Phonon dispersion of oriented DNA by inelastic x-ray scattering

Films of oriented deoxyribonucleic acid (DNA), prepared by the wet spinning method, have been studied using inelastic x-ray scattering. Spectra were recorded within the range of energy transfers $-30<\hslash \omega <30\mathrm{meV}$ at momentum transfers $\hslash \mathbf{Q}$ ranging from 2.5 to $30{\mathrm{nm}}^{-1}$ whereby the direction of $\mathbf{Q}$ essentially coincided with the helical axis. Measurements at ambient temperature cover samples in the $A$, $B$, $C$, and $D$ conformations of DNA. Within the limits of the instrumental resolution, the spectra were analyzed by the response of a damped harmonic oscillator delivering dispersion and damping of modes having displacements with nonzero projections onto $\mathbf{Q}$, i.e., essentially the compression waves traveling along the helical axis. The longitudinal speed of sound resulting from the sinusoidal dispersion varies only weakly with conformation. Our sound speed values are compared to results from Brillouin spectroscopy. The dispersion curves exhibit a minimum at about the inverse rise per residue, which—together with strong elastic scattering—reflect the large degree of disorder. Overdamping of the modes is observed for $Q>5{\mathrm{nm}}^{-1}$. The possibility that the observed large damping parameters are due to several contributing modes is discussed in terms of a simple model calculation for an idealized double helix. Whereas the quasicrystalline approximation for an effective disordered chain could well describe the sinusoidal dispersion, it fails to reproduce the observed damping by one order of magnitude. Our results indicate that the high-frequency dynamics of DNA is liquidlike and is most appropriately described by instantaneous normal modes of short correlation length.

Figure 1

Fiber diffraction pattern of the DNA samples. (a) $A$ DNA. The double arrow indicates the helical axis. The white rectangle emphasizes intense features in the $6^{\mathrm{th}}$, $7^{\mathrm{th}}$, and $8^{\mathrm{th}}$ layer line which are characteristic for the $A$ conformation. The arrow marks contamination by the $B$, and $C$ conformation. (b) $B$ DNA. The ${10}^{\mathrm{th}}$ layer line (base pair peak) is also marked, corresponding to the inverse distance of base pairs along the helical axis (double arrow). (c) $C$ DNA sample. (d) $D$ DNA. The inclined arrow marks contamination by the $B$ conformation close to the $7^{\mathrm{th}}$ layer line.

Figure 2

Elastic scans along the helical axis for the four investigated conformations. The scans represent corresponding sections through the fiber diffraction patterns shown in Fig. 1. They are displaced vertically by a constant amount for clarity. The peak positions are discussed in the text.

Figure 3

Measured and fitted (white lines) constant-$Q$ scans for $B$ DNA at room temperature. The elastic and quasielastic part is indicated by the dashed curves, whereas the solid lines represent the oscillatory part. See text for details of the fitting procedure.

Figure 4

Dispersion of eigenfrequency ${\omega }_0$ and damping $\Gamma$ of the DHO ansatz [see Eq. (1)] for $B$ DNA at ambient temperature. The initial slope, corresponding to a sound velocity of $2.84\mathrm{km}∕\sec$, is indicated by the thin, straight line. The dotted hyperbola represents results obtained by neutron scattering [26].

Figure 5

The $2\times 3$ dispersion curves of the model for a simplified double helix in the extended zone scheme [44]. The model parameters are described in the text. For the triplet of ip modes (thick lines), both strands move in phase whereas they move with different sign for the oop modes (thin lines). The experimental results for $B$ DNA at ambient temperature are shown by open circles.

Figure 6

(Color online) Left panel: Calculated dispersion and intensity for the model of a simplified double helix in the extended zone scheme [44]. From lower to higher frequency, the three branches correspond to torsional (ip1), compressional (ip2), and breathing (ip3) motion at infinite wavelength. Note that for clarity of presentation the intensity has been multiplied by the eigenvalue, and is proportional to the symbol size. Right panel: Schematic representation of the DNA. Each base pair is characterized by its radius $\rho$, angle $\phi$, and height $z$. The arrows indicate approximately the displacement patterns of the three modes, which are visible in the IXS experiment.

Figure 7

Predictions for frequency (a) and damping (b) of phononlike excitations of a disordered linear chain by the coherent potential and quasiparticle approximation [48, 49]. Disorder is represented by the relative variances $\sigma$ (NN distance) and $\lambda$ (NN force/mass). Dashed lines correspond to $\lambda =0.03$ and the experimentally observed $\sigma =0.036$. The solid lines are obtained by tripling this value of $\sigma$. Solid symbols show the fitted values for $B$ DNA at ambient temperature.

Figure 8

Dispersion of eigenfrequency ${\omega }_0$ and damping $\Gamma$ of the DHO ansatz [see Eq. (2)] resulting from the four datasets of the $A$, $B$, $C$, $D$ DNA samples at ambient temperature. The initial slopes corresponding to sound velocities of 2.8 and $3.0\mathrm{km}∕\sec$ are indicated by the two thin, straight lines. The dotted hyperbola represents results obtained by neutron scattering [26].