Dynamical transition, hydrophobic interface, and the temperature dependence of electrostatic fluctuations in proteins

Molecular dynamics simulations have revealed a dramatic increase, with increasing temperature, of the amplitude of electrostatic fluctuations caused by water at the active site of metalloprotein plastocyanin. The increased breadth of electrostatic fluctuations, expressed in terms of the reorganization energy of changing the redox state of the protein, is related to the formation of the hydrophobic protein-water interface, allowing large-amplitude collective fluctuations of the water density in the protein’s first solvation shell. On top of the monotonic increase of the reorganization energy with increasing temperature, we have observed a spike at $\simeq 220\mathrm{K}$ also accompanied by a significant slowing of the exponential collective Stokes shift dynamics. In contrast to the local density fluctuations of the hydration-shell waters, these spikes might be related to the global property of the water solvent crossing the Widom line or undergoing a weak first-order transition.

Figure 1

Stokes shift correlation function of PC (Ox) at different temperatures indicated in the plot.

Figure 2

(a) Circles indicate the water reorganization energy ${\lambda }_w$ from MD simulations of PC (Ox) and the solid line shows the fit of the simulation data to Eq. (7) with the fitting parameters: ${\lambda }_G=0.39\mathrm{eV}$, ${\lambda }_{\mathrm{eq}}=0.87+0.0084T\mathrm{eV}$, ${\tau }_{\mathrm{obs}}∕{\tau }_0=7350$, and $E_a=1867\mathrm{K}$; the dashed line assumes temperature-independent ${\lambda }_{\mathrm{eq}}$. The inset shows the exponential part of the reorganization energy ${\lambda }_E=(1-A_G){\lambda }_w$ produced by collective water fluctuations. The closed diamonds refer to half of the Stokes shift [Eq. (8)] and the open diamonds show the linear-response reorganization energy ${\lambda }_w^{\mathrm{Ox}}$ obtained from Eq. (9). (b) Exponential relaxation time of the Stokes shift correlation function obtained from MD of PC(Ox) and the Gaussian amplitude $A_G$ in Eq. (5) (inset). The dashed lines connect the points.

Figure 3

Reorganization energy obtained from the variance of the Coulomb energy gap: water component ${\lambda }_w$ [circles, Eq. (3)], the protein component ${\lambda }_P$ from the variance of $\Delta E_P^C$ (diamonds), and the total water-protein reorganization energy ${\lambda }_{\mathrm{tot}}$ from the variance of the total Coulomb energy gap $\Delta E_w^C+\Delta E_P^C$ (squares). The inset shows the Binder parameter [35] built on the Coulomb interaction of the active site with hydrating water [Eq. (13)].

Figure 4

Average number of water molecules in PC’s first solvation shell (a) and its variance (b) vs temperature. The dashed line in (b) shows a linear regression through the points and the dotted line in (a) connects the simulation points.

Figure 5

$⟨{\left(\delta M\right)}^2⟩∕N_I\left(T\right)$ vs temperature. The inset shows the second-rank orientational order parameter $p_2\left(T\right)$ [Eq. (18)]. The dotted line connects the points.

Figure 6

Exponential relaxation time ${\tau }_E^M$ extracted by fitting the correlation function $C_M\left(t\right)$ from Eq. (15) to Eq. (5). The points are the simulation results in three ranges of temperature where they are fitted to the Vogel-Fulcher temperature law (at highest temperatures) and to Arrhenius laws (at the intermediate and lowest temperature ranges). The solid lines show the results of the fits.

Figure 7

Diffusion coefficient (open circles) from the present simulations and atomic mean-squared displacements of myoglobin measured experimentally [33] (small up triangles) and obtained from MD simulations [28] (small squares). Closed diamonds show the change in the number of particles in the first solvation shell (Fig. 3). All parameters are normalized to their corresponding values at $300\mathrm{K}$.

Figure 8

Diffusion coefficient of TIP3P water calculated from all $N_w=5886$ water molecules in the simulation box (circles). The triangles indicate the results of Ref. 6 for the simulation box containing $N_w=1242$ waters; the squares denote the results from Ref. 22 with $N_w=484$ per simulation box containing two protein molecules. Closed and open points indicate temperatures above and below the dynamical transition temperature, respectively.