# Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Tight-Binding and Its Application in Nucleic Acids

#### 2.1. Wire Model

#### 2.2. Ladder Model

#### 2.3. Extended Ladder Model

#### 2.4. Fishbone Model

#### 2.5. Fishbone Ladder Model

#### 2.6. Additional Remarks

## 3. Aperiodic One-Dimensional Wires

#### 3.1. Aperiodic Substitutional Sequences

#### 3.2. Primitive Substitutions and the Perron–Frobenius Eigenvalue

#### 3.3. Induced Substitutions

#### 3.4. The Pisot Property

- (1)
- strictly quasiperiodic sequences, in which the rank of the reciprocal lattice ${n}_{r}$ is finite and larger than the dimension of the physical space of the sequence m, and
- (2)
- limit-quasiperiodic sequences, in which the rank of reciprocal lattice ${n}_{r}$ is countably infinite (in a 1–1 correspondence with the natural numbers or integers).

- (3)
- limit-periodic, i.e., a superposition of countably infinite periodic structures. Some examples are the period doubling sequence and metallic means sequences with $n=l(l+1)$ [96],
- (4)
- (5)

## 4. Energy Structure of Nucleic Acid Wires

Theorem 5.13 of Ref [108].Let $\widehat{H}$ be a Hamiltionian corresponding to the WM, where the coefficients (i.e., parameters) are determined by a primitive substitution on a finite alphabet. Then, the values of the IDOS of $\widehat{H}$ on the spectral gaps in $[0,1]$ belong to the $\mathbb{Z}\left({\lambda}_{PF}^{-1}\right)$ module generated by the components of the eigenvectors ${\overrightarrow{\mathbf{v}}}_{PF}$ and ${\overrightarrow{\mathbf{v}}}_{PF,2}$ of the substitution matrices $\mathbf{S}$ and ${\mathbf{S}}_{2}$, respectively.

## 5. Coupling Nucleic Acids with Leads: Transmission Coefficients

## 6. Current–Voltage Curves

- (a)
- The choice of the Fermi level of the leads ${E}_{F}$, which coincides with ${\u03f5}_{M}$ if one electron per site is assumed. If ${E}_{F}$ is not aligned with an allowed energy region of the segment, then no currents occur in the vicinity of $V=0$, while a metallic behavior is expected otherwise.
- (b)
- The way the external bias is applied. For example, only one of the leads’ energy bands can be shifted, so that ${\mu}_{L}={E}_{F}+eV$, and ${\mu}_{R}={E}_{F}$, or, alternatively, both leads’ bands can be symmetrically shifted so that ${\mu}_{\begin{array}{c}L\\ R\end{array}}={E}_{F}\pm \frac{eV}{2}$. This choice affects both the way the voltage drop is induced in the nucleic acid sequence and the energy limits of the conductance channel. At zero temperature, the Fermi–Dirac distributions become Heaviside step functions and determine the limits of integration. Hence, Equation (11) can be simplified to$$I\left(V\right)=\frac{2e}{h}{\int}_{{\mu}_{R}}^{{\mu}_{L}}T(E,V)dE,$$$$I\left(V\right)=\frac{2e}{h}sinh\left(\frac{eV}{2{k}_{B}T}\right){\int}_{-\infty}^{\infty}\frac{T(E,V)dE}{cosh\left(\frac{E-{E}_{F}}{{k}_{B}T}\right)+cosh\left(\frac{eV}{2{k}_{B}T}\right)},$$
- (c)
- Whether or not the transmission coefficient is considered as bias-dependent. Although assuming bias-independent transmission coefficient could be a justified choice in the small bias regime, and it is indeed less computationally costly, this assumption cannot lead, under any circumstances, to the occurrence of negative differential resistance, since an increasingly larger part (as V increases) of a nonnegative function is integrated.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DNA | Deoxyribonucleic Acid |

RNA | Ribonucleic acid |

G | Guanine |

A | Adenine |

C | Cytosine |

T | Thymine |

U | Uracil |

TB | Tight-Binding |

WM | Wire Model |

LM | Ladder Model |

ELM | Extended Ladder Model |

FM | Fishbone Model |

FLM | Fishbone Ladder Model |

DOS | Density of States |

IDOS | Integrated Density of States |

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**Figure 1.**Schematic representation of a TB model consisting of N monomers, extended at L chains. Within the model, we take into account (

**a**) the on-site energies of each site, ${\u03f5}_{n}^{l}$, and the inter-chain hopping integrals, ${t}_{n}^{l{l}^{\prime}}$, i.e., between the sites of the monomer (blue), as well as (

**b**) the inter-monomer hopping integrals, ${t}_{n{n}^{\prime}}^{l{l}^{\prime}}$, i.e., between each site of a monomer and the neighboring sites of the previous (red) and the next (green) monomers. The former are contained in the matrix ${\mathit{\epsilon}}_{n}$, while the latter in the matrices ${\mathit{\tau}}_{n-1}$ and ${\mathit{\tau}}_{n}$, respectively.

