# Theoretical Study on the Detection of Tilted Lipid Bilayers Using Surface Plasmon Resonance Techniques

Authors: Koyo Watanabe; Koji Matsuura
Journal: Journal of Sensors
 (2012)

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## Abstract

Effective refractive indices detected using surface plasmon resonance techniques are calculated as a function of the tilt angle of lipid bilayers in a multilayered model. The changes in the effective refractive indices are derived from the shift of an excitation angle of surface plasmon. To obtain effective refractive index plots, we determined refractive index changes in the lipid bilayers with 3 and 5 nm thicknesses as a function of tilt angle and obtained a relationship between the effective refractive index and tilt angle. We also showed that the effective refractive index depended on the lipid bilayers thickness and anisotropic permittivities, which vary with interchain distance.


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## Body

## 1. Introduction

Mechanical stimuli (MS) such as osmotic stress, membrane stretch, and shear force are activators for cellular functions including signaling, proliferation, impulse propagation, and gene expression [1].  MS are believed to induce formation changes in the cell membrane, leading to conformational changes in membrane proteins [2, 3]. For instance, in the cardiovascular system, transient receptor potential membrane proteins, which act as mechanosensitive ion channels, are activated by blood pressure [4]. Furthermore, in the field of mammalian reproduction, it has been confirmed that reproductive cells can be activated by shear stress caused by embryo and/or fluid motions [5]. Although unique relationships between MS and cellular activation have been reported, no study has yet provided a clear understanding of the mechanism at the molecular level.In lipid bilayer deformations, it is difficult to unravel the complex mechanisms as the ion channel functions are sensitive to the properties of the surrounding lipid bilayers [6]. If deformation of the lipid bilayers is experimentally observed after applying MS, especially shear forces or compression, the stresses on the lipid bilayer and the ion channels can be estimated and related to the function of the involved molecules. By these MS, tilt of the average molecular orientation with respect to the layer was induced [7]. It is desired that the relationship between the angle of the lipid tilting and the ion channel activity could be surveyed when the lipid tilts. To obtain greater understanding of these mechanisms, measurement technologies to evaluate the deformations of lipid layers induced by physical and MS need to be developed.Reconstructed systems of lipid bilayers are useful for investigating MS and the molecular functions because physiological molecular systems are too complex to clearly demonstrate the relationship. To study the distribution and dynamics of lipid bilayers, substrate-supported planar lipid bilayers (SPBs) have been fabricated [8, 9]. SPBs are advantageous because the cell membrane is produced on a flat substrate and can be easily observed using different measurement technologies such as atomic force microscopy (AFM), fluorescence imaging, and surface plasmon resonance (SPR) techniques because the configuration using flat substrate can be used in various biosensors. SPBs are useful systems for observing the deformation of lipid layers. Among the possible deformations occurring in lipid layers [6], we aim to detect tilted lipid bilayers on the substrate, which are related to membrane deformations such as curvature and thinning.With regard to observing lipid bilayers, fluorescence imaging techniques [10, 11], AFM [12, 13], and SPR measurements [14–16] have been frequently used to understand lipid distributions and dynamics. Fluorescence imaging techniques have the disadvantage that fluorescent molecules are added to the system, which change the lipid molecules behavior due to perturbation with the fluorescent molecules. Moreover, it would be difficult to determine the tilt of the lipid bilayers from the fluorescence signal. Concerning AFM, the cantilever impedes the behavior of molecules on the lipid bilayers, and the contact imaging mode may damage the sample surface. Although noncontact imaging using AFM is possible, when the cantilever oscillates at a frequency slightly above its resonance frequency, the fluid flow on the substrate, which is necessary for exchanging the liquid sample, contributes to the oscillation noise. We believe that SPR is the best technique for lipid tilt observations because it enables noncontact and nonlabel measurements in liquid environments, has high thickness or refractive index resolution for detecting lipid bilayers, and allows the free flow of the fluid because there are no physical interactions with probes/cantilevers.In lipid bilayer measurement using SPR, various parameters for the lipid bilayers, such as the thickness and interchain distance, affect the effective refractive index, which is detected using SPR. Therefore, to estimate the tilt angle of lipid bilayers from the experimental results and to obtain a better interpretation of the measured results, it is advantageous to know the relationship between the effective refractive index and the tilt angle for different thicknesses and interchain distances. In this paper, to derive such a relationship, we introduce a calculation model and simulate the changes in the effective refractive index due to the tilt angle of the lipid bilayers.


