A New Mathematical Model for Calculating the Electronic Coupling of a B-DNA Molecule

The charge transport properties of DNA have made this molecule very important for use in nanoscale electronics, molecular computing, and biosensoric devices. Early findings have suggested that DNA can behave as a conductor, semiconductor, or an insulator. This variation in electrical behavior is attributed to many factors such as environmental conditions, base sequence, DNA chain length, orientation, temperature, electrode contacts, and fluctuations. To better understand the charge transport characteristics of a DNA molecule, a more thorough understanding of the electronic coupling between base pairs is required. To achieve this goal, two mathematical methods for calculating the electronic interactions between base pairs of a DNA molecule have been developed, which utilize the concepts from Molecular Orbital Theory (MOT) and Electronic Band Structure Theory (EBST). The electronic coupling characteristics of a B-DNA molecule consisting of two Guanine-Cytosine base pairs have been examined for variation in the twist angle between the base pairs, the separation between base pairs, and the separation between base molecules in a given base pair, for both the HOMO and LUMO states. Comparison of results to published literature reveals similar outcomes. The electronic properties (metallic, semi-conducting, insulating) of a B-DNA molecule are also determined.


Introduction
The B-DNA molecule is considered to be a potential building block for molecular electronics due to its self-assembly and self-recognition properties. The field of B-DNA electronics is highly interdisciplinary, merging physics, biology, chemistry, computer science, and engineering. Therefore, this new field shows great promise in regards to using individual DNA molecules for producing a new range of electronic devices such as nanoscale electronics [1,2], molecular computing [3,4], and biosensoric devices [5,6], that are much smaller, faster, and more energy efficient than the present semiconductor-based electronic devices. Also, because of its self-assembly and self-recognition characteristics, DNA can easily adopt to various states and conformations, thereby providing the possibility of producing nanostructures with very high precision, beyond what is achievable with traditional silicon-based technologies [2]. Equally important is the understanding of charge transport in DNA in relation to damage and mutation throughout the macromolecule [7][8][9], the detecting, manipulating, and sequencing of DNA [10][11][12], and the transport properties of other systems with π π − interactions, such as molecular crystals and discotic materials [13,14].
Several researchers have suggested theoretically or shown experimentally, that the B-DNA molecule has electrical conducting properties. As early as 1959, Eley [15] proposed that a DNA molecule might behave as a one-dimensional aromatic crystal and illustrated electron conductivity along the helical axis. His proposal was based on the results of the electrical conductivity of crystalline organic substances ( πelectron compounds). There are many substances that behave as semiconductors with an energy gap that decreases with the number of mobile π -electrons in the molecule, thus suggesting that conductivity is associated with the intermolecular tunneling of thermally excited π -electrons. By 1960, Ladik [16] showed that the sigma coupling could be sufficient to allow for conductivity if some of the bases were in an excited or ionized state where the one-electron orbital overlap integrals for the sigma coupling between DNA base molecules are perpendicular to the helical axis. About the same time as Ladik, Pullmann and Pullmann [17] demonstrated through the use of Molecular Orbital Theory calculations that the Guanine-Cytosine (GC) base pair would be a better electron donor and electron acceptor than the Adenine-Thymine (AT) base pair. In 1962, Eley and Spivey [18], experimentally showed that π π − interactions of the stacked base pairs in DNA could lead to conducting behavior.
The interest in charge transfer in DNA took off in the early 1990s when Barton and Turro suggested that ultra-fast photo-induced charge transfer can occur over large distances between donors and acceptors that are inserted in the DNA [19][20][21]. Their hypothesis sparked a wide variety of experimental and theoretical studies into the nature of charge migration in DNA. Several charge transport experiments [22][23][24][25][26][27][28][29][30][31] have been performed on single DNA molecules revealing that different electrical characteristics exist: insulating [25,26], semiconducting [27,28], ohmic [29,30], and superconducting [31]. The variation in the charge transport properties may be due to the high sensitivity of charge propagation in DNA to extrinsic (interaction with hard substrates, metal-molecule contacts, aqueous environments) as well as intrinsic (dynamical structure fluctuations and base pair sequence) influences. This paper will be focusing only on the charge transfer in a DNA molecule which is influenced by intrinsic properties, in particular, the dynamical structure fluctuations and electronic coupling between base pairs. Two mathematical methods required for the calculation of the electronic coupling between the base pairs of a DNA molecule were developed. The first method utilizes Molecular Orbital Theory, specifically, Linear Combination of Atomic Orbitals (LCAO) overlap integrals to calculate the bond integral parameter k values, which are then used in a highly modified version of the extended Hückel method to determine the molecular orbital wave function coefficients. This more generalized form of the extended Hückel method was developed by essentially eliminating most of its assumptions and some of its limitations, resulting in more accurate values for the coefficients. The second method utilizes an amended version of the Slater-Koster relations from Electronic Structure Theory to acquire the necessary interatomic matrix elements. This amendment was required due to the non-covalent interactions involved between the B-DNA base pairs. Only the LCAO coefficients for the frontier (HOMO and LUMO) molecular orbital wave functions were considered. With known LCAO coefficients and interatomic matrix elements, the electronic coupling parameter t for a two base pair B-DNA molecular system was ascertained.
The electronic coupling parameter t varies significantly as the DNA structure changes due to mechanical influences. These changes in t are illustrated and discussed for variations in the twist angle between the base pairs, the separation between base pairs, displacement of one base pair with respect to the other in the x ± and y ± directions, and displacement between base molecules for one base pair with respect to the other in the x ± and y ± directions. There exist similarities between the results for the variations in the twist angle and separation of distance between base pairs and those of published literature [32]. The reasons for discrepancies between the values presented in this paper and those of published literature [32] are discussed.