**Figure 2.**Schematic representation of the TB models listed in Table 1. (

**a**) Wire Model (WM); (

**b**) Ladder Model (LM); (

**c**) Extended Ladder Model (ELM); (

**d**) Fishbone Model (FM); (

**e**) Fishbone Ladder Model (FLM).

**Figure 3.**Normalized IDOS of various categories of binary DNA segments with purines on the same strand, within the WM. (

**a**) Poly(GA); (

**b**) Thue–Morse; (

**c**) Fibonacci (

**d**) Period-doubling; (

**e**) Rudin–Shapiro; (

**f**) Cantor Set; (

**g**) Generalized Cantor Set (4,2); (

**h**) Kolakoski(1,2); (

**i**) Kolakoski(1,3); (

**j**) Random (50% G, 50% A). Reprinted figure from K. Lambropoulos and C. Simserides, Periodic, quasiperiodic, fractal, Kolakoski, and random binary polymers: Energy structure and carrier transport, Phys. Rev. E

**2019**, 99, 032415 [83] http://dx.doi.org/10.1103/PhysRevE.99.032415, © 2019 by the American Physical Society.

**Figure 4.**DOS of (

**a**) poly(CG), and (

**b**) poly(CT) DNA strands with diluted base-pairing at random cytosine sites with probability p. Figure reproduced with permission from F. A. B. F. de Moura, M. L. Lyra and E. L. Albuquerque, Electronic transport in poly(CG) and poly(CT) DNA segments with diluted base pairing, J. Phys. Condens. Matter

**2008**, 20, 075109 [111] http://dx.doi.org/10.1088/0953-8984/20/7/075109, © 2008 IOP Publishing. All rights reserved.

**Figure 5.**Transmission coefficient for a poly(GACT) chain within the WM, with $N=60$, ${t}_{M}=1.0$ eV, $t=0.4$ eV, and $\mathit{\tau}=0.4$ eV; i.e., $\mathit{\tau}=\sqrt{{t}_{M}t}$ (

**top**), $\mathit{\tau}=\sqrt{0.4}$ eV (

**middle**), $\mathit{\tau}=\sqrt{0.8}$ eV (

**bottom**). Reprinted figure with permission from E. Maciá, F. Triozon, and S. Roche, Contact-dependent effects and tunneling currents in DNA molecules, Phys. Rev. B

**2005**, 71, 113106 [114] http://dx.doi.org/10.1103/PhysRevB.71.113106, © 2005 by the American Physical Society.

**Figure 6.**Transmission coefficient of a periodic WM with two sites per unit cell and $N=10$ for ideal (

**top**), strong (

**middle**), and weak (

**bottom**) coupling with the leads. (Left column) Symmetric coupling. (Middle column) Asymmetric coupling with $\left|\chi \right|>1$. (Right column) Asymmetric coupling with $\left|\chi \right|<1$. The leads parameters are such that all the eigenstates of the system are contained. Reprinted from Ref. [60], K. Lambropoulos and C Simserides, Spectral and transmission properties of periodic 1D tight-binding lattices with a generic unit cell: an analysis within the transfer matrix approach, J. Phys. Commun.

**2018**, 2, 035013 [60] http://dx.doi.org/10.1088/2399-6528/aab065, CC BY 3.0.

**Figure 7.**Transmission spectra as a function of energy without (

**a**–

**c**) and with (

**d**–

**f**) the diagonal hoppings; (

**g**–

**i**) average transmission spectra as a function of energy: gray line (diagonal hoppings switched off) and black line (diagonal hoppings switched on). (Left column) ${t}_{L}={t}_{R}=0.1$ eV. (Middle column) ${t}_{L}={t}_{R}=0.5$ eV. (Right column) ${t}_{L}={t}_{R}=0.9$ eV. Reprinted from S. Malakooti, E. R. Hedin, Y. D. Kim, and Y. S. Joe, Enhancement of charge transport in DNA molecules induced by the next nearest-neighbor effects, J. Appl. Phys.

