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. 2010 Sep 24:11:480.
doi: 10.1186/1471-2105-11-480.

Using diffusion distances for flexible molecular shape comparison

Yu-Shen Liu  1 , Qi Li, Guo-Qin Zheng, Karthik Ramani, William Benjamin
Affiliations

Using diffusion distances for flexible molecular shape comparison

Yu-Shen Liu et al. BMC Bioinformatics. .

Abstract

Background: Many molecules are flexible and undergo significant shape deformation as part of their function, and yet most existing molecular shape comparison (MSC) methods treat them as rigid bodies, which may lead to incorrect shape recognition.

Results: In this paper, we present a new shape descriptor, named Diffusion Distance Shape Descriptor (DDSD), for comparing 3D shapes of flexible molecules. The diffusion distance in our work is considered as an average length of paths connecting two landmark points on the molecular shape in a sense of inner distances. The diffusion distance is robust to flexible shape deformation, in particular to topological changes, and it reflects well the molecular structure and deformation without explicit decomposition. Our DDSD is stored as a histogram which is a probability distribution of diffusion distances between all sample point pairs on the molecular surface. Finally, the problem of flexible MSC is reduced to comparison of DDSD histograms.

Conclusions: We illustrate that DDSD is insensitive to shape deformation of flexible molecules and more effective at capturing molecular structures than traditional shape descriptors. The presented algorithm is robust and does not require any prior knowledge of the flexible regions.

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Figures

Figure 1
Figure 1
Flowchart of our method. Given a molecular shape, four independent steps contain sampling (red points), calculating inner distances (green line segments) between all sample point pairs, computing diffusion distances based on diffusion maps, and building the descriptor (blue histogram). Here the input shape is the volumetric data with the simulated 8 Å resolution density map for GroEL (PDB code: 1AON).
Figure 2
Figure 2
Our diffusion distance (DD) descriptor is compared to inner distance (ID) and Euclidean distance (ED). The first row shows the input four molecules with the same main chain orientation but with different surface shapes, where the arrows mark topological changes. The second row shows the DD, ID and ED descriptors. In each plot, the horizontal x-axis denotes pairwise distances, the vertical y-axis represents distance distributions, and the scale is normalized for the comparison process. Note that DD is not sensitive to shape deformation, in particular to topological changes, so four histograms are closed; in contrast, ED is strongly sensitive to deformation and ID is sensitive to topological changes.
Figure 3
Figure 3
Illustration of the inner distance. The red dashed line denotes the inner distance, which is the shortest path within the shape boundary surface that connect two landmark points x and y. The right molecule is one deformation to the left one, and the relative change of the inner distances between the corresponding pair of points (e.g. x and y) during shape deformation are small. In contrast, the black bold line denotes the Euclidean distance defined as the length of the line segment between two landmark points x and y. Note that the Euclidean distance does not have the property of deformation invariant in contrast to the inner distance. This is because, the Euclidean distance does not consider whether the line segment crosses shape boundaries.
Figure 4
Figure 4
Illustrating variation of inner distances for shape deformation with topological changes. The red dashed lines denote the shortest paths within the shape boundary surface that connect two landmark points x and y. The object B is the deformation to the one A, and the relative changes of the inner distances between the corresponding pair of points (e.g. x and y) during the shape deformation are small. In contrast, although the object C is also a shape deformation of the object A, the topological change leads to a significant change for inner distance.
Figure 5
Figure 5
The DD descriptor is compared to ID for the morph deformations between two conformations of GroEL: 1AON and 1KP8. The first row shows the input four molecules with the same main chain orientation but with different surface shapes, where the arrows mark topological changes. The second row shows the DD and ID descriptors. Note that DD is not sensitive to shape deformation with topological changes, so four histograms are almost consistent; in contrast, ID is sensitive to topological changes.
Figure 6
Figure 6
Comparison between DD and ID. The first row shows the four morph deformations between two conformations of Ran: 1BYU and 1RRP. The second row shows the DD and ID histogram. Note that DD is not sensitive to shape deformation with topological changes, so four histograms are almost consistent; however, ID is sensitive to topological changes.
Figure 7
Figure 7
Precision-recall curves compared with some existing shape descriptors for the MolMovDB database. The results show that the DD method performs better than other shape descriptors.
Figure 8
Figure 8
Variation with increasing sampling rates for the MolMovDB database. The precision-recall curves vary with n uniform sample points on the molecular surface for testing our method, where n ∈ {50, 100, 200, 300, 400, 500, 1000}. The results suggest that the high sampling rate performs better than the low one.
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