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Manifold-based transformation of probability distributions: Application to the inverse problem of reconstructing distributions from experimental data

Tomotaka Oroguchi1,2,*, Rintaro Inoue3, and Masaaki Sugiyama3

  • 1Department of Physics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
  • 2RIKEN SPring-8 Center, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan
  • 3Institute for Integrated Radiation and Nuclear Science, Kyoto University, Kumatori, Sennan-gun, Osaka 590-0494, Japan

  • *Contact author: oroguchi@phys.keio.ac.jp
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Phys. Rev. Research 8, 023291 – Published 15 June, 2026

DOI: https://doi.org/10.1103/vkck-1pcm

Abstract

Information geometry is a mathematical framework that elucidates the manifold structure of the probability distribution space (p space), providing a systematic approach to transforming probability distributions (PDs). In this study, we utilized information geometry to address the inverse problems associated with reconstructing PDs from experimental data. Our initial finding is that the Kullback-Leibler divergence, often considered nonmetric owing to its asymmetry, can serve as a valid metric under specific geometric conditions on the manifold. Based on this finding, we formulated the manifold-based gradient descent (MBGD) method, which was employed to visualize the internal structures—represented as PDs—of two types of systems: those with static constituent elements and those with dynamic state transitions. Through the application of MBGD, we successfully reconstructed the underlying PDs for both types of systems, outperforming the standard gradient descent methods that neglect the manifold structure of p space. Therefore, the present results demonstrate the essentiality of accounting for the manifold structure of p space in the inverse problems of reconstructing PDs. The ability of MBGD to accurately reconstruct PDs for systems with dynamic state transitions underscores its potential to provide valuable physical insights by visualizing internal structures.

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References (118)

  1. S. -I. Amari and H. Nagaoka, Methods of Information Geometry (Oxford University Press, New York, 2000), Vol. 191.
  2. N. N. Chentsov, Statistical decision rules and optimal inference, Translations of Mathematical Monographs (American Mathematical Society, Providence, USA, 1982), Vol. 53.
  3. R. A. Fisher, Theory of statistical estimation, Math. Proc. Camb. Philos. Soc. 22, 700 (1925).
  4. C. R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta. Math. Soc. 37, 81 (1945).
  5. H. Nagaoka and S.-I. Amari, Differential geometry of smooth families of probability distribution s, Technical Report METR 82-7, Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo, 1982.
  6. S.-I. Amari, Differential-Geometric Methods in Statistics (Springer, Berlin, 1985), Vol. 28.
  7. S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Stat. 22, 79 (1951).
  8. S. Eguchi, Second order efficiency of minimum contrast estimators in a curved exponential family, Ann. Stat. 11, 793 (1983).
  9. S. Eguchi, A characterization of second order efficiency in a curved exponential family, Ann. Inst. Stat. Math. 36, 199 (1984).
  10. S. -I. Amari, Information Geometry and Its Applications, Applied Mathematical Sciences (Springer, Tokyo, 2016), Vol. 194.
  11. F. Nielsen, Progress in Information Geometry: Theory and Applications (Springer, Switzerland, 2021).
  12. D. Felice, C. Cafaro, and S. Mancini, Information geometric methods for complexity, Chaos 28, 032101 (2018).
  13. H. Janyszek and R. Mrugala, Riemannian geometry and the thermodynamics of model magnetic systems, Phys. Rev. A 39, 6515 (1989).
  14. G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation, Rev. Mold. Phys. 67, 605 (1995).
  15. W. Janke, D. A. Johnston, and R. P. K. C. Malmini, Information geometry of the Ising model on planar random graphs, Phys. Rev. E 66, 056119 (2002).
  16. D. C. Brondy and A. Ritz, Information geometry of finite Ising models, J. Geometry Phys. 47, 207 (2003).
  17. D. Felice, S. Mancini, and M. Pettini, Quantifying networks complexity from information geometry viewpoint, J. Math. Phys. 55, 043505 (2014).
  18. R. Franzosi, D. Felice, S. Mancini, and P. Pettini, A geometric entropy detecting the Erdös-Renyi phase transition, Eur. Phys. Lett. 111, 20001 (2015).
