A local average distance descriptor for flexible protein structure comparison
© Wang et al.; licensee BioMed Central Ltd. 2014
Received: 21 September 2013
Accepted: 22 March 2014
Published: 2 April 2014
Protein structures are flexible and often show conformational changes upon binding to other molecules to exert biological functions. As protein structures correlate with characteristic functions, structure comparison allows classification and prediction of proteins of undefined functions. However, most comparison methods treat proteins as rigid bodies and cannot retrieve similarities of proteins with large conformational changes effectively.
In this paper, we propose a novel descriptor, local average distance (LAD), based on either the geodesic distances (GDs) or Euclidean distances (EDs) for pairwise flexible protein structure comparison. The proposed method was compared with 7 structural alignment methods and 7 shape descriptors on two datasets comprising hinge bending motions from the MolMovDB, and the results have shown that our method outperformed all other methods regarding retrieving similar structures in terms of precision-recall curve, retrieval success rate, R-precision, mean average precision and F1-measure.
Both ED- and GD-based LAD descriptors are effective to search deformed structures and overcome the problems of self-connection caused by a large bending motion. We have also demonstrated that the ED-based LAD is more robust than the GD-based descriptor. The proposed algorithm provides an alternative approach for blasting structure database, discovering previously unknown conformational relationships, and reorganizing protein structure classification.
Protein structure comparison plays an important role in predicting functions of novel proteins  and several methods have been developed for pairwise [2–8] and multiple [9–16] comparisons. Most existing methods of structure comparison treat proteins as rigid bodies; however, protein structures are flexible and conformationally changeable in response to binding another molecules relating with biological functions such as immune protection, enzymatic catalysis and cellular locomotion [17, 18]. Such structural variations caused rigid-body algorithms unable to generate correct alignments or retrieve similar structures with large deformations. Therefore, flexibility of proteins should be taken into account when comparing structures and searching for similarities to a query structure.
Flexible structure comparison has received much attention in recent years. For instance, FlexProt found congruent rigid fragment pairs between two proteins and the flexible regions (hinges), and then a clustering procedure was performed to join consecutive fragment pairs into congruent domain pairs [19, 20]. FATCAT connected aligned fragment pairs based on a dynamic programming algorithm which introduced penalty scores for gaps and twists between consecutive aligned fragment pairs . Compared with FlexProt, FATCAT generates alignments with less twists but similar root mean square deviations (RMSDs) and lengths. The TOPS++FATCAT algorithm reduced the number of aligned fragment pairs during FATCAT comparison processes by applying topological constraints obtained from the alignment of secondary structure elements (SSEs) of TOPS + . Therefore, TOPS++FATCAT is more than 10 times faster compared to FATCAT. Both FlexProt and FATCAT are sequential alignment algorithms thus unable to identify non-sequential alignments. FASE  and FlexSnap  were designed to tackle the problem of non-sequential flexible structure alignment. FASE compares structures starting from aligned pairs of SSEs with an assumption that an optimal superposition of pairs of structures must have at least one pair of well-aligned SSEs. FlexSnap applies a greedy algorithm for connecting aligned fragment pairs and possesses competitive results against other state-of-the-art pairwise comparison methods. Matt, one of the most popular and accurate flexible multiple structure alignment methods, is also based on the approach of chaining aligned fragment pairs which are allowed translations and rotations during assembling [25, 26].
The alignment/superposition based comparison methods are inefficient for blasting similar structures from a structure database in real-time . Therefore, several non-alignment approaches based on different descriptors of molecular shapes were proposed. These descriptors are usually represented by histograms or vectors, and a similarity score between two molecules is calculated from corresponding descriptors without any alignment [28, 29]. For example, Daras et al. applied the spherical trace transform method to produce rotational invariant descriptor vectors constituted by weighted geometry- and attribute-based vectors for protein classification . The 3D Zernike descriptor represented a protein structure by 121 numbers based on a series expansion of 3D functions for fast retrieval of similarities, and which demonstrated that low-resolution structures were also applicable [27, 31]. Abu Deeb et al. proposed a global descriptor on protein surface, and which was constructed from local patch descriptors defined by residue-specific distance distributions between Cα atoms and the numbers of pairwise residue co-occurrences within each surface patch . Yin et al. compared local surface of proteins by geometric fingerprints of each surface patch . A fingerprint consists of 60 (4 by 15) bins corresponding to the geodesic-distance-dependent distribution of curvatures.
