Collective Estimation of Multiple Bivariate Density Functions With Application to Angular-Sampling-Based Protein Loop Modeling

REFERENCES

• Akaike, H. (1973), “Information Theory and an Extension of the Maximum Likelihood Principle,” Proceedings of the 2nd International Symposium on Information Theory, Budapest, 267–281.
   Google Scholar
• Altis, A., Nguyen, P.H., Hegger, R., and Stock, G. (2007), “Dihedral Angle Principal Component Analysis of Molecular Dynamics Simulations,” Journal of Chemical Physics, 126, 244111.
   PubMed Web of Science ®Google Scholar
• Altis, A., Otten, M., Nguyen, P.H., Hegger, R., and Stock, G. (2008), “Construction of the Free Energy Landscape of Biomolecules via Dihedral Angle Principal Component Analysis,” Journal of Chemical Physics, 128, 245102.
   PubMed Web of Science ®Google Scholar
• Archie, J., and Karplus, K. (2009), “Applying Undertaker Cost Functions to Model Quality Assessment,” Proteins, 75, 550–555.
   PubMed Web of Science ®Google Scholar
• Benkert, P., Tosatto, S. C.E., and Schomburg, D. (2008), “QMEAN: A Comprehensive Scoring Function for Model Quality Assessment,” Proteins, 71, 261–277.
   PubMed Web of Science ®Google Scholar
• Berkholz, D.S., Shapovalov, M.V., Jr, R.L., and Karplus, P.A. (2009), “Conformation Dependence of Backbone Geometry in Proteins,” Structure, 17, 1316–1325.
   PubMed Web of Science ®Google Scholar
• Bhuyan, M. S.I., and Gao, X. (2011), “A Protein-Dependent Side-Chain Rotamer Library,” BMC Bioinformatics, 12, S10.
   PubMed Web of Science ®Google Scholar
• Boomsma, W., Mardia, K.V., Taylor, C.C., Ferkinghoff-Borg, J., Krogh, A., and Hamelryck, T. (2008), “A Generative, Probabilistic Model of Local Protein Structure,” Proceedings of the National Academy of Science of the United States of America, 105, 8932–8937.
   PubMed Web of Science ®Google Scholar
• Buck, M., Bouguet-Bonnet, S., Pastor, R.W., and Jr, A.D. (2006), “Importance of the CMAP Correction to the CHARMM22 Protein Force Field: Dynamics of Hen Lysozyme,” Biophysical Journal, 90, L36–L38.
   PubMed Web of Science ®Google Scholar
• Bystroff, C., Thorsson, V., and Baker, D. (2000), “HMMSTR: A Hidden Markov Model for Local Sequence-Structure Correlations in Proteins,” Journal of Molecular Biology, 301, 173–190.
   PubMed Web of Science ®Google Scholar
• Chacón, J., and Duong, T. (2010), “Multivariate Plug-in Bandwidth Selection With Unconstrained Pilot Bandwidth Matrices,” TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 19, 375–398.
   Google Scholar
• ——— (2011), “Unconstrained Pilot Selectors for Smoothed Cross-Validation,” Australian & New Zealand Journal of Statistics, 53, 331–351.
   Web of Science ®Google Scholar
• Challis, C.J., and Schmidler, S.C. (2012), “A Stochastic Evolutionary Model for Protein Structure Alignment and Phylogeny,” Molecular Biology and Evolution, 29, 3575–3587.
   PubMed Web of Science ®Google Scholar
• Cozzetto, D., Kryshtafovych, A., and Tramontano, A. (2009), “Evaluation of CASP8 Model Quality Predictions,” Proteins, 77, 157–166.
   PubMed Web of Science ®Google Scholar
• Dahl, D.B., Bohannan, Z., Mo, Q., Vannucci, M., and Tsai, J.W. (2008), “Assessing Side-chain Perturbations of the Protein Backbone: A Knowledge Based Classification of Residue Ramachandran Space,” Journal of Molecular Biology, 378, 749–758.
   PubMed Web of Science ®Google Scholar
• DasGupta, A. (2011), Probability for Statistics and Machine Learning: Fundamentals and Advanced Topics (Springer Texts in Statistics), New York: Springer.
   Google Scholar
• Davis, I.W., Murray, L.W., Richardson, J.S., and Richardson, D.C. (2004), “MOLPROBITY: Structure Validation and All-Atom Contact Analysis for Nucleic Acids and Their Complexes,” Nucleic Acids Research, 32, W615–W619.
