Completing the theory of adhesion induced by mobile stickers: nucleation, phase-separation and exhaustion regimes, crossovers, and quantitative growth laws for vesicle adhesion patches

The 2003 Langmuir paper of de Gennes, Puech, and Brochard-Wyart derived exponent-level scaling laws ('a list of scenarios') for the growth of an adhesion patch formed when a vesicle carrying mobile stickers binds complementary receptors on a substrate across a water cushion, and closed with an explicit list of open questions: experimental tests of the predicted growth laws (linear and $t^{5/2}$ regimes for heavy vesicles; the transition between fast and slow binding reaction) and reconciliation with the observed $n=1/2$ growth; a detailed theory of the nucleation of the first sticker (barrier $\sim 10\,kT$, possibly catalyzed by defects or clusters); the regime where the sticker reservoir is exhausted and growth matches the final equilibrium; the effect of a two-dimensional gas-liquid phase separation of stickers on the growth law; the crossover when the patch radius exceeds the cushion (precontact) radius; the consequences of charged stickers (blisters) and of multiple sticker species; and the exact prefactors and logarithmic corrections beyond the exponents. This problem asks for the completion of that program as one quantitative theory linking sticker nucleation, transport, reaction, phase behavior, and reservoir finiteness to patch growth.

Background

Biomimetic model systems for cell adhesion use giant unilamellar vesicles (GUVs) - closed lipid-bilayer sacs of tens of micrometers - doped with a small fraction of mobile 'stickers': ligand molecules that diffuse freely in the fluid membrane and can bind complementary receptors fixed on a solid substrate. When such a vesicle meets a receptor-coated wall, an adhesion patch (a circular contact zone bridged by sticker-receptor bonds) nucleates and grows; its radius $R(t)$ typically follows a power law $R\sim t^{n}$ whose exponent $n$ reflects the rate-limiting transport or dissipation process.

Two earlier frameworks set the stage. Bruinsma, Behrisch and Sackmann (2000) and Boulbitch, Guttenberg and Sackmann (2001) modeled adhesion kinetics as a first-order transition controlled by the interplay of membrane thermal undulations and a double-well adhesion potential, and measured $R\propto t^{1/2}$ in a biomimetic system. Albersdörfer, Feder and Sackmann (1997) had observed adhesion-induced domain formation from the competition of short-range attraction and long-range repulsion. Brochard-Wyart and de Gennes (2002) analyzed the 'classical' case with no interfacial water film, where patch growth is limited by sticker diffusion and binding kinetics (fast or slow reaction), without hydrodynamic dewetting.

de Gennes, Puech and Brochard-Wyart (Langmuir 2003) extended this to the geometry with a water cushion: a lubricating film of thickness $h_c$ (order 100 nm) that separates the membrane from the substrate when the vesicle is made 'heavy' (denser than water by sugar additives, so that gravity presses it onto the wall) while steric Helfrich repulsion - the entropic pressure of thermally undulating membranes, with an energy per area $\sim A/(2h^{2})$ where $A=(kT)^{2}/K$ and $K$ is the membrane bending rigidity - prevents direct contact. In this geometry the patch grows by dewetting, i.e. expulsion of the water film through a moving contact line, with the viscous velocity law $V=\mathrm{d}R/\mathrm{d}t\sim(\gamma/\eta)\,\theta_{e}^{3}$, where $\gamma$ is the membrane tension, $\eta$ the water viscosity, and $\theta_{e}$ an equilibrium contact angle set by a Young equation (the classical balance of interfacial energies at a contact line) in which the osmotic pressure $kT\,\Gamma_{i}$ of the mobile stickers inside the patch (concentration $\Gamma_{i}$) does the adhesive work. Coupling this dewetting to sticker transport (free diffusion outside the patch, fast or slow binding with reaction time $\tau$, immobilization upon binding), they derived a table of scenario scaling laws: for a heavy vesicle with dilute, fast-binding stickers $R\propto t$ (linear growth); with a slow reaction ($t\ll\tau$) $R\propto t^{5/2}$; a concentrated sticker layer gives a fast, tension-driven regime; a 'sticky' vesicle (no cushion) returns to the diffusion- and reaction-limited laws of the 2002 theory; and a plateau regime assumes the sticker concentration inside the patch is pinned at a two-dimensional gas-liquid coexistence value. They also estimated the membrane-deformation barrier for nucleating the first sticker through the cushion ($\sim 10\,kT$), the line tension $\sim kT/x_{\min}$ of the patch edge (with $x_{\min}$ the inter-sticker spacing), and the angle $\theta_{c}=(2kT/(\pi K_{b}))^{1/4}$ below which the membrane tension stays constant during growth.