Nevertheless, most non-alignment methods treated proteins as rigid bodies and neglected flexibility of protein conformations required for performing biological functions. To confront the issue of flexibility, Liu and Fang et al. proposed several histogram based descriptors for flexible molecules comparison. For instance, a local diameter descriptor for depicting the local characteristics of boundary points , and another descriptor, inner distance, defined as the shortest path between landmark points [28, 35]. Both methods are sensitive to self-connection problems during molecular shape deformation. Accordingly, an improved method named Diffusion Distance Shape Descriptor (DDSD) was proposed, which is based on an average distance instead of the shortest distance between two landmark points . Although DDSD is superior to local diameter, inner distance and other descriptors in terms of retrieving similar protein structures, its performance is still unsatisfied with an F1-measure of 37.04%.
Non-alignment or descriptor based approaches are generally fast enough to search a large database in a real-time manner, but do not provide corresponding information of residues which might provide crucial information for biologists. Combining the ideas of alignment and descriptor based approaches, we propose a novel and efficient descriptor called local average distance (LAD) which is based on either geodesic distances (GDs) or Euclidean distances (EDs) for pairwise flexible protein structure comparison. Each protein structure is firstly transformed into its corresponding LAD profile, and the similarity between two proteins is calculated according to pairwise local alignment on transformed profiles. The Hinge Atlas and Hinge Atlas Gold datasets  from the MolMovDB  were employed to evaluate the performance of proposed LAD descriptors and to compare with several non-alignment and rigid/flexible structure alignment methods.
Triangular surface construction and simplification
The solvent-accessible surface (SAS)  and solvent-excluded surface [40, 41] (SES, also known as molecular surface or Connolly surface) are the most widely used definitions for protein surface analysis. Each atom of a protein is represented as a sphere with its van der Waals radius. The SAS is traced out by the center of a solvent probe sphere rolling over the spherical atoms, whereas the SES is formed by the inward-facing surface of the probe consisting of contact surface and re-entrant surface. For a more complete description of both SAS and SES please refer to . Many algorithms have been developed to build SAS and/or SES such as Gauss-Bonnet theorem , level-set , alpha shape [45, 46], beta shape , Euclidean distance transform , ray-casting et al.[50–52]. One common area-based method defines a residue as a surface residue if its surface area is greater than a specific threshold [46, 53]. The other area-based methods consider a residue with relative solvent accessibility larger than a threshold as a surface residue [54, 55]. The relative solvent accessibility is defined by taking a residue’s solvent-accessible area divided by the maximum area of that residue [56, 57]. In recent years, novel atom-depth-based approaches were proposed as alternative ways to define surface residues [58, 59]. Different algorithms employed various definitions of atom depth which could be defined as the distance of an atom from the nearest water molecule surrounding the protein, from the molecular surface, or from its closest solvent-accessible neighbor .
The input for building an LAD profile is a standard PDB file. Owing to the requirement of triangular surface meshes for GD calculation, one of the most used and fastest surface program, MSMS v2.6.1 , is applied to construct triangular surface meshes from coordinates of all backbone atoms of the protein (Figure 1a). All the parameters of MSMS are remained as default settings. This tool usually generates high resolution meshes (Figure 1b) for proteins. However, it is time-consuming and memory exhausted during the calculation of GDs among mesh vertices. To reduce the resolution of MSMS-generated meshes, an open source tool, MeshLab v1.3.2 (http://meshlab.sourceforge.net/), is adopted to downsample original meshes. The outputs of MSMS are converted into Polygon File Format (Stanford Triangle Format) as MeshLab’s inputs. The algorithm of Quadric edgecollapse, a variant of the well-known quadric error metric algorithm  , is employed to simplify meshes (Figure 1c). As a result, the face number of each MSMS-generated mesh could be reduced by 85% generally in this research.
