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The de Rham-Hodge analysis and modeling of biomolecules. (English) Zbl 1448.92158
Summary: Biological macromolecules have intricate structures that underpin their biological functions. Understanding their structure-function relationships remains a challenge due to their structural complexity and functional variability. Although de Rham-Hodge theory, a landmark of twentieth-century mathematics, has had a tremendous impact on mathematics and physics, it has not been devised for macromolecular modeling and analysis. In this work, we introduce de Rham-Hodge theory as a unified paradigm for analyzing the geometry, topology, flexibility, and Hodge mode analysis of biological macromolecules. Geometric characteristics and topological invariants are obtained either from the Helmholtz-Hodge decomposition of the scalar, vector, and/or tensor fields of a macromolecule or from the spectral analysis of various Laplace-de Rham operators defined on the molecular manifolds. We propose Laplace-de Rham spectral-based models for predicting macromolecular flexibility. We further construct a Laplace-de Rham-Helfrich operator for revealing cryo-EM natural frequencies. Extensive experiments are carried out to demonstrate that the proposed de Rham-Hodge paradigm is one of the most versatile tools for the multiscale modeling and analysis of biological macromolecules and subcellular organelles. Accurate, reliable, and topological structure-preserving algorithms for implementing discrete exterior calculus (DEC) have been developed to facilitate the aforementioned modeling and analysis of biological macromolecules. The proposed de Rham-Hodge paradigm has potential applications to subcellular organelles and the structure construction from medium- or low-resolution cryo-EM maps, and functional predictions from massive biomolecular datasets.
Reviewer: Reviewer (Berlin)
92D20 Protein sequences, DNA sequences
55M99 Classical topics in algebraic topology
53B50 Applications of local differential geometry to the sciences
Full Text: DOI
[1] Alexov, E.; Mehler, EL; Baker, N.; Baptista, AM; Huang, Y.; Milletti, F.; Erik Nielsen, J.; Farrell, D.; Carstensen, T.; Olsson, MH, Progress in the prediction of pka values in proteins, Proteins Struct Funct Bioinf, 79, 12, 3260-3275 (2011)
[2] Antosiewicz, J.; McCammon, JA; Gilson, MK, The determinants of p \(K_a\) s in proteins, Biochemistry, 35, 24, 7819-7833 (1996)
[3] Arnold, DN; Falk, RS; Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numer, 15, 1-155 (2006) · Zbl 1185.65204
[4] Atilgan, AR; Durell, S.; Jernigan, RL; Demirel, M.; Keskin, O.; Bahar, I., Anisotropy of fluctuation dynamics of proteins with an elastic network model, Biophys J, 80, 1, 505-515 (2001)
[5] Bahar, I.; Atilgan, AR; Erman, B., Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential, Fold Des, 2, 173-181 (1997)
[6] Baker, NA; Sept, D.; Joseph, S.; Holst, MJ; McCammon, JA, Electrostatics of nanosystems: application to microtubules and the ribosome, Proc Nat Acad Sci USA, 98, 18, 10037-10041 (2001)
[7] Baradaran, R.; Wang, C.; Siliciano, AF; Long, SB, Cryo-em structures of fungal and metazoan mitochondrial calcium uniporters, Nature, 559, 7715, 580-584 (2018)
