CC BY
93
NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2012, 3 (1), P. 51-55
UDC 538.9
INTERACTION OF TERAHERTZ RADIATION WITH DNA
A. N. Bugay1 1 Joint Institute for Nuclear Research, Dubna, Russia [email protected]
PACS 62.25.Fg, 87.15.ht, 87.15.hp
A nonlinear model of DNA dynamics with coupling between conformational dynamics and proton tunneling is presented. It is demonstrated that terahertz radiation can influence both vibrational excitations and proton motion in DNA hydrogen bonds. The irradiation at the edge of far IR spectral range can promote proton tunneling. If the radiation frequency matches the vibrational mode, the generation of localized excitations in form of dissipative solitons is possible. These solitons decrease the probability of proton tunneling. Keywords: DNA, terahertz radiation, proton tunneling, dissipative soliton.
1. Introduction
There has been renewed interest in understanding the dynamics and properties of DNA due to recent advances in both single-molecule experiment techniques [1], molecular dynamic simulations [2] and theoretical modeling for such nanosystems [3]. Such modeling played an important step towards nanobiotechnology [4,5].
It is believed that many biological processes depend on the nonlinear properties of DNA. A number of theoretical studies of nonlinear physics of DNA were related to localized conforma-tional excitations [3] and electronic conductivity [5]. The motion of protons in hydrogen bonds causes a great number of interesting physical effects. The importance of proton tunneling in chemical and biological systems is well known for the DNA base pairing and mutagenesis [6,7]. The proton can tunnel into less energy favorable tautomer state of base pair, which may cause incorrect pairing during DNA replication. There are a lot of theoretical studies of proton tunneling in such systems [7-9]. An important conclusion from these studies is that the vibrational excitations in conformational subsystem can significantly influence the proton tunneling.
The frequencies of vibrationalrotational transitions of biological molecules and frequencies of intermolecular interactions as well as the frequencies of tunneling transitions fall into terahertz range (0.1-10 THz). It is demonstrated that terahertz radiation can induce conformational changes in proteins and nucleic acids [10], and also can influence the number of spontaneous and induced point mutations [11]. However, the underlying physical mechanisms of this interactions remain unclear. Some theoretical models suggest that collective nonlinear excitations can be responsible for such effects [12].
In the present paper a response of DNA to external terahertz electric field is studied on a basis of nonlinear model taking account of coupling between elastic vibrational dynamics and proton tunneling.
2. Theoretical model
Let us consider the self-consistent model of conformational and proton dynamics. A movement of the proton can be considered inside the asymmetric two-well potential per each hydrogen bond, as was initially proposed by Lowdin [6]. In conditions of terahertz excitation one can neglect the upper proton states considering only ground states in the right and left wells.
52
A. N. Bugay
In this case it is useful to consider the tunneling dynamics within the framework of pseudospin formalism [13]. The corresponding proton Hamiltonian Hp can be represented as:
H — —
n,i n,i=j
2 ^ y Jiji^i
n,i,j
Z OZ I S,
n+1,íSn,j + Si
z OZ ) (1) n-1,i n,j/> \ y
where indexes 'n' correspond to n-th base pair and indexes i,j denote the number of hydrogen bonds in a base pair (from 2 in Adenine/Thymine to 3 in Guanine/Cytosine). Here pseudospin operators Sz and Sx have the sense of the electrical dipole moment operator and the tunneling operator, respectively, is the tunneling integral, J± and J^ are the exchange integrals, and A is the asymmetry parameter of the potential well.
The conformational subsystem can be treated classically [3] since it corresponds to collective movements of large groups of atoms. The Hamiltonian is taken as in a well-known Peyrard-Bishop-Dauxois model (PBD) [14]
H
E
- I)2
(2)
where rn is the relative displacement (stretching) of base pairs inside the nucleotide, m is the reduced mass of the nucleotide, coefficient K characterizes the stacking interaction, and parameters D and f3 define the potential depth and width.
The interaction of proton subsystem with vibrations and terahertz electric field E is taken into account by interaction operator
Hint — Sn,i — 9 E rn(Szn,i — Szn,i) — ^ Qi (STn,i — STn,i)
(3)
where / is the dipole moment, and g is proportional to the linear piezoeffect coefficient, Sz and Sx are the equilibrium mean values of the pseudospin operators. The last sum corresponds to modulation of tunneling integral and can be approximated as Q1 = (exp(—ar^/2) — 1), where a is the material constant which is reversely proportional to the De Boer parameter of proton in DNA.
The Heisenberg equations of motion for the mean values of the pseudospin operators supplemented by phenomenological relaxation terms in the random-phase approximation have the form:
H-
dSx,
M z s y _ (s x — Sx )(
n,i n,i V n,i n,i/\
cos a
~rY
+
sin a
rp* T2
h
IT
— Mz Sx + Mx Sz — Sy /T *
n,i n,i ' n,i n,i n,il 2
hdSU = _Mx qy _ (qz _ wCOStt sino;
' " n.i n.i n,i)\ rr-,^ -r
dt
where
Mí
d(Hp + Hmt)
dS!
