{'fraction_non_alphanumeric': 0.027457546041616838, 'fraction_numerical': 0.01152834250179383, 'mean_word_length': 4.381209781209781, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Long-range Charger transfer in DNA through Polaron Diffusion

Recent experimental evidence shows that the pi-orbitals along the stacking of base pairs can facilitate the long-range charge transfer in DNA. Proton motion in the base pair hydrogen bonds has also been found to affect the transfer rate. To explain this behavior we propose a model considering interactions of doped charges with hydrogen bonds and vibrations in DNA. The charge trapped by either protons or vibrations can cause structural distortions leading to polaron formation. By further considering polaron diffusion in DNA we find that the charge transfer rate derived from the diffusion coefficient is in agreement with the experimental results.


I. INTRODUCTION
The question on whether DNA is a conductor or an insulator is still controversial due to the different results obtained in the measurements of DNA conductivity [1][2][3][4] . The related issue of long-range charge transfer in DNA which is associated with the problem of charge mobility has also been discussed for many years with an equal amount of heated debate. It has been suggested that the overlap of the electronic π 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 the distance [5][6][7] .
A multiple-step hopping mechanism was recently proposed to explain the charge transfer behavior in DNA 8 . 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 A-T base pairs. Long-range charge transfer is accomplished by a series of incoherent charge tunneling events between two nearest G-C base pairs separated by A-T pairs. The probability of the short-range tunneling (super-exchange) strongly depends on the distance between the two G-C base pairs. However, recent experimental results for the reaction rate only showed a very slight distance dependence for the charge hole transfer through more than three A-T base pairs 9 . It is now believed that a hole can be created in a A-T base pair by thermal activation from a G-C base pair when charge tunneling becomes unlikely over a long distance [10][11][12] . The long-range hopping can, therefore, be carried out through a long A-T bridge just as by a series of shortrange tunnelings in the G-C pairs. Although this mechanism seems to give a good interpretation of the charge transfer in DNA, the cause of the incoherent charge hopping over localized states of the hole carriers with such low reaction rate (10 9 ∼ 10 6 s −1 ) has not yet been well understood.
Besides the multiple-step hopping mechanism, polaron motion has also been considered as a possible mechanism to explain the phenomena of DNA charge transfer 13,14 . The charge coupling with the DNA structural deformations can create a polaron and cause a localized state. From the study of the dynamical properties in one dimensional systems, it is known that the polaron behaves as a Brownian particle that collides with the low energy excitations of its environment which acts as a heat bath 15 . This diffusive behavior is very similar to the multiplestep hopping mechanism based on a random walk model, although the former occurs in a continuous media while the latter is considered on the discrete sites of a lattice. Thus, the behavior of the incoherent charge hopping can be understood as polaron diffusion.
For most physical systems acoustical and optical phonons are the main lattice excitations. However, when a system contains hydrogen bonds, proton motions also need to be considered. Instead of oscillatory motions in a single-minimum potential, protons can tunnel from one side of a hydrogen bond to another in a double-minimum potential 16 . This proton tunneling causes interstrand charge hopping and is important in spontaneous mutations. Since genetic information can be preserved only when a nucleotide base is matched with its complementary one by hydrogen bonding, the positions of protons are the main identification for DNA polymerases to make replications with high fidelity. However, the reaction of proton transfer could generate tautomeric base pairs and destroy the fidelity 17 . The mechanism of the proton tunneling in the isolated base pairs and its possible biological implications were widely discussed by Löwdin many years ago 18 . It has also been suggested that proton transfer and charge conduction in stacked base pairs are affected by each other [19][20][21] . The recent experimental results on the influence of mismatched base pairs show that the proton transfer is required for the long-range charge transfer 22 . Thus the effect of proton tunneling will be discussed here.
The paper is organized as follows. The model for charge motion in DNA will be introduced in the next Section, considering charge hopping, possible fluctuations in structure and their interactions. We will present the formation of a polaron in Section III, together with the study of polaron stability and proton delocalization in the hydrogen bonds. In Section IV, we calculate the polaron diffusion coefficient and the results will be compared with the experimental reaction rate of charge transfer in DNA. Section V contains our conclusions.

