Relative Free Energy Differences Research Articles

Binding Selectivity Analysis from Alchemical Receptor Hopping and Swapping Free Energy Calculations.

We present receptor hopping and receptor swapping free energy estimation protocols based on the Alchemical Transfer Method (ATM) to model the binding selectivity of a set of ligands to two arbitrary receptors. The receptor hopping protocol, where a ligand is alchemically transferred from one receptor to another in one simulation, directly yields the ligand's binding selectivity free energy (BSFE) for the two receptors, which is the difference between the two individual binding free energies. In the receptor swapping protocol, the first ligand of a pair is transferred from one receptor to another while the second ligand is simultaneously transferred in the opposite direction. The receptor swapping free energy yields the differences in binding selectivity free energies of a set of ligands, which, when combined using a generalized DiffNet algorithm, yield the binding selectivity free energies of the ligands. We test these algorithms on host-guest systems and show that they yield results that agree with experimental data and are consistent with differences in absolute and relative binding free energies obtained by conventional methods. Preliminary applications to the selectivity analysis of molecular fragments binding to the trypsin and thrombin serine protease confirm the potential of the receptor swapping technology in structure-based drug discovery. The novel methodologies presented in this work are a first step toward streamlined and computationally efficient protocols for ligand selectivity optimization between mutants and homologous proteins.

Simulating Crystallization in a Colloidal System Using State Predictive Information Bottleneck Based Enhanced Sampling.

We investigate crystal nucleation in supersaturated colloid suspensions using enhanced molecular dynamics simulations augmented with machine learning techniques. The simulations reveal that crystallization in the model colloidal system studied here, with particles interacting through a repulsive screened Coulomb Yukawa potential, proceeds from vapor to dense liquid droplet to crystalline phases across multiple high barriers. Employing a one-dimensional reaction coordinate derived from the State Predictive Information Bottleneck framework, our simulations capture back-and-forth phase transitions across multiple barriers effectively in biased metadynamics simulations. We obtain relative free energy differences between different phases and also quantify the roles of different molecular level features in driving the phase changes.

Binding characteristics of the doxepin E/Z-isomers to the histamine H1 receptor revealed by receptor-bound ligand analysis and molecular dynamics study.

Doxepin is an antihistamine and tricyclic antidepressant that binds to the histamine H1 receptor (H1R) with high affinity. Doxepin is an 85:15 mixture of the E- and Z-isomers. The Z-isomer is well known to be more effective than the E-isomer, whereas based on the crystal structure of the H1R/doxepin complex, the hydroxyl group of Thr1123.37 is close enough to form a hydrogen bond with the oxygen atom of the E-isomer. The detailed binding characteristics and reasons for the differences remain unclear. In this study, we analyzed doxepin isomers bound to the receptor following extraction from a purified H1R protein complexed with doxepin. The ratio of the E- and Z-isomers bound to wild-type (WT) H1R was 55:45, indicating that the Z-isomer was bound to WT H1R with an approximately 5.2-fold higher affinity than the E-isomer. For the T1123.37V mutant, the E/Z ratio was 89:11, indicating that both isomers have similar affinities. Free energy calculations using molecular dynamics (MD) simulations also reproduced the experimental results of the relative binding free energy differences between the isomers for WT and T1123.37V. Furthermore, MD simulations revealed that the hydroxyl group of T1123.37 did not form hydrogen bonds with the E-isomer, but with the adjacent residues in the binding pocket. Analysis of the receptor-bound doxepin and MD simulations suggested that the hydroxyl group of T1123.37 contributes to the formation of a chemical environment in the binding pocket, which is slightly more favorable for the Z-isomer without hydrogen bonding with doxepin.