[8] Bates, PW; Wei, GW; Zhao, S., Minimal molecular surfaces and their applications, J Comput Chem, 29, 3, 380-91 (2008)
[9] Bhatia, H.; Norgard, G.; Pascucci, V.; Bremer, P-T, The helmholtz-hodge decomposition-a survey, IEEE Trans Vis Comput Graphics, 19, 8, 1386-1404 (2013)
[10] Blinn, JF, A generalization of algebraic surface drawing, ACM Trans Graph, 1, 235-256 (1982)
[11] Bossavit, A., Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism, IEE Proc A (Phys Sci Meas Instrum Manag Educ Rev), 135, 8, 493-500 (1988)
[12] Bott, R.; Tu, LW, Differential forms in algebraic topology (2013), Berlin: Springer, Berlin
[13] Brooks, BR; Bruccoleri, RE; Olafson, BD; States, D.; Swaminathan, S.; Karplus, M., Charmm: a program for macromolecular energy, minimization, and dynamics calculations, J Comput Chem, 4, 187-217 (1983)
[14] Cang, ZX; Wei, GW, TopologyNet: topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLoS Comput Biol, 13, 7, e1005690 (2017)
[15] Cang, ZX; Wei, GW, Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction, Int J Numer Methods Biomed Eng (2018)
[16] Cantarella, J.; DeTurck, D.; Gluck, H., Vector calculus and the topology of domains in 3-space, Am Math Mon, 109, 5, 409-442 (2002) · Zbl 1038.53017
[17] Carlsson, G.; Zomorodian, A.; Collins, A.; Guibas, LJ, Persistence barcodes for shapes, Int J Shape Model, 11, 2, 149-187 (2005) · Zbl 1092.68688
[18] Chen, D.; Chen, Z.; Chen, C.; Geng, WH; Wei, GW, MIBPB: a software package for electrostatic analysis, J Comput Chem, 32, 657-670 (2011)
[19] Chen, J.; Geng, W., On preconditioning the treecode-accelerated boundary integral (tabi) Poisson-Boltzmann solver, J Comput Phys, 373, 750-762 (2018) · Zbl 1416.65487
[20] Chen, M.; Tu, B.; Lu, B., Triangulated manifold meshing method preserving molecular surface topology, J Mole Graph Model, 38, 411-418 (2012)
[21] Cheng, H-L; Shi, X., Quality mesh generation for molecular skin surfaces using restricted union of balls, Comput Geom, 42, 3, 196-206 (2009) · Zbl 1158.65014
[22] Cherezov, V.; Rosenbaum, DM; Hanson, MA; Rasmussen, SG; Thian, FS; Kobilka, TS; Choi, H-J; Kuhn, P.; Weis, WI; Kobilka, BK, High-resolution crystal structure of an engineered human \(\beta 2\)-adrenergic g protein-coupled receptor, Science, 318, 5854, 1258-1265 (2007)
[23] Corey, RB; Pauling, L., Molecular models of amino acids, peptides, and proteins, Rev Sci Instrum, 24, 8, 621-627 (1953)
[24] De La Torre, JG; Bloomfield, VA, Hydrodynamic properties of macromolecular complexes. i. translation, Biopolym Original Res Biomol, 16, 8, 1747-1763 (1977)
[25] Demlow, A.; Hirani, AN, A posteriori error estimates for finite element exterior calculus: the de rham complex, Found Comput Math, 14, 6, 1337-1371 (2014) · Zbl 1308.65187
[26] Desbrun M, Hirani AN, Leok M, Marsden JE (2005) Discrete exterior calculus. arXiv preprint math/0508341 · Zbl 1080.39021
[27] Dey, TK; Fan, F.; Wang, Y., An efficient computation of handle and tunnel loops via reeb graphs, ACM Trans Graph, 32, 4, 32 (2013)
[28] Dong, F.; Vijaykumar, M.; Zhou, HX, Comparison of calculation and experiment implicates significant electrostatic contributions to the binding stability of barnase and barstar, Biophys J, 85, 1, 49-60 (2003)
[29] Du, Q.; Liu, C.; Wang, X., A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J Comput Phys, 198, 2, 450-468 (2004) · Zbl 1116.74384
