# Monovalent Ions and Water Dipoles in Contact with Dipolar Zwitterionic Lipid Headgroups-Theory and MD Simulations

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## Abstract

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## 1. Introduction

## 2. Model

#### 2.1. Modified Langevin-Poisson-Boltzmann (MLPB) Model

^{2}, where n = 1.33 is the optical refractive index of water. The relative (effective) permittivity of the electrolyte solution ɛ

_{r}can be then expressed as [13,32] :

**P**| is the magnitude of the polarization vector due to a net orientation of permanent point-like water dipoles having dipole moment

**p**, ɛ

_{0}is the permittivity of the free space, while E = |

**E**| is the magnitude of the electric field strength. The absolute value of polarization P(x) is given by [13] :

_{w}(x) is the space dependency of the number density of water molecules, p

_{0}is the magnitude of the external dipole moment

**p**

_{e}= (3/(2 + n

^{2}))

**p**(see also Figure 1) [13,32], $\mathcal{L}$(u) = (coth(u)−1/u) is the Langevin function, β = 1/kT, kT is thermal energy, while γ is [13] :

_{w}

_{0}, i.e., n

_{w}(x) = n

_{0}

_{w}from where it follows :

_{r}(x) gives the classical Onsager expression:

_{r}= 78.5 for bulk solution. The parameters p

_{0}and n

_{0}

_{w}/N

_{A}are 3.1 Debye and 55 mol/l, respectively.

#### 2.2. Poisson Equation

_{0}/a

_{0}at x = 0 (see Figure 2), where a

_{0}is the area per lipid. The orientational ordering of water is taken into account assuming the spatial dependence of permittivity ɛ

_{r}(x) as described by Equation 6.

_{Zw}(x) is the macroscopic (net) volume charge density of positive charges of dipolar (zwitterionic) headgroups, ρ

_{ions}(x) is the macroscopic (net) volume charge density of co-ions (n

_{−}) and counter-ions (n

_{+}) of the electrolyte solution (see Figure 3). Since we neglect the finite volumes of the salt ions and water molecules [21,22] the co-ions and counter-ions are assumed to be distributed according to Boltzmann distribution functions [20,22–26]) :

_{0}is the unit charge and n

_{0}bulk number density of salt co-ions and counter-ions. The lipid headgroups can be oriented at various angles ω relative to the normal vector to the planar lipid layer (Figure 2), hence the volume charge density due to the positive charges of the lipid dipolar headgroups can be written in the form:

_{0}is the area per lipid, while ℘(x) is the probability density function indicating the probability that the positive charge of the dipolar lipid headgroup is located at the distance x from the negatively charged surface at x = 0 :

_{0}/a

_{0}. Note that the area per lipid a

_{0}is different in gel and liquid phase. E = |φ′|.

_{0}= 0.48nm

^{2}and a

_{0}= 0.60nm

^{2}corresponding to DPPC lipid in gel-crystalline (below 314K) and liquid-crystalline phase (above 314K), respectively [33]. Other values of model parameters were: the dipole moment of water p

_{0}= 3.1 Debye, bulk concentration of salt n

_{0}/N

_{A}= 0.1 mol/l and concentration of water n

_{0}

_{w}/N

_{A}= 55 mol/l. N

_{A}is Avogadro number.

#### 2.3. Molecular Dynamics Simulations (MD)

^{+}and 153 Cl

^{−}ions [34,35]. Chemical bonds between hydrogen and heavy atoms were constrained to their equilibrium value. Long-range electrostatic forces were taken into account using a fast implementation of the particle mesh Ewald (PME) method [36,37]. The model was examined at constant pressure (1.013 × 10

^{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.

## 3. Results

_{r}as a function of the distance from the charged planar surface (x = 0) is presented in Figure 4. The results are presented for two values of the temperature: T = 310K (a

_{0}= 0.48nm

^{2}) and T = 323K (a

_{0}= 0.60nm

^{2}). It can be seen in Figure 4 that the relative permittivity ɛ

_{r}(x) is considerably decreased in the vicinity of charged planar surface. At the charged planar surface (x = 0) the value of ɛ

_{r}(x) drops to 44 at T = 310K and to 55 at T = 323K. The effect of the charged planar surface at x = 0 is very weak already at the distance x = D. Far away from the surface (x = 0) the values of ɛ

_{r}(x) is 75.5 at temperature T = 310K and 72.6 for temperature T = 323K. The electric potential in the vicinity of the charged planar surface is considerably negative. It is −60mV for temperature T = 310K and −54mV for temperature T = 323K.

_{−}) and counter-ions (n

_{+}) of the electrolyte solution for two temperatures 310K and 323K can be seen in Figure 3. Near the negatively charged planar surface at x = 0 one can observe strong accumulation of positively charged counter-ions and depletion of the negatively charged co-ions. With increasing distance from the headgroup region (0 < x ≤ D), i.e., for x larger than D, the number density of co-ions (n

_{−}) decreases and the number density of counter-ions (n

_{+}) increases. Far away from the charged planar surface, the concentration of counter-ions (n

_{+}) equals the concentration of co-ions (n

_{−}) corresponding to electroneutrality condition in bulk solution. At higher temperature (323K) the DPPC has increased area per lipid a

_{0}= 0.60nm

^{2}resulting in lower area density of the lipid molecules and hence less negative surface charge density at x = 0 plane as in the case of lower temperature (310K). Consequently, also the calculated ion number density profiles are lower for x > D (Figure 3).

