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Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis. (English) Zbl 07301453
A protein and the connections within it are modeled by a simplicial complex which incorporates time via the oscillations of perturbed elements of the system. The time evolution of the system as oscillators move to synchronization allows the definition of a real valued function on the complex. Evolutionary homology studies the persistence barcodes of the resulting system. As a major application the ideas can be applied to the study of the flexibility of proteins. The resulting analysis of protein flexibility, which is computationally feasible, is shown to outperform other state of the art methods.
MSC:
55N31 Persistent homology and applications, topological data analysis
62R40 Topological data analysis
37N25 Dynamical systems in biology
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
Software:
javaPlex; Ripser; SW1PerS
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References:
[1] Adams, H.; Emerson, T.; Kirby, M.; Neville, R.; Peterson, C.; Shipman, P.; Chepushtanova, S.; Hanson, E.; Motta, F.; Ziegelmeier, L., Persistence images: a stable vector representation of persistent homology, J. Mach. Learn. Res., 18, 8, 1-35 (2017) · Zbl 1431.68105
[2] Adcock, A.; Carlsson, E.; Carlsson, G., The ring of algebraic functions on persistence bar codes, Homol. Homotopy Appl., 18, 1, 381-402 (2016) · Zbl 1420.55017
[3] Ahmed, M., Fasy, B.T., Wenk, C.: Local persistent homology based distance between maps. In: Proceedings of the 22nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 43-52. ACM (2014)
[4] Arai, M.; Brandt, V.; Dabaghian, Y., The effects of theta precession on spatial learning and simplicial complex dynamics in a topological model of the hippocampal spatial map, PLoS Comput. Biol., 10, 6, e1003651 (2014)
[5] Bahar, I.; Atilgan, AR; Erman, B., Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential, Fold Des., 2, 3, 173-181 (1997)
[6] Bauer, U.: Ripser: efficient computation of Vietoris-Rips persistence barcodes (2019). arXiv:1908.02518
[7] Bauer, U., Kerber, M., Reininghaus, J.: Distributed computation of persistent homology. In: 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 31-38. SIAM (2014) · Zbl 1429.68328
[8] Bendich, P.; Harer, J., Persistent intersection homology, Found. Comput. Math., 11, 3, 305-336 (2011) · Zbl 1223.55002
[9] Berwald, JJ; Gidea, M.; Vejdemo-Johansson, M., Automatic recognition and tagging of topologically different regimes in dynamical systems, Discontin. Nonlinearity Complex., 3, 4, 413-426 (2014) · Zbl 1316.70021
[10] Bubenik, P., Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16, 1, 77-102 (2015) · Zbl 1337.68221
[11] Bubenik, P.; Scott, JA, Categorification of persistent homology, Discrete Comput. Geom., 51, 3, 600-627 (2014) · Zbl 1295.55005
[12] Bubenik, P.; de Silva, V.; Scott, J., Metrics for generalized persistence modules, Found. Comput. Math., 15, 6, 1501-1531 (2015) · Zbl 1345.55006
[13] Cang, Z., Wei, G.W.: Analysis and prediction of protein folding energy changes upon mutation by element specific persistent homology. Bioinformatics 33, 3549-3557 (2017a)
[14] Cang, Z., Wei, G.W.: Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction. Int. J. Numer. Methods Biomed. Eng. 34(2), e2914 (2017b)
[15] Cang, Z., Wei, G.W.: TopologyNet: topology based deep convolutional and multi-task neural networks for biomolecular property predictions. PLoS Comput. Biol. 13(7), e1005690 (2017c). 10.1371/journal.pcbi.1005690
[16] Cang, Z.; Mu, L.; Wu, K.; Opron, K.; Xia, K.; Wei, GW, A topological approach for protein classification, Mol. Based Math. Biol., 3, 140-162 (2015) · Zbl 1347.92054
[17] Cang, Z.; Mu, L.; Wei, GW, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLoS Comput. Biol., 14, 1, e1005929 (2018)
[18] Carlsson, G., Topology and data, Bull. Am. Math. Soc., 46, 2, 255-308 (2009) · Zbl 1172.62002