Theory
The electronic coupling between two successive base pairs [33] is a summation of the products of the LCAO molecular orbital coefficients, s c ' , of both base pairs and the interatomic matrix elements, z z E , , between the base pairs. The coefficients are calculated utilizing a newly developed method, which is a more generalized method than the ordinary extended Hückel method, due to the elimination of most of its assumptions and some of its limitations, and will be referred to as the modified-extended Hückel (meH) method. The interatomic matrix elements are typically determined from the Slater-Koster relations [34], but for this work these relations were amended to include both attractive and repulsive terms in order to more accurately calculate the non-covalent interactions that exists between B-DNA base pairs.
The B-DNA model used in this article consists of two stacked GC base pairs. Each base pair contains nineteen atomic locations, which is characterized by the general molecular orbital energy equation  [35] were used; however, the k parameter was determined using the meH method because its value depends upon the distance between two atomic orbitals. By definition, , / o k β β = where β is the measured interaction energy between two atomic orbitals, and o β represents a standard β for the benzene bond distance (1.397Å). β values can be difficult to obtain; thus, the overlap integral S can be used instead as it is determined theoretically. It has been proposed that S ∝ β [36], where S is a non-energy quantity. The values of the k parameter for all the combinations of paired atomic orbitals in the B-DNA molecule were ascertained using The equations for S for all Slater type atomic orbital pairs consisting of , , σ np ns and π np atomic orbitals for n=1,2,3, and 5 have been formulated [37]. It is well known that only the z p atomic orbitals contribute to the electronic coupling; thus, only the equations for σ p 2 and π p 2 atomic orbital pairs are considered. A σ p 2 atomic orbital pair is realized when one z p atomic orbital is aligned parallel to but directly above the other z p atomic orbital, and a π p 2 atomic orbital pair is created when one z p atomic orbital is parallel to but coplanar with the other z p atomic orbital ( Figure 1).
where the parameters l A and l B are defined as for , 2 , 0 = l and . 4 Equations (6) and (7) were obtained from published literature [37,38]. p and q are described by where a µ and b µ are Slater values, and H a =Bohr radius (0.529Å). The coefficients were only determined for the frontier molecular orbitals (HOMO and LUMO).
Applying the LCAO method to solids in a rigorous manner is quite complicated. However, treating the LCAO method as an interpolation process allows simplifications to be made [34] which permit the potential energy in the Hamiltonian H to be treated as a sum of spherical potentials located on the two atoms on which the atomic orbitals are located. The energy matrix components of the Hamiltonian operator are defined using  Table I of Ref. [34]. In this form, the function n ψ is a  [39] were formulated to work well within a typical covalent bond distance but not for non-covalent interactions nor covalent interactions at distances greater than 2Å. This difficulty can be resolved by changing Harrison's ) / 1 ( d term to a standard ) / 1 ( 0 d term and by incorporating attractive and repulsive terms, which are used to scale the non-covalent interactions [33,41,42]. The attractive and repulsive terms [40,41] are Theory code like SIESTA. The electronic coupling between two adjacent base pairs represents the strength of the coupling or the potential energy between two base pairs and is written as Each z z E , makes a small contribution to the electronic coupling t . Equation (12) shows that t will vary depending on how well the base pairs are stacked. In other words, by changing the B-DNA structure, such as by twisting the base pairs with respect to each other or varying the distance between the base pairs, t may change significantly. Hence, it is very important to understand the dependency of t regarding these conformational changes because it will affect the charge transport characteristics of the B-DNA molecule.

