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Evolutionary de Rham-Hodge method
1. | Department of Mathematics, Michigan State University, MI 48824, USA |
2. | Department of Computer Science and Engineering, Michigan State University, MI 48824, USA |
3. | Department of Mathematics, Department of Biochemistry and Molecular Biology, Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA |
The de Rham-Hodge theory is a landmark of the 20$ ^\text{th} $ Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1 and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.
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show all references
References:
[1] |
M. Akram and V. Michel,
Regularisation of the Helmholtz decomposition and its application to geomagnetic field modelling, GEM-International Journal on Geomathematics, 1 (2010), 101-120.
doi: 10.1007/s13137-010-0001-y. |
[2] |
D. N. Arnold, R. S. Falk and R. Winther,
Finite element exterior calculus, homological techniques and applications, Acta Numerica, 15 (2006), 1-155.
doi: 10.1017/S0962492906210018. |
[3] |
A. R. Atilgan, S. Durell, R. L. Jernigan, M. C. Demirel, O. Keskin and I. Bahar,
Anisotropy of fluctuation dynamics of proteins with an elastic network model, Biophysical Journal, 80 (2001), 505-515.
doi: 10.1016/S0006-3495(01)76033-X. |
[4] |
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 (1997), 173-181.
doi: 10.1016/S1359-0278(97)00024-2. |
[5] |
P. W. Bates, G.-W. Wei and S. Zhao,
Minimal molecular surfaces and their applications, Journal of Computational Chemistry, 29 (2007), 380-391.
doi: 10.1002/jcc.20796. |
[6] |
P. Bendich, H. Edelsbrunner and M. Kerber,
Computing robustness and persistence for images, IEEE Transactions on Visualization and Computer Graphics, 16 (2010), 1251-1260.
doi: 10.1109/TVCG.2010.139. |
[7] |
D. Bramer and G.-W. Wei, Multiscale weighted colored graphs for protein flexibility and rigidity analysis, The Journal of Chemical Physics, 148 (2018), 054103.
doi: 10.1063/1.5016562. |
[8] |
Z. Cang, L. Mu and G.-W. Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLOS Computational Biology, 14 (2018), e1005929.
doi: 10.1371/journal.pcbi.1005929. |
[9] |
Z. Cang, L. Mu, K. Wu, K. Opron, K. Xia and G.-W. Wei, A topological approach for protein classification, Computational and Mathematical Biophysics, 1 (2015).
doi: 10.1515/mlbmb-2015-0009. |
[10] |
Z. Cang, E. Munch and G.-W. Wei, Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis, Journal of Applied and Computational Topology, (2020), 1–27. Google Scholar |
[11] |
Z. Cang and G.-W. Wei, TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLOS Computational Biology, 13 (2017), 1005690.
doi: 10.1371/journal.pcbi.1005690. |
[12] |
Z. Cang and G.-W. Wei, Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction, International Journal for Numerical Methods in Biomedical Engineering, 34 (2018), e2914.
doi: 10.1002/cnm.2914. |
[13] |
G. Carlsson, V. De Silva and D. Morozov, Zigzag persistent homology and real-valued functions, Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009, 247–256.
doi: 10.1145/1542362.1542408. |
[14] |
G. Carlsson, T. Ishkhanov, V. De Silva and A. Zomorodian,
On the local behavior of spaces of natural images, International Journal of Computer Vision, 76 (2008), 1-12.
doi: 10.1007/s11263-007-0056-x. |
[15] |
T. Cecil,
A numerical method for computing minimal surfaces in arbitrary dimension, Journal of Computational Physics, 206 (2005), 650-660.
doi: 10.1016/j.jcp.2004.12.022. |
[16] |
Z. Chen, N. A. Baker and G.-W. Wei,
Differential geometry based solvation model Ⅱ: Lagrangian formulation, Journal of Mathematical Biology, 63 (2011), 1139-1200.
doi: 10.1007/s00285-011-0402-z. |
[17] |
S. Chowdhury and F. Mémoli, Persistent path homology of directed networks, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadephia, PA, 2018, 1152–1169.
doi: 10.1137/1.9781611975031.75. |
[18] |
V. De Silva, R. Ghrist and A. Muhammad, Blind swarms for coverage in 2-D, Robotics: Science and Systems, 2005, 335–342.
doi: 10.15607/RSS.2005.I.044. |
[19] |
M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling, in Discrete Differential Geometry, Birkhäuser, Basel, 2008, 287–324.
doi: 10.1007/978-3-7643-8621-4_16. |
[20] |
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 (2011), 499-527.
doi: 10.1007/s10208-011-9100-x. |
[21] |
Q. Du, C. Liu and X. Wang,
A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198 (2004), 450-468.
doi: 10.1016/j.jcp.2004.01.029. |
[22] |
H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, RI, 2010. |
[23] |
H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological persistence and simplification, in ![]() ![]() |
[24] |
N. Foster and D. Metaxas,
Realistic animation of liquids, Graphical Models and Image Processing, 58 (1996), 471-483.
doi: 10.1006/gmip.1996.0039. |
[25] |
K. O. Friedrichs,
Differential forms on Riemannian manifolds, Communications on Pure and Applied Mathematics, 8 (1955), 551-590.
doi: 10.1002/cpa.3160080408. |
[26] |
P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognition and Image Analysis, 9 (1999), 596-603. Google Scholar |
[27] |
P. Frosini and C. Landi, Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval, in Computer Analysis of Images and Patterns, Springer, Heidelberg, 2011, 294–301.
doi: 10.1007/978-3-642-23672-3_36. |
[28] |
M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow and V. Nanda,
A topological measurement of protein compressibility, Japan Journal of Industrial and Applied Mathematics, 32 (2015), 1-17.
doi: 10.1007/s13160-014-0153-5. |
[29] |
H. Gao, M. K. Mandal, G. Guo and J. Wan, Singular point detection using discrete Hodge Helmholtz decomposition in fingerprint images, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, Dallas, TX, 2010, 1094–1097.
doi: 10.1109/ICASSP.2010.5495348. |
[30] |
J. Gomes and O. Faugeras, Using the vector distance functions to evolve manifolds of arbitrary codimension, International Conference on Scale-Space Theories in Computer Vision, Springer, 2001, 1–13. Google Scholar |
[31] |
T. Hazra, S. A. Ullah, S. Wang, E. Alexov and S. Zhao,
A super-Gaussian Poisson–Boltzmann model for electrostatic free energy calculation: Smooth dielectric distribution for protein cavities and in both water and vacuum states, Journal of Mathematical Biology, 79 (2019), 631-672.
doi: 10.1007/s00285-019-01372-1. |
[32] |
W. V. D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, Cambridge, 1989.
![]() |
[33] |
W. Humphrey, A. Dalke and K. Schulten,
VMD: visual molecular dynamics, Journal of Molecular Graphics, 14 (1996), 33-38.
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