×

zbMATH — the first resource for mathematics

Multiscale geometric modeling of macromolecules. I: Cartesian representation. (English) Zbl 1349.92016
Summary: This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems. Our approach leads to geometric and potential driven Laplace-Beltrami flows for biomolecular surface evolution and formation. The resulting surfaces are free of geometric singularities and minimize the total free energy of the biomolecular system. High order partial differential equation (PDE)-based nonlinear filters are employed for EMDB data processing. We show the efficacy of this approach in feature-preserving noise reduction. After the construction of protein multiresolution surfaces, we explore the analysis and characterization of surface morphology by using a variety of curvature definitions. Apart from the classical Gaussian curvature and mean curvature, maximum curvature, minimum curvature, shape index, and curvedness are also applied to macromolecular surface analysis for the first time. Our curvature analysis is uniquely coupled to the analysis of electrostatic surface potential, which is a by-product of our variational multiscale solvation models. As an expository investigation, we particularly emphasize the numerical algorithms and computational protocols for practical applications of the above multiscale geometric models. Such information may otherwise be scattered over the vast literature on this topic. Based on the curvature and electrostatic analysis from our multiresolution surfaces, we introduce a new concept, the polarized curvature, for the prediction of protein binding sites.

MSC:
92-08 Computational methods for problems pertaining to biology
92C40 Biochemistry, molecular biology
65D17 Computer-aided design (modeling of curves and surfaces)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
92C55 Biomedical imaging and signal processing
Software:
APBS; MIBPB; TOMOBFLOW
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N.A. Baker, Biomolecular applications of Poisson-Boltzmann methods, in: K.B. Lipkowitz, R. Larter, T.R. Cundari (Eds.), Reviews in Computational Chemistry, vol. 21, John Wiley and Sons, Hoboken, NJ, 2005.
[2] Baker, N. A.; Bashford, D.; Case, D. A., Implicit solvent electrostatics in biomolecular simulation, (Leimkuhler, B.; Chipot, C.; Elber, R.; Laaksonen, A.; Mark, A.; Schlick, T.; Schutte, C.; Skeel, R., New Algorithms for Macromolecular Simulation, (2006), Springer)
[3] Baker, N. A.; Sept, D.; Joseph, S.; Holst, M. J.; McCammon, J. A., Electrostatics of nanosystems: application to microtubules and the ribosome, Proc. Natl. Acad. Sci. USA, 98, 18, 10037-10041, (2001)
[4] Bates, P. W.; Chen, Z.; Sun, Y. H.; Wei, G. W.; Zhao, S., Geometric and potential driving formation and evolution of biomolecular surfaces, J. Math. Biol., 59, 193-231, (2009) · Zbl 1311.92212
[5] Bates, P. W.; Wei, G. W.; Zhao, S., The minimal molecular surface, (2006)
[6] Bates, P. W.; Wei, G. W.; Zhao, S., The minimal molecular surface, (Midwest Quantitative Biology Conference, Mission Point Resort, Mackinac Island, MI, September 29-October 1, (2006))
[7] Bates, P. W.; Wei, G. W.; Zhao, S., Minimal molecular surfaces and their applications, J. Comput. Chem., 29, 3, 380-391, (2008)
[8] Bertonati, C.; Honig, B.; Alexov, E., Poisson-Boltzmann calculations of nonspecific salt effects on protein-protein binding free energy, Biophys. J., 92, 1891-1899, (2007)
[9] Bobenko, A. I.; Schröder, P., Discrete Willmore flow, (Symp. on Geometry Processing, (July 2005)), 101-110
[10] Chen, D.; Chen, Z.; Chen, C.; Geng, W. H.; Wei, G. W., MIBPB: A software package for electrostatic analysis, J. Comput. Chem., 32, 657-670, (2011)
[11] Chen, Z.; Baker, N. A.; Wei, G. W., Differential geometry based solvation models I: Eulerian formulation, J. Comput. Phys., 229, 8231-8258, (2010) · Zbl 1229.92030