  19. N. Ay, Information geometry on complexity and stochastic interaction, Entropy 17, 2432 (2015).
  20. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  21. S. Ito, Stochastic thermodynamic interpretation of information geometry, Phys. Rev. Lett. 121, 030605 (2018).
  22. S. Ito, Stochastic time evolution, information geometry, and the Cramér-Rao bound, Phys. Rev. X 10, 021056 (2020).
  23. A. Uhlmann, Density operators as an arena for differential geometry, Rep. Math. Phys. 33, 253 (1993).
  24. D. Petz and C. Sudar, Geometries of quantum states, J. Math. Phys. 37, 2662 (1996).
  25. P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions, Phys. Rev. Lett. 99, 100603 (2007).
  26. P. Kumar, S. Mahapatra, P. Phukon, and T. Sarkar, Geodesics in information geometry: Classical and quantum phase transitions, Phys. Rev. E 86, 051117 (2012).
  27. A. Fujiwara, Dually flat structures induced from monotone metrics on a two-level quantum state space, Eur. Phys. J. Plus 135, 860 (2020).
  28. P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (SIAM, Philadelphia, 2010).
  29. M. Bertero, P. Boccacci, and C. De Mol, Introduction to Inverse Problems in Imaging (CRC Press, Boca Raton, FL, 2022).
  30. G. Tóth, Simultaneous Monte Carlo determination of particle size distribution and pair-correlation function of spherical colloids from a diffraction experiment, Langmuir 15, 6718 (1999).
  31. S. Martelli and P. E. D. Nunzio, Particle size distribution of nanospheres by Monte Carlo fitting of small angle X-ray scattering curves, Part. Part. Syst. Charact. 19, 247 (2002).
  32. J. Pikunic, C. Clinard, N. Cohaut, K. E. Gubbins, J. M. Guet, R. J. -M. Pellenq, I. Rannou, and J. N. Rouzaud, Structural modeling of porous carbons: Constrained reverse Monte Carlo method, Langmuir, 19, 8565 (2003).
  33. T. Sato, M. Sugiyama, K. Itoh, K. Mori, T. Fukunaga, M. Misawa, T. Otomo, and S. Takata, Structural difference between liquidlike and gaslike phases in supercritical fluid, Phys. Rev. E 78, 051503 (2008).
  34. H. Hagita, R. L. McGreevy, T. Arai, M. Inui, K. Matsuda, and K. Tamura, Frist example of multi-scale reverse Monte Carlo modeling for small-angle scattering experimental data using reverse mapping from coarse-grained particles to atoms, J. Phys.: Condens. Matter 22, 404215 (2010).
  35. J. Köfinger, L. S. Stelzl, K. Reuter, C. Allande, K. Reichel, and G. Hummer, Efficient ensemble refinement by reweighting, J. Chem. Theory Comput. 15, 3390 (2019).
  36. S. Bottaro, T. Bengsten, and K. Lindorff-Larsen, Integrating molecular simulation and experimental data: A Bayesian/maximum entropy reweighting approach, Methods Mol. Biol. 2112, 219 (2020).
  37. T. H. Edwards and S. Stoll, Optimal Tikhonov regularization for DEER spectroscopy, J. Mag. Res. 288, 58 (2018).
  38. E. J. Hustedt, R. A. Stein, and H. S. Mchaourab, Protein functional dynamics from the rigorous global analysis of DEER data: Conditions, components, and conformations, J. Gen. Physiol. 153, e201711954 (2021).
  39. M. Cordova, M. Balodis, B. S. de Almeida, M. Ceriotti, and L. Emsley, Bayesian probabilistic assignment of chemical shifts in organic solids, Sci. Adv. 7, eabk2341 (2021).
  40. R. Grau-Crespo, S. Hamad, S. R. G. Balestra, R. Issa, T. D. Sparks, A. Femandes, B. L. Griffiths, R. F. Moran, D. Mackay, and S. E. Ashbrook, Capturing local compositional fluctuations in NMR modeling of solid solutions, Chem. Sci. 16, 19357 (2025).
  41. C.-J. Feng, B. Dhayalan, and A. Tokmakoff, Refinement of peptide conformational ensembles by 2D IR spectroscopy: Application to Ala-Ala-Ala, Biophys. J. 114, 2820 (2018).
