(467) Tanizaki, S.; Feig, M. Molecular dynamics simulations of large integral membrane proteins with an implicit membrane model. *The Journal of Physical Chemistry B* **2006**, *110*, 548–556.

(468) Feig, M.; Chocholoušová, J.; Tanizaki, S. Extending the horizon: towards the efficient modeling of large biomolecular complexes in atomic detail. *Theor. Chem. Acc.* **2006**, *116*, 194–205.

(469) Gopal, S. M.; Mukherjee, S.; Cheng, Y.-M.; Feig, M. PRIMO/PRIMONA: a coarse-grained model for proteins and nucleic acids that preserves near-atomistic accuracy. *PROTEINS: Struct., Func. and Bioinf.* **2010**, *78*, 1266–1281.

(470) Heo, L.; Feig, M. One bead per residue can describe all-atom protein structures. *Structure* **2024**, *32*, 97–111.

(471) Cheng, Y. M.; Gopal, S. M.; Law, S. M.; Feig, M. Molecular dynamics trajectory compression with a coarse-grained model. *IEEE/ACM Transactions on Computational Biology and Bioinformatics* **2012**, *9*, 476–486.

(472) Kar, P.; Gopal, S. M.; Cheng, Y.-M.; Predeus, A.; Feig, M. PRIMO: a transferable coarse-grained force field for proteins. *J. Chem. Theory Comp.* **2013**, *9*, 3769–3788.

(473) MacKerell, A. D., Jr.; Feig, M.; Brooks, C. L., III. Extending the treatment of backbone energetics in protein force fields: limitations of gas-phase quantum mechanics in reproducing protein conformational distributions in molecular dynamics simulations. *J. Comput. Chem.* **2004**, *25*, 1400–1415.

(474) Kar, P.; Gopal, S. M.; Cheng, Y.-M.; Panahi, A.; Feig, M. Transferring the PRIMO coarse-grained force field to the membrane environment: simulations of membrane proteins and helix–helix association. *J. Chem. Theory Comp.* **2014**, *10*, 3459–3472.

(475) MacKerell, A. D., Jr.; Feig, M.; Brooks, C. L., III. Improved treatment of the protein backbone in empirical force fields. *J. Am. Chem. Soc.* **2004**, *126*, 698–699.

(476) Best, R. B.; Zhu, X.; Shim, J.; Lopes, P. E. M.; Mittal, J.; Feig, M.; MacKerell, A. D., Jr. Optimization of the additive CHARMM all-atom protein force field targeting improved sampling of the backbone *φ*, *ψ* and side-chain *χ*₁ and *χ*₂ dihedral angles. *J. Chem. Theory Comp.* **2012**, *8*, 3257–3273.

(477) Kar, P.; Feig, M. In *Advances in Protein Chemistry and Structural Biology*; Karabencheva-Christova, T., Ed.; Academic Press: 2014; Vol. 96, pp 143–180.

(478) Feig, M.; Gopal, S. M.; Vadivel, K.; Stumpff-Kane, A. Multiscale approaches to protein modeling: structure prediction. In *Dynamics, Thermodynamics and Macromolecular Assemblies*; Kolinski, A., Ed.; Springer: New York, 2011; pp 85–109.

(479) Predeus, A. V.; Gul, S.; Gopal, S. M.; Feig, M. Conformational Sampling of Peptides in the Presence of Protein Crowders from AA/CG-Multiscale Simulations. *J. Phys. Chem. B* **2012**, *116*, 8610–8620.

(480) Kar, P.; Feig, M. Hybrid all-atom/coarse-grained simulations of proteins by direct coupling of CHARMM and PRIMO force fields. *J. Chem. Theory Comp.* **2017**, *13*, 5753–5765.

(481) Chothia, C.; Levitt, M.; Richardson, D. Helix to helix packing in proteins. *J. Mol. Biol.* **1981**, *145*, 215–250.

(482) Im, W.; Lee, J.; Kim, T.; Rui, H. Novel free energy calculations to explore mechanisms and energetics of membrane protein structure and function. *J. Comput. Chem.* **2009**, *30*, 1622–1633.

(483) Lee, J.; Ham, S.; Im, W. Beta-hairpin restraint potentials for calculations of potentials of mean force as a function of beta-hairpin tilt, rotation, and distance. *J. Comput. Chem.* **2009**, *30*, 1334–1343.

(484) Lee, J.; Im, W. Implementation and application of helix-helix distance and crossing angle restraint potentials. *J. Comput. Chem.* **2007**, *28*, 669–680.

(485) Matthews, E. E.; Zoonens, M.; Engelman, D. M. Dynamic helix interactions in transmembrane signaling. *Cell* **2006**, *127*, 447–450.

(486) Schneider, D.; Engelman, D. M. Motifs of two small residues can assist but are not sufficient to mediate transmembrane helix interactions. *J. Mol. Biol.* **2004**, *343*, 799–804.

(487) de Planque, M. R.; Killian, J. A. Protein-lipid interactions studied with designed transmembrane peptides: role of hydrophobic

matching and interfacial anchoring. *Mol. Memb. Biol.* **2003**, *20*, 271–284.

(488) Lee, J.; Im, W. Restraint potential and free energy decomposition formalism for helical tilting. *Chem. Phys. Lett.* **2007**, *441*, 132–135.

(489) Lee, J.; Im, W. Transmembrane helix tilting: insights from calculating the potential of mean force. *Phys. Rev. Lett.* **2008**, *100*, 018103.

(490) Lee, J.; Im, W. Role of hydrogen bonding and helix-lipid interactions in transmembrane helix association. *J. Am. Chem. Soc.* **2008**, *130*, 6456–6462.

(491) Lee, J.; Chen, J.; Brooks, C. L., III; Im, W. Application of solid-state NMR restraint potentials in membrane protein modeling. *J. Magn. Reson.* **2008**, *193*, 68–76.

(492) Tycko, R.; Stewart, P. L.; Opella, S. J. Peptide plane orientations determined by fundamental and overtone nitrogen 14 NMR. *J. Am. Chem. Soc.* **1986**, *108*, 5419–5425.

(493) Woolf, T. B.; Malkin, V. G.; Malkina, O. L.; Salahub, D. R.; Roux, B. The backbone 15N chemical shift tensor of the gramicidin channel. A molecular dynamics and density functional study. *Chem. Phys. Lett.* **1995**, *239*, 186–194.

(494) Almeida, F.; Opella, S. Fd coat protein structure in membrane environments: structural dynamics of the loop between the hydrophobic trans-membrane helix and the amphipathic in-plane helix. *J. Mol. Biol.* **1997**, *270*, 481–495.

(495) Cheng, X.; Jo, S.; Qi, Y.; Marassi, F. M.; Im, W. Solid-state NMR-restrained ensemble dynamics of a membrane protein in explicit membranes. *Biophys. J.* **2015**, *108*, 1954–1962.

(496) Im, W.; Jo, S.; Kim, T. An ensemble dynamics approach to decipher solid-state NMR observables of membrane proteins. *Biochim. et Biophys. Acta Biomembranes* **2012**, *1818*, 252–262.

(497) Jo, S.; Im, W. Transmembrane helix orientation and dynamics: insights from ensemble dynamics with solid-state NMR observables. *Biophys. J.* **2011**, *100*, 2913–2921.

(498) Kim, T.; Jo, S.; Im, W. Solid-state NMR ensemble dynamics as a mediator between experiment and simulation. *Biophys. J.* **2011**, *100*, 2922–2928.

(499) Bax, A. Weak alignment offers new NMR opportunities to study protein structure and dynamics. *Protein Sci.* **2003**, *12*, 1–16.

(500) Rathinavelan, T.; Im, W. Explicit treatment of force contribution from alignment tensor using overdetermined linear equations and its application in NMR structure determination. *J. Comput. Chem.* **2007**, *28*, 1858–1864.

(501) Losonczi, J. A.; Andrec, M.; Fischer, M. W.; Prestegard, J. H. Order matrix analysis of residual dipolar couplings using singular value decomposition. *J. Magn. Reson.* **1999**, *138*, 334–342.

(502) Rathinavelan, T.; Im, W. A novel strategy to determine protein structures using exclusively residual dipolar coupling. *J. Comput. Chem.* **2008**, *29*, 1640–1649.

(503) Wereszczyński, J.; Andricioaei, I. On structural transitions, thermodynamic equilibrium, and the phase diagram of DNA and RNA duplexes under torque and tension. *Proc. Natl. Acad. Sci. USA* **2006**, *103*, 16200–16205.

(504) Rupakheti, C.; Lamoureux, G.; MacKerell, A. D., Jr.; Roux, B. Statistical mechanics of polarizable force fields based on classical Drude oscillators with dynamical propagation by the dual-thermostat extended Lagrangian. *J. Chem. Phys.* **2020**, *153*, 114108.

(505) Huang, J.; Lemkul, J. A.; Eastman, P. K.; MacKerell, A. D., Jr. Molecular dynamics simulations of explicitly solvated Drude polarizable systems on GPUs: implementation, validation, and benchmark. *J. Comput. Chem.* **2018**, *39*, 1682–1689.

(506) Lemkul, J. A.; Roux, B.; van der Spoel, D.; MacKerell, A. D., Jr. Implementation of extended Lagrangian dynamics in GROMACS for polarizable simulations using the classical Drude oscillator model. *J. Comput. Chem.* **2015**, *36*, 1473–1479.

(507) Jiang, W.; Hardy, D. J.; Phillips, J. C.; MacKerell, A. D., Jr.; Schulten, K.; Roux, B. High-performance scalable molecular dynamics simulations of a polarizable force field based on classical Drude oscillators in NAMD. *J. Phys. Chem. Lett.* **2011**, *2*, 87–92.
(508) Venable, R. M.; Luo, Y.; Gawrisch, K.; Roux, B.; Pastor, R. W. Simulations of anionic lipid membranes: development of interaction-specific ion parameters and validation using NMR data. *J. Phys. Chem. B* **2013**, *117*, 10183–10192.

(509) Han, K.; Venable, R. M.; Bryant, A. M.; Legacy, C. J.; Shen, R.; Li, H.; Roux, B.; Gericke, A.; Pastor, R. W. Graph-theoretic analysis of monomethyl phosphate clustering in ionic solutions. *J. Phys. Chem. B* **2018**, *122*, 1484–1494.

(510) Lamoureux, G.; Harder, E.; Vorobyov, I. V.; Roux, B.; MacKerell, A. D., Jr. A polarizable model of water for molecular dynamics simulations of biomolecules. *Chem. Phys. Lett.* **2006**, *418*, 245–249.

(511) Yu, W.; Lopes, P. E. M.; Roux, B.; MacKerell, A. D., Jr. Six-site polarizable model of water based on the classical Drude oscillator. *J. Chem. Phys.* **2013**, *138*, 034508.

(512) Harder, E.; Anisimov, V. M.; Whitfield, T.; MacKerell, A. D., Jr.; Roux, B. Understanding the dielectric properties of liquid amides from a polarizable force field. *J. Phys. Chem. B* **2008**, *112*, 3509–3521.

