Gigaseal Mechanics: Creep of the Gigaseal under the Action of Pressure, Adhesion, and Voltage

Abstract

Patch clamping depends on a tight seal between the cell membrane and the glass of the pipet. Why does the seal have such high electric resistance? Why does the patch adhere so strongly to the glass? Even under the action of strong hydrostatic, adhesion, and electrical forces, it creeps at a very low velocity. To explore possible explanations, we examined two physical models for the structure of the seal zone and the adhesion forces and two respective mechanisms of patch creep and electric conductivity. There is saline between the membrane and glass in the seal, and the flow of this solution under hydrostatic pressure or electroosmosis should drag a patch. There is a second possibility: the lipid core of the membrane is liquid and should be able to flow, with the inner monolayer slipping over the outer one. Both mechanisms predict the creep velocity as a function of the properties of the seal and the membrane, the pipet geometry, and the driving force. These model predictions are compared with experimental data for azolectin liposomes with added cholesterol or proteins. It turns out that to obtain experimentally observed creep velocities, a simple viscous flow in the seal zone requires ∼10 Pa·s viscosity; it is unclear what structure might provide that because that viscosity alone severely constrains the electric resistance of the gigaseal. Possibly, it is the fluid bilayer that allows the motion. The two models provide an estimate of the adhesion energy of the membrane to the glass and membrane’s electric characteristics through the comparison between the velocities of pressure-, adhesion-, and voltage-driven creep.

1. Introduction

Patch clamp moved into its dominant role in electrophysiology with the serendipitous occurrence of the gigaseal.¹ Why a membrane that is negatively charged and made of fluid lipids stick to negatively charged glass remains unclear, although van der Waals interactions seem to be the key.²⁻⁴ The mechanics of the seal give an indication of why patches can be mechanically stressed with suction without flying up the pipet. Patches do, in fact, creep under pressure³ as well as spontaneously. Those properties of patches that allow them to stick to glass are the subject of this paper.

There are several general approaches to understanding the physics of this interaction, but no matter what the model, it must permit the creation of seals with a resistance of 1–100 GΩ. The simplest model is to assume that there is a highly viscous medium between the membrane and the glass (Figure 1). In the case of lipid bilayers medium can only be saline plus the headgroups of the lipids, including the water there, which is likely to be ordered by its proximity to the glass and the membrane.^2,3 The creep rate of patches made of pure lipids is affected by the presence of proteins; for example, data suggest that some proteins might denature against the glass and thus slow the creep by serving as immobile bridges in the bilayer.^5,6 Pure lipid patches might actually have similar “stops” because no lipids are pure, and a small quantity of contaminant might alter the seal behavior drastically. A great deal of physical data is available on adhesives because they play such an important role in modern technology,⁷⁻⁹ but the mechanism of adhesion of the patch to the glass remains unclear; our goal here is to examine a few possibilities and use some of the ideas from the known physical chemistry of adhesion. We begin by considering patches made of pure lipids.

Figure 1.

Diagram of the seal zone–multilayer model. The region labeled “cell” would be saline when patching lipid vesicles.

The resistance of the seal imposes powerful constraints on any model. In what follows we use typical patch dimensions of ∼10 μm in length and a pipet radius of nominally 1 μm. If the seal is viewed as a conductive annulus filled with normal saline, the thickness would have to be on the order of angstroms to create a multi GΩ seal. The first important question is whether the seal region between the glass and the membrane (Figure 1) is an electrostatically stabilized liquid film (i.e., a common black film¹⁰ with a thickness on the order of 10 nm), or the glass and membrane are in molecular contact (a Newton black film,¹⁰ where the saline solution is in an extremely narrow film involving few hydration layers of the glass surface and the head groups of the lipids). These two cases correspond to different mechanisms of creep, different adhesion energies, and different resistances depending on the membrane and glass potential. In the case of a common black film, the motion and energy dissipation are located in the saline layer. In the case of a membrane in molecular contact with the glass, motion and dissipation are located in the bilayer.

2. Electromechanical Properties of the Seal

2.1. van der Waals Disjoining Pressure

To estimate the adhesion energy of the membrane, we will first consider the van der Waals force between the membrane and the glass. Assuming that the seal is a flat glass–seal–membrane–cell structure (Figure 1), the following formula^11,12 can be used for the van der Waals disjoining pressure, Π_vdW, and the respective van der Waals energy, σ_vdW, in the seal film

1

Here h^S and h^M are the thicknesses of the seal and the hydrophobic core of the membrane, respectively; and h^M is assumed to be equal to two extended hydrocarbon chains of the membrane lipid, ∼4 nm.¹³ The Hamaker constant A_H^S is related to the binary constants for water–water (A_H), glass–water (A_H^GW), water–membrane (A_H), and glass–membrane (A_H^GM) van der Waals interaction:^11,12

2

For A_H^WW and A_H, we use the values:^4,12A_H^WW = 3.7 × 10^–20 J and A_H = 4.1 × 10^–20 J. The other two constants were calculated using the formulas¹²

3

The data for the Hamaker constants involved were:^4,12A_H^MM = 4.5 × 10^–20 J; A_H = 5 × 10^–20 J; A_H^WMW = 0.9 × 10^–20 J; and A_H=0.83 × 10^–20 J. (A list of symbols is provided in the Supporting Information S1.) These values and eq 2 yield a Hamaker constant of A_H^S= 0.2 × 10^–20 J. Because all constants involved have a large uncertainty and A_H is a small number obtained as the difference between large numbers, not only its value but also even the sign of A_H^S is not reliable. We have chosen values of the constants in such a way that the final result for A_H corresponds to attraction (A_H^S > 0) because the presence of attractive force is essential for the formation of a gigaseal.³ Equation 1 is approximate–it neglects various effects such as electromagnetic retardation, the screening effect of the electrolyte on the Hamaker constants, and so on;^4,12 in addition, eqs 2 and 3 are rough approximations, especially for very thin films. Therefore, eq 1 and the value of A_H = 0.2 × 10^–20 J can be used only for crude estimates of the van der Waals energy of the seal.