Fast free energy estimates from λ-dynamics with bias-updated Gibbs sampling

Relative binding free energy calculations have become an integral computational tool for lead optimization in structure-based drug design. Classical alchemical methods, including free energy perturbation or thermodynamic integration, compute relative free energy differences by transforming one molecule into another. However, these methods have high operational costs due to the need to perform many pairwise perturbations independently. To reduce costs and accelerate molecular design workflows, we present a method called λ-dynamics with bias-updated Gibbs sampling. This method uses dynamic biases to continuously sample between multiple ligand analogues collectively within a single simulation. We show that many relative binding free energies can be determined quickly with this approach without compromising accuracy. For five benchmark systems, agreement to experiment is high, with root mean square errors near or below 1.0 kcal mol−1. Free energy results are consistent with other computational approaches and within statistical noise of both methods (0.4 kcal mol−1 or less). Notably, large efficiency gains over thermodynamic integration of 18–66-fold for small perturbations and 100–200-fold for whole aromatic ring substitutions are observed. The rapid determination of relative binding free energies will enable larger chemical spaces to be more readily explored and structure-based drug design to be accelerated.

Dynamics and recognition of homeodomain containing protein-DNA complex of IRX4.

Iroquois Homeobox 4 (IRX4) belongs to a family of homeobox TFs having roles in embryogenesis, cell specification, and organ development. Recently, large scale genome-wide association studies and epigenetic studies have highlighted the role of IRX4 and its associated variants in prostate cancer. No studies have investigated and characterized the structural aspect of the IRX4 homeodomain and its potential to bind to DNA. The current study uses sequence analysis, homology modeling, and molecular dynamics simulations to explore IRX4 homeodomain-DNA recognition mechanisms and the role of somatic mutations affecting these interactions. Using publicly available databases, gene expression of IRX4 was found in different tissues, including prostate, heart, skin, vagina, and the protein expression was found in cancer cell lines (HCT166, HEK293), B cells, ascitic fluid, and brain. Sequence conservation of the homeodomain shed light on the importance of N- and C-terminal residues involved in DNA binding. The specificity of IRX4 homodimer bound to consensus human DNA sequence was confirmed by molecular dynamics simulations, representing the role of conserved amino acids including R145, A194, N195, S190, R198, and R199 in binding to DNA. Additional N-terminal residues like T144 and G143 were also found to have specific interactions highlighting the importance of N-terminus of the homeodomain in DNA recognition. Additionally, the effects of somatic mutations, including the conserved Arginine (R145, R198, and R199) residues on DNA binding elucidated the importance of these residues in stabilizing the protein-DNA complex. Secondary structure and hydrogen bonding analysis showed the roles of specific residues (R145, T191, A194, N195, R198, and R199) in maintaining the homogeneity of the structure and its interaction with DNA. The differences in relative binding free energies of all the mutants shed light on the structural modularity of this protein and the dynamics behind protein-DNA interaction. We also have predicted that the C-terminal sequence of the IRX4 homeodomain could act as a potential cell-penetrating peptide, emphasizing the role these small peptides could play in targeting homeobox TFs.

Calculations of Absolute Solvation Free Energies with Transformato─Application to the FreeSolv Database Using the CGenFF Force Field.

We recently introduced transformato, an open-source Python package for the automated setup of large-scale calculations of relative solvation and binding free energy differences. Here, we extend the capabilities of transformato to the calculation of absolute solvation free energy differences. After careful validation against the literature results and reference calculations with the PERT module of CHARMM, we used transformato to compute absolute solvation free energies for most molecules in the FreeSolv database (621 out of 642). The force field parameters were obtained with the program cgenff (v2.5.1), which derives missing parameters from the CHARMM general force field (CGenFF v4.6). A long-range correction for the Lennard-Jones interactions was added to all computed solvation free energies. The mean absolute error compared to the experimental data is 1.12 kcal/mol. Our results allow a detailed comparison between the AMBER and CHARMM general force fields and provide a more in-depth understanding of the capabilities and limitations of the CGenFF small molecule parameters.

Relative binding free energy calculations with transformato: A molecular dynamics engine-independent tool.

