Estimating near-wall diffusion coefficients of arbitrarily shaped rigid macromolecules

We developed a computationally efficient approach to approximate near-wall diffusion coefficients of arbitrarily shaped rigid macromolecules. The proposed method relies on extremum principles for Stokes flows produced by the motion of rigid bodies. In the presence of the wall, the rate of energy dissipation is decreased relative to the unbounded fluid. In our approach, the position- and orientation-dependent mobility matrix of a body suspended near a no-slip plane is calculated numerically using a coarse-grained molecular model and the Rotne-Prager-Yamakawa description of hydrodynamics. Effects of the boundary are accounted for via Blake's image construction. The matrix components are scaled using ratios of the corresponding bulk values evaluated for the detailed representation of the molecule and its coarse-grained model, leading to accurate values of the near-wall diffusion coefficients. We assess the performance of the approach for two biomolecules at different levels of coarse-graining.

Figure 1

Left: hen egg-white lysozyme (cartoon representation, PDB ID 6LYZ [60]) and its van der Waals surface (mesh). Center: one-bead-per-residue (medium size spheres) and shell (small spheres) models of HEWL. Right: coarse-grained dumbbell model of HEWL (large spheres). The drawings were done using the UCSF Chimera package [61].

Figure 2

Top: 20 bp DNA B-helix (cartoon representation) and its van der Waals surface (mesh). Center: one-bead-per-residue (medium-size spheres) and shell (small spheres) models of DNA. Bottom: coarse-grained four-sphere DNA model (large spheres). The drawings were made using the UCSF Chimera package [61].

Figure 3

Mobility coefficients of a given molecule, represented either with the shell (filled silhouette) or the coarse-grained model (empty circles) are calculated in the laboratory coordinate frame consisting of three unit vectors $\{{\widehat{\mathit{e}}}_{{\parallel }_1},{\widehat{\mathit{e}}}_{{\parallel }_2},{\widehat{\mathit{e}}}_{\perp }\}$, with the boundary in the plane spanned by ${\widehat{\mathit{e}}}_{{\parallel }_1}$ and ${\widehat{\mathit{e}}}_{{\parallel }_2}$. Unit vectors $\{{\widehat{\mathit{e}}}_1,{\widehat{\mathit{e}}}_2,{\widehat{\mathit{e}}}_3\}$ correspond to principal axes of the rotational block of the model mobility tensor and define the body-fixed frame. The distance to the wall of a molecule ($h$) is measured from its center of mobility.

Figure 4

Orientation-averaged translational mobility coefficients for HEWL, for motions in directions parallel and perpendicular to the wall, as functions of the HEWL-wall distance, compared with corresponding mobility coefficients of an equivalent sphere. Error bars correspond to standard deviations calculated over an ensemble of different orientations. For a comparison, we also show results (denoted theory) obtained using Eqs. (A1, A2, A3).

Figure 5

Orientation-averaged translational mobility coefficients for DNA, for motions in directions parallel and perpendicular to the wall, as functions of the DNA-wall distance, compared with corresponding mobility coefficients of an equivalent sphere. Error bars correspond to standard deviations calculated over an ensemble of different orientations.

Figure 6

Left panel: Translational mobilities calculated for the HEWL shell model, for motions in directions parallel and perpendicular to the wall, compared with mobilities of an equivalent sphere. Right panel: HEWL translational mobilities, for motions in directions parallel and perpendicular to the wall, compared with scaled mobilities of an equivalent sphere. Data presented in both panels is for HEWL long axis oriented parallel to the wall.

Figure 7

Translational mobilities calculated for the HEWL shell model, for motions in directions parallel and perpendicular to the wall, as functions of the HEWL-wall distance, compared with scaled mobilities of a dumbbell. (a) HEWL long axis perpendicular to the wall. (b) HEWL long axis parallel to the wall (the molecule is rotated around the first principal axis of the rotational diffusion tensor, perpendicular to the long axis). (c) HEWL long axis parallel to the wall (the molecule is rotated around the second principal axis of the rotational diffusion tensor, perpendicular to the long axis). (d) HEWL long axis inclined at 45 degrees angle to the wall.

Figure 8

Translational mobilities calculated for the DNA shell model, for motions in directions parallel and perpendicular to the wall, as functions of the DNA-wall distance, compared with scaled mobilities of the four-sphere model. (a) DNA long axis perpendicular to the wall. (b) DNA long axis parallel to the wall. (c) DNA long axis inclined at 45 degrees angle to the wall.

Figure 9

Rotational mobility functions calculated for the HEWL shell model, for rotations about axes parallel and perpendicular to the wall, compared with scaled rotational mobility functions of the dumbbell. Left panel: HEWL long axis perpendicular to the wall. Right panel: HEWL long axis parallel to the wall. ${\mu }_0^r$ is the bulk rotational mobility coefficient of HEWL calculated based on its hydrodynamic radius, $R$, as ${\mu }_0^r={\left(8\pi \eta R^3\right)}^{-1}$.

Figure 10

Rotational mobility functions calculated for the DNA shell model, for rotations about axes parallel and perpendicular to the wall, compared with scaled rotational mobility functions of the four-sphere model. Left panel: DNA long axis perpendicular to the wall. Right panel: DNA long axis parallel to the wall. ${\mu }_0^r$ is the bulk rotational mobility coefficient of DNA calculated based on its hydrodynamic radius, $R$, as ${\mu }_0^r={\left(8\pi \eta R^3\right)}^{-1}$.