Application of the lambda-dynamics method to evaluate the relative binding free energies of inhibitors to HCV protease

Flexible Topology: A Dynamic Model of a Continuous Chemical Space

Ligand design problems involve searching chemical space for a molecule with a set of desired properties. As chemical space is discrete, this search must be conducted in a pointwise manner, separately investigating one molecule at a time, which can be inefficient. We propose a method called "Flexible Topology", where a ligand is composed of a set of shapeshifting "ghost" atoms, whose atomic identities and connectivity can dynamically change over the course of a simulation. Ghost atoms are guided toward their target positions using a translation-, rotation-, and index-invariant restraint potential. This is the first step toward a continuous model of chemical space, where a dynamic simulation can move from one molecule to another by following gradients of a potential energy function. This builds on a substantial history of alchemy in the field of molecular dynamics simulation, including the Lambda dynamics method developed by Brooks and co-workers [X. Kong and C.L. Brooks III, J. Chem. Phys. 105, 2414 (1996)], but takes it to an extreme by associating a set of four dynamical attributes with each shapeshifting ghost atom that control not only its presence but also its atomic identity. Here, we outline the theoretical details of this method, its implementation using the OpenMM simulation package, and some preliminary studies of ghost particle assembly simulations in vacuum. We examine a set of 10 small molecules, ranging in size from 6 to 50 atoms, and show that Flexible Topology is able to consistently assemble all of these molecules to high accuracy, beginning from randomly initialized positions and attributes.

Approaching protein design with multisite λ dynamics: Accurate and scalable mutational folding free energies in T4 lysozyme

The estimation of changes in free energy upon mutation is central to the problem of protein design. Modern protein design methods have had remarkable success over a wide range of design targets, but are reaching their limits in ligand binding and enzyme design due to insufficient accuracy in mutational free energies. Alchemical free energy calculations have the potential to supplement modern design methods through more accurate molecular dynamics based prediction of free energy changes, but suffer from high computational cost. Multisite λ dynamics (MSλD) is a particularly efficient and scalable free energy method with potential to explore combinatorially large sequence spaces inaccessible with other free energy methods. This work aims to quantify the accuracy of MSλD and demonstrate its scalability. We apply MSλD to the classic problem of calculating folding free energies in T4 lysozyme, a system with a wealth of experimental measurements. Single site mutants considering 32 mutations show remarkable agreement with experiment with a Pearson correlation of 0.914 and mean unsigned error of 1.19 kcal/mol. Multisite mutants in systems with up to five concurrent mutations spanning 240 different sequences show comparable agreement with experiment. These results demonstrate the promise of MSλD in exploring large sequence spaces for protein design.

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.

Adaptive landscape flattening in amino acid sequence space for the computational design of protein:peptide binding

For the high throughput design of protein:peptide binding, one must explore a vast space of amino acid sequences in search of low binding free energies. This complex problem is usually addressed with either simple heuristic scoring or expensive sequence enumeration schemes. Far more efficient than enumeration is a recent Monte Carlo approach that adaptively flattens the energy landscape in sequence space of the unbound peptide and provides formally exact binding free energy differences. The method allows the binding free energy to be used directly as the design criterion. We propose several improvements that allow still more efficient sampling and can address larger design problems. They include the use of Replica Exchange Monte Carlo and landscape flattening for both the unbound and bound peptides. We used the method to design peptides that bind to the PDZ domain of the Tiam1 signaling protein and could serve as inhibitors of its activity. Four peptide positions were allowed to mutate freely. Almost 75 000 peptide variants were processed in two simulations of 10⁹ steps each that used 1 CPU hour on a desktop machine. 96% of the theoretical sequence space was sampled. The relative binding free energies agreed qualitatively with values from experiment. The sampled sequences agreed qualitatively with an experimental library of Tiam1-binding peptides. The main assumption limiting accuracy is the fixed backbone approximation, which could be alleviated in future work by using increased computational resources and multi-backbone designs.