II. THEORETICAL MODEL
Although biologically functional DNA is an aperiodic system, we will focus here on a periodic DNA structure. Since the polaron state discussed here is highly localized (see below), we believed that the periodicity condition is not very constraining when we consider charge transfer over intermediate distance. In fact, in some experimental setups one looks for charge transfer between two G-C pairs which are separated by many (periodic) A-T pairs. As we can see in Fig. 1, following hole injection into the system, the charge localizes in the G-C base pair but can migrate from one unit cell to another. The essential ingredients in our model are (i) the electronic system, that realizes the charge conduction; (ii) the possible structural fluctuations, which include phonons and proton motion in hydrogen bonds; and (iii) the coupling between the charges and the structure. For a non-interacting electron system, the tightbinding model is extensively used in many theories of DNA charge transport 11,14,23 : where C i (C † i ) is the electron annihilation(creation) operator at the i-th site and t 0 is the overlap integral of electronic π orbitals between two nearest neighboring sites. (We disregard the electron spin since we are not describing magnetic phenomena here.) The energy spectrum of the electrons ishω k = −2t 0 cos (ka) giving a bandwidth 4t 0 , where a is the lattice spacing and the wave vectors k is 2πn/L with n as an integer and the system size L. In the continuum limit (a → 0), the Hamiltonian describes the kinetic energy of charge particles with a Bloch wavefunction and effective mass m , whereh 2 /2ma 2 is used to replace the overlap integral t 0 of the discrete model.
For the phonon part of the structural fluctuations in DNA, we consider that 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 25 . These modes represent lattice distortions, such as sliding, twisting or bending. The Hamiltonian that describes these modes is: where u i and v i are the lattice displacement and the internal vibration coordinate of the i-th unit cell, p i and P i are their conjugated momentum, respectively, M is the mass of a unit cell, ω 0 the oscillation frequency of the optical phonon and the dispersion relation of the acoustical motion is: ω s (k) = c s k , where c s is the sound velocity along the chain. The charge coupling to acoustical phonons is given by the Su-Schrieffer-Heeger model 24 and the interaction with optical phonons is described via the molecular crystal model of Holstein 26 . Thus the total electron-phonon interactions of DNA system is where γ s and γ v are the coupling constants. In systems with half-filled conduction band, such as an onedimensional π-conjugated polymers, the SSH term generates dimerization in the ground state (Peierls instability) and forms solitons in the excited states 27 . However, for DNA, considered here as a band insulator, both interactions can generate lattice distortions and lead to polaron formation when a charge is doped into the molecule. The second term in (3) usually induces small polarons in ionic crystals.
For the proton motion in the hydrogen bonds we use a two-level system to describe the tunneling behavior (see Fig. 2) 28 : where σ z and σ x are Pauli matrices, ε is the energy bias between the two localized proton states and t is the tunneling matrix element. The ratio t/ε provides information of the proton motion in the system. When the ratio is small, the protons are localized in one side of the hydrogen bonds. However, when it is large, the protons are delocalized. As we can see in Fig. 2(a), the normal G-C base pair is the lower energy state and its tautomeric form (G*≡C*) is an excited state. When ε ≫ t , the probability of having a tautomeric form is extremely small. However, in Fig. 2(b), the radical cation of a G-C base pair has almost the same energy as its tautomeric form 29 , i.e.,ε < ∼ t . In this case, the proton state becomes delocalized in the hydrogen bonds. To model the coupling between the protons in the hydrogen bond and the charges in the DNA strand, we use: where γ σ is the coupling intensity and n i = C † i C i is the number of charges on the i-th site. When the proton is in the lower energy state (σ z i = 1) or there is no charge around the hydrogen bond (n i = 0), the coupling vanishes. Although there are many hydrogen bonds in the base pairs of a unit cell, it is not necessary to consider all of the possible proton transfers since at room temperature the transitions to the tautomeric forms are very unlikely to occur. We only need to consider the hydrogen bond with the most probable proton transfer with charge injection into the unit cell, i.e., only one two-level system is required for each unit cell. By giving a total Hamiltonian as the sum of (1), (2), (3), (4) and (5) we have our complete model for charge motion in DNA.