We present the software package transformato for the setup of large-scale relative binding free energy calculations. Transformato is written in Python as an open source project (https://github.com/wiederm/transformato); in contrast to comparable tools, it is not closely tied to a particular molecular dynamics engine to carry out the underlying simulations. Instead of alchemically transforming a ligand L 1 directly into another L 2, the two ligands are mutated to a common core. Thus, while dummy atoms are required at intermediate states, in particular at the common core state, none are present at the physical endstates. To validate the method, we calculated 76 relative binding free energy differences for five protein–ligand systems. The overall root mean squared error to experimental binding free energies is 1.17 kcal/mol with a Pearson correlation coefficient of 0.73. For selected cases, we checked that the relative binding free energy differences between pairs of ligands do not depend on the choice of the intermediate common core structure. Additionally, we report results with and without hydrogen mass reweighting. The code currently supports OpenMM, CHARMM, and CHARMM/OpenMM directly. Since the program logic to choose and construct alchemical transformation paths is separated from the generation of input and topology/parameter files, extending transformato to support additional molecular dynamics engines is straightforward.

Galvani Offset Potential and Constant-pH Simulations of Membrane Proteins.

A central problem in computational biophysics is the treatment of titratable residues in molecular dynamics simulations of large biological macromolecular systems. Conventional simulation methods ascribe a fixed ionization state to titratable residues in accordance with their pKa and the pH of the system, assuming that an effective average model will be able to capture the predominant behavior of the system. While this assumption may be justifiable in many cases, it is certainly limited, and it is important to design alternative methodologies allowing a more realistic treatment. Constant-pH simulation methods provide powerful approaches to handle titratable residues more realistically by allowing the ionization state to vary statistically during the simulation. Extending the molecular mechanical (MM) potential energy function to a family of potential functions accounting for different ionization states, constant-pH simulations are designed to sample all accessible configurations and ionization states, properly weighted according to their Boltzmann factor. Because protonation and deprotonation events correspond to a change in the total charge, difficulties arise when the long-range Coulomb interaction is treated on the basis of an idealized infinite simulation model and periodic boundary conditions with particle-mesh Ewald lattice sums. Charging free-energy calculations performed under these conditions in aqueous solution depend on the Galvani potential of the bulk water phase. This has important implications for the equilibrium and nonequilibrium constant-pH simulation methods grounded in the relative free-energy difference corresponding to the protonated and unprotonated residues. Here, the effect of the Galvani potential is clarified, and a simple practical solution is introduced to address this issue in constant-pH simulations of the acid-sensing ion channel (ASIC).

Alchemical free energy simulations without speed limits. A generic framework to calculate free energy differences independent of the underlying molecular dynamics program.

We describe the theory of the so‐called common‐core/serial‐atom‐insertion (CC/SAI) approach to compute alchemical free energy differences and its practical implementation in a Python package called Transformato. CC/SAI is not tied to a specific biomolecular simulation program and does not rely on special purpose code for alchemical transformations. To calculate the alchemical free energy difference between several small molecules, the physical end‐states are mutated into a suitable common core. Since this only requires turning off interactions, the setup of intermediate states is straightforward to automate. Transformato currently supports CHARMM and OpenMM as back ends to carry out the necessary molecular dynamics simulations, as well as post‐processing calculations. We validate the method by computing a series of relative solvation free energy differences.

Replica-Exchange Enveloping Distribution Sampling: Calculation of Relative Free Energies in GROMOS

Molecular dynamics (MD) simulations have become an important tool to investigate biological systems. Free-energy calculations based on MD are playing an increasingly important role for computer-aided drug design and material discovery in recent years. Free-energy differences between pairs of end-states can be estimated using well-established methods such as thermodynamic integration (TI) or Bennett’s acceptance ratio (BAR). An attractive alternative is the recently developed replica-exchange enveloping distribution sampling (RE-EDS) method, which enables estimating relative free-energy differences between multiple molecules from a single simulation. Here, we provide an introduction to the principles underlying RE-EDS and give an overview of the RE-EDS pipeline. In addition, we provide a description of the two complementary tools RestraintMaker and amber2gromos. We briefly discuss the findings of three recent applications of RE-EDS to calculate relative binding or hydration free energies. In all three studies, good agreement was found between the results obtained using RE-EDS and experimental values as well as values obtained using other free-energy methods.

