Relative Binding Free Energy Calculations for Ligands with Diverse Scaffolds with the Alchemical Transfer Method

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   The Many Roles of Computation in Drug Discovery

   Jorgensen, William L.

   Science (Washington, DC, United States) (2004), 303 (5665), 1813-1818CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)

   A review. An overview is given on the diverse uses of computational chem. in drug discovery. Particular emphasis is placed on virtual screening, de novo design, evaluation of drug-likeness, and advanced methods for detg. protein-ligand binding.
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   Abel, R.; Wang, L.; Harder, E. D.; Berne, B.; Friesner, R. A. Advancing drug discovery through enhanced free energy calculations. Acc. Chem. Res. 2017, 50, 1625– 1632,  DOI: 10.1021/acs.accounts.7b00083

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   Advancing Drug Discovery through Enhanced Free Energy Calculations

   Abel, Robert; Wang, Lingle; Harder, Edward D.; Berne, B. J.; Friesner, Richard A.

   Accounts of Chemical Research (2017), 50 (7), 1625-1632CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)

   A review. A principal goal of drug discovery project is to design mols. that can tightly and selectively bind to the target protein receptor. Accurate prediction of protein-ligand binding free energies is therefore of central importance in computational chem. and computer aided drug design. Multiple recent improvements in computing power, classical force field accuracy, enhanced sampling methods, and simulation setup, have enabled accurate and reliable calcns. of protein-ligands binding free energies, and position free energy calcns. to play a guiding role in small mol. drug discovery. In this accounts, the authors outline the relevant methodol. advances, including the REST2 (Replica Exchange with Solute Temperting) enhanced sampling, the incorporation of REST2 sampling with conventional FEP (Free Energy Perturbation) through FEP/REST, the OPLS3 force fields, and the advanced simulation set up that constitute the authors' FEP+ approach, followed by the presentation of extensive comparisons with expt., demonstrating sufficient accuracy in potency prediction (better than 1 kcal/mol) to substantially impact lead optimization campaigns. The limitations of the current FEP+ implementation and best practices in drug discovery applications are also discussed followed by the future methodol. development plans to address those limitations. The authors then report results from a recent drug discovery project, in which several thousand FEP+ calcns. were successfully deployed to simultaneously optimize potency, selectivity, and soly., illustrating the power of the approach to solve challenging drug design problems. The capabilities of free energy calcns. to accurately predict potency and selectivity have led to the advance of ongoing drug discovery projects, in challenging situations where alternative approaches would have great difficulties. The ability to effectively carry out projects evaluating tens of thousands, or hundreds of thousands, of proposed drug candidates, is potentially transformative in enabling hard to drug targets to be attacked, and in facilitating the development of superior compds., in various dimensions, for a wide range of targets. More effective integration of FEP+ calcns. into the drug discovery process will ensure that the results are deployed in an optimal fashion for yielding the best possible compds. entering the clinic; this is where the greatest payoff is in the exploitation of computer driven design capabilities. A key conclusion from the work described is the surprisingly robust and accurate results that are attainable within the conventional classical simulation, fixed charge paradigm. No doubt there are individual cases that would benefit from a more sophisticated energy model or dynamical treatment, and properties other than protein-ligand binding energies may be more sensitive to these approxns. The authors conclude that an inflection point in the ability of MD simulations to impact drug discovery has now been attained, due to the confluence of hardware and software development along with the formulation of "good enough" theor. methods and models.
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   Armacost, K. A.; Riniker, S.; Cournia, Z. Novel directions in free energy methods and applications. J. Chem. Inf. Model. 2020, 60, 1,  DOI: 10.1021/acs.jcim.9b01174

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   Novel Directions in Free Energy Methods and Applications

   Armacost, Kira A.; Riniker, Sereina; Cournia, Zoe

   Journal of Chemical Information and Modeling (2020), 60 (1), 1-5CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)

   A review. Free energy changes drive the vast majority of chem. processes in nature such as protein-ligand binding, polymer formation, and reaction pathways. Being able to reliably predict free energy changes using numerical simulations has long been extremely attractive as it would enable creating new processes, model reactions, and design materials and drugs with increased efficiency. Recent developments in new methods and algorithms combined with technol. advances leading to impressive increases in computational power, have facilitated improvements in both the efficiency and accuracy of free energy calcns., making them useful for prospective applications such as the design of new mols. and modifications to chem. reactions.
4. 4

   Wang, L.; Wu, Y.; Deng, Y.; Kim, B.; Pierce, L.; Krilov, G.; Lupyan, D.; Robinson, S.; Dahlgren, M. K.; Greenwood, J.; Romero, D. L.; Masse, C.; Knight, J. L.; Steinbrecher, T.; Beuming, T.; Damm, W.; Harder, E.; Sherman, W.; Brewer, M.; Wester, R.; Murcko, M.; Frye, L.; Farid, R.; Lin, T.; Mobley, D. L.; Jorgensen, W. L.; Berne, B. J.; Friesner, R. A.; Abel, R. Accurate and Reliable Prediction of Relative Ligand Binding Potency in Prospective Drug Discovery by Way of a Modern Free-Energy Calculation Protocol and Force Field. J. Am. Chem. Soc. 2015, 137, 2695– 2703,  DOI: 10.1021/ja512751q

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   Accurate and Reliable Prediction of Relative Ligand Binding Potency in Prospective Drug Discovery by Way of a Modern Free-Energy Calculation Protocol and Force Field

   Wang, Lingle; Wu, Yujie; Deng, Yuqing; Kim, Byungchan; Pierce, Levi; Krilov, Goran; Lupyan, Dmitry; Robinson, Shaughnessy; Dahlgren, Markus K.; Greenwood, Jeremy; Romero, Donna L.; Masse, Craig; Knight, Jennifer L.; Steinbrecher, Thomas; Beuming, Thijs; Damm, Wolfgang; Harder, Ed; Sherman, Woody; Brewer, Mark; Wester, Ron; Murcko, Mark; Frye, Leah; Farid, Ramy; Lin, Teng; Mobley, David L.; Jorgensen, William L.; Berne, Bruce J.; Friesner, Richard A.; Abel, Robert

   Journal of the American Chemical Society (2015), 137 (7), 2695-2703CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)

   Designing tight-binding ligands is a primary objective of small-mol. drug discovery. Over the past few decades, free-energy calcns. have benefited from improved force fields and sampling algorithms, as well as the advent of low-cost parallel computing. However, it has proven to be challenging to reliably achieve the level of accuracy that would be needed to guide lead optimization (∼5× in binding affinity) for a wide range of ligands and protein targets. Not surprisingly, widespread com. application of free-energy simulations has been limited due to the lack of large-scale validation coupled with the tech. challenges traditionally assocd. with running these types of calcns. Here, we report an approach that achieves an unprecedented level of accuracy across a broad range of target classes and ligands, with retrospective results encompassing 200 ligands and a wide variety of chem. perturbations, many of which involve significant changes in ligand chem. structures. In addn., we have applied the method in prospective drug discovery projects and found a significant improvement in the quality of the compds. synthesized that have been predicted to be potent. Compds. predicted to be potent by this approach have a substantial redn. in false positives relative to compds. synthesized on the basis of other computational or medicinal chem. approaches. Furthermore, the results are consistent with those obtained from our retrospective studies, demonstrating the robustness and broad range of applicability of this approach, which can be used to drive decisions in lead optimization.
5. 5

   Cournia, Z.; Allen, B.; Sherman, W. Relative binding free energy calculations in drug discovery: recent advances and practical considerations. J. Chem. Inf. Model. 2017, 57, 2911– 2937,  DOI: 10.1021/acs.jcim.7b00564

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   Relative Binding Free Energy Calculations in Drug Discovery: Recent Advances and Practical Considerations

   Cournia, Zoe; Allen, Bryce; Sherman, Woody

   Journal of Chemical Information and Modeling (2017), 57 (12), 2911-2937CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)

   A review. Accurate in silico prediction of protein-ligand binding affinities has been a primary objective of structure-based drug design for decades due to the putative value it would bring to the drug discovery process. However, computational methods have historically failed to deliver value in real-world drug discovery applications due to a variety of scientific, tech., and practical challenges. Recently, a family of approaches commonly referred to as relative binding free energy (RBFE) calcns., which rely on physics-based mol. simulations and statistical mechanics, have shown promise in reliably generating accurate predictions in the context of drug discovery projects. This advance arises from accumulating developments in the underlying scientific methods (decades of research on force fields and sampling algorithms) coupled with vast increases in computational resources (graphics processing units and cloud infrastructures). Mounting evidence from retrospective validation studies, blind challenge predictions, and prospective applications suggests that RBFE simulations can now predict the affinity differences for congeneric ligands with sufficient accuracy and throughput to deliver considerable value in hit-to-lead and lead optimization efforts. Here, the authors present an overview of current RBFE implementations, highlighting recent advances and remaining challenges, along with examples that emphasize practical considerations for obtaining reliable RBFE results. The authors focus specifically on relative binding free energies because the calcns. are less computationally intensive than abs. binding free energy (ABFE) calcns. and map directly onto the hit-to-lead and lead optimization processes, where the prediction of relative binding energies between a ref. mol. and new ideas (virtual mols.) can be used to prioritize mols. for synthesis. The authors describe the crit. aspects of running RBFE calcns., from both theor. and applied perspectives, using a combination of retrospective literature examples and prospective studies from drug discovery projects. This work is intended to provide a contemporary overview of the scientific, tech., and practical issues assocd. with running relative binding free energy simulations, with a focus on real-world drug discovery applications. The authors offer guidelines for improving the accuracy of RBFE simulations, esp. for challenging cases, and emphasize unresolved issues that could be improved by further research in the field.
6. 6

   Cournia, Z.; Allen, B. K.; Beuming, T.; Pearlman, D. A.; Radak, B. K.; Sherman, W. Rigorous free energy simulations in virtual screening. J. Chem. Inf. Model. 2020, 60, 4153– 4169,  DOI: 10.1021/acs.jcim.0c00116

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   Rigorous Free Energy Simulations in Virtual Screening

   Cournia, Zoe; Allen, Bryce K.; Beuming, Thijs; Pearlman, David A.; Radak, Brian K.; Sherman, Woody

   Journal of Chemical Information and Modeling (2020), 60 (9), 4153-4169CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)

   A review. Virtual high throughput screening (vHTS) in drug discovery is a powerful approach to identify hits: when applied successfully, it can be much faster and cheaper than exptl. high-throughput screening approaches. However, mainstream vHTS tools have significant limitations: ligand-based methods depend on knowledge of existing chem. matter, while structure-based tools such as docking involve significant approxns. that limit their accuracy. Recent advances in scientific methods coupled with dramatic speedups in computational processing with GPUs make this an opportune time to consider the role of more rigorous methods that could improve the predictive power of vHTS workflows. In this Perspective, we assert that alchem. binding free energy methods using all-atom mol. dynamics simulations have matured to the point where they can be applied in virtual screening campaigns as a final scoring stage to prioritize the top mols. for exptl. testing. Specifically, we propose that alchem. abs. binding free energy (ABFE) calcns. offer the most direct and computationally efficient approach within a rigorous statistical thermodn. framework for computing binding energies of diverse mols., as is required for virtual screening. ABFE calcns. are particularly attractive for drug discovery at this point in time, where the confluence of large-scale genomics data and insights from chem. biol. have unveiled a large no. of promising disease targets for which no small mol. binders are known, precluding ligand-based approaches, and where traditional docking approaches have foundered to find progressible chem. matter.
7. 7

   Mey, A. S. J. S.; Allen, B. K.; Macdonald, H. E. B.; Chodera, J. D.; Hahn, D. F.; Kuhn, M.; Michel, J.; Mobley, D. L.; Naden, L. N.; Prasad, S.; Rizzi, A.; Scheen, J.; Shirts, M. R.; Tresadern, G.; Xu, H. Best Practices for Alchemical Free Energy Calculations [Article v1.0]. Living J. Comput. Mol. Sci. 2020, 2, 18378,  DOI: 10.33011/livecoms.2.1.18378

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   Lee, T.-S.; Allen, B. K.; Giese, T. J.; Guo, Z.; Li, P.; Lin, C.; McGee, T. D., Jr; Pearlman, D. A.; Radak, B. K.; Tao, Y.; Tsai, H.-C.; Xu, H.; Sherman, W.; York, D. M. Alchemical Binding Free Energy Calculations in AMBER20: Advances and Best Practices for Drug Discovery. J. Chem. Inf. Model. 2020, 60, 5595– 5623,  DOI: 10.1021/acs.jcim.0c00613

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   Alchemical Binding Free Energy Calculations in AMBER20: Advances and Best Practices for Drug Discovery

   Lee, Tai-Sung; Allen, Bryce K.; Giese, Timothy J.; Guo, Zhenyu; Li, Pengfei; Lin, Charles; McGee Jr., T. Dwight; Pearlman, David A.; Radak, Brian K.; Tao, Yujun; Tsai, Hsu-Chun; Xu, Huafeng; Sherman, Woody; York, Darrin M.

   Journal of Chemical Information and Modeling (2020), 60 (11), 5595-5623CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)

   A review. Predicting protein-ligand binding affinities and the assocd. thermodn. of biomol. recognition is a primary objective of structure-based drug design. Alchem. free energy simulations offer a highly accurate and computationally efficient route to achieving this goal. While the AMBER mol. dynamics package has successfully been used for alchem. free energy simulations in academic research groups for decades, widespread impact in industrial drug discovery settings has been minimal because of the previous limitations within the AMBER alchem. code, coupled with challenges in system setup and postprocessing workflows. Through a close academia-industry collaboration we have addressed many of the previous limitations with an aim to improve accuracy, efficiency, and robustness of alchem. binding free energy simulations in industrial drug discovery applications. Here, we highlight some of the recent advances in AMBER20 with a focus on alchem. binding free energy (BFE) calcns., which are less computationally intensive than alternative binding free energy methods where full binding/unbinding paths are explored. In addn. to scientific and tech. advances in AMBER20, we also describe the essential practical aspects assocd. with running relative alchem. BFE calcns., along with recommendations for best practices, highlighting the importance not only of the alchem. simulation code but also the auxiliary functionalities and expertise required to obtain accurate and reliable results. This work is intended to provide a contemporary overview of the scientific, tech., and practical issues assocd. with running relative BFE simulations in AMBER20, with a focus on real-world drug discovery applications.
9. 9

   Steinbrecher, T.; Joung, I.; Case, D. A. Soft-core potentials in thermodynamic integration: Comparing one- and two-step transformations. J. Comput. Chem. 2011, 32, 3253– 3263,  DOI: 10.1002/jcc.21909

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   Soft-core potentials in thermodynamic integration: Comparing one- and two-step transformations

   Steinbrecher, Thomas; Joung, InSuk; Case, David A.

