Effects of DC Magnetic Fields on Magnetoliposomes

The potential use of magnetic nanoparticles (MNPs) in biomedicine as magnetic resonance, drug delivery, imagenology, hyperthermia, biosensors, and biological separation has been studied in different laboratories. One of the challenges on MNP elaboration for biological applications is the size, biocompatibility, heat efficiency, stabilization in physiological conditions, and surface coating. Magnetoliposome (ML), a lipid bilayer of phospholipids encapsulating MNPs, is a system used to reduce toxicity. Encapsulated MNPs can be used as a potential drug and a gene delivery system, and in the presence of magnetic fields, MLs can be accumulated in a target tissue by a strong gradient magnetic field. Here, we present a study of the effects of DC magnetic fields on encapsulated MNPs inside liposomes. Despite their widespread applications in biotechnology and environmental, biomedical, and materials science, the effects of magnetic fields on MLs are unclear. We use a modified coprecipitation method to synthesize superparamagnetic nanoparticles (SNPs) in aqueous solutions. The SNPs are encapsulated inside phospholipid liposomes to study the interaction between phospholipids and SNPs. Material characterization of SNPs reveals round-shaped nanoparticles with an average size of 12 nm, mainly magnetite. MLs were prepared by the rehydration method. After formation, we found two types of MLs: one type is tense with SNPs encapsulated and the other is a floppy vesicle that does not show the presence of SNPs. To study the response of MLs to an applied DC magnetic field, we used a homemade chamber. Digitalized images show encapsulated SNPs assembled in chain formation when a DC magnetic field is applied. When the magnetic field is switched off, it completely disperses SNPs. Floppy MLs deform along the direction of the external applied magnetic field. Solving the relevant magnetostatic equations, we present a theoretical model to explain the ML deformations by analyzing the forces exerted by the magnetic field over the surface of the spheroidal liposome. Tangential magnetic forces acting on the ML surface result in a press force deforming MLs. The type of deformations will depend on the magnetic properties of the mediums inside and outside the MLs. The model predicts a coexistence region of oblate–prolate deformation in the zone where χ = 1. We can understand the chain formation in terms of a dipole–dipole interaction of SNP.

shows the diffractogram of iron oxide nanoparticles synthesized (top). According to peak matching with data from the PDF4 of the ICDD, conventional X-ray diffraction analysis consistently shows that the sample is composed of two phases: magnetite and maghemite. As can be seen, these two phases explain the diffraction pattern of the Iron Oxide Nanoparticles. According to the RIR Quantitative Analysis (calculated automatically by XPert HighScore software), the percentage of magnetite is approximately 63% and for maghemite 37%.

SNP in suspension under DC magnetic Fields
Control experiment to show that SNP align in the direction of the magnetic field without liposomes. We put 6.8 µl of suspended SNP in pure water at 1% and 3% dilution. Figure S2 shows a top view of the chamber experiment where the solutions of diluted SNP. After applying a DC magnetic field of 0.09 T, we observe the formation of linear chains in both concentrations.