2.2. Electrostatic Disjoining Pressure: Variation of the Surface Potential with Seal Thickness, h^S

We will investigate in this section the electrostatic characteristics of the seal film¹⁴ in Figure 1. There is a strong repulsive electrostatic contribution to the adhesion.¹⁵ Let the surface potential of the glass in contact with 150 mM NaCl be ϕ_∞^G (we assume the value^3,16 ϕ_∞ = −20 mV) and the surface potential of the outer monolayer of a free-standing bilayer be ϕ_∞^M (ϕ_∞ ≈ −50 mV^17,18). Using Gouy electroneutrality condition¹⁶ (eq A12 in the Supporting Information S2), one can calculate the respective surface charge densities: ρ_e^G = 0.018 C/m² and ρ_e = 0.052 C/m². When the membrane adheres to the glass and forms a thin liquid film (the seal), the potentials of both surfaces, ϕ^G and ϕ^M, will increase due to the interaction between the negatively charged surfaces. The effect is negligible if the Debye length is smaller than seal thickness, L_D ≪ h^S, but is strong for thin seals, where L_D > h^S. (L_D is defined with L_D² = k_BTε^S/2e²C, where k_B is Boltzmann constant, T is temperature, ε^S is the absolute dielectric permittivity of the seal, e is elementary charge, and C is electrolyte concentration [m^–3]). For 150 mM 1:1 electrolyte solution, the Debye length is L_D ≈ 1 nm. The thickness h^S of the seal region depends on the magnitude of the van der Waals, electrostatic, and other surface forces acting within the seal. In Section 2.4, we will consider the relation between the electric resistance of the seal, Res, and h^S to estimate the thickness from experimental data for Res.

The relation between the surface potentials and h^S for the case where the surface charge densities ρ_e^G and ρ_e are independent of h^S is derived in the Supporting Information S2 following Derjaguin.¹⁹ The final results for ϕ^M and ϕ^G are shown in Figure 2, together with the minimal potential ϕ_m in the seal film (the minimum of the potential ϕ(z) in the film; this quantity plays a central role in the theory of electrostatic interactions in films¹⁹). As seen in the Figure, for thick seals (e.g., h^S > 4 nm), the surface potentials are almost equal to those of free-standing membrane and glass surfaces. In such case, the known¹⁹ asymptotic formulas for the electrostatic disjoining pressure, Π_el, and the electrostatic energy, σ_el, of thick films can be used (cf. the Supporting Information S2)

4

5

Here the electrostatic factor γ_el is given by

6

For the values of the parameters cited above, γ_el = 0.087.

Figure 2.

Dependence of the potentials ϕ^M, ϕ^G, and ϕ_m [mV] on seal thickness, h^S, in the constant charge regime. For h^S > 4 nm, the surface potentials ϕ^M and ϕ^G are about the same as those of free surfaces, ϕ_∞^M and ϕ_∞. The minimal potential ϕ_m in the seal decreases roughly exponentially at large h^S, while for thin seal films, it is about equal to both ϕ^S and ϕ^G. For a very thin seal, the potentials increase significantly in absolute value. The graphs are obtained by plotting the parametric solution ϕ(ϕ_m) versus h(ϕ_m) following eqs A5, A6, and A11 in the Supporting Information S2.

Equations 4 and 5 are not correct for extremely thin films (h^S < L_D), where the surface potentials are very different from those of the free glass and membrane surface. It is seen from Figure 2 that ϕ^M, ϕ^G, and ϕ_m become equal for such thin films. The physical reason for this is that the electrostatic potential gradient has characteristic length, L_D, so if the film is thinner than L_D the potential is unable to change significantly across the film. Indeed, the difference Δϕ between the two surface potentials decreases linearly with the decrease in h^S in the limit h^S/L_D →0 (cf. Supporting Information S2 for derivation)

7

where the electrostatic factor γ_ϕ is

8

For the values of the parameters cited above, γ_ϕ = 1.47. We will use this expression when dealing with the electroosmotic creep in Section 3.1.5.

Equation 7 is approximate but perhaps predicts a qualitatively correct picture. A number of effects will significantly modify the electrostatics of very thin films, among them: charge regulation (both surface charge density and surface potential change with h^S, while the chemical potentials of ions remain constant¹⁶), charge discreteness,²⁰ effects related to the presence of lipid headgroups in the seal region such as ion expulsion and dielectric permittivity decrement, structural effects related to the state of water in film less than one nanometer thick, and so on. Because these effects can change disjoining pressure, Π_el , by an order of magnitude, we will not consider Π_el in this limiting case; we will only assume that its value is on the order of the van der Waals disjoining pressure.

Another significant simplification we are making is to neglect the presence of divalent ions in the seal. Divalent Ca²⁺ and Mg²⁺ ions assist the formation of the gigaseal.^15,21 This is either due to their specific effect on the surface potentials of the glass and the membrane (they decrease both ϕ^M and ϕ^G, which suppresses the electrostatic repulsion in the seal) or due to the formation of salt bridges connecting a negative charge at the glass surface with a negative charge at the membrane²¹ (the latter obviously favors the formation of Newton black film rather than a common black film). However, double layer made of a mixture of monovalent and divalent ions has a rather complicated structure.¹⁹ In fact, the validity of the Poisson–Boltzmann and Gouy equations for double layer containing divalent ions is doubtful because divalent ions are subject of strong image and hydration forces, which these equations neglect.²² We leave aside also the question for the ion-specific effects on the adhesion,²³ but KCl is expected to decrease the electrostatic disjoining pressure in the seal in comparison with NaCl at the same concentration due to the higher specific adsorption energy²⁴ of K⁺. These complications are not going to change the estimated orders of magnitude in what follows, but at this level of simplification, our model is unable to account for the ever-present ion-specific effects.

2.3. Adhesion Energy in the Film

Considering first the thick film limit (h^S > 4 nm), we assume that the disjoining pressure in the seal is the sum of Π_el and Π_vdW according to eqs 1 and 4. The respective sum corresponds to the DLVO^25,26 theory for Π. The sum of van der Waals and electrostatic energies in eqs 1 and 5 has a minimum at h^S = 7 nm. This equilibrium thickness of the film (electrostatically stabilized common black film¹⁰) will correspond to relatively high conductivity and high creep rates of the seal. The respective adhesion energy σ_adh = −(σ_vdW + σ_el) at h^S = 7 nm is 0.5 × 10^–3 mJ/m². The same order of σ_adh was obtained by Smith et al.,²⁷ who studied the pulling of tethers in vesicles adhered to rigid substrates.

If a common black film breaks, the lipid and the glass surface will come into close contact, keeping few hydration layers of water (Newton black film will be formed¹⁰). The respective film thickness will be on the order of few angstroms. The adhesion energy is hard to predict, first because non-DLVO forces will be present (steric, hydration, and perhaps specific interactions such as hydrogen bonding between the lipid headgroups and the glass surface^19,28,10), and second, because of the inapplicability of eqs 1 and 5 for very thin films.²⁹ The adhesive van der Waals energy for h^S = 0.5–1 nm thick seal film is, according to eq 1, on the order of σ_vdW = −0.1 to −0.2 mJ/m². Steric and electrostatic disjoining pressure will decrease the absolute value of the total adhesion energy, σ_adh, so we can assume that σ_adh is ∼50% from −σ_vdW; that is, it is on the order of σ_adh = 0.05 mJ/m². Justification for this assumption is given in the Supporting Information S4. Estimates for the thickness and the adhesion energies of common black and Newton black films are summarized in Table 1. The order of the values calculated here is lower than previous experimental estimates yielding σ_adh ≈ 1 mJ/m² by us³ and others.^30,31 The difference can be due to specific protein interactions (although Smith et al.³² found a much lower value, 10^–5 mJ/m², for the protein adhesion energy). It might be also due to an inaccurate force balance at the dome rim due to finite thickness, curvature of the membrane, and dynamic effects.