   Journal of Computational Chemistry (2011), 32 (15), 3253-3263CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)

   Mol. dynamics-based free energy calcns. allow the detn. of a variety of thermodn. quantities from computer simulations of small mols. Thermodn. integration (TI) calcns. can suffer from instabilities during the creation or annihilation of particles. This "singularity" problem can be addressed with "soft-core" potential functions which keep pairwise interaction energies finite for all configurations and provide smooth free energy curves. "One-step" transformations, in which electrostatic and van der Waals forces are simultaneously modified, can be simpler and less expensive than "two-step" transformations in which these properties are changed in sep. calcns. Here, the authors study solvation free energies for mols. of different hydrophobicity using both models. The authors provide recommended values for the two parameters αLJ and βC controlling the behavior of the soft-core Lennard-Jones and Coulomb potentials and compare one- and two-step transformations with regard to their suitability for numerical integration. For many types of transformations, the one-step procedure offers a convenient and accurate approach to free energy ests. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011.
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    Lee, T.-S.; Lin, Z.; Allen, B. K.; Lin, C.; Radak, B. K.; Tao, Y.; Tsai, H.-C.; Sherman, W.; York, D. M. Improved Alchemical Free Energy Calculations with Optimized Smoothstep Softcore Potentials. J. Chem. Theory Comput. 2020, 16, 5512– 5525,  DOI: 10.1021/acs.jctc.0c00237

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    Improved Alchemical Free Energy Calculations with Optimized Smoothstep Softcore Potentials

    Lee, Tai-Sung; Lin, Zhixiong; Allen, Bryce K.; Lin, Charles; Radak, Brian K.; Tao, Yujun; Tsai, Hsu-Chun; Sherman, Woody; York, Darrin M.

    Journal of Chemical Theory and Computation (2020), 16 (9), 5512-5525CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    Progress in the development of GPU-accelerated free energy simulation software has enabled practical applications on complex biol. systems and fueled efforts to develop more accurate and robust predictive methods. In particular, this work reexamines concerted (a.k.a., one-step or unified) alchem. transformations commonly used in the prediction of hydration and relative binding free energies (RBFEs). The authors first classify several known challenges in these calcns. into three categories: endpoint catastrophes, particle collapse, and large gradient-jumps. While endpoint catastrophes have long been addressed using softcore potentials, the remaining two problems occur much more sporadically and can result in either numerical instability (i.e., complete failure of a simulation) or inconsistent estn. (i.e., stochastic convergence to an incorrect result). The particle collapse problem stems from an imbalance in short-range electrostatic and repulsive interactions and can, in principle, be solved by appropriately balancing the resp. softcore parameters. However, the large gradient-jump problem itself arises from the sensitivity of the free energy to large values of the softcore parameters, as might be used in trying to solve the particle collapse issue. Often, no satisfactory compromise exists with the existing softcore potential form. As a framework for solving these problems, the authors developed a new family of smoothstep softcore (SSC) potentials motivated by an anal. of the derivs. along the alchem. path. The smoothstep polynomials generalize the monomial functions that are used in most implementations and provide an addnl. path-dependent smoothing parameter. The effectiveness of this approach is demonstrated on simple yet pathol. cases that illustrate the three problems outlined. With appropriate parameter selection, the authors find that a second-order SSC(2) potential does at least as well as the conventional approach and provides vast improvement in terms of consistency across all cases. Last, the authors compare the concerted SSC(2) approach against the gold-std. stepwise (a.k.a., decoupled or multistep) scheme over a large set of RBFE calcns. as might be encountered in drug discovery.
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    Kim, S.; Oshima, H.; Zhang, H.; Kern, N. R.; Re, S.; Lee, J.; Roux, B.; Sugita, Y.; Jiang, W.; Im, W. CHARMM-GUI free energy calculator for absolute and relative ligand solvation and binding free energy simulations. J. Chem. Theory Comput. 2020, 16, 7207– 7218,  DOI: 10.1021/acs.jctc.0c00884

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    CHARMM-GUI Free Energy Calculator for Absolute and Relative Ligand Solvation and Binding Free Energy Simulations

    Kim, Seonghoon; Oshima, Hiraku; Zhang, Han; Kern, Nathan R.; Re, Suyong; Lee, Jumin; Roux, Benoit; Sugita, Yuji; Jiang, Wei; Im, Wonpil

    Journal of Chemical Theory and Computation (2020), 16 (11), 7207-7218CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    Alchem. free energy simulations have long been utilized to predict free energy changes for binding affinity and soly. of small mols. However, while the theor. foundation of these methods is well established, seamlessly handling many of the practical aspects regarding the prepn. of the different thermodn. end states of complex mol. systems and the numerous processing scripts often remains a burden for successful applications. In this work, we present CHARMM-GUI Free Energy Calculator (http://www.charmm-gui.org/input/fec) that provides various alchem. free energy perturbation mol. dynamics (FEP/MD) systems with input and post-processing scripts for NAMD and GENESIS. Four submodules are available: Abs. Ligand Binder (for abs. ligand binding FEP/MD), Relative Ligand Binder (for relative ligand binding FEP/MD), Abs. Ligand Solvator (for abs. ligand solvation FEP/MD), and Relative Ligand Solvator (for relative ligand solvation FEP/MD). Each module is designed to build multiple systems of a set of selected ligands at once for high-throughput FEP/MD simulations. The capability of Free Energy Calculator is illustrated by abs. and relative solvation FEP/MD of a set of ligands and abs. and relative binding FEP/MD of a set of ligands for T4-lysozyme in soln. and the adenosine A2A receptor in a membrane. The calcd. free energy values are overall consistent with the exptl. and published free energy results (within ∼ 1 kcal/mol). We hope that Free Energy Calculator is useful to carry out high-throughput FEP/MD simulations in the field of biomol. sciences and drug discovery.
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    Zhang, H.; Kim, S.; Giese, T. J.; Lee, T.-S.; Lee, J.; York, D. M.; Im, W. CHARMM-GUI Free Energy Calculator for Practical Ligand Binding Free Energy Simulations with AMBER. J. Chem. Inf. Model. 2021, 61, 4145– 4151,  DOI: 10.1021/acs.jcim.1c00747

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    CHARMM-GUI Free Energy Calculator for Practical Ligand Binding Free Energy Simulations with AMBER

    Zhang, Han; Kim, Seonghoon; Giese, Timothy J.; Lee, Tai-Sung; Lee, Jumin; York, Darrin M.; Im, Wonpil

    Journal of Chemical Information and Modeling (2021), 61 (9), 4145-4151CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)

    Alchem. free energy methods, such as free energy perturbation (FEP) and thermodn. integration (TI), become increasingly popular and crucial for drug design and discovery. However, the system prepn. of alchem. free energy simulation is an error-prone, time-consuming, and tedious process for a large no. of ligands. To address this issue, we have recently presented CHARMM-GUI Free Energy Calculator that can provide input and postprocessing scripts for NAMD and GENESIS FEP mol. dynamics systems. In this work, we extended three submodules of Free Energy Calculator to work with the full suite of GPU-accelerated alchem. free energy methods and tools in AMBER, including input and postprocessing scripts. The BACE1 (β-secretase 1) benchmark set was used to validate the AMBER-TI simulation systems and scripts generated by Free Energy Calculator. The overall results of relatively large and diverse systems are almost equiv. with different protocols (unified and split) and with different timesteps (1, 2, and 4 fs), with R2 0.9. More importantly, the av. free energy differences between two protocols are small and reliable with four independent runs, with a mean unsigned error (MUE) below 0.4 kcal/mol. Running at least four independent runs for each pair with AMBER20 (and FF19SB/GAFF2.1/OPC force fields), we obtained a MUE of 0.99 kcal/mol and root-mean-square error of 1.31 kcal/mol for 58 alchem. transformations in comparison with expts. data. In addn., a set of ligands for T4-lysozyme was used to further validate our free energy calcn. protocol whose results are close to exptl. data (within 1 kcal/mol). In summary, Free Energy Calculator provides a user-friendly web-based tool to generate the AMBER-TI system and input files for high-throughput binding free energy calcns. with access to the full set of GPU-accelerated alchem. free energy, enhanced sampling, and anal. methods in AMBER.
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    Liu, S.; Wu, Y.; Lin, T.; Abel, R.; Redmann, J. P.; Summa, C. M.; Jaber, V. R.; Lim, N. M.; Mobley, D. L. Lead optimization mapper: automating free energy calculations for lead optimization. J. Comput.-Aided Mol. Des. 2013, 27, 755– 770,  DOI: 10.1007/s10822-013-9678-y

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    Lead optimization mapper: automating free energy calculations for lead optimization

    Liu, Shuai; Wu, Yujie; Lin, Teng; Abel, Robert; Redmann, Jonathan P.; Summa, Christopher M.; Jaber, Vivian R.; Lim, Nathan M.; Mobley, David L.

    Journal of Computer-Aided Molecular Design (2013), 27 (9), 755-770CODEN: JCADEQ; ISSN:0920-654X. (Springer)

    Alchem. free energy calcns. hold increasing promise as an aid to drug discovery efforts. However, applications of these techniques in discovery projects have been relatively few, partly because of the difficulty of planning and setting up calcns. Here, we introduce lead optimization mapper, LOMAP, an automated algorithm to plan efficient relative free energy calcns. between potential ligands within a substantial library of perhaps hundreds of compds. In this approach, ligands are first grouped by structural similarity primarily based on the size of a (loosely defined) maximal common substructure, and then calcns. are planned within and between sets of structurally related compds. An emphasis is placed on ensuring that relative free energies can be obtained between any pair of compds. without combining the results of too many different relative free energy calcns. (to avoid accumulation of error) and by providing some redundancy to allow for the possibility of error and consistency checking and provide some insight into when results can be expected to be unreliable. The algorithm is discussed in detail and a Python implementation, based on both Schroedinger's and OpenEye's APIs, has been made available freely under the BSD license.
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    Fleck, M.; Wieder, M.; Boresch, S. Dummy Atoms in Alchemical Free Energy Calculations. J. Chem. Theory Comput. 2021, 17, 4403– 4419,  DOI: 10.1021/acs.jctc.0c01328

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    Dummy Atoms in Alchemical Free Energy Calculations

    Fleck, Markus; Wieder, Marcus; Boresch, Stefan

    Journal of Chemical Theory and Computation (2021), 17 (7), 4403-4419CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    In calcns. of relative free energy differences, the no. of atoms of the initial and final states is rarely the same. This necessitates the introduction of dummy atoms. These placeholders interact with the phys. system only by bonded energy terms. We investigate the conditions necessary so that the presence of dummy atoms does not influence the result of a relative free energy calcn. On the one hand, one has to ensure that dummy atoms only give a multiplicative contribution to the partition function so that their contribution cancels from double-free energy differences. On the other hand, the bonded terms used to attach a dummy atom (or group of dummy atoms) to the phys. system have to maintain it in a well-defined position and orientation relative to the phys. system. A detailed theor. anal. of both aspects is provided, illustrated by 24 calcns. of relative solvation free energy differences, for which all four legs of the underlying thermodn. cycle were computed. Cycle closure (or lack thereof) was used as a sensitive indicator to probing the effects of dummy atom treatment on the resulting free energy differences. We find that a naive (but often practiced) treatment of dummy atoms results in errors of up to kBT when calcg. the relative solvation free energy difference between two small solutes, such as methane and ammonia. While our anal. focuses on the so-called single topol. approach to set up alchem. transformations, similar considerations apply to dual topol., at least many widely used variants thereof.
15. 15

    Zou, J.; Tian, C.; Simmerling, C. Blinded prediction of protein-ligand binding affinity using Amber thermodynamic integration for the 2018 D3R grand challenge 4. J. Comput.-Aided Mol. Des. 2019, 33, 1021– 1029,  DOI: 10.1007/s10822-019-00223-x

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    Blinded prediction of protein-ligand binding affinity using Amber thermodynamic integration for the 2018 D3R grand challenge 4

    Zou, Junjie; Tian, Chuan; Simmerling, Carlos

    Journal of Computer-Aided Molecular Design (2019), 33 (12), 1021-1029CODEN: JCADEQ; ISSN:0920-654X. (Springer)

    In the framework of the 2018 Drug Design Data Resource grand challenge 4, blinded predictions on relative binding free energy were performed for a set of 39 ligands of the Cathepsin S protein. We leveraged the GPU-accelerated thermodn. integration of Amber 18 to advance our computational prediction. When our entry was compared to exptl. results, a good correlation was obsd. (Kendall's τ: 0.62, Spearman's ρ: 0.80 and Pearson's R: 0.82). We designed a parallelized transformation map that placed ligands into several groups based on common alchem. substructures; TI transformations were carried out for each ligand to the relevant substructure, and between substructures. Our calcns. were all conducted using the linear potential scaling scheme in Amber TI because we believe the softcore potential/dual-topol. approach as implemented in current Amber TI is highly fault-prone for some transformations. The issue is illustrated by using two examples in which typical prepn. for the dual-topol. approach of Amber TI fails. Overall, the high accuracy of our prediction is a result of recent advances in force fields (ff14SB and GAFF), as well as rapid calcn. of ensemble avs. enabled by the GPU implementation of Amber. The success shown here in a blinded prediction strongly suggests that alchem. free energy calcn. in Amber is a promising tool for future com. drug design.
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    Jiang, W.; Chipot, C.; Roux, B. Computing relative binding affinity of ligands to receptor: An effective hybrid single-dual-topology free-energy perturbation approach in NAMD. J. Chem. Inf. Model. 2019, 59, 3794– 3802,  DOI: 10.1021/acs.jcim.9b00362

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    Computing Relative Binding Affinity of Ligands to Receptor: An Effective Hybrid Single-Dual-Topology Free-Energy Perturbation Approach in NAMD

    Jiang, Wei; Chipot, Christophe; Roux, Benoit

    Journal of Chemical Information and Modeling (2019), 59 (9), 3794-3802CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)

    An effective hybrid single-dual-topol. protocol is designed for the calcn. of relative binding affinities of small ligands to a receptor. The protocol was developed as an extension of the NAMD mol. dynamics program, which exclusively supports a dual-topol. framework for relative alchem. free-energy perturbation (FEP) calcns. In this protocol, the alchem. end states are represented as two sep. mols. sharing a common substructure identified through max. structural mapping. Within the substructure, an atom-to-atom correspondence is established, and each pair of corresponding atoms is holonomically constrained to share identical coordinates at all time throughout the simulation. The forces are projected and combined at each step for propagation. Following this formulation, a set of illustrative calcns. of reliable expt./simulation data, including relative solvation free energies of small mols. and relative binding affinities of drug compds. to proteins, are presented. To enhance sampling of the dual-topol. region, the FEP calcns. were carried out within a replica-exchange MD scheme supported by the multiple-copy algorithm module of NAMD, with periodically attempted swapping of the thermodn. coupling parameter λ between neighboring states. The results are consistent with expts. and benchmarks reported in the literature, lending support to the validity of the current protocol. In summary, this hybrid single-dual-topol. approach combines the conceptual simplicity of the dual-topol. paradigm with the advantageous sampling efficiency of the single-topol. approach, making it an ideal strategy for high-throughput in silico drug design.
18. 18

    Rocklin, G. J.; Mobley, D. L.; Dill, K. A. Separated topologiesA method for relative binding free energy calculations using orientational restraints. J. Chem. Phys. 2013, 138, 085104,  DOI: 10.1063/1.4792251

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    18

    Separated topologies - A method for relative binding free energy calculations using orientational restraints

    Rocklin, Gabriel J.; Mobley, David L.; Dill, Ken A.

