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Computational Drug Discovery
Methods and Applications
Edited by Vasanthanathan Poongavanam and Vijayan Ramaswamy
Volume 1
Computational Drug Discovery
Methods and Applications
Edited by Vasanthanathan Poongavanam and Vijayan Ramaswamy
Volume 2
Editors
Dr. Vasanthanathan PoongavanamUppsala UniversityDepartment of Chemistry‐BMC751 05 UppsalaSweden
Dr. Vijayan RamaswamyUniversity of Texas MD Anderson Cancer CenterInstitute for Applied Cancer ScienceTXUnited States
Cover: © Vasanthanathan Poongavanam
All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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© 2024 WILEY‐VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
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Print ISBN: 978‐3‐527‐35374‐3ePDF ISBN: 978‐3‐527‐84072‐4ePub ISBN: 978‐3‐527‐84073‐1oBook ISBN: 978‐3‐527‐84074‐8
Editors
Dr. Vasanthanathan PoongavanamUppsala UniversityDepartment of Chemistry‐BMC751 05 UppsalaSweden
Dr. Vijayan RamaswamyUniversity of Texas MD Anderson Cancer CenterInstitute for Applied Cancer ScienceTXUnited States
Cover: © Vasanthanathan Poongavanam
All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
British Library Cataloguing‐in‐Publication DataA catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2024 WILEY‐VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978‐3‐527‐35375‐0ePDF ISBN: 978‐3‐527‐84072‐4ePub ISBN: 978‐3‐527‐84073‐1oBook ISBN: 978‐3‐527‐84074‐8
Preface
Computer‐aided drug design (CADD) techniques are used in almost every stage of the drug discovery continuum, given the need to shorten discovery timelines, reduce costs, and improve the odds of clinical success. CADD integrates modeling, simulation, informatics, and artificial intelligence (AI) to design molecules with desired properties. Briefly, the application of CADD methodologies in drug discovery dates back to the 1960s, tracing its origin to the development of quantitative structure–activity relationship (QSAR) approaches. Between the 1970s and 1980s, computer graphics programs to visualize macromolecules began to take off together with advancements in computational power. This coincided with the emergence of more sophisticated techniques, including mapping energetically favorable binding sites on proteins, molecular docking, pharmacophore modeling, and modeling the dynamics of biomolecules. Since then, CADD has evolved as a powerful technique opening new possibilities, leading to increased adoption within the pharmaceutical industry and contributing to the discovery of several approved drugs.
Recent developments in CADD have been propelled by advancements in computing, breakthroughs in related fields such as structural biology, and the emergence of new therapeutic modalities. Notably, the advent of highly parallelizable GPUs and cloud computing have significantly increased computing power, while quantum computing holds promise to simulate complex systems at an unprecedented scale and speed. Advances in AI technologies, particularly generative AI for molecule design, are reducing cycle times during lead optimization. Meanwhile, the resolution revolution in cryo‐electron microscopy (cryo‐EM) and AI‐powered structure biology are shedding light on the three‐dimensional structure of many therapeutically relevant drug targets, thereby expanding our ability to carry out structure‐based drug design against these targets. Other exciting breakthroughs that offer new opportunities include the explosion in the size of "make‐on‐demand" chemical libraries that enable ultra‐large‐scale virtual screening for hit identification, the big data phenomena in medicinal chemistry with the advent of bioactivity databases like ChEMBL and GOSTAR that provide access to millions of SAR data points useful for building predictive models and for knowledge‐based compound, the emergence of new therapeutic modalities like targeted protein degradation like PROTACs and molecular glues, and viable approaches for targeting various reactive amino acid side chains beyond cysteine for developing covalent inhibitors. These developments are also now enabling drug discovery scientists to tackle high‐value drug targets previously considered undruggable.
The changing paradigm in drug discovery, complemented by technological advancements, has significantly expanded the toolbox available for computational chemists to enable drug discovery in recent years. Against this backdrop, we felt a need for a book that offers up‐to‐date information on the most important developments in the field of CADD. This book, titled “Computational Drug Discovery,” is meant to be a valuable resource for readers seeking a comprehensive account of the latest developments in CADD methods and technologies that are transforming small‐molecule drug discovery. The intended target audience for this book is medicinal chemists, computational chemists, and drug discovery professionals from industry and academia.
