Hydration-Dependent Dynamical Transition in Protein: Protein Interactions at $\approx 240\mathrm{K}$

Interprotein motions in low and fully hydrated carboxymyoglobin crystals are investigated using molecular dynamics simulation. Below $\approx 240\mathrm{K}$, the calculated dynamic structure factor exhibits a peak arising from interprotein vibration. Above $\approx 240\mathrm{K}$, the intermolecular fluctuations of the fully hydrated crystal increase drastically, whereas the low-hydration model exhibits no transition. Autocorrelation function analysis shows the transition to be dominated by the activation of diffusive intermolecular motion. The potential of mean force for the interaction remains quasiharmonic. The results indicate useful experimental avenues on protein:protein interactions to be explored using next-generation neutron sources.

Figure 1

Dynamic structure factor $S(Q,E)$ as a function of $E=\hslash \omega$ for (a) fully hydrated crystalline myoglobin for temperatures ranging from 30–300 K in 10 K steps. (b) low and fully hydrated crystalline myoglobin at 150 K. $S_{\mathrm{INC}}(\overrightarrow{q},E)$, was computed using the package nMOLDYN [29]. The spectrum was smoothed by applying a Gaussian window of the form $W(m)=\exp [-\frac{1}{2}(\alpha \frac{|m|}{N_t-1}{)}^2]$ in the time domain with $m=-(N_t-1)\dots N_t-1$. The widths in the time and frequency domains are ${\sigma }_t=\alpha /T$ and ${\sigma }_{\nu }=1/(2\pi {\sigma }_t)$, respectively, where $T$ is the total length of the simulation. $\alpha$ was chosen such that the width in the frequency domain corresponds to the instrumental Full Width at Half Maximum (FWHM) of $60\mu \mathrm{eV}$, i.e., that of Ref. 5.

Figure 2

Mean-square fluctuation of the center-of-mass of the protein as a function of temperature. Lines indicate straight line fits below (dashed line) and above (solid line) the dynamical transition.

Figure 3

Center-of-mass-distance autocorrelation function and the fitted data for the fully hydrated crystal at $T=150$ and 300 K.

Figure 4

Vibrational and Diffusional contribution to the mean-square-fluctuation of the center-of-mass of the protein in a fully hydrated crystal.

Figure 5

The potential of mean force or the effective energy landscape of fully and low hydrated crystals for different temperatures. The statistical accuracy of the PMF for fully hydrated crystal above the transition temperature was calculated by dividing the 5 ns trajectory to 10 pieces of 1 ns each and independently calculating the PMF. The error bars shown indicate the standard deviation.

Figure 6

Frequency ($E=\hslash \omega$) of the protein:protein interactions in low (top) and full hydration (bottom) crystalline carboxymyoglobin. The effective force constant $k$ was calculated using the formula $k=\sqrt{\frac{2k_{B}T\ln \frac{P_x}{P_0}}{(x-x_0{)}^2}}$ where $P_x$ and $P_0$ are the probabilities that the two protein molecules connected by a spring are at the stretched state and at the minimum energy separation $x_0$, respectively. In order to extract $P_x$ and $P_0$, the probability distribution of the distances between the center of masses of proteins was fitted by a Gaussian function given by $f(d_t)=\frac{2}{\sigma \sqrt{(2\pi )}}{e}^{-(d_t-{\overline{d}}_t{)}^2/2{\sigma }^2}$ where ${\sigma }^2$ is the variance and ${\overline{d}}_t$ the mean value of the probability distribution of the distances between the center of masses in the crystal, $d_t$. The frequency of the intermolecular interaction is $2k/m$, with m being the mass of the protein. Filled squares in the lower panel are the intermolecular underdamped translation-vibrational frequency ${\overline{\omega }}_{t-v}$ obtained from fitting the center-of-mass distance-autocorrelation function to the model described in the text.