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2015 | 3 | 1 |
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A topological approach for protein classification

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Protein function and dynamics are closely related to its sequence and structure.However, prediction of protein function and dynamics from its sequence and structure is still a fundamental challenge in molecular biology. Protein classification, which is typically done through measuring the similarity between proteins based on protein sequence or physical information, serves as a crucial step toward the understanding of protein function and dynamics. Persistent homology is a new branch of algebraic topology that has found its success in the topological data analysis in a variety of disciplines, including molecular biology. The present work explores the potential of using persistent homology as an independent tool for protein classification. To this end, we propose a molecular topological fingerprint based support vector machine (MTF-SVM) classifier. Specifically,we construct machine learning feature vectors solely fromprotein topological fingerprints,which are topological invariants generated during the filtration process. To validate the presentMTF-SVMapproach, we consider four types of problems. First, we study protein-drug binding by using the M2 channel protein of influenza A virus. We achieve 96% accuracy in discriminating drug bound and unbound M2 channels. Secondly, we examine the use of MTF-SVM for the classification of hemoglobin molecules in their relaxed and taut forms and obtain about 80% accuracy. Thirdly, the identification of all alpha, all beta, and alpha-beta protein domains is carried out using 900 proteins.We have found a 85% success in this identification. Finally, we apply the present technique to 55 classification tasks of protein superfamilies over 1357 samples and 246 tasks over 11944 samples. Average accuracies of 82% and 73% are attained. The present study establishes computational topology as an independent and effective alternative for protein classification.
Opis fizyczny
  • Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
  • Oak Ridge National Laboratory, One Bethel Valley Road, P.O. Box 2008, MS 6211, Oak Ridge, TN 37831, USA
  • Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
  • Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824, USA
  • Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
  • Mathematical Biosciences Institute, The Ohio State University, Columbus, Ohio 43210,
  • [1] P. K. Agarwal, H. Edelsbrunner, J. Harer, and Y. Wang. Extreme elevation on a 2-manifold. Discrete and Computational Geometry (DCG), 36(4):553–572, 2006. [Crossref]
  • [2] S. F. Altschul. A protein alignment scoring system sensitive at all evolutionary distances. Journal of molecular evolution, 36(3):290–300, 1993.
  • [3] I. Bahar, A. R. Atilgan, and B. Erman. Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential. Folding and Design, 2:173 – 181, 1997.
  • [4] P.W. Bates, Z. Chen, Y. H. Sun, G.W. Wei, and S. Zhao. Geometric and potential driving formation and evolution of biomolecular surfaces. J. Math. Biol., 59:193–231, 2009. [Crossref]
  • [5] P.W. Bates, G.W. Wei, and S. Zhao. Minimalmolecular surfaces and their applications. Journal of Computational Chemistry, 29(3):380–91, 2008. [Crossref]
  • [6] U. Bauer, M. Kerber, and J. Reininghaus. Distributed computation of persistent homology. Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), 2014.
  • [7] P. Bendich, H. Edelsbrunner, and M. Kerber. Computing robustness and persistence for images. IEEE Transactions on Visualization and Computer Graphics, 16:1251–1260, 2010. [Crossref]
  • [8] P. Bendich and J. Harer. Persistent intersection homology. Foundations of ComputationalMathematics (FOCM), 11(3):305– 336, 2011.
  • [9] J. Bennett, F. Vivodtzev, and V. Pascucci, editors. Topological and statistical methods for complex data: Tackling largescale, high-dimensional and multivariate data spaces. Mathematics and Visualization. Springer-Verlag Berlin Heidelberg, 2015.
  • [10] S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, and M. Spagnuolo. Describing shapes by geometrical-topological properties of real functions. ACM Computing Surveys, 40(4):12, 2008.
  • [11] P. T. Bremer, V. P. I. Hotz, and R. Peikert, editors. Topological methods in data analysis and visualization III: Theory, algorithms and applications. Mathematics and Visualization. Springer International Publishing, 2014.
  • [12] B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. States, S. Swaminathan, and M. Karplus. Charmm: A program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem., 4:187–217, 1983. [Crossref]
  • [13] C. J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2:121–167, 1998.
