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Advances in Condensed Matter Physics
Volume 2010, Article ID 380710, 28 pages
Review Article

Some Effective Tight-Binding Models for Electrons in DNA Conduction: A Review

1Yamada Physics Research Laboratory, Aoyama 5-7-14-205, Niigata 950-2002, Japan
2KazumotoIguchi Research Laboratory, 70-3 Shinhari, Hari, Anan, Tokushima 774-0003, Japan

Received 2 April 2010; Accepted 25 May 2010

Academic Editor: Victor Moshchalkov

Copyright © 2010 Hiroaki Yamada and Kazumoto Iguchi. 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.


Quantum transport for DNA conduction has been widely studied with interest in application as a candidate in making nanowires as well as interest in the scientific mechanism. In this paper, we review recent works concerning the electronic states and the conduction/transfer in DNA polymers. We have mainly investigated the energy-band structure and the correlation effects of localization property in the two- and three-chain systems (ladder model) with long-range correlation as a simple model for electronic property in a double strand of DNA by using the tight-bindingmodel. In addition, we investigated the localization properties of electronic states in several actual DNA sequences such as bacteriophages of Escherichia coli, human-chromosome 22, compared with those of the artificial disordered sequences with correlation. The charge-transfer properties for poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA polymers are also presented in terms of localization lengths within the frameworks of the polaron models due to the coupling between the charge carriers and the lattice vibrations of the double strand of DNA.

1. Introduction

Recent interests on semiconducting DNA polymers have been stimulated by successful demonstrations of the nanoscale fabrication of DNA, where current-voltage (I-V) measurements for poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA polymers have been done [14]. For such artificial periodic DNA systems, the energy-band structure is a useful starting point in order to interpret the experimental results such as semiconductivity and the metal-insulator transition [5].

On the other hand, Tran et al. measured conductivity along the double helix of lambda phage DNA (𝜆-DNA) at microwave frequencies, using the lyophilized DNA in and also without a buffer [6]. The conductivity is strongly temperature-dependent around room temperature with a crossover to a weakly temperature dependent conductivity at low temperatures [2]. Yu and Song showed that the 𝜆-DNA can be consistently modeled by considering that electrons may hop through the variable-range hopping for conduction without invoking additional ionic conduction mechanism, and that electron localization is enhanced by strong thermal structural fluctuations in DNA [7]. Indeed, the sequence of base pairs (bp) of the 𝜆-DNA is inhomogeneous as in disordered material systems.

Moreover, charge migrations in DNA have been mainly addressed in order to clarify the mechanism of damage repair which are essential to maintain the integrity of the molecule. The precise understanding of the DNA-mediated charge migration would be important in the descriptions of damage-recognition process and protein binding, or in the engineering biological processes [8, 9]. The stacked array of DNA bp provides an extended path to a long-range charge transfer although the dynamical motions or the energetic sequence-dependent heterogeneities are expected to reduce the long-range migration. Photoexcitation experiments have unveiled that charge excitations can be transferred between metallointercalators through the guanine highest occupied molecular orbitals of the DNA bridge [10]. The subsequently low-temperature experiments showed that the radiation-induced conductivity is related to the mobile charge carriers, migrating within frozen-water layers surrounding the DNA helix, rather than through the base-pair core.

Those experiments are summarized as follows: (i) band gap reduction of a double strand of DNA, (ii) transition from the tunneling hopping to the band hopping, (iii) anomalously strong temperature dependence of band gap, (iv) highly nonlinear temperature dependence of the DC conductivity, and (v) low conductivity of DNA with a complicated sequence such as 𝜆-DNA and high conductivity of DNA with a simple sequence such as poly(dG)-poly(dC) and poly(dA)-poly(dT). These results suggest that the anomalously strong temperature dependence on the physical quantities is attributed to the “self-organised” extrinsic superconductive character of DNA due to the formation of donors and accepters. Here we would like to note the following. Usually, as in solid-state physics, the extrinsic semiconductor is realized when external impurities are introduced in pure substrate materials. Such impurities produce the extrinsic semiconductive nature. However, in real DNA helices, there are no such impurities from outside; but there exists already a complicated arrangement of bases of adenine (A), guanine (G), cytosine (C), and thymine (T) inside the DNA, where among the bases each one of bases may regard other bases as impurities. Hence, a kind of “self-organized” extrinsic semiconductive nature may appear.

However, the electronic transport properties in DNA are still controversial mainly due to the complexity of the experimental environment and the molecule itself. Although theoretical explanations for the phenomena have been tried by some standard pictures used in solid-state physics such as polarons, solitons, electrons, and holes, the situation is still far from unifying the theoretical scheme.

Each DNA sequence is packed in a chromosome, varying in length from 105bp in yeast to 109bp in human. In general, the length of a mutation is relatively short (10bp) as compared to the length of a gene (103-106bp). Because the mutation rate is very low the mechanical and thermodynamic characteristics are maintained for the mutation [11]. In Section 6, we give a brief discussion about the mutation as the proton transfer between the normal and tautomer states.

The Watson-Crick (W-C) base-pair sequence is essential for DNA to fulfill its function as a carrier of the genetic code. The specific characteristics are also used extensively even in various fields such as anthropology and criminal probe. As observed in the power spectrum, the mutual information analysis and the Zipf analysis of the DNA-base sequences such as human-chromosome 22 (HCh-22), the long-range correlation exists in the total sequence, as well as the short-range periodicity. In this paper, we mainly discuss the relationship between the correlation in the DNA-base sequences and the electronic transport/transfer.

The electrical transport of DNA is closely related to the density of itinerant 𝜋-electrons because of the strong electron-lattice interaction. Resistivities of two typical DNA molecules, such as poly(dG)-poly(dC) and 𝜆-DNA, are calculated. At the half-filling state, the Peierls phase transition takes place and the poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers exhibit a large resistivity. When the density of itinerant 𝜋-electrons departs far from the half-filling state, the resistivity of poly(dG)-poly(dC) becomes small. For the 𝜆-DNA, there is no Peierls phase transition due to the aperiodicity of its base-pair arrangement. The resistivity of poly(dG)-poly(dC) decreases as the length of the molecular chain is increasing, while that of 𝜆-DNA increases as the length is increasing.

In Section 3, we introduce the Ḧuckel model of DNA molecules to treat with the 𝜋-electrons. Moreover, in Section 4, we give the effective polaron model including the electron-lattice coupling dynamics. In Section 5, we present correlation effects of localization in the ladder models and the formation of localized polarons (Holstein's polarons or solitons) due to the coupling between the charge carriers and the lattice vibrations of the double strand of DNA. These polarons in DNA act as donors and acceptors and exhibit an extrinsic semiconductor character of DNA. The results are discussed in the context of experimental observations.

The present paper essentially follows the line in [5, 1218]. In particular, it is written with a view of the localization and/or delocalization problem in the quasi-one-dimensional tight-binding models with disorder. In appendices we give some calculations and explanations related to main text.

2. Correlated DNA Sequences

As is well known, DNA has the specific binding properties; that is, only A-T and G-C pairs are possible, where the bases of nucleotide are A, T, G, C. The backbones of the bases, sugar and phosphate groups, ensure the mechanical stability of the double helix and protect the base pairs. Since the phosphate groups are negatively charged, the topology of the duplex is conserved only if it is immersed into an aqueous solution containing counterions such as Na+ and Mg+ that neutralize the phosphate groups.

The clustering of similar nucleotides can be clarified by studying the properties of the cluster size distributions on the various real DNA sequences, ranging from the viral to the higher eukaryotic sequences. It is shown that the distribution function 𝑃(𝑆) about the number 𝑆 of the consecutive C-G or A-T clusters becomes 𝑃(𝑆)exp{𝛼𝑆} [19, 20]. The values of the scaling exponent 𝛼 of CG are much larger than 𝛼 of AT. The maximum value of the A-T cluster size is found to be much larger than that of the C-G cluster size, which implies the existence of large A-T clusters.

Moreover, it has been found that the base sequences of various genes exhibit a long-range correlation, characterized by the power spectrum 𝑆(𝑓)𝑓𝛽 (0.1<𝛽<0.8) in the low-frequency limit (𝑓1) [21, 22]. As was observed in the power spectrum, the mutual information analysis and the Zipf analysis of the DNA-base sequences such as the HCh-22, the long-range structural correlation exists in the total sequence as well as the short-range periodicity. The eukaryote's DNA sequence has an apparently periodic repetition in terms of the gene duplication. The correlation length in the base sequence of genes changes from the early eukaryote to the late eukaryote as a result of evolutionary process. It is found that the long-range correlation tends to manifest in the power spectrum of the total sequence rather than in the power spectra of the exon and intron parts, separately [23].

On the other hand, in a DNA molecule, the charge carriers move along a double helix formed by two complementary sequences of four basic nucleotides: A, T, G, and C. A conduction band would form, if the DNA texts would exhibit some periodicity. The electrical resistance of the DNA molecule strongly fluctuates even if a single nucleotide in a long sequence is replaced (or removed). Quantum transport through the DNA molecule is also strongly affected by the correlations. The localization property of single-chain disordered systems with long-range correlation has been also extensively studied. The correlated disorder can lead to delocalized states within some special energy windows in the thermodynamic limit.

Accordingly, it is very interesting to compare the localization nature of the electronic states in the real DNA sequence with those in the artificial disordered sequence with long-range correlation. Recently, Krokhin et al. have reported that much longer localization length has been observed in the exon regions than in the intron regions for practically all of the allowed energies and for all randomly selected DNA sequences [24]. Through the statistical correlations in the nucleotide sequence, they suggest that the persistent difference of the localization property is related to the qualitatively different information stored by exons and introns.

In Section 5, we numerically give localization nature of the electronic states in some real DNA sequences such as bacteriophages of Escherichia coli (E. coli) and HCh-22, and so on in ladder models [15, 16, 25]. We also investigate the correlation effect on the localization property of the one-electronic states in the disordered ladder models with a long-range structural correlation that is generated by the modified Bernoulli map. Obviously, the correlation in the DNA sequences affects not only the electronic conduction but also the twist vibration of the backbones. However, such effects can be approximately ignored in the scope of this paper.


Each compositional unit of a DNA polymer is complex although the DNA polymers can be regarded as quasi-one-dimensional systems (see Figure 1). In this section, we give a brief review on the Ḧuckel theory in order to analyze the relationship between electronic structures of the separate nucleotide groups and of their infinite periodic chains. Ladik tentatively concluded that electrons hop mainly between the bases along the helical axis of DNA, so that it is enough to take into account the overlap integrals between the 𝜋-orbitals of the adjacent base pairs in the DNA duplexes [8, 9]. These studies have revealed that DNA polymers are insulators with an extremely large band gap 𝐸𝑔 (about 10–16 eV) and narrow widths of the valence and conduction bands (about 0.3–0.8 eV). This is a consequence of the orbital mixing between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO) in the base groups of nucleotide in the DNA. We can control the charge injection into DNA and the conduction in DNA, by adjusting the HOMO-LUMO of the polymer and the Fermi energy of electric leads, respectively.

Figure 1: The single bases of A, G, C, and T. Here dots () mean 𝜋-electrons.

Burnel et al. found that semiconduction in DNA is very important for such systems since the activation energies of nucleotides are lower than those of nucleoides [26]. This was the first suggestion that propagation along the sugar and phosphate groups plays a significant role in the transport properties of DNA as well as propagation along the base-stacking of the nucleotide groups in the center of the DNA molecule.

We revisit the problem of the electronic properties of individual molecules of DNA, in order to know electron transport in the double strands of DNA as a mother material for the single and double strand of DNA, by taking into account only 𝜋-electrons in the system [5]. To do so, we review the theory of 𝜋-electrons in DNA, using the Ḧuckel approximation for 𝜋-electrons in both the sugar-phosphate backbone chain and the 𝜋-stacking of the nitrogenous bases of nucleotide.

3.1. The Hückel Model

Applying the basic knowledge of quantum chemistry to biomolecules of nitrogenous bases, we can find the total number of 𝜋-orbitals and 𝜋-electrons in the bases. It is summarized in Table 1.

Table 1: The total numbers |𝑁|, |𝐶|, |𝑂|, and |𝑃| of N, C, O, and P atoms and of 𝜋-orbitals and 𝜋-electrons in the bases of A, G, C, and T, and the sugar(S) and phosphate(Ph) groups, respectively. Here there are 12, 14, 10, and 10 𝜋-electrons for the 10, 11, 8, and 8 𝜋-orbitals in the A, G, C, and T base molecules, respectively, while there are 8 and 8 𝜋-electrons for the 4 and 5 𝜋-orbitals in the sugar and phosphate groups, respectively. Here we note that the carbon atom of 𝐶𝐻3 in the T molecule does not have any 𝜋-electron since it forms the 𝑠𝑝3-hybrid orbitals. This provides 8 𝜋-electrons for the T molecule.