Ab initio metadynamics determination of temperature-dependent free-energy landscape in ultrasmall silver clusters.

Ab initio metadynamics enables the extraction of free-energy landscapes having the accuracy of first-principles electronic structure methods. We introduce an interface between the PLUMED code that computes free-energy landscapes and enhanced-sampling algorithms and the Atomic Simulation Environment (ASE) module, which includes several ab initio electronic structure codes. The interface is validated with a Lennard-Jones cluster free-energy landscape calculation by averaging multiple short metadynamics trajectories. We use this interface and analysis to estimate the free-energy landscape of Ag5 and Ag6 clusters at 10, 100, and 300K with the radius of gyration and coordination number as collective variables, finding at most tens of meV in error. Relative free-energy differences between the planar and non-planar isomers of both clusters decrease with temperature in agreement with previously proposed stabilization of non-planar isomers. Interestingly, we find that Ag6 is the smallest silver cluster where entropic effects at room temperature boost the non-planar isomer probability to a competing state. The new ASE-PLUMED interface enables simulating nanosystem electronic properties under more realistic temperature-dependent conditions.

Exploring the Effect of Enhanced Sampling on Protein Stability Prediction.

Changes in protein stability due to side-chain mutations are evaluated by alchemical free-energy calculations based on classical molecular dynamics (MD) simulations in explicit solvent using the GROMOS force field. Three proteins of different complexity with a total number of 93 single-point mutations are analyzed, and the relative free-energy differences are discussed with respect to configurational sampling and (dis)agreement with experimental data. For the smallest protein studied, a 34-residue WW domain, the starting structure dependence of the alchemical free-energy changes, is discussed in detail. Deviations from previous simulations for the two other proteins are shown to result from insufficient sampling in the earlier studies. Hamiltonian replica exchange in combination with multiple starting structures and sufficient sampling time of more than 100 ns per intermediate alchemical state is required in some cases to reach convergence.

Relative free-energy calculations for scaffold hopping-type transformations with an automated RE-EDS sampling procedure

The calculation of relative free-energy differences between different compounds plays an important role in drug design to identify potent binders for a given protein target. Most rigorous methods based on molecular dynamics simulations estimate the free-energy difference between pairs of ligands. Thus, the comparison of multiple ligands requires the construction of a “state graph”, in which the compounds are connected by alchemical transformations. The computational cost can be optimized by reducing the state graph to a minimal set of transformations. However, this may require individual adaptation of the sampling strategy if a transformation process does not converge in a given simulation time. In contrast, path-free methods like replica-exchange enveloping distribution sampling (RE-EDS) allow the sampling of multiple states within a single simulation without the pre-definition of alchemical transition paths. To optimize sampling and convergence, a set of RE-EDS parameters needs to be estimated in a pre-processing step. Here, we present an automated procedure for this step that determines all required parameters, improving the robustness and ease of use of the methodology. To illustrate the performance, the relative binding free energies are calculated for a series of checkpoint kinase 1 inhibitors containing challenging transformations in ring size, opening/closing, and extension, which reflect changes observed in scaffold hopping. The simulation of such transformations with RE-EDS can be conducted with conventional force fields and, in particular, without soft bond-stretching terms.

Dummy Atoms in Alchemical Free Energy Calculations.

In calculations of relative free energy differences, the number of atoms of the initial and final states is rarely the same. This necessitates the introduction of dummy atoms. These placeholders interact with the physical 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 calculation. 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 physical system have to maintain it in a well-defined position and orientation relative to the physical system. A detailed theoretical analysis of both aspects is provided, illustrated by 24 calculations of relative solvation free energy differences, for which all four legs of the underlying thermodynamic 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 calculating the relative solvation free energy difference between two small solutes, such as methane and ammonia. While our analysis focuses on the so-called single topology approach to set up alchemical transformations, similar considerations apply to dual topology, at least many widely used variants thereof.