## 2. SPR Measurements

Surface plasmons are quanta of collective electron oscillations excited on a metal surface. They can be excited under the Kretschmann configuration [17], which consists of a high refractive index prism, coated with a thin metal layer and adjacent to a low-refractive-index dielectric medium (shown in inset of Figure 1). In this configuration, when light with p-polarization is provided from the side of the prism, surface plasmon can be efficiently coupled with light at the optimal incident angle. The reflected light intensity as a function of incident angle R(θ) can be calculated on the basis of the multiple-reflection theory [18, 19] as follows:(1)R(θ)=|r01(θ)+r12(θ)exp⁡[2ikz(θ)]1+r01(θ)r12(θ)exp⁡[2ikz(θ)]|2,kz(θ)=2πλn1h[1-(n0n1sin⁡θ)2]-1/2,r01(θ)=n0cos⁡θ-in02sin2θ-n12n0cos⁡θ+in02sin2θ-n12,r12(θ)=in02sin2θ-n12-n2cos⁡θ′in02sin2θ-n12+n2cos⁡θ′,
where kz is the z component of the wave number for the light inside the metal; n0, n1, and n2 denote the refractive index of the prism, metal, and low dielectric medium, respectively,  λ is the wavelength of the incident light in vacuum, h denotes the thickness of the metal layer, and rij(θ), which is based on the Fresnel equations, is the amplitude reflectance at the boundary between the mediums i and j for light incident from i to j; and θ′ denotes the refraction angle at the boundary between the metal layer and the dielectric medium.Figure 1
Reflected light intensity as a function of incident angle.Figure1 shows the calculated results for the reflected light intensity.  In the present calculation, we assume an excitation at 632.8 nm, a refractive index of 1.780 for the glass prism, a refractive index of 1.330 for the dielectric medium, and a 44 nm thick gold layer with a complex refractive index of 0.300+3.089i [20]. The influence of the gold layer thickness on the sensitivity of surface plasmon sensors is discussed in [21].According to the plot, we can observe that the energy of the incident light is most efficiently absorbed at 56°, which is called the excitation angle of the surface plasmonθsp. This approach is advantageous because the excitation angle changes with the refractive index of the low-refractive-index medium n2. It is well known that such excitation angle shifts are large enough to detect the adsorption of lipid bilayers. The changes in the detected refractive index δn2 can be estimated from the relationship between θsp and n2, as expressed in [22](2)sin⁡θsp≈Real(1ng(nm2n22nm2+n22)1/2),
where ng and nm are the refractive index of glass and the complex refractive index of the metal, respectively.The electric field induced by the surface plasmon penetrates the sample medium from the metal surface and can probe the changes in refractive index. The penetration depth, which is the measurement depth, is a few hundred nanometers. Since the measurement depth is larger than the thickness of the lipid bilayers, the detected value in the lipid measurement results in an effective refractive indexneff, which can be defined as a weighted averaged refractive index over the measurement depth [23–25].


## 3. Calculation Model and Methods

### 3.1. Calculation Model

Figure2 schematically depicts the model of the lipid bilayers on the substrate [26]. It consists of six layers: glass substrate, Au film, SiO2, lower water layer, lipid bilayer, and upper water layer. The glass substrate and the Au film form the basic configuration for generating SPRs. The SiO2 layer is used for preserving the hydrophilic nature of the surface of the substrate and to form bilayer without defects on the hydrophilic surface [20]. Because the combined thickness of the SiO2  layer, water layers, and lipid bilayer is less than the measurement depth, the obtained neff is averaged across the SiO2  layer, both water layers, and the lipid bilayer.Figure 2
Model to calculate effective refractive index as a function of tilt angleα.In the model,α is the tilt of the lipid bilayers. To determine the refractive index of the lipid bilayer, the following equation is used:(3)n2=ε,
where ε denotes the permittivity of the lipid bilayers. The lipid bilayer has anisotropic permittivity, with components ε⊥ and ε∥ representing the permittivity perpendicular and parallel to the substrate, respectively, at α=0°. In the SPR measurements, permittivity parallel to the substrate can be ignored [27]. Therefore, (3) can be expressed using ε⊥ as follows:(4)nspr=ε⊥,
where nspr denotes the refractive index of the lipid bilayers detected in the SPR measurements. At α=90°, because ε⊥ is equal to ε∥, (4) is rewritten as follows:(5)nspr′=ε||.
To express nspr in terms of the tilt angle α on the basis of the results of SPR measurement for tilted anisotropic molecules (liquid crystal cells) [28, 29], we assume that the value of ε⊥ linearly changes to ε|| in the range from α=0° to α=90°, provided that ε⊥>ε||, given as(6)nspr2=εspr=ε||-ε⊥90×α+ε⊥.