[30] Duncan, BS; Olson, AD, Shape analysis of molecular surfaces, Biopolymers, 33, 2, 231-8 (1993)
[31] Edelsbrunner, H.; Harer, J., Computational topology: an introduction (2010), Providence: American Mathematical Soc, Providence · Zbl 1193.55001
[32] Edelsbrunner H, Letscher D, Zomorodian A (2000) Topological persistence and simplification. In: 41st annual symposium on foundations of computer science, 2000. Proceedings. IEEE, pp 454-463 · Zbl 1011.68152
[33] 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)
[34] Fogolari, F.; Brigo, A.; Molinari, H., The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology, J Mol Recognit, 15, 6, 377-92 (2002)
[35] Frauenfelder, H.; Sligar, SG; Wolynes, PG, The energy landscapes and motions of proteins, Science, 254, 5038, 1598-1603 (1991)
[36] Geng, W.; Krasny, R., A treecode-accelerated boundary integral poisson-boltzmann solver for electrostatics of solvated biomolecules, J Comput Phys, 247, 62-78 (2013) · Zbl 1349.78084
[37] Go, N.; Noguti, T.; Nishikawa, T., Dynamics of a small globular protein in terms of low-frequency vibrational modes, Proc Natl Acad Sci, 80, 3696-3700 (1983)
[38] Hanawa-Suetsugu, K.; Sekine, S-I; Sakai, H.; Hori-Takemoto, C.; Terada, T.; Unzai, S.; Tame, JR; Kuramitsu, S.; Shirouzu, M.; Yokoyama, S., Crystal structure of elongation factor p from thermus thermophilus hb8, Proc Nat Acad Sci, 101, 26, 9595-9600 (2004)
[39] Haslam D, Zeng T, Li R, He J (2018) Exploratory studies detecting secondary structures in medium resolution 3d cryo-em images using deep convolutional neural networks. In: Proceedings of the 2018 ACM international conference on bioinformatics, computational biology, and health informatics. ACM, pp 628-632
[40] Hekstra, DR; White, KI; Socolich, MA; Henning, RW; Šrajer, V.; Ranganathan, R., Electric-field-stimulated protein mechanics, Nature, 540, 7633, 400 (2016)
[41] Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments, Zeitschrift für Naturforschung Teil C, 28, 693-703 (1973)
[42] Hirani AN (2003) Discrete exterior calculus. PhD thesis, California Institute of Technology
[43] Hodge WVD (1989) The theory and applications of harmonic integrals. CUP Archive
[44] Honig, B.; Nicholls, A., Classical electrostatics in biology and chemistry, Science, 268, 5214, 1144-9 (1995)
[45] Im, W.; Beglov, D.; Roux, B., Continuum solvation model: electrostatic forces from numerical solutions to the Poisson-Boltzmann equation, Comput Phys Commun, 111, 1-3, 59-75 (1998) · Zbl 0935.78019
[46] Jiang, J.; Wang, Y.; Sušac, L.; Chan, H.; Basu, R.; Zhou, ZH; Feigon, J., Structure of telomerase with telomeric dna, Cell, 173, 5, 1179-1190 (2018)
[47] Juffer, A.; van Keulen, BE; van der Ploeg, A.; Berendsen, H., The electric potential of a macromolecule in a solvent: a fundamental approach, J Comput Phys, 97, 144-171 (1991) · Zbl 0743.65094
[48] Kuglstatter, A.; Stihle, M.; Neumann, C.; Müller, C.; Schaefer, W.; Klein, C.; Benz, J.; Research, RP; Development, E., Structural differences between glycosylated, disulfide-linked heterodimeric knob-into-hole fc fragment and its homodimeric knob-knob and hole-hole side products, Protein Eng Des Sel, 30, 9, 649-656 (2017)