(513) Yu, H.; Whitfield, T. W.; Harder, E.; Lamoureux, G.; Vorobyov, I.; Anisimov, V. M.; MacKerell, A. D., Jr.; Roux, B. Simulating monovalent and divalent ions in aqueous solution using a Drude polarizable force field. *J. Chem. Theory Comp.* **2010**, *6*, 774–786.

(514) Nan, Y.; MacKerell, A. D., Jr. Balancing Group I Monatomic Ion-Polar Compound Interactions for Condensed Phase Simulation in the Polarizable Drude Force Field. *J. Chem. Theory Comp.* **2024**, *20*, 3242–3257.

(515) Lin, F. Y.; Lopes, P. E.; Harder, E. D.; Roux, B.; MacKerell, A. D., Jr. Polarizable force field for molecular ions based on the classical Drude oscillator. *J. Chem. Inf. and Mod.* **2018**, *58*, 993–1004.

(516) Villa, F.; MacKerell, A. D., Jr.; Roux, B.; Simonson, T. Classical Drude polarizable force field model for methyl phosphate and its interactions with Mg(2). *J. Phys. Chem. A* **2018**, *122*, 6147–6155.

(517) Kognole, A. A.; Aytenfisu, A. H.; MacKerell, A. D., Jr. Balanced polarizable Drude force field parameters for molecular anions: phosphates, sulfates, sulfamates, and oxides. *J. Mol. Model.* **2020**, *26*, 1–11.

(518) Rupakheti, C. R.; Roux, B.; Dehez, F.; Chipot, C. Modeling induction phenomena in amino acid cation-π interactions. *Theor. Chem. Acc.* **2018**, *137*, 174.

(519) Luo, Y.; Roux, B. Simulation of osmotic pressure in concentrated aqueous salt solutions. *J. Phys. Chem. Lett.* **2010**, *1*, 183–189.

(520) Lin, B.; Lopes, P. E. M.; Roux, B.; MacKerell, A. D., Jr. Kirkwood-Buff analysis of aqueous n-methylacetamide and acetamide solutions modeled by the CHARMM additive and Drude polarizable force fields. *J. Chem. Phys.* **2013**, *139*, 084509.

(521) Li, H.; Ngo, V.; Da Silva, M. C.; Salahub, D. R.; Callahan, K.; Roux, B.; Noskov, S. Y. Representation of ion-protein interactions using the Drude polarizable force-field. *J. Phys. Chem. B* **2015**, *119*, 9401–9416.

(522) Shim, J.; Zhu, X.; Best, R. B.; MacKerell, A. D., Jr. (Ala)(4)-X-(Ala)4 as a model system for the optimization of the $\chi_1$ and $\chi_2$ amino acid side-chain dihedral empirical force field parameters. *J. Comput. Chem.* **2013**, *34*, 593–603.

(523) Khan, H. M.; Grauffel, C.; Broer, R.; MacKerell, A. D., Jr.; Havenith, R. W.; Reuter, N. Improving the force field description of tyrosine-choline cation-π interactions: QM investigation of pPhenol-N(Me)(4)(+) interactions. *J. Chem. Theory Comp.* **2016**, *12*, 5585–5595.

(524) Khan, H. M.; MacKerell, A. D., Jr.; Reuter, N. Cation-π interactions between methylated ammonium groups and tryptophan in the CHARMM36 additive force field. *J. Chem. Theory Comp.* **2019**, *15*, 7–12.

(525) Lin, F. Y.; MacKerell, A. D., Jr. Improved modeling of halogenated ligand-protein interactions using the Drude polarizable and CHARMM additive empirical force fields. *J. Chem. Inf. and Mod.* **2019**, *59*, 215–228.

(526) Croitoru, A.; Park, S. J.; Kumar, A.; Lee, J.; Im, W.; MacKerell, A. D., Jr.; Aleksandrov, A. Additive CHARMM36 force field for nonstandard amino acids. *J. Chem. Theory Comp.* **2021**, *17*, 3554–3570.

(527) Pane, A. J.; Yu, W.; Aytenfisu, A.; Tunyi, J.; Venable, R. M.; MacKerell, A. D., Jr.; Pastor, R. W. Development of CHARMM additive potential energy parameters for α-methyl amino acids. *J. Phys. Chem. B* **2021**, *125*, 11687–11696.

(528) Lopes, P. E. M.; Lamoureux, G.; MacKerell, A. D., Jr. Polarizable empirical force field for nitrogen-containing heteroaromatic compounds based on the classical Drude oscillator. *J. Comput. Chem.* **2009**, *30*, 1821–1838.

(529) Zhu, X.; MacKerell, A. D., Jr. Polarizable empirical force field for sulfur-containing compounds based on the classical Drude oscillator model. *J. Comput. Chem.* **2010**, *31*, 2330–2341.

(530) Baker, C. M.; MacKerell, A. D., Jr. Polarizability rescaling and atom-based Thole scaling in the CHARMM Drude polarizable force field for ethers. *J. Mol. Model.* **2010**, *16*, 567–576.

(531) Baker, C. M.; Lopes, P. E.; Zhu, X.; Roux, B.; MacKerell, A. D., Jr. Accurate calculation of hydration free energies using pair-specific Lennard-Jones parameters in the CHARMM Drude polarizable force field. *J. Chem. Theory Comp.* **2010**, *6*, 1181–1198.

(532) Lopes, P. E. M.; Huang, J.; Shim, J.; Luo, Y.; Li, H.; Roux, B.; MacKerell, A. D., Jr. Polarizable force field for peptides and proteins based on the classical Drude oscillator. *J. Chem. Theory Comp.* **2013**, *9*, 5430–5449.

(533) Huang, J.; MacKerell, A. D., Jr. Induction of peptide bond dipoles drives cooperative helix formation in the (AAQAA)3 peptide. *Biophys. J.* **2014**, *107*, 991–997.

(534) Lemkul, J. A.; Huang, J.; MacKerell, A. D., Jr. Induced dipole-dipole interactions influence the unfolding pathways of wild-type and mutant amyloid β-peptides. *J. Phys. Chem. B* **2015**, *119*, 15574–15582.

(535) Lin, F. Y.; Huang, J.; Pandey, P.; Rupakheti, C.; Li, J.; Roux, B. T.; MacKerell, A. D., Jr. Further optimization and validation of the classical Drude polarizable protein force field. *J. Chem. Theory Comp.* **2020**, *16*, 3221–3239.

(536) Lin, F. Y.; MacKerell, A. D., Jr. Improved modeling of cation-π and Anion-Ring interactions using the Drude polarizable empirical force field for proteins. *J. Comput. Chem.* **2020**, *41*, 439–448.

(537) Aleksandrov, A.; Lin, F. Y.; Roux, B.; MacKerell, A. D., Jr. Combining the polarizable Drude force field with a continuum electrostatic Poisson-Boltzmann implicit solvation model. *J. Comput. Chem.* **2018**, *39*, 1707–1719.

(538) Aleksandrov, A.; Roux, B.; MacKerell, A. D., Jr. pK(a) calculations with the polarizable Drude force field and Poisson-Boltzmann solvation model. *J. Chem. Theory Comp.* **2020**, *16*, 4655–4668.

(539) Denning, E. J.; Priyakumar, U. D.; Nilsson, L.; MacKerell, A. D., Jr. Impact of 2'-hydroxyl sampling on the conformational properties of RNA: Update of the CHARMM all-atom additive force field for RNA. *J. Comput. Chem.* **2011**, *32*, 1929–1943.

(540) Denning, E. J.; MacKerell, A. D., Jr. Intrinsic contribution of the 2'-hydroxyl to RNA conformational heterogeneity. *J. Am. Chem. Soc.* **2012**, *134*, 2800–2806.

(541) Hart, K.; Foloppe, N.; Baker, C. M.; Denning, E. J.; Nilsson, L.; Mackerell, A. D., Jr. Optimization of the CHARMM additive force field for DNA: improved treatment of the BI/BII conformational equilibrium. *J. Chem. Theory Comp.* **2012**, *8*, 348–362.

(542) Xu, Y.; Vanommeslaeghe, K.; Aleksandrov, A.; MacKerell, A. D., Jr.; Nilsson, L. Additive CHARMM force field for naturally occurring modified ribonucleotides. *J. Comput. Chem.* **2016**, *37*, 896–912.

(543) Baker, C. M.; Anisimov, V. M.; MacKerell, A. D., Jr. Development of CHARMM polarizable force field for nucleic acid bases based on the classical Drude oscillator model. *J. Phys. Chem. B* **2011**, *115*, 580–596.
(544) Savelyev, A.; MacKerell, A. D., Jr. All-atom polarizable force field for DNA based on the classical Drude oscillator model. *J. Comput. Chem.* **2014**, *35*, 1219–1239.

(545) Savelyev, A.; MacKerell, A. D., Jr. Balancing the interactions of ions, water and DNA in the Drude polarizable force field. *J. Phys. Chem. B* **2014**, *118*, 6742–6757.

(546) Lemkul, J. A.; Savelyev, A.; MacKerell, A. D., Jr. Induced polarization influences the fundamental forces in DNA base flipping. *J. Phys. Chem. Lett.* **2014**, *5*, 2077–2083.

(547) Savelyev, A.; MacKerell, A. D., Jr. Differential impact of the monovalent ions Li, Na, K, and Rb on DNA conformational properties. *J. Phys. Chem. Lett.* **2015**, *6*, 212–216.

(548) Savelyev, A.; MacKerell, A. D., Jr. Competition among Li, Na, K and Rb monovalent ions for DNA in molecular dynamics simulations using the additive CHARMM36 and Drude polarizable force fields. *J. Phys. Chem. B* **2015**, *119*, 4428–4440.

(549) Lemkul, J. A.; MacKerell, A. D., Jr. Polarizable force field for DNA based on the classical Drude oscillator: I. refinement using quantum mechanical base stacking and conformational energetics. *J. Chem. Theory Comp.* **2017**, *13*, 2053–2071.

(550) Lemkul, J. A.; MacKerell, A. D., Jr. Polarizable force field for DNA based on the classical Drude oscillator: II. microsecond molecular dynamics simulations of duplex DNA. *J. Chem. Theory Comp.* **2017**, *13*, 2072–2085.

(551) Lemkul, J. A.; MacKerell, A. D., Jr. Polarizable force field for RNA based on the classical Drude oscillator. *J. Comput. Chem.* **2018**, *39*, 2624–2646.

(552) Salsbury, A. M.; Lemkul, J. A. Molecular dynamics simulations of the c-kit1 promoter g-quadruplex: importance of electronic polarization on stability and cooperative ion binding. *J. Phys. Chem. B* **2019**, *123*, 148–159.

(553) Chowdhary, J.; Harder, E.; Lopes, P. E.; Huang, L.; MacKerell, A. D., Jr.; Roux, B. A polarizable force field of dipalmitoylphosphatidylcholine based on the classical Drude model for molecular dynamics simulations of lipids. *J. Phys. Chem. B* **2013**, *117*, 9142–9160.