[19] Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings 25th Annual ACM Symposium on Computational Geometry, pp. 247-256 (2009) · Zbl 1380.68385
[20] Carlsson, G.; De Silva, V., Zigzag persistence, Found. Comput. Math., 10, 4, 367-405 (2010) · Zbl 1204.68242
[21] Carlsson, G.; Verovšek, SK, Symmetric and \(r\)-symmetric tropical polynomials and rational functions, J. Pure Appl. Algebra, 220, 11, 3610-3627 (2016) · Zbl 1375.14211
[22] Carlsson, G.; Zomorodian, A., The theory of multidimensional persistence, Discrete Comput. Geom., 42, 1, 71-93 (2009) · Zbl 1187.55004
[23] Carlsson, G.; Zomorodian, A.; Collins, A.; Guibas, LJ, Persistence barcodes for shapes, Int. J. Shape Model., 11, 2, 149-187 (2005) · Zbl 1092.68688
[24] Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proceedings 25th Annual ACM Symposium on Computational Geometry, pp. 237-246 (2009) · Zbl 1380.68387
[25] Chazal, F.; Guibas, LJ; Oudot, SY; Skraba, P., Persistence-based clustering in Riemannian manifolds, J. ACM (JACM), 60, 6, 41 (2013) · Zbl 1281.68176
[26] Chazal, F.; de Silva, V.; Glisse, M.; Oudot, S., The Structure and Stability of Persistence Modules (2016), Berlin: Springer, Berlin · Zbl 1362.55002
[27] Cohen-Steiner, D.; Edelsbrunner, H.; Harer, J., Stability of persistence diagrams, Discrete Comput. Geom., 37, 1, 103-120 (2007) · Zbl 1117.54027
[28] Cohen-Steiner, D.; Edelsbrunner, H.; Harer, J., Extending persistence using Poincaré and Lefschetz duality, Found. Comput. Math., 9, 1, 79-103 (2009) · Zbl 1189.55002
[29] Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Morozov, D.: Persistent homology for kernels, images, and cokernels. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA vol. 09, pp. 1011-1020 (2009b) · Zbl 1423.55005
[30] Cohen-Steiner, D.; Edelsbrunner, H.; Harer, J.; Mileyko, Y., Lipschitz functions have \(L_p\)-stable persistence, Found. Comput. Math., 10, 2, 127-139 (2010) · Zbl 1192.55007
[31] Curto, C., What can topology tell us about the neural code?, Bull. Am. Math. Soc., 54, 1, 63-78 (2017) · Zbl 1353.92027
[32] Curto, C.; Itskov, V., Cell groups reveal structure of stimulus space, PLoS Comput. Biol., 4, 10, e1000205 (2008)
[33] Dabaghian, Y.; Mémoli, F.; Frank, L.; Carlsson, G., A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Comput. Biol., 8, 8, e1002581 (2012)
[34] de Silva, V.; Morozov, D.; Vejdemo-Johansson, M., Persistent cohomology and circular coordinates, Discrete Comput. Geom., 45, 737-759 (2011) · Zbl 1216.68322
[35] de Silva, V.; Munch, E.; Stefanou, A., Theory of interleavings on categories with a flow, Theory Appl. Categ., 33, 21, 583-607 (2018) · Zbl 1433.18002
[36] Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 345-354 (2014) · Zbl 1395.68299
[37] Di Fabio, B.; Landi, C., A Mayer-Vietoris formula for persistent homology with an application to shape recognition in the presence of occlusions, Found. Comput. Math., 11, 5, 499-527 (2011) · Zbl 1231.55004
[38] Edelsbrunner, H.; Harer, J., Computational Topology: An Introduction (2010), Providence: American Mathematical Society, Providence
[39] Edelsbrunner, H.; Letscher, D.; Zomorodian, A., Topological persistence and simplification, Discrete Comput. Geom., 28, 511-533 (2002) · Zbl 1011.68152
[40] Fasy, B.T., Wang, B.: Exploring persistent local homology in topological data analysis. In: 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6430-6434. IEEE (2016)
[41] Frosini, P., A distance for similarity classes of submanifolds of a Euclidean space, Bull. Aust. Math. Soc., 42, 3, 407-416 (1990) · Zbl 0707.53004
[42] Frosini, P.; Landi, C., Size theory as a topological tool for computer vision, Pattern Recogn. Image Anal., 9, 4, 596-603 (1999)
[43] Gabriel, P., Unzerlegbare darstellungen i, Manuscr. Math., 6, 1, 71-103 (1972) · Zbl 0232.08001
[44] Gameiro, M.; Mischaikow, K.; Kalies, W., Topological characterization of spatial-temporal chaos, Phys. Rev. E, 70, 3, 035203 (2004)