**2012**, 112, 094703 [116], http://dx.doi.org/10.1063/1.4764310, with the permission of AIP Publishing.

**Figure 8.**$I-V$ curves for poly(dA)-poly(dT) and poly(dG)-poly(dC) various disorder strengths w and pitch-size values n. For weak disorder, the cut-off voltage reduces with n, showing semiconducting behaviour. For strong disorder, the current is considerably enhanced with increasing n, giving a insulator to metal transition. Reproduced from Ref. [121], S. Kundu and S. N. Karmakar, Conformation dependent electronic transport in a DNA double-helix, AIP Adv.

**2015**, 5, 107122 [121] http://dx.doi.org/10.1063/1.4934507, CC BY 3.0.

**Table 1.**Form of the matrices ${\overrightarrow{\mathbf{\Psi}}}_{n}$, ${\mathit{\epsilon}}_{n}$, ${\mathit{\tau}}_{n}$ in the TB system of equations (Equation (1)) for several models used to describe nucleic acids and analogues: the Wire Model (WM), the Ladder Model (LM), the Extended Ladder Model (ELM), the Fishbone Model (FM) and the Fishbone Ladder Model (FLM).

Model | L | ${\overrightarrow{\mathbf{\Psi}}}_{\mathit{n}}$ | ${\mathit{\epsilon}}_{\mathit{n}}$ | ${\mathit{\tau}}_{\mathit{n}}$ |
---|---|---|---|---|

WM | 1 | ${\psi}_{n}$ | ${\u03f5}_{n}$ | ${t}_{n,n+1}$ |

LM | 2 | $\left(\begin{array}{c}{\psi}_{n}^{1}\\ {\psi}_{n}^{2}\end{array}\right)$ | $\left(\begin{array}{cc}{\u03f5}_{n}^{1}& {t}_{n}^{1,2}\\ {t}_{n}^{2,1}& {\u03f5}_{n}^{2}\end{array}\right)$ | $\left(\begin{array}{cc}{t}_{n,n+1}^{1,1}& 0\\ 0& {t}_{n,n+1}^{2,2}\end{array}\right)$ |

ELM | 2 | $\left(\begin{array}{c}{\psi}_{n}^{1}\\ {\psi}_{n}^{2}\end{array}\right)$ | $\left(\begin{array}{cc}{\u03f5}_{n}^{1}& {t}_{n}^{1,2}\\ {t}_{n}^{2,1}& {\u03f5}_{n}^{2}\end{array}\right)$ | $\left(\begin{array}{cc}{t}_{n,n+1}^{1,1}& {t}_{n,n+1}^{1,2}\\ {t}_{n,n+1}^{2,1}& {t}_{n,n+1}^{2,2}\end{array}\right)$ |

FM | 3 | $\left(\begin{array}{c}{\psi}_{n}^{1}\\ {\psi}_{n}^{2}\\ {\psi}_{n}^{3}\end{array}\right)$ | $\left(\begin{array}{ccc}{\u03f5}_{n}^{1}& {t}_{n}^{1,2}& 0\\ {t}_{n}^{2,1}& {\u03f5}_{n}^{2}& {t}_{n}^{2,3}\\ 0& {t}_{n}^{3,2}& {\u03f5}_{n}^{3}\end{array}\right)$ | $\left(\begin{array}{ccc}0& 0& 0\\ 0& {t}_{n,n+1}^{2,2}& 0\\ 0& 0& 0\end{array}\right)$ |

FLM | 4 | $\left(\begin{array}{c}{\psi}_{n}^{1}\\ {\psi}_{n}^{2}\\ {\psi}_{n}^{3}\\ {\psi}_{n}^{4}\end{array}\right)$ | $\left(\begin{array}{cccc}{\u03f5}_{n}^{1}& {t}_{n}^{1,2}& 0& 0\\ {t}_{n}^{2,1}& {\u03f5}_{n}^{2}& {t}_{n}^{2,3}& 0\\ 0& {t}_{n}^{3,2}& {\u03f5}_{n}^{3}& {t}_{n}^{3,4}\\ 0& 0& {t}_{n}^{4,3}& {\u03f5}_{n}^{4}\end{array}\right)$ | $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& {t}_{n,n+1}^{2,2}& 0& 0\\ 0& 0& {t}_{n,n+1}^{3,3}& 0\\ 0& 0& 0& 0\end{array}\right)$ |

**Table 2.**Substitutional sequences studied in the literature, together with the alphabets through which they are defined, the corresponding substitution rules, and the substitution matrices. In the last row, the subscripts o and e in the substitution rules denote substitutions that are applied on odd and even positions in the sequence, respectively.