(554) Sengul, M. Y.; MacKerell, A. D., Jr. Accurate modeling of RNA hairpins through the explicit treatment of electronic polarizability with the classical Drude oscillator force field. *J. Comput. Biophys. Chem.* **2022**, *21*, 461–471.

(555) Klauda, J. B.; Brooks, B. R.; MacKerell, A., Jr.; Venable, R. M.; Pastor, R. W. An ab initio study on the torsional surface of alkanes and its effect on molecular simulations of alkanes and a DPPC bilayer. *J. Phys. Chem. B* **2005**, *109*, 5300–5311.

(556) Levine, Z. A.; Venable, R. M.; Watson, M. C.; Lerner, M. G.; Shea, J.-E.; Pastor, R. W.; Brown, F. L. H. Determination of biomembrane bending moduli in fully atomistic simulations. *J. Am. Chem. Soc.* **2014**, *136*, 13582–13585.

(557) Sodt, A. J.; Pastor, R. W. Bending free energy from simulation: correspondence of planar and inverse hexagonal lipid phases. *Biophys. J.* **2013**, *104*, 2202–2211.

(558) Venable, R. M.; Brown, F. L. H.; Pastor, R. W. Mechanical properties of lipid bilayers from molecular dynamics simulation. *Chem. Phys. Lipids* **2015**, *192*, 60–74.

(559) Leonard, A. N.; Wang, E.; Monje-Galvan, V.; Klauda, J. B. Developing and testing of lipid force fields with applications to modeling cellular membranes. *Chem. Rev.* **2019**, *119*, 6227–6269.

(560) West, A.; Zoni, V.; Teague, W. E., Jr.; Leonard, A. N.; Vanni, S.; Gawrisch, K.; Tristram-Nagle, S.; Sachs, J. N.; Klauda, J. B. How do ethanolamine plasmalogens contribute to order and structure of neurological membranes? *J. Phys. Chem. B* **2020**, *124*, 828–839.

(561) Yuan, Y.; Yu, Y.; Klauda, J. B. Simulations of diabetic and non-diabetic peripheral nerve myelin lipid bilayers. *J. Phys. Chem. B* **2021**, *125*, 6201–6213.

(562) Yu, Y.; Klauda, J. B. Update of the CHARMM36 united atom chain model for hydrocarbons and phospholipids. *J. Phys. Chem. B* **2020**, *124*, 6797–6812.

(563) Lucker, J.; Kio, M.; Klauda, J. B. Expanding the CHARMM36 united atom chain model for the inclusion of sphingolipids. *J. Phys. Chem. B* **2024**, *128*, 4428–4439.

(564) Yu, Y.; Kramer, A.; Venable, R. M.; Simmonett, A. C.; MacKerell, A. D., Jr.; Klauda, J. B.; Pastor, R. W.; Brooks, B. R. Semi-automated optimization of the CHARMM36 lipid force field to include explicit treatment of long-range dispersion. *J. Chem. Theory Comp.* **2021**, *17*, 1562–1580.

(565) Yu, Y. L.; Kramer, A.; Venable, R. M.; Brooks, B. R.; Klauda, J. B.; Pastor, R. W. CHARMM36 lipid force field with explicit treatment of long-range dispersion: parametrization and validation for phosphatidylethanolamine, phosphatidylglycerol, and ether lipids. *J. Chem. Theory Comp.* **2021**, *17*, 1581–1595.

(566) Venable, R. M.; Kramer, A.; Pastor, R. W. molecular dynamics simulations of membrane permeability. *Chem. Rev.* **2019**, *119*, 5954–5997.

(567) Harder, E.; MacKerell, A. D., Jr.; Roux, B. Many-body polarization effects and the membrane dipole potential. *J. Am. Chem. Soc.* **2009**, *131*, 2760–2761.

(568) Li, H.; Chowdhary, J.; Huang, L.; He, X.; MacKerell, A. D., Jr.; Roux, B. Drude polarizable force field for molecular dynamics simulations of saturated and unsaturated zwitterionic lipids. *J. Chem. Theory Comp.* **2017**, *13*, 4535–4552.

(569) Chen, P.; Vorobyov, I.; Roux, B.; Allen, T. W. Molecular dynamics simulations based on polarizable models show that ion permeation interconverts between different mechanisms as a function of membrane thickness. *J. Phys. Chem. B* **2021**, *125*, 1020–1035.

(570) Ngo, V.; Li, H.; MacKerell, A. D., Jr.; Allen, T. W.; Roux, B.; Noskov, S. Polarization effects in water-mediated selective cation transport across a narrow transmembrane channel. *J. Chem. Theory Comp.* **2021**, *17*, 1726–1741.

(571) Yu, Y.; Venable, R. M.; Thirman, J.; Chatterjee, P.; Kumar, A.; Pastor, R. W.; MacKerell, A. D., Jr.; Roux, B.; Klauda, J. Drude polarizable lipid force field with explicit treatment of long-range dispersion: parametrization and validation for saturated and unsaturated zwitterionic lipids. *J. Chem. Theory Comp.* **2023**, *19*, 2590–2605.

(572) Hatcher, E.; Guvench, O.; Mackerell, A. D., Jr. CHARMM additive all-atom force field for acyclic polyalcohols, acyclic carbohydrates and inositol. *J. Chem. Theory Comp.* **2009**, *5*, 1315–1327.

(573) Hatcher, E.; Guvench, O.; Mackerell, A. D., Jr. CHARMM additive all-atom force field for aldopentofuranoses, methyl-aldopentofuranosides, and fructofuranose. *J. Phys. Chem. B* **2009**, *113*, 12466–12476.

(574) Raman, E. P.; Guvench, O.; MacKerell, A. D., Jr. CHARMM additive all-atom force field for glycosidic linkages in carbohydrates involving furanoses. *J. Phys. Chem. B* **2010**, *114*, 12981–12994.

(575) Guvench, O.; Mallajosyula, S. S.; Raman, E. P.; Hatcher, E.; Vanommeslaeghe, K.; Foster, T. J.; Jamison II, F. W.; Mackerell, A. D., Jr. CHARMM additive all-atom force field for carbohydrate derivatives and its utility in polysaccharide and carbohydrate-protein modeling. *J. Chem. Theory Comp.* **2011**, *7*, 3162–3180.

(576) Mallajosyula, S. S.; Guvench, O.; Hatcher, E.; Mackerell, A. D., Jr. CHARMM additive all-atom force field for phosphate and sulfate linked to carbohydrates. *J. Chem. Theory Comp.* **2012**, *8*, 759–776.

(577) He, X.; Lopes, P. E. M.; MacKerell, A., Jr. Polarizable empirical force field for acyclic poly-alcohols based on the classical Drude oscillator. *Biopolymers* **2013**, *99*, 724–738.

(578) Small, M. C.; Aytenfisu, A. H.; Lin, F. Y.; He, X.; MacKerell, A. D., Jr. Drude polarizable force field for aliphatic ketones and aldehydes, and their associated acyclic carbohydrates. *J. Comp. Aided Mol. Des.* **2017**, *31*, 349–363.

(579) Jana, M.; MacKerell, A. D., Jr. CHARMM Drude polarizable force field for aldopentofuranoses and methyl-aldopentofuranosides. *J. Phys. Chem. B* **2015**, *119*, 7846–7859.
(580) Patel, D. S.; He, X.; MacKerell, A. D., Jr. Polarizable empirical force field for hexopyranose monosaccharides based on the classical Drude oscillator. *J. Phys. Chem. B* **2015**, *119*, 637–652.

(581) Yang, M.; Aytenfisu, A. H.; MacKerell, A. D., Jr. Proper balance of solvent-solute and solute-solute interactions in the treatment of the diffusion of glucose using the Drude polarizable force field. *Carb. Res.* **2018**, *457*, 41–50.

(582) Aytenfisu, A. H.; Yang, M.; MacKerell, A. D., Jr. CHARMM drude polarizable force field for glycosidic linkages involving pyranoses and furanoses. *J. Chem. Theory Comp.* **2018**, *14*, 3132–3143.

(583) Pandey, P.; Aytenfisu, A. H.; MacKerell, A. D., Jr.; Mallajosyula, S. S. Drude polarizable force field parametrization of carboxylate and n-acetyl amine carbohydrate derivatives. *J. Chem. Theory Comp.* **2019**, *15*, 4982–5000.

(584) Kognole, A. A.; Aytenfisu, A. H.; MacKerell, A. D., Jr. Extension of the CHARMM classical Drude polarizable force field to N- and O-linked glycopeptides and glycoproteins. *J. Phys. Chem. B* **2022**, *126*, 6642–6653.

(585) Ruda, A.; Aytenfisu, A. H.; Angles d'Ortoli, T.; MacKerell, A. D., Jr.; Widmalm, G. Glycosidic α-linked mannopyranose disaccharides: an NMR spectroscopy and molecular dynamics simulation study employing additive and Drude polarizable force fields. *Phys. Chem. Chem. Phys.* **2023**, *25*, 3042–3060.

(586) Yu, W.; He, X.; Vanommeslaeghe, K.; MacKerell, A. D., Jr. Extension of the CHARMM general force field to sulfonyl-containing compounds and its utility in biomolecular simulations. *J. Comput. Chem.* **2012**, *33*, 2451–2468.

(587) Soteras Gutierrez, I.; Lin, F. Y.; Vanommeslaeghe, K.; Lemkul, J. A.; Armacost, K. A.; Brooks, C. L., III; MacKerell, A. D., Jr. Parametrization of halogen bonds in the CHARMM general force field: Improved treatment of ligand-protein interactions. *Bioorg. Med. Chem.* **2016**, *24*, 4812–4825.

(588) Kumar, A.; Pandey, P.; Chatterjee, P.; MacKerell, A. D., Jr. Deep neural network model to predict the electrostatic parameters in the polarizable classical Drude oscillator force field. *J. Chem. Theory Comp.* **2022**, *18*, 1711–1725.

(589) Heid, E.; Fleck, M.; Chatterjee, P.; Schröder, C.; MacKerell, A. D., Jr. Toward prediction of electrostatic parameters for force fields that explicitly treat electronic polarization. *J. Chem. Theory Comp.* **2019**, *15*, 2460–2469.

(590) Lin, F. Y.; MacKerell, A. D., Jr. Polarizable empirical force field for halogen-containing compounds based on the classical Drude oscillator. *J. Chem. Theory Comp.* **2018**, *14*, 1083–1098.

(591) Lin, F. Y.; MacKerell, A. D., Jr. Do halogen-hydrogen bond donor interactions dominate the favorable contribution of halogens to ligand-protein binding? *J. Phys. Chem. B* **2017**, *121*, 6813–6821.