_{0}in gel-crystalline phase and liquid-crystalline phase.

## 4. Discussion and Conclusions

_{r}is usually considered as a constant, it is shown in present paper that ɛ

_{r}can considerably change within the dipolar (zwitterionic) lipid headgroup region. Consequently, the electric potential in this region is substantially decreased.

## Acknowledgements

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## A. Derivation of Equations for Matlab

#### A.1. Variant A

_{r}(x) (Equation 6) and the integral ∫ exp(−ψ(x))dx and dɛ

_{r}(x)/dx in Equation 22 were calculated outside of bvp4c function. The dɛ

_{r}(x)/dx was derived from Equation 6, where the Langevin function $\mathcal{L}$(x) was expanded for small values of electric field into Taylor series up to the cubic term $\mathcal{L}$(x) ≈ x/3 − x

^{3}/45. Considering $\tilde{E}(x)=\left|\frac{\text{d}\psi (x)}{\text{d}x}\right|$, can be written :

_{r}(x) and the dɛ

_{r}(x)/dx are derived from Equation 6 with non-expanded Langevin function :

#### A.2. Variant B

## B. Average Orientation of Lipid Head-Groups

**Figure 1.**A single water molecule is considered as a sphere with permittivity n

^{2}and point-like rigid (permanent) dipole with dipole moment

**p**at the center of the sphere [13], where n is the optical refractive index of water.

**Figure 2.**Negative charges of dipolar (zwitterionic) lipid headgroups are described by negative surface charge density σ = −e

_{0}/a

_{0}at x = 0, where a

_{0}is the area per lipid. D is the distance between charges within the single dipolar lipid headgroup, while ω describes orientation angle of the dipole within the single headgroup. MD model of DPPC lipid molecule is presented at the bottom.

**Figure 3.**The calculated charge density profile of co-ions (n

_{−}) (A,B) and counter-ions (n

_{+}) (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 (a

_{0}= 0.48nm

^{2}below 314K and a

_{0}= 0.60nm

^{2}above 314K). The dipole moment of water was p

_{0}= 3.1 Debye, D = 0.42nm, bulk concentration of salt n

_{0}/N

_{A}= 0.1 mol/l and concentration of water n

_{0}

_{w}/N

_{A}= 55 mol/l, where N

_{A}is Avogadro number.

**Figure 4.**Electric potential φ and relative permittivity ɛ

_{r}as a function of the distance from the charged planar surface at x = 0 calculated for two temperatures and corresponding values of a

_{0}: T = 310K, a

_{0}= 0.48nm

^{2}(full blue lines) and T = 323K, a

_{0}= 0.60nm

^{2}(dashed red lines). These values of a

_{0}correspond to DPPC in two different phases. Relative permittivity ɛ

_{r}as the function of the distance from the charged planar surface x (panel B) was calculated from Equation 6. The MLPB equation was solved numerically as described in the Appendix A. The dipole moment of water p

_{0}= 3.1 Debye, D = 0.42nm bulk concentration of salt n

_{0}/N

_{A}= 0.1 mol/l, concentration of water n

_{0}

_{w}/N

_{A}= 55 mol/l, where N

_{A}is Avogadro number. For simplicity the second boundary condition was applied at a distance of 12nm.

**Figure 5.**Average headgroup dipole orientation angle < ω > (see also Figure 2) as a function of the temperature T. Area per lipid a

_{0}= 0.48nm

^{2}below 314K, corresponding to DPPC lipid gel-crystalline phase (full line) and a

_{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 < ω > was calculated from ℘ (x) as described in Appendix B. The values of other model parameters are the same as in Figure 4.

**Figure 6.**Probability density ℘(x) that the positive charge of the lipid dipolar headgroup (see also Figure 2) is located at the distance x from the negatively charged surface calculated from MLPB model (A,C,D,E) and obtained from MD simulations (B). The values of MLPB model parameters are the same as in Figure 4.

© 2013 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Velikonja, A.; Perutkova, Š.; Gongadze, E.; Kramar, P.; Polak, A.; Maček-Lebar, A.; Iglič, A.
Monovalent Ions and Water Dipoles in Contact with Dipolar Zwitterionic Lipid Headgroups-Theory and MD Simulations. *Int. J. Mol. Sci.* **2013**, *14*, 2846-2861.
https://doi.org/10.3390/ijms14022846

**AMA Style**

Velikonja A, Perutkova Š, Gongadze E, Kramar P, Polak A, Maček-Lebar A, Iglič A.
Monovalent Ions and Water Dipoles in Contact with Dipolar Zwitterionic Lipid Headgroups-Theory and MD Simulations. *International Journal of Molecular Sciences*. 2013; 14(2):2846-2861.
https://doi.org/10.3390/ijms14022846

**Chicago/Turabian Style**

Velikonja, Aljaž, Šarka Perutkova, Ekaterina Gongadze, Peter Kramar, Andraž Polak, Alenka Maček-Lebar, and Aleš Iglič.
2013. "Monovalent Ions and Water Dipoles in Contact with Dipolar Zwitterionic Lipid Headgroups-Theory and MD Simulations" *International Journal of Molecular Sciences* 14, no. 2: 2846-2861.
https://doi.org/10.3390/ijms14022846