Elsevier

Electrochimica Acta

Volume 299, 10 March 2019, Pages 863-874
Electrochimica Acta

Electrochemical impedance of randomly distributed defects in tethered phospholipid bilayers: Finite element analysis

https://doi.org/10.1016/j.electacta.2018.12.148Get rights and content

Abstract

Finite element analysis (FEA) reveals that random distribution of defects in tethered bilayers results in qualitatively similar electrochemical impedance spectroscopy (EIS) response as bilayers with regularly arrayed defects. Such similarity leads to a broader conclusion that the microheterogeneity of tBLMs, which is possibly always the case in real systems may be of lesser importance as far as qualitative aspects of the system are interrogated by the EIS. This means that the analytical solutions of the EIS response may be used to tentatively assess physical parameters and predict variations of EIS curves in the course of interaction between the phospholipid membranes and proteins as well as other membrane damaging agents. While qualitatively similar EIS spectra exhibit a number of quantitative differences. We found that random distribution of defects results in a decrease of the measured conductivity of tBLMs, and the upward shift of the admittance phase minimum which is always present in tBLMs containing defects, and the downshift of the position of that minimum on the frequency scale. Taking into account these features we proposed a simple algorithm that allows calculation of defect density as well as evaluation of the physical size of defects in membrane.

Introduction

Surface supported tethered bilayer membranes (tBLMs) are versatile experimental platforms for protein/membrane studies. From the electrochemical standpoint tBLM immobilized on conducting surfaces comprise a dielectric sheet separated from the solid support by a thin layer of an electrolyte [1]. Thickness of the dielectric layer is predefined by a structure of the molecular anchors which tether phospholipid bilayer to a surface [2]. The phospholipid bilayer is bathed by the conducting liquid media from both sides [3]. If void of defects or in the absence of lipophilic ions capable of electrodiffusion through the bilayer such membrane should act as an ideal capacitor [3]. Its dielectric properties can be assessed by the AC techniques such as the electrochemical impedance spectroscopy (EIS) [4], or DC techniques such as a linear sweep voltammetry.

Despite seeming simplicity tBLMs never exhibit ideal capacitive properties as documented in numerous studies. The ideal dielectric sheet with no dielectric losses should exhibit nearly perfect, single semicircular shape EIS spectra in the complex capacitance representation. Single semicircle curves were observed for freely suspended black lipid membranes [5]. However, such nearly ideal, single semicircular shape of the complex capacitance plots are not observed for tBLMs [6]. Instead, complex-shaped EIS spectra [7] exhibiting multiple semicircular and/or deformed semicircular shapes are typically reported for tBLMs [1,5,7,8].

Various physical and structural factors may determine shapes and features of the EIS spectra. For example, finite element analysis (FEA) modeling predicts the presence of the low frequency “tails” in the EIS complex capacitance spectra due to a presence of the membrane defects with ionic conductivity [9]. Similar conclusions follows from the analytical solution of the EIS spectral response [3] which predicts the development of the “tails” in the Cole Cole plots, as well as a shift of the admittance phase minimum position towards higher frequencies that follows the defect density increase.

In classical approach, the electrochemical interface is assumed to be composed of homogeneous phases in contact. Such assumption simplifies the formal modeling of the electrochemical response through the decrease of the number of the physical parameters and the utility of equivalent electric elements such as resistors, capacitors, Warburg element or a combination of thereof to approximately reproduce shapes of EIS spectra [10]. Such simplification comes at a cost of losing physical information describing the microscopic features of the electrochemical interface. These features as demonstrated in numerous works [[11], [12], [13]], may have profound effects on the electrochemical response.

Therefore, it is quite obvious that to access the microscopic properties of the interface by the intrinsically macroscopic technique such as EIS one needs by modeling to find spectral EIS features (qualitative or quantitative) that reflect underlying microscopic properties. One of the distinct examples related to tBLMs is a discovery of the extremely low (lower by three orders of a magnitude compared to a mobility of ions in the bulk of electrolyte) ionic mobility in the submembrane space separating electrode and the phospholipid bilayer [9,14]. Such phenomenon, which physical explanation still remains elusive, cannot be discovered by using the electric equivalent circuits.

Both analytical [3] and numerical FEA [9] approaches used so far to evaluate the EIS response are based on the assumption of a homogeneous distribution of defects in membranes. Those may be naturally occurring defects including transient pores [15], or artificially induced by the pore forming proteins (peptides), such as amyloid-oligomers implicated in Alzheimers disease [16], gramicidine [1] or pore-forming -hemolysin from Staphyloccocus aureus [1,3]. The lateral distribution of such defects most likely is not homogeneous, so typically assumed perfect hexagonal array of defects in phospholipid bilayers may be exceptionally rare [3]. So far question on the effect of distribution of defects on the EIS response remained elusive. Such uncertainty, obviously, preclude wider application of the EIS in biosensorics, specifically, in quantitative assessment of the activity of the pore-forming proteins in analytes [17,18].