Sequence | $\mathcal{A}$ | Substitution Rule | S |
---|---|---|---|

Fibonacci | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = AB s(B) = A | $\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$ |

Precious means | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = A${}^{n}$B s(B) = A | $\left(\begin{array}{cc}n& 1\\ 1& 0\end{array}\right)$ |

Fibonacci-class | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = B${}^{n-1}$AB s(B) = B${}^{n-1}$A | $\left(\begin{array}{cc}1& 1\\ n& n-1\end{array}\right)$ |

Mixed means | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = A${}^{n}$B${}^{m}$s(B) = A | $\left(\begin{array}{cc}n& 1\\ m& 0\end{array}\right)$ |

Metallic means | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = AB${}^{n}$s(B) = A | $\left(\begin{array}{cc}1& 1\\ n& 0\end{array}\right)$ |

Period doubling | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = AB s(B) = AA | $\left(\begin{array}{cc}1& 2\\ 1& 0\end{array}\right)$ |

Thue–Morse | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = AB s(B) = BA | $\left(\begin{array}{cc}1& 2\\ 1& 0\end{array}\right)$ |

Rudin–Shapiro | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{C},\phantom{\rule{4.pt}{0ex}}\mathrm{D}\right\}$ | $\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 1\end{array}\right)$ | |

s(A) = AB s(B) = AC | |||

s(C) = DB s(D) = DC | |||

Triadic Cantor set | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = ABA s(B) = BBB | $\left(\begin{array}{cc}2& 0\\ 1& 3\end{array}\right)$ |

Asymmetric Cantor set | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = ABAA s(B) = BBBB | $\left(\begin{array}{cc}3& 0\\ 1& 4\end{array}\right)$ |

Generalized Cantor set $(t,d)$ | $\left\{\mathrm{A},\phantom{\rule{4.pt}{0ex}}\mathrm{B}\right\}$ | s(A) = A${}^{{\textstyle \frac{t-d}{2}}}$B${}^{d}$A${}^{{\textstyle \frac{t-d}{2}}}$s(B) = B${}^{t}$ | $\left(\begin{array}{cc}t-d& 0\\ d& t\end{array}\right)$ |

Kolakoski $(p=2m,q=2n)$ | $\{\mathrm{A}=pp,\phantom{\rule{4.pt}{0ex}}\mathrm{B}=qq\}$ | s(A) = A${}^{m}$B${}^{m}$s(B) = A${}^{n}$B${}^{n}$ | $\left(\begin{array}{cc}m& n\\ m& n\end{array}\right)$ |

Kolakoski $(p=2m+1,q=2n+1)$ | $\left(\begin{array}{ccc}m& m& n\\ 1& 1& 1\\ m& n& n\end{array}\right)$ | ||

$\{\mathrm{A}=pp,\phantom{\rule{4.pt}{0ex}}\mathrm{B}=pq$, | s(A) = A${}^{m}$BC${}^{m}$s(B) = A${}^{m}$BC${}^{n}$ | ||

C $=qq\}$ | s(C) = A${}^{n}$BC${}^{n}$ | ||

Kolakoski ($p=2m,q=2m+1$) or | $\{p,q\}$ | ${s}_{o}\left(q\right)={p}^{q}$${s}_{o}\left(p\right)={p}^{p}$ | undefinable |

($p=2m+1,q=2m$) | ${s}_{e}\left(q\right)={q}^{q}$${s}_{e}\left(p\right)={q}^{p}$ |

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**MDPI and ACS Style**

Lambropoulos, K.; Simserides, C.
Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport. *Symmetry* **2019**, *11*, 968.
https://doi.org/10.3390/sym11080968

**AMA Style**

Lambropoulos K, Simserides C.
Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport. *Symmetry*. 2019; 11(8):968.
https://doi.org/10.3390/sym11080968

**Chicago/Turabian Style**

Lambropoulos, Konstantinos, and Constantinos Simserides.
2019. "Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport" *Symmetry* 11, no. 8: 968.
https://doi.org/10.3390/sym11080968