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High temperature electrical conductivity due to small polaron hopping motion in DNA molecules

We present a small polaron hopping model to interpret the high-temperature electrical conductivity measured along the DNA molecules. The model takes into account the one-dimensional character of the system and the presence of disorder in the DNA double helix. The experimental data for the lambda phage DNA (λ-DNA) and the poly(dA)-poly(dT) DNA follow nicely the theoretically predicted behavior leading to realistic values of the maximum hopping distances supporting the idea of multiphonon-assisted hopping of small polarons between next nearest neighbors of the DNA molecular "wire".


Introduction
Several mechanisms have been proposed for the interpretation of the strong temperature dependence of the conductivity measured at high temperatures. Carrier excitations across single particle gaps [1], bandlike electronic transport [2], variable range hopping [4] or small polaron transport [3,4,5] are among the mechanisms reported.
Tran and al [1] measured the conductivity and its temperature dependence along the lambda phage DNA (λ-DNA) double helix. The strong temperature dependence and the large conductivity they observed at high temperatures was attributed to carrier excitations across single particle gaps or temperature driven hopping transport processes. Alternatively, they noticed that the observed behavior would occur when phonon-assisted polaron hopping is the actual transport mechanism.
Yoo et al [6] reported measurements of electrical transport through poly(dA)-poly(dT) DNA molecules containing identical pairs. Their measured I-V characteristics at various temperatures were interpreted using a small-polaron hopping model [7]. They estimated the hopping distance to be about 16.8Å ( 5 base pairs). Their effort to interpret the temperature dependence of the conductivity observed by Tran et al [1], using the same polaron hopping model over the whole temperature range, was not quite convincing and therefore they did not exclude other possible mechanisms.
Tran et al [1] emphasized that the models proposed for the interpretation of the electrical conductivity data had not taken into account the inherent disorder associated with the random base sequences, and also the random potentials along the DNA double helix arising from the randomly positioned positively charge counterions. The influence of counterion induced disorder in DNA conduction has been recently investigated by Adessi et al [8].
Most recently Alexandre et al [9] reported ab initio calculations for poly(dC)-poly(dG) DNA, with up to 4 C-G pairs. They found a strong hole-lattice coupling and clear evidence for formation of small polarons .
Up to our knowledge, none of the models proposed takes explicitly into account the effect of disorder on the small polaron binding energy. This affects drastically the intrinsic transition rate for a hop from a molecular site i to a neighboring site j, and consequently the temperature dependence of the conductivity [10,11].
Secondly, the multiphonon-assisted character of the hopping mechanism at high temperatures, which is also reflected upon the expression of the intrinsic transition rate, has not been taken into account, either.
Thirdly, there is a lack of a systematic derivation of an analytical expression of the electrical conductivity, and the hopping distance, as functions of the temperature, appropriate for multiphonon-assisted small polaron transport through a disordered one-dimensional medium, interpreting in a consistent way charge transport along the DNA "wire".
The scope of the present work is to propose a transport model which incorporates all the above characteristics and interprets consistently the observed conductivity behavior at high temperatures.

Theory
These characteristics had been taken into account by Triberis and Friedman [10] and Triberis [11] introducing the generalized molecular crystal model (GMCM).
The GMCM model [10] is based on a generalized "hopping-model" Hamiltonian of the form: where < m|V |n > is the overlapp part , |n >= |i, [n k ] > are the eigenstates of H, H 0 is the zeroth-order (i.e. for electronic overlapp intergral J=0) Hamiltonian, and [n k ] represents the totality of the vibrational quantum mumber (...., n k , ...) for the occupation of the site with position vector r i . Assuming that the dependence on the spatial separation R ij of the two sites is exp(−2αR ij ) , α −1 being the spatial extent of the electronic wavefunction localized at a single site, we obtain the "microscopic" intrinsic transition rate γ ij , for a small polaron hopping from a site i to an empty site j and consequently, the average equilibrium transition probability , W 0 ij [10]. The transport problem is transformed to an equivalent network of impedances Z ij [10,13], given by k B is the Boltzmann's constant, and q is the carrier's charge. At high temperatures (h), the impedances Z ij are given by: Percolation takes place when The inverse quantity Z −1 c characterizes the macroscopically observed conductivity of the material which reads : Here σ h 0 = (Z h 0 ) −1 , and r h m = ξ h c /2α is the maximum hopping distance.
We apply the arguments given in reference [10] to our one-dimensional regime. We assume a lack of correlation between successive impedances in a chain, and between the distribution of R ij and site energies. We define P (Z c ) as the average number of impedances of magnitude Z c or less connected to a given site.
At high temperatures, we obtain: where N s is the concentration of sites , The indices from E i , ∆ ij and R ij , have been dropped. For the one-dimensional case, for η c we use the value 2 [14].
For a constant density of states N 0 , over an energy range E m , equation (6) gives: where T h 0 = 27 1/2 N 1/2 Thus, the conductivity of the high-temperature multiphonon assisted small polaron hopping one-dimensional regime, is given by: For a given temperature, equation (8) is expressed as: Here, i.e. the maximum hopping distance follows a T −2/3 -law. Equation (8) allow the evaluation of T h 0 from the slope of the fits of the experimental data reported for DNA samples, plotted as lnσ h − vs − T −2/3 . Using equation (10) we evaluate the maximum small polaron hopping distance at the temperature of interest.

Results and discussion
We apply our theoretical results to recently reported experimental data. Tran et al [2] data at 12 GHz show pronounced strong temperature dependent conductivity in a [227-342K]. Yoo et al [6] in poly(dA)-poly(dT) DNA molecules observed a similar behavior in a [178-306K]. The "high-temperature" region is characterized by the pronounced strong temperature dependence of the conductivity, although it is strictly determined by the conditionhω 0 ≤ k B T [15]. Figure 1 and figure 2 show the ln − vs − T −2/3 plots for these data, namely Tran et al(12 GHz)data [2] (sample(I)), and Yoo et al(poly(dA)-poly(dT))data [6] (sample II) (for α −1 = 2Å [15]). The experimental data follow nicely the T −2/3 -law.
The value of T h 0 is determined from the slope of the ln − vs − T −2/3 curves (equation (8)). Using equation (10)  lack of the analytical expression based on a transport mechanism, appropriate for the specific system, forced Yoo et al [6] to use the (fitting) parameter b to fit their experimental data as I ∼ bV . According to the authors the parameter b has a temperature dependence the physical origin of which is not clear. The above probably resulted in a non convincing attempt, as the authors [6] noticed, for the interpretation of Tran et al data [6]. The approach presented in the present work does not need any fitting parameter because the responsible transport mechanism is described in detailed, it is based on first principles, leading to analytical expressions for the conductivity as a function of the temperature, and the maximum hopping distance, as well.Our results support small polaron hopping to next neighbor sites, at high temperatures, responsible for the strong temperature dependence of the conductivity measured. They imply that as the temperature increases, in the high-temperature regime, shorter hops (or equivalently smaller impedancies) contribute to the transport process resulting in an increase of the conductivity, in accordance with experiment.

Acknowledgments
Dr F. Angenstein is appreciated for useful discussions.