Current State of Open Source Force Fields in Protein-Ligand Binding Affinity Predictions.

In drug discovery, the in silico prediction of binding affinity is one of the major means to prioritize compounds for synthesis. Alchemical relative binding free energy (RBFE) calculations based on molecular dynamics (MD) simulations are nowadays a popular approach for the accurate affinity ranking of compounds. MD simulations rely on empirical force field parameters, which strongly influence the accuracy of the predicted affinities. Here, we evaluate the ability of six different small-molecule force fields to predict experimental protein-ligand binding affinities in RBFE calculations on a set of 598 ligands and 22 protein targets. The public force fields OpenFF Parsley and Sage, GAFF, and CGenFF show comparable accuracy, while OPLS3e is significantly more accurate. However, a consensus approach using Sage, GAFF, and CGenFF leads to accuracy comparable to OPLS3e. While Parsley and Sage are performing comparably based on aggregated statistics across the whole dataset, there are differences in terms of outliers. Analysis of the force field reveals that improved parameters lead to significant improvement in the accuracy of affinity predictions on subsets of the dataset involving those parameters. Lower accuracy can not only be attributed to the force field parameters but is also dependent on input preparation and sampling convergence of the calculations. Especially large perturbations and nonconverged simulations lead to less accurate predictions. The input structures, Gromacs force field files, as well as the analysis Python notebooks are available on GitHub.

Broadening the Scope of Binding Free Energy Calculations Using a Separated Topologies Approach.

Binding free energy calculations predict the potency of compounds to protein binding sites in a physically rigorous manner and see broad application in prioritizing the synthesis of novel drug candidates. Relative binding free energy (RBFE) calculations have emerged as an industry-standard approach to achieve highly accurate rank-order predictions of the potency of related compounds; however, this approach requires that the ligands share a common scaffold and a common binding mode, restricting the methods' domain of applicability. This is a critical limitation since complex modifications to the ligands, especially core hopping, are very common in drug design. Absolute binding free energy (ABFE) calculations are an alternate method that can be used for ligands that are not congeneric. However, ABFE suffers from a known problem of long convergence times due to the need to sample additional degrees of freedom within each system, such as sampling rearrangements necessary to open and close the binding site. Here, we report on an alternative method for RBFE, called Separated Topologies (SepTop), which overcomes the issues in both of the aforementioned methods by enabling large scaffold changes between ligands with a convergence time comparable to traditional RBFE. Instead of only mutating atoms that vary between two ligands, this approach performs two absolute free energy calculations at the same time in opposite directions, one for each ligand. Defining the two ligands independently allows the comparison of the binding of diverse ligands without the artificial constraints of identical poses or a suitable atom-atom mapping. This approach also avoids the need to sample the unbound state of the protein, making it more efficient than absolute binding free energy calculations. Here, we introduce an implementation of SepTop. We developed a general and efficient protocol for running SepTop, and we demonstrated the method on four diverse, pharmaceutically relevant systems. We report the performance of the method, as well as our practical insights into the strengths, weaknesses, and challenges of applying this method in an industrial drug design setting. We find that the accuracy of the approach is sufficiently high to rank order ligands with an accuracy comparable to traditional RBFE calculations while maintaining the additional flexibility of SepTop.

Development and Benchmarking of Open Force Field 2.0.0: The Sage Small Molecule Force Field.

We introduce the Open Force Field (OpenFF) 2.0.0 small molecule force field for drug-like molecules, code-named Sage, which builds upon our previous iteration, Parsley. OpenFF force fields are based on direct chemical perception, which generalizes easily to highly diverse sets of chemistries based on substructure queries. Like the previous OpenFF iterations, the Sage generation of OpenFF force fields was validated in protein-ligand simulations to be compatible with AMBER biopolymer force fields. In this work, we detail the methodology used to develop this force field, as well as the innovations and improvements introduced since the release of Parsley 1.0.0. One particularly significant feature of Sage is a set of improved Lennard-Jones (LJ) parameters retrained against condensed phase mixture data, the first refit of LJ parameters in the OpenFF small molecule force field line. Sage also includes valence parameters refit to a larger database of quantum chemical calculations than previous versions, as well as improvements in how this fitting is performed. Force field benchmarks show improvements in general metrics of performance against quantum chemistry reference data such as root-mean-square deviations (RMSD) of optimized conformer geometries, torsion fingerprint deviations (TFD), and improved relative conformer energetics (ΔΔE). We present a variety of benchmarks for these metrics against our previous force fields as well as in some cases other small molecule force fields. Sage also demonstrates improved performance in estimating physical properties, including comparison against experimental data from various thermodynamic databases for small molecule properties such as ΔHmix, ρ(x), ΔGsolv, and ΔGtrans. Additionally, we benchmarked against protein-ligand binding free energies (ΔGbind), where Sage yields results statistically similar to previous force fields. All the data is made publicly available along with complete details on how to reproduce the training results at https://github.com/openforcefield/openff-sage.

