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The lipid bilayer is a basic building block of biological membranes and can be pictured as a barrier separating two compartments filled with electrolyte solution. Artificial planar lipid bilayers are therefore commonly used as model systems to study the physical and electrical properties of the cell membranes in contact with electrolyte solution. Among them the glycerol-based polar phospholipids which have dipolar, but electrically neutral head groups, are most frequently used in formation of artificial lipid bilayers. In this work the electrical properties of the lipid layer composed of zwitterionic lipids with non-zero dipole moments are studied theoretically. In the model, the zwitterionic lipid bilayer is assumed to be in contact with aqueous solution of monovalent salt ions. The orientational ordering of water, resulting in spatial variation of permittivity, is explicitly taken into account. It is shown that due to saturation effect in orientational ordering of water dipoles the relative permittivity in the zwitterionic headgroup region is decreased, while the corresponding electric potential becomes strongly negative. Some of the predictions of the presented mean-field theoretical consideration are critically evaluated using the results of molecular dynamics (MD) simulation.

The lipid bilayer is a basic building block of cell membranes. Although the cell membrane is a highly heterogeneous structure, composed of lipids, proteins, carbohydrates and other components [

Among others a planar lipid bilayer can be constructed on a small hole positioned on a barrier separating two compartments filled with electrolyte solution. Experimental setup usually consists of two teflon pieces with separate compartments and 100

Most frequently used lipid molecules in the procedures of forming the planar lipid bilayers are glycerol-based phospholipids [

In general, some of the lipid molecules bear net negative charge, while other lipids like glycerophospholipids are electrically neutral [

Most of the electrostatic models of electrolyte solution in contact with lipid surfaces [

In this paper the effect of the nonhomogeneous volume charge distribution in the headgroup region of the planar dipolar lipid layer on the space dependent electric potential and permittivity is presented. An analytical mean-field model, based on the previously developed Langevin-Poisson-Boltzman (LPB) model [

The Langevin-Poisson-Boltzmann (LPB) model [^{2}, where _{r}

where _{0} is the permittivity of the free space, while

where _{w}_{0} is the magnitude of the external dipole moment _{e}^{2}))

In the following, for the sake of simplicity the finite volume of ions and water molecules is not taken into account as in [_{w}_{0}, _{w}_{0}_{w}

Combining

or

In the limit of vanishing electric field strength (_{r}

At room temperature (298K) the above _{r}_{0} and _{0}_{w}_{A}

In the model the dipolar lipid headgroup is described by two charges at fixed distance _{0}_{0} at _{0} is the area per lipid. The orientational ordering of water is taken into account assuming the spatial dependence of permittivity _{r}

The corresponding Poisson equation in planar geometry can thus be written in the form (see e.g., [

where _{Zw}_{ions}_{−}) and counter-ions (_{+}) of the electrolyte solution (see

therefore

where _{0} is the unit charge and _{0} bulk number density of salt co-ions and counter-ions. The lipid headgroups can be oriented at various angles

where _{0} is the area per lipid, while

where

yields :

Using

The boundary conditions are (see for example [

where in _{0}_{0}. Note that the area per lipid _{0} is different in gel and liquid phase.

In numerical calculations the distance from the negatively charged surface _{0} = 0.48nm^{2} and _{0} = 0.60nm^{2} corresponding to DPPC lipid in gel-crystalline (below 314K) and liquid-crystalline phase (above 314K), respectively [_{0} = 3.1 Debye, bulk concentration of salt _{0}_{A}_{0}_{w}_{A}_{A}

The molecular dynamics (MD) model of DPPC planar lipid bilayer was constructed in NAMD program using all molecule performance CHARMM 36 force field. The model consists of 256 lipid units and 20174 water molecules. The solvent was 450mMKCl modeled by 153K^{+} and 153 Cl^{−} ions [^{5} Pa) and constant temperature (232K) employing Langevin dynamics and the Langevin piston method. The equations of motion were integrated using the multiple time-step algorithm. A time step of 2.0 fs was employed. Short- and long-range forces were calculated every one and two time steps, respectively.

The model was equilibrated and followed 30 ns. The last 15 ns of the simulation were used for extraction of dipole orientation angle. From the P and N atoms positions the dipole was determined for all 256 lipids in each of 1500 simulation frames and exported to Matlab2012b. The distribution of vector amplitude corresponding to distance

Electric potential _{r}_{0} = 0.48nm^{2}) and _{0} = 0.60nm^{2}). It can be seen in _{r}_{r}_{r}

Charge density profile of co-ions (_{−}) and counter-ions (_{+}) of the electrolyte solution for two temperatures 310K and 323K can be seen in _{−}) decreases and the number density of counter-ions (_{+}) increases. Far away from the charged planar surface, the concentration of counter-ions (_{+}) equals the concentration of co-ions (_{−}) corresponding to electroneutrality condition in bulk solution. At higher temperature (323K) the DPPC has increased area per lipid _{0} = 0.60nm^{2} resulting in lower area density of the lipid molecules and hence less negative surface charge density at

The average headgroup orientation angle _{0} in gel-crystalline phase and liquid-crystalline phase.

The comparison between the probability density

where Λ is determined from normalization

Although the relative permittivity _{r}_{r}

To conclude, our model shows that the effect of decreasing potential and permittivity has an impact only in the headgroup region of dipolar zwitterionic lipids and its close vicinity. The average orientation angle of the zwitterionic lipid headgroup dipole (

This work was in part supported by the Slovenian Research Agency. The research was conducted in the scope of the EBAM European Associated Laboratory (LEA). Molecular Dynamics Simulations were performed using HPC resources from Arctur Slovenia. First author was mainly supported by European social fund and SMARTEH.

_{α}

where

and

The boundary conditions (

_{r}_{r}_{r}^{3}

where the constants are :

For high values of electric field the _{r}_{r}

where constants are the same as above (

By inserting the

where the constants

A single water molecule is considered as a sphere with permittivity ^{2} and point-like rigid (permanent) dipole with dipole moment

Negative charges of dipolar (zwitterionic) lipid headgroups are described by negative surface charge density _{0}_{0} at _{0} is the area per lipid.

The calculated charge density profile of co-ions (_{−}) (A,B) and counter-ions (_{+}) (C,D) of the electrolyte solution for two temperatures 310K (full blue line) and 323K (dashed red line) and corresponding DPPC values of the area per lipid (_{0} = 0.48nm^{2} below 314K and _{0} = 0.60nm^{2} above 314K). The dipole moment of water was _{0} = 3.1 Debye, _{0}_{A}_{0}_{w}_{A}_{A}

Electric potential _{r}_{0}: _{0} = 0.48nm^{2} (full blue lines) and _{0} = 0.60nm^{2} (dashed red lines). These values of _{0} correspond to DPPC in two different phases. Relative permittivity _{r}_{0} = 3.1 Debye, _{0}_{A}_{0}_{w}_{A}_{A}

Average headgroup dipole orientation angle _{0} = 0.48nm^{2} below 314K, corresponding to DPPC lipid gel-crystalline phase (full line) and _{0} = 0.60nm^{2} above 314K, corresponding to DPPC lipid liquid-crystalline phase (dashed line). A gap near the phase transition temperature (314K) is present, because phase transition effect is not included in MLPB model. Average dipole orientation angle

Probability density