III. FORMATION OF POLARON IN DNA
Although polaron formation has already been considered very important for charge motion in DNA 13,14,30 , the effect of the proton motion in the hydrogen bonds has not yet been discussed in the literature. Here, to investigate this factor, we analytically solve our model by taking the Hamiltonians into the continuum limit. The carrier wavefunction ψ (x, t) and the phonon modes u s (x, t), u v (x, t) are considered as functions of time and a continuous coordinate x .The equations of motion for the variables in the problem can be studied in the semiclassical limit (h → 0) and by a change of variables λ = x − υt , where υ is the polaron velocity. In this case, we have 15 : where E 0 is the binding energy of the polaron, and the problem reduces to a nonlinear Schrödinger equation for the wavefunction φ 0 : where ρ is the mass density M/a , sin 2 θ represents the positions of the protons in the hydrogen bonds and is expressed by whereε = ε−γ σ φ 0 2 . The interaction energy between the carriers and protons, γ σ sin 2 θ , is significant only when the charge trapped by the hydrogen bond causes proton delocalization. Therefore, for small value of t compared to ε , we can use the approximation: From the continuity of the polaron wavefunction, equation (7) can be analytically solved and the solution is found as combinations of Jacobian elliptic functions. When the charge coupling with the protons is small, i.e., γ σ φ 0 2 max ≤ ε − t (where φ 0 max is the maximum value of φ 0 ), the polaron is purely formed by the interaction with phonons and the proton motion has no effect on the wavefunction or the binding energy. The ground state solution is simply expressed as and the binding energy is where It is easy to see that 1/g represents the size of the polaron. To calculate the binding energy and the size of a static polaron, we consider that a hole carrier is doped into a poly(G)-poly(C) DNA. The effective mass of the carrier m =h 2 /2t 0 a 2 is of order 20m e = 1.  (7), since the energy difference between the adiabatic and vertical ionization potentials of a G-C base pair is about 0.5 eV 29 , the optical part is given as about 0.5 eV ×3.38Å = 1.7 eVÅ. The acoustical part is assumed to be smaller, so we consider the value of the coupling strength as approximately 2 eVÅ. From these results, we find that the polaron is 0.75Å wide with binding energy E 0 = −1.35 eV and, therefore, it is entirely localized in a base pair. For the weak coupling to hydrogen bonds (γ σ ≤ 0.2 eV), the protons are localized in their original positions and the tautomeric base pair is unlikely to be generated.
As the coupling constant γ σ becomes larger, proton transfer can be induced since the charge trapped into a base pair reduces the bias energy ε dramatically. By considering the bias energy ε and the tunneling matrix element t of order 0.3 eVÅ −1 and 0.03 eVÅ −1 , respectively, we find that the binding energy of the polaron E 0 is −1.45 eV for γ σ = 0.226 eV, which is more stable than the above with the same values of the effective mass m and the coupling strength γs 2 ρc 2 s + γv 2 ρω 2 0 . The ground state wavefunction consists of two parts, the central and the external. For the external part, the wavefunction behaves as a hyperbolic secant function since it should vanish as λ → ∞ . In the central part of the polaron, the wavefunction is described by a Jacobian elliptic functions of second kind and the proton in the hydrogen bond is delocalized since the value of sin 2 θ is large. For the base pair in the center of the polaron, its tautomeric form can be easily induced.
The relation between the coupling constant γ σ and the binding energy is shown in Fig. 3 . When γ σ is larger than 0.24 eV , the proton becomes delocalized in the hydrogen bond and the charge binding energy increases rapidly. The local deformations from the charge interaction with phonons can also stabilize the polaron as shown in Fig. 4 with γ σ = 0.25 eV. The increase of the couplings γ s or γ v also induces proton transfer since the carrier is strongly attracted in the base pair. The effect of proton transfer has been observed in experiments on the influence of mismatches in the DNA long-range charge transfer 22 . Mismatched base pairs are found to cause a large drop in the charge transfer rate due to the weakening of hydrogen bonds. For a normal G-C base pair, the positive charge in the radical cation can be distributed by both nucleotides through the delocaliza-tion of protons. However, mismatched Guanine, which leads to a fast deprotonation to the surrounding water, might induce a bound state with lower energy. Therefore, the charge is localized in the mismatched base pair and the long-range charge transfer is interrupted.

IV. POLARON DIFFUSION IN DNA
The dynamic properties of a polaron induced by the carrier interaction with acoustical and optical phonons in one dimension were explored many years ago 15 . In the strong coupling limit, it is found that the polaron moves as a Brownian particle interacting with the light particles of the environmental heat bath. To investigate these results in the context of DNA, we first consider the case without proton coupling (i.e., γ σ = 0).
