Destabilizing giant vesicles with electric fields: an overview of current applications

(2+δ)-dimensional theory of the electromechanics of lipid membranes: Electrostatics

The coupling of electric fields to the mechanics of lipid membranes gives rise to intriguing electromechanical behavior, as, for example, evidenced by the deformation of lipid vesicles in external electric fields. Electromechanical effects are relevant for many biological processes, such as the propagation of action potentials in axons and the activation of mechanically gated ion channels. Currently, a theoretical framework describing the electromechanical behavior of arbitrarily curved and deforming lipid membranes does not exist. Purely mechanical models commonly treat lipid membranes as two-dimensional surfaces, ignoring their finite thickness. While holding analytical and numerical merit, this approach cannot describe the coupling of lipid membranes to electric fields and is thus unsuitable for electromechanical models. In a sequence of articles, we derive an effective surface theory of the electromechanics of lipid membranes, called the (2+δ)-dimensional theory, which has the advantages of surface descriptions while accounting for finite thickness effects. The present article proposes a generic dimension reduction procedure relying on low-order spectral expansions. This procedure is applied to the electrostatics of lipid membranes to obtain the (2+δ)-dimensional theory that captures potential differences across and electric fields within lipid membranes. This model is tested on different geometries relevant for lipid membranes, showing good agreement with the corresponding three-dimensional electrostatics theory.

Bilayer Charge Asymmetry and Oil Residues Destabilize Membranes upon Poration

Transmembrane asymmetry is ubiquitous in cells, particularly with respect to lipids, where charged lipids are mainly restricted to one monolayer. We investigate the influence of anionic lipid asymmetry on the stability of giant unilamellar vesicles (GUVs), minimal plasma membrane models. To quantify asymmetry, we apply the fluorescence quenching assay, which is often difficult to reproduce, and caution in handling the quencher is generally underestimated. We first optimize this assay and then apply it to GUVs prepared with the inverted emulsion transfer protocol by using increasing fractions of anionic lipids restricted to one leaflet. This protocol is found to produce highly asymmetric bilayers but with ∼20% interleaflet mixing. To probe the stability of asymmetric versus symmetric membranes, we expose the GUVs to porating electric pulses and monitor the fraction of destabilized vesicles. The pulses open macropores, and the GUVs either completely recover or exhibit leakage or bursting/collapse. Residual oil destabilizes porated membranes, and destabilization is even more pronounced in asymmetrically charged membranes. This is corroborated by the measured pore edge tension, which is also found to decrease with increasing charge asymmetry. Using GUVs with imposed transmembrane pH asymmetry, we confirm that poration-triggered destabilization does not depend on the approach used to generate membrane asymmetry.

Assessing membrane material properties from the response of giant unilamellar vesicles to electric fields

Knowledge of the material properties of membranes is crucial to understanding cell viability and physiology. A number of methods have been developed to probe membranes in vitro, utilizing the response of minimal biomimetic membrane models to an external perturbation. In this review, we focus on techniques employing giant unilamellar vesicles (GUVs), model membrane systems, often referred to as minimal artificial cells because of the potential they offer to mimick certain cellular features. When exposed to electric fields, GUV deformation, dynamic response and poration can be used to deduce properties such as bending rigidity, pore edge tension, membrane capacitance, surface shear viscosity, excess area and membrane stability. We present a succinct overview of these techniques, which require only simple instrumentation, available in many labs, as well as reasonably facile experimental implementation and analysis.

Propidium uptake and ATP release in A549 cells share similar transport mechanisms