To Design Scalable Free Energy Perturbation Networks, Optimal Is Not Enough.

Drug discovery is accelerated with computational methods such as alchemical simulations to estimate ligand affinities. In particular, relative binding free energy (RBFE) simulations are beneficial for lead optimization. To use RBFE simulations to compare prospective ligands in silico, researchers first plan the simulation experiment, using graphs where nodes represent ligands and graph edges represent alchemical transformations between ligands. Recent work demonstrated that optimizing the statistical architecture of these perturbation graphs improves the accuracy of the predicted changes in the free energy of ligand binding. Therefore, to improve the success rate of computational drug discovery, we present the open-source software package High Information Mapper (HiMap)─a new take on its predecessor, Lead Optimization Mapper (LOMAP). HiMap removes heuristics decisions from design selection and instead finds statistically optimal graphs over ligands clustered with machine learning. Beyond optimal design generation, we present theoretical insights for designing alchemical perturbation maps. Some of these results include that for n number of nodes, the precision of perturbation maps is stable at n·ln(n) edges. This result indicates that even an "optimal" graph can result in unexpectedly high errors if a plan includes too few alchemical transformations for the given number of ligands and edges. And, as a study compares more ligands, the performance of even optimal graphs will deteriorate with linear scaling of the edge count. In this sense, ensuring an A- or D-optimal topology is not enough to produce robust errors. We additionally find that optimal designs will converge more rapidly than radial and LOMAP designs. Moreover, we derive bounds for how clustering reduces cost for designs with a constant expected relative error per cluster, invariant of the size of the design. These results inform how to best design perturbation maps for computational drug discovery and have broader implications for experimental design.

Prioritizing Small Sets of Molecules for Synthesis through in-silico Tools: A Comparison of Common Ranking Methods.

Prioritizing molecules for synthesis is a key role of computational methods within medicinal chemistry. Multiple tools exist for ranking molecules, from the cheap and popular molecular docking methods to more computationally expensive molecular-dynamics (MD)-based methods. It is often questioned whether the accuracy of the more rigorous methods justifies the higher computational cost and associated calculation time. Here, we compared the performance on ranking the binding of small molecules for seven scoring functions from five docking programs, one end-point method (MM/GBSA), and two MD-based free energy methods (PMX, FEP+). We investigated 16 pharmaceutically relevant targets with a total of 423 known binders. The performance of docking methods for ligand ranking was strongly system dependent. We observed that MD-based methods predominantly outperformed docking algorithms and MM/GBSA calculations. Based on our results, we recommend the application of MD-based free energy methods for prioritization of molecules for synthesis in lead optimization, whenever feasible.

Best practices for constructing, preparing, and evaluating protein-ligand binding affinity benchmarks [Article v0.1].

Free energy calculations are rapidly becoming indispensable in structure-enabled drug discovery programs. As new methods, force fields, and implementations are developed, assessing their expected accuracy on real-world systems (benchmarking) becomes critical to provide users with an assessment of the accuracy expected when these methods are applied within their domain of applicability, and developers with a way to assess the expected impact of new methodologies. These assessments require construction of a benchmark-a set of well-prepared, high quality systems with corresponding experimental measurements designed to ensure the resulting calculations provide a realistic assessment of expected performance when these methods are deployed within their domains of applicability. To date, the community has not yet adopted a common standardized benchmark, and existing benchmark reports suffer from a myriad of issues, including poor data quality, limited statistical power, and statistically deficient analyses, all of which can conspire to produce benchmarks that are poorly predictive of real-world performance. Here, we address these issues by presenting guidelines for (1) curating experimental data to develop meaningful benchmark sets, (2) preparing benchmark inputs according to best practices to facilitate widespread adoption, and (3) analysis of the resulting predictions to enable statistically meaningful comparisons among methods and force fields. We highlight challenges and open questions that remain to be solved in these areas, as well as recommendations for the collection of new datasets that might optimally serve to measure progress as methods become systematically more reliable. Finally, we provide a curated, versioned, open, standardized benchmark set adherent to these standards (PLBenchmarks) and an open source toolkit for implementing standardized best practices assessments (arsenic) for the community to use as a standardized assessment tool. While our main focus is free energy methods based on molecular simulations, these guidelines should prove useful for assessment of the rapidly growing field of machine learning methods for affinity prediction as well.

