Principal Components of the Protein Dynamical Transition

Proteins exhibit a solvent-driven dynamical transition at 180–220 K, manifested by nonlinearity in the temperature dependence of the average mean-square displacement. Here, molecular dynamics simulations of hydrated myoglobin show that the onset of the transition at $\sim 180\mathrm{K}$ is characterized by the appearance of a single double-well principal component mode involving a global motion of two groups of helices. As the temperature is raised a few more quasiharmonic and multiminimum components successively appear. The results indicate an underlying simplicity in the protein dynamical transition.

Figure 1

Anharmonicity factor, $\rho$, for the first 20 principal component modes. For each principal component the temperature at which $\rho =2$ is reached is indicated with a black bar.

Figure 2

Standard deviation, $\sigma$, of the Gaussian fit to the probability distribution of the first 20 principal component modes. For each principal component the temperature is indicated at which $\sigma =1$ is reached. The error was defined as ${\sigma }^2=1000\overline{[P(q)-G(q){]}^2}$, taken over 3 standard deviations of the probability distribution, where $G(q)$ is the Gaussian fit.

Figure 3

Decomposition of the mass-weighted protein, $⟨u^2⟩$, into contributions arising from the harmonic, quasiharmonic, or multiminimum classes of principal component. Vertical arrows indicate the number of multiminimum modes, and italics the number of quasiharmonic modes at certain temperatures. The $⟨u^2⟩$ calculated at 300 K from the normal mode analysis using the same model system and potential function is indicated, the value of this concurs with that calculated from the “harmonic” principal component modes.

Figure 4

Free energy profiles of (a) the 1st-, (b) the 2nd-, and (c) the 5th-lowest principal component modes.

Figure 5

(a),(b) Motion of myoglobin along the first principal mode calculated at 200 K. The structures of myoglobin are shown in the two minima (black and white) of the mode. The helices are represented as cylinders. For ease of viewing the amplitude of the motion is multiplied by a factor of 6. (a) also shows the sidechain of ASP126.