Calculation of pairwise residue distances
The simplified meshes are then used to identify surface residues, and the GDs and EDs of surface residue pairs can be obtained. Each vertex of a simplified mesh belongs to the closest backbone atom of the protein. In other words, an atom could possess more than one vertex. We defined that the vertices belong to an atom as the associated vertices of that atom. A residue is regarded as a surface residue if its backbone atoms have at least one vertex.
where GD(a i , a j ) is the average GD from the ith atom to the jth atom, and represent the xth vertex of the ith atom and the yth vertex of the jth atom respectively. The symbols M and N indicate the number of vertices associated with the ith atom and the jth atom, and is the GD from vertex to vertex . The atoms possessing no associated vertices won’t be considered, hence M and N must be strictly larger than zero. In contrast to the measurement of GD, an ED between two atoms can be easily obtained from their coordinates. Once the two different distance measures between any two atoms are obtained, the distance measures between any two residues can be calculated similarly by taking an average of GDs or EDs from all associated backbone atoms.
Construction of LAD profiles
LAD is proposed to retain local characteristics of each residue in sequential relationship. The LAD profile for a protein consists of average distance values which are built by employing a sliding window scanning from N- to C-terminus. In this study, we have tried different odd window sizes ranging from 3 to 21, and the window size of 9 residues provided the best performance on the training dataset (Dataset L from ADiDoS ). Hence, a window size of 9 is applied to build all LAD profiles. We have implemented two types of LAD profiles; one is based on ED feature (LADED, Figure 1d) and the other is based on GD (LADGD, Figure 1e) feature. Given a residue at position i (residue i ) in the sequence, the LAD i for the residue i is defined by taking average distance from residue i to both side neighbouring residues within the window.
The pairwise structure comparison in this study is based on evaluating the similarities of two LAD profiles from two individual proteins. A variation of Smith-Waterman algorithm is performed to obtain the correspondence of residues between two proteins by comparing LADs instead of amino acid contents. The similarity score between two residues, residue i and residue j , for dynamic programming is inversely proportional to the absolute difference between LAD i and LAD j .
Variables D and α were trained by the Dataset L which contains 706 known domain swapping homologous pairs (Lds), 487 common homologous pairs (Lch) and 640 non-homologous pairs (Lnh) of protein structures. Both Lds and Lch were considered as a positive dataset in which each pair was anticipated possessing low LAD div values. Conversely, Lnh was considered as a negative dataset which was expected possessing high LAD div values for each pairs. Let Lds<0.5 and Lch<0.5 denote the number of pairs whose LAD div is less than 0.5 for both Lds and Lch. The Lnh≥ 0.5 represents the number of pairs whose LAD div is larger than or equal to 0.5. We have evaluated D ranging from 0.1 to 20 with an interval of 0.1, and a range of 1 to 5 with an interval of 0.5 for α. Hence, a total of 1800 (200 × 9) combinations of D and α were evaluated and the one with maximum Lds<0.5 + Lch<0.5 + Lnh≥ 0.5 was selected. Finally, (D, α) = (1, 4.5) and (D, α) = (1.1, 5) were selected for LADED and LADGD respectively.
where RMSD is the root mean square deviation of the distances between the aligned Cα atoms. Like LAD div , structural alignment of a similar structure pair tends to have both low RMSD and large N e , and low Struct div .
Comparison with structural alignment methods
Retrieval performances of 2577 queries for different methods on the Hinge Atlas dataset
Number of successful retrieval
Success rate (%)
Retrieval performances of 214 queries for different methods on the Hinge Atlas dataset
Average R-precision (%)
Mean average precision (%)
Comparison with non-alignment methods
Comparison with non-alignment methods on Liu’s dataset
Diffusion distance (DD)
Inner distance (ID)
Shape distribution (SD)
Euclidean distance (ED)
Solid angle histogram (SAH)
Geodesic distance (GD)
Spherical harmonic descriptor (SHD)
Differences between the previous and proposed ED/GD based methods
In previous studies [34–36], ED and GD were shown to be sensitive to shape deformation and not feasible for flexible molecular shape comparison. However, it is interesting that relying on the proposed LAD methods, both features become insensitive to topological changes and reveal deformation invariant properties to tackle with the flexibility problems. The reason for sensitive ED and GD features in previous studies is that both distances were computed among all global landmark points. On the contrary, the LAD exploits the characterization of local geometric features for each residue and its neighbouring residues. Therefore, ED and GD features become much less sensitive to global topological changes.