   PubMed Web of Science ®Google Scholar
• Duong, T. (2007), “ks: Kernel Density Estimation and Kernel Discriminant Analysis for Multivariate Data in R,” Journal of Statistical Software, 21, 1–16.
   Web of Science ®Google Scholar
• Fetrow, J.S. (1995), “Omega Loops: Nonregular Secondary Structures Significant in Protein Function and Stability,” The FASEB Journal, 9, 708–717.
   PubMed Web of Science ®Google Scholar
• Gao, X., Xu, J., Li, S.C., and Li, M. (2009), “Predicting Local Quality of a Sequence-Structure Alignment,” Journal of Bioinformatics and Computational Biology, 7, 789–810.
   PubMedGoogle Scholar
• Green, P., and Silverman, B. (1994), Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach, New York: Chapman & Hall/CRC.
   Google Scholar
• Gu, C. (1993), “Smoothing Spline Density Estimation: A Dimensionless Automatic Algorithm,” Journal of the American Statistical Association, 88, 495–504.
   Web of Science ®Google Scholar
• ——— (2002), Smoothing Spline ANOVA Models (Springer Series in Statistics), New York: Springer.
   Google Scholar
• Hamelryck, T., Kent, J.T., and Krogh, A. (2006), “Sampling Realistic Protein Conformations Using Local Structural Bias,” PLoS Computational Biology, 2, e131.
   PubMed Web of Science ®Google Scholar
• Hansen, M., Kooperberg, C., and Sardy, S. (1998), “Triogram Models,” Journal of the American Statistical Association, 93, 101–119.
   Web of Science ®Google Scholar
• Hooft, R.W., Sander, C., and Vriend, G. (1997), “Objectively Judging the Quality of a Protein Structure From a Ramachandran Plot,” Computer Applications in the Biosciences: CABIOS, 13, 425–430.
   PubMedGoogle Scholar
• Hovmöller, S., Zhou, T., and Ohlson, T. (2002), “Conformations of Amino Acids in Proteins,” Acta Crystallographica D Biological Crystallography, 58, 768–776.
   PubMed Web of Science ®Google Scholar
• Jacobson, M.P., Pincus, D.L., Rapp, C.S., Day, T.J., Honig, B., Shaw, D.E., and Friesner, R.A. (2004), “A Hierarchical Approach to All-Atom Protein Loop Prediction,” Proteins: Structure, Function, and Bioinformatics, 55, 351–367.
   PubMed Web of Science ®Google Scholar
• Jammalamadaka, S., and SenGupta, A. (2001), Topics in Circular Statistics (Series on Multivariate Analysis), Singapore: World Scientific.
   Google Scholar
• Jha, A., Colubri, A., Zaman, M., Koide, S., Sosnick, T., and Freed, K. (2005), “Helix, Sheet, and Polyproline II Frequencies and Strong Nearest Neighbor Effects in a Restricted Coil Library,” Biochemistry, 44, 9691–9702.
   PubMed Web of Science ®Google Scholar
• Jolliffe, I. (2002), Principal Component Analysis (Springer Series in Statistics), New York: Springer.
   Google Scholar
• Jones, M.C., Marron, J.S., and Park, B.U. (1991), “A Simple Root n Bandwidth Selector,” The Annals of Statistics, 19, 1919–1932.
   Web of Science ®Google Scholar
• Källberg, M., Margaryan, G., Wang, S., Ma, J., and Xu, J. (2014), “RaptorX Server: A Resource for Template-Based Protein Structure Modeling,” Methods in Molecular Biology, 1137, 17–27.
   PubMedGoogle Scholar
• Kendrew, J.C., Dickerson, R.E., Strandberg, B.E., Hart, R.G., Davies, D.R., Phillips, D.C., and Shore, V.C. (1960), “Structure of Myoglobin: A Three-Dimensional Fourier Synthesis at 2 A. Resolution,” Nature, 185, 422–427.
   PubMed Web of Science ®Google Scholar
• Keskin, O., Yuret, D., Gursoy, A., Turkay, M., and Erman, B. (2004), “Relationships Between Amino Acid Sequence and Backbone Torsion Angle Preferences,” Proteins, 55, 992–998.
   PubMed Web of Science ®Google Scholar
• Kleywegt, G.J., Harris, M.R., Zou, J.-Y., Taylor, T.C., Wälby, A., and Jones, T.A. (2004), “The Uppsala Electron-Density Server,” Acta Crystallographica Section D, 60, 2240–2249.