Let us consider the famous Ḧuckel model in quantum chemistry [2631]. This model concerns only the 𝜋-orbitals in the system. In this context, this theory is closely related to the so-called tight-binding model in solid-state physics, which concerns very localized orbitals at atomic sites such as Wannier's wavefunction. Therefore, this approach has been extensively applied to many polymer systems such as polyacetylene with a great success. In the so-called Ḧuckel approximation we adopt the orthogonality condition for the overlap integrals𝑆𝑟𝑟=1,𝑆𝑟𝑠=0(𝑟𝑠),(1) where 𝑆𝑟𝑠 denotes overlap integrals between atomic orbitals at 𝑟th and 𝑠th sites. We assume the special form of the resonance integrals as𝑟𝑠=12𝐾𝑆𝑟𝑠𝑟𝑟+𝑠𝑠,(2) with 𝐾=1.75 and the overlap integrals 𝑆𝑟𝑠 (𝑟=𝑠) are not necessary to be diagonal, and otherwise; almost the same procedure is kept as in the Huckel approximation. Usually the on-site (𝑟=𝑠) resonance integrals are called the Coulomb integrals denoted by 𝛼𝑟, while the off-site (𝑟𝑠) integrals are called the resonance integrals denoted by 𝛽𝑟𝑠 such that 𝛼𝑟=𝑟𝑟,𝛽𝑟𝑠=𝑟𝑠(𝑟𝑠). Here we would like to emphasize the following. In the sense of the Ḧuckel theory, the parameters are taken empirically. This means that the parameters are adjustable and feasible to give consistent results with the experimental results or the ab initio calculation results. Therefore, the exact values of the parameters neither are so important nor should be taken so seriously in this framework. This is because, once one can obtain much more precise values for the Ḧuckel parameters, one can provide the more plausible results from the Ḧuckel theory. Although many efforts of the ab initio calculations have been done for DNA systems, unfortunately at this moment there seem to be very few first principle calculations for such parameters in the DNA systems to fill out this gap. Nevertheless, we must assign some values for the Ḧuckel parameters in order to calculate the electronic properties of DNA in the framework of the Ḧuckel theory. So, we look back to the original method about time when the Ḧuckel theory was invented.

For this purpose, to use the standard Ḧuckel theory, let us adopt some simple formulae for the Ḧuckel parameters, which are defined as follows. Let 𝑋 and 𝑌 be two different atoms. Denote by 𝛼𝑋 the Coulomb integral at the 𝑋 atom and by 𝛽𝑋𝑌 the resonance integral between the 𝑋 and 𝑌 atoms:𝛼𝑋=𝛼+𝛼𝑋𝛽,𝛽𝑋𝑌=𝑙𝑋𝑌𝛽.(3) Here 𝑎𝑋 and 𝑋𝑌 are the empirical parameters that are supposed to be adopted from experimental data. And the parameters 𝛼 and 𝛽 are important. These can be thought of as the fundamental parameters in our problem of biopolymers. Conventionally we take 𝛼 as the Coulomb integral for the 2𝑝𝑥-orbital of carbon and 𝛽 as the resonance integral between the 2𝑝𝑥-orbitals of carbon, such that 𝛼=𝛼𝐶0,𝛽=𝛽𝐶𝐶1. This means that the energy level of a carbon atom is taken as the zero level, and the energy is measured in units of the resonance integral between carbon atoms. We note that the empirical values obtained from experiments are usually given by𝛼6.306.61eV,𝛽2.932.95eV.(4) Since the biomolecules consist of the atoms C, N, O, and P, let us find the plausible values of the Hückel parameters for them to apply the Hückel model to biomolecules of DNA. Pauling's electronegativities for carbon (C), nitrogen (N), oxygen (O), and phosphorus (P) are the following:𝜒𝐶2.55,𝜒𝑁3.0,𝜒𝑂3.5,𝜒𝑃2.1.(5) Then we can define the Hückel parameters for carbon. using therefore, Sandorfy's formula [32], Mulliken's formula [33, 34], and Streitwieser's formula [35], we find that 𝛼̇C=𝛼,𝛼̇N=𝛼+0.7𝛽,𝛼̇O=𝛼+1.53𝛽, and 𝛼̇P=𝛼0.724𝛽, and we obtain 𝛼̈O=𝛼+2.53𝛽,𝛼̈N=𝛼+0.276𝛽. as well as 𝛽̇C-̇C=𝛽, 𝛽̇C=̇C=1.1𝛽, 𝛽̇C-̇𝑁=𝛽̇C-̈N=0.8𝛽, 𝛽̇C=̇N=1.1𝛽, 𝛽̇C-̇O=0.9𝛽, and 𝛽̇C=̇O=1.7𝛽. Here 𝛼̇𝑋(𝛼̈𝑋) denotes the Coulomb integral of atomic state 𝑋 with one(two) electron(s) occupied. See Appendix A for the formula. And if one can get more accurate values from the ab initio calculations, then we can always replace the Coulomb integrals by the new set of values.

3.2. HOMO-LUMO of Biomolecules

For example, considering the topology of the hopping of electrons on the 𝜋-orbitals, we now find the following Hückel matrices 𝐇A for A: 𝐇𝐴=𝛼̇N𝛽̇C-̇N000𝛽̇C=̇N𝛽0000̇C-̇N𝛼̇C𝛽̇C=̇N00000000𝛽̇C=̇N𝛼̇N𝛽̇C-̇N00000000𝛽̇C-̇N𝛼̇C𝛽̇C=̇C000𝛽̇C-̈N0000𝛽̇C=̇C𝛼̇C𝛽̇C-̇C𝛽̇C-̇N𝛽000̇C=̇N000𝛽̇C-̇C𝛼̇C000𝛽̇C-̈N0000𝛽̇C-̇N0𝛼̇N𝛽̇C=̇N00000000𝛽̇C=̇N𝛼̇C𝛽̇C-̈N0000𝛽̇C-̈N000𝛽̇C-̈N𝛼̈N000000𝛽̇C-̈N000𝛼̈N,(6)Pauli's exclusion principle tells us that each state with an energy level is occupied by a pair of electrons with spin up and down. So, 𝜋-electrons occupy the energy levels in the spectrum from the bottom at low temperature. Since the lower energy levels with one half of the total number of 𝜋-electrons can be occupied by the 𝜋-electrons, there appears an energy separation between the occupied and the unoccupied states, which is called the energy gap.

The energy levels of the HOMOs and LUMOs are given by 𝜀𝐻=0.888, 𝜀𝐿=0.789 for A, where the energy is measured in units of 𝛽. We can do the same calculation for G, C, and T, respectively. Defining the energy gap between the LUMO and HOMO, Δ𝜀=𝜀𝐿𝜀𝐻. Then we obtain the result as Δ𝜀A=1.677, Δ𝜀G=1.555, Δ𝜀C=1.535, and Δ𝜀T=1.713 (see Figure 2(a)). The total energies 𝐸tot=2𝑗=occ.states𝜀𝑗 of the 𝜋-electrons of A, G, C, and T are 𝐸tot(A)=20.74, 𝐸tot(G)=26.47, 𝐸tot(C)=19.05, and 𝐸tot(T)=21.14, respectively. Therefore, 𝐸tot(C)>𝐸tot(A)>𝐸tot(T)>𝐸tot(G).(7)

Figure 2: (a) The spectrum of the 𝜋-orbitals of A, G, C, and T. (b) The spectrum of the 𝜋-orbitals of a sugar-phosphate group. PO4 stands for the phosphate group where the electron hopping between the oxygen sites is taken into account, while PO4 stands for the phosphate group where the electron hopping between the oxygen sites is forbidden. SP stands for the sugar-phosphate group where the electron hopping between the oxygen sites is taken into account. The energies are measured in units of |𝛽|. Here dots () mean 𝜋-electrons and the level with four dots means the double degeneracy of the level.

This shows that, since the lower the ground-state energy the more stable the system, the most stable molecule is G while the most unstable molecule is C.

HOMOs and LUMOs of sugar and phosphate may be required when we consider charge conduction of the DNA polymer because the sugar and phosphate group constituting the backbones of DNA polymer also has 𝜋-electrons. We give the results for the single sugar-phosphate in Figure 2(b). Moreover, the results based on the Ḧuckel approximation have been extended to the single-nucleotide systems such as A, G, C, and T with the single sugar-phosphate group, and the system of a single strand of DNA with an infinite repetition of a nucleotide group such as A, G, C, and T, respectively. See [5] for more details. This reorganization, which is difficult to calculate due to the complexity of the combined system, may lead to a smaller HOMO-LUMO gap and wider band-widths than for the bare molecule.

When the system of the nucleotide bases such as A, G, C, and T exists as an individual molecule, there is always an energy gap between the LUMO and HOMO states, where the order of the gap is about several eV’s. This means that the nucleotides of A, G, C, and T have the semiconducting character in its nature. If the 𝜋-stacking of the base is perfect, then there are two channels for 𝜋-electron hopping: one is the channel through the base stacking and the other is the one through the backbone chain of the sugar-phosphate. In this limit, the localized states of the original nucleotide bases become extended such that the levels form the energy bands.

Thus, we believe that our Ḧuckel approach in this paper fills out the gap between the simple approach of mathematical models for tight-binding calculations and the approach of the quantum chemical models for ab initio calculations from the first principle, where in the former we assume one orbital with one electron at one nucleotide while in the latter we include all electrons and atoms in the system

4. Effective Polaron Models

First principle methods are powerful enough to understand the basic electronic states of the molecules. On the other hand, the complementary model-based Hamiltonian approach is effective for understanding those of polymers as well. In this section, we give some effective 1D models and the basic properties of polymers such as transpolyacetylene and DNA polymers. In the previous section, we treated HOMOs and LUMOs and the occupation of orbitals by the 𝜋-electrons with spin. Hereafter, we omit spin of electrons when we consider the one-electron problems, for the sake of simplicity in notation.

4.1. Tight-Binding Models for Polymers

It is known that the electronic properties of planar conjugated systems are dominated by 𝜋-systems with one orbital per site. The electronic Hamiltonian for 2𝑝𝑧 orbitals through a tight-binding model with the nearest neighbors interactions only is given as𝐻el=𝑛𝐸𝑛𝐶𝑛𝐶𝑛𝑛𝑉𝑛𝑛+1𝐶𝑛𝐶𝑛+1+𝐶𝑛+1𝐶𝑛,(8) where 𝐶𝑛 and 𝐶𝑛 are creation and annihilation operators of an electron at the site 𝑛. The matrix elements are obtained from the extended Ḧuckel theory as given in the previous section:𝐸𝑛=𝛼𝑛,𝑉𝑛𝑚=𝐾2𝛼𝑛+𝛼𝑚𝑆𝑛𝑚,(9) where 𝛼𝑛 is ionization energy (Coulomb integral) of the 𝑛th 2𝑝𝑧 orbital and 𝑆𝑛𝑚 is overlap integral (resonance integral) between the 𝑛th and 𝑚th orbitals centered at neighbour sites given in Section 3. We usually treat the copolymer as a 𝜋-system with one orbital per site and represent the electronic Hamiltonian for the 2𝑝𝑧 carbon and nitrogen orbitals through a tight-binding model with nearest neighbor interactions. DNA polymers are also believed to form an effectively one-dimensional molecular wire, which is highly promising for diverse applications. Basically, the carriers mainly propagate along the aromatic 𝜋-𝜋 stacking of the strands (the interstrand coupling being much smaller), so that the one-dimensional tight-binding chain model can be a good starting point to minimally describe a DNA wire [12, 13].

Bruinsma et al. also introduced the effective tight-binding model that describes the site energies of a carrier located at the 𝑛th molecule as 𝐻=𝑛𝐸𝑛𝐶𝑛𝐶𝑛𝑛𝑡DNA𝜃cos𝑛,𝑛+1𝐶(𝑡)𝑛𝐶𝑛+1+𝐶𝑛+1𝐶𝑛,(10) where 𝐶𝑛(𝐶𝑛) is creation(annihilation) operator of the carrier at the site 𝑛 [7, 3638]. The carrier site energies 𝐸𝑛 are chosen according to the ionization potentials of respective DNA-bases as 𝜖A=8.24eV, 𝜖T=9.14eV, 𝜖C=8.87eV, and 𝜖G=7.75eV, while the hopping integral, simulating the 𝜋-𝜋 stacking between adjacent nucleotides, is taken as 𝑡DNA=0.1-0.4eV. Moreover, 𝜃𝑛,𝑛+1 denotes the relative twist angle deviated from equilibrium position between the 𝑛th and (𝑛+1)th molecules due to temperature 𝑇. Then we can estimate the order of the hopping integral. Let 𝜃𝑛,𝑛+1 be an independent random variable that follows Gaussian distribution with 𝜃𝑛,𝑛+1=0 and 𝜃2𝑛,𝑛+1=𝑘𝐵𝑇/𝐼𝜔2𝐼, where 𝐼 is the reduced moment of inertia for the relative rotation of the two adjacent bases and 𝜔𝐼 is the oscillator frequency of the mode (𝐼𝜔2𝐼=250𝐾). Therefore the fluctuation of the hopping term is about 𝑡DNAcos(𝜃𝑛,𝑛+1)𝑡DNA(1(𝜃2𝑛,𝑛+1/2))0.16eV at room temperature, which is much smaller than that (1.4eV) in the diagonal part. Accordingly, as a simple approximation, we can deal with the diagonal fluctuation with keeping the hopping integral constant.