Pairwise comparison of LAD profiles was performed by a modification of Smith-Waterman algorithm and possessed the same time complexity. The goal of a sequence alignment problem is to identify the correspondence of residues between two given proteins, while a structure alignment emphasizes on finding both an alignment and a spatial superposition. Possible combinations of corresponding residues are countable while possibilities of special superposition are innumerable. Therefore, the computational complexity of the proposed algorithm is inherently less than most commonly used structure alignment methods . The LAD algorithm was implemented by C# .NET running on an Intel Core i5-2500 3.3GHz computer with 16GB ram. According to the 551478 pairwise comparisons mentioned in the result section, it only cost an average computational time of 3.896 and 4.828 milliseconds per comparison for LADED and LADGD profiles respectively.
We proposed a novel profile-based alignment method, named LAD, for pairwise flexible protein structure comparison. It can be constructed in a sense of any kind of spatial measures of local neighbouring residues within a specific sliding window. Here, GD and ED were used to build LADGD and LADED profiles. The idea of LAD improves the ED- and GD-based descriptors which were previously shown to be sensitive to molecular shape deformation, in particular to topologically structural changes. The effectiveness of LAD descriptor has been evaluated on two datasets of hinge bending motions from the MolMovDB. Our methods are robust to deformed flexible molecules and achieve good performance regarding assignment of the queries to different classes of molecules with conformational changes, and the results have shown superior performance compared to existing alignment- and nonalignment-based tools. Finally, the reasons of LAD descriptor being insensitive to flexible proteins with self-connection circumstance was described by taking 3D domain swapping cases as examples, and further discussion of LADED possessing more robust properties than LADGD was also explained. Required computational time for pairwise LADED/LADGD profile comparisons was analyzed to demonstrate its feasibility for constructing an on-line structure comparison system. The proposed descriptor is indeed effective in retrieving deformed proteins and it could be an alternative approach for database search, discovery of previously unknown conformational relationships, and reorganization of protein structure classification.
Availability of supporting data
Local average distance
Mean average precision.
Thanks to Mr. Yueh-Lin Tsai for initiating the research topic and thanks to Professor Liu for providing the testing dataset of Hinge Atlas. This work was supported by the Center of Excellence for the Oceans from National Taiwan Ocean University and the National Science Council, Taiwan (NSC101-2627-B-019-003 and NSC102-2321-B-019-001 to Tun-Wen Pai, NSC102-2325-B-007-001 to Wen-Ching Wang).