    Journal of Chemical Physics (2013), 138 (8), 085104/1-085104/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    Orientational restraints can improve the efficiency of alchem. free energy calcns., but they are not typically applied in relative binding calcns., which compute the affinity difference been two ligands. Here, we describe a new "sepd. topologies" method, which computes relative binding free energies using orientational restraints and which has several advantages over existing methods. While std. approaches maintain the initial and final ligand in a shared orientation, the sepd. topologies approach allows the initial and final ligands to have distinct orientations. This avoids a slowly converging reorientation step in the calcn. The sepd. topologies approach can also be applied to det. the relative free energies of multiple orientations of the same ligand. We illustrate the approach by calcg. the relative binding free energies of two compds. to an engineered site in cytochrome c peroxidase. (c) 2013 American Institute of Physics.
19. 19

    Chen, W.; Deng, Y.; Russell, E.; Wu, Y.; Abel, R.; Wang, L. Accurate calculation of relative binding free energies between ligands with different net charges. J. Chem. Theory Comput. 2018, 14, 6346– 6358,  DOI: 10.1021/acs.jctc.8b00825

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    19

    Accurate Calculation of Relative Binding Free Energies between Ligands with Different Net Charges

    Chen, Wei; Deng, Yuqing; Russell, Ellery; Wu, Yujie; Abel, Robert; Wang, Lingle

    Journal of Chemical Theory and Computation (2018), 14 (12), 6346-6358CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    In drug discovery programs, modifications that change the net charge of the ligands are often considered to improve the binding potency and soly., or to address other ADME/Tox problems. Accurate calcn. of the binding free-energy changes assocd. with charge-changing perturbations remains a great challenge of central importance in computational drug discovery. The finite size effects assocd. with periodic boundary condition and lattice summation employed in common mol. dynamics simulations introduce artifacts in the electrostatic potential energy calcns., which need to be carefully handled for accurate free-energy calcns. between systems with different net charges. The salts in the buffer soln. of exptl. binding affinity assays also have a strong effect on the binding free energies between charged species, which further complicates the modeling of the charge-changing perturbations. Here, we extend our free-energy perturbation (FEP) algorithm, which has been extensively applied to many drug discovery programs for relative binding free-energy calcns. between ligands with the same net charge (charge-conserving perturbation), to enable charge-changing perturbations. We have investigated three different approaches to correct the finite size effects and tested them on 10 protein targets and 31 charge-changing perturbations. We have found that all three methods are able to successfully eliminate the box-size dependence of calcd. binding free energies assocd. with brute force FEP. Moreover, inclusion of salts matching the ionic strength of exptl. buffer soln. significantly improves the calcd. binding free energies. For ligands with multiple possible protonation states, we applied the pKa correction to account for the ionization equil. of the ligands and the results are significantly improved. Finally, the calcd. binding free energies from these methods agree with each other, and also agree well with the exptl. results. The root-mean-square error between the calcd. binding free energies and exptl. data is 1.1 kcal/mol, which is on par with the accuracy of charge-conserving perturbations. We anticipate that the outstanding accuracy demonstrated here across a broad range of target classes may have significant implications for drug discovery projects, where charge-changing modifications must be considered.
20. 20

    Rocklin, G. J.; Mobley, D. L.; Dill, K. A.; Hünenberger, P. H. Calculating the binding free energies of charged species based on explicit-solvent simulations employing lattice-sum methods: An accurate correction scheme for electrostatic finite-size effects. J. Chem. Phys. 2013, 139, 184103,  DOI: 10.1063/1.4826261

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    20

    Calculating the binding free energies of charged species based on explicit-solvent simulations employing lattice-sum methods: An accurate correction scheme for electrostatic finite-size effects

    Rocklin, Gabriel J.; Mobley, David L.; Dill, Ken A.; Huenenberger, Philippe H.

    Journal of Chemical Physics (2013), 139 (18), 184103/1-184103/32CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    The calcn. of a protein-ligand binding free energy based on mol. dynamics (MD) simulations generally relies on a thermodn. cycle in which the ligand is alchem. inserted into the system, both in the solvated protein and free in soln. The corresponding ligand-insertion free energies are typically calcd. in nanoscale computational boxes simulated under periodic boundary conditions and considering electrostatic interactions defined by a periodic lattice-sum. This is distinct from the ideal bulk situation of a system of macroscopic size simulated under non-periodic boundary conditions with Coulombic electrostatic interactions. This discrepancy results in finite-size effects, which affect primarily the charging component of the insertion free energy, are dependent on the box size, and can be large when the ligand bears a net charge, esp. if the protein is charged as well. This article studies finite-size effects on calcd. charging free energies using as a test case: the binding of the ligand 2-amino-5-methylthiazole (net charge +1 e) to a mutant form of yeast cytochrome c peroxidase in water. Considering different charge isoforms of the protein (net charges -5, 0, +3, or +9 e), either in the absence or the presence of neutralizing counterions, and sizes of the cubic computational box (edges ranging from 7.42 to 11.02 nm), the potentially large magnitude of finite-size effects on the raw charging free energies (up to 17.1 kJ mol-1) is demonstrated. Two correction schemes are then proposed to eliminate these effects, a numerical and an anal. one. Both schemes are based on a continuum-electrostatics anal. and require performing Poisson-Boltzmann (PB) calcns. on the protein-ligand system. While the numerical scheme requires PB calcns. under both non-periodic and periodic boundary conditions, the latter at the box size considered in the MD simulations, the anal. scheme only requires three non-periodic PB calcns. for a given system, its dependence on the box size being anal. The latter scheme also provides insight into the phys. origin of the finite-size effects. These two schemes also encompass a correction for discrete solvent effects that persists even in the limit of infinite box sizes. Application of either scheme essentially eliminates the size dependence of the cor. charging free energies (maximal deviation of 1.5 kJ mol-1). Because it is simple to apply, the anal. correction scheme offers a general soln. to the problem of finite-size effects in free-energy calcns. involving charged solutes, as encountered in calcns. concerning, e.g., protein-ligand binding, biomol. assocn., residue mutation, pKa and redox potential estn., substrate transformation, solvation, and solvent-solvent partitioning. (c) 2013 American Institute of Physics.
21. 21

    Dixit, S. B.; Chipot, C. Can absolute free energies of association be estimated from molecular mechanical simulations? The biotin-streptavidin system revisited. J. Phys. Chem. A 2001, 105, 9795– 9799,  DOI: 10.1021/jp011878v

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    21

    Can absolute free energies of association Be estimated from molecular mechanical simulations? The biotin-streptavidin system revisited

    Dixit, Surjit B.; Chipot, Christophe

    Journal of Physical Chemistry A (2001), 105 (42), 9795-9799CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)

    Employing state-of-the-art mol. dynamics protocols, we carried out free energy calcns. in the (N, P, T) ensemble on a fully hydrated biotin-streptavidin assembly of 27 702 atoms. The reported abs. binding free energy of -16.6±1.9 kcal/mol is in good agreement with the exptl. est. of -18.3 kcal/mol by Weber et al. [J. Am. Chem. Soc. 1992, 114, 3197-3200]. These simulations illustrate that the use of massively parallel architectures in conjunction with efficient algorithms allows us to tackle biol. relevant problems involving large mol. systems and to access key properties, like the assocn. of a protein with its ligand, under rigorous thermodn. conditions.
22. 22

    Chen, W.; Wallace, J. A.; Yue, Z.; Shen, J. K. Introducing titratable water to all-atom molecular dynamics at constant pH. Biophys. J. 2013, 105, L15– L17,  DOI: 10.1016/j.bpj.2013.06.036

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    22

    Introducing Titratable Water to All-Atom Molecular Dynamics at Constant pH

    Chen, Wei; Wallace, Jason A.; Yue, Zhi; Shen, Jana K.

    Biophysical Journal (2013), 105 (4), L15-L17CODEN: BIOJAU; ISSN:0006-3495. (Cell Press)

    Recent development of titratable coions has paved the way for realizing all-atom mol. dynamics at const. pH. To further improve phys. realism, here we describe a technique in which proton titrn. of the solute is directly coupled to the interconversion between water and hydroxide or hydronium. We test the new method in replica-exchange continuous const. pH mol. dynamics simulations of three proteins, HP36, BBL, and HEWL. The calcd. pKa values based on 10-ns sampling per replica have the av. abs. and root-mean-square errors of 0.7 and 0.9 pH units, resp. Introducing titratable water in mol. dynamics offers a means to model proton exchange between solute and solvent, thus opening a door to gaining new insights into the intricate details of biol. phenomena involving proton translocation.
23. 23

    Wang, L.; Deng, Y.; Wu, Y.; Kim, B.; LeBard, D. N.; Wandschneider, D.; Beachy, M.; Friesner, R. A.; Abel, R. Accurate modeling of scaffold hopping transformations in drug discovery. J. Chem. Theory Comput. 2017, 13, 42– 54,  DOI: 10.1021/acs.jctc.6b00991

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    23

    Accurate Modeling of Scaffold Hopping Transformations in Drug Discovery

    Wang, Lingle; Deng, Yuqing; Wu, Yujie; Kim, Byungchan; LeBard, David N.; Wandschneider, Dan; Beachy, Mike; Friesner, Richard A.; Abel, Robert

    Journal of Chemical Theory and Computation (2017), 13 (1), 42-54CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    The accurate prediction of protein-ligand binding free energies remains a significant challenge of central importance in computational biophysics and structure-based drug design. Multiple recent advances including the development of greatly improved protein and ligand mol. mechanics force fields, more efficient enhanced sampling methods, and low-cost powerful GPU computing clusters have enabled accurate and reliable predictions of relative protein-ligand binding free energies through the free energy perturbation (FEP) methods. However, the existing FEP methods can only be used to calc. the relative binding free energies for R-group modifications or single-atom modifications, and cannot be used to efficiently evaluate scaffold hopping modifications to a lead mol. Scaffold hopping or core hopping, a very common design strategy in drug discovery projects, is not only crit. in the early stages of a discovery campaign where novel active matter must be identified, but also in lead optimization where the resoln. of a variety of ADME/Tox problems may require identification of a novel core structure. In this paper, the authors introduce a method that enables theor. rigorous, yet computationally tractable, relative protein-ligand binding free energy calcns. to be pursued for scaffold hopping modifications. The authors apply the method to six pharmaceutically interesting cases where diverse types of scaffold hopping modifications were required to identify the drug mols. ultimately sent into the clinic. For these six diverse cases, the predicted binding affinities were in close agreement with expt., demonstrating the wide applicability and the significant impact Core Hopping FEP may provide in drug discovery projects.
24. 24

    Zou, J.; Li, Z.; Liu, S.; Peng, C.; Fang, D.; Wan, X.; Lin, Z.; Lee, T.-S.; Raleigh, D. P.; Yang, M.; Simmerling, C. Scaffold Hopping Transformations Using Auxiliary Restraints for Calculating Accurate Relative Binding Free Energies. J. Chem. Theory Comput. 2021, 17, 3710– 3726,  DOI: 10.1021/acs.jctc.1c00214

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    24

    Scaffold Hopping Transformations Using Auxiliary Restraints for Calculating Accurate Relative Binding Free Energies

    Zou, Junjie; Li, Zhipeng; Liu, Shuai; Peng, Chunwang; Fang, Dong; Wan, Xiao; Lin, Zhixiong; Lee, Tai-Sung; Raleigh, Daniel P.; Yang, Mingjun; Simmerling, Carlos

    Journal of Chemical Theory and Computation (2021), 17 (6), 3710-3726CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    In silico screening of drug-target interactions is a key part of the drug discovery process. Changes in the drug scaffold via contraction or expansion of rings, the breaking of rings, and the introduction of cyclic structures from acyclic structures are commonly applied by medicinal chemists to improve binding affinity and enhance favorable properties of candidate compds. These processes, commonly referred to as scaffold hopping, are challenging to model computationally. Although relative binding free energy (RBFE) calcns. have shown success in predicting binding affinity changes caused by perturbing R-groups attached to a common scaffold, applications of RBFE calcns. to modeling scaffold hopping are relatively limited. Scaffold hopping inevitably involves breaking and forming bond interactions of quadratic functional forms, which is highly challenging. A novel method for handling ring opening/closure/contraction/expansion and linker contraction/expansion is presented here. To the best of the knowledge, RBFE calcns. on linker contraction/expansion have not been previously reported. The method uses auxiliary restraints to hold the atoms at the ends of a bond in place during the breaking and forming of the bonds. The broad applicability of the method was demonstrated by examg. perturbations involving small-mol. macrocycles and mutations of proline in proteins. High accuracy was obtained using the method for most of the perturbations studied. The rigor of the method was isolated from the force field by validating the method using relative and abs. hydration free energy calcns. compared to std. simulation results. Unlike other methods that rely on λ-dependent functional forms for bond interactions, the method presented here can be employed using modern mol. dynamics software without modification of codes or force field functions.
25. 25

    Loeffler, H. H.; Bosisio, S.; Duarte Ramos Matos, G.; Suh, D.; Roux, B.; Mobley, D. L.; Michel, J. Reproducibility of free energy calculations across different molecular simulation software packages. J. Chem. Theory Comput. 2018, 14, 5567– 5582,  DOI: 10.1021/acs.jctc.8b00544

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    25

    Reproducibility of Free Energy Calculations across Different Molecular Simulation Software Packages

    Loeffler, Hannes H.; Bosisio, Stefano; Duarte Ramos Matos, Guilherme; Suh, Donghyuk; Roux, Benoit; Mobley, David L.; Michel, Julien

    Journal of Chemical Theory and Computation (2018), 14 (11), 5567-5582CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    Alchem. free energy calcns. are an increasingly important modern simulation technique to calc. free energy changes on binding or solvation. Contemporary mol. simulation software such as AMBER, CHARMM, GROMACS, and SOMD include support for the method. Implementation details vary among those codes, but users expect reliability and reproducibility, i.e., for a given mol. model and set of force field parameters, comparable free energy differences should be obtained within statistical bounds regardless of the code used. Relative alchem. free energy (RAFE) simulation is increasingly used to support mol. discovery projects, yet the reproducibility of the methodol. has been less well tested than its abs. counterpart. Here we present RAFE calcns. of hydration free energies for a set of small org. mols. and demonstrate that free energies can be reproduced to within about 0.2 kcal/mol with the aforementioned codes. Abs. alchem. free energy simulations have been carried out as a ref. Achieving this level of reproducibility requires considerable attention to detail and package-specific simulation protocols, and no universally applicable protocol emerges. The benchmarks and protocols reported here should be useful for the community to validate new and future versions of software for free energy calcns.
26. 26

    Jespers, W.; Esguerra, M.; Åqvist, J.; Gutiérrez-de Terán, H. QligFEP: an automated workflow for small molecule free energy calculations in Q. J. Cheminf 2019, 11, 26,  DOI: 10.1186/s13321-019-0348-5

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    There is no corresponding record for this reference.
27. 27

    Vilseck, J. Z.; Sohail, N.; Hayes, R. L.; Brooks, C. L., III Overcoming challenging substituent perturbations with multisite λ-dynamics: a case study targeting β-secretase 1. J. Phys. Chem. Lett. 2019, 10, 4875– 4880,  DOI: 10.1021/acs.jpclett.9b02004

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    27

    Overcoming Challenging Substituent Perturbations with Multisite λ-Dynamics: A Case Study Targeting β-Secretase 1

    Vilseck, Jonah Z.; Sohail, Noor; Hayes, Ryan L.; Brooks, Charles L.