  • [14] G. Carlsson. Topology and data. Am. Math. Soc, 46(2):255–308, 2009. [Crossref]
  • [15] G. Carlsson and V. De Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367–405, 2010. [Crossref]
  • [16] G. Carlsson, V. de Silva, and D. Morozov. Zigzag persistent homology and real-valued functions. In Proc. 25th Annu. ACM Sympos. Comput. Geom., pages 247–256, 2009.
  • [17] G. Carlsson, T. Ishkhanov, V. Silva, and A. Zomorodian. On the local behavior of spaces of natural images. International Journal of Computer Vision, 76(1):1–12, 2008. [Crossref]
  • [18] G. Carlsson, G. Singh, and A. Zomorodian. Computing multidimensional persistence. In Algorithms and computation, pages 730–739. Springer, 2009.
  • [19] G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete Computational Geometry, 42(1):71– 93, 2009. [Crossref]
  • [20] G. Carlsson, A. Zomorodian, A. Collins, and L. J. Guibas. Persistence barcodes for shapes. International Journal of Shape Modeling, 11(2):149–187, 2005.
  • [21] C.-C. Chang and C.-J. Lin. LIBSVM: A library for support vector machines. ACM Transactions on Intelligent Systems and Technology, 2:27:1–27:27, 2011. Software available at
  • [22] H. W. Chang, S. Bacallado, V. S. Pande, and G. E. Carlsson. Persistent topology and metastable state in conformational dynamics. PLos ONE, 8(4):e58699, 2013.
  • [23] F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Oudot. Proximity of persistence modules and their diagrams. In Proc. 25th ACM Sympos. on Comput. Geom., pages 237–246, 2009.
  • [24] F. Chazal, L. J. Guibas, S. Y. Oudot, and P. Skraba. Persistence-based clustering in riemannian manifolds. In Proceedings of the 27th annual ACM symposium on Computational geometry, SoCG ’11, pages 97–106, 2011.
  • [25] D. Chen, Z. Chen, C. Chen, W. H. Geng, and G. W. Wei. MIBPB: A software package for electrostatic analysis. J. Comput. Chem., 32:657 – 670, 2011.
  • [26] D. Chen, Z. Chen, and G. W. Wei. Quantum dynamics in continuum for proton transport II: Variational solvent-solute interface. International Journal for Numerical Methods in Biomedical Engineering, 28:25 – 51, 2012.
  • [27] D. Chen and G. W. Wei. Quantum dynamics in continuum for proton transport-Generalized correlation. J Chem. Phys., 136:134109, 2012. [Crossref]
  • [28] Z. Chen, N. A. Baker, and G.W. Wei. Differential geometry based solvation models I: Eulerian formulation. J. Comput. Phys., 229:8231–8258, 2010. [Crossref]
  • [29] Z. Chen, N. A. Baker, and G. W. Wei. Differential geometry based solvation models II: Lagrangian formulation. J. Math. Biol., 63:1139– 1200, 2011. [Crossref]
  • [30] Z. Chen, S. Zhao, J. Chun, D. G. Thomas, N. A. Baker, P. B. Bates, and G.W. Wei. Variational approach for nonpolar solvation analysis. Journal of Chemical Physics, 137(084101), 2012.
  • [31] J. L. Cheng, M. J. Sweredoski, and P. Baldi. DOMpro: Protein domain prediction using profiles, secondary structure, relative solvent accessibility, and recursive neural networks. Data Mining and Knowledge Discovery, 13:1–10, 2006.
  • [32] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103–120, 2007. [Crossref]
  • [33] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Extending persistence using poincaré and lefschetz duality. Foundations of Computational Mathematics, 9(1):79–103, 2009. [Crossref]
  • [34] D. Cohen-Steiner, H. Edelsbrunner, J. Harer, and D. Morozov. Persistent homology for kernels, images, and cokernels. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 09, pages 1011–1020, 2009.