The authors themselves declared the analysis 'very tentative' and listed what remains open: experimental tests of the growth laws, especially the linear and $t^{5/2}$ regimes for heavy vesicles and the fast/slow transition, and reconciliation with the observed $n=1/2$ of Boulbitch et al. and with the domain-formation experiments of Albersdörfer et al.; a detailed theory of the nucleation of the first sticker, including whether defects or clusters catalyze it; the exhaustion of the sticker reservoir when the depleted diffusion hole reaches the vesicle size, and the matching of growth to the final equilibrium; the effect of a two-dimensional gas-liquid phase separation of stickers on the growth law; the crossover when the patch radius exceeds the cushion (precontact) radius $R_c$ - the radius of the flattened zone over which the heavy vesicle, pressed by gravity, rests on the cushion at the equilibrium film thickness before any adhesion; the consequences of charged stickers, which can nucleate blisters (water-filled pockets) instead of a flat patch; multiple sticker species; and the exact prefactors and logarithmic corrections beyond the exponents, needed for quantitative comparison with experiment.

Problem Statement

Complete the program that de Gennes, Puech and Brochard-Wyart (Langmuir 19, 7112-7119, 2003) left open at the end of their scenario analysis of adhesion induced by mobile stickers: turn their exponent-level list into a complete, quantitative, experimentally tested theory of adhesion-patch growth for a vesicle carrying mobile stickers that bind receptors on a substrate across a water cushion (and, in the limit without a cushion, recovering the earlier no-cushion theory). The requested results, which jointly serve this single objective, are:

1. Nucleation. A detailed theory of the formation of the first sticker-receptor bond through the cushion: the barrier height and the nucleation rate as functions of membrane tension, bending rigidity, cushion thickness and sticker size, and whether and how nucleation is catalyzed by substrate defects or pre-formed sticker clusters.

2. Phase separation. The growth law of the patch when the stickers inside it undergo a two-dimensional gas-liquid phase separation at coexistence (the 'plateau' regime of the 2003 scenario table): how the coexistence concentration, the moving contact line, and the sticker transport couple, and what exponent - or non-power-law behavior - results.

3. Reservoir exhaustion. The late-time regime in which the sticker-depleted diffusion hole reaches the finite vesicle size: the crossover of the growth law, the approach to the final equilibrium adhesion state, and the matching between the growth dynamics and the equilibrium theory of specifically bound vesicles.

4. Crossover geometry. The crossover between cushion dewetting and the regime in which the patch radius exceeds the cushion (precontact) radius - the radius of the flattened zone over which the heavy vesicle rests on the cushion before any adhesion - so that $R>R_c$, where growth should revert to the classical no-cushion problem of Brochard-Wyart and de Gennes (2002).

5. Extensions. The effect of sticker charge (electrostatic repulsion and the possible formation of blisters - water-filled pockets - instead of a flat patch) and of multiple sticker species with different binding energies and mobilities.

6. Quantitative level. The exact prefactors and logarithmic corrections of the scaling laws of items 1-5 (including the logarithms in the sticker diffusion balance), sufficient for quantitative comparison with experiments; this item is a quantitative requirement on those laws, not an additional regime.

7. Experimental verification. Measurements of patch growth and nucleation statistics in which the control parameters - initial sticker concentration $\Gamma_{0}$, reaction time $\tau$ or two-dimensional on-rate, diffusion coefficients, cushion thickness, and membrane tension - are independently varied and measured, testing the predicted growth laws for heavy vesicles (the linear and $t^{5/2}$ regimes), the transition between fast and slow reaction, and the crossover behaviors, and settling the discrepancy between the observed $n=1/2$ growth (Boulbitch et al.) and the predicted $n=1$.

Post-2003 works cited below that treat related model classes - notably refs 22, 23 and 25, on ligand-receptor domain nucleation, dilute/dense domain coexistence, and radially growing adhesions without cushion dewetting - do not settle items 1 and 2: the barrier for the first bond through the cushion and the coexistence-limited growth law coupled to the moving contact line must be derived for the geometry and coupling defined here.

Items 1-5 and 7 are independent results, and item 6 is a quantitative requirement on the growth laws of items 1-5. In the appropriate limits the derived laws must reduce to the 2003 exponents and to the 2002 no-cushion laws.

References

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