Table 1. Estimates for the Basic Characteristics of the Seal Film: Thickness, h^S, Adhesion Energy, σ_adh, and Resistivity, Res.

film is	hS [nm]	σadh [mJ/m2]	Res [GΩ]
common black filma	5–10	0.5 × 10–3	0.17
Newton black filmb	0.5–1	∼0.05	1.2–12
experimental		∼10–5 (ref 32)	1–100
		∼10–3 (ref 27)
		0.16 (Section 4)
		∼1 (refs 3, 30, and 31)

^aValues for common black films refer to the minimum of σ_el + σ_vdW, eqs 1 and 4. Electrolyte concentration is assumed to be C = 150 mM.

^bValues for Newton black films are estimations, cf. the text.

2.4. Seal Conductivity

Typically,³ patches have seal resistances Res on the order of 1–100 GΩ. More specifically, for patches made of cells and liposomes in contact with 150 mM NaCl and a few millimolar CaCl₂, or with 200 mM KCl and 40 mM MgCl₂, we measured seal conductivities that are in the range 3–15 GΩ (with liposomes being at the higher end). These values represent the parallel combination of the pipet spanning dome and the seal. For simplicity, we will refer to seal resistance as the seal alone; that is, we neglect dome conductivity. To estimate the conductivity of the seal, we use the Stokes–Einstein relationship between the drift velocity v_i of the ith ion (of charge e_i and Stokes radius R_i) and the tangential electric field E in the seal³³

9

η^S is seal viscosity. The corresponding ion current densities, j_i, and electric current density, j = Σe_ij_i, are

10

In the first approximation, we neglect the surface conductivity^34,35 due to the double layers at both surfaces (equivalent to setting C_i(z) ≈ C_i∞). In that case, the integral current J in the seal (of cross-section 2πR_ch^S, where R_c is the radius of the capillary) is

11

where we used E = −Δϕ_out/L, whereΔϕ_out is applied voltage and L is seal length. Because we aim only at an estimate, let us assume that both the cation and the anion have the same radius R_i. The resistance Res = −Δϕ_out/J is then

12

However, surface conductivity is likely to make a significant contribution to J. The expression of Res is corrected for this effect in the Supporting Information S3; the final result for Res is

13

where the factor γ_Res is given by

14

The equation is valid only for the case where h^S > L_D; for the values of the potentials above, we obtain γ_Res = 0.46. Taking R_i = 4 Å, η^S = 0.001 Pa·s, C = 150 mM, L = 10 μm, R_c = 1 μm, and h^S = 7 nm as for a common black film, we obtain Res = 0.17 GΩ, a low value in comparison with experimental data.

Let us consider now the other limiting case, where the film is extremely thin. In this case, the double layers of the two surfaces overlap significantly, and the seal resistance is dominated by surface conductivity. We will use the result from Section 2.2 that the electrostatic potential in a very thin film is nearly constant (ϕ^M ≈ ϕ^G ≈ ϕ_m ≈ −70 mV; cf. Figure 2 and Supporting Information S2). Such a high negative value of ϕ in the seal zone means that the concentration of cations there will be much higher than the concentration of anions (this is the reason for the high cation-selectivity observed with typical gigaseals³⁶); therefore, we can neglect the conductivity due to the anions. Taking the counterion concentration as C₊ ≈ C exp(−eϕ_m/k_BT) and assuming it is about constant, we can write for the local and the integral current

15

The seal resistance is

16

Using the values ϕ_m = −70 mV, h^S = 0.5 nm, and η^S = 0.001 Pa·s, we obtain Res = 1.2 GΩ. This is about seven times higher than the resistivity of the common black film obtained from eq 13 and in much better agreement but still lower than the average experimental values, 3–15 GΩ. An even better agreement will be obtained if one accounts for the effect from the proximity of the membrane and the glass on the mobility of the ions. The seal thickness 0.5 nm is on the order of the typical diameter of a hydrated ion,³³ 0.8 nm, so the ions might roll over the glass surface, half-dipped into the membrane. Therefore, one must use not the water viscosity but something between the viscosity of water and of the membrane.³⁷ To estimate the membrane’s effective viscosity, we can use the data for the diffusion coefficient of a lipid in the membrane,²D ≈ 10^–12 m²/s, which is ∼10 times lower than the typical diffusion coefficient of a lipid in water³⁸ (10^–11 m²/s). From this result and Einstein–Stokes relation between D and η^S, one concludes that the membrane viscosity is about 10 times higher than that of water. Therefore, the effective viscosity felt by the ion must be between 0.001 and 0.01 Pa·s, the upper limit corresponding to Res = 12 GΩ.

The results for the resistivity again suggest that the seal is probably a Newton black film (cf. Table 1). We remind the reader that the calculations above concern lipid/water/glass seal, whereas biological membranes are much more complicated.³ We shall return to the effects of heterogeneity, polyvalent ions, and the ion-specific effects in a future paper.

3. Creep Rate of Patches

One way to explore the properties of the seal is to examine the rate at which the patch can creep up the pipet under the influence of a driving force. Creep can occur due to various driving forces: the adhesion energy σ_adh, sucking pressure Δp, or voltage Δϕ_out. In addition, the two configurations of the seal zone correspond to two different mechanisms of creep, with different friction forces, respectively. As previously discussed, the mechanism of motion depends on whether the seal film is a common black, highly fluid film of thickness ∼7 nm or is a stagnated Newton black film. In the first case, slippage is located entirely in the seal (of thickness h^S) and that is where the dissipation occurs. In the second case, slippage and dissipation occur in the membrane. Mixed transport using both mechanisms is also possible.

We first investigate the case where motion occurs through a shear flow in the seal film (Section 3.1). We discuss the other possibility (shear occurs between the two monolayers of the membrane) in Section 3.2.

3.1. Creep through Motion in the Seal Zone

3.1.1. Adhesion-Driven Creep

Consider the case where the driving force of creep is membrane adhesion and there is an infinite supply of lipid available from the cell (Figure 3). The adhesion energy of the membrane at the glass pulls more membrane into the capillary. The membrane is moving with velocity v_L = dL/dt, where L is the length of the seal, that is, the membrane-wetted pipet (height of the cylinder). The free energy gained from adhesion is

where A_adh = 2πR_cL is the contact area between membrane and glass and R_c is the radius of the pipet. The power (work per unit time) of the energy source is therefore

17

Figure 3.