    Journal of Physical Chemistry Letters (2019), 10 (17), 4875-4880CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)

    Alchem. free energy calcns. have made a dramatic impact upon the field of structure-based drug design by allowing functional group modifications to be explored computationally prior to exptl. synthesis and assay evaluation, thereby informing and directing synthetic strategies. In furthering the advancement of this area, a series of 21 β-secretase 1 (BACE1) inhibitors developed by Janssen Pharmaceuticals were examd. to evaluate the ability to explore large substituent perturbations, some of which contain scaffold modifications, with multisite λ-dynamics (MSλD), an innovative alchem. free energy framework. Our findings indicate that MSλD is able to efficiently explore all structurally diverse ligand end-states simultaneously within a single MD simulation with a high degree of precision and with reduced computational costs compared to the widely used approach TI/MBAR. Furthermore, computational predictions were shown to be accurate to within 0.5-0.8 kcal/mol when CM1A partial at. charges were combined with CHARMM or OPLS-AA-based force fields, demonstrating that MSλD is force field independent and a viable alternative to FEP or TI approaches for drug design.
28. 28

    Ligandswap. https://siremol.org/pages/apps/ligandswap.html (accessed September 12, 2021).

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    There is no corresponding record for this reference.
29. 29

    Wu, J. Z.; Azimi, S.; Khuttan, S.; Deng, N.; Gallicchio, E. Alchemical Transfer Approach to Absolute Binding Free Energy Estimation. J. Chem. Theory Comput. 2021, 17, 3309,  DOI: 10.1021/acs.jctc.1c00266

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    29

    Alchemical Transfer Approach to Absolute Binding Free Energy Estimation

    Wu, Joe Z.; Azimi, Solmaz; Khuttan, Sheenam; Deng, Nanjie; Gallicchio, Emilio

    Journal of Chemical Theory and Computation (2021), 17 (6), 3309-3319CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    The alchem. transfer method (ATM) for the calcn. of std. binding free energies of noncovalent mol. complexes is presented. The method is based on a coordinate displacement perturbation of the ligand between the receptor binding site and the explicit solvent bulk and a thermodn. cycle connected by a sym. intermediate in which the ligand interacts with the receptor and solvent environments with equal strength. While the approach is alchem., the implementation of the ATM is as straightforward as that for phys. pathway methods of binding. The method is applicable, in principle, with any force field, as it does not require splitting the alchem. transformations into electrostatic and nonelectrostatic steps, and it does not require soft-core pair potentials. We have implemented the ATM as a freely available and open-source plugin of the OpenMM mol. dynamics library. The method and its implementation are validated on the SAMPL6 SAMPLing host-guest benchmark set. The work paves the way to streamlined alchem. relative and abs. binding free energy implementations on many mol. simulation packages and with arbitrary energy functions including polarizable, quantum-mech., and artificial neural network potentials.
30. 30

    Gapsys, V.; Michielssens, S.; Peters, J. H.; de Groot, B. L.; Leonov, H. Molecular Modeling of Proteins; Springer, 2015; pp 173– 209.

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    There is no corresponding record for this reference.
31. 31

    Macchiagodena, M.; Pagliai, M.; Karrenbrock, M.; Guarnieri, G.; Iannone, F.; Procacci, P. Virtual Double-System Single-Box: A Nonequilibrium Alchemical Technique for Absolute Binding Free Energy Calculations: Application to Ligands of the SARS-CoV-2 Main Protease. J. Chem. Theory Comput. 2020, 16, 7160– 7172,  DOI: 10.1021/acs.jctc.0c00634

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    31

    Virtual Double-System Single-Box: A Nonequilibrium Alchemical Technique for Absolute Binding Free Energy Calculations: Application to Ligands of the SARS-CoV-2 Main Protease

    Macchiagodena, Marina; Pagliai, Marco; Karrenbrock, Maurice; Guarnieri, Guido; Iannone, Francesco; Procacci, Piero

    Journal of Chemical Theory and Computation (2020), 16 (11), 7160-7172CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    In the context of drug-receptor binding affinity calcns. using mol. dynamics techniques, we implemented a combination of Hamiltonian replica exchange (HREM) and a novel nonequil. alchem. methodol., called virtual double-system single-box, with increased accuracy, precision, and efficiency with respect to the std. nonequil. approaches. The method has been applied for the detn. of abs. binding free energies of 16 newly designed noncovalent ligands of the main protease (3CLpro) of SARS-CoV-2. The core structures of 3CLpro ligands were previously identified using a multimodal structure-based ligand design in combination with docking techniques. The calcd. binding free energies for 4 addnl. ligands with known activity (either for SARS-CoV or SARS-CoV-2 main protease) are also reported. The nature of binding in the 3CLpro active site and the involved residues besides the CYS-HYS catalytic dyad have been thoroughly characterized by enhanced sampling simulations of the bound state. We have identified several noncongeneric compds. with predicted low micromolar activity for 3CLpro inhibition, which may constitute possible lead compds. for the development of antiviral agents in Covid-19 treatment.
32. 32

    Harger, M.; Li, D.; Wang, Z.; Dalby, K.; Lagardère, L.; Piquemal, J.-P.; Ponder, J.; Ren, P. Tinker-OpenMM: Absolute and relative alchemical free energies using AMOEBA on GPUs. J. Comput. Chem. 2017, 38, 2047– 2055,  DOI: 10.1002/jcc.24853

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    Tinker-OpenMM: Absolute and relative alchemical free energies using AMOEBA on GPUs

    Harger, Matthew; Li, Daniel; Wang, Zhi; Dalby, Kevin; Lagardere, Louis; Piquemal, Jean-Philip; Ponder, Jay; Ren, Pengyu

    Journal of Computational Chemistry (2017), 38 (23), 2047-2055CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)

    The capabilities of the polarizable force fields for alchem. free energy calcns. have been limited by the high computational cost and complexity of the underlying potential energy functions. In this work, we present a GPU-based general alchem. free energy simulation platform for polarizable potential AMOEBA. Tinker-OpenMM, the OpenMM implementation of the AMOEBA simulation engine has been modified to enable both abs. and relative alchem. simulations on GPUs, which leads to a ∼200-fold improvement in simulation speed over a single CPU core. We show that free energy values calcd. using this platform agree with the results of Tinker simulations for the hydration of org. compds. and binding of host-guest systems within the statistical errors. In addn. to abs. binding, we designed a relative alchem. approach for computing relative binding affinities of ligands to the same host, where a special path was applied to avoid numerical instability due to polarization between the different ligands that bind to the same site. This scheme is general and does not require ligands to have similar scaffolds. We show that relative hydration and binding free energy calcd. using this approach match those computed from the abs. free energy approach. © 2017 Wiley Periodicals, Inc.
33. 33

    Panel, N.; Villa, F.; Fuentes, E. J.; Simonson, T. Accurate PDZ/peptide binding specificity with additive and polarizable free energy simulations. Biophys. J. 2018, 114, 1091– 1102,  DOI: 10.1016/j.bpj.2018.01.008

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    Accurate PDZ/peptide binding specificity with additive and polarizable free energy simulations

    Panel, Nicolas; Villa, Francesco; Fuentes, Ernesto J.; Simonson, Thomas

    Biophysical Journal (2018), 114 (5), 1091-1102CODEN: BIOJAU; ISSN:0006-3495. (Cell Press)

    PDZ domains contain 80-100 amino acids and bind short C-terminal sequences of target proteins. Their specificity is essential for cellular signaling pathways. Here, we studied the binding of the Tiam1 PDZ domain to peptides derived from the C-termini of its syndecan-1 and caspr4 targets. We used free energy perturbation (FEP) to characterize the binding energetics of one wild-type and 17 mutant complexes by simulating 21 alchem. transformations between pairs of complexes. Thirteen complexes had known exptl. affinities. FEP is a powerful tool to understand protein/ligand binding. It depends, however, on the accuracy of mol. dynamics force fields and conformational sampling. Both aspects require continued testing, esp. for ionic mutations. For 6 mutations that did not modify the net charge, we obtained excellent agreement with expt. using the additive, AMBER ff99SB force field, with a root mean square deviation (RMSD) of 0.37 kcal/mol. For 6 ionic mutations that modified the net charge, agreement was also good, with one large error (3 kcal/mol) and an RMSD of 0.9 kcal/mol for the other 5. The large error arose from the overstabilization of a protein/peptide salt bridge by the additive force field. Four of the ionic mutations were also simulated with the polarizable Drude force field, which represents the 1st test of this force field for protein/ligand binding free energy changes. The large error was eliminated and the RMS error for the 4 mutations was reduced from 1.8 to 1.2 kcal/mol. The overall accuracy of FEP indicated that it could be used to understand PDZ/peptide binding. Importantly, the results showed that for ionic mutations in buried regions, electronic polarization plays a significant role.
34. 34

    Beierlein, F. R.; Michel, J.; Essex, J. W. A simple QM/MM approach for capturing polarization effects in protein- ligand binding free energy calculations. J. Phys. Chem. B 2011, 115, 4911– 4926,  DOI: 10.1021/jp109054j

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    A Simple QM/MM Approach for Capturing Polarization Effects in Protein-Ligand Binding Free Energy Calculations

    Beierlein, Frank R.; Michel, Julien; Essex, Jonathan W.

    Journal of Physical Chemistry B (2011), 115 (17), 4911-4926CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)

    We present a mol. simulation protocol to compute free energies of binding, which combines a QM/MM correction term with rigorous classical free energy techniques, thereby accounting for electronic polarization effects. Relative free energies of binding are first computed using classical force fields, Monte Carlo sampling, and replica exchange thermodn. integration. Snapshots of the configurations at the end points of the perturbation are then subjected to DFT-QM/MM single-point calcns. using the B3LYP functional and a range of basis sets. The resulting quantum mech. energies are then processed using the Zwanzig equation to give free energies incorporating electronic polarization. Our approach is conceptually simple and does not require tightly coupled QM and MM software. The method has been validated by calcg. the relative free energies of hydration of methane and water and the relative free energy of binding of two inhibitors of cyclooxygenase-2. Closed thermodn. cycles are obtained across different pathways, demonstrating the correctness of the technique, although significantly more sampling is required for the protein-ligand system. Our method offers a simple and effective way to incorporate quantum mech. effects into computed free energies of binding.
35. 35

    Lodola, A.; De Vivo, M. Adv. Protein Chem. Struct. Biol.; Elsevier, 2012; Vol. 87; pp 337– 362.

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36. 36

    Hudson, P. S.; Woodcock, H. L.; Boresch, S. Use of interaction energies in QM/MM free energy simulations. J. Chem. Theory Comput. 2019, 15, 4632– 4645,  DOI: 10.1021/acs.jctc.9b00084

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    36

    Use of Interaction Energies in QM/MM Free Energy Simulations

    Hudson, Phillip S.; Woodcock, H. Lee; Boresch, Stefan

    Journal of Chemical Theory and Computation (2019), 15 (8), 4632-4645CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    The use of the most accurate (i.e., QM or QM/MM) levels of theory for free energy simulations (FES) is typically not possible. Primarily, this is because the computational cost assocd. with the extensive configurational sampling needed for converging FES is prohibitive. To ensure the feasibility of QM-based FES, the ''indirect'' approach is generally taken, necessitating a free energy calcn. between the MM and QM/MM potential energy surfaces. Ideally, this step is performed with std. free energy perturbation (Zwanzig's equation) as it only requires simulations be carried out at the low level of theory; however, work from several groups over the past few years has conclusively shown that Zwanzig's equation is ill-suited to this task. As such, many approxns. have arisen to mitigate difficulties with Zwanzig's equation. One particularly popular notion is that the convergence of Zwanzig's equation can be improved by using interaction energy differences instead of total energy differences. Although problematic numerical fluctuations (a major problem when using Zwanzig's equation) are indeed reduced, our results and anal. demonstrate that this ''interaction energy approxn.'' (IEA) is theor. incorrect, and the implicit approxn. invoked is spurious at best. Herein, we demonstrate this via solvation free energy calcns. using IEA from two different low levels of theory to the same target high level. Results from this proof-of-concept consistently yield the wrong results, deviating by ∼ 1.5 kcal/mol from the rigorously obtained value.
37. 37

    Smith, J. S.; Nebgen, B. T.; Zubatyuk, R.; Lubbers, N.; Devereux, C.; Barros, K.; Tretiak, S.; Isayev, O.; Roitberg, A. E. Approaching coupled cluster accuracy with a general-purpose neural network potential through transfer learning. Nature Commun. 2019, 10, 1– 8,  DOI: 10.1038/s41467-019-10827-4

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    Myeloid lineage enhancers drive oncogene synergy in CEBPA/CSF3R mutant acute myeloid leukemia

    Braun, Theodore P.; Okhovat, Mariam; Coblentz, Cody; Carratt, Sarah A.; Foley, Amy; Schonrock, Zachary; Smith, Brittany M.; Nevonen, Kimberly; Davis, Brett; Garcia, Brianna; LaTocha, Dorian; Weeder, Benjamin R.; Grzadkowski, Michal R.; Estabrook, Joey C.; Manning, Hannah G.; Watanabe-Smith, Kevin; Jeng, Sophia; Smith, Jenny L.; Leonti, Amanda R.; Ries, Rhonda E.; McWeeney, Shannon; Di Genua, Cristina; Drissen, Roy; Nerlov, Claus; Meshinchi, Soheil; Carbone, Lucia; Druker, Brian J.; Maxson, Julia E.

    Nature Communications (2019), 10 (1), 1-15CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)

    Acute Myeloid Leukemia (AML) develops due to the acquisition of mutations from multiple functional classes. Here, we demonstrate that activating mutations in the granulocyte colony stimulating factor receptor (CSF3R), cooperate with loss of function mutations in the transcription factor CEBPA to promote acute leukemia development. The interaction between these distinct classes of mutations occurs at the level of myeloid lineage enhancers where mutant CEBPA prevents activation of a subset of differentiation assocd. enhancers. To confirm this enhancer-dependent mechanism, we demonstrate that CEBPA mutations must occur as the initial event in AML initiation. This improved mechanistic understanding will facilitate therapeutic development targeting the intersection of oncogene cooperativity.
38. 38

    Rufa, D. A.; Macdonald, H. E. B.; Fass, J.; Wieder, M.; Grinaway, P. B.; Roitberg, A. E.; Isayev, O.; Chodera, J. D. Towards chemical accuracy for alchemical free energy calculations with hybrid physics-based machine learning/molecular mechanics potentials. bioRxiv , 2020, DOI: 10.1101/2020.07.29.227959

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39. 39

    Zhang, B.; Kilburg, D.; Eastman, P.; Pande, V. S.; Gallicchio, E. Efficient Gaussian Density Formulation of Volume and Surface Areas of Macromolecules on Graphical Processing Units. J. Comput. Chem. 2017, 38, 740– 752,  DOI: 10.1002/jcc.24745

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    39

    Efficient Gaussian density formulation of volume and surface areas of macromolecules on graphical processing units

    Zhang, Baofeng; Kilburg, Denise; Eastman, Peter; Pande, Vijay S.; Gallicchio, Emilio

    Journal of Computational Chemistry (2017), 38 (10), 740-752CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)

    The authors present an algorithm to efficiently compute accurate vols. and surface areas of macromols. on graphical processing unit (GPU) devices using an analytic model which represents at. vols. by continuous Gaussian densities. The vol. of the mol. is expressed by the inclusion-exclusion formula, which is based on the summation of overlap integrals among multiple at. densities. The surface area of the mol. was obtained by differentiation of the mol. vol. with respect to at. radii. The many-body nature of the model makes a port to GPU devices challenging. To the authors' knowledge, this is the first reported full implementation of this model on GPU hardware. To accomplish this, the authors used recursive strategies to construct the tree of overlaps and to accumulate vols. and their gradients on the tree data structures so as to minimize memory contention. The algorithm was used in the formulation of a surface area-based non-polar implicit solvent model implemented as an open source plug-in (named GaussVol) for the popular OpenMM library for mol. mechanics modeling. GaussVol is 50 to 100 times faster than the authors' best optimized implementation for the CPUs, achieving speeds >100 ns/day with 1 fs time-step for protein-sized systems on commodity GPUs.
40. 40

    Spiriti, J.; Subramanian, S. R.; Palli, R.; Wu, M.; Zuckerman, D. M. Middle-way flexible docking: Pose prediction using mixed-resolution Monte Carlo in estrogen receptor α. PloS One 2019, 14, e0215694,  DOI: 10.1371/journal.pone.0215694

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    Middle-way flexible docking: Pose prediction using mixed-resolution Monte Carlo in estrogen receptor α

    Spiriti, Justin; Subramanian, Sundar Raman; Palli, Rohith; Wu, Maria; Zuckerman, Daniel M.