  42. P. N. Akbar and R. Blümel, Inverse infrared spectral deconvolution for quantitative analysis of polymer mixtures in scattering media, Anal. Chem. 97, 23080 (2025).
  43. A. L. Wilson, C. Outeiral, S. E. Dowd, A. J. Doig, P. L. A. Popelier, J. P. Waltho, and A. Almond, Deconvolution of conformational exchange from Raman spectra of aqueous RNA nucleosides, Comm. Chem. 3, 56 (2020).
  44. A. S. Mishchenko, H. Matsui, and T. Hasegawa, Distribution of localized states from fine analysis of electron spin resonance spectra of organic semiconductors: Physical meaning and methodology, Phys. Rev. B 85, 085211 (2012).
  45. S. Tokuda, S. Souma, K. Segawa, T. Takahashi, Y. Ando, T. Nakanishi, and T. Sato, Unveiling quasiparticle dynamics of topological insulators through Bayesian modelling, Comm. Phys. 4, 170 (2021).
  46. Z. Hradil, Quantum-state estimation, Phys. Rev. A 55, R1561 (1997).
  47. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
  48. R. Blume-Kohout, Optimal, reliable estimation of quantum states, New J. Phys. 12, 043034 (2010).
  49. J. Suzuki, Y. Yang, and M. Hayashi, Quantum state estimation with nuisance parameters, J. Phys. A Math. Theor. 53, 453001 (2020).
  50. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, Phase retrieval with application to optical imaging, IEEE Signal Process. Mag. 32, 87 (2015).
  51. K. Teilum, J. G. Olsen, and B. B. Kragelund, Functional aspects of protein flexibility, Cell. Mol. Life Sci. 66, 2231 (2009).
  52. K. Teilum, J. G. Olsen, and B. B. Kragelund, Protein stability, flexibility and function, Biochim. Biophys. Acta 1814, 969 (2011).
  53. M. D. Miller and G. N. Philips, Jr., Moving beyond static snapshots: Protein dynamics and the Protein Data Bank, J. Biol. Chem. 296, 100749 (2021).
  54. R. Nussinov, Y. Liu, W. Zhang, and H. Jang, Protein conformational ensembles in function: Rules and mechanics, RSC Chem. Biol. 4, 850 (2023).
  55. T. Oroguchi, H. Hashimoto, T. Shimizu, M. Sato, and M. Ikeguchi, Intrinsic dynamics of restriction endonuclease EcoO109I studied by molecular dynamics simulations and X-ray scattering data analysis, Biophys. J. 96, 2808 (2009).
  56. R. Inoue, T. Oroguchi, T. Oda, B. Farago, A. Martel, L. Porcar, M. Sato, and M. Sugiyama, Internal dynamics of multi-domain protein as revealed by an optimized neutron spin echo measurement and all-atom molecular dynamics simulation, Phys. Rev. Res. 5, 043154 (2023).
  57. T. Oroguchi and M. Nakasako, Changes in hydration structure are necessary for collective motions of a multi-domain protein, Sci. Rep. 6, 26302 (2016).
  58. M. Karplus and J. A. McCammon, Molecular dynamics simulations of biomolecules, Nat. Struct. Biol. 9, 646 (2002).
  59. S. Bottaro and K. Lindorff-Larsen, Biophysical experiments and biomolecular simulations: A perfect match? Science 361, 355 (2018).
  60. M. Bonomi, G. T. Heller, C. Camilloni, and M. Vendruscolo, Principles of protein structural ensemble determination, Curr. Opin. Struct. Biol. 42, 106 (2017).
  61. S. -I. Amari, Differential geometry of curved exponential families–Curvature and information loss, Ann. Stat. 10, 357 (1982).
  62. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  63. K. P. Murphy, Probabilistic Machine Learning: An Introduction (The MIT Press, Cambridge, MA, 2022).
  64. D. P. Bertsekas, Convex Optimization Algorithms (Athena Scientific, Belmont, MA, 2015).
  65. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).
  66. C. Witkov and K. Zengel, Chi-Squared Data Analysis and Model Testing for Beginners, (Oxford University Press, Oxford, 2019).
