Recent advances in macromolecular hydrodynamic modeling

Abstract

The modern implementation of the boundary element method [23] has ushered unprecedented accuracy and precision for the solution of the Stokes equations of hydrodynamics with stick boundary conditions. This article begins by reviewing computations with the program BEST of smooth surface objects such as ellipsoids, the dumbbell, and cylinders that demonstrate that the numerical solution of the integral equation formulation of hydrodynamics yields very high precision and accuracy. When BEST is used for macromolecular computations, the limiting factor becomes the definition of the molecular hydrodynamic surface and the implied effective solvation of the molecular surface. Studies on 49 different proteins, ranging in molecular weight from 9 to over 400kDa, have shown that a model using a 1.1Å thick hydration layer describes all protein transport properties very well for the overwhelming majority of them. In addition, this data implies that the crystal structure is an excellent representation of the average solution structure for most of them. In order to investigate the origin of a handful of significant discrepancies in some multimeric proteins (about -20% observed in the intrinsic viscosity), the technique of Molecular Dynamics simulation (MD) has been incorporated into the research program. A preliminary study of dimeric α-chymotrypsin using approximate implicit water MD is presented. In addition I describe the successful validation of modern protein force fields, ff03 and ff99SB, for the accurate computation of solution structure in explicit water simulation by comparison of trajectory ensemble average computed transport properties with experimental measurements. This work includes small proteins such as lysozyme, ribonuclease and ubiquitin using trajectories around 10ns duration. We have also studied a 150kDa flexible monoclonal IgG antibody, Trastuzumab, with multiple independent trajectories encompassing over 320ns of simulation. The close agreement within experimental error of the computed and measured properties allows us to conclude that MD does produce structures typical of those in solution, and that flexible molecules can be properly described using the method of ensemble averaging over a trajectory. We review similar work on the study of a transfer RNA molecule and DNA oligomers that demonstrate that within 3% a simple uniform hydration model 1.1Å thick provides agreement with experiment for these nucleic acids. In the case of linear oligomers, the precision can be improved close to 1% by a non-uniform hydration model that hydrates mainly in the DNA grooves, in agreement with high resolution X-ray diffraction. We conclude with a vista on planned improvements for the BEST program to decrease its memory requirements and increase its speed without sacrificing accuracy.

Fig. 1

The least squares fit line for the viscosity factor extrapolation vs. 1/N of a rectangular cylinder of axial ratio = 40. Taken from reference .

Fig. 2

The percent difference for the average translational diffusion tensor of literature values to our accurate formulas as a function of axial ratio for the rectangular cylinder is shown. The symbols represent the works of Mansfield and Douglas [30] (MD), Ortega & Garcia de la Torre [31] (OG), Tirado and Garcia de la Torre [32] (TG), and Broersma [33] (B). Taken from reference .

Fig. 3

The % difference for the viscosity factor between our results and the literature for the rectangular cylinder as a function of axial ratio is shown. The symbols represent the referenced works of Ortega and Garcia de la Torre [31] (OG) and Mansfield and Douglas [30] (MD). Taken from reference .

Figure 4

Triangular tessellations of a rectangular cylinder, the dumbbell, and lysozyme. Note that the sharp edge of the cylinder is well defined by placing small triangles on both edge surfaces, and the polar tessellation of a dumbbell places small triangles where the beads touch

Fig. 5

Fluid stagnation in Albumin (1AO6). The solid line represents a deep pocket in the middle of the protein, the dotted line represents a medium pocket at top left of the protein, while the dashed line represents a triangle at the bottom of the protein, where there is no pocket. Mx =3 represents protein motion parallel to the vector, while Mx =1,2 constitutes motion perpendicular to the vector. Taken from reference .

Figure 6

α-chymotrypsin structures. Top Panel: crystal structure (4cha.pdb). Bottom Panel: Amber 9 typical geometry after 1 ns molecular dynamics with implicit solvent.

Fig. 7

The translational diffusion coefficient and the intrinsic viscosity of α-chymotrypsin from an MD trajectory with implicit water at pH 3.0. As the molecule shape deforms from the initial crystal structure, the transport properties evolve and settle down after 2 ns.

Fig. 8

Top panel: The translational diffusion tensor eigenvalues along the MD trajectory for lysozyme (6LYZ). Note the small difference between the eigenvalues, justyfing the use of the average. Bottom Panel: The rotational diffusion tensor eigenvalues along the MD trajectory. Note the symmetric top appearance of the eigenvalues. In both cases, the data shows only small thermal fluctuations characteristic of a globular protein.

Fig. 9

Ribbon structure of trastuzumab taken from one of the multiple MD trajectories [64]. The carbohydrate group has been removed from the lower FC region for clarity. The structural fluctuation in this flexible antibody can be appreciated in the movie mentioned in the text.

Fig. 10

Phenyl-t-RNA (1EVV) surface with 2908 triangles.

Fig. 11

Left Panel: Groove hydration produced by inflated the nitrogen atoms by 3.6 A. Middle Panel: Mixed groove and backbone hydration produced by inflating the oxygen atoms. Right Panel: Pure backbone hydration produced by inflation of phosphorous atoms.

Fig. 12

The discrepancy of the BEST computation for both translation (Dtt) and tumbling (Drr) as a function of the added radius of the nitrogen atom which is present only in the DNA bases. The best value is that which splits the error across zero between each measurement. In this case the value occurs at 3.65 A for the DNA 12mer.