Lattice Dynamics of a Protein Crystal

All-atom lattice-dynamical calculations are reported for a crystalline protein, ribonuclease A. The sound velocities, density of states, heat capacity ($C_V$) and thermal diffuse scattering are all consistent with available experimental data. $C_V\propto T^{1.68}$ for $T<35\mathrm{K}$, significantly deviating from a Debye solid. In Bragg peak vicinity, inelastic scattering of x rays by phonons is found to originate from acoustic mode scattering. The results suggest an approach to protein crystal physics combining all-atom lattice-dynamical calculations with experiments on next-generation neutron sources.

Figure 1

Dispersion relations for the acoustic [predominantly transverse (TA) or longitudinal (LA); subscripts indicate TA-LA mixing at $q_{hkl}=0.1$] and 12 lowest-frequency optical modes. Symbols are data points with connecting lines drawn for convenience. $q_{hkl}=q(2\pi /d_{hkl}{)}^{-1}$, with $d_{hkl}$ being the spacing of the real lattice. $\nu =\omega /2\pi$ is the true frequency. Arrows indicate anticrossings discussed in the text.

Figure 2

Density of states $g(\nu )$ calculated from the phonon dispersion relation $\mathit{\omega }(\mathbf{q})$ and averaged over ${\mathbf{a}}^{*}$, ${\mathbf{b}}^{*}$, and ${\mathbf{c}}^{*}$. Also shown is 120 K experimental data for hydrated (0.3 g water/g dry protein) powders of dihydrofolate reductase [27]. For comparison, the experimental data have been scaled by the ratio (3.05) of DHFR hydrogens (which dominate the experimental neutron scattering) to RNase protein atoms.

Figure 3

Heat capacity $C_V$ of the RNase crystal calculated using Eq. (1). Experimental heat capacities $C_P$ for dry RNase crystals [29] and RNase in aqueous solution [30] are also shown. The inset shows the low-temperature regime in more detail. Dotted and dashed lines are least-squares fits of the Debye $T^3$-law and a $T^b$-function, respectively, to $C_V^{\mathrm{RNase}}$ for $T\le 35\mathrm{K}$.

Figure 4

Inelastic x-ray scattering $I_{\mathrm{in}}(q)$ given by Eq. (2) versus $q$, along ${\mathbf{b}}^{*}$, calculated from all 19326 phonons, at 100 K. The $q$ range is chosen to include the first 30 Bragg peaks, corresponding to 1.28 Å real-space resolution. The left inset shows experimental data for a single 6 Å-resolution reflection for RNase [6] and the least-squares fit $y(q)=A_0+A_1{q}^{-2}$. The right inset shows the root-mean-square deviation, $\Delta =⟨[(I_{\mathrm{in}}-y)/I_{\mathrm{in}}{]}^2{⟩}^{0.5}$, for each Bragg peak.