  67. D. I. Svergun, C. Barberato, and M. H. J. Koch, CRYSOL–A program to evaluate X-ray solution scattering of biological macromolecules from atomic coordinates, J. Appl. Cryst. 28, 768 (1995).
  68. H. Kenzaki, N. Koga, N. Hori, Kanada R, W. Li, K. Okazaki, X. Yao, and S. Takada, CafeMol: A coarse-grained biomolecular simulator for simulating proteins at work, J. Chem. Theory Comput. 7, 1979 (2011).
  69. J. A. Manso, D. Nunes-Costa, S. Macedo-Ribeiro, N. Empadinhas, and P. J. B. Pereira, Molecular fingerprints for a novel enzyme family in Actinobacteria with glucosamine kinase activity, mBio 10, e00239-19 (2019).
  70. A. Guinier, G. Fourmet, C. B. Walker, and K. L. Yodowitch, Small-angle Scattering of X-rays (John Wiley and Sons, New York, 1955).
  71. See Supplemental Material at http://link.aps.org/supplemental/10.1103/vkck-1pcm for Method S1: All-atom MD simulations; Method S2: Coarse-grained MD simulations; Method S3: Calculation of SAXS data from CGMD simulations; Method S4: Q-dependent experimental errors of SAXS data; Method S5: Preparation for initial distribution models for reconstruction simulations; Note S1: Dependence of MBGD results for the particle size distribution system on step size τ; Note S2: Analyses on conformational motions of transferrin; Note S3: Determination of projection coordinates and state resolution for representing probability distribution; Note S4: Dependencies of the calculated SAXS data of transferrin on conformational changes; Note S5: Reconstruction simulation of conformational ensembles of MtEPSPS from SAXS data; Note S6: Reconstruction simulation of conformational ensembles of guanylate kinase from SAXS data; Note S7: Dependence of MBGD results for the protein conformational-ensemble systems on the step size τ; Note S8: Reconstruction simulation of conformational ensembles of guanylate kinase with increased amount of experimental data; Note S9: Reconstruction simulation of conformational ensembles of transferrin from pseudoexperimental data including noise; Note S10: Details in application of MBGD to experimental SAXS data of SjGlcNK; and Table S1, which also includes Refs. [113, 114, 115, 116, 117, 118].
  72. H. B. Curry, The method of steepest descent for non-linear minimization problems, Quart. Appl. Math. 2, 258 (1944).
  73. R. T. A. MacGillivray, S. A. Moore, J. Chen, B. F. Anderson, H. Baker, Y. Luo, M. Bewley, C. A. Smith, M. E. P. Murphy, Y. Wang, A. B. Mason, R. C. Woodworth, G. D. Brayer, and E. N. Baker, Two high-resolution crystal structures of the recombinant N-lobe of human transferrin reveal a structural change implicated in iron release, Biochemistry 37, 7919 (1998).
  74. P. D. Jeffrey, M. C. Bewley, R. T. A. MacGillivray, A. B. Mason, R. C. Woodworth, and E. N. Baker, Ligand-induced conformational change in transferrins: Crystal structure of open form of the N-terminal half-molecule of human Transferrin, Biochemistry 37, 13978 (1998).
  75. H. D. T. Mertens and D. I. Svergun, Structural characterization of proteins and complexes using small-angle X-ray solution scattering, J. Struct. Biol. 172, 128 (2010).
  76. C. A. Brosey and J. A. Tainer, Evolving SAXS versatility: Solution X-ray scattering for macromolecular architecture, functional landscapes, and integrative structural biology, Curr. Opin. Cell Biol. 58, 197 (2019).
  77. T. Ekimoto, Y. Kokabu, T. Oroguchi, and M. Ikeguchi, Combination of coarse-grained molecular dynamics simulations and small-angle X-ray scattering experiments, Biophys. Physicobiol. 16, 377 (2019).
  78. E. Valentini, A. G. Kikhney, G. Previtali, C. M. Jeffries, and D. I. Svergun, SASBDB, a repository for biological small-angle scattering data, Nucleic Acids Res. 43, D357 (2015).