The lipid bilayer of eukaryotic cells' plasma membrane is almost impermeable to small ions and large polar molecules, but its miniscule basal permeability in intact cells is poorly characterized. This report describes the intrinsic membrane permeability of A549 cells toward the charged molecules propidium (Pr²⁺) and ATP^4-. Under isotonic conditions, we detected with quantitative fluorescence microscopy, a continuous low-rate uptake of Pr (∼150 × 10^-21 moles (zmol)/h/cell, [Pr]_o = 150 μM, 32°C). It was stimulated transiently but strongly by 66% hypotonic cell swelling reaching an influx amplitude of ∼1500 (zmol/h)/cell. The progressive Pr uptake with increasing [Pr]_o (30, 150, and 750 μM) suggested a permeation mechanism by simple diffusion. We quantified separately ATP release with custom wide-field-of-view chemiluminescence imaging. The strong proportionality between ATP efflux and Pr²⁺ influx during hypotonic challenge, and the absence of stimulation of transmembrane transport following 300% hypertonic shock, indicated that ATP and Pr travel the same conductive pathway. The fluorescence images revealed a homogeneously distributed intracellular uptake of Pr not consistent with high-conductance channels expressed at low density on the plasma membrane. We hypothesized that the pathway consists of transiently formed water pores evenly spread across the plasma membrane. The abolition of cell swelling-induced Pr uptake with 500 μM gadolinium, a known modulator of membrane fluidity, supported the involvement of water pores whose formation depends on the membrane fluidity. Our study suggests an alternative model of a direct permeation of ATP (and other molecules) through the phospholipid bilayer, which may have important physiological implications.

Quantification of pulsed electric field for the rupture of giant vesicles with various surface charges, cholesterols and osmotic pressures

Electropermeabilization is a promising phenomenon that occurs when pulsed electric field with high frequency is applied to cells/vesicles. We quantify the required values of pulsed electric fields for the rupture of cell-sized giant unilamellar vesicles (GUVs) which are prepared under various surface charges, cholesterol contents and osmotic pressures. The probability of rupture and the average time of rupture are evaluated under these conditions. The electric field changes from 500 to 410 Vcm^-1 by varying the anionic lipid mole fraction from 0 to 0.60 for getting the maximum probability of rupture (i.e., 1.0). In contrast, the same probability of rupture is obtained for changing the electric field from 410 to 630 Vcm^-1 by varying the cholesterol mole fraction in the membranes from 0 to 0.40. These results suggest that the required electric field for the rupture decreases with the increase of surface charge density but increases with the increase of cholesterol. We also quantify the electric field for the rupture of GUVs containing anionic mole fraction of 0.40 under various osmotic pressures. In the absence of osmotic pressure, the electric field for the rupture is obtained 430 Vcm^-1, whereas the field is 300 Vcm^-1 in the presence of 17 mOsmL^-1, indicating the instability of GUVs at higher osmotic pressures. These investigations open an avenue of possibilities for finding the electric field dependent rupture of cell-like vesicles along with the insight of biophysical and biochemical processes.

Translocation of silica nanospheres through giant unilamellar vesicles (GUVs) induced by a high frequency electromagnetic field

Membrane model systems capable of mimicking live cell membranes were used for the first time in studying the effects arising from electromagnetic fields (EMFs) of 18 GHz where membrane permeability was observed following exposure. A present lack of understanding of the mechanisms that drive such a rapid change in membrane permeabilization as well as any structural or dynamic changes imparted on biomolecules affected by high-frequency electromagnetic irradiation limits the use of 18 GHz EMFs in biomedical applications. A phospholipid, 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) labelled with a fluorescent marker 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl) (rhodamine-DOPE) was used in constructing the giant unilamellar vesicles (GUVs). After three cycles of exposure, enhanced membrane permeability was observed by the internalisation of hydrophilic silica nanospheres of 23.5 nm and their clusters. All-atom molecular dynamics simulations of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) membranes exposed to high frequency electric fields of different field strengths showed that within the simulation timeframe only extremely high strength fields were able to cause an increase in the interfacial water dynamics characterized by water dipole realignments. However, a lower strength, high frequency EMF induced changes of the water hydrogen bond network, which may contribute to the mechanisms that facilitate membrane permeabilization in a longer timeframe.

Proof-of-Concept of Electrical Activation of Liposome Nanocarriers: From Dry to Wet Experiments

The increasing interest toward biocompatible nanotechnologies in medicine, combined with electric fields stimulation, is leading to the development of electro-sensitive smart systems for drug delivery applications. To this regard, recently the use of pulsed electric fields to trigger release across phospholipid membranes of liposomes has been numerically studied, for a deeper understanding of the phenomena at the molecular scale. Aim of this work is to give an experimental validation of the feasibility to control the release from liposome vesicles, using nanosecond pulsed electric fields characterized by a 10 ns duration and intensity in the order of MV/m. The results are supported by multiphysics simulations which consider the coupling of three physics (electromagnetics, thermal and pore kinetics) in order to explain the occurring physical interactions at the microscopic level and provide useful information on the characteristics of the train of pulses needed to obtain quantitative results in terms of liposome electropermeabilization. Finally, a complete characterization of the exposure system is also provided to support the reliability and validity of the study.