Roche investigated such a model for poly(GC) DNA polymer and 𝜆-DNA, with the on-site disorder arising from the differences in ionization potentials of the base pairs and with the bond disorder 𝑡DNAcos(𝜃𝑛,𝑛+1(𝑡)) related to the random twisting fluctuations of the nearest neighbor bases along the strand [37, 38]. While for poly(GC) the effect of disorder does not appear to be very dramatic, the situation changes when considering 𝜆-DNA.

Actually, modern development of physicochemical experimental techniques enables us to measure directly the DNA electrical transport phenomena even in single molecules. Moreover, several groups have recently performed numerical investigations of localization properties of the DNA electronic states based on the realistic DNA sequences.

4.2. Twist Polaron Models for DNA Polymers

In this subsection we give an effective 1D model with realistic parameters for the electronic conduction of the poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA polymers.

Specially, a distinctive feature of biological polymers is a complicated composition of their elementary subunits and an apparent ability of their structures to support long-living nonlinear excitations. In their polaron-like model, Hennig and coworkers studied the electron breather propagation along DNA homopolynucleotide duplexes, that is, in both poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers, and, for this purpose, they estimated the electron-vibration coupling strength in DNA, using semiempirical quantum chemistry [3942]. Chang et al. have also considered a possible mechanism to explain the phenomena of DNA charge transfer. The charge coupling with DNA structural deformations can create a polaron and thus promote a localized state. As a result, the moving electron breather may contribute to the highly efficient long-range conductivity [43]. Recent experiments seem to support the polaron mechanism for the electronic transport in DNA polymers.

The Hamiltonian for the electronic part in the DNA model is given by𝐻el(𝑡)=𝑛𝐸𝑛(𝑡)𝐶𝑛𝐶𝑛𝑛𝑉𝑛𝑛+1𝐶(𝑡)𝑛𝐶𝑛+1+𝐶𝑛+1𝐶𝑛,(11) where 𝐶𝑛 and 𝐶𝑛 are creation and annihilation operators of an electron at the site 𝑛. The on-site energies 𝐸𝑛(𝑡) are represented by𝐸𝑛(𝑡)=𝐸0+𝑘𝑟𝑛(𝑡),(12) where 𝐸0 is a constant and 𝑟𝑛 denotes the structural fluctuation caused by the coupling with the transversal Watson-Crick H-bonding stretching vibrations. The schematic illustration is given in Figure 3. The transfer integral 𝑉𝑛𝑛+1 depends on the three-dimensional distance 𝑑𝑛𝑛+1 between adjacent stacked base pairs, labeled by 𝑛 and 𝑛+1, along each strand. And it is expressed as𝑉𝑛𝑛+1(𝑡)=𝑉01𝛼𝑑𝑛𝑛+1(𝑡).(13) Parameters 𝑘 and 𝛼 describe the strengths of interaction between the electronic and vibrational variables, respectively. The 3D displacements 𝑑𝑛𝑛+1 also give rise to variation of the distances between the neighboring bases along each strand. The first-order Taylor expansion around the equilibrium positions is given by𝑑𝑛𝑛+1𝑅(𝑡)=001cos𝜃0𝑟𝑛(𝑡)+𝑟𝑛+1,(𝑡)(14) where 𝑅0 represents the equilibrium radius of the helix, 𝜃0 is the equilibrium double-helical twist angle between base pairs, and 0 is the equilibrium distance between bases along one strand given by 0=𝑎2+4𝑅20sin2𝜃021/2,(15) with 𝑎 being the distance between the neighboring base pairs in the direction of the helix axis. We adopt realistic values of the parameters obtained from the semiempirical quantum-chemical calculations (see Table 2). Further, we consider {𝑟𝑛} as independent random variables generated by a uniform distribution with the width (𝑟𝑛[𝑊,𝑊]). Accordingly, fluctuations in both the on-site energies and the off-diagonal parts in the Hamiltonian (1) are mutually correlated because they are generated by the same random sequence 𝑟𝑛 (see Figure 2(c)). The typical value of 𝑊 is 𝑊=0.1[Å], which approximately corresponds to the variance in the hydrogen bond lengths in the Watson-Crick base pairs, as seen in the X-ray diffraction experiments [44].

Table 2: Basic parameters for DNA molecules. The subscripts AT and GC for 𝑘 and 𝛼 denote the ones of the poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA polymers, respectively. The other parameters are 𝐸0=0.1eV, 𝑉0=0.1eV, 𝑎=3.4 Å, 𝑅0=10Å, and 𝜃0=36.
Figure 3: Sketch of the structure of the DNA model. The bases are represented by bullets, and the geometrical parameters 𝑅0,0,𝜃0,𝑟𝑛+1, and 𝑑𝑛𝑛+1 are indicated.
Figure 4: (a) The on-site energy 𝐸𝑛eV and (b) transfer integral 𝑉𝑛𝑛+1eV as a function of the base pair site 𝑛. The parametric plot 𝐸𝑛 versus 𝑉𝑛𝑛+1 is shown in (c). 𝑊=0.1 and the other parameters are given in text. The unit of the energy and the spatial length are eV and the number of nucleotide base pair (bp), respectively, throughout the present paper.

A quasicontinuum spectrum possessed a wealth of dynamical modes. In principle, each of these can influence the DNA charge transfer/transport. But, since there are more or less active modes, it is possible to take the whole manifold of the DNA molecular vibrations into two parts: vibrations which are most active and a stochastic bath consisting of all of the other ones. Computer simulations have pinpointed that the dynamical disorder is crucially significant for the DNA transfer/transport. Several attempts to formulate stochastical models for the interplay of the former and the latter have already appeared in the literature (see, [45, 46], e.g.). In this paper, we will deal with the polaron-like model of Hennig and coworkers as described in the works in [4042], where charge+breather propagation along DNA homopolynucleotide duplexes (i.e., in both the poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers) has been studied.

4.3. Localization due to Static Disorder

In this subsection, we assume that the fluctuations of {𝑟𝑛} are frozen (quenched disorder) and independent for different links; that is, we investigate localization properties of the poly(dA)-poly(dT) and poly(dG)-poly(dC) DNA polymers [47].

The Schr̈odinger equation 𝐻el|Φ=𝐸|Φ is written in the transfer matrix form𝜙𝑛+1𝜙𝑛=𝐸𝐸𝑛𝑉𝑛𝑛+1𝑉𝑛𝑛1𝑉𝑛𝑛+1𝜙10𝑛𝜙𝑛1,(16) where 𝜙𝑛 is the amplitude of the electronic wavefunction |Φ=𝑛𝜙𝑛|𝑛, where |𝑛=𝐶𝑛|0 at the base-pair site 𝑛 and |0 is the Fermi vacuum. We use the localization length 𝜉 and/or the Lyapunov exponent 𝛾 calculated by the mapping (16) in order to characterize the exponential localization of the wavefunction. Originally the Lyapunov exponent (the inverse localization length) is defined in the thermodynamic limit (𝑁). However, here we use the following definition of the Lyapunov exponent for the electronic wavefunction with a large system size 𝑁 [47, 48]: 𝛾(𝐸,𝑁)=𝜉1||𝜙(𝐸,𝑁)=ln𝑁||2+||𝜙𝑁1||2.2𝑁(17) We use the appropriate initial conditions 𝜙0=𝜙1=1, and for large 𝑁(𝜉) the localization length and the Lyapunov exponent are independent of the boundary condition. The energy-dependent transmission coefficient 𝑇(𝐸,𝑁) of the system between metallic electrodes is given as 𝑇(𝐸,𝑁)=exp(2𝛾𝑁) and is related to the Landauer resistance via 𝜚=(1𝑇)/𝑇 in units of the quantum resistance /2𝑒2(13[𝑘Ω𝑚]) [49, 50].

We have numerically investigated the localization properties of electronic states in the adiabatic polaron model of poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers with the realistic parameters obtained using the semiempirical quantum-chemical calculations.

We compare the localization properties of the poly(dG)-poly(dC), poly(dA)-poly(dT) DNA polymers and the mixed cases. Figures 5(a) and 5(b) show the localization length and the Lyapunov exponent in the three types of polymers with 𝑊=0.1. In low-energy regions, the localization length in the poly(dG)-poly(dC) DNA polymer is larger than that in the poly(dA)-poly(dT) DNA polymer. The system size dependence of the localization length is given in Figure 6 in relation to the resonance energy. Indeed, the localization length of the DNA polymers is larger than 𝜉>2000[bp] in almost all of the energy bands for all models; it is much larger than the system size of the oligomer used in the experiments. As is seen in Figure 6, the smaller the size of the system the more complex the resonance peaks become.

Figure 5: Comparison: (a) localization length and (b) the Lyapunov exponent in poly(dG)-poly(dC), poly(dA)-poly(dT), and the mixed DNA polymers. 𝑊=0.1, 𝜃0=36, and 𝑁=222.
Figure 6: The localization length as a function of the energy for several system sizes 𝑁(=216,219,222) in the (a) poly(dA)-poly(dT) DNA polymers and (b) poly(dG)-poly(dC) DNA polymers. 𝑊=0.1 and 𝜃0=36.
4.4. Quantum Diffusion in the Fluctuating Environment

In this subsection we numerically investigate quantum diffusion of electrons in the Hennig model of poly(dG)-poly(dC) and poly(dA)-poly(dT) with a dynamical disorder [18]. We assume that the diffusion is caused by a colored noise associated with the stochastic dynamics of distances 𝑟(𝑡) between two Watson-Crick base-pair partners: 𝑟(𝑡)𝑟(𝑡)=𝑟0exp(|𝑡𝑡|/𝜏). These fluctuations can be regarded as a stochastic process at high temperature, with phonon modes being randomly excited [51, 52]. In the model, the characteristic decay time 𝜏 of correlation can control the spread of the electronic wavepacket. Interestingly, the white-noise limit 𝜏0 can in effect correspond to a sort of motional narrowing regime (see, examples), because we find that such a regime causes ballistic propagation of the wavepacket through homogeneous DNA duplexes. Still, in the adiabatic limit 𝜏, DNA electronic states should be strongly affected by localization. The amplitude 𝑟0 of random fluctuations within the base pairs (the fluctuation in the distance between two bases in a base pair) and the correlation time 𝜏 are very critical parameters for the diffusive properties of the wavepackets. It is interesting to find the ballistic behavior in the white-noise limit versus the localization in the adiabatic limit, since there are a number of experimental works observing ballistic conductance of DNA in water solutions, which is also temperature independent. Van Zalinge et al. have tried to explain the latter effect, using a kind of acoustic phonon motions in DNA duplexes, which seems to be plausible, but not the only possible physical reason [53, 54]. We will propose an alternative explanation for the observed temperature-independent conductance, based upon our numerical results.

Generally, in the case of quantum diffusion the temporal evolution of the electronic state vector |Φ is described by the time-dependent Schr̈𝑜dinger equation 𝑖(𝜕|Φ/𝜕𝑡)=𝐻el(𝑡)|Φ, which then becomes𝑖e𝜕𝜙𝑛𝜕𝑡=𝐸𝑛(𝑡)𝜙𝑛𝑉𝑛𝑛+1(𝑡)𝜙𝑛+1𝑉𝑛1𝑛(𝑡)𝜙𝑛1,(18) where 𝜙𝑛=𝑛|Φ and the effective Planck constant is given as e=0.53. We redefined the scaled dimensionless variables 𝐸𝑛(𝑡) and 𝑉𝑛𝑛+1(𝑡) in (1) such that 𝐸𝑛/𝑉0𝐸𝑛, 𝑉𝑛𝑛+1/𝑉0𝑉𝑛𝑛+1. We used mainly the 4th-order Runge-Kutta-Gill method in the numerical simulation for the time evolution with time step 𝛿𝑡=0.01. In some cases, we ourselves confirmed that the accuracy in the obtained results is in accord with the one in the results that are gained by the help of the 6th-order symplectic integrator that is the higher-order unitary integration.