(592) Chatterjee, P.; Sengul, M. Y.; Kumar, A.; MacKerell, A. D., Jr. Harnessing deep learning for optimization of Lennard-Jones parameters for the polarizable classical Drude oscillator force field. *J. Chem. Theory Comp.* **2022**, *18*, 2388–2407.

(593) Rupakheti, C. R.; MacKerell, A. D., Jr.; Roux, B. Global optimization of the Lennard-Jones parameters for the Drude polarizable force field. *J. Chem. Theory Comp.* **2021**, *17*, 7085–7095.

(594) Boulanger, E.; Huang, L.; Rupakheti, C.; MacKerell, A. D., Jr.; Roux, B. Optimized Lennard-Jones parameters for drug-like small molecules. *J. Chem. Theory Comp.* **2018**, *14*, 3121–3131.

(595) Orr, A. A.; Sharif, S.; Wang, J.; MacKerell, A. D., Jr. Preserving the integrity of empirical force fields. *J. Chem. Inf. and Mod.* **2022**, *62*, 3825–3831.

(596) Gao, J. Methods and applications of combined quantum mechanical and molecular mechanical potentials. *Rev. Comput. Chem.* **1996**, *7*, 119–185.

(597) Garcia-Viloca, M.; Gao, J.; Karplus, M.; Truhlar, D. G. How enzymes work: analysis by modern rate theory and computer simulations. *Science* **2004**, *303*, 186–195.

(598) Gao, J.; Ma, S.; Major, D. T.; Nam, K.; Pu, J.; Truhlar, D. G. Mechanisms and free energies of enzymatic reactions. *Chem. Rev.* **2006**, *106*, 3188–3209.

(599) Senn, H. M.; Thiel, W. QM/MM studies of enzymes. *Curr. Opin. Chem. Biol.* **2007**, *11*, 182–187.

(600) Warshel, A.; Levitt, M. Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. *J. Mol. Biol.* **1976**, *103*, 227–249.

(601) Field, M. J.; Bash, P. A.; Karplus, M. A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations. *J. Comput. Chem.* **1990**, *11*, 700–733.

(602) Cui, Q. Quantum mechanical methods in biochemistry and biophysics. *J. Chem. Phys.* **2016**, *145*, 140901.

(603) Nam, K.; Shao, Y.; Major, D. T.; Wolf-Watz, M. Perspectives on computational enzyme modeling: from mechanisms to design and drug development. *ACS Omega* **2024**, *9*, 7393–7412.

(604) Bash, P. A.; Field, M. J.; Karplus, M. Free energy perturbation method for chemical reactions in the condensed phase: a dynamic approach based on a combined quantum and molecular mechanical potential. *J. Am. Chem. Soc.* **1987**, *109*, 8092–8094.

(605) Dewar, M. J. S.; Thiel, W. Ground states of molecules. 38. The MNDO method. approximations and parameters. *J. Am. Chem. Soc.* **1977**, *99*, 4899–4907.

(606) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. AM1: a new general purpose quantum mechanical molecular model. *J. Am. Chem. Soc.* **1985**, *107*, 3902–3909.

(607) Thiel, W.; Voityuk, A. A. Extension of the MNDO formalism to d orbitals: Integral approximations and preliminary numerical results. *Theor. Chim. Acta.* **1992**, *81*, 391–404.

(608) Nam, K.; Cui, Q.; Gao, J.; York, D. M. Specific reaction parameterization for the AM1/d hamiltonian for phosphoryl transfer reactions: H, O, and P atoms. *J. Chem. Theory Comp.* **2007**, *3*, 486–504.

(609) Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G. Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. *Phys. Rev. B* **1998**, *58*, 7260–7268.

(610) Stewart, J. J. P. MOPAC: a semiempirical molecular orbital program. *J. Comput.-Aided Mol. Des.* **1990**, *4*, 1–103.

(611) Lopez, X.; York, D. M. Parameterization of semiempirical methods to treat nucleophilic attacks to biological phosphates: AM1/d parameters for phosphorus. *Theor. Chem. Acc.* **2003**, *109*, 149–159.

(612) Walker, R. C.; Crowley, M. F.; Case, D. A. The implementation of a fast and accurate QM/MM potential method in Amber. *J. Comput. Chem.* **2008**, *29*, 1019–1031.

(613) Thiel, W. MNDO97, version 5.0. 1998.

(614) Vanden-Eijnden, E.; Venturoli, M. Revisiting the finite temperature string for the calculation of reaction tubes and free energies. *J. Chem. Phys.* **2009**, *130*, 194103.

(615) Woodcock, H. L., III; Hodošček, M.; Brooks, B. R. Exploring SCC-DFTB paths for mapping QM/MM reaction mechanisms. *J. Phys. Chem. A* **2007**, *111*, 5720–5728.

(616) Epifanovsky, E.; Gilbert, A. T. B.; Feng, X.; Lee, J.; Mao, Y.; Mardirossian, N.; Pokhilko, P.; White, A. F.; Coons, M. P.; Dempwolf, A. L.; et al. Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package. *J. Chem. Phys.* **2021**, *155*, 084801.

(617) Schmidt, M.; Baldridge, K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; et al. General atomic and molecular electronic structure system. *J. Comput. Chem.* **1993**, *14*, 1347–1363.

(618) Barca, G. M. J.; Bertoni, C.; Carrington, L.; Datta, D.; De Silva, N.; Deustua, J. E.; Fedorov, D. G.; Gour, J. R.; Gunina, A. O.; Guidez, E.; et al. Recent developments in the general atomic and molecular electronic structure system. *J. Chem. Phys.* **2020**, *152*, 154102.

(619) Guest, M. F.; Bush, I. J.; Van Dam, H. J. J.; Sherwood, P.; Thomas, J. M. H.; Van Lenthe, J. H.; Havenith, R. W. A.; Kendrick, J. The GAMESS-UK electronic structure package: algorithms, developments and applications. *Mol. Phys.* **2005**, *103*, 719–747.
(620) Amos, R. D.; Albert, I. L.; Andrews, J. S.; Colwell, S. M.; Handy, N. C.; Jayatilaka, D.; Knowles, P. J.; Kobayashi, R.; Laidig, K. E.; Laming, G.; et al. CADPAC, the cambridge analytic derivatives package. 1995.

(621) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; et al. Gaussian 16, revision c.01, Gaussian, Inc.: Wallingford, CT, 2016.

(622) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. Molpro: a general-purpose quantum chemistry program package. *WIREs Comput. Mol. Sci.* **2012**, 2, 242–253.

(623) Singh, U. C.; Kollman, P. A. A combined ab initio quantum mechanical and molecular mechanical method for carrying out simulations on complex molecular systems: applications to the CH 3 Cl + Cl - exchange reaction and gas phase protonation of polyethers. *J. Comput. Chem.* **1986**, 7, 718–730.

(624) König, P. H.; Hoffmann, M.; Frauenheim, T.; Cui, Q. A critical evaluation of different QM/MM frontier treatments with SCC-DFTB as the QM method. *J. Phys. Chem. B* **2005**, *109*, 9082–9095.

(625) Das, D.; Brooks, B. R. Modeling the catalytic mechanism of adenosine kinase with QM/MM methods. *Biophys. J.* **2000**, *78*, 333A.

(626) Das, D.; Eurenius, K. P.; Billings, E. M.; Sherwood, P.; Chatfield, D. C.; Hodošček, M.; Brooks, B. R. Optimization of quantum mechanical molecular mechanical partitioning schemes: gaussian delocalization of molecular mechanical charges and the double link atom method. *J. Chem. Phys.* **2002**, *117*, 10534–10547.

(627) Gao, J.; Amara, P.; Alhambra, C.; Field, M. J. A generalized hybrid orbital (GHO) method for the treatment of boundary atoms in combined QM/MM calculations. *J. Phys. Chem. A* **1998**, *102*, 4714–4721.

(628) Pu, J.; Gao, J.; Truhlar, D. Combining self-consistent-charge density-functional tight-binding (SCC-DFTB) with molecular mechanics by the generalized hybrid orbital (GHO) method. *J. Phys. Chem. A* **2004**, *108*, 5454–5463.

(629) Pu, J.; Gao, J.; Truhlar, D. Generalized hybrid orbital (GHO) method for combining ab initio Hartree–Fock wave functions with molecular mechanics. *J. Phys. Chem. A* **2004**, *108*, 632–650.

(630) Brooks, I. C. L., III; Brünger, A.; Karplus, M. Active site dynamics in protein molecules: A stochastic boundary molecular-dynamics approach. *Biopolymers* **1985**, *24*, 843–865.

(631) Beglov, D.; Roux, B. Finite representation of an infinite bulk system: solvent boundary potential for computer simulations. *J. Chem. Phys.* **1994**, *100*, 9050–9063.

(632) Rossi, I.; Truhlar, D. G. Parameterization of NDDO wavefunctions using genetic algorithms. an evolutionary approach to parameterizing potential energy surfaces and direct dynamics calculations for organic reactions. *Chem. Phys. Lett.* **1995**, *233*, 231–236.

(633) Doron, D.; Major, D. T.; Kohen, A.; Thiel, W.; Wu, X. Hybrid quantum and classical simulations of the dihydrofolate reductase catalyzed hydride transfer reaction on an accurate semi-empirical potential energy surface. *J. Chem. Theory Comp.* **2011**, *7*, 3420–3437.

(634) Zhou, Y.; Pu, J. Reaction path force matching: a new strategy of fitting specific reaction parameters for semiempirical methods in combined QM/MM simulations. *J. Chem. Theory Comp.* **2014**, *10*, 3038–3054.

(635) Kulik, H. J.; Zhang, J.; Klinman, J. P.; Martinez, T. J. How large should the QM region be in QM/MM calculations? the case of catechol O-methyltransferase. *J. Phys. Chem. B* **2016**, *120*, 11381–11394.

(636) Jindal, G.; Warshel, A. Exploring the dependence of QM/MM calculations of enzyme catalysis on the size of the QM region. *J. Phys. Chem. B* **2016**, *120*, 9913–9921.

(637) Roßbach, S.; Ochsenfeld, C. Influence of coupling and embedding schemes on QM size convergence in QM/MM approaches for the example of a proton transfer in DNA. *J. Chem. Theory Comp.* **2017**, *13*, 1102–1107.

(638) Das, S.; Nam, K.; Major, D. T. Rapid convergence of energy and free energy profiles with quantum mechanical size in quantum mechanical-molecular mechanical simulations of proton transfer in DNA. *J. Chem. Theory Comp.* **2018**, *14*, 1695–1705.

(639) Mehmood, R.; Kulik, H. J. Both configuration and QM region size matter: zinc stability in QM/MM models of DNA methyl-transferase. *J. Chem. Theory Comp.* **2020**, *16*, 3121–3134.

(640) Pérez-Barcia, A.; Cárdenas, G.; Nogueira, J. J.; Mandado, M. Effect of the QM size, basis set, and polarization on QM/MM interaction energy decomposition analysis. *J. Chem. Inf. and Mod.* **2023**, *63*, 882–897.