The book is organized into eight thematic sections, each dedicated to a cutting‐edge computational method, or a technology utilized in computational drug discovery. In total, it comprises 26 chapters authored by renowned experts from academia, pharma, and major drug discovery software providers, offering a broad overview of the latest advances in computational drug discovery.
Part I explores the role of molecular dynamics simulation and related approaches in drug discovery. It encompasses various topics such as the utilization of physics‐based methods for binding free energy estimation, the theory and application of enhanced sampling methods like Gaussian Accelerated MD to facilitate efficient sampling of the conformational space, understanding binding and unbinding kinetics of compound binding through molecular dynamics simulation, the application of computational approaches like WaterMap and 3D‐RISM framework to understand the location and thermodynamic properties of solvents that solvate the binding pocket which offers rich physical insights compound design, and the use of mixed solvent MD simulations for mapping binding hotspots on protein surfaces based on the SILCS technology.
Part II focuses on the role of quantum mechanical approaches in drug discovery, covering topics such as the use of hybrid QM/MM method for modeling reaction mechanisms and covalent inhibitor design, refinement of X‐ray and cryo‐EM structures integrating QM and QM/MM approaches for accurate assignment of tautomer, protomers, and amide flip rotamers for downstream structure‐based design, and quantifying protein–ligand interaction energies using QM methods at a reduced computational cost like the fragment molecular orbital (FMO) framework
Part III focuses on the application of AI in preclinical drug discovery, highlighting its growing importance across different stages of the drug discovery process. Given the recent advancements in AI and related technologies, we have chapters that outline advancements in deep learning for protein structure prediction, in particular the significant breakthrough achieved by AlphaFold2, the use of deep learning architectures such as Convolutional Neural Networks (CNNs), Graph Neural Networks (GNNs), and physics‐inspired neural networks for predicting protein–ligand binding affinity, the emergence of generative modeling techniques for de novo design of synthetically tractable drug‐like molecules that satisfy a defined set of constraints. In order to offer readers guidance on effectively applying machine learning (ML) models and ensuring their validity and usefulness, this section includes a chapter that discusses different approaches for evaluating the reliability and domain applicability of ML models.
Part IV of this book focuses on how the concept of chemical space and the big data phenomenon are driving drug discovery. It includes chapters describing innovative approaches in reaction‐based enumerations that enable the generation of virtual libraries containing tangible compounds, followed by computational solutions for visualizing and navigating this vast chemical space. Additionally, this section also highlights the use of SAR knowledge bases like GOSATR for extracting valuable insights and generating robust design ideas based on medicinal chemistry precedence. Wrapping up the section is a chapter highlighting how the wealth of knowledge gained by mining the data in CSD is proving valuable in various stages of drug discovery.
The ever‐expanding size of compound libraries and the advent of make‐on‐demand compound libraries have elevated virtual screening to a whole new level.
Part V focuses on ultra‐large‐scale virtual screening using approaches that scale virtual screening methods to match the size of these massively large compound libraries. Although virtual screening using docking is a well‐established approach for hit finding in drug discovery, the ability of docking programs to generate the correct binding mode and accurately estimate binding affinity is still a challenge. Hence, we have a chapter that reviews collaborative efforts within the scientific community for evaluating and comparing the performance of docking methods, establishing standardized metrics for assessing the efficiency of virtual screening techniques through rigorous competitive evaluations.
Early profiling of absorption, distribution, metabolism, excretion, and toxicity (ADMET) endpoints in early drug discovery is essential for designing and selecting compounds with superior ADMET properties. Consequently, major pharmaceutical companies have developed and implemented predictive models within their organizations for predicting multiple endpoints to enhance compound design. Part VI of the book chapter offers an overview of in silico ADMET methods and their practical applications in facilitating compound design within an industrial context. Part VII explores the role of computational techniques in accelerating the design of cutting‐edge therapeutic modalities. This section provides a comprehensive focus on two key areas: the design of molecular glues and the design of covalent inhibitors.
In addition to the aforementioned methods and approaches that revolutionize the drug discovery process, computing technologies are further accelerating drug discovery with enhanced speed and accuracy.