Adaptive Landscape Flattening Accelerates Sampling of Alchemical Space in Multisite λ Dynamics

Multisite λ dynamics (MSλD) is a powerful emerging method in free energy calculation that allows prediction of relative free energies for a large set of compounds from very few simulations. Calculating free energy differences between substituents that constitute large volume or flexibility jumps in chemical space is difficult for free energy methods in general, and for MSλD in particular, due to large free energy barriers in alchemical space. This study demonstrates that a simple biasing potential can flatten these barriers and introduces an algorithm that determines system specific biasing potential coefficients. Two sources of error, deep traps at the end points and solvent disruption by hard-core potentials, are identified. Both scale with the size of the perturbed substituent and are removed by sharp biasing potentials and a new soft-core implementation, respectively. MSλD with landscape flattening is demonstrated on two sets of molecules: derivatives of the heat shock protein 90 inhibitor geldanamycin and derivatives of benzoquinone. In the benzoquinone system, landscape flattening leads to 2 orders of magnitude improvement in transition rates between substituents and robust solvation free energies. Landscape flattening opens up new applications for MSλD by enabling larger chemical perturbations to be sampled with improved precision and accuracy.

Protein:Ligand binding free energies: A stringent test for computational protein design

A computational protein design method is extended to allow Monte Carlo simulations where two ligands are titrated into a protein binding pocket, yielding binding free energy differences. These provide a stringent test of the physical model, including the energy surface and sidechain rotamer definition. As a test, we consider tyrosyl-tRNA synthetase (TyrRS), which has been extensively redesigned experimentally. We consider its specificity for its substrate l-tyrosine (l-Tyr), compared to the analogs d-Tyr, p-acetyl-, and p-azido-phenylalanine (ac-Phe, az-Phe). We simulate l- and d-Tyr binding to TyrRS and six mutants, and compare the structures and binding free energies to a more rigorous "MD/GBSA" procedure: molecular dynamics with explicit solvent for structures and a Generalized Born + Surface Area model for binding free energies. Next, we consider l-Tyr, ac- and az-Phe binding to six other TyrRS variants. The titration results are sensitive to the precise rotamer definition, which involves a short energy minimization for each sidechain pair to help relax bad contacts induced by the discrete rotamer set. However, when designed mutant structures are rescored with a standard GBSA energy model, results agree well with the more rigorous MD/GBSA. As a third test, we redesign three amino acid positions in the substrate coordination sphere, with either l-Tyr or d-Tyr as the ligand. For two, we obtain good agreement with experiment, recovering the wildtype residue when l-Tyr is the ligand and a d-Tyr specific mutant when d-Tyr is the ligand. For the third, we recover His with either ligand, instead of wildtype Gln.

λ-Meta Dynamics Approach To Compute Absolute Solvation Free Energy

We present a new approach to combine λ dynamics with meta-dynamics (named λ-meta dynamics) to compute free energy surface with respect to λ. Particularly, the λ-meta dynamics method extends meta-dynamics to a single virtual variable λ, i.e., the coupling parameter between solute and solvent, to compute absolute solvation free energy as an exemplary application. We demonstrate that λ-meta dynamics simulations can recover the accurate potential of mean force surface with respect to λ compared to the benchmark results from traditional λ-dynamics with umbrella sampling. The solvation free energy results for five small organic molecules from λ-meta dynamics simulations using the same filling scheme show that the statistical errors are within ±0.5 kcal/mol. The new λ-meta dynamics method is general and other variables such as order parameters to describe conformational changes can be easily combined with λ-meta dynamics. This should allow for efficient samplings on high-dimension free energy landscapes.