- Hasegawa H, Holm L: Advances and pitfalls of protein structural alignment. Curr Opin Struct Biol. 2009, 19 (3): 341-348. 10.1016/j.sbi.2009.04.003.View ArticlePubMedGoogle Scholar
- Shindyalov IN, Bourne PE: Protein structure alignment by incremental combinatorial extension (CE) of the optimal path. Protein Eng. 1998, 11 (9): 739-747. 10.1093/protein/11.9.739.View ArticlePubMedGoogle Scholar
- Holm L, Park J: DaliLite workbench for protein structure comparison. Bioinformatics. 2000, 16 (6): 566-567. 10.1093/bioinformatics/16.6.566.View ArticlePubMedGoogle Scholar
- Wang S, Zheng WM: CLePAPS: fast pair alignment of protein structures based on conformational letters. J Bioinform Comput Biol. 2008, 6 (2): 347-366. 10.1142/S0219720008003461.View ArticlePubMedGoogle Scholar
- Zhu J, Weng Z: FAST: a novel protein structure alignment algorithm. Proteins. 2005, 58 (3): 618-627.View ArticlePubMedGoogle Scholar
- Zhang Y, Skolnick J: TM-align: a protein structure alignment algorithm based on the TM-score. Nucleic Acids Res. 2005, 33 (7): 2302-2309. 10.1093/nar/gki524.View ArticlePubMed CentralPubMedGoogle Scholar
- Gelly JC, Joseph AP, Srinivasan N, de Brevern AG: iPBA: a tool for protein structure comparison using sequence alignment strategies. Nucleic Acids Res. 2011, 39 (Web Server issue): W18-23.View ArticlePubMed CentralPubMedGoogle Scholar
- Shah SB, Sahinidis NV: SAS-Pro: simultaneous residue assignment and structure superposition for protein structure alignment. PloS one. 2012, 7 (5): e37493-10.1371/journal.pone.0037493.View ArticlePubMed CentralPubMedGoogle Scholar
- Guda C, Lu S, Scheeff ED, Bourne PE, Shindyalov IN: CE-MC: a multiple protein structure alignment server. Nucleic Acids Res. 2004, 32 (Web Server issue): W100-103.View ArticlePubMed CentralPubMedGoogle Scholar
- Konagurthu AS, Whisstock JC, Stuckey PJ, Lesk AM: MUSTANG: a multiple structural alignment algorithm. Proteins. 2006, 64 (3): 559-574. 10.1002/prot.20921.View ArticlePubMedGoogle Scholar
- Lupyan D, Leo-Macias A, Ortiz AR: A new progressive-iterative algorithm for multiple structure alignment. Bioinformatics. 2005, 21 (15): 3255-3263. 10.1093/bioinformatics/bti527.View ArticlePubMedGoogle Scholar
- Wang S, Peng J, Xu J: Alignment of distantly related protein structures: algorithm, bound and implications to homology modeling. Bioinformatics. 2011, 27 (18): 2537-2545.PubMed CentralPubMedGoogle Scholar
- Liu X, Zhao YP, Zheng WM: CLEMAPS: multiple alignment of protein structures based on conformational letters. Proteins. 2008, 71 (2): 728-736. 10.1002/prot.21739.View ArticlePubMedGoogle Scholar
- Shatsky M, Nussinov R, Wolfson HJ: A method for simultaneous alignment of multiple protein structures. Proteins. 2004, 56 (1): 143-156. 10.1002/prot.10628.View ArticlePubMedGoogle Scholar
- Ye Y, Godzik A: Multiple flexible structure alignment using partial order graphs. Bioinformatics. 2005, 21 (10): 2362-2369. 10.1093/bioinformatics/bti353.View ArticlePubMedGoogle Scholar
- Shealy P, Valafar H: Multiple structure alignment with msTALI. BMC bioinformatics. 2012, 13: 105-10.1186/1471-2105-13-105.View ArticlePubMed CentralPubMedGoogle Scholar
- Rashin AA, Rashin AH, Jernigan RL: Diversity of function-related conformational changes in proteins: coordinate uncertainty, fragment rigidity, and stability. Biochemistry. 2010, 49 (27): 5683-5704. 10.1021/bi100110x.View ArticlePubMed CentralPubMedGoogle Scholar