Electrohydrodynamics of Vesicles and Capsules

Giant unilamellar vesicles (GUVs) made up of phospholipid bilayer membranes (liposomes) and elastic capsules with a cross-linked, polymerized membrane, have emerged as biomimetic alternatives to investigating biological cells such as leukocytes and erythrocytes. This feature article looks at the similarities and differences in the electrohydrodynamics (EHD) of vesicles and capsules under electric fields that determines their electromechanical response. The physics of EHD is illustrated through several examples such as the electrodeformation of single and compound, spherical and cylindrical, and charged and uncharged vesicles in uniform and nonuniform electric fields, and the relevance and challenges are discussed. Both small and large deformation results are discussed. The use of EHD in understanding complex interfacial kinetics in capsules and the synthesis of nonspherical capsules using electric fields are also presented. Finally, the review looks at the large electrodeformation of water-in-water capsules and the relevance of constitutive laws in their response.

Effect of Electrical Parameters and Cholesterol Concentration on Giant Unilamellar Vesicles Electroformation

Giant unilamellar vesicles (GUVs) are used extensively as models that mimic cell membranes. The cholesterol (Chol) content in the fiber cell plasma membranes of the eye lens is extremely high, exceeding the solubility threshold in the lenses of old humans. Thus, a methodological paper pertaining to preparations of model lipid bilayer membranes with high Chol content would significantly help the study of properties of these membranes. Lipid solutions containing 1-palmitoyl-2-oleoyl-glycero-3-phosphocholine (POPC) and Chol were fluorescently labeled with phospholipid analog 1,1'-dioctadecyl-3,3,3',3'-tetramethylindocarbocyanine perchlorate (DiIC₁₈(3)) and spin-coated to produce thin lipid films. GUVs were formed from these films using the electroformation method and the results were obtained using fluorescent microscopy. Electroformation outcomes were examined for different electrical parameters and different Chol concentrations. A wide range of field frequency-field strength (ff-fs) combinations was explored: 10-10,000 Hz and 0.625-9.375 V/mm peak-to-peak. Optimal values for GUVs preparation were found to be 10-100 Hz and 1.25-6.25 V/mm, with largest vesicles occurring for 10 Hz and 3.75 V/mm. Chol:POPC mixing ratios (expressed as a molar ratio) ranged from 0 to 3.5. We show that increasing the Chol concentration decreases the GUVs size, but this effect can be reduced by choosing the appropriate ff-fs combination.

Response of an actin network in vesicles under electric pulses

We study the role of a biomimetic actin network during the application of electric pulses that induce electroporation or electropermeabilization, using giant unilamellar vesicles (GUVs) as a model system. The actin cortex, a subjacently attached interconnected network of actin filaments, regulates the shape and mechanical properties of the plasma membrane of mammalian cells, and is a major factor influencing the mechanical response of the cell to external physical cues. We demonstrate that the presence of an actin shell inhibits the formation of macropores in the electroporated GUVs. Additionally, experiments on the uptake of dye molecules after electroporation show that the actin network slows down the resealing process of the permeabilized membrane. We further analyze the stability of the actin network inside the GUVs exposed to high electric pulses. We find disruption of the actin layer that is likely due to the electrophoretic forces acting on the actin filaments during the permeabilization of the GUVs. Our findings on the GUVs containing a biomimetic network provide a step towards understanding the discrepancies between the electroporation mechanism of a living cell and its simplified model of the empty GUV.

Membrane Electroporation and Electropermeabilization: Mechanisms and Models

Exposure of biological cells to high-voltage, short-duration electric pulses causes a transient increase in their plasma membrane permeability, allowing transmembrane transport of otherwise impermeant molecules. In recent years, large steps were made in the understanding of underlying events. Formation of aqueous pores in the lipid bilayer is now a widely recognized mechanism, but evidence is growing that changes to individual membrane lipids and proteins also contribute, substantiating the need for terminological distinction between electroporation and electropermeabilization. We first revisit experimental evidence for electrically induced membrane permeability, its correlation with transmembrane voltage, and continuum models of electropermeabilization that disregard the molecular-level structure and events. We then present insights from molecular-level modeling, particularly atomistic simulations that enhance understanding of pore formation, and evidence of chemical modifications of membrane lipids and functional modulation of membrane proteins affecting membrane permeability. Finally, we discuss the remaining challenges to our full understanding of electroporation and electropermeabilization.