Cartoon of a “cell-attached” patch of bilayer. The cell (vesicle) is located at the left and serves as an effectively infinite supply of lipid relative to the area of the patch dome shown at the right.

In this section, we assume that this energy is dissipated mainly through hydrodynamic friction in the seal zone. The velocity profile in the seal zone is that of a simple shear flow

18

At z = 0 (the glass surface), velocity is zero, and at z = h^S (the membrane outer surface), the liquid is moving with velocity v_L. The corresponding local and integral dissipation rates are, respectively (cf. e.g., Batchelor³⁹),

19

where η^S is viscosity of the seal fluid. Creep velocity, v_L, can be found from the energy balance W_diss + W_adh = 0 (e.g., de Gennes⁴⁰), which yields

20

A typical creep rate for biological patches is³v_L ≈ 8–16 nm/s. If one takes L ≈ 10 μm, η^S = 0.001 Pa·s as for water, h^S ≈ 7 nm, and σ_adh = 0.5 μJ/m² as for a common black film (Table 1), one can estimate from eq 20 that v_L = 350 nm/s, two orders higher than the experimental value! In the case of Newton black film, the velocity will be even higher due to the larger value of σ_adh; if h^S = 0.5 nm and σ_adh = 50 μJ/m², then v_L ≈ 2500 nm/s. This suggests either that we are using incorrect parameters (say, it is possible that η^S is orders of magnitude higher than the viscosity of water due to the proximity of the glass and the membrane) or that the mechanism of motion is different.

Note that eq 20 is a differential equation for L(t) (since v_L = dL/dt), and its solution is

21

This parabolic dependence (velocity decreases with time ≈ t^–1/2) is in fact a version of the well-known Lucas–Washburn law.^41,42 The observed L(t) dependence for cell-attached patches seems to be quite linear^3,43 rather than following the square root formula 21; in the case of cell membranes, that may reflect the influence of cytoskeletal forces normal to the membrane and their viscoelasticity. Dome bulging seems also to be a factor (cf. Section 4).

In the patch clamp technique, pipettes usually have conical shapes rather than cylindrical. The case of adhesion driven creep in conical capillary is analyzed in the Supporting Information S5. Here we cite only the final result

22

compared with eq 20. The lengths L and L₀ are defined in Figure 4. The integral of this equation, the dependence of L on t, yields a transcendental equation

23

This can be compared with the inverse function of eq 21 for L, t = η^SL²/2h^Sσ_adh. We will use eq 23 for the interpretation of experimental data in Section 4.

Figure 4.

Diagram of a conical capillary.

3.1.2. Pressure-Driven Creep

Consider a vesicle or a cell-attached patch with an infinite supply of material, creeping under the action of an applied hydrostatic pressure, that is, the driving force is the pressure gradient Δp/L in the seal region. The flow is now more complex: it is superposition of the parabolic flat-channel Poiseuille flow and linear shear flow³⁹

24

The energy that drives the process is the mechanical work done on the system. It has two components: the work done on the “liquid” in the seal (of area 2πR_ch^S, moving with velocity, v̅_x) and the work done for moving the dome patch (of area πR_c², moving with velocity, v_L). Written as work per unit time, these are

25

where eq 24 was used; evidently, W_patch ≫ W_flow. The dissipation is concentrated in the seal, and the integral dissipation rate corresponding to the velocity profile (eq 24) is given by the expression

26

Creep velocity can be found again from the energy balance, which is W_diss + W_flow + W_patch = 0 and which yields in first approximation (neglecting terms of the order of h^S/R_c)

27

This result is similar to eq 20, with −ΔpR_c/2 instead of σ_adh. A “typical value” for the suction factor, −ΔpR_c/2, is on the order of 0.05 to 0.5 mJ/m² (R_c = 1 μm and Δp is between −100 and −1000 Pa). A comparison between adhesion-driven creep velocity and pressure-driven creep yields an estimate of σ_adh, even if η^S is unknown. This is demonstrated in Section 4.

The case of pressure-driven creep in conical capillary (Figure 4) is investigated in the Supporting Information S6; the result for v_L is

28

The result is similar to eq 22 for adhesion-driven creep, again with −ΔpR_c/2 instead of σ_adh, as it was with cylindrical capillary. However, the pressure-driven creep of the patch will follow different L(t) dependence from the adhesion driven creep because R_c depends on L in the case of conical pipet, cf. Figure 4 and eq A36 in Supporting Information S5. In addition, the applied suction pressure Δp may be time-dependent; for example, it can be⁴⁴ a linear function of t.

3.1.3. Creep-Driven Simultaneously by Pressure and Adhesion

In the experimental case, both adhesive force and pressure gradient are usually present. The difference compared with pressure-driven creep (investigated in Section 3.1.2) is that W_adh, eq 17, should be added in the power balance so that it reads W_adh + W_diss + W_flow + W_patch = 0. The respective result for the creep velocity, v_L, in a cylindrical capillary is

29

Again, this can be used to estimate h^S/η^S and σ_adh. Equation 29 suggests that if Δp = 2σ_adh/R_c (positive pressure, acting toward pushing the patch out of the capillary), the creep velocity will be zero. This can be used as a technique for measuring σ_adh by changing Δp until v_L = 0. The result for the conical capillary is similar (Supporting Information S6)

30

The conical capillary has the additional advantage that R_c increases with L: R_c = (L₀ + L) sin(α/2). For positive Δp, there exists a value of L such that R_c = 2σ_adh/Δp, at which patch dome will stand still, once again allowing for a fine determination of σ_adh.

3.1.4. Pressure-Driven Creep Motion of an Excised Patch

In the case of an excised patch of fixed area 2πR_cL_seal + A_patch (the whole patch is inside the pipet), no contribution of adhesion will be present because, in a cylindrical pipet, creep does not involve a change in the area of adhesion. Equation 29 then simplifies to

31

where v_L = dL/dt (L is the position of the patch rim) and L_seal is the length of the seal (the glass surface covered by the excised membrane). Because the right-hand side of the equation is independent of L or t, the velocity v_L is constant. The solution for L(t) is

32

The position L of the patch is now linear function of t.

3.1.5. Voltage-Driven Creep: Electroosmotic Smoluchowski Flow

For electrocapillary flow (flow under an applied voltage Δϕ_out), the Navier–Stokes equation balances viscous friction and electrostatic forces:

33

where the tangential electric field E_x = −Δϕ_out/L acts on the ions of bulk charge density ρ_e = ∑e_iC_i, with ion concentration profiles C_i = C_i∞ exp(−e_iϕ/T) according to the Boltzmann distribution; ϕ(z) is the potential distribution in the double layer (we assume that Δϕ_out ≪ ϕ^G). The Poisson equation of electrostatics states that ε^S∇²ϕ = −ρ_e, so eq 33 yields

34

After two integrations of this equation one obtains

35

The two integration constants k₀ and k₁ are determined by two boundary conditions:

(i) The electric force acting at the liquid surface (Maxwell tensor ε^SE_xE_z) is equal to the viscous force (Stokes tensor η^Sdv_x/dz):

36

which means that k₁ = 0, cf. eq 35.