### 3.2. Calculation Method

The calculation process used to determine the effective refractive index as a function of the tilt of lipid bilayers is presented here. The calculation procedure is as follows: (1) the reflected light intensities are calculated as a function of incident angle for the model shown in Figure2 with α=0°; (2) θsp is estimated and neff is derived using (2); (3) α is increased in 1° steps, and the refractive index and the thickness t of the lipid bilayers are calculated using (6) and t=t0cosα, respectively, which are set as the model parameters; (4) steps (1)–(3) are performed for tilt angles 0°–50°. The effective refractive index detected is normalized by the refractive index of water (1.330) over the metal surface. The calculation range of the tilt angle is determined on the basis of the idea that lipid bilayers with small tilt angles are typically discussed in studies on lipid functionalities [30]. Moreover, lipid bilayers with large tilt angles are structurally inconsistent with real cell membrane, and it would be difficult to keep such lipids in a bilayer form.


## 3.1. Calculation Model

Figure2 schematically depicts the model of the lipid bilayers on the substrate [26]. It consists of six layers: glass substrate, Au film, SiO2, lower water layer, lipid bilayer, and upper water layer. The glass substrate and the Au film form the basic configuration for generating SPRs. The SiO2 layer is used for preserving the hydrophilic nature of the surface of the substrate and to form bilayer without defects on the hydrophilic surface [20]. Because the combined thickness of the SiO2  layer, water layers, and lipid bilayer is less than the measurement depth, the obtained neff is averaged across the SiO2  layer, both water layers, and the lipid bilayer.Figure 2
Model to calculate effective refractive index as a function of tilt angleα.In the model,α is the tilt of the lipid bilayers. To determine the refractive index of the lipid bilayer, the following equation is used:(3)n2=ε,
where ε denotes the permittivity of the lipid bilayers. The lipid bilayer has anisotropic permittivity, with components ε⊥ and ε∥ representing the permittivity perpendicular and parallel to the substrate, respectively, at α=0°. In the SPR measurements, permittivity parallel to the substrate can be ignored [27]. Therefore, (3) can be expressed using ε⊥ as follows:(4)nspr=ε⊥,
where nspr denotes the refractive index of the lipid bilayers detected in the SPR measurements. At α=90°, because ε⊥ is equal to ε∥, (4) is rewritten as follows:(5)nspr′=ε||.
To express nspr in terms of the tilt angle α on the basis of the results of SPR measurement for tilted anisotropic molecules (liquid crystal cells) [28, 29], we assume that the value of ε⊥ linearly changes to ε|| in the range from α=0° to α=90°, provided that ε⊥>ε||, given as(6)nspr2=εspr=ε||-ε⊥90×α+ε⊥.


## 3.2. Calculation Method

The calculation process used to determine the effective refractive index as a function of the tilt of lipid bilayers is presented here. The calculation procedure is as follows: (1) the reflected light intensities are calculated as a function of incident angle for the model shown in Figure2 with α=0°; (2) θsp is estimated and neff is derived using (2); (3) α is increased in 1° steps, and the refractive index and the thickness t of the lipid bilayers are calculated using (6) and t=t0cosα, respectively, which are set as the model parameters; (4) steps (1)–(3) are performed for tilt angles 0°–50°. The effective refractive index detected is normalized by the refractive index of water (1.330) over the metal surface. The calculation range of the tilt angle is determined on the basis of the idea that lipid bilayers with small tilt angles are typically discussed in studies on lipid functionalities [30]. Moreover, lipid bilayers with large tilt angles are structurally inconsistent with real cell membrane, and it would be difficult to keep such lipids in a bilayer form.


## 4. Results and Discussion

### 4.1. Thickness and Refractive Index of Lipid Bilayers as a Function of Tilt Angle

Figures3(a) and 3(b) show the thickness and refractive index plots as a function of tilt angle, respectively. In this calculation, we used the same parameters for the refractive index that were used in Section 2. The refractive index of SiO2 is assumed to be 1.515, and the thicknesses of the Au film and the SiO2 layer are assumed to be 44 and 12 nm, respectively [20]. The thickness of the lipid bilayer, t0, is set as 5 nm because the structure of a fully hydrated lipid bilayer has been estimated at around 5 nm [31] and also because it has been experimentally confirmed by AFM [32]. The anisotropic permittivity of the lipid bilayers varies with the interchain distance d. We set d as 0.500 nm, as this value has been usually assumed for liquid paraffins [33]. The values of ε⊥ and ε∥ are 2.288 and 2.124, respectively [33].(a) Thickness of lipid bilayers as a function of tilt angle. (b) Refractive index as a function of tilt angle.
(a)(b)According to the plots in Figures3(a) and 3(b), the thickness and refractive index of the lipid bilayers decrease with increasing tilt angles. Hence, it is expected that the effective refractive index decreases with increasing tilt angles.


