Discrete Dynamics in Nature and Society

Volume 2018, Article ID 7017416, 10 pages

https://doi.org/10.1155/2018/7017416

## Continuous Approximation for Interaction Energy Transfer of DNA through Lipid Bilayers

Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Correspondence should be addressed to Mansoor H. Alshehri; as.ude.usk@irhehslahm

Received 4 September 2018; Accepted 11 October 2018; Published 1 November 2018

Academic Editor: Francisco R. Villatoro

Copyright © 2018 Mansoor H. Alshehri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this study the interaction energies for single-stranded DNA and double-stranded DNA molecules with a lipid bilayer are investigated. The 6-12 Lennard-Jones potential and continuous approximation are used to derive analytical expressions for these interaction energies. Assuming that there is a circular gap in the lipid bilayer, we determine the relationship of the molecular interaction energy, including the circular gap radius and the perpendicular distance of the single-stranded DNA and double-stranded DNA molecules from the gap. For both single-stranded and double-stranded DNA molecules, the relationship between the minimum energy location and the hole radius is calculated; in the case of the double-stranded DNA molecule, we assume that the helical phase angle is equal to . By minimizing the total interaction energies, the results demonstrate that the single-stranded DNA and double-stranded DNA molecules move through a lipid bilayer when the gap radius 10 Å and 13.8 Å, respectively. The results present in this project can be leveraged to understand the interactions between cell-penetrating peptides and biomembranes, which may improve gene and drug delivery.

#### 1. Introduction

Recently, nanomaterials have promised an unexpected growth in research and applications in many different areas, due to their nanoscales size, unique features, and distinct properties. Watson and Crick discovered the structure of DNA in 1953, one century after the discovery of the existence of DNA [1]. Owing to the self-assembling, unique physicochemical properties, and the geometric structures of DNA molecules, they have attracted attention as a promising material that has many potential applications in biotechnology and biomedicine such as stochastic biosensors, for the controlled translocation of proteins and as drug carriers across membranes [2–5]. Lipid/DNA complexes hold great promise for future medical applications such as transfect genetic material to the cell core and as novel treatment for various inherited diseases and cancers [6–9]. There are a number of research studies about the interaction behavior of the translocation of molecules into cells, which can help to explain the toxicity of nanoparticles. Maingi et al. use molecular dynamics (MD) simulations to investigate the interactions of a simple DNA nanopore with a lipid bilayer. They demonstrate the close packing of lipids around the stably inserted DNA pore and its cation selectivity [10]. He et al. determined the interaction mechanism between polyarginine peptides and asymmetric membranes by performing a coarse-grained molecular dynamics (CGMD) simulation. Their results show that peptides could be moved through a lipid bilayer by inducing a hydrophilic pore formation in the asymmetric membrane [11, 12]. Khalid et al. utilize (CGMD) simulation to investigate the transfer potential of the DNA through a DPPC/DMTAP bilayer. They also find a high energy barrier to DNA insertion into the bilayer hydrophobic core of the bilayer [13]. In addition, a number of simulations have provided good information about the interactions of DNA and lipids, such as [14–16], where the computational requirements of such a problem did not allow the exploration of the transmigration of the DNA inside the bilayer. Baowan et al. use continuous approximation together with the Lennard-Jones potential function to calculate the interaction energy of silica and carbon nanoparticles with a lipid bilayer [17–19]. Alshehri determined the interactions of boron nitride nanotubes as they moved through lipid bilayer membranes using the same technique [20]. Van der Waals energy has played an important role; hence both electrostatic interactions and van der Waals were considered as a part of the computation. In this study, classical applied mathematics is used to calculate the most important interactions of the system and provide the theoretical results which can pave the way toward further improving this area of study. Using continuous approximation and the Lennard-Jones potential function, the molecular interaction energies between ssDNA and dsDNA molecules and the dipalmitoylphosphatidylcholine bilayer (DPPC) are calculated. The penetration behavior of the DNA molecules through an assumed circular gap in the lipid bilayer is investigated. The model formulations for ssDNA and dsDNA molecules and the lipid bilayer are presented in the following section. The mathematical derivations for the ssDNA and dsDNA interaction with the lipid are detailed in Section 3. Moreover, in Section 4 the numerical results are presented, and finally in Section 5 a brief summary of the project is presented.

#### 2. Modelling Approach

We determine the molecular interatomic energy between two molecules by employing continuous approximation and the Lennard-Jones potential. We assume that the 6-12 Lennard-Jones potential is expressed aswhere is the distance between two well-defined distinct molecules, and denote the attractive and the repulsive constants, respectively, and is the interatomic distance when the potential is zero and is the energy well depth. Using a continuous approach, atoms are assumed to be evenly distributed over the molecules and the average atomic densities provided are averaged over its surface, which means that the summation over all atoms involved is replaced by surface integrals. The magnitude of total energy arising from molecular interatomic energy is then obtained mathematically by performing a surface integral over the surface of each molecule, given bywhere and denote mean volume densities or the mean surface densities of the atoms on each molecule, and is the potential function for two unbonded atoms. For convenience, we could introduce the integral asThus, the total energy can given byAt this point, we investigate the energy behavior of the molecular interaction energy of a lipid bilayer and a DNA molecule, by determining the penetration behavior of the ssDNA and dsDNA molecules through an assumed circular gap of radius in the bilayer. The ssDNA molecule is assumed to be a single helicoid, with a ruled surface having a helix as its boundary. This presumption follows the experimental studies [25–27]. A typical point on the surface of an ssDNA molecule is specified by the coordinates:where Å and Å are the unit cell length and the radius of the ssDNA helix, respectively, and the two parametric variables and are such that and . In addition, the dsDNA molecule is modelled as a surface with a double helicoid geometry, and the parametric equation is given bywhere is the helical phase angle parameter, Å is the radius of the dsDNA helix, Å is the unit cell length of the dsDNA helix, and the parametric variables and are such that and . Moreover, with reference to Figure 1 the bilayer of dipalmitoylphosphatidylcholine (DPPC) has been selected as a part of this experiment, each lipid molecule is composed of 12 particles, which are shown in the MARTINI force field as a head group consisting of choline () and phosphate () groups, an intermediate layer consisting of a glycerol group (), and a carbon tail group () [23]. We note that the phase behavior of complexes and the considerable diversity in structure stability are depending on the choice of the uncharged lipid and that the chemical type of divalent ion can impact on the stability of the complexes which involves specific interactions [28]. From the work of Bawan et el. [17], the spacing between the two layers of lipids is assumed to be 3.36 Å, and the initial value for the area per lipid is assumed to be 65 , which is taken from the work of Berger et al. [29]. The head and tail groups could be modelled as infinite boxes of thicknesses 4 and 15, respectively, following the work of Bawan et el. [18].