[49] Lanza, A.; Margheritis, E.; Mugnaioli, E.; Cappello, V.; Garau, G.; Gemmi, M., Nanobeam precession-assisted 3d electron diffraction reveals a new polymorph of hen egg-white lysozyme, IUCrJ, 6, 2, 178-188 (2019)
[50] Lee, B.; Richards, FM, The interpretation of protein structures: estimation of static accessibility, J Mol Biol, 55, 3, 379-400 (1971)
[51] Levitt, M.; Sander, C.; Stern, PS, Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme, J Mol Biol, 181, 3, 423-447 (1985)
[52] Li, L.; Li, C.; Zhang, Z.; Alexov, E., On the dielectric constant of proteins: smooth dielectric function for macromolecular modeling and its implementation in delphi, J Chem Theory Comput, 9, 4, 2126-2136 (2013)
[53] Liang, J.; Subranmaniam, S., Computation of molecular electrostatics with boundary element methods, Biophys J, 73, 1830-1841 (1997)
[54] Lim L-H (2015) Hodge laplacians on graphs. arXiv preprint arXiv:1507.05379
[55] Lu, B.; Cheng, X.; McCammon, JA, new-version-fast-multipole-method accelerated electrostatic calculations in biomolecular systems, J Comput Phys, 226, 2, 1348-1366 (2007) · Zbl 1121.92007
[56] Ma, JP, Usefulness and limitations of normal mode analysis in modeling dynamics of biomolecular complexes, Structure, 13, 373-380 (2005)
[57] Ming, D.; Kong, Y.; Lambert, MA; Huang, Z.; Ma, J., How to describe protein motion without amino acid sequence and atomic coordinates, Proc Nat Acad Sci, 99, 13, 8620-8625 (2002)
[58] Mitchell JC (1998) Hodge decomposition and expanding maps on the flat tori. PhD thesis, University of California, Berkeley
[59] Muench, SP; Huss, M.; Song, CF; Phillips, C.; Wieczorek, H.; Trinick, J.; Harrison, MA, Cryo-electron microscopy of the vacuolar atpase motor reveals its mechanical and regulatory complexity, J Mol Biol, 386, 4, 989-999 (2009)
[60] Murakami, K.; Stewart, M.; Nozawa, K.; Tomii, K.; Kudou, N.; Igarashi, N.; Shirakihara, Y.; Wakatsuki, S.; Yasunaga, T.; Wakabayashi, T., Structural basis for tropomyosin overlap in thin (actin) filaments and the generation of a molecular swivel by troponin-t, Proc Nat Acad Sci, 105, 20, 7200-7205 (2008)
[61] Natarajan, V.; Koehl, P.; Wang, Y.; Hamann, B.; Linsen, L.; Hagen, H.; Hamann, B., Visual analysis of biomolecular surfaces, Mathematical methods for visualization in medicine and life science, 237-256 (2008), Berlin: Springer, Berlin · Zbl 1255.68259
[62] Nguyen, DD; Wang, B.; Wei, GW, Accurate, robust and reliable calculations of Poisson-Boltzmann binding energies, J Comput Chem, 38, 941-948 (2017)
[63] Nguyen, DD; Xia, KL; Wei, GW, Generalized flexibility-rigidity index, J Chem Phys, 144, 234106 (2016)
[64] Nielsen, JE; McCammon, JA, Calculating pka values in enzyme active sites, Protein Sci, 12, 9, 1894-1901 (2003)
[65] Nishino, T.; Rago, F.; Hori, T.; Tomii, K.; Cheeseman, IM; Fukagawa, T., Cenp-t provides a structural platform for outer kinetochore assembly, EMBO J, 32, 3, 424-436 (2013)
[66] Opron, K.; Xia, KL; Wei, GW, Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis, J Chem Phys, 140, 234105 (2014)
[67] Richards, FM, Areas, volumes, packing, and protein structure, Ann Rev Biophys Bioeng, 6, 1, 151-176 (1977)
[68] Sander, B.; Golas, MM; Makarov, EM; Brahms, H.; Kastner, B.; Lührmann, R.; Stark, H., Organization of core spliceosomal components u5 snrna loop i and u4/u6 di-snrnp within u4/u6. u5 tri-snrnp as revealed by electron cryomicroscopy, Mol Cell, 24, 2, 267-278 (2006)
