Bioinformatics applied to macromolecules are now widely spread and in continuous expansion. In this context, representing external molecular surface such as the Van der Waals Surface or the Solvent Excluded Surface can be useful for several applications. We propose a fast and parameterizable algorithm giving good visual quality meshes representing molecular surfaces. It is obtained by isosurfacing a filtered electron density map. The density map is the result of the maximum of Gaussian functions placed around atom centers. This map is filtered by an ideal low-pass filter applied on the Fourier Transform of the density map. Applying the marching cubes algorithm on the inverse transform provides a mesh representation of the molecular surface.

The geometric structure of macromolecules, such as proteins or nucleic acids, is directly related to their function [

At first, the external surface of a molecule has to be defined. Indeed, molecules are made of atoms which have no real surface. The most frequent molecular surface representations are the Van der Waals Surface (VdWS), the Solvent Accessible Surface (SAS), and the Solvent Excluded Surface (SES) [

A cutaway view of a small molecule. The most frequent molecular surface representations are detailed. Gray discs depict the Van der Waals volumes, the gray outer line depicts the Solvent Accessible Surface (SAS) and the continuous black line depicts the Solvent Excluded Surface (SES). The SAS (resp., SES) is the limit surface for the solvent molecule center (resp., external surface). The solvent molecule is represented as a black dashed circle.

Efficient tools to represent such surfaces are the polygonal meshes, which are collection of points related by edges and faces that approximate the considered surfaces. A lot of methods have been proposed in the last few years for the generation of a molecular surface meshes.

However, the computational time remains generally high for quality meshes, and it can be a problem when there is a great amount of data to treat. In this paper, we introduce the Filtered Density Map (FDM) algorithm, which is a fast and parameterizable algorithm to generate smooth molecular surface meshes. The generated mesh is the isosurface of frequency filtered electron density map.

This paper is organized as follows. First, some other works related to molecular surface generation are succinctly described in Section

In the last few years, a lot of methods have been developed for the generation of molecular surface meshes. In 1983, Connolly [

Comparisons between the FDM method and the methods mentioned in this section are shown in Section

The FDM method is based on volumetric electron density and a frequency filtering. Each atom is seen as a Gaussian electron cloud, the dimensions of which are depending on the VdW radius. Then, the electron density map is created by taking the local maximum value of these clouds. After a Fourier Transform, it is filtered by an ideal low pass filter, in order to remove frequencies corresponding to a spatial element smaller than a solvent molecule. Finally, a marching cubes [

A Gaussian function is constructed around each atom. The value of this function at a point

For the implementation, the three-dimensional space is divided into voxels. The spacing (

The density map of the whole molecule for a point in the space is defined as the maximal value of all the Gaussian functions at this point. The maximum of the Gaussian functions is chosen instead of the summation because it is not possible to evaluate the SES using the isosurface of a summation of Gaussian functions. It can be shown by the following counterexample, in which the Gaussian affected to the atoms

In the first situation, the space between

Two situations in which the Gaussian functions would have contradictory properties. (a) Two atoms of the molecule,

In the second situation,

In order to avoid interferences, the maximum is preferred to the summation of Gaussian functions. Isosurfacing this density map returns the VdWS. This surface is not smooth and in order to compute the SES, the density map must first be filtered.

The Fourier Transform of this electron density map is computed using the FFT algorithm [

Minimal wavelength allowed on the SES. It corresponds to four times the solvent radius

Gaussian functions are preferred to balls in the spatial domain because an ideal low-pass filter makes the Gibb's phenomenon appear on sharp edges. An ideal filter is used because the cutoff frequency is exactly known and because it is numerically possible. An ideal low-pass filter in the frequency domain is equivalent to a convolution product with a

In this example, the atoms

Value of the density map before (continuous gray line) and after (dashed black line) filtering at a point belonging to both the SES and the VdWS,

Value of the density map for the example of Figure

It is important to notice that if the spacing (

The final triangular mesh is an approximation of the isosurface of the filtered electron density map. The most popular technique to extract an isosurface from a 3D image is the marching cubes algorithm [

The visual appearance of the final mesh can be improved by magnifying the number of vertices. The number of vertices is increased using a smooth interpolation scheme such as the piecewise smooth surface reconstruction of Hoppe et al. [

Some numerical results pointing out advantages and drawbacks of the FDM are shown in this section. The main characteristics to be observed are the computation time and the quality of the generated mesh. The section is divided into three parts: the analysis of the effects of the different parameters of the FDM, the results of computation time comparisons with other existing methods, and a quality measurement of the generated meshes.