(ii) At the glass surface (z = 0) where the potential is ϕ = ϕ^G, velocity is zero and therefore from eq 35

37

With these values of k₀ and k₁, for the velocity profile, we obtain the well-known general result from Smoluchowski’s approach;^16,34 the velocity profile of a flat flow is proportional to the double-layer potential profile

38

Note that the fluid velocity is equal to zero at all points, with z having the same potential ϕ(z) as the glass surface.

The creep velocity coincides with the velocity of the membrane (z = h^S); it is obtained from eq 38 by setting ϕ = ϕ^M

39

where ϕ^M is the surface potential of the outer monolayer of the membrane. From eq 39, it follows that if the membrane has a negative charge (and potential), this does not require that the membrane moves in negative direction. Actually, there are three possibilities according to eq 39: (i) If the membrane surface potential is more negative than the glass (ϕ^M < ϕ^G), then indeed it will move toward the anode. (ii) If the membrane is more positive than the glass (ϕ^M > ϕ^G), it will move toward the cathode, even though it is negatively charged; viscous drag due to counterion movement inside the seal film is stronger than the directly acting electric force, (iii) If ϕ^M = ϕ^G, then although membrane is negatively charged, it will not move; the electric force at the surface (toward the anode) is precisely compensated by the viscous force due to counterions moving toward the cathode dragging the membrane. From eq 39, we can conclude that the membrane will stop moving at a salinity and pH at which ϕ^M = ϕ^G. For example, we found³ that patches change creep direction between pH 5 and 7, which means that at pH 5 membrane has ϕ^M > ϕ^G (membrane is more “positive” than glass), but at pH 7, the situation is reversed and ϕ^M < ϕ^G.

Equation 39 is of the same form as eqs 20 and 27, but this time the “adhesion force” resulting from the outer potential is

40

Using ε of water, ϕ^G – ϕ^M ≈ 30 mV, Δϕ_out ≈ 50 mV, and h^S = 5 nm, one obtains “adhesion force” of ∼0.2 mJ/m². This is of the same order as the mechanical creep.

Note that Smoluchowski’s result is valid for unperturbed double layer only,³⁴ that is, Δϕ_out ≪ ϕ^G, while experimental Δϕ_out is on the same order as ϕ^G. This is probably the reason why eq 39 does not suggest the voltage asymmetry that was observed in patches;³ the high potential drop Δϕ_out disturbs the double layer. In such conditions, a nonlinear and probably asymmetric relation between v_L and Δϕ_out will be valid instead of eq 39. Another complication³⁴ is the possible dependences of η and ε on z, but we will not deal with that given the absence of data.

Consider now the effect of h^S on the electroosmotic creep rate.³⁹ If the film is thick (h^S > L_D as in the case of a common black film), ϕ^S and ϕ^M are almost independent of h^S (cf. Figure 2) and so is v_L according to eq 39. For a thin film (h^S ≈ L_D, as in the case of a Newton black film) at constant surface charge density, the potential difference ϕ^G – ϕ^M depends on seal thickness through eq 7; substituting it into eq 39 of Smoluchowski, we obtain how the electroosmotic creep velocity of the membrane depends on the seal thickness h^S in the case that h^S ≈ L_D and surface charge densities are fixed

41

Two features are interesting. First, v_L depends linearly on h^S (no electroosmotic creep if h^S = 0). Second, there is a multiplying effect of the “film thinness” on the potential difference; the creep velocity is very sensitive to small differences between the potential of the (freestanding) membrane and the glass surface. (Notice the difference of exponents in eq 8 for γ_ϕ instead of the Smoluchowski formula, which is linear with respect to ϕ^G – ϕ^M.) This suggests that if one could control ϕ^G, one could create a very fine method for the determination of the surface potential of a cell by varying ϕ^G until the patch ceases to creep.

3.2. Creep with Flow in the Lipid Bilayer

Consider the other limiting case of a cell- or vesicle-attached patch where the relevant flow occurs in the membrane itself (in contrast with Section 3.1, where dissipation was located entirely into the seal fluid). We assume at first that the pipet is cylindrical and we seek the velocity profile v_x^M(z) in the membrane and v_x(z) in the cell. The outer monolayer sticks to the glass, while the inner monolayer moves across the outer. The boundary conditions for such problem are, at first glance

42

where z = 0 is positioned at the outer wall of the membrane (considered fixed by the adjacent glass). This would be the simple shear flow with a profile v_x^M = v_Lz/h^M. There is a conceptual difficulty with these boundary conditions, however−the flow v_x = v_Lz/h^M will have total discharge of lipid material Q^M of

43

This amount of lipid is insufficient to coat the newly wetted pipet near the patch rim with 2πR_cv_L square meters of bilayer per second because the volume of lipid needed is obviously

44

that is, two times larger than eq 43. If the real discharge is smaller than the one given by eq 44, the flow will accumulate elastic strain in the dome region (through dilution of the lipid adsorption in the two monolayers), which will bring along a restoring Marangoni effect that will immediately “draw” the missing lipid from the adhered membrane toward the patch. There are two ways, at least, to compensate for the difference between the discharge (eq 44) needed to supply the lipid for the newly formed surface and the simple shear discharge (eq 43):

(i) The membrane can set the seal boundary into motion, until both its walls move with the same velocity, v_L, through Marangoni effect at the outer monolayer of the membrane. This case will coincide with the models in Section 3.1.

(ii) If the outer monolayer sticks tightly to the glass, the Marangoni flow will occur at the inner monolayer (the membrane–cell interface), which will result in the geometry of the flow in the membrane and in the cell shown in Figure 5. The flow in the cell will be a superposition of the simple homogeneous flow, v_x^C = 2v_L, and a backward Poiseuille flow. In this way, the inner membrane wall can move with velocity, v_x = 2v_L, while the outer monolayer stays immobile.^45,46 Additionally, in the rim region there must exist a mechanism of transfer of lipid from the inner monolayer to the outer.

Figure 5.

Diagram of the flow in the case of shear flow with dissipation in the membrane. The outer monolayer is fixed to the glass (the seal is a Newton black film), while the inner monolayer moves with velocity 2v_L. The lipid transported from the inner monolayer flip-flops in the region of the patch dome. The membrane flow excites Marangoni flow in the intracellular fluid, resulting in the profile shown (see eq 46).

