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Permeability and the hidden area of lipid bilayers

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Abstract

The passive water permeability of a lipid vesicle membrane was studied, related to the hydrostatic (not osmotic) pressure difference between the inner and the outer side of the vesicle in a water environment without additives. Each pressure difference was created by sucking a vesicle into a micropipette at a given sucking pressure. The part of the membrane sucked into the micropipette (the projection length) was measured as a function of time. The time dependence can be divided into two intervals. We put forward the idea that smoothing of membrane defects, accompanied by an increase of the membrane area, takes place during the initial time interval, which results in a faster increase of the projection length. In the second time interval the volume of the vesicle decreases due to the permeability of its membrane and the increase of the projection length is slower. The hidden area and the water permeability of a typical lipid bilayer were estimated. The measured permeability, conjugated to the hydrostatic pressure difference, is an order of magnitude higher than the known value of the permeability, conjugated to the osmotic pressure difference. A hypothesis, based on pore formation, is proposed as an explanation of this experimental result.

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Acknowledgements

The authors wish to express their gratitude to Dr M.D. Mitov for fruitful discussions. This work was carried out in the French-Bulgarian Laboratory “Vesicles and Membranes”, supported by CNRS (France) and the Bulgarian Academy of Sciences and Sofia University (Bulgaria). The contribution of the Bulgarian National Science Foundation (contract F823) is acknowledged.

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Appendix

Appendix

Appendix A: thermodynamics and definition of the passive permeability

Let us consider the system described in Materials and methods and consisting of a membrane dividing two solutions with different solute concentrations. In addition to the quantities, defined in Materials and methods, we introduce the following ones. The chemical potentials of the solvent and the solute on the first side of the membrane are \( \mu _0^1 \) and \( \mu _1^1 \), respectively, and \( \mu _0^2 \) and \( \mu _1^2 \) are the same potentials on the second side of the membrane. The molar volume of the solute is denoted by v 1. The quantity c is defined by the relation c=(c 1+c 2)/2. When the condition Δc<<c is fulfilled, the chemical potential differences \( \Delta \mu _0 = \mu _0^2 - \mu _0^1 \) and \( \Delta \mu _1 = \mu _1^2 - \mu _1^1 \) can be expressed by means of the quantities introduced above and in Materials and methods as follows (Kedem and Katchalsky 1963):

$$ \begin{array}{*{20}c} {\Delta \mu _0 = \left( {\Delta P - \Delta \pi } \right)v_0 } \hfill \\ {\Delta \mu _1 = \left( {\Delta P - \Delta \pi } \right)v_1 + \frac{1} {c}\Delta \pi } \hfill \\ \end{array} $$
(A1)

Let j 0 be the molar flow of solvent molecules and j 1 be the molar flow of solute molecules through the membrane. The volume flow \( J_0^{\text{v}} \) through the membrane (the volume flow of both the solvent and the solute) is \( J_0^{\text{v}} = j_0 v_0 + j_1 v_1 \).

Expressing Δµ 0 and Δµ 1 via ΔP and Δπ, and \( J_0^{\text{v}} \) and j 1 as linear combinations of ΔP and Δπ, one obtains (Kedem and Katchalsky 1963):

$$ \begin{array}{*{20}c} {J_0^{\text{v}} = - \left[ {L_{11} \left( {\Delta P - \Delta \pi } \right) + L_{12} \frac{{\Delta \pi }} {c}} \right]} \hfill \\ {j_1 = - \left[ {L_{21} \left( {\Delta P - \Delta \pi } \right) + L_{22} \frac{{\Delta \pi }} {c}} \right]} \hfill \\ \end{array} $$
(A2)

where L 12=L 21 according to Onsager’s relations.

When Δπ=0 (i.e. Δc=0), the flow j 1 must be proportional to c and ΔP, j 1∝cΔP. This relation imposes the following dependence of L 12 on c: L 12=L 21=cL 0. When ΔP=0, j 1 must depend linearly on c 1 and c 2, and consequently \( L_{22} = cL_{22}^0 \). Keeping the first powers with respect to c 1 and c 2, we obtain the well-known practical Kedem–Katchalsky equations in the following form:

$$ \begin{array}{*{20}c} {J_0^{\text{v}} = - \left[ {L_{11} \left( {\Delta P - \Delta \pi } \right) + L^0 \Delta \pi } \right]} \hfill \\ {j_1 = - \left( {cL^0 \Delta P + L_{22}^0 \Delta \pi } \right)} \hfill \\ \end{array} $$
(A3)

Denoting D ΔP =L 11 and D Δπ =L 11=L 0, from the first of Eq. (A3) we obtain Eq. (1).