  • [35] C. Cortes and V. Vapnik. Support-vector networks. Machine learning, 20(3):273–297, 1995. [Crossref]
  • [36] Y. Dabaghian, F. Memoli, L. Frank, and G. Carlsson. A topological paradigm for hippocampal spatial map formation using persistent homology. PLoS Comput Biol, 8(8):e1002581, 08 2012. [Crossref]
  • [37] S. J. Darnell, L. LeGault, and J. C. Mitchell. Kfc server: interactive forecasting of protein interaction hot spots. NUCLEIC ACIDS RESEARCH, 36:W265–W269, 2008. [Crossref]
  • [38] M. Dash and H. Liu. Feature selection for classification. Intelligent data analysis, 1(1):131–156, 1997.
  • [39] V. de Silva, D. Morozov, and M. Vejdemo-Johansson. Persistent cohomology and circular coordinates. Discrete and Comput. Geom., 45:737–759, 2011.
  • [40] T. K. Dey, F. Fan, and Y.Wang. Computing topological persistence for simplicialmaps. In Proc. 30th Annu. Sympos. Comput. Geom. (SoCG), pages 345–354, 2014.
  • [41] B. Di Fabio and C. Landi. A mayer-vietoris formula for persistent homology with an application to shape recognition in the presence of occlusions. Foundations of Computational Mathematics, 11:499–527, 2011.
  • [42] H. Edelsbrunner and J. Harer. Computational topology: an introduction. American Mathematical Soc., 2010.
  • [43] H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511–533, 2002.
  • [44] R.-E. Fan, P.-H. Chen, and C.-J. Lin. Working set selection using second order information for training support vector machines. The Journal of Machine Learning Research, 6:1889–1918, 2005.
  • [45] T. Fawcett. An introduction to roc analysis. Pattern recognition letters, 27(8):861–874, 2006. [Crossref]
  • [46] X. Feng, K. Xia, Y. Tong, and G.-W. Wei. Geometric modeling of subcellular structures, organelles and large multiprotein complexes. International Journal for Numerical Methods in Biomedical Engineering, 28:1198–1223, 2012.
  • [47] X. Feng, K. L. Xia, Y. Y. Tong, and G.W. Wei. Multiscale geometric modeling ofmacromolecules II: lagrangian representation. Journal of Computational Chemistry, 34:2100–2120, 2013. [Crossref]
  • [48] C. Fernandez-Lozano, E. Fernandez-Blanco, K. Dave, N. Pedreira, M. Gestal, J. Dorado, and C. R. Munteanu. Improving enzyme regulatory protein classification by means of svm-rfe feature selection. Molecular Biosystems, 10:1063–1071, 2014. [Crossref]
  • [49] P. J. Flory. Statistical thermodynamics of random networks. Proc. Roy. Soc. Lond. A,, 351:351 – 378, 1976.
  • [50] N. K. Fox, S. E. Brenner, and J.-M. Chandonia. Scope: Structural classification of proteins-extended, integrating scop and astral data and classification of new structures. Nucleic acids research, 42(D1):D304–D309, 2014. [Crossref]
  • [51] P. Frosini. A distance for similarity classes of submanifolds of a Euclidean space. BUllentin of Australian Mathematical Society, 42(3):407–416, 1990.
  • [52] P. Frosini and C. Landi. Size theory as a topological tool for computer vision. Pattern Recognition and Image Analysis, 9(4):596–603, 1999.
  • [53] P. Frosini and C. Landi. Persistent betti numbers for a noise tolerant shape-based approach to image retrieval. Pattern Recognition Letters, 34:863–872, 2013. [Crossref]
  • [54] I. Fujishiro, Y. Takeshima, T. Azuma, and S. Takahashi. Volume data mining using 3d field topology analysis. IEEE Computer Graphics and Applications, 20(5):46–51, 2000.
  • [55] M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow, and V. Nanda. Topological measurement of protein compressibility via persistence diagrams. Japan Journal of Industrial and Applied Mathematics, 32:1–17, 2014.
  • [56] R. Ghrist. Barcodes: The persistent topology of data. Bull. Amer. Math. Soc., 45:61–75, 2008.
  • [57] N. Go, T. Noguti, and T. Nishikawa. Dynamics of a small globular protein in terms of low-frequency vibrational modes. Proc. Natl. Acad. Sci., 80:3696 – 3700, 1983. [Crossref]
  • [58] S. Henikoff and J. G. Henikoff. Amino acid substitutionmatrices from protein blocks. Proceedings of the National Academy of Sciences, 89(22):10915–10919, 1992.