    PLoS One (2019), 14 (4), e0215694CODEN: POLNCL; ISSN:1932-6203. (Public Library of Science)

    There is a vast gulf between the two primary strategies for simulating protein-ligand interactions. Docking methods significantly limit or eliminate protein flexibility to gain great speed at the price of uncontrolled inaccuracy, whereas fully flexible atomistic mol. dynamics simulations are expensive and often suffer from limited sampling. We have developed a flexible docking approach geared esp. for highly flexible or poorly resolved targets based on mixed-resoln. Monte Carlo (MRMC), which is intended to offer a balance among speed, protein flexibility, and sampling power. The binding region of the protein is treated with a std. atomistic force field, while the remainder of the protein is modeled at the residue level with a G‾o model that permits protein flexibility while saving computational cost. Implicit solvation is used. Here we assess three facets of the MRMC approach with implications for other docking studies: (i) the role of receptor flexibility in cross-docking pose prediction; (ii) the use of non-equil. candidate Monte Carlo (NCMC) and (iii) the use of pose-clustering in scoring. We examine 61 co-crystd. ligands of estrogen receptor α, an important cancer target known for its flexibility. We also compare the performance of the MRMC approach with Autodock smina. Adding protein flexibility, not surprisingly, leads to significantly lower total energies and stronger interactions between protein and ligand, but notably we document the important role of backbone flexibility in the improvement. The improved backbone flexibility also leads to improved performance relative to smina. Somewhat unexpectedly, our implementation of NCMC leads to only modestly improved sampling of ligand poses. Overall, the addn. of protein flexibility improves the performance of docking, as measured by energy-ranked poses, but we do not find significant improvements based on cluster information or the use of NCMC. We discuss possible improvements for the model including alternative coarse-grained force fields, improvements to the treatment of solvation, and adding addnl. types of NCMC moves.
41. 41

    Gallicchio, E.; Xia, J.; Flynn, W. F.; Zhang, B.; Samlalsingh, S.; Mentes, A.; Levy, R. M. Asynchronous replica exchange software for grid and heterogeneous computing. Comput. Phys. Commun. 2015, 196, 236– 246,  DOI: 10.1016/j.cpc.2015.06.010

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    41

    Asynchronous replica exchange software for grid and heterogeneous computing

    Gallicchio, Emilio; Xia, Junchao; Flynn, William F.; Zhang, Baofeng; Samlalsingh, Sade; Mentes, Ahmet; Levy, Ronald M.

    Computer Physics Communications (2015), 196 (), 236-246CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)

    Parallel replica exchange sampling is an extended ensemble technique often used to accelerate the exploration of the conformational ensemble of atomistic mol. simulations of chem. systems. Inter-process communication and coordination requirements have historically discouraged the deployment of replica exchange on distributed and heterogeneous resources. Here we describe the architecture of a software (named ASyncRE) for performing asynchronous replica exchange mol. simulations on volunteered computing grids and heterogeneous high performance clusters. The asynchronous replica exchange algorithm on which the software is based avoids centralized synchronization steps and the need for direct communication between remote processes. It allows mol. dynamics threads to progress at different rates and enables parameter exchanges among arbitrary sets of replicas independently from other replicas. ASyncRE is written in Python following a modular design conducive to extensions to various replica exchange schemes and mol. dynamics engines. Applications of the software for the modeling of assocn. equil. of supramol. and macromol. complexes on BOINC campus computational grids and on the CPU/MIC heterogeneous hardware of the XSEDE Stampede supercomputer are illustrated. They show the ability of ASyncRE to utilize large grids of desktop computers running the Windows, MacOS, and/or Linux operating systems as well as collections of high performance heterogeneous hardware devices.
42. 42

    Shirts, M. R.; Chodera, J. D. Statistically optimal analysis of samples from multiple equilibrium states. J. Chem. Phys. 2008, 129, 124105,  DOI: 10.1063/1.2978177

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    Statistically optimal analysis of samples from multiple equilibrium states

    Shirts, Michael R.; Chodera, John D.

    Journal of Chemical Physics (2008), 129 (12), 124105/1-124105/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    We present a new estimator for computing free energy differences and thermodn. expectations as well as their uncertainties from samples obtained from multiple equil. states via either simulation or expt. The estimator, which we call the multistate Bennett acceptance ratio estimator (MBAR) because it reduces to the Bennett acceptance ratio estimator (BAR) when only two states are considered, has significant advantages over multiple histogram reweighting methods for combining data from multiple states. It does not require the sampled energy range to be discretized to produce histograms, eliminating bias due to energy binning and significantly reducing the time complexity of computing a soln. to the estg. equations in many cases. Addnl., an est. of the statistical uncertainty is provided for all estd. quantities. In the large sample limit, MBAR is unbiased and has the lowest variance of any known estimator for making use of equil. data collected from multiple states. We illustrate this method by producing a highly precise est. of the potential of mean force for a DNA hairpin system, combining data from multiple optical tweezer measurements under const. force bias. (c) 2008 American Institute of Physics.
43. 43

    Tan, Z.; Gallicchio, E.; Lapelosa, M.; Levy, R. M. Theory of binless multi-state free energy estimation with applications to protein-ligand binding. J. Chem. Phys. 2012, 136, 144102,  DOI: 10.1063/1.3701175

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    43

    Theory of binless multi-state free energy estimation with applications to protein-ligand binding

    Tan, Zhiqiang; Gallicchio, Emilio; Lapelosa, Mauro; Levy, Ronald M.

    Journal of Chemical Physics (2012), 136 (14), 144102/1-144102/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    The weighted histogram anal. method (WHAM) is routinely used for computing free energies and expectations from multiple ensembles. Existing derivations of WHAM require observations to be discretized into a finite no. of bins. Yet, WHAM formulas seem to hold even if the bin sizes are made arbitrarily small. The purpose of this article is to demonstrate both the validity and value of the multi-state Bennet acceptance ratio (MBAR) method seen as a binless extension of WHAM. We discuss two statistical arguments to derive the MBAR equations, in parallel to the self-consistency and max. likelihood derivations already known for WHAM. We show that the binless method, like WHAM, can be used not only to est. free energies and equil. expectations, but also to est. equil. distributions. We also provide a no. of useful results from the statistical literature, including the detn. of MBAR estimators by minimization of a convex function. This leads to an approach to the computation of MBAR free energies by optimization algorithms, which can be more effective than existing algorithms. The advantages of MBAR are illustrated numerically for the calcn. of abs. protein-ligand binding free energies by alchem. transformations with and without soft-core potentials. We show that binless statistical anal. can accurately treat sparsely distributed interaction energy samples as obtained from unmodified interaction potentials that cannot be properly analyzed using std. binning methods. This suggests that binless multi-state anal. of binding free energy simulations with unmodified potentials offers a straightforward alternative to the use of soft-core potentials for these alchem. transformations. (c) 2012 American Institute of Physics.
44. 44

    Eastman, P.; Swails, J.; Chodera, J. D.; McGibbon, R. T.; Zhao, Y.; Beauchamp, K. A.; Wang, L.-P.; Simmonett, A. C.; Harrigan, M. P.; Stern, C. D.; Wiewiora, R. P.; Brooks, B. R.; Pande, V. S. OpenMM 7: Rapid development of high performance algorithms for molecular dynamics. PLoS Comput. Biol. 2017, 13, e1005659,  DOI: 10.1371/journal.pcbi.1005659

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    OpenMM 7: Rapid development of high performance algorithms for molecular dynamics

    Eastman, Peter; Swails, Jason; Chodera, John D.; McGibbon, Robert T.; Zhao, Yutong; Beauchamp, Kyle A.; Wang, Lee-Ping; Simmonett, Andrew C.; Harrigan, Matthew P.; Stern, Chaya D.; Wiewiora, Rafal P.; Brooks, Bernard R.; Pande, Vijay S.

    PLoS Computational Biology (2017), 13 (7), e1005659/1-e1005659/17CODEN: PCBLBG; ISSN:1553-7358. (Public Library of Science)

    OpenMM is a mol. dynamics simulation toolkit with a unique focus on extensibility. It allows users to easily add new features, including forces with novel functional forms, new integration algorithms, and new simulation protocols. Those features automatically work on all supported hardware types (including both CPUs and GPUs) and perform well on all of them. In many cases they require minimal coding, just a math. description of the desired function. They also require no modification to OpenMM itself and can be distributed independently of OpenMM. This makes it an ideal tool for researchers developing new simulation methods, and also allows those new methods to be immediately available to the larger community.
45. 45

    Azimi, S.; Khuttan, S.; Wu, J. Z.; Deng, N.; Gallicchio, E. Application of the Alchemical Transfer and Potential of Mean Force Methods to the SAMPL8 Cavitand Host-Guest Blinded Challenge. arXiv.org , 2021, 2107.05155.

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46. 46

    Pal, R. K.; Gallicchio, E. Perturbation potentials to overcome order/disorder transitions in alchemical binding free energy calculations. J. Chem. Phys. 2019, 151, 124116,  DOI: 10.1063/1.5123154

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    46

    Perturbation potentials to overcome order/disorder transitions in alchemical binding free energy calculations

    Pal, Rajat K.; Gallicchio, Emilio

    Journal of Chemical Physics (2019), 151 (12), 124116/1-124116/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    The authors study the role of order/disorder transitions in alchem. simulations of protein-ligand abs. binding free energies. The authors show, in the context of a potential of mean force description, that for a benchmarking system (the complex of the L99A mutant of T4 lysozyme with 3-iodotoluene) and for a more challenging system relevant for medicinal applications (the complex of the farnesoid X receptor with inhibitor 26 from a recent D3R challenge) that order/disorder transitions can significantly hamper Hamiltonian replica exchange sampling efficiency and slow down the rate of equilibration of binding free energy ests. Further the authors' anal. model of alchem. binding combined with the formalism developed by Straub et al. for the treatment of order/disorder transitions of mol. systems can be successfully employed to analyze the transitions and help design alchem. schedules and soft-core functions that avoid or reduce the adverse effects of rare binding/unbinding transitions. The results of this work pave the way for the application of these techniques to the alchem. estn. with explicit solvation of hydration free energies and abs. binding free energies of systems undergoing order/disorder transitions. (c) 2019 American Institute of Physics.
47. 47

    Khuttan, S.; Azimi, S.; Wu, J. Z.; Gallicchio, E. Alchemical Transformations for Concerted Hydration Free Energy Estimation with Explicit Solvation. J. Chem. Phys. 2021, 154, 054103,  DOI: 10.1063/5.0036944

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    Alchemical transformations for concerted hydration free energy estimation with explicit solvation

    Khuttan, Sheenam; Azimi, Solmaz; Wu, Joe Z.; Gallicchio, Emilio

    Journal of Chemical Physics (2021), 154 (5), 054103CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    We present a family of alchem. perturbation potentials that enable the calcn. of hydration free energies of small- to medium-sized mols. in a single concerted alchem. coupling step instead of the commonly used sequence of two distinct coupling steps for Lennard-Jones and electrostatic interactions. The perturbation potentials we employ are nonlinear functions of the solute-solvent interaction energy designed to focus sampling near entropic bottlenecks along the alchem. pathway. We present a general framework to optimize the parameters of alchem. perturbation potentials of this kind. The optimization procedure is based on the λ-function formalism and the max.-likelihood parameter estn. procedure we developed earlier to avoid the occurrence of multi-modal distributions of the coupling energy along the alchem. path. A novel soft-core function applied to the overall solute-solvent interaction energy rather than individual interat. pair potentials crit. for this result is also presented. Because it does not require modifications of core force and energy routines, the soft-core formulation can be easily deployed in mol. dynamics simulation codes. We illustrate the method by applying it to the estn. of the hydration free energy in water droplets of compds. of varying size and complexity. In each case, we show that convergence of the hydration free energy is achieved rapidly. This work paves the way for the ongoing development of more streamlined algorithms to est. free energies of mol. binding with explicit solvation. (c) 2021 American Institute of Physics.
48. 48

    Gallicchio, E.; Levy, R. M. Recent Theoretical and Computational Advances for Modeling Protein-Ligand Binding Affinities. Adv. Prot. Chem. Struct. Biol. 2011, 85, 27– 80,  DOI: 10.1016/B978-0-12-386485-7.00002-8

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    Recent theoretical and computational advances for modeling protein-ligand binding affinities

    Gallicchio, Emilio; Levy, Ronald M.

    Advances in Protein Chemistry and Structural Biology (2011), 85 (Computational Chemistry Methods in Structural Biology), 27-80CODEN: APCSG7; ISSN:1876-1623. (Elsevier Ltd.)