[45] Gameiro, M.; Hiraoka, Y.; Izumi, S.; Kramar, M.; Mischaikow, K.; Nanda, V., A topological measurement of protein compressibility, Jpn. J. Ind. Appl. Math., 32, 1, 1-17 (2015) · Zbl 1320.55004
[46] Ghrist, R., Barcodes: the persistent topology of data, Bull. Am. Math. Soc., 45, 61-75 (2008) · Zbl 1391.55005
[47] Ghrist, R.: Elementary Applied Topology. Createspace Seattle (2014) · Zbl 1427.55001
[48] 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)
[49] Hatcher, A., Algebraic Topology (2002), Cambridge: Cambridge University Press, Cambridge
[50] Hu, G.; Yang, J.; Liu, W., Instability and controllability of linearly coupled oscillators: Eigenvalue analysis, Phys. Rev. E, 58, 4440-4453 (1998) · Zbl 1341.35123
[51] Kaczynski, T.; Mischaikow, K.; Mrozek, M., Computational Homology, Applied Mathematical Sciences (2004), New York: Springer, New York
[52] Kališnik, S., Tropical coordinates on the space of persistence barcodes, Found. Comput. Math. (2018) · Zbl 1423.55007
[53] Kasson, PM; Zomorodian, A.; Park, S.; Singhal, N.; Guibas, LJ; Pande, VS, Persistent voids: a new structural metric for membrane fusion, Bioinformatics, 23, 1753-1759 (2007)
[54] Khasawneh, F.A., Munch, E.: Exploring equilibria in stochastic delay differential equations using persistent homology. In: Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, 17-20 August 2014, Buffalo, NY, USA. Paper no. DETC2014/VIB-35655 (2014)
[55] Khasawneh, FA; Munch, E., Chatter detection in turning using persistent homology, Mech. Syst. Signal Process., 70-71, 527-541 (2016)
[56] Khasawneh, FA; Munch, E., Utilizing Topological Data Analysis for Studying Signals of Time-Delay Systems, 93-106 (2017), Cham: Springer International Publishing, Cham · Zbl 1387.34112
[57] Kramár, M.; Levanger, R.; Tithof, J.; Suri, B.; Xu, M.; Paul, M.; Schatz, MF; Mischaikow, K., Analysis of Kolmogorov flow and Rayleigh-Bénard convection using persistent homology, Physica D, 334, 82-98 (2016) · Zbl 1415.76582
[58] Mileyko, Y.; Mukherjee, S.; Harer, J., Probability measures on the space of persistence diagrams, Inverse Probl., 27, 12, 124007 (2011) · Zbl 1247.68310
[59] Mischaikow, K.; Nanda, V., Morse theory for filtrations and efficient computation of persistent homology, Discrete & Comput. Geom., 50, 2, 330-353 (2013) · Zbl 1278.57030
[60] Mischaikow, K.; Mrozek, M.; Reiss, J.; Szymczak, A., Construction of symbolic dynamics from experimental time series, Phys. Rev. Lett., 82, 6, 1144 (1999)
[61] Munch, E., A user’s guide to topological data analysis, J. Learn. Anal., 4, 2, 47-61 (2017)
[62] Munch, E.; Turner, K.; Bendich, P.; Mukherjee, S.; Mattingly, J.; Harer, J., Probabilistic Fréchet means for time varying persistence diagrams, Electron. J. Stat., 9, 1, 1173-1204 (2015) · Zbl 1348.68285
[63] Nanda, V.; Sazdanović, R., Simplicial Models and Topological Inference in Biological Systems, 109-141 (2014), Berlin: Springer, Berlin · Zbl 1290.92005
[64] Opron, K.; Xia, K.; Wei, GW, Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis, J. Chem. Phys., 140, 234105 (2014)
[65] Opron, K., Xia, K., Wei, G.W.: Communication: Capturing protein multiscale thermal fluctuations. J. Chem. Phys. 142(211101) (2015)
[66] Ott, E.; Grebogi, C.; Yorke, JA, Controlling chaos, Phys. Rev. Lett., 64, 11, 1196 (1990) · Zbl 0964.37501
[67] Otter, N.; Porter, MA; Tillmann, U.; Grindrod, P.; Harrington, HA, A roadmap for the computation of persistent homology, EPJ Data Sci., 6, 1, 17 (2017)
[68] Oudot, SY, Persistence Theory: From Quiver Representations to Data Analysis (Mathematical Surveys and Monographs) (2017), Providence: American Mathematical Society, Providence
[69] Oudot, SY; Sheehy, DR, Zigzag zoology: rips zigzags for homology inference, Found. Comput. Math., 15, 5, 1151-1186 (2015) · Zbl 1335.68285
[70] Park, JK; Jernigan, R.; Wu, Z., Coarse grained normal mode analysis vs. refined gaussian network model for protein residue-level structural fluctuations, Bull. Math. Biol., 75, 1, 124-160 (2013) · Zbl 1402.92330