It has been demonstrated that the motional narrowing affects the localization in the poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers. In either case of the model DNA polymers, the temperature dependence becomes virtually suppressed when the motion of the wave packet is characterized by the ballistic propagation.

We have also investigated the temporal diffusion rate in almost all of the diffusive ranges (see Figure 7). We found that the diffusion rate of the A-T model is larger than that of the G-C model at comparatively low temperatures. Interestingly enough, for relatively high temperatures in the diffusive range of the wavepacket motion the difference between the two DNA systems gets smaller.

Figure 7: Diffusion rate 𝐷(𝑡) as a function of 𝜏1 for several fluctuation strengths 𝑟𝑛0=1,3,5 at (a) A-T model and (b) G-C model, respectively.

Figure 8 shows the short-time behavior in the cases of 𝜏=1 and 𝜏=0.01 depicted on a larger time scale. It follows that in the short-time behavior the (dG)15-(dC)15 case is more diffusive than its (dA)15-(dT)15 counterpart within the range from which the spread of the wavepacket is p𝑚15.

Figure 8: Short-time behavior of 𝑚(𝑡) of A-T, G-C, and mixed models with 𝑟𝑛0=1.0 at (a) 𝜏=1 and (b) 𝜏=0.01.

We used periodic sequences, which means that 𝐸0 and 𝑉0 are constant values as the static parts of the on-site and hopping terms for poly(dG)-poly(dC) and poly(dA)-poly(dT) DNA polymers, respectively. This includes the mixed model as well. Then the motional narrowing for dynamical disorder makes time evolution of the wavepacket ballistic. However, we remark that the motional narrowing strongly localizes the wavepacket if we use a disordered sequence for the static parts of 𝐸𝑛 and/or 𝑉𝑛𝑛+1.

4.5. Quantum Diffusion Coupled with Vibrational Modes

Let us denote 𝑟𝑛(𝑡) by the stretching vibration of W-C bonds at the 𝑛 site as before. The external harmonic perturbation is equivalent to the coupling with quantum linear oscillators or with phonon modes in solid-state physics [55, 56]. We replace the disordered fluctuation 𝑟𝑛(𝑡) with the harmonic time-dependent one such as𝑟𝑛(𝑡)=𝑟𝑀𝑛0𝑖=1𝜖𝑖𝜔cos𝑖𝑡+𝜃𝑛(𝑖),(19) where 𝑀 and 𝜖𝑖 are the number of the frequency component and the strength of the perturbation, respectively. The initial phases {𝜃𝑛(𝑖)}[0,2𝜋] at each site 𝑛 are randomly chosen. In the numerical calculation of this section, we take 𝜖𝑖=𝜖/𝑀, for simplicity, and take incommensurate numbers of 𝜔𝑖𝑂(1) as the frequency set. In the limit of 𝑀, the time-dependent perturbation approaches the stochastic fluctuation as discussed in the previous subsection. In particular, we can regard the approximation as an electronic system coupled with highly excited quantum harmonic oscillators. One of the advantages of this model is that, although the number of autonomous modes is limited due to the computer power, we can freely control the number 𝑀 of frequency components in the harmonic perturbation. This is also a simple model to investigate electronic diffusion coupled with lattice vibrations.

In Figure 9(a), the MSD is shown for the poly(dG)-poly(dC), the poly(dA)-poly(dT), and the mixed DNA polymers. We find that all of the cases exhibit the normal diffusion of electron without any stochastic perturbation. As shown in Figure 9(b), the diffusion rate decreases as the perturbation strength 𝜖 increases. As a result, the coupled motion of charges and the lattice breathers connected with the localized structural vibrations may contribute to the highly efficient long-range conductivity (see Appendix D).

Figure 9: (a) MSD 𝑚(𝑡) in A-T, G-C, and mixed models perturbed by 𝑀=1,𝜖=0.5. (b) MSD 𝑚(𝑡) of A-T model perturbed by 𝑀=1 for 𝜖=0.5,0.7,1.0,2.0 from above. We set 𝑟𝑛0=1.

5. Tight-Binding Models for the Ladder Systems

In this section, to investigate the energy band structure for periodic sequences and the localization properties for the correlated disordered sequences, we introduce the ladder models of DNA polymers. Although in [12, 13, 15, 16, 25] we assumed the ladder structure consisting of sugar and phosphate groups, the model can be applicable to the DNA-like substances such as the ladder structure of base pairs without sugar-phosphate backbones. When we apply the model for the sugar and phosphate chains, the chains 𝐴 and 𝐵 are constructed by the repetition of the sugar-phosphate sites, and the interchain hopping 𝑉𝑛 at the sugar sites come from the nucleotide base pairs, that is, A-T or G-C pairs (see Figure 10(a)).

Figure 10: Models of the double strand of DNA. (a) The two-chain model and (b) the three-chain model we adapted in the main text.
5.1. The Ladder Models and the Dispersions

Since there is only one 𝜋-orbital per site, there are totally 2𝑁𝜋-orbitals in the ladder model of the DNA double chain. Let us denote by 𝜙𝐴𝑛(𝜙𝐵𝑛) the 𝜋-orbital at site 𝑛 in the chain 𝐴(𝐵). By superposition of the 𝜋-orbitals, the Schr̈odinger equation 𝐻|Ψ=𝐸|Ψ becomes 𝐴𝑛+1,𝑛𝜙𝐴𝑛+1+𝐴𝑛,𝑛1𝜙𝐴𝑛1+𝐴𝑛,𝑛𝜙𝐴𝑛+𝑉𝑛𝜙𝐵𝑛=𝐸𝜙𝐴𝑛,𝐵𝑛+1,𝑛𝜙𝐵𝑛+1+𝐵𝑛,𝑛1𝜙𝐵𝑛1+𝐵𝑛,𝑛𝜙𝐵𝑛+𝑉𝑛𝜙𝐴𝑛=𝐸𝜙𝐵𝑛,(20) where 𝐴𝑛+1,𝑛 (𝐵𝑛+1,𝑛) means the hopping integral between the 𝑛th and (𝑛+1)th sites, 𝐴𝑛,𝑛 (𝐵𝑛,𝑛) means the on-site energy at site 𝑛 in chain 𝐴 (𝐵), and 𝑉𝑛 is the hopping integral from chain 𝐴(𝐵) to chain 𝐵(𝐴) at site 𝑛, respectively. Furthermore, it can be rewritten in the matrix form: Φ𝑛+1=𝑀𝑛Φ𝑛, where Φ𝑛=(𝜙𝐴𝑛,𝜙𝐴𝑛1,𝜙𝐵𝑛,𝜙𝐵𝑛1). The transfer explicit matrix is given in Appendix B. Generally speaking, we would like to investigate the asymptotic behavior (𝑁) of the products of the matrices 𝑀𝑑=2(𝑁)=Π𝑁𝑘=1𝑀𝑘=𝑀𝑁𝑀𝑁1𝑀1.

We consider the band structure for the periodic case by setting 𝐴𝑛+1,𝑛=𝐵𝑛+1,𝑛=𝑎(𝑏) at odd(even) site 𝑛 and 𝑉𝑛=𝑣(0) at odd(even) sites 𝐴𝑛𝑛=𝐵𝑛𝑛=𝛼(𝐴𝑛𝑛=𝐵𝑛𝑛=𝛽) for odd(even) site 𝑛, for simplicity.

A simple way to solve the above equations is to use the Bloch theorem:𝜙𝐴𝑛+2=𝑒𝑖2𝑘𝑠𝜙𝐴𝑛,𝜙𝐵𝑛+2=𝑒2𝑖𝑘𝑠𝜙𝐵𝑛,(21) where 𝑠 is the length between the adjacent base groups and is assumed to be equivalent to the length between the adjacent sugar-phosphate groups and the wavenumber 𝑘 is defined as 𝜋/2𝑠𝑘𝜋/2𝑠. (We take 𝑠=1 in the following calculations.) The Schr̈odinger equation becomes Φ𝑛+1=𝑀Φ𝑛,𝑀=𝐸𝛽𝑎+𝑏𝑒𝑖𝑘00𝑎+𝑏𝑒𝑖𝑘𝐸𝛼0𝑣00𝐸𝛽𝑎+𝑏𝑒𝑖𝑘0𝑣𝑎+𝑏𝑒𝑖𝑘,𝐸𝛼(22) where Φ𝑛=(𝜙𝐴2𝑛+1,𝜙𝐴2𝑛,𝜙𝐵2𝑛+1,𝜙𝐵2𝑛)𝑡. Denote the determinant of 𝑀 by 𝐷(𝑀). Solving the equation 𝐷(𝑀)=0 for 𝐸, we obtain the energy dispersion of the system. Here we show the simple case of 𝛼=𝛽=0 as 𝐸+(±)1(𝑘)=2𝑣±𝑣2𝑎+42+𝑏2,𝐸+8𝑎𝑏cos2𝑘(±)1(𝑘)=2𝑣±𝑣2𝑎+42+𝑏2,+8𝑎𝑏cos2𝑘(23) where 𝐸±(+)(𝑘) (𝐸±()(𝑘)) stand for the upper (lower) bands in channel ±, respectively [12, 13]. The lowest and upper middle (the lower middle and highest) bands correspond to bonding (antibonding) states between the adjoint orbitals in the interchains. More details of the energy band for general case are given in Appendix C. In Figure 11(a) the energy band structure is given with varying the interchain hopping 𝑣. Figure 11(b) shows the cross-section view at 𝑣=1. There is a band gap 𝐸𝑔(𝑣) at the center in between the lower and upper middle bands in the spectrum for the whole range of 𝑣 when 𝛼=𝛽=0:𝐸𝑔(𝑣)=𝐸(+)𝜋2𝐸+()𝜋2=4(𝑎𝑏)2+𝑣2𝑣.(24) The other band gap Δ𝑔(𝑣), given asΔ𝑔1(𝑣)=𝑣+24(𝑎𝑏)2+𝑣24(𝑎+𝑏)2+𝑣2,(25) appears in between the lowest and the lower middle bands (the upper middle and highest bands) when 𝑣>𝑣𝑐2𝑎𝑏/𝑎2+𝑏2 (otherwise, it is negative and therefore semimetallic). There is a transition from semiconductor to semimetal as the 𝜋-electron hopping between the nitrogenous bases of nucleotide is increased.

Figure 11: Energy bands of the decorated ladder model. (a) The energy bands as a function of 𝑉. (b) The snap shot of the energy bands when 𝑉=1. 𝑘 means the wavevector in units of 𝜋/𝑠 such that 1.0𝑘1.0, 𝑉 means the 𝜋-electron hopping integral between the interchain sites, and E means the energy in units of 𝑉=1. Here we have taken the values 𝛼=0, 𝛽=0, 𝑎=0.9,and𝑏=1.2.

The two-chain model can be easily extended to the three-chain case (see Figure 10(b)). 𝐶𝑛+1,𝑛 means the hopping integral between the 𝑛th and (𝑛+1)th sites, 𝐶𝑛,𝑛 means the on-site energy at site 𝑛 in chain 𝐶, and 𝑉𝑛 and 𝑈𝑛 are the hopping integrals between the chains. The Schr̈odinger equation 𝐻|Ψ=𝐸|Ψ becomes 𝐴𝑛+1,𝑛𝜙𝐴𝑛+1+𝐴𝑛,𝑛1𝜙𝐴𝑛1+𝐴𝑛,𝑛𝜙𝐴𝑛+𝑉𝑛𝜙𝐶𝑛=𝐸𝜙𝐴𝑛,𝐶𝑛+1,𝑛𝜙𝐶𝑛+1+𝐶𝑛,𝑛1𝜙𝐶𝑛1+𝐶𝑛,𝑛𝜙𝐶𝑛+𝑉𝑛𝜙𝐴𝑛+𝑈𝑛𝜙𝐵𝑛=𝐸𝜙𝐶𝑛,𝐵𝑛+1,𝑛𝜙𝐵𝑛+1+𝐵𝑛,𝑛1𝜙𝐵𝑛1+𝐵𝑛,𝑛𝜙𝐵𝑛+𝑈𝑛𝜙𝐶n=𝐸𝜙𝐵𝑛.(26) It can be also rewritten in the matrix form: Φ𝑛+1=𝑀𝑑=3(𝑛)Φ𝑛, where Φ𝑛=(𝜙𝐴𝑛+1,𝜙𝐴𝑛,𝜙𝐶𝑛+1,𝜙𝐶𝑛,𝜙𝐵𝑛+1,𝜙𝐵𝑛)𝑡 and the explicit transfer matrix is given in Appendix C.