Part VIII is dedicated to exploring how cloud computing and quantum computing significantly expand the range of drug discovery opportunities. Particularly, there is great hope and excitement surrounding the potential applications of quantum computing in drug discovery. “The Quantum Computing Paradigm” provides a comprehensive review on quantum computing from the perspective of drug discovery. In addition to discussing several drug discovery applications, including peptide design, the chapter also addresses challenges associated with this emerging drug discovery technology.
In conclusion, we believe that this book provides a thorough overview of the recent advancements in computational drug discovery, making it an engaging and captivating read. We would like to express our deepest appreciation to all the authors for their invaluable contributions to this book. Their expertise, insights, and unwavering commitment have greatly enriched its content and overall significance.
14 September 2023
Vasanthanathan Poongavanam, Uppsala, SwedenVijayan Ramaswamy, Texas, USA
Acknowledgments
First and foremost, we would like to extend our sincerest gratitude and profound appreciation to all the contributing authors. Their unwavering commitment, tireless efforts, and remarkable enthusiasm have been instrumental in bringing this book to fruition. It is their willingness to share their knowledge and experience that has greatly enriched its content, resulting in a truly valuable and comprehensive book that provides an account of the latest advancements in the field of computer‐aided drug design.
We also extend our sincere gratitude to the external reviewers for their timely feedback and insightful suggestions that helped improve the quality of the book and shape the final outcome. Our special thanks to the following individuals for their invaluable contributions in reviewing the book chapters: Dr. Andreas Tosstorff (F. Hoffmann‐La Roche, Switzerland), Dr. Sagar Gore, Dr. Suneel Kumar BVS (Molecular Forecaster, Canada), Dr. Pandian Sokkar, Dr. Ono Satoshi (Mitsubishi Tanabe Pharma, Japan), Dr. Octav Caldararu (Zealand Pharma, Denmark), Dr. Sundarapandian Thangapandian (HotSpot Therapeutics, Inc, USA), Dr. Vigneshwaran Namasivayam (Dewpoint Therapeutics, Germany), Dr. Yinglong Miao (University of Kansas, USA), Nanjie Deng (Pace University, USA), and Dr. Ansuman Biswas (Ernst & Young, India).
In conclusion, we would like to express our gratitude to the publisher Wiley for entrusting us with an opportunity to edit this book and for the fruitful collaboration. Especially, we convey our appreciation to Katherine Wong (Senior Managing Editor) and Dr. Lifen Yang (Program Manager) at Wiley for their unwavering support, encouragement throughout the editing process, and their commitment to ensuring the quality and excellence of this book. The editors also extend their thanks to Prof. Jan Kihlberg and Dr. Jason B. Cross for their continuous support, which made this project possible.
About the Editors
Vasanthanathan Poongavanam is a senior scientist in the Department of Chemistry‐BMC, Uppsala University, Sweden. Before starting at Uppsala University in 2016, he was a postdoctoral fellow at the University of Vienna, Austria, and at the University of Southern Denmark. He obtained his PhD degree in Computational Medicinal Chemistry as a Drug Research Academy (DRA) Fellow at the University of Copenhagen, Denmark, on computational modeling of cytochrome P450. He has published more than 65 scientific articles, including reviews and book chapters. His scientific interests focus on in silico ADMET modeling, including cell permeability and solubility, and he has worked extensively on understanding the molecular properties that govern the pharmacokinetic profile of molecules bRo5 property space, including macrocycles and PROTACs.
Vijayan Ramaswamy (R.S.K. Vijayan) is a senior research scientist affiliated with the Structural Chemistry division at the Institute for Applied Cancer Science, The University of Texas MD Anderson Cancer Center, TX, USA. In 2016, he joined MD Anderson Cancer after a brief tenure as a scientist in computational chemistry at PMC Advanced Technologies, New Jersey, USA. He undertook postdoctoral training at Rutgers University in New Jersey, USA, and Temple University in Pennsylvania, USA. He received his PhD as a CSIR senior research fellow from the Indian Institute of Chemical Biology, Kolkata, India. He is a named co‐inventor on seven issued US patents, including an ATR kinase inhibitor that has advanced to clinical trials. He has published more than 20 scientific articles and authored one book chapter. His research focuses on applying computational chemistry methods to drive small‐molecule drug discovery programs, particularly in oncology and neurodegenerative diseases.