Added-Value Surfactants

Surfactants are ubiquitous in cellular membranes, detergents or as emulsification agents. Due to their amphiphilic properties, they cannot only mediate between two domains of very different solvent compatibility like water and organic but also show fascinating self-assembly features resulting in micelles, vesicles, or lyotropic liquid crystals. The current review article highlights some approaches towards the next generation surfactants, for example, those with catalytically active heads. Furthermore, it is shown that amphiphilic properties can be obtained beyond the classical hydrophobic-hydrophilic interplay, for instance with surfactants containing one molecular block with a special shape. Whereas, classical surfactants are static, researchers have become more interested in species that are able to change their properties depending on external triggers. The article discusses examples for surfactants sensitive to chemical (e.g., pH value) or physical triggers (temperature, electric and magnetic fields).

Exposure to high-frequency electromagnetic field triggers rapid uptake of large nanosphere clusters by pheochromocytoma cells

Background

Effects of man-made electromagnetic fields (EMF) on living organisms potentially include transient and permanent changes in cell behaviour, physiology and morphology. At present, these EMF-induced effects are poorly defined, yet their understanding may provide important insights into consequences of uncontrolled (e.g., environmental) as well as intentional (e.g., therapeutic or diagnostic) exposure of biota to EMFs. In this work, for the first time, we study mechanisms by which a high frequency (18 GHz) EMF radiation affects the physiology of membrane transport in pheochromocytoma PC 12, a convenient model system for neurotoxicological and membrane transport studies.

Methods and results

Suspensions of the PC 12 cells were subjected to three consecutive cycles of 30s EMF treatment with a specific absorption rate (SAR) of 1.17 kW kg^-1, with cells cooled between exposures to reduce bulk dielectric heating. The EMF exposure resulted in a transient increase in membrane permeability for 9 min in up to 90 % of the treated cells, as demonstrated by rapid internalisation of silica nanospheres (diameter d ≈ 23.5 nm) and their clusters (d ≈ 63 nm). In contrast, the PC 12 cells that received an equivalent bulk heat treatment behaved similar to the untreated controls, showing lack to minimal nanosphere uptake of approximately 1-2 %. Morphology and growth of the EMF treated cells were not altered, indicating that the PC 12 cells were able to remain viable after the EMF exposure. The metabolic activity of EMF treated PC 12 cells was similar to that of the heat treated and control samples, with no difference in the total protein concentration and lactate dehydrogenase (LDH) release between these groups.

Conclusion

These results provide new insights into the mechanisms of EMF-induced biological activity in mammalian cells, suggesting a possible use of EMFs to facilitate efficient transport of biomolecules, dyes and tracers, and genetic material across cell membrane in drug delivery and gene therapy, where permanent permeabilisation or cell death is undesirable.

The role of gel-phase domains in electroporation of vesicles

Transient permeabilisation of the cell membrane is a critical step to introduce drugs or DNA into living cells, yet challenging for both biological research and therapeutic applications. To achieve this, electroporation (or electropermeabilisation) has become a widely used method due to its simplicity to deliver almost any biomolecule to any cell type. Although this method demonstrates promise in the field of drug/gene delivery, the underlying physical mechanisms of the response of the heterogeneous cell membrane to strong electric pulses is still unknown. In this study, we have investigated the role of gel-phase lipids in the electroporation of binary giant unilamellar vesicles (GUVs), composed from DPPC (gel-phase) and DPhPC (fluid-phase) lipids (molar ratio 8:2 and 2:8). We have observed that the exposure to electric pulses leads to expel of fluid-phase lipids and concomitant decrease in GUV size, whereas the gel-phase domains become buckled. Based on experiments on pure fluid-phase and gel-phase GUVs, we have found that fluid-phase lipids can be expelled by electrical forces and the highly viscous gel-phase lipids cannot. Moreover, our analyses suggest that pore formation occurs primarily in fluid-phase domains and that the pore size is similar in all GUVs containing fluid-phase lipids, irrespective of the gel-phase percentage.