   PubMed Web of Science ®Google Scholar
• Kryshtafovych, A., Barbato, A., Fidelis, K., Monastyrskyy, B., Schwede, T., and Tramontano, A. (2014), “Assessment of the Assessment: Evaluation of the Model Quality Estimates in CASP10,” Proteins, 82, 112–126.
   PubMed Web of Science ®Google Scholar
• Kryshtafovych, A., Fidelis, K., and Tramontano, A. (2011), “Evaluation of Model Quality Predictions in CASP9,” Proteins, 79, 91–106.
   PubMed Web of Science ®Google Scholar
• Lai, M., and Schumaker, L. (2007), Spline Functions on Triangulations (no. v. 13 in Encyclopedia of Mathematics and Its Applications), New York: Cambridge University Press.
   Google Scholar
• Laskowski, R., MacArthur, M., Moss, D., and Thornton, J. (1993), “PROCHECK: A Program to Check the Stereochemical Quality of Protein Structures,” Journal of Applied Crystallography, 26, 283–291.
   Web of Science ®Google Scholar
• Lennox, K.P., Dahl, D.B., Vannucci, M., Day, R., and Tsai, J.W. (2010), “A Dirichlet Process Mixture of Hidden Markov Models for Protein Structure Prediction,” Annals of Applied Statistics, 4, 916–942.
   PubMed Web of Science ®Google Scholar
• Lennox, K.P., Dahl, D.B., Vannucci, M., and Tsai, J.W. (2009), “Density Estimation for Protein Conformation Angles Using a Bivariate von Mises Distribution and Bayesian Nonparametrics,” Journal of the American Statistical Association, 104, 586–596.
   PubMed Web of Science ®Google Scholar
• Maadooliat, M., Gao, X., and Huang, J.Z. (2013a), “Assessing Protein Conformational Sampling Methods Based on Bivariate Lag-Distributions of Backbone Angles,” Briefings in Bioinformatics, 14, 724–736.
   PubMed Web of Science ®Google Scholar
• ——— (2013b), “Assessing Protein Conformational Sampling Methods Based on Bivariate Lag-Distributions of Backbone Angles,” Briefings in Bioinformatics, 14, 724–736.
   PubMed Web of Science ®Google Scholar
• Mandell, D.J., Coutsias, E.A., and Kortemme, T. (2009), “Sub-Angstrom Accuracy in Protein Loop Reconstruction by Robotics-Inspired Conformational Sampling,” Nature Methods, 6, 551–552.
   PubMed Web of Science ®Google Scholar
• Mardia, K.V. (1975), “Statistics of Directional Data,” Journal of the Royal Statistical Society, 37, 349–393.
   Google Scholar
• Mardia, K.V., Taylor, C.C., and Subramaniam, G.K. (2007), “Protein Bioinformatics and Mixtures of Bivariate Von Mises Distributions for Angular Data,” Biometrics, 63, 505–512.
   PubMed Web of Science ®Google Scholar
• Miao, X., Waddell, P.J., and Valafar, H. (2008), “TALI: Local Alignment of Protein Structures Using Backbone Torsion Angles,” Journal of Bioinformatics and Computational Biology, 6, 163–181.
   PubMedGoogle Scholar
• Mu, Y., Nguyen, P.H., and Stock, G. (2005), “Energy Landscape of a Small Peptide Revealed by Dihedral Angle Principal Component Analysis,” Proteins, 58, 45–52.
   PubMed Web of Science ®Google Scholar
• Oldfield, T.J., and Hubbard, R.E. (1994), “Analysis of Cα Geometry in Protein Structures,” Proteins, 18, 324–337.
   PubMed Web of Science ®Google Scholar
• O’Sullivan, F. (1988), “Fast Computation of Fully Automated Log-Density and Log-Hazard Estimators,” SIAM Journal on Scientific and Statistical Computing, 9, 363–379.
   Web of Science ®Google Scholar
• Pertsemlidis, A., Zelinka, J., Fondon, J.W., Henderson, R.K., and Otwinowski, Z. (2005), “Bayesian Statistical Studies of the Ramachandran Distribution,” Statistical Applications in Genetics and Molecular Biology, 4, 1–18.
   Web of Science ®Google Scholar
• Qiu, J., Sheffler, W., Baker, D., and Noble, W.S. (2008), “Ranking Predicted Protein Structures With Support Vector Regression,” Proteins, 71, 1175–1182.
   PubMed Web of Science ®Google Scholar
• Ramachandran, G.N., Ramakrishnan, C., and Sasisekharan, V. (1963), “Stereochemistry of Polypeptide Chain Configurations,” Journal of Molecular Biology, 7, 95–99.