  • [59] G. E. Hinton, S. Osindero, and Y.-W. Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18:1527–1554, 2006. [Crossref]
  • [60] D. Horak, S. Maletic, and M. Rajkovic. Persistent homology of complex networks. Journal of Statistical Mechanics: Theory and Experiment, 2009(03):P03034, 2009. [Crossref]
  • [61] D. J. Jacobs, A. J. Rader, L. A. Kuhn, and M. F. Thorpe. Protein flexibility predictions using graph theory. Proteins-Structure, Function, and Genetics, 44(2):150–165, AUG 1 2001.
  • [62] S. Jo, M. Vargyas, J. Vasko-Szedlar, B. Roux, and W. Im. Pbeq-solver for online visualization of electrostatic potential of biomolecules. Nucleic Acids Research, 36:W270 –W275, 2008. [Crossref]
  • [63] M. Johnson, I. Zaretskaya, Y. Raytselis, Y. Merezhuk, S. McGinnis, and T. L. Madden. Ncbi blast: a better web interface. Nucleic acids research, 36(suppl 2):W5–W9, 2008. [Crossref]
  • [64] T. Kaczynski, K. Mischaikow, and M. Mrozek. Computational Homology, volume 157 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
  • [65] P. M. Kasson, A. Zomorodian, S. Park, N. Singhal, L. J. Guibas, and V. S. Pande. Persistent voids a new structural metric for membrane fusion. Bioinformatics, 23:1753–1759, 2007. [Crossref]
  • [66] B. Krishnamoorthy, S. Provan, and A. Tropsha. A topological characterization of protein structure. In Data Mining in Biomedicine, Springer Optimization and Its Applications, pages 431–455, 2007.
  • [67] R. A. Laskowski, J. D.Watson, and J. M. Thornton. Profunc: a server for predicting protein function from 3d structure. Nucleic acids research, 33(suppl 2):W89–W93, 2005. [Crossref]
  • [68] D. Lee, O. Redfern, and C. Orengo. Predicting protein function from sequence and structure. Nature ReviewsMolecular Cell Biology, 8(12):995–1005, 2007. [Crossref]
  • [69] H. Lee, H. Kang, M. K. Chung, B. Kim, and D. S. Lee. Persistent brain network homology from the perspective of dendrogram. Medical Imaging, IEEE Transactions on, 31(12):2267–2277, Dec 2012.
  • [70] C. S. Leslie, E. Eskin, A. Cohen, J. Weston, andW. S. Noble. Mismatch string kernels for discriminative protein classification. Bioinformtics, 20:467–476, 2004. [Crossref]
  • [71] M. Levitt, C. Sander, and P. S. Stern. Protein normal-mode dynamics: Trypsin inhibitor, crambin, ribonuclease and lysozyme. J. Mol. Biol., 181(3):423 – 447, 1985. [Crossref]
  • [72] W. Li, A. Cowley, M. Uludag, T. Gur, H. McWilliam, S. Squizzato, Y. M. Park, N. Buso, and R. Lopez. The embl-ebi bioinformatics web and programmatic tools framework. Nucleic acids research, page gkv279, 2015.
  • [73] X. Liu, Z. Xie, and D. Yi. A fast algorithm for constructing topological structure in large data. Homology, Homotopy and Applications, 14:221–238, 2012.
  • [74] J. A. McCammon, B. R. Gelin, and M. Karplus. Dynamics of folded proteins. Nature, 267:585–590, 1977. [Crossref]
  • [75] W. McCulloch and W. Pitts. A logical calculus of ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5:115–133, 1943. [Crossref]
  • [76] P. Meinicke. UProC: tools for ultra-fast protein domain classification. Bioinformtics, 31:1382–1388, 2015. [Crossref]
  • [77] K. Mischaikow, M. Mrozek, J. Reiss, and A. Szymczak. Construction of symbolic dynamics from experimental time series. Physical Review Letters, 82:1144–1147, 1999. [Crossref]
  • [78] K. Mischaikow and V. Nanda. Morse theory for filtrations and eflcient computation of persistent homology. Discrete and Computational Geometry, 50(2):330–353, 2013. [Crossref]
  • [79] V. Nanda. Perseus: the persistent homology software. Software available at perseus.