  79. M. R. Marques, A. Vaso, J. R. Neto, M. A. Fossey, J. S. Oliveira, L. A. Basso, D. S. dos Santos, W. F. de Azevedo, Jr., and M. S. Palma, Dynamics of glyphosate-induced conformational changes of Mycobacterium Tuberculosis 5-enolpyruvylshikimate-3-phosphate Synthase (EC 2.5.1.19) determined by hydrogen-deuterium exchange and electrospray mass spectrometry, Biochemistry 47, 7509 (2008).
  80. J. Blaszczyk, Y. Li, Y. Honggao, and X. Ji, Crystal structure of unligated guanylate kinase from yeast reveals GMP-induced conformational changes, J. Mol. Biol. 307, 247 (2001).
  81. T. Stehele and G. E. Schulz, Refined structure of the complex between guanylate kinase and its substrate GMP at 2.0 Å resolution, J. Mol. Biol. 224, 1127 (1992).
  82. A. Kitao, F. Hirata, and N. Gö, The effects of solvent on the conformation and the collective motions of protein: Normal mode analysis and molecular dynamics simulations of melittin in water and in vacuum, Chem. Phys. 158, 447 (1991).
  83. D. D. Boehr, R. Nussinov, and P. E. Wright, The role of dynamic conformational ensembles in biomolecular recognition, Nat. Chem. Biol. 5, 789 (2009).
  84. W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, New York, 1976).
  85. S.-I. Amari, Natural gradient works efficiently in learning, Neural Comput. 10, 251 (1998).
  86. Y. Chen, D. Z. Huang, J. Huang, S. Reich, and A. M. Stuart, Sampling via gradient flows in the space of probability measures, Math. Comp. (2026).
  87. P. Izmailov, P. Kirichenko, M. Finzi, and A. G. Wilson, Semi-supervised learning with normalizing flows, Proc. Mach. Learn. Res. 119, 4615 (2020).
  88. G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan, Normalizing flows for probabilistic modeling and inference, J. Mach. Learn Res. 22, 1 (2021).
  89. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, Beyond crystallography: Diffractive imaging using coherent x-ray light sources, Science 348, 530 (2015).
  90. L. Borcea, Electrical impedance tomography, Inverse Prob. 18, R99 (2002).
  91. Y. Hoshi and Y. Yamada, Overview of diffuse optical tomography and its clinical applications, J. Biomed. Opt. 21, 091312 (2016).
  92. L. Wang, T. J. Martinez, and V. S. Pande, Building force fields: An automatic, systematic and reproducible approach, J. Phys. Chem. Lett. 5, 1885 (2014).
  93. L. Wang, K. A. McKierman, J. Gomes, K. A. Beauchamp, T. Head-Gordon, J. E. Rice, W. C. Swope, T. J. Martínez, and V. S. Pande, Building a more predictive protein force field: A systematic and reproducible route to AMBER-FB15, J. Phys. Chem. B 121, 4023 (2017).
  94. C. Vogel, M. Bashton, N. D. Kerrison, C. Chothia, and A. Teichmann, Structure, function and evolution of multidomain proteins, Curr. Opin. Cell Biol. 14, 208 (2004).
  95. D. Ekman, A. K. Björklund, J. Frey-Skött, and A. Elofsson, Multi-domain proteins in the three kingdoms of life: Orphan domains and other unassigned regions, J. Mol. Biol. 348, 231 (2005).
  96. V. N. Uversky, Unusual biophysics of intrinsically disordered proteins, Biochim. Biophys. Acta 1834, 932 (2013).
  97. C. J. Oldfield and K. Dunker, Intrinsically disordered proteins and intrinsically disordered protein regions, Annu. Rev. Biochem. 83, 553 (2014).
  98. Y. Qi and B. Zhang, Predicting three-dimensional genome organization with chromatin states, PLoS Comput. Biol. 15, e1007024 (2019).
  99. H. Tjong, W. Li, R. Kalhor, C. Dai, S. Hao, K. Gong, Y. Zhou, H. Li, X. J. Zhou, M. A. Le Gros, C. A. Larabell, L. Chen, and F. Alber, Population-based 3D genome structure analysis reveals driving forces in spatial genome organization, Proc. Natl. Acad. Sci. USA 113, E1663 (2016).