We will now consider the second possibility in more detail for various driving forces. The membrane consists of two plates of lipids, the outer one immovably bound to the pipet surface and the inner one moving with velocity 2v_L. Instead of the boundary conditions (eq 42), we assume that (using cylindrical coordinates this time)

45

This choice of the velocity v_x^M means that the shortage of lipid due to the difference between the discharges in eqs 44 and 43 soon creates a strong gradient ∇σ^M in the inner monolayer (through a gradient of the surface density ∇Γ of lipid), yielding an additional flow of surfactant toward the patch. We assume that the lipid transfer is located entirely at the inner monolayer. Such mechanism requires two other processes to occur simultaneously. First, transfer of lipid from the inner monolayer to the outer one at the patch rim and the dome by, for example, a flip-flop mechanism, and second, if the inner monolayer moves with velocity 2v_L, then it sets into motion the adjacent cell fluid and the additional quantity of transported liquid must have a path back through the center of the pipet (cf. Figure 5). This happens in the following sequence of events: (i) Marangoni effect transports cytoplasmic fluid toward the patch; (ii) this strains the membrane and yields an increased pressure in the cell right next to the patch dome; and (iii) this increased pressure yields a backward restoring Poiseuille flow. The velocity profiles corresponding to this mechanism are v_x = 2v_L(R_c – r)/h^M in the membrane and

46

Here Δp^C is the pressure rise due to the strained patch; small terms of the order of h^M/R_c are neglected in these equations. The total discharge of lipid through the membrane is now correct (eq 44). The total discharge of cytoplasmic fluid must be πR_c²v_L. We can calculate the cytoplasmic discharge by integrating eq 46 over the cell region, which yields the balance

47

This balance determines the sought pressure rise Δp^C

48

The rise of the pressure is very small; if η^C = 0.001 Pa·s and v_L = 10 nm/s, then Δp^C = 5 × 10^–4 Pa is enough to restore the increased discharge of cytoplasmic fluid to the correct value πR_c²v_L. We can calculate now the velocity of the fluid at r = 0 by substituting eq 48 into eq 46. The result is v_x^C(r = 0) = 0; that is, the fluid in the pipet axis is immobile (cf. Figure 5). The Poiseuille flow induced by the Marangoni effect is not really important for the dissipation; the dissipation rate in the cytoplasmic fluid corresponding to the velocity profile (eq 46) is

49

while the dissipation in the membrane is

50

which is higher by many orders of magnitude compared with W_diss^C according to eq 49.

The viscous tensors acting on both sides of the inner monolayer of the membrane are given by

51

The first force is higher than the second by a factor of R_c/h^M; that is, the viscous force due to the intracellular fluid motion can be neglected in the force balance at the inner wall of the membrane. The mechanical balance at the inner monolayer equates the tangential Marangoni gradient ∇σ^M to the friction forces (eq 51)

52

From here, the tension drop from the edge of the pipet to the patch rim follows

53

where we used the value η^M = 0.01 Pa·s (which follows from the value² of D = 10^–12 m²/s; see above). As seen, the Marangoni increase in σ^M needed to produce the increased velocity v_x^M(r = R_c– h^M) = 2v_L is rather small. The difference Δσ^M is due to a slight decrease in the surface density Γ in the patch region, accumulated in the initial period of the creep (when the discharge is closer to eq 43). The shortage of lipid ΔΓ is related to Δσ^M through the Gibbs elasticity, E_G:

54

The value was calculated by assuming the order E_G ≈ 100 mJ/m². Thus, a decrease in the lipid adsorption by 0.0005% in the patch is enough to produce a very significant Marangoni flow of the inner monolayer of the membrane.

Let us investigate another point here. There are generally two mechanisms of transfer of surfactant tangentially to the membrane. First, the main convective flux 2v_LΓ, and second, a smaller diffusive flux due to ΔΓ. The latter is given approximately by DΔΓ/L; the ratio between the two fluxes is DΔΓ/2v_LΓL ≈ 10^–5. That is, the diffusive flux is negligible compared with the convective, which confirm the applicability of the approximations made in our derivation. A more intricate question is whether a bilayer can be modeled as a continual Newtonian liquid, and what precisely is the meaning of the quantity η^M. To avoid complications, we simply consider η^M to be an effective characteristic of the adhesion force between the two bilayers of the membrane. According to Amontons’ laws of friction, the larger this adhesion, the harder the slippage between the two monolayers and the higher the effective viscosity η^M.

The dissipation rate (eq 50) in the membrane determines the creep velocity; depending on the driving force, we can derive various expressions for v_L, analogously to those in Section 3.1. For example, for adhesion-driven creep where the power of the driving force is given by eq 17, from the balance W_diss + W_adh = 0 we get

55

Inserting the values η^M = 0.01 Pa·s, L = 10 μm, h^M = 4 nm, and v_L = 10 nm/s, we obtain σ_adh = 0.001 mJ/m². Compared with our Laplace-type calculations that give³ for the adhesion energy σ_adh = 1 mJ/m², this value is small. It is, however, in acceptable agreement with the estimated order of the adhesion energy in Section 2.3.

Equation 55 is easily generalized to the case of conical capillary by analogy to the derivation of eq 22

56

Formally, it differs from eq 22 for dissipation occurring in the seal only with the factor of 1/4. The same factor appears in the expression for the pressure-driven creep velocity

57

The final element of the considered mechanism of creep motion is the flip-flop transfer of the lipid molecules from the inner monolayer of the membrane to the outer one in the region of the patch dome. The influx of lipid through the inner membrane is Q^M = 2πR_ch^Mv_L. Half of it, πR_ch^Mv_L, must be transferred to the outer wall. The driving force for this transfer is the strain of the outer monolayer. In the initial period of the creep, the outer wall of the patch accumulates shortage of lipid, resulting in increased interfacial tension. The difference Δσ_⊥^M in the tension of the two monolayers is inducing a flip-flop transfer. Assuming that the flip-flop flux is linear function of Δσ_⊥, we can write

58

where j_flip-flop [m/s] is the transverse volumic flux of surfactant across the membrane, ΔΓ_⊥ is the respective difference of the surface concentrations, and Res_⊥ is the friction coefficient for flip-flop motion. The total flip-flop flux must be

59

Assuming for simplicity that the dome is a hemisphere (which is not always correct⁴⁷) so that A_patch = 2πR_c², we get for the flux, j_flip-flop , the value h^Mv_L/2R_c ≈ 2 × 10^–11 m/s. If the molecular volume is V_m ≈ 1 nm³ and the area per molecule is 1/Γ ≈ 1 nm², then a lipid molecule at the inner monolayer of the dome flip-flops once in τ = V_mΓ/j_flip-flop ≈ 100 s on the average.

