ligand concentration, their diffusion coefficients, and ligand-receptors binding energy,

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Adhesion Kinetics between a Membrane and a Flat Substrate. An

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