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Old 15-01-2009, 02:35 PM   #7
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Neighbouring nucleotide bonds nonlinearity influence on the dynamics of of conformation perturbations in the DNA molecules.
Berezin A.A, Gariaev P.P.,Maslov M.J,Reshetniak S.A.,Shaitan K.V..,Scheglov V.A.
The shugar-phosphate bonds nonlinearity in the chain of nucleotides on the dynamics on the conformative perturbations propagation process in the DNA molecule has been studied. The initial conditions for displaying considerable difference of behavior of the proposed model for linear and nonlinear bonds have been defined. Biosoliton laser hypothesis has been discussed.

In [1] there has been suggested a model describing the rotating-oscillatory perturbations in the chains of nucleotides for explaining experimental results on the hydrogen-tritium exchange in the DNA molecule. In accordance with this model the open states in a form of localized diclocations of soliton type can be generated and can propagate through DNA chain. The model of [1] (and [2,3]) describes interaction between neighboring nucleotides within the framework of linear potential. In our paper (differ from [1-3]) a principally new case is being considered, when this potential is nonlinear.
Well known that a DNA molecule represents a helix containing of adenine(A), thiamin(T), guanine(G), cytosine(C). Nucleotides in the chains display a certain genetic order and the coupling between the chains is realized through the hydrogen bonds between the complementary pairs (A T,G C).
In our study we simulate the DNA helix by an array of rovibrational (from the words rotation and vibration) vibrators, hanging on the weightless and stretchless rod. For simplify the spirality of the chain is not taken into account and the rovibrational degrees of freedom are considered "frozen".
In the considered case the hamiltonian for an "active" chain looks in the following way:
where: is the number of base pairs in a chain; is the hamiltonian, describing the own oscillations of the monomers ( rotation angles of nucleotides in a chain, moment of inertia of the basis), is the hamiltonian, characterizing nonlinear-periodical coupling between oscillators ( the elasticity constant of the chain, ), is the hamiltonian, describing the nonlinear coupling between the "active" and "frozen" ( ) chains of DNA molecule ( is the elasticity constant of the hydrogen bonds between complementary bases, coefficients in Eq. (1) are defined according to the following rule: for A T and T A pairs, for G C and C G pairs; is the parameter defined in [4] and obtained on the basis of the Sine-Gordon model and experimental data).
One should mention that at small values of the hamiltonian , which coinsides with the corresponding part of the hamiltonian, used in [2,3]. In this case the equations of motions for , obtained from (1), looks like this:
where the substitute has made.
In the case when , the system (2) can be transformed into dimensionless Sine-Gordon equation:
which represents a ”continuous analogue” of the system (2). The Eq.(3) has soliton solutions, in particular one-soliton solution, or kink, describing the dislocation propagation dynamics in the chain.
According to (1) nonlinear equations system can be given as follows:
As it can be seen, the systems (2) and (4) are considerably different. However we should mention that the numerical study of the (2) and (4) dynamics showed the following: if to take as initial conditions the single soliton solution of its "continuous analogue" (3) kink [2], one can see a similarity in the character of the solutions of (2) and (4). But when the initial conditions were taken like this
( is the ”step” function having the height of the step equal to and the inclining angle A),
the difference in the dynamics of both systems has been revealed as it can be seen in Fig.1 and Fig.2,3. Both systems (2) and (4) were integrated by the Runge-Kutta method of the 4-th power with the initial conditions looking like (5) in the interval with a step . Boundary conditions were "quasicyclic":

(poly-A-sequence). Parameter of the system . Parameter A was in variance (the inclination of the step function ).
The numerical study of the system (2) showed (Fig. 1) that the solution represents two solitary waves, moving from the right to the left at a constant velosity. The first wave has a quasikink form and the second has quasibreather form (breather is a double-soliton solution of Sine-Gordon equation). The first wave propagates faster then the second one. Both waves due to the ''quasicyclic" boundary conditions when reach the left end appear on the right end without changing their form. Quasikink propagating along the chain of pendula changes the co-ordinate of every pendulum by the angle (a pendulum makes a full turn). So passing the closed chain of pendula К times, it changes the co-ordinate of every pendulum by angle That explains the "shelf" form of the graph (Fig.1).
Fig.2 shows the results of integrating the system (4) under the same conditions. The graph displays same two solitary waves quasikink and quasibreather. Howewer it differs from already considered case. The difference is that at the very beginning the kink moves at negative accelerations wich results in its lower speed to compare with the quasibreather's one. Worth mentioning that the simulation has been done for done for homogenious poly-A-sequence so that the change of quasikink’s velocity can’t be explained by the influence if inhomogenity of the chain. This effect probably appears due to the nonlinear interaction between its monomers.
Fig. 3 illustrates the results of integrating the system (4) under the same conditions with the eonly exclusion that A=2. This time only quasikink is the case and it's negative acceleration results in reversing of the direction of it's motion. The system (2) being integrated under the same conditions results also in quasikink. Its velosity doesn't change to compare with the case shown in Fig.1.
Summing up, we would like to pay attention to the following circumstance. As it has been mentioned above in the systems of DNA type the overexcited rovibrotional states can arise. In quantum language it means that under certain conditions the overexited dynamics rotational states of nucleotides in one or both chains of the DNA molecule can be realized. A hypothesis can be put forward that there exists a principal possibility of creation a biosoliton laser on the DNA molecule. We partially proved it both experimentally and theoretically in our research on the biophoton excited luminecence of laser type in genetic strustures [5-10]). However the problem of dissipation in biopolimers makes the idea of developing a biolaser pretty problematic. At least to prove it we must fulfill the necessary conditions: where and are the width and the velosity of soliton correspondingly, is the dissipation time. If to take and (sound velosity), we'll have the following evaluation . It should be mentioned that the characteristic dissipation time due to the water hydrodynamic forces and the interenal molecular time of dissipation ( [13]).
So the dissipation effect plays a certain role in the act of dislocation forming process along the DNA molecule and it has been already considered as a viscosity parameter in DNA molecules, which depend on the presence of the structured water in the DNA molecule itself and in its microsurrounding (the dissipation magnitude) under consideration were taken 0.1 and 1 [3]. The dislocations are formed or not formed having no dependence on the values and only the ranges of parameters were taken into account. Under large values of the dislocation are being formed slower then under big ones. Much more essential point is influence of the DNA behavior through the other parameter and namely through the spirality step of the polynucleotide. As it follows from our model the change of concentration water molecules leads not only o the superspirality of the biopolimer but to the local decoupling of the DNA helix [3].
This work done on the financial support RFFI (projects N 96-02-18855 - a and N 95-04-12197 -a).

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