Multilamellarity, structure and hydration of extruded POPC vesicles by SANS

Abstract

The small-angle neutron scattering (SANS) data of 12 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) dispersions at low lipid concentration (1 mg per 100-mg heavy water) prepared by 5, 9 and 29 extrusions through filters of pores with 50, 100, 200 and 400 nm diameter are presented. They were analyzed within a theory that permits the determination of both structural and hydration parameters of the bilayers as well as the portions of multilamellar vesicles in dispersions with negligible long-range order between the vesicles. The scattering length density profile across the bilayers is approximated by assuming a central hydrocarbon core surrounded by a water-accessible coat. It is modeled by two different forms of functions. In the boat model, the scattering length density of the coat changes linearly from core to water, whereas in the strip model it is constant across the water-accessible coat. It was found that the boat model reflects the reality better than the strip model. The decrease of the multilamellar vesicle portions, either with increasing the number of extrusions at same filter size and with decreasing the filter size, was characterized quantitatively.

Appendices

Appendix 1: Derivation of Eq (1)

The neutron coherent scattering amplitude of the M-vesicle depicted in Fig. 1 is given by

$$F_{M} (q) = {\sum\limits_{m = 0}^{M - 1} {F_{m} (q)} },$$

(7)

where

$$F_m \left( q \right) = \frac{{4\pi }} {q} \cdot \int\limits_{\left( {d_L /2} \right)}^{d_{\rm L}/2} {\left( {r_m + x} \right) \cdot \sin \left[ {q \cdot \left( {r_m + x} \right)} \right] \cdot \rho \left( x \right){\rm d}x} = \frac{{4\pi }} {q} \cdot \left( {\frac{{ - \partial }} {{\partial q}}} \right)\int\limits_{\left( {d_{\rm L}/2} \right)}^{d_{\rm L}/2} {\cos \left[ {q \cdot \left( {r_m + x} \right)} \right] \cdot \rho \left( x \right){\text{d}}x} $$

(8)

is the form factor of the m-th bilayer in the vesicle. r _m =r ₀+m·d is the midpoint radius of that bilayer, and the scattering length density difference between the membrane and the solvent is described by ρ(x) at point x. As the bilayers in Fig. 1 are assumed to be symmetric, i.e.,

$$\rho (- x) = \rho (x), $$

(9)

we obtain for the integral in (8)

$$\begin{aligned} F_m \left( q \right) = \frac{{4\pi }} {q} \cdot \int\limits_{\left( {d_L /2} \right)}^{d_{\rm L}/2} {\left( {r_m + x} \right) \cdot \sin \left[ {q \cdot \left( {r_m + x} \right)} \right] \cdot \rho \left( x \right){\rm d}x} = \frac{{4\pi }} {q} \cdot \left( {\frac{{ - \partial }} {{\partial q}}} \right)\int\limits_{\left( {d_{\rm L} /2} \right)}^{d_L /2} {\cos \left[ {q \cdot \left( {r_m + x} \right)} \right] \cdot \rho \left( x \right){\text{d}}x} \\ \int\limits_{\left( {d_L /2} \right)}^{d_L /2} {\cos \left[ {q \cdot \left( {r_m + x} \right)} \right] \cdot \rho \left( x \right){\text{d}}x} = \cos \left( {qr_m } \right) \cdot \rho \left( q \right) \\ \end{aligned} $$

(10)

with

$$\rho \left( q \right) = \rho ^{{\text{strip}}} \left( q \right) = \frac{2} {q} \cdot \left[ {\left( {\rho _h - \rho _s } \right) \cdot \sin \left( {q \cdot \frac{{d_{\rm L} }} {2}} \right) + \left( {\rho _c - \rho _h } \right) \cdot \sin \left( {q \cdot \frac{c} {2}} \right)} \right] $$

(11a)

and

$$\rho \left( q \right) = \rho ^{{\text{boat}}} \left( q \right) = \frac{2} {q} \cdot \left( {\rho _c - \rho _s } \right) \cdot \left\{ {\sin \left( {q \cdot \frac{{d_{\rm L} }} {2}} \right) \cdot \frac{{\sin \left( {q \cdot h} \right)}} {{q \cdot h}} + \cos \left( {q \cdot \frac{{d_{\rm L} }} {2}} \right) \cdot \frac{{\left( {1 - \cos \left( {q \cdot \frac{{d_{\rm L} }} {2}} \right)} \right)}} {{q \cdot h}}} \right\}$$

(11b)

in case of the strip and the boat model, respectively.

The summation in (7) can be performed using the formula

$$\sum\limits_{m = 0}^{M - 1} {\cos \left( {qr_m } \right) = \cos \left[ {q \cdot \left( {r_0 + \left( {M - 1} \right) \cdot \frac{d} {2}} \right)} \right]} \cdot \frac{{\sin \left( {\frac{{qMd}} {2}} \right)}} {{\sin \frac{{qd}} {2}}}$$

(12)

giving

$$F_m \left( q \right) = \left( {\frac{{4\pi }} {q}} \right) \cdot \left( {\frac{{ - \partial }} {{\partial q}}} \right)\left\{ {\cos \left( {r_{0M} } \right) \cdot f_M \left( q \right)} \right\}$$

(13)

Here $r_{{0M}} = r_{0} + (M - 1) \cdot \frac{d}{2}$ is the radius of the vesicle consisting of M bilayers with r ₀ as the midpoint radius of the innermost bilayer in the vesicle, and

$$f_{M} (q) = \rho (q) \cdot \frac{{\sin {\left( {\frac{{qMd}}{2}} \right)}}}{{\sin {\left( {\frac{{qd}}{2}} \right)}}}.$$

The average coherent cross section of the M-vesicles is given by the averaged squared absolute value of the scattering amplitude:

$$ I^{{{\text{coh}}}}_{M} {\left( q \right)} = {\left\langle {{\left| {F_{M} {\left( q \right)}} \right|}^{2} } \right\rangle } = {\left( {\frac{{4\pi }} {q}} \right)} \cdot {\left\langle {{\left[ {r_{{0M}} \cdot \sin {\left( {qr_{{0M}} } \right)} \cdot f_{M} {\left( q \right)} + \cos {\left( {qr_{{0M}} } \right)} \cdot {\left( {\frac{{ - \partial }} {{\partial q}}} \right)}f_{M} {\left( q \right)}} \right]}^{2} } \right\rangle } $$

(14)

The second term in the brackets can be neglected with respect to the first one in case of $r_{{0M}} \gg d,$ giving (1).

Appendix 2: Averaging over vesicle-radius distribution

Using the integral (Gradshtein and Ryzhik 1971)

$$\int\limits_0^\infty {x^{\mu - 1} e^{ - \beta \cdot x} \cos \left( {\delta \cdot x} \right){\rm d}x = \frac{{\Gamma \left( \mu \right)}} {{\left( {\delta ^2 + \beta ^2 } \right)^{\frac{\mu } {2}} }} \cdot \cos \left( {\mu \cdot {\text{arcton}}\left( {\frac{\delta } {\beta }} \right)} \right)} $$

the averaging over the radius distribution in (1) can be performed readily giving

$$\left\langle {r_{0M} ^2 \cdot \sin ^2 \left( {q \cdot r_{0M} } \right)} \right\rangle = \frac{5} {8}R^2 \cdot \left\{ {1 + \xi ^3 \cdot \left[ {1 - 18 \cdot \xi + 48 \cdot \xi ^2 - 32 \cdot \xi ^3 } \right]} \right\}$$

with

$$\xi = \frac{1}{{1 + {\left( {\frac{{q \cdot R}}{2}} \right)}^{2} }}$$