- Nigham A, Hsu D: Protein conformational flexibility analysis with noisy data. J Comput Biol. 2008, 15 (7): 813-828. 10.1089/cmb.2007.0138.View ArticlePubMedGoogle Scholar
- Shatsky M, Nussinov R, Wolfson HJ: Flexible protein alignment and hinge detection. Proteins. 2002, 48 (2): 242-256. 10.1002/prot.10100.View ArticlePubMedGoogle Scholar
- Shatsky M, Nussinov R, Wolfson HJ: FlexProt: alignment of flexible protein structures without a predefinition of hinge regions. J Comput Biol. 2004, 11 (1): 83-106. 10.1089/106652704773416902.View ArticlePubMedGoogle Scholar
- Ye Y, Godzik A: Flexible structure alignment by chaining aligned fragment pairs allowing twists. Bioinformatics. 2003, 19 (2): ii246-255.PubMedGoogle Scholar
- Veeramalai M, Ye Y, Godzik A: TOPS++FATCAT: fast flexible structural alignment using constraints derived from TOPS + strings model. BMC bioinformatics. 2008, 9: 358-10.1186/1471-2105-9-358.View ArticlePubMed CentralPubMedGoogle Scholar
- Vesterstrom J, Taylor WR: Flexible secondary structure based protein structure comparison applied to the detection of circular permutation. J Comput Biol. 2006, 13 (1): 43-63. 10.1089/cmb.2006.13.43.View ArticlePubMedGoogle Scholar
- Salem S, Zaki MJ, Bystroff C: FlexSnap: flexible non-sequential protein structure alignment. Algorithms for molecular biology: AMB. 2010, 5: 12-10.1186/1748-7188-5-12.View ArticlePubMed CentralPubMedGoogle Scholar
- Menke M, Berger B, Cowen L: Matt: local flexibility aids protein multiple structure alignment. PLoS Comput Biol. 2008, 4 (1): e10-10.1371/journal.pcbi.0040010.View ArticlePubMed CentralPubMedGoogle Scholar
- Berbalk C, Schwaiger CS, Lackner P: Accuracy analysis of multiple structure alignments. Protein Sci. 2009, 18 (10): 2027-2035. 10.1002/pro.213.View ArticlePubMed CentralPubMedGoogle Scholar
- Sael L, Li B, La D, Fang Y, Ramani K, Rustamov R, Kihara D: Fast protein tertiary structure retrieval based on global surface shape similarity. Proteins. 2008, 72 (4): 1259-1273. 10.1002/prot.22030.View ArticlePubMedGoogle Scholar
- Liu YS, Wang M, Paul JC, Ramani K: 3DMolNavi: a web-based retrieval and navigation tool for flexible molecular shape comparison. BMC bioinformatics. 2012, 13: 95-10.1186/1471-2105-13-95.View ArticlePubMed CentralPubMedGoogle Scholar
- Venkatraman V, Sael L, Kihara D: Potential for protein surface shape analysis using spherical harmonics and 3D Zernike descriptors. Cell Biochem Biophys. 2009, 54 (1–3): 23-32.View ArticlePubMedGoogle Scholar
- Daras P, Zarpalas D, Axenopoulos A, Tzovaras D, Strintzis MG: Three-dimensional shape-structure comparison method for protein classification. IEEE/ACM Trans Comput Biol Bioinform. 2006, 3 (3): 193-207. 10.1109/TCBB.2006.43.View ArticlePubMedGoogle Scholar
- Sael L, Kihara D: Improved protein surface comparison and application to low-resolution protein structure data. BMC bioinformatics. 2010, 11 (11): S2-View ArticlePubMed CentralPubMedGoogle Scholar
- Abu Deeb Z, Adjeroh DA, Jiang BH: Protein surface characterization using an invariant descriptor. International journal of biomedical imaging. 2011, 2011: 918978-View ArticlePubMed CentralPubMedGoogle Scholar
- Yin S, Proctor EA, Lugovskoy AA, Dokholyan NV: Fast screening of protein surfaces using geometric invariant fingerprints. Proc Natl Acad Sci USA. 2009, 106 (39): 16622-16626. 10.1073/pnas.0906146106.View ArticlePubMed CentralPubMedGoogle Scholar
- Fang Y, Liu YS, Ramani K: Three dimensional shape comparison of flexible proteins using the local-diameter descriptor. BMC Struct Biol. 2009, 9: 29-10.1186/1472-6807-9-29.View ArticlePubMed CentralPubMedGoogle Scholar