[12] Chen, Z.; Baker, N. A.; Wei, G. W., Differential geometry based solvation models II: Lagrangian formulation, J. Math. Biol., 63, 1139-1200, (2011) · Zbl 1284.92025
[13] Chen, Z.; Wei, G. W., Differential geometry based solvation models III: quantum formulation, J. Chem. Phys., 135, 194108, (2011)
[14] Chen, Z.; Zhao, S.; Chun, J.; Thomas, D. G.; Baker, N. A.; Bates, P. B.; Wei, G. W., Variational approach for nonpolar solvation analysis, J. Chem. Phys., 137, 084101, (2012)
[15] Cheng, L. T.; Dzubiella, J.; McCammon, A. J.; Li, B., Application of the level-set method to the implicit solvation of nonpolar molecules, J. Chem. Phys., 127, 8, (2007)
[16] Cipriano, G.; Phillips, G. N.; Gleicher, M., Multi-scale surface descriptors, IEEE Trans. Vis. Comput. Graph., 15, 1201-1208, (2009)
[17] Connolly, M. L., Depth buffer algorithms for molecular modeling, J. Mol. Graph., 3, 19-24, (1985)
[18] Corey, R. B.; Pauling, L., Molecular models of amino acids, peptides and proteins, Rev. Sci. Instrum., 24, 621-627, (1953)
[19] Droske, M.; Rumpf, M., A level set formulation for Willmore flow, Interfaces Free Bound., 6, 3, 361-378, (2004) · Zbl 1062.35028
[20] Eisenhaber, F.; Argos, P., Improved strategy in analytic surface calculation for molecular systems: handling of singularities and computational efficiency, J. Comput. Chem., 14, 1272-1280, (1993)
[21] Federer, H., Curvature measures, Trans. Am. Math. Soc., 93, 418-491, (1959) · Zbl 0089.38402
[22] Feng, X.; Xia, K.; Tong, Y.; Wei, G.-W., Geometric modeling of subcellular structures, organelles and large multiprotein complexes, Int. J. Numer. Methods Biomed. Eng., 28, 1198-1223, (2012)
[23] Feng, X.; Xia, K. L.; Chen, Z.; Tong, Y. Y.; Wei, G. W., Multiscale geometric modeling of macromolecules II: Lagrangian representation, J. Comput. Chem., 34, 2100-2120, (2013)
[24] Fernandez, J., Tomobflow: feature-preserving noise filtering for electron tomography, BMC Bioinform., 178, 1-10, (2009)
[25] Fernandez, J.; Li, S.; Lucic, V., Three-dimensional anisotropic noise reduction with automated parameter tuning: application to electron cryotomography, (Current Topics in Artificial Intelligence, Lect. Notes Comput. Sci., vol. 4788, (2007)), 60-69
[26] Fernandez, J. J.; Li, S., An improved algorithm for anisotropic nonlinear diffusion for denoising cryo-tomograms, J. Struct. Biol., 144, 152-161, (2003)
[27] Frangakis, A. S.; Hegerl, R., Noise reduction in electron tomographic reconstructions using nonlinear anisotropic diffusion, J. Struct. Biol., 135, 239-250, (2001)
[28] Geng, W.; Wei, G. W., Multiscale molecular dynamics using the matched interface and boundary method, J. Comput. Phys., 230, 2, 435-457, (2011) · Zbl 1246.82028
[29] Geng, W.; Yu, S.; Wei, G. W., Treatment of charge singularities in implicit solvent models, J. Chem. Phys., 127, 114106, (2007)
[30] Gilboa, G.; Sochen, N.; Zeevi, Y. Y., Forward-and-backward diffusion processes for adaptive image enhancement and denoising, IEEE Trans. Image Process., 11, 7, 689-703, (2002)
[31] Gilboa, G.; Sochen, N.; Zeevi, Y. Y., Image sharpening by flows based on triple well potentials, J. Math. Imaging Vis., 20, 1-2, 121-131, (2004) · Zbl 1366.94051
[32] Gogonea, V.; Osawa, E. E., Implementation of solvent effect in molecular mechanics. 1. model development and analytical algorithm for the solvent-accessible surface area, Supramol. Chem., 3, 303-317, (1994)
[33] Jiang, W.; Baker, M. L.; Wu, Q.; Bajaj, C.; Chiu, W., Applications of a bilateral denoising filter in biological electron microscopy, J. Struct. Biol., 144, 114-122, (2003)
[34] G. Kindlmann, R. Whitaker, T. Tasdizen, T. Möller, Curvature-based transfer functions for direct volume rendering: methods and applications, in: Proc. IEEE Visualization, 2003.