It is usually considered in the scientific literature that the volume flow through the membrane is created by the partial pressure difference Δπ of the solute, which is due to the solute concentration difference Δc between the two sides of the membrane. This approach implicitly assumes that ΔP=0. Hence:

$$ J_0^{\text{v}} = \left( {L_{11} - L^0 } \right)\Delta \pi = D_{\Delta \pi } \Delta \pi $$
(A4)

For low enough c, \( j_0 \approx {{J_0^{\text{v}} } \mathord{\left/ {\vphantom {{J_0^{\text{v}} } {v_0 }}} \right. \kern-\nulldelimiterspace} {v_0 }} \). Taking into account the relation between Δc and Δπ, we obtain:

$$ j_0 = D_{\Delta \pi } v_0 RT\Delta c $$
(A5)

The permeability D Δc is frequently defined as:

$$ j_0 = D_{\Delta c} \Delta c $$
(A6)

Substituting Eq. (A6) in Eq. (A5), Eq. (3) is derived.

Appendix B: deduction of Eq. (9)

Let S and V be the area and the volume of the vesicle sucked into the micropipette. They are expressed by the radii R v and R p of the vesicle and of the micropipette in the following way:

$$ S = 2\pi \left( {R_{\text{v}} } \right)^2 \left[ {1 + \sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} } \right] + 2\pi R_{\text{p}} L + 2\pi \left( {R_{\text{p}} } \right)^2 $$
(B1)

and:

$$ V = \frac{2} {3}\pi \left( {R_{\text{v}} } \right)^3 + \frac{2} {3}\pi \left( {R_{\text{v}} } \right)^2 \left( {R_{\text{v}} + R_{\text{p}} } \right)\sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} + \pi \left( {R_{\text{p}} } \right)^2 L + \frac{2} {3}\pi \left( {R_{\text{p}} } \right)^3 $$
(B2)

Differentiating the above expressions we obtain:

$$ {\text{d}}S = 4\pi R_{\text{v}} \left[ {1 + \sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} - \frac{1} {2}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}\frac{1} {{\sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} }}} \right]{\text{d}}R_{\text{v}} + 2\pi R_{\text{p}} \,{\text{d}}L $$
(B3)

and:

$$ {\text{d}}V = 2\pi \left( {R_{\text{v}} } \right)^2 \left\{ {1 + \left[ {1 + \frac{1} {6}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} \right]\sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} - \frac{{\frac{1} {3}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}\left[ {1 + \frac{1} {2}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} \right]}} {{\sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} }}} \right\}{\text{d}}R_{\text{v}} + \pi \left( {R_{\text{p}} } \right)^2 {\text{d}}L $$
(B4)

The differential dR v can be eliminated from the above two equations and the result is:

$$ \frac{{{\text{d}}S - 2\pi R_{\text{p}} \,{\text{d}}L}} {{{\text{d}}V - \pi \left( {R_{\text{p}} } \right)^2 {\text{d}}L}} = \frac{1} {{R_{\text{v}} }}\phi \left( {\frac{{R_{\text{p}} }} {{R_{\text{v}} }}} \right) $$
(B5)

where the function \( \phi \left( {{{R_{\text{p}} } \mathord{\left/ {\vphantom {{R_{\text{p}} } {R_{\text{v}} }}} \right. \kern-\nulldelimiterspace} {R_{\text{v}} }}} \right) \) is defined by the expression:

$$ \phi \left( {\frac{{R_{\text{p}} }} {{R_{\text{v}} }}} \right) = \frac{{1 + \sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} - \frac{1} {2}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}\frac{1} {{\sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} }}}} {{\frac{1} {2} + \frac{1} {2}\left[ {1 + \frac{1} {6}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} \right]\sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} - \frac{1} {6}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}\left[ {1 + \frac{1} {2}\frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} \right]\frac{1} {{\sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} }}}} $$
(B6)

The Taylor series of the function φ(x) is:

$$ \phi \left( x \right) = 2 - \frac{1} {3}x^2 + \frac{1} {{18}}x^4 + O\left( {x^6 } \right) $$
(B7)