For simplicity, we set 𝐴𝑛+1,𝑛=𝐵𝑛+1,𝑛=𝑎 and 𝐶𝑛+1,𝑛=𝑐 at site 𝑛, and 𝑉𝑛=𝑈𝑛. We can analytically obtain the energy band structure for the three-chain model by using the Bloch theorem: 𝜙𝐴𝑛+1=𝑒𝑖𝑘𝑠𝜙𝐴𝑛,𝜙𝐶𝑛+1=𝑒𝑖𝑘𝑠𝜙𝐶𝑛,𝜙𝐵𝑛+1=𝑒𝑖𝑘𝑠𝜙𝐵𝑛.(27)

5.2. Localization in the Ladder Models

As was discussed in Sections 2 and 3, the sequences of the realistic DNA polymers are not periodic and accompany a variety of disorders. Generally, the randomness makes the electronic states localized because of the quasi-one-dimensional system and affects electronic conduction and optical properties, and so on. In the present section, we investigate the correlation effect on the localization properties of the one-electronic states in the disordered, two-chain (ladder) and three-chain models with a long-range structural correlation as a simple model for the electronic property in the DNA [17]. The relationship between the correlation length in the DNA sequences and the evolutionary process is suggested. Moreover, it is interesting if the localization property is related to the evolutionary process.

Figure 12 shows the DOS as a function of energy for the binary disordered systems. The sequence of the interchain hopping 𝑉𝑛 takes an alternative value 𝑊GC or 𝑊AT. We find that some gaps observed in the periodic cases close due to the disorder corresponding to the base-pair sequences.

Figure 12: DOS as a function of energy for the binary disordered cases in the two-chain model. (a)𝑊AT=2.0,𝑊GC=1.0,and𝑎=1.0,𝑏=1.0. (b)𝑊AT=2.0,𝑊GC=1.0,𝑎=1.0,and𝑏=0.5. The on-site energy is set at 𝐴𝑛𝑛=𝐵𝑛𝑛=0.

Figure 13(a) shows the energy dependence of the Lyapunov exponents (𝛾1 and 𝛾2) for some cases in the asymmetric modified Bernoulli system characterized by the bifurcation parameters 𝐵0 and 𝐵1 controlling the correlation (see Appendix G for modified Bernoulli system). They are named as follows: 𝐶𝑎𝑠𝑒(𝑖): 𝐵0=1.0,𝐵1=1.0, 𝐶𝑎𝑠𝑒(𝑖𝑖): 𝐵0=1.0,𝐵1=1.9, and 𝐶𝑎𝑠𝑒(𝑖𝑖𝑖): 𝐵0=1.7,𝐵1=1.9. Apparently, Case (i) is more localized than Cases (ii) and (iii) in the vicinity of the band center |𝐸|<1. The comparison between Case (ii) and Case (iii) shows the effect of asymmetry of the map on the localization. The ratio of the G-C pairs is 𝑅GC0.2 for Case (ii) and is 𝑅GC0.47 for Case (iii). In the energy regime |𝐸|>1, the Lyapunov exponent 𝛾2 in Case (ii) is smaller than the one in Case (iii) in spite of the same correlation strength 𝐵1. As a result, we find that, in the DNA ladder model, correlation and asymmetry enhance the localization length 𝜉(1/𝛾2) of the electronic states around |𝐸|<1, although the largest Lyapunov exponents 𝛾1 do not change effectively at all. Figures 13(b), 13(c), and 13(d) show the nonnegative Lyapunov exponents in the real DNA sequences of (b) HCh-22, (c) bacteriophages of E. coli, and (d) histone proteins. In the case of HCh-22, we used two sequences with 𝑁=105, extracted from the original large DNA sequences. The result shows that the Lyapunov exponents do not depend on the details of the difference in the sequences of HCh-22. Although the weak long-range correlation has been observed in HCh-22 as mentioned in the introduction, it does not affect the property of localization. The sequences we used are almost symmetric (𝑅GC0.5).

Figure 13: Lyapunov exponents (𝛾1,𝛾2) as a function of energy in the ladder model. (a) Modified Bernoulli model, where case (1): 𝐵0=1.0,𝐵1=1.0, Case (2): 𝐵0=1.0,𝐵1=1.9, and Case (3): 𝐵0=1.7,𝐵1=1.9 are shown. (b) HC-22. (c) Bacteriophages of E. coli (phage-𝜆, phage-186). (d) Early histone H1 and late histone H1. The on-site energy is set at 𝐴𝑛𝑛=𝐵𝑛𝑛=0, 𝑎=1.0, and 𝑏=0.5. The size of the sequence is 𝑁=105 for (a), 𝑁=105 for (b), 𝑁=48510 for the phage-𝜆 in (c), 𝑁=30624 for the phage-186 in (c), 𝑁=787 for the early histone H1 in (d), and 𝑁=1182 for the late histone H1 in (d).

In addition, in the numerical calculation we set the on-site energy as 𝐴𝑛𝑛=𝐵𝑛𝑛=𝐶𝑛𝑛=0 for simplicity. The sequence {𝐶𝑛𝑛+1} can be also generated by corresponding to the base-pair sequence. The localization properties in the simple few-chain models with the on-site disorder have been extensively investigated [57]. Figure 14 shows the energy dependence of the Lyapunov exponents in the three-chain cases. We can observe that all of the Lyapunov exponents 𝛾𝑖(𝑖𝑑) are changed by the correlation. The least nonnegative Lyapunov exponent 𝛾3 is diminished by the correlation. In particular, it is found that the correlation of the sequence enhances the localization length defined by using the least nonnegative Lyapunov exponent. We can see that the localization length diverges at the band center 𝐸=0.

Figure 14: Lyapunov exponents 𝛾𝑖(𝑖=1,2,3) as a function of energy in the correlated three-chain model. The parameters 𝑊AT=2.0,𝑊ATAT=1.0,𝑎=1.0,and𝑏=0.5 are shown. The on-site energy is set at 𝐴𝑛𝑛=𝐵𝑛𝑛=𝐶𝑛𝑛=0.

In Figure 15(a) the result for asymmetric modified Bernoulli system is shown. Apparently, the correlation and/or asymmetry of the sequence effects a change in the second and third Lyapunov exponents. In contrast, although the global feature of 𝛾1 is almost unchanged, the local structure of the energy dependence is changed by the change of 𝐵0. Figure 15(b) shows the results in phage-fd and phage-186 in the three-chain model. It is found that the structure of the energy dependence around |𝐸|<2 is different from that in the artificial sequence by the modified Bernoulli map.

Figure 15: The Lyapunov exponents (𝛾1,𝛾2,𝛾3) as a function of energy in the three-chain model. (a) Modified Bernoulli model and (b) bacteriophages of E. coli (phage-fd, phage-186). The parameters are the same as the ones in Figure 14 except for on-site energies of C-chain (Cn,n = 0).

The charge transfer efficiency based on mismatch and correlation effect for some genomic and synthetic sequences are also investigated [58].

5.3. Polaron Models for a Double Strand of DNA

The overlap of the electronic orbitals along the stacked base pairs provides a pathway for charge propagation over 50Å, and the reaction rate between electron donors and acceptors does not decay exponentially with distance. A multiple-step hopping mechanism was recently proposed to explain the long-range charge transfer behavior in DNA. In this theory, the single G-C base pair is considered as a hole donor due to its low ionization potential if compared to the one on the A-T base pairs. Besides the multiple-step hopping mechanism, polaron motion has also been considered as a possible mechanism to explain the phenomena of DNA charge transfer. The charge coupling with the DNA structural deformations may create a polaron and cause a localized state. The polaron behaves as a Brownian particle such that it collides with low-energy excitations of its environment that acts as a heat bath. For the most physical systems, the acoustical and optical phonons are the main lattice excitations as in the single-chain case given in Appendix D.

In this section, we present the formation of localized polarons due to the coupling between charge carriers and lattice vibrations of a double strand of DNA [14]. These polarons in DNA act as donors and acceptors, which exhibit an extrinsic semiconductor character of DNA. The results are discussed in the context of experimental observations.

Let 𝑥𝑛 and 𝑦𝑛 be configuration angles for rotation of the 𝑛th base group along the course of the backbone chains A and B of DNA ladder, respectively. For the sake of simplicity, we assume that all masses 𝑀𝑛 of the nucleotide groups and the bond lengths 𝑑𝑛 between 𝑆 and 𝐵𝑛 groups are identical so that 𝑀𝑛=𝑀 and 𝑑𝑛=𝑑, giving the moment of inertia 𝐼𝑛=I0=𝑀𝑑2. In this case, the lattice Hamiltonian 𝐻ph can be given by 𝐻ph𝑥𝑛,𝑦𝑛=𝑁𝑛=1𝐼02̇𝑥2𝑛+̇𝑦2𝑛+𝐴2𝑥2𝑛+𝑦2𝑛+𝐾2𝑥𝑛+1𝑥𝑛2+𝐾2𝑦𝑛+1𝑦𝑛2+𝑆2𝑥𝑛𝑦𝑛2,(28) where A, K, and S are the parameters for the rotational energy, the stacking energy, and the bonding energy of the bases, respectively. We note here that various generalizations and modifications of the model are straightforwardly possible by adapting a different combination of the interactions and the hopping integrals. In the following, we give result only for HOMO. We can obtain the result for LUMO in simple replacements.

On the other hand, generalizing the idea of Holstein for the one-dimensional molecular crystal to our case of the double strand of DNA, the tight-binding Hamiltonian for electrons is given by 𝐻𝐻el=𝑁𝑛=1𝑡𝑎𝐴(𝑛+1)𝑎𝐴𝑛𝑡𝑎𝐴(𝑛1)𝑎𝐴𝑛+𝜖𝐴𝐻𝑥𝑛𝑎𝑛𝐴𝑎𝐴𝑛+𝑣𝑛||𝑥𝑛𝑦𝑛||𝑎𝑛𝐴𝑎𝐵𝑛+𝑁𝑛=1𝑡𝑎𝐵(𝑛+1)𝑎𝐵𝑛𝑡𝑎𝐵(𝑛1)𝑎𝐵𝑛+𝜖𝐵𝐻𝑦𝑛𝑎𝑛𝐵𝑎𝐵𝑛0𝑥0200𝑑+𝑣𝑛||𝑥𝑛𝑦𝑛||𝑎𝑛𝐵𝑎𝐴𝑛.(29) For the case of LUMO, we can obtain the result by 𝑎𝑏 in 𝐻𝐿el. Here 𝑎𝑛𝐴(𝑏𝑛𝐴) is the electron creation operator in the HOMO (LUMO) at the 𝑛th nucleotide group of P, S, and 𝐵𝑛 for the chain A. 𝜖𝐴𝐻(𝑥𝑛) and 𝜖𝐵𝐻(𝑦𝑛) are the electron-lattice coupling 𝐻el-ph given by𝜖𝐴𝐻(𝑥)=𝜖𝐴𝐻+𝐹𝐻𝜖𝑥,(30)𝐵𝐻(𝑦)=𝜖𝐵𝐻𝐹𝐻𝑦,(31) respectively. We can also obtain 𝜖𝐴𝐿(𝑥𝑛), 𝜖𝐵𝐿(𝑦𝑛) for LUMO. Moreover, we assume the relation||𝜖𝐿𝜖𝐻||4|𝑡|,(32) since this condition is realized in most of the DNA systems and guarantees the semiconductivity of the DNA. The total Hamiltonian 𝐻tot is given by𝐻tot=𝐻ph+𝐻𝐻el+𝐻𝐿el,(33) while the wavefunction |Φ is given by||Φ=𝑁𝑠=𝐴,𝐵𝑛=1𝜙𝑠𝑛𝑎𝑛𝑠||0+𝜙𝑠𝑛𝑏𝑛𝑠||0+𝜑𝑠𝑛𝑎𝑛𝑠||0+𝜑𝑠𝑛𝑏𝑛𝑠||0,(34) where 𝑎𝑛𝜎 (𝑏𝑛) means the electron creation operator at site 𝑛 in chain 𝐴 (𝐵). 𝜙𝐴𝑛(𝜙𝐵𝑛) represents the HOMO and 𝜑𝐴𝑛(𝜑𝐵𝑛) represents the LUMO, at the 𝑛th nucleotide site in the chain 𝐴(𝐵), respectively. Applying to the Schr̈odinger equation 𝐻tot|Φ=𝐸tot|Φ, we obtain the following equations for the HOMO states of nucleotide groups in the double strand of DNA:𝜙𝑡𝐴𝑛+1+𝜙𝐴𝑛1+𝜖𝐴𝐻𝜙𝐴𝑛+𝐹𝐻𝑥𝑛𝜙𝐴𝑛𝑣𝑛𝜙𝐵𝑛=𝐸𝜙𝐴𝑛,𝜙𝑡𝐵𝑛+1+𝜙𝐵𝑛1+𝜖𝐵𝐻𝜙𝐵𝑛𝐹𝐻𝑦𝑛𝜙𝐵𝑛𝑣𝑛𝜙𝐴𝑛=𝐸𝜙𝐵𝑛,(35) where 𝐸𝐸tot𝐻ph(𝑥𝑛,𝑦𝑛), 𝑣𝑛𝑣(|𝑥𝑛𝑦𝑛|). For the LUMO states of nucleotide groups in the double strand of DNA, the similar equations are given by replacing 𝜙 and 𝐻 by 𝜑 and 𝐿, respectively. Moreover, according to the argument of Holstein, we find the following coupled discretized nonlinear Schr̈odinger equation (DNSE) describing the electronic behavior under the lattice vibrations in the ladder model of DNA. In Appendix E, we give the derivation of the coupled DNSEs. The single-chain version of the DNSE and some comments on the physical meanings are also given in Appendix D.