THEORY
In order to discuss the shape deformation of the magnetoliposomes under the influence of an external magnetic field, we solved the relevant magnetostatic equations for the magnetic scalar potential. Herewith we present details of the extended derivation of the model. The magnetoliposome is modeled as a spherical thin layer shell where the three zones specifically are identified via different magnetic permeability (see Figure S3). We present a particular case where the inside and outside mediums of the magnetoliposome are constructed with the same material. The general case depicted in Figure S3 can be found elsewhere ?.
MLS is modeled as a spherical thin layer shell of permeability µ 2 placed in a region of the space with permeability µ 1 . The width of the lipid membrane is l m , the radius of the outer and the inner membranes are r 1 and r 2 respectively, and the membrane radii are denoted by r m , see Figure S3. We find the field distribution in all space due to a DC magnetic field in the z direction from a scalar potential H(r, θ, φ) = −∇ϕ. The membrane is considered a lossy dielectric medium and J = 0. Furthermore, using the relation between the magnetic induction and the magnetic field B = µH and the divergence equation ∇ · B = 0, we find that ∇ 2 ϕ = 0, thus the potential φ satisfies the Laplace equation everywhere ?.
The form of the potential in spherical coordinates is well known: where k=1,2 and 3 represents the external, membrane and inner medium respectively and the constants a's and b's are found using the boundary condition that fulfills the continuity of the field. Under an external magnetic field H 0 the boundary conditions are such that H(r → ∞) → H 0 and is defined at r = 0. The continuity of the field is given by: where subscripts j represent the external (j=1) and internal (j=2) membrane surfaces.
where, since µ 1 = µ 3 , we define µ as The solution for constants a's and b's are: a 3 is related to the internal magnetic field and when µ 2 µ 1 and µ 2 µ 1 there will be a magnetic shielding i.e there is a significant decrease on the internal magnetic field. Using H(r, θ, φ) = −∇ϕ, the magnetic field is written in the next form: Densities forces over the membrane surface are generated when a magnetic field is applied, this densities forces are the result of the discontinuity of the field due to the differences of the magnetic permeability in the surfaces interfaces and can be computed using the equation (we follow a similar procedure reported by Yamamoto etal . ?, where density forces over a spherical liposome under an AC electric field are calculated to study vesicle deformations).
where the dot represents the product between a vector and a tensor, n ≡r is the normal vector of the membrane surface and T is the Maxwell Stress Tensor that physically represents the force per unit area it is defined by ?: in this definition i and j are dummy subscripts. To estimate the density forces, f j , we only take into account the magnetic term of the equation S14. The deformation of MLS are due to the normal and tangential components of the density force; the stress tensor components take the form: and The forces given by the equation S13 can deform the otherwise spheroidal MLS. The deformation can be in the direction of the DC magnetic field applied, this type of deformation is called prolate deformation. If the deformation is perpendicular to the DC magnetic field is called oblate deformation. To obtain the type and strength of deformations we calculate the work done over the membrane surface by the DC magnetic field, this is given by where: The components of the unit vector u, that keeps the local area constant are ? : where s is the deformation amplitude. For simplicity we can write the magnetic field as and η(r) k = b k r 3 + a k ; k = 1, 2, 3 we obtain the next expression for the magnetic field work Free Energy of magnetoliposomes in DC Magnetic Fields. The energies contributions that are involved in the membrane deformation are the magnetic and the bending energy, and the stables shapes acquired by the MLS is due to a competition between this two energies in the form of free energy ?: where: where s is the deformation amplitude, κ m is the membrane bending rigidity and M sp spontaneous curvature. The stables shapes acquired by the MLS are determined by minimizing the free energy respect to the parameter of the deformation, s. We obtain The MLS attains prolate-shape when s > 0 whereas if s < 0 the MLS is oblate-shape. Equation S29 can be simplified as follows where the G's terms are only functions of the vesicle geometry, the explicit form of ∆ and G's are:

Figures
Frontiers Figure S1. Chain formations inside MLSWhen MLS are subject to DC magnetic field, we can see that the small black dots (SNPs aggregates, 1 % SNP first row, and second to fourth 3 %) form chains inside the MLS. The last row shows a multilamellar vesicle. Figure S2. XRD diffractograms. X-ray diffractograms of Iron Oxide nanoparticles sample revealing experimental peaks that match with simulated magnetite and maghemite according to PDF4 of the ICDD database.  Figure S4. Model of a magnetoliposome in a DC magnetic field. MLS is immersed in a solution with permeability µ 1 . The direction of the DC is z, the interior of the MLS is filled with a material of magnetic permeability µ 1 , the lipid membrane has magnetic permeability µ 2 . The width of the lipid membrane is l m , the radius of the outer and the inner membranes are r 1 and r 2 respectively, and the membrane radii are denoted by r m .