Prediction of pKa in a system with high orthogonal barriers: Alchemical flying Gaussian method

Alchemical simulations are versatile tools for prediction of relative free energy differences. The accuracy of these methods depends critically on sampling of orthogonal (non-alchemical) degrees of freedom. Here we apply the flying Gaussian method to accelerate orthogonal degrees of freedom in peptide bond cis/trans isomerisation. The approach is demonstrated on prediction of pKa value of N-acetylproline. Isomerization of the amide bond was accelerated in this simulation by multiple orders of magnitude. Alchemical free energy was obtained by reweighting. The results show that a combination of alchemical and flying Gaussian simulations is suitable for systems with high (∼ 60 kJ/mol) energy barrier.

A Method to Calculate the Relative Binding Free Energy Differences of α-Helical Stapled Peptides.

Hydrocarbon-stapled peptides are a class of bioactive α-helical ligands developed to target protein-protein interactions. Peptide stapling has benefited from the development of several chemical reactions to modulate their membrane permeability and binding affinity. However, in most current programs, choosing the best stapling positions is usually a trial-and-error process. Here, we develop a protocol to obtain optimal stapling positions computationally. Our method is based on molecular dynamics simulations and free energy calculations with nonequilibrium approaches; here, we predict the binding poses, hot-spot residues, and binding affinity differences of a set of perfluoroarene stapled α-helical peptides of the BIM BH3 peptide to the BCLXL receptor. The prediction of the hot-spot residues within the target peptide through computational alanine scanning anticipates not only the key residues for the receptor-peptide complex formation but also which positions should be avoided when applying the stapling groups. The staple moieties introduce local conformational changes not only in the replaced positions but also on their neighbor residues of the template peptide further affecting their binding behavior. Our approach is successful at rank-ordering the binding affinities of these stapled peptides with respect to the BIM BH3 peptide.

Application of the Movable Type Free Energy Method to the Caspase-Inhibitor BindingAffinity Study.

Many studies have provided evidence suggesting that caspases not only contribute to the neurodegeneration associated with Alzheimer’s disease (AD) but also play essential roles in promoting the underlying pathology of this disease. Studies regarding the caspase inhibition draw researchers’ attention through time due to its therapeutic value in the treatment of AD. In this work, we apply the “Movable Type” (MT) free energy method, a Monte Carlo sampling method extrapolating the binding free energy by simulating the partition functions for both free-state and bound-state protein and ligand configurations, to the caspase-inhibitor binding affinity study. Two test benchmarks are introduced to examine the robustness and sensitivity of the MT method concerning the caspase inhibition complexing. The first benchmark employs a large-scale test set including more than a hundred active inhibitors binding to caspase-3. The second benchmark includes several smaller test sets studying the relative binding free energy differences for minor structural changes at the caspase-inhibitor interaction interfaces. Calculation results show that the RMS errors for all test sets are below 1.5 kcal/mol compared to the experimental binding affinity values, demonstrating good performance in simulating the caspase-inhibitor complexing. For better understanding the protein-ligand interaction mechanism, we then take a closer look at the global minimum binding modes and free-state ligand conformations to study two pairs of caspase-inhibitor complexes with (1) different caspase targets binding to the same inhibitor, and (2) different polypeptide inhibitors targeting the same caspase target. By comparing the contact maps at the binding site of different complexes, we revealed how small structural changes affect the caspase-inhibitor interaction energies. Overall, this work provides a new free energy approach for studying the caspase inhibition, with structural insight revealed for both free-state and bound-state molecular configurations.