   PubMed Web of Science ®Google Scholar
• Ramachandran, G., and Sasisekharan, V. (1968), “Conformations of Polypeptides and Proteins,” Advances in Protein Chemistry, 23, 283–438.
   PubMedGoogle Scholar
• Riccardi, L., Nguyen, P.H., and Stock, G. (2009), “Free-Energy Landscape of RNA Hairpins Constructed via Dihedral Angle Principal Component Analysis,” The Journal of Physical Chemistry B, 113, 16660–16668.
   PubMed Web of Science ®Google Scholar
• Rivest, L.P. (1988), “A Distribution for Dependent Unit Vectors,” Communications in Statistics: Theory and Methods, 17, 461–483.
   Web of Science ®Google Scholar
• Rohl, C.A., Strauss, C. E.M., Misura, K. M.S., and Baker, D. (2004), “Protein Structure Prediction Using Rosetta,” Methods in Enzymology, 383, 66–93.
   PubMed Web of Science ®Google Scholar
• Sain, S.R., Baggerly, K.A., and Scott, D.W. (1994), “Cross-Validation of Multivariate Densities,” Journal of the American Statistical Association, 89, 807–817.
   Web of Science ®Google Scholar
• Scott, D. (1992), Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley Series in Probability and Statistics), New York: Wiley.
   Google Scholar
• Sellers, B.D., Zhu, K., Zhao, S., Friesner, R.A., and Jacobson, M.P. (2008), “Toward Better Refinement of Comparative Models: Predicting Loops in Inexact Environments,” Proteins: Structure, Function, and Bioinformatics, 72, 959–971.
   PubMed Web of Science ®Google Scholar
• Shapovalov, M.V., and Jr, R.L. (2011), “A Smoothed Backbone-Dependent Rotamer Library for Proteins Derived From Adaptive Kernel Density Estimates and Regressions,” Structure, 19, 844–858.
   PubMed Web of Science ®Google Scholar
• Silverman, B. (1986), Density Estimation for Statistics and Data Analysis (Vol. 111), New York: Chapman & Hall/CRC.
   Google Scholar
• Simons, K.T., Bonneau, R., Ruczinski, I., and Baker, D. (1999), “Ab Initio Protein Structure Prediction of CASP III Targets Using Rosetta,” Proteins, 37, 171–176.
   Google Scholar
• Singh, H., Hnizdo, V., and Demchuk, E. (2002), “Probabilistic Model for Two Dependent Circular Variables,” Biometrika, 89, 719–723.
   Web of Science ®Google Scholar
• Stein, A., and Kortemme, T. (2013), “Improvements to Robotics-Inspired Conformational Sampling in Rosetta,” PloS One, 8, e63090.
   PubMed Web of Science ®Google Scholar
• Stone, C. (1990), “Large-Sample Inference for Log-Spline Models,” The Annals of Statistics, 18, 717–741.
   Web of Science ®Google Scholar
• Ting, D., Wang, G., Shapovalov, M., Mitra, R., Jordan, M.I., and Jr, R.L. (2010), “Neighbor-Dependent Ramachandran Probability Distributions of Amino Acids Developed From a Hierarchical Dirichlet Process Model,” PLOS Computational Biology, 6, e1000763.
   PubMed Web of Science ®Google Scholar
• Tuffery, P., and Derreumaux, P. (2005), “Dependency Between Consecutive Local Conformations Helps Assemble Protein Structures From Secondary Structures Using Go Potential and Greedy Algorithm,” Proteins, 61, 732–740.
   PubMed Web of Science ®Google Scholar
• Venables, W.N., and Ripley, B.D. (2002), Modern Applied Statistics With S (4th ed.), New York: Springer.
   Google Scholar
• Wahba, G. (1990), Spline Models for Observational Data, Philadelphia, PA: SIAM.
   Google Scholar
• Wand, P., and Jones, C. (1995), Kernel Smoothing (Monographs on Statistics and Applied Probability), New York: Taylor & Francis.
   Google Scholar
• Zhao, F., Peng, J., Debartolo, J., Freed, K.F., Sosnick, T.R., and Xu, J. (2010), “A Probabilistic and Continuous Model of Protein Conformational Space for Template-Free Modeling,” Journal of Computational Biology, 17, 783–798.
   PubMed Web of Science ®Google Scholar
• Zhu, K., Pincus, D.L., Zhao, S., and Friesner, R.A. (2006), “Long Loop Prediction Using the Protein Local Optimization Program,” Proteins: Structure, Function, and Bioinformatics, 65, 438–452.
   PubMed Web of Science ®Google Scholar