  100. M. Sonntag, P. K. A. Jagtap, B. Simon, M.–S. Appavou, A. Geerlof, R. Stehle, F. Gable, J. Hennig, and M. Sattler, Segmental, domain-selective perdeuteration and small-angle neutron scattering for structural analysis of multi-domain proteins, Angew. Chem. Int. Ed. 56, 9322 (2017).
  101. H. Nakagawa, T. Saio, M. Nagao, R. Inoue, M. Sugiyama, A. Ajito, T. Tominaga, and Y. Kawakita, Conformational dynamics of a multidomain protein by neutron scattering and computational analysis, Biophys. J. 120, 3341 (2021).
  102. E. Z. Eisenmesser, O. Millet, W. Labeikovsky, D. M. Korzhnev, M. Wolf-Watz, D. A. Bosco, J. J. Skalicky, L. E. Kay, and D. Kern, Intrinsic dynamics of an enzyme underlies catalysis, Nature (London) 438, 117 (2005).
  103. T. R. Alderson and L. E. Kay, NMR spectroscopy captures the essential role of dynamics in regulating biomolecular function, Cell 184, 577 (2021).
  104. A. G. Palmer, III, Enzyme dynamics from NMR spectroscopy, Acc. Chem. Res. 48, 457 (2015).
  105. L. Yu and R. Brüschweiler, Quantitative prediction of ensemble dynamics, shapes and contact propensities of intrinsically disordered proteins, PLoS Comput. Biol. 18, e1010036 (2022).
  106. A. Dashti, G. Mashayekhi, M. Shekhar, D. B. Hail, S. Salah, P. Schwander, A. des Georges, A. Singharoy, J. Frank, and A. Ourmazd, Retrieving functional pathways of biomolecules from single-particle snapshots, Nat. Comm. 11, 4734 (2020).
  107. M. Oide, T. Kato, T. Oroguchi, and M. Nakasako, Energy landscape of domain motion in glutamate dehydrogenase deduced from cryo-electron microscopy, FEBS J. 287, 3472 (2020).
  108. N. Sibille and P. Bernadó, Structural characterization of intrinsically disordered proteins by the combined use of NMR and SAXS, Biochem. Soc. Trans. 40, 955 (2012).
  109. A. G. Kikhney and D. I. Svergun, A practical guide to small angle X-ray scattering (SAXS) of flexible and intrinsically disordered proteins, FEBS Lett. 589, 2570 (2015).
  110. M. Chan-Yao-Chong, D. Durand, and T. Ha-Duong, Molecular dynamics simulations combined with nuclear magnetic resonance and/or small-angle X-ray scattering data for characterizing intrinsically disordered protein conformational ensembles, J. Chem. Inf. Model. 59, 1743 (2019).
  111. B. Vallat, et al., IHMCIF: An extension of the PDBx/mmCIF data standard for integrative structure determination methods, J. Mol. Biol. 436, 168546 (2024).
  112. T. Oroguchi, R. Inoue, and M. Sugiyama, GitHub (2025), https://github.com/TOroguchi/MBGD.
  113. R. Salomon-Ferrer, D. A. Case, and R. C. Walker, An overview of the Amber biomolecular simulation package, WIREs Comput. Mol. Sci. 3, 198 (2013).
  114. U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee, and L. G. Pedersen, Smooth particle mesh Ewald method, J. Chem. Phys. 103, 8577 (1995).
  115. J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys. 23, 327 (1977).
  116. G. P. Bourenkov, G. S. Kachalova, N. Strizhov, M. Bruning, A. Vagin, and H. D. Bartunik, Mycobacterium Tuberculosis epsp synthase in unliganded state, 2006, https://doi.org/10.2210/pdb2BJB/pdb.
  117. G. S. Kachalova, G. P. Burenkow, N. I. Strizhov, M. G. Brunning, and H. D. Bartunik, Mycobacterium Tuberculosis epsp synthase in complex with s3p, 2008, https://doi.org/10.2210/pdb2O0D/pdb.
  118. P. Chen and J. S. Hub, Interpretation of solution X-ray scattering by explicit-solvent molecular dynamics, Biophys. J. 108, 2573 (2015).

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Phys. Rev. Research 8, 023291– Published 15 June, 2026
  • Received 11 October 2024
  • Accepted 15 May 2026
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