  • [80] P. Niyogi, S. Smale, and S. Weinberger. A topological view of unsupervised learning from noisy data. SIAM Journal on Computing, 40:646–663, 2011. [Crossref]
  • [81] K. Opron, K. L. Xia, and G. W. Wei. Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis. Journal of Chemical Physics, 140:234105, 2014. [Crossref]
  • [82] K. Opron, K. L. Xia, and G. W. Wei. Communication: Capturing protein multiscale thermal fluctuations. Journal of Chemical Physics, 142(211101), 2015.
  • [83] S. Y. Oudot and D. R. Sheehy. Zigzag Zoology: Rips Zigzags for Homology Inference. In Proc. 29th Annual Symposium on Computational Geometry, pages 387–396, June 2013.
  • [84] D. Pachauri, C. Hinrichs, M. Chung, S. Johnson, and V. Singh. Topology-based kernels with application to inference problems in alzheimer’s disease. Medical Imaging, IEEE Transactions on, 30(10):1760–1770, Oct 2011.
  • [85] J. A. Perea, A. Deckard, S. B. Haase, and J. Harer. Sw1pers: Sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data. BMC Bioinformatics, 16:257, 2015.
  • [86] J. A. Perea and J. Harer. Sliding windows and persistence: An application of topological methods to signal analysis. Foundations of Computational Mathematics, 15:799–838, 2015. [Crossref]
  • [87] R. M. Pielak and J. J. Chou. Influenza m2 proton channels. Biochimica et Biophysica Acta (BBA)-Biomembranes, 1808(2):522–529, 2011.
  • [88] R. M. Pielak, K. Oxenoid, and J. J. Chou. Structural investigation of rimantadine inhibition of the am2-bm2 chimera channel of influenza viruses. Structure, 19(11):1655–1663, 2011.
  • [89] B. Rieck, H. Mara, and H. Leitte. Multivariate data analysis using persistence-based filtering and topological signatures. IEEE Transactions on Visualization and Computer Graphics, 18:2382–2391, 2012. [Crossref]
  • [90] V. Robins. Towards computing homology from finite approximations. In Topology Proceedings, volume 24, pages 503–532, 1999.
  • [91] A. Roy, A. Kucukural, and Y. Zhang. I-tasser: a unified platform for automated protein structure and function prediction. Nature protocols, 5(4):725–738, 2010. [Crossref]
  • [92] V. D. Silva and R. Ghrist. Blind swarms for coverage in 2-d. In In Proceedings of Robotics: Science and Systems, page 01, 2005.
  • [93] G. Singh, F. Memoli, T. Ishkhanov, G. Sapiro, G. Carlsson, and D. L. Ringach. Topological analysis of population activity in visual cortex. Journal of Vision, 8(8), 2008.
  • [94] P. Sonego, M. Pacurar, S. Dhir, A. Kertész-Farkas, A. Kocsor, Z. Gáspári, J. A. Leunissen, and S. Pongor. A protein classification benchmark collection for machine learning. Nucleic Acids Research, 35(suppl 1):D232–D236, 2007. [Crossref]
  • [95] G. D. Stormo, T. D. Schneider, L. Gold, and A. Ehrenfeucht. Use of the ‘perceptron’ algorithm to distinguish translational initiation sites in e. coli. Nucleic Acids Research, 10:2997–3011, 1982. [Crossref]
  • [96] M. Tasumi, H. Takenchi, S. Ataka, A. M. Dwidedi, and S. Krimm. Normal vibrations of proteins: Glucagon. Biopolymers, 21:711 – 714, 1982. [Crossref]
  • [97] A. Tausz, M. Vejdemo-Johansson, and H. Adams. Javaplex: A research software package for persistent (co)homology. Software available at, 2011.
  • [98] A. Tausz, M. Vejdemo-Johansson, and H. Adams. JavaPlex: A research software package for persistent (co)homology. In H. Hong and C. Yap, editors, Proceedings of ICMS 2014, Lecture Notes in Computer Science 8592, pages 129–136, 2014.
