Chem Phys Lipids. 2016 Dec;201:21-27. doi: 10.1016/j.chemphyslip.2016.10.006.

**Measurement of the bending elastic modulus in unilamellar vesicles membranes by fast field cycling NMR relaxometry. **

**Abstract:**

The elastic properties of lipid membranes can be conveniently characterized through the bending elastic modulus k. Elasticity directly affects the deformability of a membrane, morphological and shape transitions, fusion, lipid-protein interactions, etc. It is also a critical property for the formulation of ultradeformable liposomes, and of interest for the design of theranostic liposomes for efficient drug delivery systems and/or different imaging contrast agents. Measurements of k in liposome membranes have been made using the fast field cycling nuclear magnetic relaxometry technique. We analyze the capability of the technique to provide a consistent value of the measured quantity under certain limiting conditions. Relaxation dispersions were measured acquiring a minimal quantity of points, within a reduced Larmor frequency range and, under inferior experimental conditions (in the presence of magnetic field in-homogeneity and lower power supply stability). A simplified model is discussed, showing practical advantages when fitting the data within the reduced frequency range. Experiments are contrasted with standard measurements performed in a state-of-the-art relaxometer. The methodology was tested in samples of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) with different percentiles of cholesterol. We observe a tendency to a decrease in Îº with increasing temperature, and a tendency to increase with the cholesterol percentile.

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**Supplement: **

Lipid vesicles can be used as idealized model systems of real biomembranes. The bending elastic modulus is a quantity reflecting the amount of energy needed to modify the intrinsic curvature of a bilayer, see figure 1. It determines important biological functions of cells, like cell fusion, lipid–protein interactions and lipid-mediated protein activity.

The effects of sterols (particularly cholesterol) on the membrane flexibility was frequently characterized through the bending elastic modulus. Some applications like ultradeformable liposomes formulations, used as transdermal carriers, need to have critical elastic properties through the addition of selected additives. Today, it is clear that a close relationship exists between the elastic properties of the membrane and a myriad of processes involving both lipids and embebed proteins. How cholesterol modulates the elastic behaviour of the membrane strongly depends upon the saturation of the hidrocarbon chains of the lipids. When lipids have fully saturated chains, like DMPC, cholesterol increases k. However, it does not have mayor effects on monoinsaturated chains. Depending on the concentration, part of the lipids in the membrane will be in a cholesterol-induced ordered state [1].

However, such ordered lipids are not necessarily isolated, they may tend to agglomerate into domains (or “rafts”) in coexistence with a more “fluid” phase [2]. Plenty of questions remain on the lipid dynamics and order, and the connection between these and the mesoscopic behavior of the membrane.

Fast field-cycling (FFC) NMR relaxometry is an NMR technique already used in a series of compounds ranging from solid to liquids, and a large variety of soft materials [3]. The technique belongs to the “time-domain” NMR, since fast-switchable magnets having poor homogeneity (in terms of spectral resolution) are used. Proton relaxation rates obtained from this method are mainly driven by fluctuations of the 1H–1H dipolar couplings. It has been successfully used for the study of multilamellar vesicles and recently applied for the study of lipid molecular dynamics (strongly related with the viscoelastic properties) in large unilamellar vesicles (between 100nm and 1 um) [1, 4, 5]. In this work we particularly concentrate on the *limiting experimental conditions* and *model simplifications* that would allow a systematic measurement of k. It will turn out that k can be measured within a restricted frequency range, using a simplified physical model, from data obtained using a FFC machine having a magnet with a lower homogeneity and a lower magnetic field stability (compared to the current state-of-the-art).

** PHYSICAL MODEL**: In previous works [1, 4, 5] we showed that the spin-lattice relaxation rate dispersion R

_{1}of lipid protons in DMPC liposomes (50 nm radius, no cholesterol content) can be explained (red solid line- Figure 2) in terms of the following dynamical processes:

- Local order fluctuations due to shape fluctuations of the liposome spheroid (OF).
- Translational diffusion of the lipid molecules on a curved surface (D).
- Rotations of the lipid molecules (R).
- Fast internal motions within the lipid molecules (F).

The red solid line curve corresponding to the **complete model **(Fig. 2) was obtained after adding the contributions of each mechanism: R_{1}=R_{1}^{OF}+R_{1}^{D}+R_{1}^{R}+R_{1}^{F}.

Within the restricted frequency range marked on dark green vertical lines in Figure 2 (85 KHz to 1.3 MHz), order fluctuations become the dominant dispersive relaxation contribution while rotations only show a constant contribution therefore the physical model could be simplified. Fast motions and rotational diffusion can be replaced by a unique frequency-independent constant: **simplified model **R_{1}=R_{1}^{OF}+R_{1}^{D}+*const*. The information about the bending modulus is content on the order fluctuations term. Moreover, the sensitivity of the simplified model to each parameter was carefully analyzed concluding that we could neglect the contributions due to diffusion and again simplify the mathematical model keeping just the order fluctuations term as the only frequency dependent term: **extreme simplified model** R_{1}=R_{1}^{OF}+*const*. The experimental measurements and the optimal model curve are plotted on Figure 3.

** EXPERIMENTAL LIMITING CONDITIONS:** This work was planned with the idea of evaluating the feasibility for a small compact benchtop low-power & low-cost instrument, aimed for the measurement of the bending elastic modulus (k). We show here that even under extreme unfavorable conditions (just a few points having large errors), it is possible to measure k within an uncertainty of ±20%. It is important to mention that at normal FFC conditions (20 ppm magnet and current stability better than 1:10

^{5}) this error can be hardly decreased to less than ±10%.

Effects related to a lower homogeneity of the magnetic field on the R_{1} Larmor frequency dispersion can be analyzed by considering the extreme simplified model within the restricted frequency interval. If a certain degree of magnetic field inhomogeneity is present during the experiment, the measured relaxation rate will have contributions from a frequency interval that is determined by the effective field gradient across the sample. We applied the mean value theorem to the extreme simplified model and deduced a new model for the inhomogeneous case: **average MVT model **(see paper for further explanation). Experimental acquisitions and the optimal model curve are shown on Figure 4.

On Table 1 we present the k values obtained using the different versions of our model described before. It can be seen that all values are consistent within experimental errors.

The values of k obtained through complete, extreme simplified and average MVT models are consistent. As a consequence of the elimination of the diffusion contribution, the model sensitivity to the value of k increases.Table 1:

Graphical representation of the elastic properties of a liposome membrane. When the shape of the liposome membrane changes the local lipid order fluctuates, these variations can be characterized through the bending elastic modulus κ.Figure 1:

Spin-lattice relaxation rate dispersion of DMPC liposomes with a radius of 50nm at 303K. Experimental points are compared with the optimal model curve obtained from the complete model (red solid line). Model curve error is shown by the two dash lines beside the red solid line. Contributions from each type of motion are included: order fluctuations (solid green line), molecular rotations (solid violet line), diffusion (solid dark blue line), and fast motions (solid cyan line).Figure 2:

Figure 3:Spin-lattice relaxation rate dispersion for an equivalent sample of Figure 2 but only considering the restricted frequency range (85 KHz to 1.3 MHz). The optimal model curve using the extreme simplified model is shown by the solid red curve (only order fluctuations).

Spin-lattice relaxation rate dispersion of DMPC liposomes with a radius of 50nm at 303K measured in an inhomogeneous magnetic field. The solid red curve was calculated with the average MVT model.Figure 4:

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