Studying effects of different protonation states of His11 and His102 in ribose-5-phosphate isomerase of Trypanosoma cruzi: an example of cooperative behavior

The Trypanosoma cruzi ribose-5-phosphate isomerase B (TcRpiB) is a crucial piece in the pentose phosphate pathway and thus is a potential drug target for treatment of Chagas’ disease. TcRpiB residues, such as Cys69, Asp45, Glu149 and Pro47, have confirmed their roles in substrate recognition, catalytic reaction and binding site conformation. However, the joint performance of His11 and His102, in the D-ribose-5-phosphate (R5P) in the catalysis is not well understood. In this work, we probed the influence of different protonation states of His11 and His102 on the behavior of the ligand R5P using molecular dynamics simulations, network analysis and thermodynamic integration. Simulations revealed that a protonated His11 combined with a neutral His102 (His11+‒His102) was able to stabilize the ligand R5P in the binding site. Moreover, calculated relative free energy differences showed that when protonated His11 was coupled to a neutral His102 an exergonic process takes place. On the other hand, neutral His11 combined with a protonated His102 (His11‒His102+), sampled conformations that resembled the catalyzed product D-ribulose-5-phosphate (Ru5P). Network analysis also demonstrated some peculiarities for these systems with some negatively correlated nodes in the binding site for His11‒His102+, and exclusive suboptimal paths for His11+‒His102. Therefore, the combined approach presented in this paper proposes two suitable protonation states for the TcRpiB catalytic mechanism, where an extra proton in either histidines might favor R5P binding or influence isomerization reaction to Ru5P. Our results may guide further in silico drug discovery studies. Communicated by Ramaswamy H. Sarma

Calculation of Protein-Protein Binding Free Energies using Umbrella Sampling with Dual Resolution Water Models

The binding free energy between two proteins is a measure of how strongly they will interact under physiological conditions. All-atom statistical-mechanics-based free energy calculation methods have the potential to accurately calculate binding free energy but are completely unfeasible for large-scale applications due to their high computational cost. This motivates the development of efficient and accurate free energy calculation approaches. Dual-resolution molecular simulation approach allows one to simulate at the atomic level in the regions of interest and at a coarser level, but much faster, in less relevant parts of the biomolecular systems. This hybrid approach has shown a substantial reduction of computational cost without compromising accuracy in a classical molecular dynamics simulations. Here, we investigate whether combining the statistical-mechanics-based free energy calculation method, umbrella sampling, with dual resolution water models provides a good trade-off between speed and accuracy. We chose Barnase-Barstar protein-protein complex as a test system since it is relatively small to allow convergence of the simulations, has a known structure, and empirical relative binding free energy difference values (ΔΔG) of their observed mutations are reported in the literature. We selected eight different mutations occurring at the different sites with broadly varying empirical ΔΔG values. We then calculated ΔΔG values for each mutation using umbrella sampling simulations, where we treated protein-protein complex and one surrounding water layer at an atomic level and the bulk of the water using WT4 coarse-grained water model. We obtained a high correlation between experimental and calculated ΔΔG values. This dual resolution free energy calculation approach has the potential for future high-throughput studies of protein-protein binding.

New Generation of the Reference Interaction Site Model Self-Consistent Field Method: Introduction of Constrained Spatial Electron Density Distribution (cSED)

Abstract Quantum mechanical (QM) calculation in solution is becoming a popular and useful tool in chemistry. We have developed a hybrid method between QM and reference interaction site model (RISM). To combine RISM with QM calculations, we have to employ charge fitting approaches. By introducing constrained spatial electron density distribution (cSED) in the fitting, we developed a new generation of RISM self-consistent field (SCF). Our method (RISM-SCF-cSED) overcame the instability of the original RISM-SCF calculations. We extended RISM-SCF-cSED by coupling a variety of QM approaches. We applied our method to tautomerization reactions of cytosine, 6-chloro-2-pyridone, and isonicotinic acid in water and Stokes shift calculation of indole and 5-cyanoindole. Our method correctly evaluated relative free energy differences in the reaction and the Stokes shift change along the polarity of solvent.