Frequency-Selective Anharmonic Mode Analysis of Thermally Excited Vibrations in Proteins

Abstract

Low-frequency molecular vibrations at far-infrared frequencies are thermally excited at room temperature. As a consequence, thermal fluctuations are not limited to the immediate vicinity of local minima on the potential energy surface, and anharmonic properties cannot be ignored. The latter is particularly relevant in molecules with multiple conformations, such as proteins and other biomolecules. However, existing theoretical and computational frameworks for the analysis of molecular vibrations have so far been limited by harmonic or quasi-harmonic approximations, which are ill-suited to describe anharmonic low-frequency vibrations. Here, we introduce a fully anharmonic analysis of molecular vibrations based on a time correlation formalism that eliminates the need for harmonic or quasi-harmonic approximations. We use molecular dynamics simulations of a small protein to demonstrate that this new approach, in contrast to harmonic and quasi-harmonic normal modes, correctly identifies the collective degrees of freedom associated with molecular vibrations at any given frequency. This allows us to unambiguously characterize the anharmonic character of low-frequency vibrations in the far-infrared spectrum.

Figure 1:

Structure of the Trp-cage mini protein. Secondary structure is highlighted as a red ($\alpha$-helix) and green cartoon and chemical bonds are shown as CPK-colored sticks. Colored labels identify visible amino acid side-chains (blue: positively charged; red: negatively charged; green: polar; black: non-polar).

Figure 2:

Vibrational density of states (VDoS) of the Trp-cage protein from 0–2000 cm⁻¹ (frequencies >2000 cm⁻¹ not shown for clarity). Color illustrates the logarithm of the number of thermally populated quantum HO states at 300 K for each frequency.

Figure 3:

Analysis of frequency-dependent Eigenvalues. (A) Relative Eigenvalues (normalized by first Eigenvalue) of the frequency-dependent velocity cross-correlation matrix for selected frequencies between 0 and 7 THz (234 cm⁻¹). (B) Number of Eigenvalues (out of 852) needed to re-construct 25% (blue), 50% (green) and 75% (red) of the total VDoS (see Eq. 8).

Figure 4:

1D VDoS for projections of the simulation trajectory on FRESEAN modes obtained at 0, 1 and 2 THz. At each selected frequency, spectra are shown for the modes with the 10 largest Eigenvalues. All spectra are normalized by their integral. For 1 and 2 THz, dashed vertical lines indicate the frequency for which the modes were selected. For each spectrum, black bars indicate the full width at half maximum (FWHM) of the peak.

Figure 5:

1D VDoS for projections of the simulation trajectory on (A) harmonic (one representative set) and (B) quasi-harmonic normal modes. In each case, the spectra are shown for the first 10 modes (lowest predicted frequencies), and 10 modes with predicted frequencies closest to 1 THz and 2 THz, respectively (dashed vertical lines indicate 1 and 2 THz). All spectra are normalized by their integral and shown on the same scale as the VDoS for FRESEAN modes shown in Fig. 4. Green arrows indicate the predicted harmonic and quasi-harmonic frequencies (also given numerically as an inset). For each spectrum, black bars indicate the full width at half maximum (FWHM) of the peak.

Figure 6:

Comparison of 1D VDoS for projections of the simulation trajectory on 1 THz modes obtained with the FRESEAN mode analysis (black, modes #1–10), harmonic normal mode analysis (red, modes #19–28) and quasi-harmonic normal analysis (blue, modes #67–76). The 1D VDoS are identical to the 1 THz modes shown in Figs. 4 and 5, but the intensities are scaled by a factor of 20 and the shown frequency range is increased to 4000 cm⁻¹. The predicted frequencies used to select harmonic and quasi-harmonic normal modes for this comparison are given in red and blue, respectively.

Figure 7:

Histograms of displacement probability distributions along FRESEAN modes for (A) 0 THz, (B) 1 THz and (C) 2 THz. For 0 THz, translations and rotations were omitted and distributions are shown for vibrational modes #7–16. For 1 THz and 2 THz, distributions are shown for FRESEAN modes #1–10 with the largest Eigenvalues at the respective frequency. Each histogram is compared to the Gaussian distribution (red) of a harmonic oscillator with the frequency of the main peak in the 1D VDoS of each mode (indicated as ${\nu }_0={\omega }_0/\left(2\pi \right)$ in THz).

Figure 8:

Time auto- and cross-correlation functions (blue and red, respectively) of mass-weighted velocities projected along FRESEAN modes at (A) 0 THz, (B) 1 THz and (C) 2 THz. For 0 THz, translations and rotations were omitted and correlations were analyzed for the vibrational modes #7–16. For 1 THz and 2 THz, correlations are shown for FRESEAN modes #1–10 with the 10 largest Eigenvalues at the respective frequency.