(641) Nam, K.; Gao, J. L.; York, D. M. An efficient linear-scaling Ewald method for long-range electrostatic interactions in combined QM/MM calculations. *J. Chem. Theory Comp.* **2005**, *1*, 2–13.

(642) Nam, K. Acceleration of ab Initio QM/MM calculations under periodic boundary conditions by multiscale and multiple time step approaches. *J. Chem. Theory Comp.* **2014**, *10*, 4175–4183.

(643) Holden, Z. C.; Richard, R. M.; Herbert, J. M. Periodic boundary conditions for QM/MM calculations: Ewald summation for extended Gaussian basis sets. *J. Chem. Phys.* **2013**, *139*, 244108.

(644) Schaefer, P.; Riccardi, D.; Cui, Q. Reliable treatment of electrostatics in combined QM/MM simulation of macromolecules. *J. Chem. Phys.* **2005**, *123*, 014905.

(645) Riccardi, D.; Schaefer, P.; Yang, Y.; Yu, H.; Ghosh, N.; Prat-Resina, X.; König, P.; Li, G.; Xu, D.; Guo, H.; et al. Development of effective quantum mechanical/molecular mechanical (QM/MM) methods for complex biological processes. *J. Phys. Chem. B* **2006**, *110*, 6458–6469.

(646) Ojeda-May, P.; Li, Y.; Ovchinnikov, V.; Nam, K. Role of protein dynamics in allosteric control of the catalytic phosphoryl transfer of insulin receptor kinase. *J. Am. Chem. Soc.* **2015**, *137*, 12454–12457.

(647) Ojeda-May, P.; Mushtaq, A. U.; Rogne, P.; Verma, A.; Ovchinnikov, V.; Grundström, C.; Dulko-Smith, B.; Sauer, U. H.; Wolf-Watz, M.; Nam, K. Dynamic connection between enzymatic catalysis and collective protein motions. *Biochem.* **2021**, *60*, 2246–2258.

(648) König, G.; Hudson, P. S.; Boresch, S.; Woodcock, H. L., III. Multiscale free energy simulations: an efficient method for connecting classical MD simulations to QM or QM/MM free energies using Non-Boltzmann Bennett reweighting schemes. *J. Chem. Theory Comp.* **2014**, *10*, 1406–1419.

(649) Riccardi, D.; Schaefer, P.; Cui, Q. pKₐ calculations in solution and proteins with QM/MM free energy perturbation simulations: a quantitative test of QM/MM protocols. *J. Phys. Chem. B* **2005**, *109*, 17715–17733.

(650) Pulay, P. Convergence acceleration of iterative sequences. the case of scf iteration. *Chem. Phys. Lett.* **1980**, *73*, 393–398.

(651) Pulay, P. Improved scf convergence acceleration. *J. Comput. Chem.* **1982**, *3*, 556–560.

(652) Schlegel, H. B.; Millam, J. M.; Iyengar, S. S.; Voth, G. A.; Daniels, A. D.; Scuseria, G. E.; Frisch, M. J. Ab initio molecular dynamics: propagating the density matrix with Gaussian orbitals. *J. Chem. Phys.* **2001**, *114*, 9758–9763.

(653) Herbert, J. M.; Head-Gordon, M. Curvy-steps approach to constraint-free extended-Lagrangian ab initio molecular dynamics, using atom-centered basis functions: convergence toward Born-Oppenheimer trajectories. *J. Chem. Phys.* **2004**, *121*, 11542–11556.

(654) Niklasson, A. M. N.; Steneteg, P.; Odell, A.; Bock, N.; Challacombe, M.; Tymczak, C. J.; Holmström, E.; Zheng, G.; Weber, V. Extended Lagrangian Born-Oppenheimer molecular dynamics with dissipation. *J. Chem. Phys.* **2009**, *130*, 214109.

(655) Zheng, G.; Niklasson, A. M. N.; Karplus, M. Lagrangian formulation with dissipation of Born-Oppenheimer molecular dynamics using the density-functional tight-binding method. *J. Chem. Phys.* **2011**, *135*, 044122.

(656) Pulay, P.; Fogarasi, G. Fock matrix dynamics. *Chem. Phys. Lett.* **2004**, *386*, 272–278.
(657) Herbert, J. M.; Head-Gordon, M. Accelerated, energy-
conserving Born-Oppenheimer molecular dynamics via Fock matrix
extrapolation. *Phys. Chem. Chem. Phys.* **2005**, *7*, 3269–3275.

(658) Christensen, A. S.; Kubar, T.; Cui, Q.; Elstner, M. Semi-empirical quantum mechanical methods for non-covalent interactions for chemical and biochemical applications. *Chem. Rev.* **2016**, *116*, 5301–5337.

(659) Řežač, J.; Hobza, P. Advanced corrections of hydrogen bonding and dispersion for semiempirical quantum mechanical methods. *J. Chem. Theory Comp.* **2012**, *8*, 141–151.

(660) Gao, J.; Devi-Kesavan, L.S.; Garcia-Viloca, M. Semiempirical QM/MM potential with simple valence bond (SVB) for enzyme reactions. application to the nucleophilic addition reaction in haloalkane dehalogenase. *Theor. Chem. Acc.* **2003**, *109*, 133–139.

(661) Feynman, R. P.; Hibbs, A. R. *Quantum mechanics and path integrals*; McGraw-Hill: New York, 1965.

(662) Voth, G. A. Feynman path integral formulation of quantum mechanical transition state theory. *J. Phys. Chem.* **1993**, *97*, 8365–8377.

(663) Berne, B. J.; Thirumalai, D. On the simulation of quantum-systems - path integral methods. *Annu. Rev. Phys. Chem.* **1986**, *37*, 401–424.

(664) Hwang, J. K.; Chu, Z. T.; Yadav, A.; Warshel, A. Simulations of quantum mechanical corrections for rate constants of hydride-transfer reactions in enzymes and solutions. *J. Phys. Chem.* **1991**, *95*, 8445–8448.

(665) Hwang, J. K.; Warshel, A. A quantized classical path approach for calculations of quantum mechanical rate constants. *J. Phys. Chem.* **1993**, *97*, 10053–10058.

(666) Pollock, E. L.; Ceperley, D. M. Simulation of quantum many-body systems by path-integral methods. *Phys. Rev. B* **1984**, *30*, 2555–2568.

(667) Ceperley, D. M. Path integrals in the theory of condensed helium. *Rev. Mod. Phys.* **1995**, *67*, 279–355.

(668) Major, D. T.; Gao, J. Implementation of the bisection sampling method in path integral simulations. *J. Mol. Graph. Model.* **2005**, *24*, 121–127.

(669) Major, D. T.; Garcia-Viloca, M.; Gao, J. Path integral simulations of proton transfer reactions in aqueous solution using combined QM/MM potentials. *J. Chem. Theory Comp.* **2006**, *2*, 236–245.

(670) Sprik, M.; Klein, M. L.; Chandler, D. Staging: a sampling technique for the Monte Carlo evaluation of path integrals. *Phys. Rev. B* **1985**, *31*, 4234–4244.

(671) Azuri, A.; Engel, H.; Doron, D.; Major, D. T. Path-integral calculations of nuclear quantum effects in model systems, small molecules, and enzymes via gradient based forward corrector algorithms. *J. Chem. Theory Comp.* **2011**, *7*, 1273–1286.

(672) Vardi-Kilshtain, A.; Major, D. T.; Kohen, A.; Engel, H.; Doron, D. Hybrid quantum and classical simulations of the formate dehydrogenase catalyzed hydride transfer reaction on an accurate semiempirical potential energy surface. *J. Chem. Theory Comp.* **2012**, *8*, 4786–4796.

(673) Roston, D.; Kohen, A.; Doron, D.; Major, D. T. Simulations of remote mutants of dihydrofolate reductase reveal the nature of a network of residues coupled to hydride transfer. *J. Comput. Chem.* **2014**, *35*, 1411–1417.

(674) Doron, D.; Stojković, V.; Gakhar, L.; Vardi-Kilshtain, A.; Kohen, A.; Major, D. T. Free energy simulations of active-site mutants of dihydrofolate reductase. *J. Phys. Chem. B* **2015**, *119*, 906–916.

(675) Major, D. T.; Gao, J. An integrated path integral and free-energy perturbation-umbrella sampling method for computing kinetic isotope effects of chemical reactions in solution and in enzymes. *J. Chem. Theory Comp.* **2007**, *3*, 949–960.

(676) Gao, J.; Major, D. T.; Fan, Y.; Lin, Y.-l.; Ma, S.; Wong, K. Y. In *Molecular Modeling of Proteins*; Kukol, A., Ed.; Humana Press: 2008; Vol. 443, pp 37–62.

(677) Gao, J.; Wong, K.-Y.; Major, D. T.; Cembran, A.; Song, L.; Lin, Y.-l.; Fan, Y.; Ma, S. In *Quantum Tunnelling in Enzyme-Catalysed Reactions*; Allemann, R. K., Scrutton, N. S., Eds.; The Royal Society of Chemistry: 2009; Chapter 5, pp 105–134.

(678) Rubinstein, A.; Major, D. T. Catalyzing racemizations in the absence of a cofactor: the reaction mechanism in proline racemase. *J. Am. Chem. Soc.* **2009**, *131*, 8513–8521.

(679) Vardi-Kilshtain, A.; Azuri, A.; Major, D. T. Path-integral calculations of heavy atom kinetic isotope effects in condensed phase reactions using higher-order Trotter factorizations. *J. Comput. Chem.* **2012**, *33*, 435–441.

(680) Fan, Y.; Cembran, A.; Ma, S.; Gao, J. Connecting protein conformational dynamics with catalytic function as illustrated in dihydrofolate reductase. *Biochem.* **2013**, *52*, 2036–2049.

(681) Adamczyk, A. J.; Cao, J.; Kamerlin, S. C. L.; Warshel, A. Catalysis by dihydrofolate reductase and other enzymes arises from electrostatic preorganization, not conformational motions. *Proc. Natl. Acad. Sci. USA* **2011**, *108*, 14115–14120.

(682) Roston, D.; Demapan, D.; Cui, Q. Leaving group ability observably affects transition state structure in a single enzyme active site. *J. Am. Chem. Soc.* **2016**, *138*, 7386–7394.

(683) Roston, D.; Cui, Q. Substrate and transition state binding in alkaline phosphatase analyzed by computation of oxygen isotope effects. *J. Am. Chem. Soc.* **2016**, *138*, 11946–11957.

(684) Chin, S. A. Symplectic integrators from composite operator factorizations. *Phys. Lett. A* **1997**, *22*, 344–348.

(685) Chin, S. A.; Chen, C. R. Gradient symplectic algorithms for solving the Schrödinger equation with time-dependent potentials. *J. Chem. Phys.* **2002**, *117*, 1409–1415.

(686) Chin, S. A. Quantum statistical calculations and symplectic corrector algorithms. *Phys. Rev. E* **2004**, *69*, 046118.

