Rotational and Translational Diffusion of Proteins as a Function of Concentration

Abstract

Atomistic simulations of three different proteins at different concentrations are performed to obtain insight into protein mobility as a function of protein concentration. We report on simulations of proteins from diluted to the physiological water concentration (about 70% of the mass). First, the viscosity was computed and found to increase by a factor of 7-9 going from pure water to the highest protein concentration, in excellent agreement with in vivo nuclear magnetic resonance results. At a physiological concentration of proteins, the translational diffusion is found to be slowed down to about 30% of the in vitro values. The slow-down of diffusion found here using atomistic models is slightly more than that of a hard sphere model that neglects the electrostatic interactions. Interestingly, rotational diffusion of proteins is slowed down somewhat more (by about 80-95% compared to in vitro values) than translational diffusion, in line with experimental findings and consistent with the increased viscosity. The finding that rotation is retarded more than translation is attributed to solvent-separated clustering. No direct interactions between the proteins are found, and the clustering can likely be attributed to dispersion interactions that are stronger between proteins than between protein and water. Based on these simulations, we can also conclude that the internal dynamics of the proteins in our study are affected only marginally under crowding conditions, and the proteins become somewhat more stable at higher concentrations. Simulations were performed using a force field that was tuned for dealing with crowding conditions by strengthening the protein-water interactions. This force field seems to lead to a reproducible partial unfolding of an α-helix in one of the proteins, an effect that was not observed in the unmodified force field.

Figure 1

Structure of proteins (A) 2k57, (B) ubiquitin, and (C) 2kim.

Figure 2

Mean square fluctuations of each protein at the residue level for different concentrations for (A) 2k57, (B) ubiquitin, (C) 2kim, and (D) 2kim (using Amber ff99SB-ILDN). All plots except (D) are averages over three replica simulations of 1 μs. The shaded area shows the standard error.

Figure 3

2kim after simulating with the Amber ff99SB-ILDN force field (green-red) and the amber99sb-ws force field (cyan and yellow). The unfolding region, residue 30–50, is highlighted in red and yellow, respectively.

Figure 4

Order parameter S² at the residue level for different concentrations for (A) 2k57, (B) ubiquitin, and (C) 2kim. All plots are averages over three replica simulations of 1 μs. The shaded area shows the standard error.

Figure 5

Internal tumbling at the residue level for different concentrations for (A) 2k57, (B) ubiquitin, and (C) 2kim. All plots are averages over three replica simulations of 1 μs. The shaded area shows the standard error.

Figure 6

Concentration dependence of mobility averaged over three replica simulations. (A) Viscosity η. (B) Translational diffusion coefficient D_P/D_P,0 for proteins normalized to be one at dilute conditions (D_P,0 is the diffusion coefficient for one protein of the kind in physiological salt concentrations at infinite dilution). Data points and error bars show the average and standard error of the diffusion coefficient over all protein chains. (C) Protein tumbling time normalized to one protein of the kind in physiological salt concentration (τ_M0). Data points and error bars show the average and standard error of estimated global tumbling time over all protein chains. (D) Water diffusion coefficient D_w/D_w,0 normalized by the diffusion coefficient of water in physiological salt concentration (D_w,0). Data points and error bars (smaller than the size of symbols) show the average and standard deviation of diffusion coefficient over all water molecules.

Figure 7

Distance-based cluster analysis of three proteins, 2k57, ub, and 2kim, at three different concentrations for three different distances. In addition, a similar analysis of a ubiquitin-like hard sphere (HS-ub_1.2 and HS-ub₁, see Methods) was performed. Colors indicate the distance criterion used. Blue: 0.27 nm, red: 0.57 nm, and green: 0.87 nm.