Unfortunately, we cannot yet estimate the values of the driving force Δσ_⊥^M and the resistivity Res_⊥, because both are unknown. The knowledge of the value of both parameters is required for estimating the local and the total dissipation rates, w_flip-flop and W_flip-flop, related to flip-flopping process; these are given by the expressions

60

where we used eqs 58 and 59 and the relation A_patch = 2πR_c² (valid for hemispherical dome only) as well as the assumption that the adsorptions, Γ, in the inner and the outer monolayers are not too different (ΔΓ_⊥ ≪ Γ). If Res_⊥ (and, respectively, Δσ_⊥^M) is very large, the order of W_flip-flop will be comparable to W_diss so that the dissipation due to flip-flopping must be accounted for in the energetic balance. W_flip-flop is smaller than W_diss^M when Δσ_⊥^M < 8ΓV_mη^MLv_L/(h^M)²; for the expected order of the magnitude of the involved parameters, this inequality yields Δσ_⊥^M < 0.1 mN/m. The following argument can be given for the validity of this inequality. If Δσ_⊥^M ≈ 0.1 mN/m, it will be comparable to the order of magnitude of the adhesion energy; in such case, the adhesion will probably be unable to hold the outer monolayer against the Marangoni effect and it will creep until Δσ_⊥^M is relaxed, that is, until Δσ_⊥^M ≪ σ_adh. Nevertheless, from the inequality above, it follows that it is possible that at certain geometry of the patch (small L, large R_c) W_flip-flop and W_diss are comparable, which can be used for the experimental determination of the flip-flop resistance coefficient, Res_⊥. As a simple example, in the limiting case where W_flip-flop ≫ W_diss^M, the adhesion-driven creep velocity will be determined by the balance between W_flip-flop and W_adh, eq 17, which yields

61

Thus, in the case where the flip-flop controls the dissipation and the dome remains a hemisphere during creep, the adhesion-driven creep velocity is a constant reversely proportional to Res_⊥. An additional point here is that flip-flopping dissipation may be stronger in the excised patch configuration. It was recently reported⁴⁷ that the difference Δσ_⊥^M between the tensions of the two monolayers of a liposome patch is significantly increased in the excised patch configuration (up to 30% of the monolayer tension!) compared with the cell-attached configuration. Flip-flopping may allow Δσ_⊥^M to relax to a much lower value; nevertheless, the result is indicating that lipid-flipping should occur at a higher rate and with higher dissipation in excised patches. The question for the flip-flopping dissipation will be studied in more detail in future.

4. Comparison with Experiment

We analyzed data for pressure-driven creep of several patches of different composition based on liposomes made of azolectin: (i) pure azolectin liposomes (Azo 100%); (ii) azolectin liposomes made of 70 wt % azolectin and 30% cholesterol (Azo 70%+Chol 30%); (iii) azolectin liposomes with incorporated mechanosensitive channels of small conductivity (MscS), in weight ratio 1:100 azolectin/protein; and (iv) azolectin liposomes with incorporated two types of mechanosensitive channels of small and of large conductance (MscL+MscS) in weight ratio 1:10:1000 MscL/MscS/Azo. The bath and pipet recording solution consisted of 200 mM KCl, 40 mM MgCl₂, and 5 mM Hepes (pH 7.2 adjusted with KOH). WT-MscL-GST and WT-MscS-His6 were prepared according to published procedures.^48,49 MscS or MscS and MscL were incorporated into liposomes using either a dehydration/rehydration⁴⁸ (D/R) or sucrose⁵⁰ reconstitution method (Supporting Information S7).

Pipettes used were cones with L₀ = 10 μm and α = 10° (cf. Figure 4). All measurements were done at 24–26 °C. Details of the experimental procedure are given in the Supporting Information S7 and in ref (44). During the creep experiment, the suction pressure Δp was increased linearly with time, Δp = p_tt, until the lytic pressure was reached, at which the membrane broke (cf. the supporting information of ref (44)). Pressure Δp and displacement (defined as ΔL = L(t) – L(0), where L(0) was on the order of 5–15 μm) were monitored as a function of time (cf. Supporting Information S7, Figure S3).

In the initial period of the experiment, creep motion was accompanied by bloating of the dome as new intracellular fluid and lipid were drawn into the dome (cf. Supporting Information S7, Figure S4). We take into account only data where the dome has relaxed to a stationary shape. From the data for the displacement, ΔL, as a function of time, the creep velocity was determined through numerical differentiation using the quadratic interpolating polynomial formula

62

where indices i–1, i, and i+1 refer to three subsequent measurements.

We first assumed that the dissipation is concentrated in the seal region. We represented eq 28 in the form

63

On the right-hand side of this equation, there are only known quantities; this allows the determination of η^S/h^S. Ideally, the calculated ratio η^S/h^S should be independent of time and pressure. However, because of the dome bulging in the initial moments of the experiment, η^S/h^S is an apparent function of Δp and it relaxes to a constant value only after a relaxation time of several seconds, as illustrated in Figure 6 and in the Supporting Information S7. We calculated the relaxed value η^S/h^S for all four membranes (Azo 100%, Azo 70%+Chol 30%, MscS, and MscS+MscL) using several runs for each membrane. The deviation from the average is reasonably small. The results are given in Table 2. Assuming first that the seal thickness corresponds to a common black film in equilibrium, h^S ≈ 7 nm (cf. Section 2.3), we obtain a viscosity on the order of 10–20 Pa·s, which is four orders of magnitude higher than the viscosity of water. Such a high value in a relatively thick film is hard to explain. Viscosity of 10–20 Pa·s will yield also too high resistivity; from eq 13, one can estimate that Res would be on the order of 2000–4000 GΩ, which is incompatible with the experiment.

Figure 6.

Ratio η^S/h^S, calculated from eq 63 versus Δp. In the initial period of the experiment, the apparent value of η^S/h^S decreases steeply until it reaches a constant value, due to the bulging of the dome (cf. also the Supporting Information S7). Data for a liposome with incorporated MscS.

Table 2. Value of Ratios η^S/h^S and η^M/h^M Calculated from Pressure-Driven Creep Velocity Data for Four Different Compositions of the Membrane.

	ηS/hS [Pa·s/nm]	sd. dev. [Pa·s/nm]	ηM/hM [Pa·s/nm]	ηM [Pa·s]a	lytic pressure [kPa]b
Azo 100%	3.36	0.73	0.84	3.36	32.6
Azo 70%+Chol 30%	2.26	0.29	0.57	2.26	24.9
MscS	1.06	0.19	0.27	1.06	18.5
MscS+MscL	1.96	0.12	0.49	1.96	18.5

^aEffective viscosity of the membrane is calculated by using the value of the membrane thickness h^M = 4 nm.

^bData from ref (44).; results for η^M are correlated to the lytic pressure in Figure 7.