(687) Morrone, J. A.; Srinivasan, V.; Sebastiani, D.; Car, R. Proton momentum distribution in water: an open path integral molecular dynamics study. *J. Chem. Phys.* **2007**, *126*, 234504.

(688) Perez, A.; Tuckerman, M. E. Improving the convergence of closed and open path integral molecular dynamics via higher order trotter factorization schemes. *J. Chem. Phys.* **2011**, *135*, 064104.

(689) Engel, H.; Doron, D.; Kohen, A.; Major, D. T. Momentum distribution as a fingerprint of quantum delocalization in enzymatic reactions: open-chain path-integral simulations of model systems and the hydride transfer in dihydrofolate reductase. *J. Chem. Theory Comp.* **2012**, *8*, 1223–1234.

(690) Nomura, Y.; Roston, D.; Montemayor, E. J.; Cui, Q.; Butcher, S. E. Structural and mechanistic basis for preferential deadenylation of U6 snRNA by Usb1. *Nucleic Acids Res.* **2018**, *46*, 11488–11501.

(691) Lai, R.; Li, G.; Cui, Q. Flexibility of Binding Site is Essential to the Ca2+ Selectivity in EF-Hand Calcium-Binding Proteins. *J. Am. Chem. Soc.* **2024**, *146*, 7628–7639.

(692) Elstner, M.; Seifert, G. Density functional tight binding. *Phil. Trans. R. Soc. Lond. A* **2014**, *372*, 20120483.

(693) Gaus, M.; Cui, Q.; Elstner, M. Density functional tight binding (DFTB): Application to organic and biological molecules. *WIREs Comput. Mol. Sci.* **2014**, *4*, 49–61.

(694) Thiel, W. Perspectives on semiempirical molecular orbital theory. *Adv. Chem. Phys.* **1996**, *93*, 703–757.

(695) Gaus, M.; Goez, A.; Elstner, M. Parametrization and Benchmark of DFTB3 for Organic Molecules. *J. Chem. Theory Comp.* **2013**, *9*, 338–354.

(696) Kubar, T.; Elstner, M.; Cui, Q. Hybrid QM/MM methods for studying energy transduction in biomolecular machines. *Annu. Rev. Biophys.* **2023**, *S2*, 525–551.

(697) Hou, G.; Zhu, X.; Elstner, M.; Cui, Q. A modified QM/MM hamiltonian with the self-consistent-charge density-functional-tight-binding theory for highly charged QM regions. *J. Chem. Theory Comp.* **2012**, *8*, 4293–4304.

(698) Rowley, C. N.; Roux, B. The solvation structure of Na⁺ and K⁺ in liquid water determined from high level ab initio molecular dynamics simulations. *J. Chem. Theory Comp.* **2012**, *8*, 3526–3535.

(699) Gaus, M.; Jin, H.; Demapan, D.; Christensen, A. S.; Goyal, P.; Elstner, M.; Cui, Q. DFTB3 parametrization for copper: the
importance of orbital angular momentum dependence of hubbard parameters. *J. Chem. Theory Comput.* **2015**, *11*, 4205-4219.

(700) Watanabe, H.; Cui, Q. Quantitative analysis and correction of QM/MM boundary artifacts in adaptive QM/MM methods. *J. Chem. Theory Comp.* **2019**, *15*, 3917-3928.

(701) The PLUMED consortium. Promoting transparency and reproducibility in enhanced molecular simulations. *Nat. Meth.* **2019**, *16*, 670-673.

(702) Roston, D.; Demapan, D.; Cui, Q. Extensive free energy simulations identify water as the base in nucleotide addition by DNA polymerase. *Proc. Natl. Acad. Sci. USA* **2019**, *116*, 25048-25056.

(703) Maag, D.; Mast, T.; Elstner, M.; Cui, Q.; Kubar, T. O to bR transition in bacteriorhodopsin occurs through a proton hole mechanism. *Proc. Natl. Acad. Sci. USA* **2021**, *118*, No. e2024803118.

(704) Deng, J.; Cui, Q. Second-shell residues contribute to catalysis by predominately pre-organizing the apo state in PafA. *J. Am. Chem. Soc.* **2023**, *145*, 11333-11347.

(705) Lu, X.; Ovchinnikov, V.; Roston, D. R.; Demapan, D.; Cui, Q. Regulation and plasticity of catalysis in enzymes: insights from analysis of mechanochemical coupling in myosin. *Biochemistry* **2017**, *56*, 1482-1497.

(706) Goyal, P.; Qian, H. J.; Irle, S.; Lu, X.; Roston, D.; Mori, T.; Elstner, M.; Cui, Q. Molecular simulation of water and hydration effects in different environments: challenges and developments for DFTB based models. *J. Phys. Chem. B* **2014**, *118*, 11007-11027.

(707) Li, G.; Cui, Q. pKₐ calculations with QM/MM free energy perturbations. *J. Phys. Chem. B* **2003**, *107*, 14521-14528.

(708) Cates, M. S.; Berry, M. B.; Ho, E. L.; Li, Q.; Potter, J. D.; Phillips, G. N. Metal-ion affinity and specificity in EF-hand proteins: coordination geometry and domain plasticity in parvalbumin. *Structure* **1999**, *7*, 1269-1278.

(709) Jing, Z. F.; Liu, C. W.; Qi, R.; Ren, P. Y. Many-body effect determines the selectivity for Ca²⁺ and Mg²⁺ in proteins. *Proc. Natl. Acad. Sci. USA* **2018**, *115*, E7495-E7501.

(710) Ngo, V.; da Silva, M.; Kubillus, M.; Elstner, M.; Hui, L.; Roux, B.; Cui, Q.; Salahub, D.; Noskov, S. Quantum Effects in Cation Interactions with First and Second Coordination Shell Ligands in Metalloproteins. *J. Chem. Theory Comp.* **2015**, *11*, 4992-5001.

(711) Sherwood, P.; De Vries, A. H.; Collins, S. J.; Greatbanks, S. P.; Burton, N. A.; Vincent, M. A.; Hillier, I. H. Computer simulation of zeolite structure and reactivity using embedded cluster methods. *Faraday Discussions* **1997**, *106*, 79-92.

(712) De Vries, A. H.; Sherwood, P.; Collins, S. J.; Rigby, A. M.; Rigutto, M.; Kramer, G. J. Zeolite structure and reactivity by combined quantum-chemical-classical calculations. *J. Phys. Chem. B* **1999**, *103*, 6133-6141.

(713) Philipp, D. M.; Friesner, R. A. Mixedab initio QM/MM modeling using frozen orbitals and tests with alanine dipeptide and tetrapeptide. *J. Comput. Chem.* **1999**, *20*, 1468-1494.

(714) Amara, P.; Field, M. J. Evaluation of an ab initio quantum mechanical/molecular mechanical hybrid-potential link-atom method. *Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta)* **2003**, *109*, 43-52.

(715) Riahi, S.; Roux, B.; Rowley, C. N. QM/MM molecular dynamics simulations of the hydration of Mg (II) and Zn (II) ions. *Can. J. Chem.* **2013**, *91*, 552-558.

(716) Chen, Y.; Kale, S.; Weare, J.; Dinner, A. R.; Roux, B. Multiple time-step dual-Hamiltonian hybrid molecular dynamics - Monte Carlo canonical propagation algorithm. *J. Chem. Theory Comp.* **2016**, *12*, 1449-1458.

(717) Snyder, R.; Kim, B.; Pan, X.; Shao, Y.; Pu, J. Facilitating ab initio QM/MM free energy simulations by Gaussian process regression with derivative observations. *Phys. Chem. Chem. Phys.* **2022**, *24*, 25134-25143.

(718) Snyder, R.; Kim, B.; Pan, X.; Shao, Y.; Pu, J. Bridging semiempirical and ab initio QM/MM potentials by Gaussian process regression and its sparse variants for free energy simulation. *J. Chem. Phys.* **2023**, *159*, 054107.

(719) Rasmussen, C. E.; Williams, C. K. Gaussian processes for machine learning; MIT Press: Cambridge, MA, 2006; Vol. 2.

(720) Matthews, A. G. D. G.; van der Wilk, M.; Nickson, T.; Fujii, K.; Boukouvalas, A.; León-Villagrá, P.; Ghahramani, Z.; Hensman, J. GPflow: A Gaussian process library using TensorFlow. *J. Mach. Learn. Res.* **2017**, *18*, 1-6.

(721) Gardner, J. R.; Pleiss, G.; Bindel, D.; Weinberger, K. Q.; Wilson, A. G. GPyTorch: blackbox matrix-matrix gaussian process inference with GPU acceleration. *Advances in Neural Information Processing Systems (NeurIPS)* **2018**, *31*, 7587-7597.

(722) Pan, X.; Snyder, R.; Wang, J.-N.; Lander, C.; Wickizer, C.; Van, R.; Chesney, A.; Xue, Y.; Mao, Y.; Mei, Y.; et al. Training machine learning potentials for reactive systems: A Colab tutorial on basic models. *J. Comput. Chem.* **2024**, *45*, 638-647.

(723) Glowacki, D. R.; Orr-Ewing, A. J.; Harvey, J. N. Non-equilibrium reaction and relaxation dynamics in a strongly interacting explicit solvent: F + CD3CN treated with a parallel multi-state EVB model. *J. Chem. Phys.* **2015**, *143*, 044120.

(724) Glowacki, D. R.; Orr-Ewing, A. J.; Harvey, J. N. Product energy deposition of CN + alkane H abstraction reactions in gas and solution phases. *J. Chem. Phys.* **2011**, *134*, 214508.

(725) Harvey, J. N.; O'Connor, M.; Glowacki, D. R. *Theory and Applications of the Empirical Valence Bond Approach: From Physical Chemistry to Chemical Biology* **2017**, 93-119.

(726) Penney, W. G. The electronic structure of some polyenes and aromatic molecules. III. bonds of fractional order by the pair method. *Proc. R. Soc. Lon. A - Math. Phys. Sci.* **1937**, *158*, 0306-0324.

(727) Pauling, L. Atomic radii and interatomic distances in metals. *J. Am. Chem. Soc.* **1947**, *69*, 542-553.

(728) Johnston, H. S.; Parr, C. Activation energies from bond energies. I. hydrogen transfer reactions. *J. Am. Chem. Soc.* **1963**, *85*, 2544-2551.

(729) van Duin, A. C. T.; Dasgupta, S.; Lorant, F.; Goddard, W. A., III. ReaxFF: a reactive force field for hydrocarbons. *J. Phys. Chem. A* **2001**, *105*, 9396-9409.

(730) Nutt, D. R.; Meuwly, M. Studying reactive processes with classical dynamics: rebinding dynamics in MbNO. *Biophys. J.* **2006**, *90*, 1191-1201.

(731) Danielsson, J.; Meuwly, M. Atomistic simulation of adiabatic reactive processes based on multi-state potential energy surfaces. *J. Chem. Theory Comp.* **2008**, *4*, 1083-1093.