[69] Sharp, KA; Honig, B., Electrostatic interactions in macromolecules—theory and applications, Ann Rev Biophys Biophys Chem, 19, 301-332 (1990)
[70] Singh, AK; McGoldrick, LL; Twomey, EC; Sobolevsky, AI, Mechanism of calmodulin inactivation of the calcium-selective trp channel trpv6, Sci Adv, 4, 8, eaau6088 (2018)
[71] Tama, F.; Wriggers, W.; Brooks, CL III, Exploring global distortions of biological macromolecules and assemblies from low-resolution structural information and elastic network theory, J Mol Biol, 321, 2, 297-305 (2002)
[72] Tamstorf, R.; Grinspun, E., Discrete bending forces and their jacobians, Graph Models, 75, 6, 362-370 (2013)
[73] Tasumi, M.; Takenchi, H.; Ataka, S.; Dwidedi, AM; Krimm, S., Normal vibrations of proteins: glucagon, Biopolymers, 21, 711-714 (1982)
[74] Vorobjev, YN; Scheraga, HA, A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent, J Comput Chem, 18, 4, 569-583 (1997)
[75] Wagoner, JA; Baker, NA, Assessing implicit models for nonpolar mean solvation forces: the importance of dispersion and volume terms, Proc Nat Acad Sci USA, 103, 22, 8331-6 (2006)
[76] Wollenman, LC; Vander Ploeg, MR; Miller, ML; Zhang, Y.; Bazil, JN, The effect of respiration buffer composition on mitochondrial metabolism and function, PLoS ONE, 12, 11, e0187523 (2017)
[77] Xia K, Wei G-W (2016) A review of geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular data. arXiv preprint arXiv:1612.01735
[78] Xia, KL; Feng, X.; Tong, YY; Wei, GW, Multiscale geometric modeling of macromolecules i: Cartesian representation, J Comput Phys, 275, 912-936 (2014) · Zbl 1349.92016
[79] Xia, KL; Feng, X.; Tong, YY; Wei, GW, Persistent homology for the quantitative prediction of fullerene stability, J Comput Chem, 36, 408-422 (2015)
[80] Xia, KL; Opron, K.; Wei, GW, Multiscale multiphysics and multidomain models—flexibility and rigidity, J Chem Phys, 139, 194109 (2013)
[81] Xia, KL; Wei, GW, Persistent homology analysis of protein structure, flexibility and folding, Int J Numer Methods Biomed Eng, 30, 814-844 (2014)
[82] Yao, Y.; Sun, J.; Huang, X.; Bowman, GR; Singh, G.; Lesnick, M.; Guibas, LJ; Pande, VS; Carlsson, G., Topological methods for exploring low-density states in biomolecular folding pathways, J Chem Phys, 130, 14, 04B614 (2009)
[83] Yu, SN; Geng, WH; Wei, GW, Treatment of geometric singularities in implicit solvent models, J Chem Phys, 126, 244108 (2007)
[84] Yu, ZY; Holst, M.; Cheng, Y.; McCammon, JA, Feature-preserving adaptive mesh generation for molecular shape modeling and simulation, J Mol Graphics Model, 26, 1370-1380 (2008)
[85] Zhao, R.; Cang, Z.; Tong, Y.; Wei, G-W, Protein pocket detection via convex hull surface evolution and associated Reeb graph, Bioinformatics, 34, 17, i830-i837 (2018)
[86] Zhao R, Desbrun M, Wei G-W, Tong YY (2019) 3D hodge decompositions of edge-and face-based vector fields
[87] Zheng, Q.; Yang, S.; Wei, G-W, Biomolecular surface construction by pde transform, Int J Numer Methods Biomed Eng, 28, 3, 291-316 (2012) · Zbl 1244.92024
[88] Zhou, Y.; Lu, B.; Gorfe, AA, Continuum electromechanical modeling of protein-membrane interactions, Phys Rev E, 82, 4, 041923 (2010)
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