From Eqs. (B5) and (B6) the following relation between the time derivatives ∂L/∂t, ∂S/∂t, and ∂V/∂t is deduced:

$$ \frac{{\partial L}} {{\partial t}} = \frac{1} {{2\pi R_{\text{p}} \left[ {1 - \frac{1} {2}\frac{{R_{\text{p}} }} {{R_{\text{v}} }}\phi \left( {\frac{{R_{\text{p}} }} {{R_{\text{v}} }}} \right)} \right]}}\frac{{\partial S}} {{\partial t}} - \frac{{\phi \left( {\frac{{R_{\text{p}} }} {{R_{\text{v}} }}} \right)}} {{2\pi R_{\text{v}} R_{\text{p}} \left[ {1 - \frac{1} {2}\frac{{R_{\text{p}} }} {{R_{\text{v}} }}\phi \left( {\frac{{R_{\text{p}} }} {{R_{\text{v}} }}} \right)} \right]}}\frac{{\partial V}} {{\partial t}} $$
(B8)

The time derivative ∂V/∂t is related to the volume flow through the vesicle membrane. We denote with j out=D ΔP (p′−p out) the flow from the interior of the vesicle to the experimental cell through that part of the membrane which is out of the micropipette, and with S out the area of the same part of the membrane. Let j in=D ΔP (p′−p in) be the volume flow from the vesicle to the micropipette through that part of the membrane inside the micropipette, which is not in contact with the internal surface of the micropipette, and let S in be the area of that part of the membrane. Let Q out=j out S out and Q in=j in S in be the respective total volume flows. We will show first that the total volume flow through the cylindrical part of the membrane, which is in close contact with the micropipette, is negligible. For this aim, we will assume that the water layer between the membrane and the micropipette has a constant thickness d, determined by the interactions between the membrane and the inner surface of the micropipette. The film is enclosed between two cylindrical surfaces with radii much larger than the film thickness and we will consider it as flat in the calculations. Let x be the distance between a point in the water layer and the plane perpendicular to the axis of the micropipette and situated at its end (the distance L in Fig. 1 is measured exactly towards this plane). Evidently, 0≤xL. Let p(x) be the distribution of the hydrostatic pressure in the water film. Let q(x) be the volume flow of the water at a distance x per unit length of the perimeter of the cylindrical part of the membrane (the flow is parallel to the axis of the micropipette). Let Q cyl(x)=2πR p q(x) be the total volume flow of the water in the film at this distance x. To facilitate the calculations, first we consider the case when all the flow is directed to the interior of the micropipette. From the laminar viscous flow of a liquid between two planes it is known that the relation between q(x) and p(x) is:

$$ q\left( x \right) = - \frac{{d^3 }} {{12\eta }}\frac{{\partial p\left( x \right)}} {{\partial x}} $$
(B9)

On the other side, the change dq(x) is related to the permeability of the membrane:

$$ {\text{d}}\left[ {q\left( x \right)} \right] = D_{\Delta P} \left[ {p' - p\left( x \right)} \right]{\text{d}}x $$
(B10)

where p′ is the hydrostatic pressure inside the vesicle. From Eqs. (B9) and (B10) the following differential equation is obtained for p(x):

$$ \frac{{\partial ^2 p\left( x \right)}} {{\partial x^2 }} - \frac{{12\eta D_{\Delta P} }} {{d^3 }}\left[ {p\left( x \right) - p'} \right] = 0 $$
(B11)

The boundary conditions of the above equation are: p(L)=p in, and \( \left[ {{{\partial p\left( x \right)} \mathord{\left/ {\vphantom {{\partial p\left( x \right)} {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} \right]_{x = 0} = 0 \). The former is evident, and the latter is a consequence of the fact that q(0)=0. We denote with α the length: \( \alpha = \sqrt {d^3 /\left( {12\eta D_{\Delta P} } \right)} \). Then the solution of the differential equation of Eq. (B11) satisfying these boundary conditions is:

$$ p\left( x \right) = p' - \frac{{\exp \left( { - x/\alpha } \right) + \exp \left( {x/\alpha } \right)}} {{\exp \left( { - L/\alpha } \right) + \exp \left( {L/\alpha } \right)}}\left( {p' - p^{{\text{in}}} } \right) $$
(B12)

From this solution it follows that:

$$ - \left[ {\partial p\left( x \right)/\partial x} \right]_{x = L} = \tanh \left( {L/\alpha } \right)\left( {p' - p^{{\text{in}}} } \right)/\alpha < \left( {p' - p^{{\text{in}}} } \right)/\alpha $$
(B13)

Evidently, Q cyl(L)=2πR p q(L) is the total volume flow through the cylindrical part of the membrane. From Eqs. (B9) and (B13) it follows that:

$$ Q^{{\text{cyl}}} \left( L \right) < \frac{{d^3 }} {{12\eta D_{\Delta P} \alpha R_{\text{p}} }}2\pi \left( {R_{\text{p}} } \right)^2 \left( {p' - p^{{\text{in}}} } \right) = \frac{\alpha } {{R_{\text{p}} }}Q^{{\text{in}}} $$
(B14)