    We review recent theor. and algorithmic advances for the modeling of protein ligand binding free energies. We first describe a statistical mechanics theory of noncovalent assocn., with particular focus on deriving the fundamental formulas on which computational methods are based. The second part reviews the main computational models and algorithms in current use or development, pointing out the relations with each other and with the theory developed in the first part. Particular emphasis is given to the modeling of conformational reorganization and entropic effect. The methods reviewed are free energy perturbation, double decoupling, the Binding Energy Distribution Anal. Method, the potential of mean force method, mining min. and MM/PBSA. These models have different features and limitations, and their ranges of applicability vary correspondingly. Yet their origins can all be traced back to a single fundamental theory.
49. 49

    Gilson, M. K.; Given, J. A.; Bush, B. L.; McCammon, J. A. The Statistical-Thermodynamic Basis for Computation of Binding Affinities: A Critical Review. Biophys. J. 1997, 72, 1047– 1069,  DOI: 10.1016/S0006-3495(97)78756-3

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    The statistical-thermodynamic basis for computation of binding affinities: a critical review

    Gilson, Michael K.; Given, James A.; Bush, Bruce L.; Mccammon, J. Andrew

    Biophysical Journal (1997), 72 (3), 1047-1069CODEN: BIOJAU; ISSN:0006-3495. (Biophysical Society)

    A review with many refs. Although the statistical thermodn. of noncovalent binding has been considered in a no. of theor. papers, few methods of computing binding affinities are derived explicitly from this underlying theory. This has contributed to uncertainty and controversy in certain areas. This article therefore reviews and extends the connections of some important computational methods with the underlying statistical thermodn. A derivation of the std. free energy of binding forms the basis of this review. This derivation should be useful in formulating novel computational methods for predicting binding affinities. It also permits several important points to be established. For example, it is found that the double annihilation method of computing binding energy does not yield the std. free energy of binding, but can be modified to yield this quantity. The derivation also makes it possible to define clearly the changes in translational, rotational, configurational, and solvent entropy upon binding. It is argued that mol. mass has a negligible effect upon the std. free energy of binding for biomol. systems, and that the cratic entropy defined by Gurney is not a useful concept. In addn., the use of continuum models of the solvent in binding calcns. is reviewed, and a formalism is presented for incorporating a limited no. of solvent mols. explicitly.
50. 50

    Roux, B.; Simonson, T. Implicit Solvent Models. Biophys. Chem. 1999, 78, 1– 20,  DOI: 10.1016/S0301-4622(98)00226-9

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    Implicit solvent models

    Roux, Benoit; Simonson, Thomas

    Biophysical Chemistry (1999), 78 (1-2), 1-20CODEN: BICIAZ; ISSN:0301-4622. (Elsevier Science B.V.)

    A review with 133 refs. Implicit solvent models for biomol. simulations are reviewed and their underlying statistical mech. basis is discussed. The fundamental quantity that implicit models seek to approx. is the solute potential of mean force, which dets. the statistical wt. of solute conformations, and which is obtained by averaging over the solvent degrees of freedom. It is possible to express the total free energy as the reversible work performed in two successive steps. First, the solute is inserted in the solvent with zero at. partial charges; second, the at. partial charges of the solute are switched from zero to their full values. Consequently, the total solvation free energy corresponds to a sum of non-polar and electrostatic contributions. These two contributions are often approximated by simple geometrical models (such as solvent exposed area models) and by macroscopic continuum electrostatics, resp. One powerful route is to approx. the av. solvent d. distribution around the solute, i.e. the solute-solvent d. correlation functions, as in statistical mech. integral equations. Recent progress with semi-anal. approxns. make continuum electrostatics treatments very efficient. Still more efficient are fully empirical, knowledge-based models, whose relation to explicit solvent treatments is not fully resolved, however. Continuum models that treat both solute and solvent as dielec. continua are also discussed, and the relation between the solute fluctuations and its macroscopic dielec. const.(s) clarified.
51. 51

    Boresch, S.; Tettinger, F.; Leitgeb, M.; Karplus, M. Absolute binding free energies: A quantitative approach for their calculation. J. Phys. Chem. B 2003, 107, 9535– 9551,  DOI: 10.1021/jp0217839

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    Absolute Binding Free Energies: A Quantitative Approach for Their Calculation

    Boresch, Stefan; Tettinger, Franz; Leitgeb, Martin; Karplus, Martin

    Journal of Physical Chemistry B (2003), 107 (35), 9535-9551CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)

    The computation of abs. binding affinities by mol. dynamics (MD) based free energy simulations is analyzed, and an exact method to carry out such a computation is presented. The key to obtaining converged results is the introduction of suitable, auxiliary restraints to prevent the ligand from leaving the binding site when the native ligand-receptor interactions are turned off alchem. The authors describe a versatile set of restraints that (i) can be used in MD simulations, that (ii) restricts both the position and the orientation of the ligand, and that (iii) is defined relative to the receptor rather than relative to a fixed point in space. The free energy cost, ΔAr, for this set of restraints can be evaluated anal. Although the techniques were originally developed for the gas phase, the resulting expression is exact, since all contributions from solute-solvent interactions cancel from the final result. The value of ΔAr depends only on the equil. values and force consts. of the chosen harmonic restraint terms and, therefore, can be easily calcd. The std. state dependence of binding free energies is also investigated, and it is shown that the present approach takes this into account correctly. The anal. expression for ΔAr is verified numerically by calcns. on the complex formed by benzene with the L99A mutant of T4 lysozyme. The overall approach is illustrated by a complete binding free energy calcn. for a complex based on a simplified model for tyrosine bound to tyrosyl-tRNA-synthetase. The results demonstrate the usefulness of the proposed set of restraints and confirm that the calcd. binding free energy is independent of the details of the restraints. Comparisons are made with earlier formulations for the calcn. of binding free energies, and certain limitations of that work are described. The relationship between ΔAr and the loss of translational and rotational entropy during a binding process is analyzed.
52. 52

    Chipot; ; Pohorille, Eds. Free Energy Calculations. Theory and Applications in Chemistry and Biology; Springer Series in Chemical Physics; Springer: Berlin Heidelberg, 2007.

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    The Single-Decoupling Plugin for OpenMM. https://github.com/Gallicchio-Lab/openmm_sdm_plugin (accessed September 12, 2021).

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    SAMPL8 host–guest GDCC challenge. https://github.com/samplchallenges/SAMPL8/tree/master/host_guest/GDCC (accessed September 12, 2021).

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    Norman, B. H.; Dodge, J. A.; Richardson, T. I.; Borromeo, P. S.; Lugar, C. W.; Jones, S. A.; Chen, K.; Wang, Y.; Durst, G. L.; Barr, R. J.; Montrose-Rafizadeh, C.; Osborne, H. E.; Amos, R. M.; Guo, S.; Boodhoo, A.; Krishnan, V. Benzopyrans are selective estrogen receptor β agonists with novel activity in models of benign prostatic hyperplasia. J. Med. Chem. 2006, 49, 6155– 6157,  DOI: 10.1021/jm060491j

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    Benzopyrans Are Selective Estrogen Receptor β Agonists with Novel Activity in Models of Benign Prostatic Hyperplasia

    Norman, Bryan H.; Dodge, Jeffrey A.; Richardson, Timothy I.; Borromeo, Peter S.; Lugar, Charles W.; Jones, Scott A.; Chen, Keyue; Wang, Yong; Durst, Gregory L.; Barr, Robert J.; Montrose-Rafizadeh, Chahrzad; Osborne, Harold E.; Amos, Robert M.; Guo, Sherry; Boodhoo, Amechand; Krishnan, Venkatesh

    Journal of Medicinal Chemistry (2006), 49 (21), 6155-6157CODEN: JMCMAR; ISSN:0022-2623. (American Chemical Society)

    Benzopyran selective estrogen receptor beta agonist-1 (SERBA-1) shows potent, selective binding and agonist function in estrogen receptor β (ERβ) in vitro assays. X-ray crystal structures of SERBA-1 in ERα and β help explain obsd. β-selectivity of this ligand. SERBA-1 in vivo demonstrates involution of the ventral prostate in CD-1 mice (ERβ effect), while having no effect on gonadal hormone levels (ERα effect) at 10× the efficacious dose, consistent with in vitro properties of this mol.
56. 56

    SAMPL8 host–guest GDCC challenge submission 37. https://github.com/samplchallenges/SAMPL8/blob/master/host_guest/Analysis/Submissions/GDCC/GDCC-ATM.txt (accessed September 12, 2021).

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    Kuntz, K. W.; Campbell, J. E.; Keilhack, H.; Pollock, R. M.; Knutson, S. K.; Porter-Scott, M.; Richon, V. M.; Sneeringer, C. J.; Wigle, T. J.; Allain, C. J.; Majer, C. R.; Moyer, M. P.; Copeland, R. A.; Chesworth, R. The importance of being me: magic methyls, methyltransferase inhibitors, and the discovery of tazemetostat. J. Med. Chem. 2016, 59, 1556– 1564,  DOI: 10.1021/acs.jmedchem.5b01501

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    The Importance of Being Me: Magic Methyls, Methyltransferase Inhibitors, and the Discovery of Tazemetostat

    Kuntz, Kevin W.; Campbell, John E.; Keilhack, Heike; Pollock, Roy M.; Knutson, Sarah K.; Porter-Scott, Margaret; Richon, Victoria M.; Sneeringer, Chris J.; Wigle, Tim J.; Allain, Christina J.; Majer, Christina R.; Moyer, Mikel P.; Copeland, Robert A.; Chesworth, Richard

    Journal of Medicinal Chemistry (2016), 59 (4), 1556-1564CODEN: JMCMAR; ISSN:0022-2623. (American Chemical Society)

    Posttranslational methylation of histones plays a crit. role in gene regulation. Misregulation of histone methylation can lead to oncogenic transformation. Enhancer of Zeste homolog 2 (EZH2) methylates histone 3 at lysine 27 (H3K27) and abnormal methylation of this site is found in many cancers. Tazemetostat, an EHZ2 inhibitor in clin. development, has shown activity in both preclin. models of cancer as well as in patients with lymphoma or INI1-deficient solid tumors. Herein we report the structure-activity relationships from identification of an initial hit in a high-throughput screen through selection of tazemetostat for clin. development. The importance of several Me groups to the potency of the inhibitors is highlighted as well as the importance of balancing pharmacokinetic properties with potency.
58. 58

    Wang, J.; Wang, W.; Kollman, P. A.; Case, D. A. Automatic atom type and bond type perception in molecular mechanical calculations. J. Mol. Graphics Modell. 2006, 25, 247– 260,  DOI: 10.1016/j.jmgm.2005.12.005

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    Automatic atom type and bond type perception in molecular mechanical calculations

    Wang, Junmei; Wang, Wei; Kollman, Peter A.; Case, David A.

    Journal of Molecular Graphics & Modelling (2006), 25 (2), 247-260CODEN: JMGMFI; ISSN:1093-3263. (Elsevier Inc.)

    In mol. mechanics (MM) studies, atom types and/or bond types of mols. are needed to det. prior to energy calcns. The authors present here an automatic algorithm of perceiving atom types that are defined in a description table, and an automatic algorithm of assigning bond types just based on at. connectivity. The algorithms have been implemented in a new module of the AMBER packages. This auxiliary module, antechamber (roughly meaning "before AMBER"), can be applied to generate necessary inputs of leap-the AMBER program to generate topologies for minimization, mol. dynamics, etc., for most org. mols. The algorithms behind the manipulations may be useful for other mol. mech. packages as well as applications that need to designate atom types and bond types.
59. 59

    He, X.; Liu, S.; Lee, T.-S.; Ji, B.; Man, V. H.; York, D. M.; Wang, J. Fast, Accurate, and Reliable Protocols for Routine Calculations of Protein-Ligand Binding Affinities in Drug Design Projects Using AMBER GPU-TI with ff14SB/GAFF. ACS Omega 2020, 5, 4611– 4619,  DOI: 10.1021/acsomega.9b04233

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    59

    Fast, Accurate, and Reliable Protocols for Routine Calculations of Protein-Ligand Binding Affinities in Drug Design Projects Using AMBER GPU-TI with ff14SB/GAFF

    He, Xibing; Liu, Shuhan; Lee, Tai-Sung; Ji, Beihong; Man, Viet H.; York, Darrin M.; Wang, Junmei

    ACS Omega (2020), 5 (9), 4611-4619CODEN: ACSODF; ISSN:2470-1343. (American Chemical Society)

    Accurate prediction of the abs. or relative protein-ligand binding affinity is one of the major tasks in computer-aided drug design projects, esp. in the stage of lead optimization. In principle, the alchem. free energy (AFE) methods such as thermodn. integration (TI) or free-energy perturbation (FEP) can fulfill this task, but in practice, a lot of hurdles prevent them from being routinely applied in daily drug design projects, such as the demanding computing resources, slow computing processes, unavailable or inaccurate force field parameters, and difficult and unfriendly setting up and post-anal. procedures. In this study, we have exploited practical protocols of applying the CPU (central processing unit)-TI and newly developed GPU (graphic processing unit)-TI modules and other tools in the AMBER software package, combined with ff14SB/GAFF1.8 force fields, to conduct efficient and accurate AFE calcns. on protein-ligand binding free energies. We have tested 134 protein-ligand complexes in total for four target proteins (BACE, CDK2, MCL1, and PTP1B) and obtained overall comparable performance with the com. Schrodinger FEP+ program (). The achieved accuracy fits within the requirements for computations to generate effective guidance for exptl. work in drug lead optimization, and the needed wall time is short enough for practical application. Our verified protocol provides a practical soln. for routine AFE calcns. in real drug design projects.
60. 60

    Maier, J. A.; Martinez, C.; Kasavajhala, K.; Wickstrom, L.; Hauser, K. E.; Simmerling, C. ff14SB: improving the accuracy of protein side chain and backbone parameters from ff99SB. J. Chem. Theory Comput. 2015, 11, 3696– 3713,  DOI: 10.1021/acs.jctc.5b00255

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    ff14SB: Improving the Accuracy of Protein Side Chain and Backbone Parameters from ff99SB

    Maier, James A.; Martinez, Carmenza; Kasavajhala, Koushik; Wickstrom, Lauren; Hauser, Kevin E.; Simmerling, Carlos

    Journal of Chemical Theory and Computation (2015), 11 (8), 3696-3713CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    Mol. mechanics is powerful for its speed in atomistic simulations, but an accurate force field is required. The Amber ff99SB force field improved protein secondary structure balance and dynamics from earlier force fields like ff99, but weaknesses in side chain rotamer and backbone secondary structure preferences have been identified. Here, we performed a complete refit of all amino acid side chain dihedral parameters, which had been carried over from ff94. The training set of conformations included multidimensional dihedral scans designed to improve transferability of the parameters. Improvement in all amino acids was obtained as compared to ff99SB. Parameters were also generated for alternate protonation states of ionizable side chains. Av. errors in relative energies of pairs of conformations were under 1.0 kcal/mol as compared to QM, reduced 35% from ff99SB. We also took the opportunity to make empirical adjustments to the protein backbone dihedral parameters as compared to ff99SB. Multiple small adjustments of φ and ψ parameters were tested against NMR scalar coupling data and secondary structure content for short peptides. The best results were obtained from a phys. motivated adjustment to the φ rotational profile that compensates for lack of ff99SB QM training data in the β-ppII transition region. Together, these backbone and side chain modifications (hereafter called ff14SB) not only better reproduced their benchmarks, but also improved secondary structure content in small peptides and reprodn. of NMR χ1 scalar coupling measurements for proteins in soln. We also discuss the Amber ff12SB parameter set, a preliminary version of ff14SB that includes most of its improvements.
61. 61

    Relative Binding Free Energy Calculations for Ligands with Diverse Scaffolds with the Alchemical Transfer Method, Simulation Input Files. https://github.com/Gallicchio-Lab/ATM-relative-binding-free-energy-paper (accessed September 12, 2021).

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62. 62

    The Asynchronous Replica Exchange Framework for OpenMM. https://github.com/Gallicchio-Lab/async_re-openmm (accessed September 12, 2021).