5.4. DC-Conductivity of the Double Strand of DNA

We exclusively focused on the low-energy transport, when the charge injection energies are small compared with the molecular bandgap of the isolated molecule which is of the order of 2-3eV. In the experiment of the conductance property of the DNA, temperature dependence is important. Finite temperature can also reduce the effective system size and leads to the changes in the transport property. Moreover, the effects of the stacking energy and of temperature can be considered by introducing the fluctuation in the hopping energy such as the Su-Schrieffer-Heeger model for polyacetylene [5961].

We consider DC-conductivity of periodic DNA sequence based on the LUMO, HOMO band of 𝜋-electrons and small polaron [14]. It is known that DNA behaves as 𝑛(𝑝)-type extrinsic semiconductor, where the donors (acceptors) for the LUMO (HOMO) band are positively (negatively) charged small polarons with the total number 𝑁𝑑(𝑁𝑎) and the energy 𝜖𝑑=𝐸𝐿𝑠𝑝=𝐸𝑐𝐹2𝐿/𝐼0𝜔20, 𝜖𝑎=𝐸𝐻𝑠𝑝=𝐸𝑣+𝐹2𝐻/𝐼0𝜔20, where 𝐸𝑣(𝐸𝑐) is the valence (conduction) band edge. Following the standard argument, denote by 𝑛𝑐 and 𝑛𝐷 the numbers of electrons in the conduction (LUMO) band and in the donor levels, respectively; and denote by 𝑝𝑣 and 𝑝𝐴 the numbers of electrons in the valence (HOMO) band and in the acceptor levels, respectively. We now have the relation 𝑛𝑐+𝑛𝑑=𝑁𝑑𝑁𝑎+𝑝𝑣+𝑝𝑎, where𝑛𝑑=𝑁𝑑1/2𝑒𝛽(𝜖𝑑𝜇)+1,𝑝𝑎=𝑁𝑎1/2𝑒𝛽(𝜇𝜖𝑑),+1(36) where 𝛽1/𝐾𝐵𝑇 and 𝜇 is chemical potential of the system. If we suppose that 𝜖𝑑𝜇𝑘𝐵𝑇, 𝜇𝜖𝑎𝑘𝐵𝑇, 𝑛𝑑𝑁𝑑, and 𝑝𝑎𝑁𝑎, then𝑛𝑐=𝑒𝛽(𝜇𝜇𝑖)𝑛𝑖,(37) where 𝑛𝑖=𝑁𝑐𝑃𝑣𝑒𝛽𝐸𝑔/2 and 𝜇𝑖=(𝐸𝑐+𝐸𝑣)/2+log(𝑃𝑣/𝑁𝑐)/2𝛽.

The DC-conductivity is given as𝑒𝜎=2𝑛𝑐𝜏𝑒𝑚𝑒=𝑒2𝑝𝑣𝜏𝑚,(38) by Drude formula. This suggests that if 𝜏𝑒=𝑐𝑜𝑛𝑠𝑡 then the temperature dependence of 𝜎 comes from 𝑛𝑐, while if 𝑛𝑐=𝑐𝑜𝑛𝑠𝑡 then it comes from 𝜏𝑒. Since the temperature dependence of 𝜏𝑒 comes from other mechanisms of scattering such as the activation of polaron motion considered by Yoo et al., we can assume that 𝜏𝑒=𝜏0𝑒𝛽𝐸𝑎, where 𝐸𝑎 is the activation energy. Hence, we have𝑒𝜎=2𝑛𝑐𝜏0𝑚𝑒exp𝛽𝐸𝑎.(39) If we adopt the log𝜎 versus 1/𝑇 plots, then we find the strong temperature dependence found by Tran et al. and Yoo et al. (see Figure 16).

Figure 16: The DC-conductivity is shown for the n-type (or p-type) extrinsic semiconductor where the energy gap 𝐸𝑔=0.33eV, the hopping energy for the HOMO (or LUMO) band 𝑡=0.2eV, and the activation energy for the polaron hopping of (a) 𝐸𝑎=0.001t, (b) 𝐸𝑎=0.03t, and (c) 𝐸𝑎=0.05t are taken, respectively. 𝑁𝑐=𝑃𝑣=108 and 𝑛=𝑁𝐷𝑁𝐴=103 are assumed.

It is found that the band gap is reduced by the formation of the double strand of DNA and small polarons exist to behave as localized donors and acceptors in the DNA double helix (see Figure 17). This extrinsic semiconducting nature of DNA qualitatively explains many experimental results that are observed recently.

Figure 17: Energy gap 𝐸𝑔(𝑇) of the n-type (or p-type) extrinsic semiconductor is shown for electron conduction (or hole conduction) where the energy gap 𝐸𝑔=2eV, 𝑡=0.1eV, 𝑁𝑐=𝑃𝑣=109, and 𝑛=106 are used, respectively.

As was stated in the introduction, it is very well known that, in solid-state physics, the extrinsic semiconductor character is added by impurity atoms that are exerted from outside into host materials. For example, if the host material is a homogeneous crystal of Si or Ge with four valence bonds, then the impurities are As atoms with five valence bonds and boron (B) atoms with three valence bonds, which then provides an inhomogeneous material with impurity levels. The As atoms play a role of donors with positive charge such that the system becomes an 𝑛-type semiconductor, while the B atoms play a role of acceptors with negative charge such that the system becomes a 𝑝-type semiconductor. Such a system of n- or p-type semiconductor exhibits an extrinsic semiconductor character. On the other hand, in our systems of DNA semiconductors, there exist no such impurities exerted from outside; but already there exist inside the DNA a complicated arrangement of bases of A, G, C, and T, such that, among the bases, each one of the bases may regard other bases as impurities. Hence, a kind of extrinsic semiconductive nature may appear as a result of self-organization of the base arrangement. This is the meaning of our terminology of “extrinsic semiconductor” for the DNA systems. Therefore, it is more preferable for us to put “self-organized” in front of “extrinsic semiconductor” for DNA such as “self-organized extrinsic semiconductor” in order to represent the semiconducting character of DNA.

We would like to note that, in the 𝑑-dimensional disordered systems, the activated hopping between localized states, that is, the variable-range hopping, could be a dominant mechanism for the Dc-conductivity, and the temperature dependence is governed by𝜎=𝜎0𝑒(𝑇0/𝑇)𝑑/(𝑑+1),(40) where 𝑇0=8𝑊𝑎𝛾/𝑘𝐵. Here 𝑊 is the energy difference between the two sites, 𝛾 is the Lyapunov exponent, and 𝑎 is the distance between the nearest neighboring bases. It is reasonable to expect that the experiments of charge transport in 𝜆-DNA suggest such temperature dependence 𝜎𝑒𝑇0/𝑇 at relatively low temperature due to the 1D aperiodic base-pair sequence. The main contribution was given by the interaction with water molecules and not with counterions. Further, polaron formation was not hindered by the charge-solvent coupling. And the interaction rather increased the binding energy (self-localization) of the polaron by around 0.5 eV, which is much larger than relevant temperature scales.

Before closing this section, we give brief comments concerning the measurements of electronic conductivity in DNA again. The DNA is quasi-1D systems coupled with the environment including the surrounding substrate and contact leads. In general, the Landauer-Buttiker formula using transfer matrix method and the Kubo-Greenwood formula using linear-response theory have been used to estimate the transport properties in the quantum systems. The transfer matrix method is straightforward for the quasi-1D systems; however, it is inconvenient for the complicated cases involving the environmental effects although the two formulae are equivalent at least for single-particle cases [62, 63]. There is an indication that the transfer matrix method is also effective even for law transmission coefficient due to the contact effect, despite a very good charge transfer along the DNA sequence [64].

6. Conduction and Proton Transfer

Gene mutations sometimes cause human disorders such as cancer. Simultaneously, gene mutation can be a driving force for processes of biodiversity and evolution [65]. When a system contains hydrogen bonds, proton motion also needs to be considered. Instead of oscillatory motions in a single-well potential, protons can tunnel from one side to another in a double-well potential of the hydrogen bond. This proton tunneling causes an interstrand charge hopping. And it could generate the tautomeric base pairs and destroy the fidelity of the W-C base pairs.

Löwdin proposed that proton tunneling contributes to the formation of rare tautomeric of DNA W-C base pairs whose accumulation could result in DNA damages, point mutations, and even tumor growth [66]. The argument relies on the assumption that the rare tautomers are more stable, and once the intermolecular proton transfer occurs, lifetime of the tautomers is significantly larger than time of DNA repair. Indeed, the results from the quantum chemical and statistical mechanical calculations indicate that at room temperature at least the GC tautomers (GC) would have a sufficient lifetime to cause the DNA damage through mismatches of the W-C base pairs.

Ladik speculated that semiconductivity of DNA might be related to the origin of cancer due to transmutation of genes [8, 9]. Very recently, Shih et al. have also reported correlation between the cancerous mutation hot spots and the charge transport along DNA sequences [67].

The proton transfer causes fluctuation of the potential and more or less affects the localization/transport property of the charged carriers. At the same time, the proton transfer suffers from the lattice vibration and the electron transport along the base sequences of DNA. Accordingly, we should treat the coupling between the proton transfer and the phononic and electronic states when we investigate the stability of the tutomatic states (the excited states). Recent theoretical studies have shown that charged protonated base pairs display smaller activation barriers, which make the proton transfer and the tunneling more facile.

Chang et al. [43] have given the effective 1D model Hamiltonian 𝐻tot=𝐻elph(𝑡)+𝐻𝜎+𝐻el𝜎, where 𝐻𝜎=𝑛𝐻𝜎𝑛 represents Hamiltonian describing the proton transfers. Here Hamiltonian 𝐻elph includes the electron and phonon modes and the coupling. For the proton motion in hydrogen bonds, they used a two-level system in order to describe the tunneling behavior at molecular site 𝑛:𝐻𝜎𝑛=12𝜖𝑝𝜎𝑧𝑛+𝑡𝑝𝜎𝑥𝑛,(41) where 𝜎𝑧𝑛 and 𝜎𝑥𝑛 are Pauli matrices assigned at site 𝑛, 𝜖𝑝 is the energy bias between the two localized proton states, and 𝑡𝑝 is the tunneling matrix element between the states. They have modeled the coupling between the protons in the hydrogen bond and the charges in the DNA strand as𝐻el𝜎=𝑛𝛾𝜎𝑛𝜎𝑧𝑛𝐶1𝑛𝐶𝑛,(42) where 𝛾𝜎𝑛 is the coupling intensity. When the proton is in the lower-energy state or there is no charge around the hydrogen bond (𝐶𝑛𝐶𝑛=0), the coupling vanishes.

Indeed, the normal G-C base pair is in the lower-energy state and its tautomeric form is in an excited state. When 𝜖𝑝𝑡𝑝, the probability of having a tautomeric form is extremely small. However, the cation of a G-C base pair has almost the same energy as that of its tautomeric form 𝜖𝑝𝑡𝑝. In this case, the proton state becomes delocalized in hydrogen bonds.

The symmetry of potential well and the height of barrier essentially affect the tunneling probability between the potential wells. The probabilistic amplitude of proton could influence the transport of the charged carriers through the Coulomb interaction. In addition, the two-level approximation will be broken down if the chaotic motion occurs in the dynamics due to the coupling with lattice vibrations [68].

7. Summary and Discussion

We briefly reviewed our recent works together with concerning the electronic states and the conduction/transfer in DNA polymers. In Section 3, based on the Ḧuckel model, we have discussed the basics of quantum chemistry where the electronic states of atoms in biochemistry such as C, N, O, and P and the electronic configurations of the nitrogenous bases, the sugar-phosphate groups, the nucleotides, and the nucleotide bases are summarized, respectively. In Section 4, based on the tight-binding approximation of the DNA sequences, we have investigated the localization properties of electronic states and quantum diffusion, using a stochastic bond-vibration approach for the poly(dG)-poly(dC), the poly(dA)-poly(dT), and the mixed DNA polymers within the frameworks of the polaron models. At that time, we assumed that the dynamical disorder is caused by the DNA vibrational modes, which are caused by a noise associated with stochastic dynamics of the distances 𝑟(𝑡) between the two Watson-Crick base-pair partners. In Section 5, we have mainly investigated the energy band structure and the localization property of electrons in the disordered two- and three-chain systems (ladder model) with long-range correlation as a simple model for electronic property in a double strand of DNA by using the tight-binding model. In addition, we investigated the localization properties of electronic states in several actual DNA sequences such as bacteriophages of Escherichia coli and human chromosome 22, compared with those of the artificial disordered sequences. In Sections 2 and 4, we gave a brief explanation for DNA-bases sequences and gene mutations, respectively, which are related to the DNA carrier conduction phenomena [6972]. The relationship between the long-range correlations and the coherent charge transfer in the substitutional DNA sequences has been studied, based on the transfer matrix approaches [73, 74].

