Energy localization in the Peyrard-Bishop DNA model

We study energy localization on the oscillator chain proposed by Peyrard and Bishop to model DNA. We search numerically for conditions with initial energy in a small subgroup of consecutive oscillators of a finite chain and such that the oscillation amplitude is small outside this subgroup on a long time scale. We use a localization criterion based on the information entropy and verify numerically that such localized excitations exist when the nonlinear dynamics of the subgroup oscillates with a frequency inside the reactive band of the linear chain. We predict a mimium value for the Morse parameter $(\mu >2.25)$ (the only parameter of our normalized model), in agreement with the numerical calculations (an estimate for the biological value is $\mu =6.3$). For supercritical masses, we use canonical perturbation theory to expand the frequencies of the subgroup and we calculate an energy threshold in agreement with the numerical calculations.

Figure 1

Frequency of the nonlinear 1 system divided by $\mu$, $(\omega ∕\mu )$, plotted as a function of the energy for $\mu =2$ (dashed line), $\mu =2.25$ (solid line), and $\mu =2.5$ (dotted line). The horizontal solid line is $\omega ∕\mu =1$. Notice that at $\mu =2.25$ the frequency line is only tangent to the critical line. Arbitrary units.

Figure 2

$T_{\max }$ as a function of the energy for type (i) initial conditions at $N=100$ and $\mu =6.3$. Arbitrary units. The squares represent numerical calculations and the solid line is a spline interpolation.

Figure 3

Numerically calculated critical energy for type (i) initial conditions with $N=100$ (triangles), $N=200$ (stars), and $N=500$ (circles) as a function of the energy. Also plotted is the critical energy predicted by the 1 system (squares). Arbitrary units.

Figure 4

$T_{\max }$ as a function of the energy for type (ii) initial conditions at $N=100$ and $\mu =3.0$ (triangles). Arbitrary units. The triangles represent the numerical calculations and the solid line is a spline interpolation.

Figure 5

Numerically calculated critical energy for type (ii) initial conditions with $N=100$ as a function of $\mu$ (triangles) and critical energy predicted by the 3 system $E_c^{\left(1\right)}$ (circles), as a function of $\mu$. Arbitrary units.

Figure 6

Fourier transform of an initial condition of type (i) with $E=0.1$ (subcritical) for a lattice with $\mu =6.3$ and $N=100$. Plotted is the modulus $F\left(\omega \right)$ of the complex Fourier transform. The inset magnifies the region near $\omega =6.3$ to display that $F\left(\omega \right)$ is not zero inside the conduction band.

Figure 7

Fourier transform of an initial condition of type (i) with $E=3.0$ for a lattice with $\mu =6.3$ and $N=100$. Plotted is the modulus $F\left(\omega \right)$ of the complex Fourier transform. The inset magnifies the region near $\omega =6.0$ to display the peak of $F\left(\omega \right)$ at $\omega =6.0<\mu .$ Notice that $F\left(\omega \right)$ vanishes above $\omega =6.25<\mu .$

Figure 8

Surface of section of the symmetric 3 system at $\mu =6.3$ and $E=3.0$, showing little stochasticity. Arbitrary units.