We therefore turn to the other possible mechanism, where the creep motion occurs through flow in the membrane so that eq 57 is correct. Obviously, the result for η^M/h^M is precisely four times smaller than the one for η^S/h^S from eq 63. Because the thickness of the membrane is known,∼4 nm, we can calculate the effective membrane viscosity (Table 2). The results ranges from 1 to 3 Pa·s, which is about two orders higher than the value predicted through Stokes–Einstein relationship from the diffusive coefficient (0.01 Pa·s). This may be partly due to the fact that the geometrical factor 6π in the equation of Stokes–Einstein is smaller for 2-D diffusion.³⁷ (If D^M = k_BT/gη^MR in the membrane and D = k_BT/6πηR in water, we obtain η^M ≈ 6πηD/gD^M; if g is smaller than 6π, then η^M > 0.01 Pa·s.) In general, one must not expect that our “effective viscosity”, η^M, which is a measure of the friction between the two monolayers upon slip, has a value similar to the viscosity following from the equation of Stokes–Einstein for the lateral diffusion in the membrane.

The membrane viscosity obtained in Table 2 shows an interesting and expected correlation to the lytic pressure; the more “viscous” the membrane (more precisely, the larger the adhesion between the two monolayers; cf. the discussion below eq 54), the larger the lytic pressure (i.e., the membrane is more robust). The correlation is illustrated in Figure 7. The data demonstrate that the addition of cholesterol to the azolectin liposome makes it less robust, decreasing both lytic pressure and η^M. The presence of proteins has a more complex effect. The addition of MscS has a similar but stronger effect compared with cholesterol: the membrane containing MscS is of decreased lytic pressure and is even more mobile than Azo 70%+Chol 30%. However, the addition of MscL to a bilayer already containing MscS does not change the lytic pressure, yet it decreases the mobility of the membrane. The contrasting effects of the two proteins on η^M can be explained with their structure. There is a prevalence⁵¹ of polar and positively charged amino groups in the loop region of MscL, which are probably interacting with the negatively charged glass wall. Therefore, MscL remains electrostatically attached to the glass and serves as an obstacle for the slippage between the two monolayers of the membrane, slowing the creep (without affecting significantly lytic pressure). On the opposite, the relevant amino acid residues of MscS are polar and negatively charged,⁵² so that it is repelled by the glass; thus, this protein unsticks the membrane from the glass, making the patch more mobile.

Figure 7.

Correlation between membrane viscosities calculated from pressure-driven creep velocities and lytic pressures of the bilayers in Table 2.

The knowledge of the friction coefficient η^S/h^S (or equivalently, η^M/h^M) allows us to determine the adhesion energy from data for the adhesion driven creep from the supporting information of ref (44). We use eq 23 in the form

64

where t₀ is added because the initial moment of contact between the pipet and the cell is unknown. The creep data for MscS+MscL are given in Figure 8 in the appropriate coordinates, t versus (1 + L/L₀)² ln(1 + L/L₀) – L²/2L₀² – L/L₀. The line is a fit, and from its slope we calculated the quantity η^S/h^Sσ_adh = 1.20 × 10¹³ s/m². Since η^S/h^S is already known from the pressure-driven creep data, η^S/h^S = 1.96 Pa·s/nm (cf. Table 2); from the known value of η^S/h^Sσ_adh, we can determine the adhesion energy: σ_adh = 0.16 ± 0.02 mJ/m². The same result will be obtained if the flow is in the membrane, because changing η^S/h^S to 4η^M/h^M does not alter the final result for σ_adh. This is due to the fact that we determine σ_adh by comparing the creep velocities of adhesion-driven and pressure-driven creep. The nature of the friction force is unimportant for this comparison as far as it is the same for both driving forces. The obtained value for σ_adh is in good agreement with the estimation of the adhesion energy of the seal when it is a Newton black film (cf. Table 1).

Figure 8.

Adhesion-driven creep: the dependence of time t versus (1 + L/L₀)² ln(1 + L/L₀) – L²/2L₀² – L/L₀, cf. eq 64. From the slope, the adhesion energy was determined: σ_adh = 0.16 mJ/m². The spontaneous creeping of the MscS+MscL patch is evident in the images on the left (which are adapted and modified from ref (44)).

5. Conclusions

The two possible mechanisms of creep motion depend on whether the seal forms a Newton black film (i.e., membrane is in molecular contact with the glass) or it remains electrostatically stabilized common black film.

The assumption for common black film yields: (i) Seal of large thickness (7 nm) and very low adhesion energy (0.5 μJ/m²) according to DLVO theory, Table 1. (ii) The large seal thickness goes with resistance 0.2 GΩ (Table 1), much lower than the experimental one. (iii) The mechanism of creep motion occurs through dissipation in the seal zone. The thick seal film provides little friction, and the calculated creep velocity is larger than the experimentally observed value.

The assumption for a Newton black film (membrane sticks to the glass) yields: (i) Seal is a few angstroms thick (∼5 Å) and adhesion energy is on the order of 0.5 mJ/m², Table 1. (ii) The small h^S brings high resistance (1.2–12 GΩ, Table 1), in agreement with the experimentally observed one. (iii) The mechanism of creep motion occurs through dissipation inside the membrane and involves Marangoni flow at the inner monolayer of the membrane and flip-flop transfer of lipid at the patch dome. The creep velocity is determined by the effective viscosity η^M of the membrane (which is, in fact, a characteristic of the strength of adhesion between the two monolayers).

The analysis of the creep data points to the fact that the second hypothesis is far more plausible. The creep data allow us to determine the effective viscosity η^M (∼1 Pa·s, as it follows from pressure-driven creep data) and the adhesion energy of the bilayer to the glass (σ_adh ≈ 0.2 mJ/m², as it follows from the adhesion-driven creep data) for azolectin-based liposomes. In general, the values of the adhesion energy reported in the literature vary by many orders of magnitude (Table 1), which may reflect neglected dynamic effects at the dome rim (e.g., Young balance at the dome rim may involve significant dynamic contributions⁴⁰), finite thickness, or curvature effects. The value 0.2 mJ/m² obtained by our method from creep data compares well with the theoretical estimation for the van der Waals energy of a Newton black film.

Supporting Information Available

S1. List of symbols and abbreviations. S2. Electrostatics of the seal zone. S3. Effect of surface conductivity on seal resistivity. S4. Ratio between van der Waals attractive energy and the energy of the repulsive interactions in equilibrium thin films. S5. Adhesion-driven creep in conical capillary. S6. Pressure-driven creep rate in conical capillary. S7. Experimental data for pressure-driven creep. This material is available free of charge via the Internet at http://pubs.acs.org.

The authors declare no competing financial interest.

Funding Statement

National Institutes of Health, United States