(732) Nagy, T.; Yosa Reyes, J.; Meuwly, M. Multisurface adiabatic reactive molecular dynamics. *J. Chem. Theory Comp.* **2014**, *10*, 1366-1375.

(733) Schmid, M. H.; Das, A. K.; Landis, C. R.; Meuwly, M. Multi-state VALBOND for atomistic simulations of hypervalent molecules, metal complexes, and reactions. *J. Chem. Theory Comp.* **2018**, *14*, 3565-3578.

(734) Landis, C. R.; Cleveland, T.; Firman, T. K. Valence bond concepts applied to the molecular mechanics description of molecular shapes. 3. applications to transition metal alkyls and hydrides. *J. Am. Chem. Soc.* **1998**, *120*, 2641-2649.

(735) Firman, T. K.; Landis, C. R. Valence bond concepts applied to the molecular mechanics description of molecular shapes. 4. transition metals with pi-bonds. *J. Am. Chem. Soc.* **2001**, *123*, 11728-11742.

(736) Tubert-Brohman, I.; Schmid, M.; Meuwly, M. A molecular mechanics force field for octahedral organometallic compounds with inclusion of the trans influence. *J. Chem. Theory Comp.* **2009**, *5*, 530-539.

(737) Lammers, S.; Lutz, S.; Meuwly, M. Reactive force fields for proton transfer dynamics. *J. Comput. Chem.* **2008**, *29*, 1048-1063.

(738) Mackeprang, K.; Xu, Z.-H.; Maroun, Z.; Meuwly, M.; Kjaergaard, H. G. Spectroscopy and dynamics of double proton transfer in formic acid dimer. *Phys. Chem. Chem. Phys.* **2016**, *18*, 24654-24662.

(739) Xu, Z.-H.; Meuwly, M. Vibrational Spectroscopy and Proton Transfer Dynamics in Protonated Oxalate. *J. Phys. Chem. A* **2017**, *121*, 5389-5398.
(740) Karandashev, K.; Xu, Z.-H.; Meuwly, M.; Vanicek, J.; Richardson, J. O. Kinetic isotope effects and how to describe them. *Struct. Dyn.* **2017**, *4*, 061501.

(741) Xu, Z.-H.; Meuwly, M. Multistate reactive molecular dynamics simulations of proton diffusion in water clusters and in the bulk. *J. Phys. Chem. B* **2019**, *123*, 9846–9861.

(742) Yosa Reyes, J.; Nagy, T.; Meuwly, M. Competitive reaction pathways in vibrationally induced photodissociation of H₂SO₄. *Phys. Chem. Chem. Phys.* **2014**, *16*, 18533–18544.

(743) Yosa Reyes, J.; Brickel, S.; Unke, O. T.; Meuwly, M. HSO₃Cl: a prototype molecule for studying OH-stretching overtone induced photodissociation. *Phys. Chem. Chem. Phys.* **2016**, *18*, 6780–6788.

(744) Brickel, S.; Meuwly, M. OH-stretching overtone induced dynamics in HSO₃F from reactive molecular dynamics simulation. *J. Phys. Chem. A* **2017**, *121*, 5079–5087.

(745) Brickel, S.; Meuwly, M. Molecular determinants for rate acceleration in the claisen rearrangement reaction. *J. Phys. Chem. B* **2019**, *123*, 448–456.

(746) Rivero, U.; Unke, O. T.; Meuwly, M.; Willitsch, S. Reactive atomistic simulations of Diels-Alder reactions: The importance of molecular rotations. *J. Chem. Phys.* **2019**, *151*, 104301.

(747) Soloviov, M.; Das, A. K.; Meuwly, M. Structural interpretation of metastable states in myoglobin-NO. *Angew. Chem., Int. Ed.* **2016**, *55*, 10126–10130.

(748) Das, A. K.; Meuwly, M. Kinetic analysis and structural interpretation of competitive ligand binding for NO dioxygenation in truncated hemoglobin N. *Angew. Chem., Int. Ed.* **2018**, *57*, 3509–3513.

(749) Pearlman, D. A.; Kollman, P. A. The lag between the Hamiltonian and the system configuration in free energy perturbation calculations. *J. Chem. Phys.* **1989**, *91*, 7831–7839.

(750) Hunter, J. E.; Reinhardt, W. P.; Davis, T. F. A finite-time variational method for determining optimal paths and obtaining bounds on free energy changes from computer simulations. *J. Chem. Phys.* **1993**, *99*, 6856–6864.

(751) Jarzynski, C. Nonequilibrium equality for free energy differences. *Phys. Rev. Lett.* **1997**, *78*, 2690–2693.

(752) Crooks, G. E. Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. *J. Stat. Phys.* **1998**, *90*, 1481–1487.

(753) Hummer, G. Fast-growth thermodynamic integration: Error and efficiency analysis. *J. Chem. Phys.* **2001**, *114*, 7330–7337.

(754) Bennett, C. H. Efficient estimation of free energy differences from Monte Carlo data. *J. Chem. Phys.* **1976**, *22*, 245–268.

(755) Lead optimization in Orion using relative binding free energy (accessed August 5, 2024); https://web.archive.org/web/20240116121109, https://www.eyesopen.com/relative-binding-free-energy.

(756) Yang, W.; Cui, Q.; Min, D.; Li, H. QM/MM alchemical free energy simulations: challenges and recent developments. *Annu. Rep. Comput. Chem.* **2010**, *6*, 51–62.

(757) Heimdal, J.; Ryde, U. Convergence of QM/MM free-energy perturbations based on molecular-mechanics or semiempirical simulations. *Phys. Chem. Chem. Phys.* **2012**, *14*, 12592–12604.

(758) Hudson, P. S.; Woodcock, H. L., III; Boresch, S. Use of nonequilibrium work methods to compute free energy differences between molecular mechanical and quantum mechanical representations of molecular systems. *J. Phys. Chem. Lett.* **2015**, *6*, 4850–4856.

(759) Kearns, F. L.; Hudson, P. S.; Woodcock, H. L., III; Boresch, S. Computing converged free energy differences between levels of theory via nonequilibrium work methods: Challenges and opportunities. *J. Comput. Chem.* **2017**, *38*, 1376–1388.

(760) Schöller, A.; Kearns, F.; Woodcock, H. L., III; Boresch, S. Optimizing the calculation of free energy differences in nonequilibrium work SQM/MM switching simulations. *J. Phys. Chem. B* **2022**, *126*, 2798–2811.

(761) Schöller, A.; Woodcock, H. L., III; Boresch, S. Exploring routes to enhance the calculation of free energy differences via nonequilibrium work SQM/MM switching simulations using hybrid

charge intermediates between MM and SQM levels of theory or non-linear switching schemes. *Molecules* **2023**, *28*, 4006.

(762) Boresch, S.; Woodcock III, H. L.; Hudson, P. S. Carrying out PERT/MSCALE nonequilibrium switching simulations with CHARMM. (accessed August 5, 2024);

(763) Roux, B. The membrane potential and its representation by a constant electric field in computer simulations. *Biophys. J.* **2008**, *95*, 4205–4216.

(764) Khalili-Araghi, F.; Jogini, V.; Yarov-Yarovoy, V.; Tajkhorshid, E.; Roux, B.; Schulten, K. Calculation of the gating charge for the Kv1.2 voltage-activated potassium channel. *Biophys. J.* **2010**, *98*, 2189–2198.

(765) Li, Q.; Wanderling, S.; Paduch, M.; Medovoy, D.; Singharoy, A.; McGreevy, R.; Villalba-Galea, C. A.; Hulse, R. E.; Roux, B.; Schulten, K.; et al. Structural mechanism of voltage-dependent gating in an isolated voltage-sensing domain. *Nat. Struct. Mol. Biol.* **2014**, *21*, 244–252.

(766) Tan, K.; Sather, A.; Robertson, J. L.; Moy, S.; Roux, B.; Joachimiak, A. Structure and electrostatic property of cytoplasmic domain of ZntB transporter. *Protein Sci.* **2009**, *18*, 2043–2052.

(767) Luo, Y.; Egwolf, B.; Walters, D. E.; Roux, B. Ion selectivity of alpha-hemolysin with a beta-cyclodextrin adapter. I. Single ion potential of mean force and diffusion coefficient. *J. Phys. Chem. B* **2010**, *114*, 952–958.

(768) Egwolf, B.; Luo, Y.; Walters, D. E.; Roux, B. Ion selectivity of alpha-hemolysin with beta-cyclodextrin adapter. II. multi-ion effects studied with grand canonical Monte Carlo/Brownian dynamics simulations. *J. Phys. Chem. B* **2010**, *114*, 2901–2909.

(769) Robertson, J. L.; Palmer, L. G.; Roux, B. Multi-ion distributions in the cytoplasmic domain of inward rectifier potassium channels. *Biophys. J.* **2012**, *103*, 434–443.

(770) Roux, B. Exploring the ion selectivity properties of a large number of simplified binding site models. *Biophys. J.* **2010**, *98*, 2877–2885.

(771) Yu, H.; Noskov, S. Y.; Roux, B. Two mechanisms of ion selectivity in protein binding sites. *Proc. Natl. Acad. Sci. USA* **2010**, *107*, 20329–20334.

(772) Sompornpisut, P.; Roux, B.; Perozo, E. Structural refinement of membrane proteins by restrained molecular dynamics and solvent accessibility data. *Biophys. J.* **2008**, *95*, 5349–5361.

(773) Vasquez, V.; Sotomayor, M.; Cortes, D. M.; Roux, B.; Schulten, K.; Perozo, E. Three-dimensional architecture of membrane-embedded MscS in the closed conformation. *J. Mol. Biol.* **2008**, *378*, 55–70.

(774) Sezer, D.; Freed, J. H.; Roux, B. Multifrequency electron spin resonance spectra of a spin-labeled protein calculated from molecular dynamics simulations. *J. Am. Chem. Soc.* **2009**, *131*, 2597–2605.

(775) Roux, B.; Islam, S. M. Restrained-ensemble molecular dynamics simulations based on distance histograms from double electron-electron resonance spectroscopy. *J. Phys. Chem. B* **2013**, *117*, 4733–4739.

(776) Islam, S. M.; Stein, R. A.; McHaourab, H. S.; Roux, B. Structural refinement from restrained-ensemble simulations based on EPR/DEER data: application to T4 lysozyme. *J. Phys. Chem. B* **2013**, *117*, 4740–4754.

(777) Islam, S. M.; Roux, B. Simulating the distance distribution between spin-labels attached to proteins. *J. Phys. Chem. B* **2015**, *119*, 3901–3911.

(778) Raghuraman, H.; Islam, S. M.; Mukherjee, S.; Roux, B.; Perozo, E. Dynamics transitions at the outer vestibule of the KcsA potassium channel during gating. *Proc. Natl. Acad. Sci. USA* **2014**, *111*, 1831–1836.

(779) Kazmier, K.; Sharma, S.; Quick, M.; Islam, S. M.; Roux, B.; Weinstein, H.; Javitch, J. A.; McHaourab, H. S. Conformational dynamics of ligand-dependent alternating access in LeuT. *Nat. Struct. Mol. Biol.* **2014**, *21*, 472–479.