Part IMolecular Dynamics and Related Methods in Drug Discovery
1Binding Free Energy Calculations in Drug Discovery
Anita de Ruiter1 and Chris Oostenbrink1,2
1Institute for Molecular Modeling and Simulation, Department of Material Sciences and Process Engineering, University of Natural Resources and Life Sciences, Vienna, Muthgasse 18, 1190 Vienna, Austria
2Christian Doppler Laboratory for Molecular Informatics in the Biosciences, University of Natural Resources and Life Sciences, Vienna, Muthgasse 18, 1190 Vienna, Austria
1.1 Introduction
This chapter attempts to provide an overview of the different approaches and methods that are available to compute binding free‐energy in drug design and drug discovery. We do not provide an exhaustive list of available methods and do not rigorously derive all of the methods from first principles. Instead, we aim to give a overview of available methods and to point at the intrinsic limitations and challenges of these methods, such that researchers applying these methods can make a fair estimate of the most appropriate methods for their aims.
Numerous methods for the calculation of binding free energies have been developed over the years [1]. Which method is the best choice depends on how many free energies need to be determined, the available computational resources, the accuracy one wishes to obtain, and other specific properties of the system under study. Let us start by separating the available methods into three classes. Binding free energies can be calculated with endpoint, alchemical, or pathway methods. These methods are very different, not only in terms of their underlying theory but also in their accuracy and efficiency. The endpoint methods are very efficient in terms of computational requirements, but, unfortunately, not very accurate. Alchemical methods, on the other hand, are considered one of the most accurate but also slow methods. Pathway methods are also computationally demanding but can give important information about the binding pathways. Which method is the best choice will mostly depend on the stage at which the drug discovery/design is currently at. In the very early stages, where whole databases of compounds need to be screened, one can likely not afford the computational costs of alchemical approaches. However, since the range of binding free energies that are to be predicted may also be rather large, the faster methods will be sufficient to pick up some hit compounds. In the lead optimization stage, where rather similar compounds are studied, a more accurate method is required that can detect smaller differences in the binding free energies. Because the optimization stage also focuses on fewer leads, the higher computational demand for the more accurate method can actually be afforded.
1.1.1 Free Energy and Thermodynamic Cycles
First of all, we should look into the definition of free energy to find out what kind of property we are trying to determine. The definition of free energy in statistical mechanics is
where G is the Gibbs free energy, kB is the Boltzmann constant, T is the temperature, and QNPT is the partition function for a system with a constant number of particles, pressure, and temperature. The partition function is defined as
with r and p as the positions and momenta of all atoms in the system, respectively, and H(r,p) as the Hamiltonian of the system, giving the total energy. It is clear from the integration of all positions and momenta in Eq. (1.2) that the free energy is in trinsically a property of a statistical mechanical ensemble and not something that can be estimated from a single configuration. Free energy is a property of all (relevant) configurations of the system together. Any effort to estimate the free energy from a single configuration will likely miss out on some relevant aspects of the ensemble, such as conformational changes and their entropic effects.
In the field of drug discovery, one is not interested in the absolute free energy of a certain state, but rather in the binding free energy of, e.g. a small molecule to a protein;
where Qbound and Qfree are the partition functions for the system where the small molecule is bound to the protein and when both partners are free in solution, respectively.
Furthermore, during hit‐to‐lead or lead optimization stages, one is mostly interested in the relative binding free energy, i.e. which of the two ligands binds stronger to the protein than the other. This, together with the fact that free energy is a state function, makes it possible to design thermodynamic cycles to make it easier to calculate the free energies. Consider a thermodynamic cycle like that in Figure 1.1.
Figure 1.1 Thermodynamic cycle for the calculation of relative binding free energies of two small molecules A and B binding to a common receptor.
