Analytical tools for solitons and periodic waves corresponding to phonons on Lennard-Jones lattices in helical proteins

We study the propagation of solitons along the hydrogen bonds of an $\alpha$ helix. Modeling the hydrogen and peptide bonds with Lennard-Jones potentials, we show that the solitons can appear spontaneously and have long lifetimes. Remarkably, even if no explicit solution is known for the Lennard-Jones potential, the solitons can be characterized analytically with a good quantitative agreement using formulas for a Toda potential with parameters fitted to the Lennard-Jones potential. We also discuss and show the robustness of the family of periodic solutions called cnoidal waves, corresponding to phonons. The soliton phenomena described in the simulations of $\alpha$ helices may help to explain recent x-ray experiments on long $\alpha$ helices in Rhodopsin where a long lifetime of the vibrational modes has been observed.

Figure 1

Comparison of Lennard-Jones and Toda potential.

Figure 2

The cnoidal wave form for different wavelengths and amplitudes. (a,b,c) for $\lambda =(2,4,10)$. The curves on which the atom positions are indicated by a (small,medium,large) size dot correspond to the modulus $k^2=(0.3,0.9,0.99)$. Obviously even for $k^2=0.3$ the waveform is close to a harmonic wave form for all wavelengths, whereas for larger $k$ there are significant differences.

Figure 3

The reduced dispersion curve $\omega =2\pi \nu ∕\sqrt{ab∕m}$ as a function of $q=2∕\lambda$ for the Toda lattice for the modulus $k^2=(0.3,0.9,0.99)$. For $k^2=0.3$, the dispersion curve $\omega \left(q\right)$ is very close to the harmonic one, which peaks at $\omega \left(1\right)=2$.

Figure 4

Space-time plot of a one-dimensional lattice of elements connected by the Lennard-Jones potential. The system is initialized with a wave corresponding to a perfect Toda soliton [Eq. (10)] of low energy. The parameters of the soliton are $\kappa =0.5,E=9.39{\mathrm{cm}}^{-1}$. In this energy range, the Toda and Lennard-Jones potentials are quite similar and thus the Toda soliton survives for a long time even if the potential is not Toda.

Figure 5

Same as Fig. 4 but with a soliton of higher energy $(\kappa =2,E=1031)$. In this region, the Toda and Lennard-Jones potentials are different, but nevertheless the Toda soliton persists for a long time on the Lennard-Jones lattice.

Figure 6

Projection of the trajectories of Toda-like solitons of different energy ($30\text{–}1300{\mathrm{cm}}^{-1}$, increasing patterns) evolving on a 1D Lennard-Jones lattice. The trajectories stay near to close orbits for 8000 ps.

Figure 7

Projection of the trajectories of Toda-like cnoidal waves of different energy ($30\text{–}1300{\mathrm{cm}}^{-1}$, increasing patterns) evolving on a 1D Lennard-Jones lattice ($k^2=0.1$, 0.3, 0.6, 0.98, $\lambda =4r_0$). The trajectories stay even nearer to the close orbits for the 8000 ps than the solitonlike solutions.

Figure 8

Same as Fig. 7 for $\lambda =2r_0$.

Figure 9

Difference between Lennard-Jones and Toda potentials, as a function of the (Lennard-Jones) potential depth. The upper branch corresponds to $x<r_0$, while the lower branch to $x>r_0$. Notice that on the lower branch, the two potentials differ only a few ${\mathrm{cm}}^{-1}$ over an energy range of several hundreds of ${\mathrm{cm}}^{-1}$.

Figure 10

Three-dimensional model of an $\alpha$ helix. Amino acids are mass points (dots) connected by Lennard-Jones potentials giving hydrogen bonds (continuous line) and peptide bonds (dashed line). The helix here has open boundary conditions.

Figure 11

Energy vs speed for different strength of the peptide bond, for solitons on a 3D $\alpha$ helix with Lennard-Jones potential. The curve does not change.

Figure 12

As in Fig. 11, but changing the strength of the hydrogen bond. The curve changes.

Figure 13

Energy vs speed normalized by the sound speed of the hydrogen bond for solitons on a 3D $\alpha$ helix with Lennard-Jones potential. The continuous line is the prediction using Toda theory on the one-dimensional lattice.