Abstract
Obtaining the free energy of large molecules from quantum mechanical energy functions is a long-standing challenge. We describe a method that allows us to estimate, at the quantum mechanical level, the harmonic contributions to the thermodynamics of molecular systems of large size, with modest cost. Using this approach, we compute the vibrational thermodynamics of a series of diamond nanocrystals, and show that the error per atom decreases with system size in the limit of large systems. We further show that we can obtain the vibrational contributions to the binding free energies of prototypical protein–ligand complexes where exact computation is too expensive to be practical. Our work raises the possibility of routine quantum mechanical estimates of thermodynamic quantities in complex systems.
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Data availability
Data for all systems can be regenerated by running our stochastic Lanczos code on the geometries available at https://doi.org/10.6084/m9.figshare.22258447.v1 (ref. 35). Source Data are provided with this paper.
Code availability
The stochastic Lanczos code is available in Supplementary Software 1. The Gaussian and plane waves implementation is available through PySCF at www.pyscf.org.
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Acknowledgements
We thank S. Murlidaran for help with protein preparation. A.F.W. (theoretical analysis) was supported by the US Department of Energy (grant no. DE-SC0018140). C.L. (computational studies) was supported by the US National Science Foundation (grant no. 1931328). X.Z. (improved DFT implementation for proteins) was supported by the US Department of Energy (grant no. DE-SC0019330). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.
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A.F.W. and G.K.C. conceived the project. A.F.W., C.L. and X.Z. performed the work. All authors contributed to the writing of the paper.
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Garnet K. Chan is a co-owner of QSimulate, Inc. The other authors declare no competing interests.
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Nature Computational Science thanks Jan Jensen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Jie Pan, in collaboration with the Nature Computational Science team.
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Extended data
Extended Data Fig. 1 Harmonic binding free energies of TNKS.
Harmonic thermal contributions for the TNKS2 complex computed at the xTB level. a) An image of the TNKS2 complex. The truncated part of the protein is shown as transparent, while the remaining two protein chains and the ligand are colored orange, yellow and grey, respectively. Image rendered by VMD. b) The thermal enthalpy and entropy, and free energy of binding for the TNKS2 system for varying Lanczos order (m = 8, 16, 32 from above to below). The exact value is given by the grey vertical line. Box edges represent mean ± one standard error of 100 samples, center line represents the median, whiskers represent two standard errors, and fliers are data beyond two standard errors. c) Statistical errors in binding free energy quantities as a function of % of cost of the exact calculation (error bars denote error of error). The detailed error estimation protocol is in Supplementary Section 1.
Extended Data Fig. 2 Harmonic binding free energies of HIV protease.
Harmonic thermal contributions for the JE-2147-HIV protease complex computed at the xTB level. a) An image of the JE-2147-HIV protease complex. The two protein chains and the ligand are shown in yellow, green, and grey respectively. Image rendered by VMD. b) Statistical errors in binding free energy quantities as a function of % of cost of the exact calculation (error bars denote error of error). The detailed error estimation protocol is in Supplementary Section 1.
Supplementary information
Supplementary Information
Supplementary Sections 1–4, Figs. 1–4, Tables 1 and 2, and Discussion.
Supplementary Data 1
Source Data for Supplementary Fig. 1. Random samples of harmonic free energies.
Supplementary Software 1
Computer code for stochastic sampling of harmonic free energies.
Source data
Source Data Fig. 1
Random samples of harmonic free energies of diamonds.
Source Data Extended Data Table/Fig. 1
Random samples of harmonic free energies of TNKS.
Source Data Extended Data Table/Fig. 2
Random samples of harmonic free energies of HIV protease.
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White, A.F., Li, C., Zhang, X. et al. Quantum harmonic free energies for biomolecules and nanomaterials. Nat Comput Sci 3, 328–333 (2023). https://doi.org/10.1038/s43588-023-00432-3
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DOI: https://doi.org/10.1038/s43588-023-00432-3