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63. 63

    Rizzi, A.; Murkli, S.; McNeill, J. N.; Yao, W.; Sullivan, M.; Gilson, M. K.; Chiu, M. W.; Isaacs, L.; Gibb, B. C.; Mobley, D. L.; Chodera, J. D. Overview of the SAMPL6 host-guest binding affinity prediction challenge. J. Comp.-Aided Mol. Des. 2018, 32, 937– 963,  DOI: 10.1007/s10822-018-0170-6

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    Overview of the SAMPL6 host-guest binding affinity prediction challenge

    Rizzi, Andrea; Murkli, Steven; McNeill, John N.; Yao, Wei; Sullivan, Matthew; Gilson, Michael K.; Chiu, Michael W.; Isaacs, Lyle; Gibb, Bruce C.; Mobley, David L.; Chodera, John D.

    Journal of Computer-Aided Molecular Design (2018), 32 (10), 937-963CODEN: JCADEQ; ISSN:0920-654X. (Springer)

    Accurately predicting the binding affinities of small org. mols. to biol. macromols. can greatly accelerate drug discovery by reducing the no. of compds. that must be synthesized to realize desired potency and selectivity goals. Unfortunately, the process of assessing the accuracy of current computational approaches to affinity prediction against binding data to biol. macromols. is frustrated by several challenges, such as slow conformational dynamics, multiple titratable groups, and the lack of high-quality blinded datasets. Over the last several SAMPL blind challenge exercises, host-guest systems have emerged as a practical and effective way to circumvent these challenges in assessing the predictive performance of current-generation quant. modeling tools, while still providing systems capable of possessing tight binding affinities. Here, we present an overview of the SAMPL6 host-guest binding affinity prediction challenge, which featured three supramol. hosts: octa-acid (OA), the closely related tetra-endo-methyl-octa-acid (TEMOA), and cucurbit[8]uril (CB8), along with 21 small org. guest mols. A total of 119 entries were received from ten participating groups employing a variety of methods that spanned from electronic structure and movable type calcns. in implicit solvent to alchem. and potential of mean force strategies using empirical force fields with explicit solvent models. While empirical models tended to obtain better performance than first-principle methods, it was not possible to identify a single approach that consistently provided superior results across all host-guest systems and statistical metrics. Moreover, the accuracy of the methodologies generally displayed a substantial dependence on the system considered, emphasizing the need for host diversity in blind evaluations. Several entries exploited previous exptl. measurements of similar host-guest systems in an effort to improve their phys.-based predictions via some manner of rudimentary machine learning; while this strategy succeeded in reducing systematic errors, it did not correspond to an improvement in statistical correlation. Comparison to previous rounds of the host-guest binding free energy challenge highlights an overall improvement in the correlation obtained by the affinity predictions for OA and TEMOA systems, but a surprising lack of improvement regarding root mean square error over the past several challenge rounds. The data suggests that further refinement of force field parameters, as well as improved treatment of chem. effects (e.g., buffer salt conditions, protonation states), may be required to further enhance predictive accuracy.
64. 64

    Raman, E. P.; Paul, T. J.; Hayes, R. L.; Brooks, C. L., III Automated, accurate, and scalable relative protein-ligand binding free-energy calculations using lambda dynamics. J. Chem. Theory Comput. 2020, 16, 7895– 7914,  DOI: 10.1021/acs.jctc.0c00830

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    Automated, Accurate, and Scalable Relative Protein-Ligand Binding Free-Energy Calculations Using Lambda Dynamics

    Raman, E. Prabhu; Paul, Thomas J.; Hayes, Ryan L.; Brooks, Charles L., III

    Journal of Chemical Theory and Computation (2020), 16 (12), 7895-7914CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    Accurate predictions of changes to protein-ligand binding affinity in response to chem. modifications are of utility in small-mol. lead optimization. Relative free-energy perturbation (FEP) approaches are one of the most widely used for this goal but involve significant computational cost, thus limiting their application to small sets of compds. Lambda dynamics, also rigorously based on the principles of statistical mechanics, provides a more efficient alternative. The authors describe the development of a workflow to set up, execute, and analyze multisite lambda dynamics (MSLD) calcns. run on GPUs with CHARMM implemented in BIOVIA Discovery Studio and Pipeline Pilot. The workflow establishes a framework for setting up simulation systems for exploratory screening of modifications to a lead compd., enabling the calcn. of relative binding affinities of combinatorial libraries. To validate the workflow, a diverse data set of congeneric ligands for seven proteins with exptl. binding affinity data was examd. A protocol to automatically tailor fit biasing potentials iteratively to flatten the free-energy landscape of any MSLD system is developed, which enhances sampling and allows for efficient estn. of free-energy differences. The protocol is first validated on a large no. of ligand subsets that model diverse substituents, which shows accurate and reliable performance. The scalability of the workflow is also tested to screen >100 ligands modeled in a single system, which also resulted in accurate predictions. With a cumulative sampling time of 150 ns or less, the method results in av. unsigned errors of under 1 kcal/mol in most cases for both small and large combinatorial libraries. For the multisite systems examd., the method is more than an order of magnitude more efficient than contemporary FEP applications. The results thus demonstrate the utility of the presented MSLD workflow to efficiently screen combinatorial libraries and explore the chem. space around a lead compd. and thus are of utility in lead optimization.
65. 65

    Woods, C. J.; Malaisree, M.; Hannongbua, S.; Mulholland, A. J. A water-swap reaction coordinate for the calculation of absolute protein-ligand binding free energies. J. Chem. Phys. 2011, 134, 054114,  DOI: 10.1063/1.3519057

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    A water-swap reaction coordinate for the calculation of absolute protein-ligand binding free energies

    Woods, Christopher J.; Malaisree, Maturos; Hannongbua, Supot; Mulholland, Adrian J.

    Journal of Chemical Physics (2011), 134 (5), 054114/1-054114/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    The accurate prediction of abs. protein-ligand binding free energies is one of the grand challenge problems of computational science. Binding free energy measures the strength of binding between a ligand and a protein, and an algorithm that would allow its accurate prediction would be a powerful tool for rational drug design. Here we present the development of a new method that allows for the abs. binding free energy of a protein-ligand complex to be calcd. from first principles, using a single simulation. Our method involves the use of a novel reaction coordinate that swaps a ligand bound to a protein with an equiv. vol. of bulk water. This water-swap reaction coordinate is built using an identity constraint, which identifies a cluster of water mols. from bulk water that occupies the same vol. as the ligand in the protein active site. A dual topol. algorithm is then used to swap the ligand from the active site with the identified water cluster from bulk water. The free energy is then calcd. using replica exchange thermodn. integration. This returns the free energy change of simultaneously transferring the ligand to bulk water, as an equiv. vol. of bulk water is transferred back to the protein active site. This, directly, is the abs. binding free energy. It should be noted that while this reaction coordinate models the binding process directly, an accurate force field and sufficient sampling are still required to allow for the binding free energy to be predicted correctly. In this paper we present the details and development of this method, and demonstrate how the potential of mean force along the water-swap coordinate can be improved by calibrating the soft-core Coulomb and Lennard-Jones parameters used for the dual topol. calcn. The optimal parameters were applied to calcns. of protein-ligand binding free energies of a neuraminidase inhibitor (oseltamivir), with these results compared to expt. These results demonstrate that the water-swap coordinate provides a viable and potentially powerful new route for the prediction of protein-ligand binding free energies. (c) 2011 American Institute of Physics.
66. 66

    Procacci, P.; Macchiagodena, M. On the NS-DSSB unidirectional estimates in the SAMPL6 SAMPLing challenge. J. Comput.-Aided Mol. Des. 2021, 35, 1055,  DOI: 10.1007/s10822-021-00419-0

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    On the NS-DSSB unidirectional estimates in the SAMPL6 SAMPLing challenge

    Procacci, Piero; Macchiagodena, Marina

    Journal of Computer-Aided Molecular Design (2021), 35 (10), 1055-1065CODEN: JCADEQ; ISSN:0920-654X. (Springer)

    In the context of the recent SAMPL6 SAMPLing challenge (Rizzi et al. 2020 in J Comput Aided Mol Des 34:601-633) aimed at assessing convergence properties and reproducibility of mol. dynamics binding free energy methodologies, we propose a simple explanation of the severe errors obsd. in the nonequil. switch double-system-single-box (NS-DSSB) approach when using unidirectional ests. At the same time, we suggest a straightforward and minimal modification of the NS-DSSB protocol for obtaining reliable unidirectional ests. for the process where the ligand is decoupled in the bound state and recoupled in the bulk.
67. 67

    Öhlknecht, C.; Lier, B.; Petrov, D.; Fuchs, J.; Oostenbrink, C. Correcting electrostatic artifacts due to net-charge changes in the calculation of ligand binding free energies. J. Comput. Chem. 2020, 41, 986– 999,  DOI: 10.1002/jcc.26143

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    Correcting electrostatic artifacts due to net-charge changes in the calculation of ligand binding free energies

    Ohlknecht Christoph; Lier Bettina; Petrov Drazen; Fuchs Julian; Oostenbrink Chris; Ohlknecht Christoph; Fuchs Julian

    Journal of computational chemistry (2020), 41 (10), 986-999 ISSN:.

    Alchemically derived free energies are artifacted when the perturbed moiety has a nonzero net charge. The source of the artifacts lies in the effective treatment of the electrostatic interactions within and between the perturbed atoms and remaining (partial) charges in the simulated system. To treat the electrostatic interactions effectively, lattice-summation (LS) methods or cutoff schemes in combination with a reaction-field contribution are usually employed. Both methods render the charging component of the calculated free energies sensitive to essential parameters of the system like the cutoff radius or the box side lengths. Here, we discuss the results of three previously published studies of ligand binding. These studies presented estimates of binding free energies that were artifacted due to the charged nature of the ligands. We show that the size of the artifacts can be efficiently calculated and raw simulation data can be corrected. We compare the corrected results with experimental estimates and nonartifacted estimates from path-sampling methods. Although the employed correction scheme involves computationally demanding continuum-electrostatics calculations, we show that the correction estimate can be deduced from a small sample of configurations rather than from the entire ensemble. This observation makes the calculations of correction terms feasible for complex biological systems. To show the general applicability of the proposed procedure, we also present results where the correction scheme was used to correct independent free energies obtained from simulations employing a cutoff scheme or LS electrostatics. In this work, we give practical guidelines on how to apply the appropriate corrections easily.
68. 68

    Pan, A. C.; Xu, H.; Palpant, T.; Shaw, D. E. Quantitative characterization of the binding and unbinding of millimolar drug fragments with molecular dynamics simulations. J. Chem. Theory Comput. 2017, 13, 3372– 3377,  DOI: 10.1021/acs.jctc.7b00172

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    Quantitative Characterization of the Binding and Unbinding of Millimolar Drug Fragments with Molecular Dynamics Simulations

    Pan, Albert C.; Xu, Huafeng; Palpant, Timothy; Shaw, David E.

    Journal of Chemical Theory and Computation (2017), 13 (7), 3372-3377CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    A quant. characterization of the binding properties of drug fragments to a target protein is an important component of a fragment-based drug discovery program. Fragments typically have a weak binding affinity, however, making it challenging to exptl. characterize key binding properties including binding sites, poses, and affinities. Direct simulation of the binding equil. by mol. dynamics (MD) simulations can provide a computational route to characterize fragment binding, but this approach is so computationally intensive that it has thus far remained relatively unexplored. Here, the authors perform MD simulations of sufficient length to observe several different fragments spontaneously and repeatedly bind to, and unbind from, the protein FKBP, allowing the binding affinities, the on- and off-rates, and the relative occupancies of alternative binding sites and alternative poses within each binding site to be estd., thereby illustrating the potential of long-timescale MD as a quant. tool for fragment-based drug discovery. The data from the long-timescale fragment binding simulations reported here also provides a useful benchmark for testing alternative computational methods aimed at characterizing fragment binding properties. As an example, the authors calcd. binding affinities for the same fragments using a std. free energy perturbation (FEP) approach and found that the values agreed with those obtained from the fragment binding simulations within statistical error.
69. 69

    Lin, Y.-L.; Aleksandrov, A.; Simonson, T.; Roux, B. An overview of electrostatic free energy computations for solutions and proteins. J. Chem. Theory Comput. 2014, 10, 2690– 2709,  DOI: 10.1021/ct500195p

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    An Overview of Electrostatic Free Energy Computations for Solutions and Proteins

    Lin, Yen-Lin; Aleksandrov, Alexey; Simonson, Thomas; Roux, Benoit

    Journal of Chemical Theory and Computation (2014), 10 (7), 2690-2709CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    A review. Free energy simulations for electrostatic and charging processes in complex mol. systems encounter specific difficulties owing to the long-range, 1/r Coulomb interaction. To calc. the solvation free energy of a simple ion, it is essential to take into account the polarization of nearby solvent but also the electrostatic potential drop across the liq.-gas boundary, however distant. The latter does not exist in a simulation model based on periodic boundary conditions because there is no phys. boundary to the system. An important consequence is that the ref. value of the electrostatic potential is not an ion in a vacuum. Also, in an infinite system, the electrostatic potential felt by a perturbing charge is conditionally convergent and dependent on the choice of computational conventions. Furthermore, with Ewald lattice summation and tinfoil conducting boundary conditions, the charges experience a spurious shift in the potential that depends on the details of the simulation system such as the vol. fraction occupied by the solvent. All these issues can be handled with established computational protocols, as reviewed here and illustrated for several small ions and three solvated proteins.
70. 70

    The ATM Meta Force Plugin for OpenMM. https://github.com/Gallicchio-Lab/openmm-atmmetaforce-plugin (accessed September 12, 2021).

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    Huang, J.; Lemkul, J. A.; Eastman, P. K.; MacKerell, A. D., Jr. Molecular dynamics simulations using the drude polarizable force field on GPUs with OpenMM: Implementation, validation, and benchmarks. J. Comput. Chem. 2018, 39, 1682– 1689,  DOI: 10.1002/jcc.25339

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    Molecular dynamics simulations using the drude polarizable force field on GPUs with OpenMM: Implementation, validation, and benchmarks

    Huang, Jing; Lemkul, Justin A.; Eastman, Peter K.; MacKerell, Alexander D., Jr.

    Journal of Computational Chemistry (2018), 39 (21), 1682-1689CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)

    Presented is the implementation of the Drude force field in the open-source OpenMM simulation package allowing for access to graphical processing unit (GPU) hardware. In the Drude model, electronic degrees of freedom are represented by neg. charged particles attached to their parent atoms via harmonic springs, such that extra computational overhead comes from these addnl. particles and virtual sites representing lone pairs on electroneg. atoms, as well as the assocd. thermostat and integration algorithms. This leads to an approx. fourfold increase in computational demand over additive force fields. However, by making the Drude model accessible to consumer-grade desktop GPU hardware it will be possible to perform simulations of one microsecond or more in less than a month, indicating that the barrier to employ polarizable models has largely been removed such that polarizable simulations with the classical Drude model are readily accessible and practical.
72. 72

    Gallicchio, E.; Deng, N.; He, P.; Wickstrom, L.; Perryman, A. L.; Santiago, D. N.; Forli, S.; Olson, A. J.; Levy, R. M. Virtual Screening of Integrase Inhibitors by Large Scale Binding Free Energy Calculations: the SAMPL4 Challenge. J. Comput.-Aided Mol. Des. 2014, 28, 475– 490,  DOI: 10.1007/s10822-014-9711-9

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    Virtual screening of integrase inhibitors by large scale binding free energy calculations: the SAMPL4 challenge

    Gallicchio, Emilio; Deng, Nanjie; He, Peng; Wickstrom, Lauren; Perryman, Alexander L.; Santiago, Daniel N.; Forli, Stefano; Olson, Arthur J.; Levy, Ronald M.