Collaborative Assessment of Molecular Geometries and Energies from the Open Force Field.

Force fields form the basis for classical molecular simulations, and their accuracy is crucial for the quality of, for instance, protein-ligand binding simulations in drug discovery. The huge diversity of small-molecule chemistry makes it a challenge to build and parameterize a suitable force field. The Open Force Field Initiative is a combined industry and academic consortium developing a state-of-the-art small-molecule force field. In this report, industry members of the consortium worked together to objectively evaluate the performance of the force fields (referred to here as OpenFF) produced by the initiative on a combined public and proprietary dataset of 19,653 relevant molecules selected from their internal research and compound collections. This evaluation was important because it was completely blind; at most partners, none of the molecules or data were used in force field development or testing prior to this work. We compare the Open Force Field "Sage" version 2.0.0 and "Parsley" version 1.3.0 with GAFF-2.11-AM1BCC, OPLS4, and SMIRNOFF99Frosst. We analyzed force-field-optimized geometries and conformer energies compared to reference quantum mechanical data. We show that OPLS4 performs best, and the latest Open Force Field release shows a clear improvement compared to its predecessors. The performance of established force fields such as GAFF-2.11 was generally worse. While OpenFF researchers were involved in building the benchmarking infrastructure used in this work, benchmarking was done entirely in-house within industrial organizations and the resulting assessment is reported here. This work assesses the force field performance using separate benchmarking steps, external datasets, and involving external research groups. This effort may also be unique in terms of the number of different industrial partners involved, with 10 different companies participating in the benchmark efforts.

Chemical Space Exploration with Active Learning and Alchemical Free Energies.

Drug discovery can be thought of as a search for a needle in a haystack: searching through a large chemical space for the most active compounds. Computational techniques can narrow the search space for experimental follow up, but even they become unaffordable when evaluating large numbers of molecules. Therefore, machine learning (ML) strategies are being developed as computationally cheaper complementary techniques for navigating and triaging large chemical libraries. Here, we explore how an active learning protocol can be combined with first-principles based alchemical free energy calculations to identify high affinity phosphodiesterase 2 (PDE2) inhibitors. We first calibrate the procedure using a set of experimentally characterized PDE2 binders. The optimized protocol is then used prospectively on a large chemical library to navigate toward potent inhibitors. In the active learning cycle, at every iteration a small fraction of compounds is probed by alchemical calculations and the obtained affinities are used to train ML models. With successive rounds, high affinity binders are identified by explicitly evaluating only a small subset of compounds in a large chemical library, thus providing an efficient protocol that robustly identifies a large fraction of true positives.

Pre-Exascale Computing of Protein-Ligand Binding Free Energies with Open Source Software for Drug Design.

Nowadays, drug design projects benefit from highly accurate protein-ligand binding free energy predictions based on molecular dynamics simulations. While such calculations have been computationally expensive in the past, we now demonstrate that workflows built on open source software packages can efficiently leverage pre-exascale computing resources to screen hundreds of compounds in a matter of days. We report our results of free energy calculations on a large set of pharmaceutically relevant targets assembled to reflect industrial drug discovery projects.

Development and Benchmarking of Open Force Field v1.0.0-the Parsley Small-Molecule Force Field.

We present a methodology for defining and optimizing a general force field for classical molecular simulations, and we describe its use to derive the Open Force Field 1.0.0 small-molecule force field, codenamed Parsley. Rather than using traditional atom typing, our approach is built on the SMIRKS-native Open Force Field (SMIRNOFF) parameter assignment formalism, which handles increases in the diversity and specificity of the force field definition without needlessly increasing the complexity of the specification. Parameters are optimized with the ForceBalance tool, based on reference quantum chemical data that include torsion potential energy profiles, optimized gas-phase structures, and vibrational frequencies. These quantum reference data are computed and are maintained with QCArchive, an open-source and freely available distributed computing and database software ecosystem. In this initial application of the method, we present essentially a full optimization of all valence parameters and report tests of the resulting force field against compounds and data types outside the training set. These tests show improvements in optimized geometries and conformational energetics and demonstrate that Parsley's accuracy for liquid properties is similar to that of other general force fields, as is accuracy on binding free energies. We find that this initial Parsley force field affords accuracy similar to that of other general force fields when used to calculate relative binding free energies spanning 199 protein-ligand systems. Additionally, the resulting infrastructure allows us to rapidly optimize an entirely new force field with minimal human intervention.

