Revisiting the Temperature Dependence of Protein Diffusion inside Bacteria: Validity of the Stokes-Einstein Equation

Although the transport and mixing of proteins and other molecules inside bacteria rely on the diffusion of molecules, many aspects of the molecular diffusion in bacterial cytoplasm remain unclear or controversial, including how the diffusion-temperature relation follows the Stokes-Einstein equation. In this study, we applied single-particle tracking photoactivated localization microscopy to investigate the diffusion of histonelike nucleoid structuring (HNS) proteins and free dyes in bacterial cytoplasm at different temperatures. Although the diffusion of HNS proteins in both live and dead bacteria increased at higher temperatures and appeared to follow the Arrhenius equation, the diffusion of free dyes decreased at higher temperatures, questioning the previously proposed theories based on superthermal fluctuations. To understand the measured diffusion-temperature relations, we developed an alternative model, in which the bacterial cytoplasm is considered as a polymeric network or mesh. In our model, the Stokes-Einstein equation remains valid, while the polymeric network contributes a significant term to the viscosity experienced by the molecules diffusing in bacterial cytoplasm. Our model was successful in predicting the diffusion-temperature relations for both HNS proteins and free dyes in bacteria. In addition, we systematically examined the predicted diffusion-temperature relations with different parameters in the model, and predicted the possible existence of phase transitions.

Figure 1

Temperature dependence of HNS diffusion in live and dead E. coli. (a),(b) MSD of HNS in (a) live and (b) dead E. coli at different temperatures. Dashed lines are fitted curves using $\mathrm{MSD}=4D{\tau }^{\alpha }$. Error bars represent standard errors of the means (SEM). (c) Dependence of the generalized diffusion coefficient of HNS proteins in live (green circles) and dead (red squares) E. coli. Error bars stand for fitting errors. Red dotted lines are fittings with linear equations, while black dashed lines are fittings with the Arrhenius equation.

Figure 2

Temperature dependence of free-dye diffusion in dead E. coli. (a) MSD of dyes in dead E. coli at different temperatures. Dashed lines are fitted curves using $\mathrm{MSD}=4D{\tau }^{\alpha }$. Error bars represent SEM. (b) Dependence of the generalized diffusion coefficient of dyes in dead E. coli. Error bars stand for the fitting errors. Red dotted line is a linear fitting, while black dashed line is a fitting with the Arrhenius equation.

Figure 3

Model of additional “viscosity” experienced by molecules in a polymeric network. (a) Sketch of the model for two types of molecules: interacting (green, $E_a>0$) or noninteracting (blue, $E_a=0$) with the network. (b) Predicted temperature dependence of the diffusion coefficients from the theory for interacting (green, $E_a=15\mathrm{kJ}/\mathrm{mol}$) and noninteracting (blue, $E_a=0\mathrm{kJ}/\mathrm{mol}$) molecules. Black dashed lines are fittings using the Arrhenius equation. (c) Overlapping of model-predicted $D$- vs-$T$ curves (black dashed lines) and experimental data (colored symbols).

Figure 4

Temperature dependence of diffusion coefficients of molecules in polymeric networks with different parameters. (a) Varying $T_m$ while keeping $T_c=10\mathrm{K}$ and $E_a=15\mathrm{kJ}/\mathrm{mol}$ constant. (b) Varying $T_m$, while keeping $T_c=10\mathrm{K}$ and $E_a=0\mathrm{kJ}/\mathrm{mol}$ constant. (c) Varying $T_c$ while keeping $T_m=500\mathrm{K}$ and $E_a=15\mathrm{kJ}/\mathrm{mol}$ constant. (d) Varying $T_c$ while keeping $T_m=500\mathrm{K}$ and $E_a=0\mathrm{kJ}/\mathrm{mol}$ constant.

Figure 5

Phase diagrams based on the Pearson correlation coefficient between temperature and molecular diffusion coefficient at different activation energies: (a) $E_a=0\mathrm{kJ}/\mathrm{mol}$, (b) $E_a=15\mathrm{kJ}/\mathrm{mol}$, (c) $E_a=30\mathrm{kJ}/\mathrm{mol}$, and (d) $E_a=60\mathrm{kJ}/\mathrm{mol}$.