### 4.2. Effective Refractive Index as a Function of Tilt Angle

Based on the model in Figure2 and the calculation method described in Section 3, we calculated the effective refractive index as a function of tilt angle. In the calculation, the thickness of lipid bilayers t and that of refractive index parameters for each tilt angle α are taken from the plots in Figures 3(a) and 3(b). The calculated results when d=0.500 nm and t0=5 nm, shown in the red plot in Figure 4, indicate that the effective refractive index decreases with increasing tilt angles. When the tilt angle is between 0° and 10°, the change in the effective refractive index is 4.8×10-4. This value is large enough to distinguish differences of five degrees if we assume SPR with the refractive index resolution of 1.0×10-4 [34].Figure 4
Calculated plots of effective refractive index as function of tilt angle for thicknesst0=3.0 and 5.0 nm and interchain distance d=0.475 and 0.500 nm.To further investigate the resolution and sensitivity effects, the interchain distanced was varied slightly to 0.475 nm, a value referred to in [33]. We also varied the thickness t0 to 3 nm, which is the value measured by localized SPR microscopy [20]. The green line shows the results using t0=5 nm at d=0.475 nm, which has an anisotropic permittivity of ε⊥=2.534 and ε∥=2.245 [33].The pink and blue plots show the results for t0=3 nm at d=0.475 nm and t0=3 nm at d=0.500 nm,  respectively. These results show that the neff slopes are steeper with increasing thickness and anisotropic permittivity, that is, increasing thickness and anisotropic permittivity provides more sensitive measurements for the tilt of the lipid bilayers.


## 4.1. Thickness and Refractive Index of Lipid Bilayers as a Function of Tilt Angle

Figures3(a) and 3(b) show the thickness and refractive index plots as a function of tilt angle, respectively. In this calculation, we used the same parameters for the refractive index that were used in Section 2. The refractive index of SiO2 is assumed to be 1.515, and the thicknesses of the Au film and the SiO2 layer are assumed to be 44 and 12 nm, respectively [20]. The thickness of the lipid bilayer, t0, is set as 5 nm because the structure of a fully hydrated lipid bilayer has been estimated at around 5 nm [31] and also because it has been experimentally confirmed by AFM [32]. The anisotropic permittivity of the lipid bilayers varies with the interchain distance d. We set d as 0.500 nm, as this value has been usually assumed for liquid paraffins [33]. The values of ε⊥ and ε∥ are 2.288 and 2.124, respectively [33].(a) Thickness of lipid bilayers as a function of tilt angle. (b) Refractive index as a function of tilt angle.
(a)(b)According to the plots in Figures3(a) and 3(b), the thickness and refractive index of the lipid bilayers decrease with increasing tilt angles. Hence, it is expected that the effective refractive index decreases with increasing tilt angles.


## 4.2. Effective Refractive Index as a Function of Tilt Angle

Based on the model in Figure2 and the calculation method described in Section 3, we calculated the effective refractive index as a function of tilt angle. In the calculation, the thickness of lipid bilayers t and that of refractive index parameters for each tilt angle α are taken from the plots in Figures 3(a) and 3(b). The calculated results when d=0.500 nm and t0=5 nm, shown in the red plot in Figure 4, indicate that the effective refractive index decreases with increasing tilt angles. When the tilt angle is between 0° and 10°, the change in the effective refractive index is 4.8×10-4. This value is large enough to distinguish differences of five degrees if we assume SPR with the refractive index resolution of 1.0×10-4 [34].Figure 4
Calculated plots of effective refractive index as function of tilt angle for thicknesst0=3.0 and 5.0 nm and interchain distance d=0.475 and 0.500 nm.To further investigate the resolution and sensitivity effects, the interchain distanced was varied slightly to 0.475 nm, a value referred to in [33]. We also varied the thickness t0 to 3 nm, which is the value measured by localized SPR microscopy [20]. The green line shows the results using t0=5 nm at d=0.475 nm, which has an anisotropic permittivity of ε⊥=2.534 and ε∥=2.245 [33].The pink and blue plots show the results for t0=3 nm at d=0.475 nm and t0=3 nm at d=0.500 nm,  respectively. These results show that the neff slopes are steeper with increasing thickness and anisotropic permittivity, that is, increasing thickness and anisotropic permittivity provides more sensitive measurements for the tilt of the lipid bilayers.


## 5. Conclusion

We theoretically identified the relationship between the effective refractive index and tilt angle of lipid bilayers and demonstrated how the tilt angle of the lipid bilayers on the substrate can be detected using SPR techniques. We also showed that the detection sensitivity changes due to lipid bilayers with different thickness and anisotropic permittivity. This insight can contribute for detection of lipid tilting by shear force or compression to SPBs that contains transmembrane mechanosensing ion channel. Using this measurement technology, the relationship between the angle of the lipid tilting and the ion channel activity could be surveyed when the lipid tilts.



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