- Liu YS, Fang Y, Ramani K: IDSS: deformation invariant signatures for molecular shape comparison. BMC bioinformatics. 2009, 10: 157-10.1186/1471-2105-10-157.View ArticlePubMed CentralPubMedGoogle Scholar
- Liu YS, Li Q, Zheng GQ, Ramani K, Benjamin W: Using diffusion distances for flexible molecular shape comparison. BMC bioinformatics. 2010, 11: 480-10.1186/1471-2105-11-480.View ArticlePubMed CentralPubMedGoogle Scholar
- Flores SC, Lu LJ, Yang J, Carriero N, Gerstein MB: Hinge Atlas: relating protein sequence to sites of structural flexibility. BMC bioinformatics. 2007, 8: 167-10.1186/1471-2105-8-167.View ArticlePubMed CentralPubMedGoogle Scholar
- Flores S, Echols N, Milburn D, Hespenheide B, Keating K, Lu J, Wells S, Yu EZ, Thorpe M, Gerstein M: The database of macromolecular motions: new features added at the decade mark. Nucleic Acids Res. 2006, 34 (Database issue): D296-301.View ArticlePubMed CentralPubMedGoogle Scholar
- Lee B, Richards FM: The interpretation of protein structures: estimation of static accessibility. J Mol Biol. 1971, 55 (3): 379-400. 10.1016/0022-2836(71)90324-X.View ArticlePubMedGoogle Scholar
- Connolly ML: Analytical molecular surface calculation. J Appl Crystallogr. 1983, 16 (5): 548-558. 10.1107/S0021889883010985.View ArticleGoogle Scholar
- Richards FM: Areas, volumes, packing and protein structure. Annu Rev Biophys Bioeng. 1977, 6: 151-176. 10.1146/annurev.bb.06.060177.001055.View ArticlePubMedGoogle Scholar
- Connolly ML: Network science 1996. Molecular surfaces: a review. http://www.netsci.org/Science/Compchem/feature14.html (accessed 18 Feb. 2014)
- Tsodikov OV, Record MT, Sergeev YV: Novel computer program for fast exact calculation of accessible and molecular surface areas and average surface curvature. J Comput Chem. 2002, 23 (6): 600-609. 10.1002/jcc.10061.View ArticlePubMedGoogle Scholar
- Can T, Chen CI, Wang YF: Efficient molecular surface generation using level-set methods. J Mol Graph Model. 2006, 25 (4): 442-454. 10.1016/j.jmgm.2006.02.012.View ArticlePubMedGoogle Scholar
- Liang J, Edelsbrunner H, Fu P, Sudhakar PV, Subramaniam S: Analytical shape computation of macromolecules: I: molecular area and volume through alpha shape. Proteins. 1998, 33 (1): 1-17. 10.1002/(SICI)1097-0134(19981001)33:1<1::AID-PROT1>3.0.CO;2-O.View ArticlePubMedGoogle Scholar
- Albou LP, Schwarz B, Poch O, Wurtz JM, Moras D: Defining and characterizing protein surface using alpha shapes. Proteins. 2009, 76 (1): 1-12. 10.1002/prot.22301.View ArticlePubMedGoogle Scholar
- Ryu J, Park R, Kim D-S: Molecular surfaces on proteins via beta shapes. Comput Aided Des. 2007, 39 (12): 1042-1057. 10.1016/j.cad.2006.10.008.View ArticleGoogle Scholar
- Xu D, Zhang Y: Generating triangulated macromolecular surfaces by Euclidean Distance Transform. PloS one. 2009, 4 (12): e8140-10.1371/journal.pone.0008140.View ArticlePubMed CentralPubMedGoogle Scholar
- Decherchi S, Rocchia W: A general and robust ray-casting-based algorithm for triangulating surfaces at the nanoscale. PloS one. 2013, 8 (4): e59744-10.1371/journal.pone.0059744.View ArticlePubMed CentralPubMedGoogle Scholar
- Yu Z, Holst MJ, Cheng Y, McCammon JA: Feature-preserving adaptive mesh generation for molecular shape modeling and simulation. J Mol Graph Model. 2008, 26 (8): 1370-1380. 10.1016/j.jmgm.2008.01.007.View ArticlePubMedGoogle Scholar
- Connolly ML: The molecular surface package. J Mol Graph. 1993, 11 (2): 139-141. 10.1016/0263-7855(93)87010-3.View ArticlePubMedGoogle Scholar