T*
tga — SnA¡S
T*
))
(4)
(5)
(6)
'IX
T*, T2* are the longitudinal and transverse relaxation times of pseudospins, and the equilibrium
mean values of the pseudospin operators Sz components:
SS
<x,z n,i
is:
2M„
-th
Mn
2khT'
Ml
and Sx are defined through the molecular field
dH
dS:
'X,Z 1
n,i
Mn,i = \J (Mn,i)2 + (Mz,y
))
The equations for the conformational subsystem under conditions of low-amplitude vibrations have the following form:
d Vft -K , r» \ 2 33 n 22 7 n 223
-T-2---(r„+1 + rra_i - 2rn) + w r„ - -[3uarn + -/? w r =
at2 m 2 6
Additionally the external damping term (with damping constant jr) and Maxwell-Helmholtz pressure term are taken into account. The last one arises from the uncompensated difference in dielectric permittivity between disturbance in DNA and surrounding medium (water). The equations (4)-(7) describe the self-consistent dynamics of proton and conformational systems under the action of external terahertz electric field.
3. Parametric excitation of conformational solitons and their influence on proton dynamics
The response of the system (4)-(7) posses high frequency modes, which correspond to tunneling transitions. Unlike the typical hydrogen-bonded crystals, in DNA the tunneling integral is much smaller than asymmetry parameter of potential. This gives relatively small probability for finding the proton in the tautomer positions P0 ~ 10"4. The tunneling frequencies are determined mostly by the asymmetry parameter A, and their values are at the order of 10 — 20THz [9]. The slow conformational subsystem cannot follow such excitations, therefore the increase of tunneling probability into tautomer state occurs under irradiation. This simple conclusion is supported by experiments, where the increase of point mutations have been observed after irradiation in the far IR spectral range [11]. The exact expression for response of the system can be calculated in a usual form [13].
The excitation of conformational subsystem with low frequency (optical phonon mode) requires special consideration. Its characteristic frequency (taken at the center of Brillouin zone) is u;a = y/2D^/m ~ 2 - 3 THz [8,12].
Since the conformational system is much slower, the response of proton subsystem can be calculated using the approximation of adiabatic following [15,16]. In general case we obtain an expansion
Sn,i + (sn) + sn,i rn + sn,2 rn + ...)E + (sn) + sn,i Tn + sn,i Tn + ...)E + ... (8)
The calculation of probability for finding the proton in the tautomer state of base pair A-T or G-C in the leading order gives:
p * = n — X&l + X% E2 + xg? rnE + .... (9)
The second term is produced by the modulation of the tunneling integral, while the others result from the tunneling far from the resonant frequency. Thus, there are competing effects, which depend on conformational disturbance.
Let us consider small amplitude oscillations in conformational subsystem. The solution in a semidiscrete approximation can be cast as
r = f(0) ((—vt) exp[i(ua(q)t—qn)]+c.c.+r(i) ((—vt)+r\2) ((—vt) exp[2i(ua(q)t—qn)]+c.c..+ ....,
(10)
where q is the reciprocal lattice vector, ( is the coordinate along the DNA axis. Since the characteristic size of DNA is much lower than terahertz wavelength, the expression for the external electric field with frequency u takes form E = E0 exp[zwi] + c.c.. Therefore the resonance is possible for the nearly standing excitation (q & 0).
54
A. N. Bugay
The resulting equation for the slowly varying amplitudes in the dimensionless form can be written as
- ^ - m = ~iv!> + (11)
where ^ = (20)-1psr\0) exp(2ier), ¡3S -denotes the modulation of nonlinear coefficient by the proton response, r = uoat, £ = (/L0, L0 = (oja/a)y/2m/K is the characteristic spatial scale, a is the base spacing in DNA, 7 = The main influence of external field is given by
ponderomotive force, therefore e & a\El / (8-nmul).
The approximate solution of (11) can be found in form of perturbed soliton of nonlinear Schrodinger equation locked to the external frequency [17]
= 2V2zr] sech^e"^, (12)
$ = 2er + arccos /2 - tt/2, rf = 2e + y/e2/A - 72,
This solution has the only free parameter e, which correspond to external parametric pumping by terahertz electric field. Therefore the solution (12) has a sense of dissipative soliton. We should note that parametrically driven solitons of equation (11) can form complexes [18] or transfer to periodic or quasiperiodic structures.
The structure of (12) requires e > 27, that gives a threshold for excitation depending on the terahertz field intensity. Taking 7 ~ 109s-1, we can make an estimation of threshold intensity Ithr ~ 10 — 100 mW/cm2. This estimation agrees reasonably with experimentally observed effects of change in DNA conformation and function [10,11,19]. An experimental observation in the neuron system [20] gives even lower threshold values ~ 1 mW/cm2.