Recently, the tight-binding models for the DNA polymers have been extended to the decorated ladder models and the damaged DNA models by some groups. Furthermore, the 𝐼-𝑉 characteristics of the ladder models coupled with environments have been investigated as simple models for DNA polymers [7577].


A. Hückel Parameters

Sandorfy's formula [32], Streitwieser's formula [35], and Mulliken's formula [33, 34] are given by following relations:𝛼̇𝑋𝜒=𝛼+𝑋𝜒𝐶𝜒𝐶𝛼×4.1𝛽,(A.1)̈𝑋=𝛼̇𝑋+𝛽,(A.2)𝐶𝑋=𝑆𝐶𝑋𝑆𝐶𝐶,(A.3) respectively. See [5] for the details.

B. Transfer Matrix Method

Let us define the four-dimensional column vector in the main text as Φ𝑛(𝜙𝐴𝑛,𝜙𝐴𝑛1,𝜙𝐵𝑛,𝜙𝐵𝑛1)𝑡. Then, it can be rewritten in the following form: Φ𝑛+1=𝑀𝑛Φ𝑛, where 𝑀𝑛=𝐴𝑛𝑉𝑛𝑈𝑛𝐵𝑛.(B.1)

𝑀𝑛 is the 4×4 transfer matrix with the 2×2 matrices:𝐴𝑛𝐸𝐴𝑛,𝑛𝐴𝑛+1,𝑛𝐴𝑛,𝑛1𝐴𝑛+1,𝑛,𝐵10𝑛𝐸𝐵𝑛,𝑛𝐵𝑛+1,𝑛𝐵𝑛,𝑛1𝐵𝑛+1,𝑛,𝑈10𝑛𝑈𝑛𝐴𝑛+1,𝑛000,𝑉𝑛𝑉𝑛𝐵𝑛+1,𝑛0.00(B.2) According to the sequence of 𝑁 segments, we have to take the matrix product 𝑀(𝑁)𝑀𝑁𝑀𝑁1𝑀1, which is also a 4×4 matrix. When the double-chain system is periodic, we adopt the Bloch theorem:𝜙𝐴𝑛+𝑁=𝜌𝜙𝐴𝑛,𝜙𝐵𝑛+𝑁=𝜌𝜙𝐵𝑛,(B.3) where 𝜌=𝑒𝑖𝑘𝑁 and 𝑘 is the wavevector. Then, we find a 4×4 determinant 𝐷(𝜌) that is a fourth-order polynomial of 𝜌:𝐷𝑀(𝜌)det(𝑁)𝜌𝐼4=𝜌4𝑎1𝜌3+𝑎2𝜌2𝑎3𝜌+𝑎4=0.(B.4) It provides the wave vector 𝑘 in the system such that 𝑘=2𝜋𝑗/𝑁 for 𝑁𝑗=2,,𝑁/2. Here, if four roots are written as 𝜌1, 𝜌2, 𝜌3, and 𝜌4, then the following conditions hold:𝑎1tr𝑀(𝑁)=4𝑖=1𝜌𝑖,𝑎24𝑖<𝑗=1𝜌𝑖𝜌𝑗,𝑎34𝑖<𝑗<𝑘=1𝜌𝑖𝜌𝑗𝜌𝑘,𝑎4det𝑀(𝑁)=𝜌1𝜌2𝜌3𝜌4.(B.5) Using a physical intuition, if an electron propagation with 𝑘 along one direction in the double chain is represented by 𝜌, then the reverse propagation with 𝑘 is represented by 𝜌1. Therefore, the latter should be also accessible, since the choice of the direction of the coordinate system is arbitrary. Hence, 𝜌1 must be an eigenvalue of 𝐷(𝜌)=0 such that𝐷𝜌1=𝜌4𝑎4𝜌4𝑎3𝜌3+𝑎2𝜌2𝑎1𝜌+1=0.(B.6) This situation imposes the particular condition on the matrix 𝑀:𝑀,𝐽𝑀=𝐽,𝐽𝐉𝟎𝟎𝐉,𝐉0110(B.7) where 𝑀 means the Hermitian conjugate of 𝑀 and 𝟎 is the 2×2 zero matrix, respectively. This property is called the symplectic structure of 𝑀, where we have𝐷(𝜌)=𝜌4𝐷𝜌1,(B.8) from which we find that 𝑎1=𝑎3,𝑎4=1. Thus, 𝑀 belongs to 𝑆𝐿(4,𝐑). By using this property and dividing 𝐷(𝜌) by 𝜌2, the biquadratic equation is reduced to the quadratic equation𝑥2𝑎1𝑥+𝑎212=0,𝑥=𝜌+𝜌.(B.9) Therefore, its two roots are given as𝑥±=12𝑎1±𝐷,𝐷=𝑎214𝑎2+8.(B.10) Thus, from (B.5) and tr(𝑀2)=4𝑖=1𝜌2𝑖, we obtain 𝑎212𝑎2=tr(𝑀2).

Now we can state a simple scheme to obtain the spectrum. If an energy 𝐸 satisfies𝑥±=2cos𝑘𝑁,(B.11) then the energy is allowed; otherwise it is forbidden in channel ±, respectively. This is a generalized version of the Bloch condition for the single linear chain system with the 2×2 transfer matrix 𝑀, wheretr𝑀=2cos𝑘𝑁.(B.12) The density of states (DOS) 𝐷±(𝐸) is calculated for each channel ±, respectively:𝑑𝑘±=1𝑁𝑑cos1𝑥±(𝐸)21=𝑁𝜕𝑥±/𝜕𝜀4𝑥2±𝑑𝐸=𝐷±(𝐸)𝑑𝐸.(B.13) Therefore, the total DOS is given as the sum of 𝐷+(𝐸) and 𝐷(𝐸):𝐷(𝐸)=𝐷+(𝐸)+𝐷(𝐸),(B.14) where 𝐷(𝐸) [𝐷+(𝐸)] means the DOS contributed from the bonding (antibonding) channel (+), respectively. It agrees with the result on the tight-binding model for the ladder structure. Physically speaking, the (+) channel means the bonding (antibonding) states between two parallel strands of the DNA.

C. Transfer Matrices for Ladder Systems

In this appendix, we give the explicit expression of the energy band for the decorated ladder model given in Figure 10(a). In the unit cell, the period is taken as 𝑁=2 and it contains four 𝜋-orbitals. We apply the result in Appendix B for the case. Let the transfer matrix method be𝑀=𝐴𝑉𝑈𝐵.(C.1)𝑀 is a 4×4 transfer matrix with 2×2 matrices:𝐴=𝐴𝑛+1𝐴𝑛(𝐸𝛽)(𝐸𝛼)𝑎𝑎𝑏𝑏𝐸𝛽𝑎𝐸𝛼𝑎𝑏𝑎=𝐵,𝑉=𝐴𝑛+1𝑉𝑛𝐸𝛽𝑣𝑎𝑏𝑣0𝑎0=𝑈.(C.2) Let us calculate tr𝑀 and tr(𝑀2). We find that𝑀tr𝑀=tr(𝐴)+tr(𝐵)=2𝑃2𝑅,tr2𝐴=tr2𝐵+tr2+tr(𝑈𝑉)+tr(𝑉𝑈)=2(𝑃+𝑄)2𝑅+22,2(C.3) where𝑃=(𝐸𝛽)(𝐸𝛼)𝑎𝑏,𝑄=(𝐸𝛽)𝑣𝑎𝑎𝑏,𝑅=2+𝑏2.𝑎𝑏(C.4) The discriminant 𝐷 is given by 𝐷=2tr(𝑀2)(tr𝑀)2+8=4𝑄2. As a result, we obtain12cos2𝑘=2tr𝑀±𝐷==𝑃𝑅±𝑄(𝐸𝛽)(𝐸𝛼)𝑎𝑎𝑏2+𝑏2±𝑎𝑏(𝐸𝛽)𝑣.𝑎𝑏(C.5) Solving the above for 𝐸, we can obtain the energy bands:𝐸+(±)=12±(𝛼+𝛽)𝑣(𝛼𝛽)2+𝑣2𝑎+42+𝑏2,𝐸+2(𝛼+𝛽)𝑣+4𝛽𝑣+8𝑎𝑏cos2𝑘(±)=12±(𝛼+𝛽)+𝑣(𝛼𝛽)2+𝑣2𝑎+42+𝑏2.2(𝛼+𝛽)𝑣4𝛽𝑣+8𝑎𝑏cos2𝑘(C.6) In Figure 18(a) the energy band structure for 𝑎=𝑏=1 when 𝛼=1,𝛽=0 is given with varying the interchain hopping 𝑣. Figure 18(b) shows the cross-section view at 𝑣=1.

Figure 18: Energy bands of the decorated ladder model. (a) The energy bands as a function of 𝑉. (b) The snap shot of the energy bands when 𝑉=1. 𝑘 means the wave vector in units of 𝜋/𝑠 such that 1.0𝑘1.0, 𝑉 means the 𝜋-electron hopping integral between the interchain sites, and 𝐸 means the energy in units of 𝑉=1. Here we have taken the values 𝛼=1.0, 𝛽=0, and 𝑎=𝑏=1.

The extension to the decorated three chains (𝑑=3) is straightforward. Here we give the transfer matrix for a simple system of coupled three chains: Φ𝑛+1=𝑀𝑛Φ𝑛, where Φ𝑛=(𝜙𝐴𝑛+1,𝜙𝐴𝑛,𝜙𝐶𝑛+1,𝜙𝐶𝑛,𝜙𝐵𝑛+1,𝜙𝐵𝑛)𝑡. Furthermore, 6×6 transfer matrix with the 2×2 submatrices is given in block tridiagonal form𝑀𝑛=𝐴𝑛𝑉𝐴𝑛0𝑉𝐶𝑛𝐶𝑛𝑈𝐶𝑛0𝑈𝐵𝑛𝐵𝑛,(C.7) where 𝐶𝑛𝐸𝐶𝑛,𝑛𝐶𝑛+1,𝑛𝐶𝑛,𝑛1𝐶𝑛+1,𝑛,𝑉10𝐶𝑛𝑉𝑛𝐶𝑛+1,𝑛000,𝑉𝐴𝑛𝑉𝑛𝐴𝑛+1,𝑛0,𝑈00𝐶𝑛𝑈𝑛𝐶𝑛+1,𝑛000,𝑈𝐵𝑛𝑈𝑛𝐵𝑛+1,𝑛0.00(C.8)

For simplicity, we set 𝐴𝑛+1,𝑛=𝐵𝑛+1,𝑛=𝑎, 𝐶𝑛+1,𝑛=𝑐, 𝐴𝑛𝑛=𝐵𝑛𝑛=𝛼, 𝐶𝑛𝑛=𝜅, and 𝑉𝑛=𝑈𝑛=𝑣 at site 𝑛. Then𝑀=𝐸𝛼𝑎𝑣1𝑎𝑣000100000𝑐0𝐸𝜅𝑐𝑣1𝑐0𝑣00100000𝑎0𝐸𝛼𝑎1000010.(C.9) Even for general multichain models, some useful formulae exist in order to obtain the eigenvalues of block tridiagonal matrices and the determinants of the corresponding block tridiagonal matrices [7880].

D. Nonlinear Schrödinger Equation

In this appendix, we derive the discrete nonlinear Schr̈𝑜dinger equation for 1D electronic system coupled with lattice oscillations. The relative motions between two different base pairs can be represented by an acoustical phonon mode and the vibrational motion inside a base pair by optical phonons. These modes represent the lattice distortions such as sliding, twisting, or bending. The Hamiltonian that describes these modes is given by 𝐻ph=𝑛𝑝2𝑛+12𝑀2𝑀𝜔2𝑠𝑢𝑛+1𝑢𝑛2+𝑛𝑃2𝑛+12𝑀2𝑀𝜔2𝑜𝑣2𝑛.(D.1) Here 𝑢𝑛 and 𝑣𝑛 are lattice displacements and the internal vibration coordinates of the 𝑛th unit cell and 𝑝𝑛 and 𝑃𝑛 are their conjugated momentum, respectively. And 𝑀 is the mass of the unit cell, 𝜔𝑜 is the oscillation frequency of the optical phonon, and the dispersion relation of the acoustical motion is 𝜔𝑠(𝑘)=c𝑠𝑘, where 𝑐𝑠 is the sound velocity along the chain. Note that the two kinds of oscillations can be regarded as the dynamics of radial and angular coordinates in the polaron models in Section 4.