From the expressions for j out and j in and from Eqs. (5) it follows that Q out/Q in≈2R v/R p. Then the relative part of Q cyl (with respect to the total volume flow through the parts of the membrane not in contact with the micropipette Q out+Q in) is:

$$ \frac{{Q^{{\text{cyl}}} \left( L \right)}} {{Q^{{\text{out}}} + Q^{{\text{in}}} }} < \frac{\alpha } {{2R_{\text{v}} + R_{\text{p}} }} $$
(B15)

The analysis of our experimental data for the permeability due to the hydrostatic pressure difference gave the value D ΔP ≈3×10−11cm3/(dyn s). Substituting η=1 cP (the viscosity of the water in the film is assumed the same as that of bulk water) and d=24 Å [this is the thickness of the water layer in the lamellar L α phase in the system SOPC–water (Rand and Parsegian 1989)], we obtain α=0.62 μm. The radii of the vesicles and the pipettes used in our experiments satisfy the inequalities R v≥6.3 μm and R p≥3.5 μm. Using these minimal values, we obtain that Q cyl(L)/(Q out+Q in)<0.05. In the same way it can be shown that in the case when the total volume flow through the cylindrical part of the membrane is directed to the interior of the experimental cell, it obeys an inequality of the kind in Eq. (B15), but with right-hand side multiplied by R p/R v. The real volume flow through the cylindrical part of the membrane cannot exceed the sum of these two particular cases. Consequently, it is less than 8% of the total volume flow through the whole membrane of the vesicle, considerably less than the experimental precision of the measurements. Later on, this contribution will be neglected. Then the time derivative ∂V/∂t can be written as:

$$ \frac{{\partial V}} {{\partial t}} = - D_{\Delta P} \left[ {S^{{\text{out}}} \left( {p' - p^{{\text{out}}} } \right) + S^{{\text{in}}} \left( {p' - p^{{\text{in}}} } \right)} \right] $$
(B16)

where p out, p′, and p in are the hydrostatic pressures in the experimental cell, inside the vesicle, and inside the micropipette. Evidently:

$$ S^{{\text{out}}} = 2\pi \left( {R_{\text{v}} } \right)^2 \left[ {1 + \sqrt {1 - \frac{{\left( {R_{\text{p}} } \right)^2 }} {{\left( {R_{\text{v}} } \right)^2 }}} } \right] $$
(B17)

and:

$$ S^{{\text{in}}} = 2\pi \left( {R_{\text{p}} } \right)^2 $$
(B18)

The pressure differences (p′p out) and (p′p in) are determined by Eqs. (5).

The permeability D ΔP can be expressed from Eqs. (5), (B8), (B16), (B17), and (B18) as a function of R p, R v, dL/dt, and dS/dt. Taking into account that \( \phi \left( {{{R_{\text{p}} } \mathord{\left/ {\vphantom {{R_{\text{p}} } {R_{\text{v}} }}} \right. \kern-\nulldelimiterspace} {R_{\text{v}} }}} \right) \approx 2 \) (see Eq. B7; this approximation has been used by Olbrich et al. 2000) and replacing \( \sqrt {1 - {{\left( {R_{\text{p}} } \right)^2 } \mathord{\left/ {\vphantom {{\left( {R_{\text{p}} } \right)^2 } {\left( {R_{\text{v}} } \right)^2 }}} \right. \kern-\nulldelimiterspace} {\left( {R_{\text{v}} } \right)^2 }}} \) with \( {\left[ {1 - {{\left( {R_{{\text{p}}} } \right)}^{2} } \mathord{\left/ {\vphantom {{{\left( {R_{{\text{p}}} } \right)}^{2} } {2{\left( {R_{{\text{v}}} } \right)}^{2} }}} \right. \kern-\nulldelimiterspace} {2{\left( {R_{{\text{v}}} } \right)}^{2} }} \right]}, \) we obtain Eq. (9). The error due to these two approximations is of the order of \( {1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-\nulldelimiterspace} 6} \times {{\left( {R_{\text{p}} } \right)^2 } \mathord{\left/ {\vphantom {{\left( {R_{\text{p}} } \right)^2 } {\left( {R_{\text{v}} } \right)^2 }}} \right. \kern-\nulldelimiterspace} {\left( {R_{\text{v}} } \right)^2 }} \approx 0.02 \). It is negligible in comparison to the experimental errors of the measurements.

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Vitkova, V., Genova, J. & Bivas, I. Permeability and the hidden area of lipid bilayers. Eur Biophys J 33, 706–714 (2004). https://doi.org/10.1007/s00249-004-0415-2

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