    Journal of Computer-Aided Molecular Design (2014), 28 (4), 475-490CODEN: JCADEQ; ISSN:0920-654X. (Springer)

    As part of the SAMPL4 blind challenge, filtered AutoDock Vina ligand docking predictions and large-scale binding energy distribution anal. method binding free energy calcns. were applied to the virtual screening of a focused library of candidate binders to the LEDGF site of the HIV integrase protein. The computational protocol leveraged docking and high level atomistic models to improve enrichment. The enrichment factor of the blind predictions ranked best among all of the computational submissions, and 2nd best overall. This work represents to the authors' knowledge the 1st example of the application of an all-atom physics-based binding free energy model to large scale virtual screening. A total of 285 parallel Hamiltonian replica exchange mol. dynamics abs. protein-ligand binding free energy simulations were conducted starting from docked poses. The setup of the simulations was fully automated, calcns. were distributed on multiple computing resources and were completed in a 6-wk period. The accuracy of the docked poses and the inclusion of intramol. strain and entropic losses in the binding free energy ests. were the major factors behind the success of the method. Lack of sufficient time and computing resources to investigate addnl. protonation states of the ligands was a major cause of mispredictions. The expt. demonstrated the applicability of binding free energy modeling to improve hit rates in challenging virtual screening of focused ligand libraries during lead optimization.
73. 73

    Darden, T. A.; York, D. M.; Pedersen, L. G. Particle mesh Ewald: An NlogN method for Ewald sums in large systems. J. Chem. Phys. 1993, 98, 10089– 10092,  DOI: 10.1063/1.464397

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    Particle mesh Ewald: an N·log(N) method for Ewald sums in large systems

    Darden, Tom; York, Darrin; Pedersen, Lee

    Journal of Chemical Physics (1993), 98 (12), 10089-92CODEN: JCPSA6; ISSN:0021-9606.

    An N·log(N) method for evaluating electrostatic energies and forces of large periodic systems is presented. The method is based on interpolation of the reciprocal space Ewald sums and evaluation of the resulting convolution using fast Fourier transforms. Timings and accuracies are presented for three large cryst. ionic systems.
74. 74

    Kilburg, D.; Gallicchio, E. Analytical Model of the Free Energy of Alchemical Molecular Binding. J. Chem. Theory Comput. 2018, 14, 6183– 6196,  DOI: 10.1021/acs.jctc.8b00967

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    Analytical Model of the Free Energy of Alchemical Molecular Binding

    Kilburg, Denise; Gallicchio, Emilio

    Journal of Chemical Theory and Computation (2018), 14 (12), 6183-6196CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    We present a parametrized analytic statistical model of the thermodn. of alchem. mol. binding within the solvent potential of mean force formalism. The model describes the free energy profiles of linear single-decoupling alchem. binding free energy calcns. accurately. The parameters of the model, which are phys. motivated, are derived by max. likelihood inference from data obtained from alchem. mol. simulations. The validity of the model has been assessed on a set of host-guest complexes. The model faithfully reproduces both the binding free energy profiles and the probability densities of the perturbation energy as a function of the alchem. progress parameter. The model offers a rationalization for the characteristic shape of binding free energy profiles. The parameters obtained from the model are potentially useful descriptors of the assocn. equil. of mol. complexes. Potential applications of the model for the classification of mol. complexes and the design of alchem. mol. simulations are envisioned.
75. 75

    Gallicchio, E.; Lapelosa, M.; Levy, R. M. Binding Energy Distribution Analysis Method (BEDAM) for Estimation of Protein-Ligand Binding Affinities. J. Chem. Theory Comput. 2010, 6, 2961– 2977,  DOI: 10.1021/ct1002913

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    Binding Energy Distribution Analysis Method (BEDAM) for Estimation of Protein-Ligand Binding Affinities

    Gallicchio, Emilio; Lapelosa, Mauro; Levy, Ronald M.

    Journal of Chemical Theory and Computation (2010), 6 (9), 2961-2977CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)

    The binding energy distribution anal. method (BEDAM) for the computation of receptor-ligand std. binding free energies with implicit solvation is presented. The method is based on a well-established statistical mechanics theory of mol. assocn. In the context of implicit solvation, the theory is homologous to the test particle method of solvation thermodn. with the solute-solvent potential represented by the effective binding energy of the protein-ligand complex. Accordingly, in BEDAM the binding const. is computed by a weighted integral of the probability distribution of the binding energy obtained in the canonical ensemble in which the ligand is positioned in the binding site but the receptor and the ligand interact only with the solvent continuum. The binding energy distribution encodes all of the phys. effects of binding. The balance between binding enthalpy and entropy is seen in the authors' formalism as a balance between favorable and unfavorable binding modes which are coupled through the normalization of the binding energy distribution function. An efficient computational protocol for the binding energy distribution based on the AGBNP2 implicit solvent model, parallel Hamiltonian replica exchange sampling, and histogram reweighting is developed. Applications of the method to a set of known binders and nonbinders of the L99A and L99A/M102Q mutants of T4 lysozyme receptor are illustrated. The method is able to discriminate without error binders from nonbinders, and the computed std. binding free energies of the binders are in good agreement with exptl. measurements. Anal. of the binding affinities of these systems reflect the contributions from multiple conformations spanning a wide range of binding energies.
76. 76

    Riniker, S.; Christ, C. D.; Hansen, N.; Mark, A. E.; Nair, P. C.; van Gunsteren, W. F. Comparison of enveloping distribution sampling and thermodynamic integration to calculate binding free energies of phenylethanolamine N-methyltransferase inhibitors. J. Chem. Phys. 2011, 135, 024105,  DOI: 10.1063/1.3604534

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    Comparison of enveloping distribution sampling and thermodynamic integration to calculate binding free energies of phenylethanolamine N-methyltransferase inhibitors

    Riniker, Sereina; Christ, Clara D.; Hansen, Niels; Mark, Alan E.; Nair, Pramod C.; van Gunsteren, Wilfred F.

    Journal of Chemical Physics (2011), 135 (2), 024105/1-024105/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)

    The relative binding free energy between two ligands to a specific protein can be obtained using various computational methods. The more accurate and also computationally more demanding techniques are the so-called free energy methods which use conformational sampling from mol. dynamics or Monte Carlo simulations to generate thermodn. avs. Two such widely applied methods are the thermodn. integration (TI) and the recently introduced enveloping distribution sampling (EDS) methods. In both cases relative binding free energies are obtained through the alchem. perturbations of one ligand into another in water and inside the binding pocket of the protein. TI requires many sep. simulations and the specification of a pathway along which the system is perturbed from one ligand to another. Using the EDS approach, only a single automatically derived ref. state enveloping both end states needs to be sampled. In addn., the choice of an optimal pathway in TI calcns. is not trivial and a poor choice may lead to poor convergence along the pathway. Given this, EDS is expected to be a valuable and computationally efficient alternative to TI. In this study, the performances of these two methods are compared using the binding of ten tetrahydroisoquinoline derivs. to phenylethanolamine N-transferase as an example. The ligands involve a diverse set of functional groups leading to a wide range of free energy differences. In addn., two different schemes to det. automatically the EDS ref. state parameters and two different topol. approaches are compared. (c) 2011 American Institute of Physics.
77. 77

    Deng, Y.; Roux, B. Computations of standard binding free energies with molecular dynamics simulations. J. Phys. Chem. B 2009, 113, 2234– 2246,  DOI: 10.1021/jp807701h

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    Computations of Standard Binding Free Energies with Molecular Dynamics Simulations

    Deng, Yuqing; Roux, Benoit

    Journal of Physical Chemistry B (2009), 113 (8), 2234-2246CODEN: JPCBFK; ISSN:1520-6106. (American Chemical Society)

    A review. An increasing no. of studies have reported computations of the std. (abs.) binding free energy of small ligands to proteins using mol. dynamics (MD) simulations and explicit solvent mols. that are in good agreement with expts. This encouraging progress suggests that physics-based approaches hold the promise of making important contributions to the process of drug discovery and optimization in the near future. Two types of approaches are principally used to compute binding free energies with MD simulations. The most widely known is the alchem. double decoupling method, in which the interaction of the ligand with its surroundings are progressively switched off. It is also possible to use a potential of mean force (PMF) method, in which the ligand is phys. sepd. from the protein receptor. For both of these computational approaches, restraining potentials may be activated and released during the simulation for sampling efficiently the changes in translational, rotational, and conformational freedom of the ligand and protein upon binding. Because such restraining potentials add bias to the simulations, it is important that their effects be rigorously removed to yield a binding free energy that is properly unbiased with respect to the std. state. A review of recent results is presented, and differences in computational methods are discussed. Examples of computations with T4-lysozyme mutants, FKBP12, SH2 domain, and cytochrome P 450 are discussed and compared. Remaining difficulties and challenges are highlighted.
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    Deng, N.; Cui, D.; Zhang, B. W.; Xia, J.; Cruz, J.; Levy, R. Comparing alchemical and physical pathway methods for computing the absolute binding free energy of charged ligands. Phys. Chem. Chem. Phys. 2018, 20, 17081– 17092,  DOI: 10.1039/C8CP01524D

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    78

    Comparing alchemical and physical pathway methods for computing the absolute binding free energy of charged ligands

    Deng, Nanjie; Cui, Di; Zhang, Bin W.; Xia, Junchao; Cruz, Jeffrey; Levy, Ronald

    Physical Chemistry Chemical Physics (2018), 20 (25), 17081-17092CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)

    Accurately predicting abs. binding free energies of protein-ligand complexes is important as a fundamental problem in both computational biophysics and pharmaceutical discovery. Calcg. binding free energies for charged ligands is generally considered to be challenging because of the strong electrostatic interactions between the ligand and its environment in aq. soln. In this work, we compare the performance of the potential of mean force (PMF) method and the double decoupling method (DDM) for computing abs. binding free energies for charged ligands. We first clarify an unresolved issue concerning the explicit use of the binding site vol. to define the complexed state in DDM together with the use of harmonic restraints. We also provide an alternative derivation for the formula for abs. binding free energy using the PMF approach. We use these formulas to compute the binding free energy of charged ligands at an allosteric site of HIV-1 integrase, which has emerged in recent years as a promising target for developing antiviral therapy. As compared with the exptl. results, the abs. binding free energies obtained by using the PMF approach show unsigned errors of 1.5-3.4 kcal mol-1, which are somewhat better than the results from DDM (unsigned errors of 1.6-4.3 kcal mol-1) using the same amt. of CPU time. According to the DDM decompn. of the binding free energy, the ligand binding appears to be dominated by nonpolar interactions despite the presence of very large and favorable intermol. ligand-receptor electrostatic interactions, which are almost completely canceled out by the equally large free energy cost of desolvation of the charged moiety of the ligands in soln. We discuss the relative strengths of computing abs. binding free energies using the alchem. and phys. pathway methods.
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    Velez-Vega, C.; Gilson, M. K. Overcoming dissipation in the calculation of standard binding free energies by ligand extraction. J. Comput. Chem. 2013, 34, 2360– 2371,  DOI: 10.1002/jcc.23398

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    Overcoming dissipation in the calculation of standard binding free energies by ligand extraction

    Velez-Vega, Camilo; Gilson, Michael K.

    Journal of Computational Chemistry (2013), 34 (27), 2360-2371CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)

    This article addresses calcns. of the std. free energy of binding from mol. simulations in which a bound ligand is extd. from its binding site by steered mol. dynamics (MD) simulations or equil. umbrella sampling (US). Host-guest systems are used as test beds to examine the requirements for obtaining the reversible work of ligand extn. The authors find that, for both steered MD and US, marked irreversibilities can occur when the guest mol. crosses an energy barrier and suddenly jumps to a new position, causing dissipation of energy stored in the stretched mol(s). For flexible mols., this occurs even when a stiff pulling spring is used, and it is difficult to suppress in calcns. where the spring is attached to the mols. by single, fixed attachment points. The authors introduce and test a method, fluctuation-guided pulling, which adaptively adjusts the spring's attachment points based on the guest's at. fluctuations relative to the host. This adaptive approach is found to substantially improve the reversibility of both steered MD and US calcns. for the present systems. The results are then used to est. std. binding free energies within a comprehensive framework, termed attach-pull-release, which recognizes that the std. free energy of binding must include not only the pulling work itself, but also the work of attaching and then releasing the spring, where the release work includes an accounting of the std. concn. to which the ligand is discharged. © 2013 Wiley Periodicals, Inc.
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    Hall, R.; Dixon, T.; Dickson, A. On calculating free energy differences using ensembles of transition paths. Frontiers Mol. Biosc. 2020, 7, 106,  DOI: 10.3389/fmolb.2020.00106

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    Rizzi, A.; Jensen, T.; Slochower, D. R.; Aldeghi, M.; Gapsys, V.; Ntekoumes, D.; Bosisio, S.; Papadourakis, M.; Henriksen, N. M.; De Groot, B. L.; Cournia, Z.; Dickson, A.; Michel, J.; Gilson, M. K.; Shirts, M. R.; Mobley, D. L.; Chodera, J. D. The SAMPL6 SAMPLing challenge: Assessing the reliability and efficiency of binding free energy calculations. J. Comp. Aid. Mol. Des. 2020, 34, 601,  DOI: 10.1007/s10822-020-00290-5

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    The SAMPL6 SAMPLing challenge: assessing the reliability and efficiency of binding free energy calculations

    Rizzi, Andrea; Jensen, Travis; Slochower, David R.; Aldeghi, Matteo; Gapsys, Vytautas; Ntekoumes, Dimitris; Bosisio, Stefano; Papadourakis, Michail; Henriksen, Niel M.; de Groot, Bert L.; Cournia, Zoe; Dickson, Alex; Michel, Julien; Gilson, Michael K.; Shirts, Michael R.; Mobley, David L.; Chodera, John D.

    Journal of Computer-Aided Molecular Design (2020), 34 (5), 601-633CODEN: JCADEQ; ISSN:0920-654X. (Springer)

    In this study, we describe the concept and results for the SAMPL6 SAMPLing challenge, the first challenge from the SAMPL series focusing on the assessment of convergence properties and reproducibility of binding free energy methodologies. Participants submitted binding free energy predictions as a function of the no. of force and energy evaluations for seven different alchem. and phys.-pathway methodologies implemented with the GROMACS, AMBER, NAMD, or OpenMM simulation engines. For the two small OA binders, the free energy ests. computed with alchem. and potential of mean force approaches show relatively similar variance and bias as a function of the no. of energy/force evaluations, with the attach-pull-release, GROMACS expanded ensemble, and NAMD double decoupling submissions obtaining the greatest efficiency. Surprisingly, the results suggest that specifying force field parameters and partial charges is insufficient to generally ensure reproducibility, and we observe differences between seemingly converged predictions ranging approx. from 0.3 to 1.0 kcal/mol, even with almost identical simulations parameters and system setup. Among the conclusions emerging from the data, we found that Hamiltonian replica exchange-while displaying very small variance-can be affected by a slowly-decaying bias that depends on the initial population of the replicas, that bidirectional estimators are significantly more efficient than unidirectional estimators for nonequil.