After substitution of characteristic values of induced deformation from (12) to (9) we obtain (x(1) — xfi — X(1? )E02 > 0, thus the localized deformation decreases the probability of tautomer shift. This is confirmed by the experimental observations [11].
4. Conclusion
On the basis on simple nonlinear model of DNA dynamics it is demonstrated that terahertz radiation can influence both vibrational excitations and proton motion in DNA hydrogen bonds. It is highlighted that irradiation at the edge of far IR spectral range can promote proton tunneling and thus make the tautomer states of DNA base pairs to be more probable. If the irradiation frequency matches the vibrational mode, the parametric generation of localized excitations is possible. These dissipative solitons decrease the probability of tautomer shift due to modulation of the tunneling integral.
This preliminary theoretical results can qualitatively explain the results of experimental observations of terahertz influence on the number of spontaneous and induced point mutations [11]. The low-amplitude DNA collective breathing modes can serve as precursors to generation of the transcription bubbles and other large-scale conformational changes such as B-Z transition. This may relate to experimental observations of DNA conformation change [10] and gene transcription [19] in the presence of terahertz field. However, the precise mechanisms of these effects are too complex and require further experimental and theoretical studies of such nanosystems.
The work was conducted under the financial support of the Russian Foundation for Basic Research (grant No.09-02-00503a) and the Dynasty Foundation.
References
[1] Kumar S., Li M.S. Biomolecules under mechanical force // Physics Reports. — 2010. — 486. — 1-74.
[2] Knotts T.A., Ratore N., Shwartz D.C., de Pable J.J. A coarse grain model for DNA // The Journal of Chemical Physics. — 2007. — 126. — 084901.
[3] Yakushevich L.V. Nonlinear Physics of DNA. — Wiley-VCH, 2004.
[4] Niemeyer C.M., Mirkin C.A. (Eds.). Nanobiotechnology. Concepts, Applications and Perspectives. — Wiley-VCH, 2004.
[5] Wagenknecht H.-A. (Ed.) Charge Transfer in DNA. From Mechanism to Application. — Wiley-VCH, 2004.
[6] Lowdin P.-O. Proton tunneling in DNA and its biological implications // Reviews of Modern Physics. — 1963. — 35(3). — 724-732.
[7] Grebneva H.A. A Model for Targeted Substitution Mutagenesis During SOS Replication of Double-Stranded DNA Containing cis-syn Cyclobutane Thymine Dimers // Environmental and Molecular Mutagenesis. — 2006. — 47. — 733-745.
[8] Tolpygo K.B, Grebneva H.A. Effect of the state of h-b-1 hydrogen bond of the character of some atom vibrations in guaninecytosine pair of the DNA molecule. // Int J Quantum Chem. — 1996. — 57. — 219-227.
[9] Villani G. A time-dependent quantum dynamics investigation of the guanine-cytosine system: A six-dimensional model // The Journal of Chemical Physics. — 2008. — 128. — 114306.
[10] Fedorov V.I. et al. Study of biological effects of electromagnetic radiation of submillimeter part of terahertz range. // Biomeditsiknskaya Radioelektronika, 2011.
[11] Fedorov V.I. et al. Comparative study of the effects of infrared, submillimeter, and millimeter EM radiation //Biophysics. — 2001. — 46(2). — 293-297.
[12] Alexandrov B.S., et al. DNA breathing dynamics in the presence of a terahertz field // Physics Letters A, 2010. — 374(10). — 12141217.
[13] R. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics. — Amsterdam: North-Holland, 1975.
[14] Dauxois T., Peyrard M., Bishop A.R. Entropy-driven DNA denaturation // Phys. Rev. E. — 1993. — 47. — R44.
[15] Sazonov S.V., Electromagnetic videosolitons and breathers in the KDP-type ferroelectric // Physics of the Solid State. — 1995. — 37(6). — 1612-1622.
[16] Belonenko M.B., Nazarenko S.V. Microscopic theory of the electro-acoustic echo in order-disorder antiferroelectrics // Physics of the Solid State. — 1998. — 40(1). — 105-108.
[17] Gronbech-Jensen N., Kivshar Yu. S., Samuelsen M.R. Stabilization of breathers in parametrically driven sine-Gordon system// Phys. Rev. B. — 1991. — 43(7). — 5698.
[18] Barashenkov I.V., Zemlyanaya E.V. Stable Complexes of Parametrically Driven, Damped Nonlinear Schrödinger Solitons// Phys. Rev. Lett. — 1999. — 83. — 25682571.
[19] Bock J., et al. Mammalian stem cells reporgramming in response to Terahertz Radiation // PLoS Biol, 2010. — 5(12). — e15806.
[20] Ratushnyak A.S. et al. Influence of submillimeter range electromagnetic radiation on neuron systems // The Third International Symposium on Modern Problems of Laser Physics, Novosibirsk, 2000. — 177.
