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MODEL OF CHARGE TRANSFER ALONG DNA FRAGMENT
V.D.Lakhno

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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

    bn is the probability amplitude of the electron localization at the n -th site,
    α0n 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,
    νi,j are matrix elements of transition from the i -th to the j -th site,
    Mk is the mass of the k -th site,
    γk   is the friction coefficient,
    Kk is the elastic constant.
    All derivatives are differentiated here with time .

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

  ,  

    where

    ωn is the oscillation frequency of the n -th site,
    ηn = τα0n / , ηn,n±1 = τνn,j / ,
    ω2n = τ2Kn / Mn , ω' n = τγn / Mn ,
    κn = α'nUnT / .
    All derivatives are differentiated here with the non-dimensional variable t.

The effective mass of all sites is considered to be the same, and it is equal to Mn = 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 ω2n ≈ 10-4 - 10-6, ω'n ≈ 10-2 - 10-5.

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

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