Recently, to take into account more realistic, heterogeneous distribution of defects, an assumption was made that the heterogeneous surface can be modeled as a set of homogeneous patches with different local defect densities [19]. It was also assumed that the occurrence of particular density patch obeys log-normal distribution and the analytical solution for EIS [3] can be applied within an individual surface patch. Quite distinct deviations from the analytically predicted EIS spectra were observed upon introduction of the heterogeneity into the model. Also, it was shown that heterogeneity quantitatively affects EIS response to a pore-forming agents, therefore taking into account effect of heterogeneity is important for bioanalytical application of tBLMs [19].

Despite the fact that predicted by the analysis features may be paralleled to experimental observations [19], it remains unproved if such a synthetic approach describing the heterogeneity can in principle be applied for the real world systems. This prompted us to investigate the alternative way to describe heterogeneous distribution of defects in tBLMs. Specifically, in this study we generate lateral coordinates of the defects randomly using an appropriate random number generator. Physically this implies total randomness of the defect generation in membranes, with neither repulsive nor attractive interactions involved in the localization of an individual defect. The main objective of the present study can be formulated by the following research questions:

  • i)

    Are there qualitative features in EIS spectra that would allow distinguishing between tBLM systems with randomly and with regularly distributed defects?

  • ii)

    What are the quantitative differences in EIS spectra that are caused by the random distribution of defects?

  • iii)

    Do computer generated defect distributions result in EIS spectra as the ones calculated using experimentally measured coordinates of defects?

To compare modeling results with real systems we utilized the cholesterol dependent cytolysins (CDCs) to produce defects in tBLMs. CDCs comprise widely studied group of the pore forming toxins which after specifically binding to the cell membrane, oligomerise on the surface and insert into the bilayers producing large (up to 50 nm diameter) ion-pores [20]. Earlier studies implied that CDCs impair phospholipid membrane only after the complete protein ring structure is formed [21]. Recently, incomplete pores were reported as possible membrane damaging agents [[22], [23], [24]]. The effects of variable defect size add additional complexity to the interpretation of EIS response in tBLM systems. In the current work we do not present thorough analysis of polidispersity of the membrane pore-sizes. Nevertheless, the effect of the pore-size is discussed as well as a simple algorithm to evaluate the average size of the pores is suggested.

Section snippets

Formulation of problem

In this work we consider phospholipid bilayers located at nanometric distance from the conducting substrate, the metal. The heterogeneity of the systems under consideration stems from the presence of the microscopic (1–100 nm) size defects (holes) in bilayers. The cross-section of the phospholipid bilayer in the vicinity of the defect is shown in Fig. 1.

Under stationary (independent on time) alternating current (AC) perturbation mode, the current density and potential are related by Ohm's law

Sample preparation and experimental sequence

EIS experiments were performed using magnetron (Lesker PVD 75, UK) sputtered gold films (100 nm thick) on glass substrates with a thin layer (less than 10 nm) of Cr, sputtered to ensure adhesion of the gold film to the glass. Constant deposition parameters were used: sputtering voltage 350V, deposition rate of 0.6nms1 and sputtering pressure of a high purity (99.9999%) argon gas 3mTorr. The self-assembled monolayer for bilayer (tBLM) tethering was then formed by incubating the freshly

Effect of defect density on EIS

Defect density is the major factor that determines spectral variations of EIS in a particular tBLM system. Fig. 4, Fig. 5 show the EIS Bode plots of tBLMs containing randomly distributed (solid curves) small (rdef=1nm) and large (rdef=25nm) defects in the defect density range spanning from 0.1 μm−2–100 μm−2. These spectra are compared with the spectra calculated according to the analytical formalism developed earlier [3] for tBLM systems containing homogeneously (ideal hexagonal array)

Conclusions

In this work, we analyzed the effects of random distribution of defects in the electrochemical impedance response of tethered bilayers. Because of inability to analytically solve differential equations describing EIS spectra we used finite element analysis. Such approach allows to numerically solve Laplace's equation taking into account the microstructure of tBLMs [9]. Major finding of the current work is a qualitative similarity of the electrochemical impedance response of homogeneously and

Acknowledgements

We thank dr. Milda Pleckaityte (Life Sciences Center, Vilnius University) for purification of pore-forming protein vaginolysin, and dr. D. J. Vanderah (Institute for Biosciences and Biotechnology Research, University of Maryland) for providing molecular anchor compound diHC18 used in the current study. Raw modeling data as well as the unprocessed experimental data are available from the authors on request.

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