[71] Perea, J.A.: Persistent homology of toroidal sliding window embeddings. In: 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE (2016). 10.1109/icassp.2016.7472916
[72] Perea, JA; Harer, J., Sliding windows and persistence: an application of topological methods to signal analysis, Found. Comput. Math., 15, 3, 799-838 (2015) · Zbl 1325.37054
[73] Perea, JA; Deckard, A.; Haase, SB; Harer, J., Sw1pers: sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data, BMC Bioinform., 16, 1, 257 (2015)
[74] Perea, J.A., Munch, E., Khasawneh, F.A.: Approximating continuous functions on persistence diagrams using template functions (2019). arXiv:1902.07190
[75] Radivojac, P.; Obradovic, Z.; Smith, DK; Zhu, G.; Vucetic, S.; Brown, CJ; Lawson, JD; Dunker, AK, Protein flexibility and intrinsic disorder, Protein Sci., 13, 1, 71-80 (2004)
[76] Reininghaus, J., Huber, S., Bauer, U., Kwitt, R.: A stable multi-scale kernel for topological machine learning. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4741-4748 (2015)
[77] Robins, V., Towards computing homology from finite approximations, Topol. Proc., 24, 503-532 (1999) · Zbl 1026.55009
[78] Robins, V.; Meiss, JD; Bradley, E., Computing connectedness: an exercise in computational topology, Nonlinearity, 11, 4, 913 (1998) · Zbl 0957.54010
[79] Robins, V.; Meiss, JD; Bradley, E., Computing connectedness: disconnectedness and discreteness, Physica D, 139, 3-4, 276-300 (2000) · Zbl 1098.37546
[80] Robinson, M., Topological Signal Processing (2014), Berlin: Springer, Berlin · Zbl 1294.94001
[81] Singh, G.; Mémoli, F.; Ishkhanov, T.; Sapiro, G.; Carlsson, G.; Ringach, DL, Topological analysis of population activity in visual cortex, J. Vis., 8, 8, 11-11 (2008)
[82] Stolz, BJ; Harrington, HA; Porter, MA, Persistent homology of time-dependent functional networks constructed from coupled time series, Chaos Interdiscip. J. Nonlinear Sci., 27, 4, 047410 (2017)
[83] Tausz, A., Vejdemo-Johansson, M., Adams, H.: JavaPlex: a research software package for persistent (co)homology. Software available at http://code.google.com/p/javaplex (2011) · Zbl 1402.65186
[84] Tralie, CJ; Perea, JA, (Quasi) periodicity quantification in video data, using topology, SIAM J. Imaging Sci., 11, 2, 1049-1077 (2018) · Zbl 1401.65025
[85] Turner, K.; Mileyko, Y.; Mukherjee, S.; Harer, J., Fréchet means for distributions of persistence diagrams, Discrete Comput. Geom., 52, 1, 44-70 (2014) · Zbl 1296.68182
[86] Vejdemo-Johansson, M.; Pokorny, FT; Skraba, P.; Kragic, D., Cohomological learning of periodic motion, Appl. Algebra Eng. Commun. Comput., 26, 1-2, 5-26 (2015) · Zbl 1331.68236
[87] Wang, B.; Wei, GW, Object-oriented persistent homology, J. Comput. Phys., 305, 276-299 (2016) · Zbl 1349.55004
[88] Wei, GW; Zhan, M.; Lai, CH, Tailoring wavelets for chaos control, Phys. Rev. Lett., 89, 284103 (2002)
[89] Xia, K.; Feng, X.; Tong, Y.; Wei, GW, Persistent homology for the quantitative prediction of fullerene stability, J. Comput. Chem., 36, 6, 408-422 (2015)
[90] Xia, K.; Wei, GW, Molecular nonlinear dynamics and protein thermal uncertainty quantification, Chaos Interdiscip. J. Nonlinear Sci., 24, 013103 (2014) · Zbl 1374.92116
[91] Xia, K.; Wei, GW, Persistent homology analysis of protein structure, flexibility and folding, Int. J. Numer. Methods Biomed. Eng., 30, 814-844 (2014)
[92] Xia, K.; Wei, GW, Multidimensional persistence in biomolecular data, J. Comput. Chem., 36, 20, 1502-1520 (2015)
[93] Xia, K.; Zhao, Z.; Wei, GW, Multiresolution topological simplification, J. Comput. Biol., 22, 9, 887-891 (2015)
[94] Yang, LW; Chng, CP, Coarse-grained models reveal functional dynamics-I. elastic network models-theories, comparisons and perspectives, Bioinform. Biol. Insights, 2, 25-45 (2008)
[95] Zomorodian, A.; Carlsson, G., Computing persistent homology, Discrete Comput. Geom., 33, 2, 249-274 (2005) · Zbl 1069.55003
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