There are four states, one with ligand A bound to the protein, one with ligand A unbound from the protein, one with ligand B bound to the protein, and one with ligand B unbound from the protein. Since free energy is a state function, following the full thermodynamic cycle will lead to a free energy difference of 0:
From this, it follows that ΔGbind(B) − ΔGbind(A) = ΔGBA(prot) − ΔGBA(free). This means that we can determine the difference in binding free energy without performing a tedious simulation of the actual binding process. Although the modification of ligand A to ligand B is not something that is physically possible in the laboratory, it is possible with computer simulations and alchemical free‐energy calculations. In fact, it is often easier to obtain converged results for these unphysical processes because modifying the ligands will most likely lead to much less reorganization of the protein than the binding process would. Modifying the ligand requires intermediate states, which will be discussed in more detail in the section on alchemical methods.
1.2 Endpoint Methods
As the name implies, endpoint methods only require the simulation of the endpoints of the system of interest. For binding free energy calculations, the endpoints would be the protein–ligand complex and the separate protein and ligand. That is, we explicitly simulate the states of the corners of the thermodynamic cycle of Figure 1.1. Their efficiency and reasonable accuracy make the endpoint free energy methods very popular in the early stages of drug discovery. Here, we will discuss two kinds of endstate methods: the molecular mechanics Poisson–Boltzmann surface area (MM/PBSA) methods and methods derived from linear response theory.
1.2.1 MM/PBSA and MM/GBSA
The most commonly used methods are MM/PBSA and the closely related molecular mechanics generalized Born surface area (MM/GBSA) [2, 3].
In MM/PBSA, the free energy of a state is composed of several contributions;
Here, EMM is the molecular mechanics potential energy term, which consists of bonded interactions (Ebnd), electrostatic interactions (Eel), and van der Waals interactions (EvdW). Gpol and Gnp are the polar and nonpolar contributions to the solvation free energy, respectively. T represents the temperature of the system, and S is the entropy. Note that, although the free energy is a property of the ensemble and not an average over the ensemble, these methods assume that these terms together approximate the free energy of the state reasonably well and can be computed from individual configurations of the ensemble.
In order to calculate the absolute binding free energy of a system, the free energy of the free ligand (L), the unbound protein (P), as well as the complex (PL) needs to be computed;
Here, the angular brackets indicate an ensemble average from the simulation of the system indicated in the subscript. Equation (1.7) is the so‐called three‐average MM/PBSA (3A‐MM/PBSA) since three different simulations need to be performed. The ensembles in Eq. (1.7) are generated from snapshots of molecular dynamics simulations with an explicit solvation model. Once these snapshots are generated, they are stripped from all solvent molecules and ions, and an implicit solvation model is used for further analysis.
Gpol is determined either by solving the Poisson–Boltzmann (PB) equation or by using the generalized Born equation (in which case the method would be called MM/GBSA). GB uses an analytical expression for the polar solvation energy and is thus much faster, but also likely to be less accurate, although this is system‐dependent. Gnp is estimated by using the solvent accessible surface area (SA). The assumption that Gpol and Gnp can be approximated from an implicit solvation model means that solvent degrees of freedom are no longer treated explicitly in Eq. (1.2) and lead to the use of simple ensemble averages in Eq. (1.7). The calculation of Gpol furthermore depends strongly on the implicit solvation model that is used. Usually, the implicit solvation model requires a single dielectric constant to be chosen to describe the very complex electrostatic environment within the protein. This either makes the results unreliable, or the user can choose the constant such that the results are in agreement with known binding free energies for the system. In the latter case, MM/PBSA becomes more of an empirical method, where the parameters are optimized to reproduce experimental data. Finally, as a result of the implicit solvation model, MM/PBSA is not very well‐suited when the binding site involves a highly charged environment or when critical water molecules are within the binding site.
The second reason that the ensemble property of Eqs. (1.1) and (1.2) may be approximated by a simple ensemble average in Eq. (1.7) is the explicit separation of the protein and ligand degrees of freedom into an energetic contribution (EMM) and an entropic contribution (TS). The energetic term is computed from a force field, which is indeed well captured by an ensemble average. The entropy term is most commonly estimated with normal mode analysis (NMA). However, this method, which estimates the curvature of the energy landscape and approximates the entropy based on the expected sampling on this surface, is rather time‐consuming and therefore not suitable for larger systems. More efficient methods have been explored over the years, but especially when the interest lies with relative binding free energies (like in drug discovery), the entropy term is often simply ignored. The underlying assumption would be that similar ligands will have similar entropy terms. Effectively, however, this means that the free energy is approximated by an energy.
