Solitons in the Yakushevich model of DNA beyond the contact approximation

The Yakushevich model of DNA torsion dynamics supports soliton solutions, which are supposed to be of special interest for DNA transcription. In the discussion of the model, one usually adopts the approximation ${\ell }_0\to 0$, where ${\ell }_0$ is a parameter related to the equilibrium distance between bases in a Watson-Crick pair. Here we analyze the Yakushevich model without ${\ell }_0\to 0$. The model still supports soliton solutions indexed by two winding numbers $(n,m)$; we discuss in detail the fundamental solitons, corresponding to winding numbers (1,0) and (0,1) respectively.

Figure 1

Top: a base pair in the Yakushevich model. The only moving elements in nucleotides are the bases; these are represented as identical disks of radius $r$, which can rotate around their centers. The disk centers lie at a distance $A$ from the double-helix axis, and the rotation angles (positive in counterclockwise direction) are ${\theta }_1$ and ${\theta }_2$, respectively. The distances $A$ and $r$ are related via $A=r+{\ell }_0∕2$, and ${\ell }_0$ is the distance between the disks, ${\ell }_0=2(A-r)$. Bottom: disks represent bases and are located on the double helix; two disks as those marked in black interact via the helicoidal terms considered in the Appendix.

Figure 2

Further details of the base pair modeling in the Yakushevich model, showing in particular the points $B_1$ and $B_2$ whose spatial distance is $\ell$. For graphics convenience, $\ell$ and ${\ell }_0$ have been denoted as $L$ and $L_0$ in this plot.

Figure 3

Upper plot: the function $\beta \left(\phi \right)$, representing the “speed” $d\phi ∕dz$ [in terms of the effective dynamics description provided by Eq. (6.4)] of the $(1,0)$ soliton in adimensional units [see text and Eq. (6.7)], for $\lambda =3∕4$. Lower plot: the function $f\left(\phi \right)=\int [1∕\beta (\phi \left)\right]d\phi$, to be inverted in order to get the (1,0) soliton solution [see Eq. (6.11)], again for $\lambda =3∕4$; here $c_0$ has been chosen according to Eq. (6.10), so that $f\left(\pi \right)=0$.

Figure 4

The $(1,0)$ soliton for the Yakushevich model without the contact approximation [see Eq. (6.11)], with $\lambda =3∕4$ (solid lines) and for the Yakushevich model with the contact approximation (dotted lines). Here $\phi$ is in units of $\pi$ and $z$ in units of the distance $\delta$ between successive base pairs.

Figure 5

The $(0,1)$ soliton for the Yakushevich model without the contact approximation [see Eq. (6.5)] with $\lambda =3∕4$ (solid lines) and for the Yakushevich model with the contact approximation (dotted lines); units as in Fig. 3.

Figure 6

Half-width for the (1,0) soliton (upper) and (0,1) soliton (lower) for the Yakushevich model without the contact approximation. The curves show the half-width $\kappa$ (in units of $B^{-1}=b\delta$) of the solitons as functions of $\lambda$, with different conventions for the measure of the half-width itself; see text: $\Delta =\pi ∕20$ (solid curves), $\Delta =\pi ∕10$ (dotted curves), and $\Delta =\pi ∕5$ (dashed curves).

Figure 7

(Color online) Dispersion relations for different versions of the Yakushevich model. (a) The dispersion relations for $\omega =\omega \left(k\right)$ as described by Eqs. (A3) for the helicoidal Y model without the contact approximation. (b) Dispersion relations for the standard (i.e., ${\ell }_0=0$) Y model with helicoidal terms. (c) Dispersion relations for the standard Y model without helicoidal terms. In all plots, the solid line refers to the $\psi$ branch, the dashed one to the $\chi$ branch. We have used the values $K_s=0.13\mathrm{eV}∕{\mathrm{rad}}^2$, $K_h=0.009\mathrm{eV}∕{\mathrm{rad}}^2$ [16]; here $k$ is given in units of $\pi ∕\delta$, and $\omega$ in ${\mathrm{ps}}^{-1}$. The $\chi$ branch of the standard Y model without helicoidal terms (plot on the right) coincides with the degenerate dispersion relations of the Y model without the contact approximation and without helicoidal terms.