There are three main parameters modifiable by the user. First, the spatial spacing

With a small spatial spacing, it is possible to represent fine details. However, it drastically increases the memory space needed as well as the computation time. Indeed, reducing

Meshes generated from electron density map at a spatial spacing of

In order to generate a mesh representing the SES,

Meshes generated from electron density map filtered at a cutoff frequency of

The final mesh refinement gives foremost an esthetic advantage. The memory space needed does not increase a lot because the number of voxels remains the same. Only the size of the mesh changes and this is negligible in comparison with the space needed by the voxels representation. The computation time slightly increases but, when

Meshes refined with a factor of

In this section, computation times are compared between the FDM algorithm and algorithms found in the literature for equivalent qualities. When available, the algorithms were run on the same computer, when not, the computation times were the ones announced in the original paper. Can et al. made a comparison of their method computation time with three molecular visualization tools: UCSF Chimera [

Computation times (in s) for different methods. (

PDB code | No. atoms | LSMS | PyMol | Swiss-PDBV | Chimera | MSMS | ||
---|---|---|---|---|---|---|---|---|

1A8R | 26400 | 0.65 | 2.61 | 5.56 | 10.52 | 6.38 | 16.36 | 0.95 |

1H2I | 32318 | 0.72 | 1.83 | 6.50 | 11.37 | 5.25 | 40.04 | 3.03 |

1GTP | 34740 | 0.51 | 2.20 | 6.98 | 13.15 | 4.75 | 67.04 | 9.02 |

1FKA | 34977 | 0.75 | 2.77 | 7.89 | 26.29 | 7.36 | 77.25 | 4.50 |

1GT7 | 42700 | 0.95 | 2.43 | 7.32 | 16.10 | 6.50 | 54.39 | 3.32 |

1GAV | 43335 | 0.55 | 2.54 | 7.05 | 28.86 | 7.71 | 78.35 | 4.22 |

1G3I | 45528 | 0.54 | 2.85 | 8.18 | 19.45 | 6.21 | — | 7.67 |

1PMA | 45892 | 0.40 | 1.97 | 8.23 | 18.67 | 6.72 | — | 12.90 |

1FJG | 51995 | 0.71 | 2.88 | 8.01 | 25.18 | 8.05 | — | 15.11 |

1AON | 58870 | 0.62 | 2.64 | 8.83 | 26.36 | 8.91 | — | 10.87 |

1J0B | 60144 | 0.69 | 3.07 | 6.87 | 32.66 | 7.92 | — | 5.61 |

1OTZ | 68620 | 0.67 | 2.28 | 8.46 | 30.14 | 9.56 | — | 9.03 |

1IR2 | 77088 | 0.65 | 2.88 | 7.09 | 29.31 | 9.55 | 93.87 | 9.49 |

Computation times (in s) for different methods. (

PDB code | No. atoms | Cheng | MSMS | ||
---|---|---|---|---|---|

200D | 232 | 65 k | 0.55 | 1.35 | 0.33 |

1FG1 | 873 | 100 k | 0.85 | 2.41 | 0.65 |

3EBZ | 1651 | 200 k | 1.37 | 15.43 | 0.97 |

Mesh of the SES of a molecule with

It appears in Tables

In order to validate the quality of the results, different generated surfaces (SESs) were compared with references surfaces. These reference surfaces were generated by isosurfacing a field composed of a union of VdW balls at good resolution (spatial spacing of

RMSD and percentage of big differences with reference surfaces.

Spacing (Å) | ||||
---|---|---|---|---|

0.78 | 6.89% | 67.04% | ||

0.26 | 0% | 20.20% | ||

0.20 | 0% | 0% |

A visual comparison between the SES of 1FG1 computed with the FDM algorithm and the reference SES is shown in Figure

Visual comparison between the 1FG1 SES generated with the FDM algorithm with a spacing of

In this paper, we introduced an algorithm to compute molecular surface meshes (the FDM algorithm). It is constructed as an isosurface of a filtered electron density map (FDM). This algorithm is faster than other algorithm tested in equivalent conditions. It is slower than the MSMS algorithm for small molecules (<30000 atoms) but it returns a smooth manifold surface, which is not the case with MSMS. It makes possible to compute a precise representation of the surface with a limited number of voxels, so that the computation time and the memory space needed are reduced. Moreover, it is parameterizable on the spatial resolution, the refinement of the final mesh, and the size of the solvent molecule. Thus, the spatial resolution can be improved for a finer result but with an important computation time increase. Similarly, a smoother result can be obtained with a final refinement with a small influence on the computation time but with less precise results than reducing the spacing. Finally, the solvent molecule size can be chosen without influence on the computation time.

The refinement could be improved to be specific to molecular surface. It would enable coarse meshes to be generated rapidly and to be improved by a priori knowledge about local geometry of molecule surfaces, such that the curvature deduced from the closest atom radius. In future works, this algorithm will be used in surface-based method to detect protein hot spots [

This research is funded by the NANOTIC/TSARINE project of the Région Wallonne (Belgium).