[35] Koenderink, J. J.; van Doorn, A. J., Surface shape and curvature scales, Image Vis. Comput., 10, 8, 557-564, (Oct. 1992)
[36] Lee, B.; Richards, F. M., The interpretation of protein structures: estimation of static accessibility, J. Mol. Biol., 55, 3, 379-400, (1971)
[37] Lysaker, M.; Lundervold, A.; Tai, X. C., Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12, 12, 1579-1590, (2003) · Zbl 1286.94020
[38] Marenich, A. V.; Cramer, C. J.; Truhlar, D. G., Perspective on foundations of solvation modeling: the electrostatic contribution to the free energy of solvation, J. Chem. Theory Comput., 4, 6, 877-887, (2008)
[39] Mumford, D.; Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42, 5, 577-685, (1989) · Zbl 0691.49036
[40] Osher, S.; Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 2, 463-502, (2001) · Zbl 0988.65093
[41] Osher, S.; Sethian, J., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 1, 12-49, (1988) · Zbl 0659.65132
[42] Pantelic, R. S.; Rothnagel, R.; Huang, C. Y.; Muller, D.; Woolford, D.; Landsberg, M. J.; McDowall, A.; Pailthorpe, B.; Young, P. R.; Banks, J.; Hankamer, B.; Ericksson, G., The discriminative bilateral filter: an enhanced denoising filter for electron microscopy data, J. Struct. Biol., 155, 395-408, (2006)
[43] Perona, P.; Malik, J., Scale-space and edge-detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12, 7, 629-639, (1990)
[44] Pierotti, R. A., A scaled particle theory of aqueous and nonaqueous solutions, Chem. Rev., 76, 6, 717-726, (1976)
[45] Prabhu, N. V.; Zhu, P.; Sharp, K. A., Implementation and testing of stable, fast implicit solvation in molecular dynamics using the smooth-permittivity finite difference Poisson-Boltzmann method, J. Comput. Chem., 25, 16, 2049-2064, (2004)
[46] Richards, F. M., Areas, volumes, packing, and protein structure, Annu. Rev. Biophys. Bioeng., 6, 1, 151-176, (1977)
[47] 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., 23, 128-137, (2002)
[48] Roux, B.; Simonson, T., Implicit solvent models, Biophys. Chem., 78, 1-2, 1-20, (1999)
[49] Rudin, L. I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, (Proceedings of the Eleventh Annual International Conference of the Center for Nonlinear Studies on Experimental Mathematics: Computational Issues in Nonlinear Science, (1992), Elsevier, North-Holland, Inc. Amsterdam, The Netherlands), 259-268 · Zbl 0780.49028
[50] Sanner, M. F.; Olson, A. J.; Spehner, J. C., Reduced surface: an efficient way to compute molecular surfaces, Biopolymers, 38, 305-320, (1996)
[51] Sethian, J. A., Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comput. Phys., 169, 2, 503-555, (2001) · Zbl 0988.65095
[52] Sharp, K. A.; Honig, B., Electrostatic interactions in macromolecules - theory and applications, Annu. Rev. Biophys. Biophys. Chem., 19, 301-332, (1990)
[53] Simonett, G., The Willmore flow for near spheres, Differ. Integral Equ., 14, 1005-1014, (2001) · Zbl 1161.35429
[54] Simonson, T., Macromolecular electrostatics: continuum models and their growing pains, Curr. Opin. Struct. Biol., 11, 2, 243-252, (2001)
[55] Sochen, N.; Kimmel, R.; Malladi, R., A general framework for low level vision, IEEE Trans. Image Process., 7, 3, 310-318, (1998) · Zbl 0973.94502
[56] Soldea, O.; Elber, G.; Rivlin, E., Global segmentation and curvature analysis of volumetric data sets using trivariate b-spline functions, IEEE Trans. Pattern Anal. Mach. Intell., 28, 2, 265-278, (2006)
[57] Stillinger, F. H., Structure in aqueous solutions of nonpolar solutes from the standpoint of scaled-particle theory, J. Solution Chem., 2, 141-158, (1973)
[58] Stoschek, A.; Hegerl, R., Denoising of electron tomographic reconstructions using multiscale transformations, J. Struct. Biol., 120, 257-265, (1997)