Multi-Site λ-dynamics for simulated Structure-Activity Relationship studies

Multi-Site λ-dynamics (MSλD) is a new free energy simulation method that is based on λ-dynamics. It has been developed to enable multiple substituents at multiple sites on a common ligand core to be modeled simultaneously and their free energies assessed. The efficacy of MSλD for estimating relative hydration free energies and relative binding affinties is demonstrated using three test systems. Model compounds representing multiple identical benzene, dihydroxybenzene and dimethoxybenzene molecules show total combined MSλD trajectory lengths of ~1.5 ns are sufficient to reliably achieve relative hydration free energy estimates within 0.2 kcal/mol and are less sensitive to the number of trajectories that are used to generate these estimates for hybrid ligands that contain up to ten substituents modeled at a single site or five substituents modeled at each of two sites. Relative hydration free energies among six benzene derivatives calculated from MSλD simulations are in very good agreement with those from alchemical free energy simulations (with average unsigned differences of 0.23 kcal/mol and R(2)=0.991) and experiment (with average unsigned errors of 1.8 kcal/mol and R(2)=0.959). Estimates of the relative binding affinities among 14 inhibitors of HIV-1 reverse transcriptase obtained from MSλD simulations are in reasonable agreement with those from traditional free energy simulations and experiment (average unsigned errors of 0.9 kcal/mol and R(2)=0.402). For the same level of accuracy and precision MSλD simulations are achieved ~20-50 times faster than traditional free energy simulations and thus with reliable force field parameters can be used effectively to screen tens to hundreds of compounds in structure-based drug design applications.

Prediction of protein-ligand binding affinity by free energy simulations: assumptions, pitfalls and expectations

Many limitations of current computer-aided drug design arise from the difficulty of reliably predicting the binding affinity of a small molecule to a biological target. There is thus a strong interest in novel computational methodologies that claim predictions of greater accuracy than current scoring functions, and at a throughput compatible with the rapid pace of drug discovery in the pharmaceutical industry. Notably, computational methodologies firmly rooted in statistical thermodynamics have received particular attention in recent years. Yet free energy calculations can be daunting to learn for a novice user because of numerous technical issues and various approaches advocated by experts in the field. The purpose of this article is to provide an overview of the current capabilities of free energy calculations and to discuss the applicability of this technology to drug discovery.

Lambda-dynamics free energy simulation methods

Free energy calculations are fundamental to obtaining accurate theoretical estimates of many important biological phenomena including hydration energies, protein-ligand binding affinities and energetics of conformational changes. Unlike traditional free energy perturbation and thermodynamic integration methods, lambda-dynamics treats the conventional "lambda" as a dynamic variable in free energy simulations and simultaneously evaluates thermodynamic properties for multiple states in a single simulation. In the present article, we provide an overview of the theory of lambda-dynamics, including the use of biasing and restraining potentials to facilitate conformational sampling. We review how lambda-dynamics has been used to rapidly and reliably compute relative hydration free energies and binding affinities for series of ligands, to accurately identify crystallographically observed binding modes starting from incorrect orientations, and to model the effects of mutations upon protein stability. Finally, we suggest how lambda-dynamics may be extended to facilitate modeling efforts in structure-based drug design.

Pheromone Discrimination by the Pheromone-Binding Protein of Bombyx mori

Pheromone-binding proteins are postulated to contribute to the exquisite specificity of the insect's olfactory system, acting as a filter by preferentially binding only one of the components of the natural pheromone. Here, we investigated the possible discrimination of the two very similar components of the natural pheromone gland from the silk moth, Bombyx mori, bombykol and bombykal, by the only pheromone-binding protein (BmorPBP) known to be expressed in the pheromone-detecting sensilla. Free-energy calculations and virtual docking indicate that both bombykol and bombykal bind to BmorPBP with similar affinity. In addition, in vitro competitive binding assays showed that both bombykol and bombykal were bound by BmorPBP with nearly the same high affinity. While BmorPBP might filter out other physiologically irrelevant compounds hitting the sensillar lymph, discrimination between the natural pheromone compounds must be achieved by molecular interactions with their cognate receptors.