Then in the Hamiltonian for electrons, both the on-site potentials 𝐸𝑛 and the hopping integrals 𝑉𝑚,𝑛 depend upon these vibrations, in principle. The charge coupling to the acoustical phonons is given by the Su-Schrieffer-Heeger (SSH) model [5961], and the interaction with the optical phonons is described via the molecular crystal model of Holstein [8183]. The SSH model deals classically with the lattice degrees of freedom, while electrons are treated quantum mechanically. Thus the total Hamiltonian for the electron-phonon interactions in the DNA system is given by 𝐻elph=𝑛𝛼𝑉0+𝑢𝑛+1𝑢𝑛+𝛿𝑉𝑛𝑛+1×𝐶𝑛𝐶𝑛+1+𝐶𝑛+1𝐶𝑛+𝑛𝐸𝑛+𝛾𝑛𝑣𝑛𝐶𝑛𝐶𝑛,(D.2) where 𝛼 and 𝛾𝑛 are the coupling constants. Here 𝑉0 is the bare amplitude of the hopping term and the term 𝛿𝑉𝑛𝑛+1 is a random contribution from the conformational disorder, which we include to describe. The Anderson localization takes place when 𝐸𝑛 is a static on-site randomness. When 𝐸𝑛=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 in the system with half-filled conduction band, such as a 1D ionic crystal, the SSH term generates dimerization in the ground state (the Peierls instability) and forms solitons in the excited states.

However, DNA is considered as a band insulator. Both interactions can generate lattice distortions and lead to polaron formation when a charge is doped into the molecule.

The coupling with (𝑢𝑛+1𝑢𝑛) usually induces small polarons in ionic crystals. The calculations showed that a polaron may be built and be robust within a wide range of model parameters. The influence of random base sequences was apparently not strong enough to destroy it. Thus, polaron drifting may constitute a possible transport mechanism in DNA oligomers.

From the total Hamiltonian 𝐻tot=𝐻ph+𝐻elph of the system, we can derive the equations of motion for the variables 𝜙𝑛, 𝑢𝑛, and 𝑣𝑛 as follows: 𝑑2𝑢𝑛𝑑𝑡2+𝜔2𝑠2𝑢𝑛(𝑡)𝑢𝑛+1(𝑡)𝑢𝑛1𝜙(𝑡)=2𝛼Re𝑛+1(𝑡)𝜙𝑛(𝑡)𝜙𝑛(𝑡)𝜙𝑛1,(𝑡)(D.3)𝑑2𝑣𝑛𝑑𝑡2+𝜔2𝑜𝑣𝑛(𝑡)=𝛾𝑛||𝜙𝑛||(𝑡)2,(D.4)𝑖𝑑𝜙𝑛=𝐸𝑑𝑡𝑛+𝛾𝑛𝑣𝑛𝜙𝑛(𝑡)+𝑚𝑡𝑚𝑛𝜙𝑚(𝑡),(D.5) where 𝑡𝑚𝑛=(𝑉0+𝑢𝑚𝑢𝑛+𝛿𝑉𝑚𝑛). This means that the equilibrium position of each atom in the lattice is charged by an amount proportional to the probability for the electron to occupy that special atom. The twist polaron modifies the inter-base electronic coupling, though this effect is apparently less strong than the coupling in the Holstein model. Accordingly, we also assume that the dependence of the hopping integrals on the 𝑢𝑛 is so weak so that it can be ignored. In the following part, we deal with only the Holstein polarons.

Additionally, we assume that time scale of lattice vibrations and electron evolution are such that vibrations are slaved by electron probability. It is then possible to set time derivative in (D.4) equal to zero. Then, we obtain𝑖𝑑𝜙𝑛=𝑑𝑡𝑚𝑡𝑚𝑛𝜙𝑚||𝜙(𝑡)+𝛼𝑛||(𝑡)2𝜙𝑛(𝑡)+𝐸𝑛𝜙𝑛(𝑡),(D.6) where 𝛼=𝛾2𝑛/𝜔2𝑜. By setting 𝑡𝑚𝑛=𝑡 if |𝑚𝑛|=1 and zero otherwise, we obtain the discrete nonlinear Schr̈odinger equation (DNSE) 𝑖𝑑𝜙𝑛𝜙𝑑𝑡=𝑡𝑛+1(𝑡)+𝜙𝑛1||𝜙(𝑡)+𝑎𝑛||(𝑡)2𝜙𝑛(𝑡)+𝐸𝑛𝜙𝑛(𝑡).(D.7) The time-independent version is given as𝐸𝜙𝑛𝜙=𝑡𝑛+1+𝜙𝑛1||𝜙+𝛼𝑛||2𝜙𝑛+𝐸𝑛𝜙𝑛,(D.8)

which appears in the Holstein polaron model on the lattice. The DNSEs (40) and (41) show more various properties due to the inhomogeneity of 𝐸𝑛 and the strength 𝛼 for nonlinearity, and so forth [84, 85].

For example, localization-delocalization transition takes place, depending on the coupling strength and the initial state. The DNSE has been studied in the context of delocalization due to nonlinearity. Indeed, (D.7) describes the 1D disordered waveguide lattice, which is called the Gross-Pitaevsky (GP) equation on a discretized lattice [86]. When the absolute value of the nonlinearity parameter 𝑎 is greater than some critical value 𝛼𝑐, the excitation is self-trapped. It was mathematically proved that the DNLS has quasiperiodic self-trapped solutions called the discrete breathers [87, 88]. On the other hand, it is found that at moderate strength of nonlinearity the spreading of the wavepacket algebraically grows as pΔ𝑛)2𝑡𝜈(𝜇0.2) [89].

E. Coupled Nonlinear Schrödinger Equations

In this appendix, we derive the coupled nonlinear Schr̈odinger equations in order to describe the effect of formation of a double strand of DNA on the polarons in the HOMO band, following Holstein's argument [14]. The energy of HOMO band is expressed as 𝐸𝐻𝑥𝑛,𝑦𝑛=𝐻ph𝑥𝑛,𝑦𝑛+𝑁𝑛=1𝜖𝐻𝑥𝑛||𝜙𝐴𝑛||2+𝜖𝐻𝑦𝑛||𝜙𝐵𝑛||2𝑁𝑛=1𝑡𝜙𝐴𝑛+1+𝜙𝐴𝑛1Φ𝑛𝐵+𝜙𝐵𝑛+1+𝜙𝐵𝑛1𝜙𝑛𝐴𝑁𝑛=1𝑣𝑛𝜙𝐴𝑛Φ𝑛𝐵+𝜙𝐵𝑛𝜙𝑛𝐴.(E.1) Differentiating the energy with respect to 𝑥𝑝, 𝑦𝑝, respectively, we approximately obtain the most contributed coordinates 𝑥𝑝,𝑦𝑝 by 𝜕𝐸𝐻/𝜕𝑥𝑝=0, 𝜕𝐸𝐻/𝜕𝑦𝑝=0,𝑥𝑝||𝜙=𝜌𝐴𝑝||2𝜙𝜈𝐴𝑝𝜙𝑝𝐵+𝜙𝐵𝑝𝜙𝑝𝐴,𝑦𝑝||𝜙=𝜌𝐵𝑝||2𝜙𝜈𝐴𝑝𝜙𝑝𝐵+𝜙𝐵𝑝𝜙𝑝𝐴,(E.2) where 𝜌=𝐹𝐻/𝐼0𝜔20, 𝜈=𝛼/𝐼0𝜔20. In the derivation, we have assumed that 𝑣𝑝=𝑣𝛼|𝑥𝑝𝑦𝑝|, 𝜕𝑆𝐻𝐴𝐵/𝜕𝑥𝑝=0, where 𝜕𝑆𝐻𝐴𝐵/𝜕𝑦𝑝=0, where 𝑆𝐻𝐴𝐵=(𝜙𝐴𝑝𝜙𝑝𝐵+𝜙𝐵𝑝𝜙𝑝𝐴). Note that |Φ𝐴𝑛|2(|Φ𝐵𝑛|2) means the frontier orbital density at 𝑝th nucleotide group in chain 𝐴(𝐵) and 𝑆𝐻𝐴𝐵 means the overlapping integral of the frontier orbitals for electrons in the HOMO at 𝑝th nucleotide group between the chains 𝐴 and 𝐵. Substituting the expressions into (35) and (35), we obtain the following coupled DNSEs:𝜙𝑡𝐴𝑛+1+𝜙𝐴𝑛1+𝜖𝐻𝐴𝐵𝜙𝐴𝑛𝑣𝑛𝐴𝐵𝜙𝐵𝑛=𝐸𝜙𝐴𝑛,𝜙𝑡𝐵𝑛+1+𝜙B𝑛1+𝜖𝐻𝐵𝐴𝜙𝐵𝑛𝑣𝑛𝐴𝐵𝜙𝐴𝑛=𝐸𝜙𝐵𝑛,(E.3) where𝜖𝐴𝐵𝐻𝑛=𝜖𝐴𝐻𝜌𝐹𝐻||𝜙𝐴𝑛||2+𝜈𝐹𝐻𝑆𝐻𝐴𝐵,𝜖𝐵𝐴𝐻𝑛=𝜖𝐵𝐻𝜌𝐹𝐻||𝜙𝐵𝑛||2𝜈𝐹𝐻𝑆𝐻𝐴𝐵,𝑣𝑛𝐴𝐵=𝑣𝜈𝐹𝐻||𝜙𝐴𝑛||2+||𝜙𝐵𝑛||2.(E.4) By the same argument, similar equations can be obtained for the LUMO band case as well, just by replacing the orbitals of electrons in HOMO bands with the frontier orbitals for holes in the LUMO bands, respectively. In the limit 𝑣0,𝜈0, it becomes the decoupled DNSE in (D.8) without randomness.

F. The Lyapunov Exponents and Multichannel Conductance

The definition for energy dependence of the Lyapunov exponents is given by𝛾𝑖=lim𝑁12𝑁log𝜎𝑖𝑀𝑑(𝑁)𝑀𝑑,(𝑁)(F.1) where 𝜎𝑖() denotes the 𝑖th eigenvalue of the matrix 𝑀𝑑(𝑁)𝑀𝑑(𝑁) [47]. As the transfer matrix 𝑇𝑑(𝑁) is symplectic, the eigenvalues of 𝑀𝑑(𝑁)𝑀𝑑(𝑁) have reciprocal symmetry around the unity as 𝑒𝛾1,,𝑒𝛾𝑑𝑒𝛾𝑑,,𝑒𝛾1, where 𝛾1𝛾2𝛾𝑑0. 𝑑 denotes the number of channels; that is, 𝑑=2 in the two-chain model and 𝑑=3 in the three-chain model.

The Lyapunov exponent is related to the DOS 𝜌(𝐸) as an analogue of the Thouless relation (the generalized Thouless relation) [90]:𝑑𝑖𝛾𝑖||(𝐸)ln𝐸𝐸||𝜌𝐸𝑑𝐸.(F.2) Accordingly, we can see that the singularity in the largest Lyapunov exponent is strongly related to the singularity in the DOS.

Generally speaking, in the quasi-one-dimensional chain with the hopping disorder, the singularity in the DOS, the localization length, and the conductance at the band center depend on the parity, the bipartiteness, and the boundary condition. Since discussions on the details are out of scope of this paper, we give simple comments. Note that the parity effects appear in the odd number chains with the hopping randomness. In the odd number chain with the hopping randomness, only one mode at 𝐸=0 remained as an extended state, that is, 𝛾𝑑=0, while other exponents are positive, 𝛾𝑑1>>𝛾1>0. The behavior is seen in Figure 14 in the main text. Then, nonlocalized states with 𝛾=0 determine the conductance. Although we have ignored the bipartite structure in the three-chain models for simplicity, if we introduce bipartiteness in the intrachain hopping integral 𝑉𝑛(=𝑈𝑛), another delocalized state due to chiral symmetry appears at 𝐸=0.

Furthermore, we find that, in the thermodynamic limit (𝑛), the largest channel-dependent localization length 𝜉𝑑=1/𝛾𝑑 determines the exponential decay in the Landauer conductance 𝑔(𝑛) that is measured in units of 𝑒2/ at zero temperature and serves as the localization length of the total system of the coupled chains [49, 50]. It is given as𝑔(𝑛)=2𝑑𝑖=11cosh2𝑛/𝜉𝑖(𝑛)1exp2