  • [99] M. M. Tirion. Largeamplitude elastic motions in proteins from a single-parameter, atomic analysis. Phys. Rev. Lett., 77:1905 – 1908, 1996. [Crossref]
  • [100] B. Wang, B. Summa, V. Pascucci, and M. Vejdemo-Johansson. Branching and circular features in high dimensional data. IEEE Transactions on Visualization and Computer Graphics, 17:1902–1911, 2011. [Crossref]
  • [101] B. Wang and G. W. Wei. Objective-oriented Persistent Homology. ArXiv e-prints, Dec. 2014.
  • [102] G. W. Wei. Differential geometry based multiscale models. Bulletin of Mathematical Biology, 72:1562 – 1622, 2010. [Crossref]
  • [103] G.-W. Wei. Multiscale, multiphysics and multidomain models I: Basic theory. Journal of Theoretical and Computational Chemistry, 12(8):1341006, 2013.
  • [104] G.-W. Wei, Q. Zheng, Z. Chen, and K. Xia. Variational multiscale models for charge transport. SIAM Review, 54(4):699 – 754, 2012. [Crossref]
  • [105] W. Wu, A. Srivastava, J. Laborde, and J. F. Zhang. An eflcient multiple protein structure comparison method and its application to structure clustering and outlier detection. IEEE, BIBM, pages 69–73, 2013.
  • [106] K. L. Xia, X. Feng, Y. Y. Tong, and G.W. Wei. Persistent homology for the quantitative prediction of fullerene stability. Journal of Computational Chemsitry, 36:408–422, 2015.
  • [107] K. L. Xia, K. Opron, and G. W. Wei. Multiscale multiphysics and multidomain models - Flexibility and rigidity. Journal of Chemical Physics, 139:194109, 2013. [Crossref]
  • [108] K. L. Xia and G. W. Wei. Persistent homology analysis of protein structure, flexibility and folding. International Journal for Numerical Methods in Biomedical Engineerings, 30:814–844, 2014.
  • [109] K. L. Xia and G. W. Wei. Multidimensional persistence in biomolecular data. Journal Computational Chemistry, 36:1502– 1520, 2015.
  • [110] K. L. Xia and G. W. Wei. Persistent topology for cryo-EM data analysis. International Journal for Numerical Methods in Biomedical Engineering, 31:e02719, 2015.
  • [111] K. L. Xia, Z. X. Zhao, and G. W. Wei. Multiresolution topological simplification. Journal Computational Biology, 22:1–5, 2015.
  • [112] K. L. Xia, Z. X. Zhao, and G. W. Wei. Multiresolution persistent homology for excessively large biomolecular datasets. Journal of Chemical Physics, in press, 2015.
  • [113] Y. Yao, J. Sun, X. H. Huang, G. R. Bowman, G. Singh, M. Lesnick, L. J. Guibas, V. S. Pande, and G. Carlsson. Topological methods for exploring low-density states in biomolecular folding pathways. The Journal of Chemical Physics, 130:144115, 2009. [Crossref]
  • [114] Q. Zheng, S. Y. Yang, and G.W. Wei. Molecular surface generation using PDE transform. International Journal for Numerical Methods in Biomedical Engineering, 28:291–316, 2012.
  • [115] Y. C. Zhou, M. Feig, and G. W. Wei. Highly accurate biomolecular electrostatics in continuum dielectric environments. Journal of Computational Chemistry, 29:87–97, 2008. [Crossref]
  • [116] A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33:249–274, 2005.
  • [117] Jie Liang. Geometry of protein shape and its evolutionary pattern for function prediction and characterization. Engineering in Medicine and Biology Society, 2009. EMBC 2009. Annual International Conference of the IEEE, 2324–2327,2009.
  • [118] Liang, Jie and Kachalo, Sema and Li, Xiang and Ouyang, Zheng and Tseng, Yan-Yuan and Zhang, Jinfeng. Geometric structures of proteins for understanding folding, discriminating natives and predicting biochemical functions. The World is a Jigsaw, 2009.
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