The damping parameter of the optical polaron motion at high temperature (k B T ≫hω 0 ) is given by 15 : where M o is the classical polaron mass: and Here, we use E o as the binding energy from the optical phonons. For acoustical phonons we have the damping parameter: where the classical polaron mass for the acoustical case is and with E a the binding energy from the acoustical phonons. The random walk motion of the polaron in the long time limit (t ≫ 1/γ o or t ≫ 1/γ a ) is governed by the onedimensional diffusion equation: where n (x, t) is the density of the particles and D is the diffusion coefficient. The relation between the damping parameter and the diffusion coefficient is given by the Einstein relation: where we use D o(a) for the diffusion coefficient in the optical (acoustical) case.
As we can see from the scheme of the multiple-step hopping mechanism in Fig. 1, the hole hopping from a G-C base pair to its nearest neighbor can be viewed as an oxidation-reduction reaction with a rate constant k . According to the rate law of a first-order reaction, the concentration of the charge at the i-th G-C base pair, n i , is described by In the continuum limit, when the distance between two nearest neighboring G-C base pairs, a, is very small, we have By comparing this equation with (19), we have the correspondence from the diffusion coefficient in the continuous medium to the charge transfer rate of the discrete model: Thus, instead of a diffusion coefficient, we can use the discrete characteristic t 0 to estimate the rate constant. From (20) and (13), we have By the correspondence (23), the rate constant for the optical case is obtained as We can also derive the rate constant for the thermal activation by the acoustical phonons as For poly(G)-poly(C) DNA, the oscillation frequency is ω 0 ≈ 10 11 Hz from the theoretical and experimental results 32 . The theoretical calculation has shown the valence bandwidth of 0.04 eV 31 . By considering that the binding energy E o is about 1 eV, we have the reaction rate k o ≈ 10 9 s −1 . This estimate is in agreement with the experimental result for electron transfer in DNA which gives k o ≈ 10 8 ∼ 10 9 s −133 . This low reaction rate is generated by two factors in equation (25), the hopping probability t0 Eo and the oscillation frequency ω 0 . Due to the strong structural fluctuations at room temperature, the charge hopping could only happen when the local structure of the polaron achieves an effective configuration 30 . For each period of structural oscillation, the probability to have a successful hopping is of order t0 Eo . Thus, the charge transfer rate is determined by the frequency of reaching the effective configuration ω 0 and the probability t0 Eo . The behavior of the incoherent charge hopping is now described by the polaron motion under thermal fluctuations.
Another way to study the diffusion coefficent in the optical case is to consider the mean square displacement: and we may rewrite (24) as: Now, we can see that, for every oscillation period 2π/ω 0 , the collisions by the light particles from the environment spread the polaron with the mean-square displacement (∆x) 2 . The diffusion in the continuous medium, however, needs to be corrected when we make the correspondence since the size of the polaron is smaller than the lattice spacing. Therefore, the real reaction rate of charge transfer between the discrete base pairs should be smaller than our results. Nevertheless, the order of the charge transfer rate can still be estimated from the simple expression: We can also estimate the reaction rate k a in the acoustical case from the experimental data for the sound velocity c s ≈ 10 5 cm s 32 . Assuming γs 2 ρsc 2 s ≈ .34 eVÅ, we have k a ≈ 10 13 s −1 which is four order of magnitude larger than k o . A faster charge transfer rate is expected if the polaron is purely generated by the acoustical phonons. However, if optical phonons or proton transfer are involved, a smaller value of the reaction rate should be expected due to the increase of the binding energy in those cases.

V. CONCLUSIONS
In this paper we studied the charge conduction in periodic DNA focusing on the effect of the structural fluctuations which includes the proton motion in hydrogen bonds. We found that the charge trapped by hydrogen bonds or phonons can generate a polaron state and the coupling to the protons can induce proton transfer in a base pair. The polaron moving with small velocity is seen as a Brownian particle colliding with the light particles of the environment. This diffusion process corresponds to a multiple-step hopping mechanism in the discrete model. From this correspondence, the reaction rate of the longrange charge transfer in DNA can be derived from the diffusion coefficient, and it was found that the result predicted in the optical case is in agreement with the experimental results.
Although we did not include the effect of proton transfer on the polaron diffusion, it is expected to be crucial in the dynamics of charge hopping. From electrochemical experiments 34 , a decrease of charge transport along the DNA with mismatched base pairs is found. This behavior might be explained by the effect of the proton motions in the hydrogen bonds and we will explore this possibility in future work.