External Potential Modifies Friction of Molecular Solutes in Water

Stokes’s law for the friction of a sphere in water has been argued to work down to molecular scales, provided the effective hydrodynamic radius includes the hydration layer. In interpretations of experiments and in theoretical models, it is tacitly assumed that the solvent friction experienced by a solute does not depend on whether an external confinement potential acts on the solute. Using a novel method to extract the friction memory function from molecular dynamics simulations, we show that the solvent friction of a strongly harmonically confined methane molecule in water increases by 60% compared to its free-solution value, which is caused by an amplification of the slowest component of the memory function. The friction enhancement occurs for potential strengths typical of physical and chemical bonds and is accompanied by a significant slowing-down of the hydration water dynamics. Thus, the solvent friction acting on molecular solutes is not determined by solvent properties and solute-solvent interactions alone but results from the coupling between solute and solvent dynamics and thereby can be tuned by an external potential acting on the solute. This also explains why simulations of positionally constrained solutes do not reproduce free-solution diffusivities. Dynamic scaling arguments suggest similar effects also for macromolecular solutes provided the solution viscosity is sufficiently enhanced.

Popular Summary

All biomolecular reactions occur in water, so fully understanding how they work requires an accurate picture of how these molecules interact with their aqueous environment. A key player in this picture is friction or, more specifically, a number known as the friction coefficient, which quantifies the force needed to move a molecule at a particular speed. According to Stokes’ law (which has been shown to work down to molecular scales), the friction coefficient of an object is proportional to the viscosity of the solvent and a parameter describing the object’s shape, but it does not depend on the presence of an external confining force. Using numerical simulations of a methane molecule in water, we show that this last assumption is incorrect. In fact, when confined, the methane friction constant increases by about 60%.

We reveal the physics behind the invalidation of Stokes’ law in terms of an increased long-time tail in the memory function, which describes the time delay of the fluid response to a local perturbation in the past. We extract the memory function from our methane simulation trajectories using a novel projection method. The increased long-time tail is reflected in a slow-down of the hydration shell dynamics. Both the increase in friction and the slow-down of the hydration shell dynamics are relevant for the interpretation of spectroscopy experiments probing water dynamics around solvated molecules.

Our results not only generalize a fundamental hydrodynamics law but also provide an explanation for the experimental finding that the hydration shell dynamics around proteins depends on protein flexibility.

Figure 1

Simulation setup. (a) A single methane is solvated in water and confined in an external harmonic potential of strength $K$. (b) Radial distribution functions (RDF) of the separation between methane and water oxygens for box size $L=4.5\mathrm{nm}$ and different $K$ including the frozen limit $K=\infty$, all curves perfectly superimpose. The positions of the first two maxima, used to calculate the mean escape time ${\tau }_{\mathrm{esc}1}$, and the positions used to calculate the mean escape time ${\tau }_{\mathrm{esc}2}$, are indicated by vertical dashed lines. (c) Methane position $x(t)$ and total force $F(t)$ trajectories for two different $K$ values.

Figure 2

Simulated solvent force autocorrelations for box size $L=4.5\mathrm{nm}$. (a) Autocorrelation functions $C_{FF}^{\mathrm{sol}}(t)$ defined in Eq. (3) and (b) integrals $I_{FF}^{\mathrm{sol}}(t)$ defined in Eq. (6) for different confinement potential strengths $K$. Colored lines denote simulation results, dotted lines are analytic predictions according to Eq. (4) using the best-fit memory function $\mathrm{\Gamma }(t)$.

Figure 3

Memory kernels. (a) Best-fit memory functions $\mathrm{\Gamma }(t)$ according to the parametrization Eq. (7) for box size $L=4.5\mathrm{nm}$ and different $K$. The result for the frozen case $K=\infty$ is shown as a red line. The harmonic oscillation periods ${\tau }_0=2\pi \sqrt{m/K}$ are indicated by vertical lines. (b) Fitted decay times ${\tau }_i$ (data points) from Eq. (7) as a function of $K$. Statistical errors are smaller than the symbol size. The two time scales for the frozen limit $K=\infty$ are denoted by horizontal colored lines on the right. The harmonic oscillation period ${\tau }_0=2\pi \sqrt{m/K}$ is shown as a dashed line, the relaxation times in the $K=0$ and $K=\infty$ limits estimated from methane-water repulsive interactions are denoted by red arrows.