Expanded Ensemble Methods Can be Used to Accurately Predict Protein-Ligand Relative Binding Free Energies.

Alchemical free energy methods have become indispensable in computational drug discovery for their ability to calculate highly accurate estimates of protein-ligand affinities. Expanded ensemble (EE) methods, which involve single simulations visiting all of the alchemical intermediates, have some key advantages for alchemical free energy calculation. However, there have been relatively few examples published in the literature of using expanded ensemble simulations for free energies of protein-ligand binding. In this paper, as a test of expanded ensemble methods, we compute relative binding free energies using the Open Force Field Initiative force field (codename "Parsley") for 24 pairs of Tyk2 inhibitors derived from a congeneric series of 16 compounds. The EE predictions agree well with the experimental values (root-mean-square error (RMSE) of 0.94 ± 0.13 kcal mol-1 and mean unsigned error (MUE) of 0.75 ± 0.12 kcal mol-1). We find that while increasing the number of alchemical intermediates can improve the phase space overlap, faster convergence can be obtained with fewer intermediates, as long as acceptance rates are sufficient. We also find that convergence can be improved using more aggressive updating of biases, and that estimates can be improved by performing multiple independent EE calculations. This work demonstrates that EE is a viable option for alchemical free energy calculation. We discuss the implications of these findings for rational drug design, as well as future directions for improvement.

A Benchmark of Electrostatic Method Performance in Relative Binding Free Energy Calculations.

Relative free energy calculations are fast becoming a critical part of early stage pharmaceutical design, making it important to know how to obtain the best performance with these calculations in applications that could span hundreds of calculations and molecules. In this work, we compared two different treatments of long-range electrostatics, Particle Mesh Ewald (PME) and Reaction Field (RF), in relative binding free energy calculations using a nonequilibrium switching protocol. We found simulations using RF achieve comparable results to those using PME but gain more efficiency when using CPU and similar performance using GPU. The results from this work encourage more use of RF in molecular simulations.

Ensembler: A Simple Package for Fast Prototyping and Teaching Molecular Simulations.

Ensembler is a Python package that enables method prototyping using 1D and 2D model systems and allows deepening of the understanding of different molecular dynamics (MD) methods, starting from basic techniques to enhanced sampling and free-energy approaches. The ease of installing and using the package increases shareability, comparability, and reproducibility of scientific code developments. Here, we describe the implementation and usage of the package and provide an application example for free-energy calculation. The code of Ensembler is freely available on GitHub at https://github.com/rinikerlab/Ensembler.

Systematic Optimization of a Fragment-Based Force Field against Experimental Pure-Liquid Properties Considering Large Compound Families: Application to Saturated Haloalkanes.

Direct optimization against experimental condensed-phase properties concerning small organic molecules still represents the most reliable way to calibrate the empirical parameters of a force field. However, compared to a corresponding calibration against quantum-mechanical (QM) calculations concerning isolated molecules, this approach is typically very tedious and time-consuming. The present article describes an integrated scheme for the automated refinement of force-field parameters against experimental condensed-phase data, considering entire classes of organic molecules constructed using a fragment library via combinatorial isomer enumeration. The main steps of the scheme, referred to as CombiFF, are as follows: (i) definition of a molecule family; (ii) combinatorial enumeration of all isomers; (iii) query for experimental data; (iv) automatic construction of the molecular topologies by fragment assembly; and (v) iterative refinement of the force-field parameters considering the entire family. As a first application, CombiFF is used here to design a GROMOS-compatible united-atom force field for the saturated acyclic haloalkane family. This force field relies on an electronegativity-equalization scheme for the atomic partial charges and involves no specific terms for σ-holes and halogen bonding. A total of 749 experimental liquid densities ρliq and vaporization enthalpies ΔHvap concerning 486 haloalkanes are considered for calibration and validation. The resulting root-mean-square deviations from experiment are 49.8 (27.6) kg·m-3 for ρliq and 2.7 (1.8) kJ·mol-1 for ΔHvap for the calibration (validation) set. The values are lower for the validation set which contains larger molecules (stronger influence of purely aliphatic interactions). The trends in the optimized parameters along the halogen series and across the compound family are in line with chemical intuition based on considerations related to size, polarizability, softness, electronegativity, induction, and hyperconjugation. This observation is particularly remarkable considering that the force-field calibration did not involve any QM calculation. Once the time-consuming task of target-data selection/curation has been performed, the optimization of a force field only takes a few days. As a result, CombiFF enables an easy assessment of the consequences of functional-form decisions on the accuracy of a force field at an optimal level of parametrization.

