Pressure-dependent transition in protein dynamics at about $4\mathrm{kbar}$ revealed by molecular dynamics simulation

Molecular dynamics simulations of a crystalline protein, Staphylococcal nuclease, over the pressure range $1\text{bar}\text{to}15\mathrm{kbar}$ reveal a qualitative change in the internal protein motions at $\approx 4\mathrm{kbar}$. This change involves the existence of two linear regimes in the mean-square displacement for internal protein motion, $⟨u^2⟩\left(P\right)$ with a twofold decrease in the slope for $P>4\mathrm{kbar}$. The major effect of pressure on the dynamics is a loss, with increasing pressure of large amplitude, collective protein modes below $2\mathrm{THz}$ effective frequency, accompanied by restriction of large-scale solvent translational motion.

Figure 1

(a) Mean-square displacement $⟨u^2⟩$ averaged over all protein hydrogen atoms on two time scales $t$. The lines show linear fits performed over the ranges $P⩽3\mathrm{kbar}$ and $P⩾4\mathrm{kbar}$. The shape of $⟨u^2⟩\left(P\right)$ is similar for $t=100\mathrm{ps}$ (data not shown) and for nonhydrogen protein atoms (shown in the inset). (b) $⟨u^2⟩\left(P\right)$ for $t=1\mathrm{ps}$ averaged over all solvent hydrogens and oxygens, respectively. For the oxygens, the decomposition into translation and rotation $⟨u^2⟩={⟨u^2⟩}_{\mathrm{T}}+{⟨u^2⟩}_{\mathrm{R}}$ is also shown. The lines represent linear fits performed in the low-$P$ regime.

Figure 2

Spread of the mean-square displacements dependent on the trajectory length for two time scales $t$ and three pressure values $P$. The calculation included all protein hydrogen atoms. For clarity, for each set of $t$ and $P$ the results were shifted along the $y$ axis by an arbitrary constant.

Figure 3

Protein (a) and solvent (b) x-ray diffuse scattering intensities $I_{\mathrm{tot}}$ integrated over two ranges of the magnitude $q$ of the scattering vector. $I_{\mathrm{tot}}$ is normalized such that $I_{\mathrm{tot}}\left(1\text{bar}\right)=1$. The solvent $I_{\mathrm{tot}}$ was calculated from the full trajectory $\left(1\mathrm{ns}\right)$. Since protein x-ray diffuse scattering does not converge on the ns time scale [28], for each $P$ the protein $I_{\mathrm{tot}}$ was estimated from ten non-overlapping $100\mathrm{ps}$ subtrajectories and the error bars denote the standard deviation. The lines show linear fits in the low- and high-$P$ (for the protein: $P⩾7\mathrm{kbar}$) regimes, respectively.

Figure 4

Change in the vibrational density of states, $\Delta g\left(\nu \right)$ for selected pressure values. $\Delta g\left(\nu \right)$ was calculated as the difference $g_P\left(\nu \right)-g_{P_0}\left(\nu \right)$, with the reference pressure $P_0=1\text{bar}$. The inset shows the number of modes with $\nu ⩽2\mathrm{THz}$ with a linear fit to the high-$P$ regime. The gradient in the low- and high-$P$ regime is $-2.6\pm 0.2$ and $-1.99\pm 0.06{\mathrm{kbar}}^{-1}$, respectively. Results shown are averaged over all four proteins in each simulation.

Figure 5

Pressure dependence of the effective free energy profiles $G_m$ of selected PCA modes at various values of $P$. For clarity, in each graph the profiles are separated along the $y$ axis by a constant ($10\mathrm{kJ}{\mathrm{mol}}^{-1}$ for mode 1 and $5\mathrm{kJ}{\mathrm{mol}}^{-1}$ for the other modes).