Flow Cytometric Analysis To Evaluate Morphological Changes in Giant Liposomes As Observed in Electrofusion Experiments

Liposome fusion is a way of supplying additional components for in-liposome biochemical reactions. Electrofusion is a method that does not require the addition of fusogens, which often alter the liposome dispersion, and is therefore useful for repetitive liposome fusion. However, the details of electrofusion have not been elucidated because of the limitations surrounding observing liposomes using a microscope. Therefore, we introduced fluorescent markers and high-throughput flow cytometry to analyze the morphological changes that occur in liposome electrofusion. (i) The content mixing was evaluated by a calcein-Co²⁺-EDTA system, in which green fluorescence from dequenched free calcein is detected when the quenched calcein-Co²⁺ complex and EDTA are mixed together. (ii) Liposome destruction was evaluated from the decrease in the total membrane volume of giant liposomes. (iii) Liposome fission was evaluated from the increase in the number of giant liposomes. By applying the flow cytometric analysis, we investigated the effect of three parameters (DC pulse, AC field, and lipid composition) on liposome electrofusion. The larger numbers or higher voltages of DC pulses induced liposome fusion and destruction with higher probability. The longer application time of the AC field induced liposome fusion, fission, and destruction with higher probability. Higher content of negatively charged POPG (≥19%) strongly inhibited liposome electrofusion.

Lipid vesicles in pulsed electric fields: Fundamental principles of the membrane response and its biomedical applications

The present review focuses on the effects of pulsed electric fields on lipid vesicles ranging from giant unilamellar vesicles (GUVs) to small unilamellar vesicles (SUVs), from both fundamental and applicative perspectives. Lipid vesicles are the most popular model membrane systems for studying biophysical and biological processes in living cells. Furthermore, as vesicles are made from biocompatible and biodegradable materials, they provide a strategy to create safe and functionalized drug delivery systems in health-care applications. Exposure of lipid vesicles to pulsed electric fields is a common physical method to transiently increase the permeability of the lipid membrane. This method, termed electroporation, has shown many advantages for delivering exogenous molecules including drugs and genetic material into vesicles and living cells. In addition, electroporation can be applied to induce fusion between vesicles and/or cells. First, we discuss in detail how research on cell-size GUVs as model cell systems has provided novel insight into the basic mechanisms of cell electroporation and associated phenomena. Afterwards, we continue with a thorough overview how electroporation and electrofusion have been used as versatile methods to manipulate vesicles of all sizes in different biomedical applications. We conclude by summarizing the open questions in the field of electroporation and possible future directions for vesicles in the biomedical field.

Exploring the Applicability of Nano-Poration for Remote Control in Smart Drug Delivery Systems

Smart drug delivery systems represent an interesting tool to significantly improve the efficiency and the precision in the treatment of a broad category of diseases. In this context, a drug delivery mediated by nanosecond pulsed electric fields seems a promising technique, allowing for a controlled release and uptake of drugs by the synergy between the electropulsation and nanocarriers with encapsulated drugs. The main concern about the use of electroporation for drug delivery applications is the difference in dimension between the liposome (nanometer range) and the cell (micrometer range). The choice of liposome dimension is not trivial. Liposomes larger than 500 nm of diameter could be recognized as pathogen agents by the immune system, while liposomes of smaller size would require external electric field of high amplitudes for the membrane electroporation that could compromise the cell viability. The aim of this work is to theoretically study the possibility of a simultaneous cell and liposomes electroporation. The numerical simulations reported the possibility to electroporate the cell and a significant percentage of liposomes with comparable values of external electric field, when a 12 nsPEF is used.

Voltage-morphology coupling in biomimetic membranes: dynamics of giant vesicles in applied electric fields

An electric potential difference across the plasma membrane is common to all living cells and is essential to physiological functions such as the generation of action potentials for cell-to-cell communication. While the basics of cell electrical activity are well established (e.g. the Hodgkin-Huxley model of the action potential), the reciprocal coupling of voltage and membrane deformation has received limited attention. In recent years, studies of biomimetic membranes in externally applied electric fields have revealed a plethora of intriguing dynamics (formation of edges, pearling, and phase separation) that challenge the current understanding of membrane electromechanics.