The Conveyor Belt Umbrella Sampling (CBUS) Scheme: Principle and Application to the Calculation of the Absolute Binding Free Energies of Alkali Cations to Crown Ethers.

We recently introduced a method called conveyor belt (CB) thermodynamic integration (TI) for the calculation of alchemical free-energy differences based on molecular dynamics simulations. In the present work, the CBTI approach is generalized to conformational free-energy changes, i.e., to the determination of the potential of mean force (PMF) along a conformational coordinate ξ of interest. The proposed conveyor belt umbrella sampling (CBUS) scheme relies on the parallel simulation of K replicas k = 0,1, ..., K - 1 of the system, with K even. For each replica k, the instantaneous value of ξ is restrained to an anchor value λk. The latter anchor points are equally spaced along a forward-turn-backward-turn path (i.e., a CB) between two extreme values defining the ξ-range of interest. The rotation of the CB is controlled by a variable Λ (range from 0 to 2π) which evolves dynamically along the simulation. The evolution of Λ results from the forces exerted by the restraining potentials on the anchor points, taken equal and opposite to those they exert on the replicas. Because these forces tend to cancel out along the CB, the dynamics of Λ is essentially diffusive, and the continuous distribution of ξ-values sampled by the replica system is automatically close to homogeneous. The latter feature represents an advantage over direct counting (DCNT) and traditional umbrella sampling (TRUS), shared to some extent with replica-exchange umbrella sampling (REUS). In this work, the CBUS scheme is introduced and compared to the three latter schemes in the calculation of 45 standard absolute binding free energies. These correspond to the binding of five alkali cations to three crown ethers in three solvents. Different free-energy estimators are considered for the PMF calculation, and the calculated values are also compared to those of a previous study relying on an alchemical path, as well as to experimental data.

Overcoming Orthogonal Barriers in Alchemical Free Energy Calculations: On the Relative Merits of λ-Variations, λ-Extrapolations, and Biasing.

Alchemical free energy calculations using conventional molecular dynamics and thermodynamic integration rely on simulations performed at fixed values of the coupling parameter λ. When multiple conformers in equilibrium are separated by high barriers in the space orthogonal to λ, proper convergence may require extremely long simulations. Four main strategies can be employed to address this orthogonal-sampling problem: (a) λ-variations, where λ can change along the simulations to circumvent barriers; (b) λ-extrapolations, where statistical information is transferred between λ-points; (c) specific biasing, where orthogonal barriers are reduced using a biasing potential designed specifically for the system; and (d) generic biasing, where orthogonal barriers are reduced using a generic approach. Here, we investigate the relative merits of the first three strategies considering two benchmark systems. The KXK system involves a mutation of the central residue in a tripeptide to a glycine and the XTP system involves a hydrogen-to-bromine mutation in the base of a nucleotide. Three sampling methods are compared, the latter two involving λ-variations: molecular dynamics simulations with fixed λ-points, Hamiltonian replica exchange, and the recently introduced conveyor belt method. Two free energy estimators are applied, the second one involving λ-extrapolations: thermodynamic integration with Simpson quadrature and the multistate Bennett acceptance ratio. Finally, three different seeding schemes are considered for the generation of the initial configurations. For both benchmark systems, λ-extrapolations are found to provide little gain, whereas λ-variations can significantly enhance the convergence. They are sufficient on their own if the orthogonal barriers are low in at least one state (e.g., the glycine state in KXK). However, if the orthogonal barriers are high over the entire λ-range (e.g., the XTP system), λ-variations are only effective when applied together with a specific biasing for introducing such a low-barrier state.

Best Practices for Alchemical Free Energy Calculations [Article v1.0].