Figure 4

Friction coefficient. Results for $\gamma$ from the best-fit friction kernels are shown as a function of $K$ for three simulation box sizes $L$. Statistical errors are of the order of the symbol size. The results for $K=0$ and $K=\infty$ are obtained from free diffusion and frozen simulations and denoted by horizontal colored lines. For $L=4.5\mathrm{nm}$ an empiric fit is included (dashed line).

Figure 5

Mean escape times of water from the first hydration shell around methane. (a) Results for ${\tau }_{\mathrm{esc}1}$, defined as the mean first passage time from $r=0.37\mathrm{nm}$ to a target distance of $r=0.65\mathrm{nm}$ [indicated by dashed vertical lines in the RDF in Fig. 1], are shown as a function of $K$. Results in the limits $K=0$ and $K=\infty$ are denoted by horizontal lines. (b) The mean escape time ${\tau }_{\mathrm{esc}2}$ from $r=0.37\mathrm{nm}$ to a target distance of $r=1.37\mathrm{nm}$ as a function of $K$.

Figure 6

Orientation correlation function $C_{\mathrm{rot}}(t)$ defined in Eq. (9) of water molecules within a radius 0.4 nm around (a) a free and a frozen methane molecule and (b) around a free and a frozen water molecule. Broken lines denote exponential fits for times $t>2\mathrm{ps}$.

Figure 7

Dynamic scaling diagrams for (a) a large solute particle and for (b) a small particle. In (a) the inertial time scale ${\tau }_m$ lies above the longest water relaxation time ${\tau }_3$ and the particle relaxation time in the potential (shown as a solid line) changes from the harmonic oscillator prediction ${\tau }_0\simeq 2\pi \sqrt{m/K}$ (for $K>K^{*}$) to the overdamped prediction ${\tau }_{\gamma }\simeq 2\pi \gamma /K$ (for $K<K^{*}$) above ${\tau }_3$. In (b) ${\tau }_m$ is smaller than ${\tau }_3$ and the potential relaxation time becomes overdamped within the relaxation time range. In this case we expect the external potential of strength $K$ to modify the particle friction.

Figure 8

The free energy $\mathcal{F}(r)$ associated with the distance $r$ between the methane and surrounding water molecules together with a harmonic fit to the repulsive part (red line) and a harmonic fit around the free energy minimum (blue line).

Figure 9

Simulated autocorrelations for water in water. (a) Solvent-force autocorrelation functions $C_{FF}^{\mathrm{sol}}(t)$ and (b) integrals $I_{FF}^{\mathrm{sol}}(t)$ for a confined water molecule for different potential strengths $K$.

Figure 10

Comparison of the methane friction coefficient $\gamma ={\gamma }_1+{\gamma }_2+{\gamma }_3$ with its contributions ${\gamma }_i$ defined by Eq. (c1) as a function of $K$. We also show the sum of the short time contributions ${\gamma }_1+{\gamma }_2$.

Figure 11

The normalized autocorrelation functions $C_{FF}(t)/C_{FF}(0)$, $C_{\dot{x}\dot{x}}(t)/C_{\dot{x}\dot{x}}(0)$, and $C_{xx}(t)/C_{xx}(0)$ for the system with the size $L=4.5\mathrm{nm}$ and five different spring constants $K$ of the harmonic potential.

Figure 12

Normalized position autocorrelation function $C_{xx}(t)/C_{xx}(0)$ (blue) and normalized solvent force autocorrelation function $C_{FF}^{\mathrm{sol}}(t)/C_{FF}^{\mathrm{sol}}(0)$ (red) for spring constants (a) $250\mathrm{kJ}/(\mathrm{mol}{\mathrm{nm}}^2)$, (b) $25000\mathrm{kJ}/(\mathrm{mol}{\mathrm{nm}}^2)$, and (c) $250000\mathrm{kJ}/(\mathrm{mol}{\mathrm{nm}}^2)$.