- Rocchia W, Sridharan S, Nicholls A, Alexov E, Chiabrera A, Honig B: Rapid grid-based construction of the molecular surface and the use of induced surface charge to calculate reaction field energies: applications to the molecular systems and geometric objects. J Comput Chem. 2002, 23 (1): 128-137. 10.1002/jcc.1161.View ArticlePubMedGoogle Scholar
- Jordan RA, El-Manzalawy Y, Dobbs D, Honavar V: Predicting protein-protein interface residues using local surface structural similarity. BMC bioinformatics. 2012, 13: 41-10.1186/1471-2105-13-41.View ArticlePubMed CentralPubMedGoogle Scholar
- Ma B, Elkayam T, Wolfson H, Nussinov R: Protein-protein interactions: structurally conserved residues distinguish between binding sites and exposed protein surfaces. Proc Natl Acad Sci U S A. 2003, 100 (10): 5772-5777. 10.1073/pnas.1030237100.View ArticlePubMed CentralPubMedGoogle Scholar
- Yan C, Honavar V, Dobbs D: Identification of interface residues in protease-inhibitor and antigen-antibody complexes: a support vector machine approach. Neural computing & applications. 2004, 13 (2): 123-129.View ArticleGoogle Scholar
- Singh H, Ahmad S: Context dependent reference states of solvent accessibility derived from native protein structures and assessed by predictability analysis. BMC Struct Biol. 2009, 9: 25-10.1186/1472-6807-9-25.View ArticlePubMed CentralPubMedGoogle Scholar
- Tien MZ, Meyer AG, Sydykova DK, Spielman SJ, Wilke CO: Maximum allowed solvent accessibilites of residues in proteins. PloS one. 2013, 8 (11): e80635-10.1371/journal.pone.0080635.View ArticlePubMed CentralPubMedGoogle Scholar
- Chakravarty S, Varadarajan R: Residue depth: a novel parameter for the analysis of protein structure and stability. Structure. 1999, 7 (7): 723-732. 10.1016/S0969-2126(99)80097-5.View ArticlePubMedGoogle Scholar
- Pintar A, Carugo O, Pongor S: Atom depth as a descriptor of the protein interior. Biophys J. 2003, 84 (4): 2553-2561. 10.1016/S0006-3495(03)75060-7.View ArticlePubMed CentralPubMedGoogle Scholar
- Pintar A, Carugo O, Pongor S: Atom depth in protein structure and function. Trends Biochem Sci. 2003, 28 (11): 593-597. 10.1016/j.tibs.2003.09.004.View ArticlePubMedGoogle Scholar
- Sanner MF, Olson AJ, Spehner JC: Reduced surface: an efficient way to compute molecular surfaces. Biopolymers. 1996, 38 (3): 305-320. 10.1002/(SICI)1097-0282(199603)38:3<305::AID-BIP4>3.0.CO;2-Y.View ArticlePubMedGoogle Scholar
- Garland M, Heckbert PS: Surface simplification using quadric error metrics. ACM SIGGRAPH. 1997, 209-216.Google Scholar
- Chu CH, Lo WC, Wang HW, Hsu YC, Hwang JK, Lyu PC, Pai TW, Tang CY: Detection and alignment of 3D domain swapping proteins using angle-distance image-based secondary structural matching techniques. PloS one. 2010, 5 (10): e13361-10.1371/journal.pone.0013361.View ArticlePubMed CentralPubMedGoogle Scholar
- Lu G: TOP: a new method for protein structure comparisons and similarity searches. J Appl Crystallogr. 2000, 33 (1): 176-183. 10.1107/S0021889899012339.View ArticleGoogle Scholar
- Manning CD, Raghavan P, Schütze H: Introduction to information retrieval. 2008, New York, NY USA: Cambridge University PressView ArticleGoogle Scholar
- Poleksic A: Optimal pairwise alignment of fixed protein structures in subquadratic time. J Bioinform Comput Biol. 2011, 9 (3): 367-382. 10.1142/S0219720011005562.View ArticlePubMedGoogle Scholar
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