MODEL OF CHARGE TRANSFER ALONG DNA FRAGMENT

V.D.Lakhno

## DESCRIPTION OF THE MODEL

Quantum-classical model of excitation transfer along a DNA fragment is computed. This DNA fragment is considered to be a chain of *N* sites. Each of these sites behaves as a harmonic oscillator. It is conventional, that the base planes of nucleotides are parallel to one another, and the distance between base planes of neighbouring sites are constant; this corresponds to the B-form of DNA.

,

where

*b*_{n} is the probability amplitude of the electron localization at the *n* -th site,

α^{0}_{n} is the excitation energy at the *n* -th site,

α^{'}_{n} is coupling constant between the charge with nucleotides,

are displacements of sites from their equilibrium positions,

ν_{i,j} are matrix elements of transition from the *i* -th to the *j* -th site,

*M*_{k} is the mass of the *k* -th site,

γ_{k} is the friction coefficient,

*K*_{k} is the elastic constant.

All derivatives are differentiated here with time .

Let τ = 10^{-14} sec is the characteristic time,
*U*_{n} is the characteristic size of fluctuation of the *n* -th site,
that is = *U*_{n}u_{n}, = τ·*t*.
The corresponding system in proximity of the nearest neighbours in non-dimensional variables has the following form:

,

The effective mass of all sites is considered to be the same, and it is equal to *M*_{n} = 10^{-21} g.
High frequency intramolecular oscillations in DNA, which correspond to oscillations of bases in a separate site, have frequencies of picosecond order. Frequencies and friction coefficients on sites are considered to be the same; the corresponding non-dimensional values lay in the range
ω^{2}_{n} ≈ 10^{-4} - 10^{-6}, ω^{'}_{n} ≈ 10^{-2} - 10^{-5}.

The system is solved by the fourth-order Runge-Kutta method with constant step.