A further, very popular, approximation is to use the single‐trajectory MM/PBSA (1A‐MM/PBSA), where only the complex is simulated
GP and GL are determined from the ensemble of the complex by just removing the atoms that are not part of the state of interest, i.e. for GP, the ligand atoms are removed from the complex simulations, and for GL, the protein atoms are removed. There are two main advantages of 1A‐MM/PBSA with respect to 3A‐MM/PBSA. The most obvious one is that only a single simulation needs to be performed instead of three simulations, and therefore it is computationally more efficient. The second one comes from the fact that Ebnd and all intramolecular contributions to Eel and EvdW cancel in Eq. (1.8) because these energies for the apo protein and isolated ligand are calculated from exactly the same configuration as the complex. Also, the entropy estimate will seemingly become negligible since the ligand does not sample different conformations in the bound or in the free state. This significantly reduces the noise in the free energies, allowing for faster convergence of the results.
However, we need to consider what additional assumptions are being made with the 1A‐MM/PBSA approach. It basically assumes that the protein and ligand visit the exact same conformations when they are in complex with each other as when they are each free in solution. This is not very likely to be the case. For example, the ligand can be forced to be more rigid (and/or bend) within a tight binding site, and a protein side chain or loop can be pushed aside upon binding of the ligand. The energetic and entropic effects of such conformational changes can be significant, but are entirely absent from the 1A‐MM/PBSA approach. Unfortunately, the fact that the 1A‐MM/PBSA often leads to less noise in the calculation does not make it more appropriate.
Recent advances try to address several of the approximations in the MM/PBSA and MM/GBSA methods with some promising results, but the optimal solutions remain rather system‐dependent. For further reading, we suggest some recent reviews on the topic [3–5].
1.2.2 Linear Response Approximations
Other endpoint methods are based on the linear response theory. In the linear response approximation (LRA) framework [6, 7], two additional states need to be simulated, which are the neutralized states of the ligand when it is bound to the protein and when it is free in solution. Any partial charges of the ligand are set to 0 in these simulations. The charging free energy difference is then calculated with
where HQ is the Hamiltonian of the charged state and HN is the Hamiltonian of the neutralized state; subscripts after the angular brackets indicate which Hamiltonian was used to obtain the ensemble; and are the electrostatic interactions of the ligand with its surroundings.
In the Linear Interaction Energy (LIE) method [8], it is assumed that the electrostatic interactions of the charged ligand obtained from the ensemble of neutral states, , will average to 0. Although this assumption is reasonable for the ligand in solution, it might not hold for the ligand bound to a protein. The protein is not likely to have a random electrostatic distribution around the neutral ligand and corresponds to the preorganization energy of the protein. The free energy difference of charging a ligand using LIE is then calculated with
where β is theoretically 1/2. The nonpolar interactions are also assumed to have a linear relationship with the free energy difference, even though this is only based on observations that the free energy of solvation for nonpolar particles and the interaction energy both seem to be linearly correlated with the size of the molecule. The binding free‐energy difference based on LIE can thus be calculated with
where α and γ are empirical parameters, which can be used to fit to experimental data from a data set. Δ indicates the difference between the ensemble averages obtained from the simulation of the free ligand and when bound to the protein. Even though β has a theoretical value of ½, it is also often used as an empirical parameter. Scaling the interactions with α, β, and adding γ helps compensate for the missing factors in the LIE approach, such as intramolecular energies, entropic confinement, and desolvation effects.
As an alternative to LIE, we have developed third power fitting (TPF), in which we do not assume linearity for the charging free energy [9]. Instead, the neutral and charged states are simulated, and the curvature of the charging curve is estimated by a third‐order polynomial of a coupling parameter λ. Four constraints are used to find the best fit, which are based on the first (dG/dλ) and second (d2G/dλ2) derivatives of the free energy with respect to λ, from simulations in the N and Q states of LRA. It can be shown using the cumulant expansion that d2G/dλ2 is equal to the negative of the fluctuations of dH/dλ
With the subscript S corresponding to either N or Q. The advantage of TPF is that there is that the nonlinearity is captured without additional simulations or empirical parameters. One should keep in mind, though, that the fluctuations in Eq. (1.12) are slower to converge.