Alchemical free energy calculations are a useful tool for predicting free energy differences associated with the transfer of molecules from one environment to another. The hallmark of these methods is the use of "bridging" potential energy functions representing alchemical intermediate states that cannot exist as real chemical species. The data collected from these bridging alchemical thermodynamic states allows the efficient computation of transfer free energies (or differences in transfer free energies) with orders of magnitude less simulation time than simulating the transfer process directly. While these methods are highly flexible, care must be taken in avoiding common pitfalls to ensure that computed free energy differences can be robust and reproducible for the chosen force field, and that appropriate corrections are included to permit direct comparison with experimental data. In this paper, we review current best practices for several popular application domains of alchemical free energy calculations performed with equilibrium simulations, in particular relative and absolute small molecule binding free energy calculations to biomolecular targets.

Benchmark assessment of molecular geometries and energies from small molecule force fields.

Background: Force fields are used in a wide variety of contexts for classical molecular simulation, including studies on protein-ligand binding, membrane permeation, and thermophysical property prediction. The quality of these studies relies on the quality of the force fields used to represent the systems. Methods: Focusing on small molecules of fewer than 50 heavy atoms, our aim in this work is to compare nine force fields: GAFF, GAFF2, MMFF94, MMFF94S, OPLS3e, SMIRNOFF99Frosst, and the Open Force Field Parsley, versions 1.0, 1.1, and 1.2. On a dataset comprising 22,675 molecular structures of 3,271 molecules, we analyzed force field-optimized geometries and conformer energies compared to reference quantum mechanical (QM) data. Results: We show that while OPLS3e performs best, the latest Open Force Field Parsley release is approaching a comparable level of accuracy in reproducing QM geometries and energetics for this set of molecules. Meanwhile, the performance of established force fields such as MMFF94S and GAFF2 is generally somewhat worse. We also find that the series of recent Open Force Field versions provide significant increases in accuracy. Conclusions: This study provides an extensive test of the performance of different molecular mechanics force fields on a diverse molecule set, and highlights two (OPLS3e and OpenFF 1.2) that perform better than the others tested on the present comparison. Our molecule set and results are available for other researchers to use in testing.

Alchemical Free-Energy Calculations by Multiple-Replica λ-Dynamics: The Conveyor Belt Thermodynamic Integration Scheme.

A new method is proposed to calculate alchemical free-energy differences based on molecular dynamics (MD) simulations, called the conveyor belt thermodynamic integration (CBTI) scheme. As in thermodynamic integration (TI), K replicas of the system are simulated at different values of the alchemical coupling parameter λ. The number K is taken to be even, and the replicas are equally spaced on a forward-turn-backward-turn path, akin to a conveyor belt (CB) between the two physical end-states; and as in λ-dynamics (λD), the λ-values associated with the individual systems evolve in time along the simulation. However, they do so in a concerted fashion, determined by the evolution of a single dynamical variable Λ of period 2π controlling the advance of the entire CB. Thus, a change of Λ is always associated with K/2 equispaced replicas moving forward and K/2 equispaced replicas moving backward along λ. As a result, the effective free-energy profile of the replica system along Λ is periodic of period 2 πK-1, and the magnitude of its variations decreases rapidly upon increasing K, at least as K-1 in the limit of large K. When a sufficient number of replicas is used, these variations become small, which enables a complete and quasi-homogeneous coverage of the λ-range by the replica system, without application of any biasing potential. If desired, a memory-based biasing potential can still be added to further homogenize the sampling, the preoptimization of which is computationally inexpensive. The final free-energy profile along λ is calculated similarly to TI, by binning of the Hamiltonian λ-derivative as a function of λ considering all replicas simultaneously, followed by quadrature integration. The associated quadrature error can be kept very low owing to the continuous and quasi-homogeneous λ-sampling. The CBTI scheme can be viewed as a continuous/deterministic/dynamical analog of the Hamiltonian replica-exchange/permutation (HRE/HRP) schemes or as a correlated multiple-replica analog of the λD or λ-local elevation umbrella sampling (λ-LEUS) schemes. Compared to TI, it shares the advantage of the latter schemes in terms of enhanced orthogonal sampling, i.e. the availability of variable-λ paths to circumvent conformational barriers present at specific λ-values. Compared to HRE/HRP, it permits a deterministic and continuous sampling of the λ-range, is expected to be less sensitive to possible artifacts of the thermo- and barostating schemes, and bypasses the need to carefully preselect a λ-ladder and a swapping-attempt frequency. Compared to λ-LEUS, it eliminates (or drastically reduces) the dead time associated with the preoptimization of a biasing potential. The goal of this article is to provide the mathematical/physical formulation of the proposed CBTI scheme, along with an initial application of the method to the calculation of the hydration free energy of methanol.