[59] Tomasi, C.; Manduchi, R., Bilateral filtering for gray and color images, (Sixth International Conference on Computer Vision, (1998)), 839-846
[60] van der Heide, P.; Xu, X. P.; Marsh, B. J.; Hanein, D.; Volkmann, N., Efficient automatic noise reduction of electron tomographic reconstructions based on iterative Median filtering, J. Struct. Biol., 158, 196-204, (2007)
[61] Wang, Y.; Wei, G. W.; Yang, S.-Y., Partial differential equation transform - variational formulation and Fourier analysis, Int. J. Numer. Methods Biomed. Eng., 27, 1996-2020, (2011) · Zbl 1254.94012
[62] Wang, Y.; Wei, G. W.; Yang, S.-Y., Mode decomposition evolution equations, J. Sci. Comput., 50, 495-518, (2012) · Zbl 1457.65152
[63] Wei, G. W., Generalized perona-malik equation for image restoration, IEEE Signal Process. Lett., 6, 165-167, (1999)
[64] Wei, G. W., Differential geometry based multiscale models, Bull. Math. Biol., 72, 1562-1622, (2010) · Zbl 1198.92001
[65] Wei, G.-W., Multiscale multiphysics and multidomain models I: basic theory, J. Theor. Comput. Chem., 12, 8, 1341006, (2013)
[66] Wei, G. W.; Jia, Y. Q., Synchronization-based image edge detection, Europhys. Lett., 59, 6, 814-819, (2002)
[67] Wei, G. W.; Sun, Y. H.; Zhou, Y. C.; Feig, M., Molecular multiresolution surfaces, (2005), pp. 1-11
[68] Wei, G.-W.; Zheng, Q.; Chen, Z.; Xia, K., Variational multiscale models for charge transport, SIAM Rev., 54, 4, 699-754, (2012) · Zbl 1306.92021
[69] Willmore, T. J., Riemannian geometry, (1997), Oxford University Press USA · Zbl 0797.53002
[70] Witelski, T. P.; Bowen, M., ADI schemes for higher-order nonlinear diffusion equations, Appl. Numer. Math., 45, 2-3, 331-351, (2003) · Zbl 1061.76051
[71] A. Witkin, Scale-space filtering: A new approach to multi-scale description, in: Proceedings of IEEE International Conference on Acoustic Speech Signal Processing, vol. 9, 1984, pp. 150-153.
[72] Xia, K. L.; Wei, G. W., Three-dimensional MIB Galerkin method for elliptic interface problems, J. Comput. Phys., (2013), in press
[73] Xia, K. L.; Zhan, M.; Wei, G. W., MIB Galerkin method for elliptic interface problems, J. Comput. Phys., (2013)
[74] Xie, D.; Jiang, Y.; Brune, P.; Scott, L. R., A fast solver for a nonlocal dielectric continuum model, SIAM J. Sci. Comput., 34, B107-B126, (2012) · Zbl 1260.78016
[75] Xu, G.; Pan, Q.; Bajaj, C. L., Discrete surface modeling using partial differential equations, Comput. Aided Geom. Des., 23, 2, 125-145, (2006) · Zbl 1083.65018
[76] Yu, S. N.; Geng, W. H.; Wei, G. W., Treatment of geometric singularities in implicit solvent models, J. Chem. Phys., 126, 244108, (2007)
[77] Yu, S. N.; Wei, G. W., Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities, J. Comput. Phys., 227, 602-632, (2007) · Zbl 1128.65103
[78] Yu, Z. Y.; Holst, M.; Cheng, Y.; McCammon, J. A., Feature-preserving adaptive mesh generation for molecular shape modeling and simulation, J. Mol. Graph. Model., 26, 1370-1380, (2008)
[79] Zhang, Y.; Bajaj, C.; Xu, G., Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow, Commun. Numer. Methods Eng., 25, 1-18, (2009) · Zbl 1158.65314
[80] Zhao, S., High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces, J. Comput. Phys., 229, 3155-3170, (2010) · Zbl 1187.78044
[81] Zheng, Q.; Wei, G. W., Poisson-Boltzmann-Nernst-Planck model, J. Chem. Phys., 134, 194101, (2011)
[82] Zheng, Q.; Yang, S. Y.; Wei, G. W., Molecular surface generation using PDE transform, Int. J. Numer. Methods Biomed. Eng., 28, 291-316, (2012)
[83] Zhou, Y. C.; Wei, G. W., On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, J. Comput. Phys., 219, 1, 228-246, (2006) · Zbl 1105.65108
[84] Zhou, Y. C.; Zhao, S.; Feig, M.; Wei, G. W., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys., 213, 1, 1-30, (2006) · Zbl 1089.65117
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.