Results and Discussion
In this section, the results for a B-DNA molecule consisting of two GC base pairs will be presented along with a discussion. The data pertaining to the electronic coupling for the HOMO and LUMO energy states are provided as a function of the base pair's twist angle, separation distance between base pairs, displacement of one base pair with respect to the other base pair in the x-and y-axes, and displacement of the base molecules within a given base pair with respect to the other base pair also in the x-and y-axes. In addition, the results representing only the electronic coupling as a function of the base pair's twist angle and the separation distance between base pairs are compared to previous work [32], which reveals similar outcomes. The slight differences in the calculated versus the published data are discussed.
The electronic coupling t is very dependent on the motion of the base pairs with respect to each other (   When t becomes negative, then the π pp interactions dominate with very little σ pp interactions occurring. An interesting note is that for the equilibrium twist angle of °36 , the calculated LUMO state electronic coupling has a negative value as compared to the published value. The electronic coupling also varies as a function of distance between the base pairs ( Figure 3). Similar trends between the calculated and published data appear, with the exception of the published values for the LUMO state. The twist angle is held constant at °36 with the other parameter values remaining unchanged, except for the separation distance.
There are numerous reasons for these discrepancies (Figures 2 and 3). Firstly, according to Ref. [32], a combination of a DFT code (SIESTA), which utilized a Double -Zeta Gaussian with Polarization (DZP) basis set, and a parameterized Slater-Koster model was utilized in generating the published values. The Slater-Koster parameters σ η pp , π η pp and c R (0.87Å for B-DNA) were obtained by fitting to DFT information. Secondly, a Complete Neglect of Differential Overlap (CNDO) approximation was incorporated in order to simplify calculations by reducing the number of electron repulsive integrals to be calculated. Thirdly, a simple Hückel method was considered.
Whereas the model that was used to produce the calculated values consists of a combination of a generalized form of the extended Hückel method and a highly modified Slater-Koster method. This combination uses Coulomb and bond integral parameters, and attractive and repulsive exponential terms. In addition, no exponential cutoff distances and CNDO approximations were required due to the inclusion of the repulsive term used in Eqs. (10) and (11), which represents contributions from the electron-electron Coulomb repulsion and the ion-ion Coulomb repulsion. By not requiring an exponential cutoff distance, the exponential tails of the orbital wave functions extend to infinity, which means that the new model may provide more accurate approximations of the effects on electronic coupling due to additional σ pp interactions occurring at larger distances. In Ref. [32], the results pertaining to changes of the electronic coupling associated with rotation and separation between two base pairs only were considered, and any motion related to translations between two base pairs and between base molecules was ignored. In this paper, the additional degrees of freedom are contemplated and the data is discussed below.     The effects on the electronic coupling associated with the base molecules in base pair 2 translating in opposite directions with respect to each other and to a stationary base pair 1 are presented (Figures 6 and 7). For the HOMO state   There are similarities in the profile between the HOMO states for the x ± and y ± directions, and LUMO states for the x ± and y ± directions (Figures 4 and 6). Also, there exist similarities in profile when the twist angle is set to 36˚ (Figures 5 and 7). Additionally, the last four figures have revealed some intriguing results, which have provided further insight into the variation of the electronic coupling. The twist angle that exhibited minimum variation of the electronic coupling in regards to both the translation in the x ± and y ± directions of base pair 2 and between the base molecules in base pair 2 is °= 24 φ (Figures 8 and 9). Because t has mostly negative values it is indicative of π pp interactions dominating, and that, relatively speaking, the electronic coupling between the two base pairs in the x ± and y ± range are the same (Figures 8 and 9). In general, when the electronic coupling is at a maximum, whether due to σ pp or π pp interactions, and a significant amount of wave function overlap exists, the final electronic state is a metallic electronic state. If the electronic coupling vanishes, then an insulating electronic state exists. When the electronic coupling is between these two conditions, there exists a semiconducting electronic state. Therefore, because a DNA molecule is very dynamic, it is conceivable that for a long DNA molecular chain, these three electronic states can occur simultaneously along the length of the chain.

Conclusion
The theories that were utilized for the calculation of the electronic coupling for holes and electrons that propagate from one base pair to another were presented. In particular, Molecular Orbital Theory (MOT) and Electronic Band Structure Theory (EBST) were explained. With regard to the MOT and EBST, two new mathematical methods were presented and shown to provide good values, which were in agreement with published literature [32,33]. The slight differences in the calculated values versus the published, due to a difference in methodology, were discussed. Additionally, the equation that was utilized in the calculation of the electronic coupling between two adjacent base pairs was provided.
In this paper, the electronic coupling results for a B-DNA molecule consisting of two GC base pairs were discussed. These solutions were presented as functions of the base pairs' twist angle, separation distance between base pairs, displacement of one base pair with respect to the other base pair in the x-and y-axes, and displacement of the base molecules within a given base pair with respect to the other base pair also in the x-and y-axes, for both the HOMO and LUMO states. In summary, the electronic coupling is quite sensitive to changes in the twist angle and separation distance between the base pairs, but less sensitive to changes in the translation directions, especially at a twist angle of °24 .