(780) Kazmier, K.; Sharma, S.; Islam, S. M.; Roux, B.; Mchaourab, H. S. Conformational cycle and ion-coupling mechanism of the Na
+/hydantoin transporter Mhp1. Proc. Natl. Acad. Sci. USA **2014**, *111*, 14752–14757.

(781) Shen, R.; Han, W.; Fiorin, G.; Islam, S. M.; Schulten, K.; Roux, B. Structural refinement of proteins by restrained molecular dynamics simulations with non-interacting molecular fragments. *PLoS Comput. Biol.* **2015**, *11*, No. e1004368.

(782) Vargas, E.; Bezanilla, F.; Roux, B. In search of a consensus model of the resting state of a voltage-sensing domain. *Neuron* **2011**, *72*, 713–720.

(783) Vargas, E.; Yarov-Yarovoy, V.; Khalili-Araghi, F.; Catterall, W. A.; Klein, M. L.; Tarek, M.; Lindahl, E.; Schulten, K.; Perozo, E.; Bezanilla, F.; et al. An emerging consensus on voltage-dependent gating from computational modeling and molecular dynamics simulations. *J. Gen. Physiol.* **2012**, *140*, 587–594.

(784) Bullerjahn, J. T.; von Bülow, S.; Heidari, M.; Hénin, J.; Hummer, G. Unwrapping NPT simulations to calculate diffusion coefficients. *J. Chem. Theory Comp.* **2023**, *19*, 3406–3417.

(785) Maginn, E. J.; Messerly, R. A.; Carlson, D. J.; Roe, D. R.; Elliot, J. R. Best practices for computing transport properties 1. self-diffusivity and viscosity from equilibrium molecular dynamics. *Living Journal of Computational Molecular Science* **2019**, *1*, 6324.

(786) Allen, M.; Tildesley, D. Computer simulations of liquids; Clarendon Press: Oxford, U.K., 1987.

(787) Roe, D. R.; Cheatham, T. E., III. PTRAJ and CPPTRAJ: software for processing and analysis of molecular dynamics trajectory data. *J. Chem. Theory Comp.* **2013**, *9*, 3084–3095.

(788) Venable, R. M.; Hatcher, E.; Guvench, O.; MacKerell, A. D., Jr.; Pastor, R. W. Comparing simulated and experimental translation and rotation constants: range of validity for viscosity scaling. *J. Phys. Chem. B* **2010**, *114*, 12501–12507.

(789) von Bülow, S.; Bullerjahn, J. T.; Hummer, G. Systematic errors in diffusion coefficients from long-time molecular dynamics simulations at constant pressure. *J. Chem. Phys.* **2020**, *153*, 021101.

(790) Yeh, I. C.; Hummer, G. System-size dependence of diffusion coefficients and viscosities from molecular dynamics simulations with periodic boundary conditions. *J. Phys. Chem. B* **2004**, *108*, 15873–15879.

(791) Saffman, P. G.; Delbruck, M. Brownian motion in biological membranes. *Proc. Natl. Acad. Sci. USA* **1975**, *72*, 3111–3113.

(792) Camley, B. A.; Lerner, M. G.; Pastor, R. W.; Brown, F. L. H. Strong influence of periodic boundary conditions on lateral diffusion in lipid bilayer membranes. *J. Chem. Phys.* **2015**, *143*, 243113.

(793) Park, S.; Im, W.; Pastor, R. W. Developing initial conditions for simulations of asymmetric membranes: a practical recommendation. *Biophys. J.* **2021**, *120*, 5041–5059.

(794) Irving, J. H.; Kirkwood, J. G. The statistical mechanical theory of transport processes. IV. the equations of hydrodynamics. *J. Chem. Phys.* **1950**, *18*, 817–829.

(795) Schofield, P.; Henderson, J. R.; Rowlinson, J. S. Statistical mechanics of inhomogeneous fluids. *Proc. R. Soc. Lon. A - Math. Phys. Sci.* **1982**, *379*, 231–246.

(796) Canham, P. B. The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. *J. Theor. Biol.* **1970**, *26*, 61–81.

(797) Helfrich, W. Elastic properties of lipid bilayers - theory and possible experiments. *Zeitschrift Fur Naturforschung C* **1973**, *28*, 693–703.

(798) Campelo, F.; Arnarez, C.; Marrink, S. J.; Kozlov, M. M. Helfrich model of membrane bending: from Gibbs theory of liquid interfaces to membranes as thick anisotropic elastic layers. *Adv. Colloid Interface Sci.* **2014**, *208*, 25–33.

(799) Hossein, A.; Deserno, M. Spontaneous curvature, differential stress, and bending modulus of asymmetric lipid membranes. *Biophys. J.* **2020**, *118*, 624–642.

(800) Kozlov, M. M.; Winterhalter, M. Elastic-moduli for strongly curved monolayers - position of the neutral surface. *J. Phys. II* **1991**, *1*, 1077–1084.

(801) Sonne, J.; Hansen, F. Y.; Peters, G. H. Methodological problems in pressure profile calculations for lipid bilayers. *J. Chem. Phys.* **2005**, *122*, 124903.

(802) Sega, M.; Fábián, B.; Jedlovszky, P. Pressure profile calculation with mesh Ewald methods. *J. Chem. Theory Comp.* **2016**, *12*, 4509–4515.

(803) Lindahl, E.; Edholm, O. Spatial and energetic-entropic decomposition of surface tension in lipid bilayers from molecular dynamics simulations. *J. Chem. Phys.* **2000**, *113*, 3882–3893.

(804) Venable, R. M.; Sodt, A. J.; Rogaski, B.; Rui, H.; Hatcher, E.; MacKerell, A. D., Jr.; Pastor, R. W.; Klauda, J. B. CHARMM all-atom additive force field for sphingomyelin: elucidation of hydrogen bonding and of positive curvature. *Biophys. J.* **2014**, *107*, 134–145.

(805) Hamaneh, M. B.; Buck, M. Acceptable protein and solvent behavior in primary hydration shell simulations of hen lysozyme. *Biophys. J.* **2007**, *92*, L49–L51.

(806) Hamaneh, M. B.; Buck, M. Refinement of the primary hydration shell model for molecular dynamics simulations of large proteins. *J. Comput. Chem.* **2009**, *30*, 2635–2644.

(807) Wagoner, J. A.; Pande, V. S. Finite domain simulations with adaptive boundaries: Accurate potentials and nonequilibrium movesets. *J. Chem. Phys.* **2013**, *139*, 234114.

(808) Wu, X.; Brooks, B. R. Using the isotropic periodic sum method to calculate long-range interactions of heterogeneous systems. *J. Chem. Phys.* **2008**, *129*, 154115.

(809) Halle, B. Protein hydration dynamics in solution: a critical survey. *Philos. Trans. R. Soc. Lond. B* **2004**, *359*, 1207–1223.

(810) Steinbach, P. J.; Brooks, B. R. Protein hydration elucidated by molecular dynamics simulation. *Proc. Natl. Acad. Sci. USA* **1993**, *90*, 9135–9139.

(811) Ravikumar, K. M.; Hwang, W. Role of Hydration Force in the Self-Assembly of Collagens and Amyloid Steric Zipper Filaments. *J. Am. Chem. Soc.* **2011**, *133*, 11766–11773.

(812) Teng, X.; Hwang, W. Effect of methylation on local mechanics and hydration structure of DNA. *Biophys. J.* **2018**, *114*, 1791–1803.

(813) Meng, E. C.; Goddard, T. D.; Pettersen, E. F.; Couch, G. S.; Pearson, Z. J.; Morris, J. H.; Ferrin, T. E. UCSF ChimeraX: Tools for structure building and analysis. *Protein Sci.* **2023**, *32*, No. e4792.

(814) Shi, J.; Cho, J.-H.; Hwang, W. Heterogeneous and Allosteric Role of Surface Hydration for Protein-Ligand Binding. *J. Chem. Theory Comp.* **2023**, *19*, 1875–1887.

(815) Nguyen, C. N.; Kurtzman Young, T.; Gilson, M. K. Grid inhomogeneous solvation theory: hydration structure and thermodynamics of the miniature receptor cucurbit [7] uril. *J. Chem. Phys.* **2012**, *137*, 044101.

(816) Ramsey, S.; Nguyen, C.; Salomon-Ferrer, R.; Walker, R. C.; Gilson, M. K.; Kurtzman, T. Solvation thermodynamic mapping of molecular surfaces in AmberTools: GIST. *J. Comput. Chem.* **2016**, *37*, 2029–2037.

(817) Karplus, M.; Ichiye, T.; Pettitt, B. Configurational entropy of native proteins. *Biophys. J.* **1987**, *52*, 1083–1085.

(818) Andricioaei, I.; Karplus, M. On the calculation of entropy from covariance matrices of the atomic fluctuations. *J. Chem. Phys.* **2001**, *115*, 6289–6292.

(819) King, B.; Tidor, B. MIST: maximum information spanning trees for dimension reduction of biological data sets. *Bioinformatics* **2009**, *25*, 1165–1172.

(820) King, B.; Silver, N.; Tidor, B. Efficient Calculation of Molecular Configurational Entropies Using an Information Theoretic Approximation. *J. Phys. Chem. B* **2012**, *116*, 2891–2904.

(821) Shi, J.; Shen, Q.; Cho, J.-H.; Hwang, W. Entropy Hotspots for the Binding of Intrinsically Disordered Ligands to a Receptor Domain. *Biophys. J.* **2020**, *118*, 2502–2512.

(822) Hwang, W.; Lang, M. J.; Karplus, M. Kinesin motility is driven by subdomain dynamics. *eLife* **2017**, *6*, No. e28948.

(823) Hwang, W.; Mallis, R. J.; Lang, M. J.; Reinherz, E. L. The αfTCR mechanosensor exploits dynamic ectodomain allostery to optimize its ligand recognition site. *Proc. Natl. Acad. Sci. USA* **2020**, *117*, 21336–21345.
(824) Lange, O. F.; Grubmüller, H. Generalized correlation for biomolecular dynamics. *PROTEINS: Struct., Func. and Bioinf.* **2006**, *62*, 1053–1061.

(825) Roy, A.; Post, C. B. Detection of Long-Range Concerted Motions in Protein by a Distance Covariance. *J. Chem. Theory Comp.* **2012**, *8*, 3009–3014.

(826) Roy, A.; Hua, D. P.; Post, C. B. Analysis of Multidomain Protein Dynamics. *J. Chem. Theory Comp.* **2016**, *12*, 274–280.

(827) Brooks, B. R.; Karplus, M.; Andricioaei, I.; Archontis, G.; Best, R.; Boresch, S.; Brooks, C. L., III; Buck, M.; Caflisch, A.; Crowley, M. F.; et al. CHARMM Program Download (accessed August 5, 2024); https://academiccharmm.org/program.