Vesicle dynamics in uniform electric fields: squaring and breathing

We computationally investigate the dynamics of a vesicle exposed to uniform DC or AC electric fields. We employ the two-dimensional boundary integral method in order to simulate vesicle deformation under experimental conditions where peculiar drum-like ("squared") shapes have been observed. The vesicle membrane is modeled as an infinitely thin, capacitive, area-incompressible interface, with the surrounding fluids acting as leaky dielectrics. Our simulations capture the "squaring" phenomenon, in which vesicles deform into rectangular profiles with corner-like regions of high curvature, as vesicles undergo dynamic transitions between oblate and prolate ellipsoidal shapes.

Electroporation-mediated gene delivery

Electroporation has been used extensively to transfer DNA to bacteria, yeast, and mammalian cells in culture for the past 30 years. Over this time, numerous advances have been made, from using fields to facilitate cell fusion, delivery of chemotherapeutic drugs to cells and tissues, and most importantly, gene and drug delivery in living tissues from rodents to man. Electroporation uses electrical fields to transiently destabilize the membrane allowing the entry of normally impermeable macromolecules into the cytoplasm. Surprisingly, at the appropriate field strengths, the application of these fields to tissues results in little, if any, damage or trauma. Indeed, electroporation has even been used successfully in human trials for gene delivery for the treatment of tumors and for vaccine development. Electroporation can lead to between 100 and 1000-fold increases in gene delivery and expression and can also increase both the distribution of cells taking up and expressing the DNA as well as the absolute amount of gene product per cell (likely due to increased delivery of plasmids into each cell). Effective electroporation depends on electric field parameters, electrode design, the tissues and cells being targeted, and the plasmids that are being transferred themselves. Most importantly, there is no single combination of these variables that leads to greatest efficacy in every situation; optimization is required in every new setting. Electroporation-mediated in vivo gene delivery has proven highly effective in vaccine production, transgene expression, enzyme replacement, and control of a variety of cancers. Almost any tissue can be targeted with electroporation, including muscle, skin, heart, liver, lung, and vasculature. This chapter will provide an overview of the theory of electroporation for the delivery of DNA both in individual cells and in tissues and its application for in vivo gene delivery in a number of animal models.

Controllable electrofusion of lipid vesicles: initiation and analysis of reactions within biomimetic containers

We present a microfluidic device that is able to trap multiple giant unilamellar vesicles (GUVs) and initiate electrofusion via integrated microelectrodes. PDMS posts were designed to trap and isolate two or more vesicles. Electrodes patterned onto the glass surface of the microchannels are able to apply a short, high voltage pulse across the traps for controllable electrofusion of the GUVs. The entire array of traps and electrodes are designed such that an average of 60 individual fusion experiments can be performed on-chip. An assay based on Förster resonance energy transfer (FRET) is performed to show successful lipid mixing. Not only can the device be used to record the dynamics of lipid membrane fusion, but it can be used for reaction monitoring by fusing GUVs containing reactants. We demonstrate this by fusing vesicles encapsulating femtolitre volumes of cobalt chloride or EDTA and monitoring the amount of the complexation product over time.

Equilibrium electrodeformation of a spheroidal vesicle in an ac electric field

In this work, we develop a theoretical model to explain the equilibrium spheroidal deformation of a giant unilamellar vesicle (GUV) under an alternating (ac) electric field. Suspended in a leaky dielectric fluid, the vesicle membrane is modeled as a thin capacitive spheroidal shell. The equilibrium vesicle shape results from the balance between mechanical forces from the viscous fluid, the restoring elastic membrane forces, and the externally imposed electric forces. Our spheroidal model predicts a deformation-dependent transmembrane potential, and is able to capture large deformation of a vesicle under an electric field. A detailed comparison against both experiments and small-deformation (quasispherical) theory showed that the spheroidal model gives better agreement with experiments in terms of the dependence on fluid conductivity ratio, permittivity ratio, vesicle size, electric field strength, and frequency. The spheroidal model also allows for an asymptotic analysis on the crossover frequency where the equilibrium vesicle shape crosses over between prolate and